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1,314,259,996,463 | arxiv | \section{Introduction}
Quantum circuits based on conventional Josephson-junctions have begun to tackle real-world problems\cite{PhysRevX.6.031007}. This has been despite high decoherence produced by the loss\cite{lindstrom2009properties,macha2010losses} and noise\cite{burnett2014evidence,ramanayaka2015evidence} caused by parasitic two-level systems (TLS)\cite{phillips1972tunneling,faoro2015interacting}. In principle, superconducting nanowires can provide a route to low-decoherence quantum circuits due to their monolithic structure and lack of a TLS-hosting oxide layer. To date, superconducting nanowires with cross-sectional areas approaching the coherence length have demonstrated a variety of Josephson\cite{hao2009characteristics,levenson2011nonlinear} and phase-slip\cite{astafiev2012coherent,peltonen2016coherent,belkin2011little,webster2013nbsi} effects, but features such as their unconventional current-phase relationships\cite{likharev1979superconducting} remain unexploited in quantum circuits. Previous demonstrations of superconducting nanowire-embedded resonators exhibit unusually high dissipation, with internal quality factors ($Q_{\rm i}$) below 5$\times$10$^3$,\cite{belkin2011little,astafiev2012coherent,jenkins2014nanoscale,peltonen2016coherent}, far lower than in similar conventional Josephson-junction-based circuits\cite{de2013charge,simoen2015characterization}. In general, the performance of nanowire-embedded resonators can be limited by material quality, interface imperfections, resist residues and the measurement environment.
We demonstrate superconducting nanowire-embedded circuits with single photon $Q_{\rm i}$ up to 3.9$\times$10$^5$, comparable to or even better than conventional Josephson-junction resonators. Superconducting nanowires with widths down to 20~nm were fabricated with a neon focused ion beam (FIB). We study the loss in our devices within the well-established framework of loss mechanisms in superconducting resonators\cite{lindstrom2009properties,macha2010losses,faoro2015interacting,quintana2014characterization,goetz2016loss} to determine which factors are significant in limiting their performance. The vastly improved $Q_{\rm i}$ demonstrates that the detrimental effects can be sufficiently reduced and shows that competitive quantum circuits could be based on monolithic nanowire technology.
\section{Methods}
Superconducting 20-nm-thick NbN films were deposited on sapphire by dc magnetron sputtering from a 99.99\%-pure Nb target in a 1:1 Ar:N$_2$ atmosphere. The vacuum chamber was pumped to 6$\times$10$^{-7}$~mbar before sputtering at a pressure of 3.5$\times$10$^{-3}$~mbar and power of 200~W. The superconducting critical temperature, $T_{\rm c}$, was 10~K with a sheet resistance of 450~$\Omega/sq$. Electron-beam lithography (EBL) was used to pattern $\lambda/4$ and $\lambda/2$ coplanar microwave resonators capacitively coupled to a common microwave feed line (shown in Fig.~\ref{fig1}d). The width of the central conductor was 10~$\mu$m and the gap was 5~$\mu$m. This pattern was transferred from a 300-nm-thick-layer of polymethyl methacrylate (PMMA) into the film by a reactive ion etch (RIE) using a 2:1 ratio of SF$_{6}$:Ar, at 30~W and 30~mbar.
A neon FIB was used to directly pattern\cite{timilsina2014monte} nanowires in the central conductor of the microwave resonators at the current antinode -~see Fig.~\ref{fig1}b. With an acceleration voltage of 15~kV, the clearance dose for the NbN film is $\approx$~0.3~nC/$\mu$m$^2$. 15~kV was chosen as a compromise between minimising the spot size and minimising lateral milling of the nanowire\cite{rahman2012prospects}, leading to a few-minute mill time per $\mu$m$^2$ for a $\sim$1~pA beam current. By prior patterning of a sub-200-nm-wide precursor wire in the same EBL step as the resonator (shown in Fig.~\ref{fig1}c), we minimise the mill time and the total neon flux that the nanowire is subject to. Several devices were measured, and Table~\ref{ResTab1} shows important parameters including nanowire dimensions. The nanowire devices all feature two nanowires, configured either in parallel so that the nanowires complete a superconducting loop\cite{belkin2011little}, or in series with a wider segment in between\cite{hongisto2012single}. Here, there is no external flux- or gate-bias, so the nanowires are treated as simple constrictions within the superconductor.
Samples were enclosed within a brass box and cooled using a $^3$He refrigerator containing a heavily attenuated microwave in-line and an out-line with a cryogenic high-electron-mobility transistor (HEMT) amplifier.
\section {Results \& Discussion}
Figure~\ref{fig1}a shows the forward transmission ($S_{21}$) magnitude response of a nanowire-embedded resonator, at 307~mK and for an applied microwave drive of $-105$~dBm, demonstrating $Q_{\rm i}$~=~5.2$\times$10$^5$. This $Q_{\rm i}$ is significantly higher than in comparable nanowire-based devices\cite{belkin2011little,astafiev2012coherent,jenkins2014nanoscale,peltonen2016coherent}. This highlights the promise of the neon FIB and demonstrates that superconducting nanowires are not intrinsically lossy.
\begin{figure*}
\includegraphics[width=2\columnwidth]{fig1}
\caption{({\bf a}) The S$_{21}$ magnitude response of the nanowire-embedded resonator 3710\_1qP. The red line is a fit to Eq.~\ref{reseq}. ({\bf b}) A false-colour He FIB micrograph of a neon FIB milled nanowire (3710\_1qP) with dimensions of 27~nm by 1.2~$\mu$m. The NbN is shown in blue, while the milled region is shown in red. ({\bf c, d}) Scanning electron micrographs of ({\bf c}) the shorted end of a $\lambda/4$ resonator before milling by neon FIB. ({\bf d}) the whole $\lambda/4$ resonator. ({\bf e}) A photograph of the sample holder. In the centre is a chip, which is wirebonded to a microwave printed circuit board, the dark material is Eccosorb.}
\label{fig1}
\end{figure*}
The complex $S_{21}$ notch response of the superconducting resonators is fitted by\cite{probst2015efficient}
\begin{equation}S_{21}^{\rm}{(\nu)} = ae^{j\theta}e^{-2\pi{j}\nu\tau}\left[1-\frac{(Q_{\rm L}/|Q_{\rm c}|)e^{j\phi}}{1+2jQ_{\rm L}(\nu/\nu_{0}-1)}\right],
\label{reseq}
\end{equation}
where $\nu$ is the applied frequency, $\nu_{0}$ the resonance frequency, $Q_{\rm L}$ the loaded quality factor and $|Q_{\rm c}|$ the absolute value of the coupling quality factor; $\phi$ accounts for impedance mismatches, $a$ describes a change in amplitude, $\theta$ describes a change in phase and $\tau$ a change in the electronic delay.
The internal quality factor, $Q_{\rm i}$, is defined by $1/Q_{\rm L} = 1/Q_{\rm i} + \text{\rm Re}(1/Q_{\rm c})$ and the energy within the resonator is $W_{\rm sto} = 2P_{\rm app}S_{\rm min}Q_{\rm L}/{\omega_{0}}$, where $P_{\rm app}$ is the applied microwave power (in W) and $S_{\rm min}$ the normalized minimum of the resonator magnitude response. We describe the microwave power in terms of the average number of photons in the resonator, $\left<n\right>$, given by $\left<n\right> = W_{\rm sto}/(h\nu_0)$, where $h$ is Planck's constant.
To examine the effect of the neon FIB on the NbN film, we measured the resonator response as a function of temperature (shown in Fig.~\ref{fig4}). As temperature decreases from 2~K to 1~K, the resonant frequency increases due to changes in the complex conductivity which are described by $\frac{\Delta\nu}{\nu_{0}} = \frac{\alpha}{2}\frac{\Delta\sigma_{2}}{\sigma_{2}}$, where $\frac{\Delta\nu}{\nu_{0}}$ is the normalised resonance frequency, $\alpha$ is the kinetic inductance fraction and $\sigma_{2}$ is the imaginary part of the complex conductivity as given by Mattis-Bardeen (MB) theory\cite{mattis1958theory}. The inset of Fig.~\ref{fig4} shows the temperature dependence of the resonant frequency for all resonators on chip 1. The bunching of data points indicates a very similar $T_{\rm c}$ whether the resonator contains nanowires or not, implying that the neon FIB has not significantly suppressed the superconductivity.
Further decreasing temperature from 1~K, the resonant frequency decreases due to a thermal desaturation of TLS, which can be described by
\begin{equation}\frac{\Delta\nu}{\nu_{0}} = \frac{F\delta_{\rm TLS}^{\rm i}}{\pi}\left[{\rm Re}\Psi\left(\frac{1}{2}+\frac{1}{2\pi{j}}\frac{h\nu_{0}(T)}{k_B{T}}\right)-{\rm ln}\left(\frac{1}{2}\frac{h\nu_{0}(T_{0})}{k_B{T}}\right)\right],
\label{TLSeq2}
\end{equation}
where $F$ is the filling factor which typically relates to device geometry and electric field density, $T_0$ is a reference temperature, $\Psi$ is the complex digamma function and $F\delta_{\rm TLS}^{\rm i}$ is the intrinsic loss tangent. Fig.~\ref{fig4} shows a fit to both the MB and TLS frequency shifts, and the extracted $F\delta_{\rm TLS}^{\rm i}$ is shown in Table~\ref{ResTab1}. Barends {\it et al}.\cite{barends2009noise} have previously showed that, to determine $F\delta_{\rm TLS}^{\rm i}$ using both MB and TLS models, it is not necessary to obtain data in the temperature range covering the frequency upturn below 100mK seen in the TLS fit curve in Fig.~\ref{fig4}.
The thermal desaturation of TLS below 1~K results in absorption of microwave photons, leading to a power- and temperature-dependent resonator loss rate\cite{macha2010losses,lindstrom2009properties}. At low microwave drive, the unsaturated TLS dominate the loss, but as the microwave drive increases these TLS become saturated and therefore their loss rate decreases. At high microwave drives, where the TLS are saturated, the loss becomes dominated by residual quasiparticles, with a loss rate $\delta_{\rm qp}$ which is temperature-dependent but assumed to be independent of microwave power\cite{goetz2016loss}. The TLS and quasiparticle loss behaviour can be described by
\begin{equation}\frac{1}{Q_{\rm i}} = \delta^{\rm i}_{\rm tot} = F\delta^{0}_{\rm TLS}\frac{{\tanh}\left(\frac{h\nu_0}{2k_{\rm B}T}\right)}{\left(1+\left(\frac{\left<n\right>}{n_{\rm c}}\right)\right)^{\beta}} + \delta_{\rm qp},
\label{TLSeq}
\end{equation}
where $n_{\rm c}$ is the number of photons equivalent to the saturation field of the TLS, $\beta$ describes how quickly the TLS saturate with power and $F\delta_{\rm TLS}^{0}$ is the TLS loss tangent ($F\delta_{\rm TLS}^{0}$ is power- and temperature-independent). TLS models were originally based on the anomalous properties of glasses at low temperatures\cite{phillips1972tunneling} and assumed non-interacting TLS, which leads to a prediction of $\beta$~=~0.5. However, as superconducting circuits have improved, this model has failed to accurately describe the power dependence of dielectric losses: a weaker power dependence with $\beta <$~0.5 is frequently found\cite{macha2010losses,burnett2016analysis,wisbey2010effect,khalil2011loss}. This showed the need to consider TLS interactions \cite{faoro2012internal,faoro2015interacting,burnett2014evidence,burnett2016analysis,ramanayaka2015evidence}, changing the loss model to\cite{faoro2012internal,faoro2015interacting}
\begin{equation}
\frac{1}{Q_{i}} = FP_{\gamma}\chi{}\ln\left(\frac{Cn_{c}}{n} + \delta_{\rm qp}'\right)\tanh\left(\frac{h\nu_{0}}{2k_{B}T}\right),
\label{TLSlog}
\end{equation}
where $\chi$ is the dimensionless TLS parameter, $P_{\gamma}$ is the TLS switching rate ratio, $C$ is a large constant and $\delta_{\rm qp}'$ is the log-scaled quasiparticle loss rate
This loss is examined in more detail by fitting the resonator $S_{21}$ response as a function of microwave drive and temperature (shown in Figs.~\ref{fig3}a--c). Fig.~\ref{fig3}a (Fig.~\ref{fig3}c) show measurements of $\delta^{\rm i}_{\rm tot}$ $\left(\text{{where }}\delta^{\rm i}_{\rm tot} = 1/Q_{\rm i}\right)$ as a function of $\left<n\right>$ on bare (nanowire-embedded) resonators. Each resonator has its own symbol, with solid (hollow) symbols corresponding to measurements in a normal (Eccosorb-lined) sample box. Eccosorb CR-117 (see supplemental\cite{suppnotePRApp}) is a microwave absorber which has been shown to reduce quasiparticle excitation from stray infrared (IR) photons\cite{barends2011minimizing}, the Eccosorb lining is shown in Fig.~\ref{fig1}e. Different colours correspond to different temperatures. When analysing Figs.~\ref{fig3}a \& c with Eq.~\ref{TLSeq}, we find $\beta \approx$~0.1--0.2 (see supplemental\cite{suppnotePRApp}) implying interacting TLS. The solid lines represent fits to the interacting-TLS model, Eq.~\ref{TLSlog}. Table~\ref{ResTab1} collects fit parameters from both models.
\begin{table}
\caption{Table of resonator parameters. Resonators are named by $\nu_0$ (MHz), their chip number, $\lambda/4$ (q) or $\lambda/2$ (h) and whether they are bare resonators (B), have nanowires in series (S), have nanowires in parallel (P) or were measured in an Eccosorb-lined box (E). $\bar{w}$ refers to the nanowire widths. $\delta^{\rm i}_{\rm TLS}$ comes from fits to Eq.~\ref{TLSeq2}, while $\delta^{0}_{\rm TLS}$ \& $\delta_{\rm qp}$ come from fits to Eq.~\ref{TLSeq} and $FP_{\gamma}\chi$ come from fits to Eq.~\ref{TLSlog}.}
\begin{tabular}{m{1.3cm} m{0.9cm} m{0.9cm} m{0.9cm} m{0.9cm} m{0.9cm} m{0.9cm}}
\hline
Resonator & $\bar{w}$ (nm) & $F\delta^{\rm i}_{\rm TLS}$ ($\times$10$^{-6}$)& $FP_{\gamma}\chi$ ($\times$10$^{-6}$) &$F\delta^{0}_{\rm TLS}$ ($\times$10$^{-6}$)& $\delta_{\rm qp}$ ($\times$10$^{-7}$) \\
\hline
4094\_1qB & - & 6 & 0.57 & 6.6 & 5.6 \\
3995\_1hB & - & 6.3 & 0.61 & 6.2 & 6.9 \\
3675\_2qBE & - & 9.8 & 1.11 & 9.5 & 5.6 \\
2739\_2hBE & - & 12.6 & 1.21 & 10.2 & 13.3 \\
3710\_1qP & 27, 30 & 4.8 & 0.47 & 5.9 & 12.4 \\
3012\_1hS & 47, 48 & 6.9 & 0.42 & 7.8 & 21.1 \\
3382\_2qPE & 20, 23 & 12.9 & 1.13 & 14.3 & 5.4 \\
3468\_2hSE & 37, 34 & 13.0 & 1.27 & 14.2 & 6.8 \\
\end{tabular}
\label{ResTab1}
\end{table}
We first consider bare resonators measured in a standard sample box (solid symbols in Fig.~\ref{fig3}b). Resonators on the same chip show a fabrication-based variability, also found in the literature\cite{bruno2015reducing,goetz2016loss,lindstrom2009properties}: high-$\left<n\right>$ $Q_{\rm i}$~=~1.2--3.1$\times$10$^6$ and low-$\left<n\right>$ $Q_{\rm i}$~=~3.6--5.7$\times$10$^5$ at 307~mK. Increasing the temperature leads to an increase in low-$\left<n\right>$ $Q_{\rm i}$ because, as thermal occupation of the TLS increases, their ability to absorb microwave photons decreases\cite{bruno2015reducing,macha2010losses}, as described by the $\tanh$ temperature term. Increasing temperature also leads to a decrease in high-$\left<n\right>$ $Q_{\rm i}$. This is due to a higher quasiparticle density, meaning that more energy is lost to the quasiparticle system\cite{goetz2016loss}.
\begin{figure}
\includegraphics[width=1\columnwidth]{fig2}
\caption{Resonant frequency of resonator 3710\_1\_qP as a function of temperature. Red broken line: Variation arising from kinetic inductance changes described by MB theory. Blue dotted line: Variation arising from TLS losses. Green dash-dotted line: Fit to data including both MB and TLS effects. Inset: Normalised frequency shift as a function of temperature for all resonators on chip 1.}
\label{fig4}
\end{figure}
We next consider nanowire-embedded resonators in the standard sample box (solid symbols in Fig.~\ref{fig3}c). At 307~mK, at low $\left<n\right>$, we find $Q_{\rm i}$~=~2.7--3.9$\times$10$^5$, in good agreement with the results from the bare resonators, so the FIB-based fabrication of the nanowire has produced very little additional TLS loss. At high $\left<n\right>$, we find $Q_{\rm i}$~=~4.1--7.2$\times$10$^5$, 3--5 times lower than the bare resonators, indicating a higher residual quasiparticle density for the nanowire-embedded resonators
\begin{figure*}[t]
\includegraphics[width=2\columnwidth]{fig3}
\caption{({\bf a}) Loss tangent $\left(\delta^{\rm i}_{\rm tot} = 1/Q_{\rm i}\right)$ as a function of microwave drive for bare resonators. The colours correspond to different temperatures and the hollow symbols indicate the use of an Eccosorb CR-117 lined sample box. The solid lines in all plots represent fits to Eq.~\ref{TLSlog}. ({\bf b}) Loss tangent for the resonator 4094\_1qB with and without the Eccosorb-lined box. ({\bf c}) Loss tangent as a function of microwave drive for the nanowire-embedded resonators.}
\label{fig3}
\end{figure*}
Quasiparticles generated from pair-breaking events are an important consideration in conventional Josephson-junction devices\cite{EccosorbCorcoles}, where Eccosorb is typically used to reduce quasiparticle-based losses caused by stray IR photons\cite{barends2011minimizing,EccosorbCorcoles}. We examined whether quasiparticles generated from IR photons are important for nanowire-embedded resonators by measuring them in an Eccosorb-lined sample box. As the hollow dotted symbols in Fig.~\ref{fig3}c show, losses at high $\left<n\right>$ are much lower than for the standard sample box and $Q_{\rm i} \approx$~6--9$\times$10$^5$. This value matches that of the bare resonators for the same $\left<n\right>$, suggesting that the density of residual quasiparticles has been reduced to that of the bare resonators (see Table~\ref{ResTab1}). A saturated high-$\left<n\right>$ $Q_{\rm i}$ is not observed, due to nonlinearities in the resonance lineshape of the nanowire-embedded resonators. With the smaller quasiparticle-based loss, the TLS-based low-$\left<n\right>$ trend of loss increasing as $\left<n\right>$ decreases is once again found. The high-$\left<n\right>$ $Q_{\rm i}$ is found to increase with increasing temperature, consistent with losses from thermally generated quasiparticles as found in the bare resonators, indicating that increased quasiparticle losses in nanowire-embedded resonators in the normal sample box arose from quasiparticles excited by IR photons. As Table~\ref{ResTab1} shows, $\delta_{qp}$ of the nanowire-embedded resonators in the Eccosorb environment match those of the bare resonators (both with and without the Eccosorb environment) and are therefore limited by another mechanism which is not unique to the nanowire.
Figure 3b shows the loss for the same bare resonator with and without the Eccosorb enclosure. In contrast to nanowire-embedded resonators, the high-$\left<n\right>$ loss decreases only slightly when the Eccosorb-lined sample box is used. This is actually unsurprising since the energy gap of NbN is $\sim{10}\times$ larger than in Al. On the other hand, the reason for the sensitivity to IR photons in the nanowire-embedded resonators is not immediately obvious.
Our results demonstrate the importance of IR filtering even when nanowires have a large superconducting energy gap such as those in NbN. This is relevant to all nanowire-based devices. We note that a small suppression of $T_{\rm c}$ in our nanowire (below the precision of our $T_{\rm c}$ determination) could give some enhanced sensitivity to IR photons. Alternative explanations for the sensitivity include the nanowire exhibiting a different quasiparticle lifetime\cite{de2011number} or non-equilibrium superconductivity\cite{goldie2012non}, but these are beyond the scope of this study, although, since $Q_{\rm i}$ remains high, the number of quasiparticles created from IR photons must still be quite small\cite{barends2011minimizing}.
Finally, we compare the consistency of the TLS-loss rates (Table~\ref{ResTab1} and supplemental\cite{suppnotePRApp}) obtained from the analysis of the data shown in Figs.~\ref{fig4}~\&~\ref{fig3}. $F\delta_{\rm TLS}^{i}$ and $F\delta_{\rm TLS}^{0}$ differ by less than 20\%, this difference is because $F\delta_{\rm TLS}^{0}$ is only sensitive to near-resonant TLS, whereas $F\delta_{\rm TLS}^{\rm i}$ is also sensitive to a broad spectrum of off-resonant TLS\cite{macha2010losses,bruno2015reducing,pappas2011two}. Next, we note that $\delta^{i}_{\rm TLS} = \chi$,\cite{faoro2015interacting} so that the ratio $FP_{\gamma}\chi$/$F\delta^{i}_{\rm TLS}$ gives $P_{\gamma}$. We find an average value of $P_{\gamma} =$~0.093. This agrees well (see supplemental\cite{suppnotePRApp}) with the charge noise spectra of single-electron transistors that give $P_{\gamma} \approx $~0.10.\cite{kafanov2008charge,ratesnote} Therefore, all TLS-loss rates are consistent with each other. The TLS loss rates imply a TLS-limited $Q_{\rm i}$ up to $\approx$2$\times$10$^5$ in the quantum limit (at temperatures down to 10~mK and at single-photon energies). This is approximately 100$\times$ larger than in equivalent nanowire-embedded resonators and compares favourably with Josephson-junction-embedded resonators.
\section{Conclusion}
To conclude, we have used a neon FIB to create superconducting nanowires with widths down to 20~nm within superconducting resonators. In the low-power limit, these devices demonstrated $Q_{\rm i}$ up to 3.9$\times$10$^5$ at 300~mK, with $\delta^{\rm i}_{\rm TLS}$ and $\delta^{0}_{\rm TLS}$ corresponding to a TLS-limited $Q_{\rm i}$ up to 2$\times$10$^5$ at 10~mK. These TLS losses arise from the NbN thin-film technology rather than the neon FIB, meaning a higher $Q_{\rm i}$ should be possible with better resonator technology\cite{bruno2015reducing}. By obtaining such a high $Q_{\rm i}$ using nanowires, we have demonstrated a critical step towards realising nanowire-based, superinductance, phase-slip or Dayem-bridge circuits with coherence times comparable to conventional Josephson-junction-type devices.
\section{acknowledgements}The authors would like to acknowledge useful discussions with S.~de Graaf, E.~Dupont-Ferrier, N.~Constantino and T.~Lindstr\"{o}m. We also thank T.~Lindstr\"{o}m for the loan of equipment and critical reading of the manuscript. The authors gratefully acknowledge funding from the UK EPSRC, grant references EP/J017329/1 (JB and JCF) and EP/K024701/1 (JS and PAW), and Carl Zeiss SMT (JS and PAW). \nocite{santavicca2008impedance,faoro2015interacting,goetz2016loss,barends2011minimizing,gustavsson2016suppressing,budoyo2016effects,macha2010losses,burnett2016analysis,wisbey2010effect,khalil2011loss,faoro2012internal,lisenfeld2015observation,skacel2015probing,burnett2014evidence,burnett2016analysis,ramanayaka2015evidence,kafanov2008charge,mattis1958theory,semenov2009optical,coumou2015electrodynamics,de2014evidence,adamyan2016tunable}
|
1,314,259,996,464 | arxiv | \section{Cell algebra structures on monoid algebras}
In \cite{May}, a class of algebras called cell algebras is defined which generalize the cellular algebras of Graham and Lehrer \cite{GL}. These algebras had previously been introduced and studied as ``standardly based algebras'' by Du and Rui in \cite{DR}. Such algebras share many of the nice properties of cellular algebras. We will give conditions on a monoid $M$ and domain $R$ such that the monoid algebra $R[M]$ will be a cell algebra and will construct a standard cell basis for such algebras. We first review the definition.
Let $R$ be a commutative integral domain with unit 1 and let $A$ be an associative, unital $R$-algebra. Let $\Lambda $ be a finite set with a partial order $ \geqslant $ and for each $\lambda \in \Lambda $ let $L\left( \lambda \right),R\left( \lambda \right)$ be finite sets of ``left indices'' and ``right indices''. Assume that for each $\lambda \in \Lambda ,s \in L\left( \lambda \right),{\text{ and }}t \in R\left( \lambda \right)$ there is an element $\,_s C_t ^\lambda \in A$ such that the map $(\lambda ,s,t) \mapsto \,_s C_t ^\lambda $ is injective and $C = \left\{ {_s C_t ^\lambda :\lambda \in \Lambda ,s \in L(\lambda ),t \in R\left( \lambda \right)} \right\}$ is a free $R$-basis for $A$. Define $R$-submodules of $A$ by $A^\lambda = R{\text{ - span of }}\left\{ {_s C_t ^\mu :\mu \in \Lambda ,\mu \geqslant \lambda ,s \in L(\mu ),t \in R\left( \mu \right)} \right\}$ and $\hat A^\lambda = R{\text{ - span of }}\left\{ {_s C_t ^\mu :\mu \in \Lambda ,\mu > \lambda ,s \in L(\mu ),t \in R\left( \mu \right)} \right\}$.
\begin {definition} \label {d3.1} Given $(A,\Lambda,C)$, $A$ is a cell algebra with poset $\Lambda$ and cell basis $C$ if
\begin{description}
\item[i] For any $a \in A,\lambda \in \Lambda ,{\text{ and }}s,s' \in L\left( \lambda \right)$, there exists $r_L = r_L \left( {a,\lambda ,s,s'} \right) \in R$ such that, for any $t \in R\left( \lambda \right)$, $a \cdot \,_s C_t ^\lambda = \sum\limits_{s' \in L\left( \lambda \right)} {r_L \cdot \,_{s'} C_t ^\lambda } \, \, \bmod \hat A^\lambda $, and
\item[ii] For any $a \in A,\lambda \in \Lambda ,{\text{ and }}t,t' \in R\left( \lambda \right)$, there exists $r_R = r_R \left( {a,\lambda ,t,t'} \right) \in R$ such that, for any $s \in L\left( \lambda \right)$, $_s C_t ^\lambda \cdot a = \sum\limits_{t' \in R\left( \lambda \right)} {r_R \cdot \,_s C_{t'} ^\lambda } \, \, \bmod \hat A^\lambda $ .
\end{description}
\end {definition}
Now for each $\mathcal{D}$-class $D$ in a finite monoid $M$, choose a base element $\gamma _D $ and base $\mathcal{H}$-class $H = H_\gamma $ as above. Then define the Schutzenberger group $G_D$ of $D$ to be $G_D = G_H^R $. (Up to isomorphism, this is independent of the choice of base class $H$.) Let $R\left[ {G_D} \right]$ be the group algebra of $G_D $ over $R$.
\begin{definition} \label{d3.2}
A monoid $M$ satifies the $R$-C.A. condition for a given domain $R$ if $R\left[ {G_D } \right]$ has a cell algebra structure for every $\mathcal{D}$-class $D$ in $M$.
\end{definition}
If an $\mathcal{H}$-class $H$ contains an idempotent, then $H$ is actually a subgroup of $M$. In fact, the maximal subgroups of $M$ are just the $\mathcal{H}$-classes which contain an idempotent. In this case the group $H$ is isomophic to the Schutzenberger group $G_H^R $. If $M$ is a \emph{regular} semigroup, then every $\mathcal{D}$-class $D$ contains an idempotent $e$, so $G_D$ is isomorphic to a maximal subgroup $H_e$ of $M$. Then for regular semigroups $M$ the $R$-C.A. condition is equivalent to requiring that $R[G]$ have a cell algebra structure for every maximal subgroup $G$ of $M$.
If $M$ is a monoid (such as a transformation semigroup $\mathcal{T}_r $ or a partial transformation semigroup $\mathcal{PT}_r $) for which every group $G_D$ is a symmetric group, then the usual Murphy basis gives a cellular (and hence cell) algebra structure to $R[G_D]$ for any domain $R$. Thus $M$ satisfies the $R$-C.A. condition for any $R$.
For any finite monoid $M$, if $k$ is a field of characteristic $0$ or characteristic $p$ where $p$ does not divide the order of any $G_D$, then by Maschke's theorem, every $k[G_D]$ is semisimple. If, in addition, $k$ is algebraically closed, then each $k[G_D]$ is split semisimple. Such algebras are products of matrix algebras over $k$ and have a natural cellular (hence cell) algebra basis. Thus if $k$ is algebraically closed and of good characteristic relative to $M$, then any $M$ satisfies the $k$-C.A. condition.
Our main result is the following theorem.
\begin{theorem} \label{t3.1}
Let $M$ be a finite monoid satifying the $R$-C.A. condition for a domain $R$. Then $A = R[M]$ is a cell algebra. The choice of a cell basis for each algebra $R[G_D]$ gives rise to a standard cell basis for $A$.
\end{theorem}
\begin{proof}
For a given $D \in \mathbb{D}$, put $A_D = R[G_D ]$ and assume $\Lambda _D ,L_D ,R_D $ define a cell algebra structure on $A_D $ with cell basis \[C_D = \left\{ {_s C_t ^\lambda :\lambda \in \Lambda _D ,s \in L_D (\lambda ),t \in R_D \left( \lambda \right)} \right\}.\]
Define a poset $\Lambda $ to consist of all pairs $\left( {D,\lambda } \right)$ where $D$ is a $\mathcal{D}$-class in $M$ and $\lambda \in \Lambda _D $. Define the partial order by $(D_1 ,\lambda _1 ) > (D_2 ,\lambda _2 )$ if $D_1 < D_2 $ or $D_1 = D_2 {\text{ and }}\lambda _1 > \lambda _2 $ in $\Lambda _{D_1 } $.
For $(D,\lambda ) \in \Lambda $, define $L\left( {D,\lambda } \right)$ to be all pairs $\left( {R,s} \right)$ where $R$ is an $\mathcal{R}$-class contained in $D$ and $s \in L_D \left( \lambda \right)$. Similarly, define $R\left( {D,\lambda } \right)$ to be all pairs $\left( {L,t} \right)$ where $L$ is an $\mathcal{L}$-class contained in $D$ and $t \in R_D (\lambda )$. Finally, given $(D,\lambda ) \in \Lambda \,,\,(R,s) \in L(D,\lambda )\,,\,(L,t) \in R(D,\lambda )$, assume $R$ corresponds to row $i$ and $L$ corresponds to column $j$ in the ``egg-box'' for $D$. Then define \[_{(R,s)} C_{(L,t)}^{(D,\lambda )} = {}_i\psi _j \left( {\phi _H^R \left( {_s C_t ^\lambda } \right)} \right) = a_i \phi _H^R \left( {_s C_t ^\lambda } \right)b_j .\]
For fixed $D,R,L$, since $\left\{ {_s C_t ^\lambda :\lambda \in \Lambda _D ,s \in L_D (\lambda ),t \in R_D (\lambda )} \right\}$ is a basis for $R[G_D ]$ by assumption and ${}_i\psi _j {\text{ and }}\phi _H^R $ are bijective, \[\left\{ {_{(R,s)} C_{(L,t)}^{(D,\lambda )} :\lambda \in \Lambda _D ,s \in L_D (\lambda ),t \in R_D (\lambda )} \right\}\] will give a basis for $R[{}_iH_j ]$. Then since $A = R[M]$ is the direct sum of the free submodules $R[H]$ as $H$ varies over all $\mathcal{H}$-classes in $M$,
\[C = \left\{ {_{(R,s)} C_{(L,t)}^{(D,\lambda )} :(D,\lambda ) \in \Lambda ,(R,s) \in L(D,\lambda ),(L,t) \in R(D,\lambda )} \right\}\]
is a basis for $A$. To show $C$ is a cell basis, we must check the cell conditions (i) and (ii).
Note first that any element $m$ in a $\mathcal{D}$-class $D_m $ will be in the span of $\left\{ {_{(R,s)} C_{(L,t)}^{(D_m ,\mu )} :\mu \in \Lambda _{D_m } ,(R,s) \in L(D_m ,\mu ),(L,t) \in R(D_m ,\mu )} \right\}$. Then if $D_m < D$ for some $\mathcal{D}$-class $D$, we have $(D_m ,\mu ) > (D,\lambda )$ for any $\mu \in \Lambda _{D_m} $ and $\lambda \in \Lambda_D $, so $m \in \hat A^{(D,\lambda )} $. So $\mathop \oplus \nolimits_{D' < D} R\left[ {D'} \right] = \hat A^D \subseteq \hat A^{(D,\lambda )} $.
To prove (i), we can assume $a$ is a basis element $m \in M$. Take $(D,\lambda ) \in \Lambda \,,\,(R,s) \in L(D,\lambda )\,,\,(L,t) \in R(D,\lambda )$ and assume $R$ corresponds to row $i$ and $L$ corresponds to column $j$ in the ``egg-box'' for $D$. Then $_{(R,s)} C_{(L,t)}^{(D,\lambda )} = {}_i\psi _j \left( {\phi _H^R \left( {_s C_t ^\lambda } \right)} \right) = a_i \phi _H^R \left( {_s C_t ^\lambda } \right)b_j \in R[{}_iH_j ]$. By corollary \ref{c2.2} there are two cases to consider.
Case i: $m \cdot R\left[ {{}_iH_j } \right] \subseteq \mathop \oplus \nolimits_{D' < D} R\left[ {D'} \right]$ . Then $m \cdot _{(R,s)} C_{(L,t)}^{(D,\lambda )} \in m \cdot R[{}_iH_j ] \subseteq \mathop \oplus \nolimits_{D < D'} R[D'] \subseteq \hat A^{(D,\lambda )} $ and we can satisfy (i) by taking all coefficients $r_L $ to be zero.
Case ii: $m \cdot R\left[ {{}_iH_j } \right] = R\left[ {{}_kH_j } \right]$ for some $k$. By corollary 2.2, since $_{(R,s)} C_{(L,t)}^{(D,\lambda )} = a_i \phi _H^R \left( {_s C_t ^\lambda } \right)b_j \in R[{}_iH_j ]$, then $m\,\,_{(R,s)} C_{(L,t)}^{(D,\lambda )} = a_k \phi _H^R \left( {r_{\bar {m^ * } } \,_s C_t ^\lambda } \right)b_j $, where $m^* \equiv \bar a_k ma_j \in LT\left( H \right)$
and $\bar {m^ * } = \vartheta \circ \psi \left( {l_{m^ * } } \right) \in RT(H)$. Since $g = r_{\bar {m^ * } } \in G_D $, the cell algebra property (i) for $R[G_D ]$ gives $g \cdot \,_s C_t ^\lambda = \sum\limits_{s' \in L_D \left( \lambda \right)} {r_L \cdot \,_{s'} C_t ^\lambda } \bmod \hat A_D ^\lambda $, where $r_L $ depends on $s,s',\lambda ,{\text{ and }}m$, but is independent of $t$. Then
\[\begin{aligned}
m\,\,_{(R,s)} C_{(L,t)}^{(D,\lambda )} &= a_k \phi _H^R \left( {g\,_s C_t ^\lambda } \right)b_j \\
&= \sum\limits_{s' \in L_D (\lambda )} {r_L \cdot a_k } \phi _H^R \left( {_{s'} C_t ^\lambda } \right)b_j \text{ modulo } a_k \phi _H^R \left( {\hat A_D ^\lambda } \right)b_j
\end{aligned}\]
where \[ a_k \phi _H^R \left( {\hat A_D ^\lambda } \right)b_j \subseteq {\text{span}}\left\{ {_{(R,s)} C_{(L,t)}^{(D,\mu )} :\mu > \lambda ,s \in L_D (\mu ),t \in R_D (\mu )} \right\} \subseteq \hat A^{(D,\lambda )} .\]
But $a_k \phi _H^R \left( {_{s'} C_t ^\lambda } \right)b_j = \,_{(R',s')} C_{(L,t)}^{(D,\lambda )} $ where $R'$ is the $\mathcal{R}$-class corresponding to row $k$ of the ``egg-box''. Then $m \cdot \,_{(R,s)} C_{(L,t)}^{(D,\lambda )} = \sum\limits_{s' \in L_D (\lambda )} {r_L \cdot a_k } \phi _H^R \left( {_{s'} C_t ^\lambda } \right)b_j = \sum\limits_{s' \in L_D (\lambda )} {r_L \cdot \,_{(R',s')} C_{(L,t)}^{(D,\lambda )} } \,\,\bmod \hat A^{(D,\lambda )} $. Since $r_L $ is independent of $L$ and $t$, this yields property (i) for this case ii.
The proof of condition (ii) is parallel. Thus $C$ is a cell basis and $A = R[M]$ is a cell algebra.
\end{proof}
If $M$ is a finite monoid satisfying the $R$-C.A. condition, we will assume a fixed cell algebra structure is given to each $R[G_D]$. We will then call the cell algebra structure obtained in the proof of theorem \ref{t3.1} the \emph{standard cell algebra structure} on $R[M]$.
In \cite{East}, \cite{Wil}, and \cite{GX}, East, Wilcox, and Guo and Xi worked with finite regular semigroups with cellular algebra (hence cell algebra) structure on $R[G]$ for maximal subgroups $G$ of $M$. Their examples therefore satisfy the $R$-C.A. condition and can be seen to be cell algebras without considering the complicated involution requirements involved in showing a cellular algebra structure. In \cite{East}, East gives examples of inverse semigroups with cellular $R[G]$ for all maximal $G$ which lack an appropriate involution and therefore do not have cellular $R[M]$. These examples would be cell algebras by theorem \ref{t3.1}.
More typical examples of cell algebras that are not cellular are the algebras $R[M]$ for $M$ a transformation semigroup $\mathcal{T}_r $ or partial transformation semigroup $\mathcal{PT}_r $. These were shown to be cell algebras in \cite{May}. For these examples, a $\mathcal{D}$-class $D$ consists of mappings of a given rank $i$ and the group $G_D$ is the symmetric group $\mathfrak{G}_i$. Since, as remarked above, the symmetric groups have cellular (hence cell) structures on their group algebras, theorem \ref{t3.1} applies and provides a cell algebra structure on $M$.
\section{Properties of the cell algebra $A = R[M]$}
In this section we assume that $M$ is a finite monoid satisfying the $R$-C.A. condition, that is, such that for every $D \in \mathbb{D}$ the group algebra $R[G_D ]$ of the Schutzenberger group for $D$ is a cell algebra. Then by Theorem \ref{t3.1}, $A = R[M]$ is a cell algebra and we can apply the results in \cite{May} to $A$ with the standard cell algebra structure.
For $(D,\lambda ) \in \Lambda $, the left cell module $\,_L C^{(D,\lambda )} $ is a left $A$-module which is a free $R$-module with basis $\left\{ {_{(R,s)} C^{(D,\lambda )} :(R,s) \in L\left( {D,\lambda } \right)} \right\}$. Similarly, the right cell module $C_R ^{(D,\lambda )} $ for $(D,\lambda )$ is a right $A$-module and a free $R$-module with basis $\left\{ {C_{(L,t)} ^{(D,\lambda )} :(L,t) \in R\left( {D,\lambda } \right)} \right\}$. For each $(D,\lambda ) \in \Lambda $ there is an $R$-bilinear map $ \left\langle - , - \right\rangle \,:\,\left( {C_R ^{(D,\lambda )} ,\,_L C^{(D,\lambda )} } \right) \to R$ defined by the property $\left( {_{(R',s')} C_{(L,t)}^{(D,\lambda )} } \right) \cdot \left( {_{(R,s)} C_{(L',t')}^{(D,\lambda )} } \right) = \left\langle {C_{(L,t)}^{(D,\lambda )} ,\,_{(R,s)} C^{(D,\lambda )} } \right\rangle \,_{(R',s')} C_{(L',t')}^{(D,\lambda )} \,\bmod \,\hat A^{(D,\lambda )} $
for any choice of $R',s',L',t'$.
Right and left radicals are defined by
\[{\text{rad}}\left( {C_R^{(D,\lambda )} } \right) = \left\{ {x \in C_R^{(D,\lambda )} :\left\langle {x,y} \right\rangle = 0{\text{ for all }}y \in \,_L C^{(D,\lambda )} } \right\}\]
\[{\text{rad}}\left( {\,_L C^{(D,\lambda )} } \right) = \left\{ {y \in \,_L C^{(D,\lambda )} :\left\langle {x,y} \right\rangle = 0{\text{ for all }}x \in C_R^{(D,\lambda )} } \right\}.\]
Then define $D_R^{(D,\lambda )} = \frac{{C_R^{(D,\lambda )} }}
{{{\text{rad}}\left( {C_R^{(D,\lambda )} } \right)}}$ and $\,_L D^{(D,\lambda )} = \frac{{\,_L C^{(D,\lambda )} }}
{{{\text{rad}}\left( {\,_L C^{(D,\lambda )} } \right)}}$. Finally, define $\Lambda _0 = \left\{ {(D,\lambda ) \in \Lambda :\left\langle {x,y} \right\rangle \ne 0{\text{ for some }}x \in C_R^{(D,\lambda )} ,y \in \,_L C^{(D,\lambda )} } \right\}$. Evidently, $\lambda \in \Lambda _0 \Leftrightarrow D_R^{(D,\lambda )} \ne 0 \Leftrightarrow \,_L D^{(D,\lambda )} \ne 0$. A major result of \cite{May} is
\begin{theorem} \label{t4.1}
Assume $R = k$ is a field. Then \\
(a) $\left\{ {D_R ^{(D,\mu )} :(D,\mu ) \in \Lambda _0 } \right\}$ is a complete set of pairwise inequivalent irreducible right $A$-modules and \\ (b) $\left\{ {\,_L D^{(D,\mu )} :(D,\mu ) \in \Lambda _0 } \right\}$ is a complete set of pairwise inequivalent irreducible left $A$-modules.
\end{theorem}
To find $\Lambda _0 $ for $A$ we need to relate the bracket $\left\langle { - , - } \right\rangle $ on $A$ to the brackets $\left\langle { - , - } \right\rangle _D $ on the cell algebras $A_D = R[G_D ]$. For a given $\mathcal{D}$-class $D$, write $R_i $ for the $\mathcal{R}$-class corresponding to row $i$ of the ``egg-box'' and $L_j $ for the $\mathcal{L}$-class corresponding to column $j$.
\begin{definition} \label{d4.1}
$R_i \,,\,L_j $ are matched if there exist $x \in L_j \,,\,y \in R_i $ such that $xy \in D$. $R_i \,,\,L_j $ are unmatched if $L_j R_i \cap D = \emptyset $.
\end{definition}
\begin{lemma} \label{l4.1}
Assume $R_i \,,\,L_j $ are matched. Write $x = a_{i'} \phi _H^R (\xi )b_j \in R[L_j ]$ and $y = a_i \phi _H^R (\eta )b_{j'} \in R[R_i ]$ for any $\xi ,\eta \in R[G_D ]$. Then $xy = a_{i'} \phi _H^R \left( {\xi r_{m(i,j)} \eta } \right)b_{j'} $ where $m(i,j) \equiv b_j a_i \gamma \in RT(H)$.
\end{lemma}
\begin{proof}
If $R_i \,,\,L_j $ are matched, then for some choice of $i',j'$ and some elements $x \in {}_{i'}H_j \subseteq L_j \,,\,y \in {}_iH_{j'} \subseteq R_i $ we have $xy \in {}_{i'}H_{j'} \subseteq D$. But then (for the given $i',j'$) we have $xy \in {}_{i'}H_{j'} \subseteq D$ for \emph{any} $x \in {}_{i'}H_j \,,\,y \in {}_iH_{j'} $ by propositions \ref{p2.1} and \ref{p2.2}. In particular, for $x = a_{i'} \gamma b_j \in {}_{i'}H_j \,,\,y = a_i \gamma b_{j'} \in {}_iH_{j'} $ we get $xy = a_{i'} \gamma b_j a_i \gamma b_{j'} \in {}_{i'}H_{j'} $. Then multiplying by $\bar a_{i'} $ on the left and $\bar b_{j'} $ on the right gives $\gamma b_j a_i \gamma \in H$. But then $m(i,j) \equiv b_j a_i \gamma \in RT(H)$ by lemma \ref{l2.1}.
Now consider any $i',j'$ and any $g = r_m ,h = r_n \in G_D $ and let
\[x = a_{i'} \phi _H^R (g)b_j \in R[L_j ]\,,\,y = a_i \phi _H^R (h)b_{j'} \in R[R_i ].\]
Then
\[
\begin{aligned}
xy &= (a_{i'} \phi _H^R \left( g \right)b_j )(a_i \phi _H^R \left( h \right)b_{j'} ) = a_{i'} \gamma mb_j a_i \gamma nb_{j'} \\
&= a_{i'} \gamma m \cdot m(i,j) \cdot nb_{j'} = a_{i'} \phi _H^R \left( {r_m r_{m(i,j)} r_n } \right)b_{j'} \\
&= a_{i'} \phi _H^R \left( {gr_{m(i,j)} h} \right)b_{j'} .
\end{aligned}
\]
Then by linearity of $\phi _H^R $, $xy = a_{i'} \phi _H^R \left( {\xi r_{m(i,j)} \eta } \right)b_{j'} $ for arbitrary $\xi ,\eta \in R[G_D ]$ when $x = a_{i'} \phi _H^R (\xi )b_j \in R[L_j ]\,,\,y = a_i \phi _H^R (\eta )b_{j'} \in R[R_i ]$, proving the lemma.
\end{proof}
For $i \in \left\{ {1,2, \cdots ,n(D,R)} \right\}$, define $\left( {_L C^{(D,\lambda )} } \right)_i $
to be the free $R$-module with basis $\left\{ {_{(R_i ,s)} C^{(D,\lambda )} :\lambda \in \Lambda _D ,s \in L_D (\lambda )} \right\}$, so $\,_L C^{(D,\lambda )} = \mathop \oplus \limits_i \left( {_L C^{(D,\lambda )} } \right)_i $. Similarly, for $j \in \left\{ {1,2, \cdots ,n(D,L)} \right\}$, define $\left( {C_R ^{(D,\lambda )} } \right)_j $ to be the free $R$-module with basis $\left\{ {C_{(L_j ,t)}^{(D,\lambda )} :\lambda \in \Lambda _D ,t \in R_D (\lambda )} \right\}$, so $C_R^{(D,\lambda )} = \mathop \oplus \limits_j \left( {C_R^{(D,\lambda )} } \right)_j $. Notice that for each $i$, $\phi _i :\,_{(R_i ,s)} C^{(D,\lambda )} \mapsto \,_s C^\lambda $ gives an isomorphism (of $R$-modules) $\phi _i :\left( {_L C^{(D,\lambda )} } \right)_i \to \,_L C^\lambda $. Similarly, $\phi _j :C_{(L_j ,t)}^{(D,\lambda )} \mapsto C_t^\lambda $ gives an isomorphism $\phi _j :\left( {C_R ^{(D,\lambda )} } \right)_j \to C_R ^\lambda $.
\begin{proposition} \label{p4.1}
Take $X \in \left( {C_R ^{(D,\lambda )} } \right)_j \,,\,Y \in \left( {_L C^{(D,\lambda )} } \right)_i $. Then
\begin{enumerate} [\upshape(a)]
\item If $R_i \,,\,L_j $ are not matched, then $\left\langle {X,Y} \right\rangle = 0$,
\item If $R_i \,,\,L_j $ are matched, then
\[\left\langle {X,Y} \right\rangle = \left\langle {\phi _j (X),r_{m(i,j)} \phi _i \left( Y \right)} \right\rangle _D = \left\langle {r_{m(i,j)} \phi _j (X),\phi _i \left( Y \right)} \right\rangle _D \]
where $m(i,j) \equiv b_j a_i \gamma \in RT(H)$.
\end{enumerate}
\end{proposition}
\begin{proof}
It suffices to check (a) and (b) for basis elements $X,Y$, so assume $X = C_{(L_j ,t)}^{(D,\lambda )} \,,\,Y = \,_{(R_i ,s)} C^{(D,\lambda )} $ and put $x = _{(R_{i'} ,s')} C_{_{(L_j ,t)} }^{(D,\lambda )} = a_{i'} \phi _H^R \left( {_{s'} C_t ^\lambda } \right)b_j \in L_j $ and $y = _{(R_i ,s)} C_{_{(L_{j'} ,t')} }^{(D,\lambda )} = a_i \phi _H^R \left( {_s C_{t'} ^\lambda } \right)b_{j'} \in R_i $. Then $xy = \left\langle {X,Y} \right\rangle \,_{(R_{i'} ,s')} C_{(L_{j'} ,t')}^{(D,\lambda )} \bmod \hat A^{(D,\lambda )} $.
If $R_i \,,\,L_j $ are not matched, then $xy \in \hat A^D \subseteq \hat A^{(D,\lambda )} $, so $\left\langle {X,Y} \right\rangle = 0$, proving (a).
If $R_i \,,\,L_j $ are matched, then, by lemma \ref{l4.1},
\[ \begin{aligned}
xy &= a_{i'} \phi _H^R \left( {_{s'} C_t ^\lambda \cdot r_{m(i,j)} \cdot \,_s C_{t'} ^\lambda } \right)b_{j'} \\
&= a_{i'} \phi _H^R \left( {\left\langle {C_t^\lambda ,r_{m(i,j)} \cdot \,_s C^\lambda } \right\rangle _D \,_{s'} C_{t'} ^\lambda } \right)b_{j'} \bmod \hat A^{(D,\lambda )} \hfill \\
&= \left\langle {C_t^\lambda ,r_{m(i,j)} \cdot \,_s C^\lambda } \right\rangle _D \cdot a_{i'} \phi _H^R \left( {_{s'} C_{t'} ^\lambda } \right)b_{j'} \bmod \hat A^{(D,\lambda )} \hfill \\
&= \left\langle {\phi _j (X),r_{m(i,j)} \cdot \phi _i (Y)} \right\rangle _D \cdot _{(R_{i'} ,s')} C_{(L_{j'} ,t')}^{(D,\lambda )} \bmod \hat A^{(D,\lambda )} \hfill .
\end{aligned}
\]
This gives $\left\langle {X,Y} \right\rangle = \left\langle {\phi _j (X),r_{m(i,j)} \phi _i \left( Y \right)} \right\rangle _D = \left\langle {r_{m(i,j)} \phi _j (X),\phi _i \left( Y \right)} \right\rangle _D $, proving part (b).
\end{proof}
We note the following corollary for future use.
\begin{corollary} \label{c4.0}
Let $M$ be a finite monoid satisfying the $R$-C.A. condition and place the standard cell algebra structure on $R[M]$. Then for any $\lambda \in \Lambda _D \,,\,D \in \mathbb{D}$:
\begin{enumerate}
\item If $\rad_D \left( {\,_L C^\lambda } \right) \ne 0$, then $\rad\left( {\,_L C^{(D,\lambda )} } \right) \ne 0$,
\item If $\rad_D \left( {D_R ^\lambda } \right) \ne 0$, then $\rad\left( {D_R ^{(D,\lambda )} } \right) \ne 0$.
\end{enumerate}
\end{corollary}
\begin{proof}
Assume $\rad_D \left( {\,_L C^\lambda } \right) \ne 0$ and take a $y \ne 0$ in $\rad_D \left( {\,_L C^\lambda } \right)$. Write $y = \sum\nolimits_{s \in L(\lambda )} {c(s) \cdot \,_s C^\lambda } $ and put $Y = \sum\nolimits_{s \in L(\lambda )} {c(s)\,_{(R_i ,s)} C^{(D,\lambda )} } \in \left( {\,_L C^{(D,\lambda )} } \right)_i \subseteq \,_L C^{(D,\lambda )} $ for some $R_i \subseteq D$. Then $Y \ne 0$ and we claim $Y \in \rad\left( {\,_L C^{(D,\lambda )} } \right)$: Take any $X \in \left( {C_R ^{(D,\lambda )} } \right)_j $ . Then by the proposition, if $R_i \,,\,L_j $ are not matched we have $\left\langle {X,Y} \right\rangle = 0$, while if $R_i \,,\,L_j $ are matched, then \[\left\langle {X,Y} \right\rangle = \left\langle {r_{m(i,j)} \phi _j (X),\phi _i \left( Y \right)} \right\rangle _D = \left\langle {r_{m(i,j)} \phi _j (X),y} \right\rangle _D = 0\]
since $y \in \rad_D \left( {\,_L C^\lambda } \right)$. Then $\left\langle {X,Y} \right\rangle = 0$ for any $X \in C_R ^{(D,\lambda )} $ and $Y \in \rad\left( {\,_L C^{(D,\lambda )} } \right)$ as claimed. The proof of ii. is parallel.
\end{proof}
Write $D^2 = \left\{ {xy:x,y \in D} \right\}$ and recall that $\hat A^D = \mathop \oplus \nolimits_{D' < D} R[D'] \subseteq \hat A^{(D,\lambda )} $ for any $\lambda \in \Lambda _D $.
\begin{corollary} \label{c4.2}
For any $\mathcal{D}$-class $D$,
\begin{enumerate} [\upshape(a)]
\item If $D^2 \subseteq \hat A^D $, then $(D,\lambda ) \notin \Lambda _0 $
for any $\lambda \in \Lambda _D $
\item If $D^2 \not\subset \hat A^D $, then $(D,\lambda ) \in \Lambda _0 \Leftrightarrow \lambda \in \left( {\Lambda _D } \right)_0 $.
\end{enumerate}
\end{corollary}
\begin{proof}
$D^2 \subseteq \hat A^D $ if and only if no pair $R_i \,,\,L_j $ are matched.
(a) $D^2 \subseteq \hat A^D $ implies no pair $R_i \,,\,L_j $ is matched, so by proposition \ref{p4.1} $\left\langle {X,Y} \right\rangle = 0$ for every $X \in C_R^{(D,\lambda )} ,Y \in \,_L C_{}^{(D,\lambda )} $. Then ${\text{rad}}\left( {C_R^{(D,\lambda )} } \right) = C_R^{(D,\lambda )} $ and $(D,\lambda ) \notin \Lambda _0 $.
(b) If $D^2 \not\subset \hat A^D $ then $R_i \,,\,L_j $ are matched for at least one pair $i,j$.
For $ \Rightarrow $: If $\left( {D,\lambda } \right) \in \Lambda _0 $, then $\left\langle {X,Y} \right\rangle \ne 0$ for some $X \in \left( {C_R ^{(D,\lambda )} } \right)_j \,,\,Y \in \left( {_L C^{(D,\lambda )} } \right)_i $, where $i,j$ must be matched. Then by proposition \ref{p4.1}, $\left\langle {X,Y} \right\rangle = \left\langle {\phi _j (X),r_{m(i,j)} \phi _i \left( Y \right)} \right\rangle _D \ne 0$, where $\phi _j (X) \in C_R ^\lambda \,,\,r_{m(i,j)} \phi _i (Y) \in \,_L C^\lambda $. So ${\text{rad}}_D \left( {C_R^\lambda } \right) \ne C_R^\lambda $ and $\lambda \in \left( {\Lambda _D } \right)_0 $.
For $ \Leftarrow $: If $\lambda \in \left( {\Lambda _D } \right)_0 $, then there exist $\xi \in C_R ^\lambda \,,\,\eta \in \,_L C^\lambda $ with $\left\langle {\xi ,\eta } \right\rangle _D \ne 0$. Choose $i,j$ with $R_i \,,\,L_j $ matched and let $X = \phi _j ^{ - 1} (\xi ) \in \left( {C_R ^{(D,\lambda )} } \right)_j \,,\,Y = \phi _i ^{ - 1} (r_{m(i,j)} ^{ - 1} \cdot \eta ) \in \left( {_L C^{(D,\lambda )} } \right)_i $. (Note that $r_{m(i,j)} \in G_D $, so $r_{m\left( {i,j} \right)} ^{ - 1} \in G_D $ is well defined.) Then by proposition \ref{p4.1}, $\left\langle {X,Y} \right\rangle = \left\langle {\phi _j (X),r_{m(i,j)} \phi _i \left( Y \right)} \right\rangle _D = \left\langle {\xi ,\eta } \right\rangle _D \ne 0$. Then ${\text{rad}}\left( {C_R^{(D,\lambda )} } \right) \ne C_R^{(D,\lambda )} $ and $(D,\lambda ) \in \Lambda _0 $.
\end{proof}
Consider the question of whether a monoid algebra $k[M]$ is quasi-hereditary (as defined by \cite{CPS}). The following was proved in \cite{May}:
\begin{proposition} \label{p4.2}
Let $k$ be a field and $A$ be a cell algebra over $k$ such that $\Lambda _0 = \Lambda $. Then $A$ is quasi-hereditary.
\end{proposition}
Then by corollary \ref{c4.2} we have
\begin{corollary} \label{c4.3}
Let $k$ be a field and $M$ a finite monoid such that for every class $D \in \mathbb{D}$ , $D^2 \not\subset \hat A^D $ and the group algebra $k[G_D ]$ of the Schutzenberger group for $D$ is a cell algebra with $\left( {\Lambda _D } \right)_0 = \Lambda _D $. Then $A = k[M]$ is quasi-hereditary.
\end{corollary}
Recall that a semigroup $M$ is \emph{regular} if for every $x \in M$ there is a $y \in M$ such that $x = xyx$. In a regular semigroup, every $\mathcal{D}$-class $D$ contains an idempotent $\rho $. Then $\rho = \rho ^2 \in D \cap D^2 $ , so $D^2 \not\subset \hat A^D $. Thus
\begin{corollary} \label{c4.4}
Let $k$ be a field and $M$ a regular finite monoid such that for every class $D \in \mathbb{D}$ , the group algebra $k[G_D ]$ of the Schutzenberger group for $D$ is a cell algebra with $\left( {\Lambda _D } \right)_0 = \Lambda _D $. Then $A = k[M]$ is quasi-hereditary.
\end{corollary}
Note that in a regular semi-group the base $\mathcal{H}$-class $H$ for any class $D$ can be chosen to contain an idempotent. Then $H$ is a (maximal) subgroup of $M$ with $H$ isomorphic to $G_D $. So the condition in corollary \ref{c4.4} can be replaced by the requirement that for any maximal subgroup $G$ of $M$ the group algebra $k[G]$ must be a cell algebra with $\left( {\Lambda _G } \right)_0 = \Lambda _G $.
As a special case we obtain the following result of Putcha \cite{Pu}.
\begin{corollary} \label{c4.45}
For any regular finite monoid $M$, the complex monoid algebra $A=\mathbb{C}[M]$ is quasi-hereditary.
\end{corollary}
\begin{proof} As remarked above, each $\mathbb{C}[G_D]$ will be (split) semisimple by Maschke's theorem. As a product of complex matrix algebras it has a natural cellular basis and (being semisimple) is quasi-hereditary. Each $\mathbb{C}[G_D]$ then satisfies the conditions of \ref{c4.4}, so $A$ is quasi-hereditary.
\end{proof}
Now consider the question of semisimplicity for a monoid algebra $k[M]$. The following criterion for semisimplicity of a cell algebra follows from results in \cite{May}.
\begin{proposition} \label{p4.3}
A cell algebra $A$ over a field $k$ is semisimple if and only if for every $\lambda \in \Lambda $ we have $\,_L C^\lambda = \,_L D^\lambda $ and $C_R ^\lambda = D_R ^\lambda $.
\end{proposition}
\begin{proof}
Assume $A$ is semisimple. Then for every $\lambda \in \Lambda _0 $ we have $\,_L P^\lambda = \,_L D^\lambda $, where $\,_L P^\lambda $ is the principle indecomposable left module corresponding to the irreducible $\,_L D^\lambda $. But since $\,_L P^\lambda $ always has a filtration with $\,_L C^\lambda $ as the ``top'' quotient, we must have $\,_L P^\lambda = \,_L C^\lambda = \,_L D^\lambda $. Since each $\,_L D^\lambda $ is absolutely irreducible, the multiplicity of $\,_L P^\lambda $ as a direct summand in $A$ is just $\dim \left( {\,_L D^\lambda } \right) = \dim \left( {\,_L C^\lambda } \right) = \left| {L(\lambda )} \right|$. So $\dim \left( A \right) = \sum\nolimits_{\lambda \in \Lambda _0 } {\left| {L(\lambda )} \right|^2 } $. Similarly, for every $\lambda \in \Lambda _0 $ we have $P_R ^\lambda = C_R ^\lambda = D_R ^\lambda $ (where $P_R ^\lambda $ is the principle indecomposable right module corresponding to the irreducible $D_R ^\lambda $) and $\dim \left( A \right) = \sum\nolimits_{\lambda \in \Lambda _0 } {\left| {R(\lambda )} \right|^2 } $. But the multiplicity of $\,_L P^\lambda $ as a direct summand in $A$ and the multiplicity of $P_R ^\lambda $ as a direct summand in $A$ must be equal (as both equal the number of primitive idempotents corresponding to $\lambda \in \Lambda _0 $). So $\left| {L(\lambda )} \right| = \left| {R(\lambda )} \right|$ and we can write $\dim \left( A \right) = \sum\nolimits_{\lambda \in \Lambda _0 } {\left| {L(\lambda )} \right| \cdot \left| {R(\lambda )} \right|} $. On the other hand, a direct count of basis elements $_s C_t ^\lambda ,\,\lambda \in \Lambda \,,\,s \in L(\lambda )\,,\,t \in R(\lambda )\,,$ shows that $\dim \left( A \right) = \sum\nolimits_{\lambda \in \Lambda } {\left| {L(\lambda )} \right| \cdot \left| {R(\lambda )} \right|} . $ It follows that $\Lambda _0 = \Lambda $, so $\,_L C^\lambda = \,_L D^\lambda $ and $C_R ^\lambda = D_R ^\lambda $ for every $\lambda \in \Lambda $.
Now assume $\,_L C^\lambda = \,_L D^\lambda $ and $C_R ^\lambda = D_R ^\lambda $ for every $\lambda \in \Lambda $, so $\Lambda _0 = \Lambda $. As in \cite{May}, for $\mu \in \Lambda _0 \,,\,\lambda \in \Lambda $, let $Rd_{\lambda \mu } = [C_R ^\lambda :D_R ^\mu ]$ be the multiplicity of the irreducible $D_R ^\mu $ as a composition factor in $C_R ^\lambda $, let $Ld_{\lambda \mu } = \left[ {\,_L C^\lambda :\,_L D^\mu } \right]$ be the multiplicity of $\,_L D^\mu $ in $\,_L C^\lambda $, and write $RD = \left( {Rd_{\lambda \mu } } \right)\,,\,\,\,LD = \left( {Ld_{\lambda \mu } } \right)$ for the decomposition matrices of $A$. Then $RD$ and $LD$ are both square identity matrices. The right Cantor matrix, $RC$, and left Cantor matrix, $LC$, are the square $\left| {\Lambda _0 } \right| \times \left| {\Lambda _0 } \right|$ matrices where for $\lambda \,,\,\mu \in \Lambda _0 $, $RC_{\lambda \mu } = \left[ {P_R ^\lambda :D_R ^\mu } \right]$ and $LC_{\lambda \mu } = \left[ {\,_L P^\lambda :\,_L D^\mu } \right]$. As shown in \cite{May}, we have $RC = LD^T \cdot RD$ and $LC = RD^T \cdot LD$ (where $T$ denotes the transpose matrix). Then $RC$ and $LC$ are also identity matrices and we must have $P_R ^\lambda = D_R ^\lambda $ and $\,_L P^\lambda = \,_L D^\lambda $ for every $\lambda \in \Lambda = \Lambda _0 $. Since $A$ is a direct sum of principle indecomposable left modules isomorphic to the various$\,_L P^\lambda = \,_L D^\lambda $, it is a direct sum of irreducible modules and therefore semisimple.
\end{proof}
Consider a finite monoid $M$ satisfying the $R$-C.A. condition and place the standard cell algebra structure on $M$.
\begin{corollary} \label{c4.41}
If $k[M]$ is semisimple, then $k[G_D]$ is semisimple for every $D \in \mathbb{D} $.
\end{corollary}
\begin{proof}
If $k[G_D]$ is not semisimple for some $D$, then by the proposition either $\rad_D \left( {\,_L C^\lambda } \right) \ne 0$ or $\rad_D \left( {D_R ^\lambda } \right) \ne 0$. Assume $\rad_D \left( {\,_L C^\lambda } \right) \ne 0$ (if $\rad_D \left( {D_R ^\lambda } \right) \ne 0$ the proof is similar). Then by corollary \ref{c4.0} we have $\rad\left( {\,_L C^{(D,\lambda )} } \right) \ne 0$. But then $k[M]$ is not semisimple by proposition \ref{p4.3}.
\end{proof}
If all the algebras $k[G_D ]$ are semisimple, we would like a condition guaranteeing that $k[M]$ itself is semisimple.
For any $\mathcal{D}$-class $D$, let $\mathbb{L}(D)$ be the set of all $\mathcal{L}$-classes contained in $D$ and $\mathbb{R}(D)$ be the set of all $\mathcal{R}$-classes contained in $D$. Define the ``bijection condition'' on $D$:
\textbf{Bijection condition:} There exists a bijection $F:\mathbb{L}(D) \to \mathbb{R}(D)$ such that $L$ and $F(L)$ are matched while $L$ and $R$ are not matched if $R \ne F(L)$.
\begin{proposition} \label{p4.4}
Let $k$ be a field and $M$ a monoid such that for every $\mathcal{D}$-class $D$
\begin{enumerate}
\item $k[G_D ]$ is a semisimple cell algebra and
\item the bijection condition is satisfied for $D$.
\end{enumerate}
Then $k[M]$ is a semisimple cell algebra.
\end{proposition}
\begin{proof}
Place the standard cell algebra structure on $k[M]$ as in theorem \ref{t3.1}. We will use proposition \ref{p4.3} to show semisimplicity. Take $\left( {D,\lambda } \right) \in \Lambda $. We will show ${\text{rad}}\left( {\,_L C^{(D,\lambda )} } \right) = 0$ and therefore $\,_L C^{(D,\lambda )} = \,_L D^{(D,\lambda )} $. Take any $Y = \sum\nolimits_k {Y_k } \in {\text{rad}}\left( {\,_L C^{(D,\lambda )} } \right)$ where $Y_k \in \left( {_L C^{(D,\lambda )} } \right)_k $ . Then $\left\langle {X,Y} \right\rangle = 0$ for any $X \in C_R ^{(D,\lambda )} $. If $X \in \left( {C_R ^{(D,\lambda )} } \right)_j $, then $\left\langle {X,Y_k } \right\rangle = 0$ whenever $L_j \ne F(R_k )$ by proposition \ref{p4.1}. Then we must also have $\left\langle {X,Y_k } \right\rangle = 0$ when $L_j = F(R_k )$. But when $L_j ,R_k $ are matched, proposition \ref{p4.1} gives $\left\langle {X,Y_k } \right\rangle = \left\langle {\phi _j (X),r_{m(i,j)} \phi _i \left( {Y_k } \right)} \right\rangle _D = 0$. Since $\phi _j (X)$ is an arbitrary basis element in $C_R ^\lambda $, we have $r_{m(i,j)} \phi _i \left( {Y_k } \right) \in {\text{rad}}_D \left( {\,_L C^\lambda } \right)$. But since $k[G_D ]$ is semisimple, ${\text{rad}}_D \left( {_L C^\lambda } \right) = 0$ by proposition \ref{p4.3}. Then $r_{m(i,j)} \phi _i \left( {Y_k } \right) = 0$, and since $r_{m\left( {i,j} \right)} \in G_D $ is invertible, $Y_k = 0$. Since this is true for each $k$, we have $Y = 0$. So ${\text{rad}}\left( {_L C^{(D,\lambda )} } \right) = 0$. A parallel argument shows that ${\text{rad}}\left( {C_R ^{(D,\lambda )} } \right) = 0$, so $k[M]$ is semisimple by proposition \ref{p4.3}.
\end{proof}
Recall that an \emph{inverse} of an element $a$ in a semi-group is an element $a^{ - 1} $ such that $aa^{ - 1} a = a$ and $a^{ - 1} aa^{ - 1} = a^{ - 1} $. An \emph{inverse semi-group} is a semi-group in which each element has a unique inverse. A standard result in semi-group theory is that any inverse semi-group satisfies the bijection condition. (In fact each $\mathcal{L}$-class and each $\mathcal{R}$-class contains a unique idempotent. Given an $\mathcal{L}$-class $L$, the unique matching $\mathcal{R}$-class $R = F(L)$ is the class containing the same idempotent as $L$.) Then proposition \ref{p4.4} and corollary \ref{c4.41} yield
\begin{corollary} \label{c4.5}
Let $k$ be a field and $M$ a finite monoid which is an inverse semi-group and satisfies the $k$-C.A. condition. Then $k[M]$ is semisimple if and only if $k[G_D]$ is semisimple for every $\mathcal{D}$-class $D$.
\end{corollary}
For an inverse semi-group the base $\mathcal{H}$-class $H$ for any class $D$ can be chosen to be a (maximal) subgroup of $M$ with $H$ isomorphic to $G_D $. So the condition in corollary \ref{c4.5} can be replaced by the requirement that for any maximal subgroup $G$ of $M$ the group algebra $k[G]$ must be a semisimple cell algebra.
It is well known that if the finite monoid $M$ is an inverse semi-group and the field $k$ has characteristic not dividing the order of any $G_D$, then $k[M]$ is semisimple (see e.g. \cite{CP}, which cites \cite{Og}). In this case each $k[G_D]$ is semisimple by Maschke's theorem. If $k$ is also algebraically closed, then as remarked above, $M$ satisfies the $k$-C.A. condition. Thus corollary \ref{c4.5} yields the semisimplicity of $k[M]$ under the additional hypothesis of algebraic closure for $k$.
\section{Twisted monoid algebras}
A \emph{twisting} on a monoid $M$ (with values in a commutative domain $R$ with unit $1$) is a map $\pi :M \times M \to R$ such that (i) for all $x,y,z \in M$, \[\pi \left( {x,y} \right)\pi \left( {xy,z} \right) = \pi \left( {x,yz} \right)\pi \left( {y,z} \right)\] and (ii) for all $x \in M$, \[\pi \left( {x,id} \right) = 1 = \pi \left( {id,x} \right)\] (where $id$ is the identity in $M$).
Given a twisting $\pi $ on $M$, define an algebra $R^\pi \left[ M \right]$ to be the free $R$-module with basis $M$ and multiplication $x \circ y = \pi (x,y)\,xy$ for $x,y \in M$. Then $R^\pi \left[ M \right]$ is an associative $R$-algebra with unit $1$.
We would like conditions under which $R^\pi \left[ M \right]$ will be a cell algebra. In \cite{Wil} and \cite{GX}, Wilcox and Guo and Xi have investigated when $R^\pi \left[ M \right]$ can be a cellular algebra. Much of the difficulty in their analyses involves defining the involution anti-isomorphism $ * $ required for a cellular algebra. Since cell algebras don't require such a map, the corresponding results are both simpler to obtain and of more general applicability. We require one ``compatibility'' condition for our twisting:
\begin{definition} \label{d5.1}
A twisting $\pi $ on a monoid $M$ is \emph{compatible} if for all $a,x \in M$
\begin{enumerate}
\item If $ax \mathcal{D} x$ , then $\pi \left( {a,y} \right) = \pi \left( {a,x} \right)$ whenever $y \in H_x $,
\item If $xa\mathcal{D}x$ , then $\pi \left( {y,a} \right) = \pi \left( {x,a} \right)$ whenever $y \in H_x $.
\end{enumerate}
\end{definition}
In \cite{GL}, $\pi $ is defined to be an $\mathcal{LR}$-twisting if $x\mathcal{L}y \Rightarrow \pi \left( {x,z} \right) = \pi \left( {y,z} \right)$ and $y\mathcal{R}z \Rightarrow \pi \left( {x,y} \right) = \pi \left( {x,z} \right)$ for all $x,y,z \in M$. Clearly any $\mathcal{LR}$-twisting is compatible.
\begin{theorem} \label{t5.1}
Let $M$ be a finite monoid satisfying the $R$-C.A. condition. Let $\pi $ be a compatible twisting on $M$. Then $R^\pi \left[ M \right]$ has a cell algebra structure with the same $\Lambda ,R,L$ and cell basis $C$ as for the standard cell algebra structure on $R[M]$.
\end{theorem}
\begin{proof}
Since as $R$-modules $R^\pi \left[ M \right]$ and $R[M]$ are identical, $C$ is an $R$-basis for $R^\pi \left[ M \right]$ and we need only check that conditions (i) and (ii) for a cell algebra are satisfied for the new multiplication.
For (i), assume $a$ is a basis element, $a \in M$, and take any $_{(R,s)} C_{(L,t)}^{(D,\lambda )} \in C$. Then $_{(R,s)} C_{(L,t)}^{(D,\lambda )} \in R[H]$ is an $R$-linear combination of elements in an $\mathcal{H}$-class $H = L \cap R \subseteq D$. By corollary \ref{c2.2} we have two possibilities: I. $a \cdot R\left[ H \right] \subseteq \hat A^D = \mathop \oplus \nolimits_{D' < D} R\left[ {D'} \right]$ or II. $a \cdot R\left[ H \right] \subseteq D$ . If I. holds, $a \cdot _{(R,s)} C_{(L,t)}^{(D,\lambda )} \subseteq \hat A^D \subseteq \hat A^{(D,\lambda )} $ and we can take the coefficients $r_L $ required in (i) to be all $0$. So assume II. holds. Take an element $x \in H$ and let $c = \pi (a,x)$. Since $ax \in D = D_x $
, compatibility gives $\pi \left( {a,y} \right) = c$ for any $y \in H$. But then, by linearity, $a \circ _{(R,s)} C_{(L,t)}^{(D,\lambda )} = c \cdot a \cdot _{(R,s)} C_{(L,t)}^{(D,\lambda )} $. We can then take the coefficients $r_L $ required in (i) to be just $c$ times the corresponding coefficients in the cell algebra $R[M]$.
The proof of condition (ii) is parallel.
\end{proof}
Let $M$ be a finite monoid satisfying the $R$-C.A. condition and place the standard cell algebra structure on $M$. Suppose $R_i \,,\,L_j $ are matched classes in a $\mathcal{D}$-class $D$ of $M$. Then by definition there exist $x \in L_j \,,\,y \in R_i $ such that $xy \in D$. If $\pi $ is a compatible twisting on $M$ define $c(j,i) = \pi \left( {x,y} \right)$. Then $\pi \left( {a,b} \right) = c(j,i)$ for any $a \in L_j \,,\,b \in R_i $, so for any $X \in R[L_j ]\,,\,Y \in R[R_i ]$ we have $X \circ Y = c(j,i)XY$. It follows that for $X \in \left( {C_R ^{(D,\lambda )} } \right)_j \,,\,Y \in \left( {_L C^{(D,\lambda )} } \right)_i $ we have $\left\langle {X,Y} \right\rangle ^\pi = c(j,i)\left\langle {X,Y} \right\rangle $ where $\left\langle {X,Y} \right\rangle ^\pi $ is the bracket in the cell algebra $R^\pi \left[ M \right]$.
\begin{definition} \label{d5.2}
A twisting $\pi $ on a monoid $M$ is \emph{strongly compatible} if for all $a,x \in M$
\begin{enumerate}
\item If $ax \mathcal{D} x$ , then $\pi \left( {a,y} \right) = \pi \left( {a,x} \right) \ne 0$
whenever $y \in H_x $,
\item If $xa \mathcal{D} x$ , then $\pi \left( {y,a} \right) = \pi \left( {x,a} \right) \ne 0$
whenever $y \in H_x $.
\end{enumerate}
\end{definition}
So for a strongly compatible twisting we have $c(j,i) \ne 0$ whenever $R_i \,,\,L_j $ are matched classes. But $c(j,i) \ne 0$ in the domain $R$ yields $\left\langle {X,Y} \right\rangle ^\pi = 0 \Leftrightarrow \left\langle {X,Y} \right\rangle = 0$. Using this observation, it is easy to modify the proofs to obtain the following generalizations to twisted monoid algebras of the results in section 4.
In the following, assume $R$ is a domain, $M$ a finite monoid satisfying the $R$-C.A. condition, and $\pi $ a strongly compatible twisting from $M$ to $\pi $. Put the standard cell algebra structures on $R[M]$ and $R^\pi[M]$ as given by theorems \ref{t3.1} and \ref{t5.1}. Let $\Lambda ,\Lambda _0 ,\left\langle { - , - } \right\rangle , \rad$, etc., refer to the cell algebra $R[M]$ and $\Lambda ^\pi ,\Lambda ^\pi _0 ,\left\langle { - , - } \right\rangle ^\pi \rad^\pi$, etc., refer to the cell algebra $R^\pi (M)$.
Recall that $\Lambda ^\pi = \Lambda $.
\begin{proposition} \label{p5.1}
For any $\lambda \in \Lambda _D \,,\,D \in \mathbb{D}$:
\begin{enumerate}
\item If $\rad_D \left( {\,_L C^\lambda } \right) \ne 0$, then $\rad^\pi \left( {\,_L C^{(D,\lambda )} } \right) \ne 0$,
\item If $\rad_D \left( {D_R ^\lambda } \right) \ne 0$, then $\rad^\pi \left( {D_R ^{(D,\lambda )} } \right) \ne 0$.
\end{enumerate}
\end{proposition}
\begin{proposition} \label{p5.2}
For any $\mathcal{D}$-class $D$,
\begin{enumerate} [\upshape(a)]
\item If $D^2 \subseteq \hat A^D $, then $(D,\lambda ) \notin \Lambda^\pi _0 $ for any $\lambda \in \Lambda _D $
\item If $D^2 \not\subset \hat A^D $, then $(D,\lambda ) \in \Lambda^\pi _0 \Leftrightarrow \lambda \in \left( {\Lambda _D } \right)_0 $.
\end{enumerate}
\end{proposition}
\begin{proposition} \label{p5.3}
Let $R = k$ be a field. Assume that for every class $D \in \mathbb{D}$ , $D^2 \not\subset \hat A^D $ and $k[G_D ]$ is a cell algebra with $\left( {\Lambda _D } \right)_0 = \Lambda _D $. Then $A^\pi = k^\pi[M]$ is quasi-hereditary.
\end{proposition}
\begin{proposition} \label{p5.4}
Let $R =k $ be a field. Assume that $M$ is regular and that for every class $D \in \mathbb{D}$ , $k[G_D ]$ is a cell algebra with $\left( {\Lambda _D } \right)_0 = \Lambda _D $. Then $A^\pi = k^\pi[M]$ is quasi-hereditary.
\end{proposition}
\begin{proposition} \label{p5.5}
If $R = k$ is a field and $k^\pi[M]$ is semi-simple, then $k[G_D]$ is semi-simple for every $D \in \mathbb{D} $.
\end{proposition}
\begin{proposition} \label{p5.6}
Let $R = k$ be a field. Assume that for every $\mathcal{D}$-class $D$
\begin{enumerate}
\item the cell algebra $k[G_D ]$ is a semi-simple and
\item the bijection condition is satisfied for $D$.
\end{enumerate}
Then the cell algebra $k^\pi[M]$ is semi-simple.
\end{proposition}
\begin{proposition} \label{p5.7}
Let $R = k$ be a field. Assume the monoid $M$ is also an inverse semi-group. Then $k^\pi[M]$ is semi-simple if and only if $k[G_D]$ is semi-simple for every $\mathcal{D}$-class $D$.
\end{proposition}
The various examples such as Brauer algebras, Temperly-Lieb algebras, and other parition algebras which were studied and shown to be cellular in \cite{Wil} and \cite{GX} are all twisted monoid algebras with a compatible twisting on a monoid satisfying the $R$-C.A. condition. Thus they can be seen to be cell algebras by Theorem \ref{t5.1} without constructing the anti-isomorphism needed for the cellular structure. Related algebras which lack the anti-isomorphism $ * $, and hence are not cellular, could also be shown to be cell algebras by Theorem \ref{t5.1}. We note again that questions such as whether an algebra is quasi-hereditary or semi-simple are not much harder to answer for cell algebras than for cellular algebras.
|
1,314,259,996,465 | arxiv | \section{Introduction}
Positron emission tomography (PET) is a well established molecular imaging technique\cite{JONES2017}.
It involves administration of radiolabeled molecules containing elements emitting positrons to patients. Photons created due to the positron-electron annihilation are measured in order to localize and quantify the radiotracer~\cite{CHERRY2017A}. The principles underlying PET allow to study many biological processes e.g. metabolism (brain and cancer activity), hypoxia, apoptosis, proliferation (cancer), angiogenesis and inflammation (atherosclerotic plaque)\cite{JONES2017}. PET has been used extensively for research and clinical applications, particularly concerning imaging of brain function in neurodegenerative diseases, diagnosis and treatment of cancer (theranostic) or monitoring of radio- and pharmacotherapy progress. By choosing different markers, one can select different metabolic processes that are observed during scanning. All modern scanners, currently available on the market, detect $\gamma$-photons by usage of inorganic crystal scintillators~\cite{Karp2008a, SLOMKA2016, Van2016}. Scintillators in the form of crystal (eg. LSO or LYSO) are expensive but have undoubted advantages such as large density and high atomic number, and therefore a large cross-section for the interaction with annihilation photons through the photoelectric effect, in addition to a good energy resolution.
\\
Despite the advantages of the current PET scanners they are characterized by a number of technical and conceptual limitations. Actually, the field of view of the body that can be imaged at one shot does not typically exceed 250~mm in length~\cite{SLOMKA2016}. This means that any full-body scan has to be merged from several subsequent, not simultaneous measurements. Therefore, the information about temporal changes in radiotracer distribution is available only for the fraction of the body within the field of view of the scanner. In the present-day PET scanners less than 1 \% of the photons emitted from a patient body are detected as a consequence of a limited axial field of view (AFOV)~\cite{CHERRY2018}. For these reasons, the concept of a total-body scanner which allows almost complete detection of the radiation emitted from the body appears naturally desirable~\cite{CHERRY2017A, CHERRY2018}. Furthermore, the total-body PET will enable decreasing in the time of diagnostics or the amount of the administered radiation dose and it may also enable more effective application of shorter living tracers. Recently different designs of total-body scanners based on the standard technologies were introduced e.g. using resistive plate chamber (RPCs)~\cite{Blanco2006}, straw tubes~\cite{LWS2005,SUN2005} and
crystal scintillators~\cite{CHERRY2017A, CHERRY2018}.
The total-body PET based on crystal scintillators is already in the stage of commissioning~\cite{BERG2018} and delivering first total-body images~\cite{BADAWI2019}.
\\
The J-PET (Jagiellonian-PET) project addresses the innovative application of plastic scintillators as a detection material for the PET \cite{Moskal2011A,MOSKAL2016A}.
The application of plastic scintillators enables construction of a cost effective total-body scanner due to the less expensive detector material and the possibility of the construction of the scanner from the long axially arranged plastic strips~\cite{Moskal2011A, Moskal2014A, Moskal2015A,MOSKAL2016A,KOWALSKI2018}. Moreover the readout components are placed outside of detection chamber giving a chance for hybrid PET/MR construction. In the axial arrangement of scintillator strips, any extension of AFOV by elongation of the plastic scintillators may be achieved without significant increase of costs because the number of photomultipliers and electronic channels remains independent of the AFOV.
Due to the low light attenuation in the plastic scintillators, the length of modules could approach even 2~m. Though, this comes with deterioration of the Coincidence Resolving Time (CRT), which decreases with elongation of modules \cite{MOSKAL2016A} it can be compensated by the registration of light escaping from the scintillators with the additional layer of wavelength shifters~\cite{SYMRSKI2017}. In addition the J-PET design enables possibility of simultaneous metabolic and morphometric imaging based on the measurement of properties of positronium atoms produced inside the body during the PET diagnosis~\cite{MOSKAL2019,NATURE2019}.
\\
In this paper the prototype built out of 24 modules, forming a cylindrical diagnostic chamber with the inner diameter of 360~mm and the AFOV of 300~mm, is presented. The developed methods of synchronization and calibration of the entire setup composed of plastic scintillators, Photomultipliers (PMT) and Front-End Electronics (FEE) are described in detail. The result of a simplified image reconstruction is shown in the last chapter.
\section{General concept of the J-PET scanner}
J-PET exploits time information instead of energy to determine place of
annihilation. Scintillating signals from plastics are very “fast” (typically, 0.5 ns rise time and 1.8~ns decay time~\cite{saint,eljen,wieczorek2017}).
Such fast signals allow for superior time resolution and decrease pile-ups with respect to crystals detectors as e.g. LSO or BGO with decay times equal to 40 ns and 300 ns, respectively \cite{omega}.
In order to take advantage of these superior timing properties of plastic scintillators and to decrease the dead time due to the electronic signal processing in J-PET, the charge measurement corresponding to the deposited energy of the gamma was replaced with measurement of Time Over Threshold (TOT) ~\cite{Paka2017}.
The J-PET tomograph is constructed from axially arranged strips of plastic scintillators. Annihilation $\gamma$ photons with energy of 511 keV interact in plastic scintillators through the Compton effect \cite{MOSKAL2018} in which the deposited energy varies from event-to-event. Due to the low light attenuation plastic scintillators act as effective light-guides for these secondary photons produced by interaction of the annihilation radiation. Hence the examination chamber can be built out of long modules placed along the patient's body. Each plastic strip is read out by photomultipliers at two ends (see Fig. \ref{fig:Reconstruction}, left panel).
Since the readout is placed outside of the diagnostic chamber, the main cost of extending the AFOV of the scanner lays in cost of scintillating material.
The position of interaction with the photons in the scintillators can be determined from the time
difference of light signal arriving at photomultipliers placed at each end of detection module
\begin{equation}
\Delta I =(t_1-t_2)\times \upsilon/2
\end{equation}
where $\Delta I$ denotes the distance between the interaction point and the center of the module, t$_1$ an t$_2$ stand for times of arrival of light signal at each side and $\upsilon$ is an effective velocity of light signal within the scintillator. Then, the position of annihilation along Line Of Response (LOR) can be determined using the Time of Flight (TOF) method (see Fig. \ref{fig:Reconstruction} for pictorial description)
\begin{align}
TOF&=(t_1+t_2)/2 - (t_3+t_4)/2: & \Delta x&=TOF \times c/2,
\end{align}
where $\Delta x$ denotes distance of annihilation point from the middle of LOR, c stands for the speed of light, $t_1$ and $t_2$ are the times measured at the two ends of module A and $t_3$ and $t_4$ denote times registered with module B.
\begin{figure}[h]
\includegraphics[scale=0.75]{position.png}
\label{fig:subim1}
\includegraphics[width=4.5cm, height=3.5cm]{fig1right.png}
\label{fig:subim2}
\caption{
(Left) Schematic representation of an annihilation point reconstruction based on measured differences between arrival times $t_{i}$ of light pulses generated in two detection modules by the annihilation gamma quanta. $\Delta$x denotes distance of the annihilation point from the middle of the LOR. (Right) 24-modules full prototype of the J-PET detector.
Scintillator strips are covered with black foil and read out by photomultipliers inserted into aluminium tubes.
}
\label{fig:Reconstruction}
\end{figure}
\section{J-PET prototype electronics, time and charge measurement}
The first operating prototype of the J-PET tomograph, shown in the right panel of Fig.~\ref{fig:Reconstruction}, consists of the 24 detection modules. A basic part of the prototype is the single detection module. It consists of
(5 $\times$ 19 $\times$ 300~ mm$^3$) strip of BC-420 scintillator (Saint Gobain Crystals~\cite{saint}) read out by two R4998 Hamamatsu photomultipliers~\cite{HAMAMATSU} coupled optically to the scintillator with a silicone optical grease BC-630 (Saint Gobain Crystals). In order to increase the number of photons which can reach the photomultipliers, the scintillator is wrapped with the Vikuiti reflecting foil (3M Optical Systems~\cite{VIKUITI}). The lightproof of the detection module is additionally assured by a tight cover made of the Tedlar foil (DuPont~\cite{Dupont}).
\\
Electric signals from all detection modules are passively split into four, next amplified and sampled by a specially designed FEE board~\cite{Paka2017}.
It comprises 48 ABA-51563 amplifiers and 8 LTC2620 DACs to set individual thresholds fed into comparators implemented solely on a Field Programmable Gate Array (FPGA) device. The sampling of analog signals on an FPGA is executed by employing its Low Voltage Differential Signal (LVDS) buffers as comparators~\cite{Paka2014}. It is worth to stress that also other FPGA based designs for sampling of fast signals in the voltage domain were developed recently~\cite{WON2016,WON2016B,WON2018}.
In the prototype presented in this article, the sampling of the analog voltage signal is done at four different constant thresholds at the rising and falling edges as it is depicted schematically in the left part of Fig.~\ref{fig:signal probing}. The measurement of TOT and Time to Digital Conversion (TDC) results in the digital characteristics of the probed signal, shown in the right part of Fig.~\ref{fig:signal probing}. Combination of rising and falling edges information allows for determination of signal's charge. The time determined from the crossing of the smallest threshold allows one to estimate a start time of the signal. The times measured at higher thresholds may be used to improve the precision of the start time determination e.g. by fitting a line
to the time stamps measured at the leading edge of the signal~\cite{KIM2009B,XIE2005,XIE2009A},
or by the reconstruction of the full signal waveform which may be done by fitting a curve describing the shape of the signal using either the method of library of synchronized model signals~\cite{Moskal2015A} or e.g. the signal shape reconstruction by means of the compressive sensing theory \cite{Lech2014B, Racz2015}.
\begin{figure}[thpb]
\centering
\includegraphics[width=0.50\textwidth]{fig3.png
\caption{ Pictorial representation of electric signal probing. After signal processing, four pairs of points are acquired at four selected voltage thresholds.}
\label{fig:signal probing}
\end{figure}
The general block diagram of electronics supporting the 24 modular J-PET tomograph prototype is shown in Fig.~\ref{fig:J-PET tomographs}. For the collective power supply of 48 photomultipliers (PMT), the CAEN SY4527 card and the CAEN SY5527 power supply were used (CAEN 2015)~\cite{Caen}.
48 analog signals from the PMTs are supplied to 4 FEE modules, which are mounted on a single Trigger Readout Board v3 (TRBv3)~\cite{Traxler2011}. The platform operates in a continuous readout mode which helps maximizing the amount of collected data without preliminary selection. The board is equipped with 5 FPGA devices (Lattice ECP3), from which one operates as a master and four others as slaves being programmed with TDC firmware. The TDCs digitize the input signals and store them in buffers, which are read out at a fixed frequency of 50 kHz. A signal to initiate the readout as well as a reference signal for precise time synchronization between the TDC is provided by the master FPGA. Collected data is then sent via Gigabit Ethernet network to the storage and further analysis~\cite{Korcyl2015, Korcyl2018a}.
\begin{figure}[thpb]
\centering
\includegraphics[width=0.5\textwidth]{fig4.png}
\caption{Block diagram of electronics supporting the 24 modular J-PET tomograph prototype. The HV denotes the high voltage supplying the photomultipliers (PMT) which are read out by the Front-End electronics connected to the Trigger and Readout Board version 3 (TRBv3) and Data Acquisition System (DAQ).}
\label{fig:J-PET tomographs}
\end{figure}
\begin{figure}[thpb]
\centering
\includegraphics[width=0.52\textwidth]{calib1.png}
\caption{Block diagram of the general flow of the J-PET detector calibration. We start with the photomultipliers gains matching, then we calibrate the differential non-linearity of the FEE, threshold values and TOT measurement. This allows for the final time calibration and synchronization of the full detector.}
\label{fig:calib1}
\end{figure}
The described method of timing measurements by means of FPGA devices requires calibration due to following reasons. The internal carry-chain elements used as delay units have a different physical arrangement inside the integrated circuit and the pulse transition times of individual channels can vary up to a nanosecond. Also, the lengths and shapes of signal paths on printed circuits become important in the field of picosecond accuracy of time measurements. The applied calibration procedure assigns to various channels such a time shift that the time difference between them and an arbitrary chosen reference channel tends to zero. The typical time difference spread for one of the channels passing FEE and TRBv3 is shown in Fig.~\ref{fig:multi-threshold board}
giving time resolution of the order of 30~ps.
The general scheme of the calibration of the whole system is shown in Fig.~\ref{fig:calib1}, while in the next section we present calibration of the photomultipliers gains. The Front-End Electronics calibration and time synchronization of the whole J-PET detection system are discussed in Section V and VI, respectively.
\begin{figure}[h]
\includegraphics[width=0.50\textwidth]{figure3b.png}
\caption{
Time difference distribution obtained with a system equipped with FEE and TRBv3. The resolution of the obtained time difference amounts to $\sigma$ = 28.2 $\pm$ 2.5 ps.}
\label{fig:multi-threshold board}
\end{figure}
\section{PMT gain calibration}
PMT gain calibration is based on observing single photoelectrons. The method of recording individual photoelectrons has been used in many experiments
eg.~\cite{Ronzhin2010,Baturin2006}.
The calibrated PMT was optically coupled to a gamma irradiated scintillator. The second reference PMT, placed on the other end of the same scintillator was working in coincidence to eliminate false signals, for example, from the thermal noise of the calibrated PMT.
For a given voltage applied to the calibrated PMT waveforms of signals were collected and integrated using Riemans integral. The value of the integral was then divided by oscilloscope channels resistance used in waveforms collection, equal to 50~$\Omega$, which resulted in a charge calculation. Since the surface of the photomultiplier was obstructed in such a way that mainly one/two optical photons from the scintillator could successfully interact with its window, the charge calculated based of waveforms originates mainy from one initial photoelectron.
The histogram showing example events coming from observation of 0, 1 and 2 photoelectrons can be seen in the upper part of Fig.~\ref{fig:calibration}. Histograms of charges were fitted with a function given by Eq.~(\ref{eq3}):
\begin{eqnarray}
F(x)&=&N_0\exp{\frac{-(x-X_0)^2}{2\sigma _0^2}}+N_1\exp{\frac{-(x-X_1)^2}{2\sigma _0^2}}\nonumber \\
&+& N_2\exp{\frac{-(x-2X_1)^2}{2\sigma _0^2}},
\label{eq3}
\end{eqnarray}
where N$_i$ (i = 0, 1, 2) are normalisation constants,
X$_i$ are mean (expectation) values and $\sigma_i^2$ are Gaussian variances. According to linear scaling, it was assumed that the
expectation value for two photoelectrons (2X$_1$) is twice as large as the maximum position (X$_1$) for single photoelectron. The black curve in the upper part of Fig.~\ref{fig:calibration} is the sum of three Gaussian curves corresponding to situations when no photoelectron was registered (the first maximum),
when one photoelectron was observed (the central maximum) and when there were two photoelectrons (the submerged red line).
For charges above 0.6 pC the red line overlaps with the black one. The value of the X$_1$ parameter obtained from the fit is then used as a measure of the gain of photomultiplier.
The gain calibration curves can be obtained performing fits of
Eq.~(\ref{eq3}) to histograms measured with different voltages applied to PMT’s. The typical example
of such a curve is shown in the lower panel of Fig.\ref{fig:calibration}
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.37\textwidth]{fig6left.png}
\includegraphics[width=0.433\textwidth]{figure5b.png}
\caption{(Upper panel) Histogram of the charge measured during calibration of a photomultiplier with 0, 1 and 2 photoelectrons maxima (see text). (Lower panel) Example of a gain calibration curve. Average values of PMT output charges induced by one photoelectron detected as a function of the voltage applied to the photomultiplier is presented. The black continuous line denotes an exponential fit to the experimental points.}
\label{fig:calibration}
\end{figure}
Values of gains gathered for all 48 photomultipliers used in the J-PET prototype, operated at the voltage of 2.25 kV, are shown in Fig.~\ref{fig: Hamamatsu1}. As one can see they differ by a factor of about 3.
\begin{figure}[h]
\centering
\includegraphics[width=0.50\textwidth]{figure6.png}
\caption{Values of the gain for all the R4998 Hamamatsu PMT’s obtained from gain calibration curves for the voltage of 2250~V. Note that the histogram includes results for 51 PMTs and out of them 48 were used in the prototype described in this article.}
\label{fig: Hamamatsu1}
\end{figure}
\section{FEE calibration}
As it was mentioned in Sec. 3 the J-PET Data Acquisition System (DAQ)~\cite{Korcyl2015,Korcyl2018a} is based on the TRBv3 \cite{Traxler2011} and on specially designed FEE~\cite{Paka2014,Paka2017}. Time measurement using FPGA is based on the signal delay resulting from its propagation through the individual elements of the chain of delays. Depending on the number of elements through which the signal has passed until the instant of measurement, we obtain different time values and calculating this time interval we assume that it is proportional to the number of passed elements. This calculation is correct as long as the propagation time of the signal through each of the chain elements is the same. In general, this assumption is not fulfilled and this problem is known as Differential Non-Linearity (DNL) of time propagation of the signal in the TDC system. The level of non-linearity for individual elements is dependent on the temperature and voltage fluctuations in the electronic components during system operation. This is one of the major factors that worsen the resolution of time measurement therefore a calibration of DNL is needed. It consists in giving to the TDC input a large number of signals which are accidental, uncorrelated with an internal clock signal and homogeneously distributed within the interval of the time measurement. From these signals a histogram of time as a function of delay chain element number is created, where time is calculated as a sum of delays at all elements counting from the beginning of the delay chain till the given element. Example of DNL correction histogram is shown in
Fig.~\ref{fig:DNL}.
\begin{figure}[h]
\centering
\includegraphics[width=0.40\textwidth]{fig8top.png}
\includegraphics[width=0.40\textwidth]{fig8bottom.png}
\vspace{0.2cm}
\caption{TDC Calibration for Differential Non-Linearity (DNL). (Upper panel) Sum of signals delays for different interval number. (Lower panel) Zoom of the upper plot with time correction constant visualization.}
\label{fig:DNL}
\end{figure}
In the case of ideal elements having the same delay, the histogram should be arranged in a stepped line with the same difference in height for each subsequent interval (blue dotted line in the lower panel
of Fig.~\ref{fig:DNL}). As can be seen in this figure, however, the differences between the intervals vary which is a sign of DNL, i.e. nonlinearity on the elements of the delays chain. Such a histogram is created for each chain of delays through which signals are passed and the information contained therein is used to correct the measured signals.
In addition to the DNL calibration it is also necessary to calibrate the measurement of the width of the signals. Information about the signal size can be obtained on the basis of time measurements. With the increasing signal charge, its width increases, and thus also the time when the signal voltage exceeds some pre-set threshold. For this measurement one needs information about two times: the time when the rising edge exceeded the level of the threshold and the time when the falling edge crosses the level of this threshold. Such measurement of the TOT allows for determination of charge with very good resolution which then can be used for rejection of noise originating from registration of photons scattered in the body of the patient.
Due to the very short signals from the used scintillators and photomultipliers (order of ns), delays were deliberately introduced when measuring the falling edge. In TDC, for each channel a chain of delay elements has been implemented through which the falling edge of signal must pass. This results in an artificial extension of the signal, allowing, however, the measurement of TOT for very narrow signals. The delays on different TDC channels may slightly differ from each other and this entails the need for TOT calibration. Such calibration is made using an additional oscillator placed on the TRBv3 board providing a reference signal of 10~ns wide at the input of each channel. Thanks to this, it is possible to simultaneously measure the signal width for all channels and find the value of the calibration parameters of the falling edge delays for each channel. On the basis of the mean of the measured TOT values for each channel, the values of edge times of falling signals on the given channel are corrected. After subtraction of the calibration signal width, the corrected time should give zero values on all channels. Proof of the proper operation of this procedure is presented in Fig.~\ref{fig:TDC}.
\begin{figure}[h]
\centering
\includegraphics[width=0.40\textwidth]{figure8.png}
\caption{TOT values measured for all TDC channels of the 24-modules J-PET prototype after calibration.}
\label{fig:TDC}
\end{figure}
In the J-PET prototype dedicated FEE uses LVDS buffers to compare reference voltage (threshold level) with measured signals. The LVDS buffer work in the range from 0 V to 2 V while signals registered from photomultipliers have negative amplitudes. Therefore, to be able to apply a threshold on a signal with a negative amplitude, the base level of the signal has been shifted from zero to 2.048~V. As a result, signals of negative amplitude remain in the domain of positive voltages. The schematic representation of this procedure is presented in the upper panel of Fig.~\ref{fig:threshold level}.
\begin{figure}[h]
\includegraphics[width=0.40\textwidth]{figure9aa.png}
\includegraphics[width=0.40\textwidth]{fig10bottom.png}
\caption{(Upper panel) Pictorial representation of the calibration of threshold values applied to the J-PET photomultiplier signals. The value of real threshold equals to the difference between levels of base line and voltage at LVDS buffer.; (Lower panel) Schematic view of signals sequence used for threshold level calibration.
\label{fig:threshold level}}
\end{figure}
\\
The relation between the actual voltage threshold sampling, the recorded pulse and
the voltage applied to the LVDS comparator input has been determined in the following way.
A sequence of pulses of variable amplitude (shown schematically in the lower part of
Fig.~\ref{fig:threshold level}) was repeatedly sent from programmed generator to each input channel of readout electronics.
The parameters of this sequence were as follows:
\begin{itemize}
\item amplitude: changing from -100 mV to -600 mV with a step of 10 mV
\item rise time and fall time: 1 ns (10$\%$ \text{-} 90$\%$ of amplitude)
\item signal width at half amplitude: 4 ns
\end{itemize}
While sending the pulse sequences from the generator to the readout electronics, the voltage
level on the LVDS buffer was changed and the number of signals that exceeded this level
was counted. An example of the dependence of the number of signals accepted as a function
of the voltage level on the LVDS buffer is shown in the upper panel of Fig.~\ref{fig:registered}. Because the
baseline has been moved up to 2.048~V, for negative signals, a smaller threshold value means
the threshold applied at a higher signal amplitude, according to the upper part of
Fig.~\ref{fig:threshold level}. In order to describe dependence of the number of counts on the voltage on the buffer, a 5${^{th}}$
degree polynomial was fitted to the data. The flat part of the chart above 1.87~V level corresponds to the threshold values for which all pulses from a single sequence with the amplitude greater than 100~mV were registered.
\begin{figure}[h]
\centering
\includegraphics[width=0.40\textwidth]{figure10a.png}
\includegraphics[width=0.40\textwidth]{figure10b.png}
\caption{(Upper panel) Number of registered events for different voltage level set at LVDS buffer with the baseline set to 2.048~V. (Lower panel) Dependence between the absolute value of the real threshold and the voltage set at the LVDS buffer. A 5th order polynomials (red solid lines) were fitted to the data for the slopes description.}
\label{fig:registered}
\end{figure}
\\
By combining the information on the number of sequences, the number of pulses in the sequence and how many pulses from the sequence were registered on the threshold with a given voltage value, it is possible to convert the voltage level set on the LVDS buffer to the actual threshold applied to the signal. An example of determination of the absolute value of the real threshold (a distance from the baseline at 0~V) as a function of voltage at the LVDS buffer is shown in the lower panel of Fig.~\ref{fig:registered}.
\section{Synchronization of the J-PET prototype}
Adjustment of relative time between all elements of the detection system is necessary
in order to be able to reconstruct the place of interaction of the gamma photon with the scintillator, as well as the location of positron-electron annihilation. The time synchronization of the J-PET tomograph prototype has been divided into two stages (see Fig.~\ref{fig:calib1}):
\begin{itemize}
\item time tuning of a single detection module,
\item mutual time coordination of all detection modules.
\end{itemize}
The first stage of calibration of two photomultipliers in a single detection module is
based on the use of cosmic rays, which uniformly irradiate the scintillator over its entire
length~\cite{BAMScosm}. Time of signal of left and right photomultipliers can be expressed as:
\begin{eqnarray}
t_l = t_{hit} +\frac{z}{\upsilon}+t_{off_{l}}
\label{eq4}
\\
t_r = t_{hit}+ \frac{L - z}{\upsilon}+t_{off_{r}},
\label{eq5}
\end{eqnarray}
where $t_{hit}$ is the time in which the interaction with the scintillator occurred, $L$ is the length of the
scintillator and $z$ is the position of the gamma quantum interaction along the scintillator.
The t$_{off_l}$ and t$_{off_r}$ times are fixed time offsets for the left and right photomultiplier, respectively, resulting from propagation of the signal by FEE and cables. The speed of light $\upsilon$ in the scintillator was determined using an independent method described in~\cite{Moskal2014A}.
When calculating the difference between the two times defined in Eqs.~(\ref{eq4}) and~(\ref{eq5}):
\begin{eqnarray}
t_r - t_l = \frac{L}{\upsilon} - \frac{2z}{\upsilon} + t_{off_{r}} - t_{off_{l}}
\end{eqnarray}
\\
we get relative values of time constants for photomultipliers from given detection module (t$_{off_p}$ - t$_{off_l}$ ). Thus, by determining the value of the shift from the time difference spectrum, we can obtain the correction values of the time constants by correctly locating the central distribution position. The mean value of counts or the median in the distribution are not
good estimates because the photomultiplier efficiency at either ends of the scintillator may
differ, biasing the time calibration. Therefore, in order to determine the center
position of the two edges of the time difference spectra, two functions (known as logistic function, sigmoid or Fermi function) were adjusted
with the following formulas:
\begin{eqnarray}
f_F(x) = \frac{P_0}{\exp{\frac{x - P_1}{P_2}}+1}+P_3
\end{eqnarray}
\\
where the parameters P$_0$, P$_1$, P$_2$, P$_3$ correspond respectively to the maximum value of the function, the center of the edge, the edge inclination, and the minimum value of the function. On the basis of the position values of the centers of the two edges, it is possible to calculate how much the spectrum should be moved so that it is symmetrical with respect to the zero value. An example of the distribution of the difference in the time of registration of particles from cosmic radiation using two photomultipliers, after correction, is shown in the upper part of Fig.~\ref{fig:distribution}.
\begin{figure}[h]
\centering
\includegraphics[width=0.40\textwidth]{figure11a.png}
\includegraphics[width=0.40\textwidth]{figure11b.png}
\caption{(Upper panel) Example of distribution of the time difference between signals arrival to the two
photomultipliers located at the ends of a scintillator irradiated with cosmic rays.
The red line shows a fit of the double Fermi function. (Lower panel) Distribution of the number of photoelectrons per event observed using a $^{22}Na$ radioactive source with activity of 17.3~MBq (black solid histogram) and cosmic radiation (blue dashed histogram) registered simultaneously with the annihilation gamma quanta scaled up by a factor of 50 for better visibility.}
\label{fig:distribution}
\end{figure}
\\
An advantage of this method of synchronization is that it can be done simultaneously with a patient scan because energy of gamma photons from positron annihilation is different from the cosmic ray energies such that both components can be separated as shown in the lower part of Fig.~\ref{fig:distribution}.
\\
The concept of the time synchronization method of modules is based on the principle of transitivity, in our case on comparison with the reference point. Time synchronization between detection modules was made using a rotating sodium radioactive source together with a reference detector. The reference detector was a narrow and elongated (5 $\times$ 5 $\times$ 19~ mm$^3$) BC-420 scintillator optically coupled with an additional photomultiplier. The geometry of the reference scintillator forced the self-collimation of photons from the $^{22}Na$ source, preferring photons moving close to the longitudinal axis of the scintillator to reach the reference photomultiplier.
The system used for mutual synchronization of detection modules is schematically presented in the left part of Fig.~\ref{fig:refrence detector}.
\begin{figure}[h]
\includegraphics[width=0.34\textwidth]{figure12a.jpg}
\includegraphics[width=0.2\textwidth, angle=90]{figure12b.png}
\caption{ (Left) Schematic view of the method of the J-PET prototype synchronization using the reference detector. (Right) Reference detector mounted at the rotating arm inside the prototype barrel.}
\label{fig:refrence detector}
\end{figure}
The right part of Fig.~\ref{fig:refrence detector} shows the reference detector mounted on the rotating arm inside the J-PET tomograph prototype. The rotation allows to achieve the configuration shown in the left part of Fig.~\ref{fig:refrence detector} with respect to each of 24 detection modules.
\\
Time of signals from left and right photomultiplier can be written analogously to formulas (4) and (5). However, in this case, the reference time $t_{hit}$ is the time of emission of photons from the source located on the reference detector. Therefore, it is necessary to add the time needed to travel the path $d$ from the source to the detector module scintillator. Then the measured time consists of the following elements:
\begin{eqnarray}
t_{l1}= t_{hit}+\frac{d}{c}+\frac{z}{\upsilon}+t_{off_{l1}},
\\
t_{r1}= t_{hit}+\frac{d}{c}+\frac{L - z}{\upsilon}+t_{off_{r1}},
\end{eqnarray}
\\
The time for the reference detector can be written as:
\begin{eqnarray}
t_{ref}=t_{hit}+\frac{h}{2\upsilon}+t_{off_{ref}},
\end{eqnarray}
\\
where t$_{off_{ref}}$ are fixed time values resulting from signal propagation in FEE and cables. The factor $\frac{h}{2\upsilon}$ describes the average time it took for the light to travel inside the reference scintillator before reaching the photomultiplier. In general, the gamma photon can react in various places along the scintillator. This results in the variation of the obtained time, but does not alter the average value which was taken as a constant for all performed measurements. Taking into account the synchronization of a single module described previously, knowing the constant values of times t$_{off_{l1}}$ and t$_{off_{r1}}$ , the propagation time of the signal through FEE and cables for a single module in its entirety is following:
\begin{eqnarray}
\frac{t_{off_{r1}}+t_{off_{l1}}}{2}=t_{off_1}
\end{eqnarray}
Going into a generalization for the entire prototype of the J-PET tomograph, we label detector modules with indices $i$ and $j$. Then subtracting time from the reference detector and
time from a given detection $i^{th}$ module we obtain:
\begin{eqnarray}
t'_i=t_{ref}-\frac{t_{ri}+t_{li}}{2}=\frac{h}{2\upsilon}-\frac{d}{c}-\frac{L}{2\upsilon}+t_{off_{ref}} - t_{off_i}.
\end{eqnarray}
\\
As a result, it is possible to determine the relation between the measurement time from
any two detection modules through a reference detector, and thus to determine the relative
times and synchronization of these modules. Generally for modules $i$, $j$ it can be written as:
\begin{eqnarray}
(t_{ref}-\frac{t_{ri}+t_{li}}{2})-(t_{ref}-\frac{t_{rj}+t_{lj}}{2})=~t_{off_j} - t_{off_i}.
\end{eqnarray}
\\
The presented method of time calibration allows to synchronize each module with respect to one arbitrarily selected detector. Values of the calibration constants obtained after synchronization of the whole J-PET prototype with respect to the first detection module are presented in Fig.~\ref{fig:gaussian}~c). This allows, of course indirectly, to reconstruct the place of annihilation of a positron with an electron in the internal space of the tomograph, and thus also in the patient's body.
The parameter characterizing the precision of the tomographic image obtained on the basis of time information is Coincidence Resolving Time (CRT). It was determined by measuring the time difference of registration of annihilation gamma photons by pairs of modules located in the prototype barrel directly opposite to each other. In Fig. \ref{fig:gaussian}~a) an example of the measured time difference distribution and fitted Gaussian function with $\sigma$~=~187~ps is presented.
The obtained value corresponds to CRT~=~439~ps (equivalent to FWHM). The obtained value of $\sigma$ can be translated to the position reconstruction accuracy $\sigma(\Delta l) =$~18~mm which is fairly independent of the reconstructed position.
\begin{figure}[h]
\centering
\includegraphics[width=0.43\textwidth]{figure13a.png}
\includegraphics[width=0.43\textwidth]{figure13b.png}
\includegraphics[width=0.4\textwidth]{Fig_zonk.png}
\caption{a) Coincidence time resolution for an exemplary pair of modules. Superimposed red line indicates a fitted Gaussian function with $\sigma$ = 187~ps. b) Coincidence time resolution for all facing pairs of modules. c) Values of the calibration constants obtained after synchronization of the whole detector with respect to the first detection module for all the four thresholds used.}
\label{fig:gaussian}
\end{figure}
\\
During the measurement a collimated ${^{22}}$Na source was placed in the geometric center of the J-PET prototype, the threshold level of the trigger was set to -200~mV. Fig.~\ref{fig:gaussian}~b) shows results of the measurement of coincidence time resolution for all twelve facing pairs of modules. As a result of the Gaussian function fit to distributions of time difference for all pairs of modules, the average coincidence time resolution $\sigma=$~208~$\pm$~4~ps
(CRT~=~490~$\pm$~9~ps) was obtained, which is comparable to the best currently available scanners~\cite{SLOMKA2016,Van2016}.
\\
\section{Event selection and simplified image reconstruction}
For the analysis of acquisited signals a dedicated analysis framework was developed~\cite{Krzemien2015,KRZEMIEN2016}. The registered signals may originate from a single electron-positron annihilation events or from hits caused by gamma photons from different annihilations. This either true or accidental coincidences may be additionally influenced by scattering of photons prior to the registration in other detectors or in the radioactive source material. On the basis of an extensive modelling simulations~\cite{Kowalski2015,KOWALSKI2018}
we have selected events for further analysis using several conditions. First of all we consider events when both photomultipliers in a single detection module provide a signal. Next both of such events should occur in two detection modules above some adjusted energy threshold which was optimized for the ratio between the number of true and scattered coincidences. The absolute value of the TOF from detection modules in coincidence was required to be less than 3~ns (time of flight of the gamma photon along the diameter of the scanner) since radioactive source was in the centre of the J-PET prototype.
In order to improve selection criteria which would allow a suppression of the detector-scattered and source-scattered coincidences, we have performed modelling studies of the correlation between the detector’s identity numbers ID and the time differences between the registered signals TOF \cite{Kowalski2016}. The scintillator identifiers ID increase monotonically clockwise in the range from 1 to 24 and the differences of detection module
numbers in coincidence $\Delta ID$ were calculated as follows:
$\Delta ID = min(|ID_1 - ID_2|, 24 - |ID_1 - ID_2|)$,
where ID$_1$ and ID$_2$ denote ID of scintillator modules. For all coincidences, 2-dimensional histogram of registration time differences between subsequent scatterings TOF and scintillator identifiers differences $\Delta ID$ were calculated. This histogram is presented in the upper
part of Fig.~\ref{fig:prototypr}. The Maximum number of $\Delta ID$ is 12 which is the case when detection module lie exactly on the opposite sides relative to the detector centre. True coincidences are located in the region of low TOF and high $\Delta{ID}$.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{figure14b.png}
\includegraphics[width=0.35\textwidth]{figure14a.png}
\caption{(Upper panel) Correlation of the modules ID difference ($\Delta ID$) and Time of Flight (TOF) of gamma quanta measured between two detection modules for all types of coincidences, including accidental and scattered ones. (Lower panel) Result of the simplified point-like source image reconstruction in X-Y plane of the J-PET prototype. Size of the prototype is shown by the circle.}
\label{fig:prototypr}
\end{figure}
\\
One of the methods of reconstruction of tomographic images based on the collected data is to perform a simplified reconstruction. This can be accomplished using the idea of determining
the place of positron-electron annihilation described in Sec. 2. In this method, it is assumed that the Line Of Response (LOR) passes through the geometrical centers of cross-sections of both scintillators. No additional reconstruction data algorithms are used to build
tomographic images. It is only a collection of annihilation points, which are obtained on the
basis of the measured values of signal arrival times. In the lower part of Fig.~\ref{fig:prototypr} the
reconstructed image of a single point-like source located in the centre of the prototype detector
is shown.
\section{Conclusion}
In this paper we have described the first operating prototype of the J-PET tomography scanner built from 24 plastic scintillator strips. The 300~mm long scintillator strips were arranged in a barrel shape with 360~mm~diameter. The signals from each of the strips were recorded by two photomultipliers coupled optically with the scintillator material at the opposite ends of the strips. All signals were probed at four voltage levels by front-end boards and processed by dedicated Trigger and Readout Board providing time and TOT measurements. The prototype was built in order to progress from basic single module to a system where one has to control many units, to test electronic readout of the whole system and to develop calibration and synchronisation procedures. We found that the coincidence resolving time (CRT) for this prototype is equal to 490 $\pm$~9~ps which is comparable to the best commercial scanners. Taking only 24 detection module units turned out to be enough to reconstruct a point-like positron source. These studies demonstrate that a full scale prototype aiming for a whole human body scan is in reach.\\
Recently the first total-body PET based on crystal scintillators was taken into operation in Sacramento~\cite{BADAWI2019}. However, the high costs limits its dissemination not only to hospital facilities but even to medical research clinics~\cite{Majewski2020,Vanderberge2020}. In this article we presented a prototype of the cost-effective method to build a total body PET based on plastic scintillators. Prospects and clinical perspectives of total-body PET imagining using plastic scintillators are described in Ref.~\cite{Stepien2020}.
\section*{Acknowledgement}
The authors acknowledge technical and administrative support of A. Heczko, M. Kajetanowicz and W. Migda{\l}. This work was supported by The Polish National Center for Research and Development through grant INNOTECH-K1/IN1/64/159174/NCBR/12, the Foundation for Polish Science through the MPD and TEAM POIR.04.04.00-00-4204/17 programmes, the National Science Centre of Poland through grants no. 2016/21/B/ST2/01222, 2017/25/N/NZ1/00861, the
Ministry for Science and Higher Education through grants no. 6673/IA/SP/2016, 7150/E338/SPUB/2017/1 and 7150/E-338/M/2017, and the Austrian Science Fund FWF-P26783.
\bibliographystyle{IEEEtran}
|
1,314,259,996,466 | arxiv |
\section{Introduction}
\label{sec:Intro}
We consider the linear operator defined by the fractional Laplacian with drift,
\begin{equation}
\label{eq:Operator}
Lu(x) : = \left(-\Delta\right)^s u(x) + b(x)\dotprod\nabla u (x)+c(x) u(x),\quad \forall u \in C^{2}_c(\RR^n),
\end{equation}
where the coefficient functions $b:\RR^n\rightarrow\RR^n$ and $c:\RR^n\rightarrow\RR$ are assumed to be H\"older continuous.
The action of the fractional Laplacian operator on functions $u\in C^2_c(\RR^n)$ is given by the singular integral,
\begin{equation*}
(-\Delta)^s u(x) = c_{n,s} \hbox{ p.v.} \int_{\RR^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\, dy,
\end{equation*}
understood in the sense of the principal value. The constant $c_{n,s}$ is positive and depends only on the dimension $n\in \NN$, and on the parameter $s\in (0,1)$. The range $(0,1)$ of the parameter $s$ is particularly interesting because in this case the fractional Laplacian operator is the infinitesimal generator of the symmetric $2s$-stable process \cite[Example 3.3.8]{Applebaum}.
The fractional Laplacian plays the same paradigmatic role in the theory of non-local operators that the Laplacian plays in the theory of local elliptic operators. For this reason, the regularity of solutions to equations defined by the fractional Laplacian and its gradient perturbation is intensely studied in the literature. In this article, we study the stationary obstacle problem defined by the fractional Laplacian operator with drift \eqref{eq:Operator}, in the subcritical regime, that is, the case when the parameter $s$ belongs to the range $(1/2,1)$. Given an obstacle function, $\varphi\in C^{3s}(\RR^n)\cap C_0(\RR^n)$, we prove existence, uniqueness and optimal regularity of solutions in H\"older spaces, $u \in C^{1+s}(\RR^n)$, for the stationary obstacle problem,
\begin{equation}
\label{eq:Obstacle_problem}
\min\{\left(-\Delta\right)^s u(x) + b(x)\dotprod\nabla u (x)+c(x) u(x), u(x)-\varphi(x)\}=0,\quad\forall x\in \RR^n.
\end{equation}
Our main result is
\begin{thm}[Existence, uniqueness and optimal regularity of solutions]
\label{thm:Solutions}
Let $s \in (1/2,1)$. Assume that $b \in C^{s}(\RR^n;\RR^n)$, and $c \in C^{s}(\RR^n)$ is such that
\begin{equation}
\label{eq:Nonnegative_lower_bound_c}
c \geq 0\quad\hbox{on }\RR^n.
\end{equation}
Assume that $\varphi\in C^{3s}(\RR^n)\cap C_0(\RR^n)$ is such that
\begin{equation}
\label{eq:Boundedness_L_phi_positive_part}
(L\varphi)^+\in L^{\infty}(\RR^n).
\end{equation}
Then the obstacle problem \eqref{eq:Obstacle_problem} has a solution, $u \in C^{1+s}(\RR^n)$. If in addition the vector field $b:\RR^n\rightarrow\RR^n$ is a Lipschitz function, and there is a positive constant, $c_0$, such that
\begin{equation}
\label{eq:Lower_bound_c}
c(x)\geq c_0,\quad\forall x\in \RR^n,
\end{equation}
then there is a unique solution, $u \in C^{1+s}(\RR^n)$, to the obstacle problem \eqref{eq:Obstacle_problem}.
\end{thm}
We note that the properties of the fractional Laplacian operator with drift differ substantially depending whether the parameter $s$ takes values in the range $(0,1/2)$ (the supercritical regime), is equal to $1/2$ (the critical regime), or takes values in $(1/2,1)$ (the subcritical regime) \cite[\S 1]{Silvestre_2012a, Caffarelli_Vasseur_2010}. In the critical and subcritical regime, the fractional Laplacian operator with drift defines an elliptic pseudodifferential operator in the sense of \cite[\S 3.9]{Taylor_vol1}, and so, the drift component can be treated as a lower order term, a fact that we use extensively in our analysis of the obstacle problem \eqref{eq:Obstacle_problem}. In the supercritical regime, the operator $L$ is no longer elliptic and our analysis no longer applies. A study of the regularity of solutions in Sobolev spaces and of the Green's kernel of the stationary \emph{linear} equation defined by the fractional Laplacian with drift in the supercritical regime, can be found in \cite{Epstein_Pop_2013}.
The stationary obstacle problem defined by the fractional Laplacian operator \emph{without drift} was studied by Silvestre \cite{Silvestre_2007}, and by Caffarelli, Salsa and Silvestre \cite{Caffarelli_Salsa_Silvestre_2008}. In \cite{Silvestre_2007}, it is established the almost optimal regularity of solutions to the obstacle problem for the fractional Laplacian without drift \cite[Theorem 5.8]{Silvestre_2007}, that is, given an obstacle function, $\varphi\in C^{1+\beta}(\RR^n)$, the solution is shown to be $C^{1+\alpha}(\RR^n)$, for all $\alpha \in (0,\beta\wedge s)$. In \cite{Caffarelli_Salsa_Silvestre_2008}, the authors prove the optimal regularity of solutions \cite[Corollary 6.8]{Caffarelli_Salsa_Silvestre_2008}, that is, the solution $u$ belongs to $C^{1+s}(\RR^n)$, when the obstacle function, $\varphi$, is assumed to be in $C^{2,1}(\RR^n)$, and they establish the $C^{1,\alpha}$ regularity of the free boundary in a neighborhood of regular points \cite[Theorem 7.7]{Caffarelli_Salsa_Silvestre_2008}.
In our work, we prove existence, uniqueness and optimal regularity of solutions to the stationary obstacle problem defined by the fractional Laplacian \emph{with drift}. In proving the existence and uniqueness of solutions, we take a different approach than in \cite{Silvestre_2007, Caffarelli_Salsa_Silvestre_2008}. Specifically, to obtain existence of solutions to the obstacle problem, we first study the linear and the penalized problem defined the fractional Laplacian with drift, which we solve by establishing a priori Schauder estimates and applying perturbation arguments. To obtain uniqueness of solutions, we use a probabilistic approach and we establish the stochastic representation of solutions. The stochastic representation of solutions is especially important in mathematical finance, where the value function of American-style options are given in the form of a stochastic representation.
Our strategy in proving the optimal regularity of solutions is similar to that of \cite{Caffarelli_Salsa_Silvestre_2008}, but there are certain aspects in which the method of \cite{Caffarelli_Salsa_Silvestre_2008} is not applicable to our framework, which we now outline. In \S \ref{sec:Solutions_optimal_regularity}, we show that the obstacle problem \eqref{eq:Obstacle_problem} for the fractional Laplacian with drift can be reduced to one without drift, in which the obstacle function can only be assumed to belong to the space of functions $C^{2s+\alpha}(\RR^n)$, for all $\alpha\in (0,s)$, while in \cite{Caffarelli_Salsa_Silvestre_2008}, the obstacle function is assumed to have better regularity, i.e., it belongs to $C^{2,1}(\RR^n)$. Similarly to \cite{Caffarelli_Salsa_Silvestre_2008}, we construct a new monotonicity formula of Almgren type in Proposition \ref{prop:Monotonicity_formula}, which takes into account the limitation in regularity of the obstacle function. We then consider a suitable sequence of rescaled functions, for which we prove uniform estimates in H\"older spaces, and which we use together with the monotonicity formula, to establish the optimal regularity of solutions to the obstacle problem \eqref{eq:Obstacle_problem}. The arguments employed in \cite{Caffarelli_Salsa_Silvestre_2008} to establish the uniform estimates in H\"older spaces of the sequence of rescalings \cite[Proposition 4.3]{Caffarelli_Salsa_Silvestre_2008}, and to obtain the growth of the solution in a neighborhood of a free boundary point \cite[Lemma 6.5]{Caffarelli_Salsa_Silvestre_2008}, are not applicable to our case, due to the presence of a singular measure in the structure of our problem. Instead, our method of proof is based on a suitable application of the Moser iteration technique to obtain supremum and growth estimates (Lemma \ref{lem:Uniform_boundedness_rescalings} and Proposition \ref{prop:Growth_v_around_0}), and of a localization procedure described in \cite[Theorem 8.11.1]{Krylov_LecturesHolder} to obtain Schauder estimates (Lemma \ref{lem:Uniform_boundedness_C_1_alpha_n_ball_rescalings}). While Krylov uses the method described in \cite[Theorem 8.11.1]{Krylov_LecturesHolder} to obtain a priori Schauder estimates for solutions to a \emph{linear equation}, we apply it to obtain estimates for solutions to an \emph{obstacle problem}.
\subsection{Comparison with previous research}
\label{subsec:Previous_research}
We may compare our result on existence, uniqueness and almost optimal regularity of solutions to the stationary obstacle problem \eqref{eq:Obstacle_problem}, Proposition \ref{prop:Solutions_partial_regularity}, with the analogous results obtained by L. Silvestre, in \cite{Silvestre_2007}, for the obstacle problem defined by the fractional Laplacian operator \emph{without drift}, \cite[Complementarity conditions (1.1) and (1.2)]{Silvestre_2007}. Our results are proved in the subcritical regime, that is, the case when the parameter $s$ is contained in the range $(1/2,1)$, while the results of \cite{Silvestre_2007} hold for all $s\in (0,1)$, but the operator does not contain a drift component. The assumption used in our article that $s\in (1/2,1)$ plays an important role because it allows us to treat the drift component as a lower order term of the operator $L$. In \cite[\S 1.1]{Silvestre_2007}, existence of solutions in Sobolev spaces can be immediately obtained by variational methods, which do not seem readily applicable to our framework due to the presence of the lower order term. Instead, in \S \ref{sec:Solutions_partial_regularity}, we prove a priori Schauder estimates and we use a perturbation argument to obtain the existence of solutions in the H\"older space $C^{1+\alpha}(\RR^n)$, for some $\alpha\in (0,s)$. To improve the regularity of solutions, we begin with the fact that we know that the solutions to the obstacle problem are in $C^{1+\alpha}(\RR^n)$, and we use a bootstrap argument in conjunction to the almost optimal regularity of solutions proved in \cite[Theorem 5.8]{Silvestre_2007}, to establish in Proposition \ref{prop:Solutions_partial_regularity} the almost optimal regularity of solutions to our obstacle problem \eqref{eq:Obstacle_problem}.
We next compare our work with that of L. Caffarelli, S. Salsa and L. Silvestre \cite{Caffarelli_Salsa_Silvestre_2008}, who establish the optimal regularity of solutions to the obstacle problem defined by the fractional Laplacian operator \emph{without drift}, when the obstacle problem is assumed to belong to the space of functions $C^{2,1}(\RR^n)$ (see \cite[Corollary 6.8]{Caffarelli_Salsa_Silvestre_2008}). In \S \ref{sec:Monotonicity_formula}, we reduce our obstacle problem \eqref{eq:Obstacle_problem} to one without drift, but the obstacle function can at most be assumed to belong to $C^{2s+\alpha}(\RR^n)$, for all $\alpha\in (0,s)$, due to the presence of the drift component in the definition \eqref{eq:Operator} of the operator $L$. Analogous to \cite{Caffarelli_Salsa_Silvestre_2008}, we introduce an auxiliary `height' function, $v$, in \eqref{eq:Auxiliary_function_v}, and we consider a suitable $L_a$-extension of our nonlocal problem to a local one which satisfies conditions \eqref{eq:Upper_bound_L_a} and \eqref{eq:Equality_L_a}. Compared with \cite[Conditions (2.2)-(2.5)]{Caffarelli_Salsa_Silvestre_2008}, our extended problem has the property that $L_a v$ contains a singular measure on the set $\{y=0\}\backslash\{v=0\}$, which is not the case in \cite{Caffarelli_Salsa_Silvestre_2008}. This limitation in the regularity of the obstacle function, and the singularity appearing in our extended problem make many of the arguments used in \cite{Caffarelli_Salsa_Silvestre_2008} inapplicable to our framework. We prove a new monotonicity formula of Almgren type in \S \ref{sec:Monotonicity_formula}, and we replace the comparison arguments in \cite{Caffarelli_Salsa_Silvestre_2008} by adapting the Moser iterations method to our framework, in Lemma \ref{lem:Uniform_boundedness_rescalings} and Proposition \ref{prop:Growth_v_around_0}, and by applying the localization method of \cite[Theorem 8.11.1]{Krylov_LecturesHolder} to prove uniform Schauder estimates in Lemma \ref{lem:Uniform_boundedness_C_1_alpha_n_ball_rescalings}. A key role in our analysis plays the result of Proposition \ref{prop:Phi_at_0}, in which it is established a lower bound of the function $\Phi_v^p(r)$ defined in \eqref{eq:Phi}. Even though the definition of our function $\Phi_v^p(r)$ differs from its analogue in \cite[\S 3]{Caffarelli_Salsa_Silvestre_2008}, indirectly we appeal to the lower bounds established in \cite[Lemma 6.1]{Caffarelli_Salsa_Silvestre_2008} to derive our Proposition \ref{prop:Phi_at_0}.
\subsection{Outline of the article}
\label{subsec:Outline}
Our main result, Theorem \ref{thm:Solutions}, is proved in \S \ref{sec:Solutions}, which is organized in
four subsections. We being in \S \ref{sec:Solutions_partial_regularity} by establishing the existence and uniqueness of solutions in H\"older spaces to the linear equation \eqref{eq:Linear_equation} defined by the operator $L$ (Lemma \ref{lem:Existence_uniqueness_linear_equation}). We use this result to solve the penalized equation (Lemma \ref{lem:Existence_penalized_equation}), which leads to the proof of the existence of solutions to the obstacle problem \eqref{eq:Obstacle_problem} (Proposition \ref{prop:Existence_Holder_obstacle_problem}). Via a bootstrap argument, we prove that the solutions constructed in Proposition \ref{prop:Existence_Holder_obstacle_problem} have the almost optimal regularity, that is, they belong to the space of functions $C^{1+\alpha}(\RR^n)$, for all $\alpha\in(0,s)$, when we assume that the obstacle function is contained in $C^{1+s}(\RR^n)\cap C_0(\RR^n)$ (Proposition \ref{prop:Solutions_partial_regularity}). In \S \ref{sec:Uniqueness}, we prove the uniqueness of solutions of the obstacle problem \eqref{eq:Obstacle_problem} by establishing that they admit a suitable stochastic representation (Proposition \ref{prop:Uniqueness}). We show how to reduce the obstacle problem for the fractional Laplacian with drift \eqref{eq:Obstacle_problem} to one \emph{without} drift in \S \ref{sec:Monotonicity_formula}. Using the extended problem introduced in \cite{Caffarelli_Silvestre_2007}, we introduce a suitable `height function', $v$, in \eqref{eq:Auxiliary_function_v}, and we prove a monotonicity formula (Propositions \ref{prop:Monotonicity_formula} and \ref{prop:Phi_at_0}), which we then use in \S \ref{sec:Solutions_optimal_regularity} to establish the growth of the function $v$ is a neighborhood of a free boundary point (Proposition \ref{prop:Growth_v_around_0}). Finally, we give the proof of the optimal regularity of solutions (Theorem \ref{thm:Solutions}). In \S \ref{sec:Auxiliary_results_proofs}, we give the proof of a series of auxiliary results, and of Proposition \ref{prop:Monotonicity_formula}. In \S \ref{subsec:Function_spaces}, we state the definitions of the spaces of functions, and in \S \ref{subsec:Notation}, we introduce the notations and conventions we use throughout our article.
\subsection{Function spaces}
\label{subsec:Function_spaces}
Let $k, m$ and $n$ be positive integers, and let $\Omega\subseteq\RR^n$ be an open set. We denote by $C^{\infty}_c(\Omega;\RR^m)$ the space of smooth functions, $u:\Omega\rightarrow\RR^m$, with compact support in $\Omega$. The space $C^k(\bar\Omega;\RR^m)$ consists of functions, $u:\bar\Omega\rightarrow\RR^m$, which admit derivatives up to order $k$, such that $u$ and its derivatives up to order $k$ are continuous and bounded on $\bar\Omega$. The space $C^k(\bar\Omega;\RR^m)$ endowed with the norm
$$
\|u\|_{C^k(\bar\Omega;\RR^m)} = \sup_{\stackrel{\beta\in\NN^n}{|\beta| \leq k}}\sup_{x\in \bar\Omega} |D^{\beta}u(x)|<+\infty,\quad\forall u \in C^k(\bar\Omega;\RR^m),
$$
is a Banach space. In the preceding definition, for all multi-indices $\beta\in\NN^n$, we let $|\beta|$ denote the sum of its components.
Let $\alpha \in (0,1)$. The H\"older space $C^{k+\alpha}(\bar\Omega;\RR^m)$ consists of function $u\in C^k(\bar\Omega;\RR^m)$ satisfying the property that the seminorm
$$
\left[u\right]_{C^{k+\alpha}(\bar\Omega;\RR^m)}
:= \sup_{\stackrel{\beta\in\NN^n}{|\beta| \leq k}}
\sup_{\stackrel{x, y \in \bar\Omega}{x \neq y}} \frac{|D^{\beta} u(x)-D^{\beta} u(y)|}{|x-y|^{\alpha}} < +\infty,
\quad\forall u \in C^{k+\alpha}(\bar\Omega;\RR^m).
$$
The space $C^{k+\alpha}(\bar\Omega;\RR^m)$ endowed with the norm
$$
\|u\|_{C^{k+\alpha}(\bar\Omega;\RR^m)} = \|u\|_{C^k(\bar\Omega;\RR^m)}+ \left[u\right]_{C^{k+\alpha}(\bar\Omega;\RR^m)},
\quad\forall u \in C^{k+\alpha}(\bar\Omega;\RR^m),
$$
is a Banach space. We let $C^{k+\alpha}_0(\bar\Omega;\RR^m)$ be the closure of the space $C^{\infty}_c(\Omega;\RR^m)$ with respect to the norm $\|\cdot\|_{C^{k+\alpha}(\bar\Omega;\RR^m)}$. As usual, the space $C^{k+\alpha}_0(\bar\Omega;\RR^m)$ endowed with the norm $\|\cdot\|_{C^{k+\alpha}(\bar\Omega;\RR^m)}$ is a Banach space. The spaces $C^k_{\loc}(\RR^n;\RR^m)$ and $C^{k+\alpha}_{\loc}(\RR^n;\RR^m)$ consists of functions, $u:\RR^n\rightarrow\RR^m$, which belong to $C^k(K;\RR^m)$ and $C^{k+\alpha}(K;\RR^m)$, respectively, for all compact sets $K\subset\RR^n$. When $k=0$, we omit the superscript $k$ from the notation of the space $C^k(\bar\Omega;\RR^m)$, and when $m=1$, we write $C^k(\bar\Omega)$ instead of $C^k(\bar\Omega;\RR)$. The analogous convention applies to the spaces $C^{k+\alpha}(\bar\Omega;\RR^m)$, $C^k_{\loc}(\RR^n;\RR^m)$, and $C^{k+\alpha}_{\loc}(\RR^n;\RR^m)$.
\subsection{Notations and conventions}
\label{subsec:Notation}
We let $\cS(\RR^n)$ denote the Schwartz space \cite[Definition (3.3.3)]{Taylor_vol1} consisting of smooth functions whose
derivatives of all orders decrease faster than any polynomial at infinity, and we let $\cS'(\RR^n)$ denote its dual space, the space of tempered distributions \cite[\S 3.4]{Taylor_vol1}. We adopt the following definition of the Fourier transform of a function $u\in\cS(\RR^n)$,
$$
\widehat u (\xi) = \int_{\RR^n} e^{-ix\dotprod\xi} u(x)\ dx,\quad\forall \xi\in\RR^n.
$$
Given real numbers, $a$ and $b$, we let $a \wedge b:=\min\{a,b\}$ and $a \vee b:=\max\{a,b\}$. If $v,w\in\RR^n$, we denote by $v\cdot w$ their scalar product. For $x_0\in\RR^{n+1}$ and $r>0$, let $B_r(x_0)$ be the Euclidean ball in $\RR^{n+1}$ of radius $r$ centered at $x_0$, and for $x_0\in\RR^n$ and $r>0$, let $B'_r(x_0)$ be the Euclidean ball in $\RR^n$ of radius $r$ centered at $x_0$. We denote by $B^+_r(x^0)$ the half-ball, $B_r(x^0)\cap\left(\RR^{n}\times\RR_+\right)$, where $\RR_+:=(0,\infty)$. For brevity, when $x_0=O$, we write $B_r$, $B'_r$ and $B^+_r$ instead of $B_r(O)$, $B'_r(O)$ and $B^+_r(O)$, respectively.
For a set $S\subseteq\RR^n$, we denote its complement by $S^c:=\RR^n\backslash S$, and we let $\hbox{int } S$ denote its topological interior.
\section{Existence and optimal regularity of solutions}
\label{sec:Solutions}
In this section we prove the main result of our article, Theorem \ref{thm:Solutions}. We organize its content into three parts. In \S \ref{sec:Solutions_partial_regularity}, we prove the existence of solutions in H\"older spaces to the obstacle problem \eqref{eq:Obstacle_problem}, and we show that the solutions have the almost optimal regularity. In \S \ref{sec:Uniqueness}, we give sufficient conditions which ensure that the obstacle problem \eqref{eq:Obstacle_problem} has a unique solution in $C^{1+\alpha}(\RR^n)$, where $\alpha\in ((2s-1)\vee 0,1)$. We prove the uniqueness of solutions by establishing their stochastic representation. Even though in general we assume throughout our article that $s\in (1/2,1)$, the stochastic representation of solutions holds for all $s\in (0,1)$. In \S \ref{sec:Monotonicity_formula}, we prove a version of the monotonicity formula suitable for our operator, which is then used in \S \ref{sec:Solutions_optimal_regularity} to obtain the optimal regularity of solutions to the obstacle problem \eqref{eq:Obstacle_problem}.
\subsection{Existence and almost optimal regularity of solutions}
\label{sec:Solutions_partial_regularity}
In this section, we prove existence of solutions, $u$, to the obstacle problem \eqref{eq:Obstacle_problem} having almost optimal regularity, that is, $u \in C^{1+\beta}(\RR^n)$, for all $\beta \in (0,s)$, when the obstacle function, $\varphi$, is assumed to belong to the space of functions $C^{1+s}(\RR^n)\cap C_0(\RR^n)$. The main result of this section is
\begin{prop}[Existence, uniqueness and almost optimal regularity of solutions]
\label{prop:Solutions_partial_regularity}
Let $s \in (1/2,1)$ and $\alpha \in (0,1)$. Assume that the coefficient function $b \in C^{s}(\RR^n;\RR^n)$, and $c \in C^{s}(\RR^n)$ and satisfies condition \eqref{eq:Nonnegative_lower_bound_c}. Assume that the obstacle function $\varphi\in C^{1+\alpha}(\RR^n)\cap C_0(\RR^n)$ and satisfies condition \eqref{eq:Boundedness_L_phi_positive_part}. Then the obstacle problem \eqref{eq:Obstacle_problem} defined by the fractional Laplacian with drift has a solution, $u \in C^{1+\beta}(\RR^n)$, for all $\beta<\alpha\wedge s$.
\end{prop}
To prove Proposition \ref{prop:Solutions_partial_regularity}, we use a series of preliminary results. We first prove a maximum principle (Lemma \ref{lem:Comparison_principle}) which is used to obtain the existence and uniqueness of solutions in H\"older spaces to the linear equation \eqref{eq:Linear_equation} defined by the fractional Laplacian with drift (Lemma \ref{lem:Existence_uniqueness_linear_equation}). This result is applied to prove existence of solutions to the penalized equation \eqref{eq:Penalized_equation} defined by the fractional Laplacian with drift, which gives us the existence of solutions to the obstacle problem (Proposition \ref{prop:Existence_Holder_obstacle_problem}). The solutions we obtain at this point have less regularity than the one stated in the conclusion of Proposition \ref{prop:Solutions_partial_regularity}. A bootstrap argument is then used, together with Proposition \ref{prop:Existence_Holder_obstacle_problem}, to give the proof of Proposition \ref{prop:Solutions_partial_regularity}.
\subsubsection{The linear equation defined by the fractional Laplacian with drift}
\label{subsec:Linear_equation}
In this section we establish the existence and uniqueness in H\"older spaces of solutions to the linear equation defined by the fractional Laplacian with drift,
\begin{equation}
\label{eq:Linear_equation}
Lu=f\quad\hbox{on } \RR^n,
\end{equation}
in the case when $s\in (1/2,1)$.
\begin{lem}[Existence and uniqueness of solutions to the linear equation]
\label{lem:Existence_uniqueness_linear_equation}
Let $s\in(1/2,1)$. Let $\alpha \in (0,1)$ be such that $2s+\alpha$ is not an integer. Assume that the coefficient functions $b\in C^{\alpha}(\RR^n;\RR^n)$, and $c \in C^{\alpha}(\RR^n)$ and satisfies condition \eqref{eq:Lower_bound_c}. Then, there is a positive constant, $C=C(\alpha, \|b\|_{C^{\alpha}(\RR^n;\RR^n)}, \|c\|_{C^{\alpha}(\RR^n)}, c_0, n, s)$, such that for any source function, $f \in C^{\alpha}(\RR^n)$, there is a unique solution, $u \in C^{2s+\alpha}(\RR^n)$, to the linear equation \eqref{eq:Linear_equation}, and the function $u$ satisfies the Schauder estimate,
\begin{equation}
\label{eq:Holder_estimate}
\|u\|_{C^{2s+\alpha}(\RR^n)} \leq C\|f\|_{C^{\alpha}(\RR^n)}.
\end{equation}
\end{lem}
\begin{rmk}[Regularity of solutions to the linear equation]
The regularity of solutions in Sobolev spaces, as opposed to H\"older spaces, to the linear equation defined by the fractional Laplacian with drift in the suprecritical regime, that is the case when $s\in (0,1/2)$, has been established in \cite{Epstein_Pop_2013}, using methods specific to the theory of pseudodifferential operators. We remark that the supercritical case is more difficult to treat than the subcritical regime, $s\in (1/2,1)$, because the operator is \emph{not elliptic}. In the subcritical case, the diffusion component dominates the drift term, and so, the drift term can be treated as a lower-order perturbation. This is an important fact that we use in the proof of Lemma \ref{lem:Existence_uniqueness_linear_equation}, but which cannot be extended to the supercritical regime.
\end{rmk}
To prove Lemma \ref{lem:Existence_uniqueness_linear_equation}, we commence with
\begin{lem}[Comparison principle]
\label{lem:Comparison_principle}
Let $s\in (1/2,1)$. Assume that the coefficient function $b\in C(\RR^n;\RR^n)$, and $c\in C_{\loc}(\RR^n)$ and satisfies condition \eqref{eq:Lower_bound_c}. If $u \in C(\RR^n)\cap C^1_{\loc}(\RR^n)$ satisfies
\begin{equation}
\label{eq:Lu_nonnegative}
\left(-\Delta\right)^s u + b\dotprod\nabla u +c u \geq 0\quad\hbox{on }\RR^n,
\end{equation}
then
\begin{equation}
\label{eq:u_nonnegative}
u \geq 0 \quad\hbox{on }\RR^n.
\end{equation}
\end{lem}
\begin{proof}
We consider the auxiliary function,
\begin{equation}
\label{eq:Auxiliary_function}
v(x):=(a+|x|^2)^p,\quad\forall x\in\RR^n,
\end{equation}
where $p$ is a fixed number in the interval $(0,1/2)$, and the positive constant $a$ will be suitably chosen below. Direct calculations give us, for all $x\in\RR^n$ and $i,j=1,\ldots,n$,
\begin{equation}
\label{eq:Derivatives_aux_function}
\begin{aligned}
v_{x_i}(x) &= 2p(a+|x|^2)^{p-1}x_i,\\
v_{x_ix_j} &= 4p(p-1)(a+|x|^2)^{p-1}x_ix_j + 2p(a+|x|^2)^{p-1}\delta_{ij},
\end{aligned}
\end{equation}
where $\delta_{ij}$ denoted the Kronecker delta symbol. Because $p\in (0,1/2)$ and $a>0$, we see that the derivatives $v_{x_i}$ and $v_{x_ix_j}$ belong to $C(\RR^n)$, and so, we have that $v_{x_i}\in C^{\beta}(\RR^n)$, for all $\beta\in(0,1)$ and for all $i=1,\ldots,n$. We can find a positive constant, $C=C(p)$, such that
$$
\|v_{x_i}\|_{C^{\beta}(\RR^n)} \leq C,\quad\forall a\geq 1,\quad\forall \beta\in (0,1).
$$
By choosing $\beta>2s-1$, we have that
\begin{equation}
\label{eq:Upper_bound_frac_lap_u}
\begin{aligned}
|(-\Delta)^s v(x)| &\leq \left|\int_{B'_1}\frac{v(x+y)-v(x)-\nabla v(x)\cdot y}{|y|^{n+2s}} \ dy \right|
+ \left|\int_{(B'_1)^c}\frac{v(x+y)-v(x)-\nabla v(x)\cdot y}{|y|^{n+2s}} \ dy \right|\\
&\leq C\int_{B'_1}\frac{|y|^{1+\beta}}{|y|^{n+2s}} \ dy + C\int_{(B'_1)^c}\frac{|y|}{|y|^{n+2s}} \ dy.
\end{aligned}
\end{equation}
The first and second integral on the right-hand side of \eqref{eq:Upper_bound_frac_lap_u} are finite because we have chosen $\beta\in (2s-1,1)$ and $s\in (1/2,1)$. We obtain that there is a positive constant, $C_0$, such that
\begin{align*}
|(-\Delta)^s v(x)|&\leq C_0,\quad\forall x\in \RR^n.
\end{align*}
Using the preceding inequality, identities \eqref{eq:Auxiliary_function} and \eqref{eq:Derivatives_aux_function}, and condition \eqref{eq:Lower_bound_c}, we have that
\begin{align*}
Lv &\geq -C_0 +\frac{2pb(x)\dotprod x+c_0(a+|x^2|)}{(a+|x|^2)^{1-p}}.
\end{align*}
By choosing $a=a(\|b\|_{L^{\infty}(\RR^n)}, c_0)\geq 1$ sufficiently large, we can ensure that $Lv >0$ on $\RR^n$. For $\eps>0$, we consider the auxiliary function,
$$
w_{\eps}:=u+\eps v.
$$
Then $Lw_{\eps} >0$ on $\RR^n$, and we see that the function $w_{\eps}$ tends to $ \infty$, as $|x|\rightarrow\infty$, by definition \eqref{eq:Auxiliary_function} of $v$ and the fact that $u$ is a bounded function. If $w_{\eps}$ is not nonnegative, there is a point $x_0\in\RR^n$ where the function $w_{\eps}$ attains a global minimum. We have that
$$
w_{\eps}(x_0)<0,\ \nabla w_{\eps}(x_0)=0,\quad\hbox{and}\quad (-\Delta)^sw_{\eps}(x_0)<0.
$$
Because $c \geq 0$ on $\RR^n$, we have that $L w_{\eps}(x_0)<0$, which contradicts the fact that $Lw_{\eps} >0$ on $\RR^n$. Therefore, $w_{\eps} \geq 0$ on $\RR^n$, for all $\eps \geq 0$, and so, we obtain inequality \eqref{eq:u_nonnegative} by letting $\eps$ tend to zero.
\end{proof}
The following supremum estimate is a consequence of Lemma \ref{lem:Comparison_principle}.
\begin{lem}[Supremum estimate]
\label{lem:Sup_estimate}
Assume that the hypotheses of Lemma \ref{lem:Comparison_principle} hold. If $f \in C(\RR^n)$ and $u \in C(\RR^n)\cap C^1_{\loc}(\RR^n)$ is a solution to problem \eqref{eq:Linear_equation}, then $u$ satisfies
\begin{equation}
\label{eq:Sup_estimate}
\|u\|_{C(\RR^n)} \leq \frac{1}{c_0}\|f\|_{C(\RR^n)}.
\end{equation}
\end{lem}
\begin{proof}
Estimate \eqref{eq:Sup_estimate} follows from the observation that
$$
L\left(\pm u+\frac{1}{c_0}\|f\|_{C(\RR^n)}\right) =\pm f+\|f\|_{C(\RR^n)} \geq 0,
$$
and an application of Lemma \ref{lem:Comparison_principle}.
\end{proof}
We have the following a priori estimates in H\"older spaces.
\begin{lem}[A priori Schauder estimates]
\label{lem:Schauder_estimate}
Let $s\in(1/2,1)$. Let $\alpha \in (0,1)$ be such that $2s+\alpha$ is not an integer. Assume that the coefficient functions $b\in C^{\alpha}(\RR^n;\RR^n)$ and $c \in C^{\alpha}(\RR^n)$. Then, there is a positive constant, $C=C(\alpha, \|b\|_{C^{\alpha}(\RR^n;\RR^n)}, \|c\|_{C^{\alpha}(\RR^n)}, c_0, n, s)$, such that for any source function, $f \in C^{\alpha}(\RR^n)$, and any solution, $u \in C^{2s+\alpha}(\RR^n)$, to the linear equation \eqref{eq:Linear_equation}, the function $u$ satisfies the estimate
\begin{equation}
\label{eq:Holder_estimate_prim}
\|u\|_{C^{2s+\alpha}(\RR^n)} \leq C\left(\|f\|_{C^{\alpha}(\RR^n)}+\|u\|_{C(\RR^n)}\right).
\end{equation}
\end{lem}
\begin{proof}
By \cite[Proposition 2.8]{Silvestre_2007}, we obtain that
$$
\|u\|_{C^{2s+\alpha}(\RR^n)} \leq C\left(\|(-\Delta)^s u\|_{C^{\alpha}(\RR^n)}+\|u\|_{C(\RR^n)}\right),
$$
where $C=C(\alpha,n,s)$ is a positive constant. Using the fact that the coefficient functions $b \in C^{\alpha}(\RR^n;\RR^n)$ and $c\in C^{\alpha}(\RR^n)$, and the Interpolation inequalities \cite[Theorems 3.2.1 \& 8.8.1]{Krylov_LecturesHolder} together with the fact that $2s>1$, we obtain that, for any $\eps>0$, there is a positive constant, $C=C(\alpha, \|b\|_{C^{\alpha}(\RR^n;\RR^n)}, \|c\|_{C^{\alpha}(\RR^n)}, c_0, \eps, n, s)$, such that
$$
\|u\|_{C^{2s+\alpha}(\RR^n)} \leq \eps \|u\|_{C^{2s+\alpha}(\RR^n)} +C\left(\|L u\|_{C^{\alpha}(\RR^n)}+\|u\|_{C(\RR^n)}\right).
$$
Choosing $\eps=1/2$, we obtain the a priori Schauder estimate \eqref{eq:Holder_estimate_prim}.
\end{proof}
We can now prove the existence and uniqueness of solutions in H\"older spaces to the linear equation \eqref{eq:Linear_equation} defined by the fractional Laplacian with drift.
\begin{proof}[Proof of Lemma \ref{lem:Existence_uniqueness_linear_equation}]
Uniqueness of solutions follows from Lemma \ref{lem:Sup_estimate}. We first assume that the function $f$ is in $C^{\infty}_c(\RR^n)$, and prove the existence of solutions, $u \in C^{2s+\alpha}_0(\RR^n)$, to the simpler equation,
\begin{equation}
\label{eq:L_0}
L_0 u = (-\Delta)^s u +c_0 u =f\quad\hbox{on } \RR^n.
\end{equation}
Taking the Fourier transform in equation \eqref{eq:L_0}, and using the fact that
$$
\widehat{(\Delta)^s v}(\xi) = |\xi|^{2s} \widehat v(\xi),\quad\forall \xi\in\RR^n, \quad\forall v \in \cS(\RR^n),
$$
we set
$$
u(x):=(2\pi)^{-n} \int_{\RR^n}e^{i\xi x}\frac{1}{|\xi|^{2s}+c_0}\widehat f(\xi)\ d\xi,\quad\forall x\in\RR^n.
$$
We want to prove that $u \in C^{\infty}_0(\RR^n)$, and that $u$ solves equation \eqref{eq:L_0}. Because $f \in C^{\infty}_c(\RR^n)$, we have that $\widehat f \in \cS(\RR^n)$, and so $1/(|\xi|^{2s}+c_0)\widehat f(\xi) \in L^1(\RR^n)$. The Riemann-Lebesgue lemma \cite[Theorem 8.22 f]{Folland_realanalysis} shows that $u\in C_0(\RR^n)$. We can apply the same argument to any derivative, $D^{\alpha} u$, for all multi-indices $\alpha\in \NN^{n}$, to deduce that $u \in C^{\infty}_0(\RR^n)$, and so the function $u$ belongs to the space $C^{2s+\alpha}_0(\RR^n)$, and $u$ is a solution to equation \eqref{eq:L_0}. Using now the a priori Schauder estimate \eqref{eq:Holder_estimate_prim} together with the supremum estimate \eqref{eq:Sup_estimate}, we obtain that $u$ satisfies inequality \eqref{eq:Holder_estimate}.
We now show that equation \eqref{eq:L_0} has a solution, $u\in C^{2s+\alpha}(\RR^n)$, for any choice of the source function, $f\in C^{\alpha}(\RR^n)$. We approximate $f$ by a sequence of functions, $\{f_k\}_{k\geq 0}\subset C^{\infty}_c(\RR^n)$, in the sense that
\begin{align*}
&f_k(x)\rightarrow f(x),\quad\hbox{as } k\rightarrow\infty,\quad\forall x\in \RR^n,\\
&\sup_{k\geq 0} \|f_k\|_{C^{\alpha}(\RR^n)} <\infty.
\end{align*}
For each $k \geq 0$, we let $u_k\in C^{2s+\alpha}_0(\RR^n)$ be the unique solution to the equation $L_0 u_k =f_k$ on $\RR^n$. Using estimate \eqref{eq:Holder_estimate}, the Arzel\'a-Ascoli Theorem gives that there is a subsequence, $\{u_k\}_{k\geq 0}$, which for simplicity we denote the same as the initial sequence, which converges to a solution, $u\in C^{2s+\alpha}(\RR^n)$, to equation \eqref{eq:L_0}. The convergence takes place uniformly in $C^{2s+\beta}(K)$, for all $\beta\in (0,\alpha)$, and all compact sets $K\subset \RR^n$.
Because we assume that the coefficient functions $b\in C^{\alpha}(\RR^n;\RR^n)$ and $c\in C^{\alpha}(\RR^n)$, by \cite[Proposition 2.5]{Silvestre_2007}, we see that the operator $L:C^{2s+\alpha} \rightarrow C^{\alpha}(\RR^n)$ is well-defined. Thus, with the aid of the Schauder estimate \eqref{eq:Holder_estimate}, and the existence and uniqueness of solutions in H\"older spaces to the model equation \eqref{eq:L_0}, we can use the method of continuity to prove existence of solutions in H\"older spaces to equation \eqref{eq:Linear_equation}.
\end{proof}
\subsubsection{The penalized equation}
\label{subsec:Penalized_equation}
Before proving existence of solutions to the obstacle problem \eqref{eq:Obstacle_problem} defined by the fractional Laplacian with drift, we first prove existence of solutions to the penalized equation,
\begin{equation}
\label{eq:Penalized_equation}
L u = \beta_{\eps}(\varphi-u)\quad\hbox{on }\RR^n,
\end{equation}
where $\beta_{\eps}:\RR\rightarrow [0,\infty)$ is defined by $\beta_{\eps}(t) = t^+/\eps$, for all $t\in\RR$, and $\eps$ is any positive constant.
\begin{lem}[Existence of solutions to the penalized equation]
\label{lem:Existence_penalized_equation}
Let $s\in(1/2,1)$. Let $\alpha \in (0,1)$ be such that $2s+\alpha$ is not an integer. Assume that the obstacle function $\varphi\in C^{\alpha}(\RR^n)$, and the coefficient functions $b \in C^{\alpha}(\RR^n;\RR^n)$, and $c \in C^{\alpha}(\RR^n)$ satisfies condition \eqref{eq:Nonnegative_lower_bound_c}. Then there is a solution, $u_{\eps}\in C^{2s+\alpha}(\RR^n)$, to the penalized equation \eqref{eq:Penalized_equation}.
\end{lem}
\begin{proof}
Let $\eps>0$, and consider the operator
$$
L_{\eps} v := Lv + \frac{1}{\eps} v,\quad\forall v \in C^2(\RR^n).
$$
We also let
\begin{equation}
\label{eq:Definition_gamma_eps}
\begin{aligned}
\gamma_{\eps}(v):=\beta_{\eps}(\varphi-v)+\frac{1}{\eps}v
= \frac{1}{\eps}
\begin{cases}
\varphi,&\quad\hbox{if }\varphi>v,\\
v,&\quad\hbox{if }\varphi\leq v.
\end{cases}
\end{aligned}
\end{equation}
We notice that $\gamma_{\eps}$ is a non-decreasing function. We make use of the monotonicity of the nonlinear term $\gamma_{\eps}$, to build a sequence of functions which converges to a solution to the penalized equation \eqref{eq:Penalized_equation}. We let
$$
\underline u = 0,\quad\hbox{and}\quad\overline u = \|\varphi\|_{C(\RR^n)}.
$$
The functions $\underline u$ and $\overline u$ are chosen such that
\begin{align*}
\underline u \leq \overline u,\quad L_{\eps} \underline u \leq \gamma_{\eps}(\underline u),\quad\hbox{and}\quad L_{\eps} \overline u \geq \gamma_{\eps}(\overline u).
\end{align*}
We construct iteratively a sequence of functions, $\{u_k\}_{k\geq 0}\subset C^{2s+\alpha}(\RR^n)$, which converge to a solution to the penalized equation \eqref{eq:Penalized_equation}. Let $u_0:=\underline u$, and let $u_k\in C^{2s+\alpha}(\RR^n)$ be the unique solution given by Lemma \ref{lem:Existence_uniqueness_linear_equation} to the linear equation
\begin{equation}
\label{eq:L_eps_equation}
L_{\eps} u_k=\gamma_{\eps}(u_{k-1}),\quad \forall k\geq 1.
\end{equation}
Because we assume that $u_{k-1}\in C^{2s+\alpha}(\RR^n)$ and $\varphi \in C^{\alpha}(\RR^n)$, we see that $\gamma_{\eps}(u_{k-1})\in C^{\alpha}(\RR^n)$, and so, we can apply Lemma \ref{lem:Existence_uniqueness_linear_equation} to build the function $u_k$. Notice that the operator $L_{\eps}$ satisfies the assumptions of Lemma \ref{lem:Existence_uniqueness_linear_equation} because we assume that the coefficient functions $b\in C^{\alpha}(\RR^n;\RR^n)$, and $c\in C^{\alpha}(\RR^n)$ and satisfies condition \eqref{eq:Nonnegative_lower_bound_c}, and $\eps>0$. We use Lemma \ref{lem:Comparison_principle} to prove inductively that the sequence of solutions, $\{u_k\}_{k\geq 0}$, is non-decreasing, and
\begin{equation}
\label{eq:u_k_increasing}
\underline u=:u_0\leq u_1\leq\ldots\leq u_k \leq \overline u,\quad\forall k \in \NN.
\end{equation}
For $k=1$, we see that the following sequence of inequalities hold
$$
L_{\eps} \underline u \leq \gamma_{\eps}(\underline u) = \gamma_{\eps}(u_0) = L_{\eps} u_1 \leq \gamma_{\eps}(\overline u) \leq L_{\eps} \overline u,
$$
and so, Lemma \ref{lem:Comparison_principle} gives us that the inequality $\underline u\leq u_1 \leq \overline u$ holds. Let now $k\geq 2$, and assume that inequalities \eqref{eq:u_k_increasing} hold with $k$ replaced by $k-1$. Then the monotonicity of $\gamma_{\eps}$ implies that
\begin{equation*}
\gamma_{\eps}(\underline u)\leq \gamma_{\eps}(u_1)\leq\ldots\leq \gamma_{\eps}(u_{k-1}) \leq \gamma_{\eps}(\overline u),
\end{equation*}
and we have that $L_{\eps} u_{k-1} \leq L_{\eps} u_k \leq L_{\eps} \overline u$. The preceding inequality and Lemma \ref{lem:Comparison_principle} imply that \eqref{eq:u_k_increasing} holds, and the sequence of functions $\{u_k\}_{k\in\NN}$ satisfies
\begin{equation}
\label{eq:Uniform_bound_u_k_eps}
0\leq u_k\leq\|\varphi\|_{C(\RR^n)},\quad\forall k \in\NN.
\end{equation}
From the a priori Schauder estimates \eqref{eq:Holder_estimate}, we have that
\begin{equation}
\label{eq:Schauder_estimate_u_k_eps}
\|u_k\|_{C^{2s+\alpha}(\RR^n)} \leq C \|\gamma_{\eps}(u_{k-1})\|_{C^{\alpha}(\RR^n)},\quad\forall k \geq 1,
\end{equation}
where $C=C(\alpha, \|b\|_{C^{\alpha}(\RR^n;\RR^n)}, \|c\|_{C^{\alpha}(\RR^n)},\eps,n,s)$ is a positive constant. From definition \eqref{eq:Definition_gamma_eps} of the nonlinear term $\gamma_{\eps}$, we obtain that there is a positive constant, $C=C(\eps)$, such that
\begin{equation}
\label{eq:Holder_norm_gamma_eps}
\|\gamma_{\eps}(u_{k-1})\|_{C^{\alpha}(\RR^n)} \leq C\left(\|u_{k-1}\|_{C^{\alpha}(\RR^n)} + \|\varphi\|_{C^{\alpha}(\RR^n)}\right),\quad\forall k \geq 1,
\end{equation}
and inequalities \eqref{eq:u_k_increasing} and \eqref{eq:Uniform_bound_u_k_eps} give us that
\begin{equation}
\label{eq:Sup_norm_gamma_eps}
\|\gamma_{\eps}(u_{k-1})\|_{C(\RR^n)} \leq C\|\varphi\|_{C(\RR^n)},\quad\forall k \geq 1.
\end{equation}
From \cite[Proposition 2.9]{Silvestre_2007}, we obtain that for any $\beta\in(0,2s-1)$, there is a positive constant, $C=(\beta,n,s)$, such that
$$
\|u_{k-1}\|_{C^{1+\beta}(\RR^n)} \leq C\left(\|(-\Delta)^s u_{k-1}\|_{C(\RR^n)} + \|u_{k-1}\|_{C(\RR^n)}\right),\quad\forall k \geq 1,
$$
which combined with estimates \eqref{eq:Schauder_estimate_u_k_eps} and \eqref{eq:Uniform_bound_u_k_eps}, equation \eqref{eq:L_eps_equation}, and definition of the operator $L_{\eps}$, gives us that
\begin{align*}
\|u_{k-1}\|_{C^{1+\beta}(\RR^n)} &\leq C\left(\|\gamma_{\eps} (u_{k-1})\|_{C(\RR^n)}+\|Du_{k-1}\|_{C(\RR^n)} + \|u_{k-1}\|_{C(\RR^n)}\right),\quad\forall k \geq 1,
\end{align*}
where $C=C(\|b\|_{C(\RR^n;\RR^n)}, \beta, \|c\|_{C(\RR^n)}, c_0, \eps)$ is a positive constant. The Interpolation Inequalities \cite[Theorems 8.8.1]{Krylov_LecturesHolder} yield
\begin{align*}
\|u_{k-1}\|_{C^{1+\beta}(\RR^n)} &\leq C\left(\|\gamma_{\eps} (u_{k-1})\|_{C(\RR^n)}+ \|u_{k-1}\|_{C(\RR^n)}\right),\quad\forall k \geq 1,
\end{align*}
which combined with estimates \eqref{eq:Uniform_bound_u_k_eps}, \eqref{eq:Schauder_estimate_u_k_eps}, \eqref{eq:Holder_norm_gamma_eps} and \eqref{eq:Sup_norm_gamma_eps} give us
$$
\|u_k\|_{C^{2s+\alpha}(\RR^n)} \leq C \|\varphi\|_{C^{\alpha}(\RR^n)},\quad\forall k \in \NN,
$$
if $\alpha\leq 1+\beta$. Because $\beta$ can be chosen in the interval $(0,2s-1)$, and $2s>1$, we see that the preceding estimate holds for all $\alpha\in (0,1)$. Therefore, the sequence of functions $\{u_k\}_{k\geq 0}$ is uniformly bounded in $C^{2s+\alpha}(\RR^n)$, and we can find a subsequence (which we denote the same as the initial sequence, for simplicity) convergent on compact subsets of $\RR^n$, with respect to the $\|\cdot\|_{C^{2s+\beta}(\RR^n)}$ norm, for all $\beta\in(0,\alpha)$, to a function $u_{\eps} \in C^{2s+\alpha}(\RR^n)$. Moreover, we see that
\begin{align*}
L_{\eps} u_k &\rightarrow Lu_{\eps} + \frac{1}{\eps} u_{\eps},\quad\hbox{as } k \rightarrow \infty,\\
\gamma_{\eps} (u_k) &\rightarrow \beta_{\eps}(\varphi-u_{\eps}) + \frac{1}{\eps} u_{\eps},\quad\hbox{as } k \rightarrow \infty,
\end{align*}
from where is follows that $u_{\eps}$ is a solution to the penalized equation \eqref{eq:Penalized_equation}.
\end{proof}
Before we can apply the existence of solutions to the penalized equation \eqref{eq:Penalized_equation} to prove existence of solutions to the obstacle problem \eqref{eq:Obstacle_problem}, we need the following estimates of the penalization term and the penalization sequence. For each $\eps>0$, let $u_{\eps}\in C^{2s+\alpha}(\RR^n)$ be the solution to equation \eqref{eq:Penalized_equation} constructed in Lemma \ref{lem:Existence_penalized_equation}.
\begin{lem}[Estimates of the penalization term and the penalization sequence]
\label{lem:Estimates_penalization_equation}
In addition to the hypotheses of Lemma \ref{lem:Existence_penalized_equation}, assume that the obstacle function $\varphi\in C_0(\RR^n)$ and obeys condition \eqref{eq:Boundedness_L_phi_positive_part}. Then, for all $\eps>0$, the following estimates hold
\begin{align}
\label{eq:Uniform_bound_beta_eps}
&\|\beta_{\eps}(\varphi-u_{\eps})\|_{C(\RR^n)} \leq \|(L\varphi)^+\|_{C(\RR^n)},\\
\label{eq:Uniform_bound_u_eps}
&0\leq u_{\eps} \leq \|\varphi\|_{C(\RR^n)}.
\end{align}
\end{lem}
\begin{proof}
Estimate \eqref{eq:Uniform_bound_u_k_eps} gives us \eqref{eq:Uniform_bound_u_eps}. To prove estimate \eqref{eq:Uniform_bound_beta_eps}, we adapt the argument used to prove \cite[Lemma 1.3.1]{Friedman_1982}. Using the fact that $\varphi\in C_0(\RR^n)$ and that $u_{\eps}$ is a nonnegative function by \eqref{eq:Uniform_bound_u_eps}, we see that the nonlinear term $\beta_{\eps}(\varphi-u_{\eps})$ must attain its global maximum at some point $x_0\in\RR^n$, and $x_0$ is also a global maximum of $\varphi-u_{\eps}$. We obtain that
\begin{align*}
\varphi(x_0)-u_{\eps}(x_0)\geq 0, \quad \nabla \varphi(x_0) - \nabla u_{\eps}(x_0)=0,\quad (-\Delta)^s(\varphi-u_{\eps})(x_0) \geq 0,
\end{align*}
and using the fact that $c\geq 0$ on $\RR^n$, it follows that $L u_{\eps} (x_0) \leq L\varphi \leq (L\varphi)^+ $. Therefore, $0\leq\beta_{\eps}(\varphi-u_{\eps})(x_0) \leq (L\varphi)^+$, and inequality \eqref{eq:Uniform_bound_beta_eps} follows.
\end{proof}
\subsubsection{Existence of solutions in H\"older spaces to the obstacle problem}
\label{subsec:Existence_solutions_obstacle_problem}
We now use Lemmas \ref{lem:Existence_penalized_equation} and \ref{lem:Estimates_penalization_equation} to prove
\begin{prop}[Existence of solutions in H\"older spaces to the obstacle problem]
\label{prop:Existence_Holder_obstacle_problem}
Let $s\in(1/2,1)$. Let $\alpha\in(0,2s-1)$ be such that $\alpha+2s$ is not an integer. Assume that the obstacle function $\varphi\in C^{\alpha}(\RR^n)\cap C_0(\RR^n)$ is such that condition \eqref{eq:Boundedness_L_phi_positive_part} holds. Assume that the coefficient functions $b \in C^{\alpha}(\RR^n;\RR^n)$, and that $c \in C^{\alpha}(\RR^n)$ satisfies \eqref{eq:Nonnegative_lower_bound_c}. Then there is a solution, $u\in C^{1+\alpha}(\RR^n)$, to the obstacle problem \eqref{eq:Obstacle_problem}, and identity \eqref{eq:Obstacle_problem} holds on $\RR^n$ in the sense of distributions.
\end{prop}
\begin{rmk}
Notice that Lemma \ref{lem:Existence_penalized_equation} establishes the existence of solutions in $C^{2s+\alpha}(\RR^n)$ to the penalized equation \eqref{eq:Penalized_equation}, while Proposition \ref{prop:Existence_Holder_obstacle_problem} shows the existence of solutions in $C^{1+\alpha}(\RR^n)$ to the obstacle problem \eqref{eq:Obstacle_problem}. When $u\in C^{2s+\alpha}(\RR^n)$, by \cite[Proposition 2.6]{Silvestre_2007} it follows that $(-\Delta u)^s\in C^{\alpha}(\RR^n)$, but when $u \in C^{1+\alpha}(\RR^n)$, we make sense of $(-\Delta)^s u$ only in the distributional sense, as shown in the proof of Proposition \ref{prop:Existence_Holder_obstacle_problem}.
\end{rmk}
\begin{proof}[Proof of Proposition \ref{prop:Existence_Holder_obstacle_problem}]
For each $\eps>0$, let $u_{\eps}\in C^{2s+\alpha}(\RR^n)$ be the solution to the penalized equation \eqref{eq:Penalized_equation} constructed in the proof of Lemma \ref{lem:Existence_penalized_equation}. Using the fact that $\alpha\in(0,2s-1)$, it follows by \cite[Proposition 2.9]{Silvestre_2007} that there is a positive constant, $C=C(\alpha,n,s)$, such that
$$
\|u_{\eps}\|_{C^{1+\alpha}(\RR^n)} \leq C\left(\|u_{\eps}\|_{C(\RR^n)}+\|\beta_{\eps}(\varphi-u_{\eps})\|_{C(\RR^n)}+\|b\dotprod\nabla u_{\eps}\|_{C(\RR^n)} + \|cu_{\eps}\|_{C(\RR^n)}\right).
$$
Using the Interpolation Inequalities \cite[Theorems 8.8.1]{Krylov_LecturesHolder} and Lemma \ref{lem:Estimates_penalization_equation}, we obtain that there is a positive constant, $C=C(\alpha, \|b\|_{C^{\alpha}(\RR^n;\RR^n)}, \|c\|_{C^{\alpha}(\RR^n)}, n, s)$, such that
\begin{equation}
\label{eq:u_eps_Holder_estimate}
\|u_{\eps}\|_{C^{1+\alpha}(\RR^n)} \leq C\left(\|\varphi\|_{C(\RR^n)}+\|(L\varphi)^+\|_{C(\RR^n)}\right),\quad\forall\eps>0.
\end{equation}
Therefore, we can find a subsequence, which for simplicity we denote the same as the initial sequence, which converges locally in $C^{1+\beta}(\RR^n)$, for all $\beta\in(0,\alpha)$, to a function $u \in C^{1+\alpha}(\RR^n)$.
Notice that because $\alpha\in(0,2s-1)$, the fact that $u \in C^{1+\alpha}(\RR^n)$ does not immediately imply that the quantity $(-\Delta)^s u$ is well-defined. We now make sense of $(-\Delta)^s u$ in the sense of distributions. For this purpose, it is enough to show that $\eta(-\Delta)^s u$ is a tempered distribution, where $\eta:\RR^n\rightarrow [0,1]$ is a smooth cut-off function such that
\begin{equation}
\label{eq:Definition_eta}
\eta \equiv 1\quad\hbox{on }B'_1(x_0),\quad\hbox{and}\quad \eta \equiv 0\quad\hbox{on }(B'_2(x_0))^c,
\end{equation}
and the point $x_0\in\RR^n$ is arbitrarily chosen. We define
\begin{equation}
\label{eq:v_eps}
v:=\eta u,\quad\hbox{and}\quad v_{\eps}:=\eta u_{\eps},\quad\forall \eps>0.
\end{equation}
Direct calculations give us that
\begin{align}
\label{eq:Delta_s_v_cutoff}
(-\Delta)^s v_{\eps} = \eta (-\Delta)^s u_{\eps} + f_{\eps},
\end{align}
where the function $f_{\eps}$ is defined by
\begin{equation}
\label{eq:f_eps}
f_{\eps}(x):=u_{\eps}(x)(-\Delta)^s \eta(x)+\int_{\RR^n}\frac{(\eta(y)-\eta(x))(u_{\eps}(y)-u_{\eps}(x))}{|x-y|^{n+2s}}\ dy,\quad\forall x\in\RR^n.
\end{equation}
Multiplying equation \eqref{eq:Penalized_equation} by $\eta$, we obtain
\begin{align}
\label{eq:Penalized_equation_cutoff}
(-\Delta)^s v_{\eps} -f_{\eps} +\eta(b\dotprod\nabla u_{\eps}+c u_{\eps}) = \eta\beta_{\eps}(\varphi-u_{\eps})\quad\hbox{on }\RR^n.
\end{align}
From definition \eqref{eq:v_eps} of $v_{\eps}$, using the fact that the sequence $\{u_{\eps}\}$ converges locally in $C^{1+\beta}(\RR^n)$ to $u \in C^{1+\alpha}(\RR^n)$, for all $\beta \in (0,\alpha)$, we have the pointwise convergence on $\RR^n$,
\begin{equation}
\label{eq:f_convergence}
-f_{\eps} +\eta(b\dotprod\nabla u_{\eps}+c u_{\eps}) \rightarrow -f + \eta(b\dotprod\nabla u+c u),\quad\hbox{as }\eps\rightarrow 0,
\end{equation}
where $f$ is defined by the same formula as $f_{\eps}$ in \eqref{eq:f_eps}, but with $u_{\eps}$ replaced by $u$. From the definition \eqref{eq:v_eps} of the function $v_{\eps}$, using estimate \eqref{eq:u_eps_Holder_estimate}, and the fact that $v_{\eps}$ and $v$ have compact support contained in $B'_2(x_0)$, we have that
\begin{equation}
\label{eq:v_eps_L_2_convergence}
v_{\eps}\rightarrow v\quad\hbox{in } L^2(\RR^n),\quad\hbox{as } \eps\rightarrow 0.
\end{equation}
We now show that the convergence in \eqref{eq:v_eps_L_2_convergence} implies the convergence in the sense of distributions,
\begin{equation}
\label{eq:Delta_s_v_eps_distributions_convergence}
(-\Delta)^s v_{\eps} \rightarrow (-\Delta)^s v \quad\hbox{in } \cS'(\RR^n),\quad\hbox{as } \eps\rightarrow 0.
\end{equation}
For all $\psi\in\cS(\RR^n)$, we have that
\begin{align*}
\left\langle (-\Delta)^s v_{\eps}, \psi \right\rangle &= \left\langle \widehat{(-\Delta)^s v_{\eps}}, \widehat \psi \right\rangle\\
&=\left\langle \widehat v_{\eps}, |\xi|^{2s}\widehat \psi \right\rangle\quad\hbox{(using the fact that $\widehat{(-\Delta)^s v_{\eps}}(\xi)=|\xi|^{2s}\widehat v_{\eps}(\xi)$)},
\end{align*}
where $\left\langle\cdot,\cdot\right\rangle$ denotes the duality of $\cS'(\RR^n)$ and $\cS(\RR^n)$. From \eqref{eq:v_eps_L_2_convergence}, we know that $\widehat v_{\eps}$ converges to $\widehat v$ in $L^2(\RR^n)$, as $\eps\rightarrow 0$, and from the fact that $|\xi|^{2s}\widehat \psi\in L^2(\RR^n)$, for all $\psi\in\cS(\RR^n)$, we obtain that
\begin{align*}
\left\langle (-\Delta)^s v_{\eps}, \psi \right\rangle
&\rightarrow \left\langle \widehat v, |\xi|^{2s}\widehat \psi \right\rangle\\
&= \left\langle \widehat{(-\Delta)^s v}, \widehat \psi \right\rangle\\
&= \left\langle (-\Delta)^s v, \psi \right\rangle,
\end{align*}
from which the convergence in \eqref{eq:Delta_s_v_eps_distributions_convergence} follows. Using \eqref{eq:f_convergence} and \eqref{eq:Delta_s_v_eps_distributions_convergence} , we can let $\eps$ tend to $0$ in \eqref{eq:Penalized_equation_cutoff} to obtain that the following hold in distributional sense,
\begin{equation}
\label{eq:Obstacle_problem_cutoff}
\begin{aligned}
\begin{cases}
(-\Delta)^s (\eta u) - f + \eta(b\dotprod\nabla u+c u) =0,&\quad\hbox{if}\quad\eta\varphi<\eta u,\\
(-\Delta)^s (\eta u) - f + \eta(b\dotprod\nabla u+c u) \geq 0,&\quad\hbox{if}\quad\eta\varphi\geq \eta u,
\end{cases}
\end{aligned}
\end{equation}
where we used the fact that $v=\eta u$. Therefore, on the set $\{\eta=1\}\supseteq B'_1(x_0)$, using \eqref{eq:Delta_s_v_cutoff} applied to $v$ instead of $v_{\eps}$, we obtain
\begin{equation*}
\begin{aligned}
\begin{cases}
(-\Delta)^s u + b\dotprod\nabla u+c u =0,&\quad\hbox{if}\quad \varphi< u,\\
(-\Delta)^s u + b\dotprod\nabla u+c u \geq 0,&\quad\hbox{if}\quad\varphi\geq u.
\end{cases}
\end{aligned}
\end{equation*}
Because the point $x_0\in\RR^n$ was arbitrarily chosen, we obtain that identity \eqref{eq:Obstacle_problem} holds on $\RR^n$ in the distributional sense. Therefore, the function $u \in C^{1+\alpha}(\RR^n)$ is a solution to the obstacle problem \eqref{eq:Obstacle_problem}.
\end{proof}
Proposition \ref{prop:Existence_Holder_obstacle_problem} is the main ingredient in the proof of Proposition \ref{prop:Solutions_partial_regularity} together with \cite[Theorem 5.8]{Silvestre_2007}.
\begin{proof}[Proof of Proposition \ref{prop:Solutions_partial_regularity}]
From Proposition \ref{prop:Existence_Holder_obstacle_problem}, we may assume without loss of generality that $\alpha \in [2s-1,1)$.
We improve the regularity of solutions from $u\in C^{1+\gamma}(\RR^n)$, for all $\gamma\in (0,2s-1)$, established in Proposition \ref{prop:Existence_Holder_obstacle_problem}, to $u\in C^{1+\beta}(\RR^n)$, for all $\beta\in (0,\alpha\wedge s)$, by a bootstrapping argument.
Let $\alpha_0\in (0,2s-1)$, and let $u\in C^{1+\alpha_0}(\RR^n)$ be the solution to the obstacle problem \eqref{eq:Obstacle_problem} constructed in the proof of Proposition \ref{prop:Existence_Holder_obstacle_problem}. Then by the complementarity conditions \eqref{eq:Obstacle_problem_cutoff} and definition \eqref{eq:Definition_eta} of the cut-off function $\eta$, the function $\eta u$ is a solution to the obstacle problem,
$$
\min\{(-\Delta)^s (\eta u) - f + \eta(b\dotprod\nabla u+c u), \eta u-\eta\varphi\}=0\quad\hbox{on } \RR^n,
$$
where the function $f$ is given by the same formula as $f_{\eps}$ in \eqref{eq:f_eps}, but with $u_{\eps}$ replaced by $u$. We define the function $g$ by
\begin{equation}
\label{eq:Definition_g}
g(x):=f(x)-\eta(x)(b(x)\dotprod\nabla u(x)+c(x) u(x)),\quad\forall x\in \RR^n.
\end{equation}
Because the function $u$ belongs to $C^{1+\alpha_0}(\RR^n)$ and $\eta$ has compact support, then the function $g$ decays like $|x|^{-(n+2s)}$, as $|x|\rightarrow\infty$, and Lemma \ref{lem:Regularity_g} yields that $g$ belongs to $C^{\alpha_0\wedge (2(1-s))}(\RR^n)$. We let $w$ be defined by
\begin{equation}
\label{eq:Definition_w_existence}
w(x):=c_{n,s} \int_{\RR^n}\frac{g(y)}{|x-y|^{n-2s}}\ dy,\quad\forall x\in \RR^n.
\end{equation}
The function $w$ is a solution to the linear equation $(-\Delta)^s w = g$ on $\RR^n$. From \cite[Proposition 2.8]{Silvestre_2007} we have that $w$ belongs to $C^{2s+\alpha_0\wedge (2(1-s))}(\RR^n)$. Using the definition \eqref{eq:Definition_w_existence} of $w$, we also have that $w$ decays like $|x|^{-n}$, as $|x|\rightarrow\infty$. Therefore, we obtain that $\eta u-w$ is a continuous solution to the obstacle problem,
\begin{equation}
\label{eq:Simplified_obstacle_problem}
\begin{aligned}
(-\Delta)^s (\eta u -w) \geq 0&\quad\hbox{on }\RR^n,\\
(-\Delta)^s (\eta u -w) = 0&\quad\hbox{on } \{u>\varphi\} \cap \{\eta>0\} = \{\eta u-w>\eta\varphi-w\},\\
\eta u-w\geq \eta\varphi-w &\quad\hbox{on }\RR^n,\\
\lim_{|x|\rightarrow\infty} \eta(x)u(x)-w(x)=0.
\end{aligned}
\end{equation}
Because $w\in C^{2s+\alpha_0\wedge (2(1-s))}(\RR^n)$ and $\varphi \in C^{1+\alpha}(\RR^n)$, the obstacle function $\eta \varphi-w$ belongs to $C^{1+\gamma_1}(\RR^n)$, where
\begin{align*}
1+\gamma_1&=(2s+\alpha_0 \wedge (2(1-s)))\wedge (1+\alpha)\\
&=(2s+\alpha_0)\wedge (1+\alpha).
\end{align*}
The second equality follows from the fact that $\alpha\in (0,1)$ and $2s+2(1-s)=2>1+\alpha$. By \cite[Theorem 5.8]{Silvestre_2007}, we have that $\eta u-w\in C^{1+\beta}(\RR^n)$, for all $\beta<\alpha_1$, where $1+\alpha_1:=(2s+\alpha_0)\wedge (1+\alpha)\wedge (1+s)$. Because the center of the ball $B'_1(x_0)$ in the definition \eqref{eq:Definition_eta} of the cut-off function $\eta$ can be chosen arbitrarily in $\RR^n$, we obtain that $u \in C^{1+\beta}(\RR^n)$, for all $\beta<\alpha_1$. If $2s+\alpha_0 \geq (1+\alpha)\wedge (1+s)$, we obtain that the solution $u$ belongs to $C^{1+\beta}$, for all $\beta\in(0,\alpha\wedge s)$. Otherwise, we repeat the preceding steps, but now we notice that the H\"older exponent $\alpha_0$ can be replaced by $\alpha_0+(2s-1)$, where we recall that the increment $2s-1$ is positive, since we assume that $s\in (1/2,1)$. The fact that $w\in C^{2s+\alpha_0+(2s-1)}(\RR^n)$ gives us that $\eta \varphi-w$ belongs to $C^{1+\gamma_2}(\RR^n)$, where
$$
1+\gamma_2:=(2s+\alpha_0+(2s-1))\wedge (1+\alpha).
$$
By \cite[Theorem 5.8]{Silvestre_2007}, it follows that $\eta u-w\in C^{1+\beta}(\RR^n)$, for all $\beta<\alpha_2$, where $1+\alpha_2:=(2s+\alpha_0+(2s-1))\wedge (1+\alpha)\wedge (1+s)$, and so, the function $u$ belongs to $C^{1+\beta}(\RR^n)$, for all $\beta<\alpha_2$. We repeat this procedure $k$ times where we choose $k$ such that
\begin{align*}
1+\alpha_k&:=(2s+\alpha_0+(k-1)(2s-1))\wedge (1+\alpha)\wedge (1+s)\\
& = (1+\alpha)\wedge (1+s),
\end{align*}
and the conclusion that the function $u$ belongs to $C^{1+\beta}(\RR^n)$, for all $\beta\in (0,\alpha\wedge s)$, now follows.
\end{proof}
\subsection{Uniqueness of solutions}
\label{sec:Uniqueness}
We use a probabilistic method to prove uniqueness of solutions to the obstacle problem \eqref{eq:Obstacle_problem} by establishing their stochastic representation. Let $(\Omega, \cF, \PP)$ be a filtered probability space endowed with a filtration, $\{\cF(t)\}_{t\geq 0}$, which satisfies the usual hypotheses of completeness and right-continuity \cite[p. 72]{Applebaum}. Let $N(dt,dy)$ be a Poisson random measure on $[0,\infty)\times(\RR^n\backslash\{O\})$ with L\'evy measure,
\begin{equation}
\label{eq:Levy_measure}
\nu(dy)=\frac{c_{n,s}}{|y|^{n+2s}}\, dy,
\end{equation}
and let $\widetilde N(dt, dy)=N(dt, dy) - \nu(dy)dt$ denote its compensator. We recall the results on existence and uniqueness of solutions to the stochastic equation,
\begin{equation}
\label{eq:SDE}
dX(t)=-b(X(t))\ dt+\int_{\RR^n\backslash\{O\}} y \widetilde N(dt,dy),\quad t>0,\ X_0=x \in\RR^n.
\end{equation}
If the vector field $b:\RR^n\rightarrow\RR^n$ is a bounded, Lipschitz continuous function, then it follows by \cite[Theorem 6.2.9]{Applebaum}, that there is a unique RCLL (right-continuous with left limit) adapted solution to equation \eqref{eq:SDE}.
\begin{prop}[Uniqueness of solutions]
\label{prop:Uniqueness}
Let $s\in (0,1)$ and $\alpha\in((2s-1)\vee 0,1)$. Assume that the obstacle function $\varphi \in C(\RR^n)$, and that the coefficient function $b\in C(\RR^n;\RR^n)$ is Lipschitz continuous on $\RR^n$, and that the coefficient function $c$ is a Borel measurable function which satisfies condition \eqref{eq:Lower_bound_c}. If $u \in C^{1+\alpha}(\RR^n)$ is a solution to the obstacle problem \eqref{eq:Obstacle_problem}, then $u$ has the stochastic representation
\begin{equation}
\label{eq:Stochastic_representation}
u(x)=\sup_{\tau\in\cT} \EE^x\left[e^{-\int_0^{\tau} c(X(s))\, ds} \varphi(X(\tau))\right],\quad\forall x\in\RR^n,
\end{equation}
where $\cT$ is the set of stopping times with respect to the filtration $\{\cF(t)\}_{t\geq 0}$, and $\{X(t)\}_{t\geq 0}$ is the unique solution to the stochastic differential equation \eqref{eq:SDE}, with initial condition $X(0)=x$.
\end{prop}
\begin{proof}
Because we assume that $\alpha>2s-1$, we may apply It\^o's lemma \cite[Theorem 4.4.7]{Applebaum} to the function $u \in C^{1+\alpha}(\RR^n)$ and the unique solution, $\{X(t)\}_{t\geq 0}$, to equation \eqref{eq:SDE}, with initial condition $X(0)=x$. We obtain
\begin{align*}
d\left(e^{-\int_0^t c(X(s))\, ds} u(X(t))\right)
&= e^{-\int_0^t c(X(s))\, ds}\left[\left(-c(X(t-)) - b(X(t-))\dotprod\nabla u(X(t-))\right)\, dt\right.\\
&\quad+ \int_{\RR^n\backslash\{O\}}\left(u(X(t-)+y)-u(X(t-))\right)\widetilde N(dt,dy)\\
&\quad+ \left. \int_{\RR^n\backslash\{O\}}\left(u(X(t-)+y)-u(X(t-))-y\dotprod\nabla u(X(t-))\right) \nu(dy) dt\right].
\end{align*}
The assumptions that the function $u$ belongs to $C^{1+\alpha}(\RR^n)$, and $\alpha>2s-1$, is used to ensure that the last term in the preceding expression is well-defined. Because the function $u$ belongs to $C^1(\RR^n)$, and $s>1/2$, we see from definition \eqref{eq:Levy_measure} of the L\'evy measure, $\nu(dy)$, that there is a positive constant, $C$, such that
$$
\int_{\RR^n\backslash\{O\}} \left|u(X(t-)+y)-u(X(t-))\right|^2 \nu(dy) \leq C,\quad\forall t \geq 0,
$$
and so, it follows by the Martingale Representation Theorem \cite[Theorem 5.3.5]{Applebaum} that the process
$$
M(t):=\int_0^t\int_{\RR^n\backslash\{O\}}\left(u(X(s-)+y)-u(X(s-))\right)\widetilde N(ds,dy),\quad t\geq 0,
$$
is a martingale. We then can write
$$
e^{-\int_0^t c(X(s))\, ds} u(X(t))= u(x)- \int_0^t e^{-\int_0^s c(X(r))\, dr} Lu(X(s-))\, ds +M(t),\quad\forall t \geq 0.
$$
As usual, we define the stopping time $\tau^*$ by
$$
\tau^*:=\inf\{t \geq 0:\ u(X(t))=\varphi(X(t))\}.
$$
On the set $\{u>\varphi\}$, we have that $Lu=0$. Moreover, since we assume that the solution $u$ belongs to the H\"older space $C^{1+\alpha}$, for some $\alpha>2s-1$, it follows by \cite[Proposition 2.6]{Silvestre_2007} that the function $Lu$ is continuous on $\RR^n$, and so, we have that $Lu(x)=0$, for all $x\in\{u>\varphi\}\cup\partial\{u>\varphi\}$. We then obtain that the stopped process
$$
\left\{e^{-\int_0^{t\wedge\tau^*} c(X(s))\, ds} u(X(t\wedge\tau^*))\right\}_{t\geq 0}
$$
is a martingale, which gives us that
$$
u(x)=\EE^x\left[e^{-\int_0^{\tau^*} c(X(s))\, ds} \varphi(X(\tau^*))\right],\quad\forall x\in\RR^n,
$$
where $\EE^x$ denotes expectation with respect to the law of the unique solution, $\{X(t)\}_{t\geq 0}$, to the equation \eqref{eq:SDE}, with initial condition $X(0)=x$. The condition that the coefficient function $c$ satisfies inequality \eqref{eq:Lower_bound_c}, is used to ensure that the integrand in the preceding expression is well-defined when $\tau^*=\infty$. Because we assume that the obstacle function $\varphi$ is bounded, when $\tau^*=\infty$, the integrand in the preceding expression is zero. Because $Lu \geq 0$ on $\RR^n$, in general, we have that the process
$$
\left\{e^{-\int_0^{t} c(X(s))\, ds} u(X(t))\right\}_{t\geq 0}
$$
is a supermartingale. Together with the fact that $u\geq \varphi$ on $\RR^n$, this implies that, for all $\tau\in\cT$, we have that
$$
u(x)\geq \EE^x\left[e^{-\int_0^{\tau} c(X(s))\, ds} \varphi(X(\tau))\right].
$$
Thus we obtain the stochastic representation \eqref{eq:Stochastic_representation} of solutions $u$ to the obstacle problem \eqref{eq:Obstacle_problem}, which in particular implies that the solution is unique.
\end{proof}
\subsection{Monotonicity formula}
\label{sec:Monotonicity_formula}
In this section we prove a new Almgren-type monotonicity formula suitable for solutions to the obstacle problem defined by the fractional Laplacian with drift, \eqref{eq:Obstacle_problem}. We use the monotonicity formula to establish the optimal regularity of solutions in \S \ref{sec:Solutions_optimal_regularity}.
We assume that the hypotheses of Proposition \ref{prop:Solutions_partial_regularity} hold, and in addition that the obstacle function $\varphi$ belongs to $C^{2s+\alpha}(\RR^n)$, for all $\alpha\in (0,s)$. Proposition \ref{prop:Solutions_partial_regularity} gives us the existence of a solution $u\in C^{1+\alpha}(\RR^n)$, for all $\alpha\in (0,s)$, to the obstacle problem \eqref{eq:Obstacle_problem} which solves the ``localized" obstacle problem \eqref{eq:Simplified_obstacle_problem}. We recall that the function $w$ defined in \eqref{eq:Definition_w_existence} and appearing in \eqref{eq:Simplified_obstacle_problem}, belongs to the space $C^{2s+\alpha}(\RR^n)$, and so, the function $\eta \varphi-w$ is contained in $C^{2s+\alpha}(\RR^n)$, for all $\alpha \in (0,s)$, since we assume that $\varphi \in C^{2s+\alpha}(\RR^n)$, for all $\alpha \in (0,s)$. Therefore, using \eqref{eq:Simplified_obstacle_problem} we reduce the study of the regularity of solutions to the obstacle problem \eqref{eq:Obstacle_problem} to that of solutions to the problem,
\begin{equation}
\label{eq:Obstacle_problem_simple}
\begin{aligned}
(-\Delta)^s u \geq 0&\quad\hbox{on }\RR^n,\\
(-\Delta)^s u = 0&\quad\hbox{on } \{u>\varphi\},\\
u\geq \varphi &\quad\hbox{on }\RR^n,
\end{aligned}
\end{equation}
where we now let $u$ replace $\eta u-w$, and $\varphi$ replace $\eta\varphi-w$ in \eqref{eq:Simplified_obstacle_problem}. Thus, the natural starting assumption in proving the optimal regularity of solutions to the obstacle problem \eqref{eq:Obstacle_problem}, is that $\varphi\in C^{2s+\alpha}(\RR^n)$, for all $\alpha\in (0,s)$, and $u \in C^{1+\alpha}(\RR^n)$, for all $\alpha\in (0,s)$, is a solution to problem \eqref{eq:Obstacle_problem_simple}. We recall that the regularity of these solutions is studied in \cite{Caffarelli_Salsa_Silvestre_2008} under the assumption that $\varphi \in C^{2,1}(\RR^n)$. In our case, in general we can only assume that $\varphi\in C^{2s+\alpha}(\RR^n)$, for all $\alpha\in (0,s)$, due to the presence of the lower order terms in the expression of the operator $L$.
Let $a:=1-2s$. We consider the operator $L_a$ defined, for all $v\in C^2(\RR^n\times\RR_+)$, by
\begin{equation}
\label{eq:L_a}
L_a v(x,y) = \hbox{div }(|y|^{a}\nabla v)(x,y),\quad\forall (x,y)\in \RR^n\times\RR_+.
\end{equation}
The relation between the degenerate-elliptic operator $L_a$ and the fractional Laplacian operator, $(-\Delta)^s$, is investigated in \cite[\S 3]{Caffarelli_Silvestre_2007}, where it is established that $L_a$-harmonic functions, $u$, satisfy
\begin{equation}
\label{eq:Dirichlet_to_Neumann_map}
\lim_{y\downarrow 0} y^a u_y(x,y) = -(-\Delta)^s u(x,0),
\end{equation}
that is, the fractional Laplacian operator, $(-\Delta)^s$, is a Dirichlet-to-Neumann map for the elliptic operator $L_a$. Identity \eqref{eq:Dirichlet_to_Neumann_map} holds up to multiplication by a constant factor (see \cite[Formula (3.1)]{Caffarelli_Silvestre_2007}).
We construct the $L_a$-harmonic extensions of the functions $u(x)$ and $\varphi(x)$ from $\RR^n$ to the half-space $\RR^n\times\RR_+$ (see \cite[\S 2.4]{Caffarelli_Silvestre_2007}). For simplicity, we keep the same notation for the extensions as for the initial functions, even if the domains changed. That is, we denote the extensions of the functions $u(x)$ and $\varphi(x)$, defined for all $x\in \RR^n$, by $u(x,y)$ and $\varphi(x,y)$, defined for all points $(x,y)\in\RR^n\times\RR_+$, respectively. We assume without loss of generality that $O$ is a point on $\partial\{u=\varphi\}$. Given the fact that $\varphi \in C^{2s+\alpha}(\RR^n)$, for all $\alpha \in (0,s)$, we may consider the following auxiliary ``height" function,
\begin{equation}
\label{eq:Auxiliary_function_v}
v(x,y) := u(x,y) - \varphi(x,y) + \frac{1}{2s}(-\Delta)^s\varphi(O)|y|^{1-a},\quad\forall (x,y)\in\RR^n\times\bar\RR_+,
\end{equation}
and we extend $v$ to the whole space $\RR^{n+1}$ by even reflection, i.e. we let $v(x,y)=v(x,-y)$, for all $(x,y)\in\RR^n\times\RR_+$. Compare the definition of the function $v$ with that of $\widetilde u$ in \cite[p. 433]{Caffarelli_Silvestre_2007}, where the condition that the obstacle $\varphi$ belongs to $C^2(\RR^n)$ is required.
Because we know that the function $u \in C^{1+\alpha}(\RR^n)$ and $\varphi \in C^{2s+\alpha}(\RR^n)$, and $O\in\partial\{u=\varphi\}$, we can find a positive constant, $C$, such that
\begin{equation}
\label{eq:Ineq_v_on_R_n}
0\leq v(x) \leq C|x|^{1+\alpha},\quad\forall x\in\RR^n.
\end{equation}
In addition, the function $v$ satisfies the following properties
\begin{align}
\label{eq:Properties_v_1}
L_a v =0 &\quad\hbox{on }\RR^n\times(\RR\backslash\{0\}),\\
\label{eq:Properties_v_2}
v \geq 0&\quad\hbox{on }\RR^n\times\{0\}.
\end{align}
The integration by parts formula gives us that
\begin{align*}
L_a v(x,y) &= 2 \lim_{z\downarrow 0} |z|^a v_z(x,z) \cH^n|_{\{y=0\}},
\end{align*}
where $\cH^n|_{\{y=0\}}$ denotes the Hausdorff measure on the hyperplane $\{y=0\}$. Using now identities \eqref{eq:Dirichlet_to_Neumann_map} and \eqref{eq:Auxiliary_function_v}, we obtain
\begin{align}
\label{eq:Equation_v}
L_a v(x,y) &= 2\left(-(-\Delta)^s u(x) +(-\Delta)^s \varphi(x)-(-\Delta)^s \varphi(O)\right)\cH^n|_{\{y=0\}}.
\end{align}
Because the function $u$ solves problem \eqref{eq:Obstacle_problem_simple}, we see that
\begin{align}
\label{eq:Upper_bound_L_a}
L_a v(x,y) &\leq 2\left((-\Delta)^s \varphi(x)-(-\Delta)^s \varphi(O)\right)\cH^n|_{\{y=0\}}\quad\hbox{on } \RR^{n+1},\\
\label{eq:Equality_L_a}
L_a v(x,y) &= 2\left((-\Delta)^s \varphi(x)-(-\Delta)^s \varphi(O)\right)\cH^n|_{\{y=0\}}\quad\hbox{on } \RR^{n+1}\backslash(\{y=0\}\cap\{u=\varphi\}).
\end{align}
Notice that $L_a v$ is a singular measure supported on $\{y=0\}$. Compared to $L_a \widetilde u$, where the function $\widetilde u$ is the analogue of $v$ in \cite[p. 433]{Caffarelli_Silvestre_2007}, the singular measure $L_a v$ has nontrivial support on $\{y=0\}\backslash\{v=0\}$, while the measure $L_a\widetilde u$ is a classical function on $\{y=0\}\backslash\{\widetilde u=0\}$. This is due to the presence of the drift component in the definition of the operator $L$. This difference is one of the key points which makes the analysis of the obstacle problem for the fractional Laplacian with drift \eqref{eq:Obstacle_problem} different that the one of the obstacle problem without drift studied in \cite{Caffarelli_Salsa_Silvestre_2008}.
We denote the right-hand side in inequalities \eqref{eq:Upper_bound_L_a} and \eqref{eq:Equality_L_a} by
\begin{equation}
\label{eq:Definition_h}
h(x):=2\left((-\Delta)^s \varphi(x)-(-\Delta)^s \varphi(O)\right),\quad\forall x\in \RR^n.
\end{equation}
Because $\varphi\in C^{2s+\alpha}(\RR^n)$, we see that $h \in C^{\alpha}(\RR^n)$ by \cite[Proposition 2.6]{Silvestre_2007}, and so we have that
\begin{equation}
\label{eq:Growth_h}
|h(x)| \leq C|x|^{\alpha},\quad \forall x\in \RR^n,
\end{equation}
where $C:=2[(-\Delta)^s \varphi]_{C^{\alpha}(\RR^n)}$, for all $\alpha\in (0,s)$.
Let $U\subseteq\RR^{n+1}$ be a Borel measurable set. We say that a function $w$ belongs to the weighted Sobolev space $H^1(U,|y|^a)$, if $w$ and $Dw$ are function in $L^2_{\loc}(U)$ and
$$
\int_{U}\left(|w|^2+|\nabla w|^2\right)|y|^a <\infty.
$$
From \cite[\S 2.4]{Caffarelli_Silvestre_2007}, it follows that the auxiliary function $v$ belongs to the spaces $C(\RR^{n+1})$ and $ H^1(B_r,|y|^a)$, for all $r>0$. In particular, the following quantities are well-defined:
\begin{align}
\label{eq:F}
F_v(r)&:=\int_{\partial B_r} |v|^2 |y|^a,\\
\label{eq:Phi}
\Phi^p_v(r) &:= r\frac{d}{dr} \log\max\{F_v(r), r^{n+a+2(1+p)}\},
\end{align}
where $r>0$ and $p>0$. The function $F_v(r)$ and $\Phi^p_v(r)$ are the analogues of the functions $F_u(r)$ and $\Phi_u(r)$ given by \cite[Definitions (3.1) and (3.2)]{Caffarelli_Salsa_Silvestre_2008}, but adapted to our framework.
The main result of this section is the following analogue of \cite[Theorem 3.1]{Caffarelli_Salsa_Silvestre_2008}.
\begin{prop}[Monotonicity formula]
\label{prop:Monotonicity_formula}
Let $s\in (1/2,1)$, $\varphi\in C^{2s+\alpha}(\RR^n)$ and $u\in C^{1+\alpha}(\RR^n)$, for all $\alpha\in (0,s)$, such that $2s+\alpha$ is not an integer. Assume that the function $u$ is a solution to the obstacle problem \eqref{eq:Obstacle_problem_simple}. Then, for all $\alpha\in (2s-1,s)$ and $p\in [s,\alpha+s-1/2)$, there are positive constants, $C$ and $r_0\in (0,1)$, such that the function
\begin{equation}
\label{eq:Monotonicity_formula}
(0,r_0)\ni r\mapsto e^{Cr^{\gamma}} \Phi^p_v(r),
\end{equation}
is non-decreasing, where $\gamma:=2(\alpha+s-p)-1$, and $v$ is defined by identity \eqref{eq:Auxiliary_function_v}.
\end{prop}
The proof of Proposition \ref{prop:Monotonicity_formula} is given in \S \ref{sec:Auxiliary_results_proofs}.
Following \cite[Definition (6.1)]{Caffarelli_Salsa_Silvestre_2008} we introduce the sequence of rescalings, $\{v_r\}_{r >0}$, of the function $v$. For $r\in (0,1)$, we define
\begin{equation}
\label{eq:d_r}
d_r:=\left(\frac{1}{r^{n+a}}\int_{\partial B_r}|v|^2|y|^a\right)^{1/2},
\end{equation}
and we let
\begin{equation}
\label{eq:Rescaling}
v_r(x,y):=\frac{v(r(x,y))}{d_r},\quad\forall (x,y)\in\RR^n\times\RR,
\end{equation}
be a rescaling of the function $v$. With the aid of Proposition \ref{prop:Monotonicity_formula} we prove the following analogue of \cite[Lemma 6.1]{Caffarelli_Salsa_Silvestre_2008}.
\begin{prop}
\label{prop:Phi_at_0}
Suppose that the assumptions of Proposition \ref{prop:Monotonicity_formula} are satisfied. Then, for all $p\in [s,2s-1/2)$, the following hold.
If
\begin{equation}
\label{eq:Fraction_d_r_r_power_finite}
\liminf_{r\downarrow 0}\frac{d_r}{r^{1+p}} <\infty,
\end{equation}
then
\begin{equation}
\label{eq:Phi_at_0_p}
\Phi^p_v(0+) = n+a+2(1+p),
\end{equation}
and if
\begin{equation}
\label{eq:Fraction_d_r_r_power_infty}
\liminf_{r\downarrow 0}\frac{d_r}{r^{1+p}} =\infty,
\end{equation}
then
\begin{equation}
\label{eq:Phi_at_0}
\Phi^p_v(0+) \geq n+a+2(1+s).
\end{equation}
\end{prop}
Proposition \ref{prop:Phi_at_0} shows that the smallest value that the function $\Phi^p_v(r)$ can take is $n+a+2(1+s)$. This property is crucial in the proof of the optimal regularity of the solutions to the obstacle problem in \S \ref{sec:Solutions_optimal_regularity}.
The proof of Proposition \ref{prop:Phi_at_0} relies on the fact that the sequence of rescalings, $\{v_r\}_{r \geq 0}$, contains a subsequence strongly convergent in $H^1(B_1,|y|^a)$, as $r$ tends to $0$. To obtain this, we first prove a series of preliminary results. In Lemma \ref{lem:Uniform_boundedness_H_1_rescalings}, we prove the uniform boundedness in $H^1(B_1,|y|^a)$ of the sequence of rescalings, which is then used in Lemma \ref{lem:Uniform_boundedness_rescalings} to show the uniform boundedness in $L^{\infty}(B_{1/2})$, employing the Moser iterations technique. Lemma \ref{lem:Uniform_boundedness_H_1_rescalings} is not sufficient to conclude the strong convergence in $H^1(B_1,|y|^a)$ of the sequence of rescalings, as $r$ tends to $0$, and so, in Lemmas \ref{lem:Uniform_boundedness_C_1_alpha_n_ball_rescalings} and \ref{lem:Uniform_boundedness_C_1_alpha_n_plus_1_ball_rescalings} we improve the control we have on the sequence of rescalings by proving a uniform bound in H\"older spaces. The results of Lemmas \ref{lem:Uniform_boundedness_C_1_alpha_n_ball_rescalings} and \ref{lem:Uniform_boundedness_C_1_alpha_n_plus_1_ball_rescalings} have their analogues in \cite[Lemma 4.1 and Proposition 4.3]{Caffarelli_Salsa_Silvestre_2008}, respectively. The proofs of the latter results in \cite{Caffarelli_Salsa_Silvestre_2008} rely on the properties of the function $\widetilde u$, defined on \cite[p. 433]{Caffarelli_Salsa_Silvestre_2008}. The analogue in our case of the function $\widetilde u$ in \cite{Caffarelli_Salsa_Silvestre_2008} is the function $v$ defined in \eqref{eq:Auxiliary_function_v}. The function $v$ does not satisfy the properties of function $\widetilde u$, because $L_a v$ is a singular measure with nontrivial support on $\{y=0\}\backslash\{v=0\}$. For this reason, we cannot adapt the approach of \cite{Caffarelli_Salsa_Silvestre_2008} to our framework, and so we proceed in a different way which we outline in the sequel.
We begin with
\begin{lem}[Uniform boundedness in $H^1(B_1,|y|^a)$]
\label{lem:Uniform_boundedness_H_1_rescalings}
We assume that the hypotheses of Proposition \ref{prop:Monotonicity_formula} hold. Let $\alpha\in (1/2,s)$, and $p \in [s,\alpha+s-1/2)$, and assume that condition \eqref{eq:Fraction_d_r_r_power_infty} holds. Then there are positive constants, $C$ and $r_0$, such that
\begin{equation}
\label{eq:Uniform_boundedness_H_1_rescalings}
\|v_r\|_{H^{1}(B_1,|y|^a)} \leq C,\quad\forall r\in (0,r_0).
\end{equation}
\end{lem}
\begin{proof}
From identity \eqref{eq:Rescaling}, the following hold, for all $r>0$,
\begin{align}
\label{eq:Formula_grad_v_r}
\int_{B_1}|\nabla v_r|^2 |y|^a &=\frac{r\int_{B_r} |\nabla v|^2|y|^a}{\int_{\partial B_r} |v|^2|y|^a},\\
\label{eq:v_r_partial_B_1}
\int_{\partial B_1} |v_r|^2 |y|^a&=1.
\end{align}
From \eqref{eq:d_r} and condition \eqref{eq:Fraction_d_r_r_power_infty}, there is a positive constant, $r_0$, such that
\begin{equation}
\label{eq:Inequality_d_r}
\int_{\partial B_r}|v|^2|y|^a \geq r^{n+a+2(1+p)},\quad\forall r\in (0,r_0),
\end{equation}
and so, identities \eqref{eq:F} and \eqref{eq:Phi} give us that
$$
\Phi^p_v(r) = r\frac{d}{dr} \log \int_{\partial B_r}|v|^2|y|^a, \quad\forall r\in (0,r_0).
$$
From identities \eqref{eq:Identity_Phi_p} and \eqref{eq:v_v_nu}, it follows that
\begin{align}
\Phi^p_v(r) &= r\frac{2\int_{B_r}|\nabla v|^2|y|^a + 2\int_{B_r} v L_a v}{\int_{\partial B_r}|v|^2|y|^a}+n+a\notag\\
\label{eq:Expansion_Phi}
&= 2\int_{B_1}|\nabla v_r|^2 |y|^a + 2 \frac{r\int_{B_r} v L_a v}{\int_{\partial B_r} |v|^2|y|^a}+n+a\quad\hbox{(by identity \eqref{eq:Formula_grad_v_r}).}
\end{align}
Using \eqref{eq:Formula_grad_v_r}, \eqref{eq:Ineq_v_on_R_n}, \eqref{eq:Auxiliary_function_v}, \eqref{eq:Equality_L_a}, \eqref{eq:Definition_h} and \eqref{eq:Growth_h}, together with the preceding inequality, we see that
\begin{equation}
\label{eq:Inequality_second_term_Phi}
\frac{\left|r\int_{B_r} v L_a v\right|}{\int_{\partial B_r} |v|^2|y|^a} \leq \frac{Cr^{1+(1+\alpha)+\alpha+n}}{r^{2(1+p)+n+a}} = Cr^{2(\alpha-p+s-1/2)},\quad\forall r\in (0,r_0).
\end{equation}
From our assumption that $\alpha\in (1/2,s)$ and $p \in [s,\alpha+s-1/2)$, the right-hand side in the preceding inequality tends to zero, as $r\rightarrow 0$. By Proposition \ref{prop:Monotonicity_formula} and identity \eqref{eq:Expansion_Phi}, we obtain that there are positive constants, $C$ and $r_0$, such that
\begin{equation}
\label{eq:Uniform_bound_grad_v_r}
\int_{B_1}|\nabla v_r|^2 |y|^a \leq C,\quad\forall r\in (0,r_0).
\end{equation}
By \cite[Lemma 2.12]{Caffarelli_Salsa_Silvestre_2008}, we obtain that, for some positive constant $C=C(n,s)$, we have that
$$
\int_{\partial B_1} |v_r(x,y)-v_r(t(x,y))|^2|y|^a \leq C(1-t) \int_{B_1} |\nabla v_r|^2|y|^a,\quad\forall t \in (0,1).
$$
(Notice that on the right-hand side of the Poincar\'e inequality in \cite[Lemma 2.12]{Caffarelli_Salsa_Silvestre_2008}, the factor $(1-t)$ is missing.) Because the uniform bound \eqref{eq:v_r_partial_B_1} holds, the preceding inequality gives us that
$$
\int_{\partial B_1} |v_r(t(x,y))|^2|y|^a \leq 2C(1-t) \int_{B_1} |\nabla v_r|^2|y|^a+2,\quad\forall t \in (0,1),
$$
and multiplying by $t^a$, and integrating in the $t$-variable, we obtain
\begin{align}
\int_{B_1} |v_r|^2|y|^a &=\int_0^1\int_{\partial B_1} |v_r(t(x,y))|^2|ty|^a\notag\\
&\leq 2C \int_{B_1} |\nabla v_r|^2|y|^a\int_0^1(1-t)t^a\, dt+2\int_0^1t^a\, dt\notag\\
\label{eq:Uniform_bound_v_r}
&\leq C, \quad \forall r \in (0,r_0),
\end{align}
where $C$ is a positive constant. The last inequality follows from the uniform bound \eqref{eq:Uniform_bound_grad_v_r} and the fact that the constant $a\in (-1,0)$, since we assume that $s\in (1/2,1)$.
Inequalities \eqref{eq:Uniform_bound_v_r} and \eqref{eq:Uniform_bound_grad_v_r} give us \eqref{eq:Uniform_boundedness_H_1_rescalings}. This completes the proof.
\end{proof}
As a consequence of Lemma \ref{lem:Uniform_boundedness_H_1_rescalings} we have
\begin{lem}[Uniform boundedness in $L^{\infty}(B_{1/2})$]
\label{lem:Uniform_boundedness_rescalings}
Suppose that the assumptions of Lemma \ref{lem:Uniform_boundedness_H_1_rescalings} hold. Then there are positive constants, $C$ and $r_0$, such that
\begin{equation}
\label{eq:Uniform_boundedness_rescalings}
\|v_r\|_{L^{\infty}(B_{1/2})} \leq C,
\quad\forall r\in (0,r_0).
\end{equation}
\end{lem}
\begin{proof}
We prove the supremum estimate \eqref{eq:Uniform_boundedness_rescalings} using the Moser iterations method. We let $\eta:\RR^{n+1}\rightarrow [0,1]$ be a smooth function with compact support in $B_1$. For $r>0$, we let
\begin{equation}
\label{eq:Choice_k}
k:=\frac{r^{2s+\alpha}}{d_r},
\end{equation}
and we consider the following auxiliary functions,
\begin{align*}
q:=v_r^{\pm}+k,\quad\hbox{and}\quad w:=\eta^2(q^{\beta}-k^{\beta}),
\end{align*}
where $\beta$ is a positive constant. From \eqref{eq:Auxiliary_function_v}, \eqref{eq:Equality_L_a} \eqref{eq:Definition_h} and \eqref{eq:Rescaling}, we have that
\begin{align*}
L_a v_r &=0\quad\hbox{on } B_1\backslash B_1',\\
L_a v_r &=\frac{r^{1-a}}{d_r} h(rx) \cH^{n}|_{\{y=0\}}\quad\hbox{on } B_1\backslash \{y=0, v_r=0\}.
\end{align*}
Because $w=0$ when $v_r=0$, the preceding identities give us that
\begin{align}
\label{eq:Operator_applied_to_auxiliary_function_1}
\int_{B_1} w L_a v_r &= \frac{r^{1-a}}{d_r} \int_{B_1'} h(rx)w(x).
\end{align}
Recall that we also have that
\begin{align}
\label{eq:Operator_applied_to_auxiliary_function_2}
\int_{B_1} w L_a v_r &=-\int_{B_1}\nabla v_r \dotprod\nabla w|y|^a.
\end{align}
Using the definitions of the functions $p$ and $w$, we obtain the identities
\begin{align*}
\nabla v_r \dotprod\nabla w &= \pm \nabla q\dotprod\left(\beta\eta^2q^{\beta-1}\nabla q+2\eta\nabla\eta \left(q^{\beta}-k^{\beta}\right)\right),\\
q^{\beta-1} |\nabla q|^2 &= \frac{4}{(\beta+1)^2} \left|\nabla q^{(\beta+1)/2}\right|^2,
\end{align*}
which combined with identities \eqref{eq:Operator_applied_to_auxiliary_function_1} and \eqref{eq:Operator_applied_to_auxiliary_function_2}, and the fact that $0\leq w \leq \eta^2 q^{\beta}$, gives us that
\begin{align}
\label{eq:Operator_applied_to_auxiliary_function_int_by_parts}
\frac{4\beta}{(\beta+1)^2} \int_{\RR^{n+1}} \left|\nabla q^{(\beta+1)/2}\right|^2 \eta^2 |y|^a
&\leq \int_{\RR^{n+1}} 2\eta|\nabla\eta||\nabla q| q^{\beta} |y|^a + \frac{r^{1-a}}{d_r} \int_{B_1'} |h(rx)| q^{\beta} \eta^2.
\end{align}
We also have that
\begin{align*}
\int_{\RR^{n+1}} 2\eta|\nabla\eta||\nabla q| q^{\beta} |y|^a
&=\frac{4}{\beta+1} \int_{\RR^{n+1}}\eta|\nabla\eta| q^{(\beta+1)/2} \left|\nabla q^{(\beta+1)/2}\right||y|^a\\
&\leq \frac{4\beta\eps}{(\beta+1)^2} \int_{\RR^{n+1}}\eta^2 \left|\nabla q^{(\beta+1)/2}\right|^2|y|^a
+\frac{1}{4\eps\beta} \int_{\RR^{n+1}}|\nabla\eta|^2 q^{\beta+1} |y|^a,
\end{align*}
for all $\eps>0$. We choose $\eps=1/2$ in the preceding inequality, which we combine with inequalities \eqref{eq:Growth_h} and \eqref{eq:Operator_applied_to_auxiliary_function_int_by_parts}, the fact that $q \geq k$ and $a=1-2s$, to obtain that there is a positive constant, $C$, such that
\begin{align}
\label{eq:Estimate_grad_p_1}
\int_{\RR^{n+1}} \left|\nabla q^{(\beta+1)/2}\right|^2 \eta^2 |y|^a \leq C\int_{\RR^{n+1}} |\nabla\eta|^2 q^{\beta+1} |y|^a
+C\beta\frac{r^{2s+\alpha}}{d_r} \int_{B_1'} \frac{q^{\beta+1}}{k}\eta^2.
\end{align}
By the Trace Theorem \cite[Theorem 1.5.1.1]{Grisvard} applied with $s=1/2+\sigma$, and $\sigma \in (0,1/2)$, we obtain
\begin{equation*}
\int_{\RR^n} \left|q^{(\beta+1)/2}\eta\right|^2 \leq C\|q^{(\beta+1)/2}\eta\|^2_{H^{1/2+\sigma}(\RR^{n+1})}.
\end{equation*}
The Interpolation Inequality \cite[Theorem 1.4.3.3]{Grisvard} gives us, by choosing $\sigma:=1/4$ in the preceding estimate, that there is a positive constant, $C$, such that for all $\eps\in (0,1)$, we have
\begin{align*}
\|q^{(\beta+1)/2}\eta\|^2_{H^{1/2+\sigma}(\RR^{n+1})}
&\leq \eps\|q^{(\beta+1)/2}\eta\|^2_{H^1(\RR^{n+1})} + C\eps^{-3} \|q^{(\beta+1)/2}\eta\|^2_{L^2(\RR^{n+1})}\\
&\leq \eps\int_{B_1} \eta^2 \left|\nabla q^{(\beta+1)/2}\right|^2 + C\eps^{-3}\int_{B_1} \left(\eta^2+|\nabla\eta|^2\right)q^{\beta+1}.
\end{align*}
Combining the previous two inequalities, we obtain
\begin{equation}
\label{eq:Estimate_p_beta_plus_1}
\int_{\RR^n} \eta^2 q^{\beta+1} \leq \eps\int_{B_1} \eta^2 \left|\nabla q^{(\beta+1)/2}\right|^2 + C\eps^{-3}\int_{B_1} \left(\eta^2+|\nabla\eta|^2\right)q^{\beta+1},\quad\forall \eps>0.
\end{equation}
We choose
$$
\eps:=\frac{k d_r}{2C\beta r^{2s+\alpha}}.
$$
Using inequalities \eqref{eq:Estimate_p_beta_plus_1} and \eqref{eq:Estimate_grad_p_1}, together with the fact that $a=1-2s<0$, when $s\in (1/2,1)$, we obtain
\begin{align*}
\int_{\RR^{n+a}} \eta^2 \left|\nabla q^{(\beta+1)/2}\right|^2 |y|^a
\leq \left(\frac{C\beta r^{2s+\alpha}}{k d_r}\right)^3\int_{\RR^{n+1}} \left(\eta^2+|\nabla\eta|^2\right) q^{\beta+1} |y|^a.
\end{align*}
The choice \eqref{eq:Choice_k} of the constant $k$ now gives us that
\begin{align*}
\int_{\RR^{n+1}} \eta^2 \left|\nabla q^{(\beta+1)/2}\right|^2 |y|^a \leq C \beta^3\int_{\RR^{n+1}} \left(\eta^2+|\nabla\eta|^2\right) q^{\beta+1} |y|^a.
\end{align*}
We can now apply the Moser iteration method to conclude that
\begin{equation}
\label{eq:Sup_estimate_p}
\sup_{B_{1/2}} q \leq C \left(\int_{B_1} |q|^2|y|^a\right)^{1/2}.
\end{equation}
The Moser iteration method is applied as in \cite[p. 195-197]{GilbargTrudinger} with the observation that we replace the application of the classical Sobolev inequality \cite[Inequality (7.26)]{GilbargTrudinger} with the Sobolev inequality suitable for the weighted Sobolev space $H^1(B_1,|y|^a)$ obtained in \cite[Theorem (1.6)]{Fabes_Kenig_Serapioni_1982a}. We apply \cite[Theorem (1.6)]{Fabes_Kenig_Serapioni_1982a} to the function $\eta q^{(\beta+1)/2}$ with $p=2$, and we notice that the weight $\fw(x,y)=|y|^a$ belongs to the Muckenhoupt $A_2$ class of functions.
From definition \eqref{eq:Choice_k} of the constant $k$, definition \eqref{eq:d_r} of $d_r$ and inequality \eqref{eq:Inequality_d_r}, it follows that
$$
k \leq r^{2s+\alpha-(1+p)},
$$
and so, using the definition of the auxiliary function $q$, estimate \eqref{eq:Sup_estimate_p} gives that
$$
\sup_{B_{1/2}} v_r^{\pm} \leq C \left(\int_{B_1} |v_r|^2|y|^a\right)^{1/2} + Cr^{2s+\alpha-(1+p)}.
$$
From our assumption that $s\in (1/2, 1)$, $\alpha\in (1/2,s)$ and $p \in (s,\alpha+s-1/2)$, we see that the term $r^{2s+\alpha-(1+p)}$ tends to zero, as $r\rightarrow 0$. From estimate \eqref{eq:Uniform_boundedness_H_1_rescalings}, it follows that
$$
\sup_{B_{1/2}} v_r^{\pm} \leq C,
$$
for some positive constant $C$, and for all $r>0$ sufficiently small. The preceding estimate is equivalent to \eqref{eq:Uniform_boundedness_rescalings}.
\end{proof}
\begin{lem}[Uniform Schauder estimates on $B'_{1/4}$]
\label{lem:Uniform_boundedness_C_1_alpha_n_ball_rescalings}
Let $s\in (1/2,1)$ and $\alpha\in ((2s-1)\vee 1/2, s)$. Suppose that $\varphi\in C^{2s+\alpha}(\RR^n)$, $u\in C^{1+\alpha}(\RR^n)$, and that $u$ is a solution to problem \eqref{eq:Obstacle_problem_simple}. Let $p \in (s,\alpha+s-1/2)$, and assume that condition \eqref{eq:Fraction_d_r_r_power_infty} holds. Then for all $\beta \in (0,2s-1)$, there are positive constants, $C$ and $r_0$, such that
\begin{equation}
\label{eq:Uniform_boundedness_C_1_alpha_n_ball_rescalings}
\|v_r\|_{C^{1+\beta}(B'_{1/4})} \leq C,\quad\forall r\in (0,r_0).
\end{equation}
\end{lem}
\begin{proof}
In this proof we do not use the $L_a$-harmonic extension of the rescaling sequence, $\{v_r\}_{r>0}$. We divide the proof into several steps. In Step \ref{step:Uniform_bound_h_r_w_r}, we replace the sequence of rescalings, $\{v_r\}_{r>0}$, by a suitable modification \eqref{eq:Definition_v_r_modified} which solves the obstacle problem \eqref{eq:Obstacle_problem_w_r}, where now a non-zero source function, $h_r$, appears. We prove that the sequence of source functions, $\{h_r\}_{r>0}$, satisfies the uniform supremum estimate \eqref{eq:Uniform_bound_h_r_w_r}. In Step \ref{step:Localization_w_r}, we localize our sequence of modified rescaling functions, and we use inequality \eqref{eq:Uniform_bound_h_r_w_r} to prove the uniform global Schauder estimate \eqref{eq:Schauder_estimate_w_r_cutoff}. Finally, in Step \ref{step:Holder_continuity}, we use a localization method of Krylov \cite[Theorem 8.11.1]{Krylov_LecturesHolder} to prove the uniform Schauder estimate \eqref{eq:Uniform_boundedness_C_1_alpha_n_ball_rescalings} satisfied by the sequence of rescalings, $\{v_r\}_{r>0}$.
\begin{step}
\label{step:Uniform_bound_h_r_w_r}
We recall from \eqref{eq:Auxiliary_function_v} and \eqref{eq:Rescaling} that, restricted to the hyperplane $\{y=0\}$, the rescaling functions, $v_r$, take the following form
\begin{equation}
\label{eq:v_r_on_R_n}
v_r(x)=\frac{u(rx)-\varphi(rx)}{d_r},\quad\forall x\in\RR^n,\ \forall r>0,
\end{equation}
where $d_r$ is defined in \eqref{eq:d_r}. Because $u$ is a solution to the obstacle problem
\begin{equation}
\label{eq:Obstacle_problem_u}
\min\{(-\Delta)^s u, u-\varphi\}=0\quad\hbox{on }\RR^n,
\end{equation}
we see that $v_r$ is a solution to the obstacle problem
\begin{equation}
\label{eq:Obstacle_problem_v}
\min\{(-\Delta)^s v_r- f_r, v_r\}=0\quad\hbox{on }\RR^n,
\end{equation}
where the function $f_r$ is defined by
\begin{equation*}
f_r(x):=-\frac{r^{2s}}{d_r}(-\Delta)^s\varphi(rx),\quad\forall x\in\RR^n.
\end{equation*}
We would like to derive a uniform bound on $\|f_r\|_{L^{\infty}(B'_1)}$, for all $r>0$ sufficiently small. Because we do not have a uniform estimate on $r^{2s}/d_r(-\Delta)^s\varphi(O)$, for $r>0$ sufficiently small, we are not able to find an uniform bound on $\|f_r\|_{L^{\infty}(B'_1)}$, and so we choose a different approach.
We replace the rescaling functions \eqref{eq:v_r_on_R_n} with the following modified version,
\begin{equation}
\label{eq:Definition_v_r_modified}
w_r(x):=\frac{u(rx)-\varphi(rx)+\psi(rx)}{d_r},\quad\forall x\in\RR^n,\quad \forall r>0.
\end{equation}
We define the auxiliary function $\psi$ by
\begin{equation}
\label{eq:Definition_psi_w_r}
\psi(x):=c|x|^4\eta(x),\quad\forall x\in\RR^n,\ \forall r>0,
\end{equation}
where the positive constant $c$ will be suitably chosen below, and $\eta:\RR^n\rightarrow [0,1]$ is a smooth cut-off function with support in $B'_1$. Because $u$ solves the obstacle problem \eqref{eq:Obstacle_problem_u}, we see that the function $w_r$ solves the problem
\begin{equation}
\label{eq:Obstacle_problem_w_r}
\min\left\{(-\Delta)^s w_r(x) - h_r(x), w_r(x)-\frac{\psi(rx)}{d_r}\right\}=0,\quad\forall x\in \RR^n.
\end{equation}
where the source function $h_r$ is now given by
\begin{equation}
\label{eq:Definition_h_r_w_r}
h_r(x):=\frac{r^{2s}}{d_r}\left((-\Delta)^s\psi(rx)-(-\Delta)^s\varphi(rx)\right),\quad\forall x\in\RR^n.
\end{equation}
Our goal in this step is to show that there are positive constants, $C$ and $r_0$, such that
\begin{equation}
\label{eq:Uniform_bound_h_r_w_r}
\|h_r\|_{L^{\infty}(B'_1)} \leq C,\quad\forall r\in (0,r_0).
\end{equation}
We can rewrite $h_r$ in the form
\begin{align*}
h_r(x)
&= \frac{r^{2s}}{d_r} \left((-\Delta)^s\psi(rx) - (-\Delta)^s\psi(O)\right)
- \frac{r^{2s}}{d_r} \left((-\Delta)^s\varphi(rx) - (-\Delta)^s\varphi(O)\right)\\
&\quad+\frac{r^{2s}}{d_r} \left((-\Delta)^s\psi(O)) - (-\Delta)^s\varphi(O)\right).
\end{align*}
We choose the constant $c$, in definition \eqref{eq:Definition_psi_w_r} of $\psi$, such that
$$
(-\Delta)^s\psi(O) = (-\Delta)^s\varphi(O).
$$
Because $\varphi \in C^{2s+\alpha}(\RR^n)$ and $\psi\in C^{\infty}_c(\RR^n)$, we obtain by \cite[Proposition 2.6]{Silvestre_2007} that there is a positive constant, $C$, such that
\begin{align*}
\left|(-\Delta)^s\varphi(rx) - (-\Delta)^s\varphi(O)\right|&\leq C r^{\alpha},\\
\left|(-\Delta)^s\psi(rx) - (-\Delta)^s\psi(O)\right|&\leq C r^{\alpha},
\end{align*}
for all $x \in B'_1$ and all $r>0$. Thus, we obtain that
\begin{equation}
\label{eq:Estimate_h_r_w_r}
|h_r(x)| \leq C\frac{r^{2s+\alpha}}{d_r},\quad\forall x\in B'_1,\quad \forall r>0.
\end{equation}
From definition \eqref{eq:d_r} of $d_r$, and inequality \eqref{eq:Inequality_d_r} (implied by condition \eqref{eq:Fraction_d_r_r_power_infty}), it follows that
$$
\frac{r^{2s+\alpha}}{d_r} \leq r^{2s+\alpha-(1+p)},\quad\forall r\in (0,r_0),
$$
and, from our assumption that $s\in (1/2,1)$, $\alpha\in (1/2,s)$ and $p \in (s,\alpha+s-1/2)$, we have that the bound $r^{2s+\alpha-(1+p)}$ tends to zero, as $r\rightarrow 0$. We can now see that \eqref{eq:Uniform_bound_h_r_w_r} follows from inequality \eqref{eq:Estimate_h_r_w_r} and the preceding observation.
\end{step}
\begin{step}[Localization]
\label{step:Localization_w_r}
In Step \ref{step:Uniform_bound_h_r_w_r}, we obtained the uniform estimate \eqref{eq:Uniform_bound_h_r_w_r} on $B'_1$, but not on $\RR^n$. To be able to use this estimate in Step \ref{step:Holder_continuity}, we need to localize the sequence of rescalings, $\{w_r\}_{r>0}$, defined in \eqref{eq:Definition_v_r_modified}. We do this by simply multiplying the function $w_r$ by a suitably chosen smooth cut-off function, $\chi:\RR^n\rightarrow [0,1]$, with compact support in $B'_1$. We denote
\begin{equation}
\label{eq:Definition_w_r_cutoff}
w^{\chi}_r:=\chi w_r ,\quad\forall r>0.
\end{equation}
We next want to show that, for all $\beta\in (0,2s-1)$, there is a positive constant, $C=C(\beta)$, such that the following estimate holds, for all $r>0$,
\begin{equation}
\label{eq:Schauder_estimate_w_r_cutoff}
\|w^{\chi}_r\|_{C^{1+\beta}(\RR^n)} \leq C\left(\|g_r\|_{C(\RR^n)}+\left\|(-\Delta)^s\left(\frac{\psi(rx)}{d_r}\chi\right)\right\|_{C(\RR^n)}+\left\|\frac{\psi(rx)}{d_r}\chi\right\|_{C(\RR^n)}\right).
\end{equation}
Direct calculations give us
\begin{align*}
(-\Delta)^s w^{\chi}_r(x) = \chi(x)(-\Delta)^s w_r(x) + w_r(x) (-\Delta)^s\chi(x)-\int_{\RR^n}\frac{(\chi(x)-\chi(y))(w_r(x)-w_r(y))}{|x-y|^{n+2s}}\, dy.
\end{align*}
We let the function $g_r$ be defined by
\begin{align}
\label{eq:Definition_g_r}
g_r(x):= \chi(x)h_r(x)+ w_r(x) (-\Delta)^s\chi(x)+ w^{\chi}_r(x)-\int_{\RR^n}\frac{(\chi(x)-\chi(y))(w_r(x)-w_r(y))}{|x-y|^{n+2s}}\, dy,
\end{align}
where we recall that the function $h_r$ is defined in \eqref{eq:Definition_h_r_w_r}. Because $w_r$ solves the obstacle problem \eqref{eq:Obstacle_problem_w_r}, we see that $w^{\chi}_r$ solves the problem
\begin{equation*}
\min\left\{(-\Delta)^s w^{\chi}_r(x) +w^{\chi}_r(x) - g_r(x), w^{\chi}_r(x)-\frac{\psi(rx)}{d_r}\chi(x)\right\}=0,\quad\forall x\in \RR^n.
\end{equation*}
Because $\psi \in C^{\infty}_c(\RR^n)$ and $\varphi\in C^{2s+\alpha}(\RR^n)$, it follows from definition \eqref{eq:Definition_h_r_w_r} of $h_r$, and from \cite[Proposition 2.6]{Silvestre_2007} that $h_r$ belongs to $C^{\alpha}(\RR^n)$. From Lemma \ref{lem:Regularity_g}, we obtain that the function
$$
\RR^n\ni x\mapsto \int_{\RR^n}\frac{(\chi(x)-\chi(y))(w_r(x)-w_r(y))}{|x-y|^{n+2s}}\, dy
$$
belongs to $C^{2(1-s)}(\RR^n)$. Thus, using definition \eqref{eq:Definition_g_r} of the function $g_r$, and the fact that $u$ belongs to $C^{1+\alpha}(\RR^n)$, we obtain that $g_r \in C^{\theta}(\RR^n)$, where $\theta:=\alpha\wedge (2(1-s))$. We may now apply Lemma \ref{lem:Existence_uniqueness_linear_equation} to conclude that there is a unique solution, $t_r\in C^{2s+\theta}(\RR^n)$, to the linear equation
$$
(-\Delta)^s t_r(x) +t_r(x) = g_r(x),\quad\forall x\in \RR^n.
$$
Then the function $w^{\chi}_r-t_r$ solves the obstacle problem
\begin{equation}
\label{eq:Obstacle_problem_w_r_cutoff}
\min\left\{(-\Delta)^s (w^{\chi}_r-t_r)+(w^{\chi}_r-t_r), (w^{\chi}_r-t_r)-\left(\frac{\psi(rx)}{d_r}\chi-t_r\right)\right\}=0,\quad\hbox{on } \RR^n.
\end{equation}
Since we assume that $u\in C^{1+\alpha}(\RR^n)$ and $\varphi\in C^{2s+\alpha}(\RR^n)$, for some $\alpha \in ((2s-1)\vee 1/2,s)$, we see that the function $w^{\chi}_r-t_r$ belongs to $C^{1+\gamma}(\RR^n)$, for some $\gamma>2s-1$. It follows from Proposition \ref{prop:Uniqueness} that the function $w^{\chi}_r-t_r \in C^{1+\gamma}(\RR^n)$ is the unique solution to the obstacle problem \eqref{eq:Obstacle_problem_w_r_cutoff}. From the proof of Proposition \ref{prop:Existence_Holder_obstacle_problem}, we see from estimate \eqref{eq:u_eps_Holder_estimate}, that for all $\beta\in (0,2s-1)$, there is a positive constant $C=C(\beta)$, such that the function $w^{\chi}_r-t_r$ satisfies the Schauder estimate,
\begin{equation*}
\begin{aligned}
\|w^{\chi}_r-t_r\|_{C^{1+\beta}(\RR^n)} &\leq C\left(\|g_r\|_{C(\RR^n)}+\left\|(-\Delta)^s\left(\frac{\psi(rx)}{d_r}\chi\right)+\frac{\psi(rx)}{d_r}\chi\right\|_{C(\RR^n)}\right.\\
&\quad\left.+\left\|\frac{\psi(rx)}{d_r}\chi-t_r\right\|_{C(\RR^n)}\right),\quad\forall r>0.
\end{aligned}
\end{equation*}
By \cite[Proposition 2.9]{Silvestre_2007} and Lemma \ref{lem:Sup_estimate}, it follows that
\begin{equation*}
\|t_r\|_{C^{1+\beta}(\RR^n)} \leq C\|g_r\|_{C(\RR^n)},\quad\forall r>0,
\end{equation*}
and so, the function $w^{\chi}_r$ satisfies the Schauder estimate \eqref{eq:Schauder_estimate_w_r_cutoff}. In Step \ref{step:Holder_continuity}, we use estimate \eqref{eq:Schauder_estimate_w_r_cutoff} to obtain a uniform Schauder estimate on $\|w_r\|_{C^{1+\beta}(B'_{1/2})}$, for all $r>0$ sufficiently small.
\end{step}
\begin{step}[H\"older continuity]
\label{step:Holder_continuity}
Our goal is now to use estimate \eqref{eq:Schauder_estimate_w_r_cutoff} and prove that \eqref{eq:Uniform_boundedness_C_1_alpha_n_ball_rescalings} holds. For this purpose we employ the iteration argument used to prove \cite[Theorem 8.11.1]{Krylov_LecturesHolder}. For all $k \in \NN$, we let
$$
r_k=\frac{1}{4}\sum_{i=0}^k \frac{1}{2^i},\quad\forall k\in\NN,
$$
and we denote $B'_k:=B'_{r_k}$, for brevity. We now let $\chi_k:\RR^n\rightarrow[0,1]$ be a smooth function such that
$$
\chi_k \equiv 1,\quad\hbox{ on } B'_k,\quad\hbox{and}\quad \chi_k\equiv 0, \quad\hbox{ on } (B'_{k+1})^c.
$$
In addition, we can choose the cut-off functions $\chi_k$, such that there is a positive constant, $C$, satisfying the property,
\begin{equation}
\label{eq:Fractional_laplacian_cutoff}
\|(-\Delta)^s\chi_k\|_{C(\RR^n)} \leq C 2^{2k},\quad\forall k \in \NN.
\end{equation}
We denote
$$
\alpha_k:=\|v_r\chi_k\|_{C^{1+\beta}(\RR^n)}.
$$
We apply estimate \eqref{eq:Schauder_estimate_w_r_cutoff} to the function $w^{\chi}_r$, where we recall that $w^{\chi}_r=\chi w_r$ by \eqref{eq:Definition_w_r_cutoff}, and we choose $\chi=\chi_k$. We denote
\begin{equation}
\label{eq:Definition_f_r}
f^k_r(x):=\int_{\RR^n}\frac{(\chi_k(x)-\chi_k(y))(w_r(x)-w_r(y))}{|x-y|^{n+2s}}\, dy,\quad\forall x \in \RR^n,
\end{equation}
for all $r>0$ and $k \in \NN$. Using definition \eqref{eq:Definition_g_r} of the function $g_r$, estimate \eqref{eq:Schauder_estimate_w_r_cutoff} yields
\begin{align*}
\alpha_k &\leq C\left(\|w_r\chi_k\|_{C(\RR^n)} + \|h_r\chi_k\|_{C(\RR^n)} + \frac{1}{d_r}\|\psi(rx)\chi_k\|_{C^{1+\beta}(\RR^n)}\right.\\
&\quad\left. +\left\|(-\Delta)^s\left(\frac{\psi(rx)}{d_r}\chi_k\right)\right\|_{C(\RR^n)}
+\|w_r(-\Delta)^s\chi_k - f^k_r\|_{C(\RR^n)} \right).
\end{align*}
By Lemma \ref{lem:Uniform_boundedness_rescalings}, definitions \eqref{eq:Definition_v_r_modified} of $w_r$ and \eqref{eq:Definition_psi_w_r} of $\psi$, and estimate \eqref{eq:Uniform_bound_h_r_w_r} of $h_r$, we can find positive constants, $C$ and $r_0$ such that
\begin{align*}
\|w_r\chi_k\|_{C(\RR^n)} + \|h_r\chi_k\|_{C(\RR^n)} \leq C,\quad\forall r\in(0,r_0),\ k \in \NN,
\end{align*}
and, using the properties of the cutoff functions $\chi_k$, we obtain
$$
\frac{1}{d_r}\|\psi(rx)\chi_k\|_{C^{1+\beta}(\RR^n)}+\left\|(-\Delta)^s\left(\frac{\psi(rx)}{d_r}\chi_k\right)\right\|_{C(\RR^n)} \leq C2^{2k},\quad\forall k \in \NN.
$$
It follows that
\begin{align}
\label{eq:Estimate_alpha_k_1}
\alpha_k &\leq C\left(2^{2k} + \|w_r(-\Delta)^s\chi_k -f^k_r\|_{C(\RR^n)} \right),\quad\forall r\in (0,r_0),\ k\in\NN.
\end{align}
To evaluate the last term in the preceding inequality, we consider two cases depending on whether the point $x$ belongs to $\hbox{supp }\chi_k$ or $x$ belongs to $(\hbox{supp }\chi_k)^c$.
\begin{case}[Points $x\in \hbox{supp }\chi_k$]
\label{case:Estimate_alpha_k_x_less_1}
From definition \eqref{eq:Definition_f_r} of the function $f^k_r$, we obtain that there is a positive constant, $C$, such that
$$
|f^k_r(x)| \leq C 2^{k}\left(1+\|\nabla v_r\|_{C(B'_{k+1})}\right),\quad\forall k \in \NN.
$$
Thus, in the case when $x$ is contained in $\hbox{supp }\chi_k$, the preceding inequality together with estimates \eqref{eq:Uniform_boundedness_rescalings} and \eqref{eq:Fractional_laplacian_cutoff}, give us
\begin{align*}
|w_r(x)(-\Delta)^s\chi_k(x) -f^k_r(x)|\leq C 2^{2k}(1+\|\nabla v_r\|_{C(B'_{k+1})}),\quad\forall k \in \NN.
\end{align*}
\end{case}
\begin{case}[Points $x\notin \hbox{supp }\chi_k$]
\label{case:Estimate_alpha_k_x_greater_1}
From definition \eqref{eq:Definition_f_r} of $f_r$, we see that
\begin{align*}
w_r(x)(-\Delta)^s\chi_k(x) -f^k_r(x) = \int_{\RR^n}\frac{w_r(y)(\chi_k(x)-\chi_k(y))}{|x-y|^{n+2s}}\, dy.
\end{align*}
Because we assume that $x$ does not belong to $\hbox{supp }\chi_k$, we have that $\chi_k(x)=0$ and $\nabla\chi_k(x)=0$, and so
\begin{align*}
w_r(x)(-\Delta)^s\chi_k(x) -f^k_r(x) = -\int_{B'_1}\frac{w_r(y)(\chi_k(y)-\chi_k(x)-\nabla\chi_k(x)\dotprod(y-x))}{|x-y|^{n+2s}}\, dy.
\end{align*}
By applying Lemma \ref{lem:Uniform_boundedness_rescalings}, we obtain that there are positive constants, $C$ and $r_0$, such that
\begin{align*}
|w_r(x)(-\Delta)^s\chi_k(x) -f^k_r(x)| &\leq C2^{2k},\quad\forall k \in \NN,\quad\forall r\in (0,r_0).
\end{align*}
\end{case}
Combining Cases \ref{case:Estimate_alpha_k_x_less_1} and \ref{case:Estimate_alpha_k_x_greater_1}, we obtain that
\begin{align*}
\|w_r(-\Delta)^s\chi_k -f^k_r\|_{C(\RR^n)}\leq C 2^{2k}(1+\|\nabla v_r\|_{C(B'_{k+1})}),\quad\forall k \in \NN,\quad\forall r\in (0,r_0),
\end{align*}
and so, from inequality \eqref{eq:Estimate_alpha_k_1}, it follows that
\begin{align}
\label{eq:Estimate_alpha_k_2}
\alpha_k &\leq C 2^{2k}\left(1+\|\nabla v_r\|_{C(B'_{k+1})}\right),\quad\forall k\in\NN,\quad\forall r\in (0,r_0).
\end{align}
Applying the Interpolation inequalities \cite[Theorems 3.2.1 \& 8.8.1]{Krylov_LecturesHolder}, we have that there are positive constants, $C$ and $m$, such that for all $\eps>0$,
$$
\|\nabla v_r\|_{C(B'_{k+1})} \leq \eps \alpha_{k+2} + C\eps^{-m} \|v_r\|_{C(B'_1)},\quad\forall k\in\NN.
$$
Estimate \eqref{eq:Estimate_alpha_k_2} together with the preceding inequality and Lemma \ref{lem:Uniform_boundedness_rescalings}, give us that
\begin{align*}
\alpha_k &\leq C 2^{2k}(\eps^{-m}+\eps \alpha_{k+2}),\quad\forall k\in\NN.
\end{align*}
Because $\eps>0$ is arbitrarily chosen, we redefine it to be $\eps^2 C^{-1}2^{-2k}$. Then the preceding inequality becomes
\begin{align*}
\alpha_k &\leq \eps^2\alpha_{k+2} + C \eps^{-2m} 2^{2(m+1)k},\quad\forall k\in\NN.
\end{align*}
We multiply the inequality by $\eps^k$, and we obtain
\begin{align*}
\eps^k\alpha_k &\leq \eps^{k+2}\alpha_{k+2} + C \eps^{-2m+k} 2^{2(m+1)k},\quad\forall k\in\NN.
\end{align*}
By choosing $\eps \in (0,1)$ sufficiently small such that
$$
\sum_{k=0}^{\infty} \eps^{-2m+k} 2^{2(m+1)k} <\infty,
$$
we obtain that
\begin{align*}
\sum_{k=0}^{\infty}\eps^k\alpha_k &\leq \sum_{k=0}^{\infty}\eps^{k+2}\alpha_{k+2} + C,
\end{align*}
and so, it follows that $\alpha_0 \leq C$. Estimate \eqref{eq:Uniform_boundedness_C_1_alpha_n_ball_rescalings} now follows.
\end{step}
This concludes the proof.
\end{proof}
The uniform Schauder estimate \eqref{eq:Uniform_boundedness_C_1_alpha_n_ball_rescalings} on $B'_{1/4}$ can now be used to obtain a uniform Schauder estimate in $B^+_{1/8}$ of the rescaling sequence. We have the following consequence of Lemma \ref{lem:Uniform_boundedness_C_1_alpha_n_ball_rescalings}.
\begin{lem}[Uniform Schauder estimates on $B^+_{1/8}$]
\label{lem:Uniform_boundedness_C_1_alpha_n_plus_1_ball_rescalings}
Assume that the hypotheses of Lemma \ref{lem:Uniform_boundedness_C_1_alpha_n_ball_rescalings} hold. Then there are positive constants, $C$, $\gamma\in (0,1)$ and $r_0$, such that
\begin{equation}
\label{eq:Uniform_boundedness_C_1_alpha_n_plus_1_ball_rescalings}
\begin{aligned}
\|v_r\|_{C^{\gamma}(\bar B^+_{1/8})} &\leq C,\\
\|\partial_{x_i} v_r\|_{C^{\gamma}(\bar B^+_{1/8})} &\leq C,\quad\forall i=1,\ldots,n,\\
\||y|^a\partial_y v_r\|_{C^{\gamma}(\bar B^+_{1/8})} &\leq C,
\end{aligned}
\end{equation}
for all $r\in (0,r_0)$.
\end{lem}
\begin{proof}
By construction of the rescaling sequence, $\{v_r\}_{r>0}$, and from Lemma \ref{lem:Uniform_boundedness_H_1_rescalings}, we know that the functions $v_r$ belongs to $H^1(B^+_{1/4},|y|^a)$ and solve the equation $L_a v_r=0$ on $B^+_{1/4}$. Moreover, by Lemma
\ref{lem:Uniform_boundedness_C_1_alpha_n_ball_rescalings}, the function $v_r\upharpoonright_{B'_{1/4}}$ is H\"older continuous. By adapting the proof of \cite[Theorem 2.4.6]{Fabes_Kenig_Serapioni_1982a} to the case of non-zero boundary data, as in \cite[Theorem 8.27]{GilbargTrudinger}, we obtain that the function $v_r \in C^{\gamma}(\bar B^+_{1/8})$, for some positive constant, $\gamma\in (0,1)$. Moreover, because the sequence of functions $\{v_r\}_{r>0}$ satisfies the uniform Schauder estimate \eqref{eq:Uniform_boundedness_C_1_alpha_n_ball_rescalings}, we also obtain that there are positive constants, $C$ and $r_0$, such that
\begin{equation}
\label{eq:Holder_estimate_v_r}
\|v_r\|_{C^{\gamma}(\bar B^+_{1/8})} \leq C,\quad\forall r\in (0,r_0).
\end{equation}
A similar argument can be applied to the derivatives $\partial_{x_i} v_r$, for all $i=1,\ldots,n$. Notice that the function $\partial_{x_i} v_r$ solves the equation $L_a \partial_{x_i} v_r=0$ on $B^+_{1/4}$, and Lemma \ref{lem:Uniform_boundedness_C_1_alpha_n_ball_rescalings} gives that the derivative $\partial_{x_i} v_r\upharpoonright_{B'_{1/4}}$ is a H\"older continuous function. Because the derivative $\partial_{x_i} v_r\upharpoonright_{B^+_{1/4}}$ is a $L_a$-harmonic function, and the boundary condition on $B'_{1/4}$ is H\"older continuous, we can prove that $\partial_{x_i} v_r\in H^1(B^+_{1/4}, |y|^a)$. Therefore, we can conclude that
there are positive constants, $C$, $\gamma$ and $r_0$, such that
\begin{equation*}
\|\partial_{x_i} v_r\|_{C^{\gamma}(\bar B^+_{1/8})} \leq C,\quad \forall i =1,\ldots, n,\quad\forall r\in (0,r_0).
\end{equation*}
We now consider the case of the derivative $\partial_y v_r$. From \cite[\S 2.3]{Caffarelli_Silvestre_2007}, we know that the function $w_r:=|y|^a\partial_y v_r$ solves the conjugate equation, $L_{-a} w_r=0$ on $B^+_{1/4}$, where we recall that $a=1-2s$. From Lemma \ref{lem:Uniform_boundedness_H_1_rescalings}, we know that the function $w_r$ belongs to $L^2(B^+_{1/4}, |y|^{-a})$. From definitions \eqref{eq:Auxiliary_function_v} of the auxiliary function $v$, and \eqref{eq:Rescaling} of the rescaling $v_r$, we obtain that the boundary condition is given by
$$
\lim_{y\downarrow 0} w_r(x,y)=\frac{r^{2s}}{d_r} \left((-\Delta)^s u(rx)-(-\Delta)^s \varphi(rx)+(-\Delta)^s \varphi(O)\right).
$$
Because the function $u$ solves the obstacle problem \eqref{eq:Obstacle_problem_u} and $O\in\partial\{u=\varphi\}$, we have that $(-\Delta)^s u(O)=0$. In addition, the functions $u$ and $\varphi$ belong to $C^{1+\alpha}(B'_1)$, for all $\alpha\in (2s-1,s)$, and so, we see that there are positive constants, $\beta\in (0,1)$, $C$ and $r_0$, such that
\begin{equation*}
\|w_r\|_{C^{\beta}(B'_{1/4})} \leq C,\quad\forall r\in (0,r_0).
\end{equation*}
As in the case of the derivatives $\partial_{x_i} v_r$, we can prove that $w_r \in H^1(B^+_{1/4}, |y|^{-a})$. Then again, the argument applied in the case of the rescaling sequence $\{v_r\}_{r>0}$ to prove estimate \eqref{eq:Holder_estimate_v_r}, gives us that there are positive constants, $C$, $\gamma$ and $r_0$, such that
\begin{equation*}
\||y|^a\partial_y v_r\|_{C^{\gamma}(\bar B^+_{1/8})}=\|w_r\|_{C^{\gamma}(\bar B^+_{1/8})} \leq C,\quad\forall r\in (0,r_0).
\end{equation*}
This concludes the proof.
\end{proof}
We can now give the
\begin{proof}[Proof of Proposition \ref{prop:Phi_at_0}]
As in the proof of \cite[Lemma 6.1]{Caffarelli_Salsa_Silvestre_2008}, we consider two cases, depending on whether condition \eqref{eq:Fraction_d_r_r_power_finite} or condition \eqref{eq:Fraction_d_r_r_power_infty} is satisfied.
If condition \eqref{eq:Fraction_d_r_r_power_finite} holds, we can apply the same argument as in the proof of \cite[Lemma 6.1]{Caffarelli_Salsa_Silvestre_2008} to obtain that $\Phi^p_v(0+)=n+a+2(1+p)$, and so, identity \eqref{eq:Phi_at_0_p} holds.
Now assume that condition \eqref{eq:Fraction_d_r_r_power_infty} holds. Then, we may assume without loss of generality that
$$
\Phi^p_v(r) = r \frac{d}{dr} \log F_v(r),\quad\forall r \in (0,1).
$$
From Lemma \ref{lem:Uniform_boundedness_C_1_alpha_n_plus_1_ball_rescalings}, we can find a subsequence, $\{v_{r_k}\}_{k>0}$, which converges strongly in the space $H^1(B^+_{1/8},|y|^a)$ to a function $v_0\in H^1(B^+_{1/8},|y|^a)$. From the complementarity conditions \eqref{eq:Upper_bound_L_a} and \eqref{eq:Equality_L_a}, it follows that the rescalings $v_{r_k}$ satisfy
\begin{align*}
v_{r_k} &\geq 0 \quad\hbox{on } B'_{1/8},\\
L_a v_{r_k} &=0 \quad\hbox{on } B_{1/8}\backslash B'_{1/8},\\
L_a v_{r_k} &\leq \frac{r^{1-a}}{d_r} h(rx)\cH^{n}|_{\{y=0\}} \quad\hbox{on } B_{1/8},
\end{align*}
where the function $h$ is defined by \eqref{eq:Definition_h}. Because we assume that $\alpha\in (0,s)$ and $p\in (s,2s-1/2)$, we can choose $\alpha$ close enough to $s$, so that we have $2s+\alpha-1-p>0$. Combining this inequality with condition \eqref{eq:Fraction_d_r_r_power_infty} and estimate \eqref{eq:Growth_h}, it follows that
$$
\frac{r^{1-a}}{d_r} h(rx) \rightarrow 0,\quad\hbox{as } r\downarrow 0,\quad\forall x \in B'_{1/8},
$$
and so, the function $v_0$ satisfies
\begin{align*}
v_0 &\geq 0 \quad\hbox{on } B'_{1/8},\\
v_0(x,y)&=v_0(x,-y) \quad\forall (x,y)\in B_{1/8}\backslash B'_{1/8},\\
L_a v_0 &=0 \quad\hbox{on } B_{1/8}\backslash \left( B'_{1/8}\cap\{v_0=0\}\right),\\
L_a v_0 &\leq 0 \quad\hbox{on } B_{1/8}.
\end{align*}
By \cite[Lemma 6.1]{Caffarelli_Salsa_Silvestre_2008}, it follows that $\Phi_{v_0}(r) \geq n+a+2(1+s)$, where the function $\Phi_{v_0}(r)$ is defined in \cite[Formula (3.22)]{Caffarelli_Salsa_Silvestre_2008} by
$$
\Phi_{v_0}(r):=r(1+C_0r) \frac{d}{dr} \log\max\{F_{v_0}(r), r^{n+a+4}\},
$$
where $C_0$ is a positive constant. By \cite[Lemma 6.5]{Caffarelli_Salsa_Silvestre_2008}, we obtain that there are positive constants, $C$ and $r_0$, such that $F_{v_0}(r) \leq Cr^{n+a+2(1+s)}$, for all $r\in (0,r_0)$. We now consider two cases. If we have $F_{v_0}(r) \leq r^{n+a+2(1+p)}$, where we recall that we assume that $p\geq s$, then clearly we have that $\Phi^p_{v_0}(0+)=n+a+2(1+p)$, from the definition \eqref{eq:Phi} of the function $\Phi_{v_0}(r)$. If we have that
$$
r^{n+a+2(1+p)} \leq F_{v_0}(r) \leq Cr^{n+a+2(1+s)},
$$
then it follows from definition \eqref{eq:Phi} of the function $\Phi_{v_0}(r)$, and \cite[Formula (3.22)]{Caffarelli_Salsa_Silvestre_2008} of the function $\Phi_{v_0}(r)$ that
\begin{align}
\Phi^p_{v_0}(r) &= \frac{1}{1+C_0r} \Phi_{v_0}(r)\notag\\
\label{eq:Expansion_Phi_v_0}
& = 2\frac{r\int_{B_r}|\nabla v_0|^2|y|^a}{\int_{\partial B_r}|v_0|^2|y|^a} +n+a.
\end{align}
By letting $r$ tend to $0$ in the first identity from above, we obtain by Proposition \ref{prop:Monotonicity_formula} and \cite[Lemma 6.1]{Caffarelli_Salsa_Silvestre_2008} that
\begin{equation}
\label{eq:Lower_bound_Phi_v_second_case}
\Phi^p_{v_0}(0+) = \Phi_{v_0}(0+) \geq n+a+2(1+s).
\end{equation}
From the proof of Lemma \ref{lem:Uniform_boundedness_H_1_rescalings}, identity \eqref{eq:Expansion_Phi} and inequality \eqref{eq:Inequality_second_term_Phi} imply that
$$
\Phi^p_{v}(r) = 2\frac{r\int_{B_r}|\nabla v|^2|y|^a}{\int_{\partial B_r}|v|^2|y|^a} +n+a+ O(r^{2(\alpha-p+s-1/2)}).
$$
For all $t,r>0$, we have that
\begin{align*}
\frac{rt\int_{B_{rt}}|\nabla v|^2|y|^a}{\int_{\partial B_{rt}}|v|^2|y|^a}
&= \frac{r\int_{B_r}|\nabla v_t|^2|y|^a}{\int_{\partial B_r}|v_t|^2|y|^a},
\end{align*}
from which it follows that
$$
\Phi^p_{v}(tr) = 2\frac{r\int_{B_r}|\nabla v_t|^2|y|^a}{\int_{\partial B_r}|v_t|^2|y|^a} +n+a+ O((tr)^{2(\alpha-p+s-1/2)}).
$$
By letting $t$ tend to zero, using the strong convergence of the sequence $\{v_{r_k}\}_{k>0}$ to $v_0$ in the $H^1(B^+_{1/8})$ norm, and the fact that $\alpha-p+s-1/2>0$, we obtain that
$$
\Phi^p_{v}(0+) = 2\frac{r\int_{B_r}|\nabla v_0|^2|y|^a}{\int_{\partial B_r}|v_0|^2|y|^a} +n+a,\quad\forall r\in (0,1).
$$
Identity \eqref{eq:Expansion_Phi_v_0} gives us that
\begin{equation}
\label{eq:Homogeneity_v_0}
\Phi^p_{v}(0+) = \Phi^p_{v_0}(r) = \Phi^p_{v_0}(0+),\quad\forall r\in (0,1).
\end{equation}
The preceding identity together with \eqref{eq:Lower_bound_Phi_v_second_case}, gives us that \eqref{eq:Phi_at_0} holds.
This concludes the proof.
\end{proof}
\subsection{Optimal regularity of solutions}
\label{sec:Solutions_optimal_regularity}
In this section we prove the optimal regularity of solutions. In Proposition \ref{prop:Growth_v_around_0}, we prove a growth estimate of the auxiliary function $v$ in a neighborhood of a free boundary point, which we then use to give the optimal regularity of solutions in Lemma \ref{lem:Optimal_regularity}. We conclude with the proof of our main result, Theorem \ref{thm:Solutions}.
Analogous to \cite[Lemma 6.6]{Caffarelli_Salsa_Silvestre_2008}, we have the following consequence of Proposition \ref{prop:Phi_at_0}.
\begin{lem}
\label{lem:Upper_bound_F}
If $\Phi^p_v(r)\rightarrow \mu$, as $r\downarrow 0$, then there are positive constants, $C$ and $r_0$, such that
\begin{equation}
\label{eq:Upper_bound_F}
F_v(r) \leq C r^{\mu},\quad\forall r\in (0,r_0),
\end{equation}
where the function $F_v$ is defined by \eqref{eq:F}.
\end{lem}
\begin{proof}
To obtain estimate \eqref{eq:Upper_bound_F} we can apply exactly the same argument that was used to prove \cite[Lemma 6.6]{Caffarelli_Salsa_Silvestre_2008}.
\end{proof}
In Proposition \ref{prop:Growth_v_around_0}, we prove an estimate of the growth of the function $v$ in a neighborhood of a free boundary point. The analogue of this result can be found in \cite[Lemma 6.5]{Caffarelli_Salsa_Silvestre_2008}. While the two results are similar in spirit, their proofs are different. The proof of \cite[Lemma 6.5]{Caffarelli_Salsa_Silvestre_2008} is based on comparison arguments, which we cannot adapt to our case because the singular measure $L_a v$ is nontrivial on the set $\{y=0\}\backslash\{v=0\}$. To overcome this difficulty, in the proof of Proposition \ref{prop:Growth_v_around_0}, we construct a suitable auxiliary function, $\psi$, to compensate the singular measure, and then we apply the Moser iterations method to obtain the growth estimate \eqref{eq:Growth_v_around_O}.
\begin{prop}[Growth of $v$ in a neighborhood of a free boundary point]
\label{prop:Growth_v_around_0}
Let $s\in (1/2,1)$. Assume that the obstacle function, $\varphi \in C^{2s+\alpha}(\RR^n)$, and $u\in C^{1+\alpha}(\RR^n)$, for all $\alpha\in (0,s)$, and that $u$ is a solution to problem \eqref{eq:Obstacle_problem_simple}. Then there are positive constants, $C$ and $r_0$, such that
\begin{equation}
\label{eq:Growth_v_around_O}
0 \leq v(x,0) \leq C|x|^{1+s},\quad\forall x \in B'_{r_0}.
\end{equation}
\end{prop}
\begin{proof}
We divide the proof into several steps.
\setcounter{step}{0}
\begin{step}[Inequality satisfied by $v^+$]
\label{step:Inequality_v_+}
We want to show that there is a positive constant, $C_0$, such that the function $v^+$ satisfies the inequality
\begin{equation}
\label{eq:Inequality_v_+}
\int_{B_1} \nabla v^+\nabla\eta|y|^a \leq C_0 \int_{B_1'} |x|^{\alpha}\eta(x),
\end{equation}
for all nonnegative test functions, $\eta\in H^1_0(B_1, |y|^a)$. Because we assume that $s\in (1/2,1)$, we have that $a=1-2s \in (-1,0)$. Then any function in $H^1_0(B_1,|y|^a)$ is also contained in $H^1_0(B_1)$, and by \cite[Theorem 5.5.1]{Evans}, it has a well-defined trace on $B'_1$ and on $\partial B_1$ in the $L^2(B'_1)$ and $L^2(\partial B_1)$ sense, respectively. The space $H^1_0(B_1,|y|^a)$ is the closure of the space of smooth functions $C^{\infty}_c(B_1)$ with respect to the norm $\|\cdot\|_{H^1(B_1,|y|^a)}$.
Let $\eps>0$ and let $\phi_{\eps}:\RR\rightarrow [0,1]$ be a smooth function satisfying the properties
\begin{equation}
\label{eq:Properties_psi_eps}
\phi_{\eps}' \geq 0,\quad \phi_{\eps}(t)=0\quad\hbox{ for } t<\eps,\quad \phi_{\eps}(t)=1\quad\hbox{ for } t>2\eps.
\end{equation}
Let $\eta\in H^1_0(B_1,|y|^a)$ be a nonnegative function. Integrating the singular measure $-L_a v$ against the test function $\phi_{\eps}(v^+)\eta$, and using identity \eqref{eq:Equality_L_a}, we obtain
\begin{align*}
\int_{B_1} \nabla v\nabla (\phi_{\eps}(v^+)\eta) |y|^a = 2\int_{B_1'} \left((-\Delta)^su(x)+(-\Delta)^s\varphi(O)-(-\Delta)^s\varphi(x)\right) \phi_{\eps}(v^+)\eta.
\end{align*}
Because $(-\Delta)^s u(x)=0$, for all $x\in\{u>\varphi\}\cap B'_1$, that is, for all $x\in\{v>0\}\cap B'_1$, we see from properties \eqref{eq:Properties_psi_eps} of the function $\phi_{\eps}$, that the preceding identity becomes
\begin{align*}
\int_{B_1} \nabla v\nabla \eta \phi_{\eps}(v^+)|y|^a + \int_{B_1} \nabla v \nabla v^+\phi'_{\eps}(v^+)\eta |y|^a = 2\int_{B_1'} \left((-\Delta)^s\varphi(O)-(-\Delta)^s\varphi(x)\right) \phi_{\eps}(v^+)\eta.
\end{align*}
Using again properties \eqref{eq:Properties_psi_eps} of the functions $\phi_{\eps}$, and the fact that the function $\eta$ is assumed to be nonnegative, we obtain that
$$
\int_{B_1} \nabla v \nabla v^+\phi'_{\eps}(v^+)\eta |y|^a \geq 0,
$$
and so, by letting $\eps$ tend to zero, the preceding two formulas give us
$$
\int_{B_1} \nabla v^+\nabla\eta|y|^a \leq \int_{B_1'} h(x)\eta(x),
$$
for all nonnegative test functions, $\eta\in H^1_0(B_1,|y|^a)$, where the function $h$ is defined in \eqref{eq:Definition_h}. The preceding inequality and \eqref{eq:Growth_h} imply \eqref{eq:Inequality_v_+}.
\end{step}
\begin{step}[Construction of an auxiliary function]
\label{step:Construction_auxiliary_function_v_+}
We want to build an auxiliary function, $\psi$, to compensate the term appearing on the right-hand side of inequality \eqref{eq:Inequality_v_+}. Our goal in this step is to prove the existence of a function, $\psi \in H^1(B_1, |y|^a)$, such that there is a positive constant, $C_1$, with the properties
\begin{align}
\label{eq:Equation_extension_psi}
L_a\psi(x,y) &= 0,&\quad\forall (x,y)\in \RR^{n+1}\backslash\{y=0\},\\
\label{eq:Fractional_laplacian_psi}
(-\Delta)^s\psi(x,0) &=C_0|x|^{\alpha}\varphi(|x|),&\quad\forall x\in \RR^n,\\
\label{eq:Growth_psi_n+1_dim}
|\psi(x,y)| &\leq C_1 |(x,y)|^{1+s},&\quad\forall (x,y)\in B_1,
\end{align}
where the smooth function with compact support $\varphi:[0,\infty)\rightarrow [0,1]$ is chosen such that $\varphi(t)=1$, for all $t \in (0,1)$. We begin our construction by defining the function $\psi_1$ by (see \cite[p. 76]{Silvestre_2007})
$$
\psi_1(x):= c_{n,-s}\int_{\RR^n}\frac{C_0|z|^{\alpha}\varphi(|z|)}{|x-z|^{n-2s}}\, dz,\quad\forall x\in\RR^n,
$$
and we let $\psi_2(x):=\psi_1(x)-\psi_1(O)$, for all $x \in \RR^n$. Then the function $\psi_2$ is a solution to equation \eqref{eq:Fractional_laplacian_psi}, and by \cite[Proposition 2.8]{Silvestre_2007}, we also have that the function $\psi_2$ belongs to $C^{2s+\alpha}(\RR^n)$, since $|x|^{\alpha}\varphi(|x|)$ is contained in $C^{\alpha}(\RR^n)$. We note that we have $\psi_2(O)=0$, and because $\psi_2$ is a radial function, the gradient $\nabla \psi_2(O)=0$. We see that there is a function, $\psi_0:[0,\infty)\rightarrow\RR$, which belongs to $C^{2s+\alpha}(\bar\RR_+)$ such that $\psi_2(x)=\psi_0(|x|)$, for all $x\in\RR^n$, and $\psi_0(0)=0$ and $\psi'(0)=0$.
We extend the function $\psi_2$ from $\RR^n$ to $\RR^n\times\RR_+$ such that $\psi_2$ satisfies equation \eqref{eq:Equation_extension_psi}. We let $\psi$ denote the $L_a$-harmonic extension of the function $\psi_2$ from $\RR^n$ to $\RR^n\times\RR_+$. Using the Poisson formula \cite[\S 2.4]{Caffarelli_Silvestre_2007}, the function $\psi$ can be constructed by setting
\begin{equation}
\label{eq:Definition_psi}
\psi(x,y):=\int_{\RR^n} P(z,y) \psi_0(|x-z|) \, dz,\quad\forall (x,y)\in\RR^{n}\times\RR_+,
\end{equation}
where we recall from \cite[Formula (2.3)]{Caffarelli_Silvestre_2007} that the Poisson kernel, $P(x,y)$, is defined by
\begin{equation}
\label{eq:Poisson_kernel}
P(x,y)= C_{n,s} \frac{y^{2s}}{\left(|x|^2+y^2\right)^{(n+2s)/2}},\quad\forall (x,y)\in \RR^n\times\RR_+,
\end{equation}
where $C_{n,s}$ is a positive constant depending only on $n$ ans $s$. We extend $\psi$ from $\RR^n\times\RR_+$ to $\RR^{n+1}$ by even reflection with respect to the hyperplane $\{y=0\}$. Then the function $\psi$ satisfies conditions \eqref{eq:Equation_extension_psi} and \eqref{eq:Fractional_laplacian_psi}, and it remains to show that $\psi$ satisfies the growth condition \eqref{eq:Growth_psi_n+1_dim}. For this purpose, we prove the following
\begin{claim}[Growth of $\psi$ in the $x$ and $y$ directions]
\label{claim:Growth_psi}
There is a positive constant, $C$, such that
\begin{align}
\label{eq:Growth_x_direction}
|\psi(x,y)-\psi(0,y)| &\leq C|x|^{1+s},\quad\forall (x,y), (0,y) \in B_1,\\
\label{eq:Growth_y_direction}
|\psi(0,y)| &\leq C|y|^{1+s},\quad\forall (0,y)\in B_1.
\end{align}
\end{claim}
\begin{proof}
We prove each inequality in the following two steps.
\setcounter{step}{0}
\begin{step}[Proof of inequality \eqref{eq:Growth_x_direction}]
Taking derivative in the $x_i$ variable in \eqref{eq:Definition_psi}, we obtain
\begin{align*}
\psi_{x_i}(0,y)&=\int_{\RR^n} P(z,y) \psi_0'(|z|) \frac{z_i}{|z|}\, dz,\quad\forall y>0,\quad \forall i=1,\ldots,n.
\end{align*}
Because the function $\psi_0$ and the Poisson kernel $P(\cdot,y)$ are radial functions, we see that $\psi_{x_i}(0,y) =0$, for all $y>0$, and all $i=1,\ldots,n$, which implies that there is a positive constant, $C$, such that
\begin{align*}
|\psi_{x_i}(x,y)| &= |\psi_{x_i}(x,y)- \psi_{x_i}(0,y)|\\
&\leq\int_{\RR^n} P(z,y)|\partial_{x_i}\psi_0(|x-z|)-\partial_{x_i}\psi_0(|z|)|\, dz\\
&\leq\int_{\RR^n} P(z,y)|x|^{2s+\alpha-1}\, dz\\
&\leq |x|^{2s+\alpha-1},\quad\forall (x,y)\in\RR^{n}\times(0,\infty).
\end{align*}
where in the last two inequalities we used the fact that the function $\psi_0$ belongs to $C^{2s+\alpha}(\bar\RR_+)$, and that the Poisson kernel, $P(\cdot,y)$ is a probability density. The preceding inequality immediately implies \eqref{eq:Growth_x_direction}, since we may choose $\alpha>1-s$.
\end{step}
\begin{step}[Proof of inequality \eqref{eq:Growth_y_direction}]
From the fact that $\psi_0(0)=0$, the definition \eqref{eq:Definition_psi} of the function $\psi$, and \eqref{eq:Poisson_kernel} of the Poisson kernel, $P(x,y)$, we obtain
$$
\frac{\psi(0,y)}{y^{1+s}} = -C_{n,s} y^{s-1}\int_{\RR^n} \frac{|z|^{n+2s}}{\left(|z|^2+y^2\right)^{(n+2s)/2}} \frac{\psi_0(0)-\psi_0(|z|)}{|z|^{n+2s}}\, dz.
$$
Identity \eqref{eq:Fractional_laplacian_psi} evaluated at $x=0$, gives that
$$
\int_{\RR^n} \frac{\psi_0(0)-\psi_0(|z|)}{|z|^{n+2s}}\, dz=0,
$$
from where it follows that
$$
\frac{\psi(0,y)}{y^{1+s}} = C_{n,s} y^{s-1}\int_{\RR^n}\left(1- \frac{|z|^{n+2s}}{\left(|z|^2+y^2\right)^{(n+2s)/2}}\right) \frac{\psi_0(0)-\psi_0(|z|)}{|z|^{n+2s}}\, dz.
$$
To estimate the preceding integral, we split it into the integrals over the sets $\{|z|<2y\}$, $\{2y\leq |z|<1\}$, and $\{1\leq |z|\}$, and we denote them by $I_i$, for $i=1,2,3$, respectively. We estimate each of these integrals separately. To estimate the integral $I_1$, we use the fact that $\psi_0\in C^{2s+\alpha}(\bar\RR_+)$, and that $\psi_0(0)=0$ and $\nabla\psi_0(0)=0$. We can find a positive constant, $C$, such that
$$
|I_1| \leq C y^{s-1} \int_0^{2y} \frac{t^{2s+\alpha}t^{n-1}}{y^{n+2s}}\, dt\leq C y^{s+\alpha-1}.
$$
Because we assume that $\alpha\in (1-s,s)$, we see that the right-hand side in the preceding identity in bounded, for all $y\in (0,1)$.
To estimate the integrals $I_2$ and $I_3$, we can use the Taylor series expansion of the function $(1+y^2/|z|^2)^{-(n+2s)/2}$, since $y/|z| <1$. We have that there is a positive constant, $C$, such that
\begin{equation}
\label{eq:Taylor_approximation}
\left|1-\frac{|z|^{n+2s}}{\left(|z|^2+y^2\right)^{(n+2s)/2}}\right| \leq C\frac{y^2}{|z|^2},\quad\forall |z|\geq 2y.
\end{equation}
We use the preceding inequality to estimate $I_2$, together with the fact that $\psi_0\in C^{2s+\alpha}(\bar\RR_+)$, and that $\psi_0(0)=0$ and $\nabla\psi_0(0)=0$, and we obtain
$$
|I_2| \leq Cy^{s-1}\int_{2y}^1 \frac{y^2}{t^2} \frac{t^{2s+\alpha} t^{n-1}}{t^{n+2s}}\, dt \leq C y^{1+s}(y^{\alpha-2}-1) \leq C y^{s+\alpha-1}.
$$
As in the preceding case, the right-hand side is bounded, for all $y\in (0,1)$.
To estimate the integral $I_3$, we use the Taylor approximation \eqref{eq:Taylor_approximation} and the fact that the function $\psi_0$ is bounded. Then there is a positive constant, $C$, such that
$$
|I_3| \leq C y^{s-1} \int_{1}^{\infty} \frac{y^2}{t^2} \frac{t^{n-1}}{t^{n+2s}}\, dt \leq C y^{1+s},
$$
since the function $t\mapsto t^{-(3+2s)}$ is integrable on $(1,\infty)$. We obtain that the integral $I_3$ is also bounded, for all $y\in (0,1)$. Thus inequality \eqref{eq:Growth_y_direction} now follows.
\end{step}
This completes the proof of Claim \ref{claim:Growth_psi}.
\end{proof}
Inequalities \eqref{eq:Growth_x_direction} and \eqref{eq:Growth_y_direction} together with the fact that $\psi(O)=0$ imply that the function $\psi$ satisfies condition \eqref{eq:Growth_psi_n+1_dim}.
In conclusion, we constructed a function $\psi\in H^1(B_1, |y|^a)$ which verifies conditions \eqref{eq:Equation_extension_psi}, \eqref{eq:Fractional_laplacian_psi} and \eqref{eq:Growth_psi_n+1_dim}.
\end{step}
\begin{step}[Equality satisfied by $\psi$]
\label{step:Inequality_psi}
We now want to show that the function $\psi$ satisfies the equality
\begin{equation}
\label{eq:Inequality_psi}
\int_{B_1} \nabla \psi\nabla\eta|y|^a = 2C_0 \int_{B_1'} |x|^{\alpha}\eta(x),
\end{equation}
for all test functions, $\eta\in H^1_0(B_1,|y|^a)$. Using identity \eqref{eq:Dirichlet_to_Neumann_map}, together with \eqref{eq:Equation_extension_psi} and \eqref{eq:Fractional_laplacian_psi}, we obtain by integration by parts that
\begin{align*}
\int_{B_1} \nabla \psi\nabla\eta |y|^a &= 2\int_{B_1'} (-\Delta)^s\psi(x,0) \eta\\
&= 2C_0\int_{B_1'} |x|^{\alpha} \varphi(|x|)\eta.
\end{align*}
Because the function $\varphi$ was chosen such that $\varphi(t)=1$, for all $t\in[0,1)$, inequality \eqref{eq:Inequality_psi} now follows.
\end{step}
\begin{step}[Proof of estimate \eqref{eq:Growth_v_around_O}]
We can now prove estimate \eqref{eq:Growth_v_around_O}. Let $\eta\in H^1_0(B_1,|y|^a)$ be a nonnegative test function. From inequalities \eqref{eq:Inequality_v_+} and \eqref{eq:Inequality_psi}, we obtain that
$$
\int_{B_1} \nabla \left(v^+-\psi\right)\nabla\eta|y|^a \leq 0.
$$
Thus the function $v^+-\psi$ is $L_a$-subharmonic on $B_1$, and we can apply \cite[Theorem 2.3.1]{Fabes_Kenig_Serapioni_1982a} to obtain that there is a positive constant, $C$, such that
\begin{equation}
\label{eq:Supremum_estimate_difference_1}
\sup_{B_r} (v^+-\psi) \leq C \left(\frac{1}{r^{n+1+a}}\int_{B_{2r}}\left(|v^+|^2+|\psi|^2\right)|y|^a\right)^{1/2}, \quad\forall r\in (0,1/2).
\end{equation}
Recall that we may apply \cite[Theorem 2.3.1]{Fabes_Kenig_Serapioni_1982a} because the weight $\fw(x,y)=|y|^a$ belongs to the Muckenhoupt $A_2$ class of functions, and the analogue of the standard Sobolev inequality \cite[Inequality (7.26)]{GilbargTrudinger} in the case of our weighted space, $H^1(B_1,|y|^a)$, is proved in \cite[Theorem (1.6)]{Fabes_Kenig_Serapioni_1982a}.
Using definition \eqref{eq:F} of the function $F$, we can rewrite inequality \eqref{eq:Supremum_estimate_difference_1} in the form
\begin{align*}
\sup_{B_r} (v^+-\psi)
&\leq C r^{-(n+1+a)/2} \left(\left(\int_0^{2r} F_v(t)\, dt\right)^{1/2} + \left(\int_0^{2r} F_{\psi}(t)\, dt\right)^{1/2} \right).
\end{align*}
From Proposition \ref{prop:Phi_at_0}, we may apply inequality \eqref{eq:Upper_bound_F} with $\mu=n+a+2(1+s)$, and we obtain that there are positive constants, $C_1$ and $r_0$, such that
$$
F_v(r) \leq C_1 r^{n+a+2(1+s)},\quad\forall r\in (0,r_0),
$$
while inequality \eqref{eq:Growth_psi_n+1_dim} gives us that there is a positive constant, $C_2$, such that
$$
F_{\psi}(r) \leq C_2 r^{n+a+2(1+s)},\quad\forall r\in (0,1).
$$
Combining now the last three inequalities, and using the fact that $v \geq 0$ on $\RR^n\times\{0\}$, we find a positive constant, $C_3$, such that
\begin{align*}
\sup_{B'_r} (v-\psi) \leq C_3 r^{1+s},\quad\forall r\in(0,r_0).
\end{align*}
The preceding inequality together with \eqref{eq:Growth_psi_n+1_dim} and the fact that
$$
\sup_{B'_r} v \leq \sup_{B'_r} (v-\psi) + \sup_{B'_r} \psi,
$$
imply that inequality \eqref{eq:Growth_v_around_O} holds.
\end{step}
This concludes the proof of inequality \eqref{eq:Growth_v_around_O}.
\end{proof}
We next have the following analogue of \cite[Corollary 6.8]{Caffarelli_Salsa_Silvestre_2008}.
\begin{lem}
\label{lem:Optimal_regularity}
Let $s\in (1/2,1)$. Assume that the obstacle function, $\varphi \in C^{2s+\alpha}(\RR^n)$, and $u\in C^{1+\alpha}(\RR^n)$, for all $\alpha\in (0,s)$, and that $u$ is a solution to problem \eqref{eq:Obstacle_problem_simple}. Then the function $v\in C^{1+s}(\RR^n)$, where $v$ is defined in \eqref{eq:Auxiliary_function_v}.
\end{lem}
\begin{rmk}
Because $v \in C^{1+s}(\RR^n)$ and $\varphi \in C^{2s+\alpha}(\RR^n)$, for all $\alpha\in (0,s)$, we obtain the optimal regularity for $u$, that is $u\in C^{1+s}(\RR^n)$.
\end{rmk}
\begin{proof}[Proof of Lemma \ref{lem:Optimal_regularity}]
Recall from Proposition \ref{prop:Solutions_partial_regularity} that the function $v$ belongs to $C^{1+\alpha}(\RR^n)$, for all $\alpha\in (0,s)$. Moreover, the function $v$ solves the obstacle problem
$$
\min\{(-\Delta)^s v-(-\Delta)^s \varphi, v\}=0\quad\hbox{on }\RR^n.
$$
We denote $\Lambda :=\{v=0\}.$ By \cite[Proposition 2.8]{Silvestre_2007}, it follows that the function $v$ belongs to $C^{1+s}(\RR^n)$ if we show that the function $w:=(-\Delta)^s v$ belongs to $C^{1-s}(\RR^n)$. Because the function $v$ belongs to $C^{1+\alpha}(\RR^n)$, for all $\alpha\in (0,s)$, it follows by \cite[Proposition 2.6]{Silvestre_2007} that we also have $w \in C(\RR^n)$. It remains to consider the H\"older seminorm of $w$. We want to show that there is a positive constant, $C$, such that
\begin{equation}
\label{eq:Holder_seminorm_w}
|w(x_1)-w(x_2)|\leq C|x_1-x_2|^{1-s},\quad\forall x_1,x_2\in\RR^n.
\end{equation}
We consider the following cases.
\setcounter{case}{0}
\begin{case}[Points $x_1,x_2\notin\hbox{int }\Lambda$]
\label{case:Both_points_outside_Lambda}
For all points $x_1,x_2\notin\Lambda$, we have that $w(x_i)=-(-\Delta)^s\varphi(x_i)$, for $i=1,2.$ Because the function $\varphi$ belongs to $C^{2s+\alpha}(\RR^n)$, for all $\alpha \in (0,s)$, and $s\in (1/2,1)$, we see that $(-\Delta)^s\varphi \in C^{1-s}(\RR^n)$, and so
$$
|w(x_1)-w(x_2)|\leq \left[(-\Delta)^s\varphi\right]_{C^{1-s}(\RR^n)}|x_1-x_2|^{1-s}.
$$
Because $w$ is a continuous function, the preceding inequality holds if $x_1$ and/or $x_2$ belong to $\partial\Lambda$. Therefore, we obtain that
$$
|w(x_1)-w(x_2)|\leq \left[(-\Delta)^s\varphi\right]_{C^{1-s}(\RR^n)}|x_1-x_2|^{1-s},\quad\forall x_1,x_2\notin\hbox{int }\Lambda.
$$
\end{case}
\begin{case}[Points $x_1\in\Lambda$ and $x_2\in\partial\Lambda$]
\label{case:One_point_in_Lambda_one_point_on_boundary}
Let $x_1\in\Lambda$ and $x_2\in\partial\Lambda$. Without loss of generality, we may assume that $|x_1-x_2|=\dist(x_1,\partial\Lambda)$. Then we have that $v(x_i)=0$, for $i=1,2$. We let $\rho:=|x_1-x_2|$. Because $x_2\in\partial\Lambda$, we have by Proposition \ref{prop:Growth_v_around_0} that
\begin{equation}
\label{eq:Inequality_v_B_rho_x_2}
|v(y)| \leq C|x_2-y|^{1+s},\quad\forall y \in B'_{r_0}(x_2).
\end{equation}
Without loss of generality we may assume that $\rho<r_0/2$. Using the fact that $v(x_i)=0$, for $i=1,2$, we have
\begin{align*}
w(x_1)-w(x_2)&=-\int\frac{v(y)}{|x_1-y|^{n+2s}}\, dy + \int\frac{v(y)}{|x_2-y|^{n+2s}}\, dy\\
&=-\int_{B'_{\rho}(x_2)}\frac{v(y)}{|x_1-y|^{n+2s}}\, dy +\int_{B'_{\rho}(x_2)}\frac{v(y)}{|x_2-y|^{n+2s}}\, dy\\
&\quad+\int_{\RR^n\backslash B'_{\rho}(x_2)}v(y)\left(\frac{1}{|x_2-y|^{n+2s}}-\frac{1}{|x_1-y|^{n+2s}}\right)\, dy,
\end{align*}
which we write as a sum of three terms, $I_1$, $I_2$ and $I_3$, respectively. Using inequality \eqref{eq:Inequality_v_B_rho_x_2} and the fact that $|x_1-y|\geq |x_2-y|$, for all $y\in B'_{\rho}(x_2)$, we also have that
$$
|v(y)| \leq C|x_1-y|^{1+s},\quad\forall y \in B'_{\rho}(x_2).
$$
Then we can estimate the integral $I_1$ in the following way
\begin{align*}
|I_1| &\leq C\int_{B'_{\rho}(x_2)} \frac{|x_1-y|^{1+s}}{|x_1-y|^{n+2s}}\, dy\\
& \leq C \rho^{1-s}\\
& = C |x_1-x_2|^{1-s}
\end{align*}
We can use inequality \eqref{eq:Inequality_v_B_rho_x_2} and the preceding argument to find that
$$
|I_2| \leq C |x_1-x_2|^{1-s}.
$$
To estimate integral $I_3$, we use the Mean Value theorem and, for each $y\in \RR^n$, there is a point, $x_y \in\RR^n$, on the line connecting the points $x_1$ and $x_2$ such that
\begin{align*}
|I_3| &\leq \int_{\RR^n\backslash B'_{\rho}(x_2)}|v(y)| \frac{|x_1-x_2|}{|x_y-y|^{n+2s+1}}\, dy\\
&= |x_1-x_2|\left(\int_{B'_{r_0}(x_2)\backslash B'_{\rho}(x_2)} \frac{|v(y)|}{|x_y-y|^{n+2s+1}}\, dy
+\int_{\RR^n\backslash B'_{r_0}(x_2)}\frac{|v(y)|}{|x_y-y|^{n+2s+1}}\, dy\right)\\
&=|x_1-x_2|\left(I_3'+I_3''\right).
\end{align*}
We denote by $I'_3$ and $I''_3$ the two integrals in the parenthesis on the right-hand side of the preceding inequality. We notice that there is a positive constant, $C$, such that $|x_y-y|\geq C|x_2-y|$ on $(\RR^n\backslash B'_{\rho}(x_2))\cap \Lambda^c$, since the vector $x_y$ is a convex combination of $x_1$ and $x_2$, and we recall that $\rho=|x_1-x_2|$. Using inequality \eqref{eq:Inequality_v_B_rho_x_2}, we obtain that there is a positive constant, $C$, such that
\begin{align*}
I_3' &\leq C\int_{B'_{r_0}(x_2)\backslash B'_{\rho}(x_2)} \frac{|x_2-y|^{1+s}}{|x_2-y|^{n+2s+1}}\, dy\\
&\leq C(1+|x_1-x_2|^{-s})\quad\hbox{(recall that $\rho=|x_1-x_2|$ and $\rho \leq r_0/2$)}.
\end{align*}
Using the boundedness of the function $v$, we obtain that there is a positive constant, $C=C(r_0,s)$, such that
\begin{align*}
I_3'' \leq C\|v\|_{C(\RR^n)}\int_{r_0}^{\infty} t^{-2s-2}\, dt \leq C.
\end{align*}
Therefore, combining the estimates of the integrals $I'_3$ and $I''_3$, we obtain
\begin{align*}
|I_3| &\leq C|x_1-x_2|(|x_1-x_2|^{-s}+C)\\
&\leq C|x_1-x_2|^{1-s},
\end{align*}
when $|x_1-x_2|$ is small enough. Using the estimates for the integrals $I_1$, $I_2$, and $I_3$, it follows that
obtain that
$$
|w(x_1)-w(x_2)|\leq C|x_1-x_2|^{1-s},
$$
for all $x_1\in\Lambda$ and $x_2\in\partial\Lambda$, such that $|x_1-x_2|=\dist(x_1,\partial\Lambda)$.
\end{case}
\begin{case}[Points $x_1,x_2\in\Lambda$]
\label{case:Both_points_in_Lambda}
For $i=1,2$, let $x_i\in \Lambda$, and let $x_i'\in\partial\Lambda$ be a projection of the point $x_i$ on the set $\partial\Lambda$, in the sense that $|x_i-x'_i|=\dist(x_i,\partial\Lambda)$. We consider two subcases.
In the \emph{first} case, we assume that
$$
|x_1-x_1'| \leq 5 |x_1-x_2|\quad\hbox{or}\quad |x_2-x_2'| \leq 5 |x_1-x_2|.
$$
Without loss of generality, we may assume that the first of the preceding inequalities holds. Then we also have that
$$
|x_2-x_1'| \leq 6 |x_1-x_2|,
$$
which gives us that $|x_2-x_2'| \leq 6 |x_1-x_2|$, and so $|x_2'-x_1'| \leq 12 |x_1-x_2|$. We easily obtain
\begin{align*}
|w(x_1)-w(x_2)|&\leq |w(x_1)-w(x_1')|+|w(x_1')-w(x_2')|+|w(x_2')-w(x_2)| \\
&\leq C|x_1-x_2|^{1-s}\quad\hbox{(using Cases \ref{case:Both_points_outside_Lambda} and \ref{case:One_point_in_Lambda_one_point_on_boundary})}.
\end{align*}
We now consider the \emph{second} case. We assume that
\begin{equation}
\label{eq:Ineq_point_projection}
|x_i-x_i'| > 5 |x_1-x_2|,\quad i=1,2.
\end{equation}
As before, let $\rho=|x_1-x_2|$ and let $x:=(x_1+x_2)/2$. Using assumption \eqref{eq:Ineq_point_projection}, we see that $v\equiv0$ on $B'_{4\rho}(x)$, and so, we can write
\begin{align*}
w(x_1)-w(x_2) = \int_{\RR^n\backslash B'_{4\rho}(x)} v(y) \left(\frac{1}{|x_2-y|^{n+2s}}-\frac{1}{|x_1-y|^{n+2s}}\right)\, dy
\end{align*}
We estimate the term $|w(x_1)-w(x_2)|$ using the same argument that we applied in the case of the integral $I_3$ in Case \ref{case:One_point_in_Lambda_one_point_on_boundary}. By the Mean Value theorem, we have that
\begin{align*}
|w(x_1)-w(x_2)| &\leq |x_1-x_2|\left(\int_{B'_{r_0}(x)\backslash B'_{4\rho}(x)} \frac{|v(y)|}{|x_y-y|^{n+2s+1}}\, dy\right.\\
&\quad\left.+\int_{\RR^n\backslash B'_{r_0}(x)} \frac{|v(y)|}{|x_y-y|^{n+2s+1}}\, dy\right),
\end{align*}
where the point $x_y$ is chosen as in Case \ref{case:One_point_in_Lambda_one_point_on_boundary}. We may assume without loss of generality that $r_0\geq 8\rho$. We again use the fact that the vector $x_y$ is a convex combination of $x_1$ and $x_2$, and so we can find a positive constant, $C$, such that $|x_y-y|\geq C|x-y|$ on $B'_{r_0}(x)\backslash B'_{4\rho}(x)$. Using Proposition \ref{prop:Growth_v_around_0}, and the fact that the function $v\equiv 0$ on $B'_{4\rho}(x)$, we have that
$$
|v(y)| \leq C|x-y|^{1+s},\quad\forall y \in B'_{r_0}(x)\backslash B'_{4\rho}(x),
$$
and so, we obtain that
\begin{align*}
|w(x_1)-w(x_2)| &\leq C|x_1-x_2|(|x_1-x_2|^{-s}+C)\\
&\leq C|x_1-x_2|^{1-s},
\end{align*}
when $|x_1-x_2|$ is small enough.
\end{case}
Thus inequality \eqref{eq:Holder_seminorm_w} follows by combining the preceding cases.
\end{proof}
We can now give the
\begin{proof}[Proof of Theorem \ref{thm:Solutions}]
The existence of H\"older continuous solutions follows from Proposition \ref{prop:Solutions_partial_regularity}. Lemma \ref{lem:Optimal_regularity} gives us the optimal regularity of solutions, $u\in C^{1+s}(\RR^n)$. Proposition \ref{prop:Uniqueness} yields the uniqueness of solutions.
\end{proof}
|
1,314,259,996,467 | arxiv | \section{Introduction}
Representation of objects by linear operators (resp. square matrices) is a tool helping to understand the behaviour of algebraic structures, which often serve as a model of physical phenomena. Representations of algebraic structures bring order into their description. Particle physics offers the most spectacular example: the classification of elementary particles is based on an irreducible representation of certain Lie algebras \cite{algebra}.
The essence of combinatorics on words does not allow to expect a massive application of the representation theory. Nevertheless, in this article we try to demonstrate that a representation of the special Sturmian monoid provides a handy tool for proving some results on Sturmian sequences fixed by primitive morphisms.
Sturmian sequences were introduced more than 80 years ago by Morse and Hedlund \cite{HMo} and belong to the most explored objects in combinatorics on words.
In this article we focus on the Sturmian monoid, i.e. on the set of morphisms which map any Sturmian sequence to a Sturmian sequence. The set together with composition of morphisms forms a monoid which is very well described. Combinatorics on words has used for a long time a representation of morphisms by the so-called incidence matrices of morphisms, which in case of the Sturmian morphisms is a representation by $ (2\times 2)$-matrices with integer entries. In~\Cref{FR}, we define a new representation $\mathcal{R}$ of the special Sturmian monoid $\mathcal{M}$ by $ (3\times 3)$-matrices. Unlike the representation by incidence matrices, our representation is faithful, i.e., $\mathcal{R}$ is injective. In~\Cref{submonoidSl}, we show that the matrices assigned by $\mathcal{R}$ to Sturmian morphisms form a submonoid of the group $Sl(\mathbb{Z},3)$ and the submonoid is characterized by three convex cones in $\mathbb{R}^3$.
Using the new representation $\mathcal{R}$, we provide in Section \ref{app} new proofs of four known results.
A new result on square roots of Sturmian sequences is obtained in~\Cref{ctverce}.
More specifically, we show that the fixing morphism of the square root of a Sturmian sequence $\mathbf u$, as introduced in \cite{PeWh}, can be found among the conjugates of small powers of the morphism fixing the $\mathbf u$.
\section{Preliminaries}
Let $\mathcal{A}$ be an \emph{alphabet}, a finite set of \emph{letters}.
A \emph{(finite) word} $w$ is a finite sequence of elements of $\mathcal{A}$: $w = w_0w_1 \dots w_{n-1}$ with $w_i \in \mathcal{A}$.
The length of $w$, denoted $|w|$, equals $n$.
The \emph{empty word}, which is the unique word of length $0$, is denoted $\varepsilon$.
If $w = p f s$, i.e., the word $w$ is a concatenation of 3 words $p$, $f$ and $s$, we say that $p$ is a \emph{prefix} of $w$, $f$ is a \emph{factor} of $w$ and $s$ is a suffix of $w$.
The set of all words over $\mathcal{A}$ is denoted $\mathcal{A}^*$.
Let $\mathbf{s}= \left(s_i\right)_{i=0}^{+\infty}$ be a sequence over $\mathcal{A}$, that is, $s_i \in \mathcal{A}$ for all $i$.
Similarly, if $\mathbf{s} = pf
\mathbf{s'}$, where $p$ and $f$ are finite words, we say that $p$ is a \emph{prefix} of $\mathbf{s}$ and $f$ is a \emph{factor} of $\mathbf{s}$.
A factor $f$ of $\mathbf{s}$ is said to be \emph{left special} if $af$ and $bf$ are also factors of $\mathbf{s}$ for two distinct letters $a,b \in \mathcal{A}$.
The \emph{frequency of a letter $a \in \mathcal{A}$} in the sequence $\mathbf{s}$ equals $\lim_{i \to +\infty} \frac{| \textrm{pref}_i (\mathbf{s})|_a}{i}$, if the limit exists, where $\textrm{pref}_i (\mathbf{s})$ is the prefix of $\mathbf{s}$ of length $i$ and $|w|_a$ is the number of $a$'s in the word $w$.
A mapping $\mu: \mathcal{A}^* \to \mathcal{A}^*$ is a \emph{morphism} if for all $u,v \in \mathcal{A}^*$ we have $\mu(uv) = \mu(u)\mu(v)$.
A morphism $\mu$ is \emph{primitive} if there exists $k$ such that every $a \in \mathcal{A}$ is a factor of $\mu^k(b)$ for any $b \in \mathcal{A}$.
Morphisms are naturally used to act on infinite sequences in the following manner.
Considering a sequence $\mathbf{s} = \left(s_i\right)_{i=0}^{+\infty}$ over $\mathcal{A}$, the image of $\mathbf{s}$ by the morphism $\mu$ is the sequence $\mu(\mathbf{s}) = \mu(s_0)\mu(s_1)\mu(s_2) \dots $ over $\mathcal{A}$.
We say that a sequence $\mathbf{s}$ is a \emph{fixed point} of $\mu$ if $\mu(\mathbf{s}) = \mathbf{s}$.
\section{Sturmian sequences}
Sturmian sequences allow many equivalent definitions, see for instance~\cite{Lo2} and~\cite{Fogg}.
We present a definition relying on mechanical sequences.
Let $\alpha \in [0,1]$ and $\delta$ be real numbers.
The sequences ${\bf s}_{\alpha, \delta}$ and ${\bf s}'_{\alpha, \delta}$ defined by
$${\bf s}_{\alpha, \delta}(n) := \lfloor \alpha(n+1) +\delta\rfloor -\lfloor \alpha n +\delta\rfloor \quad \text{ for each } n \in \mathbb N $$
and
$${\bf s}'_{\alpha, \delta}(n) := \lceil\alpha(n+1) +\delta\rceil-\lceil \alpha n +\delta\rceil \quad \text{ for each } n \in \mathbb N $$
are called the \emph{lower} and the \emph{upper}, respectively, \emph{mechanical sequences} with the \emph{slope} $\alpha$ and the \emph{intercept}~$\delta$.
It is easy to see that the sequences ${\bf s}_{\alpha, \delta}$ and ${\bf s}'_{\alpha, \delta}$ have elements in the set $\{0,1\}$.
If $\alpha$ is an irrational number, then ${\bf s}_{\alpha, \delta}(n)$ is a \emph{lower Sturmian sequence} and ${\bf s}'_{\alpha, \delta}(n)$
is an \emph{upper Sturmian sequence}.
Moreover, ${\bf s}_{\alpha, \delta}(n)=0$ if the fractional part of $\alpha n +\delta$ belongs to $ [0, 1-\alpha)$, otherwise ${\bf s}_{\alpha, \delta}(n)=1$. An analogous statement is true for ${\bf s}'_{\alpha, \delta}$.
This property leads to an equivalent definition of Sturmian sequences, which relies on the two interval exchange transformation.
For given parameters $\ell_0, \ell_1 >0$, we consider two intervals of length $\ell_0$ and $\ell_1$.
To define a lower Sturmian sequence, we use the left-closed right-open intervals $I_0=[0,\ell_0)$ and $I_1=[\ell_0, \ell_0+\ell_1)$, to define an upper Sturmian sequence, we use the left-open right-closed intervals $I_0=(0,\ell_0]$ and $I_1=(\ell_0, \ell_0+\ell_1]$.
The \emph{two interval exchange transformation} (2iet) $T: I_0\cup I_1\to I_0\cup I_1$ is defined by
$$
T(x) =
\left\{
\begin{array}{ll}
x + \ell_1 & \quad \text{if} \ x \in I_0,\\
x - \ell_0 & \quad \text{if} \ x \in I_1.
\end{array}
\right.
$$
If we take an initial point $ \rho \in I_0\cup I_1$, the sequence $\mathbf u = u_0u_1u_2 \dots \in \{0,1\}^\mathbb N$ defined by
$$
u_n =
\left\{
\begin{array}{ll}
0 & \quad \text{if} \ T^n(\rho) \in I_0,\\
1 & \quad \text{if} \ T^n(\rho) \in I_1,
\end{array}
\right.
$$
i.e., a coding of the trajectory of the point $\rho$, is a \textit{2iet sequence} with the \textit{parameters} $\ell_0,\ell_1,\rho$.
We shall use the following notation for this fact:
$$ \vec{v}(\mathbf u)= (\ell_0, \ell_1, \rho)^\top,$$
and refer to $(\ell_0, \ell_1, \rho)^\top$ as a \emph{vector of parameters of $\mathbf u$}.
Clearly, collinear triples $(\ell_0, \ell_1, \rho)^\top$ and $c(\ell_0, \ell_1, \rho)^\top$ produce the same infinite sequence for any constant $c>0$.
The set of all 2iet sequences with an irrational slope $\alpha$ coincides with the set of all Sturmian sequences, see \cite{Lo2}.
The slope of the sequence $\mathbf u$ with the vector of parameters $(\ell_0, \ell_1, \rho)^\top$ equals $ \alpha = \frac{\ell_1}{\ell_0+\ell_1}$.
The language of a Sturmian sequence depends only on the slope $\alpha$ and does not depend on the intercept $\delta$. The following lemma summarizes the relation between the slope and intercept of Sturmian sequences and the assigned vectors of parameters.
\begin{lem} \label{le:parameters_Sturmian}
Each lower Sturmian sequence ${\bf s}_{\alpha, \delta}$ is generated by the transformation $T$ exchanging the intervals $I_0 = [0, 1-\alpha)$ and $I_1= [ 1-\alpha, 1)$; each upper Sturmian sequence ${\bf s'}_{\alpha, \delta}$ is generated by the transformation $T$ exchanging the intervals $I_0 = (0, 1-\alpha]$ and $I_1= ( 1-\alpha, 1]$.
Moreover,
A) if $\delta \in (0,1)$, then
\[
\vec{v}\left( {\bf s'}_{\alpha, \delta} \right) = \vec{v}\left( {\bf s}_{\alpha, \delta} \right) .
\]
B) if $\delta = 0$, then
\[
\vec{v}\left( {\bf s'}_{\alpha, 0} \right) = \left( 1- \alpha, \alpha, 1 \right)
\quad \text{ and } \quad
\vec{v}\left( {\bf s}_{\alpha, 0} \right) = \left( 1-\alpha, \alpha, 0 \right).
\]
\end{lem}
Among all Sturmian sequences with a fixed irrational slope $\alpha=\frac{\ell_1}{\ell_0+\ell_1}$, the sequence with the vector of parameters
$\vec{v}(\mathbf u)= (\ell_0, \ell_1, \ell_1)^\top$ plays a special role.
Such a sequence is called a \textit{characteristic Sturmian sequence} and it is usually denoted by ${\bf c}_\alpha$.
A Sturmian sequence $\mathbf u \in \{0,1\}^\mathbb N$ is characteristic if both sequences $0\mathbf u$ and $1\mathbf u$ are Sturmian. Equivalently, a Sturmian sequence $\mathbf u$ is characteristic, if every prefix of $\mathbf u$ is left special.
\section{Sturmian morphisms}
A morphism $\psi$ is a \emph{Sturmian morphism} if $\psi(\mathbf u)$ is a Sturmian sequence for any Sturmian sequence $\mathbf u$.
The set of Sturmian morphisms together with composition forms the so-called \emph{monoid of Sturm}, or \emph{Sturmian monoid}, which is in \cite{Lo2} denoted by ${\it St}$. Here we work with a submonoid of ${\it St}$ generated by the following Sturmian morphisms:
\begin{equation}\label{listElementary}
G: \begin{cases} 0 \to 0 \\ 1 \to 01 \end{cases} \quad
\widetilde{G}: \begin{cases} 0 \to 0 \\ 1 \to 10 \end{cases} \quad
D: \begin{cases} 0 \to 10 \\ 1 \to 1 \end{cases}
\widetilde{D}: \begin{cases} 0 \to 01 \\ 1 \to 1 \end{cases} \quad .
\end{equation}
The submonoid $\mathcal{M} = \langle G, \widetilde{G}, D, \widetilde{D} \rangle $ is also called the \emph{special Sturmian monoid}.
Any lower (resp. upper) mechanical sequence is mapped by a morphism from $\mathcal{M}$ to a lower (resp. upper) mechanical sequence.
The monoid $\mathcal{M}$ does not contain the Sturmian morphism $E: 0\mapsto 1, 1\mapsto 0$. This morphism maps a lower (resp. upper) mechanical sequence to an upper (resp. lower) mechanical sequence.
Extending the generating set of $\mathcal{M}$ by the morphism $E$ gives already the generating set of the whole Sturmian monoid~${\it St}$.
The relation between the parameters of a Sturmian sequence and its image under a Sturmian morphism can be found in \cite[Lemmas 2.2.17 and 2.2.18]{Lo2}:
\begin{equation}\label{Lot}
G({\bf s}_{\alpha, \delta}) = {\bf s}_{\frac{\alpha}{1+\alpha}, \frac{\delta}{1+\alpha}}, \quad \widetilde{G}({\bf s}_{\alpha, \delta}) = {\bf s}_{\frac{\alpha}{1+\alpha}, \frac{\alpha+ \delta}{1+\alpha}} \quad \text{and} \quad E({\bf s}_{\alpha, \delta}) = {\bf s'}_{1-\alpha, 1-\delta}.
\end{equation}
The next lemma rephrases these identities.
Figure~\ref{obrazek} illustrates the action of the morphism $G$ on a Sturmian sequence and provides a geometrical proof of the first identity in \eqref{Lot}.
\begin{lem}\label{image} Let $\mathbf u$ be a lower (resp. upper) mechanical sequence with parameters $\ell_0 >0$, $\ell_1 >0$ and $\rho \in [0, \ell_0+\ell_1)$ (resp. $\rho \in (0, \ell_0+\ell_1]$). The lower (resp. upper) mechanical sequence
\begin{itemize}
\item $G(\mathbf u)$ has the parameters $\ell_0+\ell_1$, $\ell_1$ and $\rho $;
\item $\widetilde{G}(\mathbf u)$ has the parameters $\ell_0+\ell_1$, $\ell_1$ and $\rho +\ell_1$;
\item $D(\mathbf u)$ has the parameters $\ell_0$, $\ell_0+\ell_1$ and $\rho +\ell_0$;
\item $\widetilde{D}(\mathbf u)$ has the parameters $\ell_0$, $\ell_0+\ell_1$ and $\rho $.
\end{itemize}
\end{lem}
\begin{figure}
\newcommand*\ietalpha{25}
\newcommand*\ietalphalabel{$\ell_0$}
\newcommand*\ietbeta{40}
\newcommand*\ietbetalabel{$\ell_1$}
\newcommand*\ietX{37}
\begin{tikzpicture}[scale=1,radius=2cm,>=latex]
\newcommand\colorIntHeight{0.15}
\newcommand*\ietsdist{2}
\newcommand*\ietpad{0.2}
\newcommand*\ietepsstep{1.5}
\setcounter{ietstep}{1}
\setcounter{ietstepP}{0}
\setcounter{ietstepPP}{-1}
\tikzset{disco/.style={color=black,opacity=0.7}}
\tikzset{discoAl/.style={style=disco}}
\tikzset{discoBe/.style={style=disco}}
\tikzset{discoX/.style={color=blue}}
\tikzset{hlpo/.style={very thick}}
\tikzset{hlpoA/.style={yscale=1.2}}
\tikzset{tmap/.style={ultra thick,}}
\tikzset{bmap/.style={thick,dashed}}
\tikzset{xmap/.style={style=discoX},thick}
\tikzset{tmapG/.style={thick,dashed,color=gray}}
\begin{scope}[scale=1,x=0.008\textwidth]
\node () at (103,0){};
\pgfmathsetmacro\newX{\ietX}
\oneint[yshift={(1-\number\value{ietstep})*\ietsdist cm}]{\ietalpha}{\ietalphalabel}{\ietbeta}{\ietbetalabel}{\arabic{ietstep}}{%
\ietpoint[pos=0.5,below]{discoX}{\newX}{$\rho$}{pointX\arabic{ietstep}%
}
\expandafter\draw [decorate,decoration={brace,amplitude=12pt,raise=4pt},yshift=0pt] (start\arabic{ietstep}) -- node[sloped,above=14pt] {\footnotesize $I_0$} (alpha\arabic{ietstep});
\expandafter\draw [decorate,decoration={brace,amplitude=12pt,raise=4pt},yshift=0pt] (alpha\arabic{ietstep}) -- node[sloped,above=14pt] {\footnotesize $I_1$} (end\arabic{ietstep});
}{$\ell_0+\ell_1$}
\ietstepcounters
\pgfmathsetmacro\newX{(\newX-\ietalpha)}
\oneint[yshift={(1-\number\value{ietstep})*\ietsdist cm}]{\ietalpha}{\ietalphalabel}{\ietbeta}{\ietbetalabel}{\arabic{ietstep}}{%
\ietpoint[pos=0.5,below]{discoX}{\newX}{$T(\rho)$}{pointX\arabic{ietstep}%
}
}{$\ell_0+\ell_1$}
\draw[->,bmap] (start\arabic{ietstepP}) to[] node[pos=0.55,left,inner xsep=5] {$T$} (beta\arabic{ietstep});
\draw[->,bmap] (alpha\arabic{ietstepP}) to[] node[pos=0.55,left,inner xsep=5] {$T$} (end\arabic{ietstep});
\draw[->,bmap] (alpha\arabic{ietstepP}) to[] node[pos=0.55,left,inner xsep=5] {$T$} (start\arabic{ietstep});
\draw[->,bmap] (end\arabic{ietstepP}) to[] node[pos=0.55,left,inner xsep=5] {$T$} (beta\arabic{ietstep});
\draw[->,xmap] (pointX\arabic{ietstepP}) to[] node[pos=0.55,left,inner xsep=5] {$$} (pointX\arabic{ietstep});
\pgfmathsetmacro\ietalphaOLD{\ietalpha}
\pgfmathsetmacro\ietalpha{\ietalpha+\ietbeta}
\renewcommand*\ietalphalabel{$\ell_0+\ell_1$}
\pgfmathsetmacro\ietbeta{\ietbeta}
\renewcommand*\ietbetalabel{$\ell_1$}
\ietstepcounters
\ietstepcounters
\pgfmathsetmacro\newX{\ietX}
\pgfmathsetmacro\newTX{(\newX+\ietbeta)}
\pgfmathsetmacro\newTTX{(\newTX-\ietalpha)}
\oneint[yshift={(1-\number\value{ietstep})*\ietsdist cm}]{\ietalpha}{\ietalphalabel}{\ietbeta}{\ietbetalabel}{\arabic{ietstep}}{%
\ietpoint[pos=0.5,below]{discoX}{\newX}{$\rho$}{pointX\arabic{ietstep}%
}
\ietpoint[pos=0.5,below]{discoX}{\newTX}{$T'(\rho)$}{pointTX\arabic{ietstep}%
}
\expandafter\draw [decorate,decoration={brace,amplitude=12pt,raise=4pt},yshift=0pt] (start\arabic{ietstep}) -- node[sloped,above=14pt] {\footnotesize $I_0'$} (alpha\arabic{ietstep});
\expandafter\draw [decorate,decoration={brace,amplitude=12pt,raise=4pt},yshift=0pt] (alpha\arabic{ietstep}) -- node[sloped,above=14pt] {\footnotesize $I_1'$} (end\arabic{ietstep});
\begin{scope}[decoration={raise=24pt}]
\draw [decorate,decoration={brace,amplitude=12pt}] (start\arabic{ietstep}) -- node[sloped,above=35pt] {\footnotesize $I_0$} (\ietalphaOLD,0);
\draw [decorate,decoration={brace,amplitude=12pt}] (\ietalphaOLD,0) -- node[sloped,above=35pt] {\footnotesize $I_1$} (alpha\arabic{ietstep});
\end{scope}
}{$2\ell_0+\ell_1$}
\ietstepcounters
\pgfmathsetmacro\newX{(\newX+\ietbeta)}
\oneint[yshift={(1-\number\value{ietstep})*\ietsdist cm}]{\ietalpha}{\ietalphalabel}{\ietbeta}{\ietbetalabel}{\arabic{ietstep}}{%
\ietpoint[pos=0.5,below]{discoX}{\newX}{$T'(\rho)$}{pointX\arabic{ietstep}%
}
\ietpoint[pos=0.5,below]{discoX}{\newTTX}{$T'^2(\rho) = T(\rho)$}{pointTX\arabic{ietstep}%
}
}{$2\ell_0+\ell_1$}
\draw[->,bmap] (start\arabic{ietstepP}) to[] node[pos=0.35,left,inner xsep=5] {$T'$} (beta\arabic{ietstep});
\draw[->,bmap] (alpha\arabic{ietstepP}) to[] node[pos=0.35,left,inner xsep=5] {$T'$} (end\arabic{ietstep});
\draw[->,bmap] (alpha\arabic{ietstepP}) to[] node[pos=0.35,left,inner xsep=5] {$T'$} (start\arabic{ietstep});
\draw[->,bmap] (end\arabic{ietstepP}) to[] node[pos=0.35,left,inner xsep=5] {$T'$} (beta\arabic{ietstep});
\draw[->,xmap] (pointX\arabic{ietstepP}) to[] node[pos=0.55,left,inner xsep=5] {$$} (pointX\arabic{ietstep});
\draw[->,xmap] (pointTX\arabic{ietstepP}) to[] node[pos=0.55,left,inner xsep=5] {$$} (pointTX\arabic{ietstep});
\end{scope}
\end{tikzpicture}
\caption{Illustration of the relation of parameters of $\mathbf u$ and $G(\mathbf u)$: the upper part of the figure depicts 2iet transformation $T$ with the partition $[0,\ell_0)$ and $[\ell_0,\ell_0+\ell_1)$ which produces the sequence $\mathbf u$.
The lower part contains 2iet transformation $T'$ with the partition $I_0'=[0,\ell_0+\ell_1)$ and $I_1'=[\ell_0+\ell_1,2\ell_0+\ell_1)$.
Taking $x \in I_0$, we see that $T(x) = T'(x)$, while if $x \in I_1$, we have $T(x) = T'^2(x)$, $T'(x) \in I'_1$ and $T'(x) \not \in I_0 \cup I_1$.
It follows that $G(\mathbf u)$ is produced by the transformation $T'$.
}
\label{obrazek}
\end{figure}
\section{A representation of the special Sturmian monoid}\label{FR}
A \emph{representation} of the monoid $\mathcal{M}$ (over $\mathbb R$) is a monoid homomorphism $\mu: \mathcal{M} \to \mathbb R^{n \times n}$.
A representation is \emph{faithful} if $\mu$ is injective.
A traditional representation of the monoid $\mathcal{M}$ (and in general, of a monoid of morphisms) assigns to a morphism $\varphi \in \mathcal{M}$ its \emph{incidence matrix} $M_\varphi \in \mathbb{N}^{2\times2}$ defined by
\[
(M_\varphi) _{i,j} = |\varphi(j)|_i \quad \text{ for } i,j\in \{0,1\},
\]
i.e., the entry of $M$ at the position $(i,j)$ equals the number of occurrences of the letter $i$ in the word $\varphi(j)$.
It is easy to verify that $M_{\varphi\circ \psi} = M_{\varphi}M_{\psi}$ for every pair $\varphi, \psi \in \mathcal{M}$, hence $\varphi \mapsto M_\varphi$ is a representation of the special Sturmian monoid $\mathcal{M}$.
However, the matrices assigned to $\widetilde{G}$ and $G$ coincide.
The same is true for the matrices assigned to $\widetilde{D}$ and $D$.
In other words, the representation $\varphi \mapsto M_\varphi$ is not injective and hence it is not faithful.
Let us recall that the special Sturmian monoid $\mathcal{M}$ is not free: for any $k \in \mathbb N$ we have
\begin{equation}\label{eq:relations}
GD^k\widetilde{G} = \widetilde{G}\widetilde{D}^kG \quad \text{and} \quad DG^k\widetilde{D} = \widetilde{D}\widetilde{G}^kD.
\end{equation}
Theorem 2.3.14 in~\cite{Lo2} says that \eqref{eq:relations} is the presentation of the monoid $\mathcal{M}$, i.e., no other non-trivial independent relation can be found.
To find a faithful representation of the monoid $\mathcal{M}$
we assign to each element of $\{G,\widetilde{G},D, \widetilde{D}\}$ one matrix:
\begin{equation}\label{matrices}
R_{\widetilde{G}} =\left( \begin{array}{lll}
1&1&0\\
0&1&0\\
0&1&1
\end{array}\right), \
R_G =\left( \begin{array}{lll}
1&1&0\\
0&1&0\\
0&0&1
\end{array}\right), \
R_{\widetilde{D}} =\left( \begin{array}{lll}
1&0&0\\
1&1&0\\
0&0&1
\end{array}\right), \ \text{ and } \
R_D =\left( \begin{array}{lll}
1&0&0\\
1&1&0\\
1&0&1
\end{array}\right).
\end{equation}
These matrices preserve the presentation of the special Sturmian monoid \eqref{eq:relations}.
\begin{claim}\label{aswell}
If $k \in \mathbb N$, then
\begin{equation}\label{eq:MatrixRelations}
R_{\widetilde{G}} R_{\widetilde{D}}^k R_G = R_{G} R_D^k R_{\widetilde{G}} \quad \text{ and } \quad R_{\widetilde{D}} {R_{\widetilde{G}}^k} R_D = R_{D} R_G^k {R_{\widetilde{D}}}.
\end{equation}
\end{claim}
\begin{proof} For each $k\in \mathbb N$ we have
$$
R_{\widetilde{G}}^k =\left( \begin{array}{lll}
1&k&0\\
0&1&0\\
0&k &1
\end{array}\right), \
R_{G}^k =\left( \begin{array}{lll}
1&k&0\\
0&1&0\\
0&0&1
\end{array}\right), \
R_{\widetilde{D}}^k =\left( \begin{array}{lll}
1&0&0\\
k&1&0\\
0&0&1
\end{array}\right), \ \text{ and } \
R_D^k =\left( \begin{array}{lll}
1&0&0\\
k&1&0\\
k&0&1
\end{array}\right).
$$
Hence,
\[
R_{\widetilde{G}} R_{\widetilde{D}}^k R_G =
\begin{pmatrix}
k+1&k+2&0\\
k&k+1&0\\
k&k+1&1
\end{pmatrix} = R_{G} R_D^k R_{\widetilde{G}} \quad \text{and} \quad R_{\widetilde{D}} {R_{\widetilde{G}}^k} R_D =
\begin{pmatrix}
k+1&k&0\\
k+2&k+1&0\\
k+1&k &1
\end{pmatrix} = R_{D} R_G^k {R_{\widetilde{D}}}.\qedhere
\]
\end{proof}
We can now assign a matrix to any element of $\mathcal{M}$.
\begin{defi} \label{def:R}
Let $\mathcal{R}: \mathcal{M} \mapsto \mathbb R^{3 \times 3}$ be defined
for $\psi \in \mathcal{M}$ by
\[
\mathcal{R}(\psi) = R_{\varphi_1}R_{\varphi_1}\cdots R_{\varphi_n},
\]
where $\psi= \varphi_1 \circ \varphi_2\circ \cdots \circ \varphi_n$ and $\varphi_i \in \{G,\widetilde{G},D, \widetilde{D}\}$ for every $i = 1,2, \ldots, n$.
\end{defi}
Let us note that the definition is correct. It does not depend on the decomposition of $\psi$ into the elements of $\{G,\widetilde{G},D, \widetilde{D}\}$
since by \Cref{aswell} the relations \eqref{eq:relations} of the presentation of $\mathcal{M}$ are preserved in the monoid
$\left \langle R_G,R_{\widetilde{G}},R_D, R_{\widetilde{D}} \right \rangle = \mathcal{R}(\mathcal{M})$.
Our choice of the four matrices in \eqref{matrices} enables the following matrix reformulation of \Cref{image}.
\begin{claim}\label{obrazyVektoru}
Let $\varphi \in \mathcal{M}$ and $ \vec{v}(\mathbf u)$ be the vector of parameters of a Sturmian sequence $ \mathbf u$.
The vector $\mathcal{R}(\varphi) \vec{v}(\mathbf u)$ is the vector of parameters of the Sturmian sequence $\varphi(\mathbf u)$.
\end{claim}
\begin{proof}
Let $\varphi = \varphi_1 \circ \varphi_2\circ \cdots \circ \varphi_n$ and $\varphi_i \in \{G,\widetilde{G},D, \widetilde{D}\}$ for every $i = 1,2, \ldots, n$.
We proceed by induction on $n$.
For $n = 0$, the claim trivially holds.
Set $\varphi_2 \circ \varphi_3\circ \cdots \circ \varphi_{n} = \varphi'$ and
assume that $\mathcal{R}(\varphi') \vec{v}(\mathbf u)$ is the vector of parameters of $\varphi'(\mathbf u)$.
Since $\varphi_1 \in \{G,\widetilde{G},D, \widetilde{D}\}$, then $R_{\varphi_1} \left( \mathcal{R}(\varphi') \vec{v}(\mathbf u) \right) $ is the vector of parameters of $\varphi_1 \left( \varphi'(\mathbf u) \right) $ by \Cref{image}.
The proof is finished by noticing $R_{\varphi_1} \mathcal{R}(\varphi') = \mathcal{R}(\varphi)$.
\end{proof}
As already mentioned, the traditionally used representation of the Sturmian monoid which maps a morphism to its incidence matrix is not faithful.
Nevertheless, this representation has a narrow connection to the representation $\mathcal{R}$.
By definition of the matrices in \eqref{matrices}, $\mathcal{R}$ maps the morphism $\psi$ to the matrix $ \mathcal{R}(\psi)$ of the form
\begin{equation}\label{submatrix}\mathcal{R}(\psi) = \left( \begin{array}{lll}
m_{00}&m_{01}&0\\
m_{10}&m_{11}&0\\
E&F&1
\end{array}\right), \ \text{ where } \left( \begin{array}{ll}
m_{00}&m_{01}\\
m_{10}&m_{11}
\end{array}\right) = M_\psi\ \ \text{ and } \ \ E,F \in \mathbb{Z}.
\end{equation}
In other words, the incidence matrix of $\psi$ is the top left submatrix of $\mathcal{R}(\psi)$.
\begin{proposition}\label{DefRepr}
The mapping $\mathcal{R}$ is a faithful representation of the monoid $\mathcal{M}$.
\end{proposition}
\begin{proof}
\Cref{aswell} and the presentation of $\mathcal{M}$ by \eqref{eq:relations} imply that $\mathcal{R}$ is a representation of $\mathcal{M}$.
To show injectivity of $\mathcal{R}$, we assume $\mathcal{R}(\psi) = \mathcal{R}(\varphi)$, for $\psi, \varphi \in \mathcal{M}$.
Let $\mathbf u$ be a lower Sturmian sequence.
By \Cref{obrazyVektoru} and \Cref{image} both morphisms $\psi$ and $\varphi$ map $\mathbf u$ to lower Sturmian sequences with the same parameters.
That is, $\varphi(\mathbf u) = \psi(\mathbf u)$.
By the same argument, $\varphi(\mathbf u) = \psi(\mathbf u)$ for each upper Sturmian sequence $\mathbf u$, too. In other words, the images by the morphisms $\psi$ and $\varphi$ coincide for each element of their domain. We conclude that $\varphi = \psi$. Hence, $\mathcal{R}$ is injective.
\end{proof}
We continue by listing several straightforward properties of the representation $\mathcal{R}$ of the special Sturmian monoid.
Let us remind that a \emph{convex cone} $C$ in $\mathbb{R}^n$ is a non-empty subset of $\mathbb{R}^n$ such that
$C\cap (-C) =\{0\}$ and $\mu x + \nu y \in C$ for each $x, y \in C$ and $\mu, \nu \geq 0$.
For example, the set
\[
\mathbb R_{\geq 0}^n = \{x \in \mathbb{R}^n: \text{all components of $x$ are non-negative} \}
\]
is a convex cone.
\begin{lem}\label{list}
Let $\psi \in \mathcal{M}$.
\begin{enumerate}
\item The convex cone $C_1: =\{(x,y,z)^\top \in \mathbb{R}^3: 0 \leq x, \ 0\leq y, \ 0\leq z\leq x+y\}$ is invariant under multiplication by $\mathcal{R}(\psi)$, i.e.,
$\mathcal{R}(\psi) (C_1) \subset C_1$.
\medskip
\item The convex cone $C_2 := \{(x,y,z)^\top \in \mathbb{R}^3 : 0 \leq x,\ 0\geq y, \ y\leq z\leq x\}$ is invariant under multiplication by the inverse matrix of $\mathcal{R}(\psi)$, i.e.,
$\bigl(\mathcal{R}(\psi)\bigr)^{-1} (C_2) \subset C_2$.
\medskip
\item The number $1$ is an eigenvalue of $\mathcal{R}(\psi)$ and $(0,0,1)^\top$ is its corresponding eigenvector.
In particular, the convex cone $C_3 := \{(0,0,z)^\top \in \mathbb{R}^3 : 0\leq z\}$ is invariant under multiplication by $\mathcal{R}(\psi)$.
\end{enumerate}
\end{lem}
\begin{proof}
The validity of all items for the matrices $R_G, R_{\widetilde{G}}, R_D, R_{\widetilde{D}}$ can be verified directly.
Since $\mathcal{R}(\psi)$ belongs to the monoid generated by these four matrices, $\mathcal{R}(\psi)$ satisfies these properties as well.
\end{proof}
In order to provide some applications of the representation $\mathcal{R}$, we recall three properties of a linear mapping preserving a closed convex cone. The first item mentioned in the following proposition is a consequence of the Brouwer's theorem, see for instance \cite{DuSch}; the second and the third item are a consequence of the Perron-Frobenius theory, see for instance \cite{Fiedler}.
\begin{prop}\label{PerFro} Let $A\in \mathbb{R}^{n\times n}$ and $C $ be a closed convex cone in $\mathbb{R}^n$.
\begin{enumerate}[(1)]
\item If $AC \subset C$, then at least one eigenvector of $A$ belongs to the convex cone $C$. \label{it:PF1}
\item If $A \left(\mathbb R_{\geq 0}^n\right) \subset \mathbb R_{\geq 0}^n$, then the spectral radius of $A$ is an eigenvalue corresponding to an eigenvector from $\mathbb R_{\geq 0}^n$. \label{it:PF2}
\item If all entries of the matrix $A^k$ are positive for some $k \in \mathbb{N}$, then
the spectral radius $r_A$ of $A$ is a dominant simple eigenvalue of $A$, i.e. all other eigenvalues are of modulus strictly smaller than $r_A$. The corresponding eigenvector to $r_A$ has all entries positive and an eigenvector corresponding to any other eigenvalue cannot have all entries non-negative. \label{it:PF3}
\end{enumerate}
\end{prop}
Let us note that the assumption $A\left(\mathbb R_{\geq 0}^n\right) \subset \mathbb R_{\geq 0}^n$ is equivalent to the property that all entries of $A$ are non-negative.
\begin{coro}\label{eigenvectors}
Let $\psi \in \mathcal{M}$ be a primitive morphism.
The matrix $\mathcal{R}(\psi)$ has eigenvalues $\Lambda$, $1$ and $\frac{1}{\Lambda}$, where $\Lambda >1$ is an quadratic unit. An eigenvector corresponding to $\Lambda$ can be found in the form $(x,y,z)^\top \in \bigl(\mathbb{Q}(\Lambda)\bigr)^3$ with $x>0$, $y>0$ and $z\geq 0$.
No other eigenvalue has an eigenvector with the first two components positive.
\end{coro}
\begin{proof}
The matrix $\mathcal{R}(\psi)$ has the form described in \eqref{submatrix}.
The vector $(0,0,1)^\top$ is its eigenvector corresponding to the eigenvalue $1$.
Other eigenvalues of $\mathcal{R}(\psi)$ are eigenvalues of the matrix $M_\psi$ as well.
Since all entries of $M_\psi$ are non-negative integers and $\det M_\psi = 1$, the two eigenvalues $\lambda_1$ and $\lambda_2$ of $M_\psi$ are roots of the polynomial $X^2 - pX+1$ with $p=Tr(M_\psi) >0$.
Thus $\lambda_1$, and $ \lambda_2$ are algebraic integers and $\lambda_1 \lambda_2=1$.
Primitivity of $\psi$ implies that for some integer $k\geq 1$ all entries of $M_\psi^k$ are positive.
By \Cref{it:PF3} of \Cref{PerFro}, the spectral radius of $M_\psi$ is a simple eigenvalue.
It follows, without loss of generality, $\Lambda :=\lambda_1 > 1 > \lambda_2>0$ and $\Lambda$ is the dominant eigenvalue of $\mathcal{R}(\psi)$.
Since $\lambda_2$ is an algebraic integer lying in the interval $(0,1)$, $\lambda_2$ cannot be rational.
Consequently, $\lambda_2$ and $\lambda_1$ are quadratic irrational numbers.
By \Cref{it:PF2} of \Cref{PerFro}, $\mathcal{R}(\psi)$ has a non-negative eigenvector, say $(x,y,z)^\top$, corresponding to $\Lambda$.
Since $\bigl(\mathcal{R}(\psi)-\Lambda I\bigr) (x,y,z)^\top = (0,0,0)^\top$, the entries $x,y,z$ can be chosen to belong into $\mathbb{Q}(\Lambda)$.
The vector $(x,y)^\top$ is an eigenvector of $M_\psi$ corresponding to $\Lambda$ and by \Cref{it:PF3} of \Cref{PerFro}, $x> 0, y>0$.
If $(x',y',z')$ is an eigenvector of $\mathcal{R}(\psi)$ corresponding to $\lambda_2$, then $(x',y')$ is an eigenvector of $M_\psi$ to the same eigenvalue.
It follows from \Cref{it:PF3} of \Cref{PerFro} that $x'$ and $y'$ are numbers with opposite signs.
\end{proof}
\begin{remark} The Sturmian morphism $E$ exchanging the letters $0\leftrightarrow 1$ can be associated with a matrix $R_E\in \mathbb Z^{3\times 3}$ such that $R_E \vec{v}(\mathbf u)$ is a vector of parameters of the Sturmian sequence $E(\mathbf u)$.
Using \eqref{Lot}, we see that
$$R_E = \left( \begin{array}{llc}
0&1&0\\
1&0&0\\
1&1&-1
\end{array}\right).$$
Therefore, it is possible to extend the representation $\mathcal{R}$ of the special Sturmian monoid $\mathcal{M}$ into a representation of the Sturmian monoid ${\it St}$.
Since we focus on the question when a Sturmian sequence is invariant under a primitive morphism, we can restrict ourselves to $\mathcal{M}$ only: indeed, if $\mathbf u$ is fixed by a primitive morphism $\psi\in {\it St}$, then $\mathbf u$ is fixed by the morphism $ \psi^2\in \mathcal{M}$.
Using this restriction, we avoid having negative entries in the matrices representing the morphisms and we can exploit the Perron-Frobenius theorem.
\end{remark}
\section{A submonoid of $Sl(\mathbb{Z},3)$ defined by three convex cones}\label{submonoidSl}
Let us recall the notation
$Sl(\mathbb{N},3) = \{ R \in \mathbb{N}^{3\times 3} : \det R = 1\}$ and $Sl(\mathbb{Z},3) = \{ R \in \mathbb{Z}^{3\times 3} : \det R = 1\}$. It is well known that the group $Sl(\mathbb{Z},3)$ is finitely generated. Rivat in \cite{Fogg} showed that the monoid $Sl(\mathbb{N},3)$ is not finitely generated. Clearly, our representation $\mathcal{R}$ maps the special Sturmian monoid $\mathcal{M}$ into $Sl(\mathbb{N},3)$. In particular, $\mathcal{R}(\mathcal{M})$ is a submonoid of $Sl(\mathbb{N},3)$. But $\mathcal{R}(\mathcal{M})$ is finitely generated by the four matrices $R_G, R_{\widetilde{G}}, R_D, R_{\widetilde{D}}$. Lemma \ref{list} suggests to study a new submonoid of $Sl(\mathbb{Z},3) $. We set
\begin{equation}\label{newMonoid}\mathcal{E} = \{ R \in Sl(\mathbb{Z},3) : R(C_1) \subset C_1, \ R^{-1}(C_2)\subset (C_2), \ R(C_3) \subset C_3\}, \end{equation}
where
\[
\begin{split}
C_1 &:= \{(x,y,z)^\top \in \mathbb{R}^3 \colon 0 \leq x, 0\leq y, 0\leq z\leq x+y\}, \\
C_2 &:= \{(x,y,z)^\top \in \mathbb{R}^3 \colon 0 \leq x,0\geq y, y\leq z\leq x\}, \text{ and } \\
C_3 &:= \{(0,0,z)^\top \in \mathbb{R}^3 \colon 0\leq z\}
\end{split}
\]
(in accordance with the notation in \Cref{list}).
In this section we show that the representation $\mathcal{R}$ is in fact an isomorphism between $\mathcal{M}$ and $\mathcal{E}$.
By \Cref{list}, $\mathcal{R}(\mathcal{M})\subset \mathcal{E} $. In the remaining part of the section we prove
$\mathcal{R}(\mathcal{M})=\mathcal{E} $.
To achieve this goal, we characterize the elements of $\mathcal{E}$ by inequalities.
We first give a technical lemma that shall be used to reduce the number of inequalities.
\begin{lem}\label{1enough} If $A,B,C,D,E,F \in \mathbb{N}$ satisfy \ \
$ E < A+C, \ \ F<B+D\,$ and $ AD-BC = 1$, then
$$ -A < AF -BE \leq B \quad \Longleftrightarrow\quad -C \leq CF-DE < D\,.$$
\end{lem}
\begin{proof}
$(\Longrightarrow)$ \ \ We consider validity of the left side in the form $ -A+1\leq AF -BE \leq B$.
If $C=0$, then $AD-BC =1$ implies $A=D=1$ and $E< A+C$ gives $E < 1$, i.e. $E=0$. The desired inequality $-C \leq CF-DE < D$ has now the form $0\leq 0< 1$ and it is obviously satisfied.
If $C>0$, then the assumption
$$ -A+1 \leq AF -BE \leq B \ \ \ \text{implies } \ -AC+C \leq AFC -BEC \leq BC \,.$$
We continue by substituting $BC =AD-1$
$$-AC +C \leq AFC - E(AD-1) \leq AD -1\ \Longrightarrow \
-AC +C -E\leq A(CF - ED) \leq AD -1-E. $$
As $0\leq E\leq A+C-1$, we obtain
$$-AC -A+1\leq A(CF - ED) < AD \ \ \Longrightarrow \ \ -C -\tfrac{A-1}{A}\leq CF -ED < D,
$$
where we used $A>0$ which is a simple consequence of $AD-BC = 1$.
As $A,C,D,E,F \in \mathbb{N}$ and $\tfrac{A-1}{A} \in [0,1)$, the last inequality gives the desired $-C \leq CF-DE < D$.
\bigskip
$(\Longleftarrow)$ \ \ Let us note that the relations $ E < A+C, \ \ F<B+D\,$ and $ AD-BC = 1\,$ in the assumption of the lemma are invariant under exchange of the parameters $A \leftrightarrow D$,
$C \leftrightarrow B$ and $E \leftrightarrow F$. Applying the exchange to the implication
$ -A < AF -BE \leq B \quad \Longrightarrow\quad -C \leq CF-DE < D$ \ which is already proven, we obtain the converse.
\end{proof}
\begin{lem}\label{inequalities}
If $R \in Sl(\mathbb{Z},3)$, then $R \in \mathcal{E}$ if and only if there exist $A,B,C,D,E,F \in \mathbb{N}$ such that
$$R=\begin{pmatrix}
A&B&0\\
C&D&0\\
E&F&1
\end{pmatrix}, \quad \text{where} \
AD-BC = 1 \quad \text{and}$$
\begin{equation}\label{straight}
E < A+C, \ \ F<B+D\,,
\end{equation}
\begin{equation}\label{lower}
-C \leq CF-DE < D\,.
\end{equation}
\end{lem}
\begin{proof} $(\Longrightarrow)$ \ \ Let $R=\begin{pmatrix}
A&B&G\\
C&D&H\\
E&F&J
\end{pmatrix}\in \mathcal{E}$, where $A,B,\ldots , H, J\in \mathbb{Z}$. Assuming $RC_3\subset C_3$, we have $R \left( \begin{array}{l}
0\\
0\\
1
\end{array}\right)=\left( \begin{array}{l}
G\\
H\\
J
\end{array}\right) \in C_3$. Therefore, $G=H=0$ and $J\geq 0$. The determinant of $R$ is 1 and can be now computed as $\det R = (AD-BC)J=1$. Since $ A,B,C,D, J$ are integers and $J\geq 0$, we have $AD-BC=1=J$.
\medskip
Now, assume $RC_1\subset C_1$, i.e.,
$$\left( \begin{array}{lll}
A&B&0\\
C&D&0\\
E&F&1
\end{array}\right) \left( \begin{array}{l}
x\\
y\\
z
\end{array}\right)=\left( \begin{array}{l}
Ax+By\\
Cx+Dy\\
Ex+Fy+z
\end{array}\right) \in C_1 \text{ \ \ for each }\left( \begin{array}{l}
x\\
y\\
z
\end{array}\right)\in C_1.$$
The inequalities $Ax+By\geq 0$, $Cx+Dy\geq 0$ and $Ex+Fy+z\geq 0$ for all choices $x,y\geq 0$ and $z=0$ imply $A,B, C, D,E,F\geq 0$, i.e., $R \in Sl(\mathbb{N},3)$. Moreover,
$ Ex+Fy+z \leq Ax+By +Cx+Dy$ holds for any $z$ satisfying $0 \leq z \leq x + y$. In particular, if $z = x+y$, then
$$
Ex+Fy +x+y\leq Ax+By+Cx+Dy \ \ \text{gives} \ \ 0\leq (A+C-E-1)x+ (B+D-F-1)y.
$$
Therefore, $A+C-E-1\geq 0$ and $B+D-F-1\geq 0$.
\medskip
Now assume that $R^{-1}C_2\subset C_2$. Since $R^{-1} = \begin{pmatrix}
D&-B&0\\
-C&A&0\\
FC-ED&BE-FA&1
\end{pmatrix} $, we have
$$
R^{-1} \begin{pmatrix}
x\\
y\\
z
\end{pmatrix}
= \begin{pmatrix}
Dx-By\\
-Cx+Ay\\
(FC-ED)x+(BE-FA)y+z
\end{pmatrix} \in C_2 \text{ \ \ for any choice } y\leq 0\leq x \text{\ \ and } y\leq z\leq x\,.
$$
In particular, the third coordinate satisfies two inequalities:
\begin{enumerate}[i)]
\item $-Cx+Ay \leq (FC-ED)x + (BE-FA)y +z $, and \label{it:ine1}
\item $(FC-ED)x + (BE-FA)y +z \leq Dx-By $. \label{it:ine2}
\end{enumerate}
Setting $ y=z=0$ and $x \geq 0$, the inequality \ref{it:ine1} implies $FC-ED+C \geq 0$ .
Similarly, putting $0\leq z =x $ and $y=0$ we obtain $FC-ED-D+1\leq 0$ from \ref{it:ine2}.
$(\Longleftarrow)$: We have to verify 3 conditions in the definition of $\mathcal{E}$ in \eqref{newMonoid}. Let us start with checking $R^{-1}C_2 \subset C_2$. We consider $(x,y,z)^\top \in C_2$, i.e., $x\geq 0$, $y\leq 0$, $y\leq z\leq x$. Then
$$
R^{-1} \begin{pmatrix}
x\\
y\\
z
\end{pmatrix}
= \begin{pmatrix}
Dx-By\\
-Cx+Ay\\
(FC-ED)x+(BE-FA)y+z
\end{pmatrix} = :\begin{pmatrix}
\widetilde{x}\\
\widetilde{y}\\
\widetilde{z}
\end{pmatrix} . $$
Obviously, the first component $ \widetilde{x} = Dx-By \geq 0$, as $x\geq 0\geq y$ and $B,D\geq 0$. By the same reason, the second component $ \widetilde{y} = -Cx+Ay\leq 0$. To deduce inequalities required for the third component $\widetilde{z}$ we use \eqref{lower} and by \Cref{1enough} also the inequality $ -A < AF -BE \leq B$. Hence
$$ 0\geq \underbrace{(FC-ED-D+1)}_{\leq 0}x - \underbrace{(-B-BE+FA)}_{\leq 0}y = \widetilde{z} - \widetilde{x} +\underbrace{x -z}_{\geq 0} \geq \widetilde{z} - \widetilde{x}.
$$
Analogously,
$$ 0\leq \underbrace{(FC-ED+C)}_{\geq 0}x + \underbrace{(-A+1 +BE-FA)}_{\leq 0}y = \widetilde{z} - \widetilde{y} +\underbrace{y-z}_{\leq 0} \leq \widetilde{z} - \widetilde{y}.
$$
We can conclude that $(\widetilde{x},\widetilde{y},\widetilde{z})^\top \in C_2$.
To verify that $RC_1\subset C_1$ and $RC_3\subset C_3$ is straightforward and we omit it. \end{proof}
Before we show that $\mathcal{E}$ equals the image of the special Sturmian monoid $\mathcal{M}$ under our representation~$\mathcal{R}$, we recall a property of
$Sl(\mathbb{N},2) = \{ M \in \mathbb{N}^{2\times 2} : \det M = 1\}$.
\begin{claim}\label{size2} If $M=
\begin{pmatrix}
A&B\\
C&D
\end{pmatrix}
\in Sl(\mathbb{N},2)$ and $M\neq I_2$, then either ($A\geq C$ and $B\geq D$) or ($A\leq C$ and $B\leq D$).
\end{claim}
\begin{proof}
Let us note that $A>0$ and $D>0$. Indeed, if $AD=0$, then $1=\det M= AD-CB \leq 0$ --- a contradiction.
Assume that the claim does not hold true.
First discuss the case $A< C$.
Necessarily, $B>D$. Since $1=\det M = AD-BC \leq (C-1)D - (D+1)C= -D-C \leq 0$ --- a contradiction.
Now let us discuss the case $A\geq C$. Necessarily, $B<D$ and $A> C$.
It follows that $1=\det M = AD-BC \geq (C+1)D - (D-1)C= D+C \geq 1$.
Therefore, we can replace inequalities by equalities. As $D>0$ and $ C\geq 0$, the last inequality gives $C=0$ and $D=1$.
Consequently, $A=C+1 = 1$ and $B=D-1 =0$. In other words $M=I_2$ --- a contradiction.
\end{proof}
\begin{thm}\label{rovnost}
The monoid $\mathcal{E}$ defined by \eqref{newMonoid} coincides with
$\mathcal{R}(\mathcal{M}) = \left \langle R_G, R_{\widetilde{G}}, R_D, R_{\widetilde{D}} \right \rangle$.
\end{thm}
\begin{proof}
The inclusion $\mathcal{R}(\mathcal{M})\subset \mathcal{E} $ follows from \Cref{list}.
Let us show $\mathcal{E} \subset \mathcal{R}(\mathcal{M})$.
Let $R = (R_{ij})\in \mathcal{E}$ have the form given by \Cref{inequalities}, i.e.,
\[
R=\begin{pmatrix}
A&B&0\\
C&D&0\\
E&F&1
\end{pmatrix}
\]
and all entries of $R$ are non-negative.
First assume that $C=0$. As $1=\det R =AD-BC$, we have $A=D=1$. Inequalities \eqref{straight} say $E=0$ and $0\leq F\leq B$. Therefore, $R$ has the form
$$R= \begin{pmatrix}
1&B&0\\
0&1&0\\
0&F&1
\end{pmatrix} = {\underbrace{\begin{pmatrix}
1&1&0\\
0&1&0\\
0&0&1
\end{pmatrix}}_{R_G}}^{B-F}
{\underbrace{\begin{pmatrix}
1&1&0\\
0&1&0\\
0&1&1
\end{pmatrix}}_{R_{\widetilde G}}}^F \in \mathcal{R}(\mathcal{M})\,.$$
Now assume that $B =0$. Analogously to the previous case, $\det R =1$ implies $A=D=1$. Inequalities \eqref{straight} give $0\leq E\leq C$, $F=0$ and
$$R= \begin{pmatrix}
1&0&0\\
C&1&0\\
E&0&1
\end{pmatrix} = {\underbrace{\begin{pmatrix}
1&0&0\\
1&1&0\\
0&0&1
\end{pmatrix}}_{R_{\widetilde D}}}^{C-E}
{\underbrace{\begin{pmatrix}
1&0&0\\
1&1&0\\
1&0&1
\end{pmatrix}}_{R_D}}^E \in \mathcal{R}(\mathcal{M})\,.$$
We proceed by induction on $R_{11} + R_{21} = A+C$. Note that $\det R=1$ forces $A,D >0$. Hence the case $A+C = 1$ implies $A=1,C=0$ and it is treated above. In the induction step it suffices to discuss only the situation $A,B,C,D \geq 1$.
Using \Cref{size2}, we split our discussion into the following two cases:
\begin{enumerate}[(I)]
\item $A\geq C\geq 1$ and $B\geq D\geq 1$, \label{it:pr_case_1}
\item $1\leq A\leq C$ and $1\leq B\leq D$. \label{it:pr_case_2}
\end{enumerate}
\medskip
\noindent {\bf Case \ref{it:pr_case_1}} $A\geq C\geq 1$ and $B\geq D\geq 1$:
First we show by contradiction that entries of $R$ satisfy
\begin{equation}\label{subcases}
\left(C\leq E \ \text{ and } \ D\leq F \right)
\quad \text{ or } \quad
\left (E< A \ \text{ and } \ F< B\right).
\end{equation}
Assuming that \eqref{subcases} does not hold and using the inequalities of Case~\ref{it:pr_case_1} we obtain
\[
\left ( C>E \ \text{ and } \ F\geq B\geq D \geq 1 \right)
\quad \text { or } \quad
\left ( B\geq D>F \ \text{ and } \ E\geq A\geq C \geq 1 \right).
\]
In the first case, we have $C \geq E+1$ and $D \leq F$.
Combining these inequalities with the inequality \eqref{lower} of \Cref{inequalities} we obtain
\[
D \stackrel{\eqref{lower}}{>} CF -DE \geq (E+1)F - FE = F\geq D,
\]
which is a contradiction.
In the second case, we have $F \leq B - 1$ and $E \geq A$.
Using again the inequality \eqref{lower} of \Cref{inequalities},
we obtain from \Cref{1enough} the inequality
\[-A < AF -BE.
\]
Putting all these $3$ inequalities together, we conclude
\[
-A < AF - BE \leq A(B-1) - BA = - A,
\]
which is a contradiction.
\medskip
\noindent {\bf Subcase (I.a) } $C\leq E$ and $D\leq F$:
We observe that the matrix $R'$ defined via
$$R= \begin{pmatrix}
A&B&0\\
C&D&0\\
E&F&1
\end{pmatrix} =
\underbrace{\begin{pmatrix}
1&1&0\\
0&1&0\\
0&1&1
\end{pmatrix}}_{R_{\widetilde{G}}}
\underbrace{\begin{pmatrix}
A-C&B-D&0\\
C&D&0\\
E-C&F-D&1
\end{pmatrix}}_{=:R'}$$ is in $Sl(\mathbb{N},3)$, as all entries of $R'$ are non-negative integers and $\det R' = 1$.
We continue by proving $R' \in \mathcal{E} $ using \Cref{inequalities} for $R'$.
Inequalities \eqref{straight} for the matrix $R'$ read $E-C < A-C+C$ and $F-D < B-D +D$.
These are equivalent to $E<A+C$ and $F< B+D$, which are satisfied due to $R \in \mathcal{E}$ and \Cref{inequalities}.
Inequalities \eqref{lower} for $R'$ say $-C\leq C(F-D) - (E-C)D < D$.
They are equivalent to $-C\leq CF-ED < D$, again satisfied by $R \in \mathcal{E}$ and \Cref{inequalities}.
Therefore $R' \in \mathcal{E}$.
Let us recall that we proceed by induction on $R_{11}+R_{21}$.
Since $R'_{11}+R'_{21} = A< A+C = R_{11}+R_{21}$, we know by the induction hypothesis that $R' \in \left \langle R_G, R_{\widetilde{G}}, R_D, R_{\widetilde{D}} \right \rangle$.
The relation $R=R_{\widetilde{G}}R'$ implies $R\in \left\langle R_G, R_{\widetilde{G}}, R_D, R_{\widetilde{D}} \right\rangle$ as well.
\medskip
\noindent {\bf Subcase (I.b) } $E< A$ and $F< B$:
We define the matrix $R'$ by
$$R= \begin{pmatrix}
A&B&0\\
C&D&0\\
E&F&1
\end{pmatrix} =
\underbrace{\begin{pmatrix}
1&1&0\\
0&1&0\\
0&0&1
\end{pmatrix}}_{R_G}
\underbrace{\begin{pmatrix}
A-C&B-D&0\\
C&D&0\\
E&F&1
\end{pmatrix}}_{=:R'}.$$
Obviously, $ R' \in Sl(\mathbb{N},3)$.
We use again \Cref{inequalities} to show that $R' \in \mathcal{E} $.
Inequality \eqref{straight} for $R'$ states $E< A-C+C$ and $F< B-D +D$.
These are satisfied by the specification Subcase I.b.
The inequalities \eqref{lower} for $R'$ and $R$ coincide and they follow from $R\in \mathcal{E}$ and \Cref{inequalities}.
As $R'_{11}+R'_{21}< R_{11}+R_{21}$ and $R=R_GR'$, we conclude that $R\in \langle R_G, R_{\widetilde{G}}, R_D, R_{\widetilde{D}} \rangle$.
\medskip
\noindent {\bf Case \ref{it:pr_case_2}} $1\leq A\leq C $ and $1\leq B\leq D$:
We transform this case to the Case~\ref{it:pr_case_1}.
We use the permutation matrix $P=\begin{pmatrix}
0&1&0\\
1&0&0\\
0&0&1
\end{pmatrix}$ and the following 3 facts:
\begin{itemize}
\item $P\langle R_G, R_{\widetilde{G}}, R_D, R_{\widetilde{D}} \rangle P = \langle R_G, R_{\widetilde{G}}, R_D, R_{\widetilde{D}} \rangle$.
\\ \emph{Proof:} The equality follows from the equalities $PR_{\widetilde{G}}P = R_D$, $PR_GP = R_{\widetilde{D}}$ and $P^2 =I_3$.
\item $P\mathcal{E}P =\mathcal{E}$.
\\ \emph{Proof:} The cones $C_1$, $C_2$ and $C_3$ satisfy $PC_1 = C_1$, $PC_2 =-C_2$ and $PC_3 = C_3$.
Let $R\in \mathcal{E}$.
We have $PRP \in Sl(\mathbb{Z},3)$ and $ (PRP)^{-1}C_2 = PR^{-1}P C_2 = -PR^{-1}C_2 \subset - PC_2 = C_2$.
Analogously, $PRPC_i \subset C_i $ for $i$ being $1$ or $3$.
It means that $PRP \in \mathcal{E}$. On the other hand, if $PRP \in \mathcal{E}$, then by the previous reasoning $R=P(PRP)P \in \mathcal{E}$.
\item If $R\in \mathcal{E}$ belongs to Case~\ref{it:pr_case_2}, then $PRP$ belongs to Case~\ref{it:pr_case_1}.
\end{itemize}
Proof of the theorem is now complete. \end{proof}
\section{The representation $\mathcal{R}$ and fixed points of Sturmian morphisms}\label{app}
In this section we apply the faithful representation $\mathcal{R}$ to study the parameters of fixed points of primitive morphisms.
\subsection{Parameters of a fixed point of Sturmian morphisms }
In article \cite{Peng} Peng and Tan solve the question to determine the parameters of a fixed point $\mathbf u$ of a given primitive Sturmian morphism $\psi$.
To find the slope of $\mathbf u$, i.e. the frequency of the letter $1$ in $\mathbf u$, one can use the incidence matrix of $M_\psi$.
For a primitive morphism, it is well-known that the positive components of the eigenvector corresponding to the dominant eigenvalue are proportional to the frequencies of letters.
Hence, the only non-trivial question is to determine the intercept of $\mathbf u$. \
Theorems 2.3 and 3.2 of \cite{Peng} answer this question using the structure of the words $\psi(01)$ and $\psi(10)$.
The faithful representation $\mathcal{R}(\psi)$ provides a simple algebraic method to determine the parameters of $\mathbf u$.
\begin{prop}\label{fixed}
Let $\psi\in \mathcal{M}$ be a primitive morphism and $\mathbf u$ be a Sturmian sequence with the vector of parameters $\vec{v}(\mathbf u)$.
The sequence $\mathbf u$ is fixed by $\psi$ if and only if $\vec{v}(\mathbf u)$ is an eigenvector to the dominant eigenvalue of $\mathcal{R}(\psi)$.
\end{prop}
\begin{proof}
Let the sequence $\mathbf u$ with parameters $\vec{v}(\mathbf u) = (\ell_0, \ell_1,\rho)^\top$ be fixed by $\psi $.
Since the Sturmian sequences $\mathbf u$ and $\psi(\mathbf u)$ coincide, the vectors of their parameters are collinear, i.e. there exists $\Lambda > 0$ such that $\vec{v}(\psi(\mathbf u)) = \Lambda \vec{v}(\mathbf u)$.
By \Cref{obrazyVektoru},
\[
\Lambda \vec{v}(\mathbf u) = \vec{v}(\psi(\mathbf u)) =\mathcal{R}(\psi)\vec{v}(\mathbf u).
\]
By \Cref{eigenvectors}, an eigenvector $ (\ell_0, \ell_1,\rho)^\top$ with positive components $\ell_0$ and $ \ell_1$ corresponds to the dominant eigenvalue of $\mathcal{R}(\psi)$.
To prove the converse, assume that $\vec{v}(\mathbf u)$ is an eigenvector corresponding to the dominant eigenvalue $\Lambda$ of $\mathcal{R}(\psi)$.
By \Cref{obrazyVektoru}, the sequence $\psi(\mathbf u)$ has the vector of parameters $\vec{v}(\psi(\mathbf u)) =\mathcal{R}(\psi)\vec{v}(\mathbf u) = \Lambda \vec{v}(\mathbf u)$, i.e., the vectors of parameters of $\mathbf u$ and $\psi(\mathbf u)$ are the same up to a scalar factor.
Any morphism of $\mathcal{M}$ maps a lower (resp. upper) mechanical sequence to a lower (resp. upper) mechanical sequence.
Two lower (resp. upper) mechanical sequences with the same vector of parameters up to a scalar factor coincide.
Consequently, $\psi(\mathbf u) = \mathbf u$.
\end{proof}
\subsection{Pairs ${\bf s}_{\alpha, \delta}$ and ${\bf s'}_{\alpha, \delta }$ fixed by (possibly distinct) morphisms}
Dekking \cite{Dekking} studies for which values of the slope and the intercept are both sequences ${\bf s}_{\alpha, \delta}$ and ${\bf s'}_{\alpha, \delta}$ fixed by primitive morphisms.
His result can be also proven by applying the representation $\mathcal{R}$.
Recall that the definition of ${\bf s}_{\alpha, \delta}$ and ${\bf s'}_{\alpha, \delta}$ immediately implies that ${\bf s}_{\alpha, \delta}$ and ${\bf s'}_{\alpha, \delta}$ are either identical or they differ at most on two neighbouring positions.
First we state several simple claims on invariant subspaces of matrices from $\mathcal{R}(\mathcal{M})$.
A subspace $V \subset \mathbb R^3$ is an \emph{invariant subspace} of a matrix $R$ if $RV \subset V$.
By inspecting the behaviour of the matrices assigned to the generators of $\mathcal{M}$ we obtain the following properties.
\begin{lem}\label{subspaces} Let $\psi \in \mathcal{M}$.
\begin{enumerate}
\item If $\psi \in \langle G, D \rangle $, then $\mathcal{R}(\psi)$ has an invariant subspace
$\{(x,y,z)^\top :\ y=z\} \subset \mathbb{R}^3$.
\item If $\psi \in \langle \widetilde{G}, \widetilde{D} \rangle $, then $\mathcal{R}(\psi)$ has an invariant subspace $\{(x,y,z)^\top :\ x=z\} \subset \mathbb{R}^3$.
\item If $\psi \in \langle \widetilde{G}, D \rangle $, then $\mathcal{R}(\psi)$ has an invariant subspace $\{(x,y,z)^\top :\ z=x+y\} \subset \mathbb{R}^3$.
\item If $\psi \in \langle G, \widetilde{D} \rangle $, then $\mathcal{R}(\psi)$ has an invariant subspace $\{(x,y,z)^\top :\ z=0\} \subset \mathbb{R}^3$.
\end{enumerate}
\end{lem}
The last lemma is transformed directly into the next statement in terms of parameters of a Sturmian sequence.
\begin{lem}\label{subspaces2}
Let a Sturmian sequence $\mathbf u$ with parametrs $\ell_0, \ell_1, \rho$ be fixed by a primitive morphism $\psi \in \mathcal{M}$.
\begin{enumerate}
\item If $\psi \in \langle \widetilde{G}, D \rangle $, then $\rho =\ell_0+\ell_1$.
\item If $\psi \in \langle G, D \rangle $, then $\rho =\ell_1$.
\item If $\psi \in \langle \widetilde{G}, \widetilde{D} \rangle $, then $\rho =\ell_0$.
\item If $\psi \in \langle G, \widetilde{D} \rangle $, then $\rho =0$.
\end{enumerate}
\end{lem}
\begin{proof} Item (1): Assume $\psi \in \langle \widetilde{G}, D \rangle $. By Proposition \ref{fixed}, the vector $(\ell_0, \ell_1, \rho)^\top$ is the eigenvector of $\mathcal{R}(\psi)\in \langle R_{\widetilde{D}}, R_G\rangle$ corresponding to the dominant eigenvalue.
By \Cref{subspaces}, the plane $P=\{(x,y,z)^\top :\ z=x+y\} \subset \mathbb{R}^3$ is invariant under multiplication by $R_G$ and by $R_{\widetilde{D}}$, therefore $\mathcal{R}(\psi)$ has two eigenvectors in the plane. As the eigenvector $(0,0,1)^\top$ of $\mathcal{R}(\psi)$ corresponding to $1$ does not belong to $P$, the eigenvector corresponding to the dominant eigenvalue $(\ell_0, \ell_1, \rho)^\top$ must be in $P$.
It implies $\rho = \ell_0+\ell_1$, and hence the Sturmian sequence $\mathbf u$ is an upper Sturmian sequence coding the two interval exchange transformation with the domain $(0, \ell_0+\ell_1]$.
Proofs of the other Items are analogous.
\end{proof}
Now we state the Dekking's (see \cite[Theorems 2 and 3]{Dekking}) result and provide its alternative proof.
\begin{prop}
Let $\alpha \in (0,1)$, $\alpha$ irrational, and $\delta \in [0,1)$.
Assume that both sequences ${\bf s}_{\alpha, \delta}$ and ${\bf s'}_{\alpha, \delta}$ are fixed by primitive morphisms and ${\bf s}_{\alpha, \delta}\neq {\bf s'}_{\alpha, \delta}$. Then
either
\begin {enumerate}
\item $\delta = 1-\alpha$, in which case ${\bf s}_{\alpha, \delta }$ and ${\bf s'}_{\alpha, \delta}$ are distinct fixed points of the same primitive morphism $\psi \in \langle \widetilde{G}, \widetilde{D} \rangle$; or
\item $\delta =0$, in which case ${\bf s}_{\alpha, \delta}$ is fixed by a morphism $\psi \in \langle G, \widetilde{D} \rangle$ and ${\bf s'}_{\alpha, \delta}$ is fixed by a morphism $\eta \in \langle \widetilde{G}, D \rangle$.
Moreover, if $\psi = \varphi_1\circ \varphi_2 \circ \cdots \circ \varphi_n$ with $\varphi_i\in \{G, \widetilde{D}\}$, then
$$ \eta = \xi_1\circ\xi_2\circ \cdots \circ \xi_n, \ \ \text{where} \ \ \xi_i=\begin{cases} \widetilde{G} & \text{if } \ \varphi_i =G, \\ D & \text{if } \ \varphi_i=\widetilde{D}, \end{cases} \quad \text{for every } \ i=1, \ldots, n.
$$
\end{enumerate}
\end{prop}
\begin{proof}
Without loss of generality, we assume that ${\bf s}_{\alpha, \delta}$ is fixed by a primitive morphism $\psi\in \mathcal{M}$.
Firstly, we assume that $\delta \in (0,1)$.
Due to \Cref{le:parameters_Sturmian}, $\vec{v}({\bf s}_{\alpha, \delta}) = \vec{v}({\bf s'}_{\alpha, \delta}) = (1-\alpha, \alpha, \delta)^\top$.
By \Cref{fixed}, ${\bf s'}_{\alpha, \delta}$ is fixed by $\psi $ as well. As ${\bf s}_{\alpha, \delta}\neq {\bf s'}_{\alpha, \delta}$, the primitive morphism $\psi$ has two fixed points. In particular, $\psi(0)$ has a prefix $0$ and $\psi(1)$ has a prefix $1$.
The form of morphisms $ G,D, \widetilde{G}, \widetilde{D} $ implies that the starting letters of $\psi(0)$ and $\psi(1)$ coincide whenever the morphism $G$ or $D$ occurs in the composition of $\psi$. Therefore, $\psi \in \langle \widetilde{G}, \widetilde{D} \rangle$.
By Lemma \ref{subspaces2}, $\delta = 1-\alpha$.
\medskip
Secondly, we assume that $\delta = 0$. Due to \Cref{le:parameters_Sturmian}, $\vec{v}({\bf s}_{\alpha, \delta}) = (1-\alpha, \alpha,0)^\top$ and $ \vec{v}({\bf s'}_{\alpha, \delta}) = (1-\alpha, \alpha, 1)^\top$.
Let a primitive morphism $\psi\in \mathcal{M}$ fix the sequence ${\bf s}_{\alpha,0}$ and $\mathcal{R}(\psi)$ be written in the form \eqref{submatrix}. Since $(1-\alpha, \alpha,0)^\top$ is an eigenvector of $\mathcal{R}(\psi)$, by Proposition \ref{fixed} necessarily $(1-\alpha, \alpha)^\top$ is an eigenvector of the incidence matrix $M_\psi$ and $E=F=0$. Let us recall that $M_\psi$ is the product of incidence matrices of the elementary morphisms $M_D = M_{\widetilde{D}}$ and $M_G = M_{\widetilde{G}}$. Since $E=F=0$ and the third row of both matrices $R_{G}$ and $ R_{\widetilde{D}}$ equals $( 0, 0, 1)$,
$\mathcal{R}(\psi)$ is a product of several matrices from $\{R_{G}, R_{\widetilde{D}}\}$. The representation $\mathcal{R}$ is faithful, thus $\psi \in \langle G, \widetilde{D}\rangle$.
Let $\eta $ be the morphism created in Item (2). The incidence matrices $M_\eta$ and $M_\psi$ coincide, since $M_G=M_{\widetilde{G}}$ and $M_D=M_{\widetilde{D}}$. Obviously, $
(1-\alpha, \alpha)^\top$ is the positive eigenvector of $M_\eta$ as well. By Lemma \ref{subspaces}, the positive eigenvector of $\mathcal{R}(\eta)$ belongs to the plane formed by the vectors $(x,y,z)^\top$ for which $z=x+y$. Therefore, the positive eigenvector of $\mathcal{R}(\eta)$ equals $(1-\alpha, \alpha, 1)^\top$. By Proposition \ref{fixed}, ${\bf s'}_{\alpha, \delta}$ is fixed by $\eta$.
\end{proof}
\subsection{Sturmian sequences fixed by a morphism}
Yasutomi in \cite{Ya99} gave a necessary and sufficient condition under which a Sturmian sequence is invariant under a primitive morphism. The same result is proved in \cite{BaMaPe} and \cite{BEIR}.
Using the faithful representation $\mathcal{R}$ of the Sturmian monoid we provide here a simple proof of the necessary condition.
Due to the relation $E({\bf s}_{\alpha, \delta}) = {\bf s'}_{1-\alpha, 1-\delta}$ mentioned in \eqref{Lot}, it is enough to characterize lower Sturmian sequences fixed by a primitive morphism.
\begin{proposition} \label{Yasutomi_necessary} Let $\alpha \in (0,1)$ be irrational and $\delta \in [0,1)$. If ${\bf s}_{\alpha, \delta}$ is fixed by a primitive morphism, then
\begin{enumerate}
\item
$\alpha$ and $ \delta$ belong to the same quadratic field, say $\mathbb{Q}(\sqrt{m})$;
\item $\min \{ \overline{\alpha}, 1-\overline{\alpha}\} \leq \overline{\delta} \leq \max \{ \overline{\alpha}, 1-\overline{\alpha} \}$, where the mapping $x\mapsto \overline{x}$ is the non-trivial field automorphism on $\mathbb{Q}(\sqrt{m})$ induced by $\sqrt{m} \mapsto - \sqrt{m}$.
\end{enumerate}
\end{proposition}
\begin{proof} Let $\psi$ be a primitive morphism fixing ${\bf s}_{\alpha, \delta}$. The vector $\vec{v} = (1-\alpha, \alpha, \delta)\top$ is a vector of parameters of the Sturmian sequence ${\bf s}_{\alpha, \delta}$. By Proposition \ref{fixed}, $\vec{v}$ is an eigenvector to the dominant eigenvalue $\Lambda$ of the matrix $\mathcal{R}(\psi)$. Due to Corollary \ref{eigenvectors}:
\begin{itemize}
\item $\Lambda$ is a quadratic number, i.e. $\mathbb{Q}(\Lambda) = \mathbb{Q}(\sqrt{m})$ for some $m \in \mathbb{N}$.
\item $c\vec{v} = c(1-\alpha, \alpha, \delta)^\top \in \bigl(\mathbb{Q}(\Lambda)\bigr)^3$ for some positive $c$. Consequently, $c =c(1-\alpha) + c \alpha \in \mathbb{Q}(\Lambda)$. It implies $(1-\alpha, \alpha, \delta)^\top \in \bigl(\mathbb{Q}(\Lambda)\bigr)^3$ and thus $\delta$ and $\alpha$ are in the same quadratic field $\mathbb{Q}(\sqrt{m})$.
\end{itemize}
Let us apply the field automorphism to $\mathcal{R}(\psi) (1-\alpha, \alpha, \delta)^\top = \Lambda (1-\alpha, \alpha, \delta)^\top$. Since the entries of $\mathcal{R}(\psi)$ are rational, $\mathcal{R}(\psi)$ is invariant under the automorphism and thus we get that
$(1-\overline{\alpha}, \overline{\alpha}, \overline{\delta})^\top$ is an eigenvector to the eigenvalue $\overline{\Lambda}$. The third eigenvector of $\mathcal{R}(\psi)$ is $(0,0,1)^\top$.
By Item 2) of Lemma \ref{list} and Item 1) of Proposition \ref{PerFro}, one eigenvector of $\bigl(\mathcal{R}(\psi)\bigr)^{-1}$ belongs to the cone $C_2 := \{(x,y,z)^\top \in \mathbb{R}^3 : 0 \leq x,\ 0\geq y, \ y\leq z\leq x\}$.
Now we used the fact that if a non-singular matrix has an eigenvector $\vec{d}$ to $\lambda$, then $\vec{d}$ is an eigenvector of its inverse matrix to the eigenvalue $\tfrac{1}{\lambda}$. Hence the eigenvector $ (1-\overline{\alpha}, \overline{\alpha}, \overline{\delta})$ or $ -(1-\overline{\alpha}, \overline{\alpha}, \overline{\delta})$ belongs to $C_2$.
\medskip
If $ (1-\overline{\alpha}, \overline{\alpha}, \overline{\delta})^\top \in C_2$, then $\overline{\alpha} \leq \overline{\delta} \leq 1-\overline{\alpha}$.
If $(-1+\overline{\alpha}, -\overline{\alpha}, -\overline{\delta})^\top \in C_2$, then $-\overline{\alpha} \leq -\overline{\delta} \leq -1+\overline{\alpha}$, or equivalently, $1- \overline{\alpha} \leq \overline{\delta} \leq \overline{\alpha}$.
Both cases confirm Item (2) of the proposition. \end{proof}
Let us stress that Yasutomi also proved that Items (1)
and (2) of the previous proposition are also sufficient for ${\bf s}_{\alpha, \delta}$ to be fixed by a primitive morphism.
\subsection{Conjugacy of Sturmian morphisms}
In this subsection we use the faithful representation of the special Sturmian monoid to deduce a known result on the number of morphisms which are conjugates of a given Sturmian morphism. First we recall some notions.
We say that a morphism $\varphi$
is a \emph{right conjugate} of a morphism $\psi$, or that $\psi$ is a \emph{left
conjugate} of $\varphi$, noted $\psi\triangleright\varphi$, if there
exists $w \in \mathcal A^*$ such that
\begin{equation}
w\psi(a) = \varphi(a)w, \quad \textrm{for every letter } a \in \mathcal A. \label{FirstCond}
\end{equation}
If \eqref{FirstCond} is satisfied for $\psi = \varphi$ with a non-empty $w$, then the morphism $\psi$ is called \emph{cyclic}.
A fixed point of a cyclic morphism is periodic, hence Sturmian morphisms are acyclic. To any acyclic morphism one may assign a morphism $\varphi_R$, called the \emph{rightmost conjugate of $\varphi$}, such
that the following two conditions hold:
\begin{enumerate}[\rm (i)]
\item $\varphi_R$ is a right conjugate of $\varphi$;
\item if $\xi$ is a right conjugate of $\varphi_R$, then $\xi=\varphi_R$.
\end{enumerate}
\begin{remark}\label{leftspecial}
Let us list some simple properties of the relation $\triangleright$.
\begin{enumerate}
\item If $\psi\triangleright\varphi$, then $|\varphi(a)|_b = |\psi(a)|_b$ for every $a,b \in \mathcal{A}$. Hence the incidence matrices of $\varphi$ and $\psi$ coincide, i.e., $M_{\varphi} = M_{\psi}$. In particular, $\psi$ is primitive if and only if $\varphi$ is primitive.
\item Let $\psi$ be a primitive morphism. If $\psi\triangleright\varphi$ and $\mathbf u =\varphi(\mathbf u) = \psi(\mathbf u)$, then $\varphi = \psi$.
\item If an acyclic morphism $\varphi$ acts on binary alphabet $\{0,1\}$, the last letters of $\varphi_R(0)$ and of $\varphi_R(1)$ are distinct. Consequently, if $0u$ and $1u$ occur in a fixed point of $\varphi_R$, then $0\varphi_R(u)$ and $1\varphi_R(u)$ occur in the fixed point as well.
\end{enumerate}
\end{remark}
The following result can be found in \cite[Proposition 2.3.21]{Lo2}.
\begin{theorem} \label{sturmian_conjugate} If $M=\begin{pmatrix}
A&B\\
C&D
\end{pmatrix} \in Sl(\mathbb N,2)$, then $M$ is the incidence matrix of $A+B+C+D-1$ mutually conjugate Sturmian morphisms.
\end{theorem}
\begin{proof} Fix $S \in \{ 0, 1, \ldots, A+B+C+D-2\}$. First we show that there exists a unique pair $(E,F) \in \mathbb N^2$ such that $E+F = S$ and $R= \begin{pmatrix}
A&B&0\\
C&D&0\\
E&F&1
\end{pmatrix} \in \mathcal{E}$. By \Cref{inequalities,1enough} we look for $E \in \mathbb N$ and $F=S-E$ satisfying $-A < A(S-E) -BE \leq B$. Or equivalently, $\frac{AS-B}{A+B}\leq E < \frac{AS+A}{A+B}$. Since the distance between the lower and the upper bounds on $E$ equals 1, exactly one integer $E$, namely $E = \lceil \frac{AS-B}{A+B}\rceil$, satisfies $-A < A(S-E) -BE \leq B$. Let us check that such $E$ and $F:=S-E$ satisfy \eqref{straight}.
Using the definition of $E$ and $A\geq 1$ we obtain
$$ -1<\tfrac{-B}{A+B} \leq\tfrac{AS-B}{A+B}\leq E < \tfrac{AS-B}{A+B} + 1 \leq \tfrac{A(A+B+C+D-2)-B}{A+B}= A+C -\tfrac{A-1}{A+B}\leq A+C.
$$
As $E$ is an integer, the previous inequalities confirm that $0\leq E < A+C$.
To check \eqref{straight} for $F$, we use $F:=S-E$ and write
$$
-1\leq\tfrac{-A}{A+B}\leq \tfrac{SB-A}{A+B}=S - \tfrac{AS-B}{A+B} -1< F\leq S - \tfrac{AS-B}{A+B}=\tfrac{(S+1)B}{A+B}\leq \tfrac{B(A+B+C+D-1)}{A+B} = B+D -\tfrac{1+B}{A+B}.
$$
It confirms, that $0\leq F < B+D$.
By Claim \ref{1enough}, the $E$ and $F$ satisfy \eqref{lower} as well.
By Theorem \ref{rovnost} and the fact that the representation $\mathcal{R}$ of the monoid $\mathcal{M}$ is faithful, there exist in $\mathcal{M}$ exactly $A+B+C+D-1$ Sturmian morphisms having the incidence matrix $ M$. Let us denote these morphisms by $\varphi^{(i)}$ for $i=0, 1, \ldots, A+B+C+D-2$. We need to show that these morphisms are mutually conjugate, or equivalently, to show that the rightmost conjugate $\varphi_R^{(i)}$ does not depend on the index $i$.
If the matrix $M$ is primitive, then the frequencies of letter in a fixed point of $\varphi^{(i)}$ form an eigenvector to the dominant eigenvalue of $M$. Therefore, all fixed points of the $A+B+C+D-1$ morphisms
have the same frequencies of letters, i.e., they have the same slope, say $\alpha$. By Item (3) of Remark \ref{leftspecial}, any prefix of the fixed point of $\varphi_R^{(i)}$ is a left special factor of the fixed point. In other words, the fixed point of $\varphi_R^{(i)}$ is the characteristic sequence ${\bf c}_\alpha$ for every index $i$. By Item (2) of the same remark, $\varphi_R^{(i)} = \varphi_R^{(0)}$ for every $i$.
If the matrix $M$ is not primitive, then either $B=0$ or $C=0$. Assume that $C=0$. It follows that the morphism $\varphi^{(i)}$ is of the form $0\mapsto 0, 1\mapsto 0^i10^{C-i}$, for $i = 0, 1, \ldots, C$. It is obvious that they are mutually conjugate. The case $B=0$ is analogous.
\end{proof}
\section{The square root of fixed point of characteristic Sturmian morphisms}\label{ctverce}
Saari~\cite{Sa} showed that for every Sturmian sequence $\mathbf u$ there exist 6 its
factors $w_1,\ldots,w_6$ such that \begin{equation}\label{squares} \mathbf u = w_{i_1}^2 w_{i_2}^2 w_{i_3}^2
\ldots, \qquad \text{ where $i_k \in
\{1,\ldots,6\}$ for each } k\in\mathbb{N},\end{equation} and moreover, for each $k\in \mathbb N$, the shortest square prefix of the sequence $w_{i_k}^2 w_{i_{k+1}}^2
w_{i_{k+2}}^2\dots$ is $w_{i_{k}}^2$.
This result served as inspiration for J. Peltomäki and M. Whiteland to introduce the square root $\sqrt{\mathbf u}$ of the Sturmian sequence $\mathbf u$ written in the form \eqref{squares} as $$ \sqrt{\mathbf u} = w_{i_1}
w_{i_2} w_{i_3} \ldots$$
In \cite{PeWh}, they also proved the following theorem.
\begin{theorem}\label{pelto} If $\mathbf u$ is a Sturmian sequence with the slope $\alpha$ and the intercept $\delta$, then
the sequence $\sqrt{\mathbf u}$ is a Sturmian sequence with the same slope $\alpha$ and the intercept
$\frac{1-\alpha+\delta}{2}$.
\end{theorem}
\begin{example}\label{priklad}
Let $\varphi\in\mathcal M$ such that $\varphi = DG^2: 0\mapsto 10, 1 \mapsto 10101$.
The fixed point of $\varphi$ can be written as concatenation of the squares of these 6 factors: $10, 1, 0110101, 101, 01, 01101$.
The beginning of this decomposition is as follows:
\begin{align*}
\mathbf u &=
10101101010110101011010110101011010101101010110101101010\dots\\
&=\underbrace{10}_{w_1} \underbrace{10}_{w_1} \underbrace{1}_{w_2}
\underbrace{1}_{w_2} \underbrace{01}_{w_3} \underbrace{01}_{w_3}
\underbrace{0110101}_{w_4} \underbrace{0110101}_{w_4}
\underbrace{10}_{w_1} \underbrace{10}_{w_1}
\underbrace{101}_{w_5} \underbrace{101}_{w_5}
\underbrace{01}_{w_3} \underbrace{01}_{w_3}
\underbrace{10}_{w_1} \underbrace{10}_{w_1}\dots
\end{align*}
Hence, the square root begins with
\begin{align*}
\sqrt{\mathbf u} &=
1010110110101011010101101010110101101010110101011010101101\dots\\
&=\underbrace{10}_{w_1}
\underbrace{1}_{w_2}
\underbrace{01}_{w_3}
\underbrace{0110101}_{w_4}
\underbrace{10}_{w_1}
\underbrace{101}_{w_5}
\underbrace{01}_{w_3}
\underbrace{10}_{w_1}
\underbrace{101}_{w_5}
\underbrace{0110101}_{w_4}
\underbrace{0110101}_{w_4}
\underbrace{10}_{w_1}
\underbrace{101}_{w_5}
\underbrace{01}_{w_3}
\underbrace{1}_{w_2}
\underbrace{10}_{w_1} \dots
\end{align*}
\end{example}
To state our result on the square root of characteristic Sturmian morphisms we need to recall that
a word $w = w_0w_1 \dots w_{n-1}$ is a \emph{palindrome} if it reads the same from the left as from the right, i.e., $w_k = w_{n-1-k}$ for each $k=0, 1, \ldots, n-1$.
\begin{theorem}\label{naseodmocniny} Let $\mathbf u\in\{0,1\}^\mathbb{N}$ be a characteristic Sturmian sequence
fixed by a primitive morphism $\varphi\in\mathcal M$. The square root $\sqrt{\mathbf u}$ is fixed by
a morphism $\psi$ which is a conjugate of one of the morphisms $\varphi,
\varphi^2$ or $\varphi^3$. Moreover, $\psi(0)$ and $\psi(1)$ are
palindromes of odd length. \end{theorem}
\begin{proof} Consider the characteristic Sturmian sequence $\mathbf u = {\bf s}_{\alpha, \alpha}$ fixed by $\varphi$.
It has parameters $\vec{v}(\mathbf u)=
(1-\alpha, \alpha, \alpha)^\top$. According to \Cref{fixed} the
vector $\vec{v}(\mathbf u)$ is an eigenvector of $\mathcal R(\varphi):= \begin{pmatrix}
A & B & 0 \\ C & D & 0 \\ E & F & 1 \end{pmatrix}$. From the relation
$\mathcal R (\varphi)\vec{v}(\mathbf u)=\Lambda \vec{v}(\mathbf u)$ we obtain the equality
$$C(1-\alpha)+D\alpha = \Lambda\alpha = E(1-\alpha)+F\alpha + \alpha.$$
As the values $\alpha$ and $1-\alpha$ are linearly independent over $\mathbb Q$, we conclude that
$E=C$ and $F=D-1$.
Set $P:= \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\
\frac{1}{2} & 0 & \frac{1}{2}\end{pmatrix}$ and $R:=P \mathcal R (\varphi)
P^{-1}$. The following statements hold:
\begin{enumerate}[i)]
\item $(1-\alpha, \alpha,
\frac{1}{2})^\top = P(1-\alpha, \alpha,
\alpha)^{\top}$ is an eigenvector of $R$;
\item by \Cref{pelto}, $(1-\alpha, \alpha,
\frac{1}{2})^\top$ is the vector of parameters of
$\sqrt{\mathbf u}$, or equivalently $\sqrt{\mathbf u} = {\bf s}_{\alpha, \frac12}$.
\end{enumerate}
Consider also the following statement:
\begin{enumerate}[resume*]
\item $R^{k}\in\mathcal E$ for some $k\in\{1,2,3\}$.
\end{enumerate}
\Cref{rovnost} and \Cref{fixed} together with the statements i), ii), and
iii) imply the existence of a morphism $\psi\in\mathcal M$ such that
$\sqrt{\mathbf u}$ is the fixed point of $\psi$ and $\mathcal R(\psi) = R^{k}$.
The form of the matrices $R$ and $P$ implies that the incidence matrix $M_\psi \in \mathbb Z^{2\times 2}$
of the morphism $\psi$ is equal to the matrix $M_\varphi^{k}$.
By \Cref{sturmian_conjugate}, the morphisms $\psi$ and $\varphi^{k}$ are conjugate.
To prove the first part of the theorem, it remains to show iii).
Let $M = M_\varphi =
\begin{pmatrix}
A & B \\ C & D
\end{pmatrix}$.
We have $(E,F) = (C,
D-1) = (0,1)(M - I_2)$.
By direct calculation we obtain
\[
R = P \underbrace{\begin{pmatrix} M & \vec{0} \\ (0,1)(M - I_2) & 1 \end{pmatrix}}_{ \mathcal{R}(\varphi)} P^{-1} = \begin{pmatrix} M & \vec{0} \\
\frac{1}{2}(1,1)(M-I_2) & 1
\end{pmatrix}.
\]
It is easy to show that for each $k\in \mathbb N$ we have
\[
R^{k} = \begin{pmatrix}
M^{k} & \vec{0} \\ e_k^{\top} & 1
\end{pmatrix}, \quad \quad \text{ where }e_k^{\top} = \frac{1}{2}(1,1)(M^{k}-I_2).
\]
Using $M\in\mathbb Z^{2\times2}$ and $\det M = 1$, we deduce that $M \bmod 2$ equals to one of the following 6 matrices:
$$M_1 = \begin{pmatrix}
1 & 0 \\ 0 & 1
\end{pmatrix}, \ \
M_2 = \begin{pmatrix}
0 & 1 \\ 1 & 0
\end{pmatrix}, \quad
M_3 = \begin{pmatrix}
1 & 0 \\ 1 & 1
\end{pmatrix}, \quad
M_4 = \begin{pmatrix}
1 & 1 \\ 0 & 1
\end{pmatrix}, \quad
M_5 = \begin{pmatrix}
1 & 1 \\ 1 & 0
\end{pmatrix}, \quad
M_6 = \begin{pmatrix}
0 & 1 \\ 1 & 1
\end{pmatrix}.$$
By inspection of all six matrices, we find out that for every $ i \in\{1,2,\ldots, 6\}$ there exists $k \in \{1,2,3\}$ such that $2e^\top_k \bmod 2 =(1,1)(M^{k}_i-I_2) \bmod 2 = (0,0)$. Hence for some $k\in\{1,2,3\}$, all entries of $R^{k}$ are integers, i.e., $R^k \in Sl(\mathbb{Z},3)$. We may use \Cref{inequalities} to show that for such $k$ the matrix $R^{k}$ belongs to $\mathcal E$.
Set
$$R^{k}:= \begin{pmatrix} \widetilde{A} & \widetilde{B}& 0 \\ \widetilde{C} &
\widetilde{D} & 0 \\ \widetilde{E} & \widetilde{F} & 1 \end{pmatrix},\quad \text{where}
\ \ \begin{pmatrix}
\widetilde{A} & \widetilde{B} \\ \widetilde{C} & \widetilde{D}
\end{pmatrix} :=M^{k} \quad \text{and} \ \
(\widetilde{E}, \widetilde{F}) := e_k^{\top} = \frac{1}{2}(1,1)(M^{k}-I_2). $$ It follows that
\begin{equation}\label{EF}\widetilde{E}=\frac{1}{2}(\widetilde{A}+\widetilde{C}-1)\ \ \text{ and}\ \
\widetilde{F}=\frac{1}{2}(\widetilde{B}+\widetilde{D}-1).\end{equation}
It remains to verify the following statements.
\begin{enumerate}[1)]
\item $\widetilde A \widetilde D - \widetilde B \widetilde C = 1$;
\item $\widetilde{E}<\widetilde{A}+\widetilde{C}$ and $\widetilde{F}<\widetilde{B}+\widetilde{D}$;
\item $-\widetilde C \leq \widetilde C \widetilde F - \widetilde D \widetilde E < \widetilde D$.
\end{enumerate}
The statement 1) is equivalent to $\det R^{k}=1$, which holds since $\det M^k = (\det M)^k = 1$.
The statement 2) follows from \eqref{EF}.
Since $\widetilde C$ and $\widetilde D$ are positive and cannot be simultaneously $0$,
we obtain $-2\widetilde C \leq \widetilde D - \widetilde C - 1 < 2\widetilde D$.
Using \eqref{EF}, we see that this is equivalent to 3):
\[
\widetilde C \widetilde F - \widetilde D \widetilde E
= \frac{1}{2} \left ( \widetilde D - \widetilde C - 1\right).
\]
We conclude that $R^k \in \mathcal E$.
\medskip
The second part of the theorem states that lengths of $\psi(0)$ and $\psi(1)$ are odd. Indeed, by \eqref{EF} and the fact that $M^{k}$ is the incidence matrix of the morphism $\psi$, we have
$$|\psi(0)|=\widetilde{A}+\widetilde{C} = 2 \widetilde{E} +1 \quad \text{ and } \quad |\psi(1)|=\widetilde{B}+\widetilde{D}= 2\widetilde{F}+1. $$
Finally let us show that $\psi(0)$ and $\psi(1)$ are palindromes. Let us recall
that $\sqrt{\mathbf u}={\bf s}_{\alpha,\frac{1}{2}}$. Consider the biinfinite sequence
${\bf s}_{\alpha,\frac{1}{2}} = \dots
\nu_{-3}\nu_{-2}\nu_{-1}\nu_{0}\nu_{1}\nu_{2}\dots$. Let us deduce that its left part
$ \dots
\nu_{-3}\nu_{-2}\nu_{-1}$ is the mirror image of the right part $ \nu_{0}\nu_{1}\nu_{2}\dots$.
As ${\bf s}_{\alpha,\frac{1}{2}}(n) = \lfloor\alpha(n+1) +\frac{1}{2}\rfloor-\lfloor
\alpha n +\frac{1}{2}\rfloor$ for each $n\in\mathbb Z$, we have
${\bf s}_{\alpha,\frac{1}{2}}(-n-1)=\lfloor\alpha(-n) +\frac{1}{2}\rfloor-\lfloor
\alpha (-n-1) +\frac{1}{2}\rfloor$. Using the relations
$\lfloor -x \rfloor = -\lfloor x \rfloor - 1$
and $\lfloor x + 1 \rfloor = \lfloor x \rfloor + 1$ for $x\notin\mathbb Z$
we get that ${\bf s}_{\alpha,\frac{1}{2}}(-n-1) = {\bf s}_{\alpha,\frac{1}{2}}(n)$
for each $n\in\mathbb Z$. It confirms the mirror symmetry. In particular we have that $\nu_{-1}=\nu_{0}$ and hence $\psi(\nu_0)=\psi(\nu_{-1})$.
Now we used the result of \cite{BaMaPe}, where the authors proved that a biinfinite Sturmian ${\bf s}_{\alpha, \delta}$ is fixed by a primitive morphism $\psi \in \mathcal{M}$ if and only if its right part is fixed by $\psi$.
The symmetry
of $\dots\nu_{-3}\nu_{-2}\nu_{-1} = \dots\psi(\nu_{-3})\psi(\nu_{-2})\psi(\nu_{-1})$ and $\nu_0\nu_1\nu_2 \dots= \psi(\nu_{0})\psi(\nu_{1})\psi(\nu_{2})\dots$ gives
that the mirror image of $\psi(\nu_{-1})$ is equal to $\psi(\nu_0)$.
In other words $\psi(\nu_0)$ is a palindrome. The mirror symmetry also
gives that the image under $\psi$ of the letter other than $\nu_0$
is a palindrome as well.
\end{proof}
\begin{example} (continuation of Example \ref{priklad})\ \
We observe that no morphism conjugate to $\varphi = DG^2$ creates a palindromic
image of $0$ and therefore the morphism $\psi$ fixing $\sqrt{\mathbf u}$ is not conjugate to $\varphi$.
However, $\varphi^{2}$ has a conjugate, namely
$$\psi:0\mapsto 1010101 , 1\mapsto 1010101101011010101, $$
such that both $\psi(0)$ and $\psi(1)$ are palindromes.
The prefix of $\sqrt{\mathbf u}$ displayed in Example \ref{priklad} illustrates that $\sqrt{\mathbf u}$ is fixed by $\psi$.
\end{example}
\section{Comments}
Let us list some open questions where the faithful representation $\mathcal{R}$ might be useful.
\begin{itemize}
\item \Cref{naseodmocniny} deals only with characteristic Sturmian sequences.
Yasutomi's theorem implies that if a Sturmian sequence $\mathbf u$ is fixed by a primitive morphism, then $\sqrt{\mathbf u}$ is fixed by a primitive morphism as well.
Hence, \Cref{naseodmocniny} could possible be generalized to deal with fixed points of general Sturmian morphisms.
\item Conjugate Sturmian morphisms can be ordered with respect to the relation $\triangleright$ from the leftmost conjugate morphism to the rightmost conjugate.
The representation $\mathcal{R}$ might be useful in order to describe the matrices of two subsequent elements in this chain.
\item \Cref{Yasutomi_necessary} is only the necessary condition of Yasutomi's theorem.
An open question is how to use $\mathcal{R}$ to provide a full (simpler) proof of Yasutomi's theorem.
\end{itemize}
The exhibited faithful representation relies on the geometric representation of Sturmian sequences. A natural question is finding a faithful representation of some other monoids of morphisms that fix some other families of infinite sequences, e.g., Arnoux--Rauzy sequences and sequences coding symmetric $k$-interval exchange transformations.
Let us point out that a geometrical representation of ternary Arnoux-Rauzy sequences can be found already in the article \cite{ArRo}.
Recently, a representation of Arnoux--Rauzy sequences using a generalized Ostrowski numeration system is introduced in \cite{Pelto21}.
The sequences coding $3$-interval exchange can be viewed as sequences obtained by cut-and-project sets, see \cite{MaPeSta1}.
\section*{Acknowledgements}
Jana Lep\v{s}ov\'{a} acknowledges
financial support by The French Institute in Prague and the Czech Ministry of Education, Youth and Sports through the Barrande fellowship programme
and Agence Nationale de la Recherche through the project Codys (ANR-18-CE40-0007). The research was supported by Grant Agency of Czech technical university in Prague, through the project SGS20/183/OHK4/3T/14.
Edita Pelantov\'{a} acknowledges financial support by The Ministry of Education, Youth and Sports of the Czech Republic, project no. {CZ.02.1.01/0.0/0.0/16\_019/0000778}.
\v{S}t\v{e}p\'{a}n Starosta acknowledges the support of the OP VVV MEYS funded
project {CZ.02.1.01/0.0/0.0/16\_019/0000765}.
\bibliographystyle{siam}
\IfFileExists{biblio.bib}{ |
1,314,259,996,468 | arxiv | \section{Introduction}
\label{introduction}
The deeper nature of dark matter (DM) is unknown. While the observational evidence for the existence of DM is overwhelming, its possible connection to particle physics remains poorly understood. As of today, the experimental searches of particle DM have yielded only null results \cite{Boveia:2018yeb,Hooper:2018kfv,Schumann:2019eaa}, either constraining or ruling out various models or even model paradigms of particle DM. In particular, the usual freeze-out paradigm appears less and less likely to explain the origins of DM \cite{Arcadi:2017kky}.
As the only evidence for DM are obtained through its gravitational effects -- its imprints on the Cosmic Microwave Background (CMB) and the large scale structure of the Universe, gravitational lensing and dynamics of galaxy clusters, rotational velocity curves of individual galaxies and so on --, we are motivated to ask: what if dark matter couples to the Standard Model (SM) particles only via gravity? It is clear that observationally this is not a problem, and simple and appealing mechanisms for the generation of DM have been found too. Indeed, the observed DM abundance may have been initiated purely gravitationally in the early Universe, either during or right after cosmic inflation. This idea dates back to 1980s \cite{Ford:1986sy,Turner:1987vd} and it has been gaining increasing attention recently, see e.g. Refs. \cite{Enqvist:2014zqa,Graham:2015rva,Nurmi:2015ema,Garny:2015sjg,Markkanen:2015xuw,Kainulainen:2016vzv,Bertolami:2016ywc,Heikinheimo:2016yds,Cosme:2017cxk,Enqvist:2017kzh,Cosme:2018nly,Graham:2018jyp,Alonso-Alvarez:2018tus,Ema:2018ucl,Fairbairn:2018bsw,Markkanen:2018gcw,Cosme:2018wfh,Guth:2018hsa,Ho:2019ayl,Padilla:2019fju,Tenkanen:2019aij,Tenkanen:2019wsd,AlonsoAlvarez:2019cgw,Ema:2019yrd,Laulumaa:2020pqi,Ahmed:2020fhc,Karam:2020rpa} for recent studies on gravitational DM production in different contexts.
In this paper, we will focus on a particular scenario where DM is gravitationally produced: the so-called spectator dark matter scenario, where the DM is generated by amplification of vacuum fluctuations of an energetically subdominant scalar field during inflation. The starting point is well-motivated, as weakly coupled scalar fields are typically abundant in extensions of the SM \cite{Arvanitaki:2009fg,Marsh:2015xka,Stott:2017hvl} and their dynamics during inflation is expected to provide the generic initial conditions for non-thermal production of DM after inflation \cite{Enqvist:2014zqa} -- unless the scalar field(s) themselves constitute all or part of the observed DM abundance. Previously, spectator DM scenarios have been considered in the case of free \cite{Turner:1987vd,Tenkanen:2019aij,Tenkanen:2019wsd}, self-interacting \cite{Peebles:1999fz,Nurmi:2015ema,Kainulainen:2016vzv,Heikinheimo:2016yds,Enqvist:2017kzh,Markkanen:2018gcw,Padilla:2019fju}, and non-minimally coupled cases \cite{Cosme:2017cxk,Cosme:2018nly,Cosme:2018wfh,Alonso-Alvarez:2018tus,Fairbairn:2018bsw,AlonsoAlvarez:2019cgw,Laulumaa:2020pqi}, as well as in scenarios where the DM is coupled to the inflaton \cite{Bertolami:2016ywc}, or is axion-like \cite{Beltran:2006sq,Graham:2018jyp,Guth:2018hsa,Ho:2019ayl}, but -- to the best of our knowledge -- never in the context of general non-standard (i.e. non-radiation-dominated) expansion after inflation.
In this paper we study the robustness of the spectator dark matter scenario studied in Refs. \cite{Markkanen:2018gcw,Tenkanen:2019aij} to changes in the early Universe's expansion rate. In particular, we concentrate on scenarios where the total energy density after inflation was dominated by a perfect fluid other than radiation, for example massive meta-stable particles or a fast-rolling scalar field, or where the non-radiation-dominated expansion was caused by e.g. a period of slow reheating after inflation. All these possibilities are well-motivated; for an extensive review of such scenarios, see Ref. \cite{Allahverdi:2020bys}. Compared to the standard radiation-dominated (RD) scenario, in the context of non-standard expansion two aspects in the spectator DM model will be different: the DM energy density will evolve differently as a function of time, and also the dependence of the DM perturbation spectrum on the initial spectator field value will be different from the result in the usual RD case.
These can impose sizeable changes to the values of model parameters which allow the field to constitute all DM while simultaneously satisfying all observational constraints. Here we study both free and self-interacting DM within a non-standard expansion and quantify the changes to the cases with a standard cosmological history. We will also discuss testability of the scenario through primordial DM isocurvature and non-Gaussianity, highlighting the fact that even though DM may couple to ordinary matter only via gravity, it does not mean that the scenario would not be testable.
The paper is organized as follows: in Sec. \ref{inflation}, we present the model and discuss the field dynamics during inflation, whereas Sec. \ref{dynamics} is devoted for the dynamics after inflation. In Sec. \ref{perturbations}, we compute the DM perturbation spectrum and investigate how the scenario can be contrasted with CMB observations. In Sec. \ref{results}, we present our results and discuss testability of the model. Finally, in Sec. \ref{conclusions}, we conclude with a brief outlook.
\section{Scalar field evolution during inflation}
\label{inflation}
We begin by reviewing the standard treatment for the evolution of a light scalar field during inflation. For dark matter, we consider the Lagrangian
\begin{equation}
\label{lagrangian}
\mathcal{L}_\chi=\frac12\partial^\mu\chi\partial_\mu\chi - V(\chi)\,,
\end{equation}
where
\begin{equation}
\label{potential}
V(\chi) = \frac{1}{2}m^2 \chi^2 + \frac{\lambda}{4}\chi^4\,,
\end{equation}
where $\chi$ is a real scalar field which we assume was an energetically subdominant spectator field during inflation, $m$ is its mass and $\lambda$ is a quartic self-interactions coupling. We assume that the theory is defined in a frame where the field that drives inflation, the inflaton field\footnote{The inflaton field could be, for example, the SM Higgs field \cite{Bezrukov:2007ep,Bauer:2008zj}. However, here we remain agnostic of the inflaton sector and its couplings to the SM fields and/or gravity; for reviews of Higgs-like inflation, see Refs. \cite{Rubio:2018ogq,Tenkanen:2020dge}. Likewise, we assume here that the possible couplings between the field $\chi$ and the inflaton/Higgs field are negligible and do not affect the field dynamics either during or after inflation (by e.g. rendering the $\chi$ field heavy during inflation), nor contribute to the final DM yield through any mechanism, such as production of $\chi$ quanta during reheating. For studies where these assumptions are relaxed, see e.g. Refs. \cite{Berlin:2016vnh,Adshead:2016xxj,Tenkanen:2016jic,Berlin:2016gtr}. For recent studies on scenarios where the DM field couples non-minimally to gravity, see e.g. Refs. \cite{Cosme:2017cxk,Cosme:2018nly,Cosme:2018wfh,Alonso-Alvarez:2018tus,AlonsoAlvarez:2019cgw,Laulumaa:2020pqi}.}, couples minimally to gravity and where the background during inflation space-time is well approximated by that of de Sitter, i.e. the Hubble scale during inflation is approximately constant, $H= \dot{a}/a \simeq H_{\rm inf}$. This is well motivated, as in slow-roll models of inflation that provide the best fit to data (such as plateau models, see e.g. Ref. \cite{Chowdhury:2019otk}) it is usually a very good approximation that the Hubble rate did not decrease much during inflation. Therefore, we will maintain this assumption throughout the paper. Furthermore, we assume that the inflaton field sources a major part of the primordial curvature perturbations, which eventually lead to the observed temperature fluctuations in the CMB.
Assuming that the effective mass of the spectator field $\chi$ was smaller than the Hubble rate during inflation, $V''<9H_{\rm inf}^2/4$ where the prime denotes derivative with respect to the field, it received quantum fluctuations from the rapidly expanding background. Using the stochastic approach \cite{Starobinsky:1986fx,Starobinsky:1994bd} (see also Refs. \cite{Kunimitsu:2012xx,Tokuda:2017fdh,Hardwick:2018sck,Tokuda:2018eqs,Hardwick:2019uex,Markkanen:2019kpv,Gorbenko:2019rza,Mirbabayi:2019qtx,Moreau:2019jpn,Baumgart:2019clc,Adshead:2020ijf,Markkanen:2020bfc,Bounakis:2020jdx,Moreau:2020gib,Pinol:2020cdp} for recent works), we find that the long wavelength modes of the field evolve according to the Langevin equation
\begin{equation}
\dot{\chi}(\bar{x},t) + \frac{1}{3H_{\rm inf}}V'(\chi) = f(\bar{x},t)\,,
\end{equation}
where $f(\bar{x},t)$ is a Gaussian noise term with
\begin{equation}
\langle f(\bar{x}_1,t_1)f(\bar{x}_2,t_2)\rangle = \frac{H_{\rm inf}^3}{4\pi^2}\delta(t_1-t_2)\,,
\end{equation}
and the point $\bar{x}$ is to be understood as a patch slightly larger than the Hubble volume during inflation, i.e. the field is coarse-grained over the Hubble horizon. Using standard techniques, one can turn the Langevin equation for the field into a Fokker-Planck equation for the one-point probability distribution $P(\chi(\bar{x},t))$, which reads
\begin{equation}
\frac{\partial P(\chi(\bar{x},t))}{\partial t} = D_\chi P(\chi(\bar{x},t))\,,
\end{equation}
where $D_\chi$ is the differential operator
\begin{equation}
D_\chi \equiv \frac{V''(\chi)}{3H_{\rm inf}} + \frac{V'(\chi)}{3H_{\rm inf}}\frac{\partial}{\partial\chi} + \frac{H_{\rm inf}^3}{8\pi^2}\frac{\partial^2}{\partial\chi^2}\,.
\end{equation}
One can show that there is an equilibrium solution for the one-point distribution function, which is given by
\begin{equation}
\label{eq:p}
P(\chi) =C \exp\left(-\frac{8\pi^2}{3H_{\rm inf}^4}V(\chi)\right)\,,
\end{equation}
where $C$ is a normalization factor ensuring total probability of unity. This distribution describes the ensemble of field values in patches the size of a region slightly larger than the Hubble horizon at the end of inflation, see Fig. \ref{regions}. Notably, the scalar field reaches this ``equilibrium state" in a characteristic time scale regardless of its initial distribution, after which the distribution does not evolve anymore. This is depicted in Fig. \ref{distributions}. The relaxation time scale in terms of inflationary e-folds $N\equiv \ln(a/a_0)$, where $a_0$ is the scale factor at some reference time during inflation when the field had the value $\chi_0$ over a Hubble volume, is \cite{Enqvist:2012xn}
\begin{equation}
\label{Nrel}
N_{\rm rel} \simeq
\begin{cases}
\displaystyle 11.3/\sqrt{\lambda} &\quad \lambda\chi^2 \gg m^2
,\\
\displaystyle 3H_{\rm inf}^2/m^2 & \quad \lambda\chi^2 \ll m^2\,,
\end{cases}
\end{equation}
depending on which term in Eq. \eqref{potential} dominates the spectator potential during inflation.
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.75\textwidth]{patches.png}
\par\end{centering}
\caption{When a scalar field $\chi$ is light during inflation, it acquires fluctuations which get it displaced from its initial value $\chi_0$. As inflation proceeds, the (coarse-grained) scalar field performs random walk, and the Universe ends up having an ensemble of Hubble volumes, in each of which the field has a value ($\chi_{1}$,$\chi_{2}$,...) that generically differs from the average value due to the random fluctuations. The final distribution of values $P(\chi)$ is given by Eq. \eqref{eq:p}. See also Fig. \ref{distributions}.
}
\label{regions}
\end{figure}
\begin{figure}[H]
\label{distributions}
\begin{centering}
\includegraphics[width=0.475\textwidth]{potentials_w_distributions}
\includegraphics[width=0.475\textwidth]{relaxation}
\par\end{centering}
\caption{{\it Left panel}: Equilibrium distributions $P(\chi)$ in the quartic and quadratic cases (blue and orange thick curves, respectively) with the respective potentials shown in the background (blue dashed and orange dot-dashed curves, respectively). In this figure $m^2=0.2H_{\rm inf}^2$ and $\lambda = 0.1$. {\it Right panel}: Example of the relaxation of the (unnormalized) distribution function during inflation, here assuming a narrow Gaussian initial distribution and a quadratic potential with $m^2=0.2H_{\rm inf}^2$ (shown by the dashed curve). Shown to the left of each curve is the number of e-folds elapsed since the initial state. In roughly $50$ e-folds, the distribution reaches the equilibrium state given by Eq. \eqref{eq:p}.
}
\end{figure}
Therefore, by assuming inflation lasted for long enough for the field to reach the equilibrium state\footnote{As can be seen from Eq. \eqref{Nrel}, for large enough $\lambda$ or $m/H_{\rm inf}$ the equilibrium state can be reached even within the final $\sim 60$ e-folds of inflation, i.e. between the time when the mode corresponding to our currently observable Universe exited the horizon and the end of inflation. After the equilibrium is reached, all information of the initial conditions has been erased. For scenarios where the final distribution of field values carries information of the initial state, see Refs. \cite{Hardwick:2017fjo,Torrado:2017qtr}.}, the typical $\chi$ value at the end of inflation is given by the variance of \eqref{eq:p} as
\begin{equation}
\label{variance}
\langle{\chi}_{\rm end}^2\rangle =
\begin{cases}
\displaystyle\sqrt{\frac{3}{2\pi^2}}{\frac{\Gamma(\frac{3}{4})}
{\Gamma(\frac{1}{4})}} \frac{H_{\rm inf}^2}{\sqrt{\lambda}}\approx 0.132 \frac{H_{\rm inf}^2}{\sqrt{\lambda}}\,,& \quad \lambda\chi^2 \gg m^2 \,,
\\
\displaystyle \frac{3}{8\pi^2}\frac{H_{\rm inf}^4}{m^2}\,, & \quad \lambda\chi^2 \ll m^2\,.
\end{cases}
\end{equation}
It should be noted, however, that while the variance $\langle{\chi}_{\rm end}^2\rangle $ describes the {\it typical} field value, there can be large variations in the field value in different Hubble patches at the time of photon decoupling. These constitute potentially dangerous DM isocurvature perturbations, which provide the most stringent constraints on our scenario (and also, on the flip side, the best potential for testability, as we will discuss). Before discussing the scalar field's perturbation spectrum, however, we will discuss the post-inflationary dynamics of the field and the dark matter production.
\section{Dynamics after inflation}
\label{dynamics}
\subsection{Background dynamics}
\label{bg_dyn}
As Eq. \eqref{variance} shows, in a typical situation the field is displaced away from its potential and has a finite initial value $\chi_{\rm end}\equiv \chi(\bar{x},t_{\rm end})$ over a patch slightly larger than the size of the Hubble horizon at the end of inflation, where $t_{\rm end}$ denotes the end of inflation. The equation of motion for the field describing its post-inflationary dynamics thus is\footnote{Here we neglect the gradient term, as the length scale over which the field values are correlated is typically much larger than the Hubble horizon at the end of inflation \cite{Kunimitsu:2012xx}.}
\begin{equation}
\label{eom}
\ddot{\chi} + 3H(t)\dot{\chi} + V'(\chi) = 0,
\end{equation}
where the dots denote derivatives with respect to cosmic time $t$, and
\begin{equation}
\label{Hubble}
H(t) = \frac{H_{\rm inf}}{\left(1+\frac{3(1+w)}{2}H_{\rm inf}t\right)} \simeq \frac{2}{3(1+w)}\frac{1}{t}\,,
\end{equation}
where the latter result applies shortly after inflation. Here $w\equiv p/\rho$ is the (time-averaged) equation of state parameter for the background with the energy density $\rho$ and pressure $p$. In the following, we will consider three cases: the usual radiation-dominated ($w=1/3$, $\rho\propto a^{-4}$), matter-dominated ($w=0$, $\rho\propto a^{-3}$) and kination-dominated ($w=1$, $\rho\propto a^{-6}$) Universe, so that after inflation
\begin{equation}
\label{Hscaling}
H \propto
\begin{cases}
\displaystyle a^{-3/2}, & \quad w=0, \\
\displaystyle a^{-2}, & \quad w=1/3, \\
\displaystyle a^{-3}, & \quad w=1\,.
\end{cases}
\end{equation}
If reheating is prompt, the Universe quickly becomes radiation dominated, $w=1/3$. On the other hand, an early matter-dominated epoch could arise due to e.g. slow post-inflationary reheating or massive metastable particles that began to dominate the total energy density at some early stage prior to Big Bang Nucleosynthesis (BBN); see Ref. \cite{Allahverdi:2020bys} for a recent review. Finally, scenarios with $1/3<w<1$ are encountered in models where the total energy density of the Universe is dominated by the kinetic energy of a scalar field, either through oscillations in a steep potential, e.g. $V(\phi) \propto \phi^p$ with $p>4$, or by an abrupt drop in the scalar potential in the direction of this field \cite{Turner}. The latter possibility is exactly what happens in e.g. the case of quintessential inflation \cite{Peebles:1998qn}, where the inflaton field makes a transition from potential energy domination to kinetic energy domination at the end of inflation, reaching values of $w$ close to unity. The bound $w\leq 1$ comes from the requirement that the sound speed of the dominant fluid does not exceed the speed of light. Potentially more exotic scenarios that change the expansion history of the Universe compared to the standard radiation-dominated case could also be realized, see e.g. Refs. \cite{Kasuya:2001pr,Boyle:2001du,Bento:2002ps,Cline:1999ts,Takahashi:2020car}. Here, however, we only consider the three more conservative cases above.
\subsection{Quadratic case}
\label{sec:quadratic}
Let us begin by discussing the simplest possible case where the bare mass term dominates the spectator potential both during and after inflation, $m^2 \gg \lambda\chi_{\rm end}^2$. The solution to the equation of motion \eqref{eom} is then given by
\begin{equation}
\label{phi_solution}
\chi(t) = \chi_{\rm end}\times
\begin{cases}
\displaystyle \frac{{\rm sin}(mt)}{mt}, & \quad w=0, \\
\displaystyle 2^{1/4}\Gamma\left(\frac{5}{4}\right)\frac{J_{1/4}(mt)}{\left(mt\right)^{1/4}}, & \quad w=1/3, \\
\displaystyle J_{0}(mt), & \quad w=1,
\end{cases}
\end{equation}
where $J_\nu$ is the Bessel function of rank $\nu$. This result agrees well with the usual assumption that the field starts to oscillate roughly when $H(t)\simeq m$. At late times, $mt\gg 1$, the solutions \eqref{phi_solution} oscillate rapidly with an amplitude
\begin{equation}
\label{chi0}
\chi_0(t) \propto
\begin{cases}
\displaystyle (mt)^{-1}, & \quad w=0, \\
\displaystyle (mt)^{-3/4}, & \quad w=1/3, \\
\displaystyle (mt)^{-1/2} & \quad w=1,
\end{cases}
\end{equation}
and therefore in all cases the field has the associated energy density
\begin{equation}
\label{rho_chi_scaling}
\rho_\chi = \frac12m^2\chi_0^2 \propto a^{-3} ,
\end{equation}
as can be verified by inspection of Eqs. \eqref{Hubble}, \eqref{Hscaling} and \eqref{chi0}. Therefore, regardless of the background scaling, in this case the $\chi$ field constitutes an effective cold dark matter (CDM) component from the moment it starts oscillating.
The cosmic history of our model is depicted in Fig. \ref{history_quad}.
As discussed above, the amplitude of the field remains frozen from the end of inflation
(which we denote by $a_{\mathrm{end}}$) roughly until its mass exceeds the Hubble parameter at $a_{\mathrm{osc}}$. At this moment, the field starts to oscillate and constitutes a DM component. At $a_{\mathrm{reh}}$, the $w$-dominated phase ends and the usual radiation-dominated era takes over the evolution of the Universe. In all cases considered in this paper, we assume that the dominant energy component causing the non-standard era decays only into radiation and does not affect the DM yield.
\begin{figure}[H]
\begin{centering}
\includegraphics[scale=0.55]{Cosmic_history_quad}
\par\end{centering}
\caption{Cosmological evolution of the Universe in our model, in the case where the bare mass term dominates the spectator potential both during and after inflation.}
\label{history_quad}
\end{figure}
Thus, we can write the late time energy density of the field as
\begin{eqnarray}
\label{chiscaling}
\rho_\chi(a) &=& \rho_\chi(a_{\rm end})\left(\frac{a_{\rm osc}}{a_{\rm reh}}\right)^3\left(\frac{a_{\rm reh}}{a}\right)^3 \\ \nonumber
&=&\rho_\chi(a_{\rm end})\left(\frac{a_{\rm osc}}{a_{\rm end}}\right)^3\left(\frac{a_{\rm end}}{a_{\rm reh}}\right)^3\left(\frac{a_{\rm reh}}{a}\right)^3\,,
\end{eqnarray}
where the latter form is easier to evaluate, as
\begin{equation}
\left(\frac{a_{\rm osc}}{a_{\rm end}}\right)^3 = k^{\frac{4}{3(1+w)}}\left(\frac{H_{\rm inf}}{m}\right)^{\frac{2}{1+w}}\,,
\label{aosc aend}
\end{equation}
where $k\simeq 2.1$ is a factor that accounts for the fact that the oscillations do not start exactly when $H(t)=m$ and which we have evaluated by solving Eq. \eqref{eom} numerically,
\begin{equation}
\left(\frac{a_{\rm end}}{a_{\rm reh}}\right)^3 = \left(\frac{\rho_{\rm reh}}{\rho_{\rm end}}\right)^{\frac{1}{1+w}} = \left(\frac{\pi^2 g_*(T_{\rm reh})}{90}\right)^{\frac{1}{1+w}} \left(\frac{T_{\rm reh}^2}{H_{\rm inf}M_{\rm P}}\right)^{\frac{2}{1+w}}\,,
\label{aend areh}
\end{equation}
where $\rho_{\rm end}$ and $\rho_{\rm reh}$ correspond to the total energy density of the Universe at the given times, $T_{\rm reh}$ is the radiation temperature at the time when the non-standard expansion phase ended\footnote{We assume that the decay of the dominant energy density component causing the $w$-dominated era and the subsequent thermalization of SM particles were instantaneous, so that $\rho_{\rm reh}=\pi^2/30g_*T_{\rm reh}^4$. These are both fairly safe assumptions, as earlier studies have found that in terms of scaling of the background energy density, the transition from the $w$-domination to the usual radiation domination is very quick \cite{Berlin:2016gtr,Tenkanen:2016jic,Bernal:2018kcw} and once the dominant energy density component has decayed, the SM particles generically thermalize and build up a heat bath in much less than one e-fold from their production \cite{McDonough:2020tqq}.} and $g_*$ is the corresponding effective number of degrees of freedom, and $M_{\rm P}$ is the reduced Planck mass; and
\begin{equation}
\label{areh_a}
\left(\frac{a_{\rm reh}}{a}\right)^3 = \frac{g_{*S}(T)}{g_{*S}(T_{\rm reh})}\left(\frac{T}{T_{\rm reh}}\right)^3\,,
\end{equation}
which follows from the fact that entropy is conserved after reheating. In all cases, we assume that the oscillations began before reheating, $a_{\rm osc}<a_{\rm reh}$, which amounts to requiring
\begin{equation}
\label{masscondition}
m > k^{\frac{2}{3}}\sqrt{\frac{\pi^2g_*(T_{\rm reh})}{90}}\frac{T^2_{\rm reh}}{M_{\rm P}}\,,
\end{equation}
independently of the inflationary scale $H_{\rm inf}$.
Thus, by substituting Eqs. \eqref{aosc aend}, \eqref{aend areh}, and \eqref{areh_a} into Eq. \eqref{chiscaling}, we find the present-day DM energy density
\begin{equation}
\label{DMabundance}
\Omega_\chi h^2 = \frac{k^{\frac{4}{3(1+w)}}}{2}\frac{g_{*S}(T_0)}{g_{*S}(T_{\rm reh})}\left(\frac{\pi^2g_*(T_{\rm reh})}{90}\right)^{\frac{1}{1+w}}\left(\frac{T_{\rm reh}^2}{mM_{\rm P}}\right)^{\frac{2}{1+w}}\left(\frac{T_0}{T_{\rm reh}}\right)^3\frac{m^2\chi_{\rm end}^2}{\rho_c/h^2}\,,
\end{equation}
where $T_0=2.725$ K is the present-day CMB temperature and $\rho_c/h^2 = 8.09\times 10^{-47}\, {\rm GeV}^4$ is the critical density. For suitable choices of the parameters $m\,,w\,,T_{\rm reh}$ and $\chi_{\rm end}$, the field can constitute all of the observed DM abundance, $\Omega_\chi h^2 =0.12$ \cite{Akrami:2018odb}, which in this case is produced by random fluctuations of the $\chi$ field during cosmic inflation. Note that while the local DM density is seemingly independent of the inflationary scale $H_{\rm inf}$, the typical field value (and therefore the typical DM density) is given by the variance of the field's fluctuation distribution, Eq. \eqref{variance}, which is determined by $H_{\rm inf}$. We can therefore use that equation for $\chi_{\rm end}^2$ to find the typical DM density. However, as we will show in Sec. \ref{perturbations}, maintaining the local field value $\chi_{\rm end}$ in \eqref{DMabundance} is crucial for determining the DM perturbation spectrum and therefore also in assessing the viability of the model. Finally, we note that by setting $w=1/3$, we find the result first obtained in Ref. \cite{Tenkanen:2019aij} modulo a factor 4 which was missing from Ref. \cite{Tenkanen:2019aij} but which has now been included.
\subsection{Quartic case}
\label{sec:quartic}
Let us then consider the case where both during and right after inflation the scalar field's potential was dominated by the quartic term
\begin{equation}
V\left(\chi\right)\simeq\frac{\lambda}{4}\,\chi^{4}\,,
\end{equation}
and $m^2 \ll \lambda \chi_{\rm end}^2$. As shown in Appendix \ref{appendix}, in this case the scalar field equation of motion \eqref{eom} can be expressed in terms of conformal time ${\rm d}\eta ={\rm d}t/a$ as
\begin{equation}
\label{rescaled_eom}
z'' + F(\eta, w)z + z^3 = 0\,,
\end{equation}
where $z\equiv a\sqrt{\lambda}\chi$ is the rescaled field and $F(\eta, w)$ is given by Eq. \eqref{Fterm}. As shown in the Appendix, when $w=1/3$, Eq. (\ref{rescaled_eom}) reduces to
\begin{equation}
\label{quartic_RD_eom}
z'' + z^3 = 0\,,
\end{equation}
whose solution is a well-known oscillating function: the elliptic (Jacobi) cosine function, whose exact form can be found analytically (see e.g. Refs. \cite{Ichikawa:2008ne,Kainulainen:2016vzv}) and which, besides the oscillations, has no further time-dependence in terms of $\eta$. Because $\chi \propto z/a$, this means that when the background energy density is radiation-dominated, the oscillation amplitude decays simply as $\chi \propto 1/a$, and the spectator field behaves as dark radiation. Furthermore, as discussed in Appendix \ref{appendix}, the $F$-term in Eq. \eqref{rescaled_eom} dies off very quickly regardless of $w$, and the spectator field's equation of motion always reduces to \eqref{quartic_RD_eom}. Thus, in all cases we retain the usual $\chi \propto 1/a$ scaling, which validates the following treatment of the scalar field energy density.
\subsubsection{Coherent oscillations}
As in the quadratic case discussed in Sec. \ref{sec:quadratic}, the field is in an overdamped regime roughly until its effective mass $V''(\chi) = 3\lambda\chi^{2}$ exceeds the Hubble parameter, after which $\chi$ starts to oscillate about its origin. We denote this moment by $a_{\mathrm{osc,r}}$, as the amplitude of the scalar field decays with $a^{-1}$ and the field constitutes a dark radiation component. At a time which we denote by $a_{\mathrm{osc,m}}$, the oscillation amplitude has decreased enough so that the quadratic term of the potential starts to dominate over the quartic one, and the field starts to behave as cold dark matter. For simplicity, we use the standard approximation where the energy density of $\chi$ instantaneously changes from scaling as $\rho_{\chi}\propto a^{-4}$ to $\rho_{\chi}\propto a^{-3}$ as soon as the quadratic term dominates, and assume in this subsection that the scalar field oscillations remained coherent throughout the above phases. The remaining of the cosmological history proceeds as in the scenario we studied in Sec. \ref{sec:quadratic}: the background field that is dominating the evolution of the Universe
decays at $a_{\mathrm{reh}}$ and the Universe enters into the usual radiation-dominated era, followed by a period of matter domination until the late-time dark energy domination finally takes over. The cosmic history is illustrated in Fig. \ref{fig:history}.
\begin{figure}[H]
\begin{centering}
\includegraphics[scale=0.55]{Cosmic_history_4}
\includegraphics[scale=0.55]{reheating_before2}
\par\end{centering}
\caption{{\it Upper panel}: Cosmological evolution of the Universe in a scenario where the $\chi$ field behaves like dark radiation at early times after inflation and starts to behave like CDM before reheating.
{\it Lower panel}: Same as above but in this scenario, reheating occurs before the field starts to behave like CDM. The final DM abundance is the same in both cases.
\label{fig:history}}
\end{figure}
Regardless of when the scalar field reaches the quadratic part of its potential (before or after reheating), the energy density of the field is given by
\begin{equation}
\rho_{\chi}\left(a\right)=\frac{\lambda}{4}\,\chi_{\mathrm{end}}^{4}\,\left(\frac{a_{\mathrm{osc,r}}}{a_{\mathrm{osc,m}}}\right)^{4}\,\left(\frac{a_{\mathrm{osc,m}}}{a_{\mathrm{reh}}}\right)^{3}\,\left(\frac{a_{\mathrm{reh}}}{a}\right)^{3}.\label{rho DM late times}
\end{equation}
If the field started to oscillate about the {\it quartic} part of its potential only after reheating, there is no difference in the DM abundance between this scenario and the usual radiation-dominated one studied in Ref. \cite{Markkanen:2018gcw}. Therefore, in the following we will assume reheating always happens after the field has reached the quartic part of its potential.
Let us proceed by finding an expression for the present-day energy density of the field. First, we have
\begin{equation}
\frac{a_{\mathrm{osc,r}}}{a_{\mathrm{osc,m}}}=\left(\frac{H_{\mathrm{osc,m}}}{H_{\mathrm{osc,r}}}\right)^{\frac{2}{3\left(1+w\right)}}\,,\label{relation osc r m-1}
\end{equation}
where $H_{\rm osc,r}^2 = 3\lambda \chi_{\rm end}^2$, i.e. the field starts to oscillate when it becomes effectively massive. At $a_{\mathrm{osc,m}}$, the quadratic term in the equation of motion becomes equal to the quartic term
\begin{equation}
3\lambda\chi^2\left(a_{\mathrm{osc,m}}\right)=\lambda\chi^2_{\rm end}\left(\frac{a_{\mathrm{osc,r}}}{a_{\mathrm{osc,m}}}\right)^2=m^{2}\,,
\end{equation}
which gives us a relation between the Hubble parameters:
\begin{equation}
\frac{H_{\mathrm{osc,m}}}{H_{\mathrm{osc,r}}}=\left(\frac{m}{\sqrt{3\lambda}\left|\chi_{\rm end}\right|}\right)^{\frac{3\left(1+w\right)}{2}}.\label{H osc m-1}
\end{equation}
The factor $a_{\mathrm{osc,m}}/a_{\mathrm{reh}}$ can be obtained recalling that
\begin{equation}
H(a_{\mathrm{reh}})=\sqrt{\frac{\pi^{2}\,g_{*}\left(T_{\mathrm{reh}}\right)}{90}}\,\frac{T_{\mathrm{reh}}^{2}}{M_{\mathrm{P}}}\,,
\end{equation}
so that
\begin{equation}
\left(\frac{a_{\mathrm{osc,m}}}{a_{\mathrm{reh}}}\right)^{3}=\left(\frac{m}{\sqrt{\lambda}\left|\chi_{\rm end}\right|}\right)^{-3}\,\left(\sqrt{\frac{\pi^{2}\,g_{*}\left(T_{\mathrm{reh}}\right)}{270}}\,\frac{T_{\mathrm{reh}}^{2}}{\sqrt{\lambda}\,\left|\chi_{\rm end}\right|M_{\mathrm{P}}}\right)^{\frac{2}{1+w}}. \label{aosc areh quart}
\end{equation}
Finally, for the ratio $a_{\rm reh}/a$ we can again use entropy conservation, Eq. \eqref{areh_a}.
Hence, by using the relations above, we conclude that the present DM abundance is in this case given by
\begin{equation}
\label{DMabundance_quartic}
\Omega_\chi h^2 =\frac{\sqrt{\lambda}}{4}\frac{g_{{*S}}(T_{0})}{g_{{*S}}(T_{{\rm reh}})}\,\left(\sqrt{\frac{\pi^{2}\,g_{*}\left(T_{\mathrm{reh}}\right)}{270}}\,\frac{T_{\mathrm{reh}}^{2}}{\sqrt{\lambda}\,\left|\chi_{\mathrm{end}}\right|M_{\mathrm{P}}}\right)^{\frac{2}{1+w}}\left(\frac{T_{0}}{T_{{\rm reh}}}\right)^{3}\frac{m\left|\chi_{\rm end}\right|^3}{\rho_c/h^2}\,.
\end{equation}
While this result for the DM abundance applies regardless of when the scalar field reaches the quadratic part of its potential, in deriving this result we assumed that the oscillations in the quartic part always start before reheating, $a_{\mathrm{osc,r}}<a_{\mathrm{reh}}$. This gives a lower bound on the self-coupling
\begin{equation}
\label{lambdalimit}
\lambda > \frac{\pi^2g_*(T_{\rm reh})}{270}\frac{T_{\rm reh}^4}{M_{\rm P}^2\chi_{\rm end}^2}\,,
\end{equation}
which also has to be taken into account for consistency of the calculation.
\subsubsection{Condensate evaporation}
\label{fragmentation}
If its interactions are sufficiently suppressed, the scalar field
behaves as a long-lived oscillating condensate, never fragmenting or reaching thermal equilibrium. This is the scenario considered in the previous section. However, if the self-interaction coupling $\lambda$ is large enough, the $\chi$ condensate may fragment into $\chi$ particles which thermalize into a WIMP-like
DM candidate as discussed in Refs. \cite{Ichikawa:2008ne,Nurmi:2015ema,Kainulainen:2016vzv,Heikinheimo:2016yds,Enqvist:2017kzh,Cosme:2018nly,Markkanen:2018gcw}. The condition for a
complete decay of the condensate, for quartic self-interactions, is
given by
\begin{equation}
\frac{\Gamma\left(\chi\left(a_{\mathrm{dec}}\right)\right)}{H_{\mathrm{dec}}}\simeq 0.013\lambda\left(\frac{a_{\rm dec}}{a_{\rm osc,r}}\right)^{\frac12(1+3w)}=1,\label{evaporation}
\end{equation}
where $\Gamma\left(\chi\left(a\right)\right) = 0.023\lambda^{3/2}\chi(a)$ is the
decay rate of the condensate into two $\chi$ particles \cite{Kainulainen:2016vzv}, $H_{\mathrm{dec}}$ is the Hubble parameter at the time of the decay at $a_{\mathrm{dec}}$ and the amplitude of the scalar field is $\chi\left(a\right)=\chi_{\rm end}\,\left(a_{\rm osc,r}/a\right)$. For simplicity, we assume that the decay always occurs prior to reheating, which amounts to requiring
\begin{equation}
\label{adec<areh}
\frac{a_{\rm dec}}{a_{\rm reh}} = (0.013\lambda)^{-\frac{2}{1+3w}}\left(\sqrt{\frac{\pi^2g_*(T_{\rm reh})}{270}}\frac{T_{\rm reh}^2}{\sqrt{\lambda}|\chi_{\rm end}|M_{\rm P}} \right)^{\frac{2}{3(1+w)}} < 1\,.
\end{equation}
The condensate can only fragment while in the quartic part of its potential \cite{Ichikawa:2008ne,Nurmi:2015ema,Kainulainen:2016vzv}, which in addition to Eq. \eqref{adec<areh} imposes an upper limit on the bare mass:
\begin{equation}
m^2< 3\lambda\chi^2_{\rm dec} = 3\lambda\,(0.013\lambda)^{\frac{4}{(1+3w)}}\chi^2_{\rm end}\,.
\label{bare mass lim}
\end{equation}
If the bare mass is larger than the limit (\ref{bare mass lim}), the condensate never fragments but remains oscillating until the present day, and the resulting DM abundance is given by Eq. \eqref{DMabundance_quartic}. In contrast, if the condition (\ref{bare mass lim}) is satisfied, the condensate fragments, the $\chi$ sector thermalizes with itself, and we need to compute the dark matter abundance from the freeze-out of $\chi$ particles from their internal thermal bath.
In the following, we assume that the $\chi$ particles freeze out while still relativistic. The temperature the $\chi$ particles acquire after thermalization is obtained by equating the $\chi$ condensate's energy density to the usual form of radiation energy density, which gives
\begin{equation}
T_{\chi}\left(a\right)=\left(\frac{15\lambda}{2\pi^{2}}\right)^{1/4}\left|\chi_{\mathrm{end}}\right|\left(\frac{a_{\mathrm{osc,r}}}{a}\right)\,.
\label{chi T}
\end{equation}
Note that this temperature is different from the temperature of the SM particle heat bath and can also scale differently from it in terms of $a$, see e.g. Ref. \cite{Carlson:1992fn}. However, the $\chi$ particle number density corresponding to $T_\chi$ is given by the usual expression
\begin{equation}
n_{\chi}\left(a\right) =\frac{\zeta\left(3\right)}{\pi^{2}}\,T_{\chi}^{3}(a)\,.
\label{number density FO}
\end{equation}
After freeze-out, the $\chi$ particles do not interact anymore, which means that the above relation is valid even when $\chi$ becomes non-relativistic. The energy density of $\chi$ at the present time is therefore simply $\rho_{\chi}(a_{0}) =mn_{\chi}\left(a_{0}\right)$, leading to the
following present abundance:
\begin{equation}
\label{DM_abundance_FO}
\Omega_\chi h^2 =\left(\frac{15}{2}\right)^{3/4}\frac{\zeta(3)\lambda^{3/4}}{\pi^{7/2}}\frac{g_{{*S}}(T_{0})}{g_{{*S}}(T_{{\rm reh}})}\,\left(\sqrt{\frac{\pi^{2}\,g_{*}\left(T_{\mathrm{reh}}\right)}{270}}\,\frac{T_{\mathrm{reh}}^{2}}{\sqrt{\lambda}\,\left|\chi_{\mathrm{end}}\right|M_{\mathrm{P}}}\right)^{\frac{2}{1+w}}\left(\frac{T_{0}}{T_{{\rm reh}}}\right)^{3}\frac{m\left|\chi_{\rm end}\right|^3}{\rho_c/h^2}\,,
\end{equation}
where the main difference to the result in the case of coherent oscillations \eqref{DMabundance_quartic} is that thermalization of $\chi$ particles changes the result's dependence on $\lambda$.
Finally, we note that the DM freeze-out could also occur while the DM particles are non-relativistic. In this case, the scalar field
undergoes a phase of cannibalism \cite{Carlson:1992fn}, where the $4\rightarrow2$ self-annihilations dilute the number density of $\chi$ particles before their eventual freeze-out. While performing this calculation in the standard radiation-dominated case is relatively simple \cite{Markkanen:2018gcw}, in the presence of a non-standard epoch and entropy production this calculation becomes much more involved. While the qualitative picture is quite different from the case where DM freeze-out is determined by the $2\to 2$ scatterings (as above), the quantitative difference is very modest in the standard radiation-dominated case \cite{Markkanen:2018gcw} and it is expected to be small in non-standard cases as well; see Ref. \cite{Bernal:2018ins} for an example in a matter-dominated case. Therefore, in this paper we do not consider this possibility but leave it for future work.
\section{Dark matter perturbations}
\label{perturbations}
Because the $\chi$ field is assumed to be decoupled from the SM radiation, fluctuations in the local field value necessarily generate isocurvature perturbations between the DM and radiation energy densities. Due to the non-observation of isocurvature perturbations in the CMB, this provides the most stringent observational constraints on our scenario.
More precisely, the DM isocurvature perturbation is defined as
\begin{equation}
\label{isocurvature_def}
S_{r\chi} \equiv -3H\left(\frac{\delta\rho_r}{\dot{\rho_r}} - \frac{\delta\rho_\chi}{\dot{\rho_\chi}} \right),
\end{equation}
where $\rho_i$ is the energy density of the fluid $i=r,\chi$ and perturbations are defined as deviations from the average energy density of the fluid $i$,
\begin{equation}
\delta\rho_i \equiv \frac{\rho_i(x)}{\langle\rho_i\rangle} -1\,.
\end{equation}
As discussed in Sec. \ref{inflation}, we assume that the perturbations in radiation energy density were sourced by the inflaton field, whereas the perturbations in the DM energy density were also sourced by the spectator field. Because the fluids are assumed to be decoupled from each other, we obtain
\begin{equation}
H\frac{\delta\rho_i}{\dot{\rho_i}} = \frac{\delta\rho_i}{3(1+w_i)\rho_i}\,,
\end{equation}
where $w_i \equiv p_i/\rho_i$ is the effective equation of state parameter of the fluid $i$ which relates the pressure of the fluid to its energy density, i.e. $w_r=1/3$ for radiation and $w_\chi=0$ for the spectator field at late times. The isocurvature perturbation then becomes
\begin{equation}
\label{Srchi}
S_{r\chi} = \frac{\delta f(\chi_{\rm end})}{\langle f(\chi_{\rm end})\rangle}\,,
\end{equation}
where in the quadratic case $f(\chi_{\rm end}) = \chi_{\rm end}^2$ and in the quartic case $f(\chi_{\rm end}) = |\chi_{\rm end}|^{3-2/(1+w)}$, as given in Eqs. \eqref{DMabundance} and \eqref{DMabundance_quartic}, \eqref{DM_abundance_FO}, respectively. Note that because the mean field value vanishes, $\langle \chi_{\rm end}\rangle = 0$, it would be incorrect to assume $\delta f(\chi_{\rm end}) \propto \delta \chi_{\rm end}/\chi_{\rm end}$.
The isocurvature perturbation spectrum can be found in terms of the stochastic correlation functions that describe the field fluctuations during inflation \cite{Starobinsky:1994bd,Markkanen:2019kpv}. One finds that the power spectrum of the equal-time correlator of an arbitrary function of the scalar field $f(\chi)$ is given by \cite{Markkanen:2019kpv}
\begin{equation}
\label{Pfk}
\mathcal{P}_f(k) =\mathcal{A}_f\left(\frac{k}{H_{\rm inf}}\right)^{n_f-1}\,,
\end{equation}
where
\begin{equation}
\label{Af}
\mathcal{A}_f = \frac{2}{\pi}f_n^2\Gamma\left[2-(n_f-1)\right]\sin\left(\frac{\pi (n_f-1)}{2}\right)
\end{equation}
and
\begin{equation}
\label{nf}
n_f - 1 = \frac{2\Lambda_n}{H_{\rm inf}}\,,
\end{equation}
which applies for all modes $k\ll H_{\rm inf}$, i.e. for physical distance scales much larger than the horizon at the end of inflation. Because the CMB measurements are made at scales which are exponentially larger than $H_{\rm inf}^{-1}$, the form of \eqref{Pfk} is indeed suitable for our purposes.
As discussed in Ref.~\cite{Markkanen:2019kpv} (see also Refs.~\cite{Markkanen:2018gcw,Tenkanen:2019cik}) the parameters $f_n$ and $\Lambda_n$ are related to the eigenfunctions and eigenvalues of the Schr\"odinger-like equation
\begin{equation}
\label{Schrodinger}
\left(\frac{1}{2}\frac{\partial^2}{\partial\chi^2}-\frac{1}{2}\left(v'(\chi)^2-v''(\chi)\right) \right) \psi_n(\chi)
=-\frac{4\pi^2\Lambda_n}{H_{\rm inf}^3}\psi_n(\chi)\,,
\end{equation}
where
\begin{equation}
v(\chi)=\frac{4\pi^2}{3H_{\rm inf}^4}V(\chi) =
\begin{cases}
\displaystyle \frac{\pi^2\lambda}{3}\left(\frac{\chi}{H_{\rm inf}}\right)^4\,, &\quad \lambda\chi^2 \gg m^2\,,\\
\displaystyle \frac{2\pi^2}{3}\left(\frac{m}{H_{\rm inf}}\right)^2\left(\frac{\chi}{H_{\rm inf}}\right)^2\,, &\quad \lambda\chi^2 \ll m^2\,,
\end{cases}
\end{equation}
and $f_n$ is given in terms of the eigenfunctions in \eqref{Schrodinger} as
\begin{equation}
f_n=\int d\phi \psi_0(\chi)f(\chi)\psi_n(\chi)\label{eq:fn}\,.
\end{equation}
These quantities enter the calculation of the DM isocurvature spectrum through the spectral expansion of the unequal-time correlator\footnote{By using de Sitter invariance, this result can be used to find an expression for the equal-time correlator power spectrum \eqref{Pfk}. Here we present only the most important steps; for more details on the derivation of this result, see Refs. \cite{Starobinsky:1994bd,Markkanen:2019kpv}.}
\begin{equation}
\langle f(\chi(0))f(\chi(t))\rangle = \sum_n f_n^2 e^{-\Lambda_n t}\,,
\end{equation}
where only the first non-trivial term is important, as the higher-order corrections are exponentially suppressed \cite{Markkanen:2019kpv}. In the quadratic case, $\lambda\chi^2 \ll m^2$, we find analytically
\begin{equation}
\label{Lambda_f_quadr}
\Lambda_2^{(2)} = \frac23\frac{m^2}{H_{\rm inf}}\,, \quad f^{(2)}_2 = \sqrt{2}\,,
\end{equation}
where the superscripts denote the quadratic case, whereas in the quartic case, $\lambda\chi^2 \gg m^2$, a numerical solution of the eigenvalue equation (\ref{Schrodinger}) gives
\begin{equation}
\label{Lambda_quart}
\Lambda_2^{(4)}\approx 0.289\sqrt{\lambda}H_{\rm inf}
\end{equation}
and
\begin{equation}
\label{f2_quart}
f_2^{(4)}(w)\approx
\begin{cases}
0.639 & \quad w=0\,,\\
0.867 & \quad w=1/3\,,\\
1.057 & \quad w=1\,,\\
\end{cases}
\end{equation}
where we used the fact in the quartic case $f(\chi_{\rm end}) = |\chi_{\rm end}|^{3-2/(1+w)}$, as discussed below Eq. \eqref{Srchi}.
By substituting the results \eqref{Lambda_f_quadr}--\eqref{f2_quart} into Eqs. \eqref{Af} and \eqref{nf}, we find the DM isocurvature power spectrum as
\begin{equation}
\label{PS}
\mathcal{P}_S = \mathcal{A}_S\left(\frac{k}{k_*}\right)^{n_S-1}\,,
\end{equation}
where the amplitude at $k=k_*$ is given by
\begin{equation}
\label{S_amplitude}
\mathcal{A}_S =
\begin{cases}
\displaystyle \frac{2(f_2^{(4)}(w))^2}{\pi}\Gamma\left[2-(n_S-1)\right]\sin\left(\frac{\pi (n_S-1)}{2}\right)e^{-(n_S - 1)N(k_*)}, & \quad \lambda\chi^2 \gg m^2, \\
\displaystyle \frac{4}{\pi}\Gamma\left[2-(n_S-1)\right]\sin\left(\frac{\pi (n_S-1)}{2}\right) e^{-(n_S - 1)N(k_*)}, & \quad \lambda\chi^2 \ll m^2,
\end{cases}
\end{equation}
where $N(k_*)$ is the number of e-folds between the horizon exit of a scale $k_*$ and the end of inflation, and the spectral tilt is
\begin{equation}
\label{nS}
n_S - 1 =
\begin{cases}
\displaystyle 0.579\sqrt{\lambda}, & \quad \lambda\chi^2 \gg m^2\,, \\
\displaystyle \frac{4}{3}\frac{m^2}{H_{\rm inf}^2}, & \quad \lambda\chi^2 \ll m^2\,.
\end{cases}
\end{equation}
As the reference (pivot) scale we use $k_* = 0.05\,{\rm Mpc}^{-1}$, which is also one of the pivot scales the Planck collaboration used in their analysis. Following their conventions, the result \eqref{PS} should be compared to the observational constraint for an uncorrelated DM isocurvature perturbation
\begin{equation}
\label{beta}
\mathcal{P}_S(k_*) = \frac{\beta}{1-\beta}\mathcal{P}_\zeta(k_*)\,,
\end{equation}
where $\beta < 0.38$ and $\mathcal{P}_\zeta(k_*)= 2.1\times 10^{-9}$ is the observed amplitude of the curvature power spectrum \cite{Akrami:2018odb}.
To compute the DM perturbation spectrum in terms of our free parameters, we need to know the number of e-folds between horizon exit of the pivot scale and the end of inflation. As the result \eqref{S_amplitude} shows, the DM isocurvature power spectrum is exponentially sensitive to this number and therefore the differences between different scenarios can be large. In particular, this applies to deviations from the usual radiation-dominated case, which makes it interesting to consider such scenarios and in this way also constrain them.
The number of e-folds between the horizon exit of a scale $k$ and the end of inflation is given by (see e.g. Ref. \cite{Allahverdi:2020bys})
\be
\label{Nk}
N(k) = \ln\left(\frac{a_{\rm end}}{a_{\rm reh}}\right) + \ln\left(\frac{a_{\rm reh}}{a_0}\right) + \ln\left(H_{\rm inf} k^{-1}\right)\,,
\end{equation}
where the ratio of the scale factors at the end of inflation and at the time of reheating is
\begin{eqnarray}
\label{ae arh}
\ln \left(\frac{a_{\rm end}}{a_{\rm reh}}\right) &=& \frac{1}{3(1+w)}\ln\left(\frac{\rho_{\rm reh}}{\rho_{\rm end}}\right) \\ \nonumber
&=& \frac{1}{3(1+w)}\left[\ln\left(\frac{\pi^2 g_*(T_{\rm reh})}{90}\right) + 4\ln\left(\frac{T_{\rm reh}}{M_{\rm P}}\right) - 2\ln\left(\frac{H_{\rm inf}}{M_{\rm P}} \right) \right]\,.
\end{eqnarray}
Because the result depends only logarithmically on $g_*(T_{\rm reh})$ (and is further suppressed by the $w$-dependent prefactor), this quantity should differ from the usual value ($\sim 100$) by orders of magnitude in order to affect the result. Therefore, here we simply assume $g_*(T_{\rm reh})\sim 100$, which allows to write Eq. \eqref{ae arh} as
\begin{equation}
\ln \left(\frac{a_{\rm end}}{a_{\rm reh}}\right) \simeq \frac{1}{3(1+w)}\left[4\ln\left(\frac{T_{\rm reh}}{M_{\rm P}}\right) - 2\ln\left(\frac{H_{\rm inf}}{M_{\rm P}} \right) \right]\,.
\end{equation}
We also have
\be
\ln\left(\frac{a_{\rm reh}}{a_0}\right) = \frac13\ln\left(\frac{g_{*S}(T_0)T_0^3}{g_*(T_{\rm reh})T^3_{\rm reh}} \right) \simeq -72.5 - \ln\left(\frac{T_{\rm reh}}{M_{\rm P}}\right),
\end{equation}
where we used $g_{*S}=3.909$, $T_0=2.725$ K; and
\be
\ln\left(H_{\rm inf} k^{-1}\right) \simeq 133.3 + \ln\left(\frac{H_{\rm inf}}{M_{\rm P}}\right) - \ln\left(\frac{k}{0.05\,{\rm Mpc}^{-1}}\right)\,.
\end{equation}
Thus, putting all of the above results together, we obtain for the e-fold number corresponding to the pivot scale $k_*=0.05\,{\rm Mpc}^{-1}$ the result
\be
\label{N pivot}
N(k_*) \simeq 60.8 + \frac{1}{3(1+w)}\left[4\ln\left(\frac{T_{\rm reh}}{M_{\rm P}}\right) - 2\ln\left(\frac{H_{\rm inf}}{M_{\rm P}} \right) \right] + \ln\left(\frac{H_{\rm inf}}{M_{\rm P}}\right) - \ln\left(\frac{T_{\rm reh}}{M_{\rm P}}\right)\,.
\end{equation}
It should be noted, however, that for the assumptions made in this paper (in particular about the Hubble parameter that stays roughly constant during inflation), the number of e-folds is bounded from above as $N(k_*) \mathop{}_{\textstyle \sim}^{\textstyle <} 63$ due to the BBN constraints on gravitational waves\footnote{In the presence of a stiff era, $w>1/3$, gravitational waves become enhanced compared to the case with $w\leq 1/3$ and can contribute to the $N_{\rm eff}$ parameter in a significant way, allowing one to constrain such scenarios. For recent works, see e.g. Refs. \cite{Caprini:2018mtu,Bernal:2019lpc,Figueroa:2019paj}.} \cite{Tanin:2020qjw}, which affects our results in the $w=1$ case. In general, we will use the result \eqref{N pivot} to evaluate the DM isocurvature perturbation spectrum amplitude \eqref{S_amplitude}, which we will contrast with observations through Eq. \eqref{beta}. For an illustration of the resulting $P_S(k)$ in a few example cases, see Fig.~\ref{fig:P_S}.
\begin{figure}[httb]
\begin{centering}
\includegraphics[scale=.8]{isocurvature_power_spectra}
\par\end{centering}
\caption{The DM isocurvature power spectrum $\mathcal{P}_S(k)$ as a function of $k/k_*$ in the quartic case in three different scenarios, from top to bottom: $w=0\,, 1/3\,, 1$. The horizontal dashed line shows the CMB isocurvature constraint $\mathcal{P}_S(k_*) < 1.3\times 10^{-9}$ at the CMB pivot scale $k_*=0.05\,{\rm Mpc}^{-1}$ \cite{Akrami:2018odb}. In this figure $H_{\rm inf}=10^{13}$ GeV, $T_{\rm reh}=10^8$ GeV, $\lambda = 0.4$. The figure shows that for the above parameters, the case $w=0$ is not allowed by the CMB observations, whereas the cases with $w=1/3$ or $w=1$ are viable.}
\label{fig:P_S}
\end{figure}
\section{Results}
\label{results}
Finally, we present the requirements for the field $\chi$ to constitute all DM in the Universe. In addition to satisfying the DM abundance, there are a few requisites -- either observational constraints or consistency conditions specific to our scenario -- that constrain the model, and we begin by listing them here.
\subsection{Constraints on the scenario}
\label{sec:constraints}
First, the maximum Hubble scale during inflation is
\begin{equation}
\label{H_inf}
H_{\rm inf} =\sqrt{\frac{\pi^{2}rP_{\zeta}}{2}}M_{{\rm P}} \mathop{}_{\textstyle \sim}^{\textstyle <} 6\times10^{13}\,\sqrt{\frac{r}{0.06}}\,,
\end{equation}
which follows from the definition of the tensor-to-scalar ratio $r$ and the usual slow-roll approximation, see e.g. Ref. \cite{Baumann:2009ds}. Here we have normalized $r$ to the largest value allowed by observations, $r\leq 0.06$ \cite{Akrami:2018odb}. In all our results, we assume that the field's fluctuation spectrum is not suppressed during inflation, $V''<9H_{\rm inf}^2/4$, which provides an upper limit on the parameter that characterizes the shape of the potential in each case, $\lambda$ or $m$, whereas the isocurvature constraint \eqref{beta} imposes additional constraints on them\footnote{For completeness, we note that there is also another branch of solutions to Eq. \eqref{beta}, which in the quartic case requires a very small value of the self-coupling, $\lambda \mathop{}_{\textstyle \sim}^{\textstyle <} \mathcal{O}(10^{-19})$. As this regime is phenomenologically less interesting, here we neglect this possibility. For $m/H_{\rm inf}$ the solutions corresponding to the other branch are shown in Figs. \ref{w0 quad} and \ref{w1 quad}.}. The maximum reheating temperature for a given Hubble scale during inflation is
\begin{equation}
\label{TrehMax}
T_{{\rm reh}} \leq \left(\frac{90}{\pi^{2}g_{*}(T_{{\rm reh}})}\right)^{1/4}\sqrt{H_{{\rm inf}}M_{{\rm P}}}\,,
\end{equation}
which for the SM degrees of freedom, $g_*(T_{\rm reh}) = 106.75$, gives the absolute maximum reheating temperature as $T_{{\rm reh}}^{\rm max} = 6.7\times10^{15}\left(r/0.06\right)^{1/4}\,{\rm GeV}$, which follows from Eq. \eqref{H_inf}. However, the condition \eqref{TrehMax} is more general and should be applied for each $H_{\rm inf}$ separately. On the other hand, to not interfere with the formation of light elements, the non-standard phase has to end early enough so that the Universe gets reheated to a sufficiently high temperature. In the following, we will use the requirement $T_{\rm reh}\geq 10$ MeV to account for this aspect.
In the quartic case, observations of collisions between galaxy clusters (including the Bullet Cluster) can be used to place an upper bound on the self-interaction cross-section over DM mass, $\sigma/m\le 1$ cm$^2$/g $\approx 4.6\times 10^3\,{\rm GeV}^{-3}$ ~\cite{Markevitch:2003at,Randall:2007ph,Rocha:2012jg,Peter:2012jh,Harvey:2015hha}. For our theory \cite{Heikinheimo:2016yds}
\begin{equation}
\frac{\sigma}{m} = \frac{9\lambda^2}{32\pi m^3} ,
\end{equation}
so the the galaxy collisions impose a constraint
\begin{equation}
\label{eq:sigmaDMbound}
\frac{m}{\rm GeV} > 0.027\left(\frac{\sigma/m}{{\rm cm}^2/{\rm g}}\right)^{-1/3} \lambda^{2/3}\,,
\end{equation}
which limits the quartic scenario at sub-GeV masses for detectable values of $\sigma/m$.
Finally, in deriving the results for DM abundance, Eq. \eqref{DMabundance} in the quadratic case and Eqs. \eqref{DMabundance_quartic} or \eqref{DM_abundance_FO} in the quartic case (depending on whether the condensate fragments or not), we made the assumption that the post-inflationary oscillations of the $\chi$ field always began prior to reheating\footnote{While relaxing this assumption does not change the results for the DM abundance found in Refs. \cite{Markkanen:2018gcw,Tenkanen:2019aij}, an early phase of non-standard expansion would change the isocurvature limit through the altered number of e-folds, Eq. \eqref{N pivot}. In this paper, however, we do not account for this possibility in our figures for simplicity.}, $a_{\rm osc}/a_{\rm reh}<1$, which amounts to requiring Eq. \eqref{masscondition} in the quadratic case and Eq. \eqref{lambdalimit} in the quartic case. We also assumed that the possible decay of the oscillating condensate always occurs prior to reheating, which imposes the conditions given by Eqs. \eqref{adec<areh} and \eqref{bare mass lim}. In addition to the observational constraints discussed above, these consistency conditions provide constraints on the model parameter space. They are all accounted for in the results shown in the next subsection.
\subsection{Model parameter space}
Next, we present the results by assuming in all cases that the DM abundance is given by the typical field value, $\chi_{\rm end}^2 = \langle \chi_{\rm end}^2 \rangle$, as in Eq. \eqref{variance}.
First, Figs. \ref{w0 quad} and \ref{w1 quad} show the region of the model parameter space where the quadratic scenario ($m^2 > \lambda\chi_{\rm end}^2$) explains all DM (solid colored lines) and satisfies the constraints discussed in the beginning of this section (within the shaded regions) for the cases $w=0$ (Fig.~\ref{w0 quad}) and $w=1$ (Fig.~\ref{w1 quad}). We emphasize that as the shaded regions represent the part of the parameter space where the constraints are {\it satisfied}, the regions where the $\chi$ field can successfully explain all DM are those where the solid colored lines overlap with the shaded regions.
As a few benchmark scenarios, we have considered four reheating temperatures,\linebreak $T_{\mathrm{reh}}=10^{13},\,10^{11},\,10^{5},\,10^{-2}$ GeV (shown in blue, yellow, green, and red, respectively), and assumed $g_{*}(T_{\mathrm{reh}})=100$ for all reheating temperatures except for $T_{\mathrm{reh}}=10^{-2}$ GeV, for which we used the more correct value $g_{*}(T_{\mathrm{reh}})=10$. The axis in each figure have been adjusted for each case to show only the part of the parameter space which is allowed by the constraints discussed in Sec. \ref{sec:constraints}. The dotted purple line corresponds to the DM abundance in the usual cosmological scenario with $w=1/3$ \cite{Tenkanen:2019aij}, which is shown here for comparison. Note, however, that this line shows only the DM abundance and the constraints shown in the plot do not apply to the scenario with $w=1/3$ as such but should be computed separately. Also, note that in all cases the DM isocurvature constraints have been computed assuming that the field constitutes all dark matter, and hence are applicable only along the solid lines.
We see that for fixed DM mass and reheating temperature, in the $w=0$ ($w=1$) case a higher (lower) value of $H_{\rm inf}$ than in the usual radiation-dominated scenario with $w=1/3$ is required to obtain the observed DM abundance today. This is naturally understood by the fact that the initial energy density stored in the spectator field is $\rho_\chi \propto H_{\rm inf}^4$, see Eq. \eqref{variance}, and the more the scalar field energy density becomes diluted compared to the background energy density after inflation, the higher the initial energy density of the scalar field (the value of $H_{\rm inf}$) has to be to obtain the correct DM abundance today. Note, however, that not all values of $w\,, T_{\rm reh}$ give the correct relic abundance for each set of $m\,,H_{\rm inf}$ shown in Figs. \ref{w0 quad} and \ref{w1 quad}. For example, for $T_{\rm reh}=10^{13}$ GeV there is essentially no available parameter space where the $\chi$ field has the correct DM abundance today and simultaneously satisfies both the observational constraints and also the consistency conditions discussed in the previous subsection, neither for $w=0$ nor $w=1$. Here this scenario is shown as a limiting example to highlight the volume of the allowed ($w\,, T_{\rm reh}\,,m\,,H_{\rm inf}$) parameter space.
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.48\textwidth]{matt13Log0_2}
\includegraphics[width=0.48\textwidth]{matt11Log0_2}\vspace{0.5 cm}
\includegraphics[width=0.48\textwidth]{matt5Log0_2}
\includegraphics[width=0.48\textwidth]{matt02Log0_2}
\par\end{centering}
\caption{Parameter space of the model where the scalar constitutes all DM in the Universe (colored solid lines) and satisfies all constraints described in the beginning of Sec. \ref{results} (inside the shaded regions), assuming a quadratic potential and an equation of state parameter $w=0$. The DM abundance was computed for four different reheating temperatures as shown above each panel. The axis have been adjusted for each case to show only the part of the parameter space which is allowed by the constraints discussed in Sec. \ref{sec:constraints}. The dotted purple line corresponds to the DM abundance in the usual cosmological scenario with $w=1/3$, shown here for comparison. Note that the constraints shown here are for cases with $w=0$ and do not apply to the scenario with $w=1/3$ as such.
}
\label{w0 quad}
\end{figure}
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.48\textwidth]{matt13Log1_2}
\includegraphics[width=0.48\textwidth]{matt11Log1_2}\vspace{0.5 cm}
\includegraphics[width=0.48\textwidth]{matt5Log1_2}
\includegraphics[width=0.48\textwidth]{matt02Log1_2}
\par\end{centering}
\caption{The same as Fig. \ref{w0 quad} but for $w=1$.
}
\label{w1 quad}
\end{figure}
In Figs. \ref{w0 quar} and \ref{w1 quar}, we present the allowed parameter space for the quartic scenario ($m^2 <\lambda\chi_{\mathrm{end}}^2$) in the $(\lambda,m)$ plane by fixing $H_{\mathrm{inf}}$ and varying $T_{\mathrm{reh}}$ for both $w=0$ and $w=1$. For the same reason as in the quadratic case, for fixed values of other parameters, in the $w=0$ ($w=1$) case a higher (lower) value of $H_{\rm inf}$ than in the usual radiation-dominated scenario with $w=1/3$ is required to obtain the observed DM abundance today. However, as the dimension of the parameter space is now greater than in the quadratic case, there are more combinations of parameters that allow the spectator field to successfully constitute all DM, as visualized in Figs. \ref{w0 quar} and \ref{w1 quar}. Note again that not all values of parameters shown there allow the field to constitute all DM; for example, in the right panel of Fig. \ref{w0 quar}, the case with $T_{\rm reh}=10^{11}$ GeV is ruled out. Another example is given in the right panel of Fig. \ref{w1 quar}, where the case with $T_{\rm reh}=10^{6}$ GeV is in tension with the constraints on our scenario. Note that in that figure, we also show the constraints on DM self-interactions. The purple dash-dotted curve at the bottom of the figure is a hard limit, depicting $\sigma/m= 1\,{\rm cm}^2{\rm g}^ {-1}$, whereas the blue curve assumes $\sigma/m=10^{-2}\,{\rm cm}^2{\rm g}^ {-1}$ and is shown here as a target for future observations (see Sec. \ref{testability}).
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.48\textwidth]{mLambda_13w0_3}
\includegraphics[width=0.48\textwidth]{mLambda_9w0_4_2}
\par\end{centering}
\caption{The parameter space of the model where the scalar constitutes all DM (colored solid lines) and simultaneously satisfies all constraints (inside the shaded regions), assuming a quartic potential and an equation of state parameter $w=0$. The DM abundance was computed for different reheating temperatures, as indicated in the plots. As two benchmark scenarios, we present those with $H_{\rm inf}=10^{13}$ GeV and $H_{\rm inf}=10^{9}$ GeV. Above the black dashed line the condensate remains coherent, while for masses below the dashed line the condensate evaporates. The axis have been adjusted for each case to show only the part of the parameter space which is allowed by the constraints discussed in Sec. \ref{sec:constraints}.
}
\label{w0 quar}
\end{figure}
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.48\textwidth]{mLambda_8w1_4}
\includegraphics[width=0.48\textwidth]{mLambda_6w1_4}
\par\end{centering}
\caption{The same as Fig. \ref{w0 quar} but for $w=1$, $H_{\rm inf}=10^{8}$ GeV, and $H_{\rm inf}=10^{6}$ GeV. The dash-dotted lines on the right panel correspond to constraints on the DM self-interactions that can be inferred from galaxy clusters, Eq. \eqref{eq:sigmaDMbound}. The purple dash-dotted curve corresponds to $\sigma/m= 1\,{\rm cm}^2{\rm g}^ {-1}$, while the light blue one assumes $\sigma/m=10^{-2}\,{\rm cm}^2{\rm g}^ {-1}$, shown here as a target for future observations.
}
\label{w1 quar}
\end{figure}
\subsection{Testability of the scenario}
\label{testability}
Finally, we discuss how the scenarios we have studied in this paper could be further constrained -- or supported -- by future observations.
The fact that the DM perturbations in our scenario are genuinely of isocurvature type provides probably the best avenue for testing the scenario. First, the DM isocurvature perturbations generically enhance the perturbations in the SM matter density and can lead to a sizeable enhancement in the CMB temperature and/or matter power spectrum compared to the adiabatic case \cite{Graham:2015rva,Alonso-Alvarez:2018tus,Tenkanen:2019aij}. This is due to the fact that in the presence of isocurvature, the total curvature perturbation at superhorizon scales is given by (see e.g. Ref. \cite{Wands:2000dp})
\begin{equation}
\label{zeta_tot}
\zeta = \zeta_{\rm r} + \frac{z_{\rm eq}/z}{4+3z_{\rm eq}/z}S_{\rm r\chi}\,,
\end{equation}
where $\zeta_{\rm r}$ is the contribution of the SM radiation to the curvature perturbation and $z$ ($z_{\rm eq}$) is the redshift (to the matter-radiation equality), so that at late times the prefactor of the second term is $\sim 1/3$. In our scenario deviations from the adiabatic case can be large especially at smaller physical distance scales, which is due to the fact that in our scenario the DM isocurvature spectrum is always blue-tilted (see Fig. \ref{fig:P_S}). At subhorizon scales the simple picture of Eq. \eqref{zeta_tot} does not hold but one can nevertheless show that also at those scales the effect of DM isocurvature is to increase the curvature perturbation at the linear level (see e.g. Ref. \cite{Feix:2020txt}). Furthermore, because in e.g. the usual axion DM models the corresponding power spectrum is typically nearly scale-invariant \cite{Beltran:2006sq,Visinelli:2009zm}, the spectator DM model may be distinguishable from this type of models if observations of the matter power spectrum can be extended to large enough $k$ where the data may show evidence for (scale-dependent) DM isocurvature; a potentially promising new avenue in this respect is the forthcoming Euclid satellite mission \cite{Laureijs:2011gra,Amendola:2016saw}. It is worth noting here that some recent analysis of the CMB temperature fluctuations have actually shown hints of a blue-tilted DM isocurvature contribution \cite{Chung:2017uzc,Akrami:2018odb,Feix:2020txt}, although some of these results are still preliminary and the effects of isocurvature are degenerate with the effect of other cosmological parameters.
Second, the isocurvature nature of DM in our scenario provides also an additional way to both distinguish our scenario from other models and also further test the properties of DM. That is, due to the stochastic behavior of the $\chi$ field during inflation, in all cases studied in this paper the DM isocurvature is non-Gaussian, which in practice shows not only in the form of the three-point correlation function of the DM isocurvature perturbation, $\langle S_{\rm r\chi}(\bar{x}_1)S_{\rm r\chi}(\bar{x}_2)S_{\rm r\chi}(\bar{x}_3)\rangle$ (where $S_{\rm r\chi}$ is given by Eq. \eqref{Srchi}), but also in the three-point correlator of the total curvature perturbation, $\langle \zeta(\bar{x}_1)\zeta(\bar{x}_2)\zeta(\bar{x}_3)\rangle$, which is partly determined by the former (see Eq. \eqref{zeta_tot}). Because the perturbations are uncorrelated, we have
\begin{equation}
\label{NG}
\langle \zeta(\bar{x}_1)\zeta(\bar{x}_2)\zeta(\bar{x}_3)\rangle = \langle \zeta_{\rm r}(\bar{x}_1)\zeta_{\rm r}(\bar{x}_2)\zeta_{\rm r}(\bar{x}_3)\rangle + \left(\frac{z_{\rm eq}/z}{4+3z_{\rm eq}/z}\right)^3\langle S_{\rm r\chi}(\bar{x}_1)S_{\rm r\chi}(\bar{x}_2)S_{\rm r\chi}(\bar{x}_3)\rangle\,,
\end{equation}
where the prefactor of the second term is $\sim 1/27$ at the time of last scattering and the three-point correlator for $\zeta_{\rm r}$ depends on the inflationary model \cite{Maldacena:2002vr} (and possibly also the fluctuations the SM Higgs acquired during inflation \cite{Tenkanen:2019cik}). Therefore, even if the non-Gaussianity generated during inflation was negligible, DM isocurvature perturbations can generate sizeable non-Gaussianity at late times. In the spectator DM scenario the effect is again different from the usual axion DM scenario and also many other models where non-Gaussianity can be generated, such as curvaton models (see e.g. Refs. \cite{Kawasaki:2008sn,Langlois:2008vk,Langlois:2011hn,Langlois:2012tm,Hikage:2012be,Kitajima:2017fiy}), as the non-Gaussianity in the present scenario is of non-local type\footnote{Local non-Gaussianity is defined as $\zeta(\bar{x}) = \zeta_{\rm G}(\bar{x})+\frac{3}{5}f_{\rm NL}\left(\zeta_{\rm G}^2(\bar{x})-\langle\zeta_{\rm G}^2(\bar{x})\rangle\right)$, where $f_{\rm NL}$ is the first order non-Gaussianity parameter and $\zeta_{\rm G}$ denotes the Gaussian part of the curvature perturbation (see e.g. Ref. \cite{Byrnes:2014pja}). In our scenario the curvature perturbation does not take the local form and is hence ``non-local".}. While computing the above three-point correlation functions is beyond the scope of this paper, doing so would certainly be worthwhile if primordial DM isocurvature or non-Gaussianity were discovered, as they may provide a powerful way to test the spectator DM scenario and distinguish it from other models, such as those where axions constitute the DM\footnote{For an early work that computed the three-point correlator of a stochastic scalar field in the special case of equilateral triangles, see Ref. \cite{Peebles:1999fz}. It would be interesting to generalize this calculation to other shapes as well.}.
Finally, we comment on the prospects for detecting DM self-interactions. While in the quadratic scenario studied in this paper the DM self-interactions are by definition negligible, in the opposite case where the quartic term dominated the scalar field potential already during inflation, the DM self-interactions can be sizeable. Currently observations of dynamics of celestial bodies at the galactic and galaxy cluster scales place an upper bound on the DM self-interaction cross-section over DM mass which is of the order $\sigma/m\leq 1$ cm$^2$/g, however, in the future this limit may be possible to become tightened to $\sigma/m \mathop{}_{\textstyle \sim}^{\textstyle <} \mathcal{O}(0.01)$ cm$^2$/g \cite{Tulin:2017ara}. As our results show (see Fig. \ref{w1 quar}), this will further constrain the model parameter space in the quartic case. In case of a positive detection, the quartic case can accommodate these interactions for suitable $m$ and $\lambda$ (as well as $T_{\rm reh}\, H_{\rm inf}, w$), while the quadratic case would obviously be ruled out. It should be noted that this is in contrast to the case with standard cosmology, where the spectator DM model can never account for DM self-interactions of observable size \cite{Markkanen:2018gcw}. As we have shown in this paper, however, with a modified cosmological history this is not a problem.
\section{Conclusions}
\label{conclusions}
In this paper, we have studied the spectator dark matter scenario, where the observed DM abundance is produced by amplification of quantum fluctuations of an energetically subdominant scalar field during inflation. We showed that the scenario is robust to changes in the expansion history of the early Universe in a sense that also in this case the scenario works for a wide range of DM masses and coupling values, although even relatively modest changes to the standard cosmological history can impose notable quantitative differences to the usual scenario. This is because in the presence of a non-standard expansion phase the DM energy density evolves differently as a function of time, and also the DM isocurvature perturbation spectrum turns out to be different from the result in the radiation-dominated case. We quantified these differences in both free and self-interacting DM cases and presented the refined model parameter space which allows the scalar field to constitute all DM while simultaneously satisfying all observational constraints.
While we have discussed only few example cases (early matter-domination and kination-domination encountered in e.g. quintessence models), further modifications to the early cosmological history can also be imagined. Likewise, it would be interesting to see how a non-standard phase of expansion in the early Universe can change the allowed model parameter space in scenarios where the DM field couples non-minimally to gravity, to the inflaton field, and/or to another spectator field, for instance the SM Higgs.
Finally, we discussed the prospects for testing the scenario with future observations. In particular, if primordial DM isocurvature or non-Gaussianity is ever discovered, this may provide a powerful way to test the spectator DM scenario and distinguish it from other models, such as those where axions constitute the DM. Indeed, it is worth emphasizing that despite the fact that in these models the DM field interacts with ordinary matter only via gravity, these scenarios are testable with both current and future observations of the CMB and the large scale structure of the Universe, as well as the dynamics of celestial bodies at galactic and galaxy cluster scales, as discussed in this paper. If observations ever show any deviation from the adiabatic, non-interacting cold DM paradigm, it would be interesting to see what that tells about DM candidates which are only gravitationally interacting. After all, for all we know about dark matter, this minimal scenario seems to be the one preferred by observations.
\acknowledgments
We thank D. Bettoni, M. Kamionkowski, A. Rajantie, and J. Rubio for correspondence and discussions. C.C. is supported by the Arthur B. McDonald Canadian Astroparticle Physics Reasearch Institute and T.T. by the Simons foundation. T.T. thanks the hospitality of Carleton University, where this work was initiated.
|
1,314,259,996,469 | arxiv | \section{Introduction}\label{s_intro}
For a complex matrix we present approximations for the determinant
and its logarithm, together with error bounds.
The approximations were motivated by a problem in computational
quantum field theory: the simulation of finite temperature nuclear
matter on a lattice \cite{LeeI03}. In this application, the logarithm
of the determinant is desired to 2-3 significant digits. The
matrices are sparse, and non-Hermitian.
Because the desired accuracy is low, LU decomposition with partial
pivoting \cite[\S 14.6]{Hig02}, \cite[\S 3.18]{Wil63} is too costly.
Since the matrices are not Hermitian positive-definite, sparse
approximate inverses \cite{Reu02}, Gaussian quadrature based
methods \cite{BaiG97}, and Monte Carlo methods \cite[\S 4]{Reu02}
or hybrid Monte Carlo methods \cite{DuKPR87,GoLTRS87,ScSS86}
do not apply. Monte Carlo and quadrature-based methods
can be extended to non-Hermitian matrices, however then
the sign of the determinant is usually lost, e.g. \cite[\S 3.2.3]{BaiFG96}.
To approximate the determinant $\det(M)$ we decompose $M=M_D+M_{\mathrm{off}}$
such that $M_D$ is a non-singular matrix. Then
$\det(M)=\det(M_D)\det(I+M_D^{-1}M_{\mathrm{off}})$, where $I$ is the identity matrix.
In
$$\det(I+M_D^{-1}M_{\mathrm{off}})=\mathop{\mathrm{exp}}(\mathop{\mathrm{trace}}(\mathop{\mathrm{log}}(I+M_D^{-1}M_{\mathrm{off}}))),$$
we expand $\mathop{\mathrm{log}}(I+M_D^{-1}M_{\mathrm{off}})$, obtaining
a sequence of increasingly accurate approximations.
Error bounds for these approximations depend on the spectral
radius of $M_D^{-1}M_{\mathrm{off}}$.
\subsection*{Overview}
The determinant approximations and their
error bounds are presented in \S \ref{s_main}.
Approximations from block diagonals (\S \ref{s_diag}) are extended to
a sequence of higher order approximations (\S \ref{s_higher}).
They simplify for checkerboard matrices (\S \ref{s_tridiag})
which occur in the neutron matter simulations in \cite{LeeI03}.
Comparisons with sparse inverse
approximations of determinants, which are limited to Hermitian
positive-definite matrices (\S \ref{s_si})
illustrate the competitiveness of block diagonal approximations.
As expected, the
accuracy of sparse inverse approximations increases as more matrix
elements are included.
Numerical results with matrices from nuclear matter simulations
(\S \ref{s_app}) show that determinant approximations of desired accuracy
can be obtained fast, in 1-3 iterations; and that they require
significantly less space than Gaussian elimination (with partial
or complete pivoting).
\subsection*{Notation}
The eigenvalues of a complex square matrix $A$ are $\lambda_j(A)$, and the
spectral radius is $\rho(A)\equiv\max_j{|\lambda_j(A)|}$.
The identity matrix is $I$, and $A^*$ is the conjugate transpose of $A$.
We denote by $\mathop{\mathrm{log}}(X)$ and $\mathop{\mathrm{exp}}(X)$ the logarithm and exponential
function of a matrix $X$, and by $\ln(x)$ and $e^x$
the natural logarithm and exponential function of a scalar $x$.
\section{Determinant Approximations}\label{s_main}
We present approximations to the determinant and its logarithm,
as well as error bounds.
\subsection{Diagonal Approximations}\label{s_diag}
We present relative error bounds for the approximation
of the determinant by the determinant of a block diagonal.
Let $M$ be a complex square matrix of order $n$
partitioned as a $k\times k$ block matrix
$$M=\begin{pmatrix}M_{11} & M_{12} &\ldots & M_{1k} \cr
M_{21} & M_{22} &\ldots & M_{2k} \cr
\vdots & \ddots &\ddots & \vdots\cr
M_{k1} & M_{k2} &\ldots & M_{kk}\end{pmatrix},$$
where the diagonal blocks $M_{jj}$ are square but not necessarily of the
same dimension. Analogously,
decompose $M=M_D+M_{\mathrm{off}}$ into diagonal blocks $M_D$ and off-diagonal
blocks $M_{\mathrm{off}}$,
\begin{eqnarray}\label{e_part}
M_D=\begin{pmatrix}M_{11} & & & \cr
& M_{22} & & \cr
& &\ddots & \cr
& & & & M_{kk}\end{pmatrix},\qquad
M_{\mathrm{off}}=\begin{pmatrix}0 & M_{12} &\ldots & M_{1k} \cr
M_{21} & 0 &\ldots & M_{2k} \cr
\vdots & \ddots &\ddots & \vdots\cr
M_{k1} & M_{k2} &\ldots & 0\end{pmatrix}.
\end{eqnarray}
The block diagonal matrix $M_D$ is called a pinching of $M$
\cite[\S II.5]{Bha97}.
We consider the approximation of $\det(M)$ by the determinant of a
pinching, $\det(M_D)$; and in particular bounds
of the form $\det(M)\leq \det(M_D)$. The matrices for which such bounds
are known to hold are characterized by eigenvalue monotonicity
of the following kind.
A complex square matrix $M$ is a $\tau$-matrix if \cite[pp 156-57]{EngS76}:
\begin{enumerate}
\item Each principal submatrix of $M$ has at least one real eigenvalue.
\item If $S_1$ is a principal submatrix of $M$ and $S_{11}$ a principal
submatrix of $S_1$ then $\lambda_{\mathrm{min}}(S_1)\leq \lambda_{\mathrm{min}}(S_{11})$, where
$\lambda_{\mathrm{min}}$ denotes the smallest \textit{real} eigenvalue.
\item $\lambda_{\mathrm{min}}(M)\geq 0$.
\end{enumerate}
The class of $\tau$-matrices includes Hermitian positive-definite,
M-matrices and totally non-negative matrices \cite[pp 156-57]{EngS76},
\cite[Theorem 1]{Mehr84a}.
\paragraph{Hadamard-Fischer Inequality}
If $M$ is a $\tau$-matrix then \cite[Theorem 4.3]{EngS76}
\begin{eqnarray}\label{e_fh}
\det(M)\leq \det(M_D).
\end{eqnarray}
Strictly speaking, (\ref{e_fh}) is called a Hadamard-Fischer inequality
only for $k=2$ \cite[(0.5)]{EngS76}.
If $k=2$ and $M$ is Hermitian positive-definite then (\ref{e_fh})
is Fischer's inequality \cite[Theorem 7.8.3]{HoJ85}.
If $k=n$ and $M$ Hermitian positive-definite then (\ref{e_fh}) is Hadamard's
inequality \cite[Theorem 8]{CovT88}, \cite[Theorem 7.8.1]{HoJ85}.
Extensions of (\ref{e_fh}) to generalized
Fan inequalities are derived in \cite{Mehr84b,Mehr84a}.
The Hadamard-Fischer inequality (\ref{e_fh}) implies the obvious relative
error bound for the determinant of a pinching,
$$0< {\det(M_D)-\det(M)\over \det(M_D)}\leq 1.$$
In the theorem below we tighten the upper bound.
\begin{theorem}\label{t_1a}
Let $M$ be a complex matrix of order $n$.
If $\det(M)$ is real, $M_D$ is non-singular with $\det(M_D)$ real,
and all eigenvalues $\lambda_j(M_D^{-1}M_{\mathrm{off}})$ are real
with $\lambda_j(M_D^{-1}M_{\mathrm{off}})>-1$, then
$$0< {\det(M_D)-\det(M)\over \det(M_D)}\leq
1-e^{-{n\rho^2\over 1+\lambda_{\mathrm{min}}}},$$
where $\rho\equiv\rho(M_D^{-1}M_{\mathrm{off}})$ and
$\lambda_{\mathrm{min}}\equiv\min_{1\leq j\leq n}{\lambda_j(M_D^{-1}M_{\mathrm{off}})}$.
\end{theorem}
\begin{proof}
Write $\det(M)=\det(M_D)\det(I+A)$, where $A\equiv M_D^{-1}M_{\mathrm{off}}$.
Since $I+A$ is non-singular,
\cite[Theorem 6.4.15(a)]{HoJ91} and \cite[Problem 6.2.4]{HoJ91}
imply $\det(I+A)=\mathop{\mathrm{exp}}(\mathop{\mathrm{trace}}(\mathop{\mathrm{log}}(I+A)))$. Furthermore,
$\mathop{\mathrm{log}}(I+A)=\sum_{p=1}^{\infty}{{(-1)^{p-1}\over p}A^p}$
\cite[(7) in \S 9.8]{LaT85}.
Hence
$$\det(I+A)=\mathop{\mathrm{exp}}\left(\sum_{j=1}^n{\ln(1+\lambda_j(A))}\right).$$
Because $\lambda_j(A)>-1$, $1\leq j\leq n$,
we can apply the inequality
${\lambda\over 1+\lambda}\leq\ln(1+\lambda)\leq\lambda$ \cite[4.1.33]{AbS72}
to obtain
$$\mathop{\mathrm{exp}}(\mathop{\mathrm{trace}}(A))\> e^{-{n\rho(A)^2\over 1+\lambda_{\mathrm{min}}}}
\leq\det(I+A)\leq \mathop{\mathrm{exp}}(\mathop{\mathrm{trace}}(A)).$$
At last use the fact that $M_D$ is block diagonal and
$\mathop{\mathrm{trace}}(M_{\mathrm{off}})=\mathop{\mathrm{trace}}(A)=0$.
\end{proof}
The upper bound for the relative error is small if the eigenvalues of
$M_D^{-1}M_{\mathrm{off}}$ are small in magnitude but not too close to $-1$.
The pinching $\det(M_D)$ can be a bad approximation to $\det(M)$
when $I+M_D^{-1}M_{\mathrm{off}}$ is close to singular.
If $\det(M_D)>0$ then Theorem \ref{t_1a} implies a lower bound
for $\det(M)$,
$$e^{-{n\rho^2\over 1+\lambda_{\mathrm{min}}}}\det(M_D)\leq \det(M)\leq \det(M_D).$$
In the argument of the exponential function in Theorem \ref{t_1a}
we have $\lambda_{\mathrm{min}}<0$ because $M_D^{-1}M_{\mathrm{off}}$ has a zero diagonal so that
$\mathop{\mathrm{trace}}(M_D^{-1}M_{\mathrm{off}})=0$. Hence $n\rho^2/(1+\lambda_{\mathrm{min}})>n\rho^2$.
\begin{corollary}\label{c_1}
Theorem \ref{t_1a} holds for Hermitian positive-definite matrices.
In particular, Theorem \ref{t_1a} implies
error bounds for Fischer's and Hadamard's inequalities.
\end{corollary}
The following example shows that $|\det(M)|\leq|\det(M_D)|$ may not hold
when $M_D^{-1}M_{\mathrm{off}}$ has complex eigenvalues, or real eigenvalues
smaller than $-1$.
\begin{example}
Even if all eigenvalues of $M_D^{-1}M_{\mathrm{off}}$ satisfy
$|\lambda_j(M_D^{-1}M_{\mathrm{off}})|<1$, it is still possible that
$|\det(M)|>|\det(M_D)|$ if some eigenvalues are complex.
Consider
$$M=\begin{pmatrix}1&\alpha \cr \alpha &1\end{pmatrix},\qquad
M_D=\begin{pmatrix}1&0\cr 0&1\end{pmatrix},\qquad
M_{\mathrm{off}}=\begin{pmatrix}0&\alpha\cr \alpha &0\end{pmatrix}=M_D^{-1}M_{\mathrm{off}}.$$
Then $\lambda_j(M_D^{-1}M_{\mathrm{off}})=\pm \alpha$, and $\det(M)=1-\alpha^2$.
Choose $\alpha ={1\over 2}\imath$, where $\imath=\sqrt{-1}$.
Then both eigenvalues of $M_D^{-1}M_{\mathrm{off}}$ are complex, and
$|\lambda_j(M_D^{-1}M_{\mathrm{off}})|<1$. But $\det(M)=1.25>1=\det(M_D)$.
The situation
$\det(M_D)>\det(M)$ can also occur when $M_D^{-1}M_{\mathrm{off}}$ has a real eigenvalue
that is less than $-1$. If $\alpha=3$ in the matrices above then
one eigenvalue of $M_D^{-1}M_{\mathrm{off}}$ is $- 2$, and $|\det(M)|=8>\det(M_D)=1$.
In general,
$|\det(M)|/\det(M_D)\rightarrow\infty$ as $|\alpha|\rightarrow\infty$.
\end{example}
This example illustrates that, unless the eigenvalues of $M_D^{-1}M_{\mathrm{off}}$
are real and greater than $-1$, $\det(M_D)$ is, in general, not
a bound for $\det(M)$. In the case of complex eigenvalues,
however, we can still determine how well $\det(M_D)$ \textit{approximates}
$\det(M)$. Below is a relative error bound for the case when
$M$ is 'diagonally dominant', in the sense
that the eigenvalues of $M_D^{-1}M_{\mathrm{off}}$ are small in magnitude.
\begin{theorem}[Complex Eigenvalues]\label{t_2}
Let $M$ be a complex matrix of order $n$.
If $M_D$ is non-singular and $\rho\equiv\rho(M_D^{-1}M_{\mathrm{off}})<1$
then
$${|\det(M)-\det(M_D)|\over |\det(M_D)|}\leq c\rho\>e^{c\rho},\qquad
\mathrm{where}\quad c\equiv -n\ln(1-\rho).$$
If also $c\rho<1$ then
$${|\det(M)-\det(M_D)|\over |\det(M_D)|}\leq {7\over 4}c\rho.$$
\end{theorem}
\begin{proof} This is a special case of Theorem \ref{t_3}.
\end{proof}
\begin{corollary}\label{c_2}
Theorem \ref{t_2} holds for the following classes of matrices:
M-matrices; Hermitian positive-definite matrices if $k=n$;
Hermitian positive definite block tridiagonal matrices with
equally-sized blocks of dimension $n/k$.
\end{corollary}
\begin{proof}
In all cases $\rho(M_D^{-1}M_{\mathrm{off}})<1$.
\end{proof}
In the special case of strictly diagonally dominant matrices, Theorem \ref{t_2}
leads to a bound for the approximation of $\det(M)$ by the product of
the diagonal elements.
\begin{corollary}\label{c_2a}
If the complex square matrix $M=(m_{ij})_{1\leq i,j\leq n}$ is
strictly row diagonal dominant then
$${|\det(M)-\prod_{i=1}^n{m_{ii}}|\over |\prod_{i=1}^n{m_{ii}}|}
\leq c\rho\>e^{c\rho},
\qquad\mathrm{where}\quad
\rho\leq \max_i{\sum_{j=1,j\neq i}^n{\left|\frac{m_{ij}}{m_{ii}}\right|}},
\quad c\equiv -n\ln(1-\rho).$$
If also $c\rho<1$ then
$${|\det(M)-\prod_{i=1}^n{m_{ii}}|\over |\prod_{i=1}^ni{m_{ii}}|}
\leq {7\over 4}c\rho.$$
\end{corollary}
\begin{proof} This is a consequence of Gerschgorin's theorem
\cite[Theorem 7.2.1]{GovL96}.
\end{proof}
Corollary \ref{c_2a} implies that the product of diagonal elements
is a good approximation for $\det(M)$ if $M$ is strongly diagonally dominant.
\subsection{A Sequence of General Higher Order Approximations}\label{s_higher}
We extend the diagonal approximations in \S \ref{s_diag}
to a sequence of more general approximations
that become increasingly more accurate. These approximations, called
'zone determinant approximations' in \cite{LeeI03}, are justified in the
context of nuclear matter simulations.
As before, decompose $M=M_D+M_{\mathrm{off}}$ into diagonal blocks $M_D$ and off-diagonal
blocks $M_{\mathrm{off}}$ (actually, our results hold for any decomposition
$M=M_0+M_E$ where $M_0$ is non-singular and $\rho(M_0^{-1}M_E)<1$).
Below we give a sequence of approximations $\delta_m$ for $\ln(\det(M))$
and $\Delta_m$ for $\det(M)$, as well as absolute bounds for $\delta_m$
and relative bounds for $\Delta_m$.
An absolute bound for the logarithm suffices
because $\ln(\det(M))>1$ in our applications.
\begin{theorem}\label{t_3}
Let $M=M_D+M_{\mathrm{off}}$ be a complex matrix of order $n$,
$M_D$ be non-singular and $\rho\equiv\rho(M_D^{-1}M_{\mathrm{off}})<1$. Define
$$\delta_m\equiv\ln(\det(M_D))+\>\sum_{p=1}^m
{{(-1)^{p-1}\over p}\mathop{\mathrm{trace}}((M_D^{-1}M_{\mathrm{off}})^p)},\qquad
\Delta_m\equiv e^{\delta_m},\qquad m\geq 1.$$
Then
$$|\ln(\det(M))-\delta_m|\leq c\rho^m,\qquad
{|\det(M)-\Delta_m|\over |\Delta_m|}\leq c\rho^m\>e^{c\rho^m}$$
where $c\equiv -n\ln(1-\rho)$.
If also $c\rho^m<1$ then
$${|\det(M)-\Delta_m|\over |\Delta_m|}\leq {7\over 4}\>c\>\rho^m.$$
\end{theorem}
\begin{proof}
As in the proof of Theorem \ref{t_1a}
$\det(M)=\det(M_D)\det(I+A)$, where $A\equiv M_D^{-1}M_{\mathrm{off}}$
and $\mathop{\mathrm{log}}(I+A)=\sum_{p=1}^{\infty}{{(-1)^{p-1}\over p}A^p}$.
Hence
$$\mathop{\mathrm{trace}}\left(\mathop{\mathrm{log}}(I+A)\right)=\sum_{p=1}^{\infty}
{{(-1)^{p-1}\over p}\mathop{\mathrm{trace}}(A^p)}.$$
Define the truncated sums
$$L_m\equiv\sum_{p=1}^m{{(-1)^{p-1}\over p}\mathop{\mathrm{trace}}(A^p)},
\qquad D_m\equiv e^{L_m},\qquad m\geq 1.$$
Then
$$\mathop{\mathrm{trace}}\left(\mathop{\mathrm{log}}(I+A)\right)=L_m+z,\qquad
z\equiv \sum_{i=1}^n{\left\{\ln(1+\lambda_i(A))-
\sum_{p=1}^m{{(-1)^{p-1}\over p}\lambda_i(A)^p}\right\}}.$$
Applying to each of the $n$ terms the inequality
$$\left|\ln(1+\lambda)-\sum_{p=1}^m{{(-1)^{p-1}\over p}\lambda^p}\right|
\leq -\ln(1-|\lambda|)\>|\lambda|^m$$
\cite[4.1.24]{AbS72}, \cite[4.1.38]{AbS72}
gives $|\ln(\det(I+A))-L_m|\leq c\rho^m$. The first bound follows
now with $\delta_m=\ln(\det(M_D))+L_m$.
From the first bound, the fact that $\det(I+A)=D_me^z$, and
$|e^z-1|\leq |z|\> e^{|z|}$ \cite[4.2.39]{AbS72} follows
$${|\det(I+A)-D_m|\over |D_m|}\leq c\rho^m\>e^{c\rho^m}.$$
We get the second bound from $\Delta_m=\det(M_D)\>D_m$.
If also $c\rho^m<1$ then \cite[4.2.38]{AbS72}
$${|\det(I+A)-D_m|\over |D_m|}\leq {7\over 4}c\>\rho^m.$$
\end{proof}
The accuracy of the approximations in Theorem \ref{t_3}
is determined by the spectral radius
$\rho$ of $M_D^{-1}M_{\mathrm{off}}$. In particular, the absolute error bound for the
approximation $\delta_m$ is proportional to $\rho^m$, hence
the approximations tend to improve with increasing $m$.
The numerical results in Sections \ref{s_si} and \ref{s_app} illustrate
that the pessimistic factor in the bound
$|\ln(\det(M))-\delta_m|\leq -n\ln(1-\rho)\>\rho^m$ is $n$.
We found that replacing $n$ by the number of eigenvalues whose magnitude
is close to $\rho$ makes the bound tight.
The approximations for the logarithm can be determined from successive updates
$$\delta_0\equiv \ln(\det(M_D)),\qquad\delta_m=\delta_{m-1}+
{(-1)^{m-1}\over m}\mathop{\mathrm{trace}}((M_D^{-1}M_{\mathrm{off}})^m), \qquad m\geq 1,$$
and $\Delta_m=e^{\delta_m}$. Note that $e^{\delta_0}=\det(M_D)$
is the block diagonal approximation from (\ref{e_part}). Hence
Theorem \ref{t_2} is a special case of Theorem \ref{t_3} with $m=1$.
If a block diagonal determinant approximation is sufficiently
accurate, as in \S \ref{s_app}, it can be much cheaper
to compute than a determinant via Gaussian elimination.
\subsection{Checkerboard Matrices}\label{s_tridiag}
For this particular class of matrices, which occurs in our
applications \cite{LeeI03},
every other determinant approximation $\Delta_m$ has increased accuracy.
We call a matrix $M$ with equally sized blocks $M_{ij}$ of
dimension $n/k$ in (\ref{e_part}) an \textit{odd checkerboard matrix}
(with regard to the block size $n/k$)
if $M_{ij}=0$ for $i$ and $j$ both even or both odd, $1\leq i,j \leq k$;
and an \textit{even checkerboard matrix} if $M_{ij}=0$ for $i$ odd
and $j$ even or vice versa. An odd checkerboard matrix has zero
diagonal blocks, hence its trace is zero.
\begin{theorem}\label{t_checker}
If, in addition to the conditions of Theorem \ref{t_3},
$M_{\mathrm{off}}$ is an odd checkerboard matrix then
$$\delta_0=\ln(\det(M_D)),\qquad \delta_m=
\begin{cases}\delta_{m-1} & \mathrm{if}~m~\mathrm{is~odd} \cr
\delta_{m-2}-\mathop{\mathrm{trace}}\left((M_D^{-1}M_{\mathrm{off}})^m\over m\right)
& \mathrm{if}~m~\mathrm{is~even}.\end{cases}$$
\end{theorem}
\begin{proof}
If $A$ and $B$ are odd checkerboard matrices (with regard to the
same block size) then $AB$ is an even checkerboard matrix.
If $A$ is an odd checkerboard matrix and $B$ an even checkerboard
matrix then $AB$ and $BA$ are odd checkerboard matrices.
Since $M_D^{-1}M_{\mathrm{off}}$ is an odd checkerboard matrix, so are
the powers $(M_D^{-1}M_{\mathrm{off}})^p$ for odd $p$. This means
$\mathop{\mathrm{trace}}\left((M_D^{-1}M_{\mathrm{off}})^p\right)=0$ for odd $p$.
Hence the approximations in Theorem \ref{t_3}
satisfy $\delta_{m}=\delta_{m-1}$ for $m$ odd. For $m$ even
$$\delta_m=\delta_{m-2}+\left({(-1)^{m-1}\over m}\>
\mathop{\mathrm{trace}}\left((M_D^{-1}M_{\mathrm{off}})^m\right)\right)=
\delta_{m-2}-\left({\mathop{\mathrm{trace}}\left((M_D^{-1}M_{\mathrm{off}})^m\right)\over m}\right).$$
\end{proof}
Theorem \ref{t_checker} shows that an odd-order approximation is
equal to the previous even-order approximation. Hence the even-order
approximations gain one order of accuracy.
\section{Comparison with Sparse Inverse Approximations}\label{s_si}
In the special case of Hermitian positive-definite matrices,
we illustrate that block-diagonal determinant approximations
(see Corollary \ref{c_1}) can compare favourably with approximations
based on sparse approximate inverses \cite{Reu02}.
We also show that the accuracy of
sparse inverse approximations increases when more matrix
elements are included.
\paragraph{Idea}
To understand how sparse inverse approximations work,
we first consider a representation of the determinant based on
minors of the inverse \cite[\S 0.8.4]{HoJ85}.
If $M$ is Hermitian positive-definite of order $n$,
and $M_i$ is the leading principal submatrix of order $i$ of $M$,
then \cite[\S 0.8.4]{HoJ85} $\det(M)=\det(M_{n-1})/\sigma_n$,
where $\sigma_n\equiv (M^{-1})_{nn}$ is the trailing diagonal element
of $M^{-1}$. Using this expression recursively for $\det(M_{n-1})$
gives
$$\det(M)=\prod_{i=1}^{n}{\frac{1}{\sigma_i}},\qquad
\mathrm{where}\quad \sigma_i=(M_i^{-1})_{ii}.$$
Determinant approximations based on sparse approximate inverses
replace \textit{leading} principal submatrices $M_i$
by just principal submatrices $S_i$.
Specifically \cite[\S 3.2]{Reu02}, let $M$ be Hermitian positive-definite,
and let $S_i$ be a principal submatrix of $M_i$,
such that $S_i$ includes at least row $i$ and column $i$ of $M$.
The two extreme cases are $S_i=m_{ii}$ and $S_i=M_i$.
In any case, $m_{ii}$ is the trailing diagonal element of $S_i$,
i.e. $S_{n_i,n_i}=m_{ii}$, where $n_i$ is the order of $S_i$,
$1\leq n_i\leq i$.
Let $\sigma_i$ be the trailing diagonal element of $S_i^{-1}$,
i.e. $\sigma_i=(S_i^{-1})_{n_i,n_i}$. In particular $\sigma_1=m_{11}^{-1}$.
Given $n$ such submatrices $S_i$, $1\leq i\leq n$,
the sparse inverse approximation of $\det(M)$ is defined as
\cite[Algorithm 3.3]{Reu02}.
\begin{eqnarray}\label{e_si}
\sigma\equiv \prod_{i=1}^n{\frac{1}{\sigma_i}}.
\end{eqnarray}
The sparse approximate inverse method performs
Cholesky decompositions $S_i=L_iL_i^*$, where $L_i$ is lower triangular,
$2\leq i\leq n$, and computes $1/\sigma_i=((L_{i})_{n_i,n_i})^2$.
\paragraph{Monotonicity}
We show monotonicity of the sparse inverse approximations
in the following sense: If the dimensions of the submatrices $S_i$
are increased then the determinant approximations can only become better.
\begin{lemma}\label{l_invdiag}
If
$$M=\bordermatrix{&m&k\cr m&A&B\cr k&B^* &S}$$
is Hermitian positive-definite then
$(S^{-1})_{ii}\leq (M^{-1})_{m+i,m+i}$, $1\leq i\leq k$.
\end{lemma}
\begin{proof}
The proof follows from \cite[(4)]{Cot74} and the
Shermann-Morrison formula \cite[(2.1.4)]{GovL96}.
\end{proof}
Lemma \ref{l_invdiag} implies the following lower and upper bounds for
sparse inverse approximations; the lower bound was already derived in
\cite[(3.25)]{Reu02}.
\begin{corollary}\label{c_si}
If $M$ is Hermitian positive-definite and $\sigma$ is a sparse
inverse approximation in (\ref{e_si}) then
$$\det(M)\leq \sigma\leq \prod_{i}{m_{ii}}.$$
\end{corollary}
Corollary \ref{c_si} implies that the product of diagonal elements
cannot approximate the determinant more accurately than a sparse
inverse approximation. Another consequence of
Lemma \ref{l_invdiag} is the monotonicity of the sparse inverse
approximation in the following sense: If a principal submatrix ${\hat S}_j$
is replaced by a larger principal submatrix $S_j$ then the
determinant approximation can only become better.
\begin{theorem}\label{t_si}
Let $M$ be Hermitian positive-definite of order $n$.
If for some $1<j\leq n$, $S_j$ is a principal submatrix of $M_j$,
and in turn ${\hat S}_j$ is a principal submatrix of $S_j$ then
$$\det(M)\leq \prod_{i=1}^n{1\over\sigma_i}\leq
{1\over {\hat\sigma}_j}\prod_{i=1,i\neq j}^n{1\over\sigma_i}$$
where ${\hat\sigma}_j$ is the trailing diagonal
element of ${\hat S}_j^{-1}$.
\end{theorem}
The next example of block diagonal matrices illustrates that
sparse inverse approximations can be inaccurate, even when sparsity
is exploited to full extent.
\paragraph{Block-Diagonal Matrices}
Let
$$M=\begin{pmatrix}T_3 && \\ &\ddots& \\&&T_3\end{pmatrix}$$
be a block diagonal matrix of order $n=3k$ with $n/3$ diagonal blocks
$$T_3\equiv\begin{pmatrix}3/2&-1 &\cr-1&3/2& -1\cr &-1&3/2\end{pmatrix}.$$
The obvious block diagonal approximation (\ref{e_part}) with $k=n/3$ gives the
exact determinant $\det(M_D)=\det(M)=\det(T_3)^{n/3}=(3/8)^{n/3}$.
For the sparse inverse approximation (\ref{e_si}) we choose the submatrices
$$S_{(i-1)(n/3)+1}=3/2,\qquad
S_{(i-1)(n/3)+2}=\begin{pmatrix}3/2 &-1\cr -1& 3/2\end{pmatrix}=S_{i(n/3)},
\qquad 1\leq i\leq n/3.$$
The sparse inverse approximation of $\det(T_3)$ is $\det(T_3)+2/3$.
It has no accurate digit because the relative error is $16/9$.
The sparse inverse approximation of $\det(M)$ is
$\sigma=\left(\det(T_3)+2/3)\right)^{n/3}$.
For instance, when $n=300$ then $\det(M)\approx 4\cdot10^{17}$
while the sparse inverse approximation gives $\sigma\approx 4\cdot10^{33}$.
\paragraph{Tridiagonal Toeplitz Matrices}
A block diagonal approximation can be more accurate than a
sparse inverse approximation if the dimension of the blocks is
larger than 1. Let
$$T_n=\begin{pmatrix}2 & -1 &\cr
-1 & 2 & \ddots \cr
&\ddots&\ddots&-1\cr
& & -1&2\end{pmatrix}$$
be of order $n$; then $\det(T_n)=n+1$.
In the sparse inverse approximation (\ref{e_si}) we fully exploit sparsity by choosing
$S_1=2$ and $S_i=T_2$, $2\leq i\leq n$; hence the approximation
is $\sigma=2\>(3/2)^{n-1}$.
When $M_D$ in (\ref{e_part}) consists of $k$ equally sized blocks
of dimension $n/k$ then
$\det(M_D)=\left(\det(T_{n/k})\right)^k=\left((n/k)+1\right)^k$.
For a block size $n/k\geq 4$, $\det(M_D)\leq \sigma$, so
the block diagonal approximation is more accurate than
the sparse inverse approximation.
\paragraph{2-D Laplacian}
We show that for this matrix
both, the block-diagonal and the sparse inverse
approximations are accurate to at most one digit.
The coefficient matrix from the centered finite difference discretization of
Poisson's equation is a Hermitian positive-definite block
tridiagonal matrix \cite[9.1.1]{Gre97}
$$M=\begin{pmatrix}T_m &-I_m\cr -I_m&T_m&\ddots\cr
&\ddots&\ddots &-I_m\cr &&-I_m&T_m\end{pmatrix},\qquad
\mathrm{where}\quad
T_m=\begin{pmatrix}4 &-1\cr -1&4&\ddots\cr
&\ddots&\ddots &-1\cr &&-1&4\end{pmatrix}.$$
Here $T_m$ is of order $m$, and $M$ is of order $n=m^2$
(note that the matrix considered in \cite[\S 5]{Reu02} equals $(n+1)^2M$).
The exact determinant is \cite[Theorem 9.1.2]{Gre97}
$$\det(M)=\prod_{i,j=1}^m{4\left(\sin{\left({i\pi\over 2(m+1)}\right)}^2+
\sin{\left({j\pi\over 2(m+1)}\right)}^2\right)}.$$
We compute the logarithm of this expression and compare it to the
approximations.
A block diagonal approximation (\ref{e_part}) with $k=m$ gives
$\det(M_D)=\det(T_m)^m$, where \cite[\S 28.5]{Hig02}
$$\det(T_m)=\prod_{i=1}^m{\left(4+2\cos{i\pi\over m+1}\right)}.$$
If the matrices in the sparse inverse approximation (\ref{e_si}) are
$$S_1=4,\qquad S_i=\begin{pmatrix}4&-1\cr -1&4\end{pmatrix},\quad
2\leq i\leq m+1,\qquad
S_j=\begin{pmatrix}4&0&-1\cr 0&4&-1\cr -1&-1&4\end{pmatrix},\quad
m+2\leq j\leq n,$$
then $1/\sigma_1=4$, $1/\sigma_i=15/4$, $2\leq i\leq m+1$, and
$1/\sigma_j=7/2$, $m+2\leq j\leq n$. Thus the sparse
inverse approximation is $\sigma=4(15/4)^m(7/2)^{n-m-1}$.
Table \ref{table_2} lists errors for
the block diagonal and sparse inverse approximations for
$n=900$, $n=10000$ and $n=40000$. Columns 3 and 4 represent the
relative errors
$$|\ln(\det(M_D))-\ln(\det(M))|/|\ln(\det(M))|\qquad\mathrm{and}\qquad
|\ln(\sigma)-\ln(\det(M))|/|\ln(\det(M))|,$$
while columns 5 and 6 represent the relative errors
$$|\det(M_D)^{1/n}-\det(M)^{1/n}|/|\det(M)^{1/n}|\qquad\mathrm{and}\qquad
|\sigma^{1/n}-\det(M)^{1/n}|/|\det(M)^{1/n}|.$$
We include the last two errors to allow a comparison with the
approximation of $\det\left((n+1)^2M\right)^{1/n}$ in \cite[Table 5.1]{Reu02}.
The table shows that all relative errors lie between $0.06$ and $0.2$.
Hence both approximations, block diagonal and sparse inverse,
are accurate to at most one significant digit.
To estimate the tightness of the bound
$$|\ln(\det(M_D))-\ln{\det(M)}|\leq (-n\>|\ln(1-\rho)|)\>\rho$$
in Theorem \ref{t_3},
consider the case $n=900$. Here $\rho(M_D^{-1}M_{\mathrm{off}})\approx .9898$
and $|\ln(1-\rho)|\approx 4.5845$.
The true error is
$$|\ln(\det(M_D))-\ln{\det(M)}|\approx 122.4966\approx 26\ln(1-\rho)\>\rho.$$
The matrix $M_D^{-1}M_{\mathrm{off}}$ has 26 eigenvalues with magnitude at least $0.9$.
Thus the pessimism of the bound comes from the factor $n$.
\begin{table}
\begin{center}
\begin{tabular}{|c|l|l|l|l|l|}\hline
$n$&$\ln(\det(M))$&rel. error& rel. error& rel error&rel. error\\
&&in $\ln(M_D)$& in $\ln(\sigma)$&in $M_D^{1/n}$& in $\sigma^{1/n}$\\
\hline
900& 1.0650e+03&0.1150& 0.0607& 0.1458& 0.0745\\
10000& 1.1717e+04& 0.1246& 0.0698& 0.1572& 0.0852\\
40000& 4.6761e+04& 0.1269& 0.0719& 0.1599& 0.0877\\
\hline
\end{tabular}
$$\qquad$$
\end{center}
\caption{Errors in the block diagonal approximation $M_D$ and
the sparse inverse approximation $\sigma$
for the Laplacian.}\label{table_2}
\end{table}
\section{Application to Neutron Matter Simulations}\label{s_app}
In \cite{LeeI03} we consider the quantum simulation of
nuclear matter on a lattice, and in particular how to calculate the
contribution of nucleon-nucleon-hole loops at non-zero nucleon
density. The resulting method, called zone determinant expansion,
is based on the sequence of approximations in Theorem \ref{t_3}.
Here we illustrate that 3 iterations of the zone determinant
expansion give an approximation accurate to 3 digits, and that the
method uses less space than a determinant computation based on Gaussian
elimination (with partial or complete pivoting).
In \cite{LeeI03} we derive a particle interaction matrix $M$
whose determinant $\det(M)$ is not positive, and complex in general.
Hence stochastic methods such as
hybrid Monte Carlo methods \cite{DuKPR87,GoLTRS87,ScSS86}
do not give the correct sign or phase of $\det(M)$.
This was the motivation for approximating
$\ln(\det(M))$ via a zone determinant expansion, i.e. Theorem \ref{t_3}.
Below we discuss the structure of $M$ and a physically appropriate
zone determinant expansion.
The particle interactions are considered on a 4-dimensional lattice
(3 dimensions for space and one for time).
Let the dimensions of the lattice be $L\times L\times L\times L_t$,
where $L_t$ represents the time direction.
Also let the number of particles per lattice point be $s$.
Then the interaction matrix $M$ has dimension $n\times n$
where $n=L^3L_ts$. We partition the lattice into separate spatial
zones (or cubes) of dimension $m\times m \times m$ (constraints on $m$
are discussed in \cite{LeeI03}).
Therefore particle interactions between any two zones are
represented by matrix blocks of dimension $m^3L_ts$.
As a consequence, it makes sense to approximate $\det(M)$ by the
product of principal minors associated
with particle interactions inside spatial zones.
Without loss of generality we assume that the lattice points
are ordered such that the submatrix $M_{ij}$ of order $m^3L_ts$
represents particle interactions between zones $i$ and $j$.
With $k\equiv (L/m)^3$ this gives the partitioning
$M=M_D+M_{\mathrm{off}}$ in (\ref{e_part}), where $M_D$ represents
particle interactions in the zone interiors,
while $M_{\mathrm{off}}$ represents interactions among different zones.
In \cite{LeeI03} we explain that the spectral radius
$\rho\equiv\rho(M_D^{-1}M_{\mathrm{off}})$ can be reduced
by increasing the dimension $m$ of the spatial zones.
\begin{figure}
\begin{center}
\resizebox{3in}{!}
{\includegraphics*{spy.eps}}
\end{center}
\caption{Sparsity structure of the interaction matrix $M$.}\label{f_spy}
\end{figure}
We illustrate the zone expansion on a small lattice simulation,
where we can compare the approximations to the exact determinant.
Specifically we consider the interactions between neutrons and neutral
pions, on a $4^3\times 4$ grid. The order of the interaction matrix $M$ is
$4^3\times 4\times 2=512$. Its properties are listed in Table \ref{table_3}.
\begin{table}
\begin{tabular}{|l|l|l|}
\hline
order & $n=512$ & \\
number of non-zeros & $9n$ & see Figure \ref{f_spy}\\
structure & complex non-Hermitian & see Figure \ref{f_blockspy}\\
norm & $\|M\|_F\approx 49.5$ & \\
condition number & $\|M\|_1\|M^{-1}\|_1\approx 177$ &
condest$(\cdot)$ in MATLAB 6\\
non-normality & $\|M^*M-MM^*\|_F\approx 57$ &\\
eigenvalues & complex & see Figure \ref{f_evm} \\
determinant & $\det(M)=8.5361\cdot 10^{65}+1.4168\cdot 10^{64}\imath$
& $\det(\cdot)$ in Matlab 6\\
&$\ln(\det(M))=151.81+0.016599\imath$ &\\
\hline
\end{tabular}
\caption{Properties of the interaction matrix $M$.}\label{table_3}
\end{table}
\noindent
\begin{figure}
\begin{center}
\resizebox{2in}{!}
{\includegraphics*{blockspy.eps}}
$\qquad\qquad\qquad$
\resizebox{1in}{!}
{\includegraphics*{diagspy.eps}}
\end{center}
\caption{Non-zero $8\times 8$ blocks in the interaction matrix $M$, and
sparsity structure of a single $8\times 8$ diagonal block.}\label{f_blockspy}
\end{figure}
In the context of the particular application in \cite{LeeI03},
we can partition the lattice into zones with dimension $m=1$.
The resulting partitioning has blocks
$M_{ij}\equiv M_{8(i-1)+1:8i,8(j-1)+1:j}$, $1\leq i,j\leq 64$,
of dimension $4\times 2=8$. Thus $k=64$ in the block diagonal
approximation (\ref{e_part}).
Figure \ref{f_blockspy} shows the distribution of the 448 blocks
with non-zero elements.
Each diagonal block $M_{ii}$ contains 24 non-zero elements,
its sparsity structure is shown in Figure \ref{f_blockspy}.
\begin{figure}
\begin{center}
\resizebox{3in}{!}
{\includegraphics*{evm.eps}}
\end{center}
\caption{Eigenvalue distribution of the interaction matrix $M$.}\label{f_evm}
\end{figure}
\begin{figure}
\begin{center}
\resizebox{1.5in}{!}
{\includegraphics*{xspy.eps}}
$\qquad$
\resizebox{1.5in}{!}
{\includegraphics*{fspy.eps}}
\end{center}
\caption{Sparsity structure of the matrices $M_D^{-1}M_{\mathrm{off}}$ and
$(M_D^{-1}M_{\mathrm{off}})^2$.}\label{f_zone}
\end{figure}
\begin{figure}
\begin{center}
\resizebox{1in}{!}
{\includegraphics*{checker.eps}}
\end{center}
\caption{Sparsity structure of the leading principal submatrix of order 32
of $M_D^{-1}M_{\mathrm{off}}$.}\label{f_checker}
\end{figure}
The zone partitioning is bipartite, i.e.
$M_{ij}=0$ for $i$ and $j$ both even or both odd, and $i\neq j$,
$1\leq i,j\leq k$.
Therefore $M_{\mathrm{off}}$ is an odd checkerboard matrix.
Figure \ref{f_checker} illustrates this checkerboard pattern in the
leading principal submatrix of order 32 of $M_D^{-1}M_{\mathrm{off}}$.
The sparsity structures of the matrices
$M_D^{-1}M_{\mathrm{off}}$ and $(M_D^{-1}M_{\mathrm{off}})^2$
is shown in Figure \ref{f_zone}. Because of the checkerboard structure
Theorem \ref{t_checker} implies
$\mathop{\mathrm{trace}}((M_D^{-1}M_{\mathrm{off}})^p)=0$ for odd $p$, and
$\delta_{p-1}=\delta_{p}$. Table \ref{table_1} therefore contains
only approximations of even order.
\begin{table}
\begin{center}
\begin{tabular}{|c|l|l|l|l|l|l|}\hline
$j$ &abs. error &abs. error &abs. error &$\rho^j$& rel. error &rel. error \\
&in $\Re(\delta_j)$ &in $\Im(\delta_j)$ &in $\delta_j$ &&in $\delta_j$&
in $\Delta_j$ \\
\hline
0&5.1000 & 0.0017&5.1000&& 0.0348&163.0282\\
2& 0.4817& 0.0025&0.4817&0.4374&0.0032&0.3823\\
4& 0.0909&0.0016&0.0909&0.1913&0.0006&0.0951\\
6& 0.0225&0.0008&0.0226&0.0837&0.0001&0.0223\\
8& 0.00665& 0.0003&0.0066& 0.0366&0.00004& 0.0066\\
\hline
\end{tabular}
$$\qquad$$
\end{center}
\caption{Errors in the approximations $\delta_j$ and $\Delta_j$
for the interaction matrix $M$.}\label{table_1}
\end{table}
Table \ref{table_1} shows errors in the approximations $\delta_j$
and $\Delta_j$ for approximations up to order 8.
Columns 2, 3 and 4 represent the absolute errors
$$|\Re(\ln(\det(M)))-\Re(\delta_j)|, \qquad
|\Im(\ln(\det(M)))-\Im(\delta_j)|,\qquad
|\ln(\det(M))-\delta_j|.$$
Columns 6 and 7 represent the relative errors
$$|\ln(\det(M))-\delta_j|/|\delta_j|\qquad
\mathrm{and}\qquad |\det(M)-\Delta_j|/|\Delta_j|.$$
The spectral radius $\rho\equiv\rho(M_D^{-1}M_{\mathrm{off}})\approx .6613$,
and the constant
in the error bounds of Theorems \ref{t_2} and \ref{t_3} is $c\approx 554$.
Table \ref{table_1} illustrates that
$|\ln(\det(M))-\delta_j|\approx\rho^j$,
i.e. the absolute errors in the logarithm are almost proportional
to the powers of the spectral radius of $M_D^{-1}M_{\mathrm{off}}$.
In this case the constant $c$ is too pessimistic, because
many eigenvalues of $M_D^{-1}M_{\mathrm{off}}$ have magnitude much less than $\rho$.
For instance, 160 eigenvalues of $M_D^{-1}M_{\mathrm{off}}$ have magnitude $10^{-15}$.
The imaginary parts of the logarithms appear to converge faster
than the real parts.
The block diagonal approximation $\delta_0\equiv \ln(\det(M_D))$
for $\ln(\det(M))$ has
an accuracy of 2 digits. Two more iterations give an approximation
$\delta_2$ that is accurate to 3 digits.
\begin{figure}
\begin{center}
\resizebox{1.5in}{!}
{\includegraphics*{pmqspy.eps}}
$\qquad$
\resizebox{1.5in}{!}
{\includegraphics*{lpqspy.eps}}
$\quad$
\resizebox{1.5in}{!}
{\includegraphics*{upqspy.eps}}
\end{center}
\caption{Sparsity structure of the matrices $PMQ$, $L$ and $U$ from the
LU decomposition (with complete pivoting) of $M$.}\label{f_lu}
\end{figure}
We briefly compare the computation of $\delta_0$ and $\delta_2$
to a determinant computation by Gaussian elimination of $M$.
Gaussian elimination with complete pivoting gives $PMQ=LU$, where
$P$ and $Q$ are permutation matrices, $L$ is unit lower triangular and $U$ is
upper triangular.
Figure \ref{f_lu}, which shows the sparsity
structure of the matrices $PMQ$, $L$ and $U$, illustrates that
Gaussian elimination with complete pivoting completely destroys the
sparsity structure of $M$. The matrices $L$ and $U$ together have
about $162n$ non-zeros, compared to $9n$ in $M$.
In contrast, the determinant expansion
requires no significant additional space for $\delta_0$; and $48n$
non-zeros for $M_D^{-1}M_{\mathrm{off}}$ and $n$ non-zeros for
the trace of $(M_D^{-1}M_{\mathrm{off}})^2$. That's $(48+1)n=49n$ non-zeros,
about one third of the non-zeros produced by Gaussian elimination with
complete pivoting. Gaussian elimination with partial pivoting
essentially preserves the sparsity structure of $M$ but produces
$342n$ non-zeros.
\subsection*{Acknowledgements}
We thank Gene Golub, Nick Higham, Volker Mehrmann, and Gerard Meurant for
helpful discussions.
\input{paper.bbl}
\end{document}
|
1,314,259,996,470 | arxiv | \section{Introduction}
Analytic perturbation theory tells us that if $V$ is relatively bounded to
$H_0$, then the spectrum of $H_\lambda=H_0+\lambda V$ is at a Hausdorff distance
of order $|\lambda|$ from the spectrum of $H_0$. This property is
not true for singular perturbations (like for example the magnetic perturbation
coming from a constant field), neither in the discrete nor in the
continuous case.
Maybe the first proof of spectral stability of discrete
Harper operators with respect to the variation of the intensity $b\geq
0$ of the external magnetic field is due to Elliott \cite{Ell}. The result is
refined in \cite{BEY} where it is shown that the gap boundaries are
$\frac{1}{3}$-H\"older continuous in $b$. Later results by Avron, van
Mouche and Simon \cite{AMS}, Helffer and Sj\"ostrand \cite{He-Sj1,
He-Sj2}, and Haagerup and R{\o}rdam \cite{HR}
pushed the exponent up to $\frac{1}{2}$. In fact they prove
more, they show that the Hausdorff distance between spectra behaves like
$|b-b_0|^{\frac{1}{2}}$. These results are
optimal in the sense that the H\"older constant is independent of the
length of the eventual gaps, and it is known that these gaps can close down
precisely like $|b-b_0|^{\frac{1}{2}}$
near rational values of $b_0$ \cite{He-Sj2, HKS}. Note that Nenciu
\cite{Nen3} proves a similar result for a much larger class of
discrete Harper-like operators. Many other spectral properties of
Harper operators can be found in a paper by Herrmann and Janssen \cite{HJ}.
In the continuous case, the stability of gaps for Schr\"odinger
operators was first shown by Avron
and Simon \cite{AS}, and Nenciu \cite{Nen2}. In \cite{If} a very general
result is obtained for perturbations of the anisotropic Laplacean. In \cite{AMP} spectral
continuity is proven for a large class of Hamiltonians defined by elliptic symbols. Nenciu's result implicitly
gives a $\frac{1}{2}$-H\"older continuity in $b$ for the Hausdorff
distance between spectra. Then in \cite{BC} the H\"older exponent of
gap edges was pushed up to $\frac{2}{3}$.
The first proof of Lipschitz continuity of gap edges
for discrete Harper-like operators
was given by Bellissard \cite{Bell} (later on Kotani
\cite{Ko} extended his method to more general regular lattices and
dimensions larger than two). Very recently a completely different proof was given
in \cite{Cornean}.
Our main technical result in this paper is Theorem \ref{teorema1}, extending a previous result of Nenciu \cite{Nen3} and
asserting H\"{o}lder continuity of a specific order for a class of bounded self-adjoint
operators having a locally integrable integral kernel satisfying a
weighted Schur-Holmgren estimate \eqref{novem1}. This result, combined
with the magnetic quantization \cite{KO,MP1,MPR2} and the associated
magnetic pseudodifferential calculus developped in
\cite{MPR1,IMP1,IMP2,LMS}, allow us to we prove Theorem \ref{teorema2}
stating H\"{o}lder continuity of order $1/2$ of the spectrum of
resolvents associated to a large class of elliptic Hamiltonians in a $BC^\infty$
magnetic field, with respect to the intensity of the magnetic
field. The case of unbounded operators will be considered elsewhere.
\subsection{The setting and the main result}
Consider the Hilbert space $L^2(\mathbb{R}^d)$ with $d\geq 2$. Let
$\langle x\rangle:=\sqrt{1+|{\bf x}|^2}$ and let $\alpha\geq 0$.
We consider bounded integral operators
$T\in B(L^2(\mathbb{R}^d))$ to which we can associate a locally integrable kernel
$T({\bf x},{\bf x}')$ which is continuous outside the diagonal and obeys the
following weighted Schur-Holmgren estimate:
\begin{equation}\label{novem1}
||T||_{1,\alpha}:= \max\left\{ \sup_{{\bf x}'\in
{\mathbb R}^d}\int_{{\mathbb R}^d}|T({\bf x},{\bf x}')|\langle {\bf x}-{\bf x}'\rangle^\alpha d{\bf x}, \;\sup_{{\bf x}\in
{\mathbb R}^d}\int_{{\mathbb R}^d}|T({\bf x},{\bf x}')|\langle {\bf x}-{\bf x}'\rangle^\alpha d{\bf x}'\right\}<\infty.
\end{equation}
Let us denote the set of all these operators with
$\mathcal{C}_{1,\alpha}$. When $\alpha=0$, we need to introduce a
uniformity condition.
Let $\chi$ be the characteristic function of the interval $[0,1]$ and
define
\begin{equation}\label{chichi}
\mathbb{{\mathbb R}}^d\times \mathbb{{\mathbb R}}^d\ni ({\bf x},{\bf x}')\mapsto
\chi_M({\bf x},{\bf x}'):=\chi(|{\bf x}-{\bf x}'|/M),\quad M\geq 1.
\end{equation}
If $T\in \mathcal{C}_{1,0}$ we denote by $T_M$ the operator given by the
integral kernel $\chi_M({\bf x},{\bf x}')T({\bf x},{\bf x}')$.
Then we define $\mathcal{C}_{\rm unif}$ to be the subset of $\mathcal{C}_{1,0}$
consisting of operators obeying the estimate
\begin{equation}\label{chichi2}
\lim_{M\to \infty}||T-T_M||_{1,0}=0.
\end{equation}
Note that if we only consider kernels $T({\bf x},{\bf x}')$ which are dominated
by $L^1$ functions of ${\bf x}-{\bf x}'$, then
$\mathcal{C}_{\rm unif}=\mathcal{C}_{1,0}$.
For $T\in \mathcal{C}_{1,\alpha}$, we are interested in a family of Harper-like operators
$\{T_b\}_{b\in{\mathbb R}}$ given by kernels of the form $e^{ib
\varphi({\bf x},{\bf x}')}T({\bf x},{\bf x}')$ with $\varphi: \mathbb{{\mathbb R}}^d\times
\mathbb{{\mathbb R}}^d\mapsto \mathbb{R}$ a continuous phase function satisfying the two properties:
\begin{align}\label{iunie1}
\varphi({\bf x},{\bf x}')=-\varphi({\bf x}',{\bf x})\quad {\rm and}\quad
|\varphi({\bf x},{\bf y})+\varphi({\bf y},{\bf x}')-\varphi({\bf x},{\bf x}')|\leq |{\bf x}-{\bf y}|\; |{\bf y}-{\bf x}'|.
\end{align}
Clearly, $\{T_b\}_{b\in{\mathbb R}}\subset
\mathcal{C}_{1,\alpha}$.
The Hausdorff distance between two real compact sets $A$ and $B$ is
defined as:
\begin{align}\label{kiki1}
d_H(A,B):=\max\left \{\sup_{x\in A}\inf_{y\in B}|x-y|,
\; \sup_{y\in B}\inf_{x\in A}|x-y|\right \}.
\end{align}
And here is our main technical result:
\begin{theorem}\label{teorema1} Let $H$ be self-adjoint and consider
a family of Harper-like operators $\{H_b\}_{b\in{\mathbb R}}$ as above. Then the map
$${\mathbb R}\ni b\mapsto d_H(\sigma(H_{b}),\sigma(H))\in {\mathbb R}_+$$ is continuous if
$H\in \mathcal{C}_{\rm unif}$. Moreover, if $H\in
\mathcal{C}_{1,\alpha}$ with $\alpha>0$, then the above map is
H\"older continuous with exponent $\beta:=\min\{1/2,\alpha/2\}$. More
precisely, for all $b_0$ we can find a constant $C>0$ such that:
\begin{align}\label{kiki2}
d_H(\sigma(H_{b_0+\delta}),\sigma(H_{b_0}))\leq C\;|\delta|^{\beta}.
\end{align}
\end{theorem}
\vspace{0.5cm}
{\bf Remark 1}. Denoting by $\delta =b-b_0$, then according to our notations
we have that $H_b=\left (H_{b_0}\right )_{\delta}$. It means that it
is enough to prove the theorem at $b_0=0$.
{\bf Remark 2}. It is natural to ask if the condition $H\in
\mathcal{C}_{1,\alpha}$ is
optimal in order to insure a H\"older continuity of order $\min\{1/2,\alpha/2\}$; we believe in any case that
if $\alpha$ becomes smaller and smaller, one cannot expect the H\"older coefficient to remain $1/2$. Similarly, if $\alpha=0$
it is unlikely to expect more than continuity of the Hausdorff distance.
\section{Proof of Theorem \ref{teorema1}}
Let $g\in C_0^\infty({\mathbb R}^d)$ with $0\leq g\leq 1$, $g({\bf x})=1$ if
$|{\bf x}|\leq 1/2$ and $g({\bf x})=0$ if $|{\bf x}|\geq 2$. If ${\bf y}\in{\mathbb R}^d$, denote
by $g_{{\bf y}}({\bf x})=g({\bf x}-{\bf y})$. By standard arguments, we may assume that
$\sum_{\gamma\in \mathbb{Z}^d}g_{\gamma}^2({\bf x})=1$ for all
${\bf x}\in{\mathbb R}^d$. For each $g_\gamma$ there is a finite number of neighbors
whose supports are not disjoint from ${\rm supp}(g_\gamma)$, uniformly
in $\gamma$.
Denote by $g_{{\bf y},b}({\bf x}):=g_{{\bf y}}(b^{1/2}{\bf x})=g(b^{1/2}{\bf x}-{\bf y})$. In this
way we constructed a
locally finite, quadratic partition of unity obeying
\begin{equation}\label{kiki3}
\sum_{\gamma\in \mathbb{Z}^d}g_{\gamma,b}^2({\bf x})=1,\quad {\bf x}\in{\mathbb R}^d,
\end{equation}
and if $V_{\gamma,b}$ denotes the set of functions
$g_{\gamma',b}$ whose supports are not disjoint from the support of
$g_{\gamma,b}$, then $\sup_{\gamma\in \mathbb{Z}^d}\; \#\{V_{\gamma,b}\}$
is independent of $b$. Moreover, if $\chi_{\gamma,b}$ is
the characteristic function of the support of $g_{\gamma,b}$ we have:
\begin{align}\label{kiki4}
&{\rm supp}(g_{\gamma,b})\subset \{{\bf x}\in{\mathbb R}^d:\;
|{\bf x}-b^{-1/2}\gamma|\leq 2 b^{-1/2}\},\\
\label{kiki5}
&|g_{\gamma,b}({\bf x})-g_{\gamma,b}({\bf y})|\leq ||\;|\nabla g|\;||_\infty^{\epsilon}\;
b^{\epsilon/2}|{\bf x}-{\bf y}|^\epsilon\;\{\chi_{\gamma,b}({\bf x})+
\chi_{\gamma,b}({\bf y})\},\quad 0\leq
\epsilon\leq 1.
\end{align}
\begin{lemma}\label{lemma1}
Let $\{T_\gamma\}_{\gamma\in\mathbb{Z}^d}\subset B(L^2({\mathbb R}^d))$
possibly depending on $b$ such
that
\begin{equation}\label{kiki6}
|||T|||_\infty:=\sup_{\gamma\in\mathbb{Z}^d}||T_\gamma||<\infty.
\end{equation}
Define on compactly supported functions the
maps
$$\psi\mapsto \Gamma(T)(\psi):=\sum_{\gamma\in\mathbb{Z}^d}\chi_{\gamma,b}\;
T_\gamma\;
\chi_{\gamma,b}\psi,\quad \tilde{\Gamma}(T)(\psi):=
\sum_{\gamma\in\mathbb{Z}^d}\chi_{\gamma,b}\;
\left\vert T_\gamma\;
\chi_{\gamma,b}\psi\right\vert.$$
Then both $\Gamma(T)$ and $\tilde{\Gamma}(T)$ can be extended by
continuity to
bounded maps on $L^2({\mathbb R}^d)$ and
there exists a constant $C$ independent of
$b$ such that $\max\{||\tilde{\Gamma}(T)||,\;||\Gamma(T)||\}\leq C\; |||T|||_\infty$.
\end{lemma}
\begin{proof} Let $\psi\in L^2({\mathbb R}^d)$ with compact
support. We have:
\begin{align}\label{kiki7}
||\Gamma(T)(\psi)||^2&\leq \sum_{\gamma\in\mathbb{Z}^d}
\sum_{\gamma'\in V_{\gamma,b}} |\left \langle \chi_{\gamma',b}\;
T_{\gamma'}\;
\chi_{\gamma',b}\psi ,\chi_{\gamma,b}\;
T_\gamma\;
\chi_{\gamma,b}\psi\right \rangle| \nonumber \\
&\leq \sum_{\gamma\in\mathbb{Z}^d}\sum_{\gamma'\in V_{\gamma,b}}||T_{\gamma'}\;
\chi_{\gamma',b}\psi||\; ||T_{\gamma}\;
\chi_{\gamma,b}\psi||\nonumber \\
&\leq \frac{|||T|||_\infty^2}{2}
\sum_{\gamma\in\mathbb{Z}^d}\sum_{\gamma'\in V_{\gamma,b}}\left (
||\chi_{\gamma',b}\psi||^2+
||\chi_{\gamma,b}\psi||^2\right)\leq C\;|||T|||_\infty^2||\psi||^2,
\end{align}
where in the last inequality we used:
$$\sum_{\gamma\in\mathbb{Z}^d}\sum_{\gamma'\in V_{\gamma,b}}
||\chi_{\gamma',b}\psi||^2=\int_{{\mathbb R}^d}|\psi({\bf x})|^2\left \{\sum_{\gamma\in\mathbb{Z}^d}
\sum_{\gamma'\in
V_{\gamma,b}}\chi_{\gamma',b}({\bf x})\right\}d{\bf x}\leq
C||\psi||^2.$$
The same proof also works for $\tilde{\Gamma}(T)$ since the linearity
is not used. Note that
$$||\tilde{\Gamma}(T)(\psi_1)-\tilde{\Gamma}(T)(\psi_2)||\leq
||\tilde{\Gamma}(T)(\psi_1-\psi_2)||$$
which is enough for proving continuity.
\end{proof}
\vspace{0.5cm}
\begin{lemma}\label{lemma2}
Let $A$ be a positivity preserving bounded linear operator and
define on compactly supported
functions $\psi$ the following positively homogeneous map:
$$\hat{\Gamma}_A(T)(\psi):=\sum_{\gamma\in\mathbb{Z}^d}\chi_{\gamma,b}\;
A\left\vert T_\gamma\;
\chi_{\gamma,b}\psi\right\vert.$$
Then $\hat{\Gamma}_A(T)$ can be extended by continuity to a bounded
map on the whole
space and $||\hat{\Gamma}_A(T)||\leq C\; ||A||\;|||T|||_\infty$.
\end{lemma}
\begin{proof}
We note that:
$$\left \vert
\hat{\Gamma}_A(T)(\psi_1)-\hat{\Gamma}_A(T)(\psi_2)\right\vert \leq
\sum_{\gamma\in\mathbb{Z}^d}\chi_{\gamma,b}\;
A\big\vert T_\gamma\;
\chi_{\gamma,b}(\psi_1-\psi_2)\big\vert=\hat{\Gamma}_A(T)(\psi_1-\psi_2)$$
due to the positivity preserving of $A$. Thus boundedness implies
continuity. But the proof of Lemma \ref{lemma1} can be repeated almost
identically, and the proof is over.
\end{proof}
\vspace{0.5cm}
\subsection{The case $\alpha>0$}
If $z\in \rho(H)$, denote by $R(z)=(H-z)^{-1}$. We construct the
operators
$$T_\gamma(z):=e^{ib\varphi(\cdot,b^{-1/2}\gamma)}g_{\gamma,b}R(z)g_{\gamma,b}
e^{-ib\varphi(\cdot,b^{-1/2}\gamma)},\quad T(z):=\{T_\gamma(z)\}_{\gamma\in\mathbb{Z}}.$$ Then
$|||T(z)|||_\infty\leq 1/{\rm dist}(z,\sigma(H))$. Introduce the
notation
$$fl({\bf x},{\bf y},{\bf x}'):=\varphi({\bf x},{\bf y})+\varphi({\bf y},{\bf x}')-\varphi({\bf x},{\bf x}').$$
The operator
$\Gamma(T(z))$ is bounded (see Lemma \ref{lemma1}). If ${\rm Id}$
denotes the identity operator, we can compute (use \eqref{kiki3}):
\begin{equation}\label{kiki8}
(H_b-z)\Gamma(T(z))={\rm Id}+S(z)
\end{equation}
where
\begin{align}\label{kiki9}
&(S(z)\psi)({\bf x})\nonumber \\
&:=\sum_{\gamma\in\mathbb{Z}^d}e^{ib\varphi({\bf x},b^{-1/2}\gamma)}
\int_{{\mathbb R}^d}d{\bf x}'H({\bf x},{\bf x}')
\left \{e^{ibfl({\bf x},{\bf x}',b^{-\frac{1}{2}}\gamma)}-1\right\}g_{\gamma,b}({\bf x}')
\left\{R(z)
g_{\gamma,b}e^{-ib\varphi(\cdot,b^{-\frac{1}{2}}\gamma)}\psi\right\}({\bf x}')\nonumber
\\
&+\sum_{\gamma\in\mathbb{Z}^d}e^{ib\varphi({\bf x},b^{-1/2}\gamma)}
\int_{{\mathbb R}^d}d{\bf x}'H({\bf x},{\bf x}')
\left\{g_{\gamma,b}({\bf x}')-g_{\gamma,b}({\bf x})\right\}
\left\{R(z)
g_{\gamma,b}e^{-ib\varphi(\cdot,b^{-\frac{1}{2}}\gamma)}\psi\right\}({\bf x}')\nonumber
\\
&=:(S_1(z)\psi)({\bf x})+(S_2(z)\psi)({\bf x}).
\end{align}
Let us analyze the contribution of the first term
$(S_1(z)\psi)({\bf x})$. Using the inequality (see also \eqref{iunie1})
$$\left \vert e^{ib fl({\bf x},{\bf x}',b^{-\frac{1}{2}}\gamma)}-1\right\vert \leq
2^{1-\epsilon}b^\epsilon|{\bf x}-{\bf x}'|^\epsilon
|{\bf x}'-b^{-\frac{1}{2}}\gamma|^\epsilon,\quad 0\leq \epsilon\leq 1,$$
we have:
\begin{align}\label{kiki10}
&|S_1(z)\psi({\bf x})|\\
&\leq 2^{1-\epsilon}b^\epsilon
\int_{{\mathbb R}^d}d{\bf x}'|H({\bf x},{\bf x}')|\; |{\bf x}-{\bf x}'|^\epsilon
\;\sum_{\gamma\in\mathbb{Z}^d}
g_{\gamma,b}({\bf x}')\; |{\bf x}'-b^{-1/2}\gamma|^\epsilon
\left\vert R(z)
g_{\gamma,b}e^{-ib\varphi(\cdot,b^{-1/2}\gamma)}\psi\right\vert
({\bf x}')\nonumber.
\end{align}
With the notation $L_\gamma:=g_{\gamma,b}(\cdot)
|\cdot-b^{-1/2}\gamma|^\epsilon R(z)
g_{\gamma,b}e^{-ib\varphi(\cdot,b^{-1/2}\gamma)}$ we see that the
above inequality can be written as:
$$
|S_1(z)\psi({\bf x})|\leq 2^{1-\epsilon}b^\epsilon
\int_{{\mathbb R}^d}d{\bf x}'|H({\bf x},{\bf x}')|\; |{\bf x}-{\bf x}'|^\epsilon\left\{
\tilde{\Gamma}(L)\psi\right\} ({\bf x}').
$$
Using the fact that on the support of $g_{\gamma,b}$ we have
$|{\bf x}'-b^{-1/2}\gamma|\leq 2b^{-1/2}$ it follows that
$|||L|||_\infty\leq Cb^{-\epsilon/2}||R(z)||$, thus:
\begin{align}\label{kiki11}
||S_1(z)||\leq C\;\frac{b^{\epsilon/2}}
{{\rm dist}(z,\sigma(H))}\;||H||_{1,\epsilon} .
\end{align}
Let us analyze the contribution from $S_2(z)$. Using \eqref{kiki5} we
can write:
\begin{align}\label{kiki12}
&|S_2(z)\psi|({\bf x}) \\
&\leq C\; b^{\epsilon/2}\sum_{\gamma\in\mathbb{Z}^d}
\int_{{\mathbb R}^d}d{\bf x}'|H({\bf x},{\bf x}')|\;
|{\bf x}-{\bf x}'|^\epsilon\{\chi_{\gamma,b}({\bf x})+
\chi_{\gamma,b}({\bf x}')\}
\;
\left\vert R(z)
g_{\gamma,b}e^{-ib\varphi(\cdot,b^{-1/2}\gamma)}\psi\right\vert
({\bf x}')\nonumber\\
&\leq C\; b^{\epsilon/2}\sum_{\gamma\in\mathbb{Z}^d}\chi_{\gamma,b}({\bf x})
\int_{{\mathbb R}^d}d{\bf x}'|H({\bf x},{\bf x}')|\;
|{\bf x}-{\bf x}'|^\epsilon\left\vert R(z)
g_{\gamma,b}e^{-ib\varphi(\cdot,b^{-1/2}\gamma)}\psi\right\vert
({\bf x}')\nonumber \\
&+ C\; b^{\epsilon/2}
\int_{{\mathbb R}^d}d{\bf x}'|H({\bf x},{\bf x}')|\;
|{\bf x}-{\bf x}'|^\epsilon\sum_{\gamma\in\mathbb{Z}^d}\chi_{\gamma,b}({\bf x}')\left\vert R(z)
g_{\gamma,b}e^{-ib\varphi(\cdot,b^{-1/2}\gamma)}\psi\right\vert
({\bf x}').\nonumber
\end{align}
Now denoting with $A$ the operator with integral kernel $|H({\bf x},{\bf x}')|\;
|{\bf x}-{\bf x}'|^\epsilon$ and with $L_\gamma=R(z)
g_{\gamma,b}e^{-ib\varphi(\cdot,b^{-1/2}\gamma)}$ we obtain $|S_2(z)\psi|\leq C\;
b^{\epsilon/2}\left\{\hat{\Gamma}_A(L)(\psi)
+A\tilde{\Gamma}(L)(\psi)\right\}$ thus
\begin{align}\label{kiki13}
||S_2(z)||\leq C\; \frac{b^{\epsilon/2}}
{{\rm dist}(z,\sigma(H))}\;||H||_{1,\epsilon}.
\end{align}
Going back to \eqref{kiki8} we obtain the estimate:
\begin{align}\label{kiki14}
||S(z)||\leq C\; \frac{b^{\epsilon/2}}
{{\rm dist}(z,\sigma(H))}\;||H||_{1,\epsilon}.
\end{align}
Now choose $0<\epsilon=\min\{\alpha,1\}$. It follows that
$||S(z)||\leq 1/2$ for every $z$ with
${\rm dist}(z,\sigma(H))\geq 2C\; b^{\epsilon/2}||H||_{1,\epsilon}$, and by a standard
argument it follows from \eqref{kiki8} that $z\in\rho(H_b)$. Thus for
every $x\in\sigma(H_b)$ we must have ${\rm dist}(x,\sigma(H))\leq 2C \;
b^{\epsilon/2}
||H||_{1,\epsilon}$, thus
$$\sup_{x\in \sigma(H_b)}\inf_{y\in\sigma(H)}|x-y|\;\leq \; 2C\;
b^{\min\{\alpha/2,1/2\}}
||H||_{1,\min\{\alpha,1\}}.$$
Now we can interchange $H_b$ with $H$ because
$$H({\bf x},{\bf x}')=e^{-ib\phi({\bf x},{\bf x}')}\left
\{e^{ib\phi({\bf x},{\bf x}')}H({\bf x},{\bf x}')\right\}=
e^{-ib\phi({\bf x},{\bf x}')}H_b({\bf x},{\bf x}')$$ and the $||\cdot||_{1,\alpha}$ norms
are invariant with respect to the multiplication with a unimodular
phase. Hence the Theorem is proved in the case $\alpha>0$.
\vspace{0.5cm}
\subsection{The case $\alpha=0$}
Due to our uniformity condition in \eqref{chichi2} we can approximate
$H_b$ in operator norm ({\it uniformly in $b$})
with a sequence of operators $(H_b)_M$ which have
strong localization near their diagonal. More precisely, given
$\epsilon>0$ there exists $M=M(\epsilon)$ large enough such that
$||H_b-(H_b)_M||\leq \epsilon/3$ for every $b\in\mathbb{R}$.
If $d(z,\sigma(H_b))>\epsilon/3$, then by writing
$$(H_b)_M-z=[{\rm Id}-(H_b-(H_b)_M)(H_b-z)^{-1}](H_b-z)$$
it follows that $z\notin\sigma((H_b)_M)$. It means that for every
$x\in\sigma((H_b)_M)$ we must have
$d(x,\sigma(H_b))\leq\epsilon/3$. By reversing the roles of $H_b$ and
$(H_b)_M$ we conclude that $d_H(\sigma(H_b), \sigma((H_b)_M))\leq
\epsilon/3$, uniformly in $b\geq 0$. But now both $(H_b)_M$ and $H_M$
have strong localization near the diagonal, thus we can apply the
result from $\alpha>0$, obtaining a $b(\epsilon)>0$ such that for
every $|b|\leq b(\epsilon)$ we have $d_H(\sigma(H_M), \sigma((H_b)_M))\leq
\epsilon/3$. The proof is finished by the triangle inequality.
\qed
\section{Magnetic Hamiltonians}
Let us consider in ${\mathbb R}^d$ a magnetic field $B$ with components of
class $BC^\infty({\mathbb R}^d)$, i.e. bounded, smooth and with all its
derivatives bounded. Consider a Hamiltonian given by a
real elliptic symbol $h$ of class $S^m_1({\mathbb R}^d\times{\mathbb R}^d)$ with $m>0$,
i.e. $h\in C^\infty_{\text{\sf pol}}({\mathbb R}^d\times{\mathbb R}^d)$ verifying the estimates:
$$
\forall(a,\alpha)\in\mathbb{N}^d\times\mathbb{N}^d,\ \exists C(a,\alpha)\in{\mathbb R}_+,\quad\underset{(x,\xi)\in{\mathbb R}^d\times{\mathbb R}^d}{\sup}<\xi>^{|\alpha|-m}\left|\left(\partial_x^a\partial_\xi^\alpha h\right)(x,\xi)\right|\leq C(a,\alpha),
$$
$$
\exists(R,C)\in{\mathbb R}^2_+,\quad |\xi|\geq R\Rightarrow h(x,\xi)\geq C|\xi|^m,\ \forall x\in{\mathbb R}^d.
$$
For our magnetic field $B$ we can choose a vector potential $A$ having
components of class $C^\infty_{\text{\sf pol}}({\mathbb R}^d)$; this can always
be achieved by working with the transverse gauge:
$$
A_j(x):=-\sum\limits^d_{k=1}\int_0^1ds\,B_{jk}(sx)sx_k.
$$
Let us denote by $\mathfrak{Op}^A(h)$ the magnetic quantization of $h$ defined as in \cite{MP1}. Then, this operator is self-adjoint on the magnetic Sobolev space $H^m_A({\mathbb R}^d)$ and essentially self-adjoint on the space of Schwartz test functions (see Definition 4.2 and Theorem 5.1 in \cite{IMP1}). Moreover this operator is lower semibounded and satsfies a G{\aa}rding type inequality (Theorem 5.3 in \cite{IMP1}). Thus for any $\mathfrak{z}\in\mathbb{C}\setminus[-a_0,+\infty)$, with $a_0>0$ large enough, we have that the following inverse exist
$$
\left(\mathfrak{Op}^A(h)-\mathfrak{z}\mathds{1}\right)^{-1}=\mathfrak{Op}^A\left(r^B_{\mathfrak{z}}\right)
$$
and is defined by a symbol $r^B_{\mathfrak{z}}$ of class
$S^{-m}_1({\mathbb R}^d\times{\mathbb R}^d)$ (see Proposition 6.5 in \cite{IMP2}). But
using now Lemma A.4 in \cite{MPR1} and the fact that evidently
$S^m_1({\mathbb R}^d\times{\mathbb R}^d)\subset S^m\big({\mathbb R}^d;BC_u({\mathbb R}^d)\big)$ with
$S^m\big({\mathbb R}^d;BC_u({\mathbb R}^d)\big)$ as in Definition A.3 of \cite {MPR1},
(or Proposition 1.3.3 of \cite{ABG}), we conclude that the symbol
$r^B_{\mathfrak{z}}$ has a partial Fourier transform (with respect to
the second variable) of class $L^1\big({\mathbb R}^d;BC_u({\mathbb R}^d)\big)$. In fact
looking closer to the proof of Lemma A.4 in \cite{MPR1} allows us to
conclude (see also Proposition 1.3.6. in \cite{ABG}) that the partial
Fourier transform
$\mathfrak{F}_2^{-1}\big[r^B_{\mathfrak{z}}\big]({\bf x},{\bf y})$ has rapid decay in the second variable. Now, using formulas 3.28 and 3.29 in \cite{MP1}, we conclude that $\mathfrak{Op}^A\left(r^B_{\mathfrak{z}}\right)$ is an integral operator with kernel
$$
K^A(r^B_{\mathfrak{z}})({\bf x},{\bf y}):=\left[\tilde{\Lambda}^AS^{-1}(\mathbf{1}\otimes\mathcal{F}^{-1}_{2})r^B_{\mathfrak{z}}\right]({\bf x},{\bf y})
$$
with
$$
\tilde{\Lambda}^A({\bf x},{\bf y}):=\exp\left\{-i\int_{{\bf x}}^{{\bf y}}A\right\},\quad
S^{-1}({\bf x},{\bf y}):=\left(\frac{{\bf x}+{\bf y}}{2},{\bf x}-{\bf y}\right).
$$
In conclusion:
$$
K^A(r^B_{\mathfrak{z}})({\bf x},{\bf y})=\exp\left\{-i\int_{{\bf x}}^{{\bf y}}A\right\}\left\{\left[\mathbf{1}\otimes\mathcal{F}^{-1}_{2}\right]r^B_{\mathfrak{z}}\right\}\left(\frac{{\bf x}+{\bf y}}{2},{\bf x}-{\bf y}\right).
$$
Let us also notice that if we denote by $<0,{\bf x},{\bf y}>$ the triangle of vertices $0,{\bf x},{\bf y}$, we have that
$$
\left|\int_{{\bf x}}^{{\bf y}}A+\int_{{\bf y}}^{{\bf x}'}A-\int_{{\bf x}}^{{\bf x}'}A\right|=\left|\int_{<{\bf x},{\bf y},{\bf x}'>}
B\right|\leq\|B\|_\infty|{\bf x}-{\bf y}|\;|{\bf x}'-{\bf y}|,
$$
with $\|B\|_\infty:=\underset{j,k}{\max}\underset{{\bf x}\in{\mathbb R}^d}{\sup}|B_{j,k}({\bf x})|$.
Let us conclude that for any such magnetic field and elliptic symbol
$h$, the resolvent $R=\big(\mathfrak{Op}^A(h)+a\big)^{-1}$ is a
bounded self-adjoint operator having a locally integrable integral
kernel of the form $e^{i\varphi_B({\bf x},{\bf x}')}T_B({\bf x},{\bf x}')$ with
$$
\varphi_B({\bf x},{\bf x}'):=-\int_{{\bf x}}^{{\bf x}'}A,\qquad T_B({\bf x},{\bf x}'):=
\big[S^{-1}(\mathbf{1}\otimes\mathcal{F}^{-1}_{2})r^B_{\mathfrak{z}}\big]({\bf x},{\bf x}').
$$
Now let us consider a magnetic field $B_0$ with components of class
$BC^\infty({\mathbb R}^d)$ and a small variation of it, in the same class,
$B_b({\bf x}):=B_0({\bf x})+b\mathfrak{b}({\bf x})$ with $b\in[0,1]$. Given an
elliptic symbol $h$ as before we now have two Hamiltonians
$H:=\mathfrak{Op}^{A_0}(h)$ and $H':=\mathfrak{Op}^{A}(h)$, with $A_0$
a vector potential for $B_0$ and $A$ a vector potential fos $B$. We
can write $A({\bf x})=A_0({\bf x})+b\mathfrak{a}({\bf x})$ with $\mathfrak{a}$ a vector
potential for $\mathfrak{b}$. Then we have the following result:
\begin{theorem}\label{teorema2}
For $h$, $B_0$ and $B_b$ as above, consider $H=\mathfrak{Op}^{A_0}(h)$ and
$H'=\mathfrak{Op}^{A}(h)$. For $a>0$ large enough we define the two associated resolvents as above:
$$
R:=(H+a)^{-1}=\mathfrak{Op}^{A_0}(r^{B_0}_{-a}),\ \text{with integral kernel:}\ e^{-i\left[\int_{{\bf x}}^{{\bf x}'}A_0\right]}\big[S^{-1}(\mathbf{1}\otimes\mathcal{F}^{-1}_{2})r^{B_0}_{-a}\big]({\bf x},{\bf x}'),
$$
$$
R':=(H'+a)^{-1}=\mathfrak{Op}^{A_0}(r^{B_0}_{-a}),\ \text{with
integral kernel:}\ e^{-i\left[\int_{{\bf x}}^{{\bf x}'}A\right]}\big[S^{-1}(\mathbf{1}\otimes\mathcal{F}^{-1}_{2})r^{B}_{-a}\big]({\bf x},{\bf x}').
$$
Then there exists a constant $C$ only
depending on the symbol $h$ and on the magnetic field $B_0$ such that we have the
following estimate:
$$
d_H\big(\sigma(R),\sigma(R')\big)\leq C\sqrt{b}.
$$
\end{theorem}
\begin{proof}
Let us remark that the kernels
$S^{-1}(\mathbf{1}\otimes\mathcal{F}^{-1}_{2})r^{B_0}_{-a}$ and
$S^{-1}(\mathbf{1}\otimes\mathcal{F}^{-1}_{2})r^{B}_{-a}$ are the
integral kernels of the operators given by the usual quantization
(without magnetic field)
$\mathfrak{Op}$ of the symbols $r^{B_0}_{-a}$ and resp. $r^{B}_{-a}$.
\begin{proposition}\label{b-est-symb-rez}
Being symbols of negative order, both $r^{B_0}_{-a}$ and $r^{B}_{-a}$ define bounded operators on $L^2({\mathbb R}^d)$ and we have that $\|\mathfrak{Op}(r^{B_0}_{-a})-\mathfrak{Op}(r^{B}_{-a})\|_{1,0}\leq Cb$.
\end{proposition}
\begin{proof}
Using the ideas and results in \cite{MP1} we shall use the magnetic Moyal composition $\sharp^B$ defined by the quantization associated to the field $B$. Let us compute (as tempered distributions):
$$
(h+a)\sharp^Br^{B_0}_{-a}-1=(h+a)\sharp^Br^{B_0}_{-a}-(h+a)\sharp^{B_0}r^{B_0}_{-a}=:s^b_{-a}.
$$
Due to the general theory developped in \cite{IMP1, IMP2} $s^b_{-a}$ is defined by a symbol of class $S^0_1({\mathbb R}^d\times{\mathbb R}^d)$ that can be computed by the following oscillating integral:
$$
\big[(h+a)\sharp^Br^{B_0}_{-a}-1\big]({\bf x},\xi)=(2\pi)^{-2d}\int_\Xi\int_\Xi
d{\bf y} d\eta d{\bf x}' d\zeta
$$
$$
\times
e^{-2i(<{\bf x}',\eta>-<{\bf y},\zeta>)}\big[\omega^B({\bf x},{\bf y}-{\bf x},{\bf x}'-{\bf x})-\omega^{B_0}({\bf x},{\bf y}-{\bf x},{\bf x}'-{\bf x})\big](h+a)({\bf x}-{\bf y},
\xi-\eta)r^{B_0}_{-a}({\bf x}-{\bf x}',\xi-\zeta)
$$
$$
=ib(2\pi)^{-2d}\int_\Xi\int_\Xi d{\bf y} d\eta d{\bf x}' d\zeta\
e^{-2i(<{\bf x}',\eta>-<{\bf y},\zeta>)}\omega^{B_0}({\bf x},{\bf y}-{\bf x},{\bf x}'-{\bf x})
\theta_b({\bf x},{\bf y}-{\bf x},{\bf x}'-{\bf x})
$$
$$
\times (h+a)({\bf x}-{\bf y},\xi-\eta)r^{B_0}_{-a}({\bf x}-{\bf x}',\xi-\zeta),
$$
where
$$
\theta_b({\bf x},{\bf y}-{\bf x},{\bf x}'-{\bf x})=e^{-ib\int_{<{\bf x}+{\bf y}-{\bf x}',{\bf x}+{\bf x}'-{\bf y},{\bf x}-{\bf y}-{\bf x}'>}\mathfrak{b}}-1
$$
is a function of class $BC^\infty\big({\mathbb R}^d;C^\infty_{\text{\sf pol}}({\mathbb R}^d\times{\mathbb R}^d)\big)$ and we have the following estimates for its derivatives:
$$
\left|\big(\partial_{{\bf x}}^\rho\partial_{{\bf y}}^\mu\partial_{{\bf x}'}^\nu\theta_b\big)({\bf x},{\bf y}-{\bf x},{\bf x}'-{\bf x})\right|\leq
C_{\rho,\mu,\nu}b^{1+|\rho+\mu+\nu}|{\bf y}|^{|\mu|}|{\bf x}'|^{|\nu|}.
$$
Now using Proposition 8.45 in \cite{IMP2} we conclude that $(h+a)\sharp^Br^{B_0}_{-a}-1$ is a symbol of type $S^0_1({\mathbb R}^d\times{\mathbb R}^d)$ with seminorms of order at least $b$ and using Remark 3.3 in \cite{IMP1} we conclude that it defines a bounded operator with norm of order $b$. Thus for $b$ small enough we can invert $1+s^b_{-a}$ and obtain that (using once again Proposition 8.45 in \cite{IMP2} and the Calderon-Vaillancourt Theorem \cite{Fo})
$$
r^{B}_{-a}=r^{B_0}_{-a}\sharp^B\left\{1+s^b_{-a}\right\}^{-_B},\quad r^{B}_{-a}-r^{B_0}_{-a}=-r^{B}_{-a}\sharp^Bs^b_{-a},
$$
$$
\|\mathfrak{Op}(r^{B_0}_{-a})-\mathfrak{Op}(r^{B}_{-a})\|_{1,0}\leq Cb.
$$
\end{proof}
\vspace{0.5cm}
Now we shall consider the bounded self-adjoint operators $R_b$ with
the kernel
$e^{-i\left[\int_{{\bf x}}^{{\bf x}'}A\right]}\big[S^{-1}(\mathbf{1}\otimes\mathcal{F}^{-1}_{2})r^{B_0}_{-a}\big]({\bf x},{\bf x}')$.
Due to the above Proposition, by replacing $R'$ with $R_b$ we make an
error of order $b$ in operator norm on $L^2({\mathbb R}^d)$. Now we see that
$R_b$ is a Harper-type family, for which we can apply the results of
Section 2. Here, $R=R_0$. We note that the integral kernels of $R_b$
have a common factor independent of $b$ which is of class $C_{1,\alpha}$ for any
$\alpha\geq0$. Moreover, the integral kernels of $R_b$ only differ by a unimodular exponential factor
$e^{ib\varphi({\bf x},{\bf x}')}$ where
$\varphi({\bf x},{\bf x}'):=-\int_{{\bf x}}^{{\bf x}'}\mathfrak{a}$ satisfies
\eqref{iunie1}.
Therefore Theorem \ref{teorema1} implies that
$d_H(\sigma(R_b),\sigma(R))\leq C\; \sqrt{b}$, and since
$d_H(\sigma(R_b),\sigma(R'))\leq C\; b$ it follows that
\begin{equation}\label{iunie28}
d_H(\sigma(R'),\sigma(R))\leq C\; \sqrt{b},
\end{equation}
which finishes the proof of the theorem.
\end{proof}
\vspace{0.5cm}
\noindent {\bf Acknowledgments.} H.C. acknowledges support from the Danish F.N.U. grant Mathematical
Physics, and thanks Gheorghe Nenciu for many fruitful discussions. R.P.
acknowledges support
from CNCSIS grant PCCE 8/2010 {\it Sisteme
diferentiale in analiza neliniara si aplicatii},
and thanks Aalborg University for hospitality.
|
1,314,259,996,471 | arxiv |
\section{Introduction}
\label{sec:Introduction}
Supersymmetry (SUSY) and the
Minimal Supersymmetric Standard
Model~\cite{Nappi:1982bc, Dine:1982cd,Alvarez:1982de, Dine:1993yw, Dine:1994vc,Dine:1995ag}
are extensions of the standard model (SM) that
provide explanations for several outstanding
issues with the SM. In particular, SUSY addresses
the large quantum corrections to the mass term
in the Higgs potential~\cite{Barb} and provides a viable
dark matter candidate~\cite{DM1,DM2}. Models with
general gauge-mediated (GGM) SUSY
breaking~\cite{Dimopoulos:1996fg,Martin:1997hi, Poppitz:1997gh, Meade:2008wd,Buican:2008ws, Abel:2009ve, Carpenter:2009jk, Dumitrescu:2010ef}
have the additional benefit of naturally suppressing
flavor violations in the SUSY sector. GGM models
can have a wide range of features but typically result in final states
that include the gravitino (\PXXSG) as the lightest supersymmetric particle (LSP).
The next-to-lightest supersymmetric particle (NLSP) in these models
is often taken to be a neutralino (\PSGczDo), which is a mixture
of the bino, neutral wino, and neutral higgsinos.
The conservation of
$R$ parity~\cite{FF} implies that the gravitino is stable and remains undetected.
Therefore, proton-proton (\Pp\Pp) collisions
that produce SUSY particles will have an imbalance in the
total observed transverse momentum, referred to as missing
transverse momentum \ptvecmiss and defined
as the negative vector sum
of the transverse momenta of all visible particles in an event.
Its magnitude is referred to as \ptmiss.
If the composition of the neutralino NLSP is primarily bino-like, its main decay will be to a
gravitino and a photon (\Pgg), resulting in final
states with significant missing transverse momentum and one or more photons.
This paper presents a search for GGM SUSY in final states involving two
photons and missing transverse momentum. The data sample, corresponding to an
integrated luminosity of 35.9\fbinv of $\Pp\Pp$ collisions at a
center-of-mass energy $\sqrt{s} = 13\TeV$,
was collected with the CMS detector in 2016.
The analysis described here achieves a substantial improvement in sensitivity
compared to the search performed by the CMS Collaboration on the
smaller 2015 data set~\cite{CMS:2016_anal} and is comparable in sensitivity
to similar searches from the ATLAS Collaboration~\cite{ATLAS:2016aa,ATLAS:2018_atlas}.
Two simplified model frameworks~\cite{bib-sms-1,bib-sms-2,bib-sms-3,bib-sms-4,expPaper}
are used for the interpretation of the results.
The T5gg model assumes gluino (\PSg) pair production and the T6gg model assumes
squark (\PSq) pair production. The models assume a 100\% branching fraction
for the gluinos and squarks to decay as shown in Fig.~\ref{fig:gluinoSquarkDecay}.
The squarks in the T6gg model can be either first or second generation.
We assume a 100\% branching fraction for the NLSP neutralino to decay
to a nearly massless gravitino and a photon, $\PSGczDo \to \PXXSG\Pgg$,
resulting in characteristic events with large \ptmiss and two photons.
In order for the analysis to be as model independent as possible,
we choose not to define the signal region using any hadronic variables such as jet multiplicity
or the scalar sum of the transverse momentum of the jets.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{Figure_001-a.pdf}
\includegraphics[width=0.45\textwidth]{Figure_001-b.pdf}
\caption{Diagrams showing the production of signal events in the collision
of two protons (\Pp). In gluino
(\PSg) pair production in the T5gg simplified model (left), the gluino
decays to a quark-antiquark pair (\cPq\cPaq) and a neutralino (\PSGczDo). In
squark (\PSq) pair production in the T6gg simplified model (right), the
squark decays to a quark and a neutralino. In both cases, the
neutralino subsequently decays to a photon (\Pgg) and a gravitino (\PXXSG).}
\label{fig:gluinoSquarkDecay}
\end{figure}
Standard model processes such as direct diphoton production or events with
jets produced through the strong interaction, referred to as
quantum chromodynamics (QCD) multijet events, can result in events with two photons.
If the hadronic activity in the event is
poorly measured, these processes
can mimic the signal topology even though they lack genuine \ptmiss.
For the case of QCD multijet events, there may be real photons in the event,
or jets rich in electromagnetic (EM) energy that are misreconstructed as photons.
Events with genuine \ptmiss also contribute to the composition of the candidate sample.
These events are mainly from $\PW\Pgg$ and
\PW+jet(s) production, where an electron is misidentified as a photon in $\PW\to \Pe\PGn$ decays.
A smaller background arises from $\cPZ\Pgg\Pgg$ events where the {\cPZ} boson decays to two
neutrinos, $\cPZ\to\cPgn\cPagn$.
\section{Detector, data, and simulated samples}
\label{sec:DataSimSamples}
The central feature of
the CMS apparatus is a superconducting solenoid of 6\unit{m} internal diameter,
providing a magnetic field of 3.8\unit{T}. Within the solenoid volume are a
silicon pixel and strip tracker covering the pseudorapidity region $\abs{\eta} <
2.5$, as well as a lead tungstate crystal electromagnetic calorimeter (ECAL),
and a brass and scintillator hadron calorimeter (HCAL), each composed of a
barrel and two endcap regions and covering the range $\abs{\eta} < 3.0$. Forward
calorimeters extend the coverage up to $\abs{\eta} < 5.0$. Muons are measured in
gas-ionization detectors embedded in the steel flux-return yoke outside the
solenoid and cover the range $\abs{\eta} < 2.4$. A more detailed description of the
CMS detector, together with a definition of the coordinate system used and the
relevant kinematic variables, can be found in Ref.~\cite{Chatrchyan:2008zzk}.
Events of interest are selected using a two-tiered trigger system~\cite{Khachatryan:2016bia}.
The first level is composed of custom hardware processors and uses
information from the calorimeters and muon detectors to select events at a rate of
around 100\unit{kHz}. The second level, known as the high-level trigger,
consists of a farm of processors running a version of the full event reconstruction
software optimized for fast processing. This trigger reduces the event rate to around 1\unit{kHz}
before data storage. This analysis used a diphoton trigger to collect the data.
The trigger requires a leading (subleading) photon with transverse momentum $\pt > 30$ (18)\gev,
and a combined invariant mass $m_{\Pgg\Pgg} > 95$\gev. The photons are
also required to pass isolation and cluster shape requirements.
Monte Carlo (MC) simulations are used for several purposes in this analysis. Simulations
of the signal processes are used to determine signal efficiencies;
background process simulation is used for validation of the analysis performance
and to model the contribution from $\cPZ\Pgg\Pgg\to\cPgn\cPagn\Pgg\Pgg$
events. The event generator \MGvATNLO~2.3.3~\cite{Alwall:2014hca}
is used to simulate the signal samples at leading order.
The background samples are generated at next-to-leading order using \MGvATNLO~2.4.2.
For both signal and background processes, the parton showering, hadronization,
SUSY particle decays, multiple-parton interactions, and the underlying event are described by the
\PYTHIA 8.212~\cite{Sjostrand:2007gs} program with the CUETP8M1~\cite{CMS:CUETP8M1_cite} generator tune.
The signal samples are generated with either two gluinos or two squarks and up to two additional
partons in the matrix element calculation.
The parton distribution
functions (PDFs) are obtained from the NNPDF3.0~\cite{Ball:2014uwa} set. For the background
processes, the detector response is simulated using
\GEANTfour~\cite{Agostinelli:2002hh}, while the CMS fast
simulation~\cite{Abdullin:2011zz, Giammanco:2014bza} is used for the signal events.
For both signal and background simulated events,
additional $\Pp\Pp$ interactions (pileup) are generated with \PYTHIA
and superimposed on the primary collision process.
The simulated events are reweighted to match the pileup distribution observed in data.
The signal events were generated using the T5gg and T6gg simplified models and
are characterized by the masses of the particles in the decay chain. For the
gluino (squark) mass we simulate a range of values from 1.4 to 2.5 (1.2 to
2.0)\TeV in steps of 50\GeV. These mass ranges were selected to overlap and
expand upon the mass ranges excluded by previous
searches~\cite{ATLAS:2016aa,CMS:2016_anal}.
The neutralino masses range from 10\GeV up to the mass of the gluino or squark.
The cross sections are calculated at next-to-leading-order
(NLO) accuracy including the resummation of soft gluon emission
at next-to-leading-logarithmic (NLL) accuracy~\cite{Beenakker:1996ch,Kulesza:2008jb,Kulesza:2009kq, Beenakker:2009ha,Beenakker:2011fu},
with all the unconsidered sparticles assumed to be heavy and decoupled.
The uncertainties in the cross sections are calculated as
described in Ref.~\cite{Borschensky:2014cia}.
\section{Event selection}
\label{sec:EventSelect}
Photon, electron, muon, charged and neutral hadron candidates are
reconstructed with the particle-flow event algorithm~\cite{CMS-PRF-14-001},
which reconstructs particles based on information from all detector subsystems.
The energy of photons is directly obtained from the ECAL measurement.
The energy of electrons is determined from a combination of the electron momentum
at the primary interaction vertex as determined by the tracker,
the energy of the corresponding ECAL cluster, and the energy sum of all
bremsstrahlung photons spatially compatible with originating from the electron track.
The energy of muons is obtained from the curvature of the corresponding track.
The energy of charged hadrons is determined from a combination of their momentum
measured in the tracker and the matching ECAL and HCAL energy deposits,
corrected for zero-suppression effects and for the response function of the
calorimeters to hadronic showers. Finally, the energy of neutral hadrons
is obtained from the corresponding corrected ECAL and HCAL energy.
Photon candidates are required to satisfy a series of identification criteria
to ensure a high purity~\cite{Khachatryan:2015iwa}. The shape of the energy deposit in the ECAL
must be consistent with that of an EM shower, and
the amount of energy present in the corresponding region of the HCAL must not exceed 5\% of the
ECAL energy, since EM showers are expected to be contained almost
entirely within the ECAL. To ensure high trigger efficiency,
we require all photons to satisfy $\pt > 40$\GeV.
Because the SUSY signal models used in this analysis produce photons primarily
in the central region of the detector and because the magnitude of the
background increases considerably at high $\abs{\eta}$, we consider only
photons within the barrel fiducial region of the detector ($\abs{\eta} < 1.44$).
To suppress quark and gluon jets that mimic photons, photon candidates
are required to be isolated from other reconstructed particles. Separate
requirements are made on the scalar \pt sums of charged and neutral
hadrons and EM objects in a cone of radius
$\Delta R \equiv \sqrt{\smash[b]{ (\Delta\eta)^{2} + (\Delta\phi)^{2}}} \equiv 0.3$ around the photon candidate. Each \pt
sum is corrected for the effect of pileup, and in each case the momentum of the photon candidate itself is
excluded. We further require that the photon candidate has no pixel detector track
seed, to distinguish the candidate from an electron.
For the purpose of defining the various control regions used in the analysis,
we apply an additional set of selection criteria. A misidentified ``fake'' photon ($f$) is
defined as a photon candidate that satisfies looser requirements on
photon isolation and neutral-hadron isolation and fails
either the shape requirement for the ECAL clusters or the charged-hadron
isolation requirement. In order to ensure that misidentified
photons do not differ too much from our photon selection, upper limits are
applied to both the charged-hadron isolation and cluster shape requirements.
Importantly, because of the large amount of hadronic activity expected in our SUSY signal events,
it is possible that real photons from the decay of a neutralino could fail the
charged-hadron isolation requirement and therefore fall into the misidentified photon category.
In order to avoid this potential signal contamination from SUSY events in the control regions,
we additionally require
that misidentified photons satisfy $\RNINE < 0.9$, where $\RNINE$ is defined as the ratio
of the energy deposited in a 3{$\times$}3 array of ECAL crystals to
the total energy in the cluster~\cite{Khachatryan:2015iwa}. Real photons have values
of $\RNINE$ close to unity, so by requiring $\RNINE < 0.9$ we ensure that real photons from
a possible SUSY signal will not enter our control regions.
Because of the similarity of the ECAL response to electrons and photons, $\cPZ\to\Pe\Pe$
events are used to measure the
photon identification efficiency. The selection of electron candidates is
identical to that of photons, with the exception that the candidate is required
to be matched to a pixel detector seed consistent with a track, to ensure that the electron
selection is orthogonal to that of photons.
The photon efficiency is measured via the tag-and-probe method~\cite{Khachatryan:2015iwa}.
The ratio of the observed to simulated efficiency is found to be consistent
with unity and independent of \pt and $\eta$.
The efficiency of the pixel detector seed veto for photons
is measured in $\cPZ\to\PGm\PGm\Pgg$ events and is found to
agree between data and simulation.
Events are then assigned to one of four mutually exclusive categories depending
on the selection of their highest \pt EM objects: $\Pgg\Pgg$,
$\Pe\Pe$, \ensuremath{f \! f}\xspace, and $\Pe\Pgg$. The two
EM objects are required to be separated by $\Delta R > 0.6$. Finally, because of the
trigger requirements described in Section~\ref{sec:DataSimSamples},
the invariant mass of the two
EM objects is required to be greater than 105\gev.
In addition to the requirements already described, any event with a muon
satisfying $\pt > 25\gev$ and $\abs{\eta} < 2.4$ as well as track quality
and isolation requirements is vetoed.
Similarly, we veto events with any additional electrons satisfying
$\pt > 25\gev$, $\abs{\eta} < 2.5$, and signal shape and isolation
requirements.
Events in the candidate $\Pgg\Pgg$ sample are divided into the low-\ptmiss
control region ($\ptmiss < 100$\GeV) and the high-\ptmiss signal region
($\ptmiss > 100$\GeV). The signal region is further divided into
six \ptmiss bins that were chosen such that there is a
sufficient number of events from the \ensuremath{f \! f}\xspace control sample in each bin.
\section{Estimation of backgrounds}
\label{sec:EstBkg}
As described in Section~\ref{sec:Introduction}, there are three primary backgrounds
to this analysis. QCD processes such as multijet production
can emulate the signal topology if the hadronic activity
in the event is mismeasured. A second background arises
from electroweak (EWK) processes that have genuine \ptmiss from the production of neutrinos.
There is also a small contribution from $\PZ\Pgg\Pgg\to\Pgg\Pgg\cPgn\cPagn$ events.
The contribution from the QCD background is estimated from the observed data using
the \ensuremath{f \! f}\xspace control sample. The ratio of the event yield in the candidate
$\Pgg\Pgg$ sample to that in the \ensuremath{f \! f}\xspace sample
is constructed as a function of \ptmiss. More \ensuremath{f \! f}\xspace events are observed
at high \ptmiss relative to the $\Pgg\Pgg$ sample. Different functional
forms were investigated to model the \ptmiss dependence, and an exponential
function was found to describe the data the best.
We fit the $\Pgg\Pgg$ to \ensuremath{f \! f}\xspace ratio in the
$\ptmiss < 100$\GeV control region. The predicted number of
QCD background events ($N_{\mathrm{QCD}}^i$) in bin $i$ of
the signal region is then given by the following equation, where
$N_{\ensuremath{f \! f}\xspace}^{i}$ is the number of observed \ensuremath{f \! f}\xspace events and
$g_{\mathrm{ave}}^{i}$ is the average value of the fit function $g(\ptmiss)$
in that bin:
\begin{equation}
N_{\mathrm{QCD}}^{i} = g_{\mathrm{ave}}^{i} \ N_{\ensuremath{f \! f}\xspace}^{i}
\end{equation}
In order to set a systematic uncertainty on the method,
we derive a second QCD background prediction by noting that
the \ptmiss distribution of the \ensuremath{f \! f}\xspace control sample is dependent on
the $\RNINE$ requirement on the misidentified photons.
An alternate \ensuremath{f \! f}\xspace control sample is built using photon candidates that satisfy
all of the requirements for misidentified photons as
outlined in Section~\ref{sec:EventSelect}, with
the exception that the $\RNINE$ requirement is reversed.
In the $\ptmiss < 100$\GeV control region, we perform an exponential fit
to the ratio of the event yield in the high-$\RNINE$ \ensuremath{f \! f}\xspace sample to that of the
nominal, low-$\RNINE$ \ensuremath{f \! f}\xspace sample.
This function ($h(\ptmiss)$) represents
the correction required to account for the effect of the $\RNINE$ selection
on the \ptmiss distribution. The size of the correction is between
20 and 40\% in the $\ptmiss > 100$\GeV signal region.
Multiplying the number of low-$\RNINE$ \ensuremath{f \! f}\xspace events
observed in the signal region by this function gives a proxy high-$\RNINE$ \ensuremath{f \! f}\xspace sample.
\begin{equation}
N_{\mathrm{proxy}}^{i} = h_{\mathrm{ave}}^{i} \ N_{\ensuremath{f \! f}\xspace}^{i}
\end{equation}
For $\ptmiss < 100$\GeV, the ratio of the \ptmiss distribution in the $\Pgg\Pgg$ sample to that of the
proxy \ensuremath{f \! f}\xspace sample is fit to a constant $C$.
We multiply this constant value by the proxy \ensuremath{f \! f}\xspace yield in the signal region
to get a second prediction for the QCD background in bin $i$.
\begin{equation}
N_{\mathrm{QCD}}^{i} = C \ N_{\mathrm{proxy}}^{i}
\end{equation}
The two background estimation methods give values that are consistent
within the uncertainties. All three of the fits used in the two methods
are found to represent the data well in the low-\ptmiss control region.
Several studies were performed to verify the procedure, including using a
mixed-$\RNINE$ \ensuremath{f \! f}\xspace sample with one misidentified photon
satisfying $\RNINE > 0.9$ and one satisfying $\RNINE < 0.9$
to confirm that the exponential fit continues to accurately describe the mixed-$\RNINE$ \ensuremath{f \! f}\xspace to nominal \ensuremath{f \! f}\xspace
ratio in the high-\ptmiss signal region. As an additional check, a control sample
with one photon and one misidentified photon
was used as a proxy for the $\Pgg\Pgg$ candidate sample in a closure test of the method
up to $\ptmiss = 250$\GeV. At larger values of \ptmiss,
there is the potential for signal contamination in the $\Pgg f$ control sample.
Another background for this analysis comes from EWK processes with genuine
\ptmiss. This background primarily involves $\PW\Pgg$ and {\PW}+jets events where the $\PW$ decays to an
electron and a neutrino and the electron is misidentified as photon. This leads to
final states with photons and significant \ptmiss.
To obtain an estimate of the EWK background in the signal region,
the mass peaks from the \PZ boson in the $\Pe\Pe$ control sample and
the $\Pe\Pgg$ control sample are modeled using an extended likelihood fit for
the signal plus background hypothesis. The rate at which electrons are misidentified as photons
($f_{\Pe \to \Pgg}$) is calculated using the signal fit integrals $N_{\Pe\Pgg}$ and $N_{\Pe\Pe}$
for each sample. These can be expressed in terms of the number of true
\PZ bosons, $N_{\PZ}^{\mathrm{True}}$: $N_{\Pe\Pe} = (1-f_{\Pe \to \Pgg})^2 N_{\PZ}^{\mathrm{True}}$
and $N_{\Pe\Pgg} = 2 f_{\Pe \to \Pgg} (1-f_{\Pe \to \Pgg})N_{\PZ}^{\mathrm{True}}$.
The factor of 2 in $N_{\Pe\Pgg}$ occurs because either electron in the event could be
misidentified as a photon.
Taking the ratio of
these two values, we find that the misidentification rate is given by
$f_{\Pe \to \Pgg} = N_{\Pe\Pgg}/(2N_{\Pe\Pe} + N_{\Pe\Pgg})$.
The misidentification rate is calculated as a function of several kinematic variables,
including the vertex multiplicity and \ptmiss of the event and the \pt of the EM objects.
A 30\% uncertainty is applied to cover the observed dependencies.
The final EWK background prediction is given by scaling the number of
events in the $\Pe\Pgg$ control sample by the factor
$f_{\Pe\Pgg \to \Pgg\Pgg} = f_{\Pe \to \Pgg}/ ( 1 - f_{\Pe \to \Pgg} )= (2.6 \pm 0.8)\%$.
The irreducible $\PZ\Pgg\Pgg$ background is modeled via simulation.
A 50\% uncertainty is applied to conservatively cover the effects from
the statistical uncertainty of the MC sample, the PDF uncertainty in the cross section,
NNLO corrections in the simulation, and any other sources of potential mismodeling.
\section{Sources of systematic uncertainty}
\label{sec:SysUncert}
Systematic uncertainties are calculated for each contribution
to the total background prediction. In addition,
systematic uncertainties are assigned for
the signal efficiency and the integrated luminosity. The
value of each uncertainty and the method used to calculate
it are described below.
The largest uncertainties
in the background prediction come from uncertainties associated with
the QCD background estimate.
The magnitude of each uncertainty is shown in
Table~\ref{tab:QCDerrors} for the six signal bins.
The statistical uncertainty
from the \ensuremath{f \! f}\xspace control sample
ranges from 7 to 79\% in the signal region.
The uncertainty obtained from propagating the errors in the
fit parameters to the final prediction is between 2 and 5\%.
Finally, as described in Section~\ref{sec:EstBkg}, a systematic
uncertainty in the fitting procedure is calculated by
comparing the primary prediction to the cross check prediction
derived using the high-$\RNINE$ \ensuremath{f \! f}\xspace sample. The systematic
uncertainty is taken as the difference between the two methods
or the uncertainty in that difference, whichever is larger,
and ranges between 10 and 83\% in the signal region.
\begin{table}[ht]
\centering
\topcaption{Event yield and statistical and systematic uncertainties (in numbers of events)
of the QCD background estimation for each signal \ptmiss bin for 35.9\fbinv of data at 13\TeV.}
\centering
\begin{tabular}{ c c c c c c c}
\hline
\ptmiss bin (\GeVns{}) & Expected QCD & Stat. uncert. & Fit uncert. & Cross check uncert. \\
\hline
$100-115$ & 99.0 & $+7.2, -6.7$ & $\pm 1.8 $& $\pm 9.9 $\\
$115-130$ & 32.8 & $+4.2, -3.7$ & $\pm 0.7 $& $\pm 5.5 $\\
$130-150$ & 18.8 & $+3.2, -2.7$ & $\pm 0.5 $& $\pm 4.0 $\\
$150-185$ & 9.9 & $+2.3, -1.9$ & $\pm 0.3 $& $\pm 2.8 $\\
$185-250$ & 3.1 & $+1.3, -0.9$ & $\pm 0.1 $& $\pm 1.5 $\\
$\geq$250 & 1.0 & $+0.8, -0.5$ & $\pm 0.1 $& $\pm 0.8 $\\
\hline
\end{tabular}
\label{tab:QCDerrors}
\end{table}
Uncertainties in the EWK background prediction
include the statistical uncertainty from the $\Pe\Pgg$ control
sample and the 30\% uncertainty in the rate at which
electrons are misidentified as photons. The statistical
uncertainty is less than 9\% in each of the six signal bins.
There are also several uncertainties associated with
the signal efficiency. The statistical uncertainty
from the size of the T5gg or T6gg signal scans ranges
from 2 to 44\% depending on the mass bin. The
PDF uncertainties in the cross sections for signal simulation
are between 19 and 35\% and are taken from Ref.~\cite{Borschensky:2014cia}.
Other uncertainties include how well the jet energy
scale is known (1 to 30\%) and the uncertainty in
the photon identification efficiency (2.5\%).
The uncertainty in the integrated luminosity of the data sample is 2.5\%~\cite{CMS-PAS-LUM-17-001}.
\section{Results}
\label{sec:results}
We determine 95\% confidence
level (\CL) upper limits on gluino pair production and squark
pair production cross sections using the modified frequentist \CLs
method~\cite{Junk:1999kv, Read:2002hq}. The test statistic is an
LHC-style profile likelihood ratio~\cite{CMS-NOTE-2011-005}, and its
distribution is determined using
the asymptotic approximation~\cite{Cowan:2010js}.
The likelihood function is constructed from the background and signal \ptmiss distributions
across the six bins described in Section~\ref{sec:EstBkg}. The systematic uncertainties
described in Section~\ref{sec:SysUncert} are included in the test statistic as
constrained nuisance parameters. Systematic uncertainties which
directly affect the yields of processes are assumed to follow a
log-normal probability distribution, while statistical
uncertainties from the limited size of the control samples and
the signal MC samples are modeled using gamma probability distributions.
The full background prediction
and the measured \ptmiss distribution prior to the fit are shown in Fig.~\ref{fig:Final}.
The expected and observed numbers of events for each bin in the signal region
are shown in Table~\ref{tab:ExpObs} for the pre-fit distributions and Table~\ref{tab:ExpObsv2}
for the post-fit distributions.
Notably, in the last bin we observe 12 events and expect ${ 5.4 }^{+ 1.6 }_{- 1.5 }$
background events (pre-fit). The significance of the observed data after the fit across all
six bins of the signal region is calculated using the likelihood ratio test
for each mass pair value of $m_{\PSGczDo}$ versus
$m_{\PSg}$ or $m_{\PSGczDo}$ versus $m_{\PSQ}$ for the T5gg and T6gg models, respectively.
The significance does not strongly depend on the SUSY masses, and
for all masses in both models, the significance is found to correspond to
between 2.35 and 2.45 standard deviations.
Several studies were performed to characterize the fit and the excess in the
final \ptmiss bin and to ensure that the statistical treatment of the data
is robust. In particular, the pre- and postfit distributions
were checked to make sure that the pulls are
consistent within the uncertainties.
\begin{figure}[htb!]
\centering
\includegraphics[width=0.6\textwidth]{Figure_002.pdf}
\caption{The top panel shows the observed
\ptmiss distribution in data (black points) and predicted
background distributions prior to the fit described in the text. The vertical line marks the boundary
between the control region ($\ptmiss < 100$\GeV) and the signal region ($\ptmiss > 100$\GeV).
The last bin is the overflow bin and includes all events with $\ptmiss > 250$\GeV.
The QCD background is shown in red, the EWK background
is shown in blue, and the $\PZ\Pgg\Pgg$ background is shown in green.
The \ptmiss distribution shown in pink (purple) corresponds to the T5gg
simplified model with $m_{\PSg}=1700$ (2000)\GeV and $m_{\PSGczDo} = 1000$\GeV.
The \ptmiss distributions from the T6gg simplified model are similar to the T5gg
distributions shown here.
The bottom panel shows the ratio of observed events to the expected background.
The error bars on the ratio correspond to the
statistical uncertainty in the number of observed events.
The shaded region corresponds
to the total uncertainty in the background estimate. }
\label{fig:Final}
\end{figure}
\begin{table}[ht]
\centering
\topcaption{Number of expected background and observed data events
with 35.9\fbinv of 13\TeV data
in the signal region prior to the fit defined in the text.
The uncertainty in each expected background yield includes the
statistical uncertainty and all of the systematic uncertainties described in
Section~\ref{sec:SysUncert} added in quadrature.}
\begin{tabular}{ c c c c c c}
\hline
\ptmiss bin (\GeVns{}) & QCD & EWK & $\PZ\Pgg\Pgg$ & Total background & Observed \\ [0.5ex]
\hline
$100-115$ & $99 \pm 12$ & $13.7 \pm 4.2$ & $1.3 \pm 0.6$ & $ 114 \pm 13$ & 105 \\
$115-130$ & ${32.8}^{+ 7.0}_{- 6.7}$ & $ 9.0 \pm 2.7$ & $1.1 \pm 0.6$ & ${ 42.9 }^{+ 7.5 }_{- 7.3 }$ & 39 \\
$130-150$ & ${18.8}^{+ 5.1}_{- 4.9}$ & $ 7.4 \pm 2.3$ & $1.1 \pm 0.6$ & ${ 27.3 }^{+ 5.6 }_{- 5.4 }$ & 21 \\
$150-185$ & ${9.9}^{+ 3.6}_{- 3.4}$ & $ 6.1 \pm 1.9$ & $1.3 \pm 0.7$ & ${ 17.4 }^{+ 4.1 }_{- 3.9 }$ & 21 \\
$185-250$ & ${3.1}^{+ 1.9}_{- 1.7}$ & $ 5.8 \pm 1.8$ & $1.3 \pm 0.6$ & ${ 10.2 }^{+ 2.7 }_{- 2.6 }$ & 11 \\
$\geq$250 & ${1.0}^{+ 1.1}_{- 0.9}$ & $ 3.3 \pm 1.1$ & $1.1 \pm 0.6$ & ${ 5.4 }^{+ 1.6 }_{- 1.5 }$ & 12\\
\hline
\end{tabular}
\label{tab:ExpObs}
\end{table}
\begin{table}[ht]
\centering
\topcaption{Number of expected background and observed data events
with 35.9\fbinv of 13\TeV data in the signal region after the fit defined in the text.
The stated uncertainties are the post-fit uncertainties in each expected background yield.}
\begin{tabular}{ c c c c c c}
\hline
\ptmiss bin (\GeVns{}) & QCD & EWK & $\PZ\Pgg\Pgg$ & Total background & Observed \\ [0.5ex]
\hline
$100-115$ & $92.7 \pm 7.9$ & $15.9 \pm 3.8$ & $1.6 \pm 0.8$ & $ 110.1 \pm 7.4$ & 105 \\
$115-130$ & $29.7 \pm 4.4$ & $10.4 \pm 2.5$ & $1.4 \pm 0.7$ & $ 41.5 \pm 3.9$ & 39 \\
$130-150$ & $16.0 \pm 3.2$ & $8.5 \pm 2.1$ & $1.3 \pm 0.7$ & $ 25.9 \pm 3.1$ & 21 \\
$150-185$ & $9.3 \pm 2.7$ & $7.1 \pm 1.8$ & $1.6 \pm 0.8$ & $ 18.1 \pm 2.6$ & 21 \\
$185-250$ & $2.6 \pm 1.2$ & $6.7 \pm 1.6$ & $1.6 \pm 0.8$ & $ 10.9 \pm 1.8$ & 11 \\
$\geq$250 & $0.7 \pm 0.8$ & $4.0 \pm 1.0$ & $1.4 \pm 0.7$ & $ 6.0 \pm 1.2$ & 12 \\
\hline
\end{tabular}
\label{tab:ExpObsv2}
\end{table}
In Fig.~\ref{fig:limit} we present 95\% \CL upper limits on the gluino and
squark pair production cross sections as a function of the mass pair values
for the two models considered in this analysis.
From the NLO+NLL
predicted signal cross sections and their uncertainties we derive contours
representing lower limits in the SUSY mass plane. We also show expected limit
contours based on the expected experimental cross section limits and their
uncertainties. For values of the neutralino mass between 500 and 1500\GeV,
we expect to exclude
gluino masses up to 2.02\TeV and squark masses up to 1.74\TeV. This
is an improvement of approximately 400 and 300\GeV, respectively,
upon the reach of the previous CMS result~\cite{CMS:2016_anal}.
We observe exclusions for gluino masses up to 1.86\TeV and squark masses
up to 1.59\TeV. The observed exclusions are
lower than the expected exclusions because of the observed excess in the data.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.49\textwidth]{Figure_003-a.pdf}
\includegraphics[width=0.49\textwidth]{Figure_003-b.pdf}
\caption{The 95\% confidence level upper limits on the gluino (left) and squark (right) pair
production cross sections as a function of gluino or squark and neutralino masses.
The contours show the observed and expected exclusions assuming
the NLO+NLL cross sections, with their one standard deviation
uncertainties.}
\label{fig:limit}
\end{figure}
\section{Summary}
\label{sec:summary}
The results of a search for general
gauge-mediated supersymmetry breaking
in proton-proton collisions with two photons
and missing transverse momentum
in the final state are reported.
The analysis was performed using data corresponding to 35.9\fbinv
of integrated luminosity, recorded with the CMS detector in
2016 at a proton-proton center-of-mass energy of 13\TeV.
An excess of events corresponding to 2.4 standard deviations is observed.
Limits are determined on the
masses of supersymmetric particles in two
simplified models using
data-driven background estimation methods and
NLO+NLL signal cross section calculations.
In both models, the next-to-lightest supersymmetric
particle is the neutralino, which decays with a 100\%
branching fraction to a photon and a gravitino,
the lightest supersymmetric particle. The first
simplified model assumes gluino pair production,
with each gluino decaying to a neutralino and quarks.
The second simplified model
assumes squark pair production, with each squark
decaying to a quark and a neutralino.
The expected limits on gluino and squark masses, for the respective
models, are 2.02 and 1.74\TeV at 95\% confidence level. This is an
increase in sensitivity of more than 300\GeV for each model with
respect to the analysis performed with 2.3\fbinv of integrated luminosity
collected using the CMS detector in 2015.
The observed exclusions are for gluino
masses less than 1.86\TeV and squark masses
less than 1.59\TeV, where the difference
between the expected and observed exclusions is
driven by the excess observed in the data. The analysis described
in this paper improves the observed limits by
210\GeV for gluino masses and
220\GeV for squark masses with
respect to the previous CMS result.
\begin{acknowledgments}
We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMBWF and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, FAPERGS, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES and CSF (Croatia); RPF (Cyprus); SENESCYT (Ecuador); MoER, ERC IUT, and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); NKFIA (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF (Republic of Korea); MES (Latvia); LAS (Lithuania); MOE and UM (Malaysia); BUAP, CINVESTAV, CONACYT, LNS, SEP, and UASLP-FAI (Mexico); MOS (Montenegro); MBIE (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom, RAS, RFBR, and NRC KI (Russia); MESTD (Serbia); SEIDI, CPAN, PCTI, and FEDER (Spain); MOSTR (Sri Lanka); Swiss Funding Agencies (Switzerland); MST (Taipei); ThEPCenter, IPST, STAR, and NSTDA (Thailand); TUBITAK and TAEK (Turkey); NASU and SFFR (Ukraine); STFC (United Kingdom); DOE and NSF (USA).
\hyphenation{Rachada-pisek} Individuals have received support from the Marie-Curie program and the European Research Council and Horizon 2020 Grant, contract Nos.\ 675440 and 765710 (European Union); the Leventis Foundation; the A.P.\ Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation \`a la Recherche dans l'Industrie et dans l'Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the F.R.S.-FNRS and FWO (Belgium) under the ``Excellence of Science -- EOS'' -- be.h project n.\ 30820817; the Beijing Municipal Science \& Technology Commission, No. Z181100004218003; the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the Lend\"ulet (``Momentum'') Programme and the J\'anos Bolyai Research Scholarship of the Hungarian Academy of Sciences, the New National Excellence Program \'UNKP, the NKFIA research grants 123842, 123959, 124845, 124850, 125105, 128713, 128786, and 129058 (Hungary); the Council of Science and Industrial Research, India; the HOMING PLUS program of the Foundation for Polish Science, cofinanced from European Union, Regional Development Fund, the Mobility Plus program of the Ministry of Science and Higher Education, the National Science Center (Poland), contracts Harmonia 2014/14/M/ST2/00428, Opus 2014/13/B/ST2/02543, 2014/15/B/ST2/03998, and 2015/19/B/ST2/02861, Sonata-bis 2012/07/E/ST2/01406; the National Priorities Research Program by Qatar National Research Fund; the Programa Estatal de Fomento de la Investigaci{\'o}n Cient{\'i}fica y T{\'e}cnica de Excelencia Mar\'{\i}a de Maeztu, grant MDM-2015-0509 and the Programa Severo Ochoa del Principado de Asturias; the Thalis and Aristeia programs cofinanced by EU-ESF and the Greek NSRF; the Rachadapisek Sompot Fund for Postdoctoral Fellowship, Chulalongkorn University and the Chulalongkorn Academic into Its 2nd Century Project Advancement Project (Thailand); the Welch Foundation, contract C-1845; and the Weston Havens Foundation (USA).
\end{acknowledgments}
|
1,314,259,996,472 | arxiv | \section{Statement of results}
Let $K$ be a field with $\mathrm{char}(K)\ne 2$. Let us fix an algebraic closure
$K_a$ of $K$. Let us put $\mathrm{Gal}(K):=\mathrm{Aut}(K_a/K)$. If $X$ is an
abelian variety of positive dimension
over $K_a$ then we write $\mathrm{End}(X)$ for the ring of all its
$K_a$-endomorphisms and $\mathrm{End}^0(X)$ for the corresponding
(semisimple finite-dimensional) ${\mathbb Q}$-algebra $\mathrm{End}(X)\otimes{\mathbb Q}$.
We write $\mathrm{End}_K(X)$ for the ring of all $K$-endomorphisms of $X$
and $\mathrm{End}_K^0(X)$ for the corresponding (semisimple
finite-dimensional) ${\mathbb Q}$-algebra $\mathrm{End}_K(X)\otimes{\mathbb Q}$. The
absolute Galois group $\mathrm{Gal}(K)$ of $K$ acts on $\mathrm{End}(X)$ (and
therefore on $\mathrm{End}^0(X)$) by ring (resp. algebra) automorphisms and
$$\mathrm{End}_K(X)=\mathrm{End}(X)^{\mathrm{Gal}(K)}, \ \mathrm{End}_K^0(X)=\mathrm{End}^0(X)^{\mathrm{Gal}(K)},$$
since every endomorphism of $X$ is defined over a finite separable
extension of $K$.
If $n$ is a positive integer that is not divisible by $\mathrm{char}(K)$
then we write $X_n$ for the kernel of multiplication by $n$ in
$X(K_a)$. It is well-known \cite{MumfordAV} that $X_n$ is a free
${\mathbb Z}/n{\mathbb Z}$-module of rank $2\mathrm{dim}(X)$. In particular, if $n=\ell$ is
a prime then $X_{\ell}$ is an ${\mathbb F}_{\ell}$-vector space of
dimension $2\mathrm{dim}(X)$.
If $X$ is defined over $K$ then $X_n$ is a Galois
submodule in $X(K_a)$. It is known that all points of $X_n$ are
defined over a finite separable extension of $K$. We write
$\bar{\rho}_{n,X,K}:\mathrm{Gal}(K)\to \mathrm{Aut}_{{\mathbb Z}/n{\mathbb Z}}(X_n)$ for the
corresponding homomorphism defining the structure of the Galois
module on $X_n$,
$$\tilde{G}_{n,X,K}\subset
\mathrm{Aut}_{{\mathbb Z}/n{\mathbb Z}}(X_{n})$$ for its image $\bar{\rho}_{n,X,K}(\mathrm{Gal}(K))$
and $K(X_n)$ for the field of definition of all points of $X_n$.
Clearly, $K(X_n)$ is a finite Galois extension of $K$ with Galois
group $\mathrm{Gal}(K(X_n)/K)=\tilde{G}_{n,X,K}$.
If $n=\ell$ then we get a natural faithful linear representation
$$\tilde{G}_{\ell,X,K}\subset \mathrm{Aut}_{{\mathbb F}_{\ell}}(X_{\ell})$$
of $\tilde{G}_{\ell,X,K}$ in the ${\mathbb F}_{\ell}$-vector space
$X_{\ell}$. Recall \cite{Silverberg} that all endomorphisms of $X$
are defined over $K(X_4)$; this gives rise to the natural
homomorphism $$\kappa_{X,4}:\tilde{G}_{4,X,K,} \to
\mathrm{Aut}(\mathrm{End}^0(X))$$ and $\mathrm{End}_K^0(X)$ coincides with the subalgebra
$\mathrm{End}^0(X)^{\tilde{G}_{4,X,K}}$ of $\tilde{G}_{4,X,K}$-invariants
\cite[Sect. 1]{ZarhinLuminy}.
Let $f(x)\in K[x]$ be a polynomial of degree $n\ge 3$ without
multiple roots. Let ${\mathfrak R}_f\subset K_a$ be the $n$-element set of
roots of $f$. Then $K({\mathfrak R}_f)$ is the splitting field of $f$ and
$\mathrm{Gal}(f):=\mathrm{Aut}(K({\mathfrak R}_f)/K)$ is the Galois group of $f$ (over $K$).
One may view $\mathrm{Gal}(f)$ as a group of permutations of ${\mathfrak R}_f$; it
is transitive if and only if $f(x)$ is irreducible.
Let us consider the hyperelliptic curve $C_f:y^2=f(x)$ and its
jacobian $J(C_f)$. It is well-known \cite{ZarhinTexel} that $J(C_f)$ is a
$\left[\frac{n-1}{2}\right]$-dimensional abelian variety defined
over $K$. The aim of this paper is to study $\mathrm{End}^0(J(C_f))$,
assuming that $n=q+1$ where $q$ is a power of a prime $p$ and
$\mathrm{Gal}(f)=\mathrm{PSL}_2({\mathbb F}_q)$ acts via fractional-linear transformations
on ${\mathfrak R}_f$ identified with the projective line ${\mathbb P}^1({\mathbb F}_q)$. It
follows from results of \cite{ZarhinMRL,ZarhinMMJ,ZarhinBSMF} that
for every $q$ in all characteristics there
exist $K$ and $f$ with
$\mathrm{End}^0(J(C_f))={\mathbb Q}$. On the other hand, it is known that
$\mathrm{End}^0(J(C_f))={\mathbb Q}$ \cite{ZarhinMRL,ZarhinTexel, ZarhinSbg} if
$p=2$ and $q \ge 8$. (It is also true for $q=4$ if one assumes
that $\mathrm{char}(K)\ne 3$ \cite{ZarhinBSMF}.) However, if $q=5$ then
there are examples where $\mathrm{End}^0(J(C_f))$ is a (real) quadratic
field (even in characteristic zero)
\cite{BruRankJ,HashiOnBru,WilEM,Elkin}.
Our main result is the following statement.
\begin{thm}
\label{main} Let us assume that $\mathrm{char}(K)=0$.
Suppose that $n=q+1$ where $q\ge 5$ is a prime power that is
congruent to $\pm 3$ modulo $8$. Suppose that $f(x)$ is
irreducible and $\mathrm{Gal}(f)\cong \mathrm{PSL}_2({\mathbb F}_q)$. Then one of the
following two conditions holds:
\begin{itemize}
\item[(i)]
$\mathrm{End}^0(J(C_f))={\mathbb Q}$ or a quadratic field. In particular, $J(C_f)$
is an absolutely simple abelian variety.
\item [(ii)] $q$ is congruent to $3$ modulo $8$ and $J(C_f)$ is
$K_a$-isogenous to a self-product of an elliptic curve with
complex multiplication by ${\mathbb Q}(\sqrt{-q})$.
\end{itemize}
\end{thm}
\begin{rem}
It follows from results of \cite{Katsura} (see also \cite{Shioda},
\cite{Schoen}) that if the case (ii) of Theorem \ref{main} holds
then $J(C_f)$ is isomorphic over $K_a$ to a product of mutually
isogenous elliptic curves with complex multiplication by
${\mathbb Q}(\sqrt{-q})$.
\end{rem}
The paper is organized as follows. Section \ref{luminy} contains
auxiliary results about endomorphism algebras of abelian
varieties. In Section \ref{mainp} we prove the main result. In
Section \ref{noniso} (and Section \ref{quadratic}) we prove the
absolute simplicity of $J(C_f)$ when $q\ge 11$ is congruent to $3$
modulo $8$ and $K={\mathbb Q}$. Section \ref{examples} contains examples.
\section{Endomorphism algebras of abelian varieties}
\label{luminy}
\begin{rem}
\label{minimal}
Recall \cite{FT} (see also \cite[p. 199]{ZarhinP})
that a surjective homomorphism of finite groups
$\pi:{\mathcal G}_1\twoheadrightarrow {\mathcal G}$ is called a {\sl minimal cover}
if no proper subgroup of ${\mathcal G}_1$ maps onto ${\mathcal G}$. If $H$ is a
normal subgroup of ${\mathcal G}_1$ that lies in $\ker(\pi)$ then the
induced surjection ${\mathcal G}_1/H\twoheadrightarrow {\mathcal G}$ is also a
minimal cover.
\begin{itemize}
\item[(i)] If a surjection ${\mathcal G}_2\twoheadrightarrow {\mathcal G}_1$ is also
a minimal cover then one may easily check that the composition
${\mathcal G}_2 \to{\mathcal G}$ is surjective and a minimal cover.
\item[(ii)] Clearly, if ${\mathcal G}$ is simple then every proper normal
subgroup in ${\mathcal G}_1$ lies in $\ker(\pi)$.
\item[(iii)] If ${\mathcal G}$ is perfect then its universal central
extension is a minimal cover \cite{Suzuki}.
\item[(iv)] If ${\mathcal G}'\twoheadrightarrow {\mathcal G}$ is an arbitrary
surjective homomorphism of finite groups then
there always exists a subgroup ${\mathcal H}\subset {\mathcal G}'$ such that
${\mathcal H}\to{\mathcal G}$ is surjective and a minimal cover. Clearly, if ${\mathcal G}$
is perfect then ${\mathcal H}$ is also perfect.
\end{itemize}
\end{rem}
The field inclusion
$K(X_2)\subset K(X_4)$ induces a natural surjection \cite[Sect.
1]{ZarhinLuminy}
$$\tau_{2,X}:\tilde{G}_{4,X,K}\to\tilde{G}_{2,X,K}.$$
\begin{defn} We say that $K$ is 2-{\sl balanced} with respect to
$X$ if $\tau_{2,X}$ is a minimal cover.
\end{defn}
\begin{rem}
\label{overL}
Clearly, there always exists a subgroup $H
\subset \tilde{G}_{4,X,K}$ such that $H\to\tilde{G}_{2,X,K}$ is
surjective and a minimal cover. Let us put $L=K(X_4)^H$. Clearly,
$$K \subset L \subset K(X_4), \ L\bigcap K(X_2)=K$$
and $L$ is a maximal overfield of $K$ that enjoys these
properties. It is also clear that
$$K(X_2)\subset L(X_2),\ L(X_4)=K(X_4), \ H=\tilde{G}_{4,X,L},
\tilde{G}_{2,X,L}=\tilde{G}_{2,X,K}$$
and $L$ is $2$-{\sl balanced} with respect to
$X$.
\end{rem}
The following assertion (and its proof) is (are) inspired by
Theorem 1.6 of \cite{ZarhinLuminy} (and its proof).
\begin{thm}
\label{rep}
Suppose that $E:=\mathrm{End}_K^0(X)$ is a field that contains the
center $C$ of $\mathrm{End}^0(X)$. Let $C_{X,K}$ be the centralizer of
$\mathrm{End}_K^0(X)$ in $\mathrm{End}^0(X)$.
Then:
\begin{itemize}
\item[(i)] $C_{X,K}$ is a central simple $E$-subalgebra in
$\mathrm{End}^0(X)$. In addition, the centralizer of $C_{X,K}$ in
$\mathrm{End}^0(X)$ coincides with $E=\mathrm{End}_K^0(X)$ and
$$\mathrm{dim}_E(C_{X,K})=\frac{\mathrm{dim}_C(\mathrm{End}^0(X))}{[E:C]^2}.$$
\item[(ii)] Assume that $K$ is $2$-balanced with
respect to $X$ and $\tilde{G}_{2,X,K}$ is a non-abelian simple
group. If $\mathrm{End}^0(X)\ne E$ (i.e., not all endomorphisms of $X$ are
defined over $K$) then there exist a finite perfect group $\Pi \subset
C_{X,K}^{*}$ and a surjective homomorphism $\Pi \to
\tilde{G}_{2,X,K}$ that is a minimal cover. In addition, the
induced homomorphism
$$E[\Pi] \to C_{X,K}$$
is surjective, i.e., $C_{X,K}$ is isomorphic to a direct summand of the group
algebra $E[\Pi]$.
\end{itemize}
\end{thm}
\begin{proof}
Since $E$ is a field, $C$ is a subfield of $E$ and therefore
$\mathrm{End}^0(X)$ is a central simple $C$-algebra. Now the assertion (i)
follows from Theorem 2 of Sect. 10.2 in Chapter VIII of
\cite{Bourbaki}.
Now let us prove the assertion (ii).
Recall that there is the homomorphism
$$\kappa_{X,4}:\tilde{G}_{4,X,K} \to \mathrm{Aut}(\mathrm{End}^0(X))$$ such that
$$\mathrm{End}^0(X)^{\tilde{G}_{4,X,K}}=\mathrm{End}_K^0(X)=E\supset C.$$
This implies that $$\kappa_{X,4}(\tilde{G}_{4,X,K})\subset
\mathrm{Aut}_E(\mathrm{End}^0(X))\subset \mathrm{Aut}_C(\mathrm{End}^0(X))$$ and we get a
homomorphism
$$\kappa_E:\tilde{G}_{4,X,K} \to \mathrm{Aut}_E(C_{X,K})$$ such that
\begin{equation}
C_{X,K}^{\tilde{G}_{4,X,K}}=E \label{eqn1}
\end{equation}
Assume that $E = C_{X,K}$, i.e., $E$ coincides with its own
centralizer in $\mathrm{End}^0(X)$. It follows from the Skolem-Noether
theorem
that $\mathrm{Aut}_C(\mathrm{End}^0(X))=\mathrm{End}^0(X)^{*}/C^*$. This implies that the
group $$\mathrm{Aut}_E(\mathrm{End}^0(X))=C_{X,K}^*/C^*=E^*/C^*$$ is commutative.
It follows that $\kappa_{X,4}(\tilde{G}_{4,X,K})$ is commutative.
Since $\tilde{G}_{4,X,K}$ is perfect,
$\kappa_{X,4}(\tilde{G}_{4,X,K})$ is perfect commutative and
therefore trivial, i.e., $\mathrm{End}^0(X)=\mathrm{End}_K^{0}(X)$.
Assume that $E \ne C_{X,K}$. This means that the group
$\Gamma:=\kappa_E(\tilde{G}_{4,X,K})$ is not $\{1\}$, i.e.,
$\ker(\kappa_E)\ne \tilde{G}_{4,X,K}$. Clearly, $\Gamma$ is a
finite perfect subgroup of $\mathrm{Aut}_E(C_{X,K})$.
The minimality of $\tau_{2,X}$ and the simplicity of
$\tilde{G}_{2,X,K}$ imply the existence of a minimal cover
$$\Gamma \to \tilde{G}_{2,X,K},$$
thanks to Remark \ref{minimal}.
Since $C_{X,K}$ is a central simple $E$-algebra, all its
automorphisms are inner, i.e., $\mathrm{Aut}_E(C_{X,K})=C_{X,K}^*/E^{*}$.
Let $\Delta \twoheadrightarrow \Gamma$ be the universal central
extension of $\Gamma$. It is well-known \cite[Ch. 2, Sect. 9]{Suzuki} that $\Delta$ is a finite
perfect group. The universality property implies that $\Delta
\twoheadrightarrow \Gamma$ is a minimal cover and the inclusion
map $\Gamma\subset C_{X,K}^*/E^{*}$ lifts (uniquely) to a
homomorphism $\pi:\Delta \to C_{X,K}^{*}$. Clearly, $\ker(\pi)$
lies in the kernel of $\Delta \twoheadrightarrow \Gamma$ and we
get a minimal cover
$$\pi(\Delta)\cong \Delta/\ker(\pi) \twoheadrightarrow \Gamma ,$$
thanks to Remark \ref{minimal}.
Taking the compositions of minimal
covers $\pi(\Delta)\twoheadrightarrow \Gamma$ and $\Gamma
\twoheadrightarrow \tilde{G}_{2,X,K}$, we obtain a minimal cover
$\pi(\Delta)\twoheadrightarrow \tilde{G}_{2,X,K}$. If we put
$$\Pi:=\pi(\Delta)\subset C_{X,K}^{*}.$$
then we get a minimal cover
$$\Pi\twoheadrightarrow\tilde{G}_{2,X,K}.$$
The equality \eqref{eqn1} means that
the centralizer of $\pi(\Delta)=\Pi$ in $C_{X,K}$ coincides with
$E$.
It follows that if $E[\Pi]$ is the group $E$-algebra
of $\Pi$ then the inclusion $\Pi\subset C_{X,K}^{*}$ induces the
$E$-algebra homomorphism $\omega: E[\Pi] \to C_{X,K}$ such that
the centralizer of its image in $C_{X,K}$ coincides with $E$.
We claim that $\omega(E[\Pi])=C_{X,K}$
and therefore $C_{X,K}$ is
isomorphic to a direct summand of $E[\Pi]$. This claim follows
easily from the next lemma that was proven in \cite[Lemma
1.7]{ZarhinLuminy}
\begin{lem}
\label{sur} Let $F$ be a field of characteristic zero, $T$ a
semisimple finite-dimensional $F$-algebra, $S$ a
finite-dimensional central simple $F$-algebra, $\beta: T \to S$ an
$F$-algebra homomorphism that sends $1$ to $1$. Suppose that the
centralizer of the image $\beta(T)$ in $S$ coincides with the
center $F$. Then $\beta$ is surjective, i. e. $\beta(T)=S$.
\end{lem}
\end{proof}
\begin{thm}
\label{center} Suppose that $\mathrm{End}^0(X)$ is a simple ${\mathbb Q}$-algebra,
$\tilde{G}_{2,X,K}$ is a simple non-abelian group, whose order is
not a divisor of $2\mathrm{dim}(X)$ and
$\mathrm{End}_{\tilde{G}_{2,X,K}}(X_2)\cong {\mathbb F}_4$.
Then the center $C$ of $\mathrm{End}^0(X)$ is either ${\mathbb Q}$ or a quadratic field. In
addition, there exists a finite separable field extension $L/K$ such that
$\tilde{G}_{2,X,L}=\tilde{G}_{2,X,K}$, the map $\tau_{2,X}:\tilde{G}_{4,X,L}
\twoheadrightarrow \tilde{G}_{2,X,L}$ is surjective and a minimal cover, the
${\mathbb Q}$-algebra $E:=\mathrm{End}_L^0(X)$ is either ${\mathbb Q}$ or a quadratic field and $C\subset
\mathrm{End}_L^0(X)$. In particular, $C$ is either ${\mathbb Q}$ or a quadratic field. If
$\mathrm{End}^0(X)\ne \mathrm{End}_L^0(X)$ and $C_{X,L}$ is the centralizer of $E$ in
$\mathrm{End}^0(X)$ then there exist a finite perfect group $\Pi \subset C_{X,L}^{*}$
and a surjective homomorphism $\Pi \to \tilde{G}_{2,X,K}$ that is a minimal
cover. In addition, the induced homomorphism
$$E[\Pi] \to C_{X,L}$$
is surjective, i.e., $C_{X,L}$ is isomorphic to a direct summand of the group
algebra $E[\Pi]$.
\end{thm}
\begin{proof}
Choose a field $L$ as in Remark \ref{overL}. Then
$\tilde{G}_{2,X,L}=\tilde{G}_{2,X,K}$, the map
$$\tau_{2,X}:\tilde{G}_{4,X,L}\to
\tilde{G}_{2,X,L}=\tilde{G}_{2,X,K}$$ is surjective and a minimal
cover. We have
$$\mathrm{End}_L(X)\otimes {\mathbb Z}/2{\mathbb Z} \hookrightarrow
\mathrm{End}_{\mathrm{Gal}(L)}(X_2)=\mathrm{End}_{\tilde{G}_{2,X,L}}(X_2)={\mathbb F}_4.$$ It
follows that the rank of the free ${\mathbb Z}$-module $\mathrm{End}_L(X)$ is $1$
or $2$; Lemma 1.3 of \cite{ZarhinLuminy} implies that $\mathrm{End}_L(X)$
has no zero divisors. This implies that
$\mathrm{End}_L^0(X)=\mathrm{End}_L(X)\otimes{\mathbb Q}$ is a division algebra of
${\mathbb Q}$-dimension $1$ or $2$. This means that $E=\mathrm{End}_L^0(X)$ is
either ${\mathbb Q}$ or a quadratic field.
Recall that the center $C$ of $\mathrm{End}^0(X)$ is
a number field, whose degree $[C:{\mathbb Q}]$ divides $2\mathrm{dim}(X)$. The
group $\tilde{G}_{4,X,L}$ acts via automorphisms on $C$ and
$$C^{\tilde{G}_{4,X,L}}=C\bigcap
\mathrm{End}_L^0(X)$$ is either ${\mathbb Q}$ or a quadratic field. Since
$\tilde{G}_{2,X,L}=\tilde{G}_{2,X,K}$ has no normal subgroups of
index dividing $2\mathrm{dim}(X)$ , the same is true for
$\tilde{G}_{4,X,L}$ and therefore $\tilde{G}_{4,X,L}$ acts on $C$
trivially, i.e., $C \subset \mathrm{End}_L^0(X)$.
In order to finish the proof, one has only to apply
Theorem \ref{rep} ( to $L$ instead of $K$).
\end{proof}
\begin{defn}
We say that a group is {\em FTKL-exceptional} if it is one of the following:
\begin{itemize}
\item[(i)] $\mathrm{Sp}_{2n}(q)$ for some even $q$ and $n\geq 2$, except $\mathrm{Sp}_4(2)'$ and $\mathrm{Sp}_6(2)$;
\item[(ii)] $\Omega_{2n}^\pm({\mathbb F}_q)$ for some even $q$ and $n\geq 4$, except $\Omega_8^+(2)$;
\item[(iii)] $\mL_4(q)$ for some even $q$, except $\mL_4(2)$; or
\item[(iv)] $\mG_2(q)$ for some $q=2^{2e}$, except $\mG_2(4)$.
\end{itemize}
\end{defn}
These are exactly the groups of Lie type in characteristic $2$ that are listed in the table in
Theorem 3 on p. 316 in \cite{KL}.
\begin{thm}
\label{FTKL} Let us assume that $\mathrm{char}(K)=0$.
Suppose that $\mathrm{End}^0(X)$ is a simple ${\mathbb Q}$-algebra with the
center $C$. Suppose that $\tilde{G}_{2,X,K}$ is a simple
non-abelian group, whose order is not a divisor of $2\mathrm{dim}(X)$ and
$\mathrm{End}_{\tilde{G}_{2,X,K}}(X_2)\cong {\mathbb F}_4$. Assume, in addition,
that $\tilde{G}_{2,X,K}$ is a known simple group that is not FTKL-exceptional.
Suppose also that $\mathrm{dim}(X)$
coincides with the smallest positive integers $d$ such that
$\tilde{G}_{2,X,K}$ is isomorphic to a subgroup of $\mathrm{PGL}(d,{\mathbb C})$.
Then:
\begin{itemize}
\item[(i)] The center $C$ of $\mathrm{End}^0(X)$ is either ${\mathbb Q}$ or
quadratic field.
\item[(ii)] Either $\mathrm{End}^0(X)=C$ or the following conditions
hold:
\begin{enumerate}
\item There exists a finite algebraic field extension $L/K$ such that
$\tilde{G}_{2,X,L}=\tilde{G}_{2,X,K},$, the overfield
$L$ is $2$-balanced with respect to $X$, the algebra $E:=\mathrm{End}^0_L(X)$ is either ${\mathbb Q}$
or a quadratic field, $C=E$ and the following conditions hold.
There exist a
finite perfect group $\Pi \subset \mathrm{End}^0(X)^{*}$ and a surjective
homomorphism $\Pi \to \tilde{G}_{2,X,K}$ that is a minimal cover
and a central extension. In addition, the induced homomorphism
$$E[\Pi] \to \mathrm{End}^0(X)$$
is surjective, i.e., $\mathrm{End}^0(X)$ is isomorphic to a direct summand of the group
algebra $E[\Pi]$.
\item If $C={\mathbb Q}$ then $X$ enjoys one of the following two
properties:
\begin{itemize}
\item[(a)] $X$ is isogenous over $K_a$ to a self-product of an
elliptic curve without complex multiplication.
\item[(b)] $\mathrm{dim}(X)$ is even and $X$ is isogenous over $K_a$ to a
self-product of an abelian surface $Y$ such that $\mathrm{End}^0(Y)$ is an
indefinite quaternion ${\mathbb Q}$-algebra.
\end{itemize}
\item If $C \ne{\mathbb Q}$ then
$C$ is an imaginary quadratic field and
$X$ is isogenous over $K_a$ to a
self-product of an elliptic curve with complex multiplication by
$C$.
\end{enumerate}
\end{itemize}
\end{thm}
\begin{proof}
Using Theorem \ref{center} and replacing if necessary $K$ by its
suitable extension, we may assume that $K$ is $2$-balanced with
respect to $X$, the algebra $E:=\mathrm{End}^0_K(X)$ is either ${\mathbb Q}$ or a
quadratic field, $C \subset E$ and the following conditions hold.
{\sl Either $\mathrm{End}^0(X)=E$ or there exist a finite perfect group
$\Pi \subset C_{X,K}^{*}$ and a surjective homomorphism $\Pi \to
\tilde{G}_{2,X,K}$ that is a minimal cover and such that the
induced homomorphism
$$E[\Pi] \to C_{X,K}$$
is surjective, i.e., $C_{X,K}$ is isomorphic to a direct summand of the group
algebra} $E[\Pi]$. (Here as above $C_{X,K}$ is the centralizer of $E$ in
$\mathrm{End}^0(X)$).
Assume that $\mathrm{End}^0(X)\ne E$. We are going to prove that $\Pi \to
\tilde{G}_{2,X,K}$ is a central extension, using results of
Feit-Tits and Kleidman-Liebeck \cite{FT,KL}. Without loss of
generality we may assume that there is a field embedding
$K_a\hookrightarrow {\mathbb C}$ and consider $X$ as complex abelian
variety. Let $t_X$ be the Lie algebra of $X$ that is a
$\mathrm{dim}(X)$-dimensional complex vector space. By functoriality, this
gives us the embeddings
\begin{eqnarray*}
&&j:\mathrm{End}^0(X)\hookrightarrow \mathrm{End}_{{\mathbb C}}(t_X)\cong \mathrm{M}_{\mathrm{dim}(X)}({\mathbb C}), \\
&&j:\mathrm{End}^0(X)^*\hookrightarrow \mathrm{Aut}_{{\mathbb C}}(t_X)\cong \mathrm{GL}(\mathrm{dim}(X),{\mathbb C}).
\end{eqnarray*}
Clearly, only central elements of $\Pi$ go to
scalars under $j$. It follow that there exists a central subgroup
$Z$ in $\Pi$ such that $j(Z)$ consists of scalars and
$\Pi/Z\hookrightarrow\mathrm{PGL}(\mathrm{dim}(X),{\mathbb C})$
The simplicity of
$\tilde{G}_{2,X,K}$ implies that $Z$ lies in the kernel of $\Pi
\to \tilde{G}_{2,X,K}$ and the induced map $\Pi/Z \to
\tilde{G}_{2,X,K}$ is also a minimal cover. It follows from
Theorem on p. 1092 of \cite{FT} and Theorem 3 on p. 316 of
\cite{KL} that $\Pi/Z \to \tilde{G}_{2,X,K}$ is a central
extension of $\tilde{G}_{2,X,K}$. Since $\Pi$ is a central
extension of $\Pi/Z $, it follows \cite{Suzuki} that $\Pi$ is a
central extension of $\tilde{G}_{2,X,K}$.
Now notice that $t_X$ carries a natural structure of
$E\otimes_{{\mathbb Q}}{\mathbb C}$-module. Assume that $E\ne{\mathbb Q}$, i.e., $E$ is a
quadratic field. Let $\sigma,\tau: E \hookrightarrow {\mathbb C}$ be the
two different embeddings of $E$ into ${\mathbb C}$. Then
$$E\otimes_{{\mathbb Q}}{\mathbb C}={\mathbb C}_{\sigma}\oplus{\mathbb C}_{\tau}$$ with
$${\mathbb C}_{\sigma}=E\otimes_{E,\sigma}{\mathbb C}={\mathbb C},\
{\mathbb C}_{\tau}=E\otimes_{E,\tau}{\mathbb C}={\mathbb C}$$ and
$t_X$ splits into a direct sum
$$t_X={\mathbb C}_{\sigma}t_X\oplus {\mathbb C}_{\tau}t_X.$$
Suppose that both ${\mathbb C}_{\sigma}t_X$ and ${\mathbb C}_{\tau}t_X$ do not vanish.
Then the ${\mathbb C}$-dimension $d_{\sigma}$ of non-zero ${\mathbb C}_{\sigma}t_X$ is strictly less
than dim(X). Clearly, ${\mathbb C}_{\sigma}t_X$ is $C_{X,K}$-stable and we
get a nontrivial homomorphism
$$C_{X,K}\to \mathrm{End}_{{\mathbb C}}({\mathbb C}_{\sigma}t_X)\cong \mathrm{M}_{d_{\sigma}}({\mathbb C})$$
that must be an embedding in light of the simplicity of $C_{X,K}$.
This gives us an embedding
$$C_{X,K}^{*}\to \mathrm{Aut}_{{\mathbb C}}({\mathbb C}_{\sigma}t_X)\cong\mathrm{GL}(d_{\sigma},{\mathbb C}).$$
One may easily check that all the elements of $\Pi$ that go to
scalars in $\mathrm{Aut}_{{\mathbb C}}({\mathbb C}_{\sigma}t_X)$ constitute a central
subgroup $Z_{\sigma}$ that lies in the kernel of $\Pi\to
\tilde{G}_{2,X,K}$. This gives us a central extension
$\Pi/Z_{\sigma}\twoheadrightarrow \tilde{G}_{2,X,K}$ that is a
minimal cover and an embedding $\Pi/Z_{\sigma}\hookrightarrow \mathrm{Aut}_{{\mathbb C}}({\mathbb C}_{\sigma}t_X)
\cong\mathrm{GL}(d_{\sigma},{\mathbb C})$. Since $d_{\sigma}<\mathrm{dim}(X)$, Theorem on
p. 1092 of \cite{FT} and Theorem 3 on p. 316 of \cite{KL} provide
us with a contradiction.
It follows that either ${\mathbb C}_{\sigma}t_X$
or ${\mathbb C}_{\tau}t_X$ does vanish. We may assume that
${\mathbb C}_{\tau}t_X=0$. This means that each $e\in E$ acts on $t_X$ as
multiplication by complex number $\sigma(e)$, i.e., $j(E)$
consists of scalars. Recall that the exponential map identifies
$X({\mathbb C})$ with the complex torus $t_X/\Lambda$ where $\Lambda$ is a
discrete lattice of rank $2\mathrm{dim}(X)$. In addition, $\Lambda$ is
$j(\mathrm{End}_K(X))$-stable where $\mathrm{End}_K(X)$ is an order in the
quadratic field $\mathrm{End}_K^0(X)$. Now the discreteness of $\Lambda$
implies that $E$ cannot be real and therefore is an imaginary
quadratic field. It follows easily that $X$ is isogenous over ${\mathbb C}$
to a self-product of an elliptic curve with complex multiplication
by $E$. In particular, $E=C$ and $C_{X,K}=\mathrm{End}^0(X)$.
Now let us assume that $E={\mathbb Q}$. Then $C_{X,K}=\mathrm{End}^0(X)$. Let $Y$
be an absolutely simple abelian variety such that $X$ is isogenous
to a self-product $Y^r$ for some positive integer $r$ with
$r\mid\mathrm{dim}(X)$. Then $\mathrm{End}^0(X)\cong\mathrm{M}_r(\mathrm{End}^0(Y))$.
In
particular, the center of the division algebra $\mathrm{End}^0(Y)$ is
${\mathbb Q}$. It follows from Albert's classification \cite{MumfordAV}
that $\mathrm{End}^0(Y)$ is either ${\mathbb Q}$ or a quaternion ${\mathbb Q}$-algebra.
If $\mathrm{End}^0(Y)={\mathbb Q}$ then $\mathrm{End}^0(X)\cong \mathrm{M}_r({\mathbb Q})$ and
$\Pi/Z\hookrightarrow \mathrm{PGL}(r,{\mathbb Q})\subset \mathrm{PGL}(r,{\mathbb C})$. It follows
that $r=\mathrm{dim}(X)$, i.e. $Y$ is an elliptic curve without complex
multiplication.
Suppose that $\mathrm{End}^0(Y)$ is a quaternion ${\mathbb Q}$-algebra. Since
$\mathrm{dim}(Y) =\mathrm{dim}(X)/r$ and we live in characteristic zero, $2r$
divides $\mathrm{dim}(X)$. Clearly,
$$\mathrm{End}^0(Y)\subset \mathrm{End}^0(Y)\otimes_{{\mathbb Q}}{\mathbb C}\cong \mathrm{M}_2({\mathbb C})$$ and
therefore $$\mathrm{End}^0(X)\cong\mathrm{M}_r(\mathrm{End}^0(Y))\hookrightarrow
\mathrm{M}_{2r}({\mathbb C}).$$ This implies that $\Pi \hookrightarrow \mathrm{GL}(2r,{\mathbb C})$.
It follows that $2r=\mathrm{dim}(X)$, i.e., $\mathrm{dim}(Y)=2$. It follows from
the classification of endomorphism algebras of abelian surfaces
\cite[Sect. 6]{Oort} that $\mathrm{End}^0(Y)$ is an {\sl indefinite}
quaternion ${\mathbb Q}$-algebra.
\end{proof}
\begin{thm}
\label{PSL2} Let us assume that $\mathrm{char}(K)=0$.
Suppose that $\mathrm{End}^0(X)$ is a simple ${\mathbb Q}$-algebra.
Suppose that $d:=\mathrm{dim}(X)=(q-1)/2$ where $q \ge 5$ is an odd prime power.
Suppose that $\tilde{G}_{2,X,K}\cong \mathrm{PSL}_2({\mathbb F}_q)$ and
$\mathrm{End}_{\tilde{G}_{2,X,K}}(X_2)\cong {\mathbb F}_4$. Then one of the
following two conditions holds:
\begin{itemize}
\item[(i)]
$\mathrm{End}^0(X)={\mathbb Q}$ or a quadratic field. In particular, $X$ is an
absolutely simple abelian variety.
\item[(ii)] $q$ is congruent to $3$ modulo $4$ and $X$ is
$K_a$-isogenous to a self-product of an elliptic curve with
complex multiplication by ${\mathbb Q}(\sqrt{-q})$.
\end{itemize}
\end{thm}
\begin{proof}
It is well-known \cite[Sect. 4.15]{Gor} that $\mathrm{SL}_2({\mathbb F}_q)$ is
the universal central extension of $\mathrm{PSL}_2({\mathbb F}_q)$ and therefore every
projective representation of $\mathrm{PSL}_2({\mathbb F}_q)$ lifts to a linear representation of
$\mathrm{SL}_2({\mathbb F}_q)$. The well-known list of irreducible representations of
$\mathrm{SL}_2({\mathbb F}_q)$ over complex numbers \cite[Sect. 38]{DA} tells us that the
smallest degree of a nontrivial representation of $\mathrm{SL}_2({\mathbb F}_q)$ is $(q-1)/2=d$.
This implies that we are in position to apply Theorem \ref{FTKL}. In
particular, $C$ is either ${\mathbb Q}$ or a quadratic field. We may and will assume
that $\mathrm{End}^0(X)\ne C$.
We need to rule out the following possibilities:
\begin{enumerate}
\item\label{case1} $\mathrm{dim}(X)$ is even and $X$ is isogenous over $K_a$ to a
self-product of an abelian surface $Y$ such that $\mathrm{End}^0(Y)$ is
an indefinite quaternion ${\mathbb Q}$-algebra. In particular, $\mathrm{End}^0(X)$
is a $d^2$-dimensional central simple ${\mathbb Q}$-algebra.
\item\label{case2} $q$ is congruent to $1$ modulo $4$ and
$X$ is isogenous over $K_a$ to a
self-product of an elliptic curve with complex multiplication.
In particular, $\mathrm{End}^0(X)$ is a $d^2$-dimensional central simple
algebra over the imaginary quadratic field $C$ unramified at $\infty$.
\item\label{case3} $X$ is isogenous over $K_a$ to a self-product of an
elliptic curve without complex multiplication. In particular, $\mathrm{End}^0(X)$
is a $d^2$-dimensional central simple ${\mathbb Q}$-algebra.
\end{enumerate}
By Theorem \ref{FTKL}, there exist a finite perfect
group $\Pi$ and a minimal central cover $\Pi \to \mathrm{PSL}_2({\mathbb F}_q)$ such that
$\mathrm{End}^0(X)$ is a quotient of the group algebra $E[\Pi]$ where $E=C$ is either
${\mathbb Q}$ or an imaginary quadratic field. It follows easily that $\Pi=\mathrm{PSL}_2({\mathbb F}_q)$
or $\mathrm{SL}_2({\mathbb F}_q)$, so we may always view $\mathrm{End}^0(X)$ as a simple quotient
(direct summand) $D$ of $E[\mathrm{SL}_2({\mathbb F}_q)]$. By Theorem \ref{FTKL}, $\mathrm{End}^0(X)$ is
a central simple $E$-algebra of dimension $d^2$.
Let us consider the composition
$${\mathbb Q}[\mathrm{SL}_2({\mathbb F}_q)]\subset E[\mathrm{SL}_2({\mathbb F}_q)] \twoheadrightarrow
\mathrm{End}^0(X).$$
Let $D$ be the {\sl simple} direct summand of ${\mathbb Q}[\mathrm{SL}_2({\mathbb F}_q)]$, whose image in
the {\sl simple} ${\mathbb Q}$-algebra $\mathrm{End}^0(X)$ is {\sl not} zero. We write $B\subset
\mathrm{End}^0(X)$ for the image of $D$: it is a ${\mathbb Q}$-subalgebra isomorphic to $D$. The
induced map $D \to \mathrm{End}^0(X)$ is injective, because $D$ is a simple
${\mathbb Q}$-algebra. On the other hand, $D_E=D\otimes_{{\mathbb Q}}E$ is a direct summand of
$E[\mathrm{SL}_2({\mathbb F}_q)]$ and the image of $D_E \to \mathrm{End}^0(X)$ is a non-zero ideal of
$\mathrm{End}^0(X)$. Since $\mathrm{End}^0(X)$ is simple, $D_E \to \mathrm{End}^0(X)$ is surjective. In
particular, $B$ generates $\mathrm{End}^0(X)$ as $E$-vector space and the center of $D$
embeds into the center $E$ of $\mathrm{End}^0(X)$. This implies that the center of $D$
is either ${\mathbb Q}$ or isomorphic to $E$. In addition, if the center of $D$ is
isomorphic to $E$ then $B$ contains $E$, i.e., $B$ is a $E$-vector subspace of
$\mathrm{End}^0(X)$ and therefore coincides with $\mathrm{End}^0(X)$: this implies that
$\mathrm{End}^0(X)\cong D$.
Assume that the center of $D$ is isomorphic to $E$. Then $\mathrm{End}^0(X)\cong D$
and therefore $D$ is a central simple $E$-algebra of dimension $d^2$. This
means that the simple direct summand $D$ of ${\mathbb Q}[\mathrm{SL}_2({\mathbb F}_q)]$ corresponds to an
irreducible (complex) character of $\mathrm{SL}_2({\mathbb F}_q)$ of degree $d$ as in Lemma 24.7
of \cite{DA}. These simple direct summands are described explicitly in
\cite{J,F}. In particular, if $q$ is congruent to $1$ modulo $4$ but is
not a
square then the center of $D$ is a real quadratic field ${\mathbb Q}(\sqrt{q})$, which
is not the case. This implies that $q$ is congruent to $3$ modulo $4$: in this
case the center of $D$ is an imaginary quadratic field ${\mathbb Q}(\sqrt{-q})$ and
therefore $E={\mathbb Q}(\sqrt{-q})$. It follows from Theorem \ref{FTKL} that $X$ is
$K_a$-isogenous to a self-product of an elliptic curve with complex
multiplication by ${\mathbb Q}(\sqrt{-q})$.
Now assume that the center of $D$ is {\sl not} isomorphic to $E$. Then it must
be ${\mathbb Q}$, i.e., $D$ is a central simple ${\mathbb Q}$-algebra. It follows that $D_E$ is a
central simple $E$-algebra and therefore the surjective homomorphism $D_E \to
\mathrm{End}^0(X)$ is injective. It follows that $D_E \cong \mathrm{End}^0(X)$; in particular,
the central simple ${\mathbb Q}$-algebra $D$ has ${\mathbb Q}$-dimension $d^2$. As above, this
means that the simple direct summand $D$ corresponds to an irreducible
(complex) character of $\mathrm{SL}_2({\mathbb F}_q)$ of degree $d$. Since the center of $D$ is
${\mathbb Q}$, it follows from results of \cite{J,F} that $q$ is a square, which is not
the case.
This ends the proof.
\end{proof}
\section{Hyperelliptic jacobians}
\label{mainp}
Suppose that $f(x)\in K[x]$ is a polynomial of degree $n\ge 5$
without multiple roots. Let ${\mathfrak R}_f\subset K_a $ be the set of
roots of $f$. Clearly, ${\mathfrak R}_f$ consists of $n$ elements. Let
$K({\mathfrak R}_f)\subset K_a$ be the splitting field of $f$. Clearly,
$K({\mathfrak R}_f)/K$ is a Galois extension and we write $\mathrm{Gal}(f)$ for its
Galois group $\mathrm{Gal}(K({\mathfrak R}_f)/K)$. By definition, $\mathrm{Gal}(K({\mathfrak R}_f)/K)$
permutes elements of ${\mathfrak R}_f$; further we identify $\mathrm{Gal}(f)$ with
the corresponding subgroup of $\mathrm{Perm}({\mathfrak R}_f)$,
where $\mathrm{Perm}({\mathfrak R}_f)$
is the group of permutations of ${\mathfrak R}_f$.
We write ${\mathbb F}_2^{{\mathfrak R}_f}$ for the $n$-dimensional ${\mathbb F}_2$-vector
space of maps $h:{\mathfrak R}_f \to {\mathbb F}_2$. The space ${\mathbb F}_2^{{\mathfrak R}_f}$ is
provided with a natural action of $\mathrm{Perm}({\mathfrak R}_f)$ defined as
follows. Each $s \in \mathrm{Perm}({\mathfrak R}_f)$ sends a map
$h:{\mathfrak R}_f\to {\mathbb F}_2$ to $sh:\alpha \mapsto h(s^{-1}(\alpha))$. The permutation module ${\mathbb F}_2^{{\mathfrak R}_f}$
contains the $\mathrm{Perm}({\mathfrak R}_f)$-stable hyperplane
$$({\mathbb F}_2^{{\mathfrak R}_f})^0=
\{h:{\mathfrak R}_f\to {\mathbb F}_2\mid\sum_{\alpha\in {\mathfrak R}_f}h(\alpha)=0\}$$ and the
$\mathrm{Perm}({\mathfrak R}_f)$-invariant line ${\mathbb F}_2 \cdot 1_{{\mathfrak R}_f}$ where
$1_{{\mathfrak R}_f}$ is the constant function $1$. Clearly,
$({\mathbb F}_2^{{\mathfrak R}_f})^0$ contains ${\mathbb F}_2 \cdot 1_{{\mathfrak R}_f}$ if and only if
$n$ is even.
If $n$ is even then
let us define the $\mathrm{Gal}(f)$-module
$Q_{{\mathfrak R}_f}:=({\mathbb F}_2^{{\mathfrak R}_f})^0/({\mathbb F}_2 \cdot 1_{{\mathfrak R}_f})$. If $n$ is
odd then let us put $Q_{{\mathfrak R}_f}:=({\mathbb F}_2^{{\mathfrak R}_f})^0$. If $n \ne 4$
the natural representation of $\mathrm{Gal}(f)$ is faithful, because in
this case the natural homomorphism
$\mathrm{Perm}({\mathfrak R}_f)\to\mathrm{Aut}_{{\mathbb F}_2}(Q_{{\mathfrak R}_f})$ is injective.
The canonical surjection $\mathrm{Gal}(K)\twoheadrightarrow
\mathrm{Gal}(K({\mathfrak R}_f)/K)=\mathrm{Gal}(f)$ provides $Q_{{\mathfrak R}_f}$ with a natural
structure of $\mathrm{Gal}(K)$-module. It is well-known that the
$\mathrm{Gal}(K)$-modules $J(C_f)_2$ and $Q_{{\mathfrak R}_f}$ are isomorphic (see
for instance \cite{Poonen,SPoonen,ZarhinTexel}). It follows easily
that $K(J(C_f)_2)=K({\mathfrak R}_f)$ and $\tilde{G}_{2,J(C_f),K}=\mathrm{Gal}(f)$.
Let us put $X=J(C_f)$ and $G := \tilde{G}_{2,X,K}$.
Then $G \cong \mathrm{Gal}(f)$, and the $G$-modules $X_2$ and
$Q_{{\mathfrak R}_f}$ are isomorphic. We freely interchange these two
modules throughout this section.
\begin{ex}
\label{F4c} Suppose that $n=q+1$ where $q\ge 5$ is a power of an
odd prime $p$. Suppose that $\mathrm{Gal}(f)=\mathrm{PSL}_2({\mathbb F}_q)$. Assume that
that $f(x)$ is {\sl irreducible}, i.e., $\mathrm{Gal}(f)=\mathrm{PSL}_2({\mathbb F}_q)$
acts transitively on the $(q+1)$-element set ${\mathfrak R}_f$. If $\beta\in
{\mathfrak R}_f$ then its stabilizer $\mathrm{Gal}(f)_{\beta}$ is a subgroup of
index $q+1$ and therefore contains a Sylow $p$-subgroup of
$\mathrm{PSL}_2({\mathbb F}_q)$. It follows from the classification of subgroups of
$\mathrm{PSL}_2({\mathbb F}_q)$ \cite[Theorem 6.25 on page 412]{Suzuki} and
explicit description of its Sylow $p$-subgroup and their
normalizers \cite[p. 191--192]{Huppert} that that
$\mathrm{Gal}(f)_{\beta}$ is conjugate to the (Borel) subgroup of
upper-triangular matrices and therefore the $\mathrm{PSL}_2({\mathbb F}_q)$-set
${\mathfrak R}_f$ is isomorphic to the projective line ${\mathbb P}^1({\mathbb F}_q)$ with the
standard action of $\mathrm{PSL}_2({\mathbb F}_q)$ (by fractional-linear
transformations), which is well-known to be doubly transitive \cite{Mortimer}.
Assume, in addition that $q$ is congruent to $\pm 3$ modulo $8$.
Then it is known \cite{Mortimer} that
$$\mathrm{End}_{\mathrm{Gal}(f)}(Q_{{\mathfrak R}_f})={\mathbb F}_4.$$
\end{ex}
\begin{thm}
\label{l2q} Suppose that $\mathrm{char}(K)\ne 2$ and $n=q+1$ where $q\ge
5$ is a prime power that is congruent to $\pm 3$ modulo $8$.
Suppose that $\mathrm{Gal}(f)=\mathrm{PSL}_2({\mathbb F}_q)$ acts doubly transitively on
${\mathfrak R}_f$ (where ${\mathfrak R}_f$ is identified with the projective line
${\mathbb P}^1({\mathbb F}_q)$). Then $\mathrm{End}^0(J(C_f))$ is a simple ${\mathbb Q}$-algebra,
i.e. $J(C_f)$ is either absolutely simple or isogenous to a power
of an absolutely simple abelian variety.
\end{thm}
\begin{proof}
See \cite[Theorem 3.10]{ZarhinLuminy}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{main}]
The result follows from Theorem \ref{PSL2} combined with Example
\ref{F4c} and Theorem \ref{l2q}.
\end{proof}
\section{Criteria for Absolute Simplicity}
\label{noniso}
Sometimes, it is possible to rule out the second outcome of
Theorem \ref{main}. First, recall Goursat's lemma \cite[p.
75]{LangAlg}:
\begin{lem}
Let $G_1$ and $G_2$ be finite group, and $H$ a subgroup of $G_1 \times G_2$ such that
the restrictions $p_1: H \to G_1$ and $p_2: H \to G_2$ of the projection maps are surjective.
Let $H_1$ and $H_2$ be the normal subgroups of $G_1$ and $G_2$, respectively,
such the groups $H_1 \times \{1\}$ and $\{1\} \times H_2$ are kernels of $p_2$ and $p_1$,
respectively.
Then there exists an isomorphism $\gamma: G_1/H_1 \cong G_1/H_2$
such that $H$ coincides with the preimage in $G_1\times G_2$ of the
graph of $\gamma$ in $G_1/H_1 \times G_2/H_2$.
\end{lem}
\begin{ex}\label{egprod}
Let $G_1$ be a finite simple group and $G_2$ be a finite group that
does not admit $G_1$ as a quotient. If $H$ is a subgroup of $G_1\times G_2$
that
satisfies the conditions of Goursat's lemma, then $H=G_1 \times G_2$.
Indeed, since $G_1$ is simple, $H_1=\{1\}$ or $G_1$.
We have $H_1\neq\{1\}$, since otherwise $G_1/H_1 \cong G_1$ and
no quotient of $G_2$ is isomorphic to $G_1$.
Therefore, $H_1 = G_1$, $G_1/H_1 \cong G_2/H_2 = \{1\}$,
and $H_2=G_2$.
Since $G_1/H_1\times G_2/H_2$ is a trivial group,
the graph of $\gamma$ coincides with $G_1/H_1\times G_2/H_2$,
and its preimage $H$ coincides with $G_1 \times G_2$.
\end{ex}
\begin{thm}\label{thmnoniso}
Let $K$ be a field of characteristic zero. Suppose that $f(x)\in
K[x]$ is a polynomial of degree $n \ge 5$ without multiple roots.
Let us consider the hyperelliptic curve $C_f: y^2 = f(x)$ and its
jacobian $J(C_f)$. Suppose that $h(x) \in K[x]$ is an irreducible
cubic polynomial and let us consider the elliptic curve $Y: y^2 =
h(x)$.
Let us assume that $f(x)$ and $h(x)$ enjoy the following
properties:
\begin{enumerate}
\item\label{simplecond}
$\mathrm{Gal}(K({\mathfrak R}_f)/K) = \mathrm{PSL}_2({\mathbb F}_q)$ for some odd prime power $q\equiv 3 \mod 8$
with $n=q+1$, and $\mathrm{Gal}(K({\mathfrak R}_f)/K)$ acts doubly transitively on
${\mathfrak R}_f$ (where ${\mathfrak R}_f$ is identified with the projective line ${\mathbb P}^1({\mathbb F}_q)$);
\item\label{s3cond} $\mathrm{Gal}(K({\mathfrak R}_h)/K) = {\mathbf S}_3$.
\end{enumerate}
Then $\mathrm{Hom}(J(C_f), Y)=0$ and $\mathrm{Hom}(Y, J(C_f))=0$. In particular,
$J(C_f)$ is not $K_a$-isogenous to a self-product of $Y$.
\end{thm}
\begin{proof}
First, we prove that $K({\mathfrak R}_f)$ and $K({\mathfrak R}_h)$ are linearly
disjoint over $K$. Let us put $G_1:=\mathrm{Gal}(K({\mathfrak R}_f)/K)$,
$G_2:=\mathrm{Gal}(K({\mathfrak R}_h)/K)$, and $H := \mathrm{Gal}(K({\mathfrak R}_f, {\mathfrak R}_h)/K)$, the
Galois group of the compositum of $K({\mathfrak R}_f)$ and $K({\mathfrak R}_h)$ over
$K$. By Theorem 1.14 of \cite{LangAlg}, $H$ can be considered to
be a subgroup of $G_1\times G_2$, where the Galois restriction
maps coincide with restrictions of projection maps $p_i: G_1
\times G_2 \to G_i$, with $i=1, 2$, to $H$. It follows from
Example \ref{egprod} that $H \cong G_1 \times G_2$, and $K({\mathfrak R}_f)$
and $K({\mathfrak R}_h)$ are linearly disjoint over $K$. The equalities
$\mathrm{Hom}(J(C_f), Y)=0$ and $\mathrm{Hom}(Y, J(C_f))=0$
follow from the definitions (s) and (p3) and Theorem 2.5 of
\cite{ZarhinHHJ}. Since for any positive integer $r$ we have
$\mathrm{Hom}(J(C_f), Y^r)= \prod_{i=1}^r \mathrm{Hom}(J(C_f),Y)$,
we conclude that $\mathrm{Hom}(J(C_f), Y^r) = 0$.
\end{proof}
The following assertion will be proven in Section \ref{quadratic}.
\begin{thm}
\label{class} Let $p>3$ be a prime such that $p\equiv 3 \mod 8$.
Let us put $\omega=\frac{-1+\sqrt{-p}}{2}$ and let
${\mathcal O}={\mathbb Z}+{\mathbb Z}\omega$ be the ring of integers in ${\mathbb Q}(\sqrt{-p})$. Let
${\mathcal O}_2={\mathbb Z}+2{\mathcal O}$ be the order of conductor $2$ in ${\mathbb Q}(\sqrt{-p})$.
\begin{itemize}
\item[(i)] The principal ideal $(2)$ is prime in ${\mathcal O}$.
\item[(ii)] Let ${\mathfrak b}$ be a proper fractional ${\mathcal O}_2$-ideal in
${\mathbb Q}(\sqrt{-p})$ and ${\mathfrak a}={\mathcal O}{\mathfrak b}$ be
the ${\mathcal O}$-ideal generated by ${\mathfrak b}$.
Then ${\mathfrak b}$ contains $2{\mathfrak a}$ as a subgroup of index $2$ and ${\mathfrak a}$
contains ${\mathfrak b}$ as a subgroup of index $2$.
\item[(iii)] Let ${\mathfrak a}$ is
a fractional ${\mathcal O}$-ideal in ${\mathbb Q}(\sqrt{-p})$. If ${\mathfrak b}$ is a subgroup
of index $2$ in ${\mathfrak a}$ then it is a proper ${\mathcal O}_2$-ideal in
${\mathbb Q}(\sqrt{-p})$, i.e.,
$${\mathcal O}_2=\{z\in {\mathbb Q}(\sqrt{-p})\mid z{\mathfrak b} \subset {\mathfrak b}\};$$
in addition, ${\mathfrak a}={\mathcal O}{\mathfrak b}$.
There are exactly three index $2$ subgroups in ${\mathfrak a}$;
they are mutually non-somorphic as ${\mathcal O}_2$-ideals.
\item[(iv)] If ${\mathbf h}$ is the class number of ${\mathbb Q}(\sqrt{-p})$ then
$3{\mathbf h}$ is the number of classes of proper ${\mathcal O}_2$-ideals.
\end{itemize}
\end{thm}
We write $j$ for the classical modular function \cite[Ch. 3,
Sect. 3]{LangE}.
\begin{cor} \label{jinvcor}
Let $p$ be a prime such that $p\equiv 3 \mod 8$.
Let $q\ge 11$ be an odd power of $p$. (In particular, $q
\equiv p\equiv 3 \mod 8$.) Let us put
$$\omega:=\frac{-1+\sqrt{-p}}{2}, \ \alpha: = j(\omega)\in {\mathbb C}, \ K:={\mathbb Q}(j(\omega))\subset {\mathbb C}.$$
Suppose that $f(x)\in K[x]$ is an irreducible polynomial of degree
$q+1$ such that $\mathrm{Gal}(f/K)=\mathrm{PSL}_2({\mathbb F}_q)$ acts doubly transitively
on ${\mathfrak R}_f$ (where ${\mathfrak R}_f$ is identified with the projective line
${\mathbb P}^1({\mathbb F}_q)$).
Then $J(C_f)$ is an absolutely simple abelian variety, and
$\mathrm{End}^0(J(C_f))={\mathbb Q}$ or a quadratic field.
\end{cor}
\begin{proof}
Clearly, ${\mathbb Q}(\sqrt{-p})={\mathbb Q}(\sqrt{-q})$. Since $p\equiv 3 \mod 8$,
the ring of integers ${\mathcal O}$ in ${\mathbb Q}(\sqrt{-p})$ coincides with
${\mathbb Z}+{\mathbb Z}\omega$. If $p>3$ let us consider the polynomial
$$
h_p(x):=x^3 - \dfrac{27\alpha}{4(\alpha-1728)}x -
\dfrac{27\alpha}{4(\alpha-1728)}\in K[x].
$$
If $p=3$ then $$\omega=\frac{-1+\sqrt{-3}}{2},\ \alpha =
j(\omega)=0,\ K={\mathbb Q}$$ and we put
$$h_3(x):=x^3-2\in {\mathbb Q}[x]=K[x].$$
The elliptic curve $Y: y^2=h_p(x)$ is defined over $K$ and its
$j$-invariant coincides with $\alpha=j(\omega)$, i.e.,
$Y({\mathbb C})={\mathbb C}/({\mathbb Z}+{\mathbb Z}\omega)$. Hence $Y$ admits complex multiplication
by ${\mathbb Z}+{\mathbb Z}\omega={\mathcal O}$.
The following Lemma will be proven at the end of this Section.
\begin{lem}
\label{GalS3}
The polynomial $h_p(x)$ is irreducible over $K$ and its
Galois group $\mathrm{Gal}(h_p/K) \cong {\mathbf S}_3$.
\end{lem}
Combining Lemma \ref{GalS3} and Theorem \ref{thmnoniso}, we
conclude that $J(C_f)$ is not isogenous to a self-product of $Y$.
Now the result follows from Theorem \ref{main}.
\end{proof}
\begin{proof}[Proof of Lemma \ref{GalS3}]
The case $p=3$ is easy. So, further we assume that $p>3$. First,
check that the discriminant $\Delta$ of $h_p$ is not a square in
$K$. Indeed, we have $\Delta = 1458^2\, \alpha^2/(\alpha-1728)^3$,
so $\Delta$ is a square in $K$ if and only if $\alpha-1728$ is.
According to \cite[p. 288]{Birch}, if $p\equiv 3 \mod 4$, then
$\alpha$ is real and negative. Therefore the purely imaginary
$\sqrt{\alpha-1728}$ does not lie in the real number field $K :=
{\mathbb Q}(\alpha)$. Now it suffices to check that the cubic polynomial
$h_p$ is irreducible. Suppose that this is not the case, i.e.,
$h_p$ has a root in $K$. This means that $Y$ has a $K$-rational
point of order $2$ and therefore there exists an elliptic curve
$Y'$ over $K$ and a degree $2$ isogeny $Y \to Y'$. By duality, we
get a degree $2$ isogeny $Y' \to Y$. This allows us to identify
$Y'({\mathbb C})$ with ${\mathbb C}/{\mathfrak b}$ where ${\mathfrak b}$ is a subgroup of index
$2$ in ${\mathbb Z}+{\mathbb Z}\omega={\mathcal O}$. By Theorem \ref{class}(ii), ${\mathfrak b}$ is a proper
${\mathcal O}_2$-ideal. The classical theory of complex multiplication
\cite[Th. 5.7 on p. 123]{Shimura} tells us that
${\mathbb Q}(\sqrt{-p})(j(Y'))/{\mathbb Q}(\sqrt{-p})$ is an abelian extension,
whose degree coincides with the number of
classes of proper ${\mathcal O}_2$-ideals. By Theorem \ref{class}(iv),
$[{\mathbb Q}(\sqrt{-p})(j(Y')):{\mathbb Q}(\sqrt{-p})]=3{\mathbf h}$. On the other hand,
since $Y'$ is defined over $K$, the number $j(Y')\in K$ and
therefore
$${\mathbb Q}(\sqrt{-p})(j(Y'))\subset
{\mathbb Q}(\sqrt{-p})K={\mathbb Q}(\sqrt{-p})(\alpha)={\mathbb Q}(\sqrt{-p})(j(Y)).$$
However, ${\mathbb Q}(\sqrt{-p})(j(Y))$ is the absolute class field of
${\mathbb Q}(\sqrt{-p})$ and it is well-known \cite{SerreCM,Birch},
\cite[Th. 5.7 on p. 123]{Shimura} that
$[{\mathbb Q}(\sqrt{-p})(j(Y)):{\mathbb Q}(\sqrt{-p})]={\mathbf h}$. Since ${\mathbf h}<3{\mathbf h}$, we
obtain the desired contradiction.
\end{proof}
\begin{rem}
If $p=3,11, 19, 43, 67$ or $163$ then $j(\omega)\in {\mathbb Z}$
\cite{SerreCM}, so $K ={\mathbb Q}(\alpha)= {\mathbb Q}(j(\omega))={\mathbb Q}$.
\end{rem}
\begin{thm} \label{Qjinvcor}
Let $p$ be a prime such that $p\equiv 3 \mod 8$.
Let $q\ge 11$ be an odd power of $p$. (In particular, $q \equiv
p\equiv 3 \mod 8$.) Suppose that $f(x)\in {\mathbb Q}[x]$ is an irreducible
polynomial of degree $q+1$ such that $\mathrm{Gal}(f/{\mathbb Q})=\mathrm{PSL}_2({\mathbb F}_q)$
acts doubly transitively on ${\mathfrak R}_f$ (where ${\mathfrak R}_f$ is identified
with the projective line ${\mathbb P}^1({\mathbb F}_q)$).
Then $J(C_f)$ is an absolutely simple abelian variety, and
$\mathrm{End}^0(J(C_f))={\mathbb Q}$ or a quadratic field.
\end{thm}
\begin{proof}
Let us put
$$\omega:=\frac{-1+\sqrt{-p}}{2}, \ \alpha: = j(\omega)\in {\mathbb C}, \ K:={\mathbb Q}(j(\omega))\subset {\mathbb C}.$$
Since simple non-abelian $\mathrm{PSL}_2({\mathbb F}_q)$ does not have a subgroup
of index $2$,
$$\mathrm{Gal}(f/{\mathbb Q})=\mathrm{PSL}_2({\mathbb F}_q)=\mathrm{Gal}(f/{\mathbb Q}(\sqrt{-p})).$$
Since $\mathrm{PSL}_2({\mathbb F}_q)$ is perfect and
$K{\mathbb Q}(\sqrt{-p})={\mathbb Q}(\sqrt{-p})(j(\omega))$ is abelian over
${\mathbb Q}(\sqrt{-p})$,
$$\mathrm{Gal}(f/{\mathbb Q})=\mathrm{PSL}_2({\mathbb F}_q)=\mathrm{Gal}(f/K{\mathbb Q}(\sqrt{-p})).$$
Since $$\mathrm{Gal}(f/K{\mathbb Q}(\sqrt{-p}))\subset \mathrm{Gal}(f/K)\subset
\mathrm{Gal}(f/{\mathbb Q}),$$ we conclude that
$$\mathrm{Gal}(f/K)=\mathrm{Gal}(f/{\mathbb Q})=\mathrm{PSL}_2({\mathbb F}_q)$$
acts doubly transitively on ${\mathfrak R}_f$. In order to finish the proof,
one has only to apply Corollary \ref{jinvcor}.
\end{proof}
\section{Proof of Theorem \ref{class}}
\label{quadratic} There is a positive integer $k$ such that
$p=8k+3$. It follows that $\omega^2+\omega+(2k+1)=0$. This implies
that the $4$-element algebra ${\mathcal O}/2{\mathcal O}$ contains a subalgebra
isomorphic to the finite field ${\mathbb F}_4$ and therefore coincides with
${\mathbb F}_4$. This means that $(2)$ is prime in ${\mathcal O}$. So, this proves
(i).
Suppose that ${\mathfrak b}$ is a proper ${\mathcal O}_2$-ideal in ${\mathbb Q}(\sqrt{-p})$ and
${\mathfrak a}:={\mathcal O}{\mathfrak b}$. Clearly, $2{\mathfrak a}\subset{\mathfrak b}\subset{\mathfrak a}$. Since ${\mathfrak a}$ and $2{\mathfrak a}$
are ${\mathcal O}$-ideals, ${\mathfrak b}$ does coincides neither with ${\mathfrak a}$ nor with
$2{\mathfrak a}$. Since $2{\mathfrak a}$ has index $4$ in ${\mathfrak a}$, the group ${\mathfrak b}$ has index
$2$ in ${\mathfrak a}$ and $2{\mathfrak a}$ has index $2$ in ${\mathfrak b}$. This proves (ii).
Now, suppose that ${\mathfrak a}$ is a fractional ${\mathcal O}$-ideal in
${\mathbb Q}(\sqrt{-p})$ and a subgroup ${\mathfrak b}\subset {\mathbb Q}(\sqrt{-p})$ satisfies
$2{\mathfrak a}\subset{\mathfrak b}\subset{\mathfrak a}$. If ${\mathfrak b}$ is an ${\mathcal O}$-ideal then the unique
factorization of ${\mathcal O}$-ideals and the fact that $(2)$ is prime
imply that either ${\mathfrak b}={\mathfrak a}$ or ${\mathfrak b}=2{\mathfrak a}$. So, if ${\mathfrak b}$ has index $2$
in ${\mathfrak a}$, it is neither ${\mathfrak a}$ nor $2{\mathfrak a}$ and therefore is {\sl not}
an ${\mathcal O}$-ideal.
On the other hand, it is clear that ${\mathcal O}{\mathfrak b}\subset{\mathfrak a}$ and
$2{\mathcal O}{\mathfrak b}\subset 2{\mathfrak a}\subset{\mathfrak b}$ and therefore ${\mathfrak b}$ is a proper
${\mathcal O}_2$-ideal. This proves the first assertion of (iii). We have
${\mathfrak b}\subset{\mathcal O}{\mathfrak b}\subset{\mathcal O}{\mathfrak a}={\mathfrak a}$ but ${\mathfrak b}\ne{\mathcal O}{\mathfrak b}$. Since the index of
${\mathfrak b}$ in ${\mathfrak a}$ is $2$, we conclude that ${\mathcal O}{\mathfrak b}={\mathfrak a}$. This proves the
second assertion of (iii).
Since ${\mathfrak a}$ is a free commutative
group of rank $2$, it contains exactly three subgroups of index
$2$. Let ${\mathfrak b}_1$ and ${\mathfrak b}_2$ be two distinct subgroups of index $2$
in ${\mathfrak a}$. We have
$${\mathcal O}{\mathfrak b}_1={\mathfrak a}={\mathcal O}{\mathfrak b}_2.$$
Suppose that ${\mathfrak b}_1$ and ${\mathfrak b}_2$ are isomorphic as ${\mathcal O}_2$-ideals.
This means that there exists a non-zero $\lambda\in {\mathbb Q}(\sqrt{-p})$
such that $\lambda {\mathfrak b}_1={\mathfrak b}_2$. It follows that $\lambda{\mathfrak a}={\mathfrak a}$ and
therefore $\lambda$ is a unit in ${\mathcal O}$. Since $p>3$, we have
$\lambda=\pm 1$ and therefore ${\mathfrak b}_2={\mathfrak b}_1$. This proves the last
assertion of (iii).
The assertion (iv) follows easily from (ii) and (iii). (It is also
a special case of Exercise 11 in Sect. 7 of Ch. II in \cite{BS}
and of Exercise 4.12 in Section 4.4 of \cite{Shimura}).
\section{Examples}
\label{examples}
\begin{ex}\label{ex11}
Let $S$ be a transcendental over ${\mathbb Q}$, $T=2^8 3^5/(11S^2+1)$, and put
\begin{align*}
f_{11, S}(x) :=
& (x^3 - 66x - 308)^4\\
& - 9 T (11x^5 - 44x^4 - 1573x^3 + 1892x^2 + 57358x + 103763) \\
& - 3 T^2 (x - 11).
\end{align*}
According to Table 10 of the Appendix in \cite{MMIGT},
$\mathrm{Gal}(f_{11, S}/{\mathbb Q}(S)) = \mathrm{PSL}_2({\mathbb F}_{11})$. It can be verified
using MAGMA \cite{MAGMA} that when $s = m / n$ for any nonzero
integers $-5\leq m,n \leq 5$, then $\mathrm{Gal}(f_{11, s}/{\mathbb Q}) =
\mathrm{PSL}_2({\mathbb F}_{11})$. Consider the hyperelliptic curve
$$
C_{11, s}: y^2 = f_{11, s}(x)
$$
over ${\mathbb Q}$ by any one of these $s$.
By Theorem \ref{Qjinvcor} the $5$-dimensional abelian variety
$J(C_{11, s})$ is absolutely simple. For example, if we put $s=1$,
then we obtain a hyperelliptic curve
\begin{eqnarray*}
C_{11, 1}: y^2 &=& x^{12} - 264 \,x^{10} - 1232 \,x^9 + 26136 \,x^8 + 243936 \,x^7 - 580800 \,x^6 \\
& &\ \ \ - 16612992 \,x^5 - 54104688 \,x^4 + 310712512 \,x^3 + 2391092352 \,x^2 \\
& &\ \ \ + 4956865152 \,x + 5044849216
\end{eqnarray*}
over ${\mathbb Q}$ with $J(C_{11, 1})$ absolutely simple.
\end{ex}
\begin{ex}\label{ex13}
If we define
\begin{align*}
f_{13, S}(x) :=
& (x^2 + 36) (x^3 - x^2 + 35x - 27)^4 \\
& - 4 T (7x^2 - 2x + 247) (x^2+39)^6 / 27
\end{align*}
with $T=1/(39S^2+1)$ then again \cite{MMIGT} we have
$\mathrm{Gal}(f_{13, S}/{\mathbb Q}(S)) = \mathrm{PSL}_2({\mathbb F}_{13})$.
Similarly, we checked using MAGMA
that when $s = m / n$
for any nonzero integers $-5\leq m,n \leq 5$,
then $\mathrm{Gal}(f_{13, s}/{\mathbb Q}) = \mathrm{PSL}_2({\mathbb F}_{13})$.
If we define
$$
C_{13, s}: y^2 = f_{13, s}(x)
$$
over ${\mathbb Q}$, then by Theorem \ref{FTKL} the $6$-dimensional abelian
variety $J(C_{13, s})$ is absolutely simple. As an example, take $s=-1$ to get
the hyperelliptic curve
\begin{eqnarray*}
C_{13, -1}: y^2 &=& 263/270 \,x^{14} - 539/135 \,x^{13} + 9451/54 \,x^{12} - 10114/15 \,x^{11} \\
&& \ \ \ + 376363/30 \,x^{10} - 45487 \,x^9 + 891605/2 \,x^8 - 1533844 \,x^7 \\
&& \ \ \ + 15279043/2 \,x^6 - 25943931 \,x^5 + 391472991/10 \,x^4 \\
&& \ \ \ - 896502438/5 \,x^3 - 780396201/2 \,x^2 - 365687757/5 \,x \\
&& \ \ \ - 31998670461/10
\end{eqnarray*}
defined over ${\mathbb Q}$ with $J(C_{13, -1})$ absolutely simple.
\end{ex}
See \cite{Malle} for other examples of irreducible polynomials
over ${\mathbb Q}(T)$ of degrees $n=p+1$ with $p=11, 13, 19, 29,37$, whose
Galois groups are isomorphic to $\mathrm{PSL}_2({\mathbb F}_p)$. These polynomials
can be used in a manner similar to that of Examples \ref{ex11} and
\ref{ex13}, in order to construct examples of absolutely simple
abelian varieties over ${\mathbb Q}$ of dimensions $5, 6, 9, 14,18$
respectively, whose endomorphism algebra is either ${\mathbb Q}$ or a
quadratic field.
|
1,314,259,996,473 | arxiv | \section{Introduction}
The capacity (or ``harmonic'' or ``electrostatic'' or ``Newtonian'' capacity) of a compact set $K \subset \mathbb{R}^n$, $n \geq 3$, is defined as $$\capac(K) = \inf_{\phi} \left\{\frac{1}{(n-2)\omega_{n-1}} \int_{\mathbb{R}^n} |\nabla \phi|^2 dV\; :\; \phi \text{ is Lipschitz with compact support, and } \phi \equiv 1 \text{ on } K\right\},$$
where $\omega_{n-1}$ is the hypersurface area of the unit $(n-1)$-sphere. If $\partial K$ is sufficiently regular (e.g., a $C^1$ hypersurface), then there exists a unique harmonic function on $\mathbb{R}^n \setminus K$, equaling 1 on $\partial K$ and approaching 0 at infinity, such that
$$\capac(K) = \frac{1}{(n-2)\omega_{n-1}} \int_{\mathbb{R}^n \setminus K} |\nabla u|^2 dV = - \frac{1}{(n-2)\omega_{n-1}} \int_{S} \frac{\partial u}{\partial \nu} dA,$$
for any surface $S$ enclosing $\partial K$. For example, a ball of radius $r$ has capacity equal to $r^{n-2}$. Capacity is monotone under set inclusion and enjoys nice measure-theoretic properties, such as inner and outer regularity \cite{EG}. Geometrically, it can be bounded below by the volume radius of $K$ and, if $\partial K$ is convex, bounded above in terms of the total mean curvature of $\partial K$ \cite{PS}.
Capacity also makes sense with an analogous definition in complete Riemannian manifolds, such as asymptotically flat manifolds. (Without some control on the asymptotics, however, the capacity could be zero for every compact set.) It is natural to ask how capacity behaves along a converging sequence of Riemannian manifolds. For example, it is not difficult to show that if $M$ is a smooth manifold equipped with a sequence of complete Riemannian metrics $g_i$, and $g_i$ converges uniformly (i.e, in $C^0$) on compact sets to a Riemannian metric $g$, then for any compact set $K \subset M$,
$$\limsup_{i \to \infty} \capac_{g_i}(K) \leq \capac_g(K).$$
More generally, an analogous statement holds for a sequence of complete Riemannian manifolds converging in the pointed $C^0$ Cheeger--Gromov sense. In fact, strict inequality can hold.
This upper semicontinuity of capacity contrasts sharply with two other natural notions of the ``size'' of $K$, the volume and perimeter, which are of course continuous under $C^0$ convergence of the background metrics. This different behavior of capacity springs from its non-local nature, specifically its dependence on the geometry ``at infinity.''
The aim of this paper is to study capacity in lower regularity background spaces and in particular to analyze its behavior under lower-regularity convergence.
While capacity in lower regularity (such as in metric measure spaces) has received significant attention in the analysis literature (see below), we are also interested in capacity and its continuity properties for geometric reasons. For example a recent paper by Jauregui \cite{Jau} suggests a definition of total mass in general relativity for asymptotically flat 3-manifolds that is based on the capacity--volume inequality, generalizes the well-known ADM mass, and is inspired by Huisken's isoperimetric mass \cites{Hui1,Hui2}. Several important open problems in general relativity related to the total mass seem to naturally involve Sormani--Wenger intrinsic flat (``$\mathcal{F}$'') limits (we refer the reader to \cite{Sor2} and \cite{JL}, for example), so the behavior of capacity under such convergence is of interest. For instance, Jauregui and Lee showed a lower semicontinuity Huisken's isoperimetric mass \cites{Hui1,Hui2} under pointed $\mathcal{VF}$-convergence \cite{JL} (where ``$\mathcal{VF}$'' refers to volume-preserving intrinsic flat convergence). The definition of mass in \cite{Jau} involves capacity with a negative sign, so the upper semicontinuity we prove here is supportive of lower semicontinuity of that mass. We continue this discussion to section \ref{sec_mass}.
Our approach to establishing the upper semicontinuity of capacity is inspired by Portegies' proof that certain min-max values of the Laplacian on a compact Riemannian manifold $M$ are upper semicontinuous under $\mathcal{VF}$ convergence \cite{Por}. Recall that, for example, the first such eigenvalue,
$$\lambda_1 = \inf_{f \in C^{\infty}(M)} \left\{\int_M |\nabla f|^2 dV \; : \; \int_M f^2 dV=1, \int_M fdV = 0 \right\},$$
varies continuously with respect to $C^2$ convergence of Riemannian metrics, but may only be upper semicontinuous for weaker types of convergence such as measured Gromov--Hausdorff convergence \cite{Fuk}. Since capacity is only interesting in noncompact spaces, we will specifically study the behavior of capacity under \emph{pointed} $\mathcal{VF}$-convergence. In Section \ref{sec_examples}, we provide examples to illustrate that there are essentially two distinct reasons for the capacity to jump under a $\mathcal{VF}$-limit, and both jumps ``go the same way.'' One reason is non-uniform control at infinity, which may be seen even under smooth convergence; the other is an effect of the relatively coarse nature of $\mathcal{VF}$ convergence.
Below we state one of our main theorems, that the capacity of closed balls of a fixed radius about converging points in a sequence of converging spaces cannot jump down in a limit. The natural setting for intrinsic flat convergence is the integral current spaces of Sormani and Wenger \cite{SW}, which are constructed using the integral currents on metric spaces of Ambrosio and Kirchheim \cite{AK_cur}. We will use local versions of these spaces (building on Lang and Wenger's locally integral currents \cite{LW}) and pointed convergence and make use of the definition of Dirichlet energy in these spaces appearing in \cite{Por} to define capacity. The relevant definitions will be given in Section \ref{sec_background}.
\begin{thm}
\label{thm_balls_intro}
Let $N_i = (X_i, d_i, T_i)$ and $N_\infty = (X_\infty, d_\infty, T_\infty)$ be local integral current spaces of dimension $m \geq 2$, such that $N_i \to N_\infty$ in the pointed volume-preserving intrinsic flat sense with respect to $p_i \in X_i$ and $p_\infty \in X_\infty$. Suppose the closed ball $\overline B(p_\infty,r)$ in $X$ is compact for some $r>0$. Then:
\begin{equation}
\label{eqn_limsup_balls_intro}
\limsup_{i \to \infty} \capac_{N_i} (\overline B(p_i,r)) \leq \capac_{N_\infty}(\overline B(p_\infty,r)).
\end{equation}
\end{thm}
The other main theorem, Theorem \ref{thm_sublevel}, is a version of Theorem \ref{thm_balls_intro} that replaces closed balls with sublevel sets of Lipschitz functions. Both of these theorems will be proved using the technical result Theorem \ref{thm_extrinsic}, an extrinsic version of the theorems which itself is inspired by Theorem 6.2 in \cite{Por}.
We note that there is a body of literature on capacities of sets in metric measure spaces. While we make no attempt at a comprehensive account, we discuss this partially here, referring the reader to the books by Bj\"orn and Bj\"orn \cite{BB} and Heinonen, Koskela, Shanmugalingam, and Tyson \cite{HKST} for example. One approach begins with the definition of Sobolev functions on a metric measure space, due to Haj\l asz \cite{Haj}, based on the Hardy--Littlewood maximal operator. Kinnunen and Martio used Haj\l asz's definition to develop a Sobolev $p$-capacity \cite{KM}. (Sobolev $p$-capacity, in contrast to the capacity we consider here, is found by minimizing the Sobolev norm, i.e., it includes the $L^p$ norm of the test functions.) A different approach is to consider, instead of Haj\l asz's Sobolev spaces, the Sobolev spaces defined by Shanmugalingam \cite{Sha}, called Newtonian spaces, an approach to Sobolev spaces using weak upper gradients. This is explored in detail in \cite{BB}, where the version of capacity based only on the $L^p$ norm of the weak upper gradient (as we consider in this paper) is called the variational capacity. Since the norm of the tangential differential is $\|T\|$-almost-everywhere equal to the minimal relaxed gradient (see Theorem \ref{thm_df} in the appendix, cf. \cite[Theorem 5.2]{Por}), this latter capacity is equivalent to that which we use in this paper.
We continue this discussion in the appendix and also refer the reader to \cite{GT} for additional discussion on capacities in metric measure spaces.
\begin{outline}
Section \ref{sec_background} covers the essential background material, including Ambrosio--Kirchheim currents on metric spaces, flat convergence, integral current spaces, and Sormani--Wenger intrinsic flat convergence, before moving on to local integral current spaces and pointed $\mathcal{F}$- and $\mathcal{VF}$-convergence. We also recall the definition of the Dirichlet energy of a Lipschitz function using the tangential differential (where some details are deferred to the appendix), and use that to define the capacity in a local integral current space. The main results are presented and proved in Section \ref{sec_main}, and several examples are given in Section \ref{sec_examples} to demonstrate how capacity may ``jump up'' and show that volume-preserving convergence is necessary. Section \ref{sec_mass} includes further discussion regarding how capacity on integral current spaces may be of interest for problems involving mass in general relativity.
\end{outline}
\begin{ack}
R. P. acknowledges support from CONACyT Ciencia de Frontera 2019 CF217392 grant.
\end{ack}
\section{Definitions and basic objects}
\label{sec_background}
The theory of $m$-dimensional currents on $\mathbb{R}^n$ originated with de Rham \cite{deRham} and was developed by Whitney \cite{Wh57} and especially Federer--Fleming \cite{FeFle}. This area of geometric measure theory has been instrumental for attacking a wide variety of problems in geometric analysis. However, there is interest in generalizing this theory to ambient spaces that are not smooth manifolds. This was accomplished by Ambrosio--Kirchheim by following an idea of de Giorgi: rather than viewing a current as a functional on the space of smooth differential forms, Ambrosio and Kirchheim defined an $m$-dimensional current on a metric space $X$ as a functional on $(m+1)$-tuples of Lipschitz functions $(f, \pi_1, \ldots, \pi_m)$, satisfying some additional properties \cite{AK_cur}. Special types of classical currents, such as normal and integral currents, generalize to the metric space setting.
Inspired by the Gromov--Hausdorff distance between compact metric spaces and seeking an analog of Whitney's flat convergence to be defined for metric spaces, Sormani and Wenger produced a definition of \emph{integral current spaces}, which are essentially metric spaces equipped with an integral current in the sense of Ambrosio--Kirchheim. For example, compact, connected oriented Riemannian manifolds can naturally be viewed as integral current spaces. Moreover, Sormani and Wenger defined the \emph{intrinsic flat distance} between integral current spaces, using the flat distance on metric spaces due to Wenger \cite{We2007}.
Our primary goal is to study the capacity of compact sets in an integral current space, and in particular the continuity behavior of capacity. We certainly want the theory to include ambient spaces of infinite mass (volume), such as $\mathbb{R}^n$, yet Sormani--Wenger integral current spaces (built using Ambrosio--Kirchheim currents) by definition have finite mass. To work around this, we will use Lang--Wenger's extension \cite{LW} of Ambrosio--Kirchheim \cite{AK_cur} integral currents on metric spaces to so-called \emph{locally integral currents}, then generalize Sormani--Wenger's definition of integral current space accordingly. We recall the details of locally integral currents in Section \ref{sec_currents}. In Section \ref{sec_flat_weak}, we recall local flat, weak, and flat convergence. Next, in Section \ref{sec_local_currents}, we recall the details of Sormani--Wenger intrinsic flat convergence and its pointed version. Finally, in Section \ref{sec_dirichlet}, we give the definition of capacity in local integral current spaces. This definition involves a differential of Lipschitz functions on a metric space, and there are a number of ideas from \cite{AK_rect} that will be recalled in the appendix. This approach is inspired by the work of Portegies on studying the behavior of eigenvalues of the Laplacian under $\mathcal{VF}$-convergence \cite{Por}.
\subsection{Locally integral currents}
\label{sec_currents}
The goal of this section is to arrive at the definition of a locally integral current. Below we are essentially summarizing the parts of \cite{LW} that will get us to that point, including only the minimal details, with no proofs.
First, we recall metric functionals and how they produce an outer measure. Currents of locally finite mass will be those metric functionals whose measure behaves well from the inside and outside. Locally integer rectifiable currents are defined next, and then finally locally integral currents.
Given a metric space $Z$, define:
\begin{itemize}
\item $\Lip(Z)$ as the vector space of Lipschitz functions $Z \to \mathbb{R}$,
\item $\Liploc(Z)$ as the vector space of functions $Z \to \mathbb{R}$ that are Lipschitz on bounded sets,
\item $\Lipb(Z)$ as the vector space of Lipschitz functions $Z \to \mathbb{R}$ that are bounded, and
\item $\LipB(Z)$ as the vector space of Lipschitz functions $Z \to \mathbb{R}$ that are bounded with bounded support.
\end{itemize}
The Lipschitz constant of a function $f:Z \to \mathbb{R}$ will be denoted by $\Lip(f)$.
For an integer $m \geq 0$, an $m$-dimensional metric functional $T$ will act on $(m+1)$-tuples of functions in $\LipB(Z) \times \left[\Liploc(Z)\right]^m$ and produce a real number. Such an $m$-tuple $(f, \pi_1, \ldots, \pi_m)$ (sometimes denoted more briefly by $(f,\pi)$) should be conceptually thought of as the differential form ``$f d\pi_1 \wedge \dots \wedge d\pi_m$'', so metric functionals will generalize the idea of currents. The precise definition of $T$ being a \emph{metric functional} is that
\begin{enumerate}
\item[(i)] $T$ is multilinear.
\item[(ii)] (continuity) Suppose $f \in \LipB(Z)$ and $(\pi_1, \ldots, \pi_m) \in \Liploc(Z)^m$, and that we have $m$ sequences $\{\pi_i^j\}_{j=1}^\infty$ in $\Liploc(Z)$ such that $\pi_i^j \to \pi_i$ pointwise as $j \to \infty$ for each $i=1,\ldots, m$, and the Lipschitz constants of $\pi_i^j$ are uniformly bounded in $j$ on any bounded subset of $Z$. Then
$$\lim_{j \to \infty} T(f, \pi_1^j, \ldots, \pi_m^j) = T(f,\pi_1, \ldots, \pi_m).$$
\item[(iii)] (locality) Consider $(f, \pi_1, \ldots, \pi_m) \in \LipB(Z) \times \left[\Liploc(Z)\right]^m$. Suppose that one of the $\pi_i$ is constant on the $\delta$-neighborhood of $\spt(f)$ for some $\delta > 0$. Then
$$T(f, \pi_1, \ldots, \pi_m) = 0.$$
\end{enumerate}
Metric functionals as defined here are natural analogs of the metric functionals considered by Ambrosio and Kirchheim \cite{AK_cur} (in that case, $f$ was a bounded Lipschitz function and the $\pi_i$ were required to be (globally) Lipschitz).
If $X$ and $Y$ are metric spaces and $\varphi \in \Liploc(X,Y)$ with $\varphi^{-1}(A)$ bounded for any bounded set $A \subseteq Y$, a metric functional $T$ on $X$ can be pushed forward to a metric functional on $Y$ of the same dimension as follows:
$$(\varphi_{\#}T)(f, \pi_1, \ldots, \pi_m) = T(f \circ \varphi, \pi_1 \circ \varphi, \ldots, \pi_m \circ \varphi).$$
The \emph{boundary} of an $m$-dimensional metric functional $T$, $m \geq 1$, is an $(m-1)$-dimensional metric functional $\partial T$ defined by
$$\partial T(f, \pi_1, \ldots, \pi_m) = T(\sigma, f, \pi_1, \ldots, \pi_m),$$
where $\sigma$ is any bounded Lipschitz function with bounded support that is identically 1 on the support of $f$. In \cite{LW} it is shown this definition is independent of the choice of $\sigma$ and defines a metric functional. The boundary satisfies nice properties, such as commuting with the push-forward and $\partial (\partial T)=0$.
To define currents, we require the notion of the mass measure $\|T\|$ associated to a metric functional $T$. Ambrosio--Kirchheim's definition requires finite mass, but the approach we take following Lang--Wenger will require only locally finite mass.
Given an $m$-dimensional metric functional $T$ and an open set $V \subseteq Z$, the \emph{mass of $T$ in $V$} is defined to be
$$\mathbf{M}_V(T) = \sup \sum_{\lambda} T(f^\lambda, \pi^\lambda),$$
where the supremum is taken over all finite collections
$f^\lambda \in \LipB(Z)$,
$\pi^\lambda = (\pi_1^\lambda, \ldots, \pi_m^\lambda) \in \Lip(Z)$
such that $\Lip(\pi^\lambda_i) \leq 1$, $f^\lambda$ is supported in $V$,
and $\sum_{\lambda} |f^\lambda| \leq 1$ everywhere. This
definition serves as the mass of $T$ for any open set. For an arbitrary subset $A \subseteq Z$, we define
$$\|T\|(A) = \inf\{ \mathbf{M}_V(T) \; | \; V \supseteq A \text{ is open}\},$$
so certainly $\|T\|(A) = \mathbf{M}_A(T)$ if $A$ is open.
\begin{definition}[\cite{LW}]
An \emph{$m$-dimensional metric current on $Z$ with locally finite mass}, $m \geq 0$, is an $m$-dimensional metric functional $T$ on $Z$ such that given any $\epsilon >0$ and any bounded open set $U \subseteq Z$, we have $\mathbf{M}_U(T) < \infty$ and that there exists a compact set $C \subseteq Z$ such that $\mathbf{M}_{U \setminus C}(T) < \epsilon$.
\end{definition}
The set of such objects will be denoted $\mathbf{M}_{\text{loc}, m}(Z)$. For $T \in \mathbf{M}_{\text{loc}, m}(Z)$, $\|T\|$ defined above is a Borel regular outer measure on $Z$ that is concentrated on a $\sigma$-compact set \cite[Proposition 2.4]{LW}.
We also recall the definition of two ways of identifying where the $m$-dimensional metric functional $T$ ``lives'': the support and the canonical set:
\begin{align*}
\spt(T) &= \{z \in Z \; : \; \|T\|(B(z,r))> 0 \text{ for all } r > 0\}\\
\set(T) &= \{z \in Z \; : \; \liminf_{r \to 0} \frac{\|T\|(B(z,r))}{r^m} > 0\}.
\end{align*}
The latter definition was used by Sormani and Wenger, particularly in their definition of integral current space.
Recall that metric functionals $T$ are defined as functions $\LipB(Z) \times \left[\Liploc(Z)\right]^m \to \mathbb{R}$. However, it is shown in \cite{LW} that if $T \in \mathbf{M}_{\text{loc}, m}(Z)$, then $T$ has a natural extension to the larger space of $(m+1)$-tuples $(f,\pi)$ such that $f$ is a bounded Borel function with bounded support (and the $\pi$ are as before). This allows us to restrict $T \in \mathbf{M}_{\text{loc}, m}(Z)$ to a bounded Borel set $A \subseteq Z$ as follows:
$$(T \llcorner A)(f,\pi) := T(f\chi_A, \pi).$$
Furthermore, $T \llcorner A \in \mathbf{M}_{\text{loc}, m}(Z)$ and $\|T \llcorner A\| = \|T\| \llcorner A$.
There is nice compatibility between Ambrosio--Kirchheim currents and the metric currents with locally finite mass; see \cite[Section 2.5]{LW}.
Our main interest is in the definition of locally integral currents, so we build up to that now.
A subset $C \subseteq Z$ is a \emph{compact $m$-rectifiable set} if there exist compact sets $K_1, \ldots, K_n \subset \mathbb{R}^m$ and Lipschitz maps $f_i: K_ i \to Z$ such that
$$C = \bigcup_{i=1}^n f_i(K_i).$$
\begin{definition}[\cite{LW}]
An $m$-dimensional metric functional $T$ on $Z$ is a \emph{locally integer rectifiable current} if:
\begin{enumerate}[(a)]
\item Given $\epsilon >0$ and any bounded open subset $U$ of $Z$, there is a compact $m$-rectifiable set $C \subseteq Z$ such that $\mathbf{M}_U(T) < \infty$ and $\mathbf{M}_{U \setminus C}(T) < \epsilon$.
\item For every bounded Borel set $B \subseteq Z$ and every Lipschitz map $\varphi: Z \to \mathbb{R}^m$, we have
$$\varphi_{\#} (T \llcorner B)(f, \pi) = \int_{\mathbb{R}^m} \theta f d\pi_1 \wedge \ldots \wedge d\pi_m,$$
for some integrable $\theta: \mathbb{R}^m \to \mathbb{Z}$.
\end{enumerate}
The abelian group of locally integer rectifiable $m$-currents on $Z$ will be denoted by $\mathcal{I}_{\text{loc},m}(Z)$.
\end{definition}
By (a) above, $\mathcal{I}_{\text{loc},m}(Z) \subseteq \mathbf{M}_{\text{loc}, m}(Z)$.
\subsection{Flat and weak convergence; integral currents}
\label{sec_flat_weak}
\begin{definition}[\cite{LW}]
An $m$-dimensional locally integer rectifiable current $T$ is a \emph{locally integral current} if $m=0$, or, for $m \geq 1$, if $\partial T \in \mathcal{I}_{\text{loc},m-1}(Z)$. (By \cite[Theorem 2.2]{LW}, it is sufficient to assume $\partial T \in \mathbf{M}_{\text{loc}, m-1}(Z)$.) The abelian group of locally integral currents on $Z$ will be denoted by $\mathbf{I}_{\text{loc}, m}(Z)$.
\end{definition}
\begin{definition}[\cite{LW}]
\label{def_local_flat}
A sequence $T_i$ in $\mathbf{I}_{\text{loc}, m}(Z)$ converges to $T \in \mathbf{I}_{\text{loc}, m}(Z)$ in the \emph{local flat topology} if, for every closed, bounded set $B \subseteq Z$ there exists a sequence $S_i$ in $\mathbf{I}_{\text{loc}, m+1}(Z)$ such that
$$\|T - T_i - \partial S_i\|(B) + \|S_i\|(B) \to 0$$
as $i \to \infty$.
\end{definition}
Analogous to the corresponding statement for classical currents, if $T_i \to T$ in the local flat topology, then $T_i \to T$ weakly (pointwise as functionals). Moreover, under weak convergence, $\liminf_{i \to \infty} \|T_i\|(U) \geq \|T\|(U)$ for all bounded open sets $U$. We also require the following lemma, generalizing \cite[Lemma 2.7]{Por} to the local case.
\begin{lemma}
\label{lemma_mass_convergence}
Suppose $T_i \to T$ weakly in $\mathbf{I}_{\text{loc}, m}(Z)$, and suppose there exists a bounded open set $U \subseteq Z$ such that $\|T_i\|(U) \to \|T\|(U)$ as $i \to \infty$. Then $\|T_i\| \llcorner U \to \|T\| \llcorner U$ weakly as a bounded sequence of finite Borel measures. That is, for every bounded, continuous function $f:Z \to \mathbb{R}$ supported in $U$,
$$\int_Z f d\|T_i\| \to \int_Z f d\|T\|$$
as $i \to \infty$. In particular, for every closed set $C \subset U$,
$$\limsup_{i \to \infty} \|T_i\|(C) \leq \|T\|(C).$$
\end{lemma}
This follows immediately from the proof of \cite[Lemma 2.7]{Por}, which uses the portmanteau theorem.
We conclude this section by recalling Wenger's flat distance for currents of finite mass. First, define:
$$\mathbf{M}_{m}(Z)= \{T \in \mathbf{M}_{\text{loc}, m}(Z) \;:\; \|T\|(Z) < \infty\}.$$
In Section 2.5 of \cite{LW} it is shown that $\mathbf{M}_{m}(Z)$ may be identified with the set of currents on $Z$ as originally defined by Ambrosio and Kirchheim in \cite{AK_cur}.
It is straightforward to verify that
$$\Im(Z)= \{T \in \mathbf{I}_{\text{loc}, m}(Z) \;:\; \|T\|(Z) < \infty\}$$
may be identified with the set of integral currents on $Z$ as defined in \cite{AK_cur}. (While currents and integral currents in \cite{AK_cur} were only defined on complete spaces, it is pointed out in Section 2.2 of \cite{LW} that the completeness restriction can be avoided.) We therefore may take the above equations as definitions of currents and integral currents on $Z$.
Given two integral $m$-currents $T_1$ and $T_2$ on $Z$, now taken to be a complete metric space, Wenger defined their \emph{flat distance} \cite{We2007}:
$$d_{Z}^F(T_1, T_2) = \inf_{A \in \Im(Z), B \in \mathbf{I}_{m+1}(Z)}\{\mathbf{M}(A) + \mathbf{M}(B) \; :\; T_2 - T_1 = A+\partial B\},$$
where we use $\mathbf{M}(A)$ to mean $\|A\|(Z)$, etc. This will be used below in the definition of Sormani--Wenger intrinsic flat distance in the next section.
\subsection{Local integral current spaces and pointed $\mathcal{F}$-convergence}
\label{sec_local_currents}
We first recall Sormani--Wenger's definition of an integral current space:
\begin{definition} [\cite{SW}]
An \emph{integral current space} $N=(X,d,T)$ is a metric space $(X,d)$ equipped with an integral current $T$ on the completion $(\overline X, \overline d)$, such that $\set(T)=X$.
The \emph{dimension} of $N$ is the dimension of $T$.
\end{definition}
\begin{definition} [\cite{SW}]
The \emph{Sormani--Wenger intrinsic flat distance} between integral current spaces
$N_1=(X_1, d_1, T_1)$ and $N_2=(X_2, d_2, T_2)$ of dimension $m$ is:
$$d_{\mathcal{F}}(N_1, N_2) = \inf_{Z,\varphi_1, \varphi_2} \{d_Z^F(\varphi_{1\#}(T_1),\varphi_{2\#}(T_2))\},$$
where the infimum is taken over all complete metric spaces $Z$ and isometric embeddings $\varphi_1:X_1 \to Z$ and $\varphi_2:X_2 \to Z$.
\end{definition}
Note that the $\varphi_i$ canonically extend to $\overline X_i$ as isometric embeddings, so the push-forwards are well-defined.
In \cite{SW} it is shown that $d_{\mathcal{F}}$ defines a distance on the set of equivalence classes of precompact integral current spaces of dimension $m$ (where $d_{\mathcal{F}}(N_1, N_2) =0$ if and only if there exists an isometry $\varphi: X_1 \to X_2$ so that $\varphi_\#(T_1)=T_2$).
\begin{definition}[\cite{SW}, \cite{Sor2}]
A sequence of $m$-dimensional integral current spaces $N_i$ \emph{$\mathcal{F}$-converges} to an $m$-dimensional integral current space $N$ (written $N_i \xrightarrow{\F} N$) if
$$d_{\mathcal{F}}(N_i, N) \to 0$$
and \emph{$\mathcal{VF}$-converges} to $N$ (written $N_i \xrightarrow{\VF} N$) if
$$d_{\mathcal{F}}(N_i, N) + |\mathbf{M}(N_i) - \mathbf{M}(N)| \to 0,$$
as $i \to \infty$,
where $\mathbf{M}(\cdot)$ denotes the mass of the underlying integral current.
\end{definition}
In general, if $N_i \xrightarrow{\F} N$, lower semicontinuity of mass holds: $\liminf_{i \to \infty} \mathbf{M}(N_i) \geq \mathbf{M}(N)$. Thus, the volume-preserving hypothesis in $\mathcal{VF}$-convergence simply assures there is no mass drop in the limit.
We also recall an indispensable result of Sormani and Wenger that establishes the existence of a single ``common space'' into which an $\mathcal{F}$-converging sequence embeds:
\begin{thm}[Theorem 4.2 of \cite{SW}]
\label{thm_embedding}
Suppose $N_i=(X_i, d_i, T_i)$ $\mathcal{F}$-converges to $N=(X,d,T)$. Then there exists a complete metric space $Z$ and isometric embeddings $\varphi_i : X_i \to Z$ and $\varphi: X \to Z$ such that $d_Z^F(\varphi_{i\#}(T_i), \varphi_{\#}(T)) \to 0$. Furthermore, by applying the Kuratowski embedding theorem, one may assume without loss of generality that $Z$ is a $w^*$-separable Banach space, i.e., $Z=G^*$, the dual space of $G$, where $G$ is a separable Banach space.
\end{thm}
Although the underlying spaces in an $\mathcal{F}$-converging sequence
$N_i = (X_i, d_i, T_i) \xrightarrow{\mathcal{F}} N=(X,d,T)$
may all be distinct, Sormani \cite{Sor} defines the convergence of points $x_i\in X_i$ to $x \in (\overline X, \overline d)$ as the existence of $Z$ and isometric embeddings $\varphi_i$ and $\varphi$ as in Theorem \ref{thm_embedding} for which
\begin{equation}
\label{eqn_points}
\varphi_i(x_i) \to \varphi(x) \text{ in } Z \text{ as } i \to \infty,
\end{equation}
where $\varphi$ denotes the canonical extension to $\overline X$. This concept will be used in the definition of pointed convergence below.
Next, observe that the definition of integral current space readily generalizes to the local setting:
\begin{definition} [\cite{JL}]
A \emph{local integral current space} of dimension $m \geq 0$ is a triple $N=(X,d,T)$ in which $(X,d)$ is a metric space and $T$ is a locally integral $m$-current on $(\overline X, \overline d)$ such that $\set(T)=X$.
\end{definition}
For example, any complete, connected, oriented Riemannian $m$-manifold forms a local integral current space with the usual distance function, where $T$ is given as integration \cite[Section 2.8]{LW}:
$$T(f,\pi_1, \ldots, \pi_m) = \int_M f d\pi_1 \wedge \dots \wedge d\pi _m.$$
In this case, $\|T\|$ agrees with the Riemannian volume measure (see \cite[Example 2.32]{SW} and \cite[Lemma 22(a)]{JL}).
We gather some basic known results regarding local integral current spaces:
\begin{prop}
Let $N=(X,d,T)$ be a local integral current space of dimension $m$. \begin{enumerate}[(a)]
\item Given $q \in X$, for almost all $r>0$, $T \llcorner B(q,r)$ is an integral $m$-current on $X$, so in particular, $N\llcorner B(q,r) := (\set(T \llcorner B(q,r)), d, T \llcorner B(q,r))$
forms an integral current space; see \cite[Lemma 2.34]{Sor} and \cite[Lemma 13]{JL}. Throughout the paper, the ball in the notation ``$T \llcorner B(q,r)$'' refers to a ball in the completion of $X$.)
\item $\|T\|$ is a Borel measure on $X$ that is finite on bounded sets \cite{LW}.
\item $(X,d)$ is countably $\mathcal{H}^m$-rectifiable, that is, there exist at most a countable number of Lipschitz maps $\varphi_i : A_i \subset \mathbb R^m \to X$ such that $\mathcal H^m( X \setminus \bigcup_i \varphi_i(A_i))=0$. If desired, the $A_i$ can be taken to be compact and the $\varphi_i(A_i)$ pairwise disjoint. (Claim (c) follows from (a) and \cite[Remark 2.36]{SW}.)
\item There exists a $w^*$-separable Banach space $Y$ such that $(X,d)$ embeds isometrically in $Y$. (Proof below.)
\end{enumerate}
\end{prop}
To see (d), recall that $\spt(T)$ is a $\sigma$-compact set \cite[Proposition 2.4]{LW}, and therefore separable. By the Kuratowski embedding theorem, we may isometrically embed $\spt(T)$ into $\ell^\infty$, which the dual of $\ell^1$. Since $\spt(T) \supseteq \set(T)=X$, the claim follows.
The following definition is taken from \cite{JL}, suitably generalized to possibly incomplete spaces. We also refer the reader to the paper of Takeuchi \cite{Tak} giving a different approach to pointed $\mathcal{F}$-convergence (cf. \cite[Remark 1]{JL}).
\begin{definition}
\label{def_pointed_F}
A sequence $N_i=(X_i,d_i,T_i)$ of local integral current spaces of dimension $m$ converges to a local integral current space $N=(X,d,T)$ of dimension $m$ in the \emph{pointed Sormani--Wenger intrinsic flat sense} or ``\emph{pointed $\mathcal{F}$-sense}'' (respectively, \emph{pointed volume-preserving intrinsic flat sense} or ``\emph{pointed $\mathcal{VF}$-sense}'') with respect to $p_i \in X_i$ and $q \in \overline{X}$
if
for any $r_0 > 0$, there exists $r \geq r_0$ such that $N \llcorner B(q,r)$ and $N_i \llcorner B(p_i,r)$ are precompact integral current spaces (for all $i$ sufficiently large), and $N_i \llcorner B(p_i,r) \xrightarrow{\mathcal{F}} N \llcorner B(q,r)$ (respectively, $N_i \llcorner B(p_i,r) \xrightarrow{\mathcal{VF}} N \llcorner B(q,r)$), and if $p_i \to q$ as in \eqref{eqn_points}.
\end{definition}
We verify that pointed $\mathcal{F}$-convergence is a reasonable notion, in that limits are unique:
\begin{prop}
\label{prop_pointed_limit}
Suppose $N_i=(X_i,d_i,T_i)$ is a sequence of local integral current spaces of dimension $m$ that converges in the pointed $\mathcal{F}$-sense to
a local integral current space of dimension $m$, $N=(X,d,T)$, with respect to $p_i \in X$ and $q \in \overline X$. Suppose that $N_i$ also converges in the pointed $\mathcal{F}$-sense to another local integral current space of dimension $m$, $N'=(X',d',T')$, with respect to $p_i \in X$ and $q' \in \overline X'$. Then there exists an isometry $\Phi$ from the completion of $(X,d)$ to the completion of $(X',d')$ such that $\Phi_{\#}(T) = T'$ and $\Phi(q)=q'$.
\end{prop}
A similar result appears in \cite[Proposition 3.7]{Tak}.
\begin{proof}
We will first inductively define an increasing sequence of radii $\{r_k^*\}$, diverging to infinity, and a sequence of isometries $\Phi_k : B(q, r_k^*) \to B(q', r_k^*)$ that satisfy $\Phi_k(q) = q'$ and that are current-preserving in the sense that $(\Phi_k)_\# (T \llcorner B(q, r_k^*)) = T' \llcorner B(q', r_k^*)$. Afterwards, we will use a diagonal argument to create a current-preserving isometry $\Phi : X \to X'$.
Let $r_1>0$ be arbitrary. By definition of pointed $\mathcal{F}$-convergence, there exist $r>r_1$ and $r'>r$ such that $N \llcorner B(q,r)$, $N_i \llcorner B(p_i,r)$, $N' \llcorner B(q',r')$, and $N_i \llcorner B(p_i,r')$ are all precompact integral current spaces for all $i$ sufficiently large, and that
$$N_i \llcorner B(p_i,r) \xrightarrow{\F} N \llcorner B(q,r)$$ with $p_i \to q$ and
$$N_i \llcorner B(p_i,r') \xrightarrow{\F} N' \llcorner B(q',r')$$
with $p_i \to q'$.
By \cite[Lemma 4.1]{Sor}, we may pass to a subsequence such that for almost all radii $\leq r$ we have convergence in the first equation above, and for almost all radii $\leq r'$, we have convergence in the second equation above. In particular, there exists $r_1^* > r_1$ such that
$$N_i \llcorner B(p_i,r_1^*) \xrightarrow{\F} N \llcorner B(q,r_1^*)$$
and
$$N_i \llcorner B(p_i,r_1^*) \xrightarrow{\F} N' \llcorner B(q',r_1^*)$$
as sequences of precompact integral current spaces, with $p_i \to q$ and $p_i \to q'$, respectively.
That is, there exist complete metric spaces $Z$ and $Z'$ and isometric embeddings $\varphi: B(q,r_1^*) \to Z$, $\varphi_i :B(p_i,r_1^*) \to Z$, $\varphi' : B(q',r_1^*) \to Z'$, and $\varphi_i' : B(p_i,r_1^*) \to Z'$ and such that
\[
d_F^Z((\varphi_i)_\# (T_i \llcorner B(p_i,r_1^*)), \varphi_{\#} (T \llcorner B(q,r_1^*))) \to 0
\]
and $\varphi_i(p_i) \to \varphi(q)$ in $Z$, and
\[
d_F^{Z'}((\varphi_i')_{\#} (T_i \llcorner B(p_i,r_1^*), (\varphi')_{\#} (T' \llcorner B(q',r_1^*)))) \to 0
\]
and $\varphi'_i(p_i) \to \varphi'(q')$ in $Z'$.
We apply \cite[Proposition 1.1]{LW} to the sequence of complete metric spaces given by the closed balls $\overline B(p_i,r_1^*)$ in the completions $\overline X_i$, the points $p_i$, the integral currents $T_i \llcorner B(p_i,r_1^*)$, and the isometric embeddings $\varphi_i$ and $\varphi_i'$ (which canonically extend to $\overline B(p_i,r_1^*)$). This produces an isometry $$\Phi_1:\{q\} \cup \spt(T \llcorner B(q,r_1^*)) \to \{q'\} \cup \spt(T' \llcorner B(q',r_1^*))$$ that satisfies $\Phi_1(q) = q'$ and is current-preserving: $(\Phi_1)_{\#}(T \llcorner B(q,r_1^*)) = T' \llcorner B(q',r_1^*)$. We claim that the open ball $B_{\overline X}(q,r_1^*)$ in the completion is a subset of $ \spt(T \llcorner B(q,r_1^*))$. Let $x \in B_{\overline X}(q,r_1^*)$. There exists a sequence $\{x_j\}$ in $X$ that converges in $\overline X$ to $x$. By the triangle inequality, $\overline d(x_j,q) < r_1^*$ for $j$ sufficiently large. Now, since $x_j \in X=\set(T)$, it follows that $x_j \in \set(T \llcorner B(q,r_1^*)) \subseteq \spt(T \llcorner B(q,r_1^*))$. Since the support is a closed set, we have $x \in \spt(T \llcorner B(q,r_1^*))$, which shows the claim. Thus, $\Phi_1$ restricts to an isometry $B_{\overline X}(q,r_1^*) \to B_{\overline X'}(q',r_1^*)$, which then extends to an isometry (also denoted $\Phi_1$) $\overline B(q,r_1^*) \to \overline B(q', r_1^*)$ (where again $\overline B$ refers to the closed ball in the completion). Moreover, $\Phi_1$ remains base-point-preserving and current-preserving.
Suppose $r_k^*$ and $\Phi_k$ have been defined for some $k \in \mathbb{N}$. Now take some $r_{k+1} > r_k^* + 1$ and in the same way as above find an $r_{k+1}^*> r_{k+1}$ and a current-preserving isometry $\Phi_{k+1}: B(q, r_{k+1}^*) \to B(q', r_{k+1}^*)$ that maps $q$ to $q'$. Clearly the $r_k^*$ diverge to infinity.
Now that we have created a sequence of base point- and current-preserving isometries $\{\Phi_k\}$ we will perform a diagonal argument to create a base point- and current-preserving isometry $\Phi : \overline X \to \overline X'$. For each $k$, we can restrict the maps $\Phi_{k}$ to maps from $\overline B(q,r_1^*) \to \overline B(q',r_1^*)$.
Note that $\overline B(q,r_1^*)$ and $\overline B(q',r_1^*)$ are compact metric spaces by hypothesis, so by the Arzela--Ascoli theorem for functions defined on compact metric spaces taking values in compact metric spaces \cite[Theorem 7.17]{KJ},
there exists a subsequence $n^{(1)}$ indexed by $\ell$ such that $\Phi_{n^{(1)}_\ell} : \overline B(q, r_1^*) \to \overline B(q', r_1^*)$ converges uniformly on $\overline B(q, r_1^*)$ as $\ell \to \infty$ to an isometry and clearly the resulting maps takes $q$ to $q'$. If for some $j \in \mathbb{N}$, the subsequence $n^{(j)}$ has been defined, we may select a subsequence $n^{(j+1)}$ of $n^{(j)}$ such that the restriction of the
sequence of functions $\Phi_{n^{(j+1)}_\ell}$ to isometries
$\overline B(q, r_{j+1}^*) \to \overline B(q', r_{j+1}^*)$ converges uniformly on $\overline B(q, r_{j+1}^*)$ as $\ell \to \infty$ to an isometry.
Finally, we consider the diagonal subsequence $\Phi_{n^{(\ell)}_{\ell}}$. This subsequence converges uniformly on compact sets to some map $\Phi:\overline X \to \overline X'$. It is straightforward to see that $\Phi$ is an isometry and $\Phi(q) = q'$.
We still need to show that $\Phi_\# T = T'$. It suffices to show that for every $k \in \mathbb{N}$,
\[
\Phi_\# (T \llcorner B(q, r_k^*)) = T' \llcorner B(q', r_k^*).
\]
This follows as for every $\ell$ sufficiently large we have
\[
(\Phi_{n^{(\ell)}_\ell})_\# (T \llcorner B(q, r_k^*)) = T' \llcorner B(q', r_k^*).
\]
and the isometries $\Phi_{n^{(\ell)}_\ell}$ converge uniformly to $\Phi$. Here, we are using continuity properties of metric currents with locally finite mass: see (2.4) in \cite{LW}.
\end{proof}
\subsection{Dirichlet energy and capacity}
\label{sec_dirichlet}
For integral currents on a Banach space, Portegies gave a definition of the Dirichlet energy of a Lipschitz function based on Ambrosio--Kirchheim's definition of the tangential differential of a Lipschitz function on a rectifiable set in a Banach space. By embedding a metric space in a Banach space, this allowed him to define the Dirichlet energy of a Lipschitz function on an integral current space. To make this precise, we recall from \cite{AK_rect} (with the necessary details included in the appendix) the notion of the tangential differential $d_x^S f$ of a Lipschitz function $f$ on a ${\mathcal H}^m$-countably rectifiable set $S$. We then proceed to generalize the Dirichlet energy to a local integral current space $N=(X,d,T)$. Given a Lipschitz function $f:X \to \mathbb{R}$ with bounded support, define (following \cite{Por}):
$$E_N(f) = \int_{X} |d_x ^Xf|^2 d \|T\|(x),$$
where $d_x^X f$ is the tangential differential when $X$ is embedded in some appropriate Banach space. This is well-defined and independent of the embedding. We also note $|d_x ^Xf| \leq \Lip(f)$ and $\|T\|$ is finite on bounded sets, so that $E_N(f)<\infty$.
This is consistent with the usual definition in the smooth case, again following \cite{Por}:
\begin{prop}
Let $(M,g)$ be a complete, connected, oriented Riemannian manifold, and let $N$ be the associated local integral current space. Then for a Lipschitz function $f$ of $M$ with compact support,
$$E_N(f) = \int_M |\nabla f|^2 dV,$$
where the gradient norm and volume measure are taken with respect to $g$.
\end{prop}
Now, let $N=(X,d,T)$ denote a local integral current space of dimension $m\geq 2$. Let $K \subset X$ be a closed, bounded subset. We define
\begin{equation}
\label{def_cap}
\capac_N(K) = \frac{1}{\gamma_m}\inf \{ E_N(f) \; : \; f\in \Lip_B(X), f \equiv 1 \text{ on a neighborhood of } K\},
\end{equation}
where $\gamma_m = (m-2)\omega_{m-1}$ for $m \geq 3$ and $\gamma_2 = 2\pi$. It is clear that $\capac_N(K) \in [0, \infty)$ and that $K_1 \subset K_2 $ implies $\capac_N(K_1) \leq \capac_N(K_2)$.
In Euclidean space, this agrees with the usual definition of capacity stated in the introduction. (The latter required merely that $f\equiv 1$ on $K$ itself, but in $\mathbb{R}^m$ the distinction is immaterial.) For example, a ball of radius $r$ in $\mathbb{R}^m$ has capacity $r^{m-2}$ for $m\geq 3$. Note that capacity is only interesting (i.e., not identically zero) in the case in which $X$ is unbounded, and even then it is possible $\capac_N \equiv 0$ depending on the behavior of $X$ and $\|T\|$ ``at infinity.''
\begin{remark}
Given a competitor $f$ for the capacity, replacing $f$ with its truncation between values of 0 and 1 produces another competitor whose Dirichlet energy has not increased. Thus, we may restrict to functions $f$ in \eqref{def_cap} satisfying $0 \leq f \leq 1$.
\end{remark}
\begin{remark}
One can similarly define the $p$-capacity, for $p \geq 1$, by replacing $|d_x^X f|^2$ in the definition of $E_N$ with $|d_x^X f|^p$.
\end{remark}
\section{Semicontinuity of capacity}
\label{sec_main}
The main results of this paper are Theorems \ref{thm_balls} and \ref{thm_sublevel} below, where Theorem \ref{thm_balls} is a restatement of Theorem \ref{thm_balls_intro} from the introduction. In both cases we assume pointed $\mathcal{VF}$-convergence of local integral current spaces and establish the upper semicontinuity of the capacity of sets in the spaces. In the first case, the sets are balls centered around converging points; in the second, the sets are defined as Lipschitz sublevel sets. Both theorems will ultimately be consequences of the main technical result, Theorem \ref{thm_extrinsic}.
\subsection{Corresponding regions}
Before presenting the theorems, we will recall the construction in \cite{JL} of ``corresponding regions.'' Let $N_i = (X_i, d_i, T_i) \to N = (X, d, T)$ in the pointed $\mathcal{F}$-sense as local integral current spaces of dimension $m \geq 2$
with respect to $p_i \in X_i$ and $p \in \overline X$. Suppose $K \subsetneq X$ is
nonempty and compact. The corresponding regions, to be defined, will be subsets $K_i$ of $X_i$.
Fix a function $u:X \to \mathbb{R}$ with $\{u \leq 0\} = K$, $\Lip(u)=1$ and
\begin{equation}
\label{defining_function}
u(x) = d(x, K) \text{ for }
x \in X \setminus K.
\end{equation}
Such function $u$ will be called a \emph{defining function} for $K$. For example, $u(x)= d(x, K)\geq 0$ for $x \in X$ is such a function, but the definition allows for, for example, a signed distance function to $\partial K$ if it can be well-defined and if it is a 1-Lipschitz function.
We fix $r_0$ so that $B(p,r_0) \supset K$ in $X$, and apply the definition of pointed $\mathcal{F}$-convergence to obtain $r \geq r_0$ such that $N_i \llcorner B(p_i,r) \xrightarrow{\F} N \llcorner B(p, r)$. Consequently, by Theorem \ref{thm_embedding}, there exists a $w^*$-separable Banach space $Y$ and isometric embeddings $\varphi_i: \set(T_i \llcorner B(p_i,r)) \to Y$ and $\varphi: \set(T \llcorner B(p,r)) \to Y$ such that the pushed-forward integral currents flat-converge in $Y$ and such that $\varphi_i(p_i) \to \varphi(p)$ in $Y$.
Let $U: Y \to \mathbb{R}$ be the standard 1-Lipschitz extension\footnote{When we refer to the ``standard Lipschitz extension'' of a Lipschitz function $f:A \subset Y \to \mathbb{R}$, we mean the extension $\tilde f:Y \to \mathbb{R}$ given by
$$\tilde f(y) := \inf_{a\in A} \left(f(a) + \Lip(f) d_Y(a,y)\right),$$
which has $\Lip(\tilde f)=\Lip(f)$.
} of $u\circ \varphi^{-1}$, where the latter is defined on the image of $\varphi$. We then define $u_i = U \circ \varphi_i$, a 1-Lipschitz function on $\set(T_i \llcorner B(p_i,r))$ for each $i$.
Given a sequence of nonnegative real numbers $\{\alpha_i\}$ converging to 0, the sets
\begin{equation}
\label{eqn_K_i}
K_i = u_i^{-1}(-\infty,\alpha_i] \subseteq X_i
\end{equation}
will be called a sequence of \emph{corresponding regions}; they depend on the choices of $u$, $r_0$, the space $Y$, the embeddings, and the sequence $\alpha_i$. They were essentially defined in \cite{JL}, where it was proved that (roughly --- see the proof of Proposition \ref{prop_K_i} below) $N_i \llcorner K_i$ subsequentially $\mathcal{F}$-converges to $N \llcorner K$. (We are generalizing the definition in \cite{JL} slightly by allowing the $\alpha_i$ to depend on $i$ as well as working in \emph{local} integral current spaces, though we are restricting the form of $u$.)
\medskip
Let us try to provide some intuition for the sets $K_i$. We claim that the intersection of $\varphi_i(\set(T_i \llcorner B(p_i,r)))$ with the closed $\alpha_i$-tubular neighborhood of $\varphi(K)$ is contained in $\varphi_i(K_i)$. Indeed let $x \in \set(T_i \llcorner B(p_i,r))$, define $y := \varphi_i(x)$ and assume that $d_Y(y, \varphi(K)) \leq \alpha_i$. Then
\[
\begin{split}
u_i(x) &= \inf_{a \in \varphi(\set(T \llcorner B(p,r)))}[u \circ \varphi^{-1}(a) + d_Y(a, y) ] \leq
\inf_{a \in \varphi(K)}[u\circ \varphi^{-1}(a) + d_Y(a, y) ]\\
&\leq
\inf_{a \in \varphi(K)} d_Y(a, y)
=
d_Y(y, \varphi(K) )
\leq \alpha_i,
\end{split}
\]
so $x \in K_i$; in the second inequality we used that $u$ is non-positive on $K$.
In case the defining function is chosen to be $u := d_X(\cdot, K)$, then the sets $\varphi_i(K_i)$ are \emph{exactly} the intersections of $\varphi_i(\set(T_i \llcorner B(p_i,r)))$ with the closed $\alpha_i$-neighborhood of $\varphi(K)$.
\medskip
We now establish in Proposition \ref{prop_K_i} conditions to assure that the sets $K_i$ are not ``too small'' --- a priori they could even be empty. Note that we include (b) to accommodate applications that may involve a signed distance function.
\begin{prop}
\label{prop_K_i}
As in the above setup, consider a nonempty compact set $K \subsetneq X$ with a defining function $u$ and choices of $r \geq r_0>0$ and isometric embeddings.
\begin{enumerate}
\item [(a)] There exists a subsequence $N_{i_k}$ and a positive
sequence $\alpha_{i_k} \searrow 0$ such that the corresponding regions $K_{i_k}$ for this subsequence satisfy
$$\liminf_{k \to \infty} \|T_{i_k}\|(K_{i_k}) \geq \|T\|(K).$$
\item [(b)] With the choice $\alpha_i=0$, there exists a subsequence $N_{i_k}$ such that the corresponding regions $K_{i_k}$ for this subsequence satisfy:
$$\liminf_{k \to \infty} \|T_{i_k}\|(K_{i_k}) \geq \|T\|(\{u < 0\}).$$
\end{enumerate}
\end{prop}
\begin{proof}
(a).
For $\delta \in \mathbb{R}$, let $K^\delta = \{u \leq \delta\}$. By \cite[Lemma 27]{JL}, we may pass to a subsequence (keeping the same indexing) such that the restriction of $T_i$ to $K_i^\delta= \{u_i \leq \delta\}$ $\mathcal{F}$-converges to $T \llcorner K^\delta\neq 0$. By lower semicontinuity of mass under $\mathcal{F}$-convergence, we have for $\delta \geq 0$:
\begin{equation}
\label{eqn_K_i_delta}
\liminf_{i \to \infty} \|T_i\|(K_i^\delta) \geq \|T\|(K^\delta) \geq \|T\|(K).
\end{equation}
Select $\delta_1 \in (2^{-1}, 2^0)$ for which \eqref{eqn_K_i_delta} holds with $\delta=\delta_1$. Then there exists $i_1 \geq 1$ such that
$$\|T_{i_1}\|(K_{i_1}^{\delta_1}) \geq \|T\|(K) \left(1-2^{-1}\right).$$
Using \eqref{eqn_K_i_delta} repeatedly, we iteratively select $\delta_k \in (2^{-k}, 2^{-k+1})$ and $i_k > i_{k-1}$ such that
$$\|T_{i_k}\|(K_{i_k}^{\delta_k}) \geq \|T\|(K) \left(1-2^{-k}\right).$$
Since $K_{i_k}^{\delta_k}= u_{i_k}^{-1}(-\infty, \delta_{k}]$, the claim follows with $\alpha_{i_k}=\delta_k$.\\
\medskip
\indent (b) Given $\epsilon>0$, we may select $\delta_0< 0$ such that
\begin{equation}
\label{eqn_delta-eps}
\|T\|(\{u < \delta_0\}) \geq \|T\|(\{u < 0\})-\epsilon.
\end{equation}
As in the proof of (a), by \cite[Lemma 27]{JL}, we may pass to a subsequence (keeping the same indexing) such that the restriction of $T_i$ to $K_i^\delta= \{u_i \leq \delta\}$ $\mathcal{F}$-converges to $T \llcorner K^\delta\neq 0$ for almost all $\delta \in (\delta_0,0)$. By lower semicontinuity of mass, we have
$$\liminf_{i \to \infty} \|T_i\|(K_i^\delta) \geq \|T\|(K^\delta)
$$
However, for $K_i$ defined using the sequence $\alpha_i=0$, $K_i \supseteq K_i^\delta$ and $K^\delta \supseteq \{u < \delta\}$, so by \eqref{eqn_delta-eps}
$$\liminf_{i \to \infty} \|T_i\|(K_i) \geq \|T\|(\{u < 0\})-\epsilon.$$
From this and \eqref{eqn_delta-eps}, the claim follows by letting $\epsilon \searrow 0$ and applying a diagonal argument.
\end{proof}
\begin{remark}\label{rmrk-corresponding}
We point out there is a uniform (i.e., independent of $r_0$) method to define corresponding regions in the case in which $N_i \to N$ in the pointed $\mathcal{F}$-sense ``with a common space'' with respect to $p_i \in X_i$ and $p\in \overline X$. By this, we mean when there exist a $w^*$-separable Banach space $Y$ and isometric embeddings $\varphi_i:X_i \to Y$ and $\varphi:X \to Y$ such that $\varphi_{i\#}(T_i) \to \varphi_{\#}(T)$ in the local flat topology (as in Definition \ref{def_local_flat}), and for which $\varphi_i(p_i) \to \varphi(p)$. In this case, we still define $u_i = U \circ \varphi_i$ (but note that now $u_i$ is defined on the whole $X_i$ and not only on $\set(T_i \llcorner B(p_i,r))$, and let $K_i$ be as in \eqref{eqn_K_i}. This construction will be used in the proof of Theorem \ref{thm_sublevel}.
\end{remark}
\subsection{Main results and proofs}
The following two theorems are our main results. The first is simply a restatement of Theorem \ref{thm_balls_intro} from the introduction.
\begin{thm}
\label{thm_balls}
Let $N_i = (X_i, d_i, T_i)$ and $N= (X, d, T)$ be local integral current spaces of dimension $m \geq 2$, such that $N_i \to N$ in the pointed $\mathcal{VF}$ sense with respect to $p_i \in X_i$ and $p \in X$. Suppose for some $r>0$ that the closed ball $\overline B(p,r)$ in $X$ is compact.
Then:
\begin{equation}
\label{eqn_limsup_balls}
\limsup_{i \to \infty} \capac_{N_i} (\overline B(p_i,r)) \leq \capac_{N}(\overline B(p,r)).
\end{equation}
\end{thm}
In the second main theorem, given a fixed set $K$ in the limit space $X$, we consider a sequence of corresponding regions. Since it is not clear in general if a ``common space'' exists as in Remark \ref{rmrk-corresponding}, we rely on Lang--Wenger's compactness theorem \cite{LW} to produce such a space for a subsequence.
\begin{thm}
\label{thm_sublevel}
Let $N_i = (X_i, d_i, T_i)$ and $N = (X, d, T)$ be local integral current spaces of dimension $m \geq 2$, such that $N_i \to N$ in the pointed $\mathcal{VF}$ sense with respect to $p_i \in X_i$ and $p \in \overline X$. Assume that for all $r>0$,
\begin{equation}
\label{eqn_bdry_mass_bound} \sup_{i \in \mathbb N} \|\partial T_i\| (B(p_i,r)) < \infty.
\end{equation}
Let $K \subsetneq X$ be nonempty and compact. Then, passing to a subsequence of $N_i$ that we do not relabel, one can define
corresponding regions $K_i \subseteq X_i$ as in Remark \ref{rmrk-corresponding}, and
\begin{equation}
\label{eqn_limsup_sublevel}
\limsup_{i \to \infty}\capac_{N_i} (K_i) \leq \capac_{N}(K).
\end{equation}
\end{thm}
Note the hypothesis \eqref{eqn_bdry_mass_bound} is trivially satisfied if $\partial T_i=0$ for each $i$, or, more generally, if the boundary masses are uniformly bounded.
\medskip
We will first prove an extrinsic version of these theorems, i.e. for a sequence of locally integral currents all on a fixed Banach space. Theorem \ref{thm_extrinsic}'s proof follows many of the ideas in the proof of \cite[Theorem 6.2]{Por}.
\begin{thm}\label{thm_extrinsic}
Let $Y$ be a $w^*$-separable Banach space, and let $T\neq 0$ be a locally integral $m$-dimensional current on $Y$, $m \geq 2$. Note $N_\infty=(S,d_Y,T)$ is an $m$-dimensional local integral current space, where $S=\set(T)$ and $d_Y$ is (the restriction of) the distance on $Y$. Let $K \subsetneq S$ be a nonempty compact set.
Let $u:S \to \mathbb{R}$ be a defining function for $K$ (as in \eqref{defining_function}),
and let $U: Y \to \mathbb{R}$ be the standard 1-Lipschitz extension of $u$.
Let $f \in \LipB(S)$ be given, with $0 \leq f \leq 1$, $f \equiv 1$ on a neighborhood of $K$, and $\spt(f) \subseteq B(z_0,r_0)$, where we fix any $z_0 \in Y$ and take $r_0 >3\diam(\hat K) + d_Y(K, S \setminus K)$, where $\hat K= K \cup \{z_0\}$.
Let $T_i$ be a sequence of locally integral $m$-dimensional currents on $Y$ such that
$T_i \to T$ weakly (pointwise as functionals). Assume that for any $r > 0$, there exists a bounded open set $V \supseteq B(z_0,r)$ such that $\|T_i\|(V) \to \|T\|(V)$. Let $S_i = \set(T_i)$, and note
$N_i=(S_i, d_Y, T_i)$ are $m$-dimensional local integral current spaces.
Let $\{\alpha_i\} $ be a sequence of nonnegative real numbers that converges to 0.
Define
\begin{equation}
\label{eqn_K_i_U}
K_i = U^{-1}(-\infty,\alpha_i] \cap S_i.
\end{equation}
Then each $K_i \subseteq S_i$ is a closed and bounded subset of $S_i$, and there exists a sequence $f_i \in \LipB(S_i)$, $0 \leq f_i \leq 1$, with $f_i \equiv 1$ on a neighborhood of $K_i$ (for $i$ sufficiently large), $\Lip(f_i) \leq 1+ 3\Lip(f)$, and $\spt(f_i) \subseteq B(z_0,r_0+3)$ such that
\begin{equation}
\label{USC_energy}
\limsup_{i \to \infty} E_{N_i}(f_i) \leq E_{N_\infty}(f).
\end{equation}
In particular,
\begin{equation}
\label{USC_extrinsic}
\limsup_{i \to \infty} \capac_{N_i}(K_i) \leq \capac_{N_\infty}(K).
\end{equation}
\end{thm}
\bigskip
A few remarks: 1) In many cases $d_Y(K, S \setminus K)$ is zero, but may be positive, e.g. if $K$ is a connected component of $S$. 2) The $K_i$ defined in \eqref{eqn_K_i_U} are the same as the corresponding regions in Remark \ref{rmrk-corresponding}, where here the embeddings are simply the inclusion maps.
3) Allowing for a sequence $\alpha_i \to 0$ in the definition of the corresponding regions $K_i$, we can first derive
the upper semicontinuity of capacity for a sequence of sets $C_i \subset X_i$ that are such that every tubular neighborhood of $\varphi(K)$ contains the set $\varphi_i(C_i)$, for $i$ large enough,
and then use the monotonicity of the capacity with respect to set inclusion.
\bigskip
\begin{proof}[Proof of Theorem \ref{thm_extrinsic}]
We first verify the claim that each $K_i$ (as defined in the statement of the theorem) is bounded. Choose a constant $\alpha > \sup_i \alpha_i$.
Let $p \in K_i$, so $U(p) \leq \alpha_i$. Then by definition of $U(p)$, there exists $s \in S$ such that $u(s) + d_Y(s,p) \leq \alpha$.
If $u(s) \geq 0$, we obtain that $d_Y(s,p)\leq \alpha$. If $u(s) \leq 0$,
since $K$ is compact, $u|_K$ is bounded below by $-L$ for some $L>0$. Then in both cases $d_Y(s,p) \leq \alpha+L$.
For any $\delta >0$ we define the closed and bounded subsets $K^\delta$ of $S$ by $$K^\delta : = u^{-1}(-\infty, \delta] \subseteq S.$$
Note that we have $u(s) \leq \alpha$, so $s \in K^\alpha$.
Then choosing $k \in K$ to minimize the distance from $K$ to $s$,
\begin{align*}
d_Y(p,z_0) &\leq d_Y(p,s) + d_Y(s,k) + d_Y(k,z_0)\\
&\leq \alpha + L + \alpha + \diam(\hat K) < \infty,
\end{align*}
showing $K_i$ is bounded.
\medskip
Now for a fixed $\epsilon > 0$, we show that there exists a sequence $f_i^\epsilon \in \LipB(S_i)$, $0 \leq f_i^\epsilon \leq 1$, with $f_i^\epsilon \equiv 1$ on a neighborhood of $K_i$ (for $i$ sufficiently large, depending on $\epsilon$), $\Lip(f_i^\epsilon) \leq 1+ 3\Lip(f)$, and $\spt(f_i^\epsilon) \subseteq B(z_0,r_0+3)$ such that
\begin{equation}
\label{USC_energy_with_eps}
\limsup_{i \to \infty} E_{N_i}(f_i^\epsilon) \leq E_{N_\infty}(f) + \epsilon.
\end{equation}
This will be enough to conclude the main claim, \eqref{USC_extrinsic}. For clarity of notation, we drop the index $\epsilon$ in what follows. At the very end of the proof, we show how to obtain \eqref{USC_energy} from \eqref{USC_energy_with_eps}.
Let $f:S \to \mathbb{R}$ be the Lipschitz function considered in the statement of the theorem and let $\Lambda = \Lip(f)$. Let $O \subset S$ denote the given neighborhood of $K$ on which $f \equiv 1$. The following lemma allows $f:S \to \mathbb{R}$ to be extended as a Lipschitz function to a larger subset of $Y$ that intersects $S$ inside of $O$.
\begin{lemma}\label{lem-extensionf}
For $\gamma>0$ sufficiently small, the extension of $f$ from $S$ to $S \cup U^{-1}(-\infty,\gamma]$, defined as 1 in $U^{-1}(-\infty,\gamma] \setminus S$, is a Lipschitz function, bounded between 0 and 1, with Lipschitz constant $\leq 2\Lambda$.
(We will also call the extension $f$, so now $\Lip(f) \leq 2\Lambda$.)
\end{lemma}
\begin{proof}
Since $K$ is compact, we can choose $\gamma>0$ small enough to ensure $K^{3\gamma}=u^{-1}(-\infty,3\gamma] \subset O$.
The extension $f:S \cup U^{-1}(-\infty,\gamma] \to \mathbb{R} $ obviously satisfies $0 \leq f \leq 1$. We now prove it is Lipschitz. Let $p, q \in S \cup U^{-1}(-\infty,\gamma]$. If $p,q \in S$, then $|f(p)-f(q)|$ is bounded above by $\Lip(f|_S) d_Y(p,q)=\Lambda d_Y(p,q)$. If $p,q \in (U^{-1}(-\infty,\gamma]\setminus S)\cup O$, then $f(p) - f(q) = 1 - 1 = 0$.
Therefore, the only case left to consider is that in which (say) $p \in (U^{-1}(-\infty,\gamma] \setminus S)\cup O$ and $q \in S$. In fact, since $O \subset S$ we may assume that $p \in U^{-1}(-\infty, \gamma] \setminus S$ and $q \in S \setminus O$.
By the definition of defining function, since $q \not\in K$, for any $\eta>0$, there exists $k \in K$ with $d_Y(k,q) \leq u(q)+\eta$. Then since $f(p)=1=f(k)$ and $f|_{S}$ is Lipschitz, we obtain:
\begin{align*}
|f(p) - f(q)| &= |f(k) - f(q)|\\
&\leq\Lip(f|_S) d_Y(k, q)\\
&\leq\Lambda (u(q) + \eta).
\end{align*}
We now bound $u(q)= U(q)$ in terms of $d_Y(p,q)$. Since $U$ is 1-Lipschitz, we have
$$U(q) - U(p) \leq |U(q)-U(p)| \leq d_Y(p,q),$$
so that
$$u(q) \leq d_Y(p,q) + \gamma.$$
We now show that $\gamma< d_Y(p,q)$. From the definition of $U(p)\leq \gamma$, there exists $s \in S$ such that
\begin{equation}
\label{eqn_2c}
u(s) + d_Y(s,p) < 2\gamma.
\end{equation}
Since $q \in S \setminus O$, we have $u(q)>3\gamma$, and thus
$$3\gamma - u(s) < u(q) - u(s) \leq |u(q)-u(s)| \leq d_Y(q,s) \leq d_Y(p,q) + d_Y(s,p) < d_Y(p,q) + 2\gamma - u(s),$$
having used that $u$ is 1-Lipschitz and \eqref{eqn_2c} to bound $d_Y(s,p)$. From this, it follows that $\gamma< d_Y(p,q)$. Thus,
$$|f(p) - f(q)| \leq \Lambda (2d_Y(p,q) + \eta),$$
since $\eta$ can be arbitrarily small, the proof is complete.
\end{proof}
The following technical lemma constructs many of the objects used in the rest of the proof of Theorem \ref{thm_extrinsic}. Please refer to Figures \ref{fig1} and \ref{fig2} for an illustration of some aspects of this setup.
\begin{figure}
\centering
\includegraphics[scale=0.7]{fig1.eps}
\caption{The sets $K \subset O \subset S$ are shown, along with the balls $W \subset W'$. The number $\gamma>0$ is chosen sufficiently small so that $K^{3\gamma} \subset O$. The function $f:S \to \mathbb{R}$ is identically 1 on $O$ and in Lemma \ref{lem-extensionf}, $f$ is extended by 1 to $U^{-1}(-\infty, \gamma]$.
}
\label{fig1}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.7]{fig2.eps}
\caption{The finitely many sets $A_\ell$ are subsets of $S'\setminus K^\delta$ and cover all of this set except for $\|T\|$-measure $< \epsilon_4$. The $V_\ell$ are disjoint neighborhoods of the $A_\ell$, each of which lies in $W$ or $Y \setminus \overline W$.
}
\label{fig2}
\end{figure}
\begin{lemma}\label{lem:setsAandV}
Let $\epsilon_1>0$ be given and fixed. There exist open balls $W, W' \subset Y$ about $z_0$ of radius in $(r_0, r_0+1)$ and $(r_0+2, r_0+3)$, respectively, so that
for $S'=S\cap W'$,
\begin{equation}
\label{eqn_bdry_W'}
T \llcorner S' \in \Im(Y), \quad \|T\|(\partial W)=0, \quad \text{ and } \;\; \|T\|(\partial W')=0.
\end{equation}
There exists $\delta>0$ sufficiently small so that $U^{-1}(-\infty, \delta] \subseteq W$, and moreover
$K^\delta=u^{-1}(-\infty, \delta]$ satisfies
\begin{equation}\label{eq-epsilon2}
\|T\|(K^\delta \setminus K) < \epsilon_2, \ \quad T \llcorner K^\delta \in \Im(Y), \quad \text{ and } \;\; T \llcorner (S' \setminus K^\delta) \in \Im(Y),
\end{equation}
where $\epsilon_2 = (1+3\Lambda)^{-2} \epsilon_1$.
Letting $\epsilon_3 = \epsilon_1 \left(\|T\|(W')\right)^{-1} > 0$ and $\epsilon_4 = (1+3\Lambda)^{-2} \epsilon_1$, there exists a finite collection of compact sets $A_{\ell} \subset S' \setminus K^\delta$ for $\ell=1,\ldots, N$ such that
\begin{itemize}
\item Each $A_\ell$ is the bi-Lipschitz image of a compact subset of $\mathbb{R}^m$,
\item Each $A_\ell$ is either a subset of $W$ or of $Y \setminus \overline W$,
\item The $A_{\ell}$ are pairwise disjoint,
\item For any $\gamma \in (0,\frac{\delta}{2})$, we have $d_Y(A_{\ell}, U^{-1}(-\infty, \gamma])>\delta/2$ for all $\ell$
\item Letting $\cup_\ell$ denote $\cup_{\ell=1}^N$ henceforth,
\begin{equation}\label{eq-epsilon4}
\|T\|((S'\setminus K^\delta) \setminus \cup_\ell A_\ell) < \epsilon_4,
\end{equation}
\item For all $x \in A_\ell$
\begin{equation}\label{eq-epsilon3}
(c_\ell)^2 - \epsilon_3 \leq |d_x^{S} f|^2 \leq (c_\ell)^2,
\end{equation}
where $c_\ell = \Lip(f|_{A_\ell}) \leq \Lambda$.
\end{itemize}
Furthermore, there exists $b>0$ sufficiently small so that
the open $b/10$-neighborhoods of each $A_{\ell}$ in $Y$, denoted $V_{\ell}$, satisfy
\begin{itemize}
\item Each $V_{\ell}$ are subsets of $W'$,
\item Each $V_\ell$ is either a subset of $W$ or of $Y \setminus \overline W$,
\item The $V_{\ell}$ are pairwise disjoint,
\item $d_Y(V_{\ell}, U^{-1}(-\infty, \gamma])>9b/10$ for all $\ell$,
\item For $\epsilon_5=\Lambda^{-2} \epsilon_1$,
\begin{equation}
\label{eqn_V_A}
\|T\|\left(\cup_\ell \overline V_\ell \setminus \cup A_{\ell}\right) < \epsilon_5,
\end{equation}
(If $\Lambda=0$, we take $\epsilon_5=1$).
\end{itemize}
\end{lemma}
\begin{proof}
We note that
since $\|T\|$ is a Borel measure that is finite on bounded open sets, it follows that $\|T\|$ is zero on almost all metric spheres about a given point. Choose an open ball $W \subset Y$ about $z_0$ of radius in $(r_0,r_0+1)$, with the radius chosen so that $\|T\|(\partial W)=0$.
\medskip
We claim that for $\delta>0$ sufficiently small, $U^{-1}(-\infty, \delta] \subseteq W$.
The proof is similar to the proof of each $K_i$ being bounded: Let $p \in U^{-1}(-\infty,\delta]$, i.e., $U(p) \leq \delta$. Then given $\eta > 0$, there exists $s \in S$ such that $u(s)+d_Y(s,p)\leq \delta+ \eta$. In particular, $u(s) \leq \delta+\eta,$ i.e. $s \in K^{\delta+\eta}$.
We consider two cases according to the sign of $u(s)$. If $u(s) \geq 0$, we obtain that $d_Y(s,p)\leq \delta+ \eta$.
Using
$s \in K^{\delta+\eta}$, take $k \in K$ to minimize the distance to $s$, obtaining
$$d_Y(z_0,s) \leq d_Y(z_0,k) + d_Y(k,s) \leq \diam(\hat K) + \delta +\eta.$$
Then by the triangle inequality
$$ d_Y(z_0,p) \leq d_Y(z_0,s) + d_Y(s,p)\leq \diam(\hat K) + 2\delta + 2\eta.$$
Choosing $\delta$ and $\eta$ sufficiently small, we have $\diam(\hat K) + 2\delta + 2\eta \leq 3\diam(\hat K)<r_0$. So $p \in W$ in this case.
In the other case, if $u(s) < 0$, then $s \in K$ and so $d_Y(z_0,s) \leq \diam(\hat K)$. We also have
$d_Y(s,p) \leq \delta + \eta + |u(s)|$. Let $q \in K$ and $q' \in S\setminus K$ achieve the minimum distance between $K$ and $S\setminus K$, within $\eta$:
$$d_Y(q,q') \leq d_Y(K, S \setminus K) + \eta.$$
Since $u(q') > 0$,
$$|u(s)| \leq |u(q') - u(s)| \leq d_Y(q', s) \leq d_Y(q',q) + d_Y(q,s) \leq d_Y(K, S \setminus K) + \eta + \diam(K).$$
Thus,
$$d_Y(s,p) \leq \delta + \eta + |u(s)| \leq \diam(K) + d_Y(K,S \setminus K) + \delta+2\eta.$$
Again, by the triangle inequality and choosing $\delta$ and $\eta$ sufficiently small, we obtain
$$d_Y(z_0,p) \leq d_Y(z_0,s) + d_Y(s,p) \leq \diam(\hat K) + \diam(K) + d_Y(K,S \setminus K) + \delta+2\eta < r_0,$$ so $p \in W$. It follows that $U^{-1}(-\infty, \delta] \subseteq W$.
\medskip
We now choose $\delta >0$ as small as needed so that $U^{-1}(-\infty, \delta] \subset W$ and that $K^\delta$
satisfies
\begin{equation*}
\|T\|(K^\delta \setminus K) < \epsilon_2 \quad \text{ and } \quad T \llcorner K^\delta \in \Im(Y),
\end{equation*}
where $\epsilon_2 = (1+3\Lambda)^{-2} \epsilon_1$
(cf. \cite[Lemma 24]{JL}, using Sormani's argument in \cite[Lemma 2.34]{Sor}).
\medskip
Now fix another open ball $W' \subset Y$, centered about $z_0$, of radius in $(r_0+2,r_0+3)$, so that in particular $\overline{W} \subset W'$. Let $S'=S\cap W'$; adjusting the radius if necessary,
we may ensure $T \llcorner S' \in \Im(Y)$ and
\begin{equation*}
\|T\|(\partial W')=0.
\end{equation*}
So $T \llcorner (S' \setminus K^\delta) \in \Im(Y)$ as well.
\bigskip
Apply Lemmas 3.2 and 6.1 of \cite{Por} to the integral $m$-current $T \llcorner (S' \setminus K^\delta)$ and the restriction of $f$ to $S' \setminus K^\delta$ with $\epsilon_3 = \epsilon_1 \left(\|T\|(W')\right)^{-1} > 0$, thereby obtaining a sequence of compact sets $A_{\ell} \subset S' \setminus K^\delta$ for $\ell=1,2,\ldots$ such that
\begin{itemize}
\item Each $A_\ell$ is the bi-Lipschitz image of a compact subset of $\mathbb{R}^m$.
\item The $A_{\ell}$ are pairwise disjoint.
\item $\cup_{\ell=1}^\infty A_\ell$ has zero co-measure in $S'\setminus K^\delta$ with respect to $\|T\|$.
\item For all $x \in A_\ell$
\begin{equation*}
(c_\ell)^2 - \epsilon_3 \leq |d_x^{S} f|^2 \leq (c_\ell)^2,
\end{equation*}
where $c_\ell = \Lip(f|_{A_\ell}) \leq \Lambda$.
\end{itemize}
Since $\|T\|$ is Borel regular and $\|T\|(\partial W)=0$,
we may assume without loss of generality that each $A_\ell$ is either a subset of $W$ or of $Y \setminus \overline W$.
\medskip
Now,
choose a finite subset of $\{A_{\ell}\}$, call it $A_1, \ldots, A_N$, such that
\begin{equation*}
\|T\|((S'\setminus K^\delta) \setminus \cup_\ell A_\ell) < \epsilon_4,
\end{equation*}
where, for the rest of this proof, $\cup_\ell$ will denote $\cup_{\ell=1}^N$. Let $\gamma \in (0, \frac{\delta}{2})$.
We claim that the distance from any $A_{\ell}$ to $U^{-1}(-\infty,\gamma]$ in $Y$ is at least $\delta-\gamma > \frac{\delta}{2}$.
Let $q \in A_{\ell}$. Since $A_{\ell}$ is disjoint from $K^\delta$ we have $U(q) = u(q) > \delta$. Then since $U$ is 1-Lipschitz,
if $z \in U^{-1}(-\infty,\gamma]$,
$$d_Y(q,z) \geq |U(q) - U(z)| > \delta-\gamma,$$
which proves the claim.
We know that these finitely many $A_\ell$ are pairwise disjoint and compact; so let $a>0$ be the minimum pairwise distance between them. Let $b$ be a positive real number less than $\min \{a, \frac{\delta}{2}\}>0$ and $V_{\ell}$ be the open $b/10$-neighborhood of $A_{\ell}$ in $Y$, so $V_1, \ldots, V_N$ are pairwise disjoint
and their distance to $U^{-1}(-\infty,\gamma]$ is greater than $9b/10$. Since the $A_{\ell}$ are compact subsets of $W'$, we may shrink $b>0$ if necessary to ensure the $V_{\ell}$ are subsets of $W'$. We can also shrink $b>0$ again to guarantee each $V_\ell$ is either a subset of $W$ or of $Y \setminus \overline W$. Furthermore, since $\|T\|$ is regular, we may shrink $b>0$ if necessary to ensure
\begin{equation*}
\|T\|\left(\cup_\ell \overline V_\ell \setminus \cup A_{\ell}\right) < \epsilon_5.\qedhere
\end{equation*}
\end{proof}
\medskip
Following as in (6.14)--(6.17) of \cite{Por}, we claim:
\begin{lemma}
\label{lemma_extension}
Under the assumptions of Lemma \ref{lem:setsAandV} and $\gamma>0$ small enough so that Lemma \ref{lem-extensionf} holds,
there exists a Lipschitz function $\hat f: Y \to \mathbb{R}$ such that
\begin{enumerate}[(a)]
\item $\Lip(\hat f) \leq 1+3\Lambda$
\item $0 \leq \hat f \leq 1$
\item $\hat f$ agrees with $f$ on $\cup_\ell A_\ell$ and on $U^{-1}(-\infty,\gamma]$ (in particular, $\hat f|_{U^{-1}(-\infty,\gamma]} = 1$)
\item $\Lip(\hat f|_{V_\ell}) \leq \Lip(f|_{A_\ell})$ for $\ell=1,\ldots, N$
\item $\hat f \equiv 1$ on a neighborhood of $K_i$ for all $i$ sufficiently large
\item $\spt(\hat f) \subseteq \overline{W'}$.
\end{enumerate}
In particular, $f_i:=\hat f|_{S_i}$ is Lipschitz, bounded, has bounded support, and is $1$ on a neighborhood of $K_i$ (for all $i$ sufficiently large), i.e., $f_i$
is an allowable test function for $\capac_{N_i}(K_i)$.
\end{lemma}
Again, refer to Figures \ref{fig1} and \ref{fig2} for an illustration of some aspects of this setup.
Note that in Lemma \ref{lem-extensionf} we extend $f: S \to \mathbb{R}$ to a function $f: S \cup U^{-1}(-\infty, \gamma] \to \mathbb{R}$. We now construct $\hat f$ by prescribing its values on $W' \setminus U^{-1}(-\infty, \gamma]$. We remark that $\hat f$ will not generally be an extension of $f$; they may differ on $\left( \cup V_\ell \setminus \cup A_{\ell}\right) \cap S$.
\begin{proof}
First, we define for each $\ell$ the standard Lipschitz extension of $f|_{A_\ell}$ to $V_\ell$:
\begin{equation*}
f^\ell(x) = \inf_{a\in A_\ell} \left(f(a) + c_\ell d_Y(a,x)\right),\qquad x \in V_\ell,
\end{equation*}
where, again, $c_\ell= \Lip(f|_{A_\ell})$. Note that $\Lip(f^\ell) \leq c_\ell$. Truncate these functions by defining $\hat{f}^\ell := \max\{ \min\{ f^\ell, 1 \} , 0\}$, (recalling $0 \leq f \leq 1$), and note $\Lip(\hat{f}^\ell) \leq c_\ell$.
Subsequently, define the function $\hat{f}: \left( \cup_\ell V_\ell\right)\cup U^{-1}(-\infty,\gamma] \to \mathbb{R}$ by
\begin{equation*}
\hat{f}(x) =
\begin{cases}
\hat{f}^\ell(x) & \text{ if } x \in V_\ell \\
1 & \text{ if } x \in U^{-1}(-\infty, \gamma],
\end{cases}
\end{equation*}
which satisfies $0 \leq \hat f \leq 1$.
Let us prove that $\Lip(\hat{f}) \leq 3\Lambda$. There are only two nontrivial cases. First, if $x\in V_{\ell_1}$, $y \in V_{\ell_2}$, $\ell_1 \neq \ell_2$, there exist $x_0 \in A_{\ell_1}$ and $y_0 \in A_{\ell_2}$ such that $d_Y(x,x_0) < \frac{b}{10}$ and $d_Y(y,y_0) < \frac{b}{10}$. By the triangle inequality, since $d_Y(x_0,y_0) \geq a$, we find $d_Y(x,y) \geq \tfrac{4}{5} b$.
Therefore
\begin{equation*}
\begin{split}
|\hat{f}(x) - \hat{f}(y)| &\leq |\hat{f}(x) - \hat{f}(x_0)| + |\hat{f}(x_0) - \hat{f}(y_0)| + |\hat{f}(y_0) - \hat{f}(y)| \\
&\leq c_{\ell_1} d_Y(x,x_0) + \Lip(f) d_Y(x_0, y_0) + c_{\ell_2} d_Y(y_0, y) \\
&\leq \Lip(f)\left( d_Y(x,x_0) + d_Y(x_0 , y_0) + d_Y(y_0 , y) \right) \\
&\leq \Lip(f)\left( \frac{b}{10} + d_Y(x_0 , x) + d_Y(x , y) + d_Y(y , y_0) + \frac{b}{10} \right) \\
&\leq \Lip(f)\left( \frac{4b}{10} + d_Y(x , y) \right) \\&\leq \Lip(f) \left( \frac{3}{2} d_Y(x,y) \right)\\
&= 3\Lambda d_Y(x , y).
\end{split}
\end{equation*}
Second, assume that $x\in V_{\ell}$, $y \in U^{-1}(-\infty,\gamma] $, so $d_Y(x,y) \geq \frac{9b}{10}$. There exists $x_0 \in A_{\ell}$ such that $d_Y(x,x_0) < \frac{b}{10}$ and therefore
\begin{equation*}
\begin{split}
|\hat{f}(x) - \hat{f}(y)| &\leq |\hat{f}(x) - \hat{f}(x_0)| + |\hat{f}(x_0) - \hat{f}(y)| \\
&\leq c_{\ell_1} d_Y(x, x_0) + \Lip(f) d_Y(x_0, y) \\
&\leq \Lip(f)\left( d_Y(x,x_0) + d_Y(x_0, y) \right) \\
&\leq \Lip(f)\left( d_Y(x,x_0) + d_Y(x_0, x) + d_Y(x, y) \right) \\
&\leq \Lip(f)\left( \frac{2b}{10} + d_Y(x, y) \right) \\
&\leq \Lip(f)\left( \frac{11}{9} d_Y(x,y) \right) \\
&\leq 3\Lambda d_Y(x,y).\\
\end{split}
\end{equation*}
Consequently, $\hat{f}: \left( \cup_\ell V_\ell\right)\cup U^{-1}(-\infty,\gamma] \to \mathbb{R}$ has Lipschitz constant at most $3\Lambda$. We extend it to a Lipschitz function on all $Y$ in the standard way (with the same Lipschitz constant), truncate at values of $0$ and $1$, and call the result $\hat f$.
However, $\hat f$ will not have bounded support, so we will modify $\hat f$ using a cutoff function while ensuring claims (a)--(f) will hold. Let $0\leq \rho \leq 1$ be a Lipschitz function on $Y$ that is identically one on $\overline W$ and is supported in $W'$. Since the radii of $W$ and $W'$ differ by more than 1, we may assume without loss of generality that
\begin{equation}
\label{eqn_MLip}
\Lip(\rho) \leq 1.
\end{equation}
We claim $\rho \hat f$ is the desired function. The function $\rho \hat f$ clearly has bounded support in $\overline{W'}$ and is bounded between 0 and 1, i.e. (b) and (f) hold. We see $\rho \hat f$ is Lipschitz as well:
\begin{align*}
| \rho(x) \hat f(x) - \rho(y) \hat f(y) | \leq & | \rho (x) - \rho (y) | | \hat f(x) | + | \hat f(x) - \hat f(y) | | \rho (y) | \\
\leq & \left( \Lip(\rho) + \Lip(\hat f) \right) d_Y(x , y )\\
\leq & \left(1 + 3\Lambda \right) d_Y(x , y ),
\end{align*}
having used \eqref{eqn_MLip}. That is, (a) holds.
Next, we show (c): first suppose $x \in A_\ell$. If $A_\ell \subset W$, then $\rho(x)=1$ and it follows that $\rho(x)\hat f(x)=f^\ell(x)=f(x)$. Otherwise, $A_\ell \subset Y \setminus \overline W$, which implies $f|_{A_\ell}=0$, as $W$ contains the support of $f$. Then $c_\ell=0$, so $\hat f(x)=f^\ell(x)=0$. Then $\rho \hat f$ and $f$ both vanish on $A_\ell$. Next, suppose $x \in U^{-1}(-\infty,\gamma]$. Since $W \supset U^{-1}(-\infty,\gamma]$ and $\rho \equiv 1$ on $W$, it follows that $\rho(x)\hat f(x)=1=f(x)$.
To address, (d), consider $\rho \hat f|_{V_{\ell}}$. If $V_\ell$ is a subset of $W$, then $\rho \equiv 1$ on $V_\ell$, so $\rho \hat f|_{V_{\ell}} = \hat f|_{V_\ell} = f^\ell$, whose Lipschitz constant is at most $c_\ell = \Lip(f|_{A_\ell})$. On the other hand if $V_\ell$ is a subset of $Y \setminus \overline W$, then $\rho \hat f|_{V_{\ell}}=0$. But since $A_\ell \subset Y \setminus \overline W$ and $\spt(f) \subseteq W$, we have $f|_{A_\ell}=0$, which implies $f^\ell=0$. So in this case as well, $\Lip(\rho \hat f|_{V_{\ell}}) \leq \Lip(f|_{A_\ell})$ (both are zero).
To show (e), restrict to $i$ sufficiently large so that $\alpha_i < \gamma$. Then $U^{-1}(-\infty, \alpha_i] \subset U^{-1}(-\infty, \gamma)$. By (c), $\hat f \equiv 1$ on $U^{-1}(-\infty, \gamma]$. Since $U^{-1}(-\infty, \gamma] \subset W$ and $\rho \equiv 1$ on $W$, it follows that $\rho \hat f \equiv 1$ on $U^{-1}(-\infty, \gamma]$ as well. In particular, $\rho \hat f$ is identically 1 on the neighborhood $U^{-1}(-\infty, \gamma)$ of $K_i=U^{-1}(-\infty, a_i] \cap S_i$, i.e. (e) holds.
For simplicity, we will henceforth refer to $\rho \hat f$ as simply $\hat f$.
\end{proof}
We now establish the energy estimate \eqref{USC_energy_with_eps}, which will be sufficient to obtain \eqref{USC_extrinsic}.
Note that from our hypotheses and Lemma \ref{lemma_mass_convergence}, for every closed, bounded set $C \subset Y$, we have
\begin{equation}
\label{limsup}
\limsup_{i \to \infty} \|T_i\|(C) \leq \|T\|(C).
\end{equation}
We begin with:
\begin{align}
E_{N_i}(f_i) &= \int_{Y} |d_x^{S_i} f_i|^2 d\|T_i\| \nonumber \\
&= \int_{Y\setminus W'} |d_x^{S_i} \hat f|^2 d\|T_i\| +
\int_{\cup_\ell V_\ell} |d_x^{S_i} \hat f|^2 d\|T_i\| + \int_{W' \setminus\cup_\ell V_\ell} |d_x^{S_i} \hat f|^2 d\|T_i\|.\label{ineq_3}
\end{align}
The first term is zero, since $\hat f$ vanishes outside $W'$. For the second term in \eqref{ineq_3}, by Lemma \ref{lemma_extension}(d) and \eqref{eq-epsilon3} we get
\begin{align*}
\int_{\cup_{\ell=1}^N V_\ell} |d_x^{S_i} \hat f|^2 d\|T_i\| &\leq \sum_{\ell=1}^N \Lip(\hat f|_{V_\ell})^2 \|T_i\|(V_\ell)\\
&\leq \sum_{\ell=1}^N (c_\ell)^2 \|T_i\|(V_\ell).
\end{align*}
We replace $V_{\ell}$ with its closure in the above and take the limsup to obtain, by \eqref{limsup}, \eqref{eqn_V_A}, and \eqref{eq-epsilon3}:
\begin{align}
\limsup_{i \to \infty} \int_{\cup_\ell V_\ell} |d_x^{S_i} \hat f|^2 d\|T_i\| &\leq \limsup_{i \to \infty} \sum_{\ell=1}^N (c_\ell)^2 \|T_i\|( \overline V_\ell)\nonumber\\
&\leq \sum_{\ell=1}^N (c_\ell)^2 \|T\|( \overline V_\ell)\nonumber\\
&\leq \sum_{\ell=1}^N (c_\ell)^2 \|T\|( A_\ell)+ \Lambda^2 \epsilon_5\nonumber \\
&\leq\ \int_{\cup_\ell A_\ell} \left(|d_x^{S} f|^2 +\epsilon_3 \right) d\|T\| + \epsilon_1\nonumber\\
&\leq E_{N_\infty}(f) + \|T\|(W')\epsilon_3+ \epsilon_1 \nonumber \\
&\leq E_{N_\infty}(f) + \epsilon_1 + \epsilon_1.\label{eqn_second_term}
\end{align}
\bigskip
For the third term in \eqref{ineq_3} we may omit integration over the open set $U^{-1}(-\infty, \gamma)$, since $\hat f \equiv 1$ there. It follows that:
$$ \int_{W' \setminus\cup_\ell V_\ell} |d_x^{S_i} \hat f|^2 d\|T_i\| \leq \Lip(\hat f)^2 \|T_i\|\left(\overline{W'} \setminus \left(\cup_\ell V_\ell \cup U^{-1}(-\infty,\gamma)\right)\right).$$
The set on the right is closed and bounded, so by \eqref{limsup}:
\begin{align*}
\limsup_{i\to \infty} \int_{W' \setminus\cup_\ell V_\ell} |d_x^{S_i} \hat f|^2 d\|T_i\| &\leq \Lip(\hat f)^2 \|T\|\left(\overline{W'} \setminus \left(\cup_\ell V_\ell \cup U^{-1}(-\infty,\gamma)\right)\right).
\end{align*}
Now, by considering intersections and set subtractions with $S'$ and with $K^\delta$, we find
$$\left(\overline{W'} \setminus U^{-1}(-\infty,\gamma)\right)\setminus\cup_\ell V_\ell \subseteq (\overline{W'} \setminus S') \cup \left((S' \setminus K^\delta)\setminus \cup V_{\ell} \right) \cup (K^\delta \setminus K).$$
Thus, using \eqref{eq-epsilon4} and \eqref{eq-epsilon2},
\begin{align}
\limsup_{i\to \infty} \int_{W' \setminus\cup_\ell V_\ell} &|d_x^{S_i} \hat f|^2 d\|T_i\| \nonumber \\
&\leq \Lip(\hat f)^2\left[ \|T\|\left(\overline{W'} \setminus S'\right) +\|T\|\left((S' \setminus K^\delta)\setminus \cup V_{\ell} \right)+\|T\|\left(K^\delta \setminus K\right)
\right]\nonumber\\
&\leq \Lip(\hat f)^2\left[ \|T\|\left(W' \setminus S'\right) +\|T\|(\partial W') +\epsilon_4 + \epsilon_2
\right]. \label{eqn_third_term}
\end{align}
Note that $\|T\|\left(W' \setminus S'\right) \leq \|T\|\left(Y \setminus S\right)$, where we recall $S= \set(T)$. It is shown in \cite[Theorem 4.6]{AK_cur} that an integral current's measure is concentrated on its canonical set. The same goes for locally integral currents, i.e. $\|T\|\left(Y \setminus S\right)=0$. The next term, $\|T\|(\partial W')$, vanishes by \eqref{eqn_bdry_W'}.
Combining \eqref{ineq_3}, \eqref{eqn_second_term}, and \eqref{eqn_third_term}, we have (using the definition of $\epsilon_2$ and $\epsilon_4$):
\begin{align*}
\limsup_{i \to \infty} E_{N_i}(f_i) &\leq E_{N_\infty}(f) + 2\epsilon_1 +
\left(1+3\Lambda\right)^2 (\epsilon_4 + \epsilon_2)\\
&= E_{N_\infty}(f) + 4\epsilon_1.
\end{align*}
So \eqref{USC_energy_with_eps} follows.
To then show \eqref{USC_extrinsic}, given $\epsilon_1>0$, choose $f \in \LipB(S)$, with $0 \leq f \leq 1$ and $f \equiv 1$ on a neighborhood of $K$, such that
\begin{equation}
\label{eqn_E_cap}
E_{N_\infty}(f) \leq \gamma_m\capac_{N_\infty}(K) + \epsilon_1.
\end{equation}
Then apply the above argument to produce functions $f_i$ (that are valid test functions for the capacity of $K_i$ for $i$ sufficiently large), so that
$$\limsup_{i \to \infty} \gamma_m \capac_{N_i}(K_i) \leq \limsup_{i \to \infty} E_{N_i}(f_i) \leq E_{N_\infty}(f) + 4\epsilon_1.$$
Combining this with \eqref{eqn_E_cap}, \eqref{USC_extrinsic} follows, since $\epsilon_1$ was arbitrary.
\bigskip
We conclude by establishing \eqref{USC_energy} (though this is not needed for the proof of \eqref{USC_extrinsic}).
By \eqref{USC_energy_with_eps}, we may assume that for every $j \in \mathbb{N}$ we have a sequence $\left(f^{1/j}_i\right)_i$ such that $f_i^{1/j} \in \LipB(S_i)$, $0 \leq f_i^{1/j} \leq 1$, with $f_i^{1/j} \equiv 1$ on a neighborhood of $K_i$ (for $i$ sufficiently large, depending on $1/j$), $\Lip(f_i^{1/j}) \leq 1+ 3\Lip(f)$, and $\spt(f_i^{1/j}) \subseteq B(z_0,r_0+3)$ such that
\begin{equation*}
\limsup_{i \to \infty} E_{N_i}(f_i^{1/j}) \leq E_{N_\infty}(f) + 1/j.
\end{equation*}
Then we may construct a monotonically increasing sequence $n: \mathbb{N} \to \mathbb{N}$ such that $n_1 = 1$ and for all $j \in \mathbb{N}\setminus\{1\}$ it holds that for all $i \geq n_j$, $f_i^{1/j} \equiv 1$ on a neighborhood of $K_i$ and
\[
E_{N_i}(f_i^{1/j}) \leq E_{N_\infty}(f) + 2 / j.
\]
We then define a new non-decreasing sequence $m : \mathbb{N} \to \mathbb{N}$ by
\[
m_{j} = \max\{ k \ |\ n_k \leq j \}.
\]
Note that the maximum is well-defined as $(n_k)$ is monotonically increasing. Moreover, $m_j \to \infty$ as $j \to \infty$ since for every $M_0 \in \mathbb{N}$, if we define $j_0 := n_{M_0}$, we have $m_{j_0} \geq M_0$ (in fact $m_{j_0} = M_0$ since $n: \mathbb{N} \to \mathbb{N}$ is monotonically increasing).
The sequence $(f_i)$ defined by $f_i := f^{1/m_i}_i$
then satisfies the properties mentioned in the theorem: in particular, since by construction $i \geq n_{m_i}$ for all $i \in \mathbb{N}$, we know that for all $i \in \mathbb{N} \setminus\{1\}$ $f^{1/m_i}_i\equiv 1$ on a neighborhood of $K_i$, and
\[
E_{N_i}\left(f^{1/m_i}_i\right) \leq E_{N_\infty}(f) + 2/m_i
\]
whereas $m_i \to \infty$ as $i \to \infty$.
\end{proof}
\bigskip
Now we may prove the first main theorem, Theorem \ref{thm_balls}.
\begin{proof}[Proof of Theorem \ref{thm_balls}]
Suppose the sequence $N_i = (X_i, d_i, T_i)$ converges to $N = (X, d, T)$ in the pointed $\mathcal{VF}$ sense as local integral current spaces of dimension $m \geq 2$, with respect to $p_i \in X_i$ and $p\in X$. Assume for some $r>0$ that the closed ball $K=\overline B(p, r)$ in $X$ is compact. In the first part of the proof, we assume $K \neq X$.
Given $\epsilon > 0$, take a function $f \in \Lip_B(X)$, $0\leq f \leq 1$, with $f \equiv 1$ in a neighborhood of $K$ and
\begin{equation}
E_N(f) \leq \gamma_m\capac_N(K) + \epsilon. \label{eqn_E_N_eps}
\end{equation}
Choose $r_0 > 3 \diam(K)+d(K, X \setminus K)$ sufficiently large so that $K \subseteq \spt(f) \subseteq B(p,r_0)$,
and
\begin{equation}
\label{eqn_B_X_K}
B(p,r_0) \cap (X \setminus K) \neq \emptyset.
\end{equation}
Using Definition \ref{def_pointed_F}, choose $R>r_0+4$ such that $N_i \llcorner B(p_i,R) \to N \llcorner B(p,R)$ in the $\mathcal{VF}$ sense as integral current spaces. Let $S_i = \set(T_i \llcorner B(p_i,R)) \subseteq X_i$ and $S=\set(T \llcorner B(p,R))\subseteq X$. Since $p \in X = \set(T)$, it follows $p \in \set(T\llcorner B(p,R)) = S$. Thus $S \neq \emptyset$ and $T \llcorner B(p,R) \neq 0$. It is straightforward to see $K \subseteq S$. That $K$ is a proper subset of $S$ follows from \eqref{eqn_B_X_K}.
By Theorem \ref{thm_embedding} there exists a $w^*$-separable Banach space $Y$ and isometric embeddings $\varphi_i: S_i \to Y$ and $\varphi: S \to Y$
such that the integral currents $\varphi_{i\#}(T_i \llcorner B(p_i,R))$ converge to $\varphi_{\#}(T \llcorner B(p,R))$ in the flat $d_Z^F$ sense (and therefore in the weak sense) and the masses converge,
\begin{equation}
\mathbf{M}(\varphi_{i\#}(T_i \llcorner B(p_i,R))) \to \mathbf{M}(\varphi_{\#}(T \llcorner B(p,R))), \label{eqn_masses_converge}
\end{equation}
and finally that $\varphi_i(p_i) \to \varphi(p)$ as $i \to \infty$.
Select $u(x)=d(K,x)$ as a defining function for $K$ on $S$.
Since $\varphi$ is an isometric embedding, $u \circ \varphi^{-1}:\varphi(S) \to \mathbb{R}$ is equal to $d_Y(\varphi(K), \cdot)$. It is elementary to verify that the 1-Lipschitz extension $U$ of $u \circ \varphi^{-1}$ to $Y$ is simply given by $d_Y(\varphi(K), \cdot)$.
We apply Theorem \ref{thm_extrinsic} to the Banach space $Y$;
the nonzero integral current $\varphi_{\#}(T \llcorner B(p,R))$ on $Y$, whose canonical set is $\varphi(S)$;
the nonempty compact set $\varphi(K) \subsetneq \varphi(S)$ in $Y$; the defining function $u \circ \varphi^{-1}$ of $\varphi(K)$ with standard 1-Lipschitz extension $U:Y \to \mathbb{R}$; the Lipschitz function $f \circ \varphi^{-1}: \varphi(S) \to \mathbb{R}$; the point $z_0=\varphi(p)$; the value $r_0$; the weakly converging sequence of integral currents $\varphi_{i\#}(T_i \llcorner B(p_i,R))$ on $Y$, whose canonical sets are $\varphi_i(S_i)$; and the sequence $\alpha_i=d_Y(\varphi_i(p_i), \varphi(p))$, which converges to 0 as $i \to \infty$.
A few hypotheses require verification in order to apply Theorem \ref{thm_extrinsic}. First, we claim $r_0 >3\diam(\varphi(K)) + d_Y(\varphi(K), \varphi(S) \setminus \varphi(K))$. By our choice of $r_0$ and the fact that isometries preserve diameter, it suffices to show $d(K, X \setminus K) \geq d_Y(\varphi(K), \varphi(S) \setminus \varphi(K))$. Given $\eta>0$, there exists $k \in K$ and $x \in X \setminus K$ such that
$$d(k,x) \leq d(K, X \setminus K ) + \eta.$$
Then $\varphi(k) \in \varphi(K)$, and we claim that $x \in S$ (if $\eta$ was chosen sufficiently small). By the triangle inequality,
\begin{align*}
d(p,x) &\leq d(p,k) + d(k,x)\\
&\leq \diam(K) + d(K, X \setminus K) + \eta ,
\end{align*}
which is $<R$ if $\eta$ is sufficiently small. Then $x \in B(p,R)$. Since $x \in X = \set(T),$ we have $x \in \set (T \llcorner B(p,R))= S$. Then $\varphi(x) \in \varphi(S) \setminus \varphi(K)$, so
$$d_Y(\varphi(K), \varphi(S) \setminus \varphi(K)) \leq d_Y(\varphi(k), \varphi(x)) = d(k,x) \leq d(K, X \setminus K ) + \eta.$$
Since $\eta>0$ can be arbitrarily small, the proof of the claim is complete.
Second, $f \circ \varphi^{-1}$ is clearly Lipschitz, bounded between 0 and 1, with $f \circ \varphi^{-1} \equiv 1$ in a neighborhood of $\varphi(K)$. Since $\spt(f) \subseteq B(p,r_0)$ and $\varphi$ is an isometric embedding, we have $\spt(f \circ \varphi^{-1}) \subseteq B(\varphi(p),r_0) = B(z_0,r_0)$.
Third, the mass convergence hypothesis holds by \eqref{eqn_masses_converge}, with $V$ taken to be any ball about $z_0$ of radius greater than $R$.
By Theorem \ref{thm_extrinsic}, for each $i$ sufficiently large, there exists a Lipschitz function $f_i:\varphi_i(S_i) \to \mathbb{R}$, $0 \leq f_i \leq 1$, with $f_i\equiv 1$ on a neighborhood of $K_i$ (where $K_i=U^{-1}(-\infty,\alpha_i] \cap \varphi_i(S_i)$),
and $\spt(f_i) \subseteq B(\varphi(p), r_0+3)$. Moreover,
\begin{equation}
\label{limsup_E}
\limsup_{i \to \infty} E_{\varphi_{i\#}(N_i \llcorner B(p_i,R))}(f_i) \leq E_{\varphi_{\#}(N \llcorner B(p,R))}(f \circ \varphi^{-1}),
\end{equation}
where the push-forward of an integral current space under an isometric embedding is defined in the natural way; cf. \cite[Lemma 2.39]{SW}.
Consider $f_i \circ \varphi_i:S_i \to \mathbb{R}$, which is Lipschitz, bounded between 0 and 1, equalling 1 on a neighborhood of $\varphi^{-1}(K_i)$. To control the support, we have that for $i$ large, $d_Y(\varphi_i(p_i), \varphi(p)) < 1$. From this it follows $\spt(f_i) \subseteq B(\varphi_i(p_i), r_0+4)$, and so
$$\spt(f_i \circ \varphi_i) \subseteq B(p_i, r_0+4) \subseteq B(p_i,R).$$
Thus, we may extend $f_i \circ \varphi_i$ by 0 on $X_i \setminus B(p_i,R)$ to produce a Lipschitz function on $X_i$ with the same Dirichlet energy; call it $\hat f_i$, which is a valid test function for the capacity of $K_i$.
Using \eqref{limsup_E} on the fourth line below,
\begin{align*}
\limsup_{i \to \infty} \gamma_m\capac_{N_i}(\varphi^{-1}(K_i)) &\leq \limsup_{i \to \infty} E_{N_i} (\hat f_i)
\\
&= \limsup_{i \to \infty} E_{N_i \llcorner B(p_i,R)} (f_i \circ \varphi_i) \\
&= \limsup_{i \to \infty} E_{\varphi_{i\#}(N_i \llcorner B(p_i,R))} (f_i) \\
&\leq E_{\varphi_{\#}(N \llcorner B(p,R))} (f \circ \varphi^{-1}) \\
&= E_{N \llcorner B(p,R)} (f) \\
&= E_{N} (f)\\
&\leq \gamma_m\capac_N(K) + \epsilon,
\end{align*}
having used \eqref{eqn_E_N_eps} on the last line.
Since $\epsilon$ was arbitrary, the proof will now follow from the monotonicity of capacity by showing $\overline B(p_i,r) \subseteq \varphi_i^{-1}(K_i)$. To see this, first observe that $U^{-1}(-\infty,\alpha_i] = \overline B_Y(\varphi(p), r_0 + \alpha_i)$, which follows from $Y$ being a Banach space and $\varphi$ being an isometric embedding. Intersecting both sides of this equality with $\varphi_i(S_i)$ leads to
$$K_i=\overline B_Y(\varphi(p), r_0 + \alpha_i) \cap \varphi_i(S_i).$$
Noting $\overline B_Y(\varphi_i(p_i), r_0) \subseteq \overline B_Y(\varphi(p), r_0 + \alpha_i)$, we apply $\varphi_i^{-1}$ to obtain
$$\varphi_i^{-1}(K_i) \supseteq \varphi_i^{-1}\left(\overline B_Y(\varphi_i(p_i), r_0) \cap \varphi_i(S_i)\right).$$
It is straightforward to show directly that the right-hand side is simply $\overline B(p_i,r_0)$.
The proof of the theorem is complete in the case $K=\overline B(p,r_0) \neq X$.
\medskip
To complete the proof, we consider the case $\overline B(p, r) = X$. In particular, $X$ is bounded and the function $f \equiv 1$ on $X$ is a valid test function for the capacity, i.e. $\capac_N(\overline B(p, r))=0$.
Now, choose $R>r$ so that $N_i \llcorner B(p_i,R)$ and $N_i \llcorner B(p_i,R+1)$ converge in the $\mathcal{VF}$ sense as integral current spaces to $N \llcorner B(p,R)=N=N \llcorner B(p,R+1)$. By mass convergence,
$$\|T_i\|(A(p_i,R,R+1)) \to 0 \text{ as } i \to \infty,$$
where $A(p_i,R,R+1)$ is the annulus $B(p_i,R+1) \setminus B(p_i,R)$.
Let $f_i$ be a function on $X_i$ that equals 1 on $\overline B(p_i,R)$, 0 on $X_i \setminus B(p_i,R+1)$ and has $\Lip(f_i) \leq 1$. (Such a function can easily be constructed as a radial function of $d_{i}(p_i, \cdot)$.) Then we have
\begin{align*}
\capac_{N_i}(\overline B(p_i,r)) &\leq \capac_{N_i}(\overline B(p_i,R))\\
&\leq\frac{1}{\gamma_m} \int_{X_i} |d_x^{X_i} f_i|^2 d\|T_i\|(x)\\
&\leq\frac{1}{\gamma_m} \Lip(f_i)^2 \|T_i\|(A(p_i,R,R+1)).
\end{align*}
It follows that $\limsup_{i \to \infty}\capac_{N_i}(\overline B(p_i,r))=0$, completing the proof.
\end{proof}
We conclude this section by proving the other main result, Theorem \ref{thm_sublevel}.
\begin{proof}[Proof of Theorem \ref{thm_sublevel}]
Recall that by definition each $T_i$ is a locally integral current defined on $(\overline{X}_i, \overline{d}_i)$. So, we can apply Theorem 1.1 in \cite{LW} to the currents $T_i$ and points $p_i \in X_i$. The hypothesis
$$\sup_{i \in \mathbb N} \Big( \|T_i\|(B(p_i,r))+ \|\partial T_i\| (B(p_i,r))\Big) < \infty$$
for each $r>0$ holds by \eqref{eqn_bdry_mass_bound} and the hypothesis of pointed $\mathcal{VF}$ convergence. Thus, by Theorem 1.1 in \cite{LW}, there exist a subsequence of $N_i$ (note: in this proof we will not relabel subsequences), a complete metric space $(Z,d_Z)$, a point $z\in Z$, and isometric embeddings $\varphi_i: \overline {X}_i \to Z$ such that $\varphi_i(p_i)\to z$ in $Z$ and $\varphi_{i\#} (T_i) \to T'$ in the local flat topology, for some locally integral current $T'$ on $Z$ of dimension $m$. We point out that $Z$ can be taken to be a $w^*$-separable Banach space: first, recall that integral current spaces are separable \cite[Remark 2.36]{SW}, and the same goes for local integral current spaces. The construction of $Z$ in \cite{LW} comes directly from Proposition 5.2 in \cite{We2011}. There, $Z$ is constructed as the completion of a countable union of the $X_i$ and is therefore separable. Thus, we can apply Kuratowski's embedding theorem and by replacing $Z$ with $\ell^{\infty}(Z)$ we may assume that $Z$ is a $w^*$-separable Banach space.
Let $N'=(\set(T'), d_Z, T'\llcorner \overline{ \set(T')})$. If we show that $N_i \to N'$ in the pointed $\mathcal{VF}$ sense with respect to $p_i \in X_i$ and $z \in \overline{\set(T')}$,
then by uniqueness of pointed $\mathcal{F}$ limits (Proposition \ref{prop_pointed_limit}), we would get that $N\cong N'$. Once this is done, the result would follow by applying Theorem \ref{thm_extrinsic}.
Let
$$G= \{r > 0 \;:\; N_i \llcorner B(p_i,r) \xrightarrow{\F} N \llcorner B(p,r) \text{ with } p_i \to p \text{ as } i \to \infty\}.$$
By Definition \ref{def_pointed_F}, $G$ is unbounded. Using the slicing argument in \cite[Lemma 4.1]{Sor}, we pass to a subsequence so that $\mathbb{R}^+ \setminus G$ has measure zero. Applying a similar argument in $Z$, we may pass to a further subsequence and replace $G$ with a subset, still with $\mathbb{R}^+ \setminus G$ having measure zero, such that
\begin{equation}
\label{eqn_T_prime}
\varphi_{i\#}(T_i \llcorner B(p_i,r)) \to T' \llcorner B(z,r)
\end{equation}
as integral currents in the flat sense in $Z$ for all $r \in G$.
We now verify that $z \in \overline{\set(T')}$ using Theorem 2.9 in \cite{HLP} as follows. Let $r_1 < r_2$ belong to $G$.
Take $N_i \llcorner B(p_i, r_2)$ as ``$M_i$'' and $B(p_i,r_2)$ as ``$V_i$'' in the theorem. Then
$M_i \llcorner V_i$ to $N \llcorner B(p,r_2)$ with $p_i \to p$. Thus, condition $(1)$ of that theorem is satisfied, with $N \llcorner B(p,r_2)$ playing the role of ``$N_\infty$'' and $p$ as ``$x_\infty$.''
Condition $(2)$ follows, taking $\delta= r_1$.
Since we also have $M_i=N_i \llcorner B(p_i, r_2) \to N' \llcorner B(z,r_2)$ in the flat sense in $Z$, with $\varphi_i(p_i) \to z$, Theorem 2.9 in \cite{HLP} guarantees a subsequence such that $\varphi_i(p_i)$ converges to some $x' \in \overline{\set(T' \llcorner B(z,r_2))}$ in $Z$. But we know that $\varphi_i(p_i) \to z$ in $Z$. Hence, $z=x'$, and $x' \in \overline{\set(T')}$.
Finally, we prove that $N_i \to N'$ in the pointed $\mathcal{VF}$ sense with respect to $p_i \in X_i$ and $z \in \overline{\set(T')}$. Let $r_0>0$. There exists $r \geq r_0$ with $r\in G$. Thus, from \eqref{eqn_T_prime}, $N_i \llcorner B(p_i,r) \xrightarrow{\F} N' \llcorner B(z,r)$. Since $\varphi_i(p_i) \to z$, we have shown the claim.
Putting this all together, we obtain the conclusion of the theorem.
\end{proof}
\section{Examples}
\label{sec_examples}
In this section we give examples to demonstrate that a) the capacity upper semicontinuity can be strict, i.e., the capacity can jump up in a limit (Examples 1--3), and b) \emph{volume-preserving} convergence is necessary to guarantee upper semicontinuity (Example 4). We find there are essentially two independent reasons for the upper semicontinuity phenomenon. First, even under smooth local convergence the capacity of a set can jump up due to non-uniform control at infinity, e.g. a change in the end geometry of the manifold. Second, under $\mathcal{VF}$-convergence the capacity can also jump up, even with uniform control on the geometry at infinity.
\medskip
\paragraph{\emph{Example 1: transition from cylindrical to Euclidean end geometry}}
Consider rotationally symmetric smooth Riemannian metrics on $\mathbb{R}^n$, $n \geq 2$ of the form
$$g_i = ds^2 + f_i(s)^2 d\sigma^2,$$
where each $f_i:[0,\infty) \to \mathbb{R}$, $i=1,2,\ldots$ is smooth, with $f_i(0)=0$, $f_i(s)>0$ for $s>0$, and $d\sigma^2$ is the standard metric on the unit $(n-1)$-sphere. If we assume
$$f_i(s) = \begin{cases}
s, & 0 \leq s \leq i\\
i+1,& s \geq i+1
\end{cases},$$
then the corresponding Riemannian manifold $(\mathbb{R}^n, g_i)$ is isometric to a Euclidean ball for $s \leq i$ and to a cylinder (sphere-line product) of radius $i+1$ for $s \geq i+1$. The capacity of every compact set in $(\mathbb{R}^n, g_i)$ is zero, due to the cylindrical end (explained below). However, this sequence of Riemannian manifolds converges smoothly on compact sets, and hence in the pointed $\mathcal{VF}$ sense, to Euclidean space (where of course there exist compact sets of positive capacity). This example shows we cannot expect the capacity to behave continuously even for smooth local convergence.
To verify that the capacity vanishes identically with respect to $g_i$, given $i$, consider a radial Lipschitz function $\varphi_L(s)$ on $\mathbb{R}^n$ with
$$\varphi_L(s)=\begin{cases}
1, & s \leq L\\
2-\frac{s}{L}, &L < s \leq 2L\\
0, & 2L < s
\end{cases}$$
for a parameter $L$. Taking $L > i+1$, we have
$$\int_{\mathbb{R}^n} |\nabla \varphi_L|^2 dV_{g_i} = \omega_{n-1}\int_L^{2L} \frac{1}{L^2}ds = \frac{\omega_{n-1}}{L},$$
which can be made arbitrarily small by taking $L$ large. Moreover, by taking $L$ large, we can arrange $\varphi_L=1$ on any compact set.
\medskip
\paragraph{\emph{Example 2: formation of a new end}}
Let $M=\mathbb{R} \times S^2$ be equipped with a rotationally symmetric Riemannian metric
$$g= ds^2 + f(s)^2 d\sigma^2,$$
where $f>0$ is smooth, even function. Further, assume $f^{-2}$ is integrable on $\mathbb{R}$. Let $K$ be the compact subset $\{0\} \times S^2$. We compute the capacity of $K$ in $(M,g)$ as follows.
It is elementary to verify that the function
$$\psi(s) = \begin{cases}
\int_0^s f(r)^{-2} dr & s > 0\\
\int_s^0 f(r)^{-2} dr & s < 0
\end{cases}$$
is $g$-harmonic on $M \setminus K$, equalling zero on $K$ and approaching a positive constant $C=\int_0^\infty f(r)^{-2} dr$ at $\pm \infty$. In particular, $\varphi= 1-\frac{1}{C}\psi$ is a minimizer for the capacity of $K$. (Although $\varphi$ is not 1 on a neighborhood of $K$, this discrepancy may be neglected: it is straightforward to modify $\varphi$ near $K$ so that it is 1 on a neighborhood of $K$ and such that the Dirichlet energy changes by an arbitrary small amount.) From this, we can verify that the capacity of $K$ in $(M,g)$ equals $\frac{2}{C}$:
\begin{align*}
\capac(K) &= \frac{1}{4\pi} \int_M |\nabla \varphi|^2 dV\\
&=\frac{1}{4\pi} \int_{-\infty}^\infty \frac{1}{C^2}f(s)^{-4} (4\pi f(s)^2) ds\\
&= \frac{2}{C}.
\end{align*}
Now, consider a sequence of smooth, positive functions $f_i :[-2i, \infty) \to \mathbb{R}$ such that $f_i(s) = f(s)$ for $s \geq -i $, and such that $g_i = ds^2 + f(s)^2 d\sigma^2$ is a smooth Riemannian metric with a pole at $s=-2i$, i.e. $f_i(-2i)=0$, so the underlying manifold is diffeomorphic to $\mathbb{R}^3$.
Then for every $i$, the capacity of $K$ with respect to $g_i$ equals $1/C$, i.e. is half the capacity of $K$ in $(M,g)$. This can be seen by observing the capacity of $K$ in $(M,g_i)$ is achieved by the function that is 1 for $-2i\leq s\leq 0$ and otherwise agreeing with $\varphi$ above.
But the $g_i$ converge smoothly on compact sets to $g$, and the set $K$ has capacity $2/C$ in the limit space.
\medskip
\paragraph{\emph{Example 3: capacity jump with $\mathcal{VF}$-convergence}}
Let $Y$ be Euclidean 3-space, and let $X$ be the $z=0$ subspace. Then $X$ naturally becomes a local integral current space $N$ of dimension 2 with the Euclidean metric, where the locally integral current is given by integration, oriented up. Obviously $X$ is isometrically embedded in $Y$.
Let $K =\{ (x,y,0) \; | \; x^2+y^2 \leq 1\}$. For each $i=1,2,\ldots$, define
$$X_i = K \cup \{(x,y,1/i) \; : \; x^2+y^2 \geq 1\}.$$
Letting $X_i$ have the induced Euclidean metric, $X_i$ is obviously isometrically embedded in $Y$. $X_i$ may also be equipped with the locally integral 2-current given by integration, oriented up, producing a local integral current space, $N_i$. See Figure \ref{fig3}.
\begin{figure}
\centering
\includegraphics[scale=0.8]{fig3.eps}
\caption{In Example 3, the space $X_i$ is the union of a unit disk $K$ and a plane minus a disk sitting at height $\frac{1}{i}$ above $K$. $X_i$, naturally viewed as a local integral current space, pointed $\mathcal{VF}$-converges to $X$ with respect to some sequence of points, which is a Euclidean plane. The corresponding regions for $K$ are simply $K_i=K$.
}
\label{fig3}
\end{figure}
Observe that $N_i$ converges to $N$ in the pointed $\mathcal{VF}$-sense as $i \to \infty$, where all the points are chosen to be the origin in $Y$. This can easily be seen from the fact that $X_i \to X$ in the usual local flat sense in $Y$.
Let $u$ be the defining function for $K$ in $X$ given as the signed distance in $X$ to $\partial K$, negative inside of $K$, and let $U: Y \to \mathbb{R}$ be the standard Lipschitz extension. Consider $K_i = U^{-1}(-\infty,0] \cap X_i$, a sequence of corresponding regions as in section \ref{sec_main}. We claim $K_i=K$. If $x \in K$, then $u(x)=U(x) \leq 0$. Since $K \subset X_i$, we have $x \in K_i$. On the other hand, suppose $p \in K_i$, so $U(p) \leq 0$. Then there exists $x \in X$ such that
$$u(x) + d_Y(x,p) \leq 0.$$
Clearly $u(x) \leq 0$, i.e. $x \in K$. We can see the defining function is given by $u(x) = d_Y(0,x) -1$. With the triangle inequality, we have
$$d_Y(p,0) \leq 1,$$
i.e., $x$ belongs to the closed unit ball in $Y$ about $p$. The latter only intersects $X_i$ at $K$, so $p \in K$.
Now, the capacity of $K$ in $X$ is positive, but the capacity of $K_i$ in $X_i$ is zero for all $i$. This is easy to see because $X_i$ is disconnected: the function that equals 1 on $K$ and vanishes on $X_i \setminus K$ is Lipschitz and is a valid test function for the capacity, with zero Dirichlet energy. Thus, in \eqref{eqn_limsup_sublevel} of Theorem \ref{thm_sublevel}, we have strict inequality (without needing to take a subsequence).
If desired, one can arrange a similar example with the $X_i$ connected, as follows. Join the two connected components of $X_i$ with a thin ``strip'' of area of $O(1/i^3)$ and length $O(1/i)$. Then with a Lipschitz test function $f_i$ equalling 1 on $K$, with $\Lip(f_i)$ of $O(i)$ on the strip, and 0 elsewhere, the Dirichlet energy of $f_i$ would be $O(1/i)$, i.e., the capacity of $K$ in the connected space $X_i$ would still converge to 0.
\medskip
\paragraph{\emph{Example 4: cancellation and necessity of volume-preserving $\mathcal{F}$ convergence}}
Here, we demonstrate that upper semicontinuity of capacity may fail for pointed $\mathcal{F}$-convergence, without assuming $\mathcal{VF}$-convergence. We exploit the ``cancellation'' phenomenon of intrinsic flat convergence as in \cite[Example A.19]{SW}.
Let $Y$ be Euclidean 3-space, and let $X_i$ be the union of the $z=0$ plane and an annulus sitting slightly above:
$$X_i = \{(x,y,z) \; | \; z=0\} \cup \{(x,y,1/i) \; | \; 1 \leq x^2 + y^2 \leq 4\}.$$
Equip $X_i$ with the induced metric, so that $X_i$ is isometrically embedded in $Y$. Let $T_i$ be the locally integral current on $X_i$ given by integration, oriented up on the $z=0$ plane and down on the annulus. $X_i$ with the induced metric, equipped with $T_i$, produces a sequence of local integral current spaces, $N_i$. Letting $K$ be the unit disk $$\{(x,y,0) \; | \;x^2 + y^2 \leq 1\},$$
we have $K\subset X_i$, and the capacity of $K$ in $X_i$ is a positive constant independent of $i$.
Now, $N_i$ converges in the pointed $\mathcal{F}$-sense (but not $\mathcal{VF}$-sense) to
$$X=K \cup \{(x,y,0) \; | \; x^2+y^2 \geq 4\},$$
with the induced metric and integral current given by integration, oriented up. See Figure \ref{fig4}. Here, all the base points are chosen to be the origin. Since $K$ is a compact component of $X$, we have $\capac_N(K)=0$. Using $r=1$, we have a violation of Theorem \ref{thm_balls} if $\mathcal{VF}$-convergence is not assumed.
\begin{figure}
\centering
\includegraphics[scale=0.8]{fig4.eps}
\caption{In Example 4, the space $X_i$ is the union of a plane with an oppositely-oriented annulus sitting above at height $\frac{1}{i}$. $X_i$, naturally viewed as a local integral current space, converges in the pointed $\mathcal{F}$-sense (but not $\mathcal{VF}$) to $X$ which is a Euclidean plane minus an annulus representing where the cancellation occurred.
}
\label{fig4}
\end{figure}
\section{Asymptotically flat local integral current spaces and general relativistic mass}
\label{sec_mass}
Asymptotically flat (AF) Riemannian manifolds are of particular interest in the study of general relativity. These spaces are characterized by their metric tensors (and derivatives) decaying in a precise sense to the Euclidean metric in some appropriate coordinate chart that covers all but a compact set. The ADM mass is a numerical geometric invariant of an AF manifold that is of both significant physical and geometric interest \cite{ADM}. As described in the introduction, a number of open problems seem to necessitate an understanding of asymptotic flatness and ADM mass for spaces that are neither smooth nor Riemannian (again, we refer the reader to \cite{Sor2} and \cite{JL}, for example).
In this section, we give a possible definition of asymptotic flatness for local integral current spaces and describe two possible definitions of general relativistic mass for such spaces.
\medskip
We begin with a generalization of asymptotic flatness to metric spaces:
\begin{definition}
We define a metric space $(X,d)$ to be \emph{asymptotically flat of dimension $n \geq 3$} if for any $\epsilon >0$, there exists a compact set $K \subset X$ and a bijective map $\Phi$ from $X \setminus K$ to $\mathbb{R}^n \setminus B$ (for a closed ball $B \subset \mathbb{R}^n$) that is bi-Lipschitz when $X \setminus K$ and $\mathbb{R}^n \setminus B$ are endowed with the restricted distance of $d$ and of the Euclidean distance function, respectively, such that
$$\Lip(\Phi),\Lip(\Phi^{-1}) \leq 1+\epsilon.$$
\end{definition}
It is possible to show that any AF Riemannian manifold of dimension $n$ (in the usual sense) is an AF metric space of dimension $n$ with its natural distance function.
Similarly, we define a metric measure space $(X,d,\mu)$ to be AF of dimension $n \geq 3$ if the above properties hold for $(X,d)$ and also if
$$(1+\epsilon)^{-n}\mathcal{L}^n \leq \Phi_{\#}(\mu) \leq (1+\epsilon)^n\mathcal{L}^n$$
as Borel measures on $\mathbb{R}^n \setminus B$, where $\mathcal{L}^n$ is the Lebesgue measure. For example, if $(X,d)$ is an AF metric space of dimension $n$, then equipped with Hausdorff $n$-measure, it becomes an AF metric measure space.
Now we can define a local integral current space $(X,d,T)$ of dimension $n$ to be AF if $(X,d, \|T\|)$ is an asymptotically flat metric measure space of dimension $n$. (We note other reasonable definitions are possible.) In this setting, the capacity of compact sets is well defined, as is the boundary mass of balls for almost all radii \cite[Lemma 2.34]{Sor}.
\medskip
We now proceed to discuss the concept of general relativistic mass for asymptotically flat local integral current spaces (not to be confused with the mass measure). The standard definition of ADM mass involves derivatives of the Riemannian metric coefficients and so is unsuitable for metric spaces.
A well-known approach to a ``weak'' understanding of ADM mass is due to Huisken \cites{Hui1,Hui2}: his so-called \emph{isoperimetric mass} uses only volumes and areas (perimeters) in its formulation. In dimension three, with nonnegative scalar curvature, it is known to equal the ADM mass in the smooth asymptotically flat case \cites{Hui1,Hui2,JLC0,CESY}. In \cite{JL}, Jauregui and Lee gave a definition of asymptotically flat local integral current space (more restrictive than that which we use here, essentially requiring the complement of a compact set to be a smooth manifold with a $C^0$ Riemannian metric), and used Huisken's isoperimetric mass as a substitute for ADM mass. Since the perimeters of compact sets are well defined even for $C^0$ Riemannian metrics, it was clear that Huisken's definition was well defined.
Huisken's isoperimetric mass, $m_{iso}$, is typically defined for asymptotically flat Riemannian 3-manifolds. We can generalize this concept to any 3-dimensional asymptotically flat local integral current space, using boundary mass in place of perimeter:
$$m_{iso}(X,d,T)=\sup_{\{K_j\}} \limsup_{j \to \infty} \frac{2}{\mathbb{M}(\partial(T \llcorner K_j))} \left[\|T\|(K_j)-\frac{1}{6 \sqrt{\pi}} \mathbb{M}(\partial(T \llcorner K_j))^{\frac{3}{2}}\right]\qquad \in [-\infty,\infty],$$
where $\{K_j\}$ is an exhaustion of $X$ by compact sets. Note that if $\mathbb{M}(\partial(T \llcorner K_j)) = \infty$, the expression inside the $\limsup$ is $-\infty$. In particular, we may restrict to exhaustions such that $\mathbb{M}(\partial(T \llcorner K_j))$ is finite, which is equivalent to saying $T \llcorner K_j$ is an integral $n$-current on $(X,d)$.
Huisken's definition was inspired by the isoperimetric inequality: far out in the AF end, the inequality almost holds, and the ADM mass can be detected through the deficit. Jauregui proposed a corresponding definition of mass based on the isocapacitary inequality (that the capacity of a compact set of a given volume in $\mathbb{R}^n$ is minimized by balls) \cite{Jau}. This definition of ``capacity-volume mass'' was for AF manifolds, including $C^0$ AF manifolds. However, it can be generalized to AF local integral current spaces of dimension $3$ as follows:
$$m_{CV}(X,d,T) = \sup_{\{K_j\}} \limsup_{j \to \infty} \frac{1}{4\pi\capac(K_j)^2 } \left[ \|T\|(K_j) - \frac{4\pi}{3} \capac(K_j)^{3}\right],$$
where the capacity is defined as in \eqref{def_cap}.
In \cite{Jau}, strong evidence was given for $m_{CV}$ recovering the ADM mass in the smooth case with nonnegative scalar curvature (and hence serving as a weak stand-in for the ADM mass). Furthermore, it was observed that capacity is in some ways better behaved than perimeter or boundary mass --- for example, capacity is less sensitive to perturbations, and as confirmed by our main theorems and discussed below, has a favorable semicontinuity property --- so $m_{CV}$ may ultimately be easier to work with in low-regularity ADM mass problems than $m_{iso}$.
To connect this discussion of mass with our main theorems, we conclude with a discussion of the lower semicontinuity of total mass in general relativity. In \cites{JC2,JLC0} it was shown that the ADM mass functional (and more generally, Huisken's isoperimetric mass) is lower semicontinuous on an appropriate class of asymptotically flat 3-manifolds of nonnegative scalar curvature, for pointed $C^2$, and more generally, for pointed $C^0$ Cheeger--Gromov convergence. This was further generalized to pointed $\mathcal{VF}$ convergence, under natural hypotheses, using Huisken's isoperimetric mass as a stand-in for the mass of the (potentially non-smooth) limit space \cite{JL}. Below, we argue that Theorem \ref{thm_sublevel} supports lower semicontinuity of $m_{CV}$ in dimension three.
To simplify the discussion, we recall from the appendix of \cite{Jau} that $m_{CV}$ may alternatively be written:
$$m_{CV}(X,d,T) = \sup_{\{K_j\}} \limsup_{j \to \infty} \left[ \left(\frac{\|T\|(K_j)}{4\pi }\right)^{1/3} - \capac(K_j) \right].$$
Now consider the inner expression $\left(\frac{\|T\|(K)}{4\pi }\right)^{1/3} - \capac(K)$ as a functional on compact sets $K$. To have any hope of showing $m_{CV}$ is lower semicontinuous under pointed $\mathcal{VF}$ convergence, it seems necessary to know that ``volume radius minus capacity'' itself is lower semicontinuous. Since volume is by definition continuous in $\mathcal{VF}$, this amounts to the statement that capacity is \emph{upper} semicontinuous. We demonstrated this in Theorem \ref{thm_sublevel} for example. In other words, the results of this paper are supportive of $m_{CV}$ itself being lower semicontinuous under pointed $\mathcal{VF}$-convergence, though a full proof of this is more subtle, requiring, for example, an unproven analog of the ADM mass estimate \cite[Theorem 17]{JLC0}, but for the capacity-volume mass in place of the isoperimetric mass.
\section*{Appendix: tangential differential, Dirichlet energy, Sobolev spaces, and capacity}
In this section we first review the definition of Dirichlet energy for a Lipschitz function defined on the canonical set of a current as was done by Portegies \cite{Por}, which we use in the definition of capacity. This will require the concepts of metric and $w^*$-differentials, approximate tangent spaces, and the tangential differential. After that, we relate the latter to the minimal relaxed gradient and briefly discuss several notions of Sobolev spaces on metric spaces. We conclude with a comparison of the definition of capacity we employ in this paper and other definitions appearing in the literature.
A \emph{$w^*$-separable Banach space} $Z$ is by definition a dual space $Z = G^*$, and hence Banach, of a separable Banach space $G$. The function $d_w: Z \times Z \to \mathbb{R}$ given by
\begin{equation*}\label{eq-distance-d_w}
d_w(x,y) := \sum_{j=0}^\infty 2^{-j} | \langle x-y, g_j \rangle|,
\quad \text{for } x,y \in Z,
\end{equation*}
where $\{g_j\}_{j=0}^\infty$ is a countable dense subset in the unit ball in $G$, is a distance; $d_w$ induces the $w^*$-topology on bounded subsets of $Z$, and $(Z,d_w)$ is a separable space.
One of the main examples of a $w^*$-separable Banach space is the space $\ell^\infty= (\ell^1)^*$.
\begin{definition}[{\cite[Definitions 3.1 and 3.4]{AK_rect}}]
Let $Z$ be a metric space and $g:\mathbb{R}^n \to Z$ a function.
\begin{itemize}
\item We say that $g$ is \emph{metrically differentiable} at $x \in \mathbb{R}^n$ if there is a seminorm $md_xg : \mathbb{R}^n \to \mathbb{R}$ such that
\begin{equation*}
d(g(y), g(x)) - md_x g (y - x) = o(|y - x|), \quad y \to x.
\end{equation*}
We call $md_x g$ the \emph{metric differential} of $g$ at $x$.
\item If $Z$ is a $w^*$-separable Banach space, we say that $g$ is \emph{$w^*$-differentiable} at $x\in\mathbb{R}^n$ if there is a linear map $wd_xg:\mathbb{R}^n \to Z$ such that
\begin{equation*}
\lim_{y\to x} \frac{g(y) - g(x) - wd_xg (y-x)}{|y-x|} = 0,
\end{equation*}
where the limit is understood in the $w^*$-sense.
The map $wd_x g$ is called the \emph{$w^*$-differential} of $g$ at $x$.
\end{itemize}
\end{definition}
For Lipschitz maps the following is known.
\begin{thm}[{\cite[Theorems 3.2 and 3.5]{AK_rect}}]
If $Z$ is a metric space, then any Lipschitz function $g: \mathbb{R}^n \to Z$
is metrically differentiable ${\mathcal{L}}^n$-a.e.
If additionally, $Z$ is a $w^*$-separable Banach space, then $g$
is also $w^*$-differentiable ${\mathcal{L}}^n$-a.e, and the metric and weak differential satisfy
\begin{equation*}
md_xg(v) = \|wd_xg(v)\|, \qquad \text{ for all } v \in \mathbb{R}^n \text{ and }
{\mathcal{L}}^n\text{-}a.e. \, x \in \mathbb{R}^n.
\end{equation*}
\end{thm}
\medskip
A subset $S$ of a metric space $Z$ is \emph{countably ${\mathcal H}^n$-rectifiable} if there exist Lipschitz functions $g_j: A_j \subset \mathbb{R}^n \to Z$, $j \in \mathbb N$, defined on Borel sets $A_j$ such that
\begin{equation*}
{\mathcal H}^n \left( S \backslash \bigcup_{j=1}^\infty g_j(A_j) \right) = 0.
\end{equation*}
If $Z$ is a $w^*$-separable Banach space, the \emph{approximate tangent space} to $S$ at a point $x$ is defined as
\begin{equation*}
\Tan(S,x) = wd_y g_j(\mathbb{R}^n),
\end{equation*}
whenever $y = g_j^{-1}(x)$ and $g_j$ is metrically and $w^*$-differentiable at $y$, with $J_n(wd_y g_j) > 0$, where for any linear function $L: V \to W$ between two Banach spaces, with $n=\dim V$,
$$J_n(L)= \frac{\omega_n} {{\mathcal H}^n\{v \in V\, :\, ||L(v)|| \leq 1\}}$$
denotes the $n$-Jacobian of $L$. By \cite{AK_rect}, $\Tan(S,x)$ is well defined for ${\mathcal H}^n$-almost all $x \in S$. A finite Borel measure $\mu$ is called \emph{$n$-rectifiable} if $\mu = \theta {\mathcal H}^n \llcorner S$ for a countably ${\mathcal H}^n$-rectifiable set $S$ and a Borel function $\theta:S \to (0,\infty)$.
The next theorem shows the existence of tangential differentials of Lipschitz functions on rectifiable sets.
\begin{thm}[{\cite[Theorem 8.1]{AK_rect}}]
Let $Z$ and $Z'$ be two $w^*$-separable Banach spaces, $S \subset Z$ an ${\mathcal H}^n$-countably rectifiable subset and $f: Z \to Z'$ a Lipschitz function. Let $\theta:S \to (0,\infty)$
be an $ {\mathcal H}^n $-integrable function and denote by $\mu = \theta {\mathcal H}^n \llcorner S$ the corresponding $n$-rectifiable measure.
Then for ${\mathcal H}^n$-almost every $x\in S$, there exist a Borel set $S^x \subset S$ such that
the upper $n$-dimensional density of
$\mu \llcorner S^x$ equals zero,
\begin{equation*}
\Theta_n^*( \mu \llcorner S^x, x) = 0,
\end{equation*}
and a linear and $w^*$-continuous map $L: Z \to Z'$ so that
\begin{equation*}
\lim_{y \in S \backslash S^x \to x} \frac{d_w(f(y),f(x) + L(y-x))}{|y - x|} = 0.
\end{equation*}
$\Tan(S,x)$ exists and $L$ is uniquely determined on $\Tan(S,x)$ and its restriction to $\Tan(S,x)$
is called the tangential differential to $S$ at $x$ and is denoted by
\begin{equation*}
d_x^S f: \Tan(S,x) \to Z'.
\end{equation*}
Furthermore, the tangential differential is characterized by the property that for any Lipschitz map $g: A \subset \mathbb{R}^n \to S$,
\begin{equation*}
wd_y(f \circ g) = d_{g(y)}^S f \circ wd_y g, \qquad \text{for } \mathcal{L}^n\text{-a.e. } y \in A.
\end{equation*}
\end{thm}
Note that if $d_x^S f$ is defined, then its dual norm satisfies $|d_x^S f| \leq \Lip(f)$.
If $S$ is an arbitrary separable, countably ${\mathcal H}^n$-rectifiable metric space, we isometrically embed $S$ into a $w^*$-separable Banach space $Z$, $\iota: S \to Z$. Then for ${\mathcal H}^n$-almost every $x\in S$
we can define the \emph{approximate tangent space} of $S$ at $x$ as
\begin{equation*}
\Tan(S,x) : = \Tan(\iota(S),\iota(x)).
\end{equation*}
Even if we have chosen a particular isometric embedding, $\Tan(S,x)$ is uniquely determined ${\mathcal H}^n$-a.e. up to linear isometries \cite{AK_rect}.
If additionally $\theta$ and $\mu$ are as in the previous theorem, then we define
\begin{equation*}
|d_x^S f| = |d_{\iota(x)}^{\iota(S)} (f\circ \iota^{-1})|,
\end{equation*}
for $\mu$-a.e. $x \in S$ and where the right hand side denotes the dual norm of $d_{\iota(x)}^{\iota(S)} (f\circ \iota^{-1})$. This quantity is also well defined, independent of the isometric embedding.
\begin{definition}[Definition 3.8 of \cite{Por}]\label{de:NormEn}
Let $X$ be a complete metric space, and let $T \in \mathbf{I}_{n}(X)$. Let $S = \set(T)$, and let $f: S \to \mathbb{R}$ be a Lipschitz function. Then the \emph{(Dirichlet) energy} of $f$ is given by
\begin{equation*}
E_T(f) := \int_X |d_x^S f |^2 \, d\|T\|(x).
\end{equation*}
\end{definition}
The energy of $f$ is invariant under isometric embeddings, and for any compact oriented Riemannian manifold $(M,g)$ we have that the energy is given by $\int_M |\nabla f|^2dV$, where the gradient and volume measure are taken with respect to $g$.
We next mention the relationship between $|d^S_xf|$ and the minimal relaxed gradient.
Let $X$ be a complete metric space, $T \in \mathbf{I}_{n}(X)$ and $S=\set(T)$.
The space $L^2(\|T\|)$ is the Hilbert space of equivalence classes of functions on $X$ that are square-integrable with respect to $\|T\|$ with inner product
\begin{equation*}
\langle f, g\rangle_{L^2(\|T\|)} := \int_X f g \, d\|T\|.
\end{equation*}
The space $W^{1,2}(\|T\|)$ is the completion of the set of bounded Lipschitz functions on $\spt T$ with respect to the norm $\|.\|_{W^{1,2}}$ given by
\begin{equation*}
\begin{split}
\|f \|_{W^{1,2}}^2
& = \int_X f^2 d\|T\| + \int_X |d^S_x f|^2 d\|T\|(x).
\end{split}
\end{equation*}
By definition, every $f$ in $W^{1,2}(\|T\|)$ can be represented by a Cauchy sequence $f_i$ of bounded Lipschitz functions. The limit of $d^S_xf_i$ in $\mathcal{T}^*_2(\|T\|)$,where $S$ denotes $\set(T)$, the Banach space of equivalence classes of covector fields endowed with the norm
\begin{equation*}
\| \psi \|_{\mathcal{T}^*_2(\|T\|)}^2 := \int_X | \psi(x) |^2_{[\Tan(S,x)]^*} d\|T\|(x),
\end{equation*}
is denoted by $d_xf$.
\begin{thm}[Theorem 5.2 \cite{Por}]
\label{thm_df}
Let $X$ be a complete metric space, $T \in \mathbf{I}_{n}(X)$ and
$f \in L^2(\|T\|)$. Then $f$ has a relaxed gradient in the sense of \cite[Definition 4.2]{AGS} if and only if $f \in W^{1,2}(||T||)$. Moreover, the minimal relaxed gradient equals $|d_x f|$ for $||T||$-a.e. $x \in X$.
\end{thm}
We conclude with a discussion of Sobolev spaces and capacity.
In \cites{BB, HKST} metric measure spaces $(X,d,m)$, where $(X,d)$ is separable and
$m$ is a locally finite Borel regular measure on $X$,
are considered and Newtonian
spaces of functions $N^{1,p}(X,d,m)$, $1 \leq p < \infty$, are defined. Originally defined by Shanmugalingam in her PhD thesis and subsequent paper \cite{Sha}, these are a type of Sobolev space.
Then the \emph{$p$-capacity} of a set $E \subset X$ (what was called the Sobolev
$p$-capacity in the introduction) is defined as
\begin{align*}
\capac_p(E)&= \inf\left\{ \int |u|^pdm + \int \rho_u^p dm \,\right.\\
&\qquad\qquad\left.: \, u \in N^{1,p}(X,d,m), u \geq 1 \text{ on } E \text{ outside a $p$-exceptional set of measure zero}\right\},
\end{align*}
where $\rho_u$ denotes the minimal $p$-weak upper gradient of $u$.
It is also shown that this is equivalent to \cite[Lemma 7.2.6]{HKST}
\begin{equation*}
\capac_p(E)= \inf\left\{ \int |u|^pdm + \int \rho_u^p dm \, : \, u \in N^{1,p}(X,d,m), \, 0 \leq u \leq 1 \text{ and $u=1$ on $E$}\right\}.
\end{equation*}
Other types of Sobolev spaces on metric spaces have been defined, see Theorem 10.5.1---10.5.3 in \cite{HKST}. For $1<p < \infty$, the Cheeger space,
$W_{Ch}^{1,p}$, and Newtonian space, $N^{1,p}$, are equal (up to representatives) and both norms coincide.
Provided
$m$ is a doubling measure and $X$ satisfies a $q$-Poincar\'e inequality, $1 \leq q <2$,
several Sobolev spaces coincide $$M^{1,2}=P^{1,2}=KS^{1,2}=N^{1,2}=W_{Ch}^{1,2},$$
though some norms are only comparable. Here $M^{1,2}$ is the Haj\l asz Sobolev space \cite{Haj}, $P^{1,2}$ is the Poincar\'e Sobolev space, and $KS^{1,2}$
is the Korevaar--Schoen Sobolev space.
If $m$ is only a doubling measure then $M^{1,2} \subseteq P^{1,2} \subseteq KS^{1,2} \subseteq N^{1,2}=W^{1,2}_{Ch}$.
For complete and separable metric measure spaces $(X,d,m)$, in \cite[Theorem 6.2]{AGS} (cf. \cite[Theorem 2.2.28]{GP})
it was shown that $W^{1,2}_{Ch}$ and the Sobolev space $W^{1,2}(X,d,m)$ using the minimal relaxed gradient in the sense of \cite[Definition 4.2]{AGS} (and their norms) are the same. Furthermore, any $f \in W^{1,2}(X,d,m)$ can be approximated by functions in $\Lip(X) \cap L^2(m)$.
The main difference between the capacity we use in this paper and the definition given in \cite{HKST}, is that in our definition of capacity
we only integrate the gradient term. However, it is possible to bound these capacities in terms of each other. For some constant $C$, we immediately have:
\begin{equation*}
C||T||(K) + \capac(K) \leq \capac_2(K).
\end{equation*}
On the other hand, if the space admits a Poincar\'e inequality, one can obtain $\capac_2(K) \leq C'\capac(K)$ for a constant $C'$ (see \cite[Theorem 6.16]{BB}).
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1,314,259,996,474 | arxiv | \section{Introduction}
\label{intro}
Over the past several years, studies have shown that exotic formation channels could lead to a population of highly eccentric compact binaries whose gravitational wave (GW) emission would be in the band of current ground-based detectors~\cite{2009MNRAS.395.2127O, Lee:2009ca,Wen:2002km,Kushnir:2013hpa,2013PhRvL.111f1106S,Antognini:2013lpa,Naoz:2012bx,Antonini:2013tea,Antonini:2015zsa, East:2012xq}. One such formation channel is dynamical captures in dense stellar environments, such as globular clusters and galactic nuclei~\cite{2009MNRAS.395.2127O, Lee:2009ca, East:2012xq}. In such dense environments, compact objects initially on hyperbolic trajectories can become bound after passing through closest approach due to GW emission or tidal interactions, with the subsequent bound binary having high eccentricity ($ e \sim 1$). On the other hand, the Kozai-Lidov mechanism~\cite{Wen:2002km,Naoz:2011mb,Kushnir:2013hpa,2013PhRvL.111f1106S,Antognini:2013lpa,Naoz:2012bx,Antonini:2013tea,Antonini:2015zsa, VanLandingham:2016ccd}, and other three-body interactions~\cite{Naoz:2012bx} in hierarchical triple systems, can induce resonances that drive the inner binary to the parabolic limit.
Although expected to be rare~\cite{Abadie:2010cf}, eccentric binaries could prove to be powerful probes of astrophysical dynamics. Event rates of eccentric inspirals due to dynamical captures have wide error bars, typically around two orders of magnitude~\cite{2009MNRAS.395.2127O}. Such large error is largely due to the unconstrained populations of compact objects within dense environments~\cite{Abadie:2010cf}. Likewise, the tightening of binaries due to three-body interactions in galactic nuclei are similarly uncertain~\cite{Miller:2008yw}. Hence, the detection of GWs from eccentric inspirals would provide information about the mass function of black holes (BHs) and neutron stars (NSs) in these environments, allowing us to probe astrophysics that has proven difficult to extract from electromagnetic observations~\cite{O'Leary:2007qa, Zevin:2017evb, Rodriguez:2016geq, Stevenson:2017dlk}.
Another promising area of interest for eccentric binaries is testing Einstein's theory of General Relativity (GR). For highly eccentric binaries, the distance of closest approach can be small relative to the semi-major axis of the orbit, leading to systems with pericenter velocities greater than $10\%$ the speed of light. At such high velocities, the GW luminosity \emph{in each burst} will typically be $\sim (10^{-4}-10^{-3}) L_{{\mbox{\tiny Pl}}}$, where $L_{{\mbox{\tiny Pl}}}$ is the Planck luminosity. For comparison, a BH-BH, quasi-circular binary emits radiation at $\lesssim10^{-5} L_{{\mbox{\tiny Pl}}}$ during the early phase of inspiral, increasing rapidly close to merger and eventually reaching $\sim 10^{-2} L_{{\mbox{\tiny Pl}}}$ only at merger. As such, the GWs from eccentric binaries could capture effects from the extreme gravity regime (i.e.~where gravity is dynamical, strong and non-linear) during many pericenter passages~\cite{Loutrel:2014vja}.
These GWs may be detected by current and upcoming ground-based GW detectors, and hence the study of GWs from eccentric binaries has never been more urgent. The advanced Laser Interferometer Gravitational Wave Observatory (aLIGO)~\cite{Harry:2010zz} has already achieved the first detection of GWs with the event GW150914~\cite{Abbott:2016blz}. The advanced VIRGO Interferometer (aVIRGO)~\cite{TheVirgo:2014hva} will be coming online in 2016-2017, with additional detections expected during this period. Plausible estimates of the event rates of the inspiral of compact object binaries predict that these detectors could see $\sim 5-10$ events per year in the very near future~\cite{Abadie:2010cf,Abbott:2016nhf,Abbott:2016drs}. Based on our current knowledge of the formation channels of eccentric compact binaries~\cite{2009MNRAS.395.2127O, Lee:2009ca,Wen:2002km,Kushnir:2013hpa,2013PhRvL.111f1106S,Antognini:2013lpa,Naoz:2012bx,Antonini:2013tea,Naoz:2012bx,Antonini:2015zsa}, we might expect that one or two of these events will enter the LIGO band with non-negligible orbital eccentricity. Looking toward the future, KAGRA~\cite{Uchiyama:2004vr,Somiya:2011np} in Japan is expected to come online by the end of this decade and LIGO India~\cite{Unnikrishnan:2013qwa} in the beginning of the next decade. Once these detectors are operational, the number of detected events per year will necessarily increase, thus increasing the probability of detecting eccentric inspirals.
The typical strategy for detecting well-understood GWs with ground-based interferometers is to use matched filtering~\cite{Brown:2012nn, Farr:2009pg, VanDenBroeck:2009gd, Ajith:2012az, Aasi:2012rja, Abadie:2011kd, Colaboration:2011np}. Effectively, a set of templates that best describe the signal buried in the data are used to extract the latter and estimate its physical parameters. This detection strategy hinges on having very accurate templates, as a small dephasing between the signal and a template can result in complete loss of detection~\cite{Sampson:2013jpa,Yunes:2009ke,Yagi:2013baa,Favata:2013rwa}. Hence, we must have a prior model of what to search for in the detector data output. For highly eccentric binaries, the GW emission resembles a set of discrete bursts somewhat localized in time and frequency. These bursts are centered around pericenter passage, where the orbital velocity is highest and where the binary spends the least amount of time, and thus very little GW power is contained within each individual burst. This issue alone makes matched filtering a rather impractical search strategy for such GWs, but it is further compounded by the fact that there are few, fully-analytic and accurate templates for highly eccentric binaries with which to perform matched filtering computationally efficiently~\cite{Hinder:2008kv, Huerta:2014eca}.
An alternative search strategy was presented in~\cite{Tai:2014bfa} using a power stacking method. Ideally, if one could register a set of bursts in a time-frequency decomposition of the data stream, one could then stack the power within each burst, thus creating an enhanced data product. For $N$ bursts with the same signal-to-noise ratio (SNR), the amplification in the SNR relative to the SNR in a single burst would scale as $N^{1/4}$. Although sub-optimal compared to matched filtering, the power stacking method is more robust to modeling errors and more efficient in detection of eccentric signals than current un-modeled burst methods~\cite{Tai:2014bfa}. Power stacking, however, still requires a model of where the bursts will occur in time-frequency space given some initial starting point, with which to construct a prior to search for successive bursts.
Such a burst model was developed in~\cite{Loutrel:2014vja} for tracking the bursts in time-frequency space. In general, a burst model is one that treats the bursts as N-dimensional objects in the detector's data stream and tracks the geometric centroid and volume of the bursts from one to the next as the binary inspirals. To do this,~\cite{Loutrel:2014vja} considered Keplerian orbits perturbed by quadrupolar gravitational radiation. We will refer to this model as \emph{Newtonian} in the sense that it is obtained from a fully relativistic model expanded to lowest, non-vanishing order about small orbital velocities and weak gravitational fields. The benefit of working in such a simplified scenario was that it allowed for the exploration of whether such burst signals could be used to test well-motivated deviations from GR with eccentric signals as a proof-of-concept.
The ultimate goal, of course, is to create a model that is as accurate as possible relative to the signals present in nature. The modeling of the coalescence of compact objects in full GR is an exceedingly difficult problem, which has only been solved numerically (predominantly for quasicircular binaries) in the passed several years. For eccentric binaries, numerical simulations in full GR are more computationally expensive, and thus, to obtain only a few orbits at the desired numerical accuracy requires much more computational time than that needed in the evolution of quasicircular binaries. At least for the moment, the pure numerical modeling of eccentric binaries over the last thousand orbits in full GR is currently an intractable problem. On the other hand, we could work to extend the Newtonian burst model by considering relativistic corrections to Newtonian dynamics.
The post-Newtonian (PN) formalism~\cite{Lorentz1937, Chandrasekhar:1965, Chandrasekhar:1969, Chandrasekhar:1970, Blanchet:2013haa,PW} allows for a systematic treatment of $v/c$ corrections to Newtonian dynamics, where $v$ is the orbital velocity and $c$ is the speed of light. For bound binaries, the orbital velocity is connected to the gravitational field strength through a Virial relation: a term of ${\cal{O}}(v^{2}/c^{2})$ is comparable to a term of ${\cal{O}}[GM/(R c^{2})]$, where $M$ and $R$ are the characteristic mass and orbital separation of the system, and $G$ is Newton's gravitational constant. Hence, the PN formalism for binaries is simultaneously a post-Minkowskian expansion, i.e.~it is both an expansion in $v/c \ll 1$ and an expansion in $GM/(Rc^2) \ll 1$.
The PN formalism has had a wide range of success in the modeling of binaries and their GW emission. At present, the GW emission from quasicircular binaries has been calculated to 3.5 PN order\footnote{A term proportional to $(v/c)^{2N}$ relative to its controlling factor will be said to be of N PN order.}~\cite{Blanchet:2013haa} and to 4PN order in the effective one-body Hamiltonian~\cite{Damour:2014jta}. To leading order in the mass ratio, the radiation fluxes at spatial infinity are currently known to 22 PN order~\cite{Fujita:2012cm}, due to the formulaic nature of the calculation. The PN corrections to Newtonian dynamics for eccentric binaries have proven more difficult to calculate. Damour and Deruelle~\cite{zbMATH03938612,zbMATH04001537} found a Keplerian parameterization of the solution to the 1PN order equations of motion in terms of the eccentric anomaly $u$. This quasi-Keplerian (QK) representation was extended to 2PN order in~\cite{0264-9381-12-4-009} and to 3PN order in~\cite{Memmesheimer:2004cv}.
The QK parametrization must be enhanced to include dissipation if one wishes to obtain accurate waveform models. This parameterization is a purely conservative representation of the orbital motion of eccentric binaries, because the orbital energy and angular momentum are assumed to be conserved. Dissipation occurs because GWs carry energy and angular momentum away from the system, causing the binary to inspiral and eventually merge. For eccentric binaries, the GW energy and angular momentum fluxes have been computed to full 3PN order~\cite{Arun:2007sg, Arun:2009mc}. With these fluxes at hand and assuming small eccentricities ($e \lesssim 0.1$), Refs.~\cite{Yunes:2009yz,Tessmer:2010ii,Tessmer:2010sh,Moore:2016qxz} constructed time-domain and frequency domain waveforms to 2PN order. Waveform templates also exist for higher eccentricity systems ($ e \lesssim 0.4$), for example through the hybrid time-domain x-model of~\cite{Hinder:2008kv} and the hybrid frequency-domain model of~\cite{Huerta:2014eca}. These models, however are really not applicable to highly eccentric binaries.
The burst model previously developed~\cite{Loutrel:2014vja} is currently the only purely analytic model for highly elliptic orbits. We say the orbits are highly elliptic, and not highly eccentric because the latter implies that the eccentricity could be large and potentially greater than unity. On the other hand, ``highly elliptic'' indicates that we are always considering bound orbits. The burst model has currently only been developed to Newtonian order, which is an artifact of our desire to simplify our previous analysis as much as possible to be able to consider tests of GR.
Nature, however, is not Newtonian, and thus, extending the burst model into the PN formalism serves two purposes: to improve its potential of aiding in the detection of highly elliptic binaries and to enhance its ability to perform interesting and important science. The burst model was conceived with the idea of testing GR. However, if there is an inherent modeling error within the GR model, it is possible that such modeling errors could fool us into believing we have detected a non-GR signal if we are not careful~\cite{Sampson:2013jpa, Yunes:2009ke}. Furthermore, the power stacking method is not immune to modeling error in the detection of signals and the extraction of their parameters~\cite{Tai:2014bfa}. Hence, the detection of such binaries and extraction of important astrophysics hinges on having an accurate prior to predict where the bursts occur in time-frequency space.
\subsection{Executive Summary}
We here extend the burst model developed in~\cite{Loutrel:2014vja} to higher PN order. The Newtonian burst model in~\cite{Loutrel:2014vja} focused on the bursts emitted during the inspiral of the binary only. Similarly, we here also focus on the inspiral of highly elliptic binaries within the PN framework. The motivation for developing a purely analytic model of the inspiral is the potential for the later construction of phenomenological inspiral-merger-ringdown models, a quasi-circular version of which played a pivotal role in the first gravitational wave observations by aLIGO~\cite{Abbott:2016blz}. Here, we treat the bursts as two dimensional regions of excess power in a time-frequency decomposition of a detectors data stream. We treat the bursts as boxes with characteristic time and frequency widths, which allows for a discretization of the time-frequency decomposition into tiles, with the burst being those tiles that contain excess power~\cite{Abbott:2016blz,Tai:2014bfa}. As with the Newtonian burst model, we characterize the sequence of bursts using the time and frequency centroids of the bursts, as well as the widths of the burst tiles, or alternatively the volume of the tiles, used to capture a certain amount of power within each bursts. These time-frequency observables are supplemented by a model describing the orbital evolution of the binary as a set of discrete, osculating Keplerian ellipses.
Similar to how the parameterized post-Einsteinian (ppE) burst sequence of~\cite{Loutrel:2014vja} was a parameterized deformation of a simplified GR sequence, the PN burst sequence will be a parameterized deformation of the Newtonian order sequence. The deformations will scale with an increasing power of a particular PN expansion parameter, which we choose to be the pericenter velocity $v_{p}$. The coefficients of a $k/2$-PN order term, which scales as $v_{p}^{k}$, are then a set of functions $[{\cal{V}}_{k}, {\cal{D}}_{k}, P_{k}, R_{k}]$ which are dependent on the physical parameters of the compact binary system. These functions correspond to the PN corrections to the rates of change of pericenter velocity and time eccentricity, and the expressions for the orbital period and pericenter distance, respectively. In this work, we neglect the spin of the compact objects and work in a point particle limit, such that these functions are only dependent on the time eccentricity $e_{t}$ and the symmetric mass ratio $\eta$. Hence, when working to $k/2$-PN order, one needs $4k$ functions $[{\cal{V}}_{k}(e_{t}, \eta), {\cal{D}}_{k}(e_{t}, \eta), P_{k}(e_{t}, \eta), R_{k}(e_{t}, \eta)]$ to parametrize all of the PN defomations.
We parametrize the PN burst sequence in time-frequency space by
\allowdisplaybreaks[4]
\begin{align}
\label{t-PN}
\frac{(t_{i} - t_{i-1})_{{\rm PN}}}{(t_{i} - t_{i-1})_{\rm N}} &= 1 + \vec{P}(e_{t,i}, \eta; v_{p}) \cdot \vec{X}(v_{p,i})
\\
\frac{f_{i}^{{\rm PN}}}{f_{i}^{\rm N}} &= 1 + \vec{R}^{(-1)}(e_{t,i}, \eta; v_{p}) \cdot \vec{X}(v_{p,i})
\\
\frac{\delta t_{i}^{{\rm PN}}}{\delta t_{i}^{k{\rm N}}} &= 1 + \vec{R}(e_{t,i}, \eta; v_{p}) \cdot \vec{X}(v_{p,i})
\\
\label{df-PN}
\frac{\delta f_{i}^{{\rm PN}}}{\delta f_{i}^{\rm N}} &= 1 + \vec{R}^{(-1)}(e_{t,i}, \eta; v_{p}) \cdot \vec{X}(v_{p,i})
\end{align}
where $(t_{i}, f_{i})$ are the centroid of the bursts and $(\delta t_{i}, \delta f_{i})$ are the width and height of the tiles. We create the \emph{amplitude vector fields} $[\vec{P},\vec{R}]$, whose components are the functions $[P_{k}(e_{t}, \eta; v_{p}), R_{k}(e_{t}, \eta; v_{p})]$, which we further specify as implicit functions of the PN expansion parameter $v_{p}$ since their form changes depending on the choice of expansion parameter. The components of $\vec{R}^{(-1)}$ are defined such that
\begin{equation}
\left[1 + \vec{R}(e_{t}, \eta; v_{p}) \cdot \vec{X}(v_{p})\right]^{(-1)} \doteq 1 + \vec{R}^{(-1)}(e_{t}, \eta; v_{p}) \cdot \vec{X}(v_{p})\,,
\end{equation}
where the equality $\doteq$ should be understood as working in the limit of $v_{p} \ll 1$. The \emph{state vector} $\vec{X}(v_{p})$ contains the powers of $v_{p}$ that characterized each PN order corrections, specifically $\vec{X}(v_{p}) = (v_{p}, v_{p}^{2}, ..., v_{p}^{k})$. Hence the dot products provide the complete sum of all terms in a PN expansion up to $k/2$-PN order.
These time-frequency burst parameters, specifically $(t_{i}, f_{i}, \delta t_{i}, \delta f_{i})$, are functions of the symmetric mass ratio and the total mass of the binary, which are constant in time, as well as the pericenter velocity $v_{p,i}$ and eccentricity $e_{t,i}$ during each burst, which are evolving in time under the influence of radiation reaction. Hence, we must supplement the time-frequency sequence described above with the orbital evolution of the binary. To do this, we apply an osculating approximation that assumes the bursts are emitted instantaneously at pericenter, forcing the binary to move along a discrete set of Keplerian ellipses that osculate onto one another. The parameters of the $i$-th orbit will be functions of the parameters of the previous orbit, specifically
\begin{align}
\label{dvp-PN}
\frac{(v_{p,i} - v_{p,i-1})_{{\rm PN}}}{(v_{p,i} - v_{p,i-1})_{\rm N}} &= 1 + \vec{\cal{V}}(e_{t,i-1}, \eta; v_{p}) \cdot \vec{X}(v_{p,i-1})\,,
\\
\label{det-PN}
\frac{(\delta e_{t,i} - \delta e_{t,i-1})_{\rm PN}}{(\delta e_{t,i} - \delta e_{t,i-1})_{\rm N}} &= 1+ \vec{\cal{D}}(e_{t,i-1}, \eta; v_{p}) \cdot \vec{X}(v_{p,i-1})\,.
\end{align}
where we have introduced the two new amplitude vector fields $[\vec{\cal{V}}, \vec{\cal{D}}]$. In this work, we provide explicit expressions for the amplitude vector fields $\lambda_{\rm PN burst}^{a} \equiv (\vec{P}, \vec{R}, \vec{\cal{V}}, \vec{\cal{D}})$ complete to 3PN order. Equations~\eqref{t-PN}-\eqref{df-PN} and~\eqref{dvp-PN}-\eqref{det-PN} provide the complete PN burst model, which we use to calculate the burst model to 3PN order using the results for $\lambda_{\rm PN burst}^{a}$.
How does this new burst model aid us in the detection of highly elliptic binaries? In a realistic search, the burst model acts as a prior on where the bursts will occur in time-frequency space. For example, once a search detects a burst of power within an interferometer data stream (even if this burst of power is not ``loud'' enough to allow to claim detection), the burst model can then be used to search over "future" time-frequency space for successive bursts, as well as "past" time-frequency space for bursts that may have been missed by previous searches. Physically, this amounts to searching over the parameters of the system that determine the prior, specifically the eccentricity and pericenter velocity during the initially detected burst, and the chirp mass and symmetric mass ratio of the binary.
The structure of the PN burst model should not be surprising given the structure of the ppE burst model in~\cite{Loutrel:2014vja}. The ppE model requires four exponent parameters $a_{i}$, which govern the power of $v_{p}$ of the corrections, and four amplitude parameters $\alpha_{i}$, which depend on the coupling constants of the theory and the eccentricity of the binary. In the PN formalism, the exponent parameter $k$ becomes a known quantity and only changes when one goes to higher order in the expansion variable. The amplitude parameters have now been replaced with four amplitude vector fields\footnote{These are not true vector fields, but are a set of scalar functions that have been combined into a discrete sequence. The terminology used for these functions goes along with the notation we have used to simplify some of the expressions in this work.} that parametrize the eccentricity and mass dependence of specific PN terms. As a result of this, rather than needing eight parameters as was the case in the ppE model, we require 4$k$ functions when working to $k/2$-PN order. These amplitude functions only depend on the initial eccentricity and the symmetric mass ratio, which together with the initial pericenter velocity and the chirp mass of the binary are the only parameters needed to define the model. Further, the fact that we require four vector fields to describe the burst model is a result of the fact that the model in only parametrized by four quantities: the orbital energy and angular momentum, and the energy and angular momentum fluxes of the GWs emitted by the system. Alternatively, as we will show, a different set of four parameters can be used: the orbital period, the mapping between the pericenter distance and pericenter velocity, and the rates of change of eccentricity and pericenter velocity due to radiation reaction. Working with these four quantities significantly improves the ease with which burst models can be constructed.
The remainder of this paper is dedicated to deriving the results presented above. In Section~\ref{review}, we review the Newtonian order burst model and present a simplified formalism used for the construction of the PN burst model. Section~\ref{PN} is dedicated to constructing a burst model at arbitrary PN order, which we later specialize to the cases of 1PN, 2PN, and 3PN orders. Section~\ref{discussion} discusses the results of the paper and their importance for future research. In this paper, we use geometric units where $G=c=1$.
\section{Constructing Burst Models}
\label{review}
This section is dedicated to reviewing how to create a burst model and the elements that go into such a model. We begin by reviewing the Newtonian burst model and how it was constructed. We then describe a new method of constructing burst models in general without any assumptions of the regime or theory of gravity we are working in. This new method greatly simplifies the construction of burst models, and will allow us to develop a completely generic GR PN model in the next section.
\subsection{The Newtonian Burst Model}
\label{sec:NewtBurstMod}
How do we construct a burst model? Recall that in Section~\ref{intro} we defined a burst model as a theoretical model prior to describe how the bursts evolve in time-frequency space. Such a model would tell us how the centroid and the volume of the bursts evolve in time and frequency from one burst to the next. But this evolution depends on the orbital parameters of the system, which themselves are also changing in time due to dissipative effects, such as the emission of GWs. Therefore, a complete burst model must provide a one-to-one mapping between the evolution of the system's physical parameters and how the bursts evolve in time and frequency. This requires the following ingredients:
\begin{enumerate}
\item {\textit{Orbital Evolution:}} A mapping that prescribes the evolution of the orbital parameters from one orbit to the next, including GW radiation-reaction.
\item {\textit{Centroid Mapping:}} A mapping that provides the time-frequency centroid of the burst $(t_{i}, f_{i})$, given the centroid of the previous burst $(t_{i-1}, f_{i-1})$, in terms of the orbital parameters of the system.
\item{\textit{Volume Mapping:}} A mapping that describes how the time-frequency volume of the bursts changes from one to the next, in terms of the orbital parameters of the system.
\end{enumerate}
\subsubsection{Orbit Evolution}
Let us start by reviewing how the ingredients listed above can be computed to leading (i.e.~Newtonian) order, focusing first on ingredient I (the orbital evolution). In Newtonian gravity, the orbital motion of two test particles can be described though Keplerian ellipses, which are characterized by two conserved quantities, the orbital energy $E$ and the orbital angular momentum $L$. Alternatively, one can parameterize any such orbit in terms of its pericenter distance $r_{p}$ and its orbital eccentricity $e$, which are related to $E$ and $L$ at Newtonian order in a PN expansion by
\allowdisplaybreaks[4]
\begin{align}
\label{EN}
E &= - \frac{M^{2} \eta \left(1 - e\right)}{2 r_{p}}\,,
\\
\label{LN}
L &= \eta \sqrt{M^{3} r_{p} \left(1 + e\right)}\,.
\end{align}
where $\eta = m_{1} m_{2}/M^{2}$ is the symmetric mass ratio and $M = m_{1}+m_{2}$ is the total mass of the system.
The burst model requires knowledge of how $(E,L)$ or $(r_{p},e)$ evolve from one orbit to the next. Due to the nature of the emission of gravitational radiation in highly elliptic systems, we may treat the problem as a set of Keplerian orbits that change effectively instantaneously at pericenter, allowing the orbits to \emph{osculate} onto one another. Hence, we may write
\begin{align}
E_{i} &= E_{i-1} + \Delta E_{(i,i-1)}\,,
\\
L_{i} &= L_{i-1} + \Delta L_{(i,i-1)}\,,
\end{align}
where $\Delta E_{(i,i-1)}$ and $\Delta L_{(i,i-1)}$ are the changes in orbital energy and angular momentum due to GW emission from one orbit to the next, and the labels represent which orbit the above quantities are evaluated on. By ``osculating orbits,'' we mean that the elements of the Keplerian orbit are constant throughout the orbit except at pericenter, where they change drastically and the new elements define a new Keplerian orbit. In this approximation, one thus treats the radiation, and all changes generated by it, as arising instantaneously at pericenter.
In general, the total change in energy and angular momentum between times $T_{i-1}$ and $T_{i}$ due to GW emission is given by
\begin{align}
\label{change-E}
\Delta E_{(i,i-1)} &= \int_{T_{i-1}}^{T_{i}} \dot{E}\left(r_{p}, e, \psi\right) dt\,,
\\
\label{change-L}
\Delta L_{(i,i-1)} &= \int_{T_{i-1}}^{T_{i}} \dot{L}\left(r_{p}, e, \psi\right) dt\,,
\end{align}
where $\psi$ is the true anomaly, $T_{i-1}$ and $T_{i}$ are the times of consecutive pericenter passages, and the dot refers to derivatives with respect to time. At Newtonian order, the GW energy and angular momentum fluxes, $\dot{E}$ and $\dot{L}$, are given, for example, by Eq.~(12.78) in~\cite{PW}. The fluxes are functions of the orbital elements, which are themselves functions of time through the true anomaly $\psi$. Thus, the above definitions would have to be supplemented with the time evolution of $\psi$ itself, namely $\dot{\psi}\left(r_{p}, e, \psi\right)$. In addition, since the fluxes depend on the true anomaly, they contain gauge-dependent terms~\cite{PW}. However, these terms vanish upon integration, leaving $\Delta E_{(i,i-1)}$ and $\Delta L_{(i,i-1)}$ independent of the radiation reaction gauge.
To evaluate the integrals in Eqs.~\eqref{change-E} and~\eqref{change-L}, we perform a change of variable from $t$ to the true anomaly, using $dt = d\psi / \dot{\psi}$. The new limits of integration become $\psi_{i-1}$ and $\psi_{i} = \psi_{i-1} + 2 \pi$, or more simply $[0,2\pi].$ The orbital elements now depend on the true anomaly rather than time, which simplifies the integrands, but this is still not enough to evaluate them analytically. To do so, we use the fact that the orbits are osculating and the GW emission occurs instantaneously at pericenter, which ensures that $r_{p}$ and $e$ are constant everywhere except at closest approach. With this, the integrals become
\begin{align}
\label{eq:DeltaE-int}
\Delta E_{(i,i-1)} &= \int_{0}^{2\pi} \frac{\dot{E}\left(r_{p,i-1}, e_{i-1}, \psi\right)}{\dot{\psi}\left(r_{p,i-1}, e_{i-1}, \psi\right)} d\psi
\\
\label{eq:DeltaL-int}
\Delta L_{(i,i-1)} &= \int_{0}^{2\pi} \frac{\dot{L}\left(r_{p,i-1}, e_{i-1}, \psi\right)}{\dot{\psi}\left(r_{p,i-1}, e_{i-1}, \psi\right)} d\psi
\end{align}
which can be evaluated analytically.
Alternatively, we can exploit the definition of orbital averaged GW fluxes to rewrite these changes in a simpler way. The orbital averaged energy flux, for example, is given by $\langle \dot{E} \rangle \equiv \Delta E_{(i,i-1)} / P$, where $P$ is the orbital period of the binary, and likewise for the angular momentum flux. With this definition, we are free to write
\begin{align}
\label{E-next}
E_{i} &= E_{i-1} + P_{i-1} \langle \dot{E} \rangle_{i-1}\,,
\\
\label{L-next}
L_{i} &= L_{i-1} + P_{i-1} \langle \dot{L} \rangle_{i-1}\,.
\end{align}
Indeed, we recognize the integral expressions in Eqs.~\eqref{eq:DeltaE-int} and~\eqref{eq:DeltaL-int} as simply the product of the orbital period and $\langle \dot{E} \rangle_{i-1}$ or $\langle \dot{L} \rangle_{i-1}$ by definition. It might seem odd that orbit averaged quantities appear in the above expressions since the GW emission is happening mostly during pericenter passage, and thus smearing the emission over the entire orbit would appear incorrect. However, this is purely a result of the \emph{definition} of the orbital averaged quantities, and has nothing to do with the nature of the GW emission or the validity of the orbital averaged approximation for the systems we are considering~\cite{Loutrel-avg}.
The orbital energy and angular momentum have a clear physical meaning, but $r_{p}$ and $e$ allow us to straightforwardly visualize the geometry of the system that is generating the bursts (at least at Newtonian order). At this order, it does not matter which set of quantities, $(E,L)$ or $(r_{p}, e)$, we decide to use for the orbital evolution. For the Newtonian model in~\cite{Loutrel:2014vja}, we decided to use $(r_{p}, e)$, so let us continue to do so here. We need to solve the system given by Eqs.~\eqref{EN} and~\eqref{LN} for the functionals $r_{p}\left(E, L\right)$ and $e\left(E, L\right)$. To obtain the evolution of the pericenter distance and the orbital eccentricity, we evaluate the functionals at the desired orbit, specifically $r_{p, i} = r_{p} \left[E_{i}\left(E_{i-1}, L_{i-1}\right), L_{i}\left(E_{i-1}, L_{i-1}\right)\right]$ and $e_{i} = e\left[E_{i}\left(E_{i-1}, L_{i-1}\right), L_{i}\left(E_{i-1}, L_{i-1}\right)\right]$. Evaluating the functionals with Eqs.~\eqref{E-next} and~\eqref{L-next} gives
\begin{align}
\label{burst-rp-N}
r_{p,i} &= r_{p,i-1} \left[1 - \frac{59 \pi \sqrt{2}}{24} \eta \left(\frac{M}{r_{p,i-1}}\right)^{5/2} \left(1 + \frac{121}{236} \delta e_{i-1}\right)\right]\,,
\\
\label{burst-e-N}
\delta e_{i} &= \delta e_{i-1} + \frac{85 \pi \sqrt{2}}{12} \eta \left(\frac{M}{r_{p,i-1}}\right)^{5/2} \left(1 - \frac{1718}{1800} \delta e_{i-1}\right)\,,
\end{align}
where we have kept only leading-order terms in $\delta e \equiv 1 - e \ll 1$ and in $M/r_{p} \ll 1$ in all expressions (since we are working to Newtonian order). These equations recursively describe how the orbit shrinks and circularizes as the binary inspirals, thus completely describing the orbital evolution.
\subsubsection{Centroid Mapping}
The second ingredient we need for any burst model is the centroid mapping. The centroids of the bursts are given by the set $(t_{i}, f_{i})$ at which the bursts occur. What we desire is the mapping $t_{i-1} \rightarrow t_{i}$ and $f_{i-1} \rightarrow f_{i}$. Since the orbits are osculating, the time between bursts is trivially given by the orbital period $P$ to Newtonian order, which is given by
\begin{equation}
\label{Torb}
P = \frac{2 \pi r_{p}^{3/2}}{M^{1/2} (1 - e)^{3/2}}\,.
\end{equation}
Thus to obtain the time mapping, we simply have to evaluate the orbital period at the desired orbit,
\begin{equation}
t_{i} = t_{i-1} + \frac{2 \pi}{M^{1/2}} \left[\frac{r_{p,i} \left(r_{p,i-1}, \delta e_{i-1}\right)}{\delta e_{i}\left(r_{p,i-1}, \delta e_{i-1}\right)}\right]^{3/2}
\end{equation}
where the mappings $r_{p,i}\left(r_{p,i-1}, \delta e_{i-1}\right)$ and $\delta e_{i}\left(r_{p,i-1}, \delta e_{i-1}\right)$ are given by Eqs.~\eqref{burst-rp-N} and~\eqref{burst-e-N}, respectively.
The fact that the bursts are separated by an orbital period can be seen more generally by writing $\dot{\psi} = \dot{\psi}_{\text{cons}} + \dot{\psi}_{\text{diss}}$, where $\dot{\psi}_{\text{cons}}$ is the conservative part coming from Keplerian orbital dynamics, and $\dot{\psi}_{\text{diss}}$ is the dissipative piece that comes from radiation reaction. The time between successive pericenter passages, i.e. the orbital period, is then
\begin{equation}
\label{change-t}
t_{i} - t_{i-1} = \int_{0}^{2\pi} \frac{d\psi}{\dot{\psi}}\,.
\end{equation}
The dissipative part contains terms that depend on the radiation reaction gauge, which vanish upon integration. Thus we are left with only the conservative piece of $\dot{\psi}$, and by assuming the orbits are osculating and the GW emission is instantaneous, this evaluates upon integration to the orbital period for an unperturbed, purely conservative orbit. This does not mean that the orbital period is not evolving. GW emission carries energy and angular momentum away from the binary that changes the orbital period. The dissipative part of the above integral does vanish, but the dissipative part of $P$, namely $\dot{P}$, does not. The above result simply implies that the time between pericenter passages is the orbital period of a Keplerian orbit, since radiation reaction is happening rapidly around pericenter.
The GW frequency on the other hand requires knowledge of the Fourier transform of the GWs emitted during each burst, the \emph{Fourier-domain waveform}. From Fig.~7 in~\cite{Turner:1977}, the GW power is highly peaked around
\begin{align}
\label{f-GW}
f_{\rm GW} = \frac{1}{2 \pi \tau_{\rm GW}}\,,
\end{align}
where $\tau_{\rm GW}$ is the characteristic GW time, defined by
\begin{align}
\label{tau-GW}
\tau_{\rm GW} &\equiv \frac{\text{pericenter distance}}{\text{pericenter velocity}}\,.
\end{align}
At Newtonian order, the pericenter velocity is given by
\begin{equation}
\label{vp-N}
v_{p} = \sqrt{\frac{M (1+e)}{r_{p}}}
\end{equation}
and thus $\tau_{\rm GW}$ is
\begin{align}
\tau_{\rm GW} &= \frac{r_{p}^{3/2}}{\left[M (1 + e)\right]^{1/2}}\,,
\end{align}
which roughly corresponds to the amount of time the system spends at pericenter. This time is a functional of the pericenter distance and eccentricity; hence, to obtain the frequency of the $i$-th burst $f_{i}$, one simply has to evaluate the characteristic GW time associated with the orbit $(r_{p,i}, e_{i})$, where the mapping to the parameters of the previous orbit are given by Eqs.~\eqref{burst-rp-N} and~\eqref{burst-e-N}:
\begin{equation}
f_{i} = \frac{M^{1/2} \left[2 - \delta e_{i}\left(r_{p,i-1}, \delta e_{i-1}\right)\right]^{1/2}}{2 \pi \left[r_{p,i}\left(r_{p,i-1}, \delta e_{i-1}\right)\right]^{3/2}}\,.
\end{equation}
This completes the mapping of the time-frequency centroid of the bursts.
The prescription we provide for the frequency of the bursts is dependent on the frequency domain waveform, or alternatively the GW power, peaking at $\tau_{\rm GW}^{-1}$. This intuition comes from~\cite{Turner:1977}, where for parabolic orbits, and at Newtonian order, it is shown that the GW power peaks roughly at $\tau_{\rm GW}^{-1}$. However, for circular binaries, the power peaks at twice the orbital frequency, and it can easily be checked that the prescription given above does not reproduce this result when $e=0$. This implies that there are uncontrolled remainders that depend on $\delta e$ that correct the above expression to account for this. However, because we are working in the limit where $\delta e \ll 1$, we expect such corrections to be subdominant.
\subsubsection{Volume Mapping}
The last ingredient we need is the volume mapping. The bursts are not instantaneously emitted at pericenter and are not solely peaked at one frequency. The emission is instead spread out over the full pericenter passage and over multiple frequencies. To complete the burst model, we need to determine how the time-frequency size of the bursts change from one to another. We may describe the bursts as any two dimensional objects in time and frequency. For simplicity, we chose to model the bursts as boxes with widths
\begin{align}
\label{delta-t}
\delta t = \xi_{t} \tau_{\rm GW}
\\
\label{delta-f}
\delta f= \xi_{f} f_{\rm GW}
\end{align}
where $\xi_{t}$ and $\xi_{f}$ are constants of proportionality that are chosen from data analysis considerations. For example, one can choose these constants such that a desired percentage of the GW power ($90\%$ for example) is contained in each box. More general two dimensional objects, such as ellipsoids, could be used for this construction, but boxes are the simplest. To obtain how these widths change from one burst to the next, we simply have to evaluate Eqs.~\eqref{delta-t}-\eqref{delta-f} at the parameters of the orbit $(r_{p,i}, e_{i})$:
\begin{align}
\label{eq:delta-t-eval}
\delta t_{i} &= \frac{\xi_{t} \left[r_{p,i}\left(r_{p,i-1}, \delta e_{i-1}\right)\right]^{3/2}}{M^{1/2} \left[2 - \delta e_{i-1} \left(r_{p,i-1}, \delta e_{i-1}\right)\right]^{1/2}}
\\
\label{eq:delta-f-eval}
\delta f_{i} &= \frac{\xi_{f} M^{1/2} \left[2 - \delta e_{i}\left(r_{p,i-1}, \delta e_{i-1}\right)\right]^{1/2}}{2 \pi \left[r_{p,i}\left(r_{p,i-1}, \delta e_{i-1}\right)\right]^{3/2}}\,.
\end{align}
Note that in the case of ellipsoids, the results are the same, but these quantities can instead be interpreted as the semi-minor and semi-major axes of the ellipsoids. For a realistic search, ellipsoids would actually be more appropriate choice since they are a more accurate representation of the time-frequency structure of the bursts. However, for the purposes of this work, this choice is irrelevant, as the goal is to characterize the two dimensional objects via the scales in Eqs.~\eqref{eq:delta-t-eval} and~\eqref{eq:delta-f-eval}. This completes the review of the construction of the burst model to Newtonian order.
\subsection{A Simplified Formalism}
\label{simp}
Ultimately, we are interested in a PN burst model at generic (presumably very high) PN order. Building a generic order PN model by following the construction above might at first seem like an intractable problem. To start, one would have to take the orbital energy $E_{\rm PN}$ and angular momentum $L_{\rm PN}$ at arbitrary order and invert these expressions to obtain $r_{p}(E_{\rm PN}, L_{\rm PN})$ and $e(E_{\rm PN}, L_{\rm PN})$. Then, one would need to use the energy and angular momentum fluxes to compute the evolution of the orbital energy and angular momentum. From there, the pericenter and orbital eccentricity mapping would have to be computed using the functionals $r_{p,i}\left[E_{i}(E_{i-1}, L_{i-1}), L_{i}(E_{i-1}, L_{i-1})\right]$ and $e_{i}\left[E_{i}(E_{i-1}, L_{i-1}), L_{i}(E_{i-1}, L_{i-1})\right]$. While this may actually be possible from a mathematical standpoint, it will be very non-trivial to do so at arbitrary order. Thus, in this subsection, we will instead seek a simplified formalism that is more practical to implement.
The new method we seek must be more direct than the previous method discussed, removing steps that are redundant and reducing the number of physical quantities we need to work with. We begin by noting a number of assumptions that we will use to simplify the analysis:
\begin{enumerate}
\item {\textit{Osculating Orbits:}} Any changes in the orbital parameters will be modeled as occurring instantaneously around pericenter passage, leaving the orbital parameters constant throughout the rest of the orbit.
\item {\textit{High Ellipticity:}} The orbits we consider are highly elliptical, so we define a small parameter $\delta e_{t} \equiv 1 - e_{t}$ and work perturbatively in the regime $\delta e_{t} \ll 1$.
\item{\textit{PN Orbits:}} We will work within the PN framework, expanding all expressions in the pericenter velocity $v_{p} \ll 1$.
\end{enumerate}
The first and second assumptions follow directly from the nature of the systems we consider in this paper. Note that in the second assumption we are now working with the time eccentricity from the QK parametrization. The reason for this is that in PN theory, there is no unique concept for the orbital eccentricity, as there are actually three eccentricities that enter the QK equations of motion, specifically $(e_{t}, e_{r}, e_{\phi})$. All three of these eccentricities reduce to the orbital eccentricity in the Newtonian limit, but they are distinct quantities within PN theory. We choose to work with $e_{t}$ and will express all quantities in terms of it. We discuss this in more detail in~\ref{fields}.
The final assumption is new to this analysis and replaces the previous Newtonian assumption. At Newtonian order, we worked with the pericenter distance $r_{p}$ as one of our physical parameters. We will now choose to work with the pericenter velocity $v_{p}$ instead. This change is meant to put the computation more in line with the standard PN formalism for quasi-circular inspirals, as well as to remove some difficulties that result in there being terms that depend on half-integer powers of $r_{p}$ in the dissipative sector. The mapping between the pericenter distance and velocity is given explicitly to 1PN order in Eqs.~\eqref{rp-1PN} and~\eqref{R2}, with the 2PN and 3PN corrections given in Eqs.~\eqref{R4} and~\eqref{R6}. With the above assumptions, any burst model requires the three ingredients laid out in Sec.~\ref{sec:NewtBurstMod}.
Let's start with the orbital evolution, where now we focus on the evolution of the pericenter velocity and the orbital eccentricity. Rather than starting from the orbital energy and angular momentum, we are free to write the velocity and eccentricity mappings as
\begin{align}
\label{vp-map}
v_{p,i} &= v_{p,i-1} + \Delta v_{p, (i,i-1)}\,,
\\
\label{e-map}
\delta e_{t,i} &= \delta e_{t,i-1} - \Delta e_{t,(i,i-1)}\,,
\end{align}
where $\Delta v_{p, (i,i-1)}$ and $\Delta e_{t,(i,i-1)}$ are the change in pericenter velocity and time eccentricity between two successive orbits, and we have used the fact that $\delta e_{t} = 1 - e_{t}$ to write $\Delta \delta e_{t} = - \Delta e_{t}$. These mappings are directly analogous to the mappings of energy and angular momentum in our Newtonian model, given by Eqs.~\eqref{change-E} and~\eqref{change-L}.
Expressions for $\Delta v_{p, (i,i-1)}$ and $\Delta e_{t,(i,i-1)}$ can be found in exactly the same way as in Eqs.~\eqref{eq:DeltaE-int} and~\eqref{eq:DeltaL-int}. We may thus jump ahead and directly write
\begin{align}
\Delta v_{p,(i,i-1)} &= \int_{0}^{2\pi} \frac{\dot{v}_{p}\left(v_{p,i-1}, e_{t,i-1}, \psi\right)}{\dot{\psi}\left(v_{p,i-1}, e_{t,i-1}, \psi\right)} d\psi
\\
\Delta e_{t,(i,i-1)} &= \int_{0}^{2\pi} \frac{\dot{e}_{t}\left(v_{p,i-1}, e_{t,i-1}, \psi\right)}{\dot{\psi}\left(v_{p,i-1}, e_{t,i-1}, \psi\right)} d\psi
\end{align}
where $\dot{v}_{p}$ and $\dot{e}$ are the rates of change of pericenter velocity and orbital eccentricity. These rates, once again, depend on the true anomaly and thus have gauge-dependent terms arising from the GW sector. Upon integration, these terms vanish, except now the above quantities are not necessarily gauge-invariant as they depend on the specific coordinate system one chooses to do the PN calculation in.
To our knowledge, the expressions $\dot{v}_{p}\left(v_{p}, e_{t}, \psi\right)$ and $\dot{e}_{t}\left(v_{p}, e_{t}, \psi\right)$ have not yet been explicitly computed and would not be easy to compute, which would leave something of a gap in constructing the orbit evolution for our bursts. However, we may once again exploit the definition of orbit averaging and write Eqs.~\eqref{vp-map} and~\eqref{e-map} as
\begin{align}
\label{vp-burst}
v_{p,i} &= v_{p,i-1} + P_{i-1} \langle \dot{v}_{p} \rangle_{i-1}\,,
\\
\label{e-burst}
\delta e_{t,i} &= \delta e_{t,i-1} - P_{i-1} \langle \dot{e}_{t} \rangle_{i-1}\,.
\end{align}
The orbit averaged quantities $\langle \dot{v}_{p} \rangle$ and $\langle \dot{e}_{t} \rangle$ can be easily computed from the orbital energy and angular momentum and the corresponding fluxes, which are known to full 3PN order. We have thus completely constructed the orbit evolution for our burst model.
We now focus on the centroid and volume mappings. Once again, we will treat the bursts as boxes in time and frequency, and determine the mapping between the centroids and widths of the boxes. The characteristic GW time is still given by Eq.~\eqref{tau-GW}. In the Newtonian model, we used the expression $v_{p}\left(r_{p}, e_{t}\right)$ given by Eq.~\eqref{vp-N} to write this time in terms of $(r_{p}, e_{t})$. Since we are now working with $v_{p}$ instead of $r_{p}$, we can invert the relationship between these two parameters to obtain $r_{p}\left(v_{p}, e_{t}\right)$,which is given explicitly in Eqs.~\eqref{rp-1PN}-\eqref{R2} and~\eqref{R4}-\eqref{R6}, and write $\tau_{\rm GW}$ in terms of $(v_{p}, e_{t})$. Once this time is specified, we may define the characteristic GW frequency by Eq.~\eqref{f-GW}. The centroid and volume mappings follow the exact same analysis as the Newtonian model, only parameterized by the pericenter velocity rather than the pericenter distance. Hence, we may write
\begin{align}
\label{change-t}
t_{i} &= t_{i-1} + P\left[v_{p,i}\left(v_{p,i-1}, e_{t,i-1}\right), e_{t,i} \left(v_{p,i-1}, e_{t,i-1}\right)\right],
\\
\label{change-f}
f_{i} &= f_{\rm GW} \left[v_{p,i}\left(v_{p,i-1}, e_{t,i-1}\right), e_{t,i}\left(v_{p,i-1}, e_{t,i-1}\right)\right],
\\
\label{change-dt}
\delta t_{i} &= \delta t \left[v_{p,i}\left(v_{p,i-1}, e_{t,i-1}\right), e_{t,i}\left(v_{p,i-1}, e_{t,i-1}\right)\right],
\\
\label{change-df}
\delta f_{i} &= \delta f \left[v_{p,i}\left(v_{p,i-1}, e_{t,i-1}\right), e_{t,i}\left(v_{p,i-1}, e_{t,i-1}\right)\right],
\end{align}
thus completing the last two ingredients we need for our simplified formalism.
\section{A Generic PN Formalism}
\label{PN}
With the application of assumption I, we have constructed a purely generic burst model through Eqs.~\eqref{vp-burst},~\eqref{e-burst}, and~\eqref{change-t}-\eqref{change-df} that applies in any theory of gravity. We now seek to use this formalism to create a burst model at generic PN order. We will provide explicit expressions for the burst model at 1PN, 2PN, and 3PN orders in Sec.~\ref{examples}.
The above considerations imply that, to construct our burst model, we need PN expansions for four quantities: the orbital period, the pericenter distance, the rate of change of pericenter velocity, and the rate of change of orbital eccentricity. We can write these expansions to arbitrary PN order as
\begin{align}
\label{Torb-PN}
P^{\rm PN} &= P^{\rm N}\left(v_{p}, e_{t}\right) \left[1 + \vec{P}(e_{t}, \eta; v_{p}) \cdot \vec{X}(v_{p})\right]\,,
\\
\label{vp-PN}
r_{p}^{\rm PN} &= r_{p}^{\rm N}\left(v_{p}, e_{t}\right) \left[1 + \vec{R}(e_{t}, \eta; v_{p}) \cdot \vec{X}(v_{p})\right]\,,
\\
\label{vpdot-PN}
\langle \dot{v}_{p}^{\rm PN} \rangle &= \langle \dot{v}_{p}^{\rm N} \rangle \left(v_{p}, e_{t}\right) \left[1 + \vec{V}(e_{t}, \eta; v_{p}) \cdot \vec{X}(v_{p})\right]\,,
\\
\label{edot-PN}
\langle \dot{e}_{t}^{\rm PN} \rangle &= \langle \dot{e}_{t}^{\rm N} \rangle \left(v_{p}, e_{t}\right) \left[1 + \vec{E}(e_{t}, \eta; v_{p}) \cdot \vec{X}(v_{p})\right]\,,
\end{align}
with the Newtonian order quantities
\begin{align}
\label{Porb-N}
P^{\rm N} &= \frac{2 \pi M}{v_{p}^{3}} \left(\frac{1+e_{t}}{1-e_{t}}\right)^{3/2}\,,
\\
\label{rp-N}
r_{p}^{\rm N} &= \frac{M (1+e_{t})}{v_{p}^{2}}\,,
\\
\label{vp-dot-N}
\langle \dot{v}_{p}^{\rm N} \rangle &= \frac{32}{5} \frac{\eta}{M} v_{p}^{9} \frac{\left(1 - e_{t}\right)^{3/2}}{\left(1 + e_{t}\right)^{15/2}} V_{\rm N}(e_{t})\,,
\\
\label{e-dot-N}
\langle \dot{e}_{t}^{\rm N} \rangle &= -\frac{304}{15} e_{t} \frac{\eta}{M} v_{p}^{8} \frac{\left(1 - e_{t}\right)^{3/2}}{\left(1 + e_{t}\right)^{13/2}} \left(1 + \frac{121}{304} e_{t}^{2}\right)\,,
\\
V_{\rm N}(e) &= 1 - \frac{13}{6} e_{t} + \frac{7}{8} e_{t}^{2} - \frac{37}{96} e_{t}^{3}\,.
\end{align}
We refer to the vector $\vec{X}$ as the PN \emph{state vector}, which depends on the PN expansion parameter. In our case, the PN expansion parameter is $v_{p}$ and the components of $\vec{X}$ are simply $X_{k} = v_{p}^{k}$. Furthermore, we refer to the vector fields $(\vec{P}, \vec{R}, \vec{V}, \vec{E})$ as PN \emph{amplitude vectors}, which are functions of the orbital eccentricity and the symmetric mass ratio. In Eq.~\eqref{Torb-PN}-\eqref{edot-PN}, we have chosen to include a $v_{p}$ label in the amplitude vectors to remind us that the functional form of its components depends on the parameter one expands about, i.e.~if we had chosen to work with the $x$ PN expansion parameter instead of $v_{p}$, then the eccentricity and symmetric mass ratio dependence of the PN amplitude vectors would be different. The dot products between the state vectors and the amplitude vectors take the simple form
\begin{equation}
\vec{A}(e_{t}, \eta; v_{p}) \cdot \vec{X}(v_{p}) = \sum_{k=2}^{\infty} A_{k}(e_{t}, \eta; v_{p}) v_{p}^{k}
\end{equation}
where $\vec{A} \in (\vec{P}, \vec{R}, \vec{V}, \vec{E})$. We recognize the above expression as the summation of PN corrections to the associated quantity. The summation index $k$ acts as the PN order of each term and starts at $k=2$ corresponding to the corrections at 1PN order. The components of $\vec{A}$, specifically $A_{k}(e_{t},\eta)$, we then recognize as the coefficient of the $k/2$-PN order term.
The components of the amplitude vectors $(\vec{P}, \vec{R}, \vec{V}, \vec{E})$ can be easily computed from known PN quantities. As an example, consider the orbital period. This quantity can be written as a function of the reduced energy $\varepsilon$ and angular momentum $j$ through the equation $P=2 \pi / n$, where $n$ is the mean motion, given to 3PN order by Eq.~(348a) in~\cite{Blanchet:2013haa}. In turn, the reduced energy and angular momentum can also be written in terms of the pericenter velocity and eccentricity, $\varepsilon(v_{p}, e_{t})$ and $j(v_{p}, e_{t})$, which can be inserted into the expression $P(\varepsilon, j)$ and expanded about $v_{p} \ll 1$. The coefficients of each power of $v_{p}$ are then the components of the vector field $P_{k}$. We will provide expressions for these components at specific PN orders when we construct burst models at specific PN orders.
This should be very reminiscent of the computations commonly carried out in the ppE formalism. In the latter, four deformations characterized by eight parameters (4 \emph{amplitude parameters}, which are actually fields since they depend on the eccentricity of the binary, and 4 \emph{exponent parameters}) are used to completely specify the burst model in a general theory of gravity that reduces to GR in the low-velocity/weak-field limit. Instead of the four amplitude parameters of the ppE model, we now have four amplitude vectors fields $\lambda_{\rm PN}^{a} = (\vec{P}, \vec{R}, \vec{V}, \vec{E})$, which characterize the PN corrections to the Newtonian quantities. Also, the ppE exponent parameters have been replaced by the PN exponent $k$, which is a known number. Hence, instead of the eight ppE parameters, we need $4k$ PN functions when working to $k/2$-PN order. Each of the PN vector fields is a function of the parameters of the system, which we have written solely as functions of the eccentricity and the symmetric mass ratio. This will be true at 1PN order, but at higher PN order, the functions can depend on other physical parameters, such as the spins of compact objects, or the equation of state of supranuclear matter when at least one of the binary components is a NS.
The goal of this section will be to write the PN modifications to the Newtonian mappings in terms of the set of PN functions $\lambda_{{\rm PN},k}^{a} = (P_{k}, V_{k}, R_{k}, E_{k})$. We begin with the first ingredient, the orbital evolution, specified in our simplified formalism by Eqs.~\eqref{vp-burst} and~\eqref{e-burst}. In particular, we concentrate first on the evolution of the pericenter velocity. By exploiting the definition of orbit averaging, we are able to write the change in this quantity as $P \; \langle \dot{v}_{p} \rangle$, which is exactly the second term in Eq.~\eqref{vp-burst}. Hence, to obtain the velocity mapping, we simply have to multiply Eqs.~\eqref{Torb-PN} and~\eqref{vpdot-PN} together and expand in $v_{p}$. It is not difficult to see that our expansion is a product of two sums that is equivalent to a double sum of the form
\begin{align}
\left(\vec{P} \cdot \vec{X}\right) \left(\vec{V} \cdot \vec{X}\right) &= \left(\sum_{k=2}^{\infty} P_{k} \; v_{p}^{k}\right) \left(\sum_{k=2}^{\infty} V_{k} \; v_{p}^{k}\right)
\nonumber \\
&= \sum_{k=2}^{\infty} \sum_{j=2}^{k-2}
P_{k-j} \; V_{j} \; v_{p}^{k}\,,
\nonumber \\
&= \left(\vec{P} \circ \vec{V}\right) \cdot \vec{X}\,,
\end{align}
where we have used the definition of the Cauchy product to rewrite the product of the sums as the discrete convolution of two series. When $k-2 < 2$, the convolution is exactly zero. Using this result, we write the velocity mapping as
\begin{align}
\label{vp-map-PN}
v_{p,i} &= v_{p,i-1} \left\{1 + \frac{\pi}{5} \eta v_{p, i-1}^{5} {\cal{V}}_{\rm N}\left(\delta e_{t,i-1}\right) \left[1 + \vec{{\cal{V}}}\left(\delta e_{t,i-1}, \eta; v_{p}\right) \cdot \vec{X}(v_{p,i-1})\right]\right\}
\end{align}
where the Newtonian term in this expression is
\begin{align}
\label{VN}
V_{\rm N}(\delta e_{t}) &= - \frac{65}{96} + \frac{151}{96} \delta e_{t} - \frac{9}{32} \delta e_{t}^{2} + \frac{37}{96} \delta e_{t}^{3}\,,
\\
{\cal{V}}_{\rm N}(\delta e_{t}) &= \frac{V_{\rm N}(\delta e_{t})}{\left(1 - \frac{1}{2} \delta e_{t}\right)^{6}}
\nonumber \\
&= - \frac{65}{96} - \frac{11}{24} \delta e_{t} + {\cal{O}}\left(\delta e_{t}^{2}\right)
\end{align}
and the new amplitude vector $\vec{\cal{V}}$ is
\begin{align}
\label{calV}
\vec{{\cal{V}}}(\delta e_{t}, \eta; v_{p}) &= \vec{V}(\delta e_{t}, \eta; v_{p}) + \vec{P}(\delta e_{t}, \eta; v_{p}) + \vec{P}(\delta e_{t}, \eta; v_{p}) \circ \vec{V}(\delta e_{t}, \eta; v_{p})\,,
\end{align}
which should be expanded about $\delta e_{t} \ll 1$ by Assumption II. In the above expression, the pericenter velocity is decreasing from one orbit to the next for highly elliptic orbits at Newtonian order. We refer to this behavior as \textit{pericenter braking}, which will be explored in more detail in Section~\ref{brake}.
We may follow the same procedure for the eccentricity mapping to find
\begin{align}
\label{e-map-PN}
\delta e_{t,i} &= \delta e_{t,i-1} + \frac{85 \pi}{48} \eta v_{p,i-1}^{5} {\cal{D}}_{\rm N}(\delta e_{t,i-1}) \left[1 + \vec{\cal{D}}(\delta e_{t,i-1}, \eta; v_{p}) \cdot \vec{X}(v_{p,i-1})\right]\,,
\end{align}
with the Newtonian function
\begin{align}
{\cal{D}}_{\rm N}(\delta e) &= \frac{(1 - \delta e_{t}) \left(1 - \frac{242}{425} \delta e_{t} + \frac{121}{425} \delta e_{t}^{2}\right)}{\left(1 - \frac{1}{2} \delta e_{t}\right)^{5}}\,,
\nonumber \\
&= 1 + \frac{791}{850} \delta e_{t} + {\cal{O}}\left(\delta e_{t}^{2}\right)\,,
\end{align}
and the amplitude vector
\begin{align}
\label{calD}
\vec{{\cal{D}}}(\delta e_{t}, \eta; v_{p}) &= \vec{E}(\delta e_{t}, \eta; v_{p}) + \vec{P}(\delta e_{t}, \eta; v_{p}) + \vec{P}(\delta e_{t}, \eta; v_{p}) \circ \vec{E}(\delta e_{t}, \eta; v_{p})\,.
\end{align}
We thus find that the PN amplitude vectors $(\vec{\cal{V}}, \vec{\cal{D}})$ can be expressed in terms of the known PN amplitude vectors $(\vec{P}, \vec{V}, \vec{E})$. The above expressions are purely generic within the PN formalism, allowing them to be applied at any PN order.
Now, let us consider the second ingredient of the PN burst model: the centroid mapping. The GW time is given in Eq.~\eqref{tau-GW}, while the pericenter distance is given in Eq.~\eqref{vp-PN}. We thus have that the GW time at arbitrary PN order is
\begin{equation}
\tau_{\rm GW} = \frac{M \left(2 - \delta e_{t}\right)}{v_{p}^{3}} \left[1 + \vec{R}(\delta e_{t}, \eta; v_{p}) \cdot \vec{X}(v_{p})\right]\,,
\end{equation}
and the frequency mapping between boxes is
\begin{align}
\label{f-map-PN}
f_{i} &= \frac{\left[v_{p,i} \left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]^{3}}{2 \pi M \left[2 - \delta e_{t,i} \left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]} \left\{1 + \vec{R}^{(-1)}\left[\delta e_{t,i}(r_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}\right] \cdot
\vec{X} \left[v_{p,i}\left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]\right\}\,,
\end{align}
where the functionals $v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})$ and $\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1})$ are given in Eqs.~\eqref{vp-map-PN} and~\eqref{e-map-PN}, respectively. The components of the amplitude vectors $\vec{R}^{(-1)}$ are defined recursively in~\ref{recursion}. The time mapping can trivially be constructed from Eq.~\eqref{Torb-PN} via
\begin{align}
\label{t-map-PN}
t_{i} &= t_{i-1} + \frac{2 \pi M}{\left[v_{p,i} \left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]^{3}} \frac{\left[2 - \delta e_{t,i}\left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]^{3/2}}{\left[\delta e_{t,i} \left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]^{3/2}}
\nonumber\\
&\times \left\{1 + \vec{P}\left[\delta e_{t,i}\left(v_{p,i-1}, \delta e_{t,i-1}\right), \eta; v_{p}\right] \cdot \vec{X}\left[v_{p,i} \left(v_{p,i-1}, \delta e_{t,i-1}\right)\right] \right\}\,,
\end{align}
which completes our calculation of the centroid mapping.
We have here {\emph{not}} inserted Eqs.~\eqref{vp-map-PN} and~\eqref{e-map-PN} into Eqs.~\eqref{t-map-PN} and~\eqref{f-map-PN}, and re-expanded about the pericenter velocity being small to determine the time and frequency of the bursts. The reason for this is that such an expansion results in a significant loss of accuracy compared to numerical evolutions. This results from the behavior of the orbital period in the burst model which behaves as
\begin{equation}
\frac{1}{\delta e_{t,i}^{3/2}} \sim \frac{1}{\left(\delta e_{t,i-1} + A \; v_{p,i-1}^{5}\right)^{3/2}}\,,
\end{equation}
where $A$ is a constant. Since both $\delta e_{t,i-1}$ and $v_{p,i-1}$ are assumed to be simultaneously but independently small, expanding such a function about only one of them would impose an assumption on their ratio that is not justified.
Finally, we consider the volume mapping of the bursts. Once again, we treat the bursts as boxes in time and frequency with widths defined by Eqs.~\eqref{delta-t} and~\eqref{delta-f}. Hence we simply have to evaluate these expressions within our PN formalism at $(v_{p,i}, e_{t,i})$, thus obtaining
\begin{align}
\delta t_{i} &= \frac{\xi_{t} M \left[2 - \delta e_{t,i}\left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]}{\left[v_{p,i}\left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]^{3}} \left\{1 + \vec{R}\left[\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}\right] \cdot \vec{X}\left[v_{p,i}\left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]\right\}\,,
\end{align}
\begin{align}
\delta f_{i} &= \frac{\xi_{f} \left[v_{p,i} \left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]^{3}}{2 \pi M \left[2 - \delta e_{t,i} \left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]} \left\{1 + \vec{R}^{(-1)}\left[\delta e_{t,i}(r_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}\right] \cdot \vec{X}\left[v_{p,i}\left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]\right\}\,.
\end{align}
Not surprisingly, we see that one set of corrections, specifically the PN corrections to the pericenter distance, are the PN corrections to the frequency and box size mappings. This is exactly like in the ppE burst formalism, where one ppE deformation (with two parameters: $\beta_{\rm ppE}, \bar{b}_{\rm ppE}$) characterized these mappings.
\subsection{Example PN Burst Models}
\label{examples}
We have applied the fully general formalism of Section~\ref{simp} to PN theory, developing a burst model at generic PN order. This arbitrary order model is characterized by four amplitude vector fields $\lambda^{a}_{\rm PN burst} = \left(\vec{P}, \vec{R}, \vec{\cal{V}}, \vec{\cal{D}}\right)$, which can easily be constructed from the PN corrections to the orbital period, pericenter velocity, and rate of change of pericenter velocity and orbital eccentricity. These amplitude vectors are dependent on the set of GR parameters that characterize the system, namely $\lambda^{a}_{{\mbox{\tiny GR}}} = \left(\delta e_{t}, \eta, ...\right)$. We will now apply this formalism to generate a few example burst models at specific PN orders. There are multiple coordinate systems used to calculate PN quantities. Two that are typically used within the literature are the ADM and modified harmonic coordinates. We will choose to work within the ADM coordinates. The expressions in modified harmonic coordinates can easily be obtained through the appropriate coordinate transformations, which are given for example in Eq.~(7.11) in~\cite{Arun:2007sg}.
\subsubsection{Burst Model at 1PN Order}
We begin by calculating the burst model to 1PN order. Recall that the model has three ingredients: the orbit evolution, the centroid mapping, and the volume mapping. We begin with the orbit evolution, which in our generic order model is given by Eqs.~\eqref{vp-map-PN} and~\eqref{e-map-PN}. There are no 0.5PN order corrections to any of the quantities considered here, so the state vector has only one component, specifically
\begin{equation}
\vec{X} = (v_{p}^{2})\,.
\end{equation}
To achieve a burst model at 1PN order, we simply have to compute the 1PN functions $({\cal{V}}_{2}, {\cal{D}}_{2})$. The functions $({\cal{V}}_{k}, {\cal{D}}_{k})$ are given in general by Eq.~\eqref{calV} and~\eqref{calD}, respectively. Setting $k=2$, these functions become
\begin{align}
{\cal{V}}_{2}(e_{t}, \eta; v_{p}) &= V_{2}(e_{t}, \eta; v_{p}) + P_{2}(e_{t}, \eta; v_{p})\,,
\\
{\cal{D}}_{2}(\delta e_{t}, \eta; v_{p}) &= E_{2}(e_{t}, \eta; v_{p}) + P_{2}(e_{t}, \eta; v_{p})\,.
\end{align}
where $(V_{2}, E_{2}, P_{2})$ are given in~\ref{fields}. Working to ${\cal{O}}(\delta e_{t})$, the orbit evolution becomes
\begin{align}
\label{vp-map-1PN}
\frac{\left(v_{p,i} - v_{p,i-1}\right)_{\rm 1PN}}{\left(v_{p,i} - v_{p,i-1}\right)_{\rm N}} &= 1 + {\cal{V}}_{2}(\delta e_{t,i-1}, \eta; v_{p}) v_{p,i-1}^{2} + {\cal{O}}\left(v_{p,i-1}^{3}\right)
\\
\label{et-map-1PN}
\frac{\left(\delta e_{t,i} - \delta e_{t,i-1}\right)_{\rm 1PN}}{\left(\delta e_{t,i} - \delta e_{t,i-1}\right)_{\rm N}} &= 1 + {\cal{D}}_{2}(\delta e_{t,i-1}, \eta; v_{p}) v_{p,i-1}^{2} + {\cal{O}}\left(v_{p,i-1}^{3}\right)
\end{align}
with
\begin{align}
\label{vp-map-N}
\left(v_{p,i} - v_{p,i-1}\right)_{\rm N} &= - \frac{13 \pi}{96} \eta v_{p, i-1}^{6} \left[1 + \frac{44}{65} \delta e_{t,i-1} + {\cal{O}}\left(\delta e_{t,i-1}^{2}\right) \right]
\\
\label{et-map-N}
\left(\delta e_{t,i} - \delta e_{t,i-1}\right)_{\rm N} &= \frac{85 \pi}{48} \eta v_{p,i-1}^{5} \left[1 + \frac{791}{850} \delta e_{t,i-1}+ {\cal{O}}\left(\delta e_{t,i-1}^{2}\right) \right]
\\
\label{calV-2}
{\cal{V}}_{2}(\delta e_{t,i-1}, \eta; v_{p}) &= -\frac{251}{104} \eta + \frac{8321}{2080} + \delta e_{t,i-1} \left(\frac{14541}{6760} \eta - \frac{98519}{135200}\right)+ {\cal{O}}(\delta e_{t,i-1}^{2})\,,
\\
\label{calD-2}
{\cal{D}}_{2}(\delta e_{t,i-1}, \eta; v_{p}) &= -\frac{4017}{680} \eta + \frac{4773}{800} + \delta e_{t,i-1} \left(\frac{225393}{144500} \eta - \frac{602109}{340000}\right) + {\cal{O}}(\delta e_{t,i-1}^{2})\,.
\end{align}
Let us now consider the centroid mapping. The evolution of the time centroid of the bursts is trivially given by the orbital period, so to 1PN order
\begin{align}
\frac{\left(t_{i} - t_{i-1}\right)_{\rm 1PN}}{\left(t_{i} - t_{i-1}\right)_{\rm N}} &= 1 + P_{2}\left[\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}\right] \left[v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})\right]^{2} + {\cal{O}}\left(v_{p,i}^{4}\right)\,,
\end{align}
with
\begin{align}
\left(t_{i} - t_{i-1}\right)_{\rm N} &= P^{\rm N}\left(v_{p,i}, \delta e_{t,i}\right)
\nonumber \\
&= \frac{2 \pi M}{\left[v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})\right]^{3}} \frac{\left[2 - \delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1})\right]^{3/2}}{\left[\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1})\right]^{3/2}}\,,
\\
P_{2}(\delta e_{t,i}, \eta; v_{p}) &= \frac{3}{2} \eta - \frac{3}{4} + \delta e_{t,i} \left(-\frac{5}{8} \eta + \frac{3}{4}\right) + {\cal{O}}(\delta e_{t,i}^{2})\,,
\end{align}
where $v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})$ and $\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1})$ are given by Eqs.~\eqref{vp-map-1PN} and~\eqref{et-map-1PN}, respectively.
We now move onto the frequency centroid mapping, which is characterized by the functions $R_{k}^{(-1)}$. Using the recursion method is~\ref{recursion}, $R_{2}^{(-1)} = - R_{2}$, and the frequency centroid mapping becomes
\begin{align}
\frac{f_{i}^{\rm PN}}{f_{i}^{\rm N}} &= 1 - R_{2}\left[\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}\right] \left[v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})\right]^{2} + {\cal{O}}(v_{p,i}^{4})
\end{align}
with
\begin{align}
f_{i}^{\rm N} &= \frac{\left[v_{p,i} \left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]^{3}}{2 \pi M \left[2 - \delta e_{t,i} \left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]}
\\
R_{2}(\delta e_{t,i}, \eta; v_{p}) &= \frac{7}{4} \eta - \frac{5}{2} - \frac{5}{8} \eta \delta e_{t,i} + {\cal{O}}(\delta e_{t,i}^{2})\,.
\end{align}
Finally, we focus on the volume mapping, which is trivially given by the same corrections as the frequency centroid mapping:
\begin{align}
\frac{\delta t_{i}^{\rm 1PN}}{\delta t_{i}^{\rm N}} &= 1 + R_{2}\left[\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}\right] \left[v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})\right]^{2} + {\cal{O}}(v_{p,i}^{4})\,,
\\
\frac{\delta f_{i}^{\rm 1PN}}{\delta f_{i}^{\rm N}} &= 1 - R_{2}\left[\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}\right] \left[v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})\right]^{2} + {\cal{O}}(v_{p,i}^{4})
\end{align}
where we have defined
\begin{align}
\delta t_{i}^{\rm N} &= \frac{\xi_{t} M \left[2 - \delta e_{t,i}\left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]}{\left[v_{p,i}\left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]^{3}}
\\
\delta f_{i}^{\rm N} &= \frac{\xi_{f} \left[v_{p,i} \left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]^{3}}{2 \pi M \left[2 - \delta e_{t,i} \left(v_{p,i-1}, \delta e_{t,i-1}\right)\right]}
\end{align}
This completes the burst model at 1PN order.
\subsubsection{Burst Model at 2PN Order}
Let us now calculate the burst model to 2PN order. The state vector has three components corresponding to 1PN, 1.5PN, and 2PN orders, specifically
\begin{equation}
\vec{X} = (v_{p}^{2}, v_{p}^{3}, v_{p}^{4})\,.
\end{equation}
We begin by computing the orbital evolution in the burst model. To 2PN order, the pericenter velocity and eccentricity mappings become
\begin{align}
\label{vp-map-2PN}
\frac{(v_{p,i} - v_{p,i-1})_{\rm 2PN}}{(v_{p,i} - v_{p,i-1})_{\rm N}} &= \frac{(v_{p,i} - v_{p,i-1})_{\rm 1PN}}{(v_{p,i} - v_{p,i-1})_{\rm N}} + {\cal{V}}_{3}(\delta e_{t,i-1}, \eta; v_{p}) v_{p,i-1}^{3} + {\cal{V}}_{4}(\delta e_{t,i-1}, \eta; v_{p}) v_{p,i-1}^{4}
\nonumber \\
&+ {\cal{O}}(v_{p,i-1}^{5})
\\
\label{et-map-2PN}
\frac{(\delta e_{t,i} - \delta e_{t,i-1})_{\rm 2PN}}{(\delta e_{t,i} - \delta e_{t,i-1})_{\rm N}} &= \frac{(\delta e_{t,i} - \delta e_{t,i-1})_{\rm 1PN}}{(\delta e_{t,i} - \delta e_{t,i-1})_{\rm N}} + {\cal{D}}_{3}(\delta e_{t,i-1}, \eta; v_{p}) v_{p,i-1}^{3} + {\cal{D}}_{4}(\delta e_{t,i-1}, \eta; v_{p}) v_{p,i-1}^{4}
\nonumber \\
&+ {\cal{O}}(v_{p,i-1}^{5})
\end{align}
The Newtonian and 1PN order mappings do not change from the 1PN order model, and they are given in Eqs.~\eqref{vp-map-N}-\eqref{et-map-N} and Eqs.~\eqref{vp-map-1PN}-\eqref{et-map-1PN}, respectively. Generally, the 1.5PN order and 2PN order components of the amplitude fields are given by
\begin{align}
{\cal{V}}_{3}(e_{t}, \eta; v_{p}) &= V_{3}(e_{t}, \eta; v_{p})
\\
{\cal{D}}_{3}(e_{t}, \eta; v_{p}) &= E_{3}(e_{t}, \eta; v_{p})
\\
{\cal{V}}_{4}(e_{t}, \eta; v_{p}) &= V_{4}(e_{t}, \eta; v_{p}) + P_{4}(e_{t}, \eta; v_{p}) + V_{2}(e_{t}, \eta; v_{p}) P_{2}(e_{t}, \eta; v_{p})
\\
{\cal{D}}_{4}(e_{t}, \eta; v_{p}) &= E_{4}(e_{t}, \eta; v_{p}) + P_{4}(e_{t}, \eta; v_{p}) + E_{2}(e_{t}, \eta; v_{p}) P_{2}(e_{t}, \eta; v_{p})
\end{align}
Using the results of~\ref{fields}, we obtain
\begin{align}
{\cal{V}}_{3}(\delta e_{t,i-1}, \eta; v_{p}) &= \frac{3712 \sqrt{3}}{585} + \frac{100864 \sqrt{3}}{12675} \delta e_{t,i-1} + {\cal{O}}(\delta e_{t,i-1}^{2})\,,
\\
{\cal{D}}_{3}(\delta e_{t,i-1}, \eta; v_{p}) &= \frac{10624\sqrt{3}}{3825} + \frac{1098176 \sqrt{3}}{541875} \delta e_{t,i-1} + {\cal{O}}(\delta e_{t,i-1}^{2})\,,
\\
{\cal{V}}_{4}(\delta e_{t,i-1}, \eta; v_{p}) &= \frac{119432023}{6289920} - \frac{1213031}{49920}\eta - \frac{169}{128}\eta^{2} + \delta e_{t,i-1} \left(\frac{29330909}{204422400} + \frac{816679}{202800} \eta
\right.
\nonumber \\
&\left.
- \frac{68571}{4160} \eta^{2}\right) + {\cal{O}}(\delta e_{t,i-1}^{3/2})\,,
\\
{\cal{D}}_{4}(\delta e_{t,i-1}, \eta; v_{p}) &= \frac{130397759}{4569600} - \frac{5863719}{108800}\eta + \frac{284687}{10880}\eta^{2} + \delta e_{t,i-1}^{1/2} \left(\frac{45 \sqrt{2}}{32} - \frac{9 \sqrt{2}}{16} \eta\right)
\nonumber \\
&+ \delta e_{t,i-1} \left(-\frac{26000488883}{2913120000} + \frac{887490277}{46240000}\eta - \frac{16138299}{1156000}\eta^{2}\right) + {\cal{O}}(\delta e_{t,i-1}^{3/2})\,,
\end{align}
where we have used the results of~\cite{Loutrel:2016cdw} to evaluate the tail enhancement factors.
Next, let us consider the time centroid mapping, which at 2PN order is
\begin{align}
\frac{(t_{i} - t_{i-1})_{\rm 2PN}}{(t_{i} - t_{i-1})_{\rm N}} &= \frac{(t_{i} - t_{i-1})_{\rm 1PN}}{(t_{i} - t_{i-1})_{\rm N}} + P_{4}[\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}] v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})^{4} + {\cal{O}}(v_{p,i}^{5})\,,
\end{align}
where the mappings $v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})$ and $\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1})$ are now given by Eqs.~\eqref{vp-map-2PN} and~\eqref{et-map-2PN}, respectively. There is no 1.5PN order correction to these expressions, since $P_{3} = 0$. This is a result of the fact that the orbital period comes from the conservative orbital dynamics, and is thus symmetric under time reversal. Once again, the Newtonian time centroid mapping and 1PN order correction do not change from the 1PN order burst model. The 2PN order correction is characterized solely by $P_{4}(e_{t}, \eta; v_{p})$, which is given in~\ref{fields}. Expanding about $\delta e_{t} \ll 1$, we obtain
\begin{align}
P_{4}(\delta e_{t,i}, \eta; v_{p}) &= - \frac{225}{64} + \frac{237}{64} \eta - \frac{39}{32} \eta^{2} + \delta e_{t,i}^{1/2} \left(\frac{45 \sqrt{2}}{32} - \frac{9 \sqrt{2}}{16} \eta\right)
\nonumber\\
&+ \delta e_{t,i} \left(-\frac{135}{64} + \frac{115}{64} \eta + \frac{7}{16} \eta^{2}\right) + {\cal{O}}(\delta e_{t,i}^{3/2})\,.
\end{align}
Finally, consider the frequency centroid and box size mappings, which at 2PN order are
\begin{align}
\frac{f_{i}^{\rm 2PN}}{f_{i}^{\rm N}} &= \frac{f_{i}^{\rm 1PN}}{f_{i}^{\rm N}} + R_{4}^{(-1)}[\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}] v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})^{4} + {\cal{O}}(v_{p,i}^{5})\,,
\\
\frac{\delta t_{i}^{\rm 2PN}}{\delta t_{i}^{\rm N}} &= \frac{\delta t_{i}^{\rm 1PN}}{\delta t_{i}^{\rm N}} + R_{4}[\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}] v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})^{4} + {\cal{O}}(v_{p,i}^{5})\,,
\\
\frac{\delta f_{i}^{\rm 2PN}}{\delta f_{i}^{\rm N}} &= \frac{\delta f_{i}^{\rm 1PN}}{\delta f_{i}^{\rm N}} + R_{4}^{(-1)}[\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}] v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})^{4} + {\cal{O}}(v_{p,i}^{5})\,,
\end{align}
The Newtonian and 1PN order terms are the same as those in the 1PN order burst model. Using the results of~\ref{recursion}, the field $R_{4}^{(-1)}$ is in general given by
\begin{equation}
R_{4}^{(-1)}(e_{t}, \eta; v_{p}) = -R_{4}(e_{t}, \eta; v_{p}) + R_{2}(e_{t}, \eta; v_{p})^{2}\,.
\end{equation}
Applying the expressions for $(R_{2}, R_{4})$ from~\ref{fields}, we obtain
\begin{align}
R_{4}(\delta e_{t,i}, \eta; v_{p}) &= -\frac{47}{16} + \frac{49}{16} \eta - \frac{17}{16} \eta^{2} + \delta e_{t,i} \left(-\frac{133}{64} + \frac{155}{64} \eta + \frac{13}{32} \eta^{2}\right) + {\cal{O}}(\delta e_{t,i}^{2})\,,
\\
R_{4}^{(-1)}(\delta e_{t,i}, \eta, v_{p}) &= \frac{147}{16} - \frac{189}{16} \eta + \frac{33}{8} \eta^{2} + \delta e_{t,i} \left(\frac{133}{64} + \frac{45}{64} \eta - \frac{83}{32} \eta^{2}\right) + {\cal{O}}(\delta e_{t,i}^{2})\,.
\end{align}
This completes the burst model at 2PN order.
\subsubsection{Burst Model at 3PN Order}
\label{burst-3PN}
Let us now extend the burst model to the current limits of our understanding of eccentric binaries within PN theory, i.e.~to 3PN order. The state vector will now extend to $v_{p}^{6}$, specifically
\begin{equation}
\vec{X} = (v_{p}^{2}, v_{p}^{3}, v_{p}^{4}, v_{p}^{5}, v_{p}^{6})\,.
\end{equation}
At 3PN order, the orbital evolution equations become
\begin{align}
\label{vp-map-3PN}
\frac{(v_{p,i} - v_{p,i-1})_{\rm 3PN}}{(v_{p,i} - v_{p, i-1})_{\rm N}} &= \frac{(v_{p,i} - v_{p,i-1})_{\rm 2PN}}{(v_{p,i} - v_{p, i-1})_{\rm N}}+ {\cal{V}}_{5}(\delta e_{t,i-1}, \eta; v_{p}) v_{p,i-1}^{5} + {\cal{V}}_{6}(\delta e_{t,i-1}, \eta; v_{p}) v_{p,i-1}^{6}
\nonumber \\
&+ {\cal{O}}\left(v_{p,i-1}^{7}\right)\,,
\\
\label{e-map-3PN}
\frac{(\delta e_{t,i} - \delta e_{t,i-1})_{\rm 3PN}}{(\delta e_{t,i} - \delta e_{t,i-1})_{\rm N}} &= \frac{(\delta e_{t,i} - \delta e_{t,i-1})_{\rm 2PN}}{(\delta e_{t,i} - \delta e_{t,i-1})_{\rm N}} + {\cal{D}}_{5}(\delta e_{t,i-1}, \eta; v_{p}) v_{p,i-1}^{5} + {\cal{D}}_{6}(\delta e_{t,i-1}, \eta; v_{p}) v_{p,i-1}^{6}
\nonumber \\
&+ {\cal{O}}\left(v_{p,i-1}^{7}\right)\,.
\end{align}
The new functions $[{\cal{V}}_{5}, {\cal{V}}_{6}]$ and $[{\cal{D}}_{5}, {\cal{D}}_{6}]$ give the coefficients of the 2.5PN and 3PN order corrections of the orbital evolutions. In terms of the components of the amplitude vector fields $[\vec{V}, \vec{E}, \vec{P}]$, they are given by
\begin{align}
{\cal{V}}_{5}(e_{t}, \eta; v_{p}) &= V_{5}(e_{t}, \eta; v_{p}) + V_{3}(e_{t}, \eta; v_{p}) P_{2}(e_{t}, \eta; v_{p})\,,
\\
{\cal{D}}_{5}(e_{t}, \eta; v_{p}) &= E_{5}(e_{t}, \eta; v_{p}) + E_{3}(e_{t}, \eta; v_{p}) P_{2}(e_{t}, \eta; v_{p})\,,
\\
{\cal{V}}_{6}(e_{t}, \eta; v_{p}) &= V_{6}(e_{t}, \eta; v_{p}) + P_{6}(e_{t}, \eta; v_{p}) + V_{2}(e_{t}, \eta; v_{p}) P_{4}(e_{t}, \eta; v_{p})
\nonumber \\
&+ V_{4}(e_{t}, \eta; v_{p}) P_{2}(e_{t}, \eta; v_{p})\,,
\\
{\cal{D}}_{6}(e_{t}, \eta; v_{p}) &= E_{6}(e_{t}, \eta; v_{p}) + P_{6}(e_{t}, \eta; v_{p}) + E_{2}(e_{t}, \eta; v_{p}) P_{4}(e_{t}, \eta; v_{p})
\nonumber \\
&+ E_{4}(e_{t}, \eta; v_{p}) P_{2}(e_{t}, \eta; v_{p})\,,
\end{align}
where we have used the fact that $P_{3}(e_{t}, \eta; v_{p}) = 0 = P_{5}(e_{t}, \eta; v_{p})$. Using the results of~\ref{fields}, we find for the 2.5PN order functions
\begin{align}
{\cal{V}}_{5}(\delta e_{t,i-1}, \eta; v_{p}) &= - \frac{128272 \sqrt{3}}{4095} - \frac{4832 \sqrt{3}}{117} \eta + \nu_{0} \pi + \frac{1748 \sqrt{6}}{65} \delta e_{t,i-1}^{1/2}
\nonumber \\
&+ \delta e_{t,i-1} \left(- \frac{30641528 \sqrt{3}}{266175} - \frac{1183488 \sqrt{3}}{29575} \eta + \nu_{1} \pi \right) + {\cal{O}}(\delta e_{t,i-1}^{3/2})\,,
\\
{\cal{D}}_{5}(\delta e_{t,i-1}, \eta; v_{p}) &= \frac{13072 \sqrt{3}}{8925} - \frac{241664 \sqrt{3}}{8925} \eta + \rho_{0} \pi + \frac{4544 \sqrt{6}}{425} \delta e_{t,i-1}^{1/2}
\nonumber \\
&+ \delta e_{t,i-1} \left(- \frac{81300056 \sqrt{3}}{3793125} - \frac{52270208 \sqrt{3}}{3793125} \eta + \rho_{1} \pi \right) + {\cal{O}}(\delta e_{t,i-1}^{3/2})\,,
\end{align}
where we have used the results of~\cite{Loutrel:2016cdw} and neglected the 2.5PN memory terms. The constants $[\nu_{0}, \nu_{1}, \rho_{0}, \rho_{1}]$ depend on the coefficients of the Pad\'{e} approximants created for the 2.5PN order tail enhancement factors $[\psi(e_{t}), \tilde{\psi}(e_{t})]$ in~\cite{Loutrel:2016cdw}. The exact rational form of the coefficients are too lengthy to provide here. We simply give their numeric values, which are
\begin{align}
\nu_{0} &= 34.82829720\,,
\qquad
\nu_{1} = -38.97374189\,,
\\
\rho_{0} &= 11.90237615\,,
\qquad
\rho_{1} = -36.89484102\,,
\end{align}
For the 3PN order functions, we find
\begin{align}
{\cal{V}}_{6}(\delta e_{t,i-1}, \eta; v_{p}) &= \frac{48102359171}{402554880} + \frac{385 \pi^{2}}{128} + \frac{1177 {\rm ln}(2)}{64} + \frac{1177 {\rm ln}(3)}{256}
\nonumber \\
&- \left(\frac{508363\pi^{2}}{266240} + \frac{80844193}{430080}\right) \eta + \frac{3379743}{53248} \eta^{2} + \frac{543189}{13312} \eta^{3} - \frac{1177}{256} {\rm ln}(v_{p,i-1}^{2})
\nonumber \\
&+ \delta e_{t,i-1} \left[\frac{1690426235921}{26166067200} + \frac{48839 \pi^{2}}{8320} + \frac{746539 {\rm ln}(2)}{20800} + \frac{746539 {\rm ln}(3)}{83200}
\right.
\nonumber \\
&\left.
- \left(\frac{112925149}{83865600} + \frac{80684263 \pi^{2}}{17305600}\right) \eta - \frac{753359873}{10383360} \eta^{2} + \frac{61283003}{865280} \eta^{3}
\right.
\nonumber \\
&\left.
-\frac{746539}{83200} {\rm ln}(v_{p,i-1}^{2})\right] + {\cal{O}}\left(\delta e_{t,i-1}^{3/2}\right)\,,
\\
{\cal{D}}_{6}(\delta e_{t,i-1}, \eta; v_{p}) &= \frac{7318191053}{51609600} + \frac{10549 \pi^{2}}{10880} + \frac{161249 {\rm ln}(2)}{27200} + \frac{161249 {\rm ln}(3)}{108800}
\nonumber \\
&- \left(\frac{8119255961}{21934080} + \frac{155561 \pi^{2}}{1740800}\right) \eta + \frac{11789862391}{36556800} \eta^{2} - \frac{9152141}{87040} \eta^{3} - \frac{161249}{108800} {\rm ln}(v_{p,i-1}^{2})
\nonumber \\
&+ \delta e_{t,i-1}^{1/2} \left[\frac{64557 \sqrt{2}}{5120} - \left(\frac{4120619 \sqrt{2}}{217600} - \frac{123 \pi^{2} \sqrt{2}}{4096}\right) \eta + \frac{13437 \sqrt{2}}{2720} \eta^{2} \right]
\nonumber \\
&+ \delta e_{t,i-1} \left[- \frac{11487739123}{552960000} + \frac{5805723 \pi^{2}}{4624000} + \frac{88744623 {\rm ln}(2)}{11560000} + \frac{88744623 {\rm ln}(3)}{46240000}
\right.
\nonumber \\
&\left.
+ \left(\frac{639985247281}{6991488000} + \frac{13567261 \pi^{2}}{92480000} \right)\eta - \frac{689800811001}{5178880000} \eta^{2} + \frac{1557091039}{18496000} \eta^{3}
\right.
\nonumber \\
&\left.
- \frac{88744623}{46240000} {\rm ln}(v_{p,i-1}^{2}) \right] + {\cal{O}}(\delta e_{t,i-1}^{3/2})\,.
\end{align}
This completes the orbital evolution to 3PN order.
The time centroid mapping at 3PN order becomes
\begin{align}
\label{t-map-3PN}
\frac{(t_{i} - t_{i-1})_{\rm 3PN}}{(t_{i} - t_{i-1})_{\rm N}} &= \frac{(t_{i} - t_{i-1})_{\rm 2PN}}{(t_{i} - t_{i-1})_{\rm N}} + P_{6}[\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}] v_{p,i}^{6} + {\cal{O}}(v_{p,i}^{7})\,,
\end{align}
where once again there is no 2.5PN order corrections since the orbital period comes from the conservative orbital dynamics. The 3PN order function $P_{6}(e_{t}, \eta; v_{p})$ is given in~\ref{fields}. Expanding about $\delta e_{t} \ll 1$, we obtain
\begin{align}
P_{6}(\delta e_{t,i}, \eta; v_{p}) &= - \frac{2821}{256} + \left(\frac{2123}{128} + \frac{3 \pi^{2}}{16} \right) \eta - \frac{1377}{128} \eta^{2} + \frac{73}{32} \eta^{3} + \delta e_{t,i}^{1/2} \left[ \frac{405 \sqrt{2}}{128}
\right.
\nonumber \\
&\left.
- \left(\frac{607 \sqrt{2}}{128} - \frac{123 \pi^{2} \sqrt{2}}{4096}\right) \eta + \frac{99 \sqrt{2}}{128} \eta^{2} \right] + \delta e_{t,i} \left[- \frac{213}{32} + \left(\frac{591}{128} + \frac{885 \pi^{2}}{2048} \right) \eta
\right.
\nonumber \\
&\left.- \frac{1117}{512} \eta^{2} - \frac{399}{256} \eta^{3}\right] + {\cal{O}}(\delta e_{t,i}^{3/2})\,.
\end{align}
Finally, the frequency and box widths mappings at 3PN order are
\begin{align}
\label{f-map-3PN}
\frac{f_{i}^{\rm 3PN}}{f_{i}^{\rm N}} &= \frac{f_{i}^{\rm 2PN}}{f_{i}^{\rm N}} + R_{6}^{(-1)}[\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}] v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})^{6} + {\cal{O}}(v_{p,i}^{7})\,,
\\
\frac{\delta t_{i}^{\rm 3PN}}{\delta t_{i}^{\rm N}} &= \frac{\delta t_{i}^{\rm 2PN}}{\delta t_{i}^{\rm N}} + R_{6}[\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}] v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})^{6} + {\cal{O}}(v_{p,i}^{7})\,,
\\
\frac{\delta f_{i}^{\rm 3PN}}{\delta f_{i}^{\rm N}} &= \frac{\delta f_{i}^{\rm 2PN}}{\delta f_{i}^{\rm N}} + R_{6}^{(-1)}[\delta e_{t,i}(v_{p,i-1}, \delta e_{t,i-1}), \eta; v_{p}] v_{p,i}(v_{p,i-1}, \delta e_{t,i-1})^{6} + {\cal{O}}(v_{p,i}^{7})\,,
\end{align}
where the functions $R_{6}^{(-1)}(e_{t}, \eta; v_{p})$ is
\begin{align}
R_{6}^{(-1)}(e_{t}, \eta; v_{p}) &= -R_{6}(e_{t}, \eta; v_{p}) + 2 R_{2}(e_{t}, \eta; v_{p}) R_{4}(e_{t}, \eta; v_{p}) - R_{2}(e_{t}, \eta; v_{p})^{3}\,.
\end{align}
Using the results in~\ref{fields}, we obtain
\begin{align}
R_{6}(\delta e_{t,i}, \eta; v_{p}) &= -\frac{305}{32} + \left(\frac{3131}{192} + \frac{11 \pi^{2}}{128} \right) \eta - \frac{19}{2} \eta^{2} + \frac{67}{32} \eta^{3} + \delta e_{t,i} \left[-\frac{829}{128} + \left(\frac{2245}{256} + \frac{97 \pi^{2}}{512}\right) \eta
\right.
\nonumber \\
&\left.
- \frac{333}{128} \eta^{2} - \frac{47}{32} \eta^{3} \right] + {\cal{O}}(\delta e_{t,i}^{3/2})\,,
\\
R_{6}^{(-1)}(\delta e_{t,i}, \eta; v_{p}) &= \frac{1275}{32} - \left(\frac{14345}{192} + \frac{11 \pi^{2}}{128} \right) \eta + \frac{97}{2} \eta^{2} - \frac{715}{64}\eta^{3} + \delta e_{t,i} \left[\frac{2159}{128} - \left(\frac{3267}{256} + \frac{97 \pi^{2}}{512}\right) \eta
\right.
\nonumber \\
&\left.
- \frac{179}{16} \eta^{2} + \frac{1275}{128} \eta^{3}\right] + {\cal{O}}(\delta e_{t,i}^{3/2})\,.
\end{align}
This completes the burst model at 3PN order.
\section{Properties of the PN Burst Model}
With the burst model complete to 3PN order, we complete this paper with some results that describe properties of the model. We begin by discussing the accuracy of the burst model when compared to numerical evolutions of the PN radiation reaction equations. Finally, we discuss a previously unreported phenomenon associated with the evolution of the pericenter velocity under radiation reaction.
\subsection{Accuracy of the Burst Model}
\label{accuracy}
The burst model is meant to be an accurate representation of GW bursts emitted by highly elliptic binaries in nature. Further, since this model is designed to be used as a prior in data analysis for detecting such systems, it is paramount that we characterize the accuracy of the model. The ideal test of such an analytic model would be to compare the time of arrival and frequency of eccentric bursts from a numerical relativity simulation to the those from the burst model. However, there are currently no accurate numerical relativity waveforms for the highly elliptic systems considered here. Even the second best comparison, the same as above but with accurate PN waveforms, is also currently inapplicable due to the lack of such waveforms. With the two most ideal tests out of reach, we are left with comparing the burst model to the orbital evolution of binary systems (instead of their associated waveforms) under PN radiation reaction. Such a comparison allows us to gauge the accuracy of the approximations used to construct the burst model, as well as estimate the typical error we can expect when comparing to physically accurate waveform models.
We begin by describing the method through which we obtain the numerical evolution. Ideally, the equations we would want to numerically evolve are $\langle \dot{e}_{t} \rangle (v_{p}, e_{t})$ and $\langle \dot{v}_{p} \rangle (v_{p}, e_{t})$. However, as we will explain in Sec.~\ref{brake}, there is always a point $(v_{p}, e_{t})$ where $\langle \dot{v}_{p} \rangle =0$ during the inspiral, which numerical routines will have difficulty integrating past. An alternative approach is to use a parameterization of the equations that does not present this behavior, e.g.~$\langle \dot{e}_{t} \rangle (x, e_{t})$ and $\langle \dot{x} \rangle (x,e_{t})$. The expression for $\langle \dot{e}_{t} \rangle (x, e_{t})$ to 3PN order, neglecting memory contributions, is provided in Eqs.~(6.18)-(6.19),~(6.22), and~(6.25) in~\cite{Arun:2009mc}. To obtain the expression for $\langle \dot{x} \rangle (x, e_{t})$ to 3PN order, we follow the method detailed in~\ref{fields} for $\langle \dot{v}_{p} \rangle$, which we summarize here. We begin by obtaining an expression for $x(\epsilon, j)$ by inverting Eq.~(6.5) in~\cite{Arun:2009mc}. We then take a time derivative and apply the chain rule, using the 3PN order expressions for the energy flux~\cite{Arun:2007sg} and the angular momentum flux~\cite{Arun:2009mc}. We expand the resulting expression in $x$ to obtain $\langle \dot{x} \rangle (x, e_{t})$.
For our numerical evolutions, we integrate the equations $\langle \dot{x} \rangle (x, e_{t})$ and $\langle \dot{e}_{t} \rangle (x, e_{t})$ including all of the instantaneous and tail contributions to 3PN order. For the tail enhancement factors, we use the analytic expressions provided in~\cite{Loutrel:2016cdw}. The initial conditions for the evolutions are set to guarantee the initial eccentricity is $e_{t,0} = 0.9$ and the initial GW frequency is $f_{\rm GW,0} = 10 {\rm Hz}$, i.e.~we use these initial conditions to solve for the initial value of $v_{p}$ using $f_{\rm GW}(v_{p},e_{t})$, which is provided in~\ref{fields}. We then use the expression $v_{p}(x, et)$, which is obtained from the 3PN extension of Eq.~\eqref{eq:vp} with Eq.~(7.10) in~\cite{Arun:2007sg}, to obtain the initial value of $x$. For the three systems we study, a $(1.4,1.4) M_{\odot}$ NSNS binary, a $(1.4,10) M_{\odot}$ NSBH binary, and a $(10,10) M_{\odot}$ BHBH binary, the initial conditions are listed in Table~\ref{ic}. With the initial conditions set, we numerically integrate the equations using the $\textit{NDSolve}$ routine in $\texttt{Mathematica}$ until we reach the time when
\begin{equation}
x_{f} = \frac{1}{2} \left(\frac{1 - e_{t}^{2}}{3 + e_{t}}\right)\,,
\end{equation}
which denotes the maximum value of $x$ for which test particle orbits are stable around a Schwarzschild black hole, i.e.~we require that $p > 2M (3 + e_{t})$, where $p$ is the semi-latus rectum of the orbit and we have used the Newtonian relation $x = (M/p) (1 - e_{t}^2)$. Beyond this point, we consider the inspiral to be formally over and to use the burst model one would have to extend it to include merger and ringdown.
{\renewcommand{\arraystretch}{1.2}
\begin{table}
\centering
\begin{centering}
\begin{tabular}{cccccc}
\hline
\hline
\noalign{\smallskip}
$ \text{System} $ & $ m_{1}[M_{\odot}] $ & $ m_{2}[M_{\odot}] $ & $ e_{t,0} $ & $ x_{0} $ & $ 1/x_{0} $ \\
\hline
\noalign{\smallskip}
$ \text{NSNS} $ & $ 1.4 $ & $ 1.4 $ & $ 0.9 $ & $ 7.35 \times 10^{-4} $ & $ 1360 $ \\
$ \text{NSBH} $ & $ 1.4 $ & $ 10 $ & $ 0.9 $ & $ 1.85 \times 10^{-3} $ & $ 541 $ \\
$ \text{BHBH} $ & $ 10 $ & $ 10 $ & $ 0.9 $ & $ 2.67 \times 10^{-3} $ & $ 375 $ \\
\noalign{\smallskip}
\hline
\hline
\end{tabular}
\end{centering}
\caption{\label{ic} Initial values of the PN expansion parameter $x$ for the set of compact binary systems studied. The values are obtained by requiring the initial GW frequency to be $10 {\rm Hz}$. The final column provides an estimate of the semi-major axis of the binary, since $a_{r} = M/x + {\cal{O}}(1)$ in PN theory.}
\end{table}}
For the comparison to the burst model, we use $x(t)$ and $e_{t}(t)$ to construct the pericenter velocity as a function of time $v_{p}(t)$, which we then use with the results of~\ref{fields} to obtain the orbital period and GW frequency as a function of time, specifically $P(t)$ and $f_{\rm GW}(t)$. To compute the values of these in the burst model to 3PN order, we start the model with the same initial conditions used for the numerical evolution. Once $(v_{p,0}, e_{t,0})$ are specified, all future $(v_{p,i}, e_{t,i})$ are determined from Eqs.~\eqref{vp-map-3PN}-\eqref{e-map-3PN}. From here, the orbital period and the GW frequency are determined in the burst model from Eqs.~\eqref{t-map-3PN} and~\eqref{f-map-3PN}.
Figure~\ref{compare} shows the orbital period and GW frequency as functions of time in the burst model and the numerical evolution, as well as the relative error between the two. The relative error increases as time increases, but typically the error remains below $1\%$ for the first one hundred bursts. The reason the error increases is twofold. First, the eccentricity decreases as the binary inspirals due to the loss of energy and angular momentum by GW emission. The burst model uses an expansion about $\delta e_{t} \ll 1$, and it is thus most accurate in this regime. This error can be improved by going to higher order in $\delta e_{t}$ within the burst model if one wishes.
The second reason for the increasing error is that as the binary inspirals, the GW power becomes smeared over more of the orbit. As a result, the binary's evolution resembles less a set of discrete steps. The burst model, which is only valid when $\delta e_{t} \ll 1$, hinges on the osculating behavior of highly eccentric orbits. This error is more difficult to control, but one way of improving it would be to match the evolution in the burst model to an evolution when the eccentricity is small. However, in this paper, we are only interested in highly elliptic orbits where this matching is unnecessary. Regardless, as the figure shows, the error between the burst model and the numerical evolution is sufficiently small that we can begin to test the burst model in idealized data analysis scenarios.
\begin{figure*}[ht]
\includegraphics[clip=true,scale=0.34]{TorbComparePN.eps}
\includegraphics[clip=true,scale=0.34]{fGWComparePN.eps}
\caption{\label{compare} Top panel: Comparison of the orbital period $P$ and GW frequency $f_{\rm GW}$ relative to their initial values as functions of time (in units of the initial orbital period) in the burst model (circles) and the numerical evolution (lines). The values of the pericenter velocity and time eccentricity next to each line provide the values during the 100th burst. The labels on the top axis give the value of the time eccentricity for the corresponding time for the NSBH binary. Bottom panel: Relative error between the burst model and the numerical evolutions for the orbital period and GW frequency.}
\end{figure*}
\subsection{Pericenter Braking}
\label{brake}
Let us begin by recalling that within our generic PN burst model, the change of pericenter velocity to Newtonian order is given by
\begin{align}
\label{vp-dot}
\langle \dot{v}_{p} \rangle &= \frac{32}{5} \frac{\eta}{M} v_{p}^{9} \frac{\left(1 - e_{t}\right)^{3/2}}{\left(1 + e_{t}\right)^{15/2}} V_{\rm N}(e_{t}) + {\cal{O}}(v_{p}^{11})
\end{align}
where the function $V_{\rm N}(e_{t})$ is
\begin{equation}
V_{\rm N}(e_{t}) = 1 - \frac{13}{6} e_{t} + \frac{7}{8} e_{t}^{2} - \frac{37}{96} e_{t}^{3}\,.
\end{equation}
Notice from Eq.~\eqref{VN}, which provides $V_{\rm N}(\delta e_{t})$, that to first order in $\delta e_{t}$, the above expression is negative and $v_{p}$ is thus decreasing. This seems counterintuitive considering what we know about quasi-circular binaries, i.e.~as the orbital separation $r$ decreases, the orbital velocity $v$ increases, since $v$ and $r$ are inversely related by Kepler's third law.
This behavior becomes more confusing when we consider the apocenter velocity $v_{a}$. Just as we can calculate $\langle \dot{v}_{p} \rangle$ using the method detailed in~\ref{fields}, we may also compute $\langle \dot{v}_{a} \rangle$. Following this method, and working to Newtonian order, we have
\begin{equation}
\label{va-dot}
\langle \dot{v}_{a} \rangle = \frac{32}{5} \frac{\eta}{M} v_{p}^{9} \frac{\left(1 - e_{t}\right)^{3/2}}{\left(1 + e_{t}\right)^{15/2}} V_{\rm N}(-e_{t}) + {\cal{O}}\left(v_{p}^{11}\right)\,.
\end{equation}
Notice that this expression depends on $V_{\rm N}(-e_{t})$, which is always positive. The apocenter velocity is thus always increasing as the binary inspirals, just as we would expect from quasi-circular binaries. As a result, the pericenter and apocenter velocities have very different behavior depending on the eccentricity of the system.
Let us try to understand this counter-intuitive behavior. The function $V_{\rm N}(e_{t})$ is a third order polynomial in eccentricity with an oscillating sign and with the coefficient of the ${\cal{O}}(e_{t})$ term greater than unity. This means that there will be a critical point $e_{t,{\rm crit}}$ where the function is zero, $\langle \dot{v}_{p} \rangle(e_{t,{\rm crit}}) = 0$, and due to the aforementioned behavior of the coefficients $e_{t,{\rm crit}} < 1$. Let us solve for this critical point. To Newtonian order we find
\begin{equation}
e_{t,{\rm crit}} = e_{t,{\rm crit}}^{\rm N} \equiv \frac{28}{37} - \frac{2}{111} \sigma + \frac{2672}{37} \sigma^{-1}\,,
\end{equation}
where we have defined
\begin{equation}
\sigma = \left(67770 + 222 \sqrt{1399593}\right)^{1/3}\,.
\end{equation}
The Newtonian expression for the critical eccentricity evaluates to $e_{\rm crit}^{\rm N} \approx 0.5557306$. Such a critical point also exists at 1PN order, except that now it is a function of the mass ratio and the pericenter velocity:
\begin{equation}
\label{etcrit-1PN}
e_{t,{\rm crit}} = e_{t,{\rm crit}}^{\rm N} + e_{t,{\rm crit}}^{\rm 1PN}(\eta) \; v_{p}^{2}\,,
\end{equation}
where we have defined
\begin{align}
\label{etcrit-1PN-exact}
e_{t,{\rm crit}}^{\rm 1PN}(\eta) &= \frac{1}{\sigma \left(2 \sigma^2-195 \sigma-8016\right)^2 \left(\sigma^4-4008 \sigma^2 + 16064064\right)} \left[\frac{10112970188463538176}{37} \eta
\right.
\nonumber \\
&\left.
-\frac{2775205875922795266048}{9583} + \left(\frac{149499372172271616}{37} \eta - \frac{31076515350417788928}{9583}\right) \sigma
\right.
\nonumber \\
&\left.
+ \left(-\frac{1150598736488448}{37} \eta-\frac{11994925296964608}{9583}\right) \sigma^2
\right.
\nonumber \\
&\left.
+ \left(\frac{62455763578752}{37} \eta - \frac{30800598698771712}{9583}\right) \sigma^3
\right.
\nonumber \\
&\left.
+ \left(-\frac{383487115344}{37} \eta + \frac{351826743539100}{9583}\right) \sigma^4
\right.
\nonumber \\
& \left.
+ \left(-\frac{46179281106}{37} \eta + \frac{35571374119917}{19166}\right) \sigma^5 + \left(\frac{95680418}{37} \eta - \frac{175562247275}{19166}\right) \sigma^6
\right.
\nonumber \\
&\left.
+ \left(\frac{3887918}{37} \eta - \frac{1917360308}{9583}\right) \sigma^7 + \left(\frac{53612}{111} \eta + \frac{558902}{28749}\right) \sigma^8 + \left(\frac{1738}{111} \eta - \frac{361279}{28749}\right) \sigma^9
\right.
\nonumber \\
&\left.
+ \left(-\frac{88}{333} \eta + \frac{24149}{86247}\right) \sigma^{10}\right]
\end{align}
This function evaluates to $e_{t,\rm crit}^{\rm 1PN} \approx 0.5557306 - (0.06536872 \eta + 0.3457145) v_{p}^{2}$. The overall effect of the 1PN term is to decrease the value of the Newtonian critical point, but there is no value of $v_{p} < 1$ or $\eta \in (0,1/4)$ for which $e_{t,\rm crit} = 0$ at 1PN order.
\begin{figure}[ht]
\centering
\includegraphics[clip=true,scale=0.65]{NStream.eps}
\caption{\label{braking-N} Plot of the streamlines of ($10^{4} \langle \dot{v}_{p} \rangle, 10^{3} \langle \dot{e}_{t} \rangle$) at Newtonian order. The arrows on the streamlines only indicate the direction of the flow, not the magnitude. The red dashed line displays the value of the critical eccentricity where $\langle \dot{v}_{p} \rangle = 0$ at Newtonian order. Above the critical eccentricity, the streamlines point to the left as shown in the burst model, while below, they point to the right, as is expected for quasi circular binaries.}
\end{figure}
\begin{figure*}[ht]
\includegraphics[clip=true,scale=0.65]{PNStream.eps}
\includegraphics[clip=true,scale=0.66]{PNStreamZoom.eps}
\caption{\label{braking-PN} Left: Plot of the streamlines of ($10^{4} \langle \dot{v}_{p} \rangle, 10^{3} \langle \dot{e}_{t} \rangle$) at 1PN order. The dotted line displays the value of the critical eccentricity where $\langle \dot{v}_{p} \rangle = 0$ at 1PN order, as determined numerically, while the dashed line is the same result at Newtonian order. The solid line displays the analytic result of the critical eccentricity given in Eqs.~\eqref{etcrit-1PN}-\eqref{etcrit-1PN-exact}. Right: A zoom in of the plot on the left for the region $v_{p} = (0, 0.35)$.}
\end{figure*}
Why does this behavior occur physically? The answer to this question lies in circularization. As the binary inspirals, energy and angular momentum are radiated away in such a way that the orbital eccentricity decreases, making the binary more and more circular. For quasicircular binaries, $v_{a} = v_{p} + {\cal{O}}(e_{t})$, but for highly-elliptic binaries, $v_{p} \gg v_{a}$. As a highly-elliptical binary inspirals, $v_{p}$ and $v_{a}$ will approach the same value since the eccentricity approaches zero. However, if the eccentricity is above the critical value, the two velocities will not approach the same value if they are both increasing initially. Instead, circularization causes pericenter to \emph{brake} when the eccentricity is above $e_{t,{\rm crit}}$, so that $v_{p}$ can approach $v_{a}$. In fact, one can easily show from Eqs.~\eqref{vp-dot} and~\eqref{va-dot}, that $v_{a}$ and $v_{p}$ obey the following conservation law at Newtonian order:
\begin{equation}
V_{\rm N}(-e_{t}) \langle \dot{v}_{p} \rangle - V_{\rm N}(e_{t}) \langle \dot{v}_{a} \rangle = 0\,.
\end{equation}
To further display this behavior, we plot the streamlines of $(\langle \dot{v}_{p} \rangle, \langle \dot{e}_{t} \rangle)$\footnote{We have rescaled the values of $\langle \dot{v}_{p} \rangle$ and $\langle \dot{e}_{t} \rangle$ in these plots to exemplify the behavior of the streamlines. This does not changes the results of this section.} at Newtonian order in Fig~\ref{braking-N} and at 1PN order in Fig.~\ref{braking-PN}. Notice that in both plots, systems with values of $(v_{p}, e_{t})$ above the critical value of the eccentricity, which is represented by the dashed line in Fig.~\ref{braking-N} and the dotted line in Fig.~\ref{braking-PN}, display the pericenter braking behavior that appears in the burst model. On the other hand, the pericenter velocity for systems below the critical eccentricity is always increasing.
This pericenter braking behavior is not a property of the burst model \emph{per se}, but rather it is inherited from the PN radiation-reaction equations. One may worry that this pericenter braking behavior may disappear if treating the problem exactly (for example, through a numerical treatment). The right panel of Fig.~\ref{braking-PN}, however, shows a zoom of the streamlines at small velocities, where we see that the braking behavior persists. We thus conclude that it is unlikely that pericenter braking is a artifact of the PN expansion.
Figure~\eqref{braking-PN} also allows us to compare the critical eccentricity computed at Newtonian order, at 1PN order and numerically. The latter is obtained by solving the 1PN expression for $\langle \dot{v}_{p}(e_{t}) \rangle = 0$ to find $e_{t,\rm crit}$. As expected, the numerical inversion disagrees with the its 1PN expansion at high velocities. We notice, however, that the 1PN expression is closer to the numerical inversion than the Newtonian expression is. If the numerical inversion is correct, then this implies the 1PN expansion of $e_{t,\rm crit}$ given in Eqs.~\eqref{etcrit-1PN}-\eqref{etcrit-1PN-exact} has a larger regime of validity than its Newtonian counterpart.
Finally, it is important to note that while the pericenter velocity has this unique behavior, the GW frequency and the PN parameter $x$ are both monotonically increasing, and the time eccentricity is monotonically decreasing, throughout the inspiral of the binary. In the circular case, there is a one-to-one mapping between the orbital velocity and the GW frequency, and since the orbital velocity is a monotonic function, so is the frequency. For generic eccentric inspirals, the frequency depends on both the pericenter velocity (or alternatively $x$) and the time eccentricity in such a way that it is also monotonic.
\section{Discussion}
\label{discussion}
We have constructed a generic PN order burst model. This model is characterized by four amplitude vector fields $(\vec{P}, \vec{R}, \vec{\cal{V}}, \vec{\cal{D}})$, which depend on the orbital period, pericenter distance, and rates of change of pericenter velocity and orbital eccentricity, respectively. While these quantities are not typically reported within the literature, they can be easily calculated from the quantities that are. Thus, the formalism presented here provides a formulaic means of generating burst models to any PN order. We have then applied this formalism to calculate the burst model out to the current limit to which we can compute PN quantities for eccentric binaries, i.e.~3PN order.
One direction of future research is to relax some of the assumptions used to develop this formalism. For example, we have approximated the compact objects as non-spinning point particles, which is appropriate if we are considering non-spinning BHs. However, BHs in the universe are generally considered to be spinning, while on the other hand, NSs are not well approximated by point particles. NSs will typically have small spins, however the inclusion of finite size effects and tidal perturbations would be necessary to effectively model highly elliptic NS binaries. Further, if one of the binary components is a BH, then not all of the GW power travels to spatial infinity. Instead, some of the GWs travel through the horizon of the BH, increasing its mass and spin throughout the evolution of the binary. With these considerations, we can postulate that the generic PN formalism can be extended to include such effects by writing
\begin{equation}
\vec{A} = \vec{A}_{\rm PP} + \vec{A}_{\rm Spin} + \vec{A}_{\rm FS} + \vec{A}_{\rm H} + \vec{A}_{\rm ppE}\,,
\end{equation}
where $\vec{A} \in (\vec{P}, \vec{R}, \vec{\cal{V}}, \vec{\cal{D}})$. In the above, $\vec{A}_{\rm PP}$ represents the point particle terms, computed here to 3PN order, $\vec{A}_{\rm Spin}$ are the corrections generated by the spins of the compact objects, $\vec{A}_{\rm FS}$ are generated by finite size effects of NSs, and $\vec{A}_{\rm H}$ incorporates the corrections from the GWs fluxes through BH horizons. The final term, $\vec{A}_{\rm ppE}$ represents corrections due to modified theories of gravity, which have already been considered in~\cite{Loutrel:2014vja}.
One important question to address in the future concerns the most appropriate equations one should use to obtain the numerical evolution of highly elliptic systems under radiation reaction. In this work, we have used the orbit averaged equations for $\langle \dot{e}_{t} \rangle$ and $\langle \dot{x} \rangle$. These equations are applicable when the GW emission is smeared over the entire orbit and changes to the orbital elements are small on the timescale of one orbit, as is the case in quasi-circular inspirals. However, for the highly elliptic binaries considered here, the GW emission is concentrated at pericenter passage, and changes to the orbital elements happen on timescales significantly shorter than the orbital period. The evolution of such binaries will resemble a set of discrete steps from one orbit to the next. Furthermore, it can be shown that when expressed in terms of variables that are finite in the parabolic limit, the orbit averaged fluxes of energy and angular momentum vanish for parabolic orbits. There is, of course, nothing special about the parabolic limit, and binaries on parabolic orbits will still emit GWs, which suggests a break down of the orbit-averaged formalism in this limit. Since the orbit averaged equations are currently used prolifically in the literature, it is important to determine how big of a deviation in observables is generated by considering evolutions with and without orbit-averaging in the radiation-reaction force, and what set of systems in $(f_{\rm GW}, e_{t})$ space are affected by this deviation. Such a study is currently underway~\cite{Loutrel-avg}.
Another avenue for future research is to consider how the 3PN order burst model aids in detecting highly elliptic binaries. In such a study, one would inject a waveform generated by numerically evolving the binary under radiation reaction into a simulated LIGO data stream. One could then perform an analysis to study whether the prior, specifically the burst model, is sufficient to achieve detection of such a signal given a particular noise model. One could also investigate the nature of posterior probability densities of recovered parameters and determine if such a search is accurate enough to perform parameter estimation on actual signals. With such a study completed, a follow up study could be conducted to investigate the search strategy's ability to estimate deviations from the current model, such as those from modified theories of gravity and to constrain the coupling constants of such theories. Such studies will be crucial for understanding our ability to detect and perform important astrophysics with eccentric GW signals.
\section*{Acknowledgements}
We would like to thank Frans Pretorius for several useful discussions. N.Y. acknowledges support from the NSF CAREER Grant PHY-1250636. N. L. acknowledges support from the NSF EAPSI Fellowship Award No. 1614203.
|
1,314,259,996,475 | arxiv |
\section*{Spin freezing parameter}
The spin freezing parameter $\bar q$ is defined by Eq. (2) in the main text, {\it i.e.\/},
\begin{align}
\bar{q}^2=\frac{1}{N^2}\sum_{i,j}\left[\braket{\bm{S}_i\cdot\bm{S}_j}^2\right]_J.
\label{eq:qbar}
\end{align}
This quantity takes a nonzero value in the spin ordered state, even including the spatially random one such as spin glasses, and vanishes in the paramagnetic state. Thus, it serves as the order parameter of the spin ordering or freezing. Below, we shall explain how this is the case.
Usually, the existence of the magnetic long-range order can be detected via the two-point spin correlation function $g({\bm r}_{ij})=\left[ \langle {\bf S}_i\cdot {\bm S}_j\rangle \right]_J$ by looking at its long-distance behavior, {\it i.e.\/}, whether $m^2\equiv \lim_{|r_{ij}|\rightarrow \infty} g({\bm r}_{ij})$ is zero or nonzero. In fully random systems, this quantity $m^2$ trivially vanishes after the configuration or sample average due to the sign cancellation of $\langle {\bf S}_i\cdot {\bm S}_j\rangle$, and does not work as an appropriate order parameter. This problem can be avoided simply by squaring $\langle {\bf S}_i\cdot {\bm S}_j\rangle$ before the configurational average to make it non-negative. Then, an appropriate two-point correlation function might be
\begin{align}
g^{(2)}({\bm r}_{ij})=\left[ \braket{\bm{S}_i\cdot\bm{S}_j}^2 \right]_J.
\label{eq:qbar}
\end{align}
This correlation function decays exponentially with a finite correlation length in the paramagnetic state, but tends to a nonzero value in the long-distance limit in the magnetically ordered state, even including the spatially random ordered state. Then, the long-distance limit of $g^{(2)}({\bm r}_{ij})$,
\begin{equation}
\bar q_{\infty}^2 = \lim_{|r_{ij}|\rightarrow \infty} g^{(2)}({\bm r}_{ij})
\end{equation}
yields an appropriate spin freezing parameter in the thermodynamic limit.
Equivalently, if one defines the finite-size spin freezing parameter by Eq. (\ref{eq:qbar}) above, it reduces in the thermodynamic limit $N\rightarrow \infty$ to the spin freezing parameter $\bar q_\infty$ defined above.
This spin freezing parameter can also be related to the so-called Edward-Anderson order parameter \cite{EdwardsAnderson} or the spin overlap well-known in spin-glass studies: see Ref. \cite{SGreview} for further details.
\section*{The Hams--de Raedt method}
The Hams-de Raedt method obtains the finite-temperature properties of the model with its Hamiltonian $\mathcal{H}$ based on the Taylor expansion of the Boltzmann factor $\exp(-\beta\mathcal{H})$ \cite{Hams-deRaedt,cTPQ}. The thermal average of the physical quantity $A$ can be represented as
\begin{align}
\braket{A}&=
\frac{ {\rm Tr}[\exp(-\beta\mathcal{H}/2) A \exp(-\beta\mathcal{H}/2)]} {{\rm Tr}[\exp(-\beta\mathcal{H}/2) \exp(-\beta\mathcal{H}/2)]} \nonumber
\\
&=\frac{ \overline{\Braket{0|\exp(-\beta\mathcal{H}/2) A \exp(-\beta\mathcal{H}/2)|0}}} {\overline{\Braket{0|\exp(-\beta\mathcal{H}/2) \exp(-\beta\mathcal{H}/2)|0}}} \nonumber
\\
&=\frac{ \overline{\Braket{0|(\sum_k (-\beta\mathcal{H}/2)^k/k!) A (\sum_{k'} (-\beta\mathcal{H}/2)^{k'}/k'!)|0}}} {\overline{\Braket{0|(\sum_k (-\beta\mathcal{H}/2)^k/k!) (\sum_{k'} (-\beta\mathcal{H}/2)^{k'}/k'!)|0}}},
\label{eq:HdR1}
\end{align}
where $k=0,1,2,\cdots$ is a non-negative integer and $\overline{\cdots}$ is the average over the random initial vectors $\ket{0}$. The term $(-\beta\mathcal{H}/2)^k/k!$ becomes very large at low temperatures. To control this divergent behavior, an appropriate normalization factor is introduced. Let $E_{min}$ the minimum (ground-state) energy eigenvalue, $E_{max}$ the maximum energy eigenvalue, and $E_w$ their difference $E_w=E_{max}-E_{min}$. Then, with the use of the convergence factor $h=(E_{max}-\mathcal{H})/E_w$, Eq. (\ref{eq:HdR1}) can be rewritten as
\begin{align}
\braket{A}=\frac{\overline{\sum_{k,k'} \Braket{k|A|k'}}}{\overline{\sum_{k,k'} \Braket{k|k'}}},
\label{eq:HdR2}
\end{align}
where the state $\ket{k}$ is given by
\begin{align}
\ket{k}=\exp(-\beta E_w/2) (\beta E_w/2)^k/k!\cdot h^k\ket{0}.
\label{eq:HdR3}
\end{align}
We compute the physical quantities based on Eqs. (\ref{eq:HdR2}) and (\ref{eq:HdR3}).
\begin{figure}[t]
\begin{tabular}{c}
\begin{minipage}{0.5\hsize}
\includegraphics[width=\hsize]{SSFhhl_del00N32}
\end{minipage}
\begin{minipage}{0.5\hsize}
\includegraphics[width=\hsize]{SSFhk0_del00N32}
\end{minipage}\\
\begin{minipage}{0.5\hsize}
\includegraphics[width=\hsize]{SSFhhl_del10N32}
\end{minipage}
\begin{minipage}{0.5\hsize}
\includegraphics[width=\hsize]{SSFhk0_del10N32}
\end{minipage}
\end{tabular}
\caption{\label{fig:SSF} (Color online)
Upper row: Static spin structure factor $S_q$ for the regular case
of $\Delta=0$ in the (a) $(h,h,l)$ and (b) $(h,k,0)$ plane.
Lower row: Static spin structure factor $S_q$ for the maximally random case
of $\Delta=1$ in the (c) $(h,h,l)$ and (d) $(h,k,0)$ plane.
}
\end{figure}
\section*{Static spin structure factor}
The computed ground-state spin structure factor $S_{\bm q}$ is defined by
\begin{align}
S_{\bm q}&=\frac{1}{N}\left[ \Braket{ |{\bm S}_{\bm q}|^2}\right]_J
\nonumber \\
&=\frac{1}{N} \left[ \sum_{i,j} \Braket{{\bm S}_i\cdot{\bm S}_j}
\cos{\left(\frac{2\pi}{a}{\bm q}\cdot\left({\bm r}_i-{\bm r}_j\right)\right)}
\right]_J,
\label{eq:SSF}
\end{align}
where ${\bm S}_{\bm q}=\sum_j {\bm S}_j e^{i\frac{2\pi}{a}{\bm q}\cdot{\bm r}_j}$ is the Fourier transform of the spin operator, ${\bm r}_j$ is the position vector at the site $j$, ${\bm q}$ is the wave vector, $a$ is the linear size of the cubic unit cell of the pyrochlore lattice, while $\langle \cdots \rangle$ and $[\cdots]_J$ represent the ground-state expectation value (or the thermal average at finite temperatures) and the configurational average over $J_{ij}$ realizations.
In Fig. \ref{fig:SSF}, the static spin structure factor for the regular case of $\Delta=0$ is shown in the upper row, while the one for the maximally random case of $\Delta=1$ is in the lower row.
The left side of Fig. \ref{fig:SSF} is the intensity plot in the $(h,h,l)$ plane, while the right side is in the $(h,k,0)$ plane.
As can be seen from these figures, there are no sharp peaks in any of Fig. \ref{fig:SSF}, indicating the absence of the magnetic long-range order.
The difference between the upper and the lower rows turns out to be relatively minor so that the introduction of randomness seems to give minor effects on the static spin structure factor.
This might partly be due to the finite-size effect, as the very coarse mesh size available here for $S_{\bm q}$ tends to mask the possible structure in the ${\bm q}$-space. In order to probe the possible difference between the regular and the random cases, much larger system sizes might be required. This is in contrast to the $\omega$-dependence of the dynamical structure factor $S_{\bm q}(\omega)$ where the finite-size effect tends to be more indirect and the difference between the regular and the random cases is much clearer, as shown in Fig. 2 of the main text.
In any case, a certain nontrivial structure in the ${\bm q}$-space is discernible for the static $S_{\bm q}$ even in the random case, suggesting that the underlying spin structure possesses spatial correlations extending at least to several lattice spacings. This actually seems consistent with the picture of the random-singlet state where modest length-scale of objects are expected as its basic ingredients. Recall that, in the random-singlet state, some of the singlets are formed between distant neighbors and even the resonance between distinct singlet-dimer coverings are expected, as discussed in the main text.
|
1,314,259,996,476 | arxiv | \section{Introduction}
According to QCD factorization theorem, the cross section for the inclusive production of a hadron $H$ with high transverse momentum ($p_T$) in a high-energy collision is dominated by the single parton fragmentation \cite{Collins:1989gx}, i.e.,
\begin{eqnarray}
d \sigma_{A+B \to H+X}(p_T)= && \sum_i d \hat{\sigma}_{A+B\to i+X}(p_T/z,\mu_F) \nonumber \\
&& \otimes D_{i\to H}(z,\mu_F)+{\cal O}(m_H^2/p_T^2),
\label{pqcdfact}
\end{eqnarray}
where $\otimes$ denotes a convolution in the momentum fraction $z$, the sum extends over all species of partons. $d \hat{\sigma}_{A+B \to i+X}$ indicates the partonic cross section that can be calculated in perturbation theory, while $D_{i\to H}$ indicates the fragmentation function for the parton $i$ into a hadron $H$. $\mu_F$ denotes the factorization scale which is introduced to separate the energy scales of the two parts.
The factorization formula (\ref{pqcdfact}) was first derived by Collins and Soper for light hadron production \cite{Collins:1981uw}. This factorization formula can be equally applied to the heavy quarkonium production. The proof of the factorization formula (\ref{pqcdfact}) for the quarkonium production was presented by Nayak, Qiu, and Sterman \cite{Nayak:2005rt}. The factorization formula (\ref{pqcdfact}) is called leading power (LP) factorization because it gives the LP contribution in the expansion in powers of $m_H/p_T$. The factorization formula for the next-to-leading power (NLP) correction was derived in Refs.\cite{Kang:2011zza,Kang:2011mg,Fleming:2012wy,Fleming:2013qu}, and the NLP contribution comes from the double-parton fragmentation.
Fragmentation functions play an important role in the calculation of the cross sections under the LP factorization. Unlike the fragmentation functions for the production of the light hadrons which are nonperturbative in nature, the fragmentation functions for the heavy quarkonium production can be calculated through the nonrelativistic QCD (NRQCD) factorization \cite{nrqcd}. Under NRQCD factorization, the fragmentation functions for a parton to fragment into a quarkonium can be written as
\begin{eqnarray}
D_{i\to H}(z,\mu_F)=\sum_n d_{i\to (Q\bar{Q})[n]}(z,\mu_F) \langle {\cal O}^H(n) \rangle,
\label{frag.nrqcd}
\end{eqnarray}
where $d_{i\to (Q\bar{Q})[n]}$ are short-distance coefficients (SDCs) which can be expanded as powers of $\alpha_s(m_Q)$, and $\langle {\cal O}^H(n) \rangle$ are long-distance matrix elements (LDMEs).
The fragmentation functions for quarkonia have been studied extensively. Most of the fragmentation functions for the S-wave and P-wave quarkonia are known up to $\alpha_s^2$ order \cite{Chang:1992bb, Braaten:1993jn, Braaten:1993mp, Braaten:1993rw, Braaten:1994kd, Braaten:1995cj, Chen:1993ii, Yuan:1994hn, Ma:1994zt, Ma:1995ci, Ma:1995vi, Cho:1994gb, Beneke:1995yb, Braaten:2000pc, Hao:2009fa,Sang:2009zz, Jia:2012qx, Bodwin:2014bia, Ma:2013yla, Ma:2015yka, Yang:2019gga}, and a few fragmentation functions for quarkonia were calculated up to $\alpha_s^3$ order \cite{Artoisenet:2014lpa, Sepahvand:2017gup, Artoisenet:2018dbs, Feng:2018ulg, Zhang:2018mlo, Zheng:2019dfk, Zheng:2019gnb, Feng:2017cjk}. Among these studies, the next-to-leading order (NLO) corrections to the fragmentation functions for $g \to Q\bar{Q}[^1S_0^{[1,8]}]$ have been calculated recently by three groups~\cite{Artoisenet:2014lpa, Artoisenet:2018dbs, Feng:2018ulg, Zhang:2018mlo}. The NLO fragmentation functions for $g \to Q\bar{Q}[^1S_0^{[1,8]}]$ are important in prediction of the production of the $\eta_{c,b}$ and the $h_{c,b}$ at the LHC. However, the fragmentation functions for a quark into the $\eta_{c,b}$ are only available up to $\alpha_s^2$ order. Those fragmentation functions are also important to the precision prediction of the $\eta_{c,b}$ production at the LHC. Moreover, for the production of the $\eta_{c,b}$ in $e^+e^-$ collisions, the quark fragmentation contribution is more important than the gluon fragmentation contribution due to the fact that the cross section for a quark is $\alpha_s^0$ order but the cross section for a gluon is $\alpha_s$ order. In this paper, we will calculate the fragmentation functions for a quark $q$ into the $\eta_Q$, where $Q=c,b$ but $q \neq Q$.
In the early calculations of fragmentation functions for doubly heavy mesons~\cite{Chang:1992bb, Chen:1993ii}, the fragmentation functions are determined through comparing the cross section calculated based on the NRQCD factorization with that calculated based on the factorization formula (\ref{pqcdfact}) for a process containing the doubly heavy meson being produced. In fact, the fragmentation functions can be defined through the matrix elements of nonlocal gauge-invariant operators \cite{Collins:1981uw}. The operator definition for the fragmentation functions was first applied to calculations of the fragmentation functions for doubly heavy mesons by Ma \cite{Ma:1994zt}. The calculations based on the operator definition are particularly convenient to extend to higher orders. Therefore, we will calculate the fragmentation functions based on the operator definition suggested by Collins and Soper.
The paper is organized as follows. Following the Introduction, in Sec.II, we present the definition and the analytical calculation for the fragmentation functions. In Sec.III, we present the numerical results for the fragmentation functions $D_{q \to\eta_c}(z,\mu_F)(q=u,d,s)$ and $D_{b \to \eta_c}(z,\mu_F)$, and apply the fragmentation functions to processes $Z \to \eta_c+q\bar{q}g(q=u,d,s)$ and $Z \to \eta_c+b\bar{b}g$. Section IV is reserved for a summary.
\section{The analytical calculation for the fragmentation functions}
\label{analy}
\subsection{The definition of fragmentation function}
The fragmentation functions are usually defined in the light-cone coordinate system. In this coordinate system, a d-dimensional vector $V$ is expressed as $V^{\mu}=(V^+,V^-,{\bf V}_{\perp})$, with $V^+=(V^0+V^{d-1})/\sqrt{2}$ and $V^-=(V^0-V^{d-1})/\sqrt{2}$. Then the product of two vectors is $V \cdot W= V^+ W^- + V^- W^+ - {\bf V}_{\perp} \cdot {\bf W}_{\perp}$. The gauge-invariant fragmentation function for a quark $q$ to fragment into a hadron $H$ is defined as \cite{Collins:1981uw}
\begin{widetext}
\begin{eqnarray}
D_{q\to H}(z)=&&\frac{z^{d-3}}{2\pi}\sum_{X} \int dx^- e^{-iP^+ x^-/z} \nonumber \\
&&\times \frac{1}{N_c} {\rm Tr}_{\rm color} \frac{1}{4} {\rm Tr}_{\rm Dirac} \left\lbrace \gamma^+ \langle 0 \vert \Psi(0)\bar{{\cal P}} {\rm exp}\left[ig_s \int_{0}^{\infty} dy^- A_a^+(0^+,y^-,{\bf 0}_\perp)t_a^T \right]\vert H(P^+,{\bf 0}_\perp)+X \rangle \right. \nonumber \\
&&\left. \times\langle H(P^+,{\bf 0}_\perp)+X\vert {\cal P} {\rm exp}\left[-ig_s \int_{x^-}^{\infty} dy^- A_a^+(0^+,y^-,{\bf 0}_\perp)t_a^T \right] \bar{\Psi}(x)\vert 0\rangle\right\rbrace,
\label{defrag1}
\end{eqnarray}
\end{widetext}
where $\Psi$ is the field of initial quark, $A_a^{\mu}$ is the gluon field, and $t_a(a=1 \cdots 8)$ are $SU(3)$-color matrices. The longitudinal momentum fraction is defined as $z \equiv P^+/K^+$, where $K$ is the momentum of the initial quark. The fragmentation function is defined in a reference frame in which the transverse momentum of the hadron $H$ vanishes. It is convenient to introduce a lightlike momentum whose expression is $n^{\mu}=(0,1,{\bf 0}_{\perp})$ in the reference frame where the definition of the fragmentation function carried out. Then, $z$ can be expressed as a Lorentz invariant, i.e., $z=P\cdot n/K \cdot n$. The Feynman rules can be derived from the definition (\ref{defrag1}) directly, and we have presented the Feynman rules in a previous paper \cite{Zheng:2019gnb}.
\subsection{The calculation of fragmentation function}
The definition (\ref{defrag1}) is gauge invariant. However, for the practical calculation, the gauge should be specified. We adopt the usual Feynman gauge throughout the paper. There are ultraviolet (UV) divergences in the calculation. To deal with the UV divergences, we adopt dimensional regularization with $d=4-2\epsilon$, then the UV divergences appear as the pole terms in $\epsilon$.
In the calculation, we first calculate the fragmentation function for an on-shell $Q\bar{Q}$ pair in $^1S_0^{[1]}$ state. Then the fragmentation function $D_{q \to \eta_Q}$ can be obtained from $D_{q \to (Q\bar{Q})[^1S_0^{[1]}]}$ through replacing the LDME $\langle {\cal O}^{(Q\bar{Q})[^1S_0^{[1]}]}(^1S_0^{[1]}) \rangle$ by $\langle {\cal O}^{\eta_Q}(^1S_0^{[1]}) \rangle$.
\begin{figure}[htbp]
\includegraphics[width=0.45\textwidth]{feyn}
\caption{The cut diagrams for the fragmentation function $D_{q \to (Q\bar{Q})[^1S_0^{[1]}]}$, where $q \neq Q$.
} \label{feyn}
\end{figure}
There are 16 cut diagrams for $q(K) \to (Q\bar{Q})[^1S_0^{[1]}](p_1)+g(p_2)+q(p_3)$ under the Feynman gauge, which can be collectively represented by four diagrams in Fig.\ref{feyn}. The squared amplitudes, corresponding to four diagrams in Fig.\ref{feyn}, can be written as
\begin{eqnarray}
{\cal A}_1=&& {\rm tr}\left[\slashed{n}\frac{-i}{\slashed{p_1}+\slashed{p_2}+\slashed{p_3}-m_q-i\epsilon}(ig_s\gamma_{\nu}t^b)(\slashed{p}_{3}+m_q)\right. \nonumber \\
&& \left. (-ig_s\gamma_{\mu}t^a)\frac{i}{\slashed{p_1}+\slashed{p_2}+\slashed{p_3}-m_q+i\epsilon}\right]X_{ab}^{\mu \nu},\\
{\cal A}_2=&& {\rm tr}\left[\slashed{n}\frac{-i}{\slashed{p_1}+\slashed{p_2}+\slashed{p_3}-m_q-i\epsilon}(ig_s\gamma_{\nu}t^b)(\slashed{p}_{3}+m_q)\right. \nonumber \\
&& \left. \frac{i}{(p_1+p_2)\cdot n+i\epsilon}(ig_s n_{\mu}t^a)\right]X_{ab}^{\mu \nu},\\
{\cal A}_3=&& {\rm tr}\left[\slashed{n}(-ig_s n_{\nu}t^b)\frac{-i}{(p_1+p_2)\cdot n-i\epsilon}(\slashed{p}_{3}+m_q)\right. \nonumber \\
&& \left. (-ig_s\gamma_{\mu}t^a)\frac{i}{\slashed{p_1}+\slashed{p_2}+\slashed{p_3}-m_q+i\epsilon}\right]X_{ab}^{\mu \nu},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_4=&& {\rm tr}\left[\slashed{n}(-ig_s n_{\nu}t^b)\frac{-i}{(p_1+p_2)\cdot n-i\epsilon}(\slashed{p}_{3}+m_q)\right. \nonumber \\
&& \left. \frac{i}{(p_1+p_2)\cdot n+i\epsilon}(ig_s n_{\mu}t^a)\right]X_{ab}^{\mu \nu},
\end{eqnarray}
where $\Pi_1$ is the spin-singlet projector
\begin{eqnarray}
\Pi_1=\frac{1}{(2\,m_Q)^{3/2}}(\slashed{p}_1/2-m_Q)\gamma_5(\slashed{p}_1/2+m_Q),
\end{eqnarray}
$\Lambda_1$ is the color-singlet projector
\begin{eqnarray}
\Lambda_1=\frac{\textbf{1}}{\sqrt{3}},
\end{eqnarray}
where $\textbf{1}$ is the unit matrix of the $SU(3)_c$ group. There is a common factor $X_{ab}^{\mu \nu}$ which arises from the annihilation of a virtual gluon into a $(Q\bar{Q})[^1S_0^{[1]}]$ pair and a real gluon. The factor $X_{ab}^{\mu \nu}$ can be expressed as
\begin{eqnarray}
X_{ab}^{\mu \nu}=-g_{\rho \sigma}J^{\mu \rho}_{ac}J^{\nu \sigma * }_{bc},
\end{eqnarray}
where
\begin{eqnarray}
J^{\mu \rho}_{ac}=&&{\rm tr} \left \{ \Pi_1 \Lambda_1 \left[(-ig_s\gamma_{\rho}t^c)\frac{i}{\slashed{p}_{1}/2+\slashed{p}_{2}-m_Q} (-ig_s\gamma_{\mu}t^a) \right. \right. \nonumber \\
&&\left. \left. +(-ig_s\gamma_{\mu}t^a) \frac{i}{-\slashed{p}_{1}/2-\slashed{p}_{2}-m_Q}(-ig_s\gamma_{\rho}t^c) \right] \right\}\nonumber \\
&& \cdot \frac{-i}{(p_1+p_2)^2+i \epsilon}.
\end{eqnarray}
We employ the package FeynCalc \cite{Mertig:1990an,Shtabovenko:2016sxi} to carry out the Dirac and color traces, and then the total squared amplitude (${\cal A}_{(1-4)} \equiv \sum_{i=1}^{4} {\cal A}_i$) can be written as
\begin{eqnarray}
{\cal A}_{(1-4)}=&& \frac{c_1(s_1,y,z)\, (p_3 \cdot \tilde{p})^2}{s_1^2 (s-m_q^2)^2}+\frac{c_2(s_1,y,z)\, p_1 \cdot p_3}{s_1 (s-m_q^2)^2}\nonumber \\
&& +\frac{c_3(s_1,y,z)\, p_2 \cdot p_3}{s_1 (s-m_q^2)^2}+\frac{c_4(s_1,y,z)}{(s-m_q^2)^2},
\end{eqnarray}
where
\begin{eqnarray}
&& s_1=(p_1+p_2)^2, ~~s=(p_1+p_2+p_3)^2,\nonumber \\
&& y=\frac{(p_1+p_2)\cdot n}{K\cdot n}, ~ \tilde{p}=p_1-\frac{z \,p_2}{(y-z)}.
\end{eqnarray}
The coefficients $c_i(s_1,y,z)$ can be easily extracted, and we do not list their expressions here.
The differential phase space for the fragmentation function $D_{q \to (Q\bar{Q})[^1S_0^{[1]}]}$ is\footnote{Here, we associate the scale factor $\mu^{4-d}$ with each dimensionally regulated integration in $d$ space-time dimensions. In our previous papers \cite{Zheng:2019dfk,Zheng:2019gnb}, this scale factor was put in the squared amplitudes.}
\begin{eqnarray}
d\phi_{3}(p_1,p_2,p_3)=&&2\pi \delta\left(K^+ - \sum_{i=1}^{3} p_i^+\right)\mu^{2(4-d)} \nonumber \\
&&\times \prod_{i=2,3}\frac{\theta(p_i^+)dp_i^+}{4 \pi p_i^+}\frac{ d^{d-2}\textbf{p}_{i\perp}}{(2\pi)^{d-2}}.
\end{eqnarray}
The contributions from the cut diagrams shown in Fig.\ref{feyn} can be calculated through
\begin{eqnarray}
D^{(1-4)}_{q \to (Q\bar{Q})[^1S_0^{[1]}]}(z)=N_{CS}\int d\phi_{3}(p_1,p_2,p_3) {\cal A}_{(1-4)},
\label{cut-contribution}
\end{eqnarray}
where $N_{CS}\equiv z^{d-3}/(8\pi N_c)$ is a factor from the definition of the fragmentation function. The integral on the right-hand side of Eq.(\ref{cut-contribution}) is UV divergent with $d=4$. This UV divergence is regularized by dimensional regularization with $d=4-2\epsilon$, and the integral generates $1/\epsilon$ terms. To perform the integration in Eq.(\ref{cut-contribution}), it is important to choose proper parametrization for the phase space. We present a parametrization for the phase space in Appendix \ref{Ap.phs}.
The differential phase space given in Eq.(\ref{eqa8}) can be expressed as follows
\begin{eqnarray}
&&N_{CS} d\phi_{3}(p_1,p_2,p_3)\nonumber \\
&& = N_{g}(p_1,p_2) d\phi_{2}(p_1,p_2) d\phi^{(3)}(p_1,p_2,p_3),
\label{eq.fact}
\end{eqnarray}
where $N_{g}(p_1,p_2)$ is defined as
\begin{eqnarray}
N_{g}(p_1,p_2)=\frac{(z/y)^{1-2\epsilon}}{(N_c^2-1)(2-2\epsilon)(2\pi\, y\, K\cdot n)},
\label{eq.Ng}
\end{eqnarray}
and $d\phi_{2}(p_1,p_2)$ is defined as
\begin{eqnarray}
d\phi_{2}(p_1,p_2)=&&\frac{z^{-1+\epsilon}(y-z)^{-\epsilon}\mu^{2\epsilon}}{2(4\pi)^{1-\epsilon}\Gamma(1-\epsilon)K\cdot n}\nonumber \\
&&\times \left(s_1-\frac{y}{z}4m_Q^2\right)^{-\epsilon}ds_1,\label{phsp2}
\end{eqnarray}
where the range of $s_1$ is from $(4m_Q^2 y/z)$ to $\infty$. $d\phi_{2}(p_1,p_2)$ stands for the differential phase space for a gluon with longitudinal momentum $y K\cdot n$ to fragment into a $(Q\bar{Q})[^1S_0^{[1]}]$-pair with longitudinal momentum $z K\cdot n$ at leading order (LO). According to Eqs.(\ref{eq.fact}), (\ref{eq.Ng}), (\ref{phsp2}) and (\ref{eqa8}), the expression of $d\phi^{(3)}(p_1,p_2,p_3)$ can be derived
\begin{eqnarray}
&&d\phi^{(3)}(p_1,p_2,p_3)\nonumber \\
&&=\frac{(N_c^2-1)(2-2\epsilon)\mu^{2\epsilon} K\cdot n}{16N_c(2\pi)^{3-2\epsilon}} y^{1-\epsilon}(1-y)^{-\epsilon} \nonumber \\
&&~~~\times \left[ s-s_1/y-m_q^2/(1-y)\right]^{-\epsilon}ds\, dy\, d\Omega_{3\perp}.\label{phsp4}
\end{eqnarray}
The range of $y$ is from $z$ to 1, and the range of $s$ is from $[s_1/y+m_q^2/(1-y)]$ to $\infty$.
The integrations over $\Omega_{3\perp}$ and $s$ of ${\cal A}_{(1-4)}$ can be performed using the method introduced in Ref.\cite{Zheng:2019gnb}. Then, we obtain
\begin{eqnarray}
D^{(1-4)}_{q \to (Q\bar{Q})[^1S_0^{[1]}]}(z)=&&\frac{(4\pi \mu^2)^{\epsilon}\Gamma(\epsilon)}{(4\pi)^2}\int_z^1 dy\nonumber \\
&&\int N_g d\phi_{2}(p_1,p_2) f(s_1,y,z),
\label{cut-contribution2}
\end{eqnarray}
where $N_g d\phi_{2}(p_1,p_2)\equiv N_g(p_1,p_2) d\phi_{2}(p_1,p_2)$. The expression of $f(s_1,y,z)$ is given in Appendix \ref{Ap.f}.
The contribution $D^{(1-4)}_{q \to (Q\bar{Q})[^1S_0^{[1]}]}(z)$ contains a UV pole, it should be removed through the operator renormalization \cite{Mueller:1978xu}. We carry out the renormalization using the $\overline{\rm MS}$ procedure. Then the fragmentation function under the $\overline{\rm MS}$ scheme can be obtained through
\begin{eqnarray}
&&D_{q \to (Q\bar{Q})[^1S_0^{[1]}]}(z,\mu_F)\nonumber \\
&&=D^{(1-4)}_{q \to (Q\bar{Q})[^1S_0^{[1]}]}(z)-\frac{\alpha_s}{2\pi}\left[\frac{1}{\epsilon_{UV}}- \gamma_E+ {\rm ln}~(4\pi)+{\rm ln}\frac{\mu^2}{\mu_F^2} \right]\nonumber \\
&& ~~~ \times \int_z^1 \frac{dy}{y}P_{gq}(y)D_{g\to (Q\bar{Q})[^1S_0^{[1]}]}^{\rm LO}(z/y),
\label{Dq2QQ}
\end{eqnarray}
where $\mu_F$ is the factorization scale, the expression of the splitting function $P_{gq}(y)$ is
\begin{eqnarray}
P_{gq}(y)=C_F \frac{1+(1-y)^2}{y},
\end{eqnarray}
and $D_{g\to (Q\bar{Q})[^1S_0^{[1]}]}^{\rm LO}$ is the LO fragmentation function in $d$-dimensional space-time. In the calculation, it is convenient to use the unintegrated form of $D_{g\to (Q\bar{Q})[^1S_0^{[1]}]}^{\rm LO}$, i.e.,
\begin{eqnarray}
D_{g\to (Q\bar{Q})[^1S_0^{[1]}]}^{\rm LO}(z/y)=\int N_g d\phi_{2}(p_1,p_2){\cal A}_{g \to (Q\bar{Q})[^1S_0^{[1]}]},\nonumber \\
\label{Dg2QQ}
\end{eqnarray}
where the expression of ${\cal A}_{g \to (Q\bar{Q})[^1S_0^{[1]}]}$ is
\begin{eqnarray}
&&{\cal A}_{g \to (Q\bar{Q})[^1S_0^{[1]}]}\nonumber \\
&&= \frac{16g_s^4}{3}(1-2\epsilon)(y\, K\cdot n)^2\left[\frac{(1-z/y)}{m_Q^3 s_1}-\frac{(1-z/y)}{m_Q^3 (s_1-4m_Q^2)}\right. \nonumber \\
&& ~~~ \left. +\frac{2(1-\epsilon)}{m_Q s_1^2}+\frac{4(1-z/y)^2}{m_Q (s_1-4m_Q^2)^2}\right].
\end{eqnarray}
The integral over $s_1$ in Eq.(\ref{Dg2QQ}) can be carried out easily, and we have checked that our result for $D_{g\to (Q\bar{Q})[^1S_0^{[1]}]}^{\rm LO}$ is consistent with that obtained in Refs.\cite{Braaten:1993rw,Artoisenet:2014lpa,Feng:2018ulg,Zhang:2018mlo}.
Applying Eqs.(\ref{cut-contribution2}) and (\ref{Dg2QQ}) to Eq.(\ref{Dq2QQ}), we can obtain the fragmentation function under the $\overline{\rm MS}$ scheme. It is found that the UV pole of $D^{(1-4)}_{q \to (Q\bar{Q})[^1S_0^{[1]}]}(z)$ is exactly canceled by the UV pole of the counter term from the operator renormalization. The remaining integrals no longer generate divergence, we can set $\epsilon=0$ before carrying out the integrations. Multiplying the fragmentation function $D_{q \to (Q\bar{Q})[^1S_0^{[1]}]}(z,\mu_F)$ for the $(Q\bar{Q})[^1S_0^{[1]}]$ pair by a factor $\langle {\cal O}^{\eta_Q}(^1S_0^{[1]}) \rangle/\langle {\cal O}^{(Q\bar{Q})[^1S_0^{[1]}]}(^1S_0^{[1]}) \rangle \approx \vert R_S^{(Q\bar{Q})}(0) \vert^2/(4\pi)$, we obtain the fragmentation function $D_{q \to \eta_Q}(z,\mu_F)$, i.e.,
\begin{eqnarray}
&&D_{q \to \eta_Q}(z,\mu_F)\nonumber \\
&&=\int_z^1 dy \int_{4m_Q^2 y/z}^{\infty} d s_1\, g(s_1,\mu_F,y,z),
\label{frag.q2etaQ1}
\end{eqnarray}
where
\begin{widetext}
\begin{eqnarray}
g(s_1,\mu_F,y,z)=&&\frac{\alpha_s^3\,\vert R_S^{(Q\bar{Q})}(0) \vert^2}{9\,\pi^2\, y^4 m_Q s_1^2 (s_1-4m_Q^2)^2\left[(1-y)s_1+y^2m_q^2\right]}\bigg\{ (y-1)\Big[s_1^3 \big( y^4-2y^3(z+1)+2y^2(z^2+6z+1) \nonumber \\
&& -12y\,z(z+1)+12z^2\big)+s_1^2 \big(2 y^4 m_q^2-4y^3(2m_Q^2(z+4)+z\, m_q^2)+4 y^2(4 m_Q^2(3z+2)+z^2 m_q^2) \nonumber \\
&&-48y\,z\,m_Q^2 \big)+16 s_1 y^2 m_Q^2 \big(y^2 m_Q^2-y(2m_Q^2+z\,m_q^2)+2 m_Q^2\big)+32 y^4 m_Q^4 m_q^2 \Big]-\left[(1-y)s_1+y^2m_q^2\right]\nonumber \\
&&\left. \times \left[z^2(s_1-4m_Q^2 y/z)^2+(y-z)^2s_1^2 \right]\left[\big((1-y)^2+1\big){\rm ln}\left( \frac{(1-y)s_1+y^2m_q^2}{\mu_F^2}\right) +y^2\right]\right\}.
\label{frag.q2etaQ2}
\end{eqnarray}
\end{widetext}
Here, $R_S^{(Q\bar{Q})}(0)$ is the radial wave function at the origin for the $(Q\bar{Q})$ bound state.
\section{Numerical results and discussion}
In this section, we will present the numerical results for the fragmentation functions and apply the fragmentation functions to the decay widths for the $\eta_c$ production through Z boson decays.
The input parameters for the numerical calculation are taken as follows
\begin{eqnarray}
&& m_c=1.5\,{\rm GeV},\,m_b=4.9\,{\rm GeV},\,m_{_Z}=91.1876\,{\rm GeV}, \nonumber \\
&& \alpha=1/128,{\rm sin}^2\theta_{_W}=0.231, \vert R_S^{c\bar{c}}(0)\vert ^2=0.810\,{\rm GeV}^3.
\end{eqnarray}
The value of $\vert R_S^{c\bar{c}}(0)\vert ^2$ is taken from the potential model calculation \cite{pot}. For the strong coupling constant, we adopt two-loop formula as used in our previous paper \cite{Zheng:2019dfk}, where $\alpha_s(2m_c)=0.259$.
\subsection{The fragmentation functions}
The fragmentation function for a light quark into $\eta_c$, where $\mu_F=2m_c$, $4m_c$, and $6m_c$, is shown in Fig.\ref{Dzqetac}. In the numerical calculation, the mass of the light quark is neglected, and the strong coupling is taken as $\alpha_s(2m_c)$.
\begin{figure}[htbp]
\includegraphics[width=0.45\textwidth]{Dzq2etac}
\caption{The fragmentation function $D_{q \to \eta_c}(z,\mu_F)$ as a function of $z$ for $\mu_F=2m_c$, $4m_c$, and $6m_c$, where $q$ denotes a light quark. }
\label{Dzqetac}
\end{figure}
From Fig.\ref{Dzqetac}, we can see that the fragmentation function is sensitive to the factorization scale. When $\mu_F=2m_c$, the fragmentation function increases first ($z<0.96$) and then decreases ($z>0.96$) with the increase of $z$, and the fragmentation function is less than 0 in the most $z$ region; when $\mu_F=4m_c$, the fragmentation function also increases first ($z<0.12$) and then decreases ($z>0.12$) with the increase of $z$, but the fragmentation function is greater than 0 in the most $z$ region; When $\mu_F=6m_c$, the fragmentation function decreases monotonically with the increase of $z$, and the fragmentation function is greater than 0 for $z \in (0,1)$. The fragmentation function has a singularity at $z=0$.
\begin{figure}[htbp]
\includegraphics[width=0.45\textwidth]{mufcoeline}
\includegraphics[width=0.45\textwidth]{mufcoelog}
\caption{The coefficient of ${\rm ln}(\mu_F^2/m_c^2)$ in the fragmentation function $D_{q\to \eta_c}(z,\mu_F)$, i.e., $F(z)=\frac{\alpha_s}{2\pi}\int_z^1 \frac{dy}{y}P_{gq}(y)D_{g \to \eta_c}^{LO}(z/y)$. The same curve is shown as a function of $z$ with a linear scale (upper one) and with a logarithmic scale (lower one). }
\label{mufcoe}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=0.45\textwidth]{Dzb2etac}
\caption{The fragmentation function $D_{b \to \eta_c}(z,\mu_F)$ as a function of $z$ for $\mu_F=m_b$, $m_b+2m_c$ and $m_b+4m_c$.}
\label{b2etac}
\end{figure}
In order to understand the dependence of the fragmentation function on the factorization scale, we have calculated the coefficient of ${\rm ln}(\mu_F^2/m_c^2)$ in the fragmentation function $D_{q\to \eta_c}(z,\mu_F)$. The coefficient is shown as a function of $z$ in Fig.\ref{mufcoe} with a linear scale (upper one) and with a logarithmic scale (lower one). We can see that, the coefficient decreases monotonically with the increase of $z$ and is positive for $z \in (0,1)$. Like the fragmentation function, the coefficient of ${\rm ln}(\mu_F^2/m_c^2)$ also has a singularity at $z=0$. When $\mu_F$ is very large, the fragmentation function is dominated by the ${\rm ln}(\mu_F^2/m_c^2)$ term. Therefore, the behavior of the fragmentation function is similar to that of the coefficient of ${\rm ln}(\mu_F^2/m_c^2)$ when $\mu_F$ is large enough.
In Fig.\ref{b2etac}, the fragmentation function $D_{b \to \eta_c}(z,\mu_F)$ for $\mu_F=m_b$, $m_b+2m_c$ and $m_b+4m_c$ is presented. We can see that the fragmentation function $D_{b\to \eta_c}(z,\mu_F)$ is also sensitive to the factorization scale. Actually, from Eqs.(\ref{frag.q2etaQ1}) and (\ref{frag.q2etaQ2}), we can see that the coefficient of ${\rm ln}(\mu_F^2/m_Q^2)$ in $D_{b\to \eta_c}(z,\mu_F)$ is the same as that in $D_{q\to \eta_c}(z,\mu_F)$.
\subsection{Application to the decay widths}
In this subsection, we will apply the obtained fragmentation functions to the decay widths for the processes $Z \to \eta_c+q\bar{q}g$ and $Z \to \eta_c+b\bar{b}g$.
Here, we only present the calculation formulas for $Z \to \eta_c+q\bar{q}g$, the formulas for $Z \to \eta_c+b\bar{b}g$ are similar. Under the fragmentation approach, the differential decay width $d\Gamma/dz$ for $Z \to \eta_c+q\bar{q}g$ at LO can be written as
\begin{eqnarray}
\frac{d\Gamma^{\rm Frag,LO}_{Z \to \eta_c+q\bar{q}g}}{dz}=&&2\, \Gamma_{Z \to q+\bar{q}}D_{q\to \eta_c}(z,\mu_F) \nonumber \\
&& +\int^1_z \frac{dy}{y} \frac{d\hat{\Gamma}_{Z \to g+q\bar{q}}(y,\mu_F)}{dy}\nonumber \\
&& \cdot D_{g\to \eta_c}(z/y),\label{Frag.Z2etac}
\end{eqnarray}
where the energy fraction is defined as $z\equiv E_{\eta_c}/E^{\rm max}_{\eta_c}$, and $E_{\eta_c}$ and $E^{\rm max}_{\eta_c}$ are the energy and the maximum energy of the $\eta_c$ in the rest frame of the Z boson. $\Gamma_{Z \to q+\bar{q}}$ denotes the LO decay width for the process $Z \to q+\bar{q}$, and the factor of 2 is due to the fact that the contributions from the q fragmentation and $\bar{q}$ fragmentation are the same. In Eq.(\ref{Frag.Z2etac}), we have used the fact $d\hat{\Gamma}_{Z \to q+\bar{q}}/dy=\Gamma_{Z \to q+\bar{q}}\delta(1-y)$ at LO. $d\hat{\Gamma}_{Z \to g+q\bar{q}}/dy$ is the differential decay width for the inclusive production of a gluon associated with a light-quark pair. Neglecting the light quark mass, we obtain the differential decay width under the $\overline{\rm MS}$ factorization scheme
\begin{eqnarray}
&&\frac{d\hat{\Gamma}_{Z \to g+q\bar{q}}(y,\mu_F)}{dy}\nonumber \\
&&=\Gamma_{Z \to q+\bar{q}}\frac{\alpha_s}{\pi}P_{gq}(y)\left[{\rm ln}\frac{m_{_Z}^2}{\mu_F^2}+2{\rm ln}y+{\rm ln}(1-y) \right].
\end{eqnarray}
$D_{g\to \eta_c}(z)$ and $D_{q\to \eta_c}(z,\mu_F)$ are the LO fragmentation functions. The expression of $D_{g\to \eta_c}(z)$ can be found in Ref.\cite{Braaten:1993rw}, and the expression of $D_{q\to \eta_c}(z,\mu_F)$ has been given in Eq.(\ref{frag.q2etaQ1}). It is easy to check that the logarithm terms of $\mu_F^2$ in Eq.(\ref{Frag.Z2etac}) are canceled by each other which results in that $d\Gamma^{\rm Frag,LO}_{Z \to \eta_c+q\bar{q}g}/dz$ is independent of $\mu_F$.
The physical picture of Eq.(\ref{Frag.Z2etac}) is as follows: The first term gives the contribution from that the Z boson decays into a light quark and a light antiquark with energies $m_{_Z}/2$ on a distance scale of order $1/m_{_Z}$, and one of the light quark and the light antiquark decays into an $\eta_c$ on a distance scale of order $1/m_c$. The second term gives the contribution from that the Z boson decays into a light quark-antiquark pair and a gluon on a distance scale of order $1/m_{_Z}$, and the gluon decays into an $\eta_c$ on a distance scale of order $1/m_c$. The two terms of Eq.(\ref{Frag.Z2etac}) share the same Feynman diagrams which are shown in Fig.\ref{feynZetacqqg}, but they come from different regions of the phase space. When the invariant mass of the virtual light quark (antiquark) is very small compared to $\mu_F$, the contribution is given by the first term of Eq.(\ref{Frag.Z2etac}). When the invariant mass of the virtual light quark (antiquark) is very large compared to $\mu_F$, the contribution is given by the second term of Eq.(\ref{Frag.Z2etac})
\begin{figure}[htbp]
\includegraphics[width=0.45\textwidth]{feynZetacqqg}
\caption{The Feynman diagrams for $Z \to \eta_c+q\bar{q}g$ which are responsible for the fragmentation mechanism.
} \label{feynZetacqqg}
\end{figure}
There are logarithms of $m_{Z}/m_c$ in the decay widths of $Z \to \eta_c+q\bar{q}g$, which may spoil the convergence of the perturbative expansion. These large logarithms can be resummed through the evolution of the fragmentation functions under the fragmentation approach. The decay width after the resummation of the leading logarithms (LLs) can be written as
\begin{eqnarray}
\frac{d\Gamma^{\rm Frag,LO+LL}_{Z \to \eta_c+q\bar{q}g}}{dz}=&&2\, \Gamma_{Z \to q+\bar{q}}D^{\rm LO+LL}_{q\to \eta_c}(z,\mu_F) \nonumber \\
&& +\int^1_z \frac{dy}{y} \frac{d\hat{\Gamma}_{Z \to g+q\bar{q}}(y,\mu_F)}{dy}\nonumber \\
&& \cdot D^{\rm LO+LL}_{g\to \eta_c}(z/y,\mu_F),\label{Frag.Z2etac.resum}
\end{eqnarray}
where the factorization scale is set as $\mu_F=m_{_Z}$, and the renormalization scale in the partonic decay widths is also set as $\mu_R=m_{_Z}$, so as to avoid large logarithms appearing in the partonic decay widths. The fragmentation functions $D^{\rm LO+LL}_{q\to \eta_c}(z,\mu_F=m_{_Z})$ and $D^{\rm LO+LL}_{g\to \eta_c}(z,\mu_F=m_{_Z})$ are obtained through solving the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations \cite{dglap1, dglap2, dglap3} with LO splitting functions, where the initial fragmentation functions $D_{q\to \eta_c}(z,\mu_{F0})$ and $D_{g\to \eta_c}(z,\mu_{F0})$ at $\mu_{F0}=2m_c$ \footnote{For $Z \to \eta_c+b\bar{b}g$ case, the initial factorization scale is taken as $\mu_{F0}=m_b+2m_c$.} are used as the boundary condition. We solve the DGLAP equations by using the program FFEVOL \cite{Hirai:2011si}.
In addition to the fragmentation approach, we can also calculate the decay width directly based on the NRQCD factorization, i.e.,
\begin{eqnarray}
d\Gamma^{\rm Direct,LO}_{Z \to \eta_c+q\bar{q}g}=d\tilde{\Gamma}_{Z \to (c\bar{c})[^1S_0^{[1]}]+q\bar{q}g}\langle {\cal O}^{\eta_c}(^1S_0^{[1]}) \rangle,
\end{eqnarray}
where we use ``Direct" to denote the results from the direct calculation based on the NRQCD factorization. The dominant contributions to the decay widths for $Z \to \eta_c+q\bar{q}g$ and $Z \to \eta_c+b\bar{b}g$ come from four fragmentation diagrams shown in Fig.\ref{feynZetacqqg}. The contributions from the nonfragmentation diagrams are suppressed by powers of $m_c/m_Z$ compared to the fragmentation contributions. For simplicity, under the direct NRQCD calculation, we only consider the contributions from the four fragmentation diagrams shown in Fig.\ref{feynZetacqqg}.
\begin{figure}[htbp]
\includegraphics[width=0.45\textwidth]{gammazq}
\caption{The differential decay width $d\Gamma/dz$ as a function of $z$ for the process $Z \to \eta_c+q\bar{q}g$ under the fragmentation and the direct NRQCD calculations. The contributions for $q=u,d,s$ are summed.}
\label{gammazq}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=0.45\textwidth]{gammazb}
\caption{The differential decay width $d\Gamma/dz$ as a function of $z$ for $Z \to \eta_c+b\bar{b}g$ under the fragmentation and the direct NRQCD calculations.}
\label{gammazb}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=0.45\textwidth]{gammaz_ratio}
\caption{The ratios $(d\Gamma_{\rm Frag,LO}/dz)/(d\Gamma_{\rm Direct,LO}/dz)$ as functions of $z$ for the processes $Z \to \eta_c+q\bar{q}g$ and $Z \to \eta_c+b\bar{b}g$.}
\label{gammazR}
\end{figure}
The differential decay widths for $Z \to \eta_c+q\bar{q}g$ and $Z \to \eta_c+b\bar{b}g$ under the fragmentation and the direct NRQCD approaches are presented in Figs.\ref{gammazq} and \ref{gammazb}. In order to see the difference between the ``Frag,LO" results and the ``Direct,LO" results more clearly, the ratios $(d\Gamma_{\rm Frag,LO}/dz)/(d\Gamma_{\rm Direct,LO}/dz)$ are given in Fig.\ref{gammazR}. For the ``Frag,LO" and the ``Direct,LO" calculations, the renormalization scale is fixed as $\mu_R=2m_c$ for simplicity. For the ``Frag,LO+LL" calculation, the choice of the renormalization scale and the factorization scale has been described below Eq.(\ref{Frag.Z2etac.resum}). From the figures, we can see that the differential decay widths from the ``Frag,LO" calculation are very close to those from the ``Direct,LO" calculation, especially for $0.2 \leq z \leq 0.8$.
The total decay widths can be obtained through integrating the differential decay widths $d\Gamma/dz$ over $z$. The total decay widths under the fragmentation approach and the direct NRQCD approach are given in Table \ref{tb.width}. We can see that the total decay widths obtained from the ``Frag,LO" calculation and the ``Direct,LO" calculation are also very close. Therefore, the fixed-order fragmentation approach (i.e., the ``Frag,LO" approach) provides a good approximation to the direct NRQCD calculation.
The differential and total decay widths after the resummation of the large logarithms under the fragmentation approach are also shown in Figs.\ref{gammazq} and \ref{gammazb} and Table \ref{tb.width}. We can see that, after the resummation, the differential decay widths are enhanced at smaller $z$ values but are reduced at larger $z$ values, and the total decay widths are reduced after the resummation. In fact, the fixed-order results have a big uncertainty caused by the choice of the renormalization scale. If we set the renormalization scale as $\mu_R=m_{_Z}$, the fixed-order results become $[\alpha_s(m_{_Z})/\alpha_s(2m_c)]^3=0.0958$ of those with $\mu_R=2m_c$. On the contrary, the results after the resummation have a smaller uncertainty caused by the choice of the renormalization scale. Because the renormalization scale of the initial fragmentation functions should be ${\cal O}(m_c)$, and the renormalization scale of the coefficient functions should be ${\cal O}(m_{_Z})$. Moreover, the resummed results include the leading logarithms up to all orders from the collinear radiation. Therefore, the results after the resummation are more precise than the fixed-order results.
\begin{table}[h]
\begin{tabular}{c c c}
\hline
~& $\Gamma_{Z \to \eta_c+q\bar{q}g}$ (keV) & $\Gamma_{Z \to \eta_c+b\bar{b}g}$(keV) \\
\hline
Direct & 63.0 & 19.3\\
Frag,LO & 65.6 & 20.3 \\
Frag,LO+LL & 40.3 & 15.3 \\
\hline
\end{tabular}
\caption{The decay widths of $Z \to \eta_c+q\bar{q}g$ and $Z \to \eta_c+b\bar{b}g$ under the fragmentation and the direct NRQCD approaches. For the $Z \to \eta_c+q\bar{q}g$ case, the contributions for $q=u,d,s$ are summed.}
\label{tb.width}
\end{table}
\section{Summary}
In the present paper, we have calculated the fragmentation functions for a (heavy or light) quark into a spin-singlet quarkonium, where the flavor of the initial quark is different from that of the constituent quark in the quarkonium. There are UV divergences in the phase-space integral, which are removed through the operator renormalization of the fragmentation function. We have carried out the renormalization under the $\overline{\rm MS}$ scheme. The fragmentation function $D_{q \to \eta_Q}(z,\mu_F)$ is given as a two-dimensional integral, and this two-dimensional integral can be calculated easily through numerical integration. Numerical results for a light quark or a bottom quark into the $\eta_c$ with several factorization scales are analyzed. The results show that these fragmentation functions are sensitive to the factorization scale. Especially, when $\mu_F$ is small, the fragmentation functions are negative at small $z$ values. There is a singularity at $z=0$ for these fragmentation functions.
We have applied the obtained fragmentation functions to the decay widths for the processes $Z \to \eta_c+q\bar{q}g(q=u,d,s)$ and $Z \to \eta_c+b\bar{b}g$. The differential decay widths and total decay widths are calculated under the fragmentation and the direct NRQCD approaches. It is found that the results under the fixed-order fragmentation and the direct NRQCD approaches are close to each other. Therefore, the fixed-order fragmentation approach provides a good approximation to the direct NRQCD calculation. The more precise results containing the resummation of the large logarithms under the fragmentation approach are also presented. Moreover, the fragmentation functions obtained in this paper can be used in the studies on the production of $\eta_c$ and $\eta_b$ at high-energy colliders.
\hspace{2cm}
\noindent {\bf Acknowledgments:} This work was supported in part by the Natural Science Foundation of China under Grants No. 11625520, No. 12005028 and No. 12047564, by the Fundamental Research Funds for the Central Universities under Grant No. 2020CQJQY-Z003, and by the Chongqing Graduate Research and Innovation Foundation under Grant No. ydstd1912.
|
1,314,259,996,477 | arxiv | \section{Introduction}
Discrete Mechanics first developed in [1] is a method of discretizing continuous differential equations for simulating on computers while preserving the structure of the manifold on which the dynamics evolve. The simplest discretization technique known is the Euler's technique which is used to numerically solve initial value problems by approximating the vector field to be a constant in a desired time interval 'h'. The Euler's method suffers from stability and other issues which have been to some extent addressed by a more accurate series of Runge-Kutta methods. Though the accuracy is increased, structures of the dynamics are not preserved in most cases. One such case is when a Runge-Kutta method is used to solve the Keppler problem, the energy and the angular momentum are not observed to be conserved. Discrete time models obtained via discrete mechanics are more desirable than other standard discretization schemes such as Euler’s step because they preserve certain invariance properties like kinetic energy, momentum, etc, of the system, and the computations can be done directly on the manifold, (because this discretization respects the manifold structure) thereby eliminating the problems associated with parametric representations. In the next sections, a brief description of Discrete Mechanics and its application to attitude dynamics of a rigid body on SO(3) will be provided.
\section{Discrete Mechanics}
Consider a mechanical system with the configuration space $Q$
as a smooth manifold. Then the velocity vectors lie on the tangent bundle TQ of the manifold Q and the Lagrangian for the system can be defined as $L : TQ \rightarrow R$. In discrete mechanics, the velocity phase space TQ is replaced by $Q \times Q$ which is locally isomorphic to TQ. Let us consider an integral curve q(t) in the configuration space such that q(0) = q0 and q(h) = q1, where h represents the integration step. Then, the discrete Lagrangian $L_d : Q \times Q \rightarrow R$, which is an approximation of the action integral along the integral curve segment between q0 and q1, can be defined as
\begin{equation}\label{discrete lagrangian}
L_d(q_0,q_1) \approx \int_0^h L(q(t),\dot q(t))dt
\end{equation}
Having defined the discrete Lagrangian, the action would now be the summation of all such Lagrangians along the path. The integral in the continuous case is replaced by the summation. This discrete Action is given by
\begin{equation}\label{discrete action}
A_d := \sum_{k=0}^{N-1} L_d(q_k,q_{k+1})
\end{equation}
where $q_i$ is the configuration of the system at $i$th time instant. \newline
Using the variation principles similar to the continuous case, the discretized equations of motion can be obtained as follows. [1] provides a detailed derivation of these equations.
\begin{equation}\label{discrete EL}
D_2L_d(q_{k-1},q_k)+D_1L_d(q_k,q_{k+1})=0 \; \forall \; k=0,1,...,N-1
\end{equation}
where $D_i$ is the derivative of the function with respect to the $i$th argument. \newline
Similarly, the discrete analogue of the Hamiltonian formulation can be obtained using the discrete Legendre transform. The continuous time Legendre transform is a map $\mathbb{F}L$ from the Lagrangian state space $TQ$ to the
Hamiltonian phase space $T^*Q$. Similarly, the discrete time Legendre transforms $\mathbb{F}^+L_d, \mathbb{F}^-L_d : Q \times Q \mapsto T^*Q$ [4] can be defined as
\begin{equation}\label{discrete EL1}
\mathbb{F}^{+}L_d(q_k,q_{k+1}) \mapsto (q_{k+1},p_{k+1}) = (q_k,D_2L_d(q_k,q_{k+1}))
\end{equation}
\begin{equation}\label{discrete EL2}
\mathbb{F}^{-}L_d(q_k,q_{k+1}) \mapsto (q_{k},p_{k}) = (q_k,-D_1L_d(q_k,q_{k+1}))
\end{equation}
where $p_i$ represents the corresponding conjugate momentum at $i$th time instant. \newline
In the next section, this routine will be used to develop the discretized equations of motion for the attitude dynamics of a rotating rigid body.
\section{Attitude dynamics using Discrete Mechanics}
In this section, a brief description of the attitude dynamics of a rigid body evolving on SO(3) manifold and its discrete equations of motion described in [3] will be presented. \newline
The kinetic energy of a rotating rigid body with angular velocity $\Omega$ is given by
\begin{equation}\label{eq:KE}
K = \frac{1}{2}\Omega^TJ\Omega = \frac{1}{2}tr(\widehat{\Omega}J_d\widehat{\Omega}^T)
\end{equation}
where $\widehat{\Omega} \in \mathfrak{so(3)}$ is the skew symmetric 3$\times$3 tensor of $\Omega$ formulated according to \eqref{eq:vshom}, $J$ is the body moment of inertia matrix given by
\begin{equation}
J = \frac{1}{2}\int_\textit{B} \rho (X) \widehat{X}\widehat{X}^T d^3X
\end{equation}
and $J_d$ is a matrix given by
\begin{equation}
J_d = \frac{1}{2}\int_\textit{B} \rho (X) XX^T d^3X
\end{equation}
The body moment of inertia $J$ is related to $J_d$ by
\begin{equation}\label{eq:JtoJd}
J = tr(J_d)I_{3\times3}-J_d
\end{equation}
\eqref{eq:JtoJd} can be used to solve for $J_d$ and it can be obtained as
\begin{equation}
J_d = \frac{1}{2} tr(J)I_{3\times3}-J
\end{equation}
The Lagrangian of the system would just contain the Kinetic Energy of the system and is given by
\begin{equation}
L(R,\Omega) = K = \frac{1}{2}tr(\widehat{\Omega}J_d\widehat{\Omega}^T)
\end{equation}
The rate of change of the rotation matrix $R$, which represents the orientation of the body fixed frame with respect to an inertial frame, is given by
\begin{equation}\label{eq:Rdot}
\dot R = R \widehat{\Omega}
\end{equation}
The Lagrangian of the system can be modified by replacing the angular velocity with $R^T \dot R$ from \eqref{eq:Rdot}
\begin{equation}\label{eq:AttLag}
L(R,\dot R) = K = \frac{1}{2}tr(R^T \dot R J_d \dot R^T R)
\end{equation}
From the Lagrangian of the continuous time dynamics in \eqref{eq:AttLag}, the discrete Lagrangian can be obtained by approximating $\dot R$ using Euler's scheme with $h$ as the time step.
\begin{align}
L_d(R_k,R_{k+1}) &\approx hL(R_k,\frac{R_{k+1}-R_k}{h}) \\
&= \frac{h}{2}tr(\frac{R_k^T(R_{k+1}-R_k)}{h} J_d \frac{(R_{k+1}-R_k)^T R_k}{h}) \\
&= \frac{1}{2h}tr((I_{3 \times 3}-F_k)J_d)
\end{align}
Here $R_i$ is the rotation matrix associated with the orientation of the body fixed frame with respect to the inertial frame at the $i$th time instant and $F_k = R_k^T R_{k+1}$ is used for notational brevity. Here it must be noted that $F_k$ is also a rotation matrix. \newline
The first order Hamilton's equations can be obtained using the above Lagrangian as follows. A detailed derivation of these equations can be found in [3].
\begin{equation}\label{eq:AttDyn}
Attitude \; Dynamics \;
\begin{cases}
\widehat{h\Pi} = F_kJ_d-J_dF_k^T \\
R_{k+1} = R_kF_k \\
\Pi_{k+1} = F_k^T\Pi_k+hu_k
\end{cases}
\end{equation}
where $\Pi_i$ is the conjugate momentum (the angular momentum in this case) and $u_i$ is the moment applied at the $i$th time instant. \newline
The set of equations in \eqref{eq:AttDyn} can be used to simulate the system given the time series of the moments applied at each time instant, the initial orientation and the initial angular momentum. Since the propagation of the dynamics involves finding the rotation matrix $F_k$, the implicit matrix equation $\widehat{h\Pi} = F_kJ_d-J_dF_k^T$ needs to be solved at every time instant. A numerical technique will be developed in the next section to find the roots of the nonlinear matrix equation involving $F_k$.
\section{Solution to the Implicit Nonlinear Equation in $F_k$}
In this section, a Newton-Raphson type algorithm on the Lie Group SO(3) to solve the implicit nonlinear equation $\widehat{h\Pi} = F_kJ_d-J_dF_k^T$ to find $F_k \in SO(3)$ will be developed. Given the angular momentum $\Pi_k$ at the $k$th time instant, the implicit nonlinear matrix equation needs to be solved to find the incremental rotation matrix $F_k$ at every time instant to be used in the simulation of the dynamics.To preserve the manifold structure, it is essential that $F_k$ satisfies all the properties of a rotation matrix while solving the equation. Since $F_k$ is an incremental rotation matrix, there always exists a $\widehat{w} \in \mathfrak{so(3)}$, where $\mathfrak{so(3)}$ is the Lie Algebra of SO(3). A function $g(F_k)$ will be defined as follows, the zero of which will solve the required implicit nonlinear matrix equation.
\begin{equation}\label{eq:INME}
g(F_k) = F_kJ_d-J_d F_k^T-\widehat{h\Pi_k}
\end{equation}
The above equation \eqref{eq:INME} can be expressed in terms of the vector in the lie algebra of $F_k$ as
\begin{equation} \label{eq:INMELA}
F(\widehat{w}) = e^{\widehat{w}}J_d-J_d (e^{\widehat{w}})^T-\widehat{h\Pi_k}
\end{equation}
It can be verified from the above equation \eqref{eq:INMELA} that the matrix $F(\widehat{w})$ is always a skew symmetric matrix. Therefore, a vector space homeomorphism can be established between the vector space of skew symmetric matrices and vector space $\mathbb{R}^3$ as follows
\begin{equation*}
\widehat{w} =
\begin{bmatrix}
w_1 \\
w_2 \\
w_3
\end{bmatrix}^\times
=
\begin{bmatrix}
0 & -w_3 & w_2 \\
w_3 & 0 & -w_1 \\
-w_2 & w_1 & 0
\end{bmatrix}
\end{equation*}
and
\begin{equation}\label{eq:vshom}
\begin{bmatrix}
0 & -w_3 & w_2 \\
w_3 & 0 & -w_1 \\
-w_2 & w_1 & 0
\end{bmatrix}^\vee
=
\begin{bmatrix}
w_1 \\
w_2 \\
w_3
\end{bmatrix}
\end{equation}
Using this vector space homeomorphism, the skew symmetric matrix function of $\widehat{w}$ in \eqref{eq:INMELA} can be converted to a function of $w \in \mathbb{R}^3$ in $\mathbb{R}^3$.
\begin{equation}
f(w) = [e^{\widehat{w}}J_d-J_d (e^{\widehat{w}})^T-\widehat{h\Pi_k}]^\vee
\end{equation}
The problem now boils down to finding a $w \in \mathbb{R}^3$ such that $f(w)=[0\;0\;0]^T$. A Newton-Raphson type algorithm can now be used to solve this using the jacobian of $f(w)$.But a closed form expression cannot be computed for neither $f(w)$ nor its jacobian $Df(w)$ as $f(w)$ is $F(\widehat{w})^\vee$ and hence the jacobians need to be computed jacobian of $F(\widehat{w})$. \newline
Since $F(\widehat{w})$ is always skew symmetric, its derivative with respect to $w_i$ is also skew symmetric. The derivative can be found to be
\begin{align}\label{eq:dFwdwi}
\frac{\partial F(w)}{\partial w_i} &= e^{\widehat{w}} \frac{\partial \widehat{w}}{\partial w_i} J_d-J_d (e^{\widehat{w}}\frac{\partial \widehat{w}}{\partial w_i})^T \\
&= e^{\widehat{w}} \frac{\partial \widehat{w}}{\partial w_i} J_d+J_d \frac{\partial \widehat{w}}{\partial w_i}(e^{\widehat{w}})^T
\end{align}
It can be verified that the derivative is skew symmetric and hence by using $\vee$ map defined above, the derivative $\frac{\partial f(w)}{\partial w_i}$ can be found as $[\frac{\partial F(w)}{\partial w_i}]^\vee$
\begin{equation}
\frac{\partial f(w)}{\partial w_i} = [\frac{\partial F(w)}{\partial w_i}]^\vee = [e^{\widehat{w}} \frac{\partial \widehat{w}}{\partial w_i} J_d+J_d \frac{\partial \widehat{w}}{\partial w_i}(e^{\widehat{w}})^T]^\vee
\end{equation}
These derivatives can be stacked to form the jacobian of $f(w)$ as
\begin{equation}
Df(w) = [\frac{\partial f(w)}{\partial w_1} \; \frac{\partial f(w)}{\partial w_2} \; \frac{\partial f(w)}{\partial w_3}]^T
\end{equation}
Though explicit expressions for $f(w)$ and $Df(w)$ were not obtained, it is still possible with above formulation to evaluate these at various $w$ and a numerical technique is feasible. \newline
With an initial guess $w^0$ and a step size of $\alpha$, the update equation in the Newton-Raphson algorithm is given by
\begin{equation}\label{eq:NRupdate}
w^{n+1} = w^{n} - \alpha Df(w^n)^{-1}f(w^n)
\end{equation}
The vector $w^n$ is updated until $||f(w^n)||_2$ is satisfactorily close to 0 within user defined tolerance.
\section{Conclusion}
A brief description of Discrete Mechanics and Variational integrators was provided and its application to attitude dynamics of a rigid body evolving on the Lie Group SO(3) was provided. Mainly a Newton-Raphson type algorithm was developed to solve the implicit nonlinear matrix equation arising from Discrete Mechanics to obtain the incremental rotation matrix while preserving the manifold structure.
\section*{MATLAB Package}
A package to simulate the attitude dynamics of a rigid body in MATLAB using the variational integration algorithm presented in this paper is available on request.
|
1,314,259,996,478 | arxiv | \section{Introduction}
\label{intro}
Today, machine learning has come to play an integral role
in many parts of the financial ecosystem, from portfolio management and
algorithmic trading, to fraud detection and
loan/insurance underwriting.
Time series are one of the most common data types
encountered in finance, and so time-series analysis is
one of the most widely used traditional approaches in finance and economics.
The development of machine learning algorithms has opened
a new vista for modeling the complexity of financial time series
as an alternative to the traditional econometric models, by effectively combining
diverse data and capturing nonlinear behavior.
For this reason, financial time-series modeling
has been one of the most interesting topics that has arisen in the
application of machine learning to finance.
Researchers have successfully modeled financial time series
by focusing primarily on prediction accuracy or automatic trading rules
\cite{ats09,dix15,huc09,huc10,kra17,mor14,ser13,tak13,cav16,agg17,gao16,zha15,fis17, pan18, zin18,sin17}.
Nevertheless,
financial modeling and applications remain daunting, given
the difficulties arising from the
essentially nonlinear, complex, and evolutionary
characteristics of the financial market.
On the other hand, asset allocation has been traditionally
considered an issue central to investment and risk management.
Markowitz \cite{mar52} was the first to introduce
a rigorous mathematical framework for allocation,
called modern portfolio theory (MPT).
Based on a mean--variance optimization technique,
MPT provides a method by which to assemble
assets and maximize the expected return of the portfolio for a given level of risk.
Following Markowitz's thinking, new portfolio models
have been subsequently proposed for more practical use
and to achieve a better understanding of portfolios. Examples include
the thee-factor asset pricing model \cite{fam92},
the Black--Litterman model\cite{bla92},
the resampled efficient frontier model \cite{mic98},
the global minimum variance model \cite{hau91},
the maximum diversification portfolio \cite{cho08},
and the risk-parity portfolio \cite{qia05,qia09}.
Additionally, dynamic/tactical asset allocations based on simple rules or
market anomalies
were developed to automatically adjust portfolios in response to market changes
\cite{fab14,kel14,kel14_2,kel16}.
These studies show that today there is general consensus about
the importance of effective combinations of
assets.
In this study, we propose a new method for constructing a data-driven portfolio
using recurrent neural networks (RNNs)-based future return predictions.
Throughout this study,
we will refer to this portfolio as a threshold-based portfolio (TBP),
since its properties are characterized
by the threshold levels imposed on predicted returns.
In particular, this study makes the following main contributions to the literature:
\begin{itemize}[label=$\bullet$]
\item It examines the ability of RNNs to forecast one-month-ahead stock returns.
\item It develops a new TBP portfolio method and analyzes their properties.
\begin{itemize}[label=$\bullet$]
\item
The threshold can be used for a parameter to draw a TBP frontier that
comprises the set of TBP points on a risk--return plane.
This implies that one can build a portfolio with a specific risk--return
level by selecting the appropriate threshold level.
\item The equally weighted portfolio (EWP) is the lower bound of the TBPs on the TBP frontier.
This implies that the TBPs can be characterized with respect to a reference portfolio, EW.
\end{itemize}
\item In practical application, it develops the management process for TBPs
for pursuing specific risk--return levels over multiple-periods
and for the method incorporating TBPs into MPT.
\end{itemize}
The remainder of this paper is organized as follows:
Section \ref{sec_relatedwork} discusses
some of the important work related to this area.
Section \ref{prediction models} explains the simple
recurrent neural network (S-RNN), long short-term memory (LSTM),
and gated recurrent units (GRU).
Section \ref{results} provides experimental results regarding
the prediction accuracy
of the models and the performance of TBPs. In Section \ref{sec_app},
we discuss the practical applications of TBPs. Finally, in Section \ref{conclusion},
we conclude this paper and discuss possible future extensions of our work.
\section{Related Work}\label{sec_relatedwork}
We present LSTM-based predictions and prediction-based portfolios.
\subsection{Financial Time Series Prediction Using RNNs}
Using conventional econometric models,
financial economists have found there to be
statistically significant relationship
between stock returns and lagged variables.
For example,
Campbell et al. \cite{cam93} investigated the relationship between aggregate stock market
trading volume and the serial correlation of daily stock returns.
They provide an evidence that a stock price
decline on a high-volume day is more likely than a stock price decline on a
low-volume day to be associated with an increase in the expected stock return.
Choueifaty and Coignard \cite{cho00} show that trading volume
is a significant determinant of the lead-lag patterns observed in stock returns.
For this reason, we select RNN algorithms; these
are superior for modeling time-lag effects
in multi-dimensional financial time series, by virtue of feeding
the network activations from a previous time step as inputs into
the network, to influence predictions in the current step.
In contrast, feed forward neural networks (FFNNs) are not appropriate
for capturing these time-dependent dynamics: they
operate on a fixed-size time windows, and so
they can provide only limited temporal modeling.
RNNs are less commonly applied to financial time-series
predictions, yet some recent studies
has shown promising results for use in financial time-series prediction.
Fischer and Krauss \cite{fis17}
deployed LSTM networks to predict
one-day-ahead directional movements in
a stock universe and constructed subset portfolios
by selecting constituents outperforming the
cross-sectional median return of all stocks in the next day.
They found that LSTM networks outperform memory-free
classification methods (i.e., a random forest (RAF), a deep neural net (DNN), and a logistic
regression classifier (LOG)) on measures of the mean return per
day, annualized standard deviation, annualized Sharpe ratio, and accuracy.
More recently, Bao et al. \cite{bao17} developed a hybrid model
called the WSAEs--LSTM, combined with
wavelet transforms (WT), stacked autoencoders (SAEs), and LSTM
to effectively combine diverse financial data, including
historical trading data of open price, high price, low price, closing price, and volume and
technical indicators of stock market indexes and macroeconomic variables.
The experimental results show that it produces
more accurate one-day-ahead stock price predictions
than the similar models, including WLSTM
(i.e., a combination of WT and LSTM), RNN, and LSTM.
\subsection{Machine Learning Prediction-Based Investment Portfolios}
Machine learning
has been applied to portfolio construction while
focusing on the portfolio optimization problem,
with multiple objective functions being subject to a set of constraints.
Machine learning-based prediction is a valuable tool
that can be used to mitigate difficulties inherent in traditional methods,
i.e., ranking stocks and assessing
their future potential.
Freitas et al. \cite{fre09}
present a new model of
prediction-based portfolio optimization for capturing short-term investment
opportunities.
For the universe of Brazilian stocks, they
combined their neural network predictors featuring normal prediction errors
with the mean--variance portfolio model, and show that
the resulting portfolio outperforms the mean--variance model
and beats the market index.
More recently, Mishra et al. \cite{mis16} developed a novel prediction
based mean--variance (PBMV) model, as an
alternative to the conventional Markowitz mean--variance model,
to solve the constrained portfolio optimization problem.
They present a low-complexity heuristic functional link artificial neural network (HFLANN) model
to overcome the incorrect estimation taken as the mean of the past returns
in the Markowitz mean--variance model, and carry out
the portfolio optimization task by using multi-objective evolutionary algorithms (MOEAs).
Ganeshapillai et al. \cite{gan13} propose a machine learning-based method
to build a connectedness matrix and address
the shortcomings of correlation in capturing
events such as large losses.
They show that
the matrix can be used to build portfolios
that not only ``beat the market,'' but also
outperform optimal (i.e., minimum-variance)
portfolios.
The results of these studies show that machine learning-based
estimations can be effectively used to overcome certain
limitations inherent in traditional method.
With regard to prediction-based portfolios, our work
is in line with this thinking, but is more fundamental in the sense that
we focus heavily on predicted returns by imposing thresholds
with respect to prediction
and diversification effects, by aggregating stocks rather than
adopting existing portfolio frameworks. This fact makes TVP
more data-driven than existing models.
\section{Models: S-RNN, LSTM, GRU}\label{prediction models}
S-RNNs \cite{elm90} are an extension of a conventional FFNN that adds a feedback connection
to a feedback network consisting of three layers: an input layer,
a hidden layer, and an output layer. However, \cite{ben94} found it is difficult
to train an S-RNN to capture long-term dependencies, because the gradients tend to
either vanish or explode. Alternatively, LSTM \cite{hoc97} and GRU \cite{cho14}
have been proposed to overcome the problem by using a ``gating'' approach.
The LSTM algorithm
is local in space and time \cite{hoc97}, which means that the
computational complexity of learning LSTM models per weight and time
step with the stochastic gradient descent (SGD) optimization technique is
O(1), and the learning computational complexity per time step is O(W),
where W is the number of weights. Hence, our model is capable
of handling large-scale data, as the computational complexity of our model
grows linearly with respect to the length of the input data.
\subsection{LSTM Architecture}
LSTMs can effectively learn important pieces of information that
may be found at different positions in the financial time series,
by controlling what is added and removed
from memory in the hidden layers. This is conducted by using
a combination of three gates: (1) a forget gate, (2) an input gate, and (3) an output gate.\\
\noindent {\bf Forget Gate:} An LSTM cell receives the current input $x_{t} \in \mathbb{R}^{d}$, the hidden state vectors $\mf{h}_{t-1}\in \mathbb{R}^{n}$, and a cell state $\bm{C}_{t-1} \in \mathbb{R}^{n}$ at time $t-1$. The forget gate then is then calculated as
\begin{equation}
f_{t}=\sigma(\bm{W}_{f}\mf{x}_{t}+\mf{U}_{f}\mf{h}_{t-1}+\mf{b}_{f}),
\end{equation}
where:
\begin{itemize}
\item $\mf{W}_{f}\in \mathbb{R}^{n\times d}$
is the weight matrix from the input $\mf{x}_{t}$ to the forget gate
$\mf{f}_{t}$,
\item $\mf{U}_{f}\in \mathbb{R}^{n\times n}$ is the
weight matrix from the previous hidden vector $\mf{h}_{t-1}$ to the
forget gate $\mf{f}_{t}$,
\item $\mf{b}_{f}\in \mathbb{R}^{n}$ is the forget gate bias,
\item $\mf{f}_{t} \in
\mathbb{R}^{n}$ is the output of the gate, which determines
the amount to be erased from the previous cell state, and
\item $\sigma(\cdot)$ is a sigma function.\\
\end{itemize}
\noindent {\bf Input Gate:} The input gate $\mf{i}_{t}$, which is used to scale the candidate update vector $\widetilde{\mf{C}}_{t}\in \mathbb{R}^{n}$, determines what parts of $\widetilde{\mf{C}}_{t}$ are added to the corresponding memory cell element at time $t$, based
on the recurrent connection from the hidden vector $\mf{h}_{t-1}$ and the input
at time $t$, $\mf{x}_{t}$:
\begin{eqnarray}
\mf{i}_{t}&=&\sigma(\mf{W}_{i}\mf{x}_{t}+\mf{U}_{i}\mf{h}_{t-1}+\mf{b}_{i}),\\
\widetilde{\mf{C}}_{t}&=&tanh(\mf{W}_{c}\mf{x}_{t}+\mf{U}_{c}\mf{h}_{t-1}+\mf{b}_{c}),
\label{eqn_C}
\end{eqnarray}
where:
\begin{itemize
\item $\mf{W}_{i}\in \mathbb{R}^{n \times d}$, $\mf{U}_{i}\in\mathbb{R}^{n \times n}$, and $\mf{b}_{i}\in \mathbb{R}^{n}$ are the input gate parameters,
\item $\mf{W}_{c}\in \mathbb{R}^{n \times d}$, $\mf{U}_{c}\in \mathbb{R}^{n \times n}$,
\item $\mf{b}_{c} \in \mathbb{R}^{n}$ are the parameters for selecting
a candidate state, $\widetilde{\mf{C}}_{t}$, and
\item $tanh(\cdot)$ is the tanh function.
\end{itemize}
Then, the current state of the cell $\mf{C}_{t}\in \mathbb{R}^{n}$ is given
by
\begin{equation}
\mf{C}_{t}=\mf{i}_{t}\odot\widetilde{\mf{C}}_{t}+\mf{f}_{t}\odot \mf{C}_{t-1},
\end{equation}
where $\odot$ represents the element-wise Hadamard product. \\
\noindent {\bf Output Gate:} The output gate $\mf{o}_{t} \in \mathbb{R}^{n}$, which
is used to calculate
the output
$\mf{h}_{t}\in\mathbb{R}^{n}$, determines the output from the current
cell state:
\begin{eqnarray}
\mf{o}_{t}&=&\sigma(\mf{W}_{o}\mf{x}_{t}+\mf{U}_{o}\mf{h}_{t-1}+\mf{b}_{o}), \\
\mf{h}_{t}&=& \mf{o}_{t}\odot tanh(\mf{C}_{t}),
\label{eqn_h}
\end{eqnarray}
where $\mf{W}_{o}\in \mathbb{R}^{n\times d}$, $\mf{U}_{o}\in \mathbb{R}^{n\times n}$,
and $\mf{b}_{o}\in \mathbb{R}^{n}$ are the output gate parameters.
The hidden vector $\mf{h}_{t}$ of the memory cell can be used as the final output of the network.
\subsection{GRU Architecture}
The structure of a GRU can be expressed as follows:
\begin{eqnarray}
\mf{z}_{t}&=&\sigma(\mf{W}_{z} \mf{x}_{t}+\mf{U}_{z}\mf{h}_{t-1}+\mf{b}_{z}), \\
\mf{r}_{t}&=&\sigma(\mf{W}_{r} \mf{x}_{t}+\mf{U}_{r} \mf{h}_{t-1}+\mf{b}_{r}), \\
\mf{h}_{t}&=&\mf{z}_{t}\odot \mf{h}_{t-1}+(1-\mf{z}_{t})
\odot \textrm{tanh}[\mf{W}_{h}\mf{x}_{t}+\mf{U}_{h}(\mf{r}_{t}
\odot \mf{h}_{t-1})+\mf{b}_{h}],
\end{eqnarray}
where:
\begin{itemize
\item $\mf{x}_{t},\mf{h}_{t},\mf{z}_{t}$, and $\mf{r}_{t}$
are the input vector, output vector, update gate vector,
and reset gate vector, respectively, and
\item $\mf{W},\mf{U}$, and $\mf{b}$ are forward matrices, recurrent matrices, and
biases, respectively.
\end{itemize}
\section{Experiment}\label{results}
\subsection{Data}\label{data}
\subsubsection{Universe}
The asset universe consists of the top 10 stocks in Standard and Poor's 500 index (S$\&$P500):
\begin{itemize}[label=$\bullet$]
\item Apple (AAPL), Amazon (AMZN), Bank of America Corporation (BAC), Berkshire Hathaway Inc. Class B (BRK-B), General Electric Company (GE), Johnson$\&$Johnson (JNJ), JPMorgan Chase $\&$ Co. (JPM), Microsoft Corporation (MSFT), AT$\&$T Inc. (T), and Wells Fargo $\&$ Company (WFC).
\end{itemize}
We use data from January $1997$ to December $2016$ from Yahoo Finance.
The daily stock dataset contains five attributes: open price, high price,
low price, adjust close price, and volume (OHLCV).
Figure \ref{fig_stockprice} graphically shows the normalized closed price
(i.e., subtract the mean from each original value and then divide by the standard deviation).
We convert the daily OHLCV dataset to four different monthly OHLCV datasets by calculating
the last, mean, maximum, and minimum values
of the daily OHLCV dataset per month.
Each monthly OHLCV dataset is used as a raw dataset for forecasting
one-month-ahead return at the end of each calendar month.
\begin{figure}[t]
\centering
\scalebox{0.4}
{
\includegraphics{Fig1.eps}
}
\caption{(Color online) Normalized stock prices for the 10 sample stocks over the test period
}
\label{fig_stockprice}
\end{figure}
\subsubsection{Preprocessing}
To achieve higher quality and reliable predictions, the five attributes are preprocesed as a percentage change, $(x^{(t)}-x^{(t-1)})/x^{(t-1)}$. All data were divided into
a training dataset (70$\%$) to set the model parameters
and the test set (30$\%$) for an out-of-sample model evaluation.
The 30$\%$ of the training set is used as the validation to evaluate a given model
during training.
The statistical characteristics of the data used
to train and test the deep learning model
are shown in Table \ref{StatData}.
We observe that the data are roughly in the range of $-1$ to $1$
which is a usual range of features in deep learning, save for
abnormal trading volume from max values of the volume data.
\begin{table}[htbp]
\tiny
\centering
\caption{Statistics of data (closed prices and volume) used to train, validate, and test the RNN models}
\begin{tabular}{lrrrrrrrrrr}
\toprule
& AAPL & AMZN & BAC & BRK-B & GE & JNJ & JPM & MSFT & T & WFC \\
\midrule
& \multicolumn{10}{c}{Train data} \\
Mean & 0.01, 0.45 & 0.01 0.12 & 0.01 0.124 & 0.00 0.21 & 0.00 0.17 & 0.007 0.186 & 0.00 0.16 & -0.00 0.13 & -0.00 0.20 & 0.01 0.13 \\
Std. & 0.16,2.22 & 0.20 0.59 & 0.08 0.549 & 0.06 0.79 & 0.07 0.74 & 0.069 0.901 & 0.11 0.83 & 0.12 0.73 & 0.10 0.84 & 0.07 0.62 \\
Min. & -0.57, -0.88 & -0.41 -0.75 & -0.22 -0.76 & -0.12 -0.76 & -0.17 -0.617 & -0.157 -0.66 & -0.28 -0.80 & -0.34 -0.80 & -0.18 -0.76 & -0.16 -0.84 \\
Max & 0.45,16.68 & 0.62 1.71 & 0.17 2.198 & 0.26 2.34 & 0.19 2.93 & 0.174 5.603 & 0.25 5.17 & 0.40 4.84 & 0.29 4.56 & 0.23, 2.93 \\
\hline
& \multicolumn{10}{c}{Validation data} \\
Mean & 0.06, 0.31 & -0.00 0.11 & 0.01 0.152 & 0.00 0.13 & 0.00 0.082 & 0.00 0.14 & 0.01 0.21 & 0.00 0.32 & 0.015 0.20 & 0.01, 0.12 \\
Std. & 0.12, 1.06 & 0.14 0.60 & 0.03 0.822 & 0.02 0.52 & 0.02 0.393 & 0.027 0.68 & 0.04 0.77 & 0.05 1.66 & 0.03 0.878 & 0.02, 0.58 \\
Min. & -0.15, -0.72 & -0.30 -0.70 & -0.04 -0.68 & -0.05 -0.50 & -0.065 -0.48 & -0.04 -0.65 & -0.06 -0.59 & -0.11 -0.79 & -0.06 -0.73 & -0.02, -0.67 \\
Max. & 0.35, 5.03 & 0.36 1.96 & 0.09 3.62 & 0.05 1.30 & 0.058 1.19 & 0.05 2.90 & 0.09 2.85 & 0.08 8.50 & 0.08 4.30 & 0.07, 1.41 \\
\hline
& \multicolumn{10}{c}{Test data} \\
Mean & 0.03, 0.11 & 0.04 0.36 & -0.00 0.22 & 0.00 0.27 & -0.011 0.11 & 0.00 0.10 & 0.00 0.20 & 0.00 0.15 & 0.00 0.05 & 0.00, 0.23 \\
Std. & 0.12, 0.54 & 0.14 1.40 & 0.21 0.68 & 0.06 1.06 & 0.11 0.51 & 0.04 0.51 & 0.11 0.75 & 0.08 0.70 & 0.06 0.40 & 0.14, 0.80 \\
Min. & -0.32, -0.82 & -0.25 -0.72 & -0.53 -0.60 & -0.14 -0.86 & -0.27 -0.58 & -0.12 -0.61 & -0.23 -0.73 & -0.16 -0.69 & -0.15 -0.63 & -0.35, -0.74 \\
Max. & 0.23, 1.71 & 0.54 6.45 & 0.73 2.25 & 0.12 5.64 & 0.25 1.87 & 0.07 1.49 & 0.24 2.39 & 0.24 3.05 & 0.09 1.50 & 0.40, 3.65 \\
\bottomrule
\end{tabular}%
\label{StatData}%
\end{table}%
\subsection{Experimental Design}
We build S-RNN, LSTM, and GRU architectures for one-month-ahead forecasts of stock returns.
Based on the validation set evaluation, we carried out a grid search
over their hyperparameters over the number of RNN hidden layers (1,2, or 3) and
the number of hidden units per layer (8, 16, 32, 64, or 128), and whether dropout is used
to avoid overfitting of the model.
The whole networks was trained by a backpropagation algorithm by minimizing the
the quadratic loss value, $L=\frac{1}{2}\sum_{t}^{T}(r(t)-\hat{r}(t))^{2}$
(where $\hat{r}(t)$ is the output of the last layer and $r(t)$ is the corresponding target)
on the validation set. The efficient ADAM (adaptive moment estimation)
optimization algorithm \cite{kin14} with a learning
rate of 0.001 is used to fit the models
in mini-batches of size 20. From the experiments,
we specified the topology consisting of an input layer,
an RNN layer with $h=36$ hidden neurons, a 50$\%$ dropout layer, and
an output layer with a linear activation function for regression.
The feature vectors to feed the models are overlapping sequences of 36 consecutive points (trading months in three years) for the preprocessed variables.
The sequences themselves are sliding windows shifted
by one month for each time $t\geq 36$, that is,
$\{\mf{x}_{t-35},\mf{x}_{t-34},\cdots,\mf{x}_{t} \}$.
The experimental set-up is implemented over a laboratory prototype,
equipped with an Intel quad core i7-6700 processor at 3.4GHz, Nvidia GPU (i.e., GTX 1070),
and 32GB of RAM running the Ubuntu 16.04.2 LTS x86-64 Linux
distribution. Prediction models are evaluated using Keras 2.0.4 \cite{cho16} and TensorFlow 0.11.0. In our approach, the stage of modeling and forecasting
stock returns contributes significantly to the overall processing time
and, for one asset, is obtained in an approximate processing time of
109 seconds.
\subsection{Prediction accuracy}
We evaluate the predictive ability of the three models using the hit ratio, which is defined as follows:
\begin{align}
\textrm{Hit ratio}=\frac{1}{N}\sum_{i=1}^{N} P_{i}(i=1,2,\ldots,N),
\end{align}
where $N$ is total trading months and $P_{i}$ is the prediction result
for the $i^{th}$ trading day, defined as:
\[ P_{i} =
\begin{cases}
1 & \quad \text{if } y_{t+1} \cdot \hat{y}_{t+1}
>0, \\
0 & \quad \text{ otherwise},\\
\end{cases}
\]
where $y_{t+1}$ and $\hat{y}_{t+1}$ are the realized return at the last
business day of month $t+1$
and the one-month-ahead return predicted at the last
business day of month $t$, respectively.
Table \ref{tab_hit}
shows the mean and standard deviation (SD) of the hit ratios
for the 10 assets and the use of
the last business day and the LSTM model
generates the best prediction accuracy value (0.604).
Therefore, we will use the LSTM model and the last business day OHLCV of each month
for building TBPs in the next subsections.
\begin{table}[htbp]
\centering
\caption{Mean and SD of hit ratios for the ten assets, respectively}
\begin{tabular}{lrrrr}
\toprule
& Last & Mean & Max & Min \\
\midrule
S-RNN& 0.559, $\mf{0.040}$ & 0.555, 0.066 & 0.555, 0.059 & 0.555, 0.048 \\
LSTM& $\mf{0.604}$, 0.042 & 0.536, 0.083 & 0.569, 0.048 & 0.584, 0.039 \\
GRU& 0.573, 0.053 & 0.550, 0.079 & 0.586, 0.049 & 0.575, 0.051 \\
\bottomrule
\end{tabular}%
\label{tab_hit}%
\end{table}%
\subsection{Role of Threshold and TBP
We present the three type of TBP imposing the positive and negative threshold levels
($\theta^{+}$ and $\theta^{-}$, respectively) on the predicted returns. Given
the one-month-ahead return prediction $\hat{r}_{i}$ for asset
$S_{i} (i=1,2,\ldots, n)$, the TBPs are defined as
the subset of the universe:
\begin{itemize}[label=$\bullet$]
\item Long only TBP: $\{ S_{i} \in \textrm{Universe} \textrm{ } |\textrm{ } \hat{r}_{i}\geq \theta^{+} \}$
\item Short only TBP: $\{ S_{i} \in \textrm{Universe} \textrm{ } |\textrm{ } \hat{r}_{i} < -\theta^{-} \}$
\item Long--short TBP: $\{ S_{i} \in \textrm{Universe} \textrm{ } |\textrm{ } \hat{r}_{i}\geq \theta^{+} \textrm{and } \hat{r}_{i} < -\theta^{-} \}$
\end{itemize}
To illustrate, the thresholds are used to classify assets as long and short positions.
A long (short) equity portfolio consists of assets
whose prediction is higher (lower) than $\theta^{+}$ ($\theta^{-}$).
Here, the thresholds are exogenous variables, and
as explained in the next sections,
we can determine proper threshold values for the target portfolio through backtesting.
\subsection{Portfolio Weight}
As classical portfolio models, the TBPs are built on the following underlying
assumptions for evaluating performance.
(i) all stocks are infinitely divisible;
(ii) there are no restrictions on buying and selling any selected portfolio;
(iii) there is no friction (transactions costs, taxation, commissions, liquidity, etc.); and
(iv) it is possible to buy and sell stocks at closing prices at any time $t$.
We adapt the periodic rebalancing strategy:
the investor adjusts the weights in his portfolio on the last business day of
every month, as academic research typically assumes
monthly portfolio rebalancing.
Throughout this study, we provide
the results of experiments on the long TBP (TBP in short), and other TVPs can be easily built
by adjusting the thresholds. Let $w_{i}$ denote the TBP weight on the $i^{\textrm{th}}$ asset.
The TBPs ($w_{i}\geq 0$) are subjected to the budget constraint
$\sum_{i}^{P}w_{i}=1$, where $P$ is the number of assets in the TBP.
For all the TBPs, $w_{i}$ is defined as $|w_{i}|=1/P$, that is, equally-weighted TBPs.
\subsection{Simulation Results}
\subsubsection{Experiment 1: Performance of TBPs}
Table \ref{Table_performance_TBP} provides the mean and SD
(standard deviation) values of the monthly returns of the individual assets,
the EWP of the universe, and the TBPs with different thresholds.
The explanation is as follows:
\begin{itemize}[label=$\bullet$]
\item AAPL and AMZN achieve higher returns with higher volatility.
\item The EWP achieves lower volatility by diversification effect,
which leads higher Sharpe ratio.
\item The EWP and TBPs overly outperform individual stocks in terms of
the Sharpe ratio.
\item As shown in the TBPs, an increase in $\theta$ results
in an increase in the return and volatility of TBPs.
\end{itemize}
\begin{table}[htbp]
\centering
\caption{Performance of the individual assets, EWP, and TBPs }
\begin{tabular}{rrrrrrr}
\toprule
\multicolumn{2}{c}{Asset/Portfolios} & \multicolumn{1}{c}{Threshold} & Mean & SD
& Mean/SD
& \thead{Average \\ assets}\\
\midrule
\multicolumn{2}{l}{AAPL} & \multicolumn{1}{l}{} & 0.02 & 0.075 & 0.271 &\\
\multicolumn{2}{l}{AMZN} & \multicolumn{1}{l}{} & 0.023 & 0.08 & 0.299 &\\
\multicolumn{2}{l}{BAC} & \multicolumn{1}{l}{} & 0.013 & 0.102 & 0.128 &\\
\multicolumn{2}{l}{BRK} & \multicolumn{1}{l}{} & 0.01 & 0.0378 & 0.275 &\\
\multicolumn{2}{l}{GE} & \multicolumn{1}{l}{} & 0.01 & 0.053 & 0.202 &\\
\multicolumn{2}{l}{JNJ} & \multicolumn{1}{l}{} & 0.0129 & 0.036 & 0.353 &\\
\multicolumn{2}{l}{JPN} & \multicolumn{1}{l}{} & 0.014 & 0.075 & 0.19 &\\
\multicolumn{2}{l}{MSFT} & \multicolumn{1}{l}{} & 0.017 & 0.062 & 0.275 &\\
\multicolumn{2}{l}{T} & \multicolumn{1}{l}{} & 0.01 & 0.042 & 0.239 &\\
\multicolumn{2}{l}{WFC} & \multicolumn{1}{l}{} & 0.011 & 0.048 & 0.237 &\\
\multicolumn{2}{l}{EWP} & \multicolumn{1}{l}{} & 0.014 & 0.039 & 0.368 &\\
\hline
\multicolumn{2}{l}{\multirow{5}[2]{*}{TBP}} & \multicolumn{1}{l}{0} &
0.015 & 0.04 & 0.381 & 9.072\\
\multicolumn{2}{l}{} & \multicolumn{1}{l}{0.005} & 0.017 & 0.044 & 0.381 & 6.637 \\
\multicolumn{2}{l}{} & \multicolumn{1}{l}{0.01} & 0.02 & 0.052 & 0.383 & 4.376\\
\multicolumn{2}{l}{} & \multicolumn{1}{l}{0.015} & 0.02 & 0.06 & 0.347 & 2.855\\
\multicolumn{2}{l}{} & \multicolumn{1}{l}{0.02} & 0.018 & 0.069 & 0.269 & 2.173\\
\bottomrule
\end{tabular}%
\label{Table_performance_TBP}%
\end{table}%
The remarkable fact is that the EWP serves as a benchmark
for evaluating the TBPs, in the sense that
there is a (roughly) consistent up--right shift from the point of the EWP
on the risk--return plane: as $\theta$ increases,
the return increases from 0.014 (EWP)
to 0.015 ($\theta=0.000$), and then to 0.018 ($\theta=0.02)$;
the risk increases from 0.039 (EWP) to 0.04 ($\theta=0$),
and then to 0.069 ($\theta=0.02$).
In the next section, this relationship is more clearly elucidated
as the risk--return frontier.
The EWP has been frequently used as a proxy for the risk--return ratio
of the financial market, by both academia and the financial industry
\cite{jeg01,ply01}.
It is more diversified than a value-weighted portfolio, which is
heavily concentrated into just the largest companies, so that
it is being widely traded in the real financial industry (e.g.,
the NASDAQ-100 Equal Weighted Index
allots the same weight to each stock in the index).
Therefore, the fact that such a well-known EWP serves as a benchmark
for TVPs allows us to more quantitatively characterize TBPs.
\begin{figure}[t]
\centering
\scalebox{0.3}
{
\includegraphics{Fig2.eps}
}
\caption{(Color online) Distributions of TBP monthly returns}
\label{fig_DP_system}
\end{figure}
\begin{table}[htbp]
\centering
\caption{Prediction accuracy values for assets whose predictive return is higher than $\theta$ over the test period}
\begin{tabular}{lrrr}
\toprule
$\theta$ & \thead{No. of correct \\ forecasts} & \thead{No. of total \\ forecasts } & Accuracy \\
\midrule
\multicolumn{1}{l}{0} & 343 & 562 & 0.61 \\
\multicolumn{1}{l}{0.0025} & 321 & 521 & 0.616 \\
\multicolumn{1}{l}{0.005} & 225 & 405 & 0.629 \\
\multicolumn{1}{l}{0.0075} & 204 & 326 & 0.625 \\
\multicolumn{1}{l}{0.01} & 171 & 272 & 0.628 \\
\multicolumn{1}{l}{0.0125} & 134 & 216 & 0.62 \\
\multicolumn{1}{l}{0.015} & 110 & 177 & 0.621 \\
\multicolumn{1}{l}{0.0175} & 95 & 152 & 0.625 \\
\multicolumn{1}{l}{0.02} & 83 & 134 & 0.619 \\
\multicolumn{1}{l}{0.0225} & 74 & 117 & 0.632 \\
\multicolumn{1}{l}{0.025} & 67 & 108 & 0.62 \\
\bottomrule
\end{tabular}%
\label{Threshold_Accuracy}%
\end{table}%
We examine the relationship between the magnitude of predictive returns and the prediction accuracy.
Table \ref{Threshold_Accuracy}
shows the correct forecasts among
all forecasts whose value is larger than $\theta$.
For the different $\theta$s, the prediction accuracy ranges over
$0.61 \sim 0.63$, independent of $\theta$.
We calculate the accumulated returns of TVPs
by using the closing prices on the last trading day of each month.
We rebalance all portfolios on the last trading day of each month based on the
one-month-ahead prediction; we then reinvest according
to a weight vector that divides the accumulated wealth equally among
the constituents.
The accumulated return $R_t$ is defined as:
\begin{align}
R_{t} = \prod _{i=0}^{t}(1+r_{i}),
\end{align}
where $r_{i}$
is the arithmetic return at time $i$. This is a standard performance measure
for comparing investments, and it relates the wealth at time $t$, $W_{t}$
, to the initial wealth, $W_{0}$, as
$W_{t} = W_{0}\times R_{t}$.
All experiments in this study used an initial wealth value of $W_{0} = 1$.
Figure \ref{CumulativeR_Scenario2} shows
the cumulative returns of the individual assets, EWP, and TBPs.
\begin{figure}[t]
\centering
\scalebox{0.35}
{
\includegraphics{Fig3.eps}
}
\caption{(Color online) Cumulative returns of individual assets, EWP, and TBPs}
\label{Fig_cumul_1}
\end{figure}
\subsection{Experiment 2: Robustness Test}
As a further check, we conduct a similar experiment
with the whole of the study period (i.e., January 1, 2006 to December 31, 2014).
Figure \ref{CumulativeR_Scenario2} graphically shows the cumulative returns over the test period (i.e., 30$\%$ of the period).
Over the test period, the market is more volatile than the previous one,
and the LSTM-based predictors shows a poor predictive accuracy value of 0.495.
Much of our analysis generated results similar
to those in earlier sections, but interestingly, there is
a positive relation between the magnitude of predictive return
and the prediction accuracy: that is,
the accuracy consistently increases from 0.529 ($\theta=0.00$) to 0.611 at
($\theta=0.225$), as shown in Table \ref{Threshold_Accuracy2}.
\begin{figure}[t]
\centering
\scalebox{0.35}
{
\includegraphics{Fig4.eps}
}
\caption{(Color online) Experiment 2 results: Cumulative returns of individual assets, EW, and TBPs}
\label{CumulativeR_Scenario2}
\end{figure}
\begin{table}[htbp]
\centering
\caption{Experiment 2 results: Prediction accuracies for the assets whose predictive return is higher than $\theta$ over the test period}
\begin{tabular}{lrrr}
\toprule
$\theta$ & \thead{No. of correct \\ forecasts} & \thead{No. of total\\ forecasts } & Accuracy \\
\midrule
\multicolumn{1}{l}{0} & 142 & 268 & 0.529 \\
\multicolumn{1}{l}{0.0025} & 124 & 238 & 0.521 \\
\multicolumn{1}{l}{0.005} & 115 & 218 & 0.527 \\
\multicolumn{1}{l}{0.0075} & 104 & 192 & 0.541 \\
\multicolumn{1}{l}{0.01} & 96 & 172 & 0.558 \\
\multicolumn{1}{l}{0.0125} & 79 & 143 & 0.552 \\
\multicolumn{1}{l}{0.015} & 73 & 130 & 0.561 \\
\multicolumn{1}{l}{0.0175} & 63 & 108 & 0.583 \\
\multicolumn{1}{l}{0.02} & 57 & 95 & 0.6 \\
\multicolumn{1}{l}{0.0225} & 52 & 85 & 0.611 \\
\multicolumn{1}{l}{0.025} & 46 & 73 & 0.605 \\
\bottomrule
\end{tabular}%
\label{Threshold_Accuracy2}%
\end{table}%
\section{Applications}\label{sec_app}
Regarding the practical use of TBPs, we provide illustrations
on how to manage them over multiple-periods and
and how to incorporate them into an MPT optimization portfolio.
\subsection{TBP Management}
\begin{figure}[t]
\centering
\scalebox{0.55}
{
\includegraphics{Fig5.eps}
}
\caption{(Color online) The realized monthly return versus risk of the TBPs, which are constructed using one-month-ahead return predictions, at $\theta=0.000, 0.0025,\ldots, 0.025$ on the last 10 months of the test period of experiment 1. $T$ is
the last business month. }
\label{Frontier}
\end{figure}
Figure \ref{Frontier} is a scatter plot of
the risk--return profiles of the TBPs
that we are constructed in experiment 1, at different
$\theta$s ($0.000, 0.0025,\ldots, 0.025$) over the last 10 months, along
with the lines fitted to the polynomial of degree 3.
We will refer to the line as the ``TBP frontiers.''
Note that the points indicate the realized monthly returns
and risk of the (predictive) TBPs
built using one-month-ahead predictive returns.
The TBP lines are concave, moving
upward and to the right as $\theta$ increases, thus indicating
that the greater the amount of risk by the increase of $\theta$, the greater
the realized returns.
This characteristics allows for the design of a TBP with a target risk--return.
To illustrate, let us suppose that an investor at time $T-9$ hopes to build a TBP to achieve
a target risk--return at time $T-8$.
If the target is the monthly risk of 0.05
and the monthly return of 0.016, the investor can estimate
the $\hat{\theta}$, which corresponds to the target from the TBP frontier drawn at $T-9$;
the investor can then build a TBP with the target,
using the predictive return generated and the estimated $\hat{\theta}$ at time $T-9$.
Then, at time $T-8$, the investor will obtain an approximate return of 0.017 and risk of 0.05,
as seen in the TBP frontier moving upward over the period from time $T-9$ to $T-8$.
This difference, $0.017-0.015$, is
the estimation risk of the TBP. As seen in the continual shift of TBPs as time passes,
to maintain a target risk--return, the corresponding $\hat{\theta}$
needs to be updated.
The estimation risk of TBP can be quantified by
calculating the average of the differences of realization
and expectation for both return and risk over a past period.
There is the estimation risk, but it seem to be sufficiently small to classify TBPs as
having different risk aptitude.
The TBP frontier can be more broad, and combined with riskless assets.
In summary, we illustrate the TBP management process:
\begin{enumerate}[itemindent=1cm,label=\bfseries Step \arabic*.]
\item Set a investment universe (stocks, bonds, ETFs, etc.)
\item Build forecasting models for future stock return or price
\item Select a trading position (long-only, short-only, or long--short)
and a weighting method (equal-weighted, prediction-weighed, etc.)
\item (Backtest) Draw the TBP frontiers at different thresholds
\item Select a target risk--return value and find its corresponding $\hat{\theta}$ on the TBP frontier
\item (Actual investment) Invest in the TBP with the $\hat{\theta}$
\item (Realization) Estimate the TBP at $\hat{\theta}$, and
reinvest in the TBP with the updated $\hat{\theta}$ from the realized TBP frontier
\end{enumerate}
\subsection{Mean-Variance Portfolio}
MPT is a mathematically elegant framework for building a portfolio with
specific risk-return level.
However,
it is well known that it is more difficult to estimate the means
than the covariances of asset returns \cite{mer80},
and errors in the estimates of means will have a greater impact on portfolio weights
than errors in the estimates of covariances. Furthermore,
as mean--variance optimization is extremely sensitive to expected returns, any
errors therein might make outcomes far from optimal \cite{jor85,bes92}.
For this reason, although theoretical and empirical academic studies
have examined various MPT aspects, its real-life practical applications
have mostly focused on minimum variance portfolios.
This estimation error invariably leads to inefficient portfolios,
which can be explained by considering the following three sets of portfolios \cite{bro93}.
\begin{itemize}[label=$\bullet$]
\item True efficient frontier (TEF): the efficient frontier based on true (but unknown
parameters)
\item Estimated frontier (EF): the frontier based on the estimated (and hence
incorrect) parameters
\item Actual frontier (AF): the frontier comprising the true portfolio mean and variance points
corresponding to portfolios on the estimated frontier
\end{itemize}
The use of thresholds on predicted returns
can help mitigate inefficiency by
screening a subset to be predicted more accurately,
as shown in Table \ref{Threshold_Accuracy2}.
Figure \ref{fig_threshold_MPT} shows a schematic scenario that
$\textrm{EF}_{T}$, which is estimated for the assets
screened by a threshold,
is located more closely to the TEF. (Here, for simplicity, we ignore the shift in
EF due to the change in the asset number.)
\begin{figure}[t]
\centering
\scalebox{0.3}
{
\includegraphics{Fig6.pdf}
}
\caption{(Color online) Schematic scenario for shifting EF to $\textrm{EF}_{T}$
by screening the universe at a threshold}
\label{fig_threshold_MPT}
\end{figure}
\section{Conclusion}\label{conclusion}
This study proposes a novel framework by which to construct portfolios
that target specific risk--return levels. We evaluated
the RNN networks while examining the hit ratios of the one-month-ahead forecasts
of stock returns, and then constructed TBPs
by imposing thresholds for the potential return.
The TBPs are more data-driven in building a portfolio than in existing methods,
in the sense that they are constructed purely on
the basis of data-driven models by a deep learning technique, in absence of any financial
mathematics or knowledge.
We showed that the EW of the universe plays the role of the reference portfolio to TBPs,
and thus serves to quantitatively characterize the TBP.
The TBP frontiers show that
the threshold is a parameter used to control tradeoff between the risk and the return of portfolios.
We discussed how to practically manage TBPs
to maintain a target risk--return over multiple-periods; we also
discussed the benefit of incorporating
TBPs into MPT.
The TBP is promising, since it provides a simple and straightforward way to
build portfolios with target risk--returns, using predictions alone.
Any prediction model
can be basically applied to construct TBPs.
As predictors become more accurate,
TBPs can achieve greater returns, given a certain level of risk.
In this respect, we believe that
the TBP is a valuable application of machine learning to modern-day investment practice.
\begin{acknowledgements}
This work was partly supported by the ICT R$\&$D program of MSIP/IITP [2017-0-00302, Development of Self Evolutionary AI Investing Technology] and
the ICT R$\&$D program of MSIP/IITP
[2014-0-00616, Building an Infrastructure of a Large Size Data Center].
\end{acknowledgements}
|
1,314,259,996,479 | arxiv | \section{Introduction}
The phenomenon of dispersive wave propagation is fundamental to our understanding of a wide variety of spatially extended physical systems. In such systems, the frequency, $\omega_\k$, of each wave mode is a nonlinear function of its wave-vector, $\k$ \cite{whitham_linear_1974}. Examples include gravity-capillary waves on fluid interfaces \cite{zakharov_kolmogorov_1992}, flexural waves in thin elastic plates \cite{graff_wave_1991}, drift waves in strongly magnetized plasmas \cite{horton_quasitwodimensional_1994} and Rossby waves in planetary oceans and atmospheres \cite{pedlosky_geophysical_1987}. In this article, we will be interested in Rossby waves as modeled by the Charney-Hasegawa-Mima (CHM)
equation \cite{horton_quasitwodimensional_1994} on the $\beta$-plane:
\begin{equation}
\label{eq-CHM}
\partial_t\left(\Delta \psi - F \psi \right) + \beta \partial_x \psi + J\left[\psi,\Delta \psi\right]= 0.
\end{equation}
This is the simplest two-dimensional model of the large scale dynamics of a shallow layer of fluid on the surface of a strongly rotating spehere. The surface of the sphere is approximated locally by a plane, $\mathbf{x} \in \mathbb{R}^2$, with $x$ varying in the longitudinal (meridional) direction and the $y$ varying in the latitudinal (zonal) direction. The field $\psi(\mathbf{x},t)$ is the geopotential height, $\beta$ is the Coriolis parameter measuring the variation of the Coriolis force with latitude, $F$ is the inverse of the square of the deformation radius and $J\left[f,g\right]$ denotes the Jacobian of two functions, $f$ and $g$. This equation admits harmonic solutions, $\psi(\mathbf{x},t) = \mathrm{Re}\left[ A_\k\, \exp(i\k\cdot\mathbf{x} - i\omega_\k t)\right]$ with $\k \in \mathbb{R}^2$. These solutions, known as Rossby waves, have the anisotropic dispersion relation
\begin{equation}
\label{eq-CHMdispersion}
\omega(\k) = -\frac{\beta k_x}{k^2 + F}.
\end{equation}
Since Eq.\eqref{eq-CHM} is nonlinear, modes with different wave-vectors couple together and exchange energy. If the nonlinearity is weak, one finds that this energy exchange is generally quite slow and occurs most efficiently between groups of
modes which are in {\em resonance}. For the CHM equation, such resonances involve three modes since the nonlinearity is quadratic. Four modes would be involved in the case of systems with cubic nonlinearity. Three wave-vectors $(\k_1, \k_2, \k_3)$ satisfying the resonance conditions,
\begin{equation}
\label{eq-resonance}
\left\{
\begin{aligned}
&\k_3 = \k_1 + \k_2,\\
&\omega(\k_3)-\omega(\k_1)-\omega(\k_2) =0,
\end{aligned}
\right.
\end{equation}
are referred to as a resonant triad. If one projects the spectral representation of the wave equation onto a resonant triad, one obtains a set of ordinary differential equations for the coupled time evolution of the amplitudes of the constituent modes. Such systems of equations appeared as basic models of nonlinear mode coupling in a variety of physical systems including plasma physics \cite{sagdeev_nonlinear_1969}, nonlinear optics \cite{armstrong_interactions_1962} and oceanic internal waves \cite{mccomas_resonant_1977}. An advantage of such models is that the equations of motion for a resonant triad are simple enough that explicit formulae can be obtained for both amplitudes and phases of the resonant modes \cite{craik_wave_1988, lynch2004pulsation, bustamante_effect_2009, bustamante2011resonance, harris_externally_2012}.
A disadvantage, however, is that such triads are generally not closed. Even if energy is initially mostly restricted to a single triad, other resonant modes can be generated which are not in the original triad. This process can repeat in a cascade-like fashion and result in the excitation of a large number of modes. If a large number of degrees of freedom are excited, a statistical description of energy transfer between modes is preferable. Such a description is
provided by the theory of wave turbulence \cite{nazarenko_wave_2011,zakharov_kolmogorov_1992}. This theory provides a kinetic description of energy transfer in ensembles of weakly interacting dispersive waves in which conserved quantities are redistributed along the resonant manifolds. See \cite{newell_wave_2011,newell_wave_2001} for a review.
For an infinite system, in which wave modes are indexed by a continuous wave-vector, the theory of weakly nonlinear wave turbulence becomes asymptotically exact in the weakly nonlinear limit. For finite sized systems, in which the wave modes are indexed by a discrete wave-vector, some subtleties arise. The simplest case, which is particularly relevant to numerical studies of wave turbulence, is a bi-periodic box. In this case, $\k$, is restricted to a
periodic lattice with a minimum spacing, $\Delta k$, between modes. Modulo this spacing, the components of $\k$ must be integer valued. This is an issue because if the components of $\k$ are integers, then the resonance conditions, Eq.\eqref{eq-resonance}, become a problem of Diophantine analysis. Such problems typically have far fewer solutions than their real-valued counterparts and it is generally quite difficult to find them. A complete enumeration of all solutions for the case of Eq.\eqref{eq-CHMdispersion} with $F=0$ was recently provided in \cite{bustamante_complete_2012}. This sparseness of solutions means that, in discrete systems, resonant triads can exist in isolation or in finite groups of triads known as resonant clusters. Two triads belong to the same cluster if they share at least one mode. The dynamics of small clusters consisting of two triads has been studied in considerable detail in \cite{bustamante_dynamics_2009}. Small clusters have attracted some interest in the context of atmospheric
dynamics as a possible
explanation of the unusual periods of certain observed atmospheric oscillations \cite{kartashova_model_2007}. Depending on the dispersion relation, there may or may not exist large clusters capable of distributing energy over a
large range of scales in a discrete system. In the case of Rossby waves on a sphere with infinite deformation radius, such a large cluster does exist \cite{lvov_finite-dimensional_2009}. On the other hand, for capillary waves there are no exactly resonant triads at all \cite{kartashova_wave_1998}. For the dispersion relation Eq.\eqref{eq-CHMdispersion}, numerical explorations indicate that for general values of $F$ large exactly resonant clusters are rare. Thus, in discrete systems, it is often necessary to rely on approximate resonances to account for energy transfer.
Approximate resonance is possible due to the phenomenon known as nonlinear resonance broadening. This is an effect whereby the frequency of a wave acquires a correction to its linear value which depends on the amplitude (see Chaps. 14 \& 15 of \cite{whitham_linear_1974}). Triads which are not exactly in resonance can then interact at finite amplitude if the frequency mismatch is less than this correction. Such triads are known as quasi-resonant triads and satisfy the broadened resonance conditions
\begin{equation}
\label{eq-quasiresonance}
\left\{
\begin{aligned}
&\k_3 = \k_1 + \k_2,\\
&\left|\omega(\k_3)-\omega(\k_1)-\omega(\k_2)\right| \leq \delta
\end{aligned}
\right.
\end{equation}
where $\delta$ is a characteristic value for the resonance broadening taken to be positive. Although Eq.\eqref{eq-quasiresonance} provides only a kinematic description of resonance broadening, the analogous dynamical effect can be very strikingly visualised in linear stability analyses of weakly nonlinear waves \cite{dyachenko_decay_2003,connaughton_modulational_2010} where it is found that the set of unstable perturbations lie in a neighbourhood around the set of exactly resonant perturbations. This set of quasi-resonant modes is often pictured as a ``thickened'' or broadened version of the exactly resonant manifold. For weakly nonlinear systems, this broadening is expected to be small since amplitudes are small. It may nevertheless be large enough to overcome frequency mismatches which arise when wave-vectors are restricted to a discrete grid and prevent discreteness from impeding the cascade of energy. A striking example of this effect is observed for capillary wave turbulence in a biperiodic box. For
this system, since there are no exact
resonances, as the nonlinearity is decreased the resonance broadening eventually becomes smaller than the frequency mismatches due to the grid. Direct numerical simulations illustrate that the cascade of energy to small scales stops entirely when the level of nonlinearity gets sufficiently small leading to the phenomenon of ``frozen turbulence'' \cite{pushkarev_kolmogorov_1999,pushkarev_weak_2000,connaughton_discreteness_2001}.
The interplay between exactly resonant and quasi-resonant clusters means that wave turbulence in discrete systems is nowadays believed to exhibit several regimes. If the typical resonance broadening, $\delta$, is small enough that effectively only exactly resonant clusters can interact, the dynamics are referred to as discrete wave turbulence \cite{lvov_finite-dimensional_2009,kartashova_discrete_2009,kartashova_towards_2010}. If $\delta$ is larger than the typical spacing between modes then effectively all triads can interact at least quasi-resonantly and the classical statistical theory is expected to be valid. In between is a regime consisting of a mixture of exactly resonant and quasi-resonant clusters which has been termed mesoscopic wave turbulence \cite{zakharov_mesoscopic_2005,kartashova_towards_2010}. In this intermediate regime, it has been suggested \cite{nazarenko_sandpile_2006} that forced systems could exhibit some aspects of self-organised criticality. This suggestion is motivated by the idea
that the forcing will cause the characteristic value of $\delta$ to increase until it is large enough for a large quasi-resonant
cluster to form which will then facilitate an ``avalanche'' of energy transfer to the dissipation scale thereby reducing wave amplitudes and the corresponding value of $\delta$.
In this paper we develop a kinematic concept of criticality in quasi-resonant interactions in the Eq.\eqref{eq-CHM}. Specifically, we address the question of how a large quasi-resonant cluster emerges in the CHM equation as $\delta$ is increased. Inspired by the theory of percolation on random networks \cite{stauffer_introduction_1994}, we take ``large cluster'' to mean a cluster that consists of a finite fraction of all modes in the system. We begin by analytically characterising the shape of the quasi-resonant set defined by Eq.\eqref{eq-quasiresonance} as a function of $\delta$ for a single triad in Sec. \ref{sec-singleTriad}. By expressing the boundary of the quasi-resonant set in terms of the intersection of a pair of quadratic forms, we find some surprises. In particular, we find that the area of the set diverges at a finite value of $\delta$ illustrating that the common perception of the quasi-resonant set as a ``thickened'' version of the exact resonant manifold is potentially quite misleading. In
Sec. \ref{sec-manyTriads} we numerically construct the set of quasi-resonant clusters as a function of $\delta$ for various system sizes. We show that a percolation transition occurs at a critical value, $\delta_*$, of the resonance broadening as $\delta$ is increased. At this critical value, the size of the largest cluster rapidly goes from containing a negligible fraction of the modes in the system to containing a finite fraction of them. The value of $\delta_*$ decreases as the inverse cube of the system size, a fact which we trace to quasi-resonant interactions between small scale meridional modes and large scale zonal modes. We finish with a short summary and discussion about what conclusions can be drawn about Rossby wave turbulence from our results.
\section{Characterisation of the quasi-resonant set for a single triad}
\label{sec-singleTriad}
In what follows, we shall take $\k_3$ to be fixed with $\k_2=\k_3-\k_1$. The $\delta$-detuned quasi-resonant set of $\k_3$ is the set of modes, $\k_1$, which satisfy the inequality
\begin{equation}
\label{eq-quasiresonance2}
\left|\omega(\k_3)-\omega(\k_1)-\omega(\k_3-\k_1)\right| \leq \delta.
\end{equation}
This section is devoted to determining the structure of this set as a function of the detuning, $\delta$. The boundaries of this set are given by the pair of curves
\begin{eqnarray}
\label{eq-plusBoundary}\omega(\k_3)-\omega(\k_1)-\omega(\k_3-\k_1) &=& \delta\,,\\
\label{eq-minusBoundary}\omega(\k_3)-\omega(\k_1)-\omega(\k_3-\k_1) &=& -\delta.
\end{eqnarray}
We begin by finding these curves. Clearly it suffices to solve Eq.\eqref{eq-plusBoundary} since the second boundary can be obtained from this by setting $\delta \to -\delta$. To fix notation, let us write
\begin{eqnarray*}
\k_3 &=& (p,q)\,,\\
\k_1 &=& (r,s)\,,\\
k^2 &=& p^2 + q^2.
\end{eqnarray*}
For the CHM dispersion relation, Eq.\eqref{eq-CHMdispersion}, the boundary of the quasi-resonant set, Eq.\eqref{eq-plusBoundary}, then corresponds to the curve in the $(x,y)$ plane implicitly defined by
\begin{displaymath}
-\frac{\beta\, p}{k^2 + F} + \frac{\beta\, r}{r^2 + s^2 + F} + \frac{\beta\, (p-r)}{(p-r)^2 + (q-s)^2 + F} = \delta.
\end{displaymath}
Subsequent formulae will be more compact if we shift the origin to the centre of symmetry of the curve, by defining $x = r-p/2$,
$y = s - q/2.$ Also, we rescale $\delta$ by setting $\beta = 1$ from here on. The curve we wish to study is therefore
\begin{eqnarray}
\nonumber -\frac{p}{k^2 + F} &+& \frac{x + p/2}{(x+p/2)^2 + (y+q/2)^2 + F}\\
\label{eq-curve1} &-& \frac{x - p/2}{(x-p/2)^2 + (y-q/2)^2 + F} = \delta.
\end{eqnarray}
Gathering these terms together with a common denominator we obtain the quartic curve $c(x,y)=0$, where
\begin{equation}
\label{eq-curve2}
c(x,y) = a_1 + a_2(x^2+y^2)^2 + a_3 x^2 + a_4 y^2 - a_5 x y,
\end{equation}
and the coefficients are given by
\begin{eqnarray*}
a_1 &=& \frac{1}{16}\,\left(k^2 + 4 F \right)\left[3k^2 p -\left(k^2+F\right)\left(k^2 + 4F\right)\,\delta \right]\,,\\
a_2 &=& -p - \left(k^2 +F\right)\,\delta\,,\\
a_3 &=& -\frac{1}{2}\left[p(p^2 + 3 q^2 + 6 F) - (k^2+F)(p^2-q^2 - 4F)\,\delta \right]\,,\\
a_4 &=& \frac{1}{2}\left[p(p^2 + 3 q^2 -2 F) - (k^2+F)(p^2-q^2+4F)\,\delta \right]\,,\\
a_5 &=& 2 q\left[ q^2 + F - p(k^2+F)\, \delta\right].
\end{eqnarray*}
If we introduce new variables $u = x^2$, $v = y^2$ and $w = x y$, then it becomes clear that the boundary curve, Eq.\eqref{eq-curve1},
corresponds to the intersection of 2 quadratic surfaces
\begin{eqnarray}
\label{eq-quadraticForms} a_2(u+v)^2 + a_3 u + a_4 v + a_1 &=& a_5 w\,,\\
\nonumber w^2 &=& u v\,.
\end{eqnarray}
Notice that the surface $w^2=u v$ is a cone, which is singular at the origin $x=y=0.$ We have performed a full analysis of the properties of the intersection curves for general values of the parameters $p$, $q$, $F$ and $\delta$. This is
a technical exercise which is not very illuminating. In the interests of clarity, we will restrict ourselves here to illustrating the essential qualitative features of these curves accompanied by some explicit examples.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{./ContoursWithNoIntersection.jpg}
\caption{Solutions of Eq.\eqref{eq-plusBoundary} for different values of $\delta$ with no self-intersection. Here $\k_3 =(\cos \theta, \sin \theta)$ with $\theta=\pi/6$ and $F=1/5$. The values of $\delta_1 \approx -0.721688$ and $\delta_2 \approx 1.20281$ are obtained from Eqs.\eqref{eq-delta1} and \eqref{eq-delta2} respectively.}
\label{fig-contoursNoIntersection}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{./ContoursWithIntersection.jpg}
\caption{Solutions of Eq.\eqref{eq-plusBoundary} for different values of $\delta$ which self-intersect at $\delta=\delta_2$. Here $\k_3 =(\cos \theta, \sin \theta)$ with $\theta=3\pi/7$ and $F=1/5$. The values of $\delta_1 \approx -0.185434$ and $\delta_2 \approx 0.309057$ are obtained from Eq.\eqref{eq-delta1} and Eq.\eqref{eq-delta2} respectively.}
\label{fig-contoursWithIntersection}
\end{figure}
\subsection{The curve typically exists only for a finite range of $\delta$.\\}
This is evident from studying Eq.\eqref{eq-curve1} as $\delta \to \pm \infty$. The LHS can only diverge if one of the denominators diverges. This is impossible if $F>0$. Hence if $F>0$, solutions to Eq.\eqref{eq-curve1} must cease to exist if $\delta$ gets too large by absolute value. In the exceptional case $F=0$, solutions indeed exist for all values of $\delta$ and become localised in the neighbourhood of $\pm (p/2, q/2)$ as $\delta \to \pm \infty$.
\subsection{The curve is bounded except at single critical value of $\delta$.\\}
Examining Eq.\eqref{eq-quadraticForms}, it is clear that since $u$ and $v$ are both positive, it is generally not possible to balance the quadratic terms if $u^2 + v^2 \to \infty$. Hence the curve, if it exists is bounded. The single exception to this occurs when the coefficient of the quadratic terms vanishes. The curve may therefore diverge at the single critical value of detuning given by:
\begin{equation}
\label{eq-delta1}
\delta = \delta_1 \equiv -\frac{p}{k^2 + F}.
\end{equation}
Note that this corresponds to $\delta=\omega(\k_3)$. At $\delta=\delta_1$, some algebra shows that the curve is given by the hyperbola
\begin{equation}
x^2 + 2\,\frac{q}{p} \, x y - y^2 = \frac{1}{4}\,(k^2 + F).
\end{equation}
It follows that the area of the quasi-resonant set diverges as $\delta$ tends to $\delta_1$ from below. This result illustrates that the common conception that the quasi-resonant set looks like a ``thickened'' version of the exact resonant manifold is a misconception. The special case $p\to 0$ corresponding to the case of $\k_3$ becoming zonal is discussed separately below.
\subsection{The curve may self-intersect only at a single critical value $\delta$.\\}
From Eq.\eqref{eq-quadraticForms}, self-intersection is only possible if the curve passes through $(0,0)$ in the $(u,v)$ plane. This can only happen if the coefficient $a_1 =0$. Thus we identify a second critical value of $\delta$ where self-intersection may occur:
\begin{equation}
\label{eq-delta2}
\delta = \delta_2 = \frac{3 k^2 p}{(k^2+F)(k^2 + 4 F)}.
\end{equation}
Note that $a_1=0$ does not necessarily mean the curve self-intersects. It is possible that at $\delta=\delta_2$, the curve reduces to a single
point. To probe what happens at $\delta=\delta_2$, we consider the surface $z=c(x,y)$ defined by Eq.\eqref{eq-curve2} when $\delta=\delta_2$ in the neighbourhood of the origin. Calculation of the partial derivatives indicate that this surface has a critical point at the origin. After some tedious algebra, we find that the determinant of the matrix of second derivatives is
\begin{displaymath}
\Delta = -\frac{4 (k^2+F)^2(k^4 + 4 (q^2-3p^2)F)}{k^2+4 F}.
\end{displaymath}
If $\Delta>0$ then the critical point at $(0,0)$ is a maximum or a minimum. The curve $c(x,y)=0$ is then an isolated point. On the other hand, if $\Delta<0$
the critical point is a saddle and the curve $c(x,y)=0$ has a self-intersection at $(0,0)$. The condition for self-intersection is therefore
\begin{equation}
k^4 + 4 (q^2-3p^2)F >0.
\end{equation}
We note that for $F=0$ we always have a self-intersection.
These qualitative features of the boundary of the quasi-resonant set are illustrated graphically in Figs. \ref{fig-contoursNoIntersection} and \ref{fig-contoursWithIntersection} which show the shape of the curve for different values of $\delta$. Generic parameters, specified in the captions, have been chosen with no particular symmetries. Hence the curves shown in these figures are representative of what is found from the complete analysis of Eq.\eqref{eq-curve1}. Fig. \ref{fig-contoursNoIntersection} shows an example in which no self-intersection occurs at $\delta=\delta_2$, while Fig. \ref{fig-contoursWithIntersection} shows an example in which a self-intersection occurs. The hyperbolic curves identified at $\delta=\delta_1$ are clearly visible in both cases.
\begin{figure*}
\centering
\subfigure[ $\delta=0$]{
\label{fig-p1q0delta0}
\includegraphics[width=0.6\columnwidth]{./p1q0delta0.jpg}
}
\subfigure[$\delta=1/2$]{
\label{fig-p1q0delta0p5}
\includegraphics[width=0.6\columnwidth]{./p1q0delta0p5.jpg}
}
\subfigure[$\delta=1$]{
\label{fig-p1q0delta1}
\includegraphics[width=0.6\columnwidth]{./p1q0delta1.jpg}
}
\subfigure[$\delta=2$]{
\label{fig-p1q0delta2}
\includegraphics[width=0.6\columnwidth]{./p1q0delta2.jpg}
}
\subfigure[$\delta=3$]{
\label{fig-p1q0delta3}
\includegraphics[width=0.6\columnwidth]{./p1q0delta3.jpg}
}
\subfigure[$\delta=4$]{
\label{fig-p1q0delta4}
\includegraphics[width=0.6\columnwidth]{./p1q0delta4.jpg}
}
\caption{\label{fig-meridionalSet} Shaded regions correspond to the resonant set defined by Eq.\eqref{eq-quasiresonance} with $\k_3=(1,0)$, $F=0$ and $\beta=1$ for different values of $\delta$. The exactly resonant manifold is the red solid line. }
\end{figure*}
\begin{figure*}
\centering
\subfigure[ $\delta=0$]{
\label{fig-p0q1delta0}
\includegraphics[width=0.6\columnwidth]{./p0q1delta0.jpg}
}
\subfigure[$\delta=1/10$]{
\label{fig-p0q1delta0p1}
\includegraphics[width=0.6\columnwidth]{./p0q1delta0p1.jpg}
}
\subfigure[$\delta=1/2$]{
\label{fig-p0q1delta0p5}
\includegraphics[width=0.6\columnwidth]{./p0q1delta0p5.jpg}
}
\caption{\label{fig-zonalSet}Shaded regions correspond to the resonant set defined by Eq.\eqref{eq-quasiresonance} with $\k_3=(0,1)$, $F=0$ and $\beta=1$ for different values of $\delta$. The exactly resonant manifold is the red solid line and corresponds to the two axes in this case. As explained in the remark below equation (\ref{eq-zonalCase}), this exactly resonant manifold consists of non-generic triads, i.e., triads where one or more interaction coefficients are identically zero. The case $F=0$ is chosen for simplicity only: for $F>0,$ the resonant sets have the same qualitative shape as in the case $F=0.$ }
\end{figure*}
A couple of special cases are worth noting:
\begin{itemize}
\item[(i)] {$p=1$, $q=0$ and $F=0$\\}
This corresponds to a meridional mode. In this case, $a_5=0$ so the intersection of quadratic forms given by Eq.\eqref{eq-quadraticForms} lies entirely in the $w=0$ plane and reduces to
\begin{equation}
\label{eq-meridionalCase}
-(1 + \delta) (u+v)^2 - \frac{1}{2}(1-\delta)(u-v) + \frac{1}{16}(3-\delta) = 0.
\end{equation}
We can immediately identify the critical points. The curve self-intersects at $\delta=3$ and diverges at $\delta=-1$. The point $\delta=1$ is also noteworthy, being the complementary boundary to the divergent case. For this value of $\delta$ the curve is a perfect circle.
\item[(ii)] {$p=0$, $q=1$ and $F=0$\\}
This corresponds to a zonal mode. In this case the curve simplifies to
\begin{equation}
\label{eq-zonalCase}
\delta\, (u+v)^2 + \frac{1}{2}\,\delta\,(u-v) + \frac{1}{16}\,\delta = - 2\,w.
\end{equation}
The only special value of $\delta$ for this case is $\delta=0$. We then recover the exact resonant manifold of a zonal mode, $x\, y =0$. This consists of the two coordinate axes. It is now less surprising that the boundary of the quasi-resonant set can diverge for finite $\delta$ once we appreciate that the exact resonant manifold of a zonal mode is unbounded. The divergence of the boundary of the quasi-resonant set of the non-zonal modes in some sense reflects the presence of this structure in the dispersion relation.
\end{itemize}
\noindent \textbf{Remark.} Notice that the exactly resonant manifold in case (ii) consists of non-generic triads, i.e., triads where one or more interaction coefficients are identically zero. A point on the $x$-axis corresponds to a so-called catalytic interaction ($|\k_1| = |\k_2|$), for which the zonal mode $\k_3$ does not change its energy but influences the energy exchange between the other two modes in the triad. A point on the $y$-axis corresponds to a `spurious' triad, formed by purely zonal modes, which do not interact at all: the interaction coefficients are all identically zero because the three modes are collinear. In our computations of the network of quasi-resonant modes (Section \ref{sec-manyTriads}), non-generic triads are discarded from the start.
Having given a fairly complete qualitative description of the behaviour of the curves which define the boundaries of the resonant sets as the detuning is varied, it now remains to assemble these boundary curves for postive and negative values of $\delta$ to determine the interior and exterior of the quasi-resonant set. This is again best accomplished by illustration. Figs. \ref{fig-meridionalSet} and \ref{fig-zonalSet} illustrate the quasi-resonant set for the two special cases discussed above for a range of increasing values of $\delta$. The shaded areas in these figures correspond to the quasi-resonant sets. The exact resonant curve is also shown for reference. Fig.\ref{fig-meridionalSet} illustrates one of the key points of this article: the common picture of the quasi-resonant set as a thickened version of the exact resonant manifold is appropriate only for small values of the broadening (eg Fig.\ref{fig-p1q0delta0p5}). One might counter this with the observation that it is only in the weakly nonlinear
regime that it makes sense to be discussing quasi-resonant interactions in the first place and in this regime, the broadening is necessarily small. Eq.\eqref{eq-delta1} tells us, however, that no matter how small the broadening, there are always modes close to the zonal axis, whose quasi-resonant set diverges.
\section{Structure of the network of quasi-resonant modes}
\label{sec-manyTriads}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{./Total_Clusters_var_M.jpg}
\caption{The normalised total number of clusters, $N$, plotted as a function of $\delta$ for different system sizes, $M$. Here, we have fixed $F=\beta=1$.}
\label{fig-CHM_total_numbers}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{./percolation.jpg}
\caption{The density of the largest cluster, $\rho$, as a function of resonance broadening $\delta$ for different system sizes, $M$. Here we have fixed $F=\beta=1$. The percolation threshold, $\delta_*$, decreases as a function of system size. The inset shows the same data plotted as a function of $\delta\,M^3$. The solid line is a fit of Eq.\eqref{eq-phaseTransition} to the data.}
\label{fig-CHM_density}
\end{figure}
In a turbulent system many modes are excited. The fact that a mode can be a member of more than one quasi-resonant set leads to overlap between sets. This allows modes to join together to form a network of quasi-resonant clusters analogous to the exactly resonant clusters which have been extensively studied in the literature to date. In this section we study the structure of this quasi-resonant network as the typical amount of broadening, $\delta$, in the system is varied. We consider a finite system of size $2\pi \times 2\pi$ and truncate the wavenumber space such that there are $M$ modes in the $k_x$ and $k_y$ directions, $M^2$ modes in all. The spacing between modes is $2\pi/M$. The quasi-resonant clusters were obtained by an exhaustive numerical search speeded up by incorporating symmetries of the triads. As explained in the remark below equation (\ref{eq-zonalCase}), we have eliminated from our list of triads the so-called non-generic triads, i.e., the triads for which one or more interaction
coefficients are zero. This includes triads formed out of collinear modes and triads where any two wave-vectors have the same modulus.
We first measure the total number of clusters normalised by the maximum number of possible isolated clusters which is approximately $M^4/3$. This is shown as a function of $\delta$ for several different system sizes, $M$, in Fig. \ref{fig-CHM_total_numbers}. For each system size, as $\delta$ increases, the total number of clusters first increases, then reaches a maximum at a particular value of $\delta$, which we shall denote by $\delta_*$, and then decreases.
The value, $\delta_*$, at which the maximum occurs decreases as the system size, $M$, is increased. In order to understand these results one must realise that for the CHM dispersion relation, the number of exactly resonant clusters which live on the grid of discrete wavevectors is rather small with most discrete triads necessarily exhibiting a small amount of detuning enforced by the geometry. The initial growth in the number of clusters is explained by the addition of isolated triads to the list of quasi-resonant clusters as the broadening grows large enough to incorporate the geometrical detuning generated by the grid. While one would expect this rate of increase to slow down as clusters start joining to form larger clusters, the attainment of a maximum and subsequent decrease in the total number of clusters becomes much easier to understand when we measure the size of largest cluster, $\rho$, which is simply the number of modes in the largest cluster normalised by the system size, $M^2$. This is shown in
Fig. \ref{fig-CHM_density}. We see that as $\delta$
approaches $\delta_*$ the largest cluster quickly makes a transition from including a negligible fraction of the total number of modes in the system to including almost all of them. Thus the network of quasi-resonant modes undergoes a percolation transition at $\delta=\delta_*$. While the results shown in Figs. \ref{fig-CHM_total_numbers} and \ref{fig-CHM_density} were obtained for $\beta=F =1$, we checked that the results are not very sensitive to this choice. Although this is an entirely kinematic study, we expect that the emergence of a percolating quasi-resonant cluster is the explanation for the dynamical transition from discrete wave turbulence to Kolmogorov turbulence in the CHM equation as the nonlinearity of the system is increased.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{./critical_points.jpg}
\caption{Percolation threshold, $\delta_*$, plotted against system size and for different values of $F$. Here, we have fixed $\beta=1$. For large system sizes, $\delta_*$ exhibits the scaling $\delta_* \sim M^{-\sigma_1}$, where we measure $\sigma_1 = 3.00\;(2.97,3.03)$. The brackets show the 95\% confidence interval.}
\label{fig-critical_points}
\end{figure}
The decrease of the percolation threshold, $\delta_*$, as the system size, $M$, increases is clearly due to the fact that as $M$ increases, the spacing
between discrete modes decreases. The amount of broadening required to overcome the geometrical detuning is therefore smaller and hence clusters can
form more easily. We would like to quantify this decrease. We have already learned that the area of the quasiresonant set of any particular mode, $\k$, diverges at a finite value of $\delta=\omega(\k)$ as indicated in Eq.\eqref{eq-delta1}. It seems plausible that as soon as the broadening is sufficiently large to allow any mode in the system to have a divergent quasi-resonant set, then the largest cluster must be of the size of the system. Thus one might estimate
\begin{equation}
\label{delta_sat1}
\delta_*\approx\min_{\k}|\omega(\k)|.
\end{equation}
By this argument, Eq.\eqref{eq-CHMdispersion} tells us that for large systems we should see the scaling, $\delta_* \sim M^{-2}$. A numerical measurement of how $\delta_*$ scales with system size for a range of values of $F$, is plotted in figure \ref{fig-critical_points}. We see that the value of $F$ becomes redundant for large $M$ as already mentioned. Each of the curves converge to the straight line indicated by the scaling
\begin{equation}
\delta_*\sim M^{-\sigma_1},
\end{equation}
with exponent $\sigma_1 = 3.00$ with a 95\% confidence intervals if $(2.97,3.03)$, which was obtained by bias-corrected bootstrapping. The connected component therefore forms much more easily than the naive argument above would suggest. The origin of this $M^{-3}$ scaling can be traced to the
fact that zonal modes require very little detuning in order to interact with high $\k$ almost-meridional modes. Take $F=0$ for simplicity, although the following argument works for any $F>0.$ Let us consider the largest-scale
zonal mode in the system, $\k=(0,1).$ We recall that the exactly resonant manifold $x y = 0$ of this zonal mode, figure \ref{fig-zonalSet}(a), gives rise to non-generic triads which do not interact efficiently and therefore must be discarded. So we ask what is the minimum value of the detuning so that the quasiresonant set of the mode $\k=(0,1)$ contains some new modes. From Fig. \ref{fig-zonalSet}(b) it is clear that the first new mode to join the quasi-resonant set will be $\k_1 = (M,0),$ corresponding to $x=M, y = -1/2.$ By directly replacing these values of $x,y$ into Eq.\eqref{eq-zonalCase}, we obtain
\begin{displaymath}
\left[ \left(M^2+\frac{1}{4}\right)^2 + \frac{1}{2}\left(M^2 - \frac{1}{4}\right) + \frac{1}{16}\right] \, \delta_* = M\,.
\end{displaymath}
For large $M$, this has solution $\delta_* \sim {M^{-3}}$, in agreement with Fig. \ref{fig-critical_points} and the inset of Fig. \ref{fig-CHM_density}. This suggests that the percolation transition is driven by interactions between large scale zonal modes and small scale meridional modes. This is consistent with the known scale-nonlocality of wave turbulence in the CHM equation (see \cite{connaughton_feedback_2011} and the references therein) and provides further evidence, albeit circumstantial, that the percolation transition is associated with the onset of turbulence in the CHM model.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{./M256_missing_nodes.jpg}
\caption{Color map in $\k$-space showing the smallest value of broadening required for each mode to become a member of a cluster of any size. The darker regions are resilient to becoming members of clusters in the sense that large values of broadening are required. Lighter regions join clusters easily.}
\label{fig-M256_missing_nodes}
\end{figure}
In order to further illustrate this point, we can ask which modes have the greatest or least tendency to become part of a quasi-resonant cluster. This is done in figure (\ref{fig-M256_missing_nodes}), where we have coloured each mode according to the minimal amount of resonance broadening required for this mode to join a cluster of any size. Blue modes become active very easily whereas red modes are resilient to becoming part of any cluster. Rather than appearing homogeneous and random, we see the appearance of definite structure. We see a circular region containing modes with a strong propensity to join clusters including a narrow large-scale region of zonal modes accumulating near the $k_x=0$ axis with very low interaction thresholds as expected from the discussion above. We also remark upon the group of large-scale meridional scales reluctant to form any quasi-resonant connections. The relative reluctance of large-scale meridional modes to exchange energy has already been remarked up in the literature
and suggested as an explanation of the inherent anisotropy of Rossby waves turbulence. For example in \cite{vallis_atmospheric_2006}, a wave-turbulence boundary was computed by comparing the CHM dispersion relation to the inverse of the eddy-turnover time. It is then argued that inside this region Rossby waves dominate, but with a frequency incommensurate with that of the surrounding turbulence so that energy cannot penetrate into this region. The boundary thus obtained seems similar in position and shape to the dark region in Fig. \ref{fig-M256_missing_nodes}. This anisotropic energy distribution at large scales has been documented in numerical simulations of Rossby wave turbulence (see \cite{huang_anisotropic_2001} and the references therein). It is somewhat surprising to see it emerging again here from purely kinematic considerations.
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{./fitting.jpg}
\caption{Fitting the data in Fig.\ref{fig-CHM_density} to a standard phase transition profile. The lefthand side of Eq.\eqref{eq-phaseTransition} is plotted as a function of $\delta$. The fit (solid line) is taken over the range $\left[\delta_*:2\right]$ with $\delta_*=0.815$. It has slope 2.03. which suggests a value of $z\approx \frac{1}{2}$.}
\label{fig-fitting}
\end{figure}
Finally we might ask whether the density of the giant cluster illustrated in Fig. \ref{fig-CHM_density} shows the generic profile for a second order phase transition,
\begin{equation}
\label{eq-phaseTransition}
\rho(\delta) = \left\{ \begin{aligned}
&0 &\mbox{$\delta\leq \delta_*$}\\
&c\,(\delta-\delta_*)^z & \mbox{$\delta > \delta_*$},
\end{aligned}
\right.
\end{equation}
and, if so, what is the exponent $z.$ Notice that the system size $M$ has been absorbed after appropriate rescalings of $\delta$ and $\delta_*$ by the factor $M^3.$ Eq.\eqref{eq-phaseTransition} contains 3 adjustable parameters, $\delta_*$, $c$ and $z$, which makes it difficult to unambiguously determine the exponent $z$. As pointed out in \cite{connaughton_warm_2004}, if Eq.\eqref{eq-phaseTransition} holds, then
\begin{equation}
\rho\ \left(\dd{\rho}{\delta}\right)^{-1} = \frac{1}{z}\left( \delta-\delta_*\right)
\end{equation}
for $\delta>\delta_*$. Plotting this quantity against $\delta$ produces an easier fitting problem because the amplitude $c$, cancels out, the fitting becomes linear and the value of $\delta_*$ corresponds to the point at which the fitted line crosses zero. The values of $\dd{\rho}{\delta}$ would ideally be obtained independently from the values for $\rho$. In our case, this was not possible so we obtained them by locally interpolating the measured values of $\rho$ using $Mathematica$ and differentiating the result. The outcome of this analysis is shown in Fig. \ref{fig-fitting}. We can see that a case can be made for a linear fit in the neighbourhood of $\delta_*$. Taking $\delta_*=0.815$ (the point at which the straight line fit obviously starts to fail), and fitting the data over the range $\left[\delta_*:2\right]$ gives the fit shown in the figure. The slope gives a value of $z\approx \frac{1}{2}$. This is standard Landau value for the mean field theory of a second order phase transition of a scalar field.
These results, while suggestive, are far from definitive. A more detailed numerical study in the vicinity of $\delta_*$ will be required before we can start putting estimates of uncertainty on these values. For the purposes of comparision, Eq.\eqref{eq-phaseTransition} with the best fit values of the parameters, is plotted with the original data in the inset of Fig.\ref{fig-CHM_density} (solid line).
\section{Conclusions and outlook}
\label{sec-conclusion}
To conclude, we have presented a kinematic analysis of the properties of quasi-resonant triads in the CHM equation. We described the analytic form of the quasi-resonant set defined by the quasi-resonance conditions, Eq. \eqref{eq-quasiresonance} as a function of the resonance broadening, $\delta$. We found that they have non-trivial geometric shape and are not well described as simply thickened versions of the exact resonant manifold as is commonly assumed. In particular, we found that the quasi-resonant set becomes unbounded for above a critical value of $\delta$ and that this can occur for arbitrarily small values of $\delta$ as we consider modes approaching the zonal axis. We then conducted an in-depth numerical study of the structure of quasi-resonant clusters as a function of $\delta$ and identified a percolation transition as $\delta$ is increased. At the transition, a large cluster is formed which contains a finite fraction of all the modes in the system. For a system containing $M^2$ modes, the value
of the percolation threshold decreases as $M^{-3}$. This scaling results from the ease with which large scale zonal modes interact with small scale meridional modes, a reflection of the nonlocality of Rossby wave turbulence.
We speculate that the percolation transition corresponds dynamically to the transition from mesoscopic to classical wave turbulence. The fact that a percolation transition exists is consistent with earlier work on capillary waves \cite{connaughton_discreteness_2001}. In fact, we believe that this transition is not a special feature of the CHM dispersion relation and is generic \cite{harris_kinematics_2013}. Furthermore our results provide circumstantial support for the sandpile picture of mesoscopic wave turbulence suggested in \cite{nazarenko_sandpile_2006} since a small change in resonance broadening in the vicinity of $\delta_*$ can trigger a transition from a state which cannot support an energy cascade to one which can. In order to better understand these issues, we believe that it is important to move beyond the kinematic picture of resonance broadening and attempt to devise methods of studying these effects dynamically.
\begin{acknowledgments}
C.C. acknowledges the support of the EPSRC grant EP/H051295/1. \, M.D.B. acknowledges support from University College Dublin, Seed Funding Projects SF564 and SF652.
\end{acknowledgments}
|
1,314,259,996,480 | arxiv | \section{Introduction}
As is known the collective behavior of many-particle systems can be effectively described
within the framework of a one-particle marginal distribution function governed by the
kinetic equation which is a scaling limit of underlying dynamics \cite{CGP97}-\cite{Sp91}.
The statement of a problem of the derivation of the Boltzmann kinetic equation from the
Hamiltonian dynamics goes back to D. Hilbert \cite{H}. The sequential analysis of this
problem within the framework of the perturbation theory was realized by N.N. Bogolyubov
\cite{B46}. The approach to the derivation of the Boltzmann equation as a result of the
scaling limit of the BBGKY hierarchy (Bogolyubov--Born--Green--Kirkwood--Yvon) was originated
in H. Grad's work \cite{GH}.
At present significant progress in the problem solving of the rigorous derivation of the
Boltzmann equation for a system of hard spheres in the Boltzmann--Grad scaling limit is
observed \cite{C72}-\cite{V}. Nowadays these results were extended on many-particle
systems interacting via a short-range potential \cite{SR12},\cite{PSS}.
The conventional approach to the derivation of the Boltzmann equation with hard sphere
collisions from underlying dynamics (see \cite{CGP97}-\cite{Sp91} and references cited
therein) is based on the construction of the Boltzmann--Grad asymptotic behavior of a
solution of the Cauchy problem of the BBGKY hierarchy for hard spheres represented in
the form of the perturbation (iteration) series \cite{U01}.
Another approach to the description of the many-particle evolution is given within the
framework of marginal observables governed by the dual BBGKY hierarchy \cite{BGer}. One
of the aims of the present paper consists in the description of the kinetic evolution
of hard spheres in terms of the evolution of observables (the Heisenberg picture of the
evolution). The Heisenberg picture representation of dynamics is in fact the best
mathematically fully consistent formulation, since the notion of state in more subtle,
in particular it depends on the reference frame. For example, for this reason it is useful
in relativistic quantum field theory.
One more approach to the description of the kinetic evolution of hard spheres is based
on the non-Markovian generalization of the Enskog equation. In paper \cite{GG} in case
of initial states specified in terms of a one-particle distribution function the equivalence
of the description of the evolution of hard sphere states by the Cauchy problem of the BBGKY
hierarchy and in terms of a one-particle distribution function governed by the generalized
Enskog kinetic equation was established. The Boltzmann--Grad asymptotic behavior of a
non-perturbative solution of the Cauchy problem of the generalized Enskog equation is
described by the Boltzmann equation with hard sphere collisions \cite{GG04}. We note that
within the framework of the perturbation theory a particular case of this approach reduces
to the Bogolyubov's method \cite{B46} of the derivation of the Boltzmann equation and its
generalizations.
We remark also that the rigorous method of the description of the kinetic evolution of a
tracer hard sphere and an environment of hard spheres was developed in paper \cite{GG12}
and a rigorous derivation of the Brownian motion in the hydrodynamic limit was realized
in \cite{BGS}. The approaches to the rigorous derivation in a mean field scaling limit of
quantum kinetic equations from underlying many-particle quantum dynamics were considered
in reviews \cite{G09},\cite{G12}.
The paper is organized as follows.
In section 2 we formulate necessary preliminary facts about evolution equations of a system
of hard spheres and review recent rigorous results on the derivation of the Boltzmann kinetic
equation from underlying dynamics. In section 3 we develop an approach to the description of
the kinetic evolution of infinitely many hard spheres within the framework of the evolution
of marginal observables. For this purpose we establish the Boltzmann--Grad asymptotic behavior
of a solution of the Cauchy problem of the dual BBGKY hierarchy for marginal observables of
hard spheres. The constructed scaling limit is governed by the set of recurrence evolution
equations, namely by the dual Boltzmann hierarchy. Furthermore, in this section the relationships
of the dual Boltzmann hierarchy for the limit marginal observables with the Boltzmann kinetic
equation are established. In section 4 for the description of the kinetic evolution of hard
spheres we develop one more approach based on the non-Markovian generalization of the Enskog
kinetic equation. We prove that the Boltzmann--Grad scaling limit of a non-perturbative solution
of the Cauchy problem of the generalized Enskog kinetic equation is governed by the Boltzmann
equation and the property on the propagation of initial chaos is established. Moreover, the
Boltzmann--Grad asymptotic behavior of a one-dimensional system of hard spheres is outlined.
Finally, in section 5 we conclude with some observations and perspectives for future research.
\section{Evolution equations of many hard spheres}
It is well known that a description of many-particle systems is formulated in terms
of two sets of objects: observables and states. The functional of the mean value of
observables defines a duality between observables and states and as a consequence
there exist two approaches to the description of the evolution. Usually the evolution
of many-particle systems is described within the framework of the evolution of states
by the BBGKY hierarchy for marginal distribution functions. An equivalent approach
to the description of the evolution of many-particle systems is given in terms of
marginal observables governed by the dual BBGKY hierarchy.
\subsection{The BBGKY hierarchy for hard spheres}
We consider a system of identical particles of a unit mass with the diameter $\sigma>0$,
interacting as hard spheres with elastic collisions. Every hard sphere is characterized
by the phase space coordinates:
$(q_{i},p_{i})\equiv x_{i}\in\mathbb{R}^{3}\times\mathbb{R}^{3},\,i\geq1.$
Let $L^{1}_{\alpha}=\oplus^{\infty}_{n=0}\alpha^n L^{1}_{n}$ be the space of
sequences $f=(f_0,f_1,\ldots,f_n,\ldots)$ of integrable functions $f_n(x_1,\ldots,x_n)$
defined on the phase space of $n$ hard spheres, that are symmetric with respect
to permutations of the arguments $x_1,\ldots,x_n$, equal to zero on the set of the
forbidden configurations: $\mathbb{W}_n\equiv\{(q_1,\ldots,q_n)\in\mathbb{R}^{3n}\big|
|q_i-q_j|<\sigma\,\,\,\text{for at least one pair}\,\, (i,j):i\neq j\in(1,\ldots,n)\}$ and
equipped with the norm: $\|f\|_{L^{1}_{\alpha}}=\sum_{n=0}^{\infty}\alpha^n \|f_n\|_{L^{1}_n}
=\sum_{n=0}^{\infty}\alpha^n \int dx_1\ldots dx_n|f_n(x_1,\ldots,x_n)|$,
where $\alpha>1$ is a real number. We denote by $L_{0}^1\subset L^{1}_{\alpha}$
the everywhere dense set in $L^{1}_{\alpha}$ of finite sequences of continuously
differentiable functions with compact supports.
The evolution of all possible states of a system of a non-fixed, i.e. arbitrary but
finite, number of hard spheres is described by the sequence
$F(t)=(1,F_{1}(t,x_1),\ldots,F_{s}(t,x_{1},\ldots,x_{s}),\ldots)\in L^{1}_{\alpha}$
of the marginal ($s$-particle) distribution functions $F_s(t,x_1,\ldots,x_s),\,s\geq1$,
governed by the Cauchy problem of the weak formulation of the BBGKY hierarchy \cite{CGP97}:
\begin{eqnarray}
\label{NelBog1}
&&\hskip-8mm\frac{\partial}{\partial t}F_s(t)=\big(\sum\limits_{j=1}^{s}\mathcal{L}^\ast(j)+
\epsilon^{2}\sum\limits_{j_{1}<j_{2}=1}^{s}
\mathcal{L}_{\mathrm{int}}^\ast(j_{1},j_{2})\big)F_{s}(t)+\\
&&+\epsilon^{2}\sum_{i=1}^{s}\,\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{s+1}
\mathcal{L}^\ast_{\mathrm{int}}(i,s+1)F_{s+1}(t),\nonumber\\ \nonumber\\
\label{eq:NelBog2}
&&\hskip-8mmF_s(t)_{\mid t=0}=F_s^{0,\epsilon}, \quad s\geq1.
\end{eqnarray}
In the hierarchy of evolution equations (\ref{NelBog1}) represented in a dimensionless
form the coefficient $\epsilon>0$ is a scaling parameter (the ratio of the diameter
$\sigma>0$ to the mean free path of hard spheres) and, if $t\geq0$, the operators
$\mathcal{L}^{\ast}(j)$ and $\mathcal{L}_{\mathrm{int}}^{\ast}(j_1,j_{2})$ are defined
on the subspace $L_{n,0}^1\subset L_{n}^1$ by the formulas:
\begin{eqnarray}\label{aL}
&&\hskip-7mm\mathcal{L}^\ast(j)f_n\doteq -\langle p_{j},
\frac{\partial}{\partial q_{j}}\rangle f_n,\\
\label{aLint}
&&\hskip-7mm\mathcal{L}_{\mathrm{int}}^\ast(j_{1},j_{2})f_{n}
\doteq\int_{\mathbb{S}_{+}^2}d\eta\langle\eta,(p_{j_{1}}-p_{j_{2}})\rangle
\big(f_n(x_1,\ldots,q_{j_{1}},p_{j_{1}}^\ast,\ldots,\\
&&q_{j_{2}},p_{j_{2}}^\ast,\ldots,x_n)\delta(q_{j_{1}}-q_{j_{2}}+\epsilon\eta)-
f_n(x_1,\ldots,x_n)\delta(q_{j_{1}}-q_{j_{2}}-\epsilon\eta)\big),\nonumber
\end{eqnarray}
where the following notations are used: the symbol $\langle \cdot,\cdot \rangle$
means a scalar product, $\delta$ is the Dirac measure,
$\mathbb{S}_{+}^{2}\doteq\{\eta\in\mathbb{R}^{3}\big|\left|\eta\right|=1,\,
\langle\eta,(p_{j_{1}}-p_{j_{2}})\rangle\geq0\}$ and the pre-collision momenta
$p_{j_{1}}^\ast,p_{j_{2}}^\ast$ are determined by the expressions:
\begin{eqnarray}\label{momenta}
&&p_{j_{1}}^\ast=p_{j_{1}}-\eta\left\langle\eta,\left(p_{j_{1}}-p_{j}\right)\right\rangle,\\
&&p_{j_{2}}^\ast=p_{j_{2}}+\eta\left\langle\eta,\left(p_{j_{1}}-p_{j_{2}}\right)\right\rangle.\nonumber
\end{eqnarray}
The adjoint Liouville operator $\mathcal{L}^\ast_s=\sum_{i=1}^{s}\mathcal{L}^\ast(i)+
\epsilon^{2}\sum_{i<j=1}^{s}\mathcal{L}_{\mathrm{int}}^\ast(i,j)$ is an infinitesimal generator
of the group of operators of $s$ hard spheres: $S_{s}^\ast(t)\equiv S_{s}^\ast(t,1,\ldots,s)$,
which is adjoint to the group of operators $S_{s}(t)$ defined almost everywhere on the phase
space $\mathbb{R}^{3s}\times(\mathbb{R}^{3s}\setminus \mathbb{W}_s)$ as the shift operator
of phase space coordinates along the phase space trajectories of $s$ hard spheres \cite{CGP97}.
The adjoint group of operators $S_{s}^\ast(t)$ coincides with the group of operators of hard
spheres $S_{s}(-t)$ \cite{CGP97}.
In case of $t\leq0$ the generator of the BBGKY hierarchy with hard sphere collisions is determined
by the corresponding operator \cite{CGP97}.
If initial data $F(0)=(1,F_1^{0,\epsilon},\ldots,F_n^{0,\epsilon},\ldots)\in L^{1}_{\alpha}$ and
$\alpha>e$, then for arbitrary $t\in\mathbb{R}$ a unique non-perturbative solution of the Cauchy
problem (\ref{NelBog1}),(\ref{eq:NelBog2}) of the BBGKY hierarchy with hard sphere collisions
exists and it is represented by a sequence of the functions \cite{GRS04}:
\begin{eqnarray}\label{F(t)}
&&\hskip-8mm F_{s}(t,x_1,\ldots,x_{s})=\sum\limits_{n=0}^{\infty}\frac{1}{n!}
\int_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{s+1}\ldots dx_{s+n}
\mathfrak{A}_{1+n}(-t,\{Y\},X\setminus Y)F_{s+n}^{0,\epsilon},\quad s\geq1,
\end{eqnarray}
where the generating operator of the $n$ term of series expansion (\ref{F(t)})
is the $(n+1)th$-order cumulant of adjoint groups of operators of hard spheres:
\begin{eqnarray}\label{nLkymyl}
&&\hskip-8mm \mathfrak{A}_{1+n}(-t,\{Y\},X\setminus Y)=
\sum\limits_{\texttt{P}:\,(\{Y\},X\setminus Y)={\bigcup\limits}_i X_i}
(-1)^{|\texttt{P}|-1}(|\texttt{P}|-1)!
\prod_{X_i\subset \texttt{P}}S_{|\theta(X_i)|}(-t,\theta(X_i)),
\end{eqnarray}
and the following notations are used: $\{Y\}$ is a set consisting of one element
$Y\equiv(1,\ldots,s)$, i.e. $|\{Y\}|=1$, $\sum_\texttt{P}$ is a sum over all possible
partitions $\texttt{P}$ of the set $(\{Y\},X\setminus Y)\equiv(\{Y\},s+1,\ldots,s+n)$
into $|\texttt{P}|$ nonempty mutually disjoint subsets $X_i\in(\{Y\},X\setminus Y)$,
the mapping $\theta$ is the declusterization mapping defined by the formula:
$\theta(\{Y\},X\setminus Y)=X$. The simplest examples of cumulants (\ref{nLkymyl}) are
given by the following expansions:
\begin{eqnarray*}
&&\mathfrak{A}_{1}(-t,\{Y\})=S_{s}(-t,Y),\\
&&\mathfrak{A}_{2}(-t,\{Y\},s+1)=S_{s+1}(-t,Y,s+1)-S_{s}(-t,Y)S_{1}(-t,s+1).
\end{eqnarray*}
For initial data $F(0)\in L^{1}_{\alpha,0}\subset L^{1}_{\alpha}$ sequence (\ref{F(t)})
is a strong solution of the Cauchy problem (\ref{NelBog1}),(\ref{eq:NelBog2}) and for
arbitrary initial data from the space $L^{1}_{\alpha}$ it is a weak solution \cite{GRS04}.
We remark, as a result of the application to cumulants (\ref{nLkymyl}) of analogs of
the Duhamel equations, solution series (\ref{F(t)}) reduces to the iteration series
of the BBGKY hierarchy with hard sphere collisions (\ref{NelBog1}).
In order to describe the evolution of infinitely many hard spheres we must construct
the solutions for initial data from more general Banach spaces. In the capacity
of such Banach space in \cite{CGP97}-\cite{Sp91},\cite{L} it was used the space
$L^{\infty}_\xi$ of sequences $f=(f_0,f_1,\ldots,f_n,\ldots)$ of continuous functions
$f_n(x_1,\ldots,x_n)$ defined on the phase space of hard spheres, that are symmetric
with respect to permutations of the arguments $x_1,\ldots,x_n$, equal to zero on the
set of the forbidden configurations $\mathbb{W}_n$ and equipped with the norm:
\begin{eqnarray*}
&&\|f\|_{L^{\infty}_\xi}=\sup\limits_{n\geq0}\xi^{-n}
\sup\limits_{x_1,\ldots,x_{n}}|f_n(x_1,\ldots,x_{n})|\exp(\frac{\beta}{2}\sum_{i=1}^n p_i^2),
\end{eqnarray*}
where $\beta>0$ and $\xi>0$ are some parameters.
In the space $L^{\infty}_\xi$ for series expansion (\ref{F(t)}) the following statement is true.
If $F(0)\in L^{\infty}_\xi$, every term of series expansion (\ref{F(t)}) exists and this series
converges uniformly on each compact almost everywhere for finite time interval. The sequence of
marginal distribution functions (\ref{F(t)}) is a unique weak solution of the Cauchy problem
(\ref{NelBog1}),(\ref{eq:NelBog2}) of the BBGKY hierarchy for hard spheres.
\subsection{On the Boltzmann--Grad asymptotic behavior}
To consider the conventional approach to the derivation of the Boltzmann kinetic equation with
hard sphere collisions from underlying dynamics \cite{PG90} (see also \cite{CGP97} and references
cited therein) we shall represent a solution of the Cauchy problem of the BBGKY hierarchy for hard
spheres in the form of the perturbation (iteration) series:
\begin{eqnarray}\label{BBGKYpert}
&&\hskip-9mm F_{s}(t,x_1,\ldots,x_{s})=\sum\limits_{n=0}^{\infty}\epsilon^{2n}\int_0^tdt_{1}\ldots
\int_0^{t_{n-1}}dt_{n}\int_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{s+1}\ldots dx_{s+n}\,
S_{s}(-t+t_{1})\\
&&\times\sum\limits_{i_{1}=1}^{s}\mathcal{L}_{\mathrm{int}}^{\ast}(i_{1},s+1)
S_{s+1}(-t_{1}+t_{2})\ldots\nonumber\\
&&\times S_{s+n-1}(-t_{n}+t_{n})\sum\limits_{i_{n}=1}^{s+n-1}
\mathcal{L}_{\mathrm{int}}^{\ast}(i_{n},s+n)
S_{s+n}(-t_{n})F_{s+n}^{0,\epsilon},\quad s\geq1,\nonumber
\end{eqnarray}
where the notations from formula (\ref{aLint}) are used. If $F(0)\in L^{\infty}_\xi$, every
term of series expansion (\ref{BBGKYpert}) exists \cite{GP85},\cite{PG90} and the iteration
series converges uniformly on each compact almost everywhere for finite time interval:
$|t|<t_0(\beta,\xi)$. In paper \cite{GP85} for initial data close to equilibrium states,
namely locally perturbed equilibrium distribution functions, and in paper \cite{Ger} for
arbitrary initial data from the space $L^{\infty}_\xi$ in a one-dimensional hard sphere
(rod) system it were proved the existence of series expansion (\ref{BBGKYpert}) for arbitrary
time interval.
The sequence of marginal distribution functions (\ref{BBGKYpert}) is a unique weak solution
of the Cauchy problem (\ref{NelBog1}),(\ref{eq:NelBog2}) of the BBGKY hierarchy for hard
spheres \cite{GP85}.
Let the set $\mathbb{K}_s^0\subset\mathbb{R}^{3s}\setminus \mathbb{W}_s$ be an arbitrary
compact set of the admissible configurations of a system of $s$ hard spheres consisting from
the configurations: $|q_i-q_j|\geq\epsilon_0(\epsilon)+\epsilon, i\neq j\in(1,\ldots,s)$, where
$\epsilon_0(\epsilon)$ is a fixed value such that: $\lim_{\epsilon\rightarrow 0}\epsilon_0(\epsilon)=0$
and $\lim_{\epsilon\rightarrow 0}\frac{\epsilon}{\epsilon_0(\epsilon)}=0$.
Then the Boltzmann--Grad asymptotic behavior of perturbative solution (\ref{BBGKYpert}) is described
by the following statement \cite{PG89},\cite{PG90}.
\begin{theorem}
If for initial data $F(0)=(1,F_1^{0,\epsilon},\ldots,F_n^{0,\epsilon},\ldots)\in L^{\infty}_\xi$
uniformly on every compact set from the phase space
$\mathbb{R}^{3n}\times(\mathbb{R}^{3n}\setminus\mathbb{W}_n)$ it holds:
$\lim_{\epsilon\rightarrow 0}|\epsilon^{2n}F_{n}^{0,\epsilon}(x_1,\ldots,x_{n})-f_{n}^0(x_1,\ldots,x_{n})|=0,$
then for any finite time interval the function $\epsilon^{2s}F_{s}(t,x_1,\ldots,x_{s})$
defined by series (\ref{BBGKYpert}) converges in the Boltzmann--Grad limit uniformly with
respect to configuration variables from the set $\mathbb{K}_s^0$ and in a weak
sense with respect to momentum variables to the limit marginal distribution function
$f_s(t,x_1,\ldots,x_{s})$ given by the series expansion
\begin{eqnarray}\label{Iter2}
&&\hskip-9mm f_{s}(t,x_1,\ldots,x_{s})=
\sum\limits_{n=0}^{\infty}\int_0^tdt_{1}\ldots\hskip-2mm\int_0^{t_{n-1}}dt_{n}
\int_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{s+1}\ldots dx_{s+n}
\prod\limits_{i_1=1}^{s}S_{1}(-t+t_{1},i_1)\\
&&\times\sum\limits_{k_{1}=1}^{s}\mathcal{L}_{\mathrm{int}}^{0,\ast}(k_{1},s+1)
\prod\limits_{j_1=1}^{s+1}S_{1}(-t_{1}+t_{2},j_1)\ldots \nonumber\\
&&\times\prod\limits_{i_n=1}^{s+n-1} S_{1}(-t_{n}+t_{n},i_n)
\sum\limits_{k_{n}=1}^{s+n-1}\mathcal{L}_{\mathrm{int}}^{0,\ast}(k_{n},s+n)
\prod\limits_{j_n=1}^{s+n}S_{1}(-t_{n},j_n)f_{s+n}^0.\nonumber
\end{eqnarray}
\end{theorem}
In series expansion (\ref{Iter2}) the following operator is introduced:
\begin{eqnarray}\label{aLint0}
&&\hskip-9mm(\mathcal{L}_{\mathrm{int}}^{0,\ast}(j_1,j_{2})f_n)(x_1,\ldots,x_n)\doteq
\int_{\mathbb{S}_{+}^2}d\eta\langle\eta,(p_{j_1}-p_{j_2})\rangle \big(f_n(x_1,\ldots\\
&&q_{j_1},p_{j_1}^\ast,\ldots q_{j_2},p_{j_2}^\ast,\ldots,x_n)-f_n(x_1,\ldots,x_n)\big)
\delta(q_{j_1}-q_{j_2}),\nonumber
\end{eqnarray}
where the notations accepted in formula (\ref{aLint}) are used and the momenta
$p_{j_1}^{\ast},p_{j_2}^{\ast}$ are given by expressions (\ref{momenta}) as above.
If $f(0)=(1,f_1^0,\ldots,f_n^0,\ldots)\in L^{\infty}_\xi$, every term of series expansion
(\ref{Iter2}) exists and this series converges uniformly on each compact almost everywhere
for finite time interval $t<t_0(\beta,\xi)$.
We note that for $t\geq0$ sequence (\ref{Iter2}) is a weak solution of the Cauchy problem
of the limit BBGKY hierarchy known as the Boltzmann hierarchy with hard sphere collisions:
\begin{eqnarray}
\label{Bl1}
&&\hskip-9mm\frac{\partial}{\partial t}f_s(t)=\sum\limits_{j=1}^{n}\mathcal{L}^\ast(j)f_{s}(t)+
\sum_{i=1}^{s}\,\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{s+1}
\mathcal{L}^{0,\ast}_{\mathrm{int}}(i,s+1)f_{s+1}(t),\\ \nonumber\\
\label{Bi2}
&&\hskip-9mm f_s(t)_{\mid t=0}=f_s^0, \quad s\geq1,
\end{eqnarray}
where the operators $\mathcal{L}^\ast(j)$ and $\mathcal{L}^{0,\ast}_{\mathrm{int}}(i,s+1)$ are
defined by formulae (\ref{aL}) and (\ref{aLint0}), respectively.
We remark that the same statement takes place concerning the Boltzmann--Grad behavior of
non-perturbative solution (\ref{F(t)}) of the Cauchy problem of the BBGKY hierarchy for
hard spheres.
To derive the Boltzmann kinetic equation \cite{V} we will consider initial data (\ref{eq:NelBog2})
specified by a one-particle distribution function, namely initial data satisfying a chaos condition
\cite{CGP97}
\begin{eqnarray}\label{eq:Bog2_haos}
&&\hskip-7mm F^{0,\epsilon}_{s}(x_1,\ldots,x_{s})
=\prod_{i=1}^{s}F_{1}^{0,\epsilon}(x_i)
\mathcal{X}_{\mathbb{R}^{3s}\setminus\mathbb{W}_s^{\epsilon}},\quad s\geq1,
\end{eqnarray}
where $\mathcal{X}_{\mathbb{R}^{3s}\setminus\mathbb{W}_s^{\epsilon}}$ is a characteristic
function of the set $\mathbb{R}^{3s}\setminus\mathbb{W}_s^{\epsilon}$ of allowed configurations.
Such assumption about initial data is intrinsic for the kinetic description of a gas,
because in this case all possible states are characterized only by a one-particle
marginal distribution function.
Since the initial limit marginal distribution functions satisfy a chaos condition too, i.e.
\begin{eqnarray}\label{lih2}
&&\hskip-5mmf_s^0(x_1,\ldots,x_{s})=\prod \limits_{i=1}^{s}f_{1}^0(x_i), \quad s\geq2,
\end{eqnarray}
perturbative solution (\ref{Iter2}) of the Cauchy problem (\ref{Bl1}),(\ref{Bi2}) of the
Botzmann hierarchy has the following property (the propagation of initial chaos):
\begin{eqnarray*}
&&\hskip-5mm f_{s}(t,x_1,\ldots,x_{s})=\prod\limits_{i=1}^{s}f_{1}(t,x_i), \quad s\geq2,
\end{eqnarray*}
where for $t\geq0$ the limit one-particle distribution function is determined by series
expansion (\ref{Iter2}) in case of $s=1$ and initial data (\ref{lih2}). If $t\geq0$, the
limit one-particle distribution function is governed by the Boltzmann kinetic equation with
hard sphere collisions \cite{CGP97}
\begin{eqnarray*}
&&\hskip-8mm\frac{\partial}{\partial t}f_{1}(t,x_1)=
-\langle p_1,\frac{\partial}{\partial q_1}\rangle f_{1}(t,x_1)+
\int_{\mathbb{R}^3\times\mathbb{S}^2_+}d p_2\, d\eta\,\langle\eta,(p_1-p_2)\rangle\\
&&\hskip+5mm\times\big(f_1(t,q_1,p_1^{\ast})f_1(t,q_1,p_2^{\ast})-
f_1(t,x_1)f_1(t,q_1,p_2)\big), \nonumber
\end{eqnarray*}
where the momenta $p_{1}^{\ast}$ and $p_{2}^{\ast}$ are given by expressions (\ref{momenta}).
The Boltzmann--Grad asymptotic behavior of the equilibrium state of infinitely many hard spheres
and the systems of particles interacting via a short-range potential is established in paper \cite{PG88}
(see also \cite{CGP97}). It was proved that the equilibrium marginal distribution functions in the
Boltzmann--Grad scaling limit converge uniformly on any compacts to the Maxwell distribution functions.
\subsection{The dual BBGKY hierarchy for hard spheres}
Let $C_{\gamma}$ be the space of sequences $b=(b_0,b_1,\ldots,b_n,\ldots)$ of continuous functions
$b_n\in C_n$ equipped with the norm: $\|b\|_{C_{\gamma}}=\max_{n\geq 0}\,\frac{\gamma^{n}}{n!}\,
\|b_n\|_{C_n}=\max_{n\geq 0}\,\frac{\gamma^{n}}{n!}\sup_{x_1,\ldots,x_{n}}|b_n(x_1,\ldots,x_n)|$,
and $C_{\gamma}^0\subset C_{\gamma}$ is the set of finite sequences of infinitely differentiable
functions with compact supports.
If $t\geq0$, the evolution of marginal observables of a system of a non-fixed number of hard spheres
is described by the Cauchy problem of the weak formulation of the dual BBGKY hierarchy \cite{BGer}:
\begin{eqnarray}\label{dh}
&&\hskip-9mm\frac{\partial}{\partial t}B_{s}(t)=\big(\sum\limits_{j=1}^{s}\mathcal{L}(j)+
\epsilon^{2}\sum\limits_{j_1<j_{2}=1}^{s}\mathcal{L}_{\mathrm{int}}(j_1,j_{2})\big)B_{s}(t)+\\
&&+\epsilon^{2}\sum_{j_1\neq j_{2}=1}^s
\mathcal{L}_{\mathrm{int}}(j_1,j_{2})B_{s-1}(t,x_1,\ldots,x_{j_1-1},x_{j_1+1},\ldots,x_s),
\nonumber\\
\nonumber\\
\label{dhi}
&&\hskip-9mmB_{s}(t,x_1,\ldots,x_s)_{\mid t=0}=B_{s}^{\epsilon,0}(x_1,\ldots,x_s),\quad s\geq1.
\end{eqnarray}
In recurrence evolution equations (\ref{dh}) represented in a dimensionless form, as above,
the coefficient $\epsilon>0$ is a scaling parameter (the ratio of the diameter $\sigma>0$
to the mean free path of hard spheres) and operators (\ref{aL}) and (\ref{aLint}) are the adjoint
operators to the operators $\mathcal{L}(j)$ and $\mathcal{L}_{\mathrm{int}}(j_1,j_{2})$ defined
on $C_{s}^0$ as follows:
\begin{eqnarray}\label{com}
&&\hskip-5mm\mathcal{L}(j)b_n\doteq\langle p_{j},\frac{\partial}{\partial q_{j}}\rangle b_n,\\
\label{comint}
&&\hskip-5mm(\mathcal{L}_{\mathrm{int}}(j_1,j_{2})b_n)(x_1,\ldots,x_n)\doteq
\int_{\mathbb{S}_+^2}d\eta\langle\eta,(p_{j_1}-p_{j_2})\rangle
\big(b_n(x_1,\ldots,
\end{eqnarray}
\begin{eqnarray*}
&&q_{j_1},p_{j_1}^\ast,\ldots,q_{j_2},p_{j_2}^\ast,\ldots,x_n)
-b_n(x_1,\ldots,x_n)\big)\delta(q_{j_1}-q_{j_2}+\epsilon\eta),\nonumber
\end{eqnarray*}
where the the post-collision momenta $p_{j_1}^\ast,p_{j_2}^\ast$ are determined by
expressions (\ref{momenta}) and $\mathbb{S}_{+}^{2}\doteq\{\eta\in\mathbb{R}^{3}
\big|\left|\eta\right|=1,\,\langle\eta,(p_{j_2}-p_{j_2})\rangle\geq0\}$. If $t\leq0$, a generator of the dual BBGKY hierarchy
is determined by corresponding operator \cite{BGer}.
Let $Y\equiv(1,\ldots,s), Z\equiv (j_1,\ldots,j_{n})\subset Y$ and $\{Y\setminus Z\}$ is the
set consisting from one element $Y\setminus Z=(1,\ldots,j_{1}-1,j_{1}+1,\ldots,j_{n}-1,j_{n}+1,\ldots,s)$.
The non-perturbative solution $B(t)=(B_{0},B_{1}(t,x_1),\ldots,B_{s}(t,x_1,\ldots,x_s),\ldots)$
of the Cauchy problem (\ref{dh}),(\ref{dhi}) is represented by the sequence of marginal ($s$-particle)
observables:
\begin{eqnarray}\label{sdh}
&&\hskip-12mm B_{s}(t,x_1,\ldots,x_s)=\sum_{n=0}^s\,
\frac{1}{n!}\sum_{j_1\neq\ldots\neq j_{n}=1}^s
\mathfrak{A}_{1+n}\big(t,\{Y\setminus Z\},Z\big) B_{s-n}^{\epsilon,0}(x_1,\\
&&\ldots x_{j_1-1},x_{j_1+1},\ldots,x_{j_n-1},x_{j_n+1},\ldots,x_s), \quad s\geq1,\nonumber
\end{eqnarray}
where the generating operator of $n$ term of this expansion is the $(1+n)th$-order cumulant
of the groups of operators of hard spheres defined by the formula:
\begin{eqnarray}\label{cumulant}
&&\hskip-5mm\mathfrak{A}_{1+n}(t,\{Y\setminus Z\},Z)\doteq
\sum\limits_{\mathrm{P}:\,(\{Y\setminus Z\},Z)={\bigcup}_i X_i}
(-1)^{\mathrm{|P|}-1}({\mathrm{|P|}-1})!\prod_{X_i\subset \mathrm{P}}
S_{|\theta(X_i)|}(t,\theta(X_i)),
\end{eqnarray}
and notations accepted in formula (\ref{nLkymyl}) are used. The simplest examples of marginal
observables (\ref{sdh}) are given by the following expansions:
\begin{eqnarray*}
&&B_{1}(t,x_1)=\mathfrak{A}_{1}(t,1)B_{1}^{\epsilon,0}(x_1),\\
&&B_{2}(t,x_1,x_2)=\mathfrak{A}_{1}(t,\{1,2\})B_{2}^{\epsilon,0}(x_1,x_2)+
\mathfrak{A}_{2}(t,1,2)(B_{1}^{\epsilon,0}(x_1)+B_{1}^{\epsilon,0}(x_2)).
\end{eqnarray*}
If $\gamma<e^{-1}$, then for $t\in\mathbb{R}$ in case of
$B(0)=(B_{0},B_{1}^{\epsilon,0},\ldots,B_{s}^{\epsilon,0},\ldots)\in C_{\gamma}^0\subset C_{\gamma}$
the sequence of marginal observables (\ref{sdh}) is a classical solution and
for arbitrary initial data $B(0)\in C_{\gamma}$ it is a generalized solution.
We remark that expansion (\ref{sdh}) can be also represented in the form of the perturbation
(iteration) series of the dual BBGKY hierarchy (\ref{dh}) as a result of applying of analogs
of the Duhamel equations to cumulants (\ref{cumulant}) of the groups of operators of hard spheres \cite{BGer}.
The single-component sequences of marginal observables correspond to observables of certain structure,
namely the marginal observables $b^{(1)}=(0,b_{1}^{(1)}(x_1),0,\ldots)$ correspond to the additive-type
observable, and the marginal observables $b^{(k)}=(0,\ldots,0,b_{k}^{(k)}(x_1,\ldots,x_k),0,\ldots)$
corresponds to the $k$-ary-type observables \cite{BGer}. If in capacity of initial data (\ref{dhi})
we consider the additive-type marginal observable, then the structure of solution expansion \eqref{sdh}
is simplified and attains the form
\begin{eqnarray}\label{af}
&&\hskip-8mm B_{s}^{(1)}(t,x_1,\ldots,x_s)=
\mathfrak{A}_{s}(t,1,\ldots,s)\sum_{j=1}^sB_{1}^{(1),\epsilon}(0,x_j), \quad s\geq 1.
\end{eqnarray}
We note that the mean value of the marginal observable $B(t)\in C_{\gamma}$ at
$t\in \mathbb{R}$ in the initial marginal state
$F(0)=(1,F_{1}^{\epsilon,0},\ldots,F_{n}^{\epsilon,0},\ldots)\in L^{1}=\bigoplus_{n=0}^{\infty}L^{1}_{n}$
is defined by the following functional:
\begin{eqnarray}\label{avmar-1}
&&\hskip-5mm\big\langle B(t)\big|F(0)\big\rangle=\sum\limits_{s=0}^{\infty}\,
\frac{1}{s!}\int_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{s}}
dx_{1}\ldots dx_{s}\,B_{s}(t,x_1,\ldots,x_s)F_{s}^{\epsilon,0}(x_1,\ldots,x_s).
\end{eqnarray}
Owing to the estimate: $\|B(t)\|_{C_{\gamma}}\leq e^2(1-\gamma e)^{-1}\|B(0)\|_{C_{\gamma}}$,
functional (\ref{avmar-1}) exists under the condition that: $\gamma<e^{-1}$. In case of
$F(0)\in L^{\infty}_\xi$ the existence of mean value functional (\ref{avmar-1}) is proved
in a one-dimensional space in paper \cite{R10}.
In a special case functional (\ref{avmar-1}) of mean values of the additive-type marginal
observables $B^{(1)}(0)=(0,B_{1}^{(1),\epsilon}(0,x_1),0,\ldots)$ takes the form:
\begin{eqnarray*}\label{avmar-11}
&&\hskip-5mm\big\langle B^{(1)}(t)\big|F(0)\big\rangle=
\big\langle B^{(1)}(0)\big|F(t)\big\rangle=\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{1}\,
B_{1}^{(1),\epsilon}(0,x_1)F_{1}(t,x_1),\nonumber
\end{eqnarray*}
where the sequence $B^{(1)}(t)$ is represented by (\ref{af}) and the one-particle distribution function
$F_{1}(t)$ is given by series expansion (\ref{F(t)}).
In the general case for mean values of marginal observables the following equality is true:
\begin{eqnarray}\label{eqos}
&&\hskip-5mm\big\langle B(t)\big|F(0)\big\rangle=\big\langle B(0)\big|F(t)\big\rangle,
\end{eqnarray}
where the sequence $F(t)$ is given by formula (\ref{F(t)}). This equality signify the
equivalence of two pictures of the description of the evolution of hard spheres by means
of the BBGKY hierarchy (\ref{NelBog1}) and the dual BBGKY hierarchy for hard spheres (\ref{dh}).
\subsection{The generalized Enskog kinetic equation}
In paper \cite{GG} it was proved that, if initial states of a hard sphere system is
specified in terms of a one-particle distribution function on allowed configurations, then
at arbitrary moment of time the evolution of states governed by the BBGKY hierarchy can
be completely described within the framework of the one-particle marginal distribution
function $F_{1}(t)$ governed by the generalized Enskog kinetic equation. In this case all
possible correlations, generating by hard sphere dynamics, are described in terms of the
explicitly defined marginal functionals of the state $F_{s}\big(t\mid F_{1}(t)\big),\,s\geq2$.
If $t\geq 0$, the one-particle distribution function is governed by the Cauchy
problem of the following generalized Enskog equation \cite{GG}:
\begin{eqnarray}\label{gke1}
&&\hskip-9mm\frac{\partial}{\partial t}F_{1}(t,x_1)=
-\langle p_1,\frac{\partial}{\partial q_1}\rangle F_{1}(t,x_1)+
\epsilon^2\int_{\mathbb {R}^3\!\times\mathbb{S}_{+}^{2}}
d p_{2}d\eta\,\langle\eta,(p_1-p_{2})\rangle\times\\
&&\times\big(F_{2}(t,q_1,p_1^\ast,q_1-\epsilon\eta,p_{2}^\ast\mid F_{1}(t))
-F_{2}(t,x_1,q_1+\epsilon\eta,p_{2}\mid F_{1}(t))\big),\nonumber\\ \nonumber\\
\label{2}
&&\hskip-9mm F_1(t,x_1)_{\mid t=0}=F_1^{\epsilon,0}(x_1).
\end{eqnarray}
In kinetic equation (\ref{gke1}) represented in a dimensionless form, as above, the coefficient
$\epsilon>0$ is a scaling parameter, the pre-collision momenta $p_{1}^\ast,p_{2}^\ast$
are determined by expressions (\ref{momenta}), $\mathbb{S}_{+}^{2}\doteq\{\eta\in\mathbb{R}^{3}\big|\left|\eta\right|=1,\,\langle\eta,(p_{1}-p_{2})\rangle\geq0\}$,
and the collision integral is defined in terms of the marginal functional of the state in case of $s=2$.
The the marginal functionals of the state $F_{s}(t,x_1,\ldots,x_s\mid F_{1}(t)),\,s\geq 2,$
are represented by the following series expansion:
\begin{eqnarray}\label{f}
&&\hskip-9mm F_{s}(t,x_1,\ldots,x_s\mid F_{1}(t))\doteq\\
&& \sum_{n=0}^{\infty}\frac{1}{n!}\,\int_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}
dx_{s+1}\ldots dx_{s+n}\,\mathfrak{V}_{1+n}(t,\{Y\},X\setminus Y)\prod_{i=1}^{s+n}F_{1}(t,x_i),\nonumber
\end{eqnarray}
where the notations accepted above are used: $Y\equiv(1,\ldots,s),\,X\equiv(1,\ldots,s+n)$, and
the $(n+1)th$-order generating evolution operator $\mathfrak{V}_{1+n}(t),\,n\geq0$, is defined by
the expansion:
\begin{eqnarray}\label{skrrn}
&&\hskip-12mm\mathfrak{V}_{1+n}(t,\{Y\},X\setminus Y)\doteq\\
&&\hskip-9mm \sum_{k=0}^{n}(-1)^k\,
\sum_{m_1=1}^{n}\ldots\sum_{m_k=1}^{n-m_1-\ldots-m_{k-1}}\frac{n!}{(n-m_1-\ldots-m_k)!}
\widehat{\mathfrak{A}}_{1+n-m_1-\ldots-m_k}(t)\nonumber\\
&&\hskip-9mm \prod_{j=1}^k\,\sum_{k_2^j=0}^{m_j}\ldots
\sum_{k^j_{n-m_1-\ldots-m_j+s}=0}^{k^j_{n-m_1-\ldots-m_j+s-1}}\prod_{i_j=1}^{s+n-m_1-\ldots-m_j}
\frac{1}{(k^j_{n-m_1-\ldots-m_j+s+1-i_j}-k^j_{n-m_1-\ldots-m_j+s+2-i_j})!}\nonumber\\
&&\hskip-9mm\times\widehat{\mathfrak{A}}_{1+k^j_{n-m_1-\ldots-m_j+s+1-i_j}-k^j_{n-m_1-\ldots-m_j+s+2-i_j}}
(t,i_{j},s+n-m_1-\ldots-m_j+1+\nonumber \\
&&\hskip-9mm +k^j_{s+n-m_1-\ldots-m_j+2-i_j},\ldots,s+n-m_1-\ldots-m_j+k^j_{s+n-m_1-\ldots-m_j+1-i_j}).\nonumber
\end{eqnarray}
In expression (\ref{skrrn}) it means that: $k^j_1\equiv m_j$ and $k^j_{n-m_1-\ldots-m_j+s+1}\equiv 0$
and we denote by the evolution operator $\widehat{\mathfrak{A}}_{1+n-m_1-\ldots-m_k}(t)
\equiv \widehat{\mathfrak{A}}_{1+n-m_1-\ldots-m_k}(t,\{Y\},s+1,\ldots,s+n-m_1-\ldots-m_k)$
the $(n-m_1-\ldots-m_k)th$-order scattering cumulant, namely
\begin{eqnarray*}\label{scacu}
&&\hskip-8mm\widehat{\mathfrak{A}}_{1+n}(t,\{Y\},X\setminus Y)\doteq
\mathfrak{A}_{1+n}(-t,\{Y\},X\setminus Y)
\mathcal{X}_{\mathbb{R}^{3(s+n)}\setminus \mathbb{W}^{\epsilon}_{s+n}}
\prod_{i=1}^{s+n}\mathfrak{A}_{1}(t,i), \quad n\geq1,\nonumber
\end{eqnarray*}
where the operator $\mathfrak{A}_{1+n}(-t)$ is $(1+n)th$-order cumulant
(\ref{nLkymyl}) of the adjoint groups of operators of hard spheres. We give
several simplest examples of generating evolution operators (\ref{skrrn}):
\begin{eqnarray*}
&&\hskip-9mm\mathfrak{V}_{1}(t,\{Y\})=\widehat{\mathfrak{A}}_{1}(t,\{Y\})\doteq
S_s(-t,1,\ldots,s)\mathcal{X}_{\mathbb{R}^{3s}\setminus \mathbb{W}^{\epsilon}_{s}}\prod_{i=1}^{s}S_1(t,i),\\
&&\hskip-9mm\mathfrak{V}_{2}(t,\{Y\},s+1)=\widehat{\mathfrak{A}}_{2}(t,\{Y\},s+1)-
\widehat{\mathfrak{A}}_{1}(t,\{Y\})\sum_{i_1=1}^s
\widehat{\mathfrak{A}}_{2}(t,i_1,s+1).\nonumber
\end{eqnarray*}
If $\|F_{1}(t)\|_{L^{1}(\mathbb{R}^{3}\times\mathbb{R}^{3})}<e^{-(3s+2)}$, series expansion
\ref{f}) converges in the norm of the space $L^{1}_{s}$ for arbitrary $t\in\mathbb{R}$ \cite{GG},
and thus, the series expansion of the collision integral of kinetic equation (\ref{gke1})
converges under the condition that: $\|F_1(t)\|_{L^1(\mathbb{R}\times\mathbb{R})}<e^{-8}$.
A solution of the Cauchy problem (\ref{gke1}),(\ref{2}) is represented by the series expansion \cite{GG}:
\begin{eqnarray}\label{F(t)1}
&&\hskip-12mm F_{1}(t,x_1)=\sum\limits_{n=0}^{\infty}\frac{1}{n!}
\int_{(\mathbb{R}^3\times\mathbb{R}^3)^n}dx_2\ldots dx_{n+1}\,
\mathfrak{A}_{1+n}(-t)\prod_{i=1}^{n+1}F_{1}^{\epsilon,0}(x_i)
\mathcal{X}_{\mathbb{R}^{3(n+1)}\setminus \mathbb{W}^{\epsilon}_{n+1}},
\end{eqnarray}
where the generating operator $\mathfrak{A}_{1+n}(-t)\equiv\mathfrak{A}_{1+n}(-t,1,\ldots,n+1)$
is the $(1+n)th$-order cumulant (\ref{nLkymyl}) of adjoint groups of operators of hard spheres.
If initial one-particle distribution function $F_{1}^{\epsilon,0}$ is a continuously
differentiable integrable function with compact support, then function (\ref{F(t)1}) is a strong
solution of the Cauchy problem (\ref{gke1}),(\ref{2}) and for the arbitrary integrable function
$F_{1}^{\epsilon,0}$ it is a weak solution.
If initial one-particle marginal distribution function satisfies the following condition:
\begin{eqnarray}\label{G_1}
&&|F_{1}^{\epsilon,0}(x_1)|\leq ce^{\textstyle-\frac{\beta}{2}{p^{2}_1}},
\end{eqnarray}
where $\beta>0$ is a parameter, $c<\infty$ is some constant, then every term of series
expansion (\ref{F(t)1}) exists, series (\ref{F(t)1}) converges uniformly on each compact almost
everywhere with respect to $x_1$ for finite time interval and function (\ref{F(t)1}) is a unique
weak solution of the Cauchy problem (\ref{gke1}),(\ref{2}) of the generalized Enskog kinetic equation.
The proof of the last statement is based on analogs of the Duhamel equations for cumulants
(\ref{nLkymyl}) of groups of adjoint operators of hard spheres and the estimates established
for the iteration series of the BBGKY hierarchy with hard sphere collisions \cite{CGP97}.
We point out the relationship of the description of the evolution of many hard spheres in terms
of the marginal observables and by the one-particle marginal distribution function governed by
the generalized Enskog kinetic equation (\ref{gke1}).
For mean value functional (\ref{avmar-1}) the following equality holds:
\begin{eqnarray}\label{w}
&&\big\langle B(t)\big|F^c(0)\big\rangle=\big\langle B(0)\big|F(t\mid F_{1}(t))\big\rangle,
\end{eqnarray}
where the sequence $B(t)$ is a sequence of marginal observables defined by expansions (\ref{sdh}),
the sequence $F^c(0)=\big(1,F_{1}^{0,\epsilon}(x_1),\ldots,
\prod_{i=1}^{n}F_{1}^{0,\epsilon}(x_i)\mathcal{X}_{\mathbb{R}^{3n}\setminus\mathbb{W}^{\epsilon}_n}\big)$
is a sequence of initial marginal distribution functions and
$F(t\mid F_{1}(t))=\big(1,F_1(t),F_2(t\mid F_{1}(t)),\ldots,F_s(t\mid F_{1}(t))\big)$ is the
sequence which consists from solution expansion (\ref{F(t)1}) of the generalized Enskog kinetic equation
and marginal functionals of the state (\ref{f}).
In particular case of the $s$-ary initial marginal observable
$B^{(s)}(0)=(0,\ldots,0,B_{s}^{(s),\epsilon}(0,x_1,\ldots,$ $x_s),0,\ldots),\,s\geq2$,
equality (\ref{w}) takes the form
\begin{eqnarray*}\label{avmar-12}
&&\hskip-9mm\big\langle B^{(s)}(t)\big|F^c(0)\big\rangle=
\big\langle B^{(s)}(0)\big|F(t\mid F_{1}(t))\big\rangle=\\
&&=\frac{1}{s!}\int_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{s}}dx_{1}\ldots dx_{s}\,
B_{s}^{(s),\epsilon}(0,x_1,\ldots,x_s)F_s(t,x_1,\ldots,x_s\mid F_{1}(t)),\nonumber
\end{eqnarray*}
where the marginal functional of the state $F_{s}(t\mid F_{1}(t))$ is determined by series expansion
(\ref{f}). We emphasize that in fact functionals (\ref{f}) characterize the correlations
generated by dynamics of a hard sphere system with elastic collisions.
Correspondingly, in case of the additive-type marginal observable
$B^{(1)}(0)=(0,B_{1}^{(1),\epsilon}(0,x_1),$ $0,\ldots)$ equality (\ref{w}) takes the form
\begin{eqnarray*}\label{avmar-11}
&&\hskip-5mm\big\langle B^{(1)}(t)\big|F^c(0)\big\rangle=
\big\langle B^{(1)}(0)\big|F(t\mid F_{1}(t))\big\rangle=\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{1}\,
B_{1}^{(1),\epsilon}(0,x_1)F_{1}(t,x_1),
\end{eqnarray*}
where the one-particle marginal distribution function $F_{1}(t)$ is represented by series expansion
(\ref{F(t)1}). Therefore for the additive-type marginal observables the generalized Enskog
kinetic equation (\ref{gke1}) is dual to the dual BBGKY hierarchy for hard spheres (\ref{dh})
with respect to bilinear form (\ref{avmar-1}).
Thus, if the initial state is specified by the one-particle distribution function
on allowed configurations, then the evolution of hard spheres governed by the dual BBGKY
hierarchy (\ref{dh}) for marginal observables of hard spheres can be completely described
within the framework of the generalized Enskog kinetic equation (\ref{gke1}) by the sequence
of marginal functionals of the state (\ref{f}).
\section{The kinetic evolution within the framework of marginal observables}
In this section we consider the problem of the rigorous description of the kinetic
evolution within the framework of many-particle dynamics of observables by giving
an example of the Boltzmann--Grad asymptotic behavior of a solution of the dual
BBGKY hierarchy with hard sphere collisions. Furthermore, we establish the links of
the dual Boltzmann hierarchy for the Boltzmann--Grad limit of marginal observables
with the Boltzmann kinetic equation.
\subsection{The Boltzmann--Grad asymptotics of the dual BBGKY hierarchy}
The Boltzmann--Grad scaling limit of non-perturbative solution (\ref{sdh}) of the Cauchy
problem (\ref{dh}),(\ref{dhi}) of the dual BBGKY hierarchy is described by the following
statement.
\begin{theorem}\label{3.1}
Let for $B_{n}^{\epsilon,0}\in C_n,\,n\geq1,$ it holds:
$\mathrm{w^{\ast}-}\lim_{\epsilon\rightarrow 0}(\epsilon^{-2n}B_{n}^{\epsilon,0}-b_{n}^0)=0,$
then for arbitrary finite time interval the Boltzmann--Grad limit of solution (\ref{sdh}) of
the Cauchy problem (\ref{dh}),(\ref{dhi}) of the dual BBGKY hierarchy exists in the
sense of the $\ast$-weak convergence on the space $C_s$
\begin{eqnarray}\label{asymto}
&&\mathrm{w^{\ast}-}\lim\limits_{\epsilon\rightarrow 0}\big(\epsilon^{-2s}B_{s}(t)
-b_{s}(t)\big)=0,
\end{eqnarray}
and it is determined by the following expansion:
\begin{eqnarray}\label{Iterd}
&&\hskip-12mm b_{s}(t)=\sum\limits_{n=0}^{s-1}\,\int_0^tdt_{1}\ldots\int_0^{t_{n-1}}dt_{n}
S_{s}^{0}(t-t_{1})\sum\limits_{i_{1}\neq j_{1}=1}^{s}\mathcal{L}_{\mathrm{int}}^0(i_{1},j_{1})
S_{s-1}^{0}(t_{1}-t_{2})\ldots\\
&&\hskip-8mm S_{s-n+1}^{0}(t_{n-1}-t_{n})
\hskip-3mm\sum\limits^{s}_{\mbox{\scriptsize $\begin{array}{c}i_{n}\neq j_{n}=1,\\
i_{n},j_{n}\neq (j_{1},\ldots,j_{n-1})\end{array}$}}\hskip-4mm
\mathcal{L}_{\mathrm{int}}^0(i_{n},j_{n})S_{s-n}^{0}(t_{n})b_{s-n}^0((x_1,\ldots,x_s)
\setminus(x_{j_{1}},\ldots,x_{j_{n}})),\nonumber\\
&&\hskip-8mm s\geq1.\nonumber
\end{eqnarray}
\end{theorem}
In expansion (\ref{Iterd}) for groups of operators of noninteracting particles
the following notations are used:
\begin{eqnarray*}
&&S_{s-n+1}^{0}(t_{n-1}-t_{n})\equiv S_{s-n+1}^{0}(t_{n-1}-t_{n},Y \setminus (j_{1},\ldots,j_{n-1}))=
\prod\limits_{j\in Y \setminus (j_{1},\ldots,j_{n-1})}S_{1}(t_{n-1}-t_{n},j),
\end{eqnarray*}
and we denote by $\mathcal{L}_{\mathrm{int}}^{0}(j_1,j_{2})$ the operator:
\begin{eqnarray}\label{int0}
&&\hskip-9mm(\mathcal{L}_{\mathrm{int}}^{0}(j_1,j_{2})b_n)(x_1,\ldots,x_n)\doteq
\int_{\mathbb{S}_+^2}d\eta\langle\eta,(p_{j_1}-p_{j_2})\rangle
\big(b_n(x_1,\ldots,q_{j_1},p_{j_1}^\ast,\ldots,\\
&&\hskip+5mm q_{j_2},p_{j_2}^\ast,\ldots,x_n)-b_n(x_1,\ldots,x_n)\big)\delta(q_{j_1}-q_{j_2}),\nonumber
\end{eqnarray}
where
${\Bbb S}_{+}^{2}\doteq\{\eta\in\mathbb{R}^{3}\big|\,|\eta|=1,\langle\eta,(p_{j_1}-p_{j_2})\rangle\geq0\}$
and the momenta $p_{j_1}^{\ast},p_{j_2}^{\ast}$ are determined by expressions (\ref{momenta}).
Before to consider the proof scheme of the theorem we give some comments.
If $b^0\in C_{\gamma}$, then the sequence $b(t)=(b_0,b_1(t),\ldots,b_{s}(t),\ldots)$ of limit
marginal observables (\ref{Iterd}) is a generalized global solution of the Cauchy problem of
the dual Boltzmann hierarchy with hard sphere collisions
\begin{eqnarray}\label{vdh}
&&\hskip-9mm \frac{d}{dt}b_{s}(t,x_1,\ldots,x_s)=
\sum\limits_{j=1}^{s}\mathcal{L}(j)\,b_{s}(t,x_1,\ldots,x_s)+\\
&&+\sum_{j_1\neq j_{2}=1}^s\mathcal{L}_{\mathrm{int}}^{0}(j_1,j_{2})
b_{s-1}(t,x_1,\ldots,x_{j_1-1},x_{j_1+1},\ldots,x_s),\nonumber\\ \nonumber\\
\label{vdhi}
&&\hskip-9mm b_{s}(t,x_1,\ldots,x_s)_{\mid t=0}=b_{s}^0(x_1,\ldots,x_s), \quad s\geq1.
\end{eqnarray}
It should be noted that equations set (\ref{vdh}) has the structure of recurrence evolution equations.
We give several examples of the evolution equations of the dual Boltzmann hierarchy (\ref{vdh})
\begin{eqnarray*}
&&\hskip-9mm\frac{\partial}{\partial t}b_{1}(t,x_1)=
\langle p_1,\frac{\partial}{\partial q_{1}}\rangle\,b_{1}(t,x_1),\\
&&\hskip-9mm\frac{\partial}{\partial t}b_{2}(t,x_1,x_2)=
\sum\limits_{j=1}^{2}\langle p_j,\frac{\partial}{\partial q_{j}}\rangle\,b_{2}(t,x_1,x_2)+\int_{\mathbb{S}_{+}^2}d\eta\langle\eta,(p_{1}-p_{2})\rangle\\
&&\times\big(b_1(t,q_{1},p_{1}^\ast)-b_1(t,x_1)+b_1(t,q_{2},p_{2}^\ast)-b_1(t,x_2)\big)\delta(q_{1}-q_{2}).
\end{eqnarray*}
The proof of the limit theorem for the dual BBGKY hierarchy is based on formulas for cumulants
of asymptotically perturbed groups of operators of hard spheres.
For arbitrary finite time interval the asymptotically perturbed group of operators of hard
spheres has the following scaling limit in the sense of the $\ast$-weak convergence on the
space $C_s$:
\begin{eqnarray}\label{Kato}
&&\mathrm{w^{\ast}-}\lim\limits_{\epsilon\rightarrow 0}\big(S_s(t)b_s-
\prod\limits_{j=1}^{s}S_{1}(t,j)b_s\big)=0.
\end{eqnarray}
Taking into account analogs of the Duhamel equations for cumulants of asymptotically
perturbed groups of operators, in view of formula (\ref{Kato}) we have
\begin{eqnarray*}\label{apc}
&&\hskip-8mm \mathrm{w^{\ast}-}\lim\limits_{\epsilon\rightarrow 0}
\Big(\epsilon^{-2n}\frac{1}{n!}
\mathfrak{A}_{1+n}\big(t,\{Y\setminus X\},j_1,\ldots,j_{n}\big)\,b_{s-n}-\\
&& -\int_0^tdt_{1}\ldots\int_0^{t_{n-1}}dt_{n}
\, S_{s}^{0}(t-t_{1})\sum\limits^{s}_{\mbox{\scriptsize $\begin{array}{c}i_{1}=1,\\
i_{1}\neq j_{1}\end{array}$}}\mathcal{L}_{\mathrm{int}}^0(i_{1},j_{1})\,
S_{s-1}(t_{1}-t_{2})\ldots\\
&& S_{s-n+1}^{0}(t_{n-1}-t_{n})
\sum\limits^{s}_{\mbox{\scriptsize $\begin{array}{c}i_{n}=1,\\
i_{n}\neq (j_{1},\ldots,j_{n})\end{array}$}}\mathcal{L}_{\mathrm{int}}^0(i_{n},j_{n})
S_{s-n}^{0}(t_{n})\,b_{s-n}\Big)=0,
\end{eqnarray*}
where we used notations accepted in formula (\ref{Iterd}) and
$b_{s-n}\equiv b_{s-n}((x_1,\ldots,x_s)\setminus(x_{j_{1}},\ldots,x_{j_{n}}))$. As a result
of this equality we establish the validity of statement (\ref{asymto}) for solution expansion
(\ref{sdh}) of the dual BBGKY hierarchy with hard sphere collisions (\ref{dh}).
We consider the Boltzmann--Grad limit of a special case of marginal observables, namely the
additive-type marginal observables. As it was noted above in this case solution (\ref{sdh})
of the dual BBGKY hierarchy (\ref{dh}) is represented by formula (\ref{af}).
If for the initial additive-type marginal observable $B_{1}^{(1),\epsilon}(0)$ the following
condition is satisfied:
\begin{eqnarray*}
&&\hskip-5mm \mathrm{w^{\ast}-}\lim\limits_{\epsilon\rightarrow 0}\big( \epsilon^{-2}
B_{1}^{(1),\epsilon}(0)-b_{1}^{(1)}(0)\big)=0,
\end{eqnarray*}
then, according to statement (\ref{asymto}), for additive-type marginal observable (\ref{af})
we derive
\begin{eqnarray*}
&&\hskip-5mm \mathrm{w^{\ast}-}\lim\limits_{\epsilon\rightarrow 0}
\big(\epsilon^{-2s}B_{s}^{(1),\epsilon}(t)-b_{s}^{(1)}(t)\big)=0,
\end{eqnarray*}
where the limit marginal observable $b_{s}^{(1)}(t)$ is determined as a special case
of expansion (\ref{Iterd}):
\begin{eqnarray}\label{itvad}
&&\hskip-10mm b_{s}^{(1)}(t,x_1,\ldots,x_s)=\int_0^t dt_{1}\ldots\int_0^{t_{s-2}}dt_{s-1}\,
S_{s}^{0}(t-t_{1})\sum\limits_{i_{1}\neq j_{1}=1}^{s}
\mathcal{L}_{\mathrm{int}}^0(i_{1},j_{1})\\
&&\times S_{s-1}^{0}(t_{1}-t_{2})\ldots S_{2}^{0}(t_{s-2}-t_{s-1})\hskip-5mm
\sum\limits^{s}_{\mbox{\scriptsize $\begin{array}{c}i_{s-1}\neq j_{s-1}=1,\\
i_{s-1},j_{s-1}\neq (j_{1},\ldots,j_{s-2})\end{array}$}}\hskip-5mm
\mathcal{L}_{\mathrm{int}}^0(i_{s-1},j_{s-1})\nonumber\\
&&\times S_{1}^{0}(t_{s-1})\,b_{1}^{(1)}(0,(x_1,\ldots,x_s)
\setminus (x_{j_{1}},\ldots,x_{j_{s-1}})),\quad s\geq1.\nonumber
\end{eqnarray}
We make several examples of expansions (\ref{itvad}) of the limit additive-type marginal
observable:
\begin{eqnarray*}
&&\hskip-8mm b_{1}^{(1)}(t,x_1)=S_{1}(t,1)\,b_{1}^{(1)}(0,x_1),\\
&&\hskip-8mm b_{2}^{(1)}(t,x_1,x_2)=\int_0^t dt_{1}\prod\limits_{i=1}^{2}S_{1}(t-t_{1},i)\,
\mathcal{L}_{\mathrm{int}}^0(1,2)\sum\limits_{j=1}^{2}S_{1}(t_{1},j)\,b_{1}^{(1)}(0,x_j).
\end{eqnarray*}
Thus, in the Boltzmann--Grad scaling limit the kinetic evolution of hard spheres
is described in terms of limit marginal observables (\ref{Iterd}) governed by the
dual Boltzmann hierarchy (\ref{vdh}). Similar approach to the description of the
mean field asymptotic behavior of quantum many-particle systems was developed in
paper \cite{G11}.
\subsection{The derivation of the Boltzmann kinetic equation}
We consider links of the constructed Boltzmann--Grad asymptotic behavior of the additive-type
marginal observables with the nonlinear Boltzmann kinetic equation. Furthermore, the relations
between the evolution of observables and the description of the kinetic evolution of states in
terms of a one-particle marginal distribution function are discussed.
Indeed, for the additive-type marginal observables the Boltzmann--Grad scaling limit gives an
equivalent approach to the description of the kinetic evolution of hard spheres in terms of the
Cauchy problem of the Boltzmann equation with respect to the Cauchy problem (\ref{vdh}),(\ref{vdhi})
of the dual Boltzmann hierarchy. In the case of the $k$-ary marginal observable a solution of the
dual Boltzmann hierarchy (\ref{vdh}) is equivalent to the property of the propagation of initial
chaos for the $k$-particle marginal distribution function in the sense of equality (\ref{eqos}).
If $b(t)\in C_{\gamma}$ and $f_1^0\in L^{1}(\mathbb{R}^{3}\times\mathbb{R}^{3})$, then
under the condition that: $\|f_1^0\|_{L^{1}(\mathbb{R}^{3}\times\mathbb{R}^{3})}<\gamma$,
there exists the Boltzmann--Grad limit of mean value functional (\ref{avmar-1}) which is
determined by the series expansion
\begin{eqnarray*}
&&\hskip-7mm \big\langle b(t)\big|f^{(c)}\big\rangle=\sum\limits_{s=0}^{\infty}\,\frac{1}{s!}\,
\int_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{s}}
dx_{1}\ldots dx_{s}\,b_{s}(t,x_1,\ldots,x_s)\prod\limits_{i=1}^{s} f_1^0(x_i),
\end{eqnarray*}
where we assumed that the initial state is specified by a one-particle marginal distribution
function, namely it represents by the sequence
$f^{(c)}\equiv(1,f_1^0(\textbf{u}_1),\ldots,{\prod\limits}_{i=1}^{s}f_{1}^0(\textbf{u}_i),\ldots)$.
Consequently, for the limit additive-type marginal observables (\ref{itvad}) the following
equality is true:
\begin{eqnarray*}\label{avmar-2}
&&\hskip-7mm \big\langle b^{(1)}(t)\big|f^{(c)}\big\rangle=
\sum\limits_{s=0}^{\infty}\,\frac{1}{s!}\,\int_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{s}}
dx_{1}\ldots dx_{s}\,b_{s}^{(1)}(t,x_1,\ldots,x_s)\prod \limits_{i=1}^{s} f_{1}^0(x_i)=\nonumber\\
&&\hskip+7mm=\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}dx_{1}\,b_{1}^{(1)}(0,x_1)f_{1}(t,x_1),\nonumber
\end{eqnarray*}
where the function $b_{s}^{(1)}(t)$ is represented by expansion (\ref{itvad}) and the limit
marginal distribution function $f_{1}(t,x_1)$ is represented by the series expansion
\begin{eqnarray}\label{viter}
&&\hskip-9mm f_{1}(t,x_1)=\sum\limits_{n=0}^{\infty}\int_0^tdt_{1}\ldots\int_0^{t_{n-1}}dt_{n}\,
\int_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{n}}dx_{2}\ldots dx_{n+1}S_{1}(-t+t_{1},1)\\
&&\times\mathcal{L}_{\mathrm{int}}^{0,\ast}(1,2)
\prod\limits_{j_1=1}^{2}S_{1}(-t_{1}+t_{2},j_1)\ldots\nonumber\\
&&\times\prod\limits_{i_{n}=1}^{n}S_{1}(-t_{n}+t_{n},i_{n})
\sum\limits_{k_{n}=1}^{n}\mathcal{L}_{\mathrm{int}}^{0,\ast}(k_{n},n+1)
\prod\limits_{j_n=1}^{n+1}S_{1}(-t_{n},j_n)\prod\limits_{i=1}^{n+1}f_1^0(x_i).\nonumber
\end{eqnarray}
In series (\ref{viter}) the operator (\ref{aLint0}) adjoint to operator (\ref{int0}) in the
sense of functional (\ref{avmar-1}) is used.
If the function $f_1^0$ is continuous, every term of series expansion (\ref{viter}) exists and
this series converges uniformly on each compact almost everywhere for finite time interval.
For $t\geq0$ limit marginal distribution function (\ref{viter}) is a weak solution of the Cauchy
problem of the Boltzmann kinetic equation with hard sphere collisions:
\begin{eqnarray}
\label{Bolz}
&&\hskip-5mm\frac{\partial}{\partial t}f_{1}(t,x_1)=
-\langle p_1,\frac{\partial}{\partial q_1}\rangle f_{1}(t,x_1)+\\
&&\hskip+5mm+\int_{\mathbb{R}^3\times\mathbb{S}^2_+}d p_2\,d\eta\,
\langle\eta,(p_1-p_2)\rangle\big(f_1(t,q_1,p_1^{\ast})f_1(t,q_1,p_2^{\ast})-
f_1(t,x_1)f_1(t,q_1,p_2)\big), \nonumber\\ \nonumber\\
\label{Bolzi}
&&\hskip-5mm f_1(t,x_1)_{\mid t=0}=f_{1}^0(x_1),
\end{eqnarray}
where the momenta $p_{1}^{\ast}$ and $p_{2}^{\ast}$ are determined by expressions (\ref{momenta}).
Thus, we establish that the dual Boltzmann hierarchy with hard sphere collisions (\ref{vdh})
for additive-type marginal observables and initial states specified by one-particle marginal
distribution function (\ref{lih2}) describes the evolution of a hard sphere system
just as the Boltzmann kinetic equation with hard sphere collisions (\ref{Bolz}).
\subsection{On the propagation of initial chaos}
We prove that within the framework of the evolution of marginal observables of hard
spheres in the Boltzmann--Grad scaling limit a chaos property for states is fulfilled
at arbitrary instant.
The property of the propagation of initial chaos is a consequence of the validity
of the following equality for the mean value functionals of the limit $k$-ary marginal
observables in the case of $k\geq2$:
\begin{eqnarray}\label{dchaos}
&&\hskip-7mm \big\langle b^{(k)}(t)\big|f^{(c)}\big\rangle=\sum\limits_{s=0}^{\infty}\,\frac{1}{s!}\,
\int_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{s}}dx_{1}\ldots dx_{s}\,b_{s}^{(k)}(t,x_1,\ldots,x_s)
\prod \limits_{i=1}^{s} f_1^0(x_i)\\
&&\hskip+7mm =\frac{1}{k!}\int_{(\mathbb{R}^{3}\times\mathbb{R}^{3})^{k}}
dx_{1}\ldots dx_{k}\,b_{k}^{(k)}(0,x_1,\ldots,x_k)
\prod\limits_{i=1}^{k} f_{1}(t,x_i),\nonumber
\end{eqnarray}
where the limit one-particle distribution function $f_{1}(t,x_i)$ is defined by series
expansion (\ref{viter}) and therefore it is governed by the Cauchy problem
(\ref{Bolz}),(\ref{Bolzi}) of the Boltzmann kinetic equation with hard sphere collisions.
Thus, in the Boltzmann--Grad scaling limit an equivalent approach to the description
of the kinetic evolution of hard spheres in terms of the Cauchy problem (\ref{Bolz}),(\ref{Bolzi})
of the Boltzmann kinetic equation is given by the Cauchy problem (\ref{vdh}),(\ref{vdhi})
of the dual Boltzmann hierarchy for the additive-type marginal observables.
In the general case of the $k$-ary marginal observables a solution of the dual Boltzmann
hierarchy (\ref{vdh}) is equivalent to the validity of a chaos property for the $k$-particle
marginal distribution functions in the sense of equality (\ref{dchaos}) or in other words
the Boltzmann--Grad scaling dynamics does not create correlations.
\section{The Boltzmann--Grad asymptotics of the generalized Enskog equation}
In this section we consider an approach to the rigorous derivation of the Boltzmann
equation with hard sphere collisions from the generalized Enskog kinetic equation.
\subsection{The Boltzmann--Grad limit theorem}
For a solution of the generalized Enskog kinetic equation (\ref{gke1}) the following
Boltzmann--Grad scaling limit theorem is true \cite{GG04}.
\begin{theorem}
If the initial one-particle marginal distribution function $F_{1}^{\epsilon,0}$ satisfies
condition (\ref{G_1}) and there exists its limit in the sense of a weak convergence:
$\mathrm{w-}\lim_{\epsilon\rightarrow 0}(\epsilon^{2}F_{1}^{\epsilon,0}(x_1)-f_{1}^0(x_1))=0,$
then for finite time interval the Boltzmann--Grad limit of solution (\ref{F(t)1}) of the Cauchy
problem (\ref{gke1}),(\ref{2}) of the generalized Enskog equation exists in the same sense
\begin{eqnarray}\label{asymt}
&&\mathrm{w-}\lim\limits_{\epsilon\rightarrow 0}\big(\epsilon^{2}F_{1}(t,x_1)-f_{1}(t,x_1)\big)=0,
\end{eqnarray}
where the limit one-particle distribution function is represented by uniformly convergent
on arbitrary compact set series expansion (\ref{viter}).
If $f_{1}^0$ satisfies condition (\ref{G_1}), then for $t\geq 0$ the limit one-particle
distribution function represented by series (\ref{viter}) is a weak solution of the Cauchy
problem (\ref{Bolz}),(\ref{Bolzi}) of the Boltzmann kinetic equation with hard sphere collisions.
\end{theorem}
The proof of this theorem is based on formulas for cumulants (\ref{nLkymyl}) of asymptotically
perturbed groups of adjoint operators of hard spheres. Namely, in the sense of a weak convergence
the following equality holds:
\begin{eqnarray*}\label{lemma2}
&&\hskip-5mm\mathrm{w-}\lim\limits_{\epsilon\rightarrow 0}\Big(\epsilon^{-2n}\frac{1}{n!}
\mathfrak{A}_{1+n}(-t,1,\ldots,n+1)f_{n+1}-\\
&&\hskip-5mm -\int_0^tdt_{1}\ldots\int_0^{t_{n-1}}dt_{n}S_{1}(-t+t_{1},1)
\mathcal{L}_{\mathrm{int}}^{0,\ast}(1,2)\prod\limits_{j_1=1}^{2}S_{1}(-t_{1}+t_{2},j_1)\ldots\\
&&\hskip-5mm\prod\limits_{i_{n}=1}^{n}S_{1}(-t_{n-1}+t_{n},i_{n})
\sum\limits_{k_{n}=1}^{n}\mathcal{L}_{\mathrm{int}}^{0,\ast}(k_{n},n+1)
\prod\limits_{j_n=1}^{n+1}S_{1}(-t_{n},j_n)f_{n+1}\Big)=0, \quad n\geq1,
\end{eqnarray*}
where notations accepted in formula (\ref{viter}) are used.
Thus, the Boltzmann--Grad scaling limit of solution (\ref{F(t)1}) of the generalized Enskog
equation is governed by the Boltzmann kinetic equation with hard sphere collisions (\ref{Bolz}).
We note that one of the advantage of the developed approach to the derivation of the Boltzmann
equation is the possibility to construct of the higher-order corrections to the Boltzmann--Grad
evolution of many-particle systems with hard sphere collisions.
\subsection{A scaling limit of marginal functionals of the state}
As we note above the all possible correlations of a hard sphere system are described
by marginal functionals of the state (\ref{f}). Taking into consideration the fact of
the existence of the Boltzmann--Grad scaling limit (\ref{asymt}) of solution (\ref{F(t)1})
of the generalized Enskog kinetic equation (\ref{gke1}), for marginal functionals of the
state (\ref{f}) the following statement holds.
\begin{theorem}
Under the conditions of the Boltzmann--Grad limit theorem for the generalized Enskog kinetic
equation for finite time interval the following Boltzmann--Grad limits of marginal functionals
of the state (\ref{f}) exist in the sense of a weak convergence on the space of bounded functions:
\begin{eqnarray*}
&&\mathrm{w-}\lim\limits_{\epsilon\rightarrow 0}\big(\epsilon^{2s}
F_{s}\big(t,x_1,\ldots,x_s\mid F_{1}(t)\big)-\prod\limits_{j=1}^{s}f_{1}(t,x_j)\big)=0,
\end{eqnarray*}
where the limit one-particle distribution function $f_{1}(t)$ is represented
by series expansion (\ref{viter}).
\end{theorem}
The proof of this limit theorem is based on the validity of the following formulas for generating
evolution operators (\ref{skrrn}) of scattering cumulants of asymptotically perturbed adjoint groups
of operators of hard spheres:
\begin{eqnarray*}
&&\mathrm{w-}\lim\limits_{\epsilon\rightarrow 0}\big(\mathfrak{V}_{1}(t,\{Y\})f_{s}-If_{s}\big)=0,\\
&&\mathrm{w-}\lim\limits_{\epsilon\rightarrow 0}\epsilon^{-2n}
\mathfrak{V}_{1+n}(t,\{Y\},X\setminus Y)f_{s+n}=0, \quad n\geq 1,
\end{eqnarray*}
where for $f=(1,f_1,\ldots,f_n\ldots)\in L^{\infty}_\xi$ these limits exist in the sense of
a weak convergence.
Thus, the Boltzmann--Grad scaling limits of marginal functionals of the state (\ref{f}) are
the products of solution (\ref{viter}) of the Boltzmann equation with hard sphere collisions
(\ref{Bolz}) that means the property of the propagation of initial chaos \cite{CGP97},\cite{CIP}.
\subsection{Remark: a one-dimensional system of hard spheres}
We consider the Boltzmann--Grad asymptotic behavior of a solution of the generalized
Enskog equation in a one-dimensional space. In this case the dimensionless collision
integral $\mathcal{I}_{GEE}$ has the structure \cite{Ger},\cite{GP83}:
\begin{eqnarray*}
&&\hskip-9mm\mathcal{I}_{GEE}=\int_0^\infty dP\,P \big(F_{2}(t,q_1,p_1-P,
q_1-\epsilon,p_1\mid F_{1}(t))-F_{2}(t,q_1,p_1,q_1-\epsilon,p_1+P\mid F_{1}(t)+\\
&&+F_{2}(t,q_1,p_1+P,q_1+ \epsilon,p_1\mid F_{1}(t))
-F_{2}(t,q_1,p_1,q_1+\epsilon,p_1-P\mid F_{1}(t))\big),\nonumber
\end{eqnarray*}
where $\epsilon>0$ is a scaling parameter and the marginal functional of the state
$F_{2}(t\mid F_{1}(t))$ is represented by series expansion (\ref{f}) in the case of $s=2$
in a one-dimensional space.
As we can see in the Boltzmann--Grad limit the collision integral of the generalized
Enskog equation in a one-dimensional space vanishes, i.e. in other words dynamics of
a one-dimensional system of elastically interacting hard spheres is trivial (a free
molecular motion or the Knudsen flow).
We note that in paper \cite{BG12} it was established that the Boltzmann--Grad asymptotic
behavior of inelastically interacting hard rods is not trivial in contrast to a one-dimensional
hard rod system with elastic collisions and it is governed by the Boltzmann-type kinetic equation
for granular gases.
\section{Conclusion and outlook}
In the paper two new approaches to the description of the kinetic evolution of
many-particle systems with hard sphere collisions were developed. One of them is
a formalism for the description of the evolution of infinitely many hard spheres
within the framework of marginal observables in the Boltzmann--Grad scaling limit.
Another approach to the description of the kinetic evolution of hard spheres is
based on the non-Markovian generalization of the Enskog kinetic equation.
In particular, it was established that a chaos property of the Boltzmann--Grad
scaling behavior of the $s$-particle marginal distribution function of infinitely
many hard spheres is equivalent in the sense of equality (\ref{dchaos}) to a solution
of the dual Boltzmann hierarchy (\ref{vdh}) in the case of the $s$-ary marginal observable.
In other words the Boltzmann--Grad scaling dynamics does not create correlations.
One of the advantage of the considered approaches is the possibility to construct
the kinetic equations in scaling limits, involving correlations at initial time,
which can characterize the condensed states of a hard sphere system.
We emphasize that the approach to the derivation of the Boltzmann equation from
underlying dynamics governed by the generalized Enskog kinetic equation enables
to construct also the higher-order corrections to the Boltzmann--Grad evolution
of many-particle systems with hard sphere collisions.
\addcontentsline{toc}{section}{References}
{\small
\renewcommand{\refname}{References}
|
1,314,259,996,481 | arxiv | \subsection*{Polyominoes}
A \emph{polyomino} is an edge-connected set of unit squares,
called \emph{cells}, embedded in the integer lattice.
Two cells are adjacent if, and only if, they share a common edge.
Edge-connected means that every pair of cells is connected by
a path through adjacent cells.
Polyominoes are often classified by area and referred to as
$n$-ominoes when they contain $n$ cells.
For example, the games of dominoes and Tetris are played with
$2$-ominoes and $4$-ominoes (tetrominoes), respectively (see
Fig. \ref{fig:tetrominoes}).
\begin{figure}
\centerline{\includegraphics[scale=1.0]{tetrominoes}} \caption{All
possible $4$-ominoes (tetrominoes)} \label{fig:tetrominoes}
\end{figure}
Polyominoes have been extensively studied and have a wide-range of
applications in mathematics and the physical sciences
\cite{Golomb, Klarner}.
The problem of counting $n$-ominoes has garnered considerable
interest \cite{Jensen,Mertens,Redelmeier}, and although
counts up to $47$-ominoes are known (see sequence $A001168$
\cite{Sloane}), the problem of finding a formula
for the number remains open.
Several other subclasses of polyominoes have been defined.
\emph{Free} polyominoes treat polyominoes that are translations,
rotations, or reflections of each other to be equivalent whereas
\emph{fixed} polyominoes only consider translations as being
equivalent.
For example, Fig. \ref{fig:tetrominoes} shows the $19$
equivalence classes of fixed tetrominoes and $5$ equivalence
classes (a,b,c,d, and e), of free tetrominoes.
If every column (row) of a polyomino is a contiguous strip of
cells then the polyomino is called \emph{column-convex}
(\emph{row-convex}).
A \emph{convex} polyomino is one that is both column and row
convex (see Fig. \ref{fig:convexivity}).
No closed-form formula is known for the number, $a(n)$, of
fixed column-convex $n$-ominoes; however, P\'{o}lya \cite{Polya}
derived the recurrence relation
$a(n)=5a(n-1)-7a(n-2)+4a(n-3)$ with $a(1)=1$, $a(2)=2$, $a(3)=6$,
and $a(4)=19$.
This recurrence relation has the rational generating function
$g(x) = \frac{x(1-x)^3}{1-5x+7x^2-4x^3}$ (see sequence $A001169$
\cite{Sloane}).
\begin{figure}
\centerline{\includegraphics[scale=1.0]{convexivity}}
\caption{$10$-ominoes that exhibit different convexivity
properties.} \label{fig:convexivity}
\end{figure}
\subsection*{Minimum Area \polyVenns{n}}
An \polyVenn{n}\ is a Venn diagram comprised of $n$ curves,
each of which is the perimeter of some polyomino.
In particular, each polyomino must be free of holes in order for
the perimeter to be a simple, closed curve, and when placed on top
of another polyomino, may not partially cover any of the bottom
polyomino's cells (i.e., the corners of the curves must have
unit coordinates).
Referring to the examples in Fig. \ref{fig:venn_polyominoes}, we
see that an \polyVenn{n}\ can be drawn by tracing the curves on
the lines of a piece of graph paper;
in the (combinatorial) graph
drawing community, this is referred to as an
\emph{orthogonal grid drawing} \cite{Battista}.
In fact, any orthogonal grid drawing of a Venn diagram will
produce curves that are the perimeters of polyominoes.
Since each bounded region must contain at least one cell and
there is exactly one unbounded region, the minimum area for such a
diagram is $2^n - 1$ cells.
In addition, since each curve encloses $2^{n-1}$ regions, it must
be the perimeter of at least a $2^{n-1}$-omino.
This leads us to the following definition of a minimum area
\polyVenn{n}:
\vspace{12pt}
\textsc{Definition} \linebreak
\indent A \textit{minimum area \polyVenn{n}} is an orthogonal
unit-grid drawing of a Venn diagram with area $2^n - 1$.
\vspace{12pt}
By necessity, each curve of a minimum area
\polyVenn{n} has area $2^{n-1}$.
All the Venn diagrams in Fig. \ref{fig:venn_polyominoes} are
minimum area congruent \polyVenns{n}.
By trial-and-error, we have also found minimum area
non-congruent \linebreak[4]
\polyVenns{n} for $n=6,7$ (see Figs. \ref{fig:min6},\ref{fig:min7}).
It is unknown if minimum area \polyVenns{n}\ exist
for $n \geq 8$, although we suspect
there is an upper limit due to the rigid constraints of
orthogonal grid drawings.
Orthogonal grid drawings of Venn diagrams were first
studied by Eloff and van Zijl \cite{Eloff};
they developed a heuristic algorithm based on a
greedy incremental approach.
An optimization step in the algorithm attempted to
reduce the overall area of the diagram, but
there was no upper bound.
In addition, their algorithm produced polyominoes
with holes, so the resulting diagrams
would not be considered Venn diagrams in
the formal sense (because the sets were not
represented by simple, closed curves).
In the following sections, we present algorithms
for approximating minimum area \polyVenns{n}.
The first algorithm is trivial and produces \polyVenns{n}\ with
less than $3/2$ times the minimum area.
The second algorithm improves upon the first by using symmetric
chain decompositions of the Boolean lattice and produces
\polyVenns{n}\ whose areas are asymptotically minimum
(i.e., the ratio of total cells to required cells tends to one
as $n$ increases).
\begin{figure}
\centerline{\includegraphics[scale=0.8]{min6}} \caption[]{A
minimum area \polyVenn{6}.} \label{fig:min6}
\end{figure}
\begin{figure}
\centerline{\includegraphics[scale=0.675]{min7}} \caption[]{A
minimum area \polyVenn{7}.} \label{fig:min7}
\end{figure}
There is another definition of area
based on the $w \times h$ bounding box that contains an
\polyVenn{n}; such a box must also have at least one cell to
represent the empty set.
For example, the \polyVenns{n}\ in Fig.
\ref{fig:venn_polyominoes} are contained by
$4 \times 1$,
$2 \times 5$,
$5 \times 5$, and
$7 \times 7$ bounding boxes, respectively.
Since an \polyVenn{n}\ must be comprised of at least
$2^n-1$ cells, a bounding box must have area at least
$2^n$.
This leads us to the following definition of a minimum bounding
box \polyVenn{n}:
\vspace{12pt}
\textsc{Definition} \linebreak \indent A
\textit{minimum bounding box \polyVenn{n}} is an
orthogonal unit-grid drawing of a Venn diagram that is
enclosed by a $2^s \times 2^t$ rectangle \linebreak
where $s+t=n$.
\vspace{12pt}
Of the congruent \polyVenns{n}\ in Fig. \ref{fig:venn_polyominoes},
only (a) is a minimum bounding box \polyVenn{n}.
Figure \ref{fig:minbbox} shows some examples of minimum bounding
box non-congruent \polyVenns{n}.
\begin{figure}
\centerline{\includegraphics[scale=0.8]{minbbox}} \caption[
]{Minimum bounding box \polyVenns{n}\ for $2 \le n \le 5$.}
\label{fig:minbbox}
\end{figure}
At present, we leave minimum bounding box \polyVenns{n}\ and
focus the rest of this paper on minimum area \polyVenns{n}.
\subsection*{A $3/2$-APPROX Algorithm}
This algorithm is best explained by way of an example.
Suppose we
wish to draw a \polyVenn{5}\ with the curves
$\{A,B,C,D,E\}$.
We begin by drawing a $1 \times 14$ rectangle and labelling it as
region $ABCDE$;
in other words, the curves are $1 \times 14$ rectangles stacked
on top of each other.
We now place $30$ cells around the perimeter of $ABCDE$ and
uniquely label them with the $30$ remaining non-empty regions;
the result is shown in Fig. \ref{fig:venn5_naive}.
After adding the perimeter cells, each curve becomes a polyomino
formed by the original $1 \times 14$ rectangle with ``bumps''
wherever the curve encloses a perimeter cell.
\begin{figure}
\centerline{\includegraphics[scale=0.875]{venn5_naive}} \caption[
]{A na\"{i}ve approximation for a minimum area \polyVenn{5}; curve
$A$ is highlighted} \label{fig:venn5_naive}
\end{figure}
In the general case, this algorithm will produce an $n$-Venn
polyomino beginning with a $1 \times (2^{n-1}-2)$ rectangle that has
a perimeter of $2^n - 2$ (for the $2^n$ regions less the empty and
full sets).
The resulting diagrams have an area of $2^n + 2^{n-1} - 4$ which
is less than $3/2$ times the minimum area of $2^n - 1$.
\subsection*{An Asymptotically Optimal Algorithm}
The previous algorithm can be significantly improved by noting
that not all regions need to be placed adjacent to the initial
rectangle;
instead, if region $X$ is a subset of region $Y$ then
$X$ can be placed directly above or below $Y$ (depending on if $Y$
is above or below the initial rectangle), and the curves will
remain as polyomino perimeters.
This chaining of regions can continue as long as the subset
property is maintained.
Figure \ref{fig:venn5_chains} shows an example
of \polyVenn{5}\ that chains regions as much as possible.
Note also that the resulting polyominoes are column-convex.
\begin{figure}
\centerline{\includegraphics[scale=1.0]{venn5_chains}} \caption[
]{An approximation for a minimum area \polyVenn{5}\ using
column-convex polyominoes and symmetric chains; curve $A$ is
highlighted.} \label{fig:venn5_chains}
\end{figure}
When regions are chained, a smaller perimeter is needed for the
initial rectangle and so the total area of the diagram is reduced.
A smaller area diagram is created by minimizing the number of
chains, so the question arises as to the best way to decompose the
regions into chains;
for this question, we need to use a result from the theory of
partially ordered sets.
Given a set $S$ with powerset $\powerset{S}$, we define the
partially ordered set (poset) $\lattice{S}$ with elements
$\powerset{S}$ ordered by inclusion.
Since $\lattice{S}$ is closed under union, intersection, and
complement, it is a Boolean lattice.
Figure \ref{fig:poset}(a) shows an example of
$\lattice{\{A,B,C,D\}}$.
\begin{figure}
\centering \scalebox{0.75}{\input{poset.pstex_t}} \caption[]{(a) A
Hasse diagram of the poset $\lattice{\{A,B,C,D\}}$, (b) one of its
symmetric chain decompositions, and (c) the resulting
\polyVenn{4}.} \label{fig:poset}
\end{figure}
Let $|S|=n$. A symmetric chain decomposition (SCD) of $\lattice{S}$
is a partition of $S$ into $\binom{n}{\lfloor n/2 \rfloor)}$
symmetric chains.
Each symmetric chain is a sequence of subsets
$x_1, x_2, \ldots, x_t$ with the following properties:
\pagebreak
\begin{equation}
x_i \subset x_{i+1} \mbox{ for all } 1 \leq i < t,
\label{eq:subsetprop}
\end{equation}
\begin{equation}
|x_i|=n-|x_{t-i+1}| \mbox{ for all } 1 \leq i \leq \lceil t/2 \rceil.
\end{equation}
Symmetric chain decompositions form an essential ingredient of the
recent proof of Griggs, Killian and Savage \cite{Griggs} that
symmetric Venn diagrams exist if and only if the number of curves
is prime.
Several algorithms exist for decomposing $\lattice{S}$ into
symmetric chains; we describe two of these algorithms below.
The first, due to de Bruijn, van Ebbenhorst Tengbergen, and
Kruyswijk \cite{DeBruijn} is called the \emph{Christmas tree pattern}
by Knuth \cite{Knuth7216}.
It is an inductive construction that creates a set $T_n$ of
${n \choose \lfloor n/2 \rfloor}$ chains.
Initially $T_1 = \{ \emptyset \subset \{1\} \}$. To obtain $T_{n}$ from $T_{n-1}$,
take each chain $x_1 \subset x_2 \subset \cdots \subset x_t$ in
$T_n$ and replace it with the two chains
$x_2 \subset \cdots \subset x_t$ and
$x_1 \subset x_1 \cup \{n\} \subset x_2 \cup \{n\} \subset \cdots
\subset x_t \cup \{n\}$ in $T_{n+1}$ if $t > 1$.
If $t=1$ the first chain is empty and is ignored.
A second method, due to Aigner \cite{Aigner}, can be described as
a greedy lexicographic algorithm.
It is efficient and easy-to-implement, and is the method that we
used in creating the example diagrams.
Let $m(x,y)$ be the smallest element in a set $x$ that is not in the
set $y$, where $m(x,y) = -\infty$ if $x \subset y$.
We say that $x$ is \emph{lexicographically smaller} than $y$
if $m(x,y) < m(y,x)$.
In Aigner's algorithm, the following process is repeated until every
element of $\lattice{\{1,2,\ldots,n\}}$ is contained in some chain.
For $k = 0,1,2,\ldots,n$, denote by $R(k)$ the set of subsets of $\{1,2,\ldots,n\}$
size $k$ that are not yet in any chain.
Let $j$ be the smallest value for which $R(j)$ is non-empty and
let $x$ be the lexicographically smallest set in $R(j)$.
The set $x$ becomes the smallest set in a new chain
$x = x_1 \subset x_2 \subset \cdots \subset x_t$.
The successive elements of this chain are obtained by taking
$x_{i+1} \in R(i+1)$ to be the lexicographically smallest set that
contains $x_{i}$.
It is by no means obvious that this algorithm is correct, but
indeed it is!
Because of their subset property (\ref{eq:subsetprop}),
the symmetric chains can be
directly used to layout the regions of an \polyVenn{n}.
Figure \ref{fig:poset}(b) shows the SCD of $\lattice{\{A,B,C,D\}}$
that is produced by Aigner's algorithm, and
Fig. \ref{fig:poset}(c) shows the resulting \polyVenn{4}.
The \polyVenn{5}\ in Fig. \ref{fig:venn5_chains} was also
produced from Aigner's SCD of $\lattice{\{A,B,C,D,E\}}$.
In the general case, this algorithm will produce an
\polyVenn{n} beginning with a
$1 \times (\binom{n}{\lfloor n/2 \rfloor}-2)/2$
rectangle that has a perimeter of
$\binom{n}{\lfloor n/2 \rfloor}$.
The resulting diagrams have an area of
$(\binom{n}{\lfloor n/2 \rfloor}-2)/2 + 2^n - 2$.
The lower bound
$\binom{2n}{n} < \frac{2^{2n}}{\sqrt{\pi}(n^2+n/2+3/32)^{1/4}}$
\cite{Grosswald} can be used to show that the algorithm
produces diagrams whose area is
$1 + O(1/\sqrt{n})$
times the minimum area of
$2^n - 1$;
therefore, as $n$ increases, the approximation gets
asymptotically close to optimal.
\subsection*{Open Problems and Final Remarks}
To close the paper, we list some open problems that are inspired by the
examples in this paper.
With the exception of the congruent \polyVenns{n}, the
examples in this paper were constructed by hand,
and it is very likely that relatively na\"ive
programs will be able to extend them.
Such extension would be interesting, but even more interesting would
be general results that apply for arbitrary numbers of curves.
\begin{enumerate}
\item Are there congruent \polyVenns{n}\ for $n \geq 6$?
Figure \ref{fig:venn_polyominoes} shows that they exist for $n = 2,3,4,5$.
\item Is there a \polyVenn{5}\ whose curves are convex polyominoes?
(The curves in Figure \ref{fig:venn_polyominoes}(d) are not both
row-convex and column-convex polyominoes.)
\item Are there minimum bounding box \polyVenns{n}\ for $n \geq 6$?
Figure \ref{fig:minbbox} shows that they exist for $n = 2,3,4,5$.
\item Are there minimum area \polyVenns{n}\ for $n \geq 8$?
Figure \ref{fig:min7} shows one for $n = 7$.
\item One problem for which we have not attempted solutions
is the construction of \polyVenns{n}\ that fill a $w \times h$ box,
where $wh = 2^n-1$.
Of course, a necessary condition is that $2^n-1$ not
be a Mersenne prime.
For example, is there a \polyVenn{4}\ that fits in a
$3 \times 5$ rectangle or a \polyVenn{6}\ that fits
in a $7 \times 9$ or $3 \times 27$ rectangle?
\end{enumerate}
\emph{AUTHORS' NOTE:}
Since submitting the original manuscript of
this paper, Bette Bultena has discovered a \polyVenn{6} with an
$8 \times 8$ bounding box (see problem $3$ above).
Figure \ref{fig:min6} has also been used to represent
the results of experiments in plant genetics \cite{Casimiro}.
|
1,314,259,996,482 | arxiv |
\section{Conclusion}
We presented an algorithm, designed using hybrid system tools, that unites Nesterov's accelerated algorithm and the heavy ball algorithm to ensure fast convergence and uniform global asymptotic stability of the unique minimizer for $\mathcal{C}^1$, nonstrongly convex objective functions $L$. The hybrid convergence rate for nonstrongly convex $L$ is $\frac{1}{(t+2)^2}$ globally and exponential locally. In simulation, we showed performance improvement not only over the individual heavy ball and Nesterov algorithms, but also over the HAND-1 algorithm in \cite{poveda2019inducing}. In the process, we proved the existence of solutions for the individual heavy ball and Nesterov algorithms, and we extended the convergence rate results for Nesterov's algorithm in \cite{muehlebach2019dynamical} to functions $L$ with generic $z_1^*$, $L^*$, and $\zeta > 0$. Additionally, we established uniform global attractivity of the minimizer for Nesterov's algorithm, when $L$ is $\mathcal{C}^1$, nonstrongly convex, and has a unique minimizer. Future work will extend the uniting algorithm to a general framework, allowing the local and global algorithms to be any accelerated gradient algorithm. We will also extend the uniting algorithm to learning applications.
\section{Derivation of Sets $\U_0$, $\mathcal{T}_{1,0}$, and $\mathcal{T}_{0,1}$ from \Cref{sec:NonstronglyConvexNest}}
\label{sec:DerivOfSets}
\section{{\color{red}Proofs of Heavy Ball Results}}
\label{sec:DynamicalHBF}
\section{{\color{red}Proofs of Results on Nesterov's Accelerated Gradient Descent}}
\label{sec:DynamicalNesterov}
\section{General Results for Hybrid Systems}
\label{sec:GenHybridResults}
The following proposition, from \cite{65}, is used to prove the existence of solutions to the hybrid closed-loop system.
\IfConf{\begin{prop}[Basic existence of solutions]}{\begin{proposition}(Basic existence of solutions):} \label{prop:SolnExistence}
Let $\HS = (C,F,D,G)$ satisfy Definition \ref{def:HBCs}. Take an arbitrary $\xi \in C \cup D$. If $\xi \in D$ or\\
\noindent
(VC) there exists a neighborhood $U$ of $\xi$ such that for every $x \in U \cap C$,
$$F(x) \cap T_C(x) \neq \emptyset,$$
then there exists a nontrivial solution $x$ to $\HS$ with $x(0,0) = \xi$. If (VC) holds for every $\xi \in C \setminus D$, then there exists a nontrivial solution to $\HS$ from every initial point in $C \cup D$, and every $x \in \mathcal{S}_{\HS}$ satisfies exactly one of the following conditions:
\begin{enumerate}[label={(\alph*)}]
\item \label{item:SolnExistenceItemA} $x$ is complete;
\item \label{item:SolnExistenceItemB} $\mathrm{dom} \ x$ is bounded and the interval $I^J$, where $J = \mathrm{sup}_j \ \mathrm{dom} \ x$, has nonempty interior and $t \mapsto x(t,J)$ is a maximal solution to $\dot{z} \in F(z)$, in fact\\ $\mathrm{lim}_{t \mapsto T} \lvert x(t,J) \rvert = \infty$, where $T = \mathrm{sup}_t \ \mathrm{dom} \ x$;
\item \label{item:SolnExistenceItemC} $x(T,J) \not\in C \cup D$, where $(T,J) = \mathrm{sup} \ \mathrm{dom} \ x$.
\end{enumerate}
Furthermore, if $G(D) \subset C \cup D$, then (c) above does not occur.
\IfConf{\end{prop}}{\end{proposition}}
The following definition, from \cite[Definition~3.17]{220}, describes the basic properties that a function must satisfy to serve as a Lyapunov function for the hybrid closed-loop algorithm $\HS$.
\IfConf{\begin{defn}[Lyapunov function candidate]}{\begin{definition}[Lyapunov function candidate]} \label{def:LyapCandidate}
The sets $\mathcal{U}$, $\mathcal{A} \subset \reals^n$, and the function $V : \dom V \rightarrow \reals$ define a \underline{Lyapunov function candidate} on $\mathcal{U}$ with respect to $\mathcal{A}$ for the hybrid closed-loop system $\HS = \left(C, F, D, G\right)$ if the following conditions hold:
\begin{enumerate}
\item $\left(\overline{C} \cup D \cup G(D) \right) \cup \mathcal{U} \subset \dom V$;
\item $\mathcal{U}$ contains an open neighborhood of\IfConf{\\}{} $\mathcal{A} \cap \left(C \cup D \cup G(D)\right)$;
\item $V$ is continuous on $\mathcal{U}$ and locally Lipschitz on an open set containing $\overline{C} \cap \mathcal{U}$;
\item $V$ is positive definite on $\overline{C} \cup D \cup G(D)$ with respect to $\mathcal{A}$.
\end{enumerate}
\IfConf{\end{defn}}{\end{definition}}
The following theorem is used to prove the uniform global asymptotic stability of the hybrid closed-loop system, via Lyapunov stability and an invariance principle.
\IfConf{\begin{thm}[Hybrid Lyapunov theorem]}{\begin{theorem}(Hybrid Lyapunov theorem):} \label{thm:hybrid Lyapunov theorem}
Given sets $\mathcal{U}, \mathcal{A} \subset \reals^n$ and a function $V : \mathrm{dom} \ V \rightarrow \reals$ defining a Lyapunov candidate on $\mathcal{U}$ with respect to $\mathcal{A}$ for the closed-loop hybrid system $\HS = (C,F,D,G)$, suppose
\begin{itemize}
\item $\HS$ satisfies the hybrid basic conditions;
\item $\mathcal{A}$ is compact and $\mathcal{U}$ contains a nonzero open neighborhood of $\mathcal{A}$;
\item $\dot{V}$ and $\Delta V$ satisfy
\begin{align}
& \dot{V}(x) = \max_{\xi \in F(x)} \langle \nabla V(x),\xi \rangle \leq 0 \qquad \qquad \forall x \in C \cap \mathcal{U} \label{eqn:CTLyapunov}\\
& \Delta V(x) := \max_{\xi \in G(x)} V(\xi) - V(x) \leq 0 \quad \quad \ \forall x \in D \cap \mathcal{U} \label{eqn:DTLyapunov}
\end{align}
\end{itemize}
Then $\mathcal{A}$ is stable. Furthermore, $\mathcal{A}$ is attractive and, hence, pre-asymptotically stable if any of the following conditions hold:
\begin{enumerate}
\item Strict decrease during flows and jumps:
\begin{align}
& \dot{V}(x) < 0 \qquad \qquad \qquad \quad \forall x \in (C \cap \mathcal{U}) \backslash \mathcal{A} \\
& \Delta V(x) < 0 \qquad \qquad \qquad \; \forall x \in (D \cap \mathcal{U}) \backslash \mathcal{A}
\end{align}
\item Strict decrease during flows and no instantaneous Zeno:
\begin{enumerate}[label={(\alph*)}]
\item $\dot{V}(x) < 0$ for each $x \in (C \cap \mathcal{U}) \backslash \mathcal{A}$,
\item any instantaneous Zeno solution $x$ to $\HS$ where $\mathrm{rge} \ x \subset \mathcal{U}$ converges to $\mathcal{A}$;
\end{enumerate}
\item Strict decrease during jumps and no complete continuous solution:
\begin{enumerate}[label={(\alph*)}]
\item $\Delta V(x) < 0$ for each $x \in (D \cap \mathcal{U}) \backslash \mathcal{A}$,
\item any complete continuous solution $x$ to $\HS$ where $\mathrm{rge} \ x \subset \mathcal{U}$ converges to $\mathcal{A}$;
\end{enumerate}
\item \label{item:HybridLyapunovWeak} Weak decrease during flows and jumps: for each $\chi \in \mathcal{U}$ with $r := V(\chi) > 0$ there is no complete solution $x$ to $\HS$, $x(0,0) = \chi$ such that
\begin{equation}
\mathrm{rge} \ x \subset \defset{x}{V(x) = r} \cap \mathcal{U}
\end{equation}
and the set $\mathcal{U}$ is the subset of the basin of pre-attraction.
\end{enumerate}
\IfConf{\end{thm}}{\end{theorem}}
Observe that, if the set $\mathcal{A}$ is pre-asymptotically stable via Theorem \ref{thm:hybrid Lyapunov theorem} and the Lyapunov function $V$ also has compact sublevel sets, namely, for each $c_V > 0$, $\defset{x}{V(x) \leq c_V}$ is compact, then the origin is {\em globally pre-asymptotically stable}.
The following result is used to show that, when a hybrid closed-loop algorithm $\HS$ has a set $\mathcal{A}$ globally asymptotically stable, then when $\HS$ satisfies the hybrid basic conditions, the set $\mathcal{A}$ is also uniformly globally asymptotically stable\footnote{Uniform global asymptotic stability allows an equivalent characterization involving a class-$\mathcal{KL}$ function \cite{65}.} for $\HS$.
\IfConf{\begin{thm}[pAS implies $\mathcal{KL}$ pAS]}{\begin{theorem}(Pre-asymptotic stability implies $\mathcal{KL}$ pre-asymptotic stability):}
\label{thm:GASImpliesUGAS}
Suppose that the hybrid closed-loop system $\HS$ satisfies the hybrid basic conditions and that a compact set $\mathcal{A}$ is pre-asymptotically stable with basin of pre-attraction $\mathcal{B}^p_{\mathcal{A}}$. Then, $\mathcal{B}^p_{\mathcal{A}}$ is open and $\mathcal{A}$ is $\mathcal{KL}$ pre-asymptotically stable on $\mathcal{B}^p_{\mathcal{A}}$ for $\HS$; namely, there exists a function $\beta \in \mathcal{KL}$ such that
\begin{equation}
\left|x(0,0)\right|_{\mathcal{A}} \beta \left(\left|x(0,0)\right|_{\mathcal{A}}, t+j \right) \quad \forall (t,j) \in \dom x
\end{equation}
for each $x \in \mathcal{S}_{\HS}(\mathcal{B}^p_{\mathcal{A}})$.
\IfConf{\end{thm}}{\end{theorem}}
For Proposition \ref{prop:GAS-HBF} and Theorem \ref{thm:HybridInvariancePrinciple} we use the following definition of weak invariance, from \cite{65}.
\IfConf{\begin{defn}[Weak invariance]}{\begin{definition}[Weak invariance]}
Given a hybrid system $\HS$, a set $S \subset \reals^n$ is said to be
\begin{itemize}
\item weakly forward invariant if for every $\xi \in S$ there exists at least one complete $x \in \mathcal{S}_{\HS}(\xi)$ with $\mathrm{rge} \ x \subset S$;
\item weakly backward invariant if for every $\xi \in S$ and every $T> 0$, there exists at least one $x \in \mathcal{S}_{\HS}(S)$ such that for some $(t^*, j^*) \in \dom x$, $t^* + j^* \geq T$, it is the case that $x(t^*, j^*) = \xi$ and $x(t,j) \in S$ for all $(t,j) \in \dom x$ with $t + j \leq t^* + j^*$;
\item weakly invariant if it is both weakly forward invariant and weakly backward invariant.
\end{itemize}
\IfConf{\end{definition}}{\end{definition}}
The following {\em hybrid invariance principle}, from \cite[Theorem~3.23]{220}, is used to establish attractivity when only a ``weak'' Lyapunov function is available -- meaning that the function does not strictly decrease along both flows and jumps of the hybrid system. It is also useful to check where particular solutions of interest converge to.
\IfConf{\begin{thm}[Hybrid Invariance Principle]}{\begin{theorem}(Hybrid Invariance Principle):} \label{thm:HybridInvariancePrinciple}
Given a hybrid closed-loop system $\HS = (C,F,D,G)$ with state $x \in \reals^n$ satisfying the hybrid basic conditions, nonempty $\mathcal{U} \subset \reals^n$, and a function $V : \dom V \rightarrow \reals$, suppose that \ref{def:LyapCandidate} is satisfied, and that \eqref{eqn:CTLyapunov} and \eqref{eqn:DTLyapunov} hold. With $X := C \cup D \cup G(D)$, we empoly the following definitions:
\begin{align}
V^{-1}(r) & := \defset{x \in X}{V(x) = r\!\!}\\
\dot{V}^{-1}(0) & := \defset{x \in C}{\dot{V}(x) = 0\!\!}\\
\Delta V^{-1}(0) & := \defset{x \in D}{\Delta V(x) = 0\!\!}
\end{align}
Let $x$ be a precompact solution to $\HS$ with $\overline{\rge x} \subset \mathcal{U}$. Then, for some $r \in V(\mathcal{U} \cap X)$, the following hold:
\begin{enumerate}
\item The solution $x$ converges to the largest weakly invariant set in
\begin{equation}
V^{-1}(r) \cap \mathcal{U} \cap \left[\dot{V}^{-1}(0) \cup \left(\Delta V^{-1}(0) \cap G\left(\Delta V^{-1}(0)\right) \right) \right];
\end{equation}
\item The solution $x$ converges to the largest weakly invariant set in
\begin{equation}
V^{-1}(r) \cap \mathcal{U} \cap \Delta V^{-1}(0) \cap G\left(\Delta V^{-1}(0)\right)
\end{equation}
if in addition the solution $X$ is Zeno;
\item The solution $x$ converges to the largest weakly invariant set in
\begin{equation}
V^{-1}(r) \cap \mathcal{U} \cap \dot{V}^{-1}(0)
\end{equation}
if, in addition, the solution $x$ is such that, for some $a > 0$ and some $J \in N$, $t_{j + 1} - t_j > a$ for all $j \geq J$; i.e., the given solution $x$ is such that the elapsed time between consecutive jumps is eventually bounded below by a positive constant $a$.
\end{enumerate}
\IfConf{\end{thm}}{\end{theorem}}
\section{Uniting Specific Accelerated Gradient Methods}
\label{sec:SpecificCases}
In this section, we present specific cases of uniting algorithms, which fit into the framework described in \Cref{sec:ModelingUF} and which use combinations of the heavy ball algorithm and Nesterov's algorithm. For each case, we present the data of the hybrid closed-loop system $\HS$ resulting from the specific combination of optimization algorithms, and then we present the main result for such a combination. We start first with the simplest case, which unites local and global heavy ball algorithms.
\subsection{Uniting Heavy Ball Algorithms}
\label{sec:HBF-HBF}
In this section, we present a specific case of the framework in \Cref{sec:ModelingUF} which unites two standard heavy ball algorithms, namely, one heavy ball algorithm with small $\lambda$ used far from the minimizer and the other with large $\lambda$ used near the minimizer. For this specific case, \cref{ass:LisNSCVX} holds.
The static state-feedback laws used for $\HS_{q}$ are
\begin{equation} \label{eqn:HBF_HCQ}
u = \kappa_{q}(h_{q}(z)) = -\lambda_{q} z_2 - \gamma_{q} \nabla L(z_1)
\end{equation}
For each $q \in Q := \{0,1\}$. {\color{red}The state $\tau$ is not needed because the heavy ball algorithm is time-invariant, as shown in \cref{eqn:HBF}.} The parameters $\lambda_{q} > 0$ and $\gamma_{q} > 0$ should be designed for each $q \in Q$, to achieve fast convergence without oscillations nearby the minimizer.
For the individual heavy ball algorithms, $h_{q}$ is defined as
\begin{equation} \label{eqn:HBF-HBFFunctionH}
h_0(z) = h_1(z) := \matt{z_2\\ \nabla L(z_1)}
\end{equation}
where $h_0$ corresponds to the output for the local heavy ball algorithm and $h_1$ corresponds to the output for the global heavy ball algorithm.
The design of the logic and parameters of the individual algorithms is done using the Lyapunov functions
\begin{equation} \label{eqn:VfunctionHBF}
V_{q}(z) := \gamma_{q} \left(L(z_1) - L^* \right) + \frac{1}{2} \left|z_2\right|^2
\end{equation}
defined for each $q \in Q$ and each $z \in \reals^{2n}$. As evidenced by \cref{def:Convexity} in \Cref{sec:Preliminaries}, the set of nonstrongly convex functions includes a subset of functions $L$ which have a connected continuum of minimizers, with the most extreme case being linear functions \cite{boyd2004convex} \cite{nocedal2006numerical}. Therefore, for nonstrongly convex functions\footnote{Since the value of $L$ is the same for all $z_1 \in \A_1$, $L(\A_1)$ is a singleton.}, $L^* := L(\A_1) \in \reals$ is its minimum value.
{\color{red}
To define the closed set $\mathcal{T}_{0,1}$ and ensure Lipschitz continuity of the static state-feedback laws $\kappa_{q}$ in \cref{eqn:HBF_HCQ}, We impose the following assumption\footnote{In the case where $L$ is $\mathcal{C}^2$ and strongly convex, this implies there exists some $M > 0$ such that $\nabla L$ is Lipschitz continuous.} on $\nabla L$.
\begin{assumption}[Lipschitz Continuity of $\nabla L$] \label{ass:Lipschitz}
the function $\nabla L$ is Lipschitz-continuous with constant $M > 0$, namely,
\begin{enumerate}[label={(LC\arabic*)},leftmargin=*]
\item \label{item:LC1} $\left|\nabla L(z_1) - \nabla L(u_1)\right| \leq M \left|z_1 - u_1\right|$;
\item \label{item:LC2} $L(z_1) \leq L(u_1) + \left\langle \nabla L(u_1), z_1 - u_1 \right\rangle + \frac{M}{2}\left|z_1 - u_1\right|^2$.
\end{enumerate}
for all $z_1, u_1 \in \reals^n$.
\end{assumption}
Then, using \cref{def:Convexity}, the suboptimality condition in \cref{lemma:ConvexSuboptimality}, the Lyapunov functions in \cref{eqn:VfunctionHBF}, letting $\varepsilon_0 > 0$, $\alpha > 0$, $c_0 > 0$, and $\gamma_0 > 0$ from $\kappa_0$ be such that
\begin{subequations} \label{eqn:UTilde0SetEquations}
\begin{align}
\tilde{c}_0 & := \varepsilon_0 \alpha > 0\label{eqn:UTilde0SetEquationsA}\\
d_0 & := c_0 - \gamma_0 \left(\frac{\tilde{c}_0^2}{\alpha}\right) > 0. \label{eqn:UTilde0SetEquationsB}
\end{align}
\end{subequations}
and letting $\varepsilon_{1,0} \in (0,\varepsilon_0)$, $\alpha > 0$, and $c_{1,0} \in (0, c_0)$ be such that
\begin{subequations} \label{eqn:TTilde10SetEquations}
\begin{align}
\tilde{c}_{1,0} & := \varepsilon_{1,0} \alpha \in (0, \tilde{c}_0) \label{eqn:TTilde10SetEquationsA}\\
d_{1,0} & := 2c_{1,0} - 2\gamma_1 \left(\frac{\tilde{c}_{1,0}^2}{\alpha} \right) \in \left(0,d_0 \right) \label{eqn:HatD10}
\end{align}
\end{subequations}
we define the sets $\U_0$ and $\mathcal{T}_{1,0}$ as in \cref{eqn:U0} and \cref{eqn:T10}, respectively. The parameters $\tilde{c}_0$, $d_0$, $\lambda_0$, and $\gamma_0$ are designed so that $\U_0$ is in the region where $\kappa_0$ is to be used. In this design, $\lambda_0$ is large to avoid oscillations when converging to the minimum. By construction, $\mathcal{T}_{1,0}$ is in the interior of $\U_0$.
Using \cref{ass:Lipschitz}, \cref{lemma:ConvexSuboptimality}, and choosing $\hat{c}_0 > c_0$, we define the set
\begin{equation} \label{eqn:T01}
\mathcal{T}_{0,1} \subset \defset{z \in \reals^{2n}}{\gamma_0 \left(\frac{\alpha}{M^2} \right) \left|\nabla L(z_1)\right|^2 + \frac{1}{2}\left|z_2\right|^2 \geq \hat{c}_0}.
\end{equation}
This set defines the (closed) complement of a sublevel set with level larger than $c_0$. It is used for the design of the set $D_0$, which triggers the jumps from using $\kappa_0$ to using $\kappa_1$, so that when, in particular, the state $z_1$ is far from the set $\A_1$, then $\kappa_1$ is used to steer it back to nearby it. For details on the derivations of these sets, see \cref{sec:DerivOfSets}.
We can apply our main result from \Cref{sec:BasicProperties} to this special case as follows. \Cref{lemma:HBCC} and \cref{prop:ExistenceC} hold for the hybrid closed-loop algorithm $\HS$, defined via \cref{eqn:HS-TimeVarying}-\cref{eqn:CAndDGradientsNestNSC}, omitting the state $\tau$ and with static state-feedback $\kappa_{q}$ in \cref{eqn:HBF_HCQ}, since the following conditions are satisfied.
\begin{enumerate}
\item Due to $L$ satisfying \cref{ass:LisNSCVX,ass:QuadraticGrowth,ass:Lipschitz}, the sets $\U_0$, $\mathcal{T}_{1,0}$, and $\mathcal{T}_{0,1}$, defined via \cref{eqn:U0}, \cref{eqn:T10}, and \cref{eqn:T01} are closed by construction. Therefore \cref{item:C1} of \cref{ass:AssumpsForHBC} also holds.
\item Due to $L$ satisfying \cref{ass:Lipschitz}, \cref{item:C2} of \cref{ass:AssumpsForHBC} also holds for $\dot{x} = F(x)$.
\end{enumerate}
\Cref{thm:GASGen} holds due to the following additional conditions being satisfied.
\begin{enumerate}
\item We impose \cref{ass:CompactAndConnected} on $\A_1$.
\item The closed-loop algorithm $\HS_1$ in \cref{eqn:Hq}, resulting from $\kappa_1$ in \cref{eqn:HBF_HCQ}, has the set $\A_1 \times \{0\}$ uniformly globally asymptotically stable as stated in \cref{thm:GAS-HBF}. This is a stronger condition than that in \cref{item:UC1} of \cref{ass:UnitingControl}.
\item The closed-loop algorithm $\HS_0$ in \cref{eqn:Hq}, resulting from $\kappa_0$ in \cref{eqn:HBF_HCQ}, has the set $\A_1 \times \{0\}$ globally asymptotically stable as stated in \cref{thm:GAS-HBF}. Therefore, \cref{item:UC3} of \cref{ass:UnitingControl} holds.
\item The existence of the constants $\tilde{c}_{1,0} \in (0, \tilde{c}_0)$ and $d_{1,0} \in (0, d_0)$, defined via \cref{eqn:UTilde0SetEquations} and \cref{eqn:TTilde10SetEquations}, which, combined with the condition in \cref{eqn:V1Bound}, leads to every $z \in \mathcal{T}_{1,0}$ being in a $c_{1,0}$-sublevel set of $V_1$ via \cref{def:Convexity} and \cref{lemma:ConvexSuboptimality}, contained in the interior of $\U_0$, implies that there exist $\tilde{c}_1 \in (0,\tilde{c}_{1,0})$ and $d_1 \in (0,d_{1,0})$ such that \cref{eqn:c1d1Set} holds. Moreover, since \cref{item:UC3} holds, then each solution $z$ to $\HS_0$ with initial condition in $\mathcal{T}_{1,0}$, resulting from applying $\kappa_0$, remains in $\U_0$. Therefore, \cref{item:UC2} holds.
\item The construction of $\U_0$ and $\mathcal{T}_{0,1}$, in \cref{eqn:U0} and \cref{eqn:T01}, respectively, with $\tilde{c}_0 > 0$ and $d_0 > 0$ defined via \cref{eqn:UTilde0SetEquations} and with $\hat{c}_0 > c_0 > 0$, satisfies \cref{item:UC4} of \cref{ass:UnitingControl}.
\end{enumerate}}
{\color{red}Under \cref{ass:LisNSCVX,ass:QuadraticGrowth,ass:Lipschitz}, the hybrid closed-loop system $\HS$, resulting from $\kappa_{q}$ defined via \cref{eqn:HBF_HCQ} has a hybrid convergence rate that is exponential. To prove such a rate in the following theorem, we use \cref{thm:HBFConvergenceRate}.
\begin{theorem}[Convergence rate for $\HS$]
Let $L$ satisfy \cref{ass:LisNSCVX,ass:QuadraticGrowth,ass:Lipschitz}. Additionally, let $\lambda_{q} > 0$, $\gamma_{q} > 0$, $\varepsilon_{1,0} \in (0,\varepsilon_0)$, $\alpha > 0$ from \cref{ass:QuadraticGrowth}, $c_{1,0} \in (0, c_0)$, $\tilde{c}_{1,0} \in (0, \tilde{c}_0)$ in \cref{eqn:UTilde0SetEquationsA} and \cref{eqn:TTilde10SetEquationsA}, $d_{1,0} \in (0, d_0)$ in \cref{eqn:UTilde0SetEquationsB} and \cref{eqn:HatD10}, $\hat{c}_0 > c_0$, $\U_0$ in \cref{eqn:U0}, $\mathcal{T}_{1,0}$ in \cref{eqn:T10}, and $\mathcal{T}_{0,1}$ in \cref{eqn:T01}. Then, each maximal solution $(t,j) \mapsto x(t,j) = (z_1(t,j), z_2(t,j),q(t,j))$ to the hybrid closed-loop system $\HS$ satisfies
\begin{equation} \label{eqn:HBFHybridConvergenceRate}
L(z_1(t,j) - L^*) \leq \left(L(z_1(0,0)) - L^*\right) \exp \left(-\frac{\psi}{2} t\right)
\end{equation}
for all $(t,j) \in \dom x$, where $\psi > 0$, $\psi - \lambda_{q} < 0$ comes from \cref{thm:HBFConvergenceRate}.
\end{theorem}
\begin{proof}
To prove the exponential convergence bound in \cref{eqn:HBFHybridConvergenceRate}, since $L$ is nonstrongly convex and has quadratic growth away from $\A_1$, by \cref{ass:LisNSCVX,ass:QuadraticGrowth}, then by \cref{thm:HBFConvergenceRate} each maximal solution $t \mapsto (z_1(t),z_2(t))$ to the individual heavy ball algorithms $\HS_0$ and $\HS_1$ satisfy \cref{eqn:HBFHybridConvergenceRate} for all $t \geq 0$, where $\psi > 0$, $\psi - \lambda_{q} < 0$. Due to the construction of $G$, the state $z$ does not change in $D$. Moreover, due to the construction of $D$, in \cref{eqn:CAndDGradientsNestNSC}, with $\U_0$ in \cref{eqn:U0}, $\mathcal{T}_{1,0}$ in \cref{eqn:T10}, and $\mathcal{T}_{0,1}$ in \cref{eqn:T01}, with $\tilde{c}_{1,0} \in (0, \tilde{c}_0)$ in \cref{eqn:UTilde0SetEquationsA} and \cref{eqn:TTilde10SetEquationsA}, with $d_{1,0} \in (0, d_0)$ in \cref{eqn:UTilde0SetEquationsB} and \cref{eqn:HatD10}, and with $\hat{c}_0 > c_0$, as described in the proof of \cref{thm:GASGen} solutions to $\HS$ are guaranteed to jump no more than twice. Therefore, due to the construction of $G$ and $D$, and due to the convergence rates of $\HS_1$ and $\HS_0$, then each maximal solution $(t,j) \mapsto x(t,j) = (z_1(t,j), z_2(t,j),q(t,j))$ to the hybrid closed-loop system $\HS$ starting from $C_1$ satisfies \cref{eqn:HBFHybridConvergenceRate} for all $(t,j) \in \dom x$, where $\psi > 0$, $\psi - \lambda_{q} < 0$ comes from \cref{thm:HBFConvergenceRate}.
\end{proof}}
\begin{example} \label{ex:exampleHBF}
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
%
\begin{picture}(30.8,12.1)(0,0)
\footnotesize
\put(0,0.5){\includegraphics[scale=0.4,trim={1.2cm 6cm 1.2cm 0.6cm},clip,width=30.8\unitlength]{Figures/HBFGradientsPlots.eps}}
\put(0,6.2){$z_1$}
\put(16,0){$t[s]$}
\end{picture}
\caption{A comparison of the evolution of $z_1$ over time for $\HS_0$, $\HS_1$, and $\HS$, defined via \cref{eqn:HS-TimeVarying} with $C$ and $D$ defined in \cref{eqn:CAndDGradientsNestNSC}, for a function $L(z_1) := z_1^2$ with a single minimizer at $\A_1 = \{0\}$. Heavy ball $\HS_1$ with $\lambda_1 = \frac{1}{5}$, shown in purple, settles to within $1\%$ of $\A_1$ in about 60.7 seconds. The heavy ball algorithm $\HS_0$ with $\lambda_0 = 30$, shown in green, settles to within $1\%$ of $\A_1$ in about 186.3 seconds. The hybrid closed-loop system $\HS$, shown in blue, settles to within $1\%$ of $\A_1$ in about 2.9 seconds.}
\label{fig:UnitingGradientsHBF}
\end{figure}
To show the effectiveness of the uniting algorithm, we compare it in simulation to the individual heavy ball algorithms $\HS_0$ and $\HS_1$. For this comparison, we use the objective function $L(z_1) := z_1^2$, with a single minimizer at $\A_1 = \{0\}$. For simulation, we used the heavy ball parameter values $\lambda_0 = 30$, $\lambda_1 = \frac{1}{5}$, and $\gamma_0 = \gamma_1 = \frac{1}{2}$. The parameter values for the uniting algorithm are $c_0 = 1200$, $c_{1,0} = 925$, $\varepsilon_0 = 12.5$, $\varepsilon_{1,0} = 6.3$, and $\alpha = 1$, which yield the values $\tilde{c}_0 = 12.5$, $\tilde{c}_{1,0} = 6.3$, $d_0 = 1.1219 \times 10^3$, and $d_{1,0} = 1810.31$, which are calculated via \cref{eqn:UTilde0SetEquations} and \cref{eqn:TTilde10SetEquations}. We also pick $\hat{c}_0 = c_0 + 1$. Initial conditions for $\HS_0$, $\HS_1$, and $\HS$ are $z_1(0,0) = -50$, $z_2(0,0) = 0$, and $q(0,0) = 0$.
\Cref{fig:UnitingGradientsHBF} shows the $z_1$component over time for each of the algorithms\footnote{Code at \url{gitHub.com/HybridSystemsLab/UnitingGradientsHBF}}. Black dots with times labeled in seconds denote when each solution settles within $1\%$ of $\A_1$. Algorithm $\HS_1$, shown in purple, reaches the set $\A_1$ quickly with a rise time of about 2.9 seconds. However, it overshoots to about 36.4 at a peak time of 3.1 seconds. Then it continues oscillating until it settles within $1\%$ of $\A_1$ in about 60.7 seconds. Algorithm $\HS_0$, shown in green, slowly settles within $1\%$ of $\A_1$ in about 186.3 seconds, which is the same as its rise time. The hybrid closed-loop system $\HS$, shown in blue, settles within $1\%$ of $\A_1$ in about 2.9 seconds, which is also the same as its rise time. To show percent improvement of the uniting algorithm over the individual algorithms, we use the following formula:
\begin{equation} \label{eqn:PercentImprovement}
\left(\frac{\text{Time of other algorithm} - \text{Time of } \HS}{\text{Time of other algorithm}}\right) \times 100 \%
\end{equation}
Using this formula, the hybrid closed-loop algorithm shows a $95.3\%$ improvement over $\HS_1$ and a $98.5\%$ improvement over $\HS_0$.
\end{example}
\begin{example} \label{ex:exampleHBFNoise}
To show that the uniform global asymptotic stability of $\A$, established in \cref{thm:GASGen}, is robust to small perturbations, we simulate the hybrid algorithm uniting local and global heavy ball algorithms, with zero-mean Gaussian noise added to measurements of the gradient.
The objective function $L$, heavy ball parameter values $\lambda_{q}$ and $\gamma_{q}$, uniting algorithm parameter values $c_0$, $c_{1,0}$, $\varepsilon_0$, $\varepsilon_{1,0}$, $\alpha$, $\tilde{c}_0$, $\tilde{c}_{1,0}$, $d_0$, $d_{1,0}$, and $\hat{c}_0$, and initial conditions $z_1(0,0)$, $z_2(0,0)$, and $q(0,0)$ are the same as those used in \cref{ex:exampleHBF}. However, noise is generated for trajectories of the perturbed system by drawing random values from the standard normal distribution with zero mean and standard deviations of $\sigma = 0.1$, $\sigma = 0.5$, and $\sigma = 1$, and interpolating intermediary values at intervals of $0.01$ between these random numbers, using \texttt{interp1} in MATLAB.
\Cref{fig:HBFNoise} compares the trajectory of the nominal system with the three trajectories of the perturbed system. On the right hand side of \Cref{fig:HBFNoise}, a close-up of the local portion of the convergence is shown. The nominal system, which is shown in blue, converges to within $1\%$ of $\A_1$ in about $2.9$ seconds, as it does in \cref{fig:UnitingGradientsHBF}. The perturbed system with standard deviation of $\sigma = 0.1$, shown in purple, converges to within $1\%$ of $\A_1$ in about $33.3$ seconds, due to a slightly earlier switch to $\kappa_0$, which was caused by the noise in measurements of the gradient. The perturbed systems with $\sigma = 0.5$ and $\sigma = 1$, shown in red and green, respectively, converge to within $1\%$ of $\A_1$ in about $1.71$ and $1.73$ seconds due to slightly later switches to $\kappa_0$.
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
\begin{picture}(30.8,14)(0,0)
\footnotesize
\put(0,0.5){\includegraphics[scale=0.4,trim={0.7cm 0.6cm 1cm 0.5cm},clip,width=30.8\unitlength]{Figures/HeavyBallGradientsNoise.eps}}
\put(10,0){$z_1$}
\put(25,0){$z_1$}
\put(0,8){$z_2$}
\end{picture}
\caption{A comparison of the nominal and perturbed systems, with different values of $\sigma$. The nominal system, which converges to within a distance of $1\%$ of $\A_1$ in $2.9$ seconds, is in blue. The perturbed system with $\sigma = 0.1$, which converges to within $1\%$ of $\A_1$ in about $33.3$ seconds, is in purple. The perturbed system with $\sigma = 0.5$, which converges to within $1\%$ of $\A_1$ in about $1.71$ seconds, is in red. And the perturbed system with $\sigma = 1$, which converges to within $1\%$ of $\A_1$ in about $1.73$ seconds, is in green. Left: the full trajectory of each system. Right: a close up of the local convergence.}
\label{fig:HBFNoise}
\end{figure}
\end{example}
\section{Introduction}
\label{sec:Intro}
\subsection{Background and Motivation}
\label{sec:Background}
There has been growing interest in analyzing accelerated gradient methods from a dynamical systems perspective \cite{franca2018dynamical}, \cite{muehlebach2019dynamical}. A dynamical systems perspective permits the use of well established analysis tools, such as Lyapunov theory, to study convergence and stability properties of accelerated algorithms \cite{polyak2017lyapunov}, \cite{attouch2000heavy}, \cite{su2016differential}, \cite{krichene2015accelerated}, \cite{wibisono2016variational}, \cite{wilson2016lyapunov}, \cite{kolarijani2019continuous}. The {\em heavy ball method} is an accelerated gradient method that guarantees convergence to the minimizer of a nonstrongly convex function $L$ \cite{polyak1964some}, and that achieves a faster convergence rate than classical gradient descent by adding a ``velocity'' term to the gradient. The dynamical system characterization for this method is
\begin{equation}\label{eqn:HBF}
\ddot{\xi} + \lambda \dot{\xi} + \gamma \nabla L(\xi) = 0
\end{equation}
where $\lambda$ and $\gamma$ are positive tunable parameters that represent friction and gravity, respectively; see \cite{attouch2000heavy}, \cite{polyak2017lyapunov}.
In \cite{ghadimi2015global}, it is shown that the heavy ball method converges exponentially when $L$ is strongly convex and converges with rate $\frac{1}{t}$ when $L$ is nonstrongly convex. For the case when $L$ is strongly convex, and inspired by the heavy ball algorithm, two algorithms with a resettable velocity term are proposed in \cite{le2021hybrid} and shown to guarantee exponential convergence. In \cite{sebbouh2020convergence}, however, it was demonstrated that the heavy ball algorithm converges exponentially for nonstrongly convex $L$ when such an objective function also has the property of quadratic growth away from its minimizer. Global asymptotic stability of the minimizer, which is the property that all solutions that start close to the minimizer stay close, and solutions from all initial conditions converge to the minimizer, is demonstrated in \cite{michalowsky2014multidimensional}, \cite{michalowsky2016extremum}, when $L$ is nonstrongly convex and smooth. The work in \cite{polyak2017lyapunov} provides several Lyapunov functions to establish global asymptotic stability of the minimizer and convergence rates for the heavy ball method, both when $L$ is strongly convex and when $L$ is nonstrongly convex.
Another powerful accelerated method is {\em Nesterov's accelerated gradient descent}.
One characterization of the dynamical system for Nesterov's method, for nonstrongly convex $L$, proposed in \cite{muehlebach2019dynamical}, is
\begin{equation} \label{eqn:MJODENCVX}
\ddot{\xi} + 2\bar{d}(t)\dot{\xi} + \frac{1}{M\zeta^2}\nabla L(\xi + \bar{\beta}(t) \dot{\xi}) = 0,
\end{equation}
where $M > 0$ is the Lipschitz constant of the gradient of $L$ and where the constant $\zeta > 0$ rescales time in solutions to \eqref{eqn:MJODENCVX}. The dynamical system in \eqref{eqn:MJODENCVX} resembles the model of a mass-spring-damper, with a curvature-dependent damping term where the total damping is a linear combination of $\bar{d}(t)$ and $\bar{\beta}(t)$. In \cite{muehlebach2019dynamical}, the convergence rate of Nesterov's method is characterized as $\frac{1}{(t+2)^2}$ for \eqref{eqn:MJODENCVX} (for $t \geq 1$), when $\zeta = 1$, and when the minimizer is the origin, at which $L$ is zero.
Whereas \cite{muehlebach2019dynamical} started with the ODE in \eqref{eqn:MJODENCVX}
and subsequently showed that the discrete-time analog of Nesterov's method arises from discretizing \eqref{eqn:MJODENCVX} with a semi-implicit Euler integration scheme, one of the earliest analyses of a dynamical system characterization for Nesterov's method, in \cite{su2016differential}, started with the discrete-time analog of Nesterov's method and showed that for a vanishing step size the trajectories of such an accelerated gradient scheme approach the solutions of the ODE
\begin{equation} \label{eqn:SuODE}
\ddot{\xi} + \frac{3}{t}\dot{\xi} + \nabla L(\xi) = 0
\end{equation}
for all $t > 0$. Such an ODE does not have a local curvature dependent damping term, as \eqref{eqn:MJODENCVX} does, and which \cite{muehlebach2019dynamical} argues is instrumental to the intuition behind the acceleration phenomenon. The development in \cite{su2016differential} includes the analysis of a variation of this dynamical system for higher friction, and show that their dynamical system characterizations have a convergence rate of $\frac{1}{t^2}$. In \cite{krichene2015accelerated}, the analysis of the dynamical system in \cite{su2016differential} is extended to include optimization of objective functions $L$ with non-Euclidean geometries, using a Bregman divergence to characterize the distance of the state $\xi$ from the minimizer. The work in \cite{krichene2015accelerated} combines this dynamical system with mirror descent to design an accelerated mirror descent ODE, with a convergence rate of $\frac{1}{t^2}$. In \cite{wibisono2016variational}, a dynamical system, consisting of an Euler-Lagrange equation, is derived for Nesterov's algorithm via a Bregman Lagrangian. In \cite{wibisono2016variational} an exponential rate of convergence for such a system under ideal scaling is provided, and, for a polynomial class of dynamical systems, a convergence rate of $\frac{1}{t^p}$ with $p \geq 2$ is shown.
In \cite{kolarijani2018fast} and \cite{kolarijani2019continuous}, two hybrid algorithms based on the ODE in \eqref{eqn:SuODE} are presented: one with a state-dependent, time-invariant damping input and another with an input that controls the magnitude of the gradient term. The algorithms require the objective function to satisfy the Polyak-\L ojasiewicz inequality, which includes a subclass of nonconvex functions in which all stationary points are global minimizers. The authors in \cite{poveda2019inducing} propose two hybrid reset algorithms based on the ODE in \eqref{eqn:SuODE}, HAND-1 and HAND-2, which yield an exponential convergence rate for strongly convex $L$ and a rate of $\frac{1}{t^2}$ for nonstrongly convex $L$, with the latter rate only assured until the first reset.
While the results in \cite{muehlebach2019dynamical}, \cite{su2016differential}, \cite{krichene2015accelerated}, \cite{wibisono2016variational}, \cite{kolarijani2018fast}, and \cite{kolarijani2019continuous} characterize the convergence properties of Nesterov's method (or a variation of) the stability properties of the method are not revealed. While stability properties for such methods were studied in \cite{polyak2017lyapunov}, a particularly useful property for optimization algorithms, called {\em uniform global asymptotic stability} (UGAS), requires that solutions reach a neigborhood of the minimizer in time that is uniform on the set of initial conditions. After finite time, the error of such solutions becomes smaller than a given threshold \cite{220}. Due to such a guarantee for solutions, UGAS is typically useful for certifying robustness to small perturbations in time-varying dynamical and hybrid systems \cite{65}, \cite{220}. Remarkably, the algorithms with resets in the velocity term proposed in \cite{le2021hybrid} and \cite{teel2019first} can be shown to induce UGAS of the minimizer (with zero velocity term) and reduced oscillations, for the particular case when L is strongly convex. Unfortunately, as shown in \cite{poveda2019inducing}, via a counterexample, Nesterov-like algorithms do not necessarily assure UGAS of the minimizer when $L$ is nonstrongly convex. In response to this, \cite{poveda2019inducing} proposes the HAND-1 and HAND-2 reset algorithms, and prove UGAS of the minimizer for both algorithms. The exponential convergence rate of HAND-2, however, only applies to strongly convex $L$, and the convergence rate of $\frac{1}{t^2}$ for HAND-1, for nonstrongly convex $L$, only holds up until the first reset.
\IfConf{
\vspace{-0.2cm}
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
\begin{picture}(20,30.2)(0,0)
\footnotesize
\put(0.5,15){\includegraphics[scale=0.3,trim={0.8cm 0.4cm 0.6cm 0.5cm},clip,width=20\unitlength]{Figures/MotivationalPlot.eps}}
\put(0.5,0.5){\includegraphics[scale=0.3,trim={0.8cm 0.4cm 0.6cm 0.5cm},clip,width=20\unitlength]{Figures/LogScaleMotivationalPlot.eps}}
\put(0.8,18.1){$\xi$}
\put(0.8,23){$\xi$}
\put(0.8,27.8){$\xi$}
\put(10.3,0.3){$t [s]$}
\put(0,6.5){\rotatebox{90}{$L(\xi) - L^*$}}
\put(8.2,27.5){slow convergence}
\put(8.2,26.7){without oscillations}
\put(9.9,22){oscillations}
\put(6.2,24.2){fast convergence}
\put(7,17){no oscillations}
\put(6.2,19.5){fast convergence}
\end{picture}
\vspace{-0.3cm}
\caption{Comparison of the performance of the heavy ball method, with large $\lambda$, Nesterov's accelerated gradient descent, and the proposed logic-based algorithm. The objective function is $L(\xi) = \xi^2$. Top: the heavy ball algorithm, with large $\lambda$, converges very slowly. Top inset: zoomed out view of heavy ball. Second from top: Nesterov's accelerated gradient descent converges quickly, but with oscillations. Third from top: our proposed logic-based algorithm yields fast convergence, with no oscillations. Bottom: comparison of the value of $L(\xi) - L^*$ (in log scale) versus time for each algorithm. Different tunings of the logic-based algorithm's parameters leads to modifications of the solution's profile.}
\label{fig:MotivationalPlot}
\end{figure}
}{
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
\begin{picture}(30.8,12)(0,0)
\footnotesize
\put(0.7,0.5){\includegraphics[scale=0.3,trim={0.9cm 0.25cm 1.25cm 0.5cm},clip,width=14\unitlength]{Figures/MotivationalPlot2.eps}}
\put(16.5,0.5){\includegraphics[scale=0.3,trim={0.6cm 0cm 1.25cm 0.5cm},clip,width=14\unitlength]{Figures/LogScaleMotivationalPlot2.eps}}
\put(0,3){$\xi$}
\put(0,6.6){$\xi$}
\put(0,10.1){$\xi$}
\put(7.7,0.2){$t [s]$}
\put(23.7,0.3){$t [s]$}
\put(15.5,4.8){\rotatebox{90}{$L(\xi) - L^*$}}
\put(6.4,9.9){slow convergence}
\put(6.4,9.2){without oscillations}
\put(7.4,5.9){oscillations}
\put(4.8,7.4){fast convergence}
\put(5.5,2){no oscillations}
\put(4.8,3.8){fast convergence}
\end{picture}
\caption{Comparison of the performance of the heavy ball method, with large $\lambda$, Nesterov's accelerated gradient descent, and the proposed logic-based algorithm. The objective function is $L(\xi) = \xi^2$. Top left: the heavy ball algorithm, with large $\lambda$, converges very slowly. Top inset: zoomed out view of heavy ball. Middle left: Nesterov's accelerated gradient descent converges quickly, but with oscillations. Bottom left: our proposed logic-based algorithm yields fast convergence, with no oscillations. Right: comparison of the value of $L(\xi) - L^*$ (in log scale) versus time for each algorithm. Different tunings of the logic-based algorithm's parameters leads to modifications of the solution's profile.}
\label{fig:MotivationalPlotTR}
\end{figure}}
The work in this paper is motivated by the lack of an accelerated gradient algorithm assuring UGAS, with a convergence rate that holds for all time and that resembles that of Nesterov’s method (at least far from the minimizer), when the objective function is nonstrongly convex. However, attaining such a rate is expected to lead to oscillations, which are typically seen in accelerated gradient methods. The performance of the heavy ball method, for instance, depends highly on the choice of $\lambda$ and $\gamma$.
In particular, for a fixed value of $\gamma$,
the choice of the friction parameter $\lambda$
significantly affects the asymptotic behavior of the solutions
to \eqref{eqn:HBF}. For rather simple choices of the function $L$,
the literature on this method indicates that
large values of $\lambda$ are seen to give rise to slowly converging
solutions resembling solutions yielded by steepest descent \cite{attouch2000heavy}. The top plot\footnote{Code at\IfConf{\\}{} \texttt{gitHub.com/HybridSystemsLab/UnitingMotivation}}\IfConf{}{ on the left} in \IfConf{Fig. \ref{fig:MotivationalPlot}}{Figure \ref{fig:MotivationalPlotTR}} demonstrates such behavior. In contrast, smaller values of $\lambda$ give rise
to fast solutions with oscillations getting wilder as $\lambda$ decreases
\cite{attouch2000heavy}. Nesterov's method converges quickly but also suffers from oscillations \cite{su2016differential}. The oscillatory behavior of Nesterov's method, with $\zeta = 2$, is shown in the \IfConf{second plot from the top}{middle plot on the left} in \IfConf{Fig. \ref{fig:MotivationalPlot}}{Figure \ref{fig:MotivationalPlotTR}}.
Due to its implications on robustness, we are particularly interested in an algorithm that assures uniform global asymptotic stability of the minimizer of $L$ with a rate of convergence that holds for all time, and without the undesired oscillations. As pointed out in Section \ref{sec:Background}, these properties are not guaranteed by Nesterov's method. The behavior shown in the \IfConf{first and second}{top and middle} plots in \IfConf{Fig. \ref{fig:MotivationalPlot}}{Figure \ref{fig:MotivationalPlotTR}} motivates the logic-based algorithm proposed in this paper. The proposed algorithm exploits the main features of heavy ball and Nesterov's method to achieve fast convergence and UGAS of the minimizer. More precisely, without knowledge of the location of the minimizer, it selects Nesterov's method to converge quickly to nearby the minimizer and, once solutions reach a neighborhood of the minimizer, switches to the heavy ball method with large $\lambda$ to avoid oscillations.
An example solution to our proposed logic-based algorithm, shown in the \IfConf{third plot from the top in}{bottom plot on the left of} \IfConf{Fig. \ref{fig:MotivationalPlot}}{Figure \ref{fig:MotivationalPlotTR}}, demonstrates the improvement obtained by using Nesterov's method globally and the heavy ball method locally, under relatively mild assumptions on the objective function $L$. The proposed algorithm guarantees UGAS and a (hybrid) convergence rate that holds for all $t \geq 1$.
\IfConf{
\vspace{-0.2cm}
}{}
\subsection{Contributions}
\label{sec:Contributions}
\IfConf{
\vspace{-0.2cm}
}{}
The main contributions of this paper are as follows. \IfConf{
\vspace{-0.45cm}
\begin{enumerate}[label={\arabic*)},leftmargin=*]
\item {\em A uniting algorithm for fast convergence and UGAS of the minimizer:}
In Section \ref{sec:NonstronglyConvexNest} we propose an algorithm,
designed using hybrid system tools (see Section \ref{sec:Preliminaries}), which unites Nesterov's method globally and the heavy ball method with large $\lambda$ locally to guarantee fast convergence with UGAS of the minimizer $\xi^*$ of a nonstrongly convex objective function $L$, without knowledge of $L^* := L(\xi^*)$ or $\xi^*$; see Sections \ref{sec:NonstronglyConvexNest} and \ref{sec:ProofMainResult}.
\item {\em Well-posedness and existence of solutions:} In Section \ref{sec:NonstronglyConvexNest} we prove well-posedness and existence of solutions for the proposed hybrid closed-loop algorithm.
It is important for our algorithm to be well-posed as we want to ensure robustness to small noise in measurements of the gradient of $L$.
\item {\em Robustness to small perturbations:} Due to the well-posedness of the proposed hybrid uniting algorithm, we show, via numerical simulations, that the established UGAS property is robust to small perturbations in measurements of the gradient of $L$ \cite[Theorem~7.21]{65}; see Section \ref{sec:Examples}.
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
\vspace{-0.2cm}
\begin{picture}(20,15)(0,0)
\footnotesize
\put(0,0.5){\includegraphics[trim={0.5cm 0.3cm 1cm 0.6cm},clip,width=20\unitlength]{Figures/ComparisonPlotsLogScale.eps}}
\put(0,6.2){\rotatebox{90}{$L(\xi) - L^*$}}
\put(10.2,0.4){$t[s]$}
\end{picture}
\vspace{-0.3cm}
\caption{A comparison of the evolution of $L$ over time for Nesterov's method in \eqref{eqn:MJODE_ZetaNum}, heavy ball, HAND-1 from \cite{poveda2019inducing}, and our proposed uniting algorithm, for a function $L(\xi) := \xi^2$, with a single minimizer at $\xi^* = 0$. Nesterov's method, shown in purple, settles to within $1\%$ of $\xi^*$ in about 8.8 seconds. The heavy ball algorithm, shown in green, settles to within $1\%$ of $\xi^*$ in about 138.1 seconds. HAND-1, shown in orange, settles to within $1\%$ of $\xi^*$ in about 14.3 seconds. The hybrid closed-loop system $\HS$, shown in blue, settles to within $1\%$ of $\xi^*$ in about 2.4 seconds. As opposed to Fig. \ref{fig:MotivationalPlot}, which uses $\zeta = 2$ for $\HS_1$, this example uses $\zeta = 1$, which results in slower convergence of solutions to $\HS$ and $\HS_1$ than in Fig. \ref{fig:MotivationalPlot}.}
\label{fig:NSCLargerZeta}
\end{figure}
\item {\em A (hybrid) convergence rate preserving the rates of Nesterov's method and heavy ball:} In Sections \ref{sec:NonstronglyConvexNest} and \ref{sec:ProofMainResult} we show that our uniting algorithm attains a rate of $\frac{1}{(t+2)^2}$ for the global algorithm and $\exp \left(-(1-m)\psi t\right)$, where $m \in (0,1)$ and $\alpha > 0$ are such that $\psi := \frac{m\alpha\gamma}{\lambda} > 0$ and $\nu := \psi (\psi - \lambda) < 0$, for the local algorithm.
Whereas the convergence rate $\frac{1}{t^2}$ for the HAND-1 algorithm in \cite{poveda2019inducing} holds only until the first reset, the algorithm we propose has a (hybrid) convergence rate that preserves the rates of the individual optimization algorithms for all (hybrid) time such that $t \geq 1$. In \IfConf{Fig. \ref{fig:NSCLargerZeta}}{Figure \ref{fig:NSCLargerZetaTR}} and Section \ref{sec:Examples}, our uniting algorithm is shown via numerical simulations\footnote{Code at \texttt{gitHub.com/HybridSystemsLab/UnitingTradeoff} \label{foot:Tradeoff}} to have improved performance over the HAND-1 algorithm in \cite{poveda2019inducing}.
\item {\em Extension of the results on Nesterov's method in \cite{muehlebach2019dynamical}:}
In particular, while the convergence rate results in \cite{muehlebach2019dynamical} assume that $L(\xi_1^*) = 0$ at $\xi^* = 0$, and $\zeta = 1$ for \eqref{eqn:MJODENCVX},
here we prove uniform global attractivity (UGA) of the minimizer, with a convergence rate of $\frac{1}{(t+2)^2}$, for cost functions with a minimum value that is not necessarily zero, which holds for a generic parameter $\zeta > 0$. We achieve the relaxation on $\zeta$ by moving it into the numerator of the coefficient of the gradient, effectively decoupling $\zeta$ and $M$, namely,
\begin{equation} \label{eqn:MJODE_ZetaNum}
\ddot{\xi} + 2\bar{d}(t)\dot{\xi} + \frac{\zeta^2}{M}\nabla L(\xi + \bar{\beta}(t) \dot{\xi}) = 0.
\end{equation}
\end{enumerate}}{
\begin{enumerate}[label={\arabic*)},leftmargin=*]
\item {\em A uniting algorithm for fast convergence and UGAS of the minimizer:}
In Section \ref{sec:NonstronglyConvexNest} we propose a uniting algorithm that solves optimization problems of the form $\min_{\xi \in \reals^n} L(\xi)$
with accelerated gradient methods.
Designed using hybrid system tools (see Section \ref{sec:Preliminaries}), the algorithm unites Nesterov's method in \eqref{eqn:MJODE_ZetaNum_TR} globally and the heavy ball method in \eqref{eqn:HBF} with large $\lambda$ locally to guarantee fast convergence with UGAS of the minimizer $\xi^*$ of a nonstrongly convex objective function $L$; see Sections \ref{sec:NonstronglyConvexNest} and \ref{sec:ProofMainResult}. The establishment of UGAS solves the difficult problem of achieving such a property for Nesterov-like algorithms \cite{poveda2019inducing}, \cite{poveda2020heavy}.
The algorithm we propose exploits measurements of $\nabla L$ and requires no knowledge of $L^* := L(\xi^*)$ or $\xi^*$. In practice, such measurements of $\nabla L$ are typically approximated from measurements of $L$. The algorithm, however, does not require measurements of the Hessian of $L$.
\item {\em Well-posedness and existence of solutions:} In Section \ref{sec:NonstronglyConvexNest} we prove well-posedness and existence of solutions for the proposed hybrid closed-loop algorithm. Hybrid systems that are {\em well-posed} are defined to be those hybrid systems, vaguely speaking, for which graphical limits of graphically convergent sequences of solutions, with no perturbations and with vanishing perturbations, respectively, are still solutions \cite[Chapter~6]{65}. It is important for our algorithm to be well-posed as we want to ensure robustness to small noise in measurements of the gradient of $L$.
\item {\em Robustness to small perturbations:} Due to the well-posedness of the proposed hybrid uniting algorithm, we show that the established UGAS property is robust to small perturbations in measurements of the gradient of $L$ \cite[Theorem~7.21]{65}. We illustrate this robustness in Section \ref{sec:Examples} via numerical simulations that include small noise in measurements of the gradient.
\item {\em A (hybrid) convergence rate preserving the rates of Nesterov's method and heavy ball:} In Sections \ref{sec:NonstronglyConvexNest} and \ref{sec:ProofMainResult} we show that our uniting algorithm attains a rate of $\frac{1}{(t+2)^2}$ for the global algorithm and $\exp \left(-(1-m)\psi t\right)$, where $m \in (0,1)$ and $\alpha > 0$ are such that $\psi := \frac{m\alpha\gamma}{\lambda} > 0$ and $\nu := \psi (\psi - \lambda) < 0$, for the local algorithm. The latter rate holds under the mild assumption on $L$ of quadratic growth away from the minimizer. As mentioned in Section \ref{sec:Background}, Nesterov-like algorithms do not necessarily assure UGAS of the minimizer. The HAND-1 algorithm for nonstrongly convex $L$, proposed in \cite{poveda2019inducing}, provides UGAS via a hybrid restarting mechanism that yields a convergence rate $\frac{1}{t^2}$. However, this convergence rate holds only until the first reset. The algorithm we propose not only renders the minimizer UGAS, but also has a (hybrid) convergence rate that preserves the rates of the individual optimization algorithms for all (hybrid) time such that $t \geq 1$. Moreover, the global rate of our algorithm is commensurate with that of HAND-1. In \IfConf{Fig. \ref{fig:NSCLargerZeta}}{Figure \ref{fig:NSCLargerZetaTR}} and Section \ref{sec:Examples}, our uniting algorithm is shown via numerical simulations\footnote{Code at \texttt{gitHub.com/HybridSystemsLab/UnitingTradeoff} \label{foot:Tradeoff_TR}} to have improved performance over the HAND-1 algorithm in \cite{poveda2019inducing}.
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
%
\begin{picture}(24,17.5)(0,0)
\footnotesize
\put(0,0.5){\includegraphics[trim={0.5cm 0.2cm 1cm 0.3cm},clip,width=24\unitlength]{Figures/ComparisonPlotsLogScale.eps}}
\put(0,8){\rotatebox{90}{$L(\xi) - L^*$}}
\put(12.3,0.5){$t[s]$}
\end{picture}
\caption{A comparison of the evolution of $L$ over time for Nesterov's method in \eqref{eqn:MJODE_ZetaNum_TR}, heavy ball, HAND-1 from \cite{poveda2019inducing}, and our proposed uniting algorithm, for a function $L(\xi) := \xi^2$, with a single minimizer at $\xi^* = 0$. Nesterov's method, shown in purple, settles to within $1\%$ of $\xi^*$ in about 8.8 seconds. The heavy ball algorithm, shown in green, settles to within $1\%$ of $\xi^*$ in about 138.1 seconds. HAND-1, shown in orange, settles to within $1\%$ of $\xi^*$ in about 14.3 seconds. The hybrid closed-loop system $\HS$, shown in blue, settles to within $1\%$ of $\minSet$ in about 2.4 seconds. As opposed to Figure \ref{fig:MotivationalPlotTR}, which uses $\zeta = 2$ for $\HS_1$, this example uses $\zeta = 1$, which results in slower convergence of solutions to $\HS$ and $\HS_1$ than in Figure \ref{fig:MotivationalPlotTR}.}
\label{fig:NSCLargerZetaTR}
\end{figure}
\item {\em Extension of the results on Nesterov's method in \cite{muehlebach2019dynamical}:} In the process, in Section \ref{sec:ProofMainResult}, we extend the properties and convergence results for Nesterov's method in \cite{muehlebach2019dynamical}. In particular, while the convergence rate results in \cite{muehlebach2019dynamical} assume that $L(\xi_1^*) = 0$ at $\xi^* = 0$, and $\zeta = 1$ for \eqref{eqn:MJODENCVX},
here we prove uniform global attractivity (UGA) of the minimizer, with a convergence rate of $\frac{1}{(t+2)^2}$, for cost functions with a minimum value that is not necessarily zero, which holds for a generic parameter $\zeta > 0$. We achieve the relaxation on $\zeta$ by moving it into the numerator of the coefficient of the gradient, effectively decoupling $\zeta$ and $M$. This leads to the ODE
\begin{equation} \label{eqn:MJODE_ZetaNum_TR}
\ddot{\xi} + 2\bar{d}(t)\dot{\xi} + \frac{\zeta^2}{M}\nabla L(\xi + \bar{\beta}(t) \dot{\xi}) = 0.
\end{equation}
Such a modification leads to faster convergence as $\zeta$ increases, and slower convergence as $\zeta \rightarrow 0$.
\end{enumerate}}
Preliminary work in \cite{hustig2021uniting} proposed an algorithm uniting Nesterov's method globally and heavy ball locally for $\mathcal{C}^2$, strongly convex objective functions $L$, and included different results and examples that reflect such conditions, with proofs omitted due to space considerations. The uniting algorithm proposed in this paper relaxes the conditions in \cite{hustig2021uniting} to $\mathcal{C}^1$, nonstrongly convex $L$ with a unique minimizer. Such a relaxation is reflected in the results, examples, and proofs presented here. \IfConf{More technical details can be found in \cite{dhustigs2022unitingNSC}, which is available online.}{}
\subsection{Notation}
The sets of real, positive real, and natural numbers are denoted by $\reals$, $\reals_{>0}$, and $\naturals$, respectively. The closed unit ball, of appropriate dimension, in the Euclidean norm is denoted as $\ball$. The set $\mathcal{C}^n$ represents the family of $n$-th continuously differentiable functions. For vectors $v \in \reals^n$ and $w \in \reals^n$, $\left|v\right| = \sqrt{v^{\top} v}$ denotes the Euclidean vector norm of $v$, and $\langle v,w \rangle = v^{\top} w$ the inner product of $v$ and $w$. For any $x \in \reals^n$ and $y \in \reals^m$, $\left(x,y\right) := [x^{\top},y^{\top}]^{\top}$. The closure of a set $S$ is denoted $\overline{S}$ and the set of interior points of $S$ is denoted $\mathrm{int}(S)$. Given a set $S \subset \reals^n \times \reals^m$, the projection of $S$ onto $\reals^n$ is defined as $\Pi(S) := \defset{x \in \reals^n}{\exists y \ \text{such that } (x,y) \in S}$. The distance of a point $x \in \reals^n$ to a set $S \in \reals^n$ is defined by $\left|x\right|_S = \inf_{y \in S} \left| y - x \right|$. Given a set-valued mapping $M : \reals^m \rightrightarrows \reals^n$, the domain of $M$ is the set $\mathrm{dom} M = \defset{x \in \reals^m}{M(x) \neq \emptyset}$, and the range of $M$ is the set $\mathrm{rge} \ M = \defset{y \in \reals^n}{ \exists x \in \reals^m \mbox{ such that } y\in M(x)\!\!}$. A function $\alpha : \reals_{\geq 0} \rightarrow \reals_{\geq 0}$ is a class-$\mathcal{K}_{\infty}$ function, also written $\alpha \in \mathcal{K}_{\infty}$, if $\alpha$ is zero at zero, continuous, strictly increasing, and unbounded. A function $\beta : \reals_{\geq 0} \times \reals_{\geq 0} \rightarrow \reals_{\geq 0}$ is a class-$\mathcal{KL}$ function, also written $\beta \in \mathcal{KL}$, if it is nondecreasing in its first argument, nonincreasing in its second argument, $\lim_{r \rightarrow 0^+} \beta(r,s) = 0$ for each $s \in \reals_{\geq 0}$, and $\lim_{s \rightarrow \infty} \beta(r,s) = 0$ for each $r \in \reals_{\geq 0}$.
\section{Uniting Optimization Algorithm}
\label{sec:NonstronglyConvexNest}
\subsection{Problem Statement} \label{sec:ProblemStatement}
As illustrated in \IfConf{Fig. \ref{fig:MotivationalPlot}}{Figure \ref{fig:MotivationalPlotTR}}, the performance of Nesterov's accelerated gradient descent commonly suffers from oscillations near the minimizer. This is also the case for the heavy ball method when $\lambda > 0$ is small. However, when $\lambda$ is large, the heavy ball method converges slowly, albeit without oscillations. In Section \ref{sec:Intro} we discussed how Nesterov's algorithm guarantees a rate of $\frac{1}{(t+2)^2}$ for nonstrongly convex $L$. We also discussed how the heavy ball algorithm guarantees a rate of $\frac{1}{t}$ for nonstrongly convex $L$, although it was demonstrated in \cite{sebbouh2020convergence} that the heavy ball algorithm converges exponentially for nonstrongly convex $L$ when such an objective function also has the property of quadratic growth away from its minimizer. We desire to attain the rate $\frac{1}{(t+2)^2}$ globally and an exponential rate locally,
while avoiding oscillations via the heavy ball algorithm with large $\lambda$. We state the problem to solve as follows:
\IfConf{\begin{prob}}{\begin{problem}}\label{problem:ProbStatement}
Given a scalar, real-valued, continuously differentiable, and nonstrongly convex objective function $L$ with a unique minimizer,
design an optimization algorithm that, without knowing the function $L$ or the location of its minimizer, has the minimizer uniformly globally asymptotically stable, with a convergence rate of $\frac{1}{(t+2)^2}$ globally and an exponential convergence rate locally,
and with robustness to arbitrarily small noise in measurements of $\nabla L$.
\IfConf{\end{prob}}{\end{problem}}
\subsection{Modeling}
In this section, we present an algorithm that solves \IfConf{Problem \ref{problem:ProbStatement}}{Problem $(\star)$}. We interpret the ODEs in \eqref{eqn:HBF} and \IfConf{\eqref{eqn:MJODE_ZetaNum}}{\eqref{eqn:MJODE_ZetaNum_TR}} as control systems consisting of a plant and a control algorithm \cite{191} \cite{220}. Defining $z_1$ as $\xi$ and $z_2$ as $\dot{\xi}$, the plant associated to these ODEs is given by
the double integrator
\begin{equation}\label{eqn:HBFplant-dynamicsTR}
\matt{\dot{z}_1\\ \dot{z}_2\\} = \matt{z_2\\ u} =: F_P(z,u) \ \ \ (z,u) \in \reals^{2n} \times \reals^n =: C_P
\end{equation}
With this model, the optimization algorithms that we consider assign $u$ to a function of the state that involves the cost function, and such a function of the state may be time dependent. The control algorithm leading to \eqref{eqn:HBF} assigns $u$ to $-\lambda z_2 - \gamma \nabla L(z_1)$
where $\gamma > 0$ and $\lambda > 0$, and the control algorithm leading to \IfConf{\eqref{eqn:MJODE_ZetaNum}}{\eqref{eqn:MJODE_ZetaNum_TR}} assigns $u$ to $-2\bar{d}(t)z_2 - \frac{\zeta^2}{M}\nabla L(z_1 + \bar{\beta}(t)z_2)$
where $\zeta > 0$, $M >0$ is the Lipschitz constant for $\nabla L$, and where $\bar{d}(t)$ and $\bar{\beta}(t)$ are defined, for all $t \geq 0$, as
\begin{equation} \label{eqn:dBarBetaBar}
\bar{d}(t) := \frac{3}{2(t+2)}, \quad \bar{\beta}(t) := \frac{t - 1}{t + 2}.
\end{equation}
The proposed logic-based algorithm ``unites'' the two optimization algorithms modeled by $\kappa_{q}$, where the logic variable $q \in Q := \{0,1\}$ indicates which algorithm is currently being used. The local and global algorithms, respectively, are defined as
\begin{subequations} \label{eqn:StaticStateFeedbackLawsNSC}
\begin{align}
\kappa_0(h_0(z)) & = -\lambda z_2 - \gamma \nabla L(z_1) \label{eqn:StaticStateFeedbackLawLocal}\\
\kappa_1(h_1(z,t),t) & = -2\bar{d}(t)z_2 - \frac{\zeta^2}{M}\nabla L(z_1 + \bar{\beta}(t)z_2)\label{eqn:StaticStateFeedbackLawGlobalNSCVX}
\end{align}
\end{subequations}
where the algorithm defined by $\kappa_1$ plays the role of the global algorithm in uniting control (see, e.g., {\cite[Chapter~4]{220}), while the algorithm defined by $\kappa_0$ plays the role of the local algorithm.
The outputs $h_0$ corresponding to the output for the heavy ball algorithm and $h_1$ corresponding to the output for Nesterov's algorithm are defined as
\begin{equation} \label{eqn:H0H1NSCNesterovHBF}
h_0(z)\! :=\! \matt{z_2\\ \nabla L(z_1)}\!, h_1(z,t)\! :=\! \matt{z_2\\ \nabla L(z_1 + \bar{\beta}(t) z_2)}.
\end{equation}
Namely, the algorithm exploits measurements of $\nabla L$, which in practice are typically approximated using measurements of $L$. The parameters $\lambda > 0$ and $\gamma > 0$ should be designed to achieve convergence without oscillations nearby the minimizer.
Since the ODE in \IfConf{\eqref{eqn:MJODE_ZetaNum}}{\eqref{eqn:MJODE_ZetaNum_TR}} is time varying, and since solutions to hybrid systems are parameterized by $(t,j) \in \reals_{\geq 0} \times \naturals$, we employ the state $\tau$ to capture ordinary time as a state variable, in this way, leading to a time-invariant hybrid system. To encapsulate the plant, static state-feedback laws, and the time-varying nature of the ODE in \IfConf{\eqref{eqn:MJODE_ZetaNum}}{\eqref{eqn:MJODE_ZetaNum_TR}}, we define a hybrid closed-loop system $\HS$ with state $x := \left(z, q, \tau\right) \in \reals^{2n} \times Q \times \reals_{\geq 0}$
as \IfConf{
\vspace{-0.8cm}
}{}
\begin{subequations} \label{eqn:HS-TimeVarying}
\begin{equation}
\left.\begin{aligned}
\dot{z} & = \matt{z_2 \\ \kappa_{q}(h_{q}(z,\tau),\tau)} \label{eqn:FlowMap}\\
\dot{q} & = 0 \\
\dot{\tau} & = q
\end{aligned}\right\} =: F(x) \qquad x \in C := C_0 \cup C_1
\end{equation}
\begin{equation}
\left.\begin{aligned}
z^+ & = \matt{z_1 \\ z_2}\\
q^+ & = 1-q\\
\tau^+ & = 0
\end{aligned}\right\} =: G(x) \qquad x \in D := D_0 \cup D_1 \label{eqn:JumpMap}
\end{equation}
\end{subequations}
The sets $C_0$, $C_1$, $D_0$, and $D_1$ are defined as
\begin{subequations} \label{eqn:CAndDGradientsNestNSC}
\begin{align}
&C_0 := \mathcal{U}_0 \times \{0\} \times \{0\}, \ C_1 := \overline{\reals^{2n}\setminus \mathcal{T}_{1,0}} \times \{1\} \times \reals_{\geq 0}\\
&D_0 := \mathcal{T}_{0,1} \times \{0\} \times \{0\}, \ D_1 := \mathcal{T}_{1,0}\times \{1\} \times \reals_{\geq 0}.
\end{align}
\end{subequations} \IfConf{
\vspace{-0.7cm}
}{}The sets $\mathcal{U}_0$, $\mathcal{T}_{1,0}$, and $\mathcal{T}_{0,1}$ are precisely defined in Section \ref{sec:AssDesign}, but
the idea behind their construction is as follows. The switch between $\kappa_0$ and $\kappa_1$ is governed by a {\em supervisory algorithm} implementing switching logic; see \IfConf{Fig. \ref{fig:FeedbackDiagram}}{Figure \ref{fig:FeedbackDiagramTR}}. The supervisor selects between these two optimization algorithms, based on the output of the plant and the optimization algorithm currently applied. When $z \in \mathcal{U}_0$, $q = 0$, and $\tau = 0$ (i.e., $x \in C_0$), due to the design of $\mathcal{U}_0$ in Section \ref{sec:U0}, then the state $z$ is near the minimizer, which is denoted $z_1^*$, and the supervisor allows flows of \eqref{eqn:HS-TimeVarying} using $\kappa_0$ and $\dot{\tau} = q = 0$ to avoid oscillations. Conversely, when $z \in \overline{\reals^{2n}\setminus \mathcal{T}_{1,0}}$ and $q = 1$ (i.e., $x \in C_1$), due to the design of $\mathcal{T}_{1,0}$ in Section \ref{sec:DesignT10}, then the state $z$ is far from the minimizer and the supervisor allows flows of \eqref{eqn:HS-TimeVarying} using $\kappa_1$ and $\dot{\tau} = q = 1$ to converge quickly to the neighborhood of the minimizer. When $z \in \mathcal{T}_{1,0}$ and $q = 1$ (i.e., $x \in D_1$), then this indicates that the state $z$ is near the minimizer, and the supervisor assigns $u$ to $\kappa_0$, resets $q$ to $0$, and resets $\tau$ to $0$. Conversely, when $z \in \mathcal{T}_{0,1}$, $q = 0$, and $\tau = 0$ (i.e., $x \in D_0$), due to the design of $\mathcal{T}_{0,1}$ in Section \ref{sec:T01}, then this indicates that the state $z$ is far from the minimizer and the supervisor assigns $u$ to $\kappa_1$ and resets $q$ to $1$. The complete algorithm, defined in \eqref{eqn:HS-TimeVarying}-\eqref{eqn:CAndDGradientsNestNSC}, is summarized in Algorithm \ref{algo:HS-Uniting}.
\begin{algorithm}[thpb]
\caption{Uniting algorithm}
\begin{algorithmic}[1]
\label{algo:HS-Uniting}
\IfConf{\State}{\STATE} Set $q(0,0)$ to $0$, $\tau(0,0)$ to $0$, and set $z(0,0)$ as an initial condition with an arbitrary value.\\
\IfConf{\While{true}}{\WHILE{true}}
\IfConf{\If{$z \in \mathcal{T}_{0,1}$, $q = 0$, and $\tau = 0$}}{\IF{$z \in \mathcal{T}_{0,1}$, $q = 0$, and $\tau = 0$}}
\IfConf{\State}{\STATE} Reset $q$ to $1$.
\IfConf{\ElsIf{$z \in \mathcal{T}_{1,0}$ and $q = 1$}}{\ELSIF{$z \in \mathcal{T}_{1,0}$ and $q = 1$}}
\IfConf{\State}{\STATE} Reset $q$ to $0$ and $\tau$ to $0$.
\IfConf{\ElsIf{$z \in \mathcal{U}_0$, $q = 0$, and $\tau = 0$}}{\ELSIF{$z \in \mathcal{U}_0$, $q = 0$, and $\tau = 0$}}
\IfConf{\State}{\STATE} Assign $u$ to $\kappa_0(h_0(z))$ and update $z$, $q$, and $\tau$ according to \eqref{eqn:FlowMap}.
\IfConf{\ElsIf{$z \in \overline{\reals^{2n}\setminus \mathcal{T}_{1,0}}$ and $q = 1$}}{\ELSIF{$z \in \overline{\reals^{2n}\setminus \mathcal{T}_{1,0}}$ and $q = 1$}}
\IfConf{\State}{\STATE} Assign $u$ to $\kappa_1(h_1(z,\tau),\tau)$ and update $z$, $q$, and $\tau$ according to \eqref{eqn:FlowMap}.
\IfConf{\EndIf}{\ENDIF}
\IfConf{\EndWhile}{\ENDWHILE}
\end{algorithmic}
\end{algorithm} \IfConf{\vspace{-0.5cm}}{}
The reason that the state $\tau$ in \eqref{eqn:HS-TimeVarying} changes at the rate $q$ during flows and is reset to $0$ at jumps is that when the state $x$ is in $C_1$, then $\dot{\tau} = q = 1$, which implies that $\tau$ behaves as ordinary time, so it is used to represent time in the time-varying algorithm $\kappa_1$. On the other hand, when the state $x$ is in $C_0$, then $\dot{\tau} = q = 0$ causes the state $\tau$ to stay at zero, which is an appropriate value for $\tau$ as it is not required by the time-invariant algorithm $\kappa_0$. Such an evolution ensures that the set to asymptotically stabilize is compact.
\IfConf{Fig. \ref{fig:FeedbackDiagram}}{Figure \ref{fig:FeedbackDiagramTR}} shows the feedback diagram of this hybrid closed-loop system $\HS$. We denote the closed-loop system resulting from $\kappa_0$ as $\HS_0$, which is given by \IfConf{
\vspace{-0.5cm}
}{}
\begin{equation}\label{eqn:H0}
\dot{z} = \matt{z_2 \\ \kappa_0(h_0(z))} \qquad z \in \reals^{2n}
\end{equation}
and we denote the closed-loop system resulting from $\kappa_1$ as $\HS_1$, which is given by \IfConf{
\vspace{-0.9cm}
}{}
\begin{equation} \label{eqn:H1}
\dot{z} = \matt{z_2 \\ \kappa_1(h_1(z,\tau),\tau)}, \ \dot{\tau} = 1 \qquad \left(z,\tau\right) \in \reals^{2n} \times \reals_{\geq 0}.
\end{equation}
\IfConf{
\vspace{-0.3cm}
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
\begin{picture}(20,13)(0,0)
\footnotesize
\put(0,0.5){\includegraphics[scale=0.4,trim={0cm 0cm 0cm 0cm},clip,width=20\unitlength]{Figures/FeedbackDiagram2.eps}}
\put(8,4.5){{\bf Supervisor}}
\put(1,3.2){$\ \ \dot{q} = 0 \qquad \quad \ \dot{\tau} = q \ \ \ (z, q, \tau) \in C := C_0 \cup C_1$}
\put(1,1.8){$q^+ = 1- q \ \ \ \tau^+ = 0 \ \ \ (z, q, \tau) \in D := D_0 \cup D_1$}
\put(13.8,7.9){\bf plant}
\put(13.5,10.5){$\dot{z}_1 = z_2$}
\put(13.5,9.5){$\dot{z}_2 = u$}
\put(2.3,11.8){$\kappa_1(h_1(z,\tau),\tau)$}
\put(2.3,10.8){$\dot{\tau} = 1, \tau^+ = 0$}
\put(2.3,9.6){\bf global ($q = 1$)}
\put(3.2,7.9){$\kappa_0(h_0(z))$}
\put(2.5,6.2){\bf local ($q = 0$)}
\put(10,7){$q$}
\put(11.2,10.4){$u$}
\put(18,10.4){$h_{q}$}
\put(0.2,10.4){$h_{q}$}
\end{picture}
\vspace{-0.3cm}
\caption{Feedback diagram of the hybrid closed-loop system $\HS$, in \eqref{eqn:HS-TimeVarying}, uniting global and local optimization algorithms. Measurements of the gradient are used for the input of $\kappa_{q}$.}
\label{fig:FeedbackDiagram}
\end{figure}}{
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
\begin{picture}(30.8,12)(0,0)
\footnotesize
\put(0,2){\includegraphics[scale=0.2,trim={1.5cm 0cm 0cm 0.8cm},clip,width=11.5\unitlength]{Figures/Surface4.eps}}
\put(11.5,0.5){\includegraphics[scale=0.4,trim={0cm 0cm 0cm 0cm},clip,width=19\unitlength]{Figures/FeedbackDiagram2.eps}}
\put(18.5,4.5){{\bf Supervisor}}
\put(12.1,3.2){$\ \ \dot{q} = 0 \qquad \quad \ \dot{\tau} = q \ \ \ (z, q, \tau) \in C := C_0 \cup C_1$}
\put(12.1,1.8){$q^+ = 1- q \ \ \ \tau^+ = 0 \ \ \ (z, q, \tau) \in D := D_0 \cup D_1$}
\put(24.4,7.4){\bf plant}
\put(24.1,9.9){$\dot{z}_1 = z_2$}
\put(24.1,8.9){$\dot{z}_2 = u$}
\put(13.5,11.2){$\kappa_1(h_1(z,\tau),\tau)$}
\put(13.5,10.2){$\dot{\tau} = 1, \tau^+ = 0$}
\put(13.3,9.1){\bf global ($q = 1$)}
\put(14.3,7.5){$\kappa_0(h_0(z))$}
\put(13.6,5.9){\bf local ($q = 0$)}
\put(21,7){$q$}
\put(22.1,10.1){$u$}
\put(28.4,10.1){$h_{q}$}
\put(11.6,10.1){$h_{q}$}
\put(7.6,5.3){$\nabla L(z_1)$}
\put(5.5,3.8){$z_1^*$}
\put(6.7,3.3){$z_1$}
\put(8.8,4){$z_{1_{\circ}}$}
\end{picture}
\caption{Feedback diagram of the hybrid closed-loop system $\HS$ (on the right), in \eqref{eqn:HS-TimeVarying}, uniting global and local optimization algorithms. An example optimization problem to solve is shown on the left and, for this example optimization problem, measurements of the gradient are used for the input of $\kappa_{q}$.}
\label{fig:FeedbackDiagramTR}
\end{figure}}
\subsection{Design of the Hybrid Algorithm} \label{sec:AssDesign}
In order for the supervisor to determine when the state component $z_1$ is close to the minimizer of $L$, denoted $\minSet$, without knowledge of $\minSet$ or $L^* := L(z_1)$, we impose the following assumptions on $L$.
\IfConf{\begin{assum}}{\begin{assumption}} \label{ass:LisNSCVX}
The function $L$ is $\mathcal{C}^1$, (nonstrongly) convex\footnote{A function $L \!:\! \reals^n \rightarrow \reals$ is (nonstrongly) convex if, for all $u_1, w_1 \in \reals^n$, $L(u_1) \geq L(w_1) + \left\langle \nabla L(w_1), u_1 - w_1 \right\rangle $ \cite{boyd2004convex}. \label{foot:Convexity}}, and has a single minimizer $z_1^*$.
\IfConf{\end{assum}}{\end{assumption}}
\IfConf{\begin{assum}[Quadratic growth of $L$]}{\begin{assumption}[Quadratic growth of $L$]} \label{ass:QuadraticGrowth}
The function $L$ has quadratic growth away from its minimizer $\minSet$; i.e., there exists $\alpha > 0$ such that \IfConf{$L(z_1) - L^* \geq \alpha \left| z_1 - z_1^* \right|^2$, for all $z_1 \in \reals^{n}$.}{
\begin{equation} \label{eqn:QuadraticGrowth}
L(z_1) - L^* \geq \alpha \left| z_1 - z_1^* \right|^2 \quad \forall z_1 \in \reals^{n}.
\end{equation}}
\IfConf{\end{assum}}{\end{assumption}}
\IfConf{\begin{rem}}{\begin{remark}}
Assumption \ref{ass:LisNSCVX}, which is a common assumption used in the analysis of optimization algorithms \cite{boyd2004convex} \cite{nesterov2004introductory}, ensures that the objective function is continuously differentiable, which is necessary for well-posedness of $\HS$, as was explained in Section \ref{sec:Contributions}. Additionally, the nonstrongly convex property and the restriction that $L$ has a single minimizer $z_1^*$ in Assumption \ref{ass:LisNSCVX} rules out the possibility of the objective function having a continuum of minimizers or multiple isolated minimizers.
Assumption \ref{ass:QuadraticGrowth}, which is used for the construction of $\mathcal{U}_0$, $\mathcal{T}_{1,0}$, and $\mathcal{T}_{0,1}$, is employed as a means of determining when the state $z$ is near the minimizer of $L$, via measurements of \IfConf{$\nabla L$}{the gradient}. Such an assumption is also commonly used in the analysis of convex optimization algorithms; see, e.g., \cite{drusvyatskiy2018error}, \cite{karimi2016linear}.
\IfConf{\end{rem}}{\end{remark}}
To make the switch back to $\kappa_1$, we impose the following assumption on $L$.
\IfConf{\begin{assum}[Lipschitz Continuity of $\nabla L$]}{\begin{assumption}[Lipschitz Continuity of $\nabla L$]} \label{ass:Lipschitz}
The function $\nabla L$ is Lipschitz continuous with constant $M > 0$, namely, for all $w_1, u_1 \in \reals^n$, \IfConf{$\left|\nabla L(w_1) - \nabla L(u_1)\right| \leq M \left|w_1 - u_1\right|$.}{\\ $\left|\nabla L(w_1) - \nabla L(u_1)\right| \leq M \left|w_1 - u_1\right|$.}
\IfConf{\end{assum}}{\end{assumption}}
\IfConf{\begin{rem}}{\begin{remark}}
Assumption \ref{ass:Lipschitz} is used in the forthcoming construction of $\mathcal{T}_{0,1}$. Additionally, Assumption \ref{ass:Lipschitz} is commonly used in nonlinear analysis to ensure that the differential equations of the individual optimization algorithms, for example, those in \IfConf{\eqref{eqn:MJODE_ZetaNum}}{\eqref{eqn:MJODE_ZetaNum_TR}} and \eqref{eqn:HBF}, do not have solutions that escape in finite time, which is used to guarantee existence and completeness of maximal solutions to $\HS_{q}$ \cite[Theorem~3.2]{khalil2002nonlinear}.
\IfConf{\end{rem}}{\end{remark}}
Under Assumptions \ref{ass:LisNSCVX} and \ref{ass:QuadraticGrowth}, the following lemma, used in some of the results to follow, relates the size of the gradient at a point to the distance from the point to $\minSet$.
\IfConf{\begin{lem}[Suboptimality]}{\begin{lemma}(Suboptimality):} \label{lemma:ConvexSuboptimality}
Let $L$ satisfy Assumptions \ref{ass:LisNSCVX} and \ref{ass:QuadraticGrowth}, and let $\alpha > 0$ come from Assumption \ref{ass:QuadraticGrowth}. For some $\varepsilon > 0$, if $z_1 \in \reals^{n}$ is such that
$\left| \nabla L(z_1) \right| \leq \varepsilon \alpha$, then $\left| z_1 - z_1^* \right| \leq \varepsilon$.
\IfConf{\end{lem}}{\end{lemma}}
\IfConf{
\vspace{-0.2cm}
}{}
\IfConf{\begin{pf}}{\begin{proof}}
Combining Assumption \ref{ass:LisNSCVX} and\IfConf{}{ \eqref{eqn:QuadraticGrowth} from} Assumption \ref{ass:QuadraticGrowth} with $u_1 = z_1^*$ and $w_1 = z_1$ yields
\begin{align} \label{eqn:NearOptimalityC}
\IfConf{\alpha \left| z_1 - z_1^* \right|^2 & \leq \left| L(z_1) - L^* \right| \leq \left| \left\langle \nabla L(z_1), z_1^* - z_1 \right\rangle \right|\nonumber\\ & \leq \left| \nabla L(z_1) \right| \left| z_1 - z_1^* \right|}{\alpha \left| z_1 - z_1^* \right|^2 \leq \left| L(z_1) - L^* \right| \leq \left| \left\langle \nabla L(z_1), z_1^* - z_1 \right\rangle \right| \leq \left| \nabla L(z_1) \right| \left| z_1 - z_1^* \right|}
\end{align}
where the first inequality holds since $L(z_1) \geq L^*$.
Then,
\begin{equation}\label{eqn:Suboptimality}
\left| z_1 - z_1^* \right| \leq \frac{1}{\alpha} \left| \nabla L(z_1) \right|.
\end{equation}
From \eqref{eqn:Suboptimality}, we can deduce that $\left| \nabla L(z_1) \right| \leq \varepsilon \alpha$ implies $\left| z_1 - z_1^* \right| \leq \frac{1}{\alpha} \left(\varepsilon \alpha \right) = \varepsilon$.\IfConf{\hfill{} \qed}{}
\IfConf{\end{pf}}{\end{proof}} \IfConf{
\vspace{-0.3cm}
}{}
The suboptimality condition from Lemma \ref{lemma:ConvexSuboptimality} is typically used as a stopping condition for optimization, as it indicates that the argument of $L$ is close enough to the minimizer $\minSet$ \cite{boyd2004convex}. We exploit Lemma \ref{lemma:ConvexSuboptimality} to determine when the state component $z_1$ of the hybrid closed-loop system $\HS$ is close enough to the minimizer $\minSet$ so as to switch to the local optimization algorithm, $\kappa_0$, in this way activating $\HS_0$; see \IfConf{Fig. \ref{fig:FeedbackDiagram}}{Figure \ref{fig:FeedbackDiagramTR}}. \IfConf{
\vspace{-0.1cm}
}{}
\subsubsection{Design of the Set $\mathcal{U}_0$} \label{sec:U0}
Recall from lines 7-8 of Algorithm \ref{algo:HS-Uniting} that the objective is to design $\mathcal{U}_0$ such that when $z \in \mathcal{U}_0$, $q = 0$, and $\tau = 0$, the state component $z_1$ is near $\minSet$ and the uniting algorithm allows flows of \eqref{eqn:HS-TimeVarying} with $\kappa_0$ and $q = 0$. For such a design, we use Assumptions \ref{ass:LisNSCVX} and \ref{ass:QuadraticGrowth} and the Lyapunov function
\begin{equation} \label{eqn:LyapunovHBF}
V_0(z) := \gamma \left(L(z_1) - L^* \right) + \frac{1}{2} \left|z_2\right|^2
\end{equation}
defined for each $z \in \reals^{2n}$, where $\gamma > 0$.
Given $\varepsilon_0 > 0$, $c_0 > 0$, and $\gamma > 0$ from $\kappa_0$ in \eqref{eqn:StaticStateFeedbackLawLocal}, let $\alpha > 0$ come from Assumption \ref{ass:QuadraticGrowth} such that
\begin{equation} \label{eqn:UTilde0SetEquations}
\tilde{c}_0 := \varepsilon_0 \alpha > 0, \quad
d_0 := c_0 - \gamma \left(\frac{\tilde{c}_0^2}{\alpha}\right) > 0.
\end{equation}
Then, $V_0$ in \eqref{eqn:LyapunovHBF} can be upper bounded, using Assumption \ref{ass:LisNSCVX} as done to arrive to \eqref{eqn:NearOptimalityC}, as follows: for each $z \in \reals^{2n}$
\begin{align} \label{eqn:c0SublevelSet}
\IfConf{V_0(z) = \gamma \left(L(z_1) - L^*\right) + \frac{1}{2}\left|z_2\right|^2 \leq & \gamma \left|\nabla L(z_1)\right| \left| z_1 - z_1^* \right|\nonumber\\ & + \frac{1}{2}\left|z_2\right|^2.}{V_0(z) = \gamma \left(L(z_1) - L^*\right) + \frac{1}{2}\left|z_2\right|^2 \leq \gamma \left|\nabla L(z_1)\right| \left| z_1 - z_1^* \right| + \frac{1}{2}\left|z_2\right|^2.}
\end{align}
Then,
due to $L$ being $\mathcal{C}^1$, nonstrongly convex, and having a single minimizer $z_1^*$ by Assumption \ref{ass:LisNSCVX}, and due to $L$ having quadratic growth away from $\minSet$ by Assumption \ref{ass:QuadraticGrowth}, when $\left|\nabla L(z_1)\right| \leq \tilde{c}_0$, the suboptimality condition in Lemma \ref{lemma:ConvexSuboptimality} implies $\left| z_1 - z_1^* \right| \leq \frac{\tilde{c}_0}{\alpha}$, from where we get
\begin{equation} \label{eqn:V0Bound}
V_0(z) \leq \gamma \left(\frac{\tilde{c}_0^2}{\alpha}\right) + \frac{1}{2}\left|z_2\right|^2
\end{equation}
Then, by defining the set $\mathcal{U}_0$ as
\begin{equation}\label{eqn:U0}
\mathcal{U}_0 := \defset{z \in \reals^{2n}\!\!}{\!\!\left| \nabla L(z_1) \right| \leq \tilde{c}_0,\frac{1}{2} \left|z_2\right|^2 \leq d_0\!\!},
\end{equation}
every $z \in \mathcal{U}_0$ belongs to the $c_0$-sublevel set of $V_0$. In fact, using the conditions in \eqref{eqn:UTilde0SetEquations} and \eqref{eqn:V0Bound}, we have that for each $z \in \mathcal{U}_0$, \IfConf{$V_0(z) \leq \gamma \left(\frac{\tilde{c}_0^2}{\alpha}\right) + \frac{1}{2}\left|z_2\right|^2 \leq c_0$.}{
\begin{equation} \label{eqn:V0c0SublevelSet}
V_0(z) \leq \gamma \left(\frac{\tilde{c}_0^2}{\alpha}\right) + \frac{1}{2}\left|z_2\right|^2 \leq c_0.
\end{equation}}
Since $\kappa_0$ in \eqref{eqn:StaticStateFeedbackLawLocal} is such that the set $\{\minSet\} \times \{0\}$ is globally asymptotically stable for the closed-loop system resulting from controlling \eqref{eqn:HBFplant-dynamicsTR} by $\kappa_0$, as we show in the forthcoming Proposition \ref{prop:GAS-HBF}, the set $\mathcal{U}_0$ is contained in the basin of attraction induced by $\kappa_0$.
\subsubsection{Design of the Set $\mathcal{T}_{1,0}$}
\label{sec:DesignT10}
Recall from lines 5-6 of Algorithm \ref{algo:HS-Uniting} that the objective is to design $\mathcal{T}_{1,0}$ such that when $z \in \mathcal{T}_{1,0}$ and $q = 1$, the state component $z_1$ is near $\minSet$ and the supervisor resets $q$ to $0$, resets $\tau$ to $0$, and assigns $u$ to $\kappa_0(h_0(z))$. For such a design, we use Assumptions \ref{ass:LisNSCVX} and \ref{ass:QuadraticGrowth} and the Lyapunov function
\begin{equation} \label{eqn:LyapunovNesterovNSCVX}
V_1(z,\tau)\! :=\! \frac{1}{2}\left|\bar{a}(\tau)\left(z_1 - z_1^*\right) \!+\! z_2\right|^2 + \frac{\zeta^2}{M} (L(z_1) - L^*)\!
\end{equation}
defined for each $z \in \reals^{2n}$ and each $\tau \geq 0$, where $\zeta > 0$, $M > 0$ is the Lipschitz constant of $\nabla L$, and the function $\bar{a}$ is defined as
\begin{equation} \label{eqn:BarA}
\bar{a}(\tau) := \frac{2}{\tau+2}.
\end{equation}
Given $c_{1,0} \in (0, c_0)$ and $\varepsilon_{1,0} \in (0,\varepsilon_0)$, where $c_0 > 0$ and $\varepsilon_0 > 0$ come from Section \ref{sec:U0}, let $\tilde{c}_0$ and $d_0$ be given in \eqref{eqn:UTilde0SetEquations}, and let $\alpha > 0$ come from Assumption \ref{ass:QuadraticGrowth} such that
\begin{subequations} \label{eqn:TTilde10SetEquations}
\begin{align}
\tilde{c}_{1,0} & := \varepsilon_{1,0} \alpha \in (0, \tilde{c}_0) \\
d_{1,0} & := c_{1,0} - \left(\frac{\tilde{c}_{1,0}}{\alpha}\right)^2 - \frac{\zeta^2}{M}\left(\frac{\tilde{c}_{1,0}^2}{\alpha}\right) \in (0, d_0)
\end{align}
\end{subequations}
where $\zeta > 0$ comes from \IfConf{\eqref{eqn:MJODE_ZetaNum}}{\eqref{eqn:MJODE_ZetaNum_TR}}. Note that $\bar{a}$, defined via \eqref{eqn:BarA}, which is in $V_1$, equals $1$ when $\tau = 0$ and, as $\tau \rightarrow \infty$, $\bar{a}(\tau)$ decreases such that $\bar{a}(\tau) \rightarrow 0$. Namely, $\bar{a}$ is upper bounded by $1$.
Then, with $V_1$ given in \eqref{eqn:LyapunovNesterovNSCVX} and using Assumption \ref{ass:LisNSCVX} with $u_1 = z_1^*$ and $w_1 = z_1$, \IfConf{$V_1(z,\tau) \leq \left| z_1 - z_1^* \right|^2 + \left|z_2\right|^2 + \frac{\zeta^2}{M} \left| \nabla L(z_1) \right|\left| z_1 - z_1^* \right|$.}{
\begin{equation} \label{eqn:c10SublevelSetNestNSC}
V_1(z,\tau) \leq \left| z_1 - z_1^* \right|^2 + \left|z_2\right|^2 + \frac{\zeta^2}{M} \left| \nabla L(z_1) \right|\left| z_1 - z_1^* \right|.
\end{equation}}
Then, due to $L$ being $\mathcal{C}^1$, nonstrongly convex, and having a single minimizer $z_1^*$ by Assumption \ref{ass:LisNSCVX}, and due to $L$ having quadratic growth away from $\minSet$ by Assumption \ref{ass:QuadraticGrowth}, when $\left|\nabla L(z_1)\right| \leq \tilde{c}_{1,0}$, the suboptimality condition in Lemma \ref{lemma:ConvexSuboptimality} implies $\left| z_1 - z_1^* \right| \leq \frac{\tilde{c}_{1,0}}{\alpha}$, from where we get
\begin{equation} \label{eqn:V1Bound}
V_1(z,\tau) \leq \left(\frac{\tilde{c}_{1,0}}{\alpha}\right)^2 + \left|z_2\right|^2 + \frac{\zeta^2}{M} \left(\frac{\tilde{c}_{1,0}^2}{\alpha}\right).
\end{equation}
Then, by defining $\mathcal{T}_{1,0}$ as
\begin{equation} \label{eqn:T10}
\mathcal{T}_{1,0} := \defset{z \in \reals^{2n}\!\!}{\!\!\left| \nabla L(z_1) \right| \leq \tilde{c}_{1,0}, \left|z_2\right|^2 \leq d_{1,0}\!\!}
\end{equation}
which, by construction, is contained in the interior of $\mathcal{U}_0$ defined in \eqref{eqn:U0}, every $z \in \mathcal{T}_{1,0}$ belongs to the $c_{1,0}$-sublevel set of $V_1$. In fact, using the conditions in \eqref{eqn:TTilde10SetEquations} and \eqref{eqn:V1Bound}, we have for each $z \in \mathcal{T}_{1,0}$, \IfConf{$V_1(z,\tau) \leq \left(\frac{\tilde{c}_{1,0}}{\alpha}\right)^2 + \left|z_2\right|^2 + \frac{\zeta^2}{M} \left(\frac{\tilde{c}_{1,0}^2}{\alpha}\right) \leq c_{1,0}$.}{
\begin{equation} \label{eqn:V1BoundNSCNest}
V_1(z,\tau) \leq \left(\frac{\tilde{c}_{1,0}}{\alpha}\right)^2 + \left|z_2\right|^2 + \frac{\zeta^2}{M} \left(\frac{\tilde{c}_{1,0}^2}{\alpha}\right) \leq c_{1,0}.
\end{equation}}
The constants $\tilde{c}_0$, $\tilde{c}_{1,0}$, $d_0$, and $d_{1,0}$ in \eqref{eqn:UTilde0SetEquations} and \eqref{eqn:TTilde10SetEquations} comprise the hysteresis necessary to avoid chattering at the switching boundary. The idea behind these hysteresis boundaries is as follows. When $z \in \mathcal{U}_0$ and $q = 1$, we have that $z \in \overline{\reals^{2n}\setminus \mathcal{T}_{1,0}}$, and it is not yet time to switch to $\kappa_0$ but to continue to flow using $\kappa_1$. But once $z \in \mathcal{T}_{1,0}$ then $z$ is close enough to $\{\minSet\} \times \{0\}$, and the supervisor switches to $\kappa_0$. Note that $\mathcal{T}_{0,1} \cap \mathcal{T}_{1,0} = \emptyset$. \IfConf{Fig. \ref{fig:Hysteresis}}{Figure \ref{fig:HysteresisTR}} illustrates the hysteresis mechanism in the design of $\mathcal{U}_0$ and $\mathcal{T}_{1,0}$.
\IfConf{
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
\begin{picture}(20,9.7)(0,0)
\footnotesize
\put(0,0){\includegraphics[scale=0.2,trim={0cm 0cm 0cm 0cm},clip,width=20\unitlength]{Figures/Hysteresis.eps}}
\put(4,9){$q = 0$}
\put(14,9){$q = 1$}
\put(0.1,0.4){$\gamma \left(\frac{\alpha}{M^2} \right) \left|\nabla L(z_1)\right|^2 + \frac{1}{2}\left|z_2\right|^2 = c_0$}
\put(10,7.9){$\left|\nabla L(z_1)\right| = \tilde{c}_{1,0}, \frac{1}{2}\left|z_2\right|^2 = d_{1,0}$}
\put(5,6.5){$\mathcal{U}_0$}
\put(1,8){$\mathcal{T}_{0,1}$}
\put(14.5,4.5){$\mathcal{T}_{1,0}$}
\put(12,0.3){$\overline{\reals^{2n} \setminus \mathcal{T}_{1,0}}$}
\put(3.3,2.3){$\{\minSet\} \times \{0\}$}
\put(16.3,0.6){$\{\minSet\} \times \{0\}$}
\end{picture}
\vspace{-0.2cm}
\caption{An illustration of hysteresis in the design of the sets $\mathcal{U}_0$, $\mathcal{T}_0$, and $\mathcal{T}_{0,1}$ on $\reals^{2n}$, via the constants $\tilde{c}_{1,0} \in (0, \tilde{c}_0)$, $d_{1,0} \in (0,d_0)$, and $c_0 > 0$. Left: due to the design of $\mathcal{U}_0$ in \eqref{eqn:U0}, every $z \in \mathcal{U}_0$ belongs to the $c_0$-sublevel set of the Lyapunov function $V_0$, where $V_0$ is defined via \eqref{eqn:LyapunovHBF}. Hence, the same value of $c_0 > 0$ is also used to define $\mathcal{T}_{0,1}$ as the closed complement of a sublevel set of $V_0$ with level equal to $c_0$. Right: the constants $\tilde{c}_{1,0} \in (0, \tilde{c}_0)$ and $d_{1,0} \in (0,d_0)$, defined via \eqref{eqn:TTilde10SetEquations}, are chosen such that the set $\mathcal{T}_{1,0}$ in \eqref{eqn:T10} is contained in the interior of $\mathcal{U}_0$.}
\label{fig:Hysteresis}
\end{figure}}{
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
\begin{picture}(24,11.5)(0,0)
\footnotesize
\put(0,0){\includegraphics[scale=0.2,trim={0cm 0cm 0cm 0cm},clip,width=24\unitlength]{Figures/Hysteresis.eps}}
\put(5,10.8){$q = 0$}
\put(17,10.8){$q = 1$}
\put(0.3,0.5){$\gamma \left(\frac{\alpha}{M^2} \right) \left|\nabla L(z_1)\right|^2 + \frac{1}{2}\left|z_2\right|^2 = c_0$}
\put(13,9.5){$\left|\nabla L(z_1)\right| = \tilde{c}_{1,0}, \frac{1}{2}\left|z_2\right|^2 = d_{1,0}$}
\put(5,7){$\U_0$}
\put(1,9){$\mathcal{T}_{0,1}$}
\put(17,5){$\mathcal{T}_{1,0}$}
\put(13,0.5){$\overline{\reals^{2n} \setminus \mathcal{T}_{1,0}}$}
\put(4.1,2.7){$\{\minSet\} \times \{0\}$}
\put(19,1){$\{\minSet\} \times \{0\}$}
\end{picture}
\caption{An illustration of hysteresis in the design of the sets $\U_0$, $\mathcal{T}_0$, and $\mathcal{T}_{0,1}$ on $\reals^{2n}$, via the constants $\tilde{c}_{1,0} \in (0, \tilde{c}_0)$, $d_{1,0} \in (0,d_0)$, and $c_0 > 0$. Left: due to the design of $\U_0$ in \eqref{eqn:U0}, every $z \in \U_0$ belongs to the $c_0$-sublevel set of the Lyapunov function $V_0$, where $V_0$ is defined via \eqref{eqn:LyapunovHBF}. Hence, the same value of $c_0 > 0$ is also used to define $\mathcal{T}_{0,1}$ as the closed complement of a sublevel set of $V_0$ with level equal to $c_0$. Right: the constants $\tilde{c}_{1,0} \in (0, \tilde{c}_0)$ and $d_{1,0} \in (0,d_0)$, defined via \eqref{eqn:TTilde10SetEquations}, are chosen such that the set $\mathcal{T}_{1,0}$ in \eqref{eqn:T10} is contained in the interior of $\U_0$.}
\label{fig:HysteresisTR}
\end{figure}}
\subsubsection{Design of the Set $\mathcal{T}_{0,1}$} \label{sec:T01}
Recall from lines 3-4 of Algorithm \ref{algo:HS-Uniting} that the objective is to design $\mathcal{T}_{0,1}$ such that when $z \in \mathcal{T}_{0,1}$, $q = 0$, and $\tau = 0$, the state component $z_1$ is far from $\minSet$ and the supervisor resets $q$ to $1$ and assigns $u$ to $\kappa_1(h_1(z,\tau),\tau)$ so that $\kappa_1$ steers $z_1$ back to nearby $\minSet$.
Given $c_0 > 0$, let $\alpha > 0$ come from Assumption \ref{ass:QuadraticGrowth}, and let $M > 0$ come from Assumption \ref{ass:Lipschitz}. Then, using Assumption \ref{ass:Lipschitz} with $u_1 = z_1^*$ and $w_1 = z_1$ yields $\left|\nabla L(z_1)\right| \leq M \left| z_1 - z_1^* \right|$
for all $z_1 \in \reals^n$.
Since $L$ has quadratic growth away from $\minSet$ by Assumption \ref{ass:QuadraticGrowth}, then dividing both sides of $\left|\nabla L(z_1)\right| \leq M \left| z_1 - z_1^* \right|$
by $M$ and substituting into \IfConf{Assumption \ref{ass:QuadraticGrowth}}{\eqref{eqn:QuadraticGrowth}} leads to $L(z_1) - L^* \geq \frac{\alpha}{M^2} \left|\nabla L(z_1)\right|^2$,
where $\alpha > 0$ comes from Assumption \ref{ass:QuadraticGrowth}.
Then, $V_0$ in \eqref{eqn:LyapunovHBF} is lower bounded as follows: for each $z \in \reals^{2n}$, \IfConf{$V_0(z) = \gamma \left(L(z_1) - L^*\right) + \frac{1}{2}\left|z_2\right|^2 \geq \gamma \left(\frac{\alpha}{M^2} \right) \left|\nabla L(z_1)\right|^2 + \frac{1}{2}\left|z_2\right|^2$. Using such a lower bound}{
\begin{equation} \label{eqn:LowerBoundV0}
V_0(z) = \gamma \left(L(z_1) - L^*\right) + \frac{1}{2}\left|z_2\right|^2 \geq \gamma \left(\frac{\alpha}{M^2} \right) \left|\nabla L(z_1)\right|^2 + \frac{1}{2}\left|z_2\right|^2.
\end{equation}
Using the right-hand side of \eqref{eqn:LowerBoundV0}} and the same $c_0 > 0$ as in Section \ref{sec:U0}, we define the set
\begin{equation} \label{eqn:T01}
\!\!\!\!\!\mathcal{T}_{0,1} := \defset{\!z \in \reals^{2n}\!\!\!}{\!\!\!\gamma \left(\frac{\alpha}{M^2} \right)\! \left|\nabla L(z_1)\right|^2 + \frac{1}{2}\left|z_2\right|^2 \geq c_0\!\!\!}.
\end{equation}
The set in \eqref{eqn:T01} defines the (closed) complement of a sublevel set of the Lyapunov function $V_0$ in \eqref{eqn:LyapunovHBF} with level equal to $c_0$. The constant $c_0$ is also a part of the hysteresis mechanism, as shown in \IfConf{Fig. \ref{fig:Hysteresis}}{Figure \ref{fig:HysteresisTR}}. When $z \in \mathcal{U}_0$, $q = 0$, and $\tau = 0$, then the supervisor does not need to switch to $\kappa_1$, as the state component $z$ is close enough to the minimizer to keep using $\kappa_0$. But if $z \in \mathcal{T}_{0,1}$ while $q = 0$ and $\tau = 0$, then $z$ is far enough from the minimizer, and the supervisor then switches to $\kappa_1$.
\subsection{Design of the Parameter $\lambda$}
\label{sec:DesignOfLambda}
The heavy ball parameter $\lambda > 0$ should be made large enough to avoid oscillations near the minimizer, as stated in Sections \ref{sec:Background}, \ref{sec:Contributions}, and \ref{sec:ProblemStatement}. To gain some intuition on how to tune $\lambda$, consider the quadratic objective function $L(z_1) = \frac{1}{2} a_1 z_1^2$, $a_1 > 0$, which was analyzed in detail in \cite{attouch2000heavy}. For such a case, solutions to the heavy ball algorithm are overdamped (i.e., converge slowly with no oscillations) when $\lambda > 2\sqrt{a_1}$, critically damped (i.e., the fastest convergence possible with no oscillations) when $\lambda = 2\sqrt{a_1}$, and underdamped (fast convergence with oscillations) when $\lambda < 2\sqrt{a_1}$. Therefore, setting $\lambda \geq 2\sqrt{a_1}$ gives the desired behavior of solutions to $\HS_0$, for such an objective function.
More generally, setting $\lambda$ sufficiently large to avoid oscillations suffices, in practice. Numerically, $\lambda$ can be tuned as follows. Choose an arbitrarily large value of $\lambda$. If there is still oscillations or overshoot locally, despite the switch from $\kappa_1$ to $\kappa_0$ being made near the minimizer, then gradually increase $\lambda$ until the oscillations and overshoot disappear. See \IfConf{Examples \ref{ex:Robustness} and \ref{ex:NSC}}{Examples \ref{ex:Robustness}, \ref{ex:NSC}, and \ref{ex:tradeOff}} where $\lambda$ was tuned in such a way.
\subsection{Well-posedness of the Hybrid Closed-Loop System $\HS$}
When $L$ satisfies Assumptions \ref{ass:LisNSCVX}, \ref{ass:QuadraticGrowth}, and \ref{ass:Lipschitz}, the hybrid closed-loop system $\HS$ in \eqref{eqn:HS-TimeVarying} satisfies the hybrid basic conditions, listed in Definition \ref{def:HBCs}, as demonstrated in the following lemma. A hybrid closed-loop system $\HS$ that satisfies the hybrid basic conditions is said to be well-posed in the sense that the limit of a graphically convergent sequence of solutions to $\HS$ having a mild boundedness property is also a solution to $\HS$ \cite{65}.
\IfConf{\begin{lem}[Well-posedness of $\HS$]}{\begin{lemma}(Well-posedness of $\HS$):}
\label{lemma:HBC}
Let the function $L$ satisfy Assumptions \ref{ass:LisNSCVX}, \ref{ass:QuadraticGrowth}, and \ref{ass:Lipschitz}. Let the sets $\mathcal{U}_0$, $\mathcal{T}_{1,0}$, and $\mathcal{T}_{0,1}$ be defined via \eqref{eqn:T10}, and \eqref{eqn:T01}, respectively. Let the functions $\bar{d}$ and $\bar{\beta}$ be defined as in \eqref{eqn:dBarBetaBar}. Let $\kappa_0$ and $\kappa_1$ be defined via \eqref{eqn:StaticStateFeedbackLawsNSC}. Then, the hybrid closed-loop system $\HS$ in \eqref{eqn:HS-TimeVarying} satisfies the hybrid basic conditions.
\IfConf{\end{lem}}{\end{lemma}}
\IfConf{\begin{pf}}{\begin{proof}}
The objective function $L$ is $\mathcal{C}^1$, nonstrongly convex, and has a single minimizer by Assumption \ref{ass:LisNSCVX}.
Therefore, since $\nabla L$ is continuous, the following hold: the set $\mathcal{U}_0$, defined via \eqref{eqn:U0}, is closed since it is a sublevel set of the continuous function $V_0$; due to $\bar{a}$ in \eqref{eqn:BarA} being continuous, the set $\mathcal{T}_{1,0}$, defined via \eqref{eqn:T10}, is closed since it is a sublevel set of the continuous function $V_1$; the set $\mathcal{T}_{0,1}$, defined via \eqref{eqn:T01}, is closed since it is the closed complement of a set. Therefore, since the sets $\mathcal{U}_0$, $\mathcal{T}_{1,0}$, and $\mathcal{T}_{0,1}$ are closed, then the sets $D_0$, $D_1$, $C_0$, and $C_1$ are closed. Since $C$ and $D$ are both finite unions of finite and closed sets, then $C$ and $D$ are also closed.
Since $\bar{d}$ and $\bar{\beta}$, defined via \eqref{eqn:dBarBetaBar}, are continuous, and since by Assumption \ref{ass:LisNSCVX}, $L$ is $\mathcal{C}^1$, then $h_{q}$ in \eqref{eqn:H0H1NSCNesterovHBF} and $\kappa_{q}$ in \eqref{eqn:StaticStateFeedbackLawsNSC} are continuous. In turn, the map $z \mapsto$ $F_P(z,\kappa_{q}(h_{q}(z,\tau),\tau))$ is also continuous since $F_P$ in \eqref{eqn:HBFplant-dynamicsTR} is a $\mathcal{C}^1$ function of $\kappa_{q}$ and $h_{q}$. Therefore, $x \mapsto F(x)$ is continuous. The map $G$ satisfies \ref{item:A3} by construction since it is continuous.\IfConf{\hfill{} \qed}{}
\IfConf{\end{pf}}{\end{proof}} \IfConf{
\vspace{-0.3cm}
}{}
In Theorem \ref{thm:GASNestNSC} we show that $\HS$ has a compact pre-asymptotically stable set. In light of this property, Lemma \ref{lemma:HBC} is key as it leads to pre-asymptotic stability that is robust to small perturbations \cite[Theorem~7.21]{65}.
In the case of gradient-based algorithms, for instance, such perturbations can take the form of small noise in measurements of the gradient.
\subsection{Existence of solutions to $\HS$}
Under Assumptions \ref{ass:LisNSCVX}, \ref{ass:QuadraticGrowth}, and \ref{ass:Lipschitz}, every maximal solution to $\HS$ is complete and bounded, as stated in the following lemma. Such a property is useful since it guarantees that nontrivial solutions to $\HS$ exist from each initial point in $C \cup D$, and that such solutions do not escape $C \cup D$. When every maximal solution is complete, then uniform global pre-asymptotic stability\footnote{Uniform global pre-asymptotic stability indicates the possibility of a maximal solution that is not complete, even though it may be bounded.} of the set $\mathcal{A}$ becomes uniform global asymptotic stability. The following lemma also states that $\Pi(C_0) \cup \Pi(D_0) = \reals^{2n}$ and $\Pi(C_1) \cup \Pi(D_1) = \reals^{2n}$. Such a property ensures that nontrivial solutions to $\HS$, which exist from each initial point in $C \cup D$, also exist from any initial point in $\reals^{2n} \times Q \times \reals_{\geq 0}$.
\IfConf{\begin{prop}[Existence of solutions to $\HS$]}{\begin{proposition}(Existence of solutions to $\HS$):}
\label{prop:Existence}
Let the function $L$ satisfy Assumptions \ref{ass:LisNSCVX}, \ref{ass:QuadraticGrowth}, and \ref{ass:Lipschitz}. Let the sets $\mathcal{U}_0$, $\mathcal{T}_{1,0}$, and $\mathcal{T}_{0,1}$ be defined via \eqref{eqn:T10}, and \eqref{eqn:T01}, respectively. Let the functions $\bar{d}$ and $\bar{\beta}$ be defined as in \eqref{eqn:dBarBetaBar}. Let $\kappa_0$ and $\kappa_1$ be defined via \eqref{eqn:StaticStateFeedbackLawsNSC}. Then, $\Pi(C_0) \cup \Pi(D_0) = \reals^{2n}$, $\Pi(C_1) \cup \Pi(D_1) = \reals^{2n}$, and each maximal solution $(t,j) \mapsto x(t,j) = (z(t,j),q(t,j),\tau(t,j))$ to $\HS$ in \eqref{eqn:HS-TimeVarying} is bounded and complete.
\IfConf{\end{prop}}{\end{proposition}}
\IfConf{\begin{pf}}{\begin{proof}}
Since Assumptions \ref{ass:LisNSCVX}, \ref{ass:QuadraticGrowth}, and \ref{ass:Lipschitz} hold, then $\HS$ satisfies the hybrid basic conditions by Lemma \ref{lemma:HBC}. With $\tilde{c}_0 > 0$ and $d_0 > 0$ defined via \eqref{eqn:UTilde0SetEquations}, since $L$ is $\mathcal{C}^1$, nonstrongly convex, has a single minimizer by Assumption \ref{ass:LisNSCVX}, and has quadratic growth away from $\minSet$ by Assumption \ref{ass:QuadraticGrowth}, from the arguments below \eqref{eqn:c0SublevelSet}, every $z \in \mathcal{U}_0$ belongs to the $c_0$-sublevel set of $V_0$; recall that $\mathcal{U}_0$ is defined in in \eqref{eqn:U0} and that $V_0$ is defined via \eqref{eqn:LyapunovHBF}. Additionally, since by Assumption \ref{ass:QuadraticGrowth} $L$ has quadratic growth away from $\minSet$, then $\mathcal{T}_{0,1}$ in \eqref{eqn:T01}, defines the closed complement of a sublevel set of $V_0$ with level equal to $c_0$. Therefore, due to the definitions of $\mathcal{U}_0$ in \eqref{eqn:U0} and $\mathcal{T}_{0,1}$ in \eqref{eqn:T01}, $\Pi(C_0) \cup \Pi(D_0) = \reals^{2n}$. Furthermore, since $\mathcal{T}_{1,0}$ is defined via \eqref{eqn:T10}, and since by the definitions of $C_1$ and $D_1$ in \eqref{eqn:CAndDGradientsNestNSC}, $C_1$ is the closed complement of $D_1$, then $\Pi(C_1) \cup \Pi(D_1) = \reals^{2n}$.
Due to the definitions of $C_0$, $D_0$, $C_1$, and $D_1$ in \eqref{eqn:CAndDGradientsNestNSC}, $\mathcal{U}_0$ in \eqref{eqn:U0}, $\mathcal{T}_{1,0}$ in \eqref{eqn:T10}, and $\mathcal{T}_{0,1}$ in \eqref{eqn:T01}, then $C \setminus D$ is equal to $\mathrm{int}(C)$. Hence, for each point $x \in C \setminus D$, the tangent cone to $C$ at $x$ is
\begin{equation} \label{eqn:TangentCone}
T_C(x) := \begin{cases} \reals^{2n} \times \{0\} \times \{0\} & \ \ \text{if } x \in C_0 \setminus D_0,\\
\reals^{2n} \times \{1\} \times \reals_{\geq 0} & \ \ \text{if } x \in C_1 \setminus D_1.
\end{cases}
\end{equation}
Therefore, $F(x) \cap T_C(x) \neq \emptyset$, satisfying (VC) of for each point $x \in C \setminus D$, and nontrivial solutions exist for every initial point in $\left(C_0 \cup C_1\right) \cup \left(D_0 \cup D_1\right)$, where $\Pi(C_0) \cup \Pi(D_0) = \reals^{2n}$ and $\Pi(C_1) \cup \Pi(D_1) = \reals^{2n}$. To prove that item (c) of \IfConf{\cite[Proposition~A.1]{dhustigs2022unitingNSC}}{Proposition \ref{prop:SolnExistence}} does not hold, we need to show that $G(D) \subset C \cup D$. With $D$ defined in \eqref{eqn:CAndDGradientsNestNSC}, \IfConf{$G(D) = \left(\mathcal{T}_{0,1} \times \{1\} \times \{0\} \right) \cup \left(\mathcal{T}_{1,0} \times \{0\} \times \{0\} \right)$.}{
\begin{equation} \label{eqn:G(D)}
G(D) = \left(\mathcal{T}_{0,1} \times \{1\} \times \{0\} \right) \cup \left(\mathcal{T}_{1,0} \times \{0\} \times \{0\} \right).
\end{equation}}
Notice that $\mathcal{T}_{1,0} \times \{0\} \times \{0\} \subset C_0$ and $\mathcal{T}_{0,1} \times \{1\} \times \{0\} \subset C_1$. Therefore, $G(D) \subset C$; hence $G(D) \subset C \cup D$. Therefore, item (c) of \IfConf{\cite[Proposition~A.1]{dhustigs2022unitingNSC}}{Proposition \ref{prop:SolnExistence}} does not hold. Then it remains to prove that item (b) does not happen.
To this end, we show first that $\HS_0$, defined via \eqref{eqn:H0}, has no finite time escape\footnote{{\em Finite escape time} describes when there exists a solution $t \mapsto x(t)$ to a continuous-time nonlinear system that satisfies $\lim\limits_{t \nearrow t_e} \left|x(t)\right| = \infty$ for some finite time $t_e$.}, and has unique and bounded solutions. Since $L$ is $\mathcal{C}^1$ by Assumption \ref{ass:LisNSCVX}, and $\nabla L$ is Lipschitz continuous by Assumption \ref{ass:Lipschitz}, then $h_0$ in \eqref{eqn:H0H1NSCNesterovHBF} and $\kappa_0$ in \eqref{eqn:StaticStateFeedbackLawLocal} are Lipschitz continuous, which, since $F_P$ is a $\mathcal{C}^1$ function of $h_0$ and $\kappa_0$, means the map $z \mapsto F_P(z,\kappa_0(h_0(z)))$ is also Lipschitz continuous. Therefore, by \cite[Theorem~3.2]{khalil2002nonlinear}, $\dot{z} = F_P(z,\kappa_0(h_0(z)))$ has no finite time escape and each maximal solution to $\HS_0$ is unique. To show that each maximal solution to $\HS_0$ is bounded, we use the Lyapunov function in \eqref{eqn:LyapunovHBF}, defined for each $z \in \reals^{2n}$. Then, solutions to $\dot{z} = F_P(z,\kappa_0(h_0(z)))$ starting from any $c_V$-sublevel set $W := \defset{z \in \reals^{2n}}{V_0(z) \leq c_V}$, $c_V \geq 0$, remains in such a set for all time since $V_0$ satisfies
\begin{equation}\label{eqn:VdotHBF}
\dot{V}_0(z)
\!=\!
\langle \nabla V_0(z),F_P(z,\kappa_0(h_0(z))) \rangle
\!=\!
-\lambda \left|z_2\right|^2 \leq 0
\end{equation}
for each $z \in \reals^{2n}$, since $\lambda$ is positive. Then, to show that $V_0$ is radially unbounded, we derive class-$\mathcal{K}_{\infty}$ functions $\alpha_1$ and $\alpha_2$ such that\IfConf{\footnote{Since $L$ has quadratic growth away from $z_1^*$ by Assumption \ref{ass:QuadraticGrowth}, then the choice of $\alpha_1$ comes from lower bounding $L(z_1) - L^*$ in $V_0$ via Assumption \ref{ass:QuadraticGrowth}. The choice of $\alpha_2$ comes from the following: since $L$ is $\mathcal{C}^1$, nonstrongly convex, and has a single minimizer by Assumption \ref{ass:LisNSCVX}, then the expression $L(z_1) - L^*$ in $V_0$ is upper bounded using the definition of nonstrong convexity in Footnote \ref{foot:Convexity}, by the same process that $L(z_1) - L^*$ is upper bounded in \eqref{eqn:NearOptimalityC}, to get \eqref{eqn:c0SublevelSet}. Then, $\left|\nabla L(z_1)\right|$ in \eqref{eqn:c0SublevelSet} is upper bounded via Assumption \ref{ass:Lipschitz} with $u_1 = z_1^*$ and $w_1 = z_1$.},}{,} for all $z \in \reals^{2n}$, with $z^* := (z_1^*,0)$,\IfConf{
\vspace{-0.8cm}
\begin{align} \label{eqn:RadiallyUnboundedV0}
\!\!\alpha_1(\left|z - z^*\right|) & := \min \left\{\!\alpha \gamma,\frac{1}{2}\! \right\} \left|z - z^*\right|^2\\
& \leq V_0(z) \nonumber\\
& \leq \alpha_2(\left|z - z^*\right|) := \left(\!M \gamma + \frac{1}{2}\!\right) \left|z - z^*\right|^2. \nonumber
\end{align}
\vspace{-0.5cm}
}{\begin{align} \label{eqn:RadiallyUnboundedV0TR}
\alpha_1(\left|z - z^*\right|) := \min \left\{\!\alpha \gamma,\frac{1}{2}\! \right\} \left|z - z^*\right|^2 & \leq V_0(z) \nonumber\\
&\leq \alpha_2(\left|z - z^*\right|) := \left(\!M \gamma + \frac{1}{2}\!\right) \left|z - z^*\right|^2.
\end{align}}
\IfConf{}{Since $L$ has quadratic growth away from $z_1^*$ with constant $\alpha > 0$ by Assumption \ref{ass:QuadraticGrowth}, then the choice of $\alpha_1$ comes from lower bounding $V_0$ as follows
\begin{align}
V_0(z) = \gamma \left(L(z_1) - L^*\right) + \frac{1}{2} \left|z_2\right|^2 & \geq \alpha \gamma \left|z_1 - z_1^*\right|^2 + \frac{1}{2} \left|z_2\right|^2 \nonumber\\
& \geq \min \left\{\!\alpha \gamma,\frac{1}{2}\! \right\} \left|z - z^*\right|^2 = \alpha_1(\left|z - z^*\right|)
\end{align}
for all $z_1 \in \reals^n$.
The choice of $\alpha_2$ comes from the following. Since $L$ is $\mathcal{C}^1$, nonstrongly convex, has a single minimizer by Assumption \ref{ass:LisNSCVX}, and since $\nabla L$ is Lipschitz continuous with constant $M > 0$ by Assumption \ref{ass:Lipschitz}, we upper bound $V_0$ in the following manner, using the definition of nonstrong convexity in Footnote \ref{foot:Convexity} to get \eqref{eqn:c0SublevelSet}, and then using the Lipschitz bound in Assumption \ref{ass:Lipschitz} with $u_1 = z_1^*$ and $w_1 = z_1$ to upper bound \eqref{eqn:c0SublevelSet}, yielding
\begin{align} \label{eqn:UpperBoundV0}
V_0(z) = \gamma \left(L(z_1) - L^*\right) + \frac{1}{2}\left|z_2\right|^2 & \leq \gamma \left|\nabla L(z_1)\right| \left| z_1 - z_1^* \right| + \frac{1}{2}\left|z_2\right|^2 \nonumber\\
& \leq M \gamma \left|z_1 - z_1^*\right|^2 + \frac{1}{2}\left|z_2\right|^2 \nonumber\\
& \leq \left(\!M \gamma + \frac{1}{2}\!\right) \left|z - z^*\right|^2 = \alpha_2(\left|z_1 - z_1^*\right|)
\end{align}
for all $z_1 \in \reals^n$.} Since \IfConf{\eqref{eqn:RadiallyUnboundedV0}}{\eqref{eqn:RadiallyUnboundedV0TR}} is satisfied for $V_0$ in \eqref{eqn:LyapunovHBF} for all $z \in \reals^{2n}$, then $V_0$ is radially unbounded (in $z$, relative to $\{\minSet\} \times \{0\}$). Therefore, $W$ is compact and, due to \eqref{eqn:VdotHBF}, forward invariant for $\HS_1$, that is, any nontrivial solution starting in the subset $W$ is complete and stays in $W$. Therefore, each maximal solution to $\HS_0$, defined via \eqref{eqn:H0}, is bounded. \IfConf{\vspace{-0.1cm}}{}
Next, we show that $\HS_1$ in \eqref{eqn:H1} has no finite time escape from $\reals^{2n} \times \reals_{\geq 0}$, and has unique solutions. Since $\bar{d}$ and $\bar{\beta}$, defined via \eqref{eqn:dBarBetaBar}, are continuous, and since by Assumption \ref{ass:LisNSCVX}, $L$ is $\mathcal{C}^1$, then $h_1$ in \eqref{eqn:H0H1NSCNesterovHBF} and $\kappa_1$ in \eqref{eqn:StaticStateFeedbackLawsNSC} are also continuous. Furthermore, since by Assumption \ref{ass:Lipschitz} $\nabla L$ is Lipschitz continuous, then $h_1$ in \eqref{eqn:H0H1NSCNesterovHBF} and $\kappa_1$ in \eqref{eqn:StaticStateFeedbackLawsNSC} are Lipschitz continuous which, in turn, means the map $z \mapsto F_P(z,\kappa_1(h_1(z,\tau),\tau))$ is Lipschitz continuous. Consequently, since the map $z \mapsto F_P(z,\kappa_1(h_1(z,\tau),\tau))$ is Lipschitz continuous and since the solution component $\tau$ of $\HS_1$ increases linearly, then by \cite[Theorem~3.2]{khalil2002nonlinear}, $\HS_1$ in \eqref{eqn:H1} has no finite escape time from $\reals^{2n} \times \reals_{\geq 0}$ and each maximal solution to $\HS_0$ is unique. Therefore, each maximal solution to $\HS_1$, defined via \eqref{eqn:H1}, is complete and unique. \IfConf{\vspace{-0.1cm}}{}
Since $\HS_0$ has no finite time escape from $\reals^{2n}$ and $\HS_1$ has no finite time escape from $\reals^{2n} \times \reals_{\geq 0}$, then this means $\dot{x} = F(x)$ has no finite time escape from $C$ for $\HS$, as $q$ does not change in $C$ and as the state component $\tau$ is bounded in $C$, namely, the state component $\tau$ -- which is always reset to $0$ in $D$ -- increases linearly in $C_1$ and remains at $0$ in $C_0$. Therefore, there is no finite time escape from $C \cup D$, for solutions $x$ to $\HS$. Therefore, item (b) from \IfConf{\cite[Proposition~A.1]{dhustigs2022unitingNSC}}{Proposition \ref{prop:SolnExistence}} does not hold.\IfConf{\hfill{} \qed}{}
\IfConf{\end{pf}}{\end{proof}} \IfConf{\vspace{-0.3cm}}{}
\subsection{Main Result}
\label{sec:MainResult}
In this section, we present a result that establishes UGAS of the set
\begin{align} \label{eqn:SetOfMinimizersHS-NSCVX}
\IfConf{\mathcal{A} := & \defset{z \in \reals^{2n}}{\nabla L(z_1) = z_2 = 0} \times \{0\} \times \{0\}\nonumber\\
= & \{\minSet\} \times \{0\} \times \{0\} \times \{0\}}{\mathcal{A} := \defset{z \in \reals^{2n}\!\!\!}{\!\!\!\nabla L(z_1) = z_2 = 0\!\!} \times \{0\} \times \{0\}
= \{\minSet\} \times \{0\} \times \{0\} \times \{0\}}
\end{align}
and a hybrid convergence rate that, globally, is equal to $\frac{1}{(t+2)^2}$ while locally, is exponential, for the hybrid closed loop algorithm $\HS$ in \eqref{eqn:HS-TimeVarying} and \eqref{eqn:CAndDGradientsNestNSC}. Recall that the state $x := \left(z, q, \tau\right) \in \reals^{2n} \times Q \times \reals_{\geq 0}$. In light of this, the first component of $\mathcal{A}$, namely, $\{\minSet\}$, is the minimizer of $L$. The second component of $\mathcal{A}$, namely, $\{0\}$, reflects the fact that we need the velocity state $z_2$ to equal zero in $\mathcal{A}$ so that solutions are not pushed out of such a set. The third component in $\mathcal{A}$, namely, $\{0\}$, is due to the logic state ending with the value $q = 0$, namely using $\kappa_0$ as the state $z$ reaches the set of minimizers of $L$. The last component in $\mathcal{A}$ is due to $\tau$ being set to, and then staying at, zero when the supervisor switches to $\kappa_0$.
\IfConf{\begin{thm}[UGAS of $\mathcal{A}$ for $\HS$]}{\begin{theorem}(Uniform global asymptotic stability of $\mathcal{A}$ for $\HS$):} \label{thm:GASNestNSC}
Let the function $L$ satisfy Assumptions \ref{ass:LisNSCVX}, \ref{ass:QuadraticGrowth}, and \ref{ass:Lipschitz}. Let $\zeta > 0$, $\lambda > 0$, $\gamma > 0$, $c_{1,0} \in (0,c_0)$, and $\varepsilon_{1,0} \in (0,\varepsilon_0)$ be given. Let $\alpha > 0$ be generated by Assumption \ref{ass:QuadraticGrowth}, and let $M >0$ be generated by Assumption \ref{ass:Lipschitz}. Let $\tilde{c}_{1,0} \in (0,\tilde{c}_0)$ and $d_{1,0} \in (0,d_0)$ be defined via \eqref{eqn:UTilde0SetEquations} and \eqref{eqn:TTilde10SetEquations}. Let the sets $\mathcal{U}_0$, $\mathcal{T}_{1,0}$, and $\mathcal{T}_{0,1}$ be defined via \eqref{eqn:T10}, and \eqref{eqn:T01}, respectively. Let the functions $\bar{d}$ and $\bar{\beta}$ be defined as in \eqref{eqn:dBarBetaBar}, and let $\kappa_0$ and $\kappa_1$ be defined via \eqref{eqn:StaticStateFeedbackLawsNSC}.Then, the set $\mathcal{A}$, defined via \eqref{eqn:SetOfMinimizersHS-NSCVX}, is uniformly globally asymptotically stable for $\HS$ given in \eqref{eqn:HS-TimeVarying}-\eqref{eqn:CAndDGradientsNestNSC}. Furthermore, each maximal solution $(t,j) \mapsto x(t,j) = (z(t,j), q(t,j), \tau(t,j))$ of the hybrid closed-loop algorithm $\HS$ starting from $C_1$ with $\tau(0,0)=0$ satisfies the following:
\begin{enumerate}[label={\arabic*)},leftmargin=*]
\item \label{item:1} The domain $\dom x$ of the solution $x$ is of the form $\cup_{j=0}^{1} (I^j \times \{j\})$, with $I^0$ of the form $[t_0,t_1]$ and with $I^1$ of the form $[t_1,\infty)$ for some $t_1 \geq 0$ defining the time of the first jump;
\item \label{item:2} For each $t \in I^0$ such that\footnote{Note that at each $t \in I^0$, $q(t,0) = 1$, and at each $t \in I^1$, $q(t,1) = 0$.} $t \geq 1$
\begin{align} \label{eqn:UnitingConvergenceRateNSCHS1}
\IfConf{& L(z_1(t,0)) - L^* \nonumber\\
& \leq \frac{9cM}{\zeta^2(t+2)^2} \left(\left|z_1(0,0) - z_1^* \right|^2 + \left| z_2(0,0) \right|^2 \right)}{L(z_1(t,0)) - L^* \leq \frac{9cM}{\zeta^2(t+2)^2} \left(\left|z_1(0,0) - z_1^* \right|^2 + \left| z_2(0,0) \right|^2 \right)}
\end{align}
where $c := \left(1 + \zeta^2\right)\exp \left(\sqrt{\frac{13}{4} + \frac{\zeta^4}{M}}\right)$. Namely, $L(z_1(t,0)) - L^*$ is \IfConf{$\mathcal{O}\!\left(\!\frac{9cM}{\zeta^2(t+2)^2}\!\right)$;}{\\ $\mathcal{O}\!\left(\!\frac{9cM}{\zeta^2(t+2)^2}\!\right)$;}
\item \label{item:3} For each $t \in I^1$
\begin{equation} \label{eqn:UnitingConvergenceRateNSCHS0}
L(z_1(t,1)) - L^* = \mathcal{O}\left(\exp \left(-(1-m)\psi t\right)\right)
\end{equation}
where $m \in (0,1)$ is such that
$\psi := \frac{m\alpha\gamma}{\lambda} > 0$ and $\nu := \psi (\psi - \lambda) < 0$.
\end{enumerate}
\IfConf{\end{thm}}{\end{theorem}}
As will be shown in the forthcoming proof of Theorem \ref{thm:GASNestNSC} in Section \ref{sec:ProofMainResult}, solutions starting from $C_1$ jump no more than once.
The UGAS of the hybrid closed-loop algorithm $\HS$ in Theorem \ref{thm:GASNestNSC} is proved as follows. First, in the forthcoming Proposition \ref{prop:GAS-HBF}, we establish UGAS of the set $\{\minSet\} \times \{0\}$ for the closed-loop algorithm $\HS_0$ in \eqref{eqn:H0} via Lyapunov theory and the application of an invariance principle. Then, in the forthcoming Proposition \ref{prop:UGANestrovNSCVX}, we prove uniform global attractivity (UGA) of the set $\{\minSet\} \times \{0\} \times \reals_{\geq 0}$ for the closed-loop algorithm $\HS_1$ in \eqref{eqn:H1} via Lyapunov theory and a comparison principle. Then, UGAS of $\mathcal{A}$ for $\HS$ and item \ref{item:1} in Theorem \ref{thm:GASNestNSC} follow from a trajectory-based proof employing the UGAS of $\HS_0$, the UGA of $\HS_1$, and the construction of the sets $\mathcal{U}_0$, $\mathcal{T}_{1,0}$, and $\mathcal{T}_{0,1}$. The hybrid convergence rate of the closed-loop algorithm $\HS$ in items \ref{item:2} and \ref{item:3} of Theorem \ref{thm:GASNestNSC} is proved in the forthcoming Propositions \ref{prop:HBFConvergenceRate}, \ref{prop:ConvergenceNSCVXNesterov}, and \ref{prop:UpperBoundofV}, \ref{prop:ConvergenceNSCVXNesterov}, and \ref{prop:UpperBoundofV}.
\section{Numerical Examples}
\label{sec:Examples}
In this section, we present multiple numerical examples to illustrate the hybrid closed-loop algorithm in \eqref{eqn:HS-TimeVarying} and \eqref{eqn:CAndDGradientsNestNSC}. Example \ref{ex:Robustness} first illustrates the operation of the nominal hybrid closed-loop system $\HS$, and then demonstrates the robustness of $\HS$ to different amounts of noise in measurements of $\nabla L$. Example \ref{ex:NSC} compares solutions to the hybrid closed-loop algorithm in \eqref{eqn:HS-TimeVarying} and \eqref{eqn:CAndDGradientsNestNSC} with solutions to $\HS_0$, $\HS_1$, and HAND-1 from \cite{poveda2019inducing}, with parameters chosen such that HAND-1 and $\HS$ are compared on equal footing. Example \ref{ex:NSC} then compares multiple solutions of $\HS$, starting from different initial values of $z_1$, to multiple solutions of HAND-1 from such initial values of $z_1$, to show that $\HS$ has a consistent percentage of improvement over HAND-1 for different solutions. \IfConf{}{Example \ref{ex:tradeOff} Illustrates the trade-off between speed of convergence and the resulting values of parameters for the uniting algorithm $\HS$, for different tunings of $\zeta > 0$. As in Example \ref{ex:NSC}, the parameter values for Example \ref{ex:tradeOff} are chosen such that HAND-1 and $\HS$ are compared on equal footing.}
\IfConf{\begin{exmp}}{\begin{example}} \label{ex:Robustness}
In this example, we simulate a solution to the nominal hybrid closed-loop system $\HS$ to illustrate how the uniting algorithm works. Then, we compare that same solution to solutions with different amounts of noise in measurements of $\nabla L$. For both the nominal system and the perturbed system, the choice of objective function, parameter values, and initial conditions are as follows. We use the objective function $L(z_1) := z_1^2$, the gradient of which is Lipschitz continuous with $M = 2$, and which has a single minimizer at $\minSet = 0$. This choice of objective function is made so that we can easily tune $\lambda$, as described in Section \ref{sec:DesignOfLambda}.
We arbitrarily chose the heavy ball parameter value $\gamma = \frac{2}{3}$ and we tuned $\lambda$ to $200$ by choosing a value arbitrarily larger than $2\sqrt{a_1}$, where $a_1$ comes from Section \ref{sec:DesignOfLambda}., and gradually increasing it until there is no overshoot in the hybrid algorithm.
The parameter values for the uniting algorithm are $c_0 = 7000$, $c_{1,0} \approx 6819.68$, $\varepsilon_0 = 10$, $\varepsilon_{1,0} = 5$, and $\alpha = 1$, which yield the values $\tilde{c}_0 = 10$, $\tilde{c}_{1,0} = 5$, $d_0 = 6933$, and $d_{1,0} = 6744$, which are calculated via \eqref{eqn:UTilde0SetEquations} and \eqref{eqn:TTilde10SetEquations}. These values are chosen for proper tuning of the algorithm, in order to get nice performance, and the value of $c_{1,0}$ is chosen to exploit the properties of Nesterov's method for a longer time, so that the nominal solution gets closer to the minimizer faster.
Initial conditions for $\HS$ are $z_1(0,0) = 50$, $z_2(0,0) = 0$, $q(0,0) = 1$, and $\tau(0,0) = 0$. The plot \IfConf{on the top}{on the top} in \IfConf{Fig. \ref{fig:NominalNoise}}{Figure \ref{fig:NominalNoise_TR}} shows the solution to the nominal hybrid closed-loop algorithm\footnote{\label{foot:RobustnessURL}Code at \IfConf{\\}{} \texttt{gitHub.com/HybridSystemsLab/UnitingRobustness}} $\HS$, namely, the value of $z_1$ over time, with the time it takes for the solution to settle to within $1\%$ of $\minSet$ marked with a black dot and labeled in seconds. The jump at which the switch from $\HS_1$ to $\HS_0$ occurs is labeled with an asterisk. The solution converges quickly, without oscillations near the minimizer. \IfConf{
\begin{figure}
\vspace{-0.15cm}
\centering
\setlength{\unitlength}{1.0pc}
\centering
\subfloat{
\begin{picture}(20,5.5)(0,0)
\footnotesize
\put(0,0.2){\includegraphics[scale=0.4,trim={0cm 0cm 0cm 0cm},clip,width=20\unitlength]{Figures/PlotsNominalTR.eps}}
\put(0,2.75){$z_1$}
\put(10.6,0){$t$}
\end{picture}}\\
\subfloat{
\footnotesize
\begin{tabular}{|c|c|c|}
\hline
$\sigma$ & {\scriptsize $\lim\limits_{t+j \rightarrow \infty} \sup \left|z_1(t,j) - z_1^*\right|$} & {\scriptsize $\lim\limits_{t+j \rightarrow \infty} \sup \left|L(z_1(t,j)) - L^*\right|$} \\
\hline
\hline
$0.01$ & $8.857 \times 10^{-6}$ & $7.844 \times 10^{-11}$\\
\hline
$0.1$ & $8.011 \times 10^{-4}$ & $6.418 \times 10^{-7}$\\
\hline
$0.5$ & $9.039 \times 10^{-4}$ & $8.171 \times 10^{-7}$\\
\hline
$1$ & $6.982 \times 10^{-3}$ & $4.875 \times 10^{-5}$ \\
\hline
$5$ & $9.459 \times 10^{-3}$ & $8.947 \times 10^{-5}$ \\
\hline
$10$ & $1.450 \times 10^{-2}$ & $2.103 \times 10^{-4}$ \\
\hline
$15$ & $4.938 \times 10^{-2}$ & $2.438 \times 10^{-3}$ \\
\hline
$20$ & $5.992 \times 10^{-2}$ & $3.591 \times 10^{-3}$ \\
\hline
$25$ & $6.663 \times 10^{-2}$ & $4.439 \times 10^{-3}$ \\
\hline
\end{tabular}}
\caption{Top: The evolution over time of $z_1$, for the nominal hybrid closed-loop algorithm $\HS$, for a function $L(z_1) := z_1^2$ with a single minimizer at $\minSet = 0$. The time at which the solution settles to within $1\%$ of $\minSet$ is marked with a dot and labeled in seconds. The jump is labeled with an asterisk. Bottom: Simulation results for perturbed solutions using zero mean Gaussian noise, with each simulation using a different value of the standard deviation $\sigma$. Results listed are for a large value of $t+j$.}
\label{fig:NominalNoise}
\end{figure}
}{\begin{figure}
\centering
\setlength{\unitlength}{1.0pc}
\centering
\subfloat{
\begin{picture}(25,6)(0,0)
\footnotesize
\put(0,0.2){\includegraphics[scale=0.4,trim={0cm 0cm 0cm 0cm},clip,width=25\unitlength]{Figures/PlotsNominalTR.eps}}
\put(0,3.5){$z_1$}
\put(13.2,0){$t$}
\end{picture}}\\
\subfloat{
\begin{tabular}{|c|c|c|}
\hline
$\sigma$ & $\lim\limits_{t+j \rightarrow \infty} \sup \left|z_1(t,j) - z_1^*\right|$ & $\lim\limits_{t+j \rightarrow \infty} \sup \left|L(z_1(t,j)) - L^*\right|$ \\
\hline
\hline
$0.01$ & $8.857 \times 10^{-6}$ & $7.844 \times 10^{-11}$\\
\hline
$0.1$ & $8.011 \times 10^{-4}$ & $6.418 \times 10^{-7}$\\
\hline
$0.5$ & $9.039 \times 10^{-4}$ & $8.171 \times 10^{-7}$\\
\hline
$1$ & $6.982 \times 10^{-3}$ & $4.875 \times 10^{-5}$ \\
\hline
$5$ & $9.459 \times 10^{-3}$ & $8.947 \times 10^{-5}$ \\
\hline
$10$ & $1.450 \times 10^{-2}$ & $2.103 \times 10^{-4}$ \\
\hline
$15$ & $4.938 \times 10^{-2}$ & $2.438 \times 10^{-3}$ \\
\hline
$20$ & $5.992 \times 10^{-2}$ & $3.591 \times 10^{-3}$ \\
\hline
$25$ & $6.663 \times 10^{-2}$ & $4.439 \times 10^{-3}$ \\
\hline
\end{tabular}}
\caption{Top: The evolution over time of $z_1$, for the nominal hybrid closed-loop algorithm $\HS$, for a function $L(z_1) := z_1^2$ with a single minimizer at $\minSet = 0$. The time at which the solution settles to within $1\%$ of $\minSet$ is marked with a dot and labeled in seconds. The jump is labeled with an asterisk. Bottom: Simulation results for perturbed solutions using zero mean Gaussian noise, with each simulation using a different value of the standard deviation $\sigma$. Results listed are for a large value of $t+j$.}
\label{fig:NominalNoise_TR}
\end{figure}}
\IfConf{
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
%
\begin{picture}(20,13.5)(0,0)
\footnotesize
\put(0.4,0.5){\includegraphics[scale=0.3,trim={0.5cm 0.4cm 0.8cm 0.4cm},clip,width=9\unitlength]{Figures/PlotsNoisy.eps}}
\put(10,0.5){\includegraphics[scale=0.3,trim={0.5cm 0.4cm 0.7cm 0.4cm},clip,width=9\unitlength]{Figures/PlotsNoisyLog.eps}}
\put(0.0,2.3){$z_1$}
\put(0.0,5.45){$z_1$}
\put(0.0,8.6){$z_1$}
\put(0.0,11.8){$z_1$}
\put(4.7,0.2){$t [s]$}
\put(14.3,0.15){$t [s]$}
\put(9.2,5.2){\rotatebox{90}{$L(z_1) - L^*$}}
\put(4,3.6){{\scriptsize $\sigma = 25$}}
\put(14,3.6){{\scriptsize $\sigma = 25$}}
\put(4,6.7){{\scriptsize $\sigma = 20$}}
\put(14,6.65){{\scriptsize $\sigma = 20$}}
\put(4,10){{\scriptsize $\sigma = 10$}}
\put(14,9.9){{\scriptsize $\sigma = 10$}}
\put(4,13.1){{\scriptsize $\sigma = 5$}}
\put(14,13.05){{\scriptsize $\sigma = 5$}}
\end{picture}
\vspace{-0.25cm}
\caption{Simulation results for hybrid closed-loop algorithm $\HS$, for a function $L(z_1) := z_1^2$ with a single minimizer at $\minSet = 0$, with zero-mean Gaussian noise added to measurements of the gradient. Each subplot is labeled with the standard deviation used. Left subplots: the value of $z_1$ over time for each perturbed solution, with the jump in each solution labeled by an asterisk. Right subplots: the corresponding value of $L$ over time for each perturbed solution.}
\label{fig:NoisyLog}
\end{figure}}{
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
%
\begin{picture}(30.2,20.7)(0,0)
\footnotesize
\put(0.2,0.5){\includegraphics[scale=0.3,trim={0.5cm 0.4cm 0.8cm 0.4cm},clip,width=14\unitlength]{Figures/PlotsNoisy.eps}}
\put(16,0.5){\includegraphics[scale=0.3,trim={0.5cm 0.4cm 0.7cm 0.4cm},clip,width=14\unitlength]{Figures/PlotsNoisyLog.eps}}
\put(0.0,3.4){$z_1$}
\put(0.0,8.25){$z_1$}
\put(0.0,13.3){$z_1$}
\put(0.0,18.2){$z_1$}
\put(7.1,0.35){$t [s]$}
\put(22.9,0.3){$t [s]$}
\put(15,1.4){\rotatebox{90}{$L(z_1) - L^*$}}
\put(15,6.3){\rotatebox{90}{$L(z_1) - L^*$}}
\put(15,11.2){\rotatebox{90}{$L(z_1) - L^*$}}
\put(15,16.1){\rotatebox{90}{$L(z_1) - L^*$}}
\put(6.8,5.3){$\sigma = 25$}
\put(22.6,5.3){$\sigma = 25$}
\put(6.6,10.2){$\sigma = 20$}
\put(22.4,10.2){$\sigma = 20$}
\put(6.6,15.2){$\sigma = 10$}
\put(22.4,15.2){$\sigma = 10$}
\put(6.4,20.1){$\sigma = 5$}
\put(22.2,20.1){$\sigma = 5$}
\end{picture}
\caption{Simulation results for hybrid closed-loop algorithm $\HS$, for a function $L(z_1) := z_1^2$ with a single minimizer at $\minSet = 0$, with zero-mean Gaussian noise added to measurements of the gradient. Each subplot is labeled with the standard deviation used. Left subplots: the value of $z_1$ over time for each perturbed solution, with the jump in each solution labeled by an asterisk. Right subplots: the corresponding value of $L$ over time for each perturbed solution.}
\label{fig:NoisyLogTR}
\end{figure}}
To show that the uniform global asymptotic stability of $\mathcal{A}$, established in Theorem \ref{thm:GASNestNSC}, is robust to small perturbations, due to the hybrid closed-loop system $\HS$ satisfying the hybrid basic conditions by Lemma \ref{lemma:HBC}. we simulate the hybrid algorithm, using the objective function, parameter values, and initial conditions listed in the first paragraph of this example, with zero-mean Gaussian noise added to measurements of the gradient.
Separate simulations were run for each of the following standard deviations: $\sigma \in \{0.01,0.1,0.5,1,5,10,15,20,25\}$. \IfConf{Fig. \ref{fig:NoisyLog}}{Figure \ref{fig:NoisyLogTR}} shows some of these perturbed solutions, with each subplot labeled with the corresponding standard deviation used\footnote{Code found at same link as in Footnote \ref{foot:RobustnessURL}.}. The subplots on the left side of \IfConf{Fig. \ref{fig:NoisyLog}}{Figure \ref{fig:NoisyLogTR}} show the value of $z_1$ over time for different standard deviations, and the subplots on the right side of \IfConf{Fig. \ref{fig:NoisyLog}}{Figure \ref{fig:NoisyLogTR}} show the corresponding value of $L$ over time for such standard deviations. Note that, while all perturbed solutions shown in \IfConf{Fig. \ref{fig:NoisyLog}}{Figure \ref{fig:NoisyLogTR}} get close to the minimizer quickly, such perturbed solutions do not get as close to the minimizer as the solution to the nominal algorithm does; see the plot on the \IfConf{top}{top} in \IfConf{Fig. \ref{fig:NominalNoise}}{Figure \ref{fig:NominalNoise_TR}}. Also note that as the standard deviation gets larger, the corresponding perturbed solution stays slightly farther away from the minimizer. The results for all standard deviations are listed in the table \IfConf{on the bottom in}{in} \IfConf{Fig. \ref{fig:NominalNoise}}{Figure \ref{fig:NominalNoise_TR}}, showing the neighborhood of $\minSet$ that each solution settles to, for a large value of $t+j$, along with the corresponding value of $L$.
\IfConf{\end{exmp}}{\end{example}}
\IfConf{\begin{exmp}}{\begin{example}} \label{ex:NSC}
In this example, to show the effectiveness of the uniting algorithm, we compare the hybrid closed-loop algorithm $\HS$, defined via \eqref{eqn:HS-TimeVarying} and \eqref{eqn:CAndDGradientsNestNSC}, with the individual closed-loop optimization algorithms $\HS_0$ and $\HS_1$ and with the HAND-1 algorithm from \cite{poveda2019inducing} which, in \cite{poveda2019inducing}, is designed and analyzed for nonstrongly convex functions $L$ satisfying Assumptions \ref{ass:LisNSCVX} and \ref{ass:Lipschitz}. \IfConf{The bound for HAND-1 is $L(z_1(t,0)) - L^* \leq \frac{B}{t^2}$ for all $\left(t,j\right) \in \dom (z, \tau)$ such that $j = 0$, $z_1(0,0) = z_2(0,0)$, $\tau(0,0) = T_{\min}$, $z_1(0,0) \in K_0 := \{z_1^*\} + r\ball$, where $B := \frac{r^2}{2c_1} + T^2_{\min} \left(L(z_1(0,0)) - L^* \right) > 0$, $r \in \reals_{> 0}$, $c_1 > 0$. Such a rate is only guaranteed until the first jump.}{First, we compare the convergence rates of $\HS$ and HAND-1 analytically. Using an alternate state space representation, namely, $z_1 := \xi$ and $z_2 := \xi + \frac{\tau}{2}\dot{\xi}$, the HAND-1 algorithm has state $(z, \tau) \in \reals^{2n+1}$ and data $(C,F,D,G)$
\begin{equation} \label{eqn:HANDFAndG}
F(z, \tau) := \matt{\frac{2}{\tau}(z_2 - z_1)\\-2c_1\tau \nabla L(z_1) \\1} \ \forall (z,\tau) \in C, \quad G(z, \tau) := \matt{z\\mathcal{T}_{\min}} \ \forall (z,\tau) \in D
\end{equation}
where $c_1 > 0$ and the flow and jump sets are\\ $C := \defset{(z, \tau) \in \reals^{2n+1}\!\!\!}{\!\!\!\tau \in [T_{\min}, T_{\max}]\!\!}$ and\\ $D := \defset{(z, \tau) \in \reals^{2n+1}\!\!\!}{\!\!\!\tau \in [T_{\text{med}},T_{\max}]\!\!}$,
with $0 < T_{\min} < T_{\text{med}} < T_{\max} < \infty$, and $T_{\text{med}} \geq \sqrt{\frac{B}{\delta_{\text{med}}}} + T_{\min} > 0$, $\delta_{\text{med}} > 0$. It is shown in \cite{poveda2019inducing} that each maximal solution $(t,j) \mapsto (z(t,j),\tau(t,j))$ to the HAND-1 algorithm satisfies
\begin{equation}\label{eqn:PNRateHAND-1}
L(z_1(t,0)) - L^* \leq \frac{B}{t^2}
\end{equation}
for all $\left(t,j\right) \in \dom (z, \tau)$ such that $j = 0$, $z_1(0,0) = z_2(0,0)$, $\tau(0,0) = T_{\min}$, $z_1(0,0) \in K_0 := \{z_1^*\} + r\ball$, where $B := \frac{r^2}{2c_1} + T^2_{\min} \left(L(z_1(0,0)) - L^* \right) > 0$, $r \in \reals_{> 0}$, $c_1 > 0$.
For the hybrid closed-loop algorithm $\HS$, the coefficient of the bound on $\HS_1$ from \eqref{eqn:UnitingConvergenceRateNSCHS1}, namely,
\begin{equation} \label{eqn:BoundH1Again}
L(z_1(t,0)) - L^* \leq \frac{9cM}{\zeta^2(t+2)^2} \left(\left|z_1(0,0) - z_1^* \right|^2 + \left| z_2(0,0) \right|^2 \right)
\end{equation}
for each $t \in I^0$, $t \geq 1$, at which $q(t,0) = 1$, and for each $\zeta > 0$, and $M > 0$, is $\frac{9cM}{\zeta^2}\left(\left|z_1(0,0) - z_1^* \right|^2 + \left| z_2(0,0) \right|^2 \right)$, where $c := \left(1 + \zeta^2\right)\exp \left(\sqrt{\frac{13}{4} + \frac{\zeta^4}{M}}\right)$. The coefficient of the bound in HAND-1 is $B := \frac{r^2}{2c_1} + T^2_{\min} \left(L(z_1(0,0)) - L^* \right)$. Since, as $t \rightarrow \infty$, $\frac{1}{(t+2)^2} \rightarrow \frac{1}{t^2}$, then, comparing the coefficients of the bounds, the bound in \eqref{eqn:BoundH1Again} is slightly better than \eqref{eqn:PNRateHAND-1} since $\frac{r^2}{2c_1}$ is very large for small $t$. Neglecting the $\frac{r^2}{2c_1}$ term, however, the bound on $\HS_1$ \eqref{eqn:BoundH1Again} matches \eqref{eqn:PNRateHAND-1}. The rate for HAND-1, nevertheless, is only guaranteed until the first jump. After this, there is no characterized bound for HAND-1. In contrast, $\HS$ has a characterized bound for the domain of every solution such that $t \geq 1$. Namely, it has rate $\frac{1}{(t+2)^2}$ until the state $z$ is within a small neighborhood of the minimizer -- where the rate then switches to $\exp \left(-(1-m)\psi t\right)$, where, given $\gamma > 0$ and $\lambda > 0$, $m \in (0,1)$ is such that $\psi = \frac{m\alpha\gamma}{\lambda} > 0$ and $\nu = \psi (\psi - \lambda) < 0$.}
Next, we compare $\HS_0$, $\HS_1$, $\HS$, and HAND-1 in simulation. To compare these algorithms, we use the same objective function $L$, heavy ball parameter values $\lambda$ and $\gamma$, Lipschitz parameter $M$, Nesterov parameter $\zeta$,
and uniting algorithm parameter values $c_0$, $c_{1,0}$, $\varepsilon_0$, $\varepsilon_{1,0}$, $\alpha$, $\tilde{c}_0$, $\tilde{c}_{1,0}$, $d_0$, and $d_{1,0}$ as in Example \ref{ex:Robustness}. Given $\zeta = 2$, the HAND-1 parameters $c_1 = 0.5$ and $T_{\min} = \frac{1+\sqrt{7}}{2}$ are chosen such that the resulting gain coefficients for $z_1$ and $z_2$ are the same for both $\HS$ and HAND-1, so that these algorithms are compared on equal footing\footnote{Although there exist parameter values for which HAND-1 has faster, oscillation-free performance, due to the way $\HS$ and HAND-1 relate to each other, they are compared fairly for a particular set of parameters.}. The remaining HAND-1 parameters, $r$ and $\delta_{\text{med}}$, have different values depending on the initial conditions $z_1(0,0) = z_2(0,0)$, listed in \IfConf{\cite[Table~2]{dhustigs2022unitingNSC}}{Table \ref{table:PercentImprovementNSCTR}}, which leads to different values of $T_{\text{med}}$ and $T_{\max}$, for each solution. Such values are chosen such that $T_{\text{med}} \geq \sqrt{\frac{B}{\delta_{\text{med}}}} + T_{\min} > 0$. Additionally, we choose $T_{\max} = T_{\text{med}} + 1$. The parameter values for the uniting algorithm are $\varepsilon_0 = 10$, $\varepsilon_{1,0} = 5$, and $\alpha = 1$. The remaining parameter values $c_0$ and $c_{1,0}$ are different depending on the initial condition $z_1(0,0)$ and are listed in \IfConf{\cite[Table~2]{dhustigs2022unitingNSC}}{Table \ref{table:PercentImprovementNSCTR}}, which leads to different values of which leads to different values of $d_0$, calculated via \eqref{eqn:UTilde0SetEquations}, and $d_{1,0}$ calculated via \eqref{eqn:TTilde10SetEquations}. These values are chosen for proper tuning of the algorithm, in order to get nice performance, and for exploiting the properties of Nesterov's method as long as we want. Initial conditions for all solutions to $\HS$ are $z_2(0,0) = 0$, $q(0,0) = 1$, and $\tau(0,0) = 0$, with values of $z_1(0,0)$ listed in \IfConf{\cite[Table~2]{dhustigs2022unitingNSC}}{Table \ref{table:PercentImprovementNSCTR}}. Initial conditions for all solutions to HAND-1 are $\tau(0,0) = T_{\min}$, with values of $z_1(0,0) = z_2(0,0)$ listed in \IfConf{\cite[Table~2]{dhustigs2022unitingNSC}}{Table \ref{table:PercentImprovementNSCTR}}.
\begin{table}[thpb]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
& { Average time} & { Average $\%$}\\
Algorithm & { to converge (s)} & { improvement} \\
\hline
\hline
$\HS$ & $0.811$ & -- \\
\hline
$\HS_0$ & $690.759$ & $99.9$ \\
\hline
$\HS_1$ & $4.409$ & $81.6$ \\
\hline
HAND-1 & $8.649$ & $90.6$ \\
\hline
\end{tabular}
\IfConf{
\caption{Average times for which $\HS$, $\HS_0$, $\HS_1$, and HAND-1 settle to within $1\%$ of $\minSet$, and the average percent improvement of $\HS$ over each algorithm. Percent improvement is calculated via \eqref{eqn:PercentImprovement}. The objective function used for this table is $L(z_1) := z_1^2$.}
}{
\caption{Average times for which $\HS$, $\HS_0$, $\HS_1$, and HAND-1 settle to within $1\%$ of $\minSet$, and the average percent improvement of $\HS$ over each algorithm. Percent improvement is calculated via \eqref{eqn:PercentImprovement}. The objective function used for this table is $L(z_1) := z_1^2$.}
}
\label{table:TimePercentImpNSC}
\end{center}
\end{table}
\IfConf{Table \ref{table:TimePercentImpNSC}}{Table \ref{table:TimePercentImpNSC}} shows the time that each algorithm takes to settle within\footnote{Code at \texttt{gitHub.com/HybridSystemsLab/UnitingNSC}} $1\%$ of $\minSet$, averaged over solutions starting from ten different values\footnote{Code at\IfConf{\\}{} \texttt{gitHub.com/HybridSystemsLab/UnitingDifferentICs}\label{foot:DifferentICsCode}} of $z_1(0,0)$ (listed in the first column of \IfConf{\cite[Table~2]{dhustigs2022unitingNSC}}{Table \ref{table:PercentImprovementNSCTR}}), and the average percent improvement of $\HS$ over $\HS_0$, $\HS_1$, and HAND-1, which is calculated using the following formula
\begin{equation} \label{eqn:PercentImprovement}
\left(\frac{\text{Time of } \HS_0, \HS_1, \text{ or HAND-1} - \text{Time of } \HS}{\text{Time of } \HS_0, \HS_1, \text{ or HAND-1}}\right) \times 100 \% .
\end{equation}
As can be seen in
\IfConf{Table \ref{table:TimePercentImpNSC}}{Table \ref{table:TimePercentImpNSC}}, $\HS$ converges faster than the other algorithms, and the average percent improvement of $\HS$ over each of the other algorithms in \IfConf{Table \ref{table:TimePercentImpNSC}}{Table \ref{table:TimePercentImpNSC}} is $99.9\%$ over $\HS_0$, $81.6\%$ over $\HS_1$, and $90.6\%$ over HAND-1. \IfConf{
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
%
\begin{picture}(20,9.5)(0,0)
\footnotesize
\put(0,0.5){\includegraphics[scale=0.4,trim={0.7cm 0.1cm 1cm 0.3cm},clip,width=20\unitlength]{Figures/UnitingNSCTrajectoriesLog.eps}}
\put(0,3.3){\rotatebox{90}{$L(z_1) - L^*$}}
\put(5.3,0.2){$t[s]$}
\put(15,0.2){$t[s]$}
\end{picture}
\vspace{-0.25cm}
\caption{The evolution of $L$ over time, from different initial conditions, for $\HS$ (left) and HAND-1 (right). All solutions are for the objective function $L(z_1) := z_1^2$, and the parameters used for HAND-1 and $\HS$ are listed in \cite[Table~2]{dhustigs2022unitingNSC}, with different values of $c_0$
and $c_{1,0}$ for each solution of $\HS$, leading to different values of $d_0$ calculated via \eqref{eqn:UTilde0SetEquations} and $d_{1,0}$ calculated via \eqref{eqn:TTilde10SetEquations}, and different values of $r$ and $\delta_{\text{med}}$ for each solution of HAND-1, leading to different values of $T_{\text{med}}$ and $T_{\max}$.}
\label{fig:NSCTrajectoriesPlotsLog}
\end{figure}
}{
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
%
\begin{picture}(30.8,15)(0,0)
\footnotesize
\put(0,0.5){\includegraphics[scale=0.4,trim={0.8cm 0.1cm 1cm 0.3cm},clip,width=30.8\unitlength]{Figures/UnitingNSCTrajectoriesLog.eps}}
\put(0,6){\rotatebox{90}{$L(z_1) - L^*$}}
\put(8.3,0.4){$t[s]$}
\put(23.3,0.4){$t[s]$}
\end{picture}
\caption{The evolution of $L$ over time, from different initial conditions, for $\HS$ (left) and HAND-1 (right). All solutions are for the objective function $L(z_1) := z_1^2$, and the parameters used for HAND-1 and $\HS$ are listed in Table \ref{table:PercentImprovementNSCTR}, with different values of $c_0$
and $c_{1,0}$ for each solution of $\HS$, leading to different values of $d_0$ calculated via \eqref{eqn:UTilde0SetEquations} and $d_{1,0}$ calculated via \eqref{eqn:TTilde10SetEquations}, and different values of $r$ and $\delta_{\text{med}}$ for each solution of HAND-1, leading to different values of $T_{\text{med}}$ and $T_{\max}$.}
\label{fig:NSCTrajectoriesPlotsLogTR}
\end{figure}}
\IfConf{}{
\begin{table}[thpb]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
& & & & & \multicolumn{2}{|c|}{{\footnotesize Time to converge (s)\par}} & {\footnotesize$\%$ Improve-}\\
\cline{6-7}
$z_1(0,0)$ & $c_0$ & $c_{1,0}$ & $r$ & $\delta_{\text{med}}$ & $\HS$ & HAND-1 & {\footnotesize ment} \\
\hline
\hline
$110$ & $34000$ & $32719.231$ & $111$ & $240700$ & 0.811 & 8.649 & 90.6 \\
\hline
$100$ & $28000$ & $27053.704$ & $101$ & $199000$ & 0.811 & 8.65 & 90.6 \\
\hline
$90$ & $23000$ & $21927.75$ & $91$ & $161300$ & 0.811 & 8.648 & 90.6 \\
\hline
$80$ & $18000$ & $17341.37$ & $81$ & $127550$ & 0.811 & 8.65 & 90.6 \\
\hline
$70$ & $14000$ & $13294.565$ & $71$ & $97700$ & 0.811 & 8.649 & 90.6 \\
\hline
$60$ & $10500$ & $9787.333$ & $61$ & $71875$ & 0.811 & 8.648 & 90.6 \\
\hline
$50$ & $7000$ & $6819.676$ & $51$ & $50000$ & 0.810 & 8.65 & 90.6 \\
\hline
$40$ & $5000$ & $4391.593$ & $41$ & $32075$ & 0.811 & 8.65 & 90.6 \\
\hline
$30$ & $3000$ & $2503.083$ & $31$ & $18110$ & 0.811 & 8.648 & 90.6 \\
\hline
$20$ & $2000$ & $1154.148$ & $21$ & $8112$ & 0.811 & 8.648 & 90.6 \\
\hline
\end{tabular}
\caption{Times for which $\HS$ and HAND-1 settle to within $1\%$ of $\minSet$, and percent improvement of $\HS$ over HAND-1, for solutions from different initial conditions, shown in Figure \ref{fig:NSCTrajectoriesPlotsLogTR}. The objective function used for this table is $L(z_1) := z_1^2$.}
\label{table:PercentImprovementNSCTR}
\end{center}
\end{table}
}
\IfConf{Fig. \ref{fig:NSCTrajectoriesPlotsLog}}{Figure \ref{fig:NSCTrajectoriesPlotsLogTR}} compares different solutions for $\HS$ and HAND-1, from different values of $z_1(0,0)$,
for the objective function\footnote{Code found at same link as in Footnote \ref{foot:DifferentICsCode}.} $L(z_1) := z_1^2$. \IfConf{\cite[Table~2]{dhustigs2022unitingNSC}}{Table \ref{table:PercentImprovementNSCTR}} lists the times for which each solution settles to within $1\%$ of $\minSet$ for both $\HS$ and HAND-1, and shows the percent improvement of $\HS$ over HAND-1.
As can be seen in \IfConf{Fig. \ref{fig:NSCTrajectoriesPlotsLog}}{Figure \ref{fig:NSCTrajectoriesPlotsLogTR}} and in \IfConf{\cite[Table~2]{dhustigs2022unitingNSC}}{Table \ref{table:PercentImprovementNSCTR}}, the percent improvement of $\HS$ over HAND-1 for all solutions is $90.6\%$, which shows consistency in the performance of $\HS$ versus HAND-1.
\IfConf{}{The bound for HAND-1, shown in \eqref{eqn:PNRateHAND-1} and which holds only until the first reset, is only guaranteed when $z_1(0,0) = z_2(0,0)$. This leads to a required nonzero velocity for HAND-1 in most scenarios, which leads to overshoot. In contrast, $\HS$ has no such constraint on $z_2(0,0)$, which can be set to zero in all scenarios. The lack of such a constraint on the initial condition $z_2(0,0)$ for the hybrid closed-loop algorithm $\HS$ is essential to its improved performance over HAND-1, as the overshoot in solutions to HAND-1 due to $z_1(0,0) = z_2(0,0)$ leads to a slower convergence time than for $\HS$, as seen in Table \ref{table:TimePercentImpNSC}. Moreover, as described previously in this example, no bound for HAND-1 is characterized after the first reset, whereas the (hybrid) convergence bound characterized for $\HS$ holds for the domain of every solution such that $t \geq 1$.}
\IfConf{\end{exmp}}{\end{example}}
\IfConf{
\vspace{-0.2cm}
For an illustration of the trade-off between speed of convergence and the resulting values of parameters for the uniting algorithm $\HS$, for different values of $\zeta > 0$, see \cite[Example~4.3]{dhustigs2022unitingNSC}.
\vspace{-0.15cm}
}{
\begin{example} \label{ex:tradeOff}
This example explores the trade-off that results from using different values of $\zeta > 0$ for the uniting algorithm. Particularly, for $\zeta = 1$, we first compare the uniting algorithm in simulation with the individual optimization algorithms $\HS_0$, $\HS_1$, and the HAND-1 algorithm from \cite{poveda2019inducing}, using the same objective function as in Example \ref{ex:NSC}, and next we compare the resulting solutions with those in Table \ref{table:TimePercentImpNSC}. Recall that the objective function in Example \ref{ex:NSC} is $L(z_1) := z_1^2$, the gradient of which is Lipschitz continuous with $M = 2$, and which has a single minimizer at $\minSet = 0$. Since the gain coefficient of $\nabla L$ is proportional to $\zeta^2$, we choose different parameters for the HAND-1 algorithm for the simulation depicted in\footnote{Code found at same link as in Footnote \IfConf{\ref{foot:Tradeoff}}{\ref{foot:Tradeoff_TR}}} Figure \ref{fig:NSCLargerZetaTR}, so that the gain coefficients of $z_1$ and $z_2$ are the same for HAND-1 and $\HS$ in this simulation. Namely, given $\zeta = 1$, for HAND-1 we choose $T_{\min} = 3$ and $c_1 = 0.25$. For the other HAND-1 parameters, we choose $r = 51$ and $\delta_{\text{med}} = 4010$ such that $T_{\text{med}} \geq \sqrt{\frac{B}{\delta_{\text{med}}}} + T_{\min} > 0$, and we again choose $T_{\max} = T_{\text{med}} + 1$ to ensure resets happen at the proper times.
We arbitrarily choose $\gamma = \frac{2}{3}$, and we tuned $\lambda$ to $40$ by choosing a value arbitrarily larger than $2\sqrt{a_1}$ and gradually increasing until there was no overshoot in the hybrid algorithm.
The uniting algorithm parameters are $c_0 = 320$, $c_{1,0} \approx 271.584$, $\varepsilon_0 = 10$, $\varepsilon_{1,0} = 5$, and $\alpha = 1$, which yield the values $\tilde{c}_0 = 10$, $\tilde{c}_{1,0} = 5$, $d_0 \approx 253.333$, and $d_{1,0} \approx 234.084$, which are calculated via \eqref{eqn:UTilde0SetEquations} and \eqref{eqn:TTilde10SetEquations}. These values are chosen for proper tuning of the algorithm, in order to get nice performance, and for exploiting the properties of Nesterov's method as long as we want. Initial conditions for $\HS$ are $z_1(0,0) = 50$, $z_2(0,0) = 0$, $q(0,0) = 1$, and $\tau(0,0) = 0$, and for HAND-1 are $z_1(0,0) = z_2(0,0) = 50$ and $\tau(0,0) = T_{\min}$.
First, we compare solutions to each algorithm within Figure \ref{fig:NSCLargerZetaTR} itself.
Table \ref{table:TimePercentImpNSCZeta1} shows the time that each algorithm takes to settle within $1\%$ of $\minSet$, averaged over solutions starting from ten different values of $z_1(0,0)$ (listed in the first column of Table \ref{table:PercentImprovementNSCTR}), and the percent improvement of $\HS$ over $\HS_0$, $\HS_1$, and HAND-1, which is calculated using \eqref{eqn:PercentImprovement}.
While the closed-loop algorithm $\HS$ still converges faster than all the other algorithms in Figure \ref{fig:NSCLargerZetaTR} and Table \ref{table:TimePercentImpNSCZeta1}, the improvement over $\HS_0$, $\HS_1$, and HAND-1 is smaller than it is in Table \ref{table:TimePercentImpNSC}.
\begin{table}[thpb]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Algorithm & Average time to converge (s) & Average $\%$ improvement \\
\hline
\hline
$\HS$ & $2.387$ & -- \\
\hline
$\HS_0$ & $138.066$ & $98.3$ \\
\hline
$\HS_1$ & $8.782$ & $72.8$ \\
\hline
HAND-1 & $14.343$ & $83.4$ \\
\hline
\end{tabular}
\caption{Times for which $\HS$, $\HS_0$, $\HS_1$, and HAND-1 settle to within $1\%$ of $\minSet$, and percent improvement of $\HS$ over each algorithm, as shown in Figure \ref{fig:NSCLargerZetaTR}. Percent improvement is calculated via \eqref{eqn:PercentImprovement}. The objective function used for this table is $L(z_1) := z_1^2$.}
\label{table:TimePercentImpNSCZeta1}
\end{center}
\end{table}
Next, we compare solutions using $\zeta = 1$, in Figure \ref{fig:NSCLargerZetaTR}, with solutions using $\zeta = 2$, in Table \ref{table:TimePercentImpNSC}. Since $\zeta > 0$ scales time in solutions to \IfConf{\eqref{eqn:MJODE_ZetaNum}}{\eqref{eqn:MJODE_ZetaNum_TR}}, Then smaller values of $\zeta$ result in slower settling to within $1\%$ of $\minSet$ for $\HS_1$ with less frequent oscillations, as seen in Figure \ref{fig:NSCLargerZetaTR} with $\zeta = 1$ (about $8.8$ seconds), while larger values of $\zeta$ result in settling to within $1\%$ of $\minSet$ for $\HS_1$ faster, with more frequent oscillations, as seen in Figure \ref{fig:MotivationalPlotTR} and Table \ref{table:TimePercentImpNSC} with $\zeta = 2$ (about $4.5$ seconds). For the uniting algorithm, this translates to faster settling to within $1\%$ of $\minSet$ with $\zeta = 2$ (about $0.8$ seconds), in Figure \ref{fig:MotivationalPlotTR} and Table \ref{table:TimePercentImpNSC}, compared with slower settling to within $1\%$ of $\minSet$ with $\zeta = 1$ (about $2.4$ seconds), in Figure \ref{fig:NSCLargerZetaTR}, but with no oscillations, in both cases, due to the switch to $\HS_0$. In both Figure \ref{fig:NSCLargerZetaTR}, and Table \ref{table:TimePercentImpNSC}, the uniting algorithm converges more quickly than the HAND-1 algorithm, when both algorithms are tuned to have the same gain coefficients for the $z_1$ and $z_2$ terms. Although larger $\zeta$ results in faster convergence, the trade-off is that even though the $z_2$ (velocity) term generally reduces quickly as it approaches the neighborhood of the minimizer for any size of $\zeta$, the $z_2$ still ends up relatively larger near the minimizer than it is when $\zeta$ is smaller. The consequence is that, when $\zeta$ is larger, $d_{1,0}$ needs to be set much larger so that the uniting algorithm can still make the switch to $\HS_0$ at the proper time. This also means that $c_{1,0}$ needs to be set much larger, due to the definition of $d_{1,0}$ in \eqref{eqn:TTilde10SetEquations}. Additionally, $c_0$ and $d_0$, also need to be set larger to ensure the algorithm still has adequate hysteresis. Recall that, in Example \ref{ex:NSC}, for $\zeta = 2$, we have the parameter values $c_0 = 7000$, $c_{1,0} \approx 6819.676$, $d_0 = 6933$, and $d_{1,0} = 6744$, which are quite large, while for the simulation shown in Figure \ref{fig:NSCLargerZetaTR} these same parameters have much smaller values, as listed in the second paragraph of this example.
\end{example}}
\subsection{Uniting Other Gradient Algorithms}
In \Cref{sec:HBF-HBF,sec:NonstronglyConvexNest}, we illustrated the framework in \Cref{sec:ModelingUF} using specific cases involving different combinations of the heavy ball algorithm and Nesterov's algorithm, for $\kappa_0$ and $\kappa_1$. Other gradient-based algorithms, however, could be used as $\kappa_0$ and $\kappa_1$ for the general framework. One example includes classic gradient descent, which has the ODE
\begin{equation}
\dot{\xi} + \gamma \nabla L(\xi) = 0
\end{equation}
where $\gamma > 0$ is tunable, and which is commonly known to have a convergence rate of $\frac{1}{t}$. As gradient descent does not have a ``velocity'' term, it tends to converge slowly, without oscillations near the minimizer -- similar behavior as seen for heavy ball with large $\lambda$. Due to this behavior, gradient descent could be used as the local algorithm $\kappa_0$ in the general framework in \Cref{sec:ModelingUF}, with either heavy ball with small $\lambda$ or Nesterov's algorithm as $\kappa_1$.
Another example includes the triple momentum method. The triple momentum method was first proposed in \cite{van2017fastest} as a discrete-time accelerated gradient method. A characterization of the continuous-time, high-resolution dynamical system, derived in \cite{sun2020high}, is
\begin{subequations} \label{eqn:TMODE}
\begin{align}
& \ddot{\xi} + 2\sqrt{\eta \left(\gamma,\lambda \right)}\dot{\xi} + \left(1+\sqrt{\eta\left(\gamma,\lambda \right) \gamma}\right) \nabla L(w) = 0\\
& w := \xi + \sqrt{\gamma}\sigma \dot{\xi}\\
& y := \xi + \sqrt{\gamma}\delta \dot{\xi}
\end{align}
\end{subequations}
where $y \in \reals^n$ is the output, where the gradient is applied to $w \in \reals^n$, where $\gamma > 0$, $\lambda > 0$, $\sigma > 0$, and $\delta > 0$ are tunable parameters, and where $\eta(\gamma, \lambda)$ is
\begin{equation}
\eta(\gamma, \lambda) := \left(\frac{1-\lambda}{\sqrt{\gamma}\left(1+\lambda \right)}\right)^2 \in (0,M].
\end{equation}
where $M > 0$ is the Lipschitz constant of $\nabla L$.
The authors in \cite{van2017fastest} and \cite{sun2020high} also give an ideal tuning of of the parameters as follows:
\begin{equation} \label{eqn:IdealTMTuning}
\left(\gamma,\lambda,\sigma,\delta \right) := \left(\frac{1+\rho}{M}, \frac{\rho^2}{2-\rho}, \frac{\rho^2}{\left(1+\rho\right)\left(2-\rho\right)}, \frac{\rho^2}{1-\rho^2}\right)
\end{equation}
where
\begin{equation}
\rho := 1 - \frac{1}{\kappa}
\end{equation}
where $\kappa := \frac{M}{\mu}$, $\mu > 0$, is the condition number of $L$. The authors in \cite{sun2020high} characterize the convergence rate for \cref{eqn:TMODE} to be exponential, both for the general parameters and for the optimal tuning in \cref{eqn:IdealTMTuning}, and numerically found \cref{eqn:TMODE} to converge more quickly than Nesterov's algorithm, for given values of the condition number $\kappa$. Since \cref{eqn:TMODE} has an exponential convergence rate, it would be ideal for use as the global optimization algorithm $\kappa_1$ in the general framework, defined in \cref{sec:ModelingUF}, with either gradient descent or heavy ball with large $\lambda$ as $\kappa_0$.
\section{Preliminaries}
\label{sec:Preliminaries}
\subsection{Hybrid Systems}
In this paper, {we use the hybrid systems framework to design our proposed uniting algorithm since such a framework allows for the combination of continuous-time behavior, such as behavior of the individual heavy ball and Nesterov methods, with discrete-time events, such as the switch between these two methods.} A hybrid system $\HS$ has data $(C,F,D,G)$ and is defined as \cite[Definition~2.2]{65} \IfConf{
\vspace{-0.6cm}
}{}
\begin{equation}\label{eqn:GeneralH}
\HS =
\begin{cases}
\dot{x} & \!\!\!\!\! \in F(x) \quad x \in C \\
x^+ & \!\!\!\!\! \in G(x) \quad x \in D
\end{cases}
\end{equation}
where $x \in \reals^n$ is the system state, $F : \reals^n \rightrightarrows \reals^n$ is the flow map, $C \subset \reals^n$ is the flow set, $G : \reals^n \rightrightarrows \reals^n$ is the jump map, and $D \subset \reals^n$ is the jump set. The notation $\rightrightarrows$ indicates that $F$ and $G$ are set-valued maps. A solution $x$ to $\HS$ is parameterized by $(t,j) \in \reals_{\geq 0} \times \naturals$, where $t$ is the amount of time that has passed and $j$ is the number of jumps that have occurred. The domain of $x$, namely, $\mathrm{dom} x \subset \reals_{\geq 0} \times \naturals$, is a hybrid time domain, which is a set such that for each $(T,J) \in \mathrm{dom} x$, $\mathrm{dom} x \cap \left( \left[0,T \right] \times \{0,1,\ldots, J\} \right) = \cup_{j=0}^{J} ([t_j,t_{j+1}],j)$ for a finite sequence of times $0 = t_0 \leq t_1 \leq t_2 \leq \ldots \leq t_{J+1}$. A hybrid arc $x$ is a function on a hybrid time domain that, for each $j \in \naturals$, $t \mapsto x(t,j)$ is locally absolutely continuous on the interval $I^j := \defset{t\!\!}{\!\!(t,j) \in \dom x\!\!}$. A solution $x$ to $\HS$ is called maximal if it cannot be extended further. The set $\mathcal{S}_{\HS}$ contains all maximal solutions to $\HS$. A solution is called complete if its domain is unbounded.
The following definitions, from \cite{65} and \cite{220}, will be used in the analysis of the hybrid closed-loop system, obtained with the proposed hybrid control algorithm.
\IfConf{\begin{defn}[Hybrid basic conditions]}{\begin{definition}[Hybrid basic conditions]} \label{def:HBCs} A hybrid system $\HS$ is said to satisfy the hybrid basic conditions if its data $(C,F,D,G)$ is such that
\begin{enumerate}[label={(A\arabic*)},leftmargin=*]
\item $C$ and $D$ are closed subsets of $\reals^n$;
\item $F : \reals^n \rightrightarrows \reals^n$ is outer semicontinuous and locally bounded relative to $C$, $C \subset \mathrm{dom} \ F$, and $F(x)$ is convex for every $x \in C$;
\item \label{item:A3} $G : \reals^n \rightrightarrows \reals^n$ is outer semicontinuous and locally bounded relative to $D$, and $D \subset \mathrm{dom} \ G$.
\end{enumerate}
\IfConf{\end{defn}}{\end{definition}}
The notions of stability, uniform global stability, pre-attractivity, uniform global pre-attractivity, and uniform global pre-asymptotic stability (UGpAS) are listed in the following definition, from \cite{220} and \cite{65}.
\IfConf{\begin{defn}[Stability and attractivity notions]}{\begin{definition}[Stability and attractivity notions]} \label{def:UGpAS}
Given a hybrid \IfConf{closed-loop}{\\closed-loop} system $\HS$ as in \eqref{eqn:GeneralH}, a nonempty set $\mathcal{A} \subset \reals^n$ is said to be
\begin{itemize}
\item \underline{Stable} for $\HS$ if for each $\eps > 0$ there exists $\delta > 0$ such that each solution $x$ to $\HS$ with $\left|x(0,0)\right|_{\mathcal{A}} \leq \delta$ satisfies $\left|x(t,j)\right|_{\mathcal{A}} \leq \eps$ for all $(t,j) \in \dom x$;
\item \underline{Uniformly globally stable} for $\HS$ if there exists a class-$\mathcal{K}_{\infty}$ function $\alpha$ such that any solution $x$ to $\HS$ satisfies $\left|x(t,j)\right|_{\mathcal{A}} \leq \alpha\left(\left|x(0,0)\right|_{\mathcal{A}}\right)$ for all $(t,j) \in \dom x$;
\item \underline{Pre-attractive} for $\HS$ if there exists $\mu > 0$ such that every solution $x$ to $\HS$ with $\left|x(0,0)\right|_{\mathcal{A}} \leq \mu$ is such that $(t,j) \mapsto \left|x(t,j)\right|_{\mathcal{A}}$ is bounded and if $x$ is complete then $\lim\limits_{(t,j) \in \dom x, \ t+j \rightarrow \infty} \left|x(t,j)\right|_{\mathcal{A}} = 0$;
\item \underline{Uniformly globally pre-attractive} for $\HS$ if for each $\eps > 0$ and $\delta > 0$ there exists $T > 0$ such that, for any solution $x$ to $\HS$ with $\left|x(0,0)\right|_{\mathcal{A}} \leq \delta$, $(t,j) \in \dom x$ and $t + j \leq T$ imply $\left|x(t,j)\right|_{\mathcal{A}} \leq \eps$;
\item \underline{Uniformly globally pre-asymptotically stable (UGpAS)} for $\HS$ if it is both\\ uniformly globally stable and uniformly globally pre-attractive.
\end{itemize}
\IfConf{\end{defn}}{\end{definition}}
In the notions involving convergence in Definition \ref{def:UGpAS}, when every maximal solution is complete, then the prefix ``pre'' is dropped to obtain attractivity, uniform global attractivity (UGA), and uniform global asymptotic stability (UGAS).
The prefix ``pre'' is in the notions involving convergence in Definition \ref{def:UGpAS} to allow for maximal solutions that are not complete. When every maximal solution is complete, such a property guarantees that nontrivial solutions exist from each initial point in $C \cup D$ to the hybrid system resulting from using our proposed uniting algorithm.
As was mentioned in Section \ref{sec:Background}, establishing UGAS for Nesterov's algorithm is a difficult problem to solve, due to its time-varying nature, as some solutions converge in a non-uniform way. We show in Section \ref{sec:ProofMainResult} that our proposed uniting algorithm overcomes such a difficulty. \IfConf{
\vspace{-0.1cm}
}{}
\section{Proof of Theorem \ref{thm:GASNestNSC}}
\label{sec:ProofMainResult}
This section provides a proof of Theorem \ref{thm:GASNestNSC} from Section \ref{sec:MainResult}. The proof consists of the following steps.
\begin{itemize}
\item Section \ref{sec:PropertiesOfH0} establishes uniform global asymptotic stability of $\{\minSet\} \times \{0\}$, and an exponential convergence rate for the closed-loop algorithm $\HS_0$;
\item Section \ref{sec:PropertiesOfH1} establishes uniform global attractivity of $\{\minSet\} \times \{0\} \times \reals_{\geq 0}$, and a convergence rate $\frac{1}{(t+2)^2}$ for the closed-loop algorithm $\HS_1$;
\item Section \ref{sec:UGAS} uses the properties in Sections \ref{sec:PropertiesOfH0} and \ref{sec:PropertiesOfH1} and a trajectory-based approach to prove uniform global asymptotic stability of $\mathcal{A}$, defined via \eqref{eqn:SetOfMinimizersHS-NSCVX}, for $\HS$;
\item Section \ref{sec:ConvRateH} proves the convergence rate of $\HS$ using the convergence rates of the individual closed-loop algorithms $\HS_0$ and $\HS_1$ established in Sections \ref{sec:PropertiesOfH0} and \ref{sec:PropertiesOfH1}, respectively.
\end{itemize}
\subsection{Properties of $\HS_0$}
\label{sec:PropertiesOfH0}
The following result establishes that the closed-loop algorithm $\HS_0$ in \eqref{eqn:H0} has the set $\{\minSet\} \times \{0\}$ uniformly globally asymptotically stable. To prove it, we use an invariance principle.
\IfConf{\begin{prop}[UGAS of $\{\minSet\} \times \{0\}$ for $\HS_0$]}{\begin{proposition}(Uniform global asymptotic stability of $\{\minSet\} \times \{0\}$ for $\HS_0$):} \label{prop:GAS-HBF}
Let $L$ satisfy Assumptions \ref{ass:LisNSCVX}, \ref{ass:QuadraticGrowth}, and \ref{ass:Lipschitz}. For each $\lambda > 0$ and $\gamma > 0$, the set $\{\minSet\} \times \{0\}$ is uniformly globally asymptotically stable for the closed-loop algorithm $\HS_0$ in \eqref{eqn:H0}.
\IfConf{\end{prop}}{\end{proposition}} \IfConf{
\vspace{-0.3cm}
}{}\IfConf{\begin{pf}}{\begin{proof}}
By Proposition \ref{prop:Existence}, each maximal solution to the closed-loop algorithm $\HS_0$, defined via \eqref{eqn:H0}, is bounded, complete, and unique. Recall that, in the proof of Proposition \ref{prop:Existence}, it was shown that $V_0$ in \eqref{eqn:LyapunovHBF} satisfies \eqref{eqn:VdotHBF} for all $z \in \reals^{2n}$, since $\lambda$ is positive. Therefore, by an application of \IfConf{\cite[Theorem~A.3]{dhustigs2022unitingNSC}}{Theorem \ref{thm:hybrid Lyapunov theorem}}, since $\gamma > 0$ and $\lambda > 0$, the set $\{\minSet\} \times \{0\}$ is stable for the closed-loop algorithm $\HS_0$.
Since by Lemma \ref{lemma:HBC} $\HS_0$ satisfies the hybrid basic conditions, then, using the invariance principle in \IfConf{\cite[Theorem~A.6]{dhustigs2022unitingNSC}}{Theorem \ref{thm:HybridInvariancePrinciple}}, every maximal solution that is complete and bounded approaches the largest weakly invariant set for $\HS_0$ in \eqref{eqn:H0} that is contained in
\begin{equation}\label{eqn:LaSalleSetVHBF}
\defset{z \in \reals^{2n}\!\!}{\!\!\dot{V}_0(z) = 0\!\!} \cap \defset{z \in \reals^{2n}\!\!}{\!\!V_0(z) = r\!\!}, \ r \geq 0.
\end{equation}
Such a set is nonempty only when $r = 0$ and, precisely, is equal to $\{\minSet\} \times \{0\}$. This property can be seen by noticing that
$\defset{z \in \reals^{2n}}{\dot{V}_0(z) = 0\!\!}
=
\defset{z \in \reals^{2n}}{z_2 = 0\!\!}$, and that after setting $z_2$ to zero in
\eqref{eqn:H0} we obtain $\matt{\dot{z}_1\\ 0\\} = \matt{0\\ -\gamma \nabla L(z_1)}$.
For any solution to this system, its $z_1$ component satisfies $0 = \gamma \nabla L(z_1)$, which, since $\gamma > 0$ and since $\nabla L(z_1) = 0$ only when $z_1$ is the minimizer of $L$,
leads to $z_1 = \minSet$. Then, the only maximal solution that starts and stays in \eqref{eqn:LaSalleSetVHBF} is the solution from $\{\minSet\} \times \{0\}$, for which $r = 0$. Then, every bounded and complete solution to the closed-loop algorithm $\HS_0$ converges to $\{\minSet\} \times \{0\}$. The arguments above involving the Lyapunov theorem in \IfConf{\cite[Theorem~A.3]{dhustigs2022unitingNSC}}{Theorem \ref{thm:hybrid Lyapunov theorem}} and the invariance principle in \IfConf{\cite[Theorem~A.6]{dhustigs2022unitingNSC}}{Theorem \ref{thm:HybridInvariancePrinciple}} yield global pre-asymptotic stability of $\{\minSet\} \times \{0\}$ for $\HS_0$. Since by Proposition \ref{prop:Existence}, each maximal solution to $\HS_0$ is complete, then $\{\minSet\} \times \{0\}$ is globally asymptotically stable for the closed-loop algorithm $\HS_0$. Since $\HS_0$ satisfies the hybrid basic conditions by Lemma \ref{lemma:HBC}, then, by \IfConf{\cite[Theorem~A.4]{dhustigs2022unitingNSC}}{Theorem \ref{thm:GASImpliesUGAS}}, $\{\minSet\} \times \{0\}$ is uniformly globally asymptotically stable for $\HS_0$.\IfConf{\hfill{} \qed}{}
\IfConf{\end{pf}}{\end{proof}} \IfConf{
\vspace{-0.5cm}
}{}Next, we establish the convergence rate of the closed-loop algorithm $\HS_0$. To do so, we use the following Lyapunov function, proposed in \cite[Lemma~4.2]{sebbouh2020convergence}, for $\HS_0$: \IfConf{\vspace{-0.2cm}}{}
\begin{align} \label{eqn:AlternateVHBF}
\IfConf{V(z) := & \gamma \left(L(z_1) - L^*\right) + \frac{1}{2}\left|\psi(z_1 - z_1^*) + z_2\right|^2 \nonumber\\
& + \frac{\nu}{2}\left|z_1 - z_1^*\right|^2}{V(z) := \gamma \left(L(z_1) - L^*\right) + \frac{1}{2}\left|\psi(z_1 - z_1^*) + z_2\right|^2 + \frac{\nu}{2}\left|z_1 - z_1^*\right|^2}
\end{align}
where, given $\lambda > 0$, $\psi > 0$ is chosen such that $\nu := \psi \left(\psi - \lambda\right) < 0$. When $L$ satisfies Assumption \ref{ass:LisNSCVX}, the following lemma, which is a version of \cite[Lemma~4.2]{sebbouh2020convergence} tailored for
the unperturbed heavy ball algorithm in \eqref{eqn:H0}, gives an upper bound on the change of the Lyapunov function in \eqref{eqn:AlternateVHBF}.
\IfConf{\begin{lem}}{\begin{lemma}} \label{lemma:IntermediateDotV}
Let $L$ satisfy Assumption \ref{ass:LisNSCVX}, and let $\lambda > 0$ and $\gamma > 0$, which come from $\HS_0$ in \eqref{eqn:H0}, be given. For each $\psi > 0$ such that $\nu := \psi (\psi - \lambda) < 0$, the following bound is satisfied for each $z \in \reals^{2n}$: \IfConf{\vspace{-0.1cm}}{}
\begin{equation} \label{eqn:dotVHBFAlternateLemma}
\dot{V}(z) \leq -\psi \left(a(z_1) + 2\nu c(z_1)\right) + 2(\psi - \lambda)b(z)
\end{equation}
where $V$ is defined in \eqref{eqn:AlternateVHBF}, $a(z_1) := \gamma \left(L(z_1) - L^*\right)$, $b(z) := \frac{1}{2}\left|\psi(z_1 - z_1^*) + z_2\right|^2$, and $c(z_1) := \frac{1}{2}\left| z_1 - z_1^* \right|^2$.
\IfConf{\end{lem}}{\end{lemma}} \IfConf{\vspace{-0.5cm}}{}
\IfConf{\begin{pf}}{\begin{proof}} Since $L$ is $\mathcal{C}^1$, nonstrongly convex, and has a single minimizer $z_1^*$, and
since \IfConf{$\nabla V(z) = \left[\gamma \nabla L(z_1) + \psi \left( \psi \left( z_1 - z_1^* \right) + z_2 \right) + \nu \left( z_1 - z_1^* \right) \right.$\\$\left. \psi \left( z_1 - z_1^* \right) + z_2 \right]$}{$\nabla V(z) =$\\$\left[\gamma \nabla L(z_1) + \psi \left( \psi \left( z_1 - z_1^* \right) + z_2 \right) + \nu \left( z_1 - z_1^* \right) \ \; \psi \left( z_1 - z_1^* \right) + z_2 \right]$}, then we evaluate the derivative of $V$, defined via \eqref{eqn:AlternateVHBF}, using the map $z \mapsto F_P(\kappa_0(h_0(z)))$, where $F_P$ is defined via \eqref{eqn:HBFplant-dynamicsTR}, $\kappa_0$ is defined in \eqref{eqn:StaticStateFeedbackLawLocal}, and $h_0$ is defined via \eqref{eqn:H0H1NSCNesterovHBF}. For each $z \in \reals^{2n}$, we obtain \IfConf{
\begin{align} \label{eqn:AlternateDotVHBF}
\dot{V}(z) = & \left\langle \nabla V(z),F_P(\kappa_0(h_0(z))) \right\rangle \\
& = -\gamma \psi \left\langle \nabla L(z_1),z_1 - z_1^* \right\rangle\nonumber\\
& + \left(\nu + \psi(\psi - \lambda) \right) \left\langle z_2, z_1 - z_1^*\right\rangle + (\psi - \lambda)\left|z_2\right|^2.\nonumber
\end{align}
}{\begin{align} \label{eqn:AlternateDotVHBF_TR}
\dot{V}(z) = & \left\langle \nabla V(z),F_P(\kappa_0(h_0(z))) \right\rangle = \left\langle \nabla V(z), \matt{z_2 \\ \kappa_0(h_0(z))} \right\rangle \\
= & \gamma \left\langle \nabla L(z_1), z_2 \right\rangle + \psi \left\langle z_2, \psi \left( z_1 - z_1^* \right) + z_2 \right\rangle + \nu \left\langle z_2, z_1 - z_1^* \right\rangle \nonumber\\
& - \lambda \left\langle z_2, \psi \left( z_1 - z_1^* \right) + z_2 \right\rangle - \gamma \left\langle \nabla L(z_1), \psi \left( z_1 - z_1^* \right) + z_2 \right\rangle \nonumber\\
= & -\gamma \psi \left\langle \nabla L(z_1),z_1 - z_1^* \right\rangle + \left(\nu + \psi(\psi - \lambda) \right) \left\langle z_2, z_1 - z_1^*\right\rangle + (\psi - \lambda)\left|z_2\right|^2.\nonumber
\end{align}}
Note that $\left|\psi\left( z_1 - z_1^* \right) + z_2\right|^2 = \left|z_2\right|^2 + 2\psi \left\langle z_2, z_1 - z_1^* \right\rangle + \psi^2 \left| z_1 - z_1^* \right|^2$, from where we obtain \IfConf{$\left|z_2\right|^2 = $\\$\left|\psi\left( z_1 - z_1^* \right) + z_2\right|^2 - 2\psi \left\langle z_2, z_1 - z_1^* \right\rangle - \psi^2 \left| z_1 - z_1^* \right|^2$}{$\left|z_2\right|^2 = \left|\psi\left( z_1 - z_1^* \right) + z_2\right|^2 - 2\psi \left\langle z_2, z_1 - z_1^* \right\rangle - \psi^2 \left| z_1 - z_1^* \right|^2$}.
Substituting the expression for $\left|z_2\right|^2$ into \IfConf{\eqref{eqn:AlternateDotVHBF}}{\eqref{eqn:AlternateDotVHBF_TR}}, we arrive at, for all $z \in \reals^{2n}$, \IfConf{
\begin{align} \label{eqn:Substituting}
\dot{V}(z) = & -\gamma\psi \left\langle \nabla L(z_1),z_1 - z_1^* \right\rangle + 2(\psi - \lambda)b(z)\nonumber\\
& - 2\psi \nu c(z_1)
\end{align}
}{\begin{align} \label{eqn:Substituting_TR}
\dot{V}(z) = & -\gamma \psi \left\langle \nabla L(z_1),z_1 - z_1^* \right\rangle + (\psi - \lambda) \left|\psi\left( z_1 - z_1^* \right) + z_2\right|^2\nonumber\\
& + \left(\nu - \psi(\psi - \lambda) \right) \left\langle z_2, z_1 - z_1^*\right\rangle - \psi^2 (\psi - \lambda) \left| z_1 - z_1^* \right|^2\nonumber\\
= & -\gamma\psi \left\langle \nabla L(z_1),z_1 - z_1^* \right\rangle + 2(\psi - \lambda)b(z) - 2\psi \nu c(z_1)
\end{align}}
since $\nu = \psi(\psi - \lambda)$,
where $b(z) = \frac{1}{2}\left|\psi\left( z_1 - z_1^* \right) + z_2\right|^2$ and $c(z_1) = \frac{1}{2}\left| z_1 - z_1^* \right|^2$.
Since $L$ is $\mathcal{C}^1$, nonstrongly convex, and has a unique minimizer by Assumption \ref{ass:LisNSCVX}, then using the definition of nonstrong convexity in Footnote \ref{foot:Convexity} with $u_1 = z_1^*$ and $w_1 = z_1$, we get $- \left(L(z_1) - L^*\right) \geq - \left\langle \nabla L(z_1), z_1 - z_1^* \right\rangle$.
Substituting it into \IfConf{\eqref{eqn:Substituting}}{\eqref{eqn:Substituting_TR}} yields, for all $z \in \reals^{2n}$, $\dot{V}(z) \leq -\psi a(z_1) + 2(\psi - \lambda)b(z) - 2\psi \nu c(z_1)$,
where $a(z_1) = \gamma \left(L(z) - L^*\right)$, and \eqref{eqn:dotVHBFAlternateLemma} is satisfied.\IfConf{\hfill{} \qed}{}
\IfConf{\end{pf}}{\end{proof}} \IfConf{
\vspace{-0.25cm}
}{}We employ Lemma 5.2 to show that when $L$ satisfies Assumptions \ref{ass:LisNSCVX} and \ref{ass:QuadraticGrowth}, the convergence rate of the closed-loop algorithm $\HS_0$ in \eqref{eqn:H0} is exponential. This is supported by the following proposition, which is a version of \cite[Theorem~3.2]{sebbouh2020convergence} tailored for the unperturbed heavy ball algorithm $\HS_0$ in \eqref{eqn:H0}.
\IfConf{\begin{prop}[Convergence rate for $\HS_0$]}{\begin{proposition}(Convergence rate for $\HS_0$):} \label{prop:HBFConvergenceRate}
Let $L$ satisfy Assumptions \ref{ass:LisNSCVX} and \ref{ass:QuadraticGrowth}, let $\alpha > 0$ come from \IfConf{Assumption \ref{ass:QuadraticGrowth}}{\eqref{eqn:QuadraticGrowth}}, and let $\lambda > 0$ and $\gamma > 0$ come from $\HS_0$ in \eqref{eqn:H0}. For each $m \in (0,1)$ such that $\psi := \frac{m\alpha\gamma}{\lambda} > 0$ and $\nu := \psi (\psi - \lambda) < 0$,
each maximal solution $t \mapsto z(t)$ to the closed-loop algorithm $\HS_0$ satisfies
\begin{equation} \label{eqn:ConvergenceRateHBFNSCVX}
\IfConf{L(z_1(t)) - L^* = \mathcal{O}\left(\exp \left(-(1-m)\psi t\right)\right)}{L(z_1(t)) - L^* = \mathcal{O}\left(\exp \left(-(1-m)\psi t\right)\right) \quad \forall t \in \dom z \ (= \reals_{\geq 0}).}
\end{equation}
\IfConf{for all $t \in \dom z \ (= \reals_{\geq 0})$.}{}
\IfConf{\end{prop}}{\end{proposition}} \IfConf{\vspace{-0.45cm}}{}
\IfConf{\begin{pf}}{\begin{proof}}
By Lemma \ref{lemma:IntermediateDotV}, the bound in \eqref{eqn:dotVHBFAlternateLemma} is satisfied for $V$ in \eqref{eqn:AlternateVHBF} for each $z \in \reals^{2n}$ since, by Assumption \ref{ass:LisNSCVX}, $L$ is $\mathcal{C}^1$, nonstrongly convex, and has a single minimizer $z_1^*$. Then, since $\psi = \frac{m\alpha\gamma}{\lambda} > 0$ is such that $\nu = \psi (\psi - \lambda) < 0$ and $c$ is nonnegative, this leads to
\begin{equation} \label{eqn:NuIsNeg}
V(z) = a(z_1) + b(z) + \nu c(z_1) \leq a(z_1) + b(z) \quad \forall z \in \reals^{2n}
\end{equation}
where $a$, $b$, and $c$ are defined below \eqref{eqn:dotVHBFAlternateLemma}.
By Assumption \ref{ass:QuadraticGrowth}, $L$ has quadratic growth away from $\minSet$. Therefore, we have, for all $z \in \reals^{2n}$, \IfConf{
\vspace{-1cm}
\begin{align} \label{eqn:InequalityFollowingFromQG}
a(z_1) + 2\nu c(z_1) = a(z_1) - 2\left|\nu\right| c(z_1) \geq \left(1 - \frac{\left|\nu\right|}{\alpha \gamma}\right) a(z_1).
\end{align}
\vspace{-0.9cm}
}{\begin{align} \label{eqn:InequalityFollowingFromQG_TR}
a(z_1) + 2\nu c(z_1) = & a(z_1) - 2\left|\nu\right| c(z_1) = \gamma \left(L(z_1) - L^*\right) - \left|\nu\right| \left| z_1 - z_1^* \right|^2 \\
\geq & \gamma \left(L(z_1) - L^*\right) - \frac{\left|\nu\right|\left(L(z_1) - L^*\right)}{\alpha}
= \left(1 - \frac{\left|\nu\right|}{\alpha \gamma}\right) a(z_1).\nonumber
\end{align}}
Observe that, for each $m \in (0,1)$ such that $\psi = \frac{m\alpha\gamma}{\lambda} > 0$ and $\nu = \psi (\psi - \lambda) < 0$,
we have \IfConf{$\left|\nu\right| = \psi\left(\lambda - \psi\right) \leq \lambda \psi = m \alpha \gamma$.}{
\begin{equation} \label{eqn:NuUpperBound}
\left|\nu\right| = \psi\left(\lambda - \psi\right) \leq \lambda \psi = m \alpha \gamma
\end{equation}}
It follows from \IfConf{this upper bound on $\left|\nu\right|$ and}{} \IfConf{\eqref{eqn:InequalityFollowingFromQG}}{\eqref{eqn:InequalityFollowingFromQG_TR} and \eqref{eqn:NuUpperBound}} that
\begin{equation} \label{eqn:LowerBoundAandC}
a(z_1) + 2\nu c(z_1) \geq (1-m)a(z_1) \IfConf{\vspace{0.2cm}}{}
\end{equation}
for all $z \in \reals^{2n}$. Noticing that from \eqref{eqn:NuIsNeg} we have $a(z_1) + \frac{1}{(1-m)}b(z) \geq a(z_1) + b(z) \geq V(z)$, substituting \eqref{eqn:LowerBoundAandC} into \eqref{eqn:dotVHBFAlternateLemma} we have \IfConf{ \vspace{-0.15cm}
\begin{align} \label{eqn:dotVContainingV}
\dot{V}(z) & \leq - (1 - m) \psi a(z_1) + \psi b(z) + (\psi - 2\lambda) b(z) \nonumber\\
& \leq - (1-m)\psi a(z_1) + \psi b(z)\nonumber\\
& \leq -(1-m)\psi V(z)
\end{align}
}{\begin{align} \label{eqn:dotVContainingV_TR}
\dot{V}(z) & \leq - (1-m) \psi a(z_1) + 2(\psi - \lambda)b(z) \nonumber\\
& \leq - (1 - m) \psi a(z_1) + \psi b(z) + (\psi - 2\lambda) b(z) \nonumber\\
& \leq - (1-m)\psi a(z_1) + \psi b(z)\nonumber\\
& \leq -(1-m)\psi \left(a(z_1) + \frac{1}{(1-m)}b(z)\right)\nonumber\\
& \leq -(1-m)\psi V(z)
\end{align}}
for all $z \in \reals^{2n}$. The \IfConf{second}{third} inequality comes from the fact that we choose $\psi = \frac{m\alpha\gamma}{\lambda} > 0$ such that $\psi - \lambda < 0$ and, consequently, $\psi - 2\lambda < 0$. Applying Gr\"{o}nwall's inequality to \IfConf{\eqref{eqn:dotVContainingV}}{\eqref{eqn:dotVContainingV_TR}} shows that every maximal solution $t \mapsto z(t)$ to \eqref{eqn:HBF} satisfies $V(z(t)) \leq V(z(0)) \exp \left(-(1-m)\psi t\right)$
for all $t \ \in \dom z \ (= \reals_{\geq 0})$. Therefore, each maximal solution $t \mapsto z(t)$ to the closed-loop algorithm $\HS_0$ in \eqref{eqn:H0} satisfies \eqref{eqn:ConvergenceRateHBFNSCVX} for all $t \ \in \dom z \ (= \reals_{\geq 0})$. \IfConf{\hfill{} \qed}{}
\IfConf{\end{pf}}{\end{proof}}\IfConf{
\vspace{-0.6cm}
}{}\subsection{Properties of $\HS_1$}
\label{sec:PropertiesOfH1}
When $L$ satisfies Assumptions \ref{ass:LisNSCVX} and \ref{ass:Lipschitz}, then we can derive an upper bound, for all $t \geq 1$, on the Lyapunov function $V_1$ in \eqref{eqn:LyapunovNesterovNSCVX} along solutions to $\HS_1$. To derive such a bound, we extend \cite[Proposition~3.2]{muehlebach2019dynamical}, which assumes $L^* = 0$ and $z_1^*=0$, to the general case of $L^* \in \reals$ and a single minimizer $z_1^* \in \reals^n$, in the following proposition. \IfConf{Its proof is in \cite{dhustigs2022unitingNSC}.}{}
\IfConf{\begin{prop}}{\begin{proposition}} \label{prop:ConvergenceNSCVXNesterov} Let $L$ satisfy Assumptions \ref{ass:LisNSCVX} and \ref{ass:Lipschitz}. Then, each maximal solution $t \mapsto (z(t), \tau(t))$ to the closed-loop algorithm $\HS_1$ in \eqref{eqn:H1} with $\tau(0) = 0$ satisfies
\begin{equation} \label{eqn:ConvergenceRateNCVX}
V_1(z(t),t) \leq \frac{9}{(t+2)^2}V_1(z(1),1)
\end{equation}
for all $t \geq 1$, where $V_1$ is defined via \eqref{eqn:LyapunovNesterovNSCVX}.
\IfConf{\end{prop}}{\end{proposition}}
\IfConf{}{\begin{proof}
See Section \ref{sec:ProofProp54}.
\end{proof}}
The following proposition establishes that the closed-loop algorithm $\HS_1$ has a convergence rate $\frac{1}{(t+2)^2}$ for all $t \geq 1$. To prove it, we use Proposition \ref{prop:ConvergenceNSCVXNesterov}. This theorem is a new result, which was not analyzed in \cite{muehlebach2019dynamical}.
\IfConf{\begin{prop}[Convergence rate for $\HS_1$]}{\begin{proposition}(Convergence rate for $\HS_1$):} \label{prop:UpperBoundofV}
Let $L$ satisfy Assumptions \ref{ass:LisNSCVX} and \ref{ass:Lipschitz}. Let $\zeta > 0$ and $M > 0$ come from Assumption \ref{ass:Lipschitz}. Then, for each maximal solution $t \mapsto (z(t), \tau(t))$ to the closed-loop algorithm $\HS_1$ in \eqref{eqn:H1} with $\tau(0)=0$, the following holds:
\begin{align} \label{eqn:BoundOnL}
&\frac{\zeta^2}{M}(L(z_1(t)) - L^*) \\
& \leq V_1(z(t),t)
\leq \frac{9c}{(t + 2)^2}\left(\left|z_1(0) - z_1^* \right|^2 + \left|z_2(0)\right|^2\right) \nonumber
\end{align}
for all $t \geq 1$, where $c := \left(1 + \zeta^2\right)\exp \left(\sqrt{\frac{13}{4} + \frac{\zeta^4}{M}}\right)$.
\IfConf{\end{prop}}{\end{proposition}} \IfConf{\vspace{-0.4cm}}{}
\IfConf{\begin{pf}}{\begin{proof}}
The proof consists of the following steps.
\begin{enumerate}[label={\arabic*)},leftmargin=*]
\item First, we use the definition of nonstrong convexity in Footnote \ref{foot:Convexity} and the Lipschitz continuity of $\nabla L$ in Assumption \ref{ass:Lipschitz}, to show that $V_1$ satisfies $V_1(z,\tau) \leq \alpha_2 \left|z\right|_{\mathcal{A}_2}^2$ with $\alpha_2 := (1 + \zeta^2) > 0$;
\item Then, we use the Lipschitz continuity of $\nabla L$ in Assumption \ref{ass:Lipschitz} and the comparison principle to show that the bound in step 1) along $t \mapsto z(t)$ satisfies \IfConf{$V_1(z(t),t) \leq \alpha_2 \exp \left(\!\!\sqrt{\frac{13}{4} + \frac{\zeta^4}{M}}t \right)\! \left|z(0)\right|_{\mathcal{A}_2}^2$}{$V_1(z(t),t) \leq \alpha_2 \exp \left(2\left(\sqrt{\frac{13}{4} + \frac{\zeta^4}{M}}\right)t \right) \left(\left|z_1(0) - z_1^* \right|^2 + \left|z_2(0)\right|^2\right)$} for all $t \geq 1$;
\item Next, we show that at $t = 1$, $V_1(z(1),1)$ is upper bounded by\\ $c \left(\left|z_1(0) - z_1^* \right|^2 + \left|z_2(0)\right|^2\right)$, where \IfConf{$c = $\\$ \left(1 + \zeta^2\right)\exp \left(\sqrt{\frac{13}{4} + \frac{\zeta^4}{M}}\right)$}{$c = \left(1 + \zeta^2\right)\exp \left(\sqrt{\frac{13}{4} + \frac{\zeta^4}{M}}\right)$};
\item Finally, we combine the bound in 3) with \eqref{eqn:ConvergenceRateNCVX} to get \eqref{eqn:BoundOnL} for all $t \geq 1$.
\end{enumerate}
Proceeding with step 1), the Lyapunov function $V_1$, defined via \eqref{eqn:LyapunovNesterovNSCVX}, can be upper bounded by a class-$\mathcal{K}_{\infty}$ function, namely, defining the set \IfConf{$\mathcal{A}_2 := \{\minSet\} \times \{0\}$,}{
\begin{equation} \label{eqn:SetAForConvRate}
\mathcal{A}_2 := \{\minSet\} \times \{0\}
\end{equation}}
then, $V_1$ satisfies \IfConf{$V_1(z,\tau) \leq \alpha_2 \left|z\right|_{\mathcal{A}_2}^2$}{
\begin{equation} \label{eqn:UpperBoundOnV}
V_1(z,\tau) \leq \alpha_2 \left|z\right|_{\mathcal{A}_2}^2
\end{equation}}
for all $(z, \tau) \in \reals^{2n} \times \reals_{\geq 0}$,
and with $\alpha_2$ derived as follows. Since $\bar{a}$, defined via \eqref{eqn:BarA}, equals $1$ at $\tau = 0$ and $\bar{a}$ is monotonically decreasing to zero as $\tau$ tends to $\infty$, then $\bar{a}$ is upper bounded by $1$, and, consequently, the first term of $V_1$ can be upper bounded, for all $(z, \tau) \in \reals^{2n} \times \reals_{\geq 0}$, as follows:
\begin{equation} \label{eqn:UpperBound1}
\frac{1}{2} \left|\frac{2}{(\tau+2)} \left(z_1 - z_1^*\right) + z_2\right|^2 \leq \left| z_1 - z_1^* \right|^2 + \left|z_2\right|^2.
\end{equation}
The second term of $V_1$ can be bounded as follows. Since by Assumption \ref{ass:LisNSCVX}, $L$ is $\mathcal{C}^1$, (nonstrongly) convex, and has a single minimizer $z_1^*$, then, since $\nabla L(z_1^*) = 0$, we can upper bound $L(z_1) - L^*$ in the following manner, using the definition of nonstrong convexity in Footnote \ref{foot:Convexity} and the Lipschitz continuity of $\nabla L$ in Assumption \ref{ass:Lipschitz}, using $u_1 = z_1^*$ and $w_1 = z_1$: $\left| L(z_1) - L^* \right| \leq \left| \left\langle \nabla L(z_1), z_1^* - z_1 \right\rangle \right| \leq \left| \nabla L(z_1) \right| \left| z_1 - z_1^* \right| \leq M \left| z_1 - z_1^* \right|^2$,
for all $z_1 \in \reals^n$. Therefore, since $L(z_1) \geq L^*$, we can upper bound the second term of $V_1$ as follows:
\begin{equation} \label{eqn:UpperBound2}
\frac{\zeta^2}{M}(L(z_1) - L^*) \leq \zeta^2 \left| z_1 - z_1^* \right|^2 \leq \zeta^2\left(\left| z_1 - z_1^* \right|^2 + \left|z_2\right|^2 \right)
\end{equation}
for all $z \in \reals^{2n}$. Using \eqref{eqn:UpperBound1} and \eqref{eqn:UpperBound2} $V_1(z,\tau)$ is upper bounded by $V_1(z,\tau) \leq \left(1+\zeta^2\right) \left|z \right|_{\mathcal{A}_2}^2 =:\alpha_2 \left|z \right|_{\mathcal{A}_2}^2$.
Next, for step 2), in order to apply the comparison principle,
we define the system
\begin{equation}\label{eqn:MJ_CTDynamicsNCVX}
\IfConf{\matt{z_2 \\ -2\bar{d}(t)z_2 - \frac{\zeta^2}{M} \nabla L(z_1 + \bar{\beta}(t) z_2)} =: f(z,t) \ \ z \in \reals^{2n}.}{
\matt{\dot{z_1} \\ \dot{z_2}} = \matt{z_2 \\ -2\bar{d}(t)z_2 - \frac{\zeta^2}{M} \nabla L(z_1 + \bar{\beta}(t) z_2)} =: f(z,t) \quad z \in \reals^{2n}.}
\end{equation}
Since $\nabla L$ is Lipschitz continuous with constant $M > 0$ by Assumption \ref{ass:Lipschitz}, then using Assumption \ref{ass:Lipschitz} with $w_1 = z_1 + \bar{\beta}(t)z_2$ and $u_1 = z_1^*$ yields, for each $z_1, z_2 \in \reals^n$ and each $t \in \reals_{\geq 0}$, \IfConf{$\left|\nabla L(z_1 + \bar{\beta}(t)z_2)\right| \leq M \left|z_1 - z_1^* + \bar{\beta}(t)z_2 \right|$.}{
\begin{equation}\label{eqn:LipschitzMJGradL}
\left|\nabla L(z_1 + \bar{\beta}(t)z_2)\right| \leq M \left|z_1 - z_1^* + \bar{\beta}(t)z_2 \right|.
\end{equation}}
Then, since $\left|\bar{d}(t)\right| \leq \frac{3}{4}$ and $\left|\bar{\beta}(t)\right| \leq 1$ for all $t \geq 0$, we have \IfConf{$\left|f(z,t)\right|^2 \leq \left|z_2\right|^2 + \frac{9}{4} \left|z_2\right|^2 + \frac{\zeta^4}{M^2} \left|\nabla L(z_1 + \bar{\beta}(t)z_2)\right|^2 \leq \left(\frac{13}{4} + \frac{\zeta^4}{M} \right) \left|z\right|^2_{\mathcal{A}_2}$
}{\begin{align} \label{eqn:Squaredf_TR}
\left|f(z,t)\right|^2 & = \left|z_2\right|^2 + \left|-2\bar{d}(t)z_2 - \frac{\zeta^2}{M} \nabla L(z_1 + \bar{\beta}(t)z_2)\right|^2 \nonumber\\
& \leq \left|z_2\right|^2 + \frac{9}{4} \left|z_2\right|^2 + \frac{\zeta^4}{M^2} \left|\nabla L(z_1 + \bar{\beta}(t)z_2)\right|^2 \nonumber\\
& \leq \frac{13}{4} \left|z_2\right|^2 + \frac{\zeta^4}{M} \left(\left|z_1 - z_1^*\right|^2 + \left|z_2\right|^2\right) \nonumber\\
& = \frac{\zeta^4}{M} \left|z_1 - z_1^*\right|^2 + \left(\frac{13}{4} + \frac{\zeta^4}{M} \right) \left|z_2\right|^2 \nonumber\\
& \leq \left(\frac{13}{4} + \frac{\zeta^4}{M} \right) \left|z\right|^2_{\mathcal{A}_2}
\end{align}}
for all $z \in \reals^{2n}$ and all $t \in \reals_{\geq 0}$, where \IfConf{$\mathcal{A}_2 = \{\minSet\} \times \{0\}$}{$\mathcal{A}_2$ is defined via \eqref{eqn:SetAForConvRate}}. \IfConf{}{The second inequality in \IfConf{\eqref{eqn:Squaredf}}{\eqref{eqn:Squaredf_TR}} comes from applying \eqref{eqn:LipschitzMJGradL}.}
The comparison principle \cite[Lemma~3.4]{khalil2002nonlinear}, leads to the following bound of the norm of the solution to \eqref{eqn:MJ_CTDynamicsNCVX}: \IfConf{$\left|z(t)\right|_{\mathcal{A}_2} \leq \exp \left(\frac{1}{2}\sqrt{\frac{13}{4} + \frac{\zeta^4}{M}} t \right) \left|z(0) \right|_{\mathcal{A}_2}$,}{
\begin{equation}
\left|z(t)\right|_{\mathcal{A}_2} \leq \exp \left(\frac{1}{2}\sqrt{\frac{13}{4} + \frac{\zeta^4}{M}} t \right) \left|z(0) \right|_{\mathcal{A}_2}
\end{equation}}
for all $t \geq 0$.
Then, \IfConf{$V_1(z,\tau) \leq \alpha_2 \left|z\right|_{\mathcal{A}_2}^2$}{\eqref{eqn:UpperBoundOnV}} along $t \mapsto \left(z(t) \right)$ reduces to, for all $t \geq 0$\IfConf{, $V_1(z(t),t) \leq \left(1+\zeta^2\right) \left|z(t)\right|_{\mathcal{A}_2}^2 \leq \left(1+\zeta^2\right) \exp \left(\sqrt{\frac{13}{4} + \frac{\zeta^4}{M}} t \right) \left|z(0) \right|^2_{\mathcal{A}_2}$.}{
\begin{align}
V_1(z(t),t) & \leq \left(1+\zeta^2\right) \left|z(t)\right|_{\mathcal{A}_2}^2 \nonumber\\
& \leq \left(1+\zeta^2\right) \exp \left(\sqrt{\frac{13}{4} + \frac{\zeta^4}{M}} t \right) \left( \left|z_1(0) - z_1^* \right|^2 + \left|z_2(0)\right|^2 \right).
\end{align}}
In step 3), we evaluate this bound at $t = 1$.
Finally, for step 4), taking $c = \left(1 + \zeta^2\right)\exp \left(\!\sqrt{\frac{13}{4} + \frac{\zeta^4}{M}}\right)$, combining \eqref{eqn:ConvergenceRateNCVX} with 3) at $t = 1$ yields \eqref{eqn:BoundOnL}
for all $t \geq 1$.\IfConf{\hfill{} \qed}{}
\IfConf{\end{pf}}{\end{proof}} \IfConf{\vspace{-0.2cm}}{}
The following proposition establishes that the closed-loop system $\HS_1$ in \eqref{eqn:H1} has the set
\begin{equation} \label{eqn:setA_UGA}
\mathcal{A}_1 := \{\minSet\} \times \{0\} \times \reals_{\geq 0}
\end{equation}
uniformly globally attractive. To prove it, we use Proposition \ref{prop:UpperBoundofV}.
This corollary is a new result, which was not analyzed in \cite{muehlebach2019dynamical}.
\IfConf{\begin{prop}[UGA of $\mathcal{A}_1$ in \eqref{eqn:setA_UGA} for $\HS_1$]}{\begin{proposition}(Uniform global attractivity of $\mathcal{A}_1$ in \eqref{eqn:setA_UGA} for $\HS_1$):} \label{prop:UGANestrovNSCVX}
Let $L$ satisfy Assumptions \ref{ass:LisNSCVX} and \ref{ass:Lipschitz}. Let $\zeta > 0$ and let $M > 0$ come from Assumption \ref{ass:Lipschitz}. Then, for every $\varepsilon > 0$ and every compact set of initial conditions $K \subset \reals^{2n}$, there exists $t^* > 0$ such that for all maximal solutions $t \mapsto (z(t),\tau(t))$ to the closed-loop algorithm $\HS_1$ in \eqref{eqn:H1} from $(z(0),\tau(0)) \in K \times \{0\}$ we have \IfConf{\vspace{-0.15cm}}{}
\begin{equation} \label{eqn:UpperBoundOnL}
\frac{\zeta^2}{M}(L(z_1(t)) - L^*) \leq \varepsilon \qquad \forall t \geq t^*.
\end{equation}
\IfConf{\end{prop}}{\end{proposition}} \IfConf{\vspace{-0.74cm}}{}
\IfConf{\begin{pf}}{\begin{proof}}
Pick a solution $t \mapsto (z(t),\tau(t))$ to $\HS_1$, such that $\tau(0)\IfConf{\!\!\!}{}=\IfConf{\!\!\!}{}0$.
By Assumptions \ref{ass:LisNSCVX} and \ref{ass:Lipschitz}, $\nabla L$ is Lipschitz continuous and $L$ is nonstrongly convex. Hence, by Proposition \ref{prop:UpperBoundofV}, $\frac{\zeta^2}{M}(L(z_1(t)) - L^*)$ is bounded by the right-hand side of \eqref{eqn:BoundOnL} for all $t \geq 1$. When $t=1$, the right-hand side of \eqref{eqn:BoundOnL} equals $c \left(\left|z_1(0) - z_1^* \right|^2 + \left|z_2(0)\right|^2\right)$,
where $c := $\IfConf{\\}{}$ \left(1 + \zeta^2\right)\exp \left(\sqrt{\frac{13}{4} + \frac{\zeta^4}{M}}\right)$. Then, given $\eps > 0$, $t^* = $\IfConf{\\}{}$ \sqrt{\frac{9c}{\eps}\left(\left|z_1(0) - z_1^* \right|^2 + \left|z_2(0)\right|^2\right)} + 2 > 0$ is such that \eqref{eqn:UpperBoundOnL} holds, namely, \IfConf{$\frac{\zeta^2}{M}(L(z_1(t^*)) - L^*) \leq $\\$ \frac{9c}{(t^* + 2)^2}\left(\left|z_1(0) - z_1^* \right|^2 + \left|z_2(0)\right|^2\right) \leq \eps$,}{
\begin{equation}
\frac{\zeta^2}{M}(L(z_1(t^*)) - L^*) \leq \frac{9c}{(t^* + 2)^2}\left(\left|z_1(0) - z_1^* \right|^2 + \left|z_2(0)\right|^2\right) \leq \eps
\end{equation}}
where the first inequality comes from \eqref{eqn:BoundOnL}.
Then, $\mathcal{A}_1$, defined via \eqref{eqn:setA_UGA}, is uniformly globally pre-attractive for \eqref{eqn:H1}. Furthermore, since
by Proposition \ref{prop:Existence} each maximal solution to the closed-loop algorithm $\HS_1$, defined via \eqref{eqn:H1}, is complete, then $\mathcal{A}_1$ in \eqref{eqn:setA_UGA} is UGA for \eqref{eqn:H1}.\IfConf{\hfill{} \qed}{}
\IfConf{\end{pf}}{\end{proof}} \IfConf{
\vspace{-0.4cm}
}{}\subsection{Uniform Global Asymptotic Stability of $\mathcal{A}$ for $\HS$}
\label{sec:UGAS}
The hybrid closed-loop algorithm $\HS$ satisfies the hybrid basic conditions by Lemma \ref{lemma:HBC}, satisfying the first assumption of \IfConf{\cite[Theorem~A.3]{dhustigs2022unitingNSC}}{Theorem \ref{thm:hybrid Lyapunov theorem}}.
Furthermore, $\Pi(C_0) \cup \Pi(D_0) = \reals^{2n}$, $\Pi(C_1) \cup \Pi(D_1) = \reals^{2n}$, and each maximal solution $(t,j) \mapsto x(t,j) = (z(t,j), q(t,j),\tau(t,j))$ to $\HS$ in \eqref{eqn:HS-TimeVarying}-\eqref{eqn:CAndDGradientsNestNSC} is complete and bounded by Proposition \ref{prop:Existence}. Since by Assumption \ref{ass:LisNSCVX}, $L$ has a unique minimizer $\minSet$, then $\mathcal{A}$, defined via \eqref{eqn:SetOfMinimizersHS-NSCVX}, is compact by construction, and $\mathcal{U} = \reals^{2n} \times Q \times \reals_{\geq 0}$ contains a nonzero open neighborhood of $\mathcal{A}$, satisfying the second assumption of \IfConf{\cite[Theorem~A.3]{dhustigs2022unitingNSC}}{Theorem \ref{thm:hybrid Lyapunov theorem}}.
To prove attractivity of $\mathcal{A}$, we proceed by contradiction. Suppose there exists a complete solution $x$ to $\HS$ such that $\lim\limits_{t+j \rightarrow \infty}\left|x(t,j)\right|_{\mathcal{A}} \neq 0$. Since Proposition \ref{prop:Existence} guarantees completeness of maximal solutions, we have the following cases:
\begin{enumerate}[label={\alph*)},leftmargin=*]
\item There exists $(t',j') \in \dom x$ such that $x(t,j) \in C_1 \setminus D_1$ for all $(t,j) \in \dom x, t+j \geq t' + j'$;
\item There exists $(t',j') \in \dom x$ such that $x(t,j) \in C_0 \setminus (\mathcal{A} \cup D_0)$ for all $(t,j) \in \dom x, t+j \geq t' + j'$;
\item There exists $(t',j') \in \dom x$ such that $x(t,j) \in D$ for all $(t,j) \in \dom x, t+j \geq t' + j'$.
\end{enumerate}
Case a) contradicts the fact that, by Proposition \ref{prop:UGANestrovNSCVX}, the set $\mathcal{A}_1$, defined via \eqref{eqn:setA_UGA}, is uniformly globally attractive for $\HS_1$, for all $t \geq t^*$. Such uniform global attractivity of $\mathcal{A}$, guaranteed by Proposition \ref{prop:UGANestrovNSCVX}, implies there exist $\tilde{c}_1 \in (0,\tilde{c}_{1,0})$ and $d_1 \in (0,d_{1,0})$ such that the state $z$ reaches $\left(\{\minSet\} + \tilde{c}_1 \ball \right) \times \left(\{0\} + d_1 \ball\right) \subset \mathcal{T}_{1,0}$ at some finite flow time $t \geq t^*$ or as $t \rightarrow \infty$. In turn, due to the construction of $C_1$ and $D_1$ in \eqref{eqn:CAndDGradientsNestNSC}, with $\mathcal{T}_{1,0}$ defined via \eqref{eqn:T10}, the solution $x$ must reach $D_1$ at some $(t,j) \in \dom x, t+j \geq t' + j'$. Therefore, case a) does not happen.
Case b) contradicts the fact that, by Proposition \ref{prop:GAS-HBF}, $\{\minSet\} \times \{0\}$ is uniformly globally asymptotically stable for $\HS_0$. In fact, $\lim\limits_{t+j \rightarrow \infty} \left|x(t,j)\right|_{\mathcal{A}} = 0$, and since $\mathcal{A} \subset C_0$, case b) does not happen.
Case c) contradicts the fact that, due to the construction of $\mathcal{T}_{1,0}$ in \eqref{eqn:T10} and $\mathcal{T}_{0,1}$ in \eqref{eqn:T01}, we have $G(D) \cap D := \left(\left(\mathcal{T}_{0,1} \times \{1\} \times \{0\} \right) \cup \left(\mathcal{T}_{1,0} \times \{0\} \times \{0\} \right)\right)$\\$\cap \left(\left(\mathcal{T}_{0,1} \times \{0\} \times \{0\} \right) \cup \left(\mathcal{T}_{1,0}\times \{1\} \times \reals_{\geq 0} \right) \right) = \emptyset$
where \IfConf{}{$G(D)$ is defined via \eqref{eqn:G(D)} and} $D$ is defined in \eqref{eqn:CAndDGradientsNestNSC}. Such an equality holds since $\mathcal{T}_{1,0} \cap \mathcal{T}_{0,1} = \emptyset$; see the end of Section \ref{sec:DesignT10}. Therefore, case c) does not happen.
Therefore, cases a)-c) do not happen, and each maximal and complete solution $x = (z, q, \tau)$ to $\HS$ with $\tau(0,0) = 0$ converges to $\mathcal{A}$. Consequently, by the construction of $C$ and $D$ in \eqref{eqn:CAndDGradientsNestNSC}, the uniform global attractivity of $\mathcal{A}_1$ (defined via \eqref{eqn:setA_UGA}) for $\HS_1$ established in Proposition \ref{prop:UGANestrovNSCVX}, the uniform global asymptotic stability of $\{\minSet\} \times \{0\}$ for $\HS_0$ established in Proposition \ref{prop:GAS-HBF}, and since each maximal solution to $\HS$ is complete by Proposition \ref{prop:Existence}, the set $\mathcal{A}$ is uniformly globally asymptotically stable for $\HS$.
To show that each maximal and complete solution $x$ to $\HS$ jumps no more than twice, we proceed by contradiction. Without loss of generality, suppose there exists a maximal and complete solution that jumps three times. We have the following possible cases:
\begin{enumerate}[label={\roman*)},leftmargin=*]
\item The solution first jumps at a point in $D_0$, then jumps at a point in $D_1$, and then jumps at a point in $D_0$; or
\item The solution first jumps at a point in $D_1$, then jumps at a point in $D_0$, and then jumps at a point in $D_1$.
\end{enumerate}
Case i) does not hold since, once the jump in $D_1$ occurs, the solution $x$ is in $(\mathcal{T}_{1,0} \times \{0\} \times \{0\}) \subset C_0$. Due to the construction of $\mathcal{T}_{1,0}$ in \eqref{eqn:T10} and $\mathcal{T}_{0,1}$ in \eqref{eqn:T01} such that $\mathcal{T}_{1,0} \cap \mathcal{T}_{0,1} = \emptyset$, as described in the contradiction of case c) above, and due to the uniform global asymptotic stability of $\minSet \times \{0\}$ for $\HS_0$ by Proposition \ref{prop:GAS-HBF}, the solution $x$ will never return to $D_0$. Therefore, case i) does not happen. Case ii) leads to a contradiction for the same reason, and in this case, once the first jumps in $D_1$ occurs, no more jumps happen. Therefore, since cases i)-ii) do not happen, each maximal and complete solution $x$ to $\HS$ with $\tau(0,0)=0$ has no more than two jumps.
\subsection{Convergence Rate of $\HS$}
\label{sec:ConvRateH}
Finally, we prove the hybrid convergence rate of $\HS$. Letting $\zeta > 0$ and letting $M > 0$ come from Assumption \ref{ass:Lipschitz}, then by Proposition \ref{prop:UpperBoundofV}, since $L$ satisfies Assumptions \ref{ass:LisNSCVX} and \ref{ass:Lipschitz}, each maximal solution $t \mapsto (z(t), \tau(t))$ to the closed-loop algorithm $\HS_1$ with $\tau(0,0) = 0$ satisfies \eqref{eqn:BoundOnL}, for all $t \geq 1$, where $c := \left(1 + \zeta^2\right)\exp \left(\sqrt{\frac{13}{4} + \frac{\zeta^4}{M}}\right)$. By Proposition \ref{prop:HBFConvergenceRate}, since $L$ satisfies Assumptions \ref{ass:LisNSCVX} and \ref{ass:QuadraticGrowth}, then, given $\gamma > 0$ and $\lambda > 0$, for each $m \in (0,1)$ such that $\psi := \frac{m\alpha\gamma}{\lambda} > 0$ and $\nu := \psi (\psi - \lambda) < 0$,
each maximal solution $t \mapsto z(t)$ to the closed-loop algorithm $\HS_0$ satisfies \eqref{eqn:ConvergenceRateHBFNSCVX} for all $t \ \in \dom z \ (= \reals_{\geq 0})$.
Since maximal solutions $(t,j) \mapsto x(t,j) = (z(t,j),q(t,j),\tau(t,j))$ to $\HS$ starting from $C_1$ are guaranteed to jump no more than once, as implied by the contradiction in cases i)-ii) above, then the domain of each maximal solution $x$ to $\HS$ starting from $C_1$ is $\cup_{j=0}^{1} (I^j,j)$, with $I^0$ of the form $[t_0,t_1]$ and with $I^1$ of the form $[t_1,\infty)$. Therefore, given $\zeta > 0$, $\lambda > 0$, $\gamma > 0$, $c_{1,0} \in (0,c_0)$, $\varepsilon_{1,0} \in (0,\varepsilon_0)$, $\alpha > 0$ from Assumption \ref{ass:QuadraticGrowth}, and $M >0$ from Assumption \ref{ass:Lipschitz}, due to the construction of $\mathcal{U}_0$, $\mathcal{T}_{1,0}$, and $\mathcal{T}_{0,1}$ in \eqref{eqn:U0}, \eqref{eqn:T10}, and \eqref{eqn:T01}, with $\tilde{c}_{1,0} \in (0,\tilde{c}_0)$ and $d_{1,0} \in (0,d_0)$ defined via \eqref{eqn:UTilde0SetEquations} and \eqref{eqn:TTilde10SetEquations}, and due to the individual convergence rates of $\HS_1$ and $\HS_0$, each maximal solution $(t,j) \mapsto x(t,j) = (z(t,j),q(t,j),\tau(t,j))$ to the hybrid closed-loop algorithm $\HS$ that starts in $C_1$, such that $\tau(0,0)=0$, satisfies \eqref{eqn:UnitingConvergenceRateNSCHS1} for each $t \in I^0$ at which $q(t,0)$ is equal to 1 and $t \geq 1$, and satisfies \eqref{eqn:UnitingConvergenceRateNSCHS0} for each $t \in I^1$ at which $q(t,1)$ is equal to 0.
\IfConf{}{
\section{Extensions}
Some possible extensions of Theorem \ref{thm:GASNestNSC}, Proposition \ref{prop:GAS-HBF}, Lemma \ref{lemma:IntermediateDotV}, and Propositions \ref{prop:HBFConvergenceRate}, \ref{prop:ConvergenceNSCVXNesterov}, \ref{prop:UpperBoundofV}, and \ref{prop:UGANestrovNSCVX} are as follows. It is possible to extend Theorem \ref{thm:GASNestNSC}, Proposition \ref{prop:GAS-HBF}, Lemma \ref{lemma:IntermediateDotV}, and Propositions \ref{prop:HBFConvergenceRate}, \ref{prop:ConvergenceNSCVXNesterov}, \ref{prop:UpperBoundofV}, and \ref{prop:UGANestrovNSCVX} to include $\mathcal{C}^1$, nonstrongly convex objective functions $L$ with a compact and connected set of minimizers. Such an extension could be achieved via
the use of Clarke's generalized derivative (see \cite{clarke1990optimization}). Additionally, Clarke's generalized derivative could be utilized to extend the analysis of the hybrid closed-loop algorithm to include nonsmooth nonstrongly convex objective functions $L$ with a compact and connected set of minimizers.}
\section{Proof of Proposition \ref{prop:ConvergenceNSCVXNesterov}}
\label{sec:ProofProp54}
The Lyapunov function $V_1$, defined via \eqref{eqn:LyapunovNesterovNSCVX}, is positive definite with respect to $\mathcal{A}_1$, defined via \eqref{eqn:setA_UGA},
since, by Assumption \ref{ass:LisNSCVX}, $L$ is $\mathcal{C}^1$, nonstrongly convex, and has a unique minimizer $z_1^*$.
Then, letting
\begin{equation} \label{eqn:BarV1}
\bar{v}_1(z,\tau) := z_1 + \bar{\beta}(\tau)z_2,
\end{equation}
letting $\varphi(z,\tau) := \bar{a}(\tau)\left(\bar{a}(\tau)\left( z_1 - z_1^* \right) + z_2\right) +\frac{\zeta^2}{M}\nabla L(z_1)$, and since \IfConf{$\nabla V_1(z,\tau) $$ = \left[\varphi(z,\tau) \ \ \left(\bar{a}(\tau)\left( z_1 - z_1^* \right) + z_2\right) \right.$\\$\left. \diff{\bar{a}(\tau)}{\tau} \left\langle z_1 - z_1^*,\left(\bar{a}(\tau)\left( z_1 - z_1^* \right) + z_2\right) \right\rangle \right]$}{$\nabla V_1(z,\tau) $\\$ = \left[\varphi(z,\tau) \ \ \left(\bar{a}(\tau)\left( z_1 - z_1^* \right) + z_2\right) \quad \diff{\bar{a}(\tau)}{\tau} \left\langle z_1 - z_1^*,\left(\bar{a}(\tau)\left( z_1 - z_1^* \right) + z_2\right) \right\rangle \right]$}, we evaluate the derivative of $V_1$, using the map $z \mapsto$ $F_P(z,\kappa_1(h_1(z,\tau),\tau))$, where $F_P$ is defined in \eqref{eqn:HBFplant-dynamicsTR}, $\kappa_1$ is defined via \eqref{eqn:StaticStateFeedbackLawGlobalNSCVX}, and $h_1$ is defined in \eqref{eqn:H0H1NSCNesterovHBF}, to yield \IfConf{
\begin{align} \label{eqn:dotVNSCVX}
\dot{V}_1(z,\tau) = & \left\langle \nabla V_1(z,\tau),F_P(z,\kappa_1(h_1(z,\tau),\tau)) \right\rangle \nonumber\\
= & -\frac{\bar{a}({\tau})\zeta^2}{M}\left\langle z_1 - z_1^*,\nabla L(\bar{v}_1(z,\tau))\right\rangle \nonumber\\
& + \bar{a}(\tau) \diff{\bar{a}(\tau)}{\tau}\left| z_1 - z_1^* \right|^2 \nonumber\\
&+ \left(\bar{a}(\tau) - 2\bar{d}(\tau)\right)\left|z_2\right|^2 \nonumber\\
&+ \left(\bar{a}^2(\tau) - 2\bar{d}(\tau)\bar{a}(\tau) + \diff{\bar{a}(\tau)}{\tau}\right)\left\langle z_1 - z_1^*,z_2 \right\rangle \nonumber\\
&- \frac{\zeta^2}{M} \left\langle z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle
\end{align}
}{\begin{align} \label{eqn:dotVNSCVX_TR}
\dot{V}_1(z,\tau) = & \left\langle \nabla V_1(z,\tau),F_P(z,\kappa_1(h_1(z,\tau),\tau)) \right\rangle \nonumber\\
= & \left\langle \nabla V_1(z,\tau), \matt{\matt{z_2 \\ - 2\bar{d}(\tau)z_2 - \frac{\zeta^2}{M}\nabla L(\bar{v}_1(z,\tau))} \\ 1} \right\rangle \nonumber\\
= & \bar{a}(\tau) \left\langle \bar{a}(\tau)\left(z_1 - z_1^*\right) + z_2, z_2 \right\rangle + \frac{\zeta^2}{M}\left\langle z_2,\nabla L(z_1) \right\rangle - 2\bar{d}(\tau) \left|z_2\right|^2 \nonumber\\
&- 2\bar{d}(\tau)\bar{a}(\tau)\left\langle z_1 - z_1^*,z_2 \right\rangle -\frac{\bar{a}(\tau)\zeta^2}{M} \left\langle z_1 - z_1^*,\nabla L(\bar{v}_1(z,\tau)) \right\rangle \nonumber\\
& -\frac{\zeta^2}{M} \left\langle z_2,\nabla L(\bar{v}_1(z,\tau)) \right\rangle +\bar{a}(\tau)\diff{\bar{a}(\tau)}{\tau}\left|z_1 - z_1^*\right|^2 +\diff{\bar{a}(\tau)}{\tau}\left\langle z_1 - z_1^*,z_2 \right\rangle \nonumber\\
= & -\frac{\bar{a}({\tau})\zeta^2}{M}\left\langle z_1 - z_1^*,\nabla L(\bar{v}_1(z,\tau))\right\rangle + \bar{a}(\tau) \diff{\bar{a}(\tau)}{\tau}\left| z_1 - z_1^* \right|^2 \nonumber\\
&+ \left(\bar{a}(\tau) - 2\bar{d}(\tau)\right)\left|z_2\right|^2 + \left(\bar{a}^2(\tau) - 2\bar{d}(\tau)\bar{a}(\tau) + \diff{\bar{a}(\tau)}{\tau}\right)\left\langle z_1 - z_1^*,z_2 \right\rangle \nonumber\\
&- \frac{\zeta^2}{M} \left\langle z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle
\end{align}
}
for all $(z, \tau) \in \reals^{2n} \times \reals_{\geq 0}$. Since $L$ is $\mathcal{C}^1$, nonstrongly convex, and has a unique minimizer by Assumption \ref{ass:LisNSCVX}, then using the definition of nonstrong convexity in Footnote \ref{foot:Convexity} with $u_1 = z_1^*$ and $w_1 = \bar{v}_1(z,\tau)$, where $\bar{v}_1$ is defined via \eqref{eqn:BarV1}, we get
\begin{align} \label{eqn:ConvexityZstarBarV}
- \left\langle \bar{v}_1(z,\tau) - z_1^*, \nabla L(\bar{v}_1(z,\tau)) \right\rangle & \leq -\left(L(\bar{v}_1(z,\tau)) - L^*\right)
\end{align}
for each $z \in \reals^{2n}$ and $\tau \in \reals_{\geq 0}$. Using the definition of nonstrong convexity in Footnote \ref{foot:Convexity} with $u_1 = \bar{v}_1(z,\tau)$, where $\bar{v}_1$ is defined via \eqref{eqn:BarV1}, and $w_1 = z_1$ yields
\begin{align} \label{eqn:ConvexityZstarZ}
\left\langle \nabla L(z_1), \bar{\beta}(\tau)z_2 \right\rangle & \leq L(\bar{v}_1(z,\tau)) - L(z_1)
\end{align}
for each $z \in \reals^{2n}$ and $\tau \in \reals_{\geq 0}$.
Combining \eqref{eqn:ConvexityZstarBarV} and \eqref{eqn:ConvexityZstarZ} yields\\ $- \left\langle \bar{v}_1(z,\tau) - z_1^*,\! \nabla L(\bar{v}_1(z,\tau)) \right\rangle + \left\langle \nabla L(z_1), \bar{\beta}(\tau)z_2 \right\rangle \leq \!\! -L(\bar{v}_1(z,\tau)) + L(\bar{v}_1(z,\tau)) - L(z_1) + L^*$.
Then, rearranging terms gives, for all $z \in \reals^{2n}$ and $\tau \in \reals_{\geq 0}$,
\begin{align}\label{eqn:BoundOnQAndGradFNSCVX}
\IfConf{& - \langle z_1 - z_1^*, \nabla L(\bar{v}_1(z,\tau)) \rangle \\
& \leq - \left(L(z_1) - L^*\right)+ \left\langle \bar{\beta}(\tau) z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle . \nonumber}{& - \langle z_1 - z_1^*, \nabla L(\bar{v}_1(z,\tau)) \rangle \\
& \leq - \left(L(z_1) - L^*\right)
+ \left\langle \bar{\beta}(\tau) z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle . \nonumber}
\end{align}
Substituting the bound in \eqref{eqn:BoundOnQAndGradFNSCVX} into \IfConf{\eqref{eqn:dotVNSCVX}}{\eqref{eqn:dotVNSCVX_TR}} yields
\begin{align} \label{eqn:SubsV1Bound}
\IfConf{\dot{V}_1(z,\tau) \leq & - \frac{\bar{a}(\tau)\zeta^2}{M} \left(L(z_1) - L^*\right) \nonumber\\
& + \frac{\bar{a}(\tau)\zeta^2}{M} \left\langle \bar{\beta}(\tau) z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle \nonumber\\
& + \bar{a}(\tau) \diff{\bar{a}(\tau)}{\tau}\left| z_1 - z_1^* \right|^2 + \left(\bar{a}(\tau) - 2\bar{d}(\tau)\right)\left|z_2\right|^2\nonumber\\
& + \left(\bar{a}^2(\tau) - 2\bar{d}(\tau)\bar{a}(\tau) + \diff{\bar{a}(\tau)}{\tau}\right)\left\langle z_1 - z_1^*,z_2 \right\rangle \nonumber\\
&- \frac{\zeta^2}{M} \left\langle z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle}{
\dot{V}_1(z,\tau) \leq & - \frac{\bar{a}(\tau)\zeta^2}{M} \left(\left(L(z_1) - L^*\right)
- \left\langle \bar{\beta}(\tau) z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle\right) \nonumber\\
& + \bar{a}(\tau) \diff{\bar{a}(\tau)}{\tau}\left| z_1 - z_1^* \right|^2 + \left(\bar{a}(\tau) - 2\bar{d}(\tau)\right)\left|z_2\right|^2\nonumber\\
& + \left(\bar{a}^2(\tau) - 2\bar{d}(\tau)\bar{a}(\tau) + \diff{\bar{a}(\tau)}{\tau}\right)\left\langle z_1 - z_1^*,z_2 \right\rangle \nonumber\\
&- \frac{\zeta^2}{M} \left\langle z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle}
\end{align}
for all $(z, \tau) \in \reals^{2n} \times \reals_{\geq 0}$. Then, noticing that $\frac{\bar{a}(\tau)}{2}\left|\bar{a}(\tau)\left(z_1 - z_1^*\right) + z_2 \right|^2 =$\IfConf{}{\\}$\frac{\bar{a}^3(\tau)}{2}\left| z_1 - z_1^* \right|^2$\IfConf{\\}{}$+ \bar{a}^2(\tau)\left\langle z_1 - z_1^*,z_2 \right\rangle + \frac{\bar{a}(\tau)}{2}\left|z_2\right|^2$,
adding it to and subtracting it from \eqref{eqn:SubsV1Bound}, and rearranging terms, yields \IfConf{
\begin{align} \label{eqn:PreVanishBound}
& \dot{V}_1(z,\tau) \\
& \leq -\bar{a}(\tau)V_1(z,\tau) + \left(\frac{\bar{a}^3(\tau)}{2} + \bar{a}(\tau)\diff{\bar{a}(\tau)}{\tau}\right) \left| z_1 - z_1^* \right|^2\nonumber\\
& +\left(\frac{3\bar{a}(\tau)}{2} - 2\bar{d}(\tau)\right)\left|z_2\right|^2\nonumber\\
& + \left(2\bar{a}^2(\tau) - 2\bar{d}(\tau)\bar{a}(\tau) + \diff{\bar{a}(\tau)}{\tau}\right)\left\langle z_1 - z_1^*,z_2 \right\rangle \nonumber\\
& - \frac{\zeta^2}{M}\left(1 - \bar{\beta}(\tau)\bar{a}(\tau)\right)\left\langle z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle \nonumber
\end{align}}{
\begin{align} \label{eqn:PreVanishBoundTR}
\dot{V}_1(z,\tau) \leq & -\bar{a}(\tau)V_1(z,\tau) + \bar{a}(\tau) \diff{\bar{a}(\tau)}{\tau}\left| z_1 - z_1^* \right|^2 + \left(\bar{a}(\tau) - 2\bar{d}(\tau)\right)\left|z_2\right|^2 \nonumber\\
& + \left(\bar{a}^2(\tau) - 2\bar{d}(\tau)\bar{a}(\tau) + \diff{\bar{a}(\tau)}{\tau}\right) \left\langle z_1 - z_1^*,z_2 \right\rangle + \frac{\bar{a}^3(\tau)}{2}\left| z_1 - z_1^* \right|^2 \nonumber\\
& + \frac{\bar{a}(\tau)}{2}\left|z_2\right|^2 + \bar{a}^2(\tau)\left\langle z_1 - z_1^*,z_2 \right\rangle \nonumber\\
& - \frac{\zeta^2}{M} \left(1 - \bar{\beta}(\tau)\bar{a}(\tau)\right) \left\langle z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle \nonumber\\
\leq & -\bar{a}(\tau)V_1(z,\tau) + \left(\frac{\bar{a}^3(\tau)}{2} + \bar{a}(\tau)\diff{\bar{a}(\tau)}{\tau}\right) \left| z_1 - z_1^* \right|^2 \nonumber\\
& +\left(\frac{3\bar{a}(\tau)}{2} - 2\bar{d}(\tau)\right)\left|z_2\right|^2\nonumber\\
& + \left(2\bar{a}^2(\tau) - 2\bar{d}(\tau)\bar{a}(\tau) + \diff{\bar{a}(\tau)}{\tau}\right)\left\langle z_1 - z_1^*,z_2 \right\rangle \nonumber\\
& - \frac{\zeta^2}{M}\left(1 - \bar{\beta}(\tau)\bar{a}(\tau)\right)\left\langle z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle
\end{align}}
for all $(z, \tau) \in \reals^{2n} \times \reals_{\geq 0}$.
Due to the definitions of the functions $\bar{a}$ and $\bar{d}$, in \eqref{eqn:BarA} and \eqref{eqn:dBarBetaBar}, respectively, the cross term $\left\langle z_1 - z_1^*,z_2 \right\rangle$ vanishes since $2\bar{a}^2(\tau) - 2\bar{d}(\tau)\bar{a}(\tau) + \diff{\bar{a}(\tau)}{\tau} = 2\left(\frac{2}{\tau + 2}\right)^2 - 2\left(\frac{3}{2(\tau+2)}\right)\left(\frac{2}{\tau + 2}\right) - \frac{2}{\left(\tau + 2\right)^2} = 0$
Moreover, the definitions of the functions $\bar{d}$ and $\bar{a}$ lead to the $\left| z_1 - z_1^* \right|^2$ and $\left|z_2\right|^2$ terms in \IfConf{\eqref{eqn:PreVanishBound}}{\eqref{eqn:PreVanishBoundTR}} vanishing due to $\frac{\bar{a}^3(\tau)}{2} + \bar{a}(\tau)\diff{\bar{a}(\tau)}{\tau} = \frac{\left(\frac{2}{\tau + 2}\right)^3}{2} + \left(\frac{2}{\tau + 2}\right)\left(-\frac{2}{(\tau + 2)^2}\right) = 0$
and $\frac{3\bar{a}(\tau)}{2} - 2\bar{d}(\tau) = \frac{3\left(\frac{2}{\tau + 2}\right)}{2} - 2\left(\frac{3}{2\left(\tau + 2\right)}\right) = 0$.
The bound in \IfConf{\eqref{eqn:PreVanishBound}}{\eqref{eqn:PreVanishBoundTR}} reduces to
\begin{align} \label{eqn:finalDotVNSCVX}
& \IfConf{\dot{V}_1(z,\tau) \\
& \leq -\bar{a}(\tau)V_1(z,\tau) \nonumber\\
& - \frac{\zeta^2}{M}\left(1 - \bar{\beta}(\tau)\bar{a}(\tau)\right)\left\langle z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle \nonumber}{
\dot{V}_1(z,\tau) \leq -\bar{a}(\tau)V_1(z,\tau) - \frac{\zeta^2}{M}\left(1 - \bar{\beta}(\tau)\bar{a}(\tau)\right)\left\langle z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle}
\end{align}
for all $(z, \tau) \in \reals^{2n} \times \reals_{\geq 0}$.
By Assumption \ref{ass:Lipschitz}, $\nabla L$ is Lipschitz continuous with Lipschitz constant $M > 0$. An equivalent characterization of the Lipschitz continuity of $\nabla L$, from \cite[Theorem~2.1.5]{nesterov2004introductory}, for all $z_1, u_1 \in \reals^n$, is $\left\langle \nabla L(z_1) - \nabla L(u_1), z_1 - u_1 \right\rangle \leq M \left|z_1 - u_1\right|^2$.
Then, using such a bound
with $w_1 = \bar{v}_1(z,\tau)$, where $\bar{v}_1$ is defined in \eqref{eqn:BarV1}, and $u_1 = z_1$, we get, for all $z \in \reals^{2n}$ and $\tau \in \reals_{\geq 0}$,\IfConf{}{\\}
$\left\langle \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1), \bar{\beta}(\tau)z_2\right\rangle =\left\langle \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1), \bar{v}_1(z,\tau) - z_1\right\rangle \leq$\IfConf{}{\\}$M \bar{\beta}^2(\tau) \left|z_2\right|^2$.
Therefore, we use $\left\langle z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1)\right\rangle \leq M \bar{\beta}(\tau) \left|z_2\right|^2$
to upper bound the last term of \eqref{eqn:finalDotVNSCVX} as follows:\IfConf{
\begin{align}
& - \frac{\zeta^2}{M}\left(1 - \bar{\beta}(\tau)\bar{a}(\tau)\right)\left\langle z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle \nonumber\\
& \leq - \zeta^2\left(1 - \bar{\beta}(\tau)\bar{a}(\tau)\right)\bar{\beta}(\tau)\left|z_2\right|^2.
\end{align}}{
\begin{align}
& - \frac{\zeta^2}{M}\left(1 - \bar{\beta}(\tau)\bar{a}(\tau)\right)\left\langle z_2, \nabla L(\bar{v}_1(z,\tau)) - \nabla L(z_1) \right\rangle \nonumber\\
& \leq - \zeta^2\left(1 - \bar{\beta}(\tau)\bar{a}(\tau)\right)\bar{\beta}(\tau)\left|z_2\right|^2.
\end{align}}
Let $\bar{v}_2(\tau) := \zeta^2\left(1 - \bar{\beta}(\tau)\bar{a}(\tau)\right)\bar{\beta}(\tau)$. Since the function $\bar{\beta}$ is defined as in \eqref{eqn:dBarBetaBar}, then $\bar{\beta}(\tau) < 0$ for $\tau \in [0,1)$ and $\bar{\beta}(\tau) \geq 0$ for $\tau \geq 1$. Consequently, since, additionally, $\bar{a}$ is defined as in \eqref{eqn:BarA}, then $\bar{v}_2(\tau) < 0$ for $\tau \in [0,1)$ and $\bar{v}_2(\tau) \geq 0$ for $\tau \geq 1$. This leads to the following cases:
\begin{equation} \label{eqn:MJ_dotVNCVX}
\dot{V}_1(z,\tau) \leq
\begin{cases}
-\bar{a}(\tau)V_1(z(\tau),\tau) - \bar{v}_2(\tau)\left|z_2(\tau)\right|^2, & \!\!\!\! \tau \in [0,1)\\
-\bar{a}(\tau)V_1(z(\tau),\tau), & \!\!\!\! \tau \geq 1
\end{cases}
\end{equation}
Applying Gr\"{o}nwall's Inequality to the second case of \eqref{eqn:MJ_dotVNCVX}, namely,\IfConf{ \vspace{-0.2cm}
\begin{align*}
V_1(z(t),t) & \leq V_1(z(1),1) \exp \left(- \int_{1}^{t} \bar{a}(\tau) d\tau\right) \\
& = V_1(z(1),1) \exp \left(-2\ln\left(t+2\right) -2\ln\left(3\right)\right) \\
& = V_1(z(1),1)\exp \left(-\ln \left(\frac{t+2}{3}\right)^2\right) \\
& = V_1(z(1),1)\left(\frac{1}{\exp \left(\ln \left(\frac{t+2}{3}\right)^2\right)}\right) \\
& = \frac{9}{(t+2)^2}V_1(z(1),1)
\end{align*}}{
\begin{align*}
V_1(z(t),t) & \leq V_1(z(1),1) \exp \left(- \int_{1}^{t} \bar{a}(\tau) d\tau\right) \\
& = V_1(z(1),1) \exp \left(-2\ln\left(t+2\right) -2\ln\left(3\right)\right) \\
& = V_1(z(1),1)\exp \left(-\ln \left(\frac{t+2}{3}\right)^2\right) \\
& = V_1(z(1),1)\left(\frac{1}{\exp \left(\ln \left(\frac{t+2}{3}\right)^2\right)}\right) \\
& = \frac{9}{(t+2)^2}V_1(z(1),1)
\end{align*}}
shows that each maximal solution $t \mapsto (z(t),\tau(t))$ to the closed-loop algorithm $\HS_1$, such that $\tau(0) = 0$, satisfies \eqref{eqn:ConvergenceRateNCVX}, for all $t \geq 1$.
\section{Results on Suboptimality}
\label{sec:Suboptimality}
\section{Problem Statement and General Uniting Framework}
\label{sec:ProblemStatement}
\subsection{Problem Statement}
As illustrated in \cref{fig:MotivationalPlotTR}, the performance of Nesterov's accelerated gradient descent commonly suffers from oscillations near the minimizer. This is also the case for the heavy ball method when $\lambda > 0$ is small. However, when $\lambda$ is large, the heavy ball method converges slowly, albeit without oscillations. In \Cref{sec:Intro} we discussed how the heavy ball algorithm guarantees an exponential rate for strongly convex $L$ and a rate of $\frac{1}{t}$ for nonstrongly convex $L$, although it was demonstrated in \cite{sebbouh2019nesterov} that the heavy ball algorithm converges exponentially for nonstrongly convex $L$ when such an objective function also has the property of quadratic growth away from its minimizer. We also discussed how Nesterov's algorithm guarantees an exponential convergence rate for strongly convex $L$ and a rate of $\frac{1}{t^2}$ for nonstrongly convex $L$. We desire to attain such rates, while avoiding oscillations via the heavy ball algorithm with large $\lambda$. We state the following general problem to solve as follows:
\begin{problem}\label{problem:ProbStatement}
Given a scalar, real-valued, continuously differentiable objective function $L$ with a set of minimizers that is a singleton or a connected continuum set, and given two optimization algorithms $\kappa_0$ and $\kappa_1$, with $\kappa_0$ to be used locally and $\kappa_1$ to be used globally, design a uniting optimization framework that preserves the convergence rates of $\kappa_0$ and $\kappa_1$, without oscillations and with uniformity with respect to the compact sets of initial conditions, without knowing the function $L$ or the location of its minimizer, and with robustness to arbitrarily small noise in measurements of $\nabla L$.
\end{problem}
In this paper, we propose a framework for logic-based algorithms, which unite any two continuous-time, gradient-based optimization algorithms $\kappa_0$ and $\kappa_1$ to solve \cref{problem:ProbStatement}. The central idea is that the global optimization algorithm $\kappa_1$ provides fast convergence to the neighborhood of the set of minimizers and the local optimization algorithm $\kappa_0$ provides stable convergence in the neigborhood of the set of minimizers, without oscillations. A logic variable is used to indicate which algorithm -- either $\kappa_0$ or $\kappa_1$ -- is currently in use, and the switch between local and global algorithms is based on sublevel sets of the Lyapunov functions of $\kappa_0$ and $\kappa_1$.
One difficulty in designing such a uniting framework is that the objective function $L$ and the set of minimizers are unknown, so the algorithm must be able to detect when to switch, and do so in a way that avoids chattering.
\subsection{Modeling of the Uniting Framework}
\label{sec:ModelingUF}
We interpret the ODEs in \cref{eqn:MJODE}, \cref{eqn:MJODENCVX}, and \cref{eqn:HBF} as control systems consisting of a plant and a control algorithm \cite{191} \cite{220}. Then, defining $z_1$ as $\xi$ and $z_2$ as $\dot{\xi}$, the plant for these ODEs is given by
the double integrator
\begin{equation}\label{eqn:HBFplant-dynamicsTR}
\matt{\dot{z}_1\\ \dot{z}_2\\} = \matt{z_2\\ u} =: F_P(z,u) \qquad (z,u) \in \reals^{2n} \times \reals^n =: C_P
\end{equation}
{\color{red}With this model, the class of optimization algorithms that we consider assign $u$ to a function of the state that involves the cost function, and such a function of the state may be time dependent. For instance, if the ODE in \cref{eqn:MJODE} is used, we have $u = \kappa(h(z)) = -2dz_2 - \frac{1}{M_{\zeta}} \nabla L(z_1 + \beta z_2)$, where $M_{\zeta} := M\zeta^2$, $M > 0$, $\zeta > 0$, and where $d$ and $\beta$ are defined a subsequent section. If the ODE in \cref{eqn:MJODENCVX} is used, we have $u = \kappa(h(z,t)) := -2\bar{d}(t)z_2 - \frac{1}{M_{\zeta}}\nabla L(z_1 + \bar{\beta}(t)z_2)$, where $\bar{d}$ and $\bar{\beta}$ are defined in a subsequent section. If the ODE in \cref{eqn:HBF} is used, then we have $u = \kappa(h(z)) = -\lambda_{q} z_2 - \gamma_{q} \nabla L(z_1)$, with tunable parameters $\lambda > 0$ and $\gamma > 0$.}
For the framework presented in this section, we cope with the trade-off between damping oscillations and converging fast by uniting two control algorithms $\kappa_{q}$, where the logic variable $q \in Q := \{0,1\}$
indicates which algorithm is currently being used. The algorithm defined by $\kappa_1$, which plays the role of the global algorithm in uniting control (see, e.g., \cite{220}), is used far from the minimizer and is designed to quickly get close to the critical point. The algorithm defined by $\kappa_0$, which plays the role of the local algorithm, is used near the minimizer and is designed to avoid oscillations. The switch between $\kappa_0$ and $\kappa_1$ is governed by a {\em supervisory} algorithm implementing switching logic. The supervisor selects between these two optimization algorithms, based on the plant's output and the optimization algorithm currently applied. The design of the logic and parameters of the individual algorithms is done using Lyapunov functions $V_{q}$, which take different forms depending on the specific optimization methods used for $\kappa_0$ and $\kappa_1$. Since the ODE in \cref{eqn:MJODENCVX} is time varying, and since solutions to hybrid systems are parameterized by $(t,j) \in \reals_{\geq 0} \times \naturals$, we employ the state $\tau$ to {\color{red}capture} ordinary time {\color{red}as a state variable, in this way, leading to a time-invariant hybrid system}.
To encapsulate the plant, static state-feedback laws, and the time-varying nature of the ODE in \cref{eqn:MJODENCVX}, we define a hybrid closed-loop system $\HS$ with state $x := \left(z, q, \tau\right) \in \reals^{2n} \times Q \times \reals_{\geq 0}$
{\color{red}as follows:
\begin{subequations} \label{eqn:HS-TimeVarying}
\begin{equation}
\left.\begin{aligned}
\dot{z} & = \matt{z_2 \\ \kappa_{q}(h_{q}(z,\tau))}\\
\dot{q} & = 0 \\
\dot{\tau} & = q
\end{aligned}\right\} \forall x \in C := C_0 \cup C_1
\end{equation}
\begin{equation}
\left.\begin{aligned}
z^+ & = \matt{z_1 \\ z_2}\\
q^+ & = 1-q\\
\tau^+ & = 0
\end{aligned}\right\} \forall x \in D := D_0 \cup D_1
\end{equation}
\end{subequations}
Its data is $(C,F,D,G)$ with $F(x) := \left(z_2,\kappa_{q}(h_{q}(z,\tau)),0,q \right)$ and $G(x) := \left(z_1,z_2,1-q,0\right)$.}
The reason that the state $\tau$ increases at the rate $q$ during flows and is reset to $0$ during jumps is that when the global algorithm is being used, $q = 1$ causes ordinary time to increase linearly, starting from $0$, and when the local algorithm is being used, $q = 0$ causes the time to stay at zero. Such an evolution prevents finite escape time during flows and also ensures that the set of interest is compact.
The sets $C_0$, $C_1$, $D_0$, and $D_1$ are defined as
\begin{subequations} \label{eqn:CAndDGradientsNestNSC}
\begin{align}
&C_0 := \U_0 \times \{0\} \times \reals_{\geq 0}, \ C_1 := \overline{\reals^{2n}\setminus \mathcal{T}_{1,0}} \times \{1\} \times \reals_{\geq 0}\\
&D_0 := \mathcal{T}_{0,1} \times \{0\} \times \reals_{\geq 0}, \ D_1 := \mathcal{T}_{1,0}\times \{1\} \times \reals_{\geq 0},
\end{align}
\end{subequations}
and where $\U_0$, $\mathcal{T}_{1,0}$, and $\mathcal{T}_{0,1}$ are defined differently, depending on the specific optimization algorithms employed for $\kappa_0$ and $\kappa_1$.
The outputs $h_{q}$ are also defined differently, based on the specific optimization methods used. \Cref{fig:FeedbackDiagram} shows the feedback diagram of this hybrid closed-loop system $\HS$. We denote, for each $q \in Q = \{0,1\}$, the closed-loop systems resulting from the individual optimization algorithms as $\HS_{q}$ {\color{red}with state $\left(z,\tau\right)$, which are given by
\begin{equation} \label{eqn:Hq}
\left.\begin{aligned}
\dot{z} & = \matt{z_2 \\ \kappa_{q}(h_{q}(z,\tau))}\\
\dot{\tau} & = 1
\end{aligned}\right\} \forall \left(z,\tau\right) \in \reals^{2n} \times \reals_{\geq 0}.
\end{equation}}
\begin{figure}[thpb]
\centering
\setlength{\unitlength}{1.0pc}
\begin{picture}(30.8,12)(0,0)
\normalsize
\put(0,2){\includegraphics[scale=0.2,trim={1.5cm 0cm 0cm 0.8cm},clip,width=11.5\unitlength]{Figures/Surface4.eps}}
\put(11.5,0.5){\includegraphics[scale=0.4,trim={0cm 0cm 0cm 0cm},clip,width=19\unitlength]{Figures/FeedbackDiagram2.eps}}
\put(18.5,4.5){{\bf Supervisor}}
\put(12.1,3.2){$\ \ \dot{q} = 0 \qquad \quad \ \dot{\tau} = q \ \ \ (z, q, \tau) \in C := C_0 \cup C_1$}
\put(12.1,1.8){$q^+ = 1- q \ \ \ \tau^+ = 0 \ \ \ (z, q, \tau) \in D := D_0 \cup D_1$}
\put(24.4,7.4){\bf plant}
\put(24.1,9.9){$\dot{z}_1 = z_2$}
\put(24.1,8.9){$\dot{z}_2 = u$}
\put(13.9,10.6){$\kappa_1(h_1(z,\tau))$}
\put(13.3,9.1){\bf global ($q = 1$)}
\put(13.9,7.5){$\kappa_0(h_0(z,\tau))$}
\put(13.6,5.9){\bf local ($q = 0$)}
\put(21,7){$q$}
\put(22.1,10.1){$u$}
\put(28.4,10.1){$h_{q}$}
\put(11.6,10.1){$h_{q}$}
\put(7.6,5.3){$\nabla L(z_1)$}
\put(5.5,3.8){$z_1^*$}
\put(6.7,3.3){$z_1$}
\put(8.8,4){$z_{1_{\circ}}$}
\end{picture}
\caption{Feedback diagram of the hybrid closed-loop system $\HS$ (on the right), in \eqref{eqn:HS-TimeVarying}, uniting global and local optimization algorithms. An example optimization problem to solve is shown on the left and, for this example optimization problem, measurements of the gradient are used for the input of $\kappa_{q}$.}
\label{fig:FeedbackDiagram}
\end{figure}
\subsection{Design}
The sets $\U_0$ and $\mathcal{T}_{1,0}$ need to be designed such that the supervisor can determine when the state component $z_1$ is close to the set of minimizers of $L$, which we denote as
\begin{equation} \label{eqn:setOfMinimizers}
\A_1 := \defset{z_1 \in \reals^n}{\nabla L(z_1) = 0\!\!},
\end{equation}
without knowledge of $z_1^*$ or $L^*$. To facilitate such a design,
we impose the following assumptions\footnote{For the case of strongly convex $L$, nonstrong convexity and quadratic growth is implied, since these are weaker properties than strong convexity \cite{drusvyatskiy2018error} \cite{necoara2019linear} \cite{boyd2004convex}} on $L$.
\begin{assumption} \label{ass:LisNSCVX}
$L$ is $\mathcal{C}^1$ and (nonstrongly) convex.
\end{assumption}
\begin{assumption}[Quadratic growth of $L$] \label{ass:QuadraticGrowth}
{\color{red}The function} $L$ has quadratic growth away from its minima $\A_1$; i.e., there exists $\alpha > 0$ such that\footnote{\Cref{ass:QuadraticGrowth} can be relaxed to a ball $\mathcal{B}^L_{\nu} = \defset{z_1 \in \reals^{n}\!\!}{\!\!L(z_1) - L^* < \nu\!\!}$ for some $\nu > 0$.}
\begin{equation} \label{eqn:QuadraticGrowth}
L(z_1) - L^* \geq \alpha \left| z_1 \right|_{\A_1}^2 \ \forall z_1 \in \reals^{n}.
\end{equation}
\end{assumption}
\begin{remark}
\Cref{ass:LisNSCVX}, which is a common assumption used in the analysis of optimization algorithms \cite{boyd2004convex} \cite{nesterov2004introductory}, ensures that the objective function is continuously differentiable, which is necessary for well-posedness of $\HS$. Additionally, the nonstrongly convex property in \cref{ass:LisNSCVX} restricts the objective function to having either a unique minimizer or a continuum of minimizers, rather than multiple isolated minimizers, and is used to construct sets $\U_0$ and $\mathcal{T}_{1,0}$ (defined below).
\Cref{ass:QuadraticGrowth} is also used for the construction of $\U_0$ and $\mathcal{T}_{1,0}$, as a means of determining when the state $z$ is near the minimizer of $L$, via measurements of the gradient. Such an assumption is also commonly used in the analysis of convex optimization algorithms \cite{drusvyatskiy2018error}, \cite{karimi2016linear}.
\end{remark}
The set $\U_0$ is defined, via \cref{def:Convexity}, and\\ \cref{lemma:ConvexSuboptimality}, as follows:
\begin{equation} \label{eqn:U0}
\U_0 := \defset{z \in \reals^{2n}}{\left| \nabla L(z_1) \right| \leq \tilde{c}_0,\frac{1}{2} \left|z_2\right|^2 \leq d_0}
\end{equation}
where the parameters $\tilde{c}_0 > 0$ and $d_0 > 0$ are designed so that $\U_0$ is in the region where $\kappa_0$ is used.
The set $\mathcal{T}_{1,0}$ is defined, via \cref{def:Convexity}, and \cref{lemma:ConvexSuboptimality}, as follows:
\begin{equation} \label{eqn:T10}
\mathcal{T}_{1,0} := \defset{z \in \reals^{2n}}{\left| \nabla L(z_1) \right| \leq \tilde{c}_{1,0}, \left|z_2\right|^2 \leq d_{1,0}}
\end{equation}
where the parameters $\tilde{c}_{1,0} \in (0, \tilde{c}_0)$ and $d_{1,0} \in (0, d_0)$ are designed such that $\mathcal{T}_{1,0}$ is contained in the interior of $\U_0$. When $q = 1$, $\left| \nabla L(z_1) \right| \leq \tilde{c}_{1,0}$, and $\left|z_2\right|^2 \leq d_{1,0}$, the supervisor will switch from the global algorithm $\kappa_1$ to the local algorithm $\kappa_0$. The constants $\tilde{c}_0$, $\tilde{c}_{1,0}$, $d_0$, and $d_{1,0}$ comprise the hysteresis necessary to avoid chattering at the switching boundary. Examples illustrating the design of $\U_0$ and $\mathcal{T}_{1,0}$ for specific cases of $\kappa_0$ and $\kappa_1$ are presented in \Cref{sec:SpecificCases}.
The set $\mathcal{T}_{0,1}$ should be designed such that the supervisor can determine that the state $z$ is far from the minimizer, when $z \in \mathcal{T}_{0,1}$ and $q = 0$, and make the switch back to $\kappa_1$ to ensure that the state $z$ reaches the neighborhood of the minimizer in finite time. Additionally, the set $\mathcal{T}_{0,1}$ should be designed such that, when used in combination with $\U_0$ and $\mathcal{T}_{1,0}$ to define \cref{eqn:CAndDGradientsNestNSC}, each solution $x$ to the hybrid closed-loop algorithm $\HS$ jumps no more than twice, and the set of interest is guaranteed to be at least weakly forward invariant.
To ensure that the hybrid closed-loop system $\HS$ in \cref{eqn:HS-TimeVarying}, with $C$ and $D$ defined via \cref{eqn:CAndDGradientsNestNSC}, is well-posed, and to ensure that every maximal solution to the hybrid closed-loop system $\HS$ is complete, we impose the following assumptions on the set $\mathcal{T}_{0,1}$ and the static state-feedback laws $\kappa_{q}(h_{q}(z,\tau))$.
\begin{assumption}[Closed sets and continuous static state-feedback laws]\label{ass:AssumpsForHBC}
\begin{enumerate}[label={(C\arabic*)},leftmargin=*]
\item \label{item:C1} The set $\mathcal{T}_{0,1}$ is closed;
\item \label{item:C2} {\color{red}The map $z \mapsto F_P(z,\kappa_{q}(h_{q}(z,\tau)))$, in \cref{eqn:HBFplant-dynamicsTR} with $u = \kappa(h(z,\tau))$, is Lipschitz continuous with constant $M_{{\color{red}\kappa}} > 0$, namely,
\begin{equation}
\left|F_P(z,\kappa_{q}(h_{q}(z,\tau_1))) - F_P(v,\kappa_{q}(h_{q}(v,\tau_2))) \right| \leq M_{\kappa} \left|z - v\right|
\end{equation}
for all $z,v \in \reals^{2n}$ and all $\tau_1,\tau_2 \in \reals_{\geq 0}$.}
\end{enumerate}
\end{assumption}
\begin{remark}
Whereas $C$ is closed by construction, \cref{item:C1} is needed to ensure that the set $D$ is closed. \Cref{item:C2} is used to ensure that the map $x \mapsto F(x)$ is continuous. The closure of $C$ and $D$ and the continuity of $F$ and $G$ are required for $\HS$ to be well-posed which, in turn, leads to robustness of the asymptotic stability our framework guarantees, as stated in results to follow. Additionally, \cref{item:C2} is an assumption commonly used in nonlinear analysis to ensure that the differential equations of the individual optimization algoriths, for example those in \cref{eqn:MJODE,eqn:MJODENCVX,eqn:HBF}, do not have solutions that escape in finite time, which is used to guarantee the existence of solutions to $\HS_{q}$ \cite[Theorem~3.2]{khalil2002nonlinear}.
\end{remark}
\subsection{Basic Properties of $\HS$} \label{sec:BasicProperties}
Under \cref{ass:LisNSCVX,ass:AssumpsForHBC}, the hybrid closed-loop system $\HS$ in \cref{eqn:HS-TimeVarying}, with $C$ and $D$ defined via \cref{eqn:CAndDGradientsNestNSC}, is well-posed as it satisfies the hybrid basic conditions, as demonstrated in the following lemma.
\begin{lemma}[Well-posedness of $\HS$]
\label{lemma:HBCC}
Let $L$ satisfy \cref{ass:LisNSCVX,ass:AssumpsForHBC}. Then the hybrid closed-loop system $\HS$ satisfies the hybrid basic conditions, as listed in \cref{def:HBCs}.
\end{lemma}
\begin{proof}
The sets $\U_0$ and $\mathcal{T}_{1,0}$ are closed by construction, and by \cref{item:C1} $\mathcal{T}_{0,1}$ is closed. Therefore, the sets $D_0$, $D_1$, $C_0$, and $C_1$ are closed. Since $D$ is a finite union of finite sets, then $D$ is closed, and since $C$ is a finite union of finite and closed sets, then $C$ is also closed.
By construction the map $x \mapsto F(x)$ is continuous since, by \cref{item:C2},
{\color{red}$z \mapsto F_P(z,\kappa_{q}(h_{q}(z,\tau)))$}
is Lipschitz continuous. The map $G$ satisfies \cref{item:A3} by construction.
\end{proof}
As mentioned above, a hybrid closed-loop system $\HS$ that satisfies the hybrid basic conditions is well-posed. Such a property is important because when the hybrid closed-loop system $\HS$ is well-posed and has a compact pre-asymptotically stable set, we can show that such pre-asymptotic stability is robust to small perturbations \cite[Theorem~7.21]{65}. In the case of gradient-based algorithms, for instance, such perturbations can take the form of small noise in measurements of the gradient.
When \cref{ass:LisNSCVX,ass:QuadraticGrowth,ass:AssumpsForHBC} hold, then every maximal solution to the hybrid closed-loop system $\HS$ is complete and bounded.
\begin{proposition}[Existence of solutions to $\HS$]
\label{prop:ExistenceC}
Let $L$ satisfy \cref{ass:LisNSCVX,ass:QuadraticGrowth} and let the set $\mathcal{T}_{0,1}$ and the
{map \color{red}$z \mapsto F_P(z,\kappa_{q}(h_{q}(z,\tau)))$} {\color{red}, $q \in \{0,1\}$,} satisfy \cref{ass:AssumpsForHBC}. Furthermore, let $\U_0$ and $\mathcal{T}_{1,0}$ be defined via \cref{eqn:U0} and \cref{eqn:T10}, respectively. Then, each {\color{red}maximal solution\\ $(t,j) \mapsto x(t,j) = (z_1(t,j),z_2(t,j),q(t,j),\tau(t,j))$ to $\HS$ is bounded and complete.}
\end{proposition}
\begin{proof}
Since \cref{ass:LisNSCVX,ass:AssumpsForHBC} hold, then $\HS$ is well-posed. For points $x \in C \setminus D$, the tangent cone $T_C(x) \in \reals^{2n}$. Therefore $F$ satisfies (VC) of \cref{prop:SolnExistence}, and nontrivial solutions exist for every initial point in $C \cup D$.
To prove that item (c) of \cref{prop:SolnExistence} does not hold, we need to show that $G(D) \subset C \cup D$. For $D$, as defined in \cref{eqn:CAndDGradientsNestNSC}, $G(D)$ is as follows
\begin{equation} \label{eqn:CG(D)}
G(D) = \left(\mathcal{T}_{0,1} \times \{1\} \times \reals_{\geq 0} \right) \cup \left(\mathcal{T}_{1,0} \times \{0\} \times \reals_{\geq 0} \right)
\end{equation}
Notice that $\left(\mathcal{T}_{1,0} \times \{0\} \times \reals_{\geq 0} \right) \subset C_0$ and $\left(\mathcal{T}_{0,1} \times \{1\} \times \reals_{\geq 0} \right) \subset C_1$. Therefore,\\ $G(D) \subset C$, and this means also that $G(D) \subset C \cup D$. Therefore, item (c) of \cref{prop:SolnExistence} does not hold. Then it remains to prove that item (b) does not apply.
Since by \cref{item:C2},
{\color{red}$z \mapsto F_P(z,\kappa_{q}(h_{q}(z,\tau)))$} is Lipschitz continuous, {\color{red}and since $\tau$ increases linearly},
then by \cite[Theorem~3.2]{khalil2002nonlinear},
{\color{red}there is} no finite time escape for $\HS_{q}${\color{red}, defined via \cref{eqn:Hq}}. Therefore, every maximal solution {\color{red}$(z,\tau)$} to the individual optimization algorithms $\HS_{q}$ is complete, with unique solutions. Therefore, this means $\dot{x} = F(x)$ has no finite time escape from $C$ for $\HS$, as $q$ does not change in $C$ and as $\tau$ is bounded in $C$, namely, $\tau$ -- which is always reset to $0$ in $D$ -- increases linearly in $C_1$ and remains at $0$ in $C_0$. Moreover, the jump map $G$, defined via \cref{eqn:HS-TimeVarying}, shows that when the system is in $D$, $z$ does not change, $q$ is reset from $0$ to $1$ or vice versa, and $\tau$ is always reset to $0$. Therefore, by the construction of $G$, and by the properties of solutions to
{\color{red}$\HS_{q}$},
there is no finite time escape from $C \cup D$, for solutions to $\HS$. Therefore, item (b) from \cref{prop:SolnExistence} does not hold. This means only item (a) is true, and every maximal solution $x$ to $\HS$ is {\color{red}bounded and} complete.
\end{proof}
As evidenced by \cref{def:Convexity} in \Cref{sec:Preliminaries}, the set of nonstrongly convex functions includes a subset of functions $L$ which have a connected continuum of minimizers, with the most extreme case being linear functions \cite{boyd2004convex} \cite{nocedal2006numerical}. Therefore, we make the following assumption\footnote{When $L$ is strongly convex, then $L$ has a single minimizer, which implies compact and connected $\A_1$.}, used in the results to follow.
\begin{assumption}[Compact and connected $\A_1$]
\label{ass:CompactAndConnected}
The set $\A_1$, defined in \cref{eqn:setOfMinimizers}{\color{red},} is compact and connected.
\end{assumption}
To establish uniform global asymptotic stability of the set of interest for the hybrid closed-loop algorithm $\HS$, we impose the following assumptions on the closed-loop algorithms $\HS_{q}$ in \cref{eqn:Hq}.
We also make assumptions about the sets $\U_0$, $\mathcal{T}_{1,0}$, and $\mathcal{T}_{0,1}$, to ensure that each solution $x$ to the hybrid closed-loop algorithm $\HS$ jumps no more than twice, and ensure that the set of interest is guaranteed to be at least weakly forward invariant for $\HS$.
\begin{assumption}[{\color{red}Assumptions on asymptotic stability and attractivity}] \label{ass:UnitingControl}\\
Given $\A_1 \times \{0\} {\color{red}\times \reals_{\geq 0}} \in \reals^{2n} {\color{red}\times \reals_{\geq 0}}$ and a plant defined via \cref{eqn:HBFplant-dynamicsTR} with $y = h{\color{red}_{q}}(z)$, {\color{red}given $\tilde{c}_0 > 0$ and $d_0 > 0$ defining the closed set $\U_0 \in \reals^{2n}$ via} \cref{eqn:U0}, the interior of which contains an open neighborhood of $\A_1 \times \{0\} {\color{red}\times \reals_{\geq 0}} \in \reals^{2n} {\color{red}\times \reals_{\geq 0}}$, and {\color{red}given $\tilde{c}_{1,0} \in (0,\tilde{c}_0)$ and $d_{1,0} \in (0,d_0)$ defining the closed set $\mathcal{T}_{1,0} \subset \U_0$ via} \cref{eqn:T10}, the following conditions {\color{red}hold:}
\begin{enumerate}[label={(UC\arabic*)},leftmargin=*]
\item \label{item:UC1} The closed-loop algorithm $\HS_1$, resulting from $\kappa_1$, has the set $\A_1 \times \{0\} {\color{red}\times \reals_{\geq 0}}$ uniformly globally attractive, namely, every maximal solution {\color{red}$\chi := (z,\tau)$} to $\HS_1${\color{red}, given by \cref{eqn:Hq} with $u = \kappa_1(h_1(z,\tau))$,} is complete and for each $\eps > 0$ and $r > 0$ there exists $T > 0$ such that, for any solution {\color{red}$\chi$} to $\HS_1$ with $\left|{\color{red}\chi}(0)\right|_{\A_1 \times \{0\} {\color{red}\times \reals_{\geq 0}}} \leq r$, $t \in \dom {\color{red}\chi}$ and $t \geq T$ imply\footnote{{\color{red} Although we employ $\tau$ as a state to capture ordinary time, we can use just $t$ to parameterize solutions.}} $\left|{\color{red}\chi}(t)\right|_{\A_1 \times \{0\} {\color{red}\times \reals_{\geq 0}}} \leq \eps$;
\item \label{item:UC2} There exist positive constants $\tilde{c}_1 \in (0,\tilde{c}_{1,0})$ and $d_1 \in (0,d_{1,0})$ such that the closed set $\mathcal{T}_{1,0}$ satisfies
\begin{equation} \label{eqn:c1d1Set}
\left(\A_1 + \tilde{c}_1 \ball \right) \times \left(\{0\} + d_1 \ball\right) \subset \mathcal{T}_{1,0}
\end{equation}
and each solution to {\color{red}$\HS_0$} in \cref{eqn:Hq} with initial condition in $\mathcal{T}_{1,0} {\color{red}\times \reals_{\geq 0}}$, resulting from applying $\kappa_0$, remains in $\U_0$;
\item \label{item:UC3} The closed-loop algorithm $\HS_0$, resulting from $\kappa_0$, has a set $\A_1 \times \{0\} {\color{red}\times \reals_{\geq 0}}$ uniformly globally asymptotically stable, namely, every maximal solution {\color{red}$\chi$} to $\HS_0${\color{red}, given by \cref{eqn:HBFplant-dynamicsTR} with $u = \kappa_0(h_0(z,\tau))$,} is complete, $\A_1 \times \{0\} {\color{red}\times \reals_{\geq 0}}$ is uniformly globally attractive for $\HS_0$, and there exists a class-$\mathcal{K}_{\infty}$ function $\alpha$ such that any solution {\color{red}$\chi$} to $\HS_0$ satisfies $\left|{\color{red}\chi}(t)\right|_{\A_1 \times \{0\} {\color{red}\times \reals_{\geq 0}}} \leq \alpha \left(\left|{\color{red}\chi}(0)\right|_{\A_1 \times \{0\} {\color{red}\times \reals_{\geq 0}}}\right)$ for all $t \in \dom {\color{red}\chi}$.
\item \label{item:UC4} There exists a constant $\delta_0 > 0$ and a closed set $\mathcal{T}_{0,1} + 2\delta_0 \ball \subset \overline{\reals^{2n}\setminus \U_0}$ such that when $z \in \mathcal{T}_{0,1}$ and $\kappa_0$ is currently being used, {\color{red}the hybrid closed-loop algorithm $\HS$ assigns $u$ to $\kappa_1$}.
\end{enumerate}
\end{assumption}
\begin{remark}
\Cref{ass:CompactAndConnected} ensures that the set $\A$ (defined below) is compact, which is required in the forthcoming analysis of the stability properties of $\HS$. \Cref{item:UC1,item:UC3} of \cref{ass:UnitingControl} ensure that $\HS_1$ and $\HS_0$ have the desired attractivity and stability properties, respectively, to establish the stability of the hybrid closed-loop algorithm $\HS$. Such assumptions are reasonable in light of the numerous results in the literature for gradient-based optimization algorithms, cited in \Cref{sec:Intro}. Furthermore, the conditions in \Cref{ass:UnitingControl} are similar to conditions imposed in \cite{220} for general uniting control algorithms.
\Cref{item:UC1} ensures that solutions {\color{red}$\left(z,\tau\right)$ starting with $z$ in $\U_0$, with $q = 0$ and $\tau \in \reals_{\geq 0}$, stay in $\U_0$ and converge to $\A_1 \times \{0\} \times \reals_{\geq 0}$} under the affect of $\kappa_0$.
\Cref{item:UC4} guarantees that solutions starting with the state $z \in T_{0,1}$ and with $q = 0$ triggers a jump resetting $q$ to $1$. \Cref{item:UC3} guarantees that, after such a jump, $z$ reaches $\mathcal{T}_{1,0}$ in finite time with $\kappa_1$ applied.
\Cref{item:UC2} ensures that solutions from $\mathcal{T}_{1,0}$ under the effect of $\kappa_0$ cannot reach the boundary of $\U_0$.
\end{remark}
The following theorem establishes that the hybrid closed-loop system $\HS$ in \cref{eqn:HS-TimeVarying}-\cref{eqn:CAndDGradientsNestNSC} has the set
\begin{equation} \label{eqn:SetOfMinimizersHS-NSCVX}
\A := \defset{z \in \reals^{2n}}{\nabla L(z_1) = z_2 = 0} \times \{0\} \times \{0\} = \A_1 \times \{0\} \times \{0\} \times \{0\}
\end{equation}
uniformly globally asymptotically stable. The penultimate component $\{0\}$ in $\A$ is due to the logic state ending with the value $q = 0$, namely using $\kappa_0$ as the state $z$ reaches the set of minimizers of $L$. The last component $\{0\}$ in $\A$ is due to $\tau$ being set to and then staying at zero when the supervisor switches to $\kappa_0$. To prove this result, we use an invariance principle.
\begin{theorem}[Uniform global asymptotic stability of $\A$ for $\HS$] \label{thm:GASGen}
Let $L$ satisfy \cref{ass:LisNSCVX,ass:QuadraticGrowth,ass:CompactAndConnected}, let the
{\color{red}map $z\mapsto F_P(z,\kappa_{q}(h_{q}(z,\tau)))$, $q \in \{0,1\}$, in \cref{eqn:HBFplant-dynamicsTR}}
satisfy \cref{ass:AssumpsForHBC}, let the closed-loop optimization algorithms $\HS_{q}$ {\color{red}in \cref{eqn:Hq}} and the sets $\U_0$, $\mathcal{T}_{1,0}$, and $\mathcal{T}_{0,1}$ satisfy \cref{ass:UnitingControl}. Additionally, let $\delta_0 > 0$, $\tilde{c}_{1,0} \in (0,\tilde{c}_0)$, $\tilde{c}_1 \in (0,\tilde{c}_{1,0})$, $d_{1,0} \in (0,d_0)$, and $d_1 \in (0,d_{1,0})$. Then, the set $\A$, defined via \eqref{eqn:SetOfMinimizersHS-NSCVX}, is uniformly globally asymptotically stable for $\HS$ {\color{red}given in \cref{eqn:HS-TimeVarying}-\cref{eqn:CAndDGradientsNestNSC}}.
\end{theorem}
\begin{proof}
The hybrid closed-loop algorithm $\HS$ satisfies the hybrid basic conditions by \cref{lemma:HBCC}, satisfying the first assumption of \cref{thm:hybrid Lyapunov theorem}. Furthermore, every maximal solution $(t,j) \mapsto x(t,j) = (z_1(t,j), z_2(t,j), q(t,j),\tau(t,j))$ to $\HS$ for \cref{eqn:HS-TimeVarying}-\cref{eqn:CAndDGradientsNestNSC} is complete and bounded by \cref{prop:ExistenceC}. By \cref{ass:CompactAndConnected}, $\A_1$ is compact and connected.
Therefore, $\A$, defined via \cref{eqn:SetOfMinimizersHS-NSCVX}, is compact by construction, and $\U = \reals^{2n}$ contains a nonzero open neighborhood of {\color{red}$\A_1$}.
To prove the attractivity of $\A$, we use a trajectory based approach which uses the properties of the individual optimization algorithms $\HS_0$ and $\HS_1$. The possible trajectory types for $\HS$ are:
\begin{enumerate}[label={\arabic*)},leftmargin=*]
\item Trajectories that start in $C_0$, namely, such that $z(0,0) \in \U_0$, $q(0,0) = 0$, and $\tau(0,0) = 0$;
\item Trajectories that start in the interior of $D_1$, namely, such that $q(0,0) = 1$, $\tau(0,0) = 0$, and either $\left|\nabla L(z_1(0,0))\right| < \tilde{c}_{1,0}$, $\left|z_2(0,0)\right|^2 \leq d_{1,0}$ or $\left|\nabla L(z_1(0,0))\right| \leq \tilde{c}_{1,0}$, $\left|z_2(0,0)\right|^2 < d_{1,0}$;
\item Trajectories that start in the interior of $C_1$, namely, such that $q(0,0) = 1$, $\tau(0,0) = 0$, and either $\left|\nabla L(z_1(0,0))\right| > \tilde{c}_{1,0}$, $\left|z_2(0,0)\right|^2 \geq d_{1,0}$ or $\left|\nabla L(z_1(0,0))\right| \geq \tilde{c}_{1,0}$, $\left|z_2(0,0)\right|^2 > d_{1,0}$;
\item Trajectories that start in $D_0$, namely, such that $z(0,0) \in \mathcal{T}_{0,1}$, $q(0,0) = 0$, $\tau(0,0) = 0$.
\end{enumerate}
First we analyze trajectories of type 1). By \cref{item:UC4}, the set $\mathcal{T}_{0,1} + 2\delta_0 \ball \subset \overline{\reals^{2n}\setminus\U_0}$. Therefore, $\U_0$ does not share a boundary with $\mathcal{T}_{0,1}$ and, when $z(0,0) \in \U_0$, including on the boundary of $\U_0$, then by \cref{item:UC1} of \cref{ass:UnitingControl} solutions only flow using $\kappa_0$ for all future hybrid time in its domain.
Next we analyze trajectories of type 2). Due to the construction of $G$, defined via \cref{eqn:HS-TimeVarying}, when $z(0,0) \in D_1$ and $q = 1$, then the algorithm jumps, resetting $q$ to $0$ and assigning $u$ to $\kappa_0$ while $z$ does not change. After such a jump, due to the construction of $D_1$, defined via \cref{eqn:CAndDGradientsNestNSC} with $0 < \tilde{c}_{1,0} < \tilde{c}_0$ and $0 < d_{1,0} < d_0$, and due to \cref{item:UC1,item:UC2} of \cref{ass:UnitingControl}, the state $z$ remains in $\U_0$ due to the affect of $\kappa_0$ and from such a point the solution exhibits the same behavior as for trajectories of type 1).
Then, we analyze trajectories of type 3). When $z(0,0)$ starts in $C_1$, then the algorithm flows using $\kappa_1$. Then, the state $z$ reaches $D_1$ in finite time since, by \cref{item:UC3}, $\A_1 \times \{0\} {\color{red}\times \reals_{\geq 0}}$ is uniformly globally attractive for $\HS_1$ and therefore the state $z$ reaches $\left(\A_1 + \tilde{c}_1 \ball \right) \times \left(\{0\} + d_1 \ball\right)$ after finite flow time or as $t$ tends to $\infty$, from points starting in $\overline{\reals^{2n}\setminus \left(\left(\A_1 + \tilde{c}_1 \ball \right) \times \left(\{0\} + d_1\ball\right)\right)}$. After such a finite flow time, the algorithm jumps, with the solution exhibiting the same behavior as for trajectories of type 2).
Finally, we analyze trajectories of type 4). By \cref{item:UC4}, when $z(0,0)$ is in $\mathcal{T}_{0,1}$, and $q = 0$, then the algorithm jumps, resetting $q$ to $1$ and updating $u$ to $\kappa_1$. Then, the state $x$ is in $C_1$ and from such a point the solution exhibits exhibits the same behavior as for trajectories of type 3).
Due to the behaviors of trajectories 1)-5) as described above, and due to the properties in \cref{ass:UnitingControl}, every maximal solution $x$ to $\HS$ exhibits no more than two jumps. Since, additionally, every trajectory of the hybrid closed-loop algorithm $\HS$ ends with $q = 0$, and since $\tau$ is set to and then stays at zero when the supervisor switches to $\kappa_0$, then every bounded and complete solution to $\HS$ converges to $\A$. Generally, the Lyapunov theorem and the invariance principle prove global pre-asymptotic stability. But since by \cref{prop:ExistenceC} every maximal solution is complete, then $\A$, defined via \cref{eqn:SetOfMinimizersHS-NSCVX} is uniformly globally asymptotically stable for $\HS$.
\end{proof}
The framework defined in \cref{eqn:HS-TimeVarying}-\cref{eqn:CAndDGradientsNestNSC} allows for combinations of different methods, including Nesterov's algorithm, the heavy ball method, classic gradient descent, and the triple momentum method \cite{van2017fastest} \cite{sun2020high}, to name a few examples. In the next section, we present specific cases utilizing combinations of Nesterov's algorithm and the heavy ball algorithm.
|
1,314,259,996,483 | arxiv | \section{Introduction}
Long gamma-ray bursts (GRBs) are powerful explosions occurring at
cosmological distances, linked to the death of massive stars. The
gamma-ray (or prompt) event is followed by an afterglow at longer
wavelengths, which is crucial in order to understand the physics of
these sources, but also to investigate the nature of the interstellar
medium (ISM) of high redshift galaxies. Before GRBs, such studies made
use of Lyman-break galaxies (LBGs, see e.g. Steidel et al. 1999) and
galaxies that happen to be along the lines of sight to bright
background quasars, commonly referred to as QSO-Damped Lyman Alpha
(DLA) systems. However, both classes are entangled by selection
effects. In fact, LBGs fall in the bright end of the galaxy luminosity
function and may not entirely represent typical high-redshift
galaxies. On the other hand, QSO sightlines preferentially probe
galaxy halos, rather than bulges or discs, for cross-section effects
(Fynbo et al. 2008). Indeed, Savaglio et al. (2004; 2005) studied the
ISM of a sample of faint $K$-band selected galaxies at $1.4 < z <
2.0$, finding {Mg}{II} and {Fe}{II} abundances much higher than in QSO
systems but similar to those in gamma-ray burst hosts. Unfortunately,
these galaxies are too faint to be spectroscopically studied up to
higher redshifts, using 8m class telescopes. In this context, long
GRBs can be used as torchlights to illuminate the high-redshift ISM,
and thus represent an independent tool to study high-redshift
galaxies.
Several papers report a metallic content in GRB host galaxies in the
range $10^{-2} - 1 $ with respect to solar values (see e.g., Fynbo et
al. 2006; Savaglio 2006; Prochaska et al. 2007). The GRB host
metallicity is thus on average higher than in QSO-DLA systems,
supporting the notion that GRBs explode well within their hosts. Since
long GRBs are linked to the death of massive stars, they are though to
originate in molecular clouds. In this scenario, absorption from
ground-state and vibrationally excited levels of H$_2$ and other
molecules is expected, but not observed (Vreeswijk et al. 2004;
Tumlinson et al. 2007). The non-detection of these molecular states
(with the exception of GRB\,080607, see Prochaska et al. 2009; Sheffer
et al. 2009) could be a consequence of the intense UV flux from the
GRB afterglow, which photo-dissociates the molecules. However,
molecular hydrogen is not detected in QSO-DLA either (e.g., Noterdaeme
et al. 2008; Tumlinson et al. 2007), possibly indicating that these
molecules are just hard to see at high redshift.
This is just an example of how a GRB can modify its surrounding
medium. The most impressive manifestation of the transient nature of
GRBs in optical spectroscopy is the detection of strong absorption
features related to the excited levels of the {{O}{I}}, {{Fe}{II}},
{{Ni}{II}}, {{Si}{II}} and {{C}{II}} species and their time
variability (Vreeswijk et al. 2007). This variability can not be
explained assuming infrared excitation or collisional processes
(Prochaska, Chen \& Bloom 2006; Vreeswijk et al. 2007; D'Elia et
al. 2009a), thus excitation by the intense GRB UV flux is the leading
mechanism to produce these features. In this framework, the
GRB/absorber distance can be evaluated comparing the observed
ground state and excited level abundances with that predicted by
time-dependent photo-excitation codes. This distance turns out to
be in the range $\sim 0.1-1$ kpc (Vreeswijk et al. 2007; D'Elia et
al. 2009a,b; Ledoux et al. 2009).
Within the described framework, the best and most complete tool to
perform these kind of studies is high resolution spectroscopy. In
fact, it is the only way to disentangle the GRB interstellar medium in
components and to separate the contribution to the absorption coming
from the excited levels from the ground state ones. In addition, a
high spectral resolution allows us to check for saturation of
lines a few km s$^{-1}$ wide that may appear unsaturated in lower
resolution spectra (see e.g. Penprase et al. 2010).
In this paper we present data on GRB\,081008, observed both in high
and low resolution UVES and FORS2 at the VLT.
The paper is organized as follows. Section $2$ summarizes the
GRB\,081008 detection and observations from the literature; Sect. $3$
presents the UVES observations and data reduction; Sect. $4$ is
devoted to the study of the features from the host galaxy, in
particular their metallicity and distance from the GRB explosion
site; Sect. $5$ presents the FORS2 data and makes a comparison with
the UVES ones; finally in Sect. $6$ the results are discussed and
conclusions are drawn. We assume a concordance cosmology with $H_0=70$
km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\rm m} = 0.3$, $\Omega_\Lambda =
0.7$. Hereafter, with [X/H] we refer to the X element abundance
relative to solar values.
\section{GRB\,081008}
GRB\,081008 was discovered by {\it Swift}/BAT on October 8, 2008, at
19:58:29 UT, and was detected by both the XRT and the UVOT instruments
(Racusin et al. 2008). The UVOT magnitude in the white filter was
reported to be $15.0$ at $96$ s from the trigger. The afterglow was
also detected in all filters (from {\it B} to {\it K}) by SMARTS/ANDICAM $\sim 4$
hr post burst (Cobb 2008). The redshift was secured by the
Gemini-South/GMOS, which observed the afterglow $5$ hr after the {\it
Swift} trigger, reporting a redshift of $z=1.967$ (Cucchiara et
al. 2008a). This value was later confirmed by our VLT/UVES+FORS2 data
(D'Avanzo et al. 2008). The host galaxy was identified in the
Gemini-South/GMOS acquisition image, and spectroscopically confirmed
to be at the GRB redshift. The host of GRB\,081008 has $R=20.75\pm
0.01$, which corresponds to an absolute AB magnitude of $-21.5$
(Cucchiara et al. 2008b). A multi-wavelength study of the prompt event
and the early afterglow phase of GRB\,081008 is reported by Yuan et
al. (2010, hereafter Y10), which present {\it Swift} (BAT+XRT+UVOT),
ROTSE-III and GROND data.
\begin{table}
\begin{center}
\caption{Rest frame equivalent widths of the UVES features.}
{\footnotesize
\smallskip
\begin{tabular}{|l|c|c|c|}
\hline
Species &Transition & W$_r$ (\AA) & $\Delta$ W$_r$ (\AA, $1\sigma$) \\
\hline
OI$^3P_{2}$ (g.s) &$1302$ &$0.57$ &$0.05$ \\
\hline
AlII$^1S_0$ (g.s) &$1670$ &$0.67$ &$0.01$ \\
\hline
AlIII$^2S_{1/2}$ (g.s) &$1854$ &$0.22$ &$0.01$ \\
&$1862$ &$0.13$ &$0.02$ \\
\hline
SiII$^2P^{0}_{1/2}$ (g.s) &$1260$ &$0.63$ &$0.07$ \\
&$1808$ &$0.20$ &$0.02$ \\
\hline
SiII$^2P^{0}_{3/2}$ (1*) &$1264$ &$0.63$ &$0.07$ \\
&$1816$ &$0.06$ &$0.02$ \\
\hline
CrII$^2S_{1/2}$ (g.s.) &$2056$ &$0.20$ &$0.02$ \\
&$2062$ &$0.14$ &$0.02$ \\
&$2066$ &$0.11$ &$0.02$ \\
\hline
FeII$a^6D_{9/2}$ (g.s) &$2249$ &$0.12$ &$0.01$ \\
&$2260$ &$0.20$ &$0.02$ \\
\hline
FeII$a^6D_{7/2}$ (1*) &$1618$ &$0.05$ &$0.01$ \\
&$1621$ &$0.11$ &$0.01$ \\
\hline
FeII$a^6D_{5/2}$ (2*) &$1629$ &$0.03$ &$0.01$ \\
\hline
FeII$a^6D_{3/2}$ (3*) &$1634$ &$0.03$ &$0.01$ \\
&$1636$ &$0.03$ &$0.01$ \\
\hline
FeII5s$a^4F_{9/2}$ (5*) &$1637$ &$0.04$ &$0.01$ \\
&$1612$ &$0.12$ &$0.01$ \\
&$1702$ &$0.21$ &$0.02$ \\
\hline
FeII$a^4D_{7/2}$ (9*) &$1635$ &$0.03$ &$0.01$ \\
\hline
NiII$^2D_{5/2}$ (g.s.) &$1741$ &$0.07$ &$0.02$ \\
\hline
NiII$^4F_{9/2}$ (2*) &$2166$ &$0.19$ &$0.01$ \\
&$2217$ &$0.27$ &$0.01$ \\
&$2223$ &$0.09$ &$0.02$ \\
\hline
ZnII$^2S_{1/2}$ (g.s.) &$2026$ &$0.19$ &$0.02$ \\
&$2062$ &$0.09$ &$0.02$ \\
\hline
\end{tabular}
}
\end{center}
\end{table}
\section{UVES observations and data reduction}
The GRB\,081008 afterglow was observed with the high resolution
UV-visual echelle spectrograph (UVES, Dekker et al. 2000), mounted at
the VLT-UT2 telescope, in the framework of the ESO program
082.A-0755. Observations began on the $9^{th}$ October 2008 at
00:16:43 UT ($\sim 4.25$ hr after the {\it Swift}/BAT trigger), when
the magnitude of the afterglow was $R\sim18.5$. Data were acquired
under good observing conditions, with seeing $\sim 0.7$. Only the
UVES-dichroic-1 (red and blue arm) was used due to observational and
scheduling constraints. The net exposure time of the observation is 30
minutes. The slit width was set to be $1''$ (corresponding to a
resolution of $R=40000$) and the read-out mode was rebinned to 2
$\times$ 2 pixels. The spectral range of our observation is
$\sim$3300\AA\ to $\sim$3870\AA, $\sim$4780\AA\ to $\sim$5750\AA, and
$\sim$5830\AA\ to $\sim$6810\AA. Table 1 makes a summary of our
observations.
\begin{figure*}
\centering
\includegraphics[angle=90,width=18cm]{Fig0_New.ps}
\caption{The full, {\bf smoothed and} normalized UVES spectrum. Solid lines
indicate the noise level as a function of wavelength. }
\end{figure*}
The data reduction was performed using the UVES pipeline (version
2.9.7, Ballester et al. 2000). The signal-to-noise ratio per pixel is
$\sim 3-5$ in the blue arm and $\sim 5-8$ in the red one. The noise
spectrum, used to determine the errors in the best-fit line
parameters, was calculated from the real, background-subtracted
spectrum, using line-free regions to evaluate the standard deviation
of continuum pixels. Since the noise spectrum has been produced after
the pipeline processing and the background subtraction, it takes into
account possible systematic errors coming from the data reduction
process. Fig. 1 shows the full, {\bf smoothed and} normalized UVES spectrum.
\begin{figure}
\centering
\includegraphics[angle=-0,width=9cm]{Fig1aNew.ps}
\includegraphics[angle=-0,width=9cm]{Fig1bNew.ps}
\caption{The {Fe}{II} ground and excited absorption features. Solid
lines represent the two Voigt components, best-fit model. Vertical
lines identify the component velocities. The zero point has been
arbitrarily placed at the redshift of the redmost component
($z=1.9683$). g.s. and n* indicate ground state and n-th excited
transitions, respectively.}
\label{spe1}
\end{figure}
\section{UVES data analysis}
The gas residing in the GRB host galaxy is responsible for many
features observed in the GRB\,081008 afterglow spectrum. Metallic
features are apparent from neutral ({{O}{I}}) and low-ionization
({{Al}{II}}, {{Al}{III}}, {{Si}{II}}, {{Cr}{II}}, {{Fe}{II}},
{{Ni}{II}}, {{Zn}{II}}) species. In addition, strong absorption lines
from the fine structure levels of {{Si}{II}}, {{Fe}{II}} and from the
metastable levels of {{Fe}{II}} and {{Ni}{II}} are identified,
suggesting that the intense radiation field from the GRB excites such
features. Table~2 gives a summary of all the absorption lines due to
the host galaxy gas and report their rest frame equivalent widths
(W$_r$). The spectral features were analyzed with FITLYMAN (Fontana
\& Ballester 1995), using the atomic parameters given in Morton
(2003). The probed ISM of the host galaxy is resolved into two main
components separated by $20$ km s$^{-1}$ (Figs. 2 and 3). The wealth
of metal-line transitions allows us to precisely determine the
redshift of the GRB host galaxy. This yields a vacuum-heliocentric
value of $z=1.9683 \pm 0.0001$, setting the reference point to the
redmost component (hereafter component I). The absorption
features have been fitted with Voigt profiles, fixing the redshift
of the two components when studying different lines. All
transitions appear to be nicely lined-up in redshift, with the
exception of component II of {Si}{II}$\lambda$1808. We attribute
this misalignment to a contamination of another feature, and fit
just the redmost side of component II. All ground state and
metastable species present absorption features in both components,
while fine structure levels in component I only.
Two sharp features can be seen at $v=\pm80$ km s$^{-1}$ from the
{Fe}{II} a$^6$D$_{5/2}$ line (Fig. 2). They can not be separated in
the FORS2 spectrum, thus we can not safely assess if they are real or
not. The Doppler $b$ parameter has been linked between different
excited transitions belonging to the same species. A small variation
of the Doppler parameter is allowed among different species, but the
fits are quite good even fixing it. The values for components I and II
are $\sim 10$ and $\sim 20$ km s$^{-1}$, respectively. An exception to
this behaviour is represented by component II of {Zn}{II}. In order to
obtain a good fit, a $b \sim 4$ and $\sim 50$ km s$^{-1}$ is required
for component I and II, respectively. This large $b$ value in
component II is necessary to adequately fit what appears as a low
level of the continuum in particular in the {Zn}{II}$\lambda$2026
feature. {\bf The large difference between the b parameters deduced
for {Zn}{II} and {Cr}{II} is odd, but we do not have a simple
explanation for it.} The column densities and b parameters for all
the elements and ions of the host galaxy's absorbing gas are reported
in Table 3. A third component is actually identified at $-78$ km
s$^{-1}$ for some low-ionization lines only, i.e., the
{O}{I}$\lambda$1302, {Al}{II}$\lambda$1670 and {Si}{II}$\lambda$1260,
and for the fine structure level {Si}{II}$\lambda$1264 (see
Fig. 4). These lines are however heavily saturated, and their column
densities reported in the table just set a lower limit to the true
values. The reported upper limits are at the 90\% confidence level.
\begin{figure}
\centering
\includegraphics[angle=-0,width=8.5cm,height=7cm]{Fig2aNew.ps}
\includegraphics[angle=-0,width=8.5cm,height=7cm]{Fig2bNew.ps}
\includegraphics[angle=-0,width=8.5cm,height=7cm]{Fig2cNew.ps}
\caption{The {Ni}{II} (top panel), {Al}{III} and {Si}{II}, (middle
panel), {Cr}{II} and {Zn}{II} (bottom panel) absorption
features. Solid lines represent the two Voigt components, best-fit
model. Vertical lines identify the component velocities. The zero
point has been arbitrarily placed at the redshift of the redmost
component ($z=1.9683$). g.s. and n* indicate ground state and n-th
excited transitions, respectively.}
\label{spe1}
\end{figure}
\begin{table*}
\begin{center}
\caption{Absorption line logarithmic column densities for the three components
of the main system, derived from the UVES spectrum.}
{\footnotesize
\smallskip
\begin{tabular}{|lc|cc|cc|cc|}
\hline
\hline
Species & Observed transitions & \multicolumn{6}{c}{N (cm$^{-2}$)} \\
\hline
{H}{I} $^2S_{1/2}$ & Ly$\alpha$ (UVES) &\multicolumn{6}{c}{$21.33 \pm 0.12$} \\
\hline
{H}{I} $^2S_{1/2}$ & Ly$\alpha$ (FORS2) &\multicolumn{6}{c}{$21.11 \pm 0.10$}\\
& Components &$20.82 \pm 0.14$ &($z_1=1.944$) &&&$20.79 \pm 0.12$ &($z_2=1.975$) \\
\hline
\hline
Metals & Components &\multicolumn{2}{c}{I ($0$ km s$^{-1}$)}&\multicolumn{2}{c}{II ($-20$ km s$^{-1}$)}&\multicolumn {2}{c}{III ($-78$ km s$^{-1}$)}\\
\hline
Species & Observed transitions & N (cm$^{-2}$)& b (km s$^{-1}$) & N (cm$^{-2}$)& b (km s$^{-1}$) & N (cm$^{-2}$)& b (km s$^{-1}$) \\
\hline
{O}{I} $^3P_{2}$ & $\lambda$1302 & $>14.34$ &SAT & $> 14.54$ &SAT& $> 14.88$ &SAT \\
\hline
{Al}{II} $^1S_0$ & $\lambda$1670 & $>12.82$ &SAT & $> 13.26$ &SAT& $> 12.78$ &SAT \\
\hline
{Al}{III} $^2S_{1/2}$ & $\lambda$1854, $\lambda$1862 & $12.96 \pm 0.04$ &$12$&$ 13.04\pm0.04$ &$17$& $< 12.3$ &N/A\\
\hline
{Si}{II} $^2P^{0}_{1/2}$ & $\lambda$1260, $\lambda$1808 & $15.44 \pm 0.03$ &$9$ &$15.08 \pm 0.18$ &$4$ & $>13.44$ &SAT \\
\hline
{Si}{II} $^2P^{0}_{3/2}$ & $\lambda$1264, $\lambda$1816 & $15.21 \pm 0.05$ &$9$ &$<15.0 $ &N/A & $< 15.0$ &N/A \\
\hline
{Cr}{II} $^2S_{1/2}$ & $\lambda$2056, $\lambda$2062, $\lambda$2066 & $13.51 \pm 0.05$ &$16$ &$13.55 \pm 0.04$ &$21$& $< 13.3$ &N/A\\
\hline
{Fe}{II} $a^6D_{9/2}$ & $\lambda$2249, $\lambda$2260 & $15.09 \pm 0.03$ &$12$ &$14.95 \pm 0.04$ &$20$& $< 14.7$ &N/A\\
\hline
{Fe}{II} $a^6D_{7/2}$ & $\lambda$1618, $\lambda$1621 & $13.95 \pm 0.05$ &$12$ & $ < 13.7$ &N/A & $< 13.7$ &N/A \\
\hline
{Fe}{II} $a^6D_{5/2}$ & $\lambda$1629 & $13.71 \pm 0.09$ &$12$ & $ < 13.7$ &N/A & $< 13.7$ &N/A\\
\hline
{Fe}{II} $a^6D_{3/2}$ & $\lambda$1634, $\lambda$1636 & $13.59 \pm 0.09$ &$12$ & $ < 13.5$ &N/A & $< 13.5$ &N/A\\
\hline
{Fe}{II} $a^4F_{9/2}$ & $\lambda$1612, $\lambda$1637, $\lambda$1702 & $14.19 \pm 0.05$ &$12$ &$13.76 \pm 0.07$ &$20$& $< 13.2$ &N/A\\
\hline
{Fe}{II} $a^4D_{7/2}$ & $\lambda$1635 & $13.35 \pm 0.12$ &$12$ &$< 13.15$ &$20$& $< 13.15$&N/A\\
\hline
{Ni}{II} $^2D_{5/2}$ & $\lambda$1741 & $13.52 \pm 0.08$ &$12$ &$13.33 \pm 0.13$ &$11$& $< 13.3$ &N/A\\
\hline
{Ni}{II} $^4F_{9/2}$ & $\lambda$2166, $\lambda$2217, $\lambda$2223 & $13.57 \pm 0.02$ &$12$ &$13.27 \pm 0.04$ &$11$& $< 12.9$ &N/A\\
\hline
{Zn}{II} $^2S_{1/2}$ & $\lambda$2026, $\lambda$2062 & $12.82 \pm 0.06$ &$4$ &$12.87 \pm 0.04$ &$54$& $< 12.4$ &N/A\\
\hline
\hline
\end{tabular}
}
\end{center}
\end{table*}
\begin{figure}
\centering
\includegraphics[angle=-0,width=9cm]{Fig3New.ps}
\caption{The {O}{I}$\lambda$1302, {Al}{II}$\lambda$1670,
{Si}{II}$\lambda$1260 ground state and {Si}{II}$\lambda$1264 fine
structure transitions. These lines need a three Voigt component
model to be fitted and are heavily saturated. The zero point has
been arbitrarily placed at the redshift of the redmost component
($z=1.9683$). g.s. and n* indicate ground state and n-th excited
transitions, respectively. }
\label{spe1}
\end{figure}
\subsection{Abundances}
The GRB\,081008 redshift was high enough to allow the hydrogen
Ly$\alpha$ line to enter the UVES spectral window. Unfortunately, the
UVES spectrum is extremely noisy in this region, and the derived
hydrogen column density, $\log (N_{\rm HI}/{\rm cm}^{-2}) = 21.33 \pm
0.12$ is quite uncertain. The fit is plotted in Fig. 5 (top panel)
superimposed to the smoothed UVES spectrum. The fit is particularly
poor on the wings, possibly because more than one component is needed
to model the absorption. To obtain a better estimate of the column
density, we used the FORS2 spectrum, that has a better S/N (see next
section for details). The two component fit shown in Fig. 5 (bottom
panel) gives a better representation of the data. {\bf The two
components are centered at $z_1=1.944$ and $z_2=1.975$,
respectively, and have column densities of $\log (N_{\rm HI}/{\rm
cm}^{-2}) = 20.82 \pm 0.14$ and $\log (N_{\rm HI}/{\rm cm}^{-2}) =
20.79 \pm 0.12$, respectively. The FORS2 total} column
density is $\log (N_{\rm HI}/{\rm cm}^{-2}) = 21.11 \pm 0.10$. {\bf
This is our best fit result for $N_H$ and will be used in the
following}. The metallicity has been derived summing all non
saturated component, excited level and ionic contributions belonging
to the same atom, dividing these values by $N_{\rm H}$ and comparing
them to the corresponding solar values given in Asplund et al. (2009).
The upper limits of component III result in an increment of the total
column densities of $< 20$\% in the worst cases, so they were not
included in the computation.
The results are listed in Table 4. Column 2 reports the total
abundance of each atom, while columns 3 and 4 report the absolute and
solar-scaled $N_X$/$N_{\rm H}$ ratios, respectively, with $X$ the
corresponding element in column 1. Lower limits are reported whenever
saturation does not allow us to securely fit the metallic column
densities (see e.g., Fig. 4, where the line profiles reach the zero
value of the normalized flux). In particular, for {O}{I} and {Al}{II}
we considered also the values of the third, saturated component, while
for {Si}{II} this has not been considered, since the fit to the
{Si}{II}$\lambda$1260 resulted in a $N$ value of component III which
is considerably lower than that of components I and II. We derived
metallicity values between $0.3$ and $0.05$ with respect to the solar
ones. We caution, however, that many transitions belonging to other
ionization states, which are commonly observed in GRB afterglow
spectra could not be taken into account, because they are outside the
UVES-dichroic-1 spectral range. In addition, the dust depletion can
prevent the observation of part of the metallic content of the
GRB\,081008 host galaxy. The reported relative abundances should then
be considered as lower limits to the true GRB\,081008
metallicity. However, some considerations on higher ionization states
are possible analyzing the FORS2 spectra, and dust content can be
investigated through the study of the depletion pattern (see sects. 5
and 6).
\begin{figure}
\centering
\includegraphics[angle=-0,width=8.5cm]{Fig4a.ps}
\includegraphics[angle=-0,width=8.5cm]{Fig4b.ps}
\smallskip
\caption{The Ly$\alpha$ absorption feature at the GRB\,081008
redshift. Top panel shows the single Voigt component, best-fit model
for the UVES spectrum. Bottom panel shows the double component,
best-fit model for the FORS2 spectrum. {\bf The UVES fit is poor,
while the FORS2 one gives a more reliable description of $N_H$.} }
\label{spe1}
\end{figure}
\subsection{Excited levels}
The level structure of an atom or ion is characterized by a principal
quantum number $n$, which defines the atomic level, and by the
spin-orbit coupling (described by the quantum number $j$), which
splits these levels into fine structure sub-levels. Excited features
are routinely detected in GRB absorption spectroscopy, at the host
redshift, due to the population of both $n>1$ and/or $n=1$ fine
structure levels. GRB\, 081008 behaves the same way. In fact,
component I features the first and second fine structure levels of the
{Fe}{II} ground state ($a^6D$), the first fine structure level of
the {Si}{II} $^2P^{0}$, and the {Fe}{II} $a^4F_{9/2}$,
{Fe}{II} $a^4D_{7/2}$, {{Ni}{II}} $^4F_{9/2}$ metastable
levels (the subscript represents the spin-orbit quantum number
$j$). Moreover, the {Fe}{II} $a^4F_{7/2}$ and {{Ni}{II}}
$^4F_{9/2}$ excited states are also detected in component II (see
Table 3 for details).
There is conspicuous literature on the population of excited states in
GRB surrounding medium and their detection in the afterglow spectra
(see e.g. Prochaska, Chen \& Bloom 2006; Vreeswijk et al. 2007; D'Elia
et al. 2010 and references therein). There is general consensus that
these features are produced by indirect UV pumping by the afterglow,
i.e., through the population of higher levels followed by the
depopulation into the states responsible for the absorption
features. This has been demonstrated both by the detection of
variability of fine structure lines in multi-epoch spectroscopy
(Vreeswijk et al. 2007; D'Elia et al. 2009a), and through the column
density ratios of different excited levels when multiple spectra were
not available (Ledoux et al. 2009; D'Elia et al. 2009b).
Concerning GRB\,081008, the high column density of the first
metastable level of {Fe}{II} ($a^4F_{9/2}$) with respect to the fine
structure levels of the ground state can hardly be explained with a
level population distribution given by a Boltzmann function (Vreeswijk
et al. 2007), meaning that collisional excitations can be safely
rejected. The lack of multi-epoch spectroscopy does not allow us
to completely rule out the possibility that the exciting UV flux
come from regions of high star-formation rates and not from the
GRB. In fact, fine structure emission lines are present in
Lyman-break, high redshift galaxies (see Shapley et al. 2003). If
we assume that this flux comes from the GRB, we can estimate the
GRB/absorber distance, comparing observed column densities to those
predicted by a time-dependent, photo-excitation code for the time when
the spectroscopic observations were acquired. The photo-excitation
code is that used by Vreeswijk et al. (2007) and D'Elia et
al. (2009a), to which we refer the reader for more details. Our
equations take into account the $(4\pi)^{-1}$ correction factor to
the flux experienced by the absorbing gas described by Vreeswijk
(2011). We assume that the species for which we are running the
code are at the ground state before the GRB blast wave reaches the
gas. The GRB flux behavior before the UVES observation was estimated
using the data in Y10 (lightcurve and spectral index), with no
spectral variation assumed during the time interval between the burst
and our observation. We concentrate on {Fe}{II} and {Si}{II} levels
because the {Ni}{II} ground state has column densities not far from
the $90\%$ confidence level of $\log (N_{Ni}{II}/{\rm cm}^{-2}) =
13.3$, and thus the uncertainties on such values are high (Table
3). In addition, {Ni}{II} ground state is detected only through the
$\lambda 1741$ transition, because the lower oscillator strengths of
the $\lambda 1709$ and $1751$ lines prevents a detection of these
features above the $90\%$ level. The initial column densities of the
ground states were computed from the observed column densities of all
the levels of each ion, i.e., we are assuming that the species are not
excited at $t=0$. The initial values for {Fe}{II} and {Si}{II} are
$\log (N_{SiII}/{\rm cm}^{-2}) = 15.63 \pm 0.03$ and $\log
(N_{FeII}/{\rm cm}^{-2}) = 15.21 \pm 0.02$ for component I, and
$\log (N_{FeII}/{\rm cm}^{-2}) = 14.98 \pm 0.04$ for component
II. Finally, the Doppler parameter used as input of this model has
been left free to vary between $10$ and $20$ km s$^{-1}$, i.e. the
range of values that best fit the absorption features of components I
and II.
\begin{figure}
\centering
\includegraphics[angle=-0,width=9cm]{Fig5a_New.ps}
\includegraphics[angle=-0,width=9cm]{Fig5b_New.ps}
\caption{Top panel: {Fe}{II} column densities for the ground level
(open circle), fine structure levels of the ground state (filled
circles), first metastable (open square) and second metastable (open
triangle) transitions for component I in the spectrum of
GRB\,081008. Column density predictions from our time-dependent
photo-excitation code are also shown. They refer to the ground level
(dotted line), fine structure level (solid lines), first and second
excited level (dashed and thick solid lines, respectively)
transitions, in the case of an absorber placed at $50$ pc from the
GRB. Bottom panel: the reduced $\chi^2$ as a function of the
distance for the model reproduced in the upper panel. Dashed lines
indicate the best fit distance and enclose the 90\% confidence
range.}
\label{spe1}
\end{figure}
\begin{figure}
\centering
\includegraphics[angle=-0,width=9cm]{Fig6_New.ps}
\caption{The {Si}{II} column densities for the ground level (open
circle) and first fine structure level (filled circle) transitions
for component I in the spectrum of GRB\,081008. Column density
predictions from our time-dependent photo-excitation code are also
shown. They refer to the ground level (dotted line) and first fine
structure level (thick solid line) transitions, in the case of an
absorber placed at $52$ pc from the GRB. The two thin solid lines
display the models which enclose the fine structure level data at
the 90\% confidence level (error bars for this transition are drawn
both at $1\sigma$ and 90\% confidence levels).}
\label{spe1}
\end{figure}
\begin{table}
\caption{Metallicity computed from the UVES data.}
{\footnotesize
\smallskip
\begin{tabular}{|l|ccc|}
\hline
Element $X$& $\log N_X /{\rm cm}^{-2}$ &$\log N_X$/$N_{\rm H}$ & $[X/{\rm H}]$ \\
\hline
O & $>15.12\pm0.06$ & $>-5.99\pm0.13$ & $>-2.68\pm0.11$ \\
Al & $>13.70\pm0.04$ & $>-7.41\pm0.13$ & $>-1.86\pm0.11$ \\
Si & $ 15.75\pm0.04$ & $ -5.32\pm0.12$ & $ -0.87\pm0.10$ \\
Cr & $ 13.83\pm0.03$ & $ -7.28\pm0.08$ & $ -0.92\pm0.10$ \\
Fe & $ 15.42\pm0.04$ & $ -5.69\pm0.13$ & $ -1.19\pm0.11$ \\
Ni & $ 13.74\pm0.07$ & $ -7.37\pm0.13$ & $ -1.29\pm0.12$ \\
Zn & $ 13.15\pm0.04$ & $ -7.96\pm0.13$ & $ -0.52\pm0.11$ \\
\hline
\end{tabular}
}
\end{table}
\begin{figure}
\centering
\includegraphics[angle=-0,width=9cm]{Fig7_New.ps}
\caption{The {Fe}{II} column densities for the ground level (open
circle), first fine structure level (upper limit) and first excited
level (open square) transitions for component II in the spectrum of
GRB\,081008. Column density predictions from our time-dependent
photo-excitation code are also shown. They refer to the ground level
(dotted line), first fine structure level (solid line) and
first excited level (thick dashed line) transitions, in the case of an
absorber placed at $200$ pc from the GRB. The two thin dashed lines
display the models which enclose the excited level data at the 90\%
confidence level (error bars for this transition are drawn both at
$1\sigma$ and 90\% confidence levels).}
\label{spe1}
\end{figure}
Fig. 6 (top) shows the model that best fits the {{Fe}{II}} data,
obtained for a distance of $50$ pc and a Doppler parameter of $20$ km
s$^{-1}$. Fig. 6 (bottom) reproduces the behaviour of the reduced
$\chi^2$ as a function of the distance GRB/absorber. The distance of
component I from the GRB explosion site results $d_{I,
FeII}=51^{+21}_{-11}$ pc at the 90\% confidence level. The same
calculation was performed using the {{Si}{II}} atomic data. The
results are displayed in Fig. 7, and the estimated distance is
$d_{I,SiII}=52\pm 6$ pc, which is consistent with what was estimated
using the {{Fe}{II}} data.
For component II, we have much less excited transitions. Fig. 8 shows
the model which best fits the {{Fe}{II}} data and the two
theoretical curves compatible within the error bars for the
{Fe}{II} $a^4F_{9/2}$ excited level column density, which is
actually the only one with a positive detection in component II. The
resulting distance between the GRB and this absorbing component is
$d_{II, FeII} = 200^{+60}_{-80}$ pc (90\% confidence level), a
larger value than $d_{I}$, as expected given the lack of excited
transition in component II.
\section{FORS2 spectroscopy}
In the framework of the ESO program 082.A-0755, we observed the
afterglow of GRB\,081008 also with the FORS2 low resolution
spectrograph ($R=780$), mounted on VLT/UT1. We took three spectra of
$900$ s each, starting around Oct 09 at 00:20 UT (about 4.4 hours
after the burst). We used the 600B grism, whose spectral coverage is
$330-630$ nm. The extraction of the spectra was performed within the
MIDAS environment. Wavelength and flux calibration of the three
spectra were obtained by using the helium-argon lamp and observing
spectrophotometric stars. Tab. 1 reports a summary of our FORS2
observations. We searched for variability in the W$_r$ of the FORS2
absorption lines, but we found none at the $2 \sigma$ level. This is
not surprising, since fine structure and excited lines are expected to
vary by less than $\sim 0.05$ decades (in column density) during the
acquisition time of the FORS2 spectra, which is $\sim 15$ min rest
frame (see Figs. 6-8). Since no variability is detected, we co-added
the three spectra, to improve the signal-to-noise ratio of our data,
to obtain a value of $\sim 60 - 80$ at $\lambda > 4000$\AA. The
resulting spectrum is presented in Fig. 9, together with the spectral
features identified at $z=1.97$, a redshift consistent with that
estimated using the UVES data. A list of the features detected in the
FORS2 spectrum is reported in the first column of Table 5.
\begin{figure*}
\centering
\includegraphics[width=10cm,height=20cm,angle=-90]{Fig8.ps}
\caption{The flux-calibrated, co-added FORS2 spectrum of the
GRB\,081008 afterglow, together with the spectral features
identified at $z=1.9683$.}
\label{spe1}
\end{figure*}
\begin{table}
\begin{center}
\caption{GRB081008 absorption features detected in the FORS2 spectrum, together
with their W$_r$ and column densities.
UVES data are shown for comparison.}
{\footnotesize
\smallskip
\begin{tabular}{|lcc|c|}
\hline
Transition & W$_r$ (\AA)$^a$ & N$^b$ &UVES N$^b$ \\
\hline
{C}{II}$\lambda$1334 & BLEND & - & - \\
\hline
{C}{IV}$\lambda$1548 & BLEND & - & - \\
\hline
{C}{IV}$\lambda$1550 & BLEND & - & - \\
\hline
{O}{I}$\lambda$1302 & BLEND & - & SAT \\
\hline
{O}{I}$\lambda$1304 & BLEND & - & - \\
\hline
{O}{I}$\lambda$1306 & BLEND & - & - \\
\hline
{Al}{II}$\lambda$1670 & $0.74$ & $14.60^{+0.26}_{-0.20}$ & SAT \\
\hline
{Al}{III}$\lambda$1854 & $0.30$ & $13.42 \pm 0.02$ &$13.30\pm0.03$ \\
\hline
{Al}{III}$\lambda$1862 & $0.17$ & $13.42 \pm 0.02$ &$13.30\pm0.03$ \\
\hline
{Si}{II}$\lambda$1260 & BLEND & $15.74^{+0.15}_{-0.11}$ &$15.60 \pm 0.04$ \\
\hline
{Si}{II}$\lambda$1304 & BLEND & $15.74^{+0.15}_{-0.11}$ &$15.60 \pm 0.04$ \\
\hline
{Si}{II}$\lambda$1526 & $0.67$ & $15.74^{+0.15}_{-0.11}$ &$15.60 \pm 0.04$ \\
\hline
{Si}{II}$\lambda$1808 & $0.25$ & $15.74^{+0.15}_{-0.11}$ &$15.60 \pm 0.04$ \\
\hline
{Si}{II} $\lambda$1264 & BLEND & $15.17 \pm 0.01$ &$15.21 \pm 0.05$ \\
\hline
{Si}{II} $\lambda$1309 & BLEND & $15.17 \pm 0.01$ &$15.21 \pm 0.05$ \\
\hline
{Si}{II} $\lambda$1533 & BLEND & $15.17 \pm 0.01$ &$15.21 \pm 0.05$ \\
\hline
{Si}{II} $\lambda$1816 & $0.07$ & $15.17 \pm 0.01$ &$15.21 \pm 0.05$ \\
\hline
{Si}{IV} $\lambda$1393 & $0.47$ & - & - \\
\hline
{Si}{IV} $\lambda$1402 & $0.40$ & - & - \\
\hline
{Cr}{II}$\lambda$2056 & $0.27$ & $13.94 \pm 0.02$ &$13.83 \pm 0.03$ \\
\hline
{Cr}{II}$\lambda$2062 & BLEND & $13.94 \pm 0.02$ &$13.83 \pm 0.03$ \\
\hline
{Cr}{II}$\lambda$2066 & $0.17$ & $13.94 \pm 0.02$ &$13.83 \pm 0.03$ \\
\hline
{Fe}{II}$\lambda$1608 & $0.54$ & $15.29^{+0.13}_{-0.10}$ &$15.33 \pm 0.02$ \\
\hline
{Fe}{II}$\lambda$1621 & BLEND & - &$14.95 \pm 0.04$ \\
\hline
{Fe}{II}$\lambda$1629 & BLEND & - &$14.95 \pm 0.04$ \\
\hline
{Fe}{II}$\lambda$1702 & $0.24$ & $14.16 \pm 0.02$ &$14.33 \pm 0.05$ \\
\hline
{Zn}{II}$\lambda$2026 & $0.24$ & $13.22 \pm 0.02$ &$13.15 \pm 0.04$ \\
\hline
{Zn}{II}$\lambda$2062 & BLEND & $13.22 \pm 0.02$ &$13.15 \pm 0.04$ \\
\hline
\end{tabular}
}
\end{center}
$^a$ The error on W$_r$ is $0.01$\AA ($1\sigma$).
$^b$ All values of the column densities are logarithmic (in cm$^{-2}$).
\end{table}
The Voigt fitting procedure is not adequate to compute the column
densities of metallic species in low resolution spectroscopy. In
this case, the Curve of Growth (COG) analysis (see e.g. Spitzer 1978)
must be applied. For weak absorption lines, with width $W_r < 0.1$\AA,
and for Doppler parameters $b>20$ km s$^{-1}$, $W_r$ is proportional
to the column density $N$, and virtually insensitive to the Doppler
parameter itself. For stronger lines this does not hold any more, and
the relation between $W_r$ and $N$ is described by a COG, which is a
function of $b$. In order to fit the correct COG to the data and to
estimate $b$, different transitions (with different oscillator
strengths $f$) of the same species are needed. Following Spitzer
(1978), we built up a code to perform this fit on our FORS2 data. To
test our code, we compute $W_r$ for all the UVES transitions featuring
two components, and apply our fitting program. The result of the fit
is shown in Fig. 10 (top panel), and the estimated column densities
are reported in Table 6 (errors are given at the $1\sigma$ level).
\begin{figure}
\centering
\includegraphics[angle=-0,width=9cm]{Fig9.ps}
\includegraphics[angle=-0,width=9cm]{Fig9a.ps}
\caption{Top panel: the COG analysis tested using the UVES species
featuring two components. Bottom panel: COG analysis applied to the
FORS2 lines with a measured $W_r$. Solid lines represent the best
fit obtained using the reported $b$ values. Dashed lines show
the $b=\infty$ curve for comparison. The COG fits component I and
II together in both plots.}
\label{spe1}
\end{figure}
The effective Doppler parameter evaluated from the fit, $b=23$ km
s$^{-1}$, is compatible with that estimated using the line fitting
profile. To compare the column densities estimated with the two
methods, we sum for each species the contribution coming from the two
components using the line fitting method (see values in Table 3), and
report the results in Table 6. The agreement between line fitting and
COG analyses is very good: each column density is within $1\sigma$
from the corresponding value estimated using the other method. The
only exception is the {Fe}{II5s}, whose column density values
however overlap at the $2\sigma$ level.
\begin{table}
\begin{center}
\caption{Comparison between UVES column densities evalutated with the line fitting and COG methods.}
{\footnotesize
\smallskip
\begin{tabular}{|l|cc|}
\hline
Specie & N (COG analysis) &N (Line fitting) \\
\hline
{Al}{III} & $13.29^{+0.05}_{-0.10}$ & $13.30 \pm 0.03$ \\
\hline
{Si}{II} & $15.66^{+0.06}_{-0.12}$ & $15.60 \pm 0.03$ \\
\hline
{Cr}{II} & $13.83^{+0.03}_{-0.07}$ & $13.83 \pm 0.03$ \\
\hline
{Fe}{II} (g.s.) & $15.31^{+0.01}_{-0.07}$ & $15.33 \pm 0.02$ \\
\hline
{Fe}{II} $a^4F_{9/2}$ & $14.13^{+0.07}_{-0.11}$ & $14.33 \pm 0.05$ \\
\hline
{Ni}{II} (g.s.) & $13.81^{+0.01}_{-0.03}$ & $13.74 \pm 0.07$ \\
\hline
{Ni}{II} $a^4F_{9/2}$ & $13.73^{+0.05}_{-0.09}$ & $13.75 \pm 0.02$ \\
\hline
{Zn}{II} & $13.14^{+0.05}_{-0.10}$ & $13.15 \pm 0.04$ \\
\hline
\end{tabular}
All values are logarithmic (in cm$^{-2}$).
}
\end{center}
\end{table}
We now apply the COG analysis to the FORS2 spectrum. First of all, we
compute the $W_r$ from the data. As shown in Table 5, despite the
identification of nearly $30$ transitions, reliable $W_r$ can be
evaluated only for $13$ (second column of the table). This is because
the lower FORS2 resolution does not enable to separate many of these
transitions which are blended with each other. We then run the COG
code using the FORS2 $W_r$, and evaluate the corresponding column
densities. The results are shown in the third column of Table 5, while
the last column shows the UVES column densities for comparison. Errors
are again at the $1\sigma$ level, and Fig. 10 (bottom panel) shows the
graphical output of the fit. The effective Doppler parameter
estimated ($b=31 \pm 2$ km s$^{-1}$) reproduces quite well the
combination of the values computed for component I and II ($\sim 10$
and $\sim 20$ km s$^{-1}$, respectively, separated by $\sim 20$ km
s$^{-1}$) using the line fitting method. The FORS2 and UVES spectra
give consistent column densities, with the $3\sigma$ confidence
regions overlapping in the worst cases.
\section{Conclusions and discussion}
In this paper we present high and low resolution spectroscopy of the
optical afterglow of GRB\,081008, observed using UVES and FORS2
spectrographs at the VLT $\sim 5$ hr after the trigger. We detect
several absorption features (both neutral and excited) at the common
redshift of $z=1.9683$. The spectra show that the gas absorbing the
GRB afterglow light can be described with three components identified
in this paper as I, II and III, according to their decreasing velocity
values.
We estimated the distances between the GRB and the absorbers. We find
a distance for component I of $d_{FeII,I}=51^{+21}_{-11}$ pc and
$d_{SiII,I}=52 \pm 6$, using {{Fe}II} and {{Si}II}, respectively. The
{{Si}II} leads to a smaller uncertainty because its fine structure
level is more sensitive to the flux experienced by the absorber. Other
papers mainly use {{Fe}{II}} as distance estimator, so for a safer
comparison is better to consider our {{Fe}{II}} value. For component
II, this distance is greater, $d_{II}=200^{+60}_{-80}$ pc. We stress
that these values are obtained assuming a three component
absorber. However, we can not exclude a higher number of components,
because our spectrum has a low S/N and a limited resolution.
Component II is far away from GRB than component I, as expected given
the lack of fine structure lines in this absorber. Component III does
not show excited levels at all, and only shows low ionization
states. Therefore, this is produced by an absorber located even
farther from the GRB, in a region which is not significantly
influenced by the prompt/afterglow emission.
Component I of GRB\,081008 is the closest to a GRB ever recorded. In
fact, for the 6 other GRBs for which the GRB/absorber distance have
been estimated, the closest components are at $d=80 - 700$ pc from the
GRB (Vreeswijk et al. 2007; D'Elia et al. 2009,a,b; Ledoux et al. 2009;
D'Elia et al. 2010; Th\"one et al. 2011). The values reported in
literature have been corrected for the $4(\pi)^{-1/2}$ factor
discussed by Vreeswijk (2011). This behaviour can be interpreted as
due to a dense environment close to the GRB explosion site. This high
density is possibly witnessed by the a non negligible dust amount (see
below) and by the metal content of the GRB surrounding medium. In
fact, the GRB\,081008 surroundings have the highest metallicity and
the highest abundances of, e.g., {{Fe}{II}} and {{Ni}{II}}, among this
sub-sample of GRBs. This high density in the GRB surroundings could
constitute a barrier to the GRB prompt/afterglow emission, that is not
able to strongly excite the interstellar medium up to the distances
reached by the other GRBs.
The neutral hydrogen column density is $\log (N_{\rm H, opt}/{\rm
cm}^{-2}) = 21.11 \pm 0.10$, while that estimated from {\it Swift}
XRT data is $\log{N_{\rm H,X}}/{\rm cm}^{-2}=21.66^{+0.14}_{-0.26}$
(Campana et al. 2010). The latter value is for a solar abundance
medium. Using $N_{\rm H, opt}$ we evaluate the GRB\,081008 host
galaxy's metallicity. The values we find are in the range [X/H] $=
-1.29$ to $-0.52$ with respect to the solar abundances. This value
lies in the middle of the GRB distribution, (Savaglio 2006; Prochaska
et al. 2007; Savaglio, Glazebrook \& Le Borgne, 2009). From X--ray
data a limit of [X/H] $> -1.83$ ($90\%$ confidence limit) can be set
assuming a solar abundance pattern and requiring that the absorbing
medium is not Thomson thick. If we set the metallicity to [X/H] =
$-0.5$, the absorbing column density in the X-rays is higher, namely,
$\log{N_{\rm H,X}}/{\rm cm}^{-2}=22.24^{+0.19}_{-0.30}$ and higher for
lower metallicities. Fynbo et al. (2009) and Campana et al. (2010)
show that in GRBs with a detectable Ly$\alpha$ feature (i.e., those at
$z>2$) $N_{\rm H,X}$ is on average a factor of $10$ higher than
$N_{\rm H,opt}$, and GRB\,081008 follows this trend. The intense GRB
flux, which ionizes the hydrogen and prevents part of it to be
optically detected, is the common explanation for this discrepancy
(Fynbo et al. 2009; Campana et al. 2010; Schady et al. 2011).
It is worth noting that observed abundances of {{Fe}{II}} and
{{Zn}{II}} are significantly different ([Fe/H]$= -1.29\pm 0.11$ and
[Zn/H]$= -0.52 \pm 0.11$). This can be ascribed to the different
refractory properties of the two elements, with the former that
preferentially tends to produce dust grains while the latter prefers
the gas phase. The comparison between these `opposite' elements can
thus provide information on the dust content in the GRB
environments. In order to be more quantitative, we derive the dust
depletion pattern for the GRB\,081008 environment, following the
method described in Savaglio (2000). We consider the four depletion
patterns observed in the Milky Way, namely, those in the warm halo
(WH), warm disk + halo (WHD), warm disk (WD) and cool disk (CD) clouds
(Savage \& Sembach 1996). We find that the best fit to our data is
given by the WH cloud pattern, with a metallicity of
$logZ_{GRB}/Z_{\odot} \sim -0.5$ and a GRB dust-to-metal ratio
comparable to that of the WH environment, e.g., $d/d_{WH}=1$
(Fig. 11). This metallicity value is consistent with our [Zn/H]
measurement. This agreement is self-consistent with the use of zinc as
a good indicator of metallicity. Since the latter quantity is linked
to the extinction (see e.g., Savaglio, Fall \& Fiore 2003) we derive
$A_V \sim 0.19$ mag along the GRB\,081008 line of sight. We check this
value by modeling the flux-calibrated FORS2 spectrum. The SED is
dominated by the Ly$\alpha$ which is difficult to model given the
high fluctuations and other absorption lines. Anyway, the inferred $A_V$
value is low and compatible with that evaluated from the dust
depletion.
\begin{figure}
\centering
\includegraphics[angle=-0,width=10cm]{Fig10_New2.eps}
\caption{Depletion patterns in the absorbing gas of
GRB\,081008. Filled squares are taken from average gas-phase
abundance measurements in warm halo (blue), warm disk + halo
(green), warm disk (red) and cool disk (cyan) clouds of the Milky
Way (Savage \& Sembach 1996). Filled circles represent our data
points, which are best fitted by the warm halo cloud pattern.}
\label{spe1}
\end{figure}
Another hint of dust is the non detection of {{Fe}{II}} in the third
component. The {{Fe}{II}} column densities in components I and II are
very similar, and this lets us believe that in component III the iron
is present as well, but in the dust form. The higher presence of dust
in components far away from the GRB has already been pointed out by
D'Elia et al. (2007). They report a possible presence of dust in
component III of GRB\,050730, while the closer component II (featuring
{{Fe}{II}} fine structure lines) shows more iron in the gas state. The
detection of more dust far away from the GRB can be explained since
dust grains containing iron tends to be efficiently destroyed during a
blast wave occurring after a GRB explosion (Perna, Lazzati \& Fiore
2003).
The analysis of the FORS2 spectra extends our surveyed wavelength
range, allowing the detection of higher ionization species, such as
{{C}{IV}} and {{Si}{IV}}. Anyway, line profiles of high and low
ionization species rarely match in redshift space and often if they
do, it is because the line blending cannot be resolved in a spectrum,
regardless of resolution and S/N.
We stress that the availability of simultaneous high and low
resolution spectra of a GRB afterglow is an extremely rare event. In
this context, the comparison of the column densities obtained fitting
the line profile of a high resolution spectrum with that estimated by
the Curve of Growth analysis applied to a low resolution one could be
extremely important. In fact, this can help to determine a range of
column densities for which it is safe to apply the Curve of Growth
analysis when high resolution data are missing. This is because high
column densities can result in the saturation effect, a problem that
is difficult to address using low resolution spectra only (see
e.g. Penprase et al. 2010). Prochaska (2006) widely discuss the
limits and perils of the COG analysis applied to low resolution
data. They find that this kind of analysis tends to underestimate the
column densities of the absorbing species. This is because strong
transitions drive the COG fit since the relative error associated to
their W$_r$ is smaller than that for weak ones. Nevertheless, strong
transitions are more affected by saturation, and in order to match
their observed column densities, the COG fit is forced towards high
values of the effective Doppler parameter. High resolution data often
show that the main contribution to the column density of strong
transitions comes from one narrow component. On the other hand, the
main contribution to the W$_r$ comes from other components which
account for a small fraction of the column density. These inferred
high values for the effective Doppler parameter are thus mimicking a
more complex situation, with the result of underestimating the real
column densities. For what concerns GRB\,081008 the UVES observations
show no or just mild saturation even for the strongest transitions,
and the two main components give a similar contribution to the total
column densities. This is the reason why there is a good agreement
between COG analysis of low resolution data and line fitting analysis
of high resolution ones for this particular GRB (within $3\sigma$ in
the worst cases).
Finally, we detect two weak intervening systems in our spectra. The
first one is a {{C}{IV}} absorber in the FORS2 spectrum at $z =1.78$,
and the second one is a {{Mg}{II}} system in the UVES spectrum at
$z=1.286$. This last system has $W_r({{Mg}{II}\lambda 2796})=0.3$\AA,
the detection limit being $0.1$\AA\, at the $2\sigma$ confidence
level. The redshift path analyzed for {{Mg}{II}} is $z=0.18-0.38$ and
$z=0.71-1.43$ for the UVES spectrum, and $z=0.36-1.21$ for the FORS2
one.
\section*{Acknowledgments}
We thank an anonymous referee for a deep and critical reading of the
paper, which strongly increased its quality. This work was partially
supported by ASI (I/l/011/07/0).
|
1,314,259,996,484 | arxiv | \section{Supplementary material}
\subsection{Derivation of the $\gamma$GL equation}
It is convenient to recast the self-consistency equation [i.e., Eq.~(1) of the article] for a two-band superconductor in the form
\begin{equation}
\left(
\begin{array}{c}
\Delta_1({\bf x})\\
\Delta_2({\bf x})
\end{array}
\right)=
\left(
\begin{array}{cc}
\lambda_{11}& \lambda_{12}\\
\lambda_{21}=\lambda_{12}& \lambda_{22}
\end{array}
\right)
\left(
\begin{array}{c}
R_1[\Delta_1({\bf x})]\\
R_2[\Delta_2({\bf x})]
\end{array}
\right),
\label{matr_self_A}
\end{equation}
where $R_i[\Delta_i({\bf x})]$ is a functional of $\Delta_i({\bf x})$ related to the anomalous Green function as [see Eq. (2) in the article]
\begin{equation}
R_i[\Delta_i({\bf x})] = \frac{\langle {\hat\psi}_{i\uparrow}({\bf x}){\hat
\psi}_{i\downarrow}({\bf x})\rangle}{N(0)}
\label{Ri}
\end{equation}
and $\lambda_{ij}=\lambda_{ji}=N(0)g_{ij}$, with $N(0)$ the total density of states. Then, Eq.~(\ref{matr_self_A}) can further be rearranged as
\begin{equation}
\left(
\begin{array}{c}
R_1[\Delta_1({\bf x})]\\
R_2[\Delta_2({\bf x})]
\end{array}
\right)=
\frac{1}{\eta}
\left(
\begin{array}{cc}
\lambda_{22}& -\lambda_{12}\\
-\lambda_{12}& \lambda_{11}
\end{array}
\right)
\left(
\begin{array}{c}
\Delta_1({\bf x})\\
\Delta_2({\bf x})
\end{array}
\right),
\label{matr_self_B}
\end{equation}
with $\eta=\lambda_{11}\lambda_{22}-\lambda^2_{12}$. Based on Eq.~(2) of the article and Eq.~(\ref{matr_self_B}) given above, one finds
\begin{equation}
\Big(\frac{\lambda_{11}}{\eta} -\alpha_2\Big)\Delta_2 + \beta_2\Delta_2|\Delta_2|^2 -{\cal K}_2{\bf D}^2\Delta_2 - \frac{\lambda_{12}}{\eta} \Delta_1= 0,
\label{gammaGL_A}
\end{equation}
where $\alpha_2$, $\beta_2$ and ${\cal K}_2$ are defined in Eq.~(3) of the article. Now, assuming that
$$
T=T_{c2}=\frac{2e^{\Gamma}}{\pi}\hbar\omega_c\,e^{-1/(n_2\lambda_{22})},
$$
with $n_i=N_i(0)/N(0)$ the partial density of states, and keeping only the terms linear in $\lambda_{12}$, Eq.~(\ref{gammaGL_A}) yields
\begin{equation}
\beta_2\Delta_2|\Delta_2|^2 -{\cal K}_2{\bf D}^2\Delta_2 - \frac{\lambda_{12}}{\lambda_{11}\lambda_{22}} \Delta_{1,\lambda_{12}\to 0}= 0,
\label{gammaGL_B}
\end{equation}
with $\beta_2$ and ${\cal K}_2$ taken at $T = T_{c2}$, which is the $\gamma$GL equation [$\gamma=\lambda_{12}/(\lambda_{11}\lambda_{22})$] given by Eq.~(4) of the article.
\subsection{Derivation of Eq. (7) in the article}
Let us again consider Eq.~(\ref{gammaGL_A}) but now for $T_{c2} < T < T_{c1}$ and {\it in the homogeneous case}. In this temperature domain $\Delta_2 \to 0$ for $\lambda_{12} \to 0$ and $\Delta_{1,\lambda_{12}\to 0} \not= 0$. From Eq.~(\ref{gammaGL_A}) we have
\begin{equation}
\Big[\frac{\lambda_{11}}{\eta} -n_2\ln\Big(\frac{2e^{\Gamma}\hbar\omega_c}{\pi T} \Big)\Big]\Delta_2 + n_2\frac{7\zeta(3)}{8\pi^2T^2}\Delta_2|\Delta_2|^2 = \frac{\lambda_{12}}{\eta} \Delta_1.
\label{gammaGL_C}
\end{equation}
Note that the expansion given by Eq.~(2) in the article and therefore also Eq.~(\ref{gammaGL_A}) given above is not justified for {\it a spatially nonuniform solution} at $T_{c2} < T < T_{c1}$ because the coherence length $\xi_2$ associated with a spatial variation of the condensate in a weaker band does not diverge as $\lambda_{12}$ goes to zero. In other words, one cannot invoke the gradient expansion in the Gor'kov derivation of Eq.~(2) in the article. However, for homogeneous case Eq.~(2) of the article is simply an expansion in powers of the order parameter $\Delta_2$ and, so, is fully correct. Keeping only terms linear in $\lambda_{12}$, from Eq.~(\ref{gammaGL_C}) one finds
\begin{equation}
n_2\ln(T/T_{c2})\Delta_2=\frac{\lambda_{12}}{\lambda_{11}\lambda_{22}}
\Delta_{1,\lambda_{12}\to 0},
\label{gammaGL_D}
\end{equation}
which is Eq.~(7) in the article.
We remark that at first sight Eq.~(\ref{gammaGL_D}) prescribes that $\Delta_2$ is infinite when $T \to T_{c2}$ and $\lambda_{12}= {\rm const}$. This is not true because the validity of the expansion in powers of $\lambda_{12}$ given in Eq.~(\ref{gammaGL_D}) requires that $\lambda_{12} \lesssim \lambda^{({\rm lim})}_{12}(T)$, and $\lambda^{({\rm lim})}_{12}(T) \to 0$ for $T \to T_{c2}$. Hence, $\lambda_{12}$ can not be considered as constant when using Eq.~(\ref{gammaGL_D}) for $T \to T_{c2}$. In particular, during the derivation of Eq.~(\ref{gammaGL_D}) we first assumed that
\begin{align}
\frac{\lambda_{11}}{\eta}-n_2\ln\Big(\frac{2e^{\Gamma}\hbar\omega_c}{\pi T} \Big) = & \,n_2 \ln\big(T/T_{c2}\big)\notag \\
&+\frac{\lambda^2_{12}}{\lambda_{11}\lambda^2_{22}} + {\cal O}(\lambda^4_{12})
\label{lambda_exp}
\end{align}
and, then, we ignored contributions of the order $\lambda^2_{12}$ and higher. However, when the first and second terms in the right-hand-side of Eq.~(\ref{lambda_exp}) become of the same order of magnitude, then the contributions $\propto \lambda^2_{12}$ can not be neglected any more. This makes it possible to estimate $\lambda^{({\rm lim})}_{12}(T)$ as
\begin{equation}
\lambda^{({\rm lim})}_{12}(T) \sim \lambda_{22} \Big[\lambda_{11} n_2 \ln\big( T/T_{c2}\big)\Big]^{1/2}.
\label{lambda_lim}
\end{equation}
which proves that $\lambda^{({\rm lim})}_{12}(T) \to 0$ when $T \to T_{c2}$. Notice that Eq.~(\ref{gammaGL_D}) does not hold for $T \geq T_{c1}$, where $\Delta_{1,\lambda_{12}\to 0} =0$. Formally, it gives $\Delta_2 = 0$ which means that $\Delta_2$ is not linear in $\lambda_{12}$ for small interband couplings any longer. This is directly related to the fact that $T=T_{c1}$ is one more hidden critical point for the system of interest, however it is now associated with a stronger band. As pointed out in the article, the hidden criticality associated with $T_{c1}$ is overshadowed by the usual critical behavior around $T_c$.
|
1,314,259,996,485 | arxiv | \section{Introduction}\label{sec:intro}
After the immense progress achieved in the last three decades, observational cosmology is about to undergo transformational changes once again. A number of high-precision, wide-field experiments across the electromagnetic spectrum will soon start operations. Examples include the Rubin Observatory Legacy Survey of Space and Time (LSST)\footnote{\url{https://www.lsst.org/}.}, Euclid\footnote{\url{https://www.euclid-ec.org/}.} and the Roman Telescope\footnote{\url{https://roman.gsfc.nasa.gov/}.} in the optical, as well as the Simons Observatory\footnote{\url{https://simonsobservatory.org/}.} (SO) and CMB Stage 4 (S4) in the microwave, which will deliver galaxy samples of unprecedented size as well as high-precision measurements of Cosmic Microwave Background (CMB) anisotropies, respectively. As the data volume of cosmological surveys increases, these experiments will become increasingly dominated by systematic rather than statistical uncertainties, which will require the development of novel analysis methods.
Galaxy clusters constitute the most massive bound objects in the Universe and their abundance as a function of mass is a powerful probe of cosmology, which has the potential to tightly constrain the amplitude of matter fluctuations, $\sigma_{8}$, and the fractional matter density today, $\Omega_{m}$ (see e.g. \cite{Voit:2005, Allen:2011}). However, this exciting cosmological probe has so far received less attention compared to e.g. cosmic shear or galaxy clustering, as it has been limited by systematic uncertainties related to the determination of cluster masses (see e.g. Refs.~\cite{Carlstrom:2002, Voit:2005, Allen:2011} for a discussion). Galaxy clusters can be detected by several different techniques: (i) in the optical by looking for large overdensities in the galaxy distribution, (ii) in the microwave, through their imprint on the observed CMB temperature anisotropies, the thermal Sunyaev-Zel'dovich (tSZ) effect \cite{Sunyaev:1970}, and finally (iii) in the X-ray through the emission of the hot gas trapped inside these clusters. All of these methods measure an observable that is connected to mass, such as richness $\lambda$, tSZ decrement $Y$ and gas temperature and density $T, \rho$. The uncertainty in the mass-observable relation is the largest systematic uncertainty in cosmological analyses of galaxy clusters and needs to be calibrated using external data. Weak gravitational lensing is sensitive to all matter in the Universe and therefore, the lensing signal for galaxies located behind a given cluster can be used to infer cluster halo masses and calibrate the mass-observable relation (e.g. \cite{Allen:2011}). Examples of recent cosmological analyses of galaxy clusters include Refs.~\cite{Hasselfield:2013, Bocquet:2019, Planck:2016XXIV, Zubeldia:2019}, which use CMB data from the Atacama Cosmology Telescope\footnote{\url{https://act.princeton.edu/}.} (ACT), the South Pole Telescope\footnote{\url{https://pole.uchicago.edu/}.} (SPT) and Planck respectively, as well as Refs.~\cite{Mantz:2014, Abbott:2020}, which use X-ray data from Chandra and optical data from the Dark Energy Survey\footnote{\url{https://www.darkenergysurvey.org/}.} (DES), respectively.
In addition, several recent works have investigated joint constraints on cosmology and cluster mass calibration: for example Ref.~\cite{Madhavacheril:2017} forecasted constraints from a joint analysis of CMB S4 cluster abundances and LSST weak lensing, Ref.~\cite{Salcedo:2020} focused on a combination of cluster weak lensing with galaxy clustering and the cross-correlation between cluster and galaxy overdensity and finally Ref.~\cite{Shirasaki:2020} took a different approach: focusing only on power spectra, the authors investigated the potential of multi-wavelength analyses to jointly constrain cosmology and properties of the intracluster medium.
In this work, we focus on the abundance of galaxy clusters detected through the tSZ effect in CMB temperature anisotropy maps. Building on previous work \cite{Oguri:2011, Shirasaki:2015, Krause:2017}, we propose a new method for joint cosmological parameter inference and cluster mass calibration from a combination of weak lensing measurements and tSZ cluster abundances. Specifically, we combine cluster number counts with the spherical harmonic cosmic shear power spectrum and the cross-correlation between cluster overdensity and cosmic shear. We use a halo model \cite{Ma:2000, Peacock:2000, Seljak:2000, Cooray:2002} framework for modeling the observables and their full non-Gaussian covariance. Using this framework, we forecast constraints on cosmological and mass calibration parameters for a combination of LSST and SO and investigate the different sources of cosmological and astrophysical information. Finally, we compare our results to those obtained with more traditional tSZ mass calibration methods, which are based on stacked measurements of cluster weak lensing (for a summary of the method, the reader is referred to e.g. Ref.~\cite{Madhavacheril:2017}, for examples of stacked weak lensing analyses, see e.g. Refs.~\cite{Medezinski:2018, Miyatake:2019}). Although we focus on forecasting the constraining power of future experiments in this work, the methods presented here are equally applicable to joint analyses of current surveys, such as ACT, SPT and DES.
This paper is organized as follows. In Sec.~\ref{sec:obs}, we present the cosmological observables used in our analysis. Section \ref{sec:theory} outlines the theoretical modeling of the observables within the halo model and in Sec.~\ref{sec:covariance}, we derive expressions for the joint covariance between the probes considered. Sec.~\ref{sec:lsstxso} describes our fiducial assumptions for forecasting joint constraints from LSST and SO and Sec.~\ref{sec:forecasts} describes the forecasting methodology. We present our results in Sec.~\ref{sec:results} and conclude in Sec.~\ref{sec:conclusions}. Implementation details are deferred to the Appendices
\section{Observables}\label{sec:obs}
In this work, we investigate the potential of joint analyses of tSZ cluster number counts and cosmic shear to simultaneously calibrate cluster masses and constrain cosmological parameters. To this end, we focus on combining cluster number counts $\mathcal{N}_{\mathrm{cl}}$ with cosmic shear power spectra $C_{\ell}^{\gamma \gamma}$ and cross-correlations between cluster overdensity $\delta_{\mathrm{cl}}$ and cosmic shear, $C_{\ell}^{\delta_{\mathrm{cl}} \gamma}$. In the following, we describe these observables in more detail. Unless stated otherwise, all theoretical predictions in this work assume a flat cosmological model, i.e. $\Omega_{k} = 0$.
\subsection{tSZ cluster number counts}
\subsubsection{Cluster detection}\label{subsubsec:obs.tSZ.detection}
The modeling of both the thermal Sunyaev-Zel'dovich signal and cluster detection in this work closely follows Ref.~\cite{Madhavacheril:2017}. We give a brief summary below but refer the reader to Ref.~\cite{Madhavacheril:2017} for more details.
The thermal Sunyaev-Zel'dovich effect is a secondary anisotropy of the CMB due to inverse Compton scattering of CMB photons with energetic, free electrons in galaxy clusters (for a review of tSZ cosmology, see e.g. \cite{Carlstrom:2002}). The tSZ effect leads to a characteristic spectral distortion of the CMB blackbody spectrum that is proportional to the integrated pressure along a given direction $\boldsymbol{\theta}$, given by (see e.g. \cite{Carlstrom:2002, Planck:2016XXII})
\begin{equation}
\frac{\Delta T}{T_{\mathrm{CMB}}}(\nu, \boldsymbol{\theta}) = f(\nu) \frac{\sigma_{T}}{m_{e}c^{2}}\int \mathrm{d}l \; P_{e}(l, \boldsymbol{\theta}) \equiv f(\nu)y(\boldsymbol{\theta}).
\end{equation}
In this equation, $f(\nu)$ is defined as $f(\nu) = x \coth{\sfrac{x}{2}} - 4$ with $x=\sfrac{h \nu}{k_{B}T_{\mathrm{CMB}}}$, where $T_{\mathrm{CMB}}$ denotes the CMB temperature, $h$ and $k_{B}$ are the Planck and Boltzmann constants, respectively. Furthermore, $m_{e}$ denotes electron mass, $\sigma_{T}$ is the Thompson cross-section, $P_{e}(l, \boldsymbol{\theta})$ denotes the three-dimensional cluster pressure profile and $\mathrm{d}l$ is the line-of-sight distance in direction $\boldsymbol{\theta}$. Finally, we have defined the dimensionless Compton-y parameter $y(\boldsymbol{\theta})$, which determines the amplitude of the tSZ signal. We model $P_{e}(l, \boldsymbol{\theta})$ following Ref.~\cite{Madhavacheril:2017}, adopting the analytic pressure profile from Ref.~\cite{Arnaud:2010} with the parameter values given in Ref.~\cite{Madhavacheril:2017}.
Following Ref.~\cite{Madhavacheril:2017}, we assume that a matched-filter applied to a CMB map is used to define a cluster. For each detected cluster, we define the spherical aperture tSZ flux as \cite{Alonso:2016}
\begin{equation}
Y_{500} = \frac{4 \pi}{D_{A}^{2}(z)} \int_{0}^{R_{500}} \mathrm{d}^{2}r\; r^{2} \frac{\sigma_{T}}{m_{e}c^{2}} P_{e}(r),
\end{equation}
where $D_{A}(z)$ denotes the physical angular diameter distance and $R_{500}$ is the radius where the density equals 500 times the critical density of the Universe at the cluster redshift $z$\footnote{We note that $Y_{500}$ is not a directly observable quantity but can be related to any measurement of the integrated Compton-y parameter.}. For a given multi-frequency CMB experiment, the uncertainties in measuring $Y_{500}$, denoted $\sigma_{N}$, are determined by the noise and resolution of the different frequency maps. In order to compute these uncertainties, we again follow Ref.~\cite{Madhavacheril:2017} and refer the reader to that work for further details.
\subsubsection{Mass-observable relation}\label{subsubsec:obs.tSZ.ym}
As the quantity $Y_{500}$ is obtained by integrating the Compton-y parameter over the cluster's extent, it is a measure for the total thermal energy of the cluster. We thus expect $Y_{500}$ to be a measure for the cluster halo mass $M$\footnote{Here $M$ denotes a generic mass definition and we transform between definitions as needed. The procedure chosen to transform between mass definitions is outlined in Appendix \ref{ap:sec:mass-trans}.}. The relation between the mean flux $\bar{Y}_{500}$ and the underlying halo mass $M$ is the main systematic uncertainty in tSZ cluster cosmology. In this work, we follow Refs.~\cite{Planck:2014XX, Alonso:2016, Madhavacheril:2017} and model this relation as
\begin{equation}
\begin{aligned}
\bar{Y}_{500}(M_{500}, z) = & Y_{*}\left[\frac{M_{500}}{M_{*}}\right]^{\alpha_{Y}} e^{\beta_{Y}\log^{2}{\left(\sfrac{M_{500}}{M_{*}}\right)}}(1+z)^{\gamma_{Y}} \times \\ &E^{\sfrac{2}{3}}(z)\left[\frac{D_{A}(z)}{100 \; \sfrac{\mathrm{Mpc}}{h}}\right]^{-2},
\end{aligned}
\label{eq:Y-M-mean}
\end{equation}
where $M_{500}$ denotes the mass enclosed within the radius where the density equals 500 times the critical density of the Universe at the cluster redshift. The quantities $\alpha_{Y}$ and $\beta_{Y}$ account for the first and second order mass dependence and $\gamma_{Y}$ parameterizes a redshift dependence, additional to that expected from self-similar evolution. Furthermore, $Y_{*}$ and $M_{*}$ are constants and $E(z) = \sfrac{H(z)}{H_{0}}$. The quantities $H(z)$ and $H_{0}$ denote the Hubble parameter and its present day value, respectively. The distribution of true tSZ fluxes is usually assumed to take a log-normal form around their mean $\bar{Y}_{500}$, i.e. (e.g. \cite{Alonso:2016})
\begin{equation}
\begin{aligned}
p(Y_{500}^{\mathrm{true}}|M_{500}, z) = \frac{1}{\sqrt{2\pi}\sigma_{\log Y_{500}}(M, z)} \times \\e^{-\sfrac{(\log{Y_{500}^{\mathrm{true}}}-\log{\bar{Y}_{500}(M_{500}, z)})^{2}}{2\sigma^{2}_{\log Y_{500}}(M, z)}},
\end{aligned}
\end{equation}
where we have introduced the intrinsic mass- and redshift-dependent scatter $\sigma_{\log Y_{500}}(M, z)$, which we model as \cite{Madhavacheril:2017}
\begin{equation}
\sigma_{\log Y_{500}}(M, z) = \sigma_{\log Y_{0}}\left[\frac{M_{500}}{M_{*}}\right]^{\alpha_{\sigma}}(1+z)^{\gamma_{\sigma}}.
\label{eq:Y-M-scatter}
\end{equation}
In the above equation, $\alpha_{\sigma}$ and $\gamma_{\sigma}$ parametrize the mass- and redshift-dependence of the intrinsic scatter, respectively.
\subsubsection{Cluster number counts}
The probability to observe a galaxy cluster at redshift $z$ with mass $M$, true tSZ amplitude $Y_{500}^{\mathrm{true}}$ and observed tSZ amplitude $Y_{500}^{\mathrm{obs}}$ is given by
\begin{equation}
p(M, z, Y_{500}^{\mathrm{true}}, Y_{500}^{\mathrm{obs}}) = p(Y_{500}^{\mathrm{obs}}) p(Y_{500}^{\mathrm{true}}|Y_{500}^{\mathrm{obs}}) p(M, z|Y_{500}^{\mathrm{true}}).
\label{eq:cluster_prob}
\end{equation}
Using
\begin{equation}
p(M, z|Y_{500}^{\mathrm{true}}) = \frac{p(M, z)}{p(Y_{500}^{\mathrm{true}})} p(Y_{500}^{\mathrm{true}}|M, z),
\end{equation}
we can rewrite Eq.~\ref{eq:cluster_prob} as
\begin{widetext}
\begin{equation}
p(M, z, Y_{500}^{\mathrm{true}}, Y_{500}^{\mathrm{obs}}) = \frac{p(Y_{500}^{\mathrm{obs}})}{p(Y_{500}^{\mathrm{true}})} p(Y_{500}^{\mathrm{true}}|Y_{500}^{\mathrm{obs}}) p(M, z) p(Y_{500}^{\mathrm{true}}|M, z).
\label{eq:cluster-prob}
\end{equation}
\end{widetext}
Here $p(M, z)$ denotes the normalized halo mass function (as we are computing the probability to observe a cluster), $p(Y_{500}^{\mathrm{true}}|M, z)$ is the probability that a cluster of $Y_{500}^{\mathrm{true}}$ at redshift $z$ has halo mass $M$ and finally $p(Y_{500}^{\mathrm{obs}}|Y_{500}^{\mathrm{true}}) = \sfrac{p(Y_{500}^{\mathrm{obs}})}{p(Y_{500}^{\mathrm{true}})} p(Y_{500}^{\mathrm{true}}|Y_{500}^{\mathrm{obs}})$ denotes the survey-specific cluster selection function. The selection function quantifies the probability of measuring $Y_{500}^{\mathrm{obs}}$ for a true tSZ flux $Y_{500}^{\mathrm{true}}$ and is determined by the experimental uncertainties discussed in Sec.~\ref{subsubsec:obs.tSZ.detection}.
If we instead set $p(M, z)$ to the unnormalized halo mass function, i.e. $p(M, z) = \sfrac{\mathrm{d}n}{\mathrm{d}M}$, then Eq.~\ref{eq:cluster-prob} gives us the number of detected clusters with $M, z, Y_{500}^{\mathrm{obs}}, Y_{500}^{\mathrm{true}}$. Therefore, the observed number of thermal Sunyaev-Zel'dovich detected galaxy clusters in redshift bin $i$ with $z \in [z_{i, \mathrm{min}}, z_{i, \mathrm{max}}]$ and tSZ signal amplitude bin $\alpha$ with $Y_{500}^{\mathrm{obs}} \in [Y_{500, \alpha}^{\mathrm{obs}, \mathrm{min}}, Y_{500, \alpha}^{\mathrm{obs}, \mathrm{max}}]$ becomes
\begin{widetext}
\begin{equation}
\mathcal{N}^{i}_{\mathrm{cl}, \alpha}\coloneqq \mathcal{N}_{\mathrm{cl}}(\Delta Y_{500, \alpha}^{\mathrm{obs}}, \Delta z_{i}) = \Omega_{s} \int_{z_{i, \mathrm{min}}}^{z_{i, \mathrm{max}}} \mathrm{d}z \frac{c}{H(z)} \frac{\mathrm{d}V}{\mathrm{d}\chi} \int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M}
\int \mathrm{d}Y_{500}^{\mathrm{true}} \int_{Y_{500, \alpha}^{\mathrm{obs}, \mathrm{min}}}^{Y_{500, \alpha}^{\mathrm{obs}, \mathrm{max}}} \mathrm{d}Y_{500}^{\mathrm{obs}} \; p(Y_{500}^{\mathrm{obs}}|Y_{500}^{\mathrm{true}}) p(Y_{500}^{\mathrm{true}}|M, z),
\end{equation}
\end{widetext}
where we have integrated over halo mass and $Y_{500}^{\mathrm{true}}$, which are not directly observable.
Here, $\sfrac{\mathrm{d}V}{\mathrm{d}\chi} = \chi^{2}$ denotes the comoving volume element in comoving distance and we have performed the integration over solid angle, which for a survey covering a sky fraction $f_{\mathrm{sky}}$, yields $\Omega_{s} = 4\pi f_{\mathrm{sky}}$.
Defining the integrated survey selection function for $Y_{500}^{\mathrm{obs}}$ bin $\alpha$ as
\begin{equation}
S_{\alpha}(Y_{\mathrm{true}}, M, z) = \int_{Y_{500, \alpha}^{\mathrm{obs}, \mathrm{min}}}^{Y_{500, \alpha}^{\mathrm{obs}, \mathrm{max}}} \mathrm{d}Y_{500}^{\mathrm{obs}} \; p(Y_{500}^{\mathrm{obs}}|Y_{500}^{\mathrm{true}}),
\end{equation}
we finally obtain
\begin{equation}
\begin{aligned}
\mathcal{N}^{i}_{\mathrm{cl}, \alpha} = \Omega_{s} \int_{z_{i, \mathrm{min}}}^{z_{i, \mathrm{max}}} \mathrm{d}z \frac{c}{H(z)} \frac{\mathrm{d}V}{\mathrm{d}\chi} \int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M} \times \\ \int \mathrm{d}Y_{500}^{\mathrm{true}}\; p(Y_{500}^{\mathrm{true}}|M, z) S_{\alpha}(Y_{500}^{\mathrm{true}}, M, z).
\label{eq:cluster-counts}
\end{aligned}
\end{equation}
Using the results derived in Sec.~\ref{subsubsec:obs.tSZ.detection}, we can obtain an expression for $S_{\alpha}(Y_{500}^{\mathrm{true}}, M, z)$. Let us assume a detection threshold for clusters given by $q \sigma_{N}(M, z)$, where $\sigma_{N}(M, z)$ denotes the noise in the $Y$ measurement for a cluster of halo mass $M$ at redshift $z$ and $q$ is the detection level\footnote{In this work, we set $q=5$, which corresponds to a $5\sigma$ detection threshold and is typical for CMB tSZ detections.}. This leads to \cite{Alonso:2016}
\begin{equation}
\begin{aligned}
S_{\alpha}(Y_{500}^{\mathrm{true}}, M, z) = \int_{\mathrm{max}(q \sigma_{N}, Y_{500, \alpha}^{\mathrm{obs}, \mathrm{min}})}^{Y_{500, \alpha}^{\mathrm{obs}, \mathrm{max}}} \mathrm{d}Y_{500}^{\mathrm{obs}}\; p(Y_{500}^{\mathrm{obs}}|Y_{500}^{\mathrm{true}}).
\end{aligned}
\end{equation}
Assuming a Gaussian distribution for $p(Y_{500}^{\mathrm{obs}}|Y_{500}^{\mathrm{true}})$ given by \cite{Alonso:2016}
\begin{equation}
p(Y_{500}^{\mathrm{obs}}|Y_{500}^{\mathrm{true}}) = \frac{1}{\sqrt{2\pi}\sigma_{N}(M, z)}e^{-\sfrac{(Y_{500}^{\mathrm{obs}} - Y_{500}^{\mathrm{true}})^{2}}{2 \sigma^{2}_N(M, z)}},
\end{equation}
we finally arrive at \cite{Alonso:2016}
\begin{equation}
\begin{aligned}
S_{\alpha}(Y_{500}^{\mathrm{true}}, M, z) = \frac{1}{2}\left[\mathrm{erf} \left(\frac{Y_{500, \alpha}^{\mathrm{obs}, \mathrm{max}} - Y_{500}^{\mathrm{true}}}{\sqrt{2}\sigma_{N}(M, z)}\right)\right. - \\
\left.\mathrm{erf} \left(\frac{\mathrm{max}(q \sigma_{N}, Y_{500, \alpha}^{\mathrm{obs}, \mathrm{min}}) - Y_{500}^{\mathrm{true}}}{\sqrt{2}\sigma_{N}(M, z)}\right)\right],
\end{aligned}
\end{equation}
where $\sigma_{N}(M, z)$ is fully determined by experimental uncertainties.
\subsection{Power spectra}\label{subsec:obs.ps}
We combine cluster number counts with two different power spectra: the cosmic shear power spectrum and the cross-power spectrum between cluster overdensity and cosmic shear.
Let us consider two tracers $a, b \in [\gamma_{i}, \delta_{\mathrm{cl}, \alpha}^{j}]$, where $\gamma$ denotes cosmic shear and $\delta_{\mathrm{cl}}$ denotes cluster overdensity. Furthermore $i, j$ label the respective redshift bins and $\alpha$ the tSZ amplitude bin. Employing the Limber approximation \cite{Limber:1953, Kaiser:1992, Kaiser:1998}, we can write their spherical harmonic power spectrum as
\begin{equation}
\begin{aligned}
C_{\ell}^{ab}=\int \mathrm{d} z \; \frac{c}{H(z)} \; \frac{W^{a}\boldsymbol{\left(}\chi(z)\boldsymbol{\right)}W^{b}\boldsymbol{\left(}\chi(z)\boldsymbol{\right)}}{\chi^{2}(z)} \times \\
P_{ab}\left(k=\frac{\ell+\sfrac{1}{2}}{\chi(z)}, z\right),
\end{aligned}
\end{equation}
where $c$ is the speed of light, $\chi(z)$ is the comoving distance and $P_{ab}(k, z)$ denotes the three-dimensional power spectrum between probes $a$ and $b$. The quantity $W^{a}\boldsymbol{\left(}\chi(z)\boldsymbol{\right)}$ is a probe-specific window function, which we discuss next for cosmic shear and cluster overdensity.
\subsubsection{Cosmic shear power spectrum}
Cosmic shear is sensitive to the integrated matter distribution between source galaxies and the observer, and the cosmic shear kernel $W^{\gamma}\boldsymbol{\left(}\chi(z)\boldsymbol{\right)}$ is given by
\begin{equation}
W^{i}_{\gamma}\boldsymbol{\left(}\chi(z)\boldsymbol{\right)} = \frac{3}{2} \frac{\Omega_{m} H^{2}_{0}}{c^{2}} \frac{\chi(z)}{a} \int_{\chi(z)}^{\chi_{h}} \mathrm{d} z' n^{i}(z') \frac{\chi(z')-\chi(z)}{\chi(z')},
\label{eq:gammawindow}
\end{equation}
where $n^{i}(z)$ denotes the normalized redshift distribution of source galaxies in redshift bin $i$, $\chi_{h}$ is the comoving distance to the horizon and $a$ denotes the scale factor. As cosmic shear is sensitive to all gravitationally interacting matter in the Universe, we further set $P_{\gamma \gamma}(k, z) = P_{mm}(k, z)$, where $P_{mm}(k, z)$ denotes the matter power spectrum.
The observed cosmic shear auto-power spectrum receives an additional contribution due to shape noise from intrinsic galaxy ellipticities. We model the shape noise power spectrum of redshift bin $i$ as $N^{i}_{\gamma \gamma} = \sfrac{\sigma_{\epsilon, i}^{2}}{\bar{n}^{i}_{\mathrm{source}}}$, where $\bar{n}^{i}_{\mathrm{source}}$ denotes the mean angular galaxy number density and $\sigma_{\epsilon, i}$ is the standard deviation of the intrinsic ellipticity in each component.
\subsubsection{Cross-correlation between cluster overdensity and cosmic shear}
Galaxy clusters are a biased tracer of the matter distribution and their clustering properties can therefore be analyzed analogously to galaxy clustering. In this work, we focus on the angular power spectrum between cluster overdensity and cosmic shear, which can be computed by cross-correlating maps of cluster overdensity and galaxy ellipticity. The redshift distribution of galaxy clusters with tSZ amplitudes in $\Delta Y_{500, \alpha}^{\mathrm{obs}}$, detectable by a given survey, is determined by their number density as a function of redshift (see e.g. \cite{Fedeli:2009}). From Eq.~\ref{eq:cluster-counts} we thus obtain
\begin{equation}
\begin{aligned}
\mathcal{N}_{\mathrm{cl}, \alpha}(z) \coloneqq \mathcal{N}_{\mathrm{cl}}(z, \Delta Y_{500, \alpha}^{\mathrm{obs}}) = \Omega_{s} \frac{c}{H(z)} \frac{\mathrm{d}V}{\mathrm{d}\chi} \int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M} \times \\ \int \mathrm{d}Y_{500}^{\mathrm{true}}\; p(Y_{500}^{\mathrm{true}}|M, z) S_{\alpha}(Y_{500}^{\mathrm{true}}, M, z).
\label{eq:nclust_z}
\end{aligned}
\end{equation}
Finally, normalizing Eq.~\ref{eq:nclust_z} to unity by dividing by the total number of observable clusters in tSZ bin $\alpha$ $\mathcal{N}_{\mathrm{cl}, \alpha}=\int \mathrm{d}z \; \mathcal{N}_{\mathrm{cl}, \alpha}(z)$, we obtain the redshift distribution of galaxy clusters as
\begin{equation}
n_{\mathrm{cl}, \alpha}(z) = \frac{\mathcal{N}_{\mathrm{cl}, \alpha}(z)}{\mathcal{N}_{\mathrm{cl}, \alpha}}.
\end{equation}
In addition to considering bins in tSZ amplitude, we can subdivide the galaxy cluster distribution into redshift bins. We denote the resulting distributions by $n^{i}_{\mathrm{cl}, \alpha}(z)$ and the window function $W^{i}_{\delta_{\mathrm{cl}}, \alpha}\boldsymbol{\left(}\chi(z)\boldsymbol{\right)}$ thus becomes
\begin{equation}
W^{i}_{\delta_{\mathrm{cl}}, \alpha}\boldsymbol{\left(}\chi(z)\boldsymbol{\right)} = \frac{H(z)}{c} n^{i}_{\mathrm{cl}, \alpha}(z).
\end{equation}
While the cross-correlation between cosmic shear and cluster overdensity $C_{\ell}^{\gamma \delta_{\mathrm{cl}}}$ is free from observational noise, the auto-correlation of the cluster overdensity $C_{\ell}^{\delta_{\mathrm{cl}}\delta_{\mathrm{cl}}}$ is subject to Poisson noise. In this analysis, we model this noise power spectrum as $N^{i}_{\delta_{\mathrm{cl}, \alpha} \delta_{\mathrm{cl}, \alpha}} = \sfrac{1}{\bar{n}^{i}_{\mathrm{cl}, \alpha}}$, where $\bar{n}^{i}_{\mathrm{cl}, \alpha}$ denotes the mean angular density of galaxy clusters in tSZ amplitude bin $\alpha$ and redshift bin $i$.
\subsubsection{Systematics modeling}\label{subsec:obs.ps.syst}
We account for potential systematic uncertainties in the cosmic shear measurement by including simple models for these systematics in our theoretical predictions\footnote{The main systematic uncertainty for tSZ cluster number counts is the $Y-M$ relation, which we discuss in Sec.~\ref{subsubsec:obs.tSZ.ym}. We note that we do not account for possible halo assembly bias when modeling the cluster overdensity, as the magnitude and significance of the effect are currently a matter of investigation (see e.g. Ref.~\cite{Sunayama:2019}).}. The most important systematics for cosmic shear are photometric redshift uncertainties and multiplicative biases in measured galaxy shapes.
\paragraph{Photometric redshift uncertainties}
For each tomographic redshift bin $i$, we parameterize the impact of photo-$z$ uncertainties as
\begin{equation}
n_{i}(z) \propto \hat{n}_{i}(z + \Delta z_{i}),
\end{equation}
where $n_{i}$ denotes the true, underlying redshift distribution, while $\hat{n}_{i}$ is estimated from the galaxy photo-$z$s. The parameter $\Delta z_{i}$ allows us to marginalize over potential biases in the mean of the redshift distributions.
\paragraph{Multiplicative shear bias}
The estimated weak lensing shear $\boldsymbol{\hat{\gamma}}$ is prone to multiplicative calibration uncertainties, which we model as (e.g. \cite{Heymans:2006})
\begin{equation}
\boldsymbol{\hat{\gamma}} = (1 + m_{i}) \boldsymbol{\gamma}.
\end{equation}
In the above equation, $\boldsymbol{\gamma}$ is the true galaxy shear and $m_{i}$ denotes the multiplicative bias parameter for tomographic redshift bin $i$.
\section{Theoretical modeling}\label{sec:theory}
In this work, we compute nonlinear matter power spectra $P_{mm}(k, z)$ using the \texttt{Halofit} fitting function \cite{Smith:2003} with the revisions by Ref.~\cite{Takahashi:2012}\footnote{This choice is motivated by the fact that the halo model described below is not able to accurately model power spectra in the transition regime between the 1- and 2-halo term \cite{Mead:2015}.}. We compute theoretical predictions for all other three-dimensional power spectra $P_{ab}(k, z)$ using the halo model \cite{Ma:2000, Peacock:2000, Seljak:2000, Cooray:2002}. In this model, the power spectrum is split into two distinct terms, the 1-halo and the 2-halo term. The 1-halo term quantifies clustering within a single halo, while the 2-halo term accounts for the contributions to $P_{ab}(k, z)$ coming from the relative clustering of tracers in different halos. These two quantities can be written as
\begin{equation}
\begin{aligned}
P^{1h}_{ab}(k, z) &= I^{0}_{ab}(k, k, z), \\
P^{2h}_{ab}(k, z) &= I^{1}_{a}(k, z) I^{1}_{b}(k, z) P_{\mathrm{lin}}(k, z),
\label{eq:pk_1h_2h}
\end{aligned}
\end{equation}
and the total power spectrum then becomes
\begin{equation}
P_{ab}(k, z) = P^{1h}_{ab}(k, z) + P^{2h}_{ab}(k, z).
\label{eq:pk_hm}
\end{equation}
In Equations \ref{eq:pk_1h_2h} and \ref{eq:pk_hm} we have used the general notation (see e.g. \cite{Cooray:2001, Krause:2017})
\begin{equation}
I^{n}_{a_1\cdots a_m}(k_1,\cdots,k_m) = \int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M}b_{h, n}(M) \left\langle\prod_{i=1}^m \left[\tilde{u}_{a_{i}}(k_{i}, M) \right]\right\rangle,
\label{eq:halo-mod-intg}
\end{equation}
where $b_{h, n}(M)$ is the $n$-th order halo bias and we define $b_{h,1}(M)\equiv b_{h}(M)$, $b_{h,0}\equiv1$. The quantity $\tilde{u}_{a_{i}}(k_{i}, M)$ is the Fourier transform of the normalized profile of the distribution of a given tracer within a halo of mass $M$ and $\langle \cdots \rangle$ denotes an ensemble average.
In order to model $P_{ab}(k, z)$, we additionally need expressions for the normalized density profiles for all probes considered, which we will discuss next.
\subsection{Cosmic shear}
Cosmic shear is sensitive to all matter in the Universe and we can therefore employ the halo model quantities for the matter distribution when predicting the statistical properties of cosmic shear. We define $\tilde{u}_{m}(k, M) \equiv \sfrac{M}{\bar{\rho_{m}}} \; u_{m}(k, M)$, where $\bar{\rho}_{m}$ denotes the comoving matter density, and set $\tilde{u}_{\gamma}(k, M) = \tilde{u}_{m}(k, M)$. We further assume a Navarro-Frenk-White profile \cite{Navarro:1996} for the Fourier transform of the matter distribution inside a halo of mass $M$, i.e. \cite{Navarro:1996}
\begin{widetext}
\begin{equation}
u_{m}(k, M) = \left[{\rm ln}(1+c)-\frac{c}{1+c}\right]^{-1} \left\{\sin x\left[{\rm Si}\left((1+c)\,x\right)-{\rm Si}(x)\right]+ \cos x\left[{\rm Ci}\left((1+c)x\right)-{\rm Ci}(x)\right]-\frac{\sin(cx)}{(1+c)x}\right\},
\end{equation}
\end{widetext}
where $x=\sfrac{k R_\Delta}{c}$, $R_\Delta$ denotes the halo radius, $c=c(M)$ is the concentration parameter, and ${\rm Si}/{\rm Ci}$ denote the sine and cosine integral functions.
\subsection{Galaxy cluster overdensity}
We follow Refs.~\cite{Huetsi:2008, Krause:2017} and assume that each halo of mass $M$ contains at most one galaxy cluster, which is located at its center. In order to derive the Fourier transform of the normalized cluster density profile, we first consider the number density of galaxy clusters in redshift bin $i$ and tSZ amplitude bin $\alpha$ as a function of position $\mathbf{r}$. This can be written as
\begin{equation}
\begin{aligned}
n^{i}_{\mathrm{cl}, \alpha}(\mathbf{r}) = \sum_{\substack{z \in \Delta z_{i},\\ j}} \int \mathrm{d}Y_{500}^{\mathrm{true}}\; p(Y_{500}^{\mathrm{true}}|M, z) \times \\ S_{\alpha}(Y_{500}^{\mathrm{true}}, M, z) \delta_{\mathcal{D}}(\mathbf{r}_{j}),
\end{aligned}
\end{equation}
where $\delta_{\mathcal{D}}(\mathbf{r})$ denotes the Dirac delta function. Switching from discrete to continuous variables, we obtain the mean cluster density in the tSZ and redshift bin as
\begin{equation}
\bar{n}^{i}_{\mathrm{cl}, \alpha} = \int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M} \int \mathrm{d}Y_{500}^{\mathrm{true}}\; p(Y_{500}^{\mathrm{true}}|M, z) S_{\alpha}(Y_{500}^{\mathrm{true}}, M, z).
\end{equation}
Finally, using the fact that the Fourier transform of the Dirac delta function equals unity, we obtain
\begin{equation}
\tilde{u}^{i}_{\delta_{\mathrm{cl}, \alpha}}(k, M) = \frac{\int \mathrm{d}Y_{500}^{\mathrm{true}}\; p(Y_{500}^{\mathrm{true}}|M, z) S_{\alpha}(Y_{500}^{\mathrm{true}}, M, z)}{\bar{n}^{i}_{\mathrm{cl}, \alpha}}.
\end{equation}
\subsection{Halo model implementation}
We compute the halo mass function $\sfrac{\mathrm{d}n}{\mathrm{d}M}$ and the halo bias $b_{h}(M)$ using the fitting functions derived in Ref.~\cite{Sheth:1999}. We further assume the concentration-mass relation of halos $c(M)$ to follow the fitting function derived in Ref.~\cite{Duffy:2008}. Unless noted otherwise (e.g. $M_{500}$), halo masses are defined with respect to the mean matter density $\bar{\rho}_{m}$ and we assume a virial collapse density contrast as given by Ref.~\cite{Bryan:1998}\footnote{We note that we transform $\Delta_{c}$ as given in Ref.~\cite{Bryan:1998} to be relative to the matter density instead of the critical density.}.
We further note that the 2-halo term for matter converges to $P_{\mathrm{lin}}(k, z)$ as $k \rightarrow 0$. This imposes a nontrivial constraint on $I^{1}_{m}(k, z)$ as
\begin{equation}
\int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M}b_{h}(M) \frac{M}{\bar{\rho}_{m}} = 1.
\end{equation}
We enforce this constraint by adding a constant, correcting for the finite minimal mass cutoff in our halo model integrals. This correction is not necessary for other tracers considered in this work, as these have a physical minimal mass cutoff in all halo model integrals.
In this work, we compute theoretical predictions for cosmological observables using the LSST Dark Energy Science Collaboration (DESC) Core Cosmology Library (CCL\footnote{\url{https://github.com/LSSTDESC/CCL}.}) \cite{Chisari:2019}.
\section{Covariance matrix}\label{sec:covariance}
We compute the joint covariance matrix of cosmic shear, tSZ cluster number counts and the cross-correlation between cosmic shear and cluster overdensity analytically using the halo model. The resulting expressions for all possible combinations between these probes are discussed below. With the exception of the Gaussian covariance of angular power spectra, which does not include mode-coupling effects due to observing only a fraction of the sky (see e.g. Ref.~\cite{Garcia:2019}), these expressions will be useful for both forecasts as well as analyses using real data.
\subsection{Cluster number counts}
The auto-covariance of cluster number counts in redshift bins $i, j$ and tSZ $Y$ bins $\alpha, \beta$ can be subdivided into a Poissonian and a super-sample covariance (SSC) part, i.e.
\begin{equation}
\mathrm{Cov}(\mathcal{N}^{i}_{\mathrm{cl}, \alpha} , \mathcal{N}^{j}_{\mathrm{cl}, \beta}) = \mathrm{Cov}_{\mathrm{P}}(\mathcal{N}^{i}_{\mathrm{cl}, \alpha} , \mathcal{N}^{j}_{\mathrm{cl}, \beta}) + \mathrm{Cov}_{\mathrm{SSC}}(\mathcal{N}^{i}_{\mathrm{cl}, \alpha} , \mathcal{N}^{j}_{\mathrm{cl}, \beta}).
\end{equation}
The Poissonian part of the total covariance accounts for the fact that clusters are discrete tracers. The SSC on the other hand quantifies correlations between cluster number counts in different $Y$ bins caused by the presence of long, unresolvable wavelength modes, larger than the survey volume (see e.g. Refs.~\cite{Hamilton:2006, Takada:2013}).
In this work, we follow Refs.~\cite{Schaan:2014, Krause:2017} and estimate the Poissonian contribution to the total covariance as
\begin{equation}
\mathrm{Cov}_{\mathrm{P}}(\mathcal{N}^{i}_{\mathrm{cl}, \alpha} , \mathcal{N}^{j}_{\mathrm{cl}, \beta}) = \delta^{\mathcal{D}}_{\alpha \beta} \delta^{\mathcal{D}}_{ij} \;\mathcal{N}^{i}_{\mathrm{cl}, \alpha} ,
\end{equation}
where we assume non-overlapping cluster number count bins in tSZ amplitude and redshift and set cross-correlations between cluster number counts at different redshifts to zero.
The super-sample covariance can be estimated as \cite{Takada:2014, Schaan:2014, Krause:2017}
\begin{widetext}
\begin{equation}
\begin{aligned}
\mathrm{Cov}_{\mathrm{SSC}}&(\mathcal{N}^{i}_{\mathrm{cl}, \alpha} , \mathcal{N}^{j}_{\mathrm{cl}, \beta}) = \delta_{ij} \Omega_{s}^{2} \int_{z_{i, \mathrm{min}}}^{z_{i, \mathrm{max}}} \mathrm{d}z \frac{c}{H(z)} \left[\frac{\mathrm{d}V}{\mathrm{d}\chi}\right]^{2}
\left[\int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M} b_{h}(M) \int \mathrm{d}Y_{500}^{\mathrm{true}}\; p(Y_{500}^{\mathrm{true}}|M, z) S_{\alpha}(Y_{500}^{\mathrm{true}}, M, z)\right] \times \\
&\left[\int \mathrm{d}M' \frac{\mathrm{d}n}{\mathrm{d}M'} b_{h}(M') \int \mathrm{d}Y_{500}^{\prime, \mathrm{true}}\; p(Y_{500}^{\prime, \mathrm{true}}|M', z) S_{\beta}(Y_{500}^{\prime, \mathrm{true}}, M', z)\right] \sigma_b^2(z).
\end{aligned}
\end{equation}
\end{widetext}
The quantity $\sigma_b^2(z)$ is the variance of the long wavelength background mode $\delta_{\rm LS}$ over the survey footprint, given by
\begin{equation}
\sigma_b^2(z) = \int \frac{\mathrm{d}k_\perp^2}{(2\pi)^2}P_{\rm lin}(k_\perp,z)\left|\tilde{W}(k_\perp,z)\right|^2.
\end{equation}
In the above equation, $\tilde{W}(k_\perp,z)$ denotes the Fourier transform of the survey footprint, which we approximate as a compact circle with an area matched to our data set:
\begin{equation}
\tilde{W}(k_\perp,z)=\frac{2 J_1(k_\perp\chi(z)\theta_s)}{k_\perp \chi(z)\theta_s},\hspace{12pt} \theta_s={\rm arccos}(1-2f_{\rm sky}),
\end{equation}
where $J_1(x)$ is the cylindrical Bessel function of order 1.
\subsection{Angular power spectra}
The covariance of two angular power spectra $C^{ab}_{\ell}$ and $C^{cd}_{\ell'}$ can be written as the sum of a Gaussian, non-Gaussian and super-sample covariance (SSC) part, i.e.
\begin{equation}
\begin{aligned}
\mathrm{Cov}(C^{ab}_{\ell}, C^{cd}_{\ell'}) = &\mathrm{Cov}_{\mathrm{G}}(C^{ab}_{\ell}, C^{cd}_{\ell'}) + \mathrm{Cov}_{\mathrm{NG}}(C^{ab}_{\ell}, C^{cd}_{\ell'}) +\\ &\mathrm{Cov}_{\mathrm{SSC}}(C^{ab}_{\ell}, C^{cd}_{\ell'}).
\end{aligned}
\end{equation}
The non-Gaussian covariance accounts for mode-coupling due to the non-Gaussianity of the fields being cross-correlated. In analogy to cluster number counts, the SSC quantifies the coupling of small-scale modes due to the presence of long, super-survey modes.
The Gaussian covariance matrix is given by (see e.g. \cite{Hu:2004, Krause:2017})
\begin{equation}
\begin{aligned}
\mathrm{Cov}_{\mathrm{G}}(C_{\ell}^{ab}, C_{\ell'}^{cd}) = \frac{\delta_{\ell \ell'}}{(2\ell+1)\Delta \ell f_{\mathrm{sky}}} \times \\
\left [(C_{\ell}^{ac} + \delta^{\mathcal{D}}_{ac}N^{ac})(C_{\ell}^{bd} + \delta^{\mathcal{D}}_{bd}N^{bd}) \times \right. \\
+ \left. (C_{\ell}^{ad} + \delta^{\mathcal{D}}_{ad}N^{ad})(C_{\ell}^{bc} + \delta^{\mathcal{D}}_{bc}N^{bc})\right ],
\label{eq:theorycovmat}
\end{aligned}
\end{equation}
where $\Delta \ell$ accounts for possible binning of the angular power spectra $C_{\ell}^{ab}$ into bandpowers. The quantities $N^{ab}$ denote the noise power spectra, which are nonzero only for auto-correlations. The expressions for these noise power spectra for the probes considered in our analysis are given in Sec.~\ref{subsec:obs.ps}.
The non-Gaussian covariance is given by the angular projection of the three-dimensional trispectrum\footnote{The trispectrum is the connected part of the four-point function.} $T^{abcd}(k_{1}, k_{2}, k_{3}, k_{4})$ as (see e.g. \cite{Krause:2017})
\begin{widetext}
\begin{equation}
\mathrm{Cov}_{\mathrm{NG}}(C^{ab}_{\ell}, C^{cd}_{\ell'}) = \frac{1}{\Omega_{s}} \int_{\vert \boldsymbol{\ell} \vert \in \ell_{1}} \int_{\vert \boldsymbol{\ell}' \vert \in \ell_{2}} \int \frac{\mathrm{d}^{2}\boldsymbol{\ell}}{A(\ell_{1})} \; \frac{\mathrm{d}^{2}\boldsymbol{\ell}'}{A(\ell_{2})} \; \mathrm{d}\chi \; \frac{W^{a}(\chi)W^{b}(\chi)W^{c}(\chi)W^{d}(\chi)}{\chi^{6}} T^{abcd}(\sfrac{\boldsymbol{\ell}}{\chi}, \sfrac{-\boldsymbol{\ell}}{\chi}, \sfrac{\boldsymbol{\ell}'}{\chi}, \sfrac{-\boldsymbol{\ell}'}{\chi}).
\end{equation}
\end{widetext}
The quantity $A(\ell_{i})$ denotes the area of an annulus of width $\Delta \ell_{i}$ around $\ell_{i}$, i.e. $A(\ell_{i}) \equiv \int_{\vert \boldsymbol{\ell} \vert \in \ell_{i}} \mathrm{d}^{2}\boldsymbol{\ell}$, which is approximately given by $A(\ell_{i}) \approx 2 \pi \Delta \ell_{i} \ell_{i}$ for $\ell_{i} \gg \Delta \ell_{i}$.
Using the halo model, the trispectrum $T^{abcd}$ can be written as (e.g. \cite{Takada:2013}):
\begin{equation}
T^{abcd} = T^{abcd, 1h} + (T^{abcd, 2h}_{22} + T^{abcd, 2h}_{13}) + T^{abcd, 3h} + T^{abcd, 4h},
\end{equation}
where
\begin{widetext}
\begin{equation}
\begin{aligned}
T^{abcd, 1h}(\mathbf{k}_{a}, \mathbf{k}_{b}, \mathbf{k}_{c}, \mathbf{k}_{d}) &= I^{0}_{abcd}(k_{a}, k_{b}, k_{c}, k_{d}), \\
T^{abcd, 2h}_{22}(\mathbf{k}_{a}, \mathbf{k}_{b}, \mathbf{k}_{c}, \mathbf{k}_{d}) &= P_{\mathrm{lin}}(k_{ab})I^{1}_{ab}(k_{a}, k_{b})I^{1}_{cd}(k_{c}, k_{d}) + 2 \; \mathrm{perm.}, \\
T^{abcd, 2h}_{13}(\mathbf{k}_{a}, \mathbf{k}_{b}, \mathbf{k}_{c}, \mathbf{k}_{d}) &= P_{\mathrm{lin}}(k_{a})I^{1}_{a}(k_{a})I^{1}_{bcd}(k_{b}, k_{b}, k_{c}) + 3 \; \mathrm{perm.}, \\
T^{abcd, 3h}(\mathbf{k}_{a}, \mathbf{k}_{b}, \mathbf{k}_{c}, \mathbf{k}_{d}) &= B^{\mathrm{PT}}(\mathbf{k}_{a}, \mathbf{k}_{b}, \mathbf{k}_{cd})I^{1}_{a}(k_{a})I^{1}_{b}(k_{b})I^{1}_{cd}(k_{c}, k_{d}) + 5 \;\mathrm{perm.},\\
T^{abcd, 4h}(\mathbf{k}_{a}, \mathbf{k}_{b}, \mathbf{k}_{c}, \mathbf{k}_{d}) &= T^{\mathrm{PT}}(\mathbf{k}_{a}, \mathbf{k}_{b}, \mathbf{k}_{c}, \mathbf{k}_{d})I^{1}_{a}(k_{a})I^{1}_{b}(k_{b})I^{1}_{c}(k_{c})I^{1}_{d}(k_{d}).
\label{eq:halo-mod-trisp}
\end{aligned}
\end{equation}
\end{widetext}
Here, ${\bf k}_{ab}\equiv {\bf k}_a+{\bf k}_b$, and the quantities $B^{\mathrm{PT}}$ and $T^{\mathrm{PT}}$ denote the matter bi- and trispectrum respectively, as estimated using tree-level perturbation theory. The full expressions for these terms can be found in Ref.~\cite{Takada:2013}. For simplicity, we follow \cite{Krause:2017} and approximate the 2- to 4-halo trispectrum as the linearly biased matter trispectrum and only include a probe-specific 1-halo trispectrum contribution. Specifically, we set
\begin{equation}
T^{abcd} = T^{abcd, 1h} + b_{a}b_{b}b_{c}b_{d}T^{m, 2h+3h+4h},
\end{equation}
where $T^{abcd, 1h}$ and $T^{m, 2h+3h+4h}$ are computed following Equations \ref{eq:halo-mod-trisp}. For $T^{abcd, 1h}$, we evaluate Eq.~\ref{eq:halo-mod-intg} for probes $a, b, d, c$, while for $T^{m, 2h+3h+4h}$, we use the corresponding expressions for the matter distribution. Finally, $b_{a}$ denotes the linear bias of tracer $a$ predicted using the halo model, i.e.
\begin{equation}
b_{a} = \int \mathrm{d}M\,\frac{\mathrm{d}n}{\mathrm{d}M}b_{h}(M) \tilde{u}_{a}(0, M),
\end{equation}
and we set $b_{\gamma}(M) = 1$. From Eq.~\ref{eq:halo-mod-intg}, we see that the 1-halo trispectrum is given by
\begin{widetext}
\begin{equation}
T^{abcd, 1h}(\mathbf{k}_{a}, \mathbf{k}_{b}, \mathbf{k}_{c}, \mathbf{k}_{d}) = \int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M}
\left\langle \tilde{u}_{a}(k_{a}, M) \tilde{u}_{b}(k_{b}, M) \tilde{u}_{c}(k_{c}, M) \tilde{u}_{d}(k_{d}, M)\right\rangle.
\end{equation}
\end{widetext}
A special case arises when $T^{abcd, 1h}(\mathbf{k}_{a}, \mathbf{k}_{b}, \mathbf{k}_{c}, \mathbf{k}_{d})$ contains two cluster number count tracers $\delta_{\mathrm{cl}, \alpha}^{i}, \delta_{\mathrm{cl}, \beta}^{j}$ (set to tracers $c, d$ w.l.o.g.), as a halo can at most contain a single cluster. Accounting for this fact, we then obtain
\begin{widetext}
\begin{equation}
T^{abcd, 1h}(\mathbf{k}_{a}, \mathbf{k}_{b}, \mathbf{k}_{c}, \mathbf{k}_{d}) = \delta_{ij}\delta_{\alpha \beta} \int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M}
\tilde{u}_{a}(k_{a}, M) \tilde{u}_{b}(k_{b}, M) \frac{\tilde{u}^{i}_{\delta_{\mathrm{cl}, \alpha}}(k_{c}, M)}{(\bar{n}^{i}_{\mathrm{cl}, \alpha})^{2}}.
\end{equation}
\end{widetext}
Finally, we compute the super-sample covariance contribution following the treatment of \cite{Krause:2017}, i.e.:
\begin{widetext}
\begin{equation}
\begin{aligned}
\mathrm{Cov}_{\mathrm{SSC}}(C^{ab}_{\ell}, C^{cd}_{\ell'}) = \int \mathrm{d}\chi \;\frac{W^{a}(\chi)W^{b}(\chi)W^{c}(\chi)W^{d}(\chi)}{\chi^{4}} \frac{\partial P_{ab}(\sfrac{\ell}{\chi}, z(\chi))}{\partial \delta_{\rm LS}}\frac{\partial P_{cd}(\sfrac{\ell'}{\chi}, z(\chi))}{\partial \delta_{\rm LS}}\sigma^{2}_{b}(z(\chi)).
\end{aligned}
\end{equation}
\end{widetext}
The quantity $\partial P_{ab}(k, z)/\partial \delta_{\rm LS}$ denotes the response of the power spectrum $P_{ab}$ to a large-scale density fluctuation, which we estimate using the halo model and results from perturbation theory as (e.g. \cite{Krause:2017}):
\begin{widetext}
\begin{equation}
\begin{aligned}
\frac{\partial P_{ab}(k, z)}{\partial \delta_{\rm LS}} = \left( \frac{68}{21} - \frac{1}{3}\frac{\mathrm{d}\log{k^{3} P_{\mathrm{lin}}}(k, z)}{\mathrm{d}\log k} \right) I_{a}^{1}(k)I_{b}^{1}(k)P_{\mathrm{lin}}(k, z) + I_{ab}^{1}(k, k) - (b_{a, a \neq \gamma} + b_{b, b \neq \gamma})P_{ab}(k, z).
\label{eq:ps-resp}
\end{aligned}
\end{equation}
\end{widetext}
The last term in Eq.~\ref{eq:ps-resp} accounts for the fact that observed overdensity fields are computed using the mean density estimated inside the survey volume.
For consistency with our implementation of the trispectrum, we compute the response function $\sfrac{\partial P_{ab}(k, z)}{\partial \delta_{\rm LS}} $ for a given probe as the linearly biased response of the matter field\footnote{In order to test the robustness of our results to this approximation, we also compute the SSC contribution to the covariance using the probe-specific halo model quantities in Eq.~\ref{eq:ps-resp}. We find our forecasted constraints to be unaffected by this change and therefore resort to the approach described above for consistency.}.
\subsection{Cross-correlations between cluster number counts and angular power spectra}
Finally, the cross-covariance between cluster number counts and angular power spectra vanishes for purely Gaussian fields, but it receives both non-Gaussian and SSC contributions, i.e.
\begin{equation}
\begin{aligned}
\mathrm{Cov}(\mathcal{N}^{\alpha}_{\mathrm{cl}, i}, C^{ab}_{\ell}) = \mathrm{Cov}_{\mathrm{NG}}(\mathcal{N}^{\alpha}_{\mathrm{cl}, i}, C^{ab}_{\ell}) + \mathrm{Cov}_{\mathrm{SSC}}(\mathcal{N}^{\alpha}_{\mathrm{cl}, i}, C^{ab}_{\ell}).
\end{aligned}
\end{equation}
Following Refs.~\cite{Takada:2007, Schaan:2014}, we can write the non-Gaussian part of this cross-covariance as
\begin{widetext}
\begin{equation}
\begin{aligned}
\mathrm{Cov}&_{\mathrm{NG}}(\mathcal{N}^{\alpha}_{\mathrm{cl}, i}, C^{ab}_{\ell}) = \Omega_{s} \int_{z_{i, \mathrm{min}}}^{z_{i, \mathrm{max}}} \mathrm{d}z \frac{c}{H(z)} \frac{W^{a}(\chi(z))W^{b}(\chi(z))}{\chi^{4}(z)} \frac{\mathrm{d}V}{\mathrm{d}\chi} \times \\
&\left\{\int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M} \tilde{u}_{a}(k, M) \tilde{u}_{b}(k, M) \int \mathrm{d}Y_{500}^{\mathrm{true}}\; p(Y_{500}^{\mathrm{true}}|M, z) S_{\alpha}(Y_{500}^{\mathrm{true}}, M, z)+ \right. \\
&\left. \left(\left[\int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M} b_{h}(M) \tilde{u}_{a}(k, M) \int \mathrm{d}Y_{500}^{\mathrm{true}}\; p(Y_{500}^{\mathrm{true}}|M, z) S_{\alpha}(Y_{500}^{\mathrm{true}}, M, z)\right]
\left[\int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M} b_{h}(M) \tilde{u}_{b}(k, M)\right] + \right. \right. \\
&\left. \left. \left[\int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M} b_{h}(M) \tilde{u}_{b}(k, M) \int \mathrm{d}Y_{500}^{\mathrm{true}}\; p(Y_{500}^{\mathrm{true}}|M, z) S_{\alpha}(Y_{500}^{\mathrm{true}}, M, z)\right]
\left[\int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M} b_{h}(M) \tilde{u}_{a}(k, M)\right]\right)P_{\mathrm{lin}}(k, z)\right\}.
\end{aligned}
\end{equation}
\end{widetext}
Furthermore, the SSC covariance is given by (see e.g. \cite{Schaan:2014, Krause:2017})
\begin{widetext}
\begin{equation}
\begin{aligned}
\mathrm{Cov}_{\mathrm{SSC}}(\mathcal{N}^{\alpha}_{\mathrm{cl}, i}, C^{ab}_{\ell}) &= \Omega_{s} \int_{z_{i, \mathrm{min}}}^{z_{i, \mathrm{max}}} \mathrm{d}\chi \frac{W^{a}(\chi)W^{b}(\chi)}{\chi^{2}} \frac{\mathrm{d}V}{\mathrm{d}\chi} \left[\int \mathrm{d}M \frac{\mathrm{d}n}{\mathrm{d}M} b_{h}(M) \int \mathrm{d}Y_{500}^{\mathrm{true}}\; p(Y_{500}^{\mathrm{true}}|M, z) S_{\alpha}(Y_{500}^{\mathrm{true}}, M, z)\right] \times \\
&\frac{\partial P_{ab}(\sfrac{\ell}{\chi}, z(\chi))}{\partial \delta_{\rm LS}}\sigma^{2}_{b}(z(\chi)).
\end{aligned}
\end{equation}
\end{widetext}
\section{Combination of LSST and SO}\label{sec:lsstxso}
We assess the potential of a joint analysis of tSZ number counts, cosmic shear and the cross-correlation between cluster overdensity and cosmic shear to simultaneously infer cosmology and mass calibration by performing a Fisher matrix forecast for a combination of LSST and SO\footnote{We note that a similar analysis could be performed for current surveys, such as ACT, SPT and DES.}. The survey specifications assumed for each survey and probe are detailed below.
\subsection{LSST specifications}
We follow Ref.~\cite{Madhavacheril:2017} and model an LSST-like survey assuming a sky coverage of $18'000$ square degrees (corresponding to $f_{\mathrm{sky}}=0.4$), an angular galaxy number density for the weak lensing sample of $\bar{n}_{\mathrm{source}} = 20$ arcmin$^{-2}$ and standard deviation of the intrinsic ellipticity in each component of $\sigma_{\epsilon} = 0.3$. We further assume the redshift distribution of these galaxies to follow the functional form given in Ref.~\cite{Smail:1994}
\begin{equation}
n(z) \propto z^{2} e^{\frac{z}{z_{0}}},
\end{equation}
where we set $z_{0} = 0.3$. The assumed redshift distribution roughly matches the one outlined in the LSST DESC Science Requirements Document \cite{LSST-SRD:2018}, while both the intrinsic ellipticity and angular galaxy number density are more conservative and are derived by extrapolating results from the Hyper-Suprime Cam (HSC) survey \cite{Aihara:2018}. We subdivide the galaxies into four tomographic redshift bins of approximately equal galaxy number between redshift $z_{\mathrm{min}}=0$ and $z_{\mathrm{max}}=3$\footnote{This leads to the following redshift bin edges $z_{\mathrm{min}, i}, z_{\mathrm{max}, i} = [0., 0.57], [0.57, 0.89], [0.89, 1.41], [1.41, 3.]$ for $i = 0, \cdots 3$.} and estimate the true redshift distribution in each photometric redshift bin $i$ using (e.g. \cite{Amara:2007})
\begin{equation}
n_{i}(z_{t}) = \int_{z_{\mathrm{min}, i}}^{z_{\mathrm{max}, i}} \mathrm{d} z_{p} \; p(z_{p}|z_{t}) n(z_{t}),
\end{equation}
where $z_{p}$ denotes photometric and $z_{t}$ true redshift, respectively. Finally, we model $p(z_{p}|z_{t})$ assuming $z_{p}$ to be Gaussian distributed around $z_{t}$ with $\sigma_{p} = 0.05$ \cite{Schaan:2017}.
We compute spherical harmonic power spectra for all auto- and cross-correlations between those redshift bins in 13 angular multipole bins between $\ell_{\mathrm{min}}=100$ and $\ell_{\mathrm{max}}=4600$\footnote{The maximal angular multipole is chosen in accordance with previous LSST forecasts, see e.g. Refs.~\cite{Krause:2017, Schaan:2017}. Furthermore, we choose the bin centers as $\ell_{\mathrm{mean}} = \{100, \allowbreak 200, \allowbreak 300, \allowbreak 400, \allowbreak 600, \allowbreak 800, \allowbreak 1000, \allowbreak 1400, \allowbreak 1800, \allowbreak 2200, \allowbreak 3000, \allowbreak 3800, \allowbreak 4600\}$.}.
\subsection{SO specifications}
We model the expected survey specifications for SO following Ref.~\cite{Ade:2019}, focusing only on the Large Aperture Telescope (LAT). We assume observations in six frequency bandpasses with beam full-width half-maxima (FWHM) and white noise levels for a sky coverage of $f_{\mathrm{sky}}=0.4$ as given in Tab.~\ref{tab:SOspecs} (c.f. Tab. 1 in Ref.~\cite{Ade:2019}). We additionally model the atmospheric noise contribution following Ref.~\cite{Ade:2019} and refer the reader to their Sec. 2.2 for more details.
\subsubsection{Cluster number counts}
We subdivide the cluster number counts into five bins in redshift between $z_{\mathrm{min}}=0$ and $z_{\mathrm{max}}=1.5$. The maximal cluster redshift is chosen in order to ensure a large enough source sample for mass calibration. Furthermore, photometric redshift uncertainties for LSST are expected to increase significantly at high redshift, which will further limit the usage of high redshift galaxies for mass calibration. We subdivide each of these redshift bins into roughly 15 tSZ amplitude bins between $Y^{\mathrm{obs}}_{500, \mathrm{min}} = 4\times 10^{-13}$ and $Y^{\mathrm{obs}}_{500, \mathrm{max}} = 3\times 10^{-8}$. The exact bin edges and bin numbers considered depend on the cluster redshift bin, as we follow observational analyses (see e.g. \cite{Haan:2016}) and ensure that each bin contains at least a single galaxy cluster\footnote{We note that not applying this cut results in significantly tighter constraints on mass-calibration parameters. However, we choose to not include low cluster number count bins for two reasons: (i) these bins mainly correspond to the high mass end of the mass function, where the approximations made for computing the covariance matrix in this work might break down, and (ii) including bins with very few objects can cause numerical instabilities in Fisher matrix computations.}. The exact bin configurations are given in Appendix \ref{ap:sec:implementation.counts}.
\subsubsection{Cluster lensing}
In order to measure the cluster lensing cross-correlation $C_{\ell}^{\gamma \delta_{\mathrm{cl}}}$, we subdivide the cluster overdensity field into four redshift bins between $z_{\mathrm{min}}=0$ and $z_{\mathrm{max}}=1.41$ and four tSZ amplitude bins between $Y^{\mathrm{obs}}_{500, \mathrm{min}} = 4\times 10^{-13}$ and $Y^{\mathrm{obs}}_{500, \mathrm{max}} = 1.4\times 10^{-9}$. We remove five bins from this subdivision, as they contain less than one cluster, which leaves us with 11 cluster overdensity bins\footnote{The exact bin configurations are given in Appendix \ref{ap:sec:implementation.clust-lens}.}. Furthermore, we only include cross-correlations between galaxy cluster overdensity and cosmic shear for bin combinations for which the lenses are located behind the clusters. These specifications leave us with 20 cross-power spectra $C_{\ell}^{\gamma^{i} \delta^{j}_{\mathrm{cl}, \alpha}}$, which we compute for 16 angular multipole bins between $\ell_{\mathrm{min}}=100$ and $\ell_{\mathrm{max}}=9400$\footnote{This choice of maximal angular multipole ensures that we include a significant amount of information coming from the 1-halo term and is similar to earlier analyses, e.g. \cite{Krause:2017}. Furthermore, the bin centers are chosen as $\ell_{\mathrm{mean}} = \{100, \allowbreak 200, \allowbreak 300, \allowbreak 400, \allowbreak 600, \allowbreak 800, \allowbreak 1000, \allowbreak 1400, \allowbreak 1800, \allowbreak 2200, \allowbreak 3000, \allowbreak 3800, \allowbreak 4600, \allowbreak 6200, \allowbreak 7800, \allowbreak 9400\}$.}.
Finally, when combining LSST and SO, we assume full overlap between the two surveys over a fraction of the sky $f_{\mathrm{sky}}=0.4$. Fig.~\ref{fig:observables} shows an example for each of the three observables considered in our analysis, computed according to the survey and binning specifications given above.
\begin{table}
\caption{Summary of assumed survey specifications for SO LAT (see also Tab. 1 in Ref.~\cite{Ade:2019}).} \label{tab:SOspecs}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{ccc}
Frequency [GHz] & FWHM [arcmin] & Noise (goal) [$\mu$K arcmin] \\ \hline \Tstrut
27 & 7.4 & 52 \\
39 & 5.1 & 27 \\
93 & 2.2 & 5.8 \\
145 & 1.4 & 6.3 \\
225 & 1.0 & 15 \\
280 & 0.9 & 37
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
\begin{figure*}
\begin{center}
\includegraphics[width=0.98\textwidth]{fig1.pdf}
\caption{Examples of the observables considered in this analysis. The leftmost panel shows the cosmic shear auto-power spectrum for redshift bin $i=1$ ($z_{\mathrm{min}} = 0.57$, $z_{\mathrm{max}} = 0.89$), the middle panel shows the cross-correlation between cosmic shear bin $i=3$ ($z_{\mathrm{min}} = 1.41$, $z_{\mathrm{max}} = 3.$) and cluster overdensity bin $i=1$, $\alpha=1$ ($z_{\mathrm{min}} = 0.35$, $z_{\mathrm{max}} = 0.7$, $Y_{\mathrm{min}} = 3.08 \times 10^{-12}$, $Y_{\mathrm{max}} = 2.4 \times 10^{-11}$) and finally the last panel shows the cluster number counts for redshift bin $i=2$ ($z_{\mathrm{min}} = 0.5$, $z_{\mathrm{max}} = 0.75$). We have subdivided the cluster lensing power spectrum into its 1-halo and 2-halo contribution. In all panels, the shaded regions show the 1 $\sigma$ uncertainties.}
\label{fig:observables}
\end{center}
\end{figure*}
\section{Methodology for joint cosmology and mass calibration}\label{sec:mass-calib}
We forecast constraints on cosmological and mass calibration parameters from a joint analysis of cluster number counts, cosmic shear and cluster lensing power spectra, assuming a Gaussian likelihood given by
\begin{equation}
\mathscr{L}(\mathbf{D}^{\mathrm{obs}} \vert \theta) = \frac{1}{[(2\pi)^{d}\det{\mathbf{C}}]^{\sfrac{1}{2}}} e^{-\frac{1}{2}(\mathbf{D}^{\mathrm{obs}}-\mathbf{D}^{\mathrm{theor}})^{\mathrm{T}}\mathbf{C}^{-1}(\mathbf{D}^{\mathrm{obs}}-\mathbf{D}^{\mathrm{theor}})},
\label{eq:likelihood}
\end{equation}
where $\mathbf{C}$ denotes the non-Gaussian covariance matrix, computed as outlined in Sec.~\ref{sec:covariance}\footnote{We note that when computing the inverse covariance matrix, we first invert the correlation matrix and then transform back to the inverse covariance matrix. This avoids numerical instabilities due to the large dynamic range in the covariance matrix elements.} . Furthermore, $\mathbf{D}^{\mathrm{obs}}$ is the observed data vector, given by
\begin{multline}
\mathbf{D}^{\mathrm{obs}} = (C^{\gamma_{i1} \gamma_{j1}}_{\ell}, \cdots, C^{\gamma_{in} \gamma_{jn}}_{\ell}, \; C_{\ell}^{\gamma_{i1} \delta^{k1}_{\mathrm{cl}, \alpha 1}}, \cdots, C_{\ell}^{\gamma_{im} \delta^{km}_{\mathrm{cl}, \alpha m}}, \\
\mathcal{N}^{l1}_{\mathrm{cl}, \beta 1}, \cdots, \mathcal{N}^{lo}_{\mathrm{cl}, \beta o})_{\mathrm{obs}},
\label{eq:psvector}
\end{multline}
and $\mathbf{D}^{\mathrm{theor}}$ denotes the corresponding theoretical prediction. The correlation matrix obtained in our analysis for the experimental specifications given in Sec.~\ref{sec:lsstxso} is shown in Fig.~\ref{fig:covariance-fid}\footnote{The correlation matrix $\mathsf{Corr}$ is obtained from the covariance matrix $\mathbf{C}$ as $\mathsf{Corr}_{ij} = \sfrac{\mathbf{C}_{ij}}{\sqrt{\mathbf{C}_{ii} \mathbf{C}_{jj}}}$.}. The full matrix has dimensions $(n, n) = (519, 519)$ and consists of $130$ $C^{\gamma \gamma}_{\ell}$ measurements, $320$ $C_{\ell}^{\gamma \delta_{\mathrm{cl}}}$ measurements and $69$ $\mathcal{N}_{\mathrm{cl}}$ measurements. As can be seen, the different probes are significantly correlated and the importance of non-Gaussian contributions to the covariance, which give rise to the off-diagonal elements, increase with angular multipole $\ell$ and tSZ amplitude $Y^{\mathrm{obs}}_{500}$.
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\textwidth]{fig2.pdf}
\caption{Joint correlation matrix of tSZ cluster number counts, cosmic shear and the cross-correlation between cluster overdensity and cosmic shear obtained in this analysis.}
\label{fig:covariance-fid}
\end{center}
\end{figure*}
Traditionally, tSZ cluster mass calibration has been performed in a two step process: in a first step, cosmic shear, CMB lensing or X-ray measurements are used to derive prior constraints on cluster masses or mass calibration. In a second step, these prior constraints are folded into the cluster number counts likelihood to derive constraints on cosmological and mass calibration parameters. A number of different approaches exist in the literature (see e.g. \cite{Sehgal:2011, Bocquet:2015, Haan:2016, Alonso:2016, Louis:2017}), which vary in the data used to derive priors on mass calibration and their derivation. In order to further validate the mass calibration method proposed in this work, we compare its forecasted constraints to those obtained in such a stacking analysis. For the stacked cluster number counts likelihood, we closely follow the approach outlined in Ref.~\cite{Madhavacheril:2017}: we compute uncertainties on inferred weak lensing masses assuming measurements of the real-space cluster lensing signal for all clusters in the sample. These constraints are used to derive cluster number counts binned in redshift $z$, tSZ signal-to-noise $q$ and weak lensing mass $M_{\mathrm{WL}}$. The measurements are finally used to compute constraints on cosmological and mass calibration parameters assuming Poisson noise (i.e. neglecting the non-Gaussian covariance discussed above\footnote{We have made this choice in order to maintain consistency with the original analysis in Ref.~\cite{Madhavacheril:2017}}).
\section{Forecasting methods}\label{sec:forecasts}
We use a Fisher matrix formalism to forecast constraints on cosmological and mass calibration parameters for both methods outlined above. The Fisher matrix allows for propagation of experimental uncertainties to uncertainties on model parameters. Under the assumption that the dependence of the data covariance matrix on the parameters of interest $\theta_{\alpha}$ can be neglected, the Fisher matrix for a given experiment, measuring a data vector $\mathbf{D}$, is given by (see e.g. \cite{Fisher:1935, Kendall:1979, Tegmark:1997})
\begin{equation}
F_{\alpha \beta} = \frac{\partial \mathbf{D}}{\partial \theta_{\alpha}}\mathbf{C}^{-1}\frac{\partial \mathbf{D}}{\partial \theta_{\beta}}.
\end{equation}
The Cram\'er-Rao bound states that the uncertainty on $\theta_{\alpha}$, marginalized over all other $\theta_{\beta}$ satisfies
\begin{equation}
\Delta \theta_{\alpha} \geq \sqrt{(F^{-1})_{\alpha \alpha}}.
\end{equation}
Computing the Fisher matrix requires the assumption of a fiducial model. In this work, we choose cosmological parameter values close to those derived by the Planck Collaboration in their 2015 data release using only temperature data \cite{Planck:2016} (c.f. the first column of Tab. 4 in Ref.~\cite{Planck:2016}). The fiducial values assumed for all parameters are summarized in Tab.~\ref{tab:fiducial-model}.
We assess the potential of a combination of LSST and SO to simultaneously constrain cosmology and mass calibration by mainly investigating its constraining power on the time evolution of the dark energy equation of state parameter $w(a)$, parametrized as $w(a) = w_{0} + (1 - a)w_{a}$ \cite{Chevallier:2001, Linder:2003}\footnote{We note, however, that we expect the methods presented here to be useful for constraining any cosmological parameter affecting late-time structure growth, such as e.g. the sum of neutrino masses, $\sum_{i} m_{\nu, i}$.}. We therefore focus on $w_{0}w_{a}$CDM and forecast constraints on the set of cosmological and systematics parameters given by $\boldsymbol{\theta} = \{H_{0}, \allowbreak \Omega_{b}h^{2}, \allowbreak \Omega_{c}h^{2}, \allowbreak A_{s}, \allowbreak n_{s}, \allowbreak w_{0}, \allowbreak w_{a}, \allowbreak Y_{*}, \allowbreak \sigma_{\log{Y_{0}}}, \allowbreak \alpha_{\sigma}, \allowbreak \gamma_{\sigma}, \allowbreak \alpha_{Y}, \allowbreak \beta_{Y}, \allowbreak \gamma_{Y}, \allowbreak \Delta z_{i}, \allowbreak m_{i} \}$, $i \in [0, \cdots, 3]$, where $H_{0}$ is the Hubble parameter, $\Omega_{b}h^{2}$ is the physical baryon density today, $\Omega_{c}h^{2}$ is the physical cold dark matter density today, $n_{\mathrm{s}}$ denotes the scalar spectral index, $A_{s}$ is the primordial power spectrum amplitude at pivot wave vector $k_{0}=0.05$ Mpc$^{-1}$\footnote{We note that for consistency with Ref.~\cite{Madhavacheril:2017} we choose to parametrize the power spectrum amplitude in terms of $A_{s}$ instead of $\sigma_{8}$, which denotes the r.m.s. of linear matter fluctuations in spheres of comoving radius 8 $h^{-1}$ Mpc.} and $w_{0}, w_{a}$ parametrize the equation of state of dark energy. We compute derivatives of the observables with respect to these parameters numerically using a five-point stencil with step $\epsilon = 0.01\theta$, where $\theta$ denotes any parameter considered in our analysis\footnote{For parameters with fiducial value of zero, we set $\epsilon = 0.01$.}. We test the stability of our results by varying the parameter $\epsilon$ and find our results to be largely insensitive to this choice.
Unless stated otherwise, we combine our constraints with prior information from the Planck power spectrum following Ref.~\cite{Madhavacheril:2017}. Specifically, we include Planck temperature information from angular scales $2< \ell <30$ from the full Planck angular sky coverage ($f_{\mathrm{sky}}= 0.6$), temperature and polarization information from $30< \ell <100$ from the part of sky in which Planck and SO overlap ($f_{\mathrm{sky}}= 0.4$) and finally temperature and polarization information from $30< \ell <2500$ from the part of sky covered by Planck but not by SO ($f_{\mathrm{sky}}= 0.2$). Including the full Planck angular range and sky coverage, or the forecasted SO primary CMB information was found to not significantly impact forecasted constraints on $w_{0}$ and $w_{a}$ \cite{Ade:2019}, which are the primary focus of this work. We further follow Ref.~\cite{Krause:2017} and assume Gaussian priors on $\Delta z_{i}$ and $m_{i}$ with standard deviations $\sigma(\Delta z_{i})=0.002$ and $\sigma(m_{i})=0.004$ respectively. However, we do not assume any priors on the mass calibration parameters.
\begin{table*}
\caption{Summary of assumed fiducial model and parameters considered in the Fisher analysis.} \label{tab:fiducial-model}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{cccc}
Parameter & Fiducial value & Prior & Description \\ \hline \Tstrut
$H_{0}$ & 69. & Planck\footnote{See description in Sec.~\ref{sec:forecasts}.} & cosmology \\
$\Omega_{b}h^{2}$ & 0.02222 & Planck & cosmology \\
$\Omega_{c}h^{2}$ & 0.1197 & Planck & cosmology \\
$A_{s}$ & $2.1955 \times 10^{-9}$ & Planck & cosmology \\
$n_{s}$ & 0.9655 & Planck & cosmology \\
$w_{0}$ & -1. & Planck & cosmology \\
$w_{a}$ & 0. & Planck & cosmology \\ \\
$Y_{*}$ & $2.42 \times 10^{-10}$ & - & mean of $Y-M$ relation\footnote{See Eq.~\ref{eq:Y-M-mean}.} \\
$\alpha_{Y}$ & 1.79 & - & mean of $Y-M$ relation\\
$\beta_{Y}$ & 0. & - & mean of $Y-M$ relation\\
$\gamma_{Y}$ & 0. & - & mean of $Y-M$ relation \\
$\sigma_{\log{Y_{0}}}$ & 0.127 & - & scatter of $Y-M$ relation\footnote{See Eq.~\ref{eq:Y-M-scatter}.} \\
$\alpha_{\sigma}$ & 0. & - & scatter of $Y-M$ relation\\
$\gamma_{\sigma}$ & 0. & - & scatter of $Y-M$ relation\\ \\
$\Delta z_{i}$ & 0. & $\mathcal{N}(\mu = 0, \sigma=0.002)$\footnote{Here, $\mathcal{N}$ denotes a 1-dimensional Gaussian distribution.} & photo-$z$ uncertainties \\
$m_{i}$ & 0. & $\mathcal{N}(\mu = 0, \sigma=0.004)$ & multiplicative shear bias \\
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table*}
\section{Results}\label{sec:results}
Fig.~\ref{fig:constraints-data-splits-cosmo-H0-As-w0-wa} shows our fiducial forecasted constraints on a subset of cosmological parameters\footnote{The full panel is shown in Fig.~\ref{fig:constraints-data-splits-cosmo} in the Appendix.} for a combination of LSST and SO, denoted $\texttt{gg+gdc+nc}$\footnote{Here, $\texttt{gg}$ denotes cosmic shear, $\texttt{gdc}$ denotes the cross-correlation between cluster overdensity and cosmic shear and finally $\texttt{nc}$ denotes cluster number counts.} in the figure. These constraints are obtained from a joint analysis of SO tSZ cluster number counts, LSST cosmic shear and the cross-correlation between cosmic shear and cluster overdensity, combined with prior information from Planck as described in Sec.~\ref{sec:forecasts}. The corresponding constraints on mass calibration parameters are shown in Fig. ~\ref{fig:constraints-data-splits-Y-M}. As can be seen, the combination of SO clusters with LSST cosmic shear has the potential to provide rather tight constraints on both cosmological and mass calibration parameters. As an example, the dark energy equation of state parameters $w_{0}$ and $w_{a}$ are constrained to a level of $\sim 8 \%$ and $\sigma(w_{a})\sim 0.3$, respectively. This constitutes an improvement in the Dark Energy Task Force (DETF) Figure of Merit \cite{Albrecht:2006} with respect to LSST cosmic shear alone of approximately a factor of two. In addition, we find that this combination provides tight constraints on $H_{0}$ and $A_{s}$, improving the uncertainties on the primordial power spectrum amplitude by a factor of two, again compared to LSST cosmic shear. These results also imply tighter constraints on $\sigma_{8}$, which is directly constrained by low-redshift large-scale structure observables. Comparing our fiducial constraints to those obtained from the Planck CMB prior alone, we find significant improvements in the constraints on $H_{0}$, $A_{s}$, $w_{0}$ and $w_{a}$, with the dark energy figure of merit increasing by a factor of approximately $1400$. Looking at the mass calibration parameters, we find a $\sim 3 \%$ constraint on the amplitude of the $Y-M$ relation, $Y_{*}$. Comparing this constraint to existing measurements is complicated by the fact that the respective analyses significantly differ in both methodology and constrained parameter set. We note however that this constraint constitutes a significant improvement compared to current constraints, which are at the level of $~ 17 \%$ (see e.g. Ref.~\cite{Bocquet:2019}). These results are especially remarkable, as the cosmological constraints are fully and self-consistently marginalized over uncertainties in the tSZ $Y-M$-relation and cosmic shear measurement systematics and are derived accounting for the full non-Gaussian covariance between cluster number counts and the various cosmic shear observables. Similarly, the constraints on mass calibration shown in Fig.~\ref{fig:constraints-data-splits-Y-M} illustrate the constraining power of LSST and SO when self-consistently marginalizing over cosmic shear systematics.
In order to disentangle the contribution of separate probes to these constraints, we compute forecasted constraints for two subsets of our full data vector: in the first case, we combine only cosmic shear and cluster number counts (denoted $\texttt{gg+nc}$), and in the second case we combine cluster number counts and the cluster lensing power spectrum (denoted $\texttt{gdc+nc}$). The obtained constraints are shown in Figures \ref{fig:constraints-data-splits-cosmo-H0-As-w0-wa} and \ref{fig:constraints-data-splits-Y-M} alongside our fiducial ones. From these figures we see that the combination $\texttt{gg+nc}$ yields cosmological parameter constraints comparable to those obtained from our fiducial case, while leading to significantly weaker constraints on mass calibration. The combination $\texttt{gdc+nc}$ on the other hand, shows the opposite behavior, i.e. the cosmological constraints are weaker while the constraints on mass calibration are comparable to the fiducial case. These results suggest that adding cosmic shear to cluster number counts mainly affects the cosmological constraining power. Combining cluster lensing and number counts on the other hand, allows for precise mass calibration and breaks some of the degeneracies between cosmology and the $Y-M$ relation, inherent to cluster counts alone.
It is interesting to ask which angular scales in $C_{\ell}^{\gamma \delta_{\mathrm{cl}}}$ contribute most to the constraints on the $Y-M$ relation. To this end, we forecast constraints for $\texttt{gdc+nc}$ restricting the angular multipole range for the cluster lensing cross-correlation to $\ell \leq 3000$ as compared to our fiducial case with $\ell \leq 9400$. Somewhat surprisingly, we find almost identical constraints on both cosmological and mass calibration parameters in both cases\footnote{As the constraints are almost indistinguishable, we do not show them in any of the figures. In addition, further reducing the multipole range to $\ell \leq 1000$ only leads to modest increases in parameter uncertainties.}. This suggests that the constraints on mass calibration are driven by the large and intermediate angular scales rather than the smallest scales considered in our analysis. As can be seen from Fig.~\ref{fig:observables}, these scales receive contributions from both the 1- and the 2-halo term of the power spectrum. For the intermediate redshift bin shown in Fig.~\ref{fig:observables}, the 2- to 1-halo transition occurs at $\ell \sim 300$, while for the highest redshift bin, it is pushed to $\ell \sim 600$. Our results thus suggest that the amplitude of $C_{\ell}^{\gamma \delta_{\mathrm{cl}}}$ on relatively large angular scales contains some information on mass calibration, as also seen in Ref.~\cite{Majumdar:2004}. The large-scale amplitude of the cluster lensing signal is predominantly determined by the cluster bias, which depends on mass, and is therefore sensitive to mass calibration parameters, thus allowing for constraining the mass-observable relation. This is different from traditional mass calibration methods, which solely focus on the 1-halo term and thus use information from smaller scales to constrain the $Y-M$ relation\footnote{A potential concern about using information from the large-scale amplitude of the cluster lensing signal for mass calibration is the uncertainty on cluster bias models. In order to test the robustness of our results to these uncertainties, we forecast constraints from $\texttt{gdc+nc}$ accounting for a $10 \%$ uncertainty in the amplitude of the cluster lensing power spectrum, finding only modest increases in parameter constraints.}. This complementarity therefore suggests an interesting way to test for systematics in mass calibration by comparing the results obtained with both methods.
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\textwidth]{fig3.pdf}
\caption{Forecasted constraints on a subset of cosmological parameters obtained in a joint analysis of LSST and SO for three different data splits. The constraints are marginalized over mass calibration and cosmic shear systematics parameters. The inner (outer) contour shows the $68 \%$ confidence limit (c.l.) ($95 \%$ c.l.).}
\label{fig:constraints-data-splits-cosmo-H0-As-w0-wa}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\textwidth]{fig4.pdf}
\caption{Forecasted constraints on mass calibration parameters obtained in a joint analysis of LSST and SO for three different data splits. The constraints are marginalized over cosmological and cosmic shear systematics parameters. The inner (outer) contour shows the $68 \%$ c.l. ($95 \%$ c.l.).}
\label{fig:constraints-data-splits-Y-M}
\end{center}
\end{figure*}
We further test the methodology presented in this analysis by comparing the obtained forecasted constraints to those obtained performing a traditional stacking analysis, as described in Sec.~\ref{sec:mass-calib}. As the stacking analysis does not contain cosmic shear information, we only perform this comparison for the $\texttt{gdc+nc}$ data split. We constrain the same parameter set and apply identical priors to both methods, except that for consistency with existing analyses we do not account for cosmic shear systematics when forecasting constraints from the stacking method. The resulting constraints are shown in Figures \ref{fig:constraints-cosmo-cls-vs-stacking-H0-As-w0-wa}\footnote{The full panel for the cosmological parameter constraints is shown in Fig.~\ref{fig:constraints-cosmo-cls-vs-stacking} in the Appendix.} and \ref{fig:constraints-Y-M-cls-vs-stacking}. As opposed to the constraints obtained from the stacking method, the constraints from the cross-correlation method are fully marginalized over cosmic shear systematic uncertainties and are derived taking into account the full non-Gaussian covariance between cluster counts and cosmic shear. From Figures \ref{fig:constraints-cosmo-cls-vs-stacking-H0-As-w0-wa} and \ref{fig:constraints-Y-M-cls-vs-stacking} we see that the cross-correlation method nevertheless yields significantly tighter constraints on cosmological parameters, especially $H_{0}$ and $A_{s}$, where we find a reduction in the $1\sigma$ uncertainties of approximately $30\%$ and $40\%$ respectively. For the mass calibration, we find the cross-correlation method to yield comparable or tighter constraints on the parameters entering the mean of the $Y-M$ relation (see Eq.~\ref{eq:Y-M-mean}), e.g. $\beta_{Y}$. In contrast however, the obtained constraints on the scatter in the $Y-M$ relation (see Eq.~\ref{eq:Y-M-scatter}) are weaker. From Fig.~\ref{fig:constraints-Y-M-cls-vs-stacking} we see that the larger uncertainties on these parameters are mainly driven by increased parameter degeneracies obtained for the cross-correlation method. This suggests that these differences are not due to the mass calibration method itself but rather due to the different treatment of cluster number counts in both analyses: while the stacking method allows for binning the cluster number counts in both $M_{\mathrm{WL}}$ and $Y^{\mathrm{obs}}_{500}$, the number counts in the cross-correlation method are only binned in $Y^{\mathrm{obs}}_{500}$. This lack of explicit mass information in the cluster number counts can lead to larger degeneracies and thus enhanced correlations between the different mass calibration parameters. Further confirmation comes from the fact that we find the derivatives of the stacked cluster number counts with respect to the parameters of $\sigma_{\log Y_{500}}(M, z)$ marginalized over $Y^{\mathrm{obs}}_{500}$ to be significantly larger than the derivatives obtained when marginalizing the cluster number counts over $M_{\mathrm{WL}}$. Another way of seeing this is that we find a loss of most of the constraining power on the scatter of the $Y-M$ relation when using the stacked cluster number counts marginalized over $M_{\mathrm{WL}}$. As discussed above, an additional reason for these differences might be the fact that the constraints derived using the cross-correlation method are fully marginalized over systematics in the cosmic shear and take into account the cross-correlation between cluster number counts and cosmic shear, in contrast to the stacking method.
Despite the somewhat weaker constraints on the scatter in the $Y-M$ relation, these results show that the cluster lensing power spectrum provides a promising alternative to traditional tSZ mass calibration methods, as it allows for both precise mass calibration and additionally provides cosmological information complementary to cluster number counts (as can be seen from the fact that the cosmological constraints from $\texttt{gdc+nc}$ are tighter than those obtained with the stacking method).
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\textwidth]{fig5.pdf}
\caption{Comparison of the forecasted constraints on a subset of cosmological parameters obtained using the two methods outlined in Sec.~\ref{sec:mass-calib}. The constraints are marginalized over mass calibration and cosmic shear systematics parameters. The inner (outer) contour shows the $68 \%$ c.l. ($95 \%$ c.l.).}
\label{fig:constraints-cosmo-cls-vs-stacking-H0-As-w0-wa}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\textwidth]{fig6.pdf}
\caption{Comparison of the forecasted constraints on mass calibration parameters obtained using the two methods outlined in Sec.~\ref{sec:mass-calib}. The constraints are marginalized over cosmological and cosmic shear systematics parameters. The inner (outer) contour shows the $68 \%$ c.l. ($95 \%$ c.l.).}
\label{fig:constraints-Y-M-cls-vs-stacking}
\end{center}
\end{figure*}
\section{Summary and conclusions}\label{sec:conclusions}
In this work we present a novel method for joint cosmological parameter inference and cluster mass calibration from a combination of weak lensing measurements and thermal Sunyaev-Zel'dovich cluster abundances. We focus on a combination of cluster number counts, angular cosmic shear power spectra and the angular cross-correlation between cluster overdensity and cosmic shear, which acts as the main cluster mass calibrator in our analysis. Using a halo model approach, we derive and compute theoretical estimates for all observables as well as their full non-Gaussian covariance. We then forecast constraints for a joint analysis of LSST and SO on both cosmological and mass calibration parameters in a Fisher analysis, fully marginalizing over systematic uncertainties in cosmic shear measurements. Our results show that the method presented here yields competitive constraints on both cosmological and mass calibration parameters. Furthermore, we find most of the mass calibration information to be contained in the large and intermediate angular scales of the cross correlation between cosmic shear and cluster overdensity.
We then compare our constraints to those obtained in a more traditional stacked cluster weak lensing analysis. Generally, we find the method presented here to yield tighter constraints on cosmological parameters and comparable or tighter constraints on the mean of the mass-observable relation. However, we find the scatter in the mass-observable relation to be more strongly constrained with the traditional method. We attribute this not to the mass calibration method itself but rather to different treatments of cluster number counts in both methods: the traditional methods allow for binning the cluster number counts in mass $M_{\mathrm{WL}}$ and tSZ amplitude $Y^{\mathrm{obs}}_{500}$ while the cluster counts in the method presented here are solely binned in $Y^{\mathrm{obs}}_{500}$. The additional mass binning in traditional methods allows to break degeneracies between the parameters of the mass-observable relation and therefore leads to tighter constraints on its scatter.
Therefore, our analysis shows that the cross-correlation between cluster overdensity and cosmic shear provides a promising alternative to traditional mass calibration methods, offering several advantages compared to traditional approaches. First of all, the constraints derived using the method presented here are fully and consistently marginalized over cosmic shear measurement systematics and are derived taking into account the full non-Gaussian covariance between cluster counts and cosmic shear. Secondly, computing the cross-correlation between cosmic shear and cluster overdensity amounts to performing a statistical mass calibration. In contrast, traditional mass calibration methods require measuring the cluster lensing signal for each cluster in the sample, which might become prohibitively expensive for future surveys. Finally, the joint cluster count and cosmic shear likelihood derived in this work can be readily combined with other probes of the large-scale structure, such as galaxy clustering.
We envisage several possible extensions of the present work. On the one hand it will be interesting to test the method presented here by applying it to combinations of current CMB and large-scale structure surveys, such as ACT, SPT or DES. Due to the lower signal-to-noise in these data, as compared to LSST and SO, we however expect to constrain only a subset of the parameters considered in this work, especially those entering the mass calibration. Furthermore, applying this method to data will necessitate the inclusion of additional systematics, such as baryon feedback effects on the matter power spectrum (see e.g. Refs.~\cite{Rudd:2008, vanDaalen:2011}). On the theoretical side, we aim to investigate the potential of the cross-correlation method to constrain non-parametric mass-observable relations, which would remove the need of assuming uncertain functional forms for both the mean and scatter of the $Y-M$ relation.
The analysis presented in this work shows that the cross-correlation method provides a promising and self-consistent way for jointly analyzing thermal Sunyaev-Zel'dovich cluster counts and cosmic shear. This bodes well for paving the way for multi-probe analyses including tSZ cluster number counts and harnessing the full potential of galaxy clusters as a precision cosmological probe.
\begin{acknowledgments}
AN would especially like to thank An\v{z}e Slosar and David Alonso: An\v{z}e Slosar for pulling one of his many Eastern European tricks when AN was stuck on this project and for comments on an earlier version of this manuscript. David Alonso for encouragement and many helpful discussions and comments on an earlier version of this manuscript. We would further like to thank Mat Madhavacheril and Nick Battaglia for many useful discussions regarding stacked weak lensing mass calibration and for help with using their code \texttt{szar}\footnote{\url{https://github.com/nbatta/szar}.}. We would also like to thank Mat Madhavacheril and Colin Hill for comments on an earlier version of this manuscript. Finally, we would like to thank Elisabeth Krause for helpful discussions regarding covariance matrices and Emmanuel Schaan for helpful discussions and for code comparison.
JD and AN acknowledge support from National Science Foundation Grant No. 1814971. The Flatiron Institute is supported by the Simons Foundation.
This is not an official SO collaboration paper.
The color palettes employed in this work are taken from $\tt{http://colorpalettes.net}$. The contour plots have been created using $\tt{corner.py}$ \cite{ForemanMackey:2016}.
\end{acknowledgments}
|
1,314,259,996,486 | arxiv | \section{Results}
In order to bring out the role of NQEs and exchange symmetry, we performed simulations of solid deuterium using three different methods, treating deuterium as 1) a classical particle (MD), 2) a distinguishable quantum particle (PIMD), and 3) an indistinguishable boson (PIMD-B) (Fig. \ref{fig:density_nqe} and \ref{fig:geo}). The average density $n(r)$ is greatly affected by exchange processes (Fig. \ref{fig:density_nqe}).
Even for distinguishable deuterium the NQEs make the atomic density distribution of neighbouring atoms overlap (Fig. \ref{fig:density_nqe}a-c). This overlap suggests the possible role of exchange processes. Indeed, as the exchange of deuterium atoms is allowed \textit{via} PIMD-B simulation, it is difficult to spot the precise equilibrium positions of deuterium $I4_1/amd$ phase due to active exchange (Fig. \ref{fig:density_nqe}d-f). This implicates that the connected ring polymers of deuterium atoms emerge at low temperatures (Fig. \ref{fig:density_nqe}d-f). At first sight (Fig. \ref{fig:geo}a), it would appear that the $n(r)$ would correspond to that of a glassy system, however our analysis shows that the $I4_1/amd$ symmetry is hidden but not lost (Fig. \ref{fig:geo}b,c). To show it, we evaluated the structure factor,
\begin{equation}
\label{eq:sq}
S(q) =\frac{1}{PN}\sum^{P}_\tau\sum^{N}_{j,k}e^{-i\mathbf{q}(\mathbf{R}^{(\tau)}_j-\mathbf{R}^{(\tau)}_k)},
\end{equation}
where $P, N, \mathbf{R}^{(\tau)}_j$ are the number of beads, the number of particles and position of atom $j$ at $\tau$ imaginary time, respectively. Bragg peaks can be clearly seen with and without exchange at the same positions in reciprocal space (Fig. \ref{fig:geo}b and Supplemental Fig. 8). Thus, the result indicates that this peculiar exchange of deuterium does not break the solid long-range order. Also, the pair correlation of solid phase is preserved under exchange interactions as evidenced by the radial distribution function $g(r)$ of MD, PIMD and PIMD-B simulations (Fig. \ref{fig:geo}c). Even in the active exchange regime, the system still remains metallic as the solid phase. This can be understood given that this anomalous deuterium phase preserves the solid long-range order. Thus, the density of states (Fig. \ref{fig:geo}d) is similar to that of solid (Supplemental Fig. 2b). The presence of disorder in a supersolid phase might introduce the localisation of electronic states\cite{Anderson58}. However, our analysis based on inverse participation ratio (IPR) shows that the electronic states of supersolid phase are delocalised (Fig. \ref{fig:geo}d and Supplemental section III).
The fact that one can reconcile long range order and a very active exchange regime remains puzzling also in the Feynman isomorphism. In order to get insight into how this is possible, we look at the beads’ spatial arrangement as it evolves during the simulation where all permutations contribute to the forces on atoms at each time step\cite{Hirshberg2019}. This can be measured by a structure factor of the beads system considered as a set of independent particles $S_{rel}(q) =\frac{1}{PN}\sum^{P}_{\tau,\tau^{\prime}}\sum^{N}_{j,k}e^{-iq(\mathbf{R}^{(\tau)}_j-\mathbf{R}^{(\tau^{\prime})}_k)}$. While the beads distribution changes dynamically from one time step to another, the overall long-range order of $(P \times N)$ configuration is still preserved (Supplemental Fig. 9). This points to a highly coherent exchange mechanism.
An elegant way of measuring whether a system is superfluid is to compute its winding number\cite{Pollock1987}. This quantity reflects the number of paths that, due to exchange, are so long that they wrap around the periodic boundary conditions\cite{Pollock1987}. In our approach, in which all permutation are sampled at every time step, standard methods to evaluate it cannot be applied. Therefore, we have developed an approximate but highly accurate approach to measure the winding number in PIMD-B simulations (Supplemental Material section III and Supplemental Fig. 11). The result obtained is presented in Fig. \ref{fig:permutation_prob}. It shows that at $T<1.0$ K a superfluid condensate is formed. The analysis of probability of observing longer rings also confirms this picture (Supplemental Fig. 12). Our calculation shows that for high pressure deuterium a defect-free pathway to supersolidty is possible.
Experiments on such thermodynamic conditions will be feasible in near future given the rapid advancement of diamond anvil cell techniques at cryogenic temperature\cite{Dias715,Dalladay-Simpson2016,Ji2019}, and verifying this prediction in experiments will be a fascinating challenge to undertake.
\begin{acknowledgments}
We are grateful to L. Bonati, M. Yang, V. Rizzi, D. Frenkel, V. Kapil, C. Schran and K. Trachenko for helpful discussions. This research was supported by the European Union (Grant No. ERC-2014-ADG-670227/VARMET) and the NCCR MARVEL, funded by the Swiss National Science Foundation. Computational resources were provided by the Euler cluster at ETH Zürich and the Swiss National Supercomputing Centre (CSCS) under project ID s1052. C.W.M. acknowledges the support from Korea Institute of Science and Technology Information (KISTI) for the Nurion cluster (KSC-2019-CRE-0139 and KSC-2019-CRE-0248). Part of this work was performed under the gracious hospitality of ETH Zürich and Universit\`a della Svizzera italiana, Lugano.
All the implementations of the isotropic and full-cell NPT simulations of PIMD-B are freely available in the \texttt{LAMMPS Github} repository. All the necessary input files of this computational study are also available in the author's \href{https://github.com/changwmyung/H_PIMD}{\texttt{Github}} repository.
\end{acknowledgments}
\section{Density functional calculations and machine learning potential}
\subsection{Convergence of density functional theory calculations}
\floatsetup[figure]{style=plain,subcapbesideposition=top}
\begin{figure}[h]
\includegraphics[width=0.45\linewidth, height=5cm]{figs-si/conv-ecut.png}
\label{fig:ecut}
\includegraphics[width=0.45\linewidth, height=5cm]{figs-si/conv-kp.png}
\label{fig:kp}
\caption{\textbf{The convergence test of DFT total energy.} The convergence of the total energy per atoms with respect to (a) the planewave energy cutoff and (b) the number of k-points.}
\label{fig:conv}
\end{figure}
Since the interatomic distance between deuterium atoms in high-pressure solid phases is small ($<1.0 \AA$), the reliability of density functional theory (DFT) calculation is affected by the pseudisation radius ($r_c$) of PAW pseudopotential. In the pressure range of $p = 800-1200 \ \text{GPa}$, the Wigner-Sitz radius of system ($r_s$) is $\sim 0.5$ \AA ($\sim 0.94 \ a.u.$). Previous DFT study showed that the $r_c$ of $0.5 a.u.$ is required to ensure the convergence of total energy of high-pressure hydrogen solid\cite{mcmahon11}. Therefore, we employ a PAW pseudopotential of $r_c =0.5 \ a.u.$ following the previous work.
As the long-range van der Waals (vdW) interactions play important roles in high-pressure hydrogen/deuterium systems\cite{li2013,cui20}, we calculated the energy, forces and stress tensors using the non-local vdW functional, vdW-DF2 functional\cite{Thonhauser2007, Berland2015, Sabatini2012}, implemented in \texttt{Quantum Espresso} package (v6.6)\cite{PGiannozzi2017}. A previous study of $I4_1/amd$ phase showed that the long-range dispersion contribution to the enthalpy is sensitive to the size of a supercell. They observed that the long-range contribution starts to converge from a supercell of 72 atoms\cite{cui20}. Therefore, we used a supercell of 128 atoms to eliminate any finite size effects related to the long-range dispersion interactions.
Because the high-pressure deuterium system is highly compressed with a small unit-cell, rigorous tests are needed to ensure the total energy convergence depending on planwave energy cutoff (Supplemental Fig. \ref{fig:conv}a) and the number of k-points(Supplemental Fig. \ref{fig:conv}b). We choose the planewave energy cutoff of $100 \ \text{Ry}$ that shows $0.8 \ \text{meV/atom}$ error compared to the fully converged case of $200 \ \text{Ry}$. We use the k-point mesh of ($10\times10\times8$) that shows $0.01 \ \text{meV}$ error compared to the fully converged case of k-mesh ($12\times12\times12$) in Supplemental Fig. \ref{fig:conv}b. The corresponding $k$-grid spacing is 2$\pi \times$ 0.0136 \AA$^{-1}$. This is finer than the k-mesh spacing of previous studies of metallic hydrogen (2$\pi \times$ 0.04 \AA$^{-1}$\cite{Pickard2007} or 2$\pi \times$ 0.05 \AA$^{-1}$ \cite{Cudazzo08}) to ensure the convergence of Fermi surface. Therefore, we train the machine learning (ML) potentials for molecular dynamics simulations based on the DFT calculations with above parameters.
\subsection{Electronic and vibrational structures of deuterium $I4_1/amd$ phase}
Our DFT calculations of band structure (Supplemental Fig. \ref{fig:elec}) and density of states (Supplemental Fig. \ref{fig:elec}) at the vdW-DF2 level are consistent with the previous works in which the $I4_1/amd$ phase is metallic .
\floatsetup[figure]{style=plain,subcapbesideposition=top}
\begin{figure}[h]
\includegraphics[width=0.45\linewidth, height=5cm]{figs-si/band.png}
\label{fig:band}
\includegraphics[width=0.45\linewidth, height=5cm]{figs-si/dos.png}
\label{fig:dos}
\caption{\textbf{The electronic structure of deuterium $I4_1/amd$ solid phase.} (a) The band structure of deuterium $I4_1/amd$ solid phase of the primitive cell, where $S'=S|S_0$, $\Gamma'=\Gamma|X$ and $R'=R|G$ and high-symmetry points $\Gamma$=(0.0,0.0,0.0), X=(0.0,0.0,0.5), P=(0.25,0.25,0.25), N=(0.0,0.5,0.0), M=(0.5, 0.5,-0.5), S=(0.28, 0.72, -0.28), S$_0$=(-0.28, 0.28, 0.28), R=(-0.06, 0.06, 0.5), G=(0.5, 0.5, -0.06) of the BZ. (b) The density of states of hydrogen $I4_1/amd$ solid phase of ($4\times4\times2$) supercell. The fermi energy ($E_F$) is indicated as cyan dashed line.}
\label{fig:elec}
\end{figure}
It is also important to ensure that our DFT calculation describes the vibrational property of $I4_1/amd$ phase accurately without any unstable modes. This is crucial for building accurate ML potentials to perform molecular dynamics simulations. The phonon dispersion of $I4_1/amd$ phase has no imaginary phonon branch across the Brillouin zone (BZ) (Supplemental Fig. \ref{fig:ph}).
\floatsetup[figure]{style=plain,subcapbesideposition=top}
\begin{figure}[h]
\includegraphics[height=6cm]{figs-si/ph-band.png}
\caption{\textbf{The phonon dispersion of deuterium $I4_1/amd$ phase.} The phonon band structure of hydrogen $I4_1/amd$ solid phase of $(4\times4\times4)$ supercell of the primitive cell, where high-symmetry points are $\Gamma$=(0.0,0.0,0.0), X=(0.0,0.0,0.5), P=(0.25,0.25,0.25), N=(0.0,0.5,0.0), M=(0.5, 0.5,-0.5), S=(0.28, 0.72, -0.28), R=(-0.06, 0.06, 0.5), G=(0.5, 0.5, -0.06) of the BZ.}
\label{fig:ph}
\end{figure}
\subsection{Machine learning potential}
\floatsetup[figure]{style=plain,subcapbesideposition=top}
\begin{figure}[h]
\includegraphics[height=6cm]{figs-si/error-e.png}
\caption{\textbf{The total energy root mean squared error (RMSE) of ML potential for the testing set.} The total energy RMSE per atom (meV/atom) is plotted by bars as a function of total energy (meV/atom) shifted by its average, 12.687 eV/atom.}
\label{fig:errore}
\end{figure}
\floatsetup[figure]{style=plain,subcapbesideposition=top}
\begin{figure}[h]
\includegraphics[height=6cm]{figs-si/error-f.png}
\caption{\textbf{The comparison of forces between DFT and ML potential for the testing set.} The $x,y,z$ components of forces $(f_x,f_y,f_z)$ of DFT and ML potential for the testing set are plotted.}
\label{fig:errorf}
\end{figure}
\floatsetup[figure]{style=plain,subcapbesideposition=top}
\begin{figure}[h]
\includegraphics[height=6cm]{figs-si/error-v.png}
\caption{\textbf{The comparison of stress tensor virials between DFT and ML potential for the testing set.} The comparison of stress tensor virial $(v_{xx}, v_{yy}, v_{zz}, v_{yz}, v_{xz}, v_{xy})$ of DFT and ML potential for the testing set are plotted for the case of $p = 800$ GPa.}
\label{fig:errorv}
\end{figure}
\captionsetup{format=naturestyle,labelformat=naturestyle-table,justification=just,singlelinecheck=false,labelsep=naturestyle}
\begin{table}[]
\begin{tabular}{cccc}
\hline
\hline
Pressure (GPa) & Energy ($meV/$atom) & Force ($meV/$\AA) & Virial ($eV$) \\ \hline
800 & 0.7 & 60.0 & 2.48 \\
1000 & 0.8 & 86.6 & 1.93 \\
1200 & 1.0 & 91.1 & 2.05 \\ \hline \hline
\end{tabular}
\caption{\label{tab:mlp-error}The test errors of neural network potential for potential energy ($meV/atom$), force ($meV/\AA$) and pressure virial (eV) at $p=800, 1000, 1200$ GPa.}
\end{table}
\captionsetup{format=naturestyle,labelformat=naturestyle,justification=just,singlelinecheck=false,labelsep=naturestyle}
We sampled the configurations of solid and liquid phases of high-pressure deuterium by using Born-Oppenheimer DFT (vdW-DF2 functional\cite{Thonhauser2007, Berland2015, Sabatini2012}) NPT MD simulations at the temperature range of $0.5 \ \text{K}<T<600 \ \text{K}$ and the pressure range of $800-1200 \ \text{GPa}$ with the $k$-mesh of $(6\times6\times4)$. By randomly choosing $\sim 50k$ configurations from the DFT MD trajectories, we construct a first ML potential. We used this potential to run preliminary PIMD-B simulations. We selected from these simulations $\sim 50k$ configurations (including long exchange ones) for which we recalculated energy, forces and stress virials using a denser $k$-mesh of $(10\times10\times8)$. We trained the final ML potential using these data.
We trained ML neural network potential for each pressure (800, 1000 and 1200 GPa) at the whole temperature range using the \texttt{DeepMD-kit} package\cite{Wang2018} with smooth edition (SE) descriptor\cite{Zhang2018}. The SE descriptor is constructed to represent the atomic environment by $(32\times64\times128)$ neural network with 16 axis neurons\cite{Zhang2018}. The energy, forces and tensor virials are predicted by the fitting neural network of 4 hidden layers ($512\times256\times128\times64\times32$) over $1-10$ million iterations. The test errors at $p=800, 1000, 1200$ GPa are found in Table 1. For instance, the test errors of 800 GPa case for the total energy (Supplemental Fig. \ref{fig:errore}), force (Supplemental Fig. \ref{fig:errorf}) and stress tensor virial (Supplemental Fig. \ref{fig:errorv}) are 0.7 meV/atom 60.0 meV/$\AA$ and 2.48 eV, respectively.
\subsection{Defect}
From the early stage, defect has been considered as the most plausible pathway in forming a supersolid phase\cite{THOULESS96, Lifshitz69, Chester1970,Pederiva97,Ceperley04,day07}. In $^4$He solid, however, PIMC simulations showed that the formations of vacancy and interstitial are thermodynamically unfavourable\cite{Pederiva97,Ceperley04}. Since a defect-free supersolid phase was not observed in a PIMC simulation, the origin of a $^4$He supersolid phase is still under debate\cite{day07}. In light of previous studies, the role of defect as a pathway to supersolid should not be overlooked in high-pressure deuterium. Therefore, we investigate the thermodynamic stability of various defect types in high-pressure deuterium solid.
Because the atomic positions of $I4_1/amd$ phase are all symmetrically equivalent, only a single type of mono vacancy exists. On the other hands, there exist the two types of interstitials, $D_{3h}$ and $D_{4h}$.
The single point DFT calculation shows that the formation energies of the above defects are too high to be formed compared to the melting temperature of high-pressure deuterium $\sim 120$ K. The mono deuterium vacancy defect has the formation energy of $E[\text{V}^0_{\text{D}}] = 647 \ \text{meV}$. The $D_{4h}$ interstitial formation energy is $E[\text{D}_i] = 280 \ \text{meV}$. And the $D_{3h}$ interstitial is highly unstable that any local minimum configuration is not found. Finally, the formation energy of vacancy-interstitial pair is $E[\text{V}^0_{\text{D}}+\text{D}_i] = 286 \ \text{meV}$. Therefore, we conclude that the mono and pair defects of the $I4_1/amd$ phase does not exist at $T\sim 1$ K.
\section{\label{sec:level1} NPT implementation of bosonic path integral molecular dynamics}
Our implementation of NPT PIMD-B simulation follows the NPT PIMD algorithm of Martyna et al. where we adopt the Nose-Hoover chain thermostat/barostat\cite{Martyna99,martyna96}. The major revision of PIMD-B simulation on the algorithm is the inclusion of bosonic exchange to the pressure estimator.
\subsection{\label{sec:level3} Primitive pressure estimator}
The primitive pressure estimator $p^{(prim)}_{\alpha\beta}$ of PIMD is given by
\begin{eqnarray}
\nonumber
p^{(prim)}_{\alpha\beta} = &\frac{NPk_BT}{det(\overrightarrow{h})} \delta_{\alpha\beta} + \frac{1}{det(\mathbf{h})} \sum_i^N \Big(\Phi_{spring}(\textbf{r}^{(\tau)}_i) \\
&+\frac{1}{P}\sum^P_\tau (\textbf{f}^{(\tau)}_i)_\alpha (\textbf{r}^{(\tau)}_i)_\beta \Big)- \sum_\mu^d \sum^P_\tau h_{\beta\mu} \frac{\phi(\mathbf{h}, \mathbf{r}^{(\tau)})}{\partial h_{\alpha\mu}},
\end{eqnarray}
where $\alpha(\beta), N,P,T,\mathbf{r}^{(\tau)}_i,\mathbf{f}^{(\tau)}_i$ and $\mathbf{h}$ are Cartesian axis, the number of particles, the number of beads, temperature, the position of the particle $i$ at imaginary time $\tau$, the force on the particle $i$ at imaginary time $\tau$ and the cell matrix. The interactions between beads at different imaginary times $\tau$ are given as the spring term $\Phi_{spring}(\mathbf{r}^{(\tau)}_i)$. In PIMD NPT simulation without bosonic exchange, $\Phi_{spring}(\mathbf{r}^{(\tau)}_i)= -m_i\omega^2_P\sum^P_\tau (r^{(\tau)}_i-r^{(\tau)}_{i+1})_\alpha (r^{(\tau)}_i-r^{(\tau)}_{i+1})_\beta$. If the nuclei follow Bose statistics, the pressure virial should be modified since the forces on atoms need to include exchange effects, which we discuss in the following subsection \ref{sec:level3BosonP}.
\subsection{\label{sec:level3BosonP} Derivation of pressure estimator of indistinguishable Bosonic NPT simulation}
The quantum partition function $Q_P$ of the 3D system of $N$ particles with $P$ beads,
reciprocal temperature $1/k_BT$ and cell shape $\textbf{h}$ is given by
\begin{eqnarray}
\nonumber
Q_P(N,\beta,h) &= \Big[ \prod^N_{i} \Big(\frac{m_i P k_B T}{2\pi\hbar^2}\Big)^{3P/2} \Big] \int dr^{(1)}_i ... dr^{(P)}_i \\
& e^{-\beta[\sum^{N,P}_{i,\tau}\Phi_{spring}(r^{(\tau)}_i) + \frac{1}{P}\sum^P_\tau \phi(r^{(\tau)}_i,V)]}
\end{eqnarray}
The spring term under bosonic exchange is\cite{Hirshberg2019}
\begin{eqnarray}
\Phi_{spring} = V^{(N)}_B = -\frac{1}{\beta} ln \Big[ \frac{1}{N} \sum^N_{k=1} e^{-\beta(E^{(k)}_N+V^{(N-k)}_B)} \Big]
\end{eqnarray}
As $Q_P$ depends the cell matrix $\mathbf{h}$, the pressure tensor \(p_{\alpha\beta}\) under full-fledged cell fluctuations is
\begin{eqnarray}
p_{\alpha\beta} = \frac{1}{\beta det[h]} \sum^{x,y,z}_{\gamma} h_{\beta\gamma} \Big( \frac{\partial ln Q_M}{\partial h_{\alpha\gamma}} \Big)_{N,T}
\end{eqnarray}
We adapt the scaled variable $\mathbf{s}_i$ of atom $i$ in the cell $\mathbf{h}$ as $\mathbf{r}_i = \mathbf{h} \cdot \mathbf{s}_i$. In the summation form, the $\alpha$ component position of atom $i$ is $r_{i,\alpha} = \sum_\beta h_{\alpha\beta}s_{i,\beta}$. Since we are interested in the spring term, which now considers the
bosonic symmetry, we focus on the spring contribution to the pressure
tensor.
The spring term contribution to the pressure tensor becomes (in scaled coordinates),
\begin{eqnarray}
p_{\alpha\beta, spring} = \sum^{x,y,z}_{\gamma} \sum^{N,P}_{i,\tau} \frac{\partial \Phi_{spring}}{\partial(h \cdot s^{(\tau)}_i)}_{\alpha} h_{\beta\gamma} s^{(\tau)}_{i,\gamma}.
\end{eqnarray}
where $\mathbf{h \cdot s^{(\tau)}_i} =
r^{(\tau)}_i$. If we consider distinguishable particles, converting it into the Cartesian coordinates results in
\begin{eqnarray}
p_{\alpha\beta, spring} = \sum^{N,P}_{i,\tau} - m_i\omega_P (r^{(\tau+1)}_i-r^{(\tau)}_i)_\alpha (r^{(\tau+1)}_i-r^{(\tau)}_i)_\beta.
\end{eqnarray}
With the bosonic symmetry, the spring contribution becomes
\begin{eqnarray}
\label{eq:p_force}
p_{\alpha\beta, spring}^{(N)} &= \sum^{N,P}_{i,\tau} \frac{\partial V^{(N)}_B}{\partial r^{(\tau)}_{i,\alpha}} r^{(\tau)}_{i,\beta}\\
\label{eq:p_iter}
&= \frac{\sum^N_{k=1} \Big[ \sum^{N,P}_{i,\tau} \Big( \frac{\partial E^{(k)}_N}{\partial r^{(\tau)}_{i,\alpha}} r^{(\tau)}_{i,\beta} + \frac{\partial V^{(N-k)}_B}{\partial r^{(\tau)}_{i,\alpha}} r^{(\tau)}_{i,\beta} \Big) \Big] e^{-\beta(E^{(k)}_N+V^{(N-k)}_B)}}{\sum^N_{k=1} e^{-\beta(E^{(k)}_N+V^{(N-k)}_B)}}
\end{eqnarray}
And we note that
\begin{eqnarray}
\sum^{N,P}_{i,\tau} \frac{\partial E^{(k)}_N}{\partial r^{(\tau)}_{i,\alpha}}r^{(\tau)}_{i,\beta} = -m_i\omega_P \sum^N_{i=N-k+1}\sum^P_{\tau} (r^{(\tau+1)}_i-r^{(\tau)}_i)_\alpha (r^{(\tau+1)}_i-r^{(\tau)}_i)_\beta
\end{eqnarray}
It is implied that $r^{(M+1)}_N = r^1_{N-k+1}$ or otherwise $r^{M+1}_i = r^{(1)}_{i+1}$. The spring contribution to the pressure tensor is calculated through the iterative equation (\ref{eq:p_iter}).
However, since we calculate the forces of spring term $\frac{\partial V^{(N)}_B}{\partial r^{(\tau)}_{i,\alpha}}$ already, the spring contribution to the pressure virial is obtained by equation (\ref{eq:p_force}). Although we have implemented both isotropic and full-cell Parrinello-Rahman NPT PIMD-B, we only present the isotropic NPT. All the features are implemented in a development version of \texttt{LAMMPS} and be found in the author's \href{https://github.com/changwmyung/H_PIMD}{\texttt{Github}} repository.
\subsection{\label{sec:level3}Equations of motion for NPT PIMD-B simulation}
We follow the NPT equations of motion of Martyna et al. where each Cartesian degree of freedom of the system, $dN$, couples to the Nose-Hoover chain (NHC)\cite{Martyna99,martyna96}, where $d$ is the dimension of system. Each Nose-Hoover chain couples to the each degree of freedom $dNPN_{nhc}$, where $N_{nhc}$ is the number of NHC. The default number of NHC for thermostat/barostat are $N_{nhc}=3$ for the whole PIMD-B calculations in this work. The only difference of PIMD-B NPT simulation compared to PIMD is that now the bosonic pressure estimator is used to measure the pressure of system. And although NPT PIMD simulation usually uses center of mass (centroid) pressure estimator and the corresponding equations of motion, the definition of centriod of ring polymers becomes elusive and ill-defined with the bosonic symmetry. Therefore, NPT PIMD-B simulation calculates primitive pressure estimator of equation (\ref{eq:p_force}) and the corresponding equations of motion without introducing the centroid.
\begin{eqnarray}
\dot{u}_i^{(\tau)} = &\frac{\mathbf{p}_i^{(\tau)}}{m_i^{(\tau)}} + \frac{\mathbf{p}_\epsilon}{W} u_i^{(\tau)},\\
\dot{\mathbf{p}}_i^{(\tau)} = &\mathbf{g}_i^{(\tau)} + \frac{1}{P} \mathbf{f}_i^{(\tau)} - (1+\frac{1}{NP})\frac{p_\epsilon}{W} \mathbf{p}_i^{(\tau)} - \frac{\mathbf{b}_{\xi_{i1, i}}^{(\tau)}}{Q_{i,1}^{(\tau)}},\\
\dot{V} = &\frac{dVp_\epsilon}{W},\\
\dot{p}_\epsilon = & dV(p_{int}-p_{ext})+\frac{1}{NP}\sum_I^N\sum_\tau^P\frac{(\mathbf{p}_i^{(\tau)})^2}{m_i^{(\tau)}}-\frac{p_{\eta_1}}{Q_1^{\epsilon}}p_\epsilon,\\
\dot{\xi}_{i,k}^{(\tau)}= &\frac{\mathbf{p}_{\xi_{i,k}}^{(\tau)}}{Q_{i,k}^{(\tau)}},\\
\dot{\eta}_k^{(\tau)}=&\frac{p_{\eta_k}}{Q_k^{(\epsilon)}},\\
\dot{\mathbf{p}}_{\xi_{i,1}}^{(\tau)}=&\big[ \frac{\mathbf{b}_{i,i}^{(\tau)}}{m_i^{(\tau)}} - k_B T \hat{n} \big] - \frac{\mathbf{b}_{\xi_{i,1},\xi_{i,2}}^{(\tau)}}{Q_{i,2}^{(\tau)}},\\
\dot{\mathbf{p}}_{\xi_{i,2}}^{(\tau)}=& \big[ \frac{\mathbf{b}_{\xi_{i,1},\xi_{i,1}}^{(\tau)}}{m_{i,1}^{(\tau)}} - k_B T \hat{n} \big],\\
\dot{p}_{\eta_1}=& \big[ \frac{p^2_\epsilon}{W} -k_B T] - \frac{p_{\eta_2}}{Q_2^{(\epsilon)}} p_{\eta_1},\\
\dot{p}_{\eta_2}=&\big[ \frac{(p_{\eta_{1}})^2}{Q^{(\epsilon)}_{1}} -k_BT].
\end{eqnarray}
where $\mathbf{u}^{(\tau)}_i$ is the normal mode transformation of the position of particle $i$ in the imaginary time $\tau$, $p_\epsilon$ is the momentum conjugate to $\epsilon=lnV$, $\eta_k$ is the $k$th element of the volume thermostat whose momentum conjugate is $p_{\eta k}$. $\xi^{(\tau)}_{i,k}$ is the $k$th element of the thermostat chain of the particle $i$ in the imaginary time $\tau$. And $\mathbf{p}^{(\tau)}_{\xi_{i,k}}$ is its conjugate momentum. The barostat mass parameters are $W=d(NM+1)k_BT/\omega^2_b$ and $Q_k^{(\epsilon)}=\frac{k_BT}{\omega^2_b}$, where $\omega_b$ is the damping frequency. The vectors $\mathbf{b}^{(\tau)}_{\xi,\xi'}$ and $\mathbf{n}$ are [($p^{(\tau)}_{\xi,x}p^{(\tau)}_{\xi',x}$), ($p^{(\tau)}_{\xi,y}p^{(\tau)}_{\xi',y}$), ($p^{(\tau)}_{\xi,z}p^{(\tau)}_{\xi',z}$)] and [1,1,1], respectively.
In PIMD-B NPT equations of motion, we account for the bosonic symmetry by calculating the spring forces of beads $\mathbf{g}^{(\tau)}_i=-\nabla_{u_i^{(\tau)}} \Phi_{spring}(\mathbf{u}_i^{(\tau)})$ and bosonic pressure estimator $p_{int}$ that includes bosonic exchange effects.
\subsection{Convergence of Bosonic NPT path integral molecular dynamics}
The Bosonic PIMD NPT simulations at various thermodynamics conditions ($0.6 \ \text{TPa} < \text{P} < 1.2 \ \text{TPa}$ and $0.1 \ \text{K} < \text{T} < 500 \ \text{TPa}$) are performed with the time step of $\Delta t = 0.5 \ fs$. Throughout the whole simulations, we used ensemble sampling frequency (damping parameter) for thermostat $\omega = 100\times \Delta t = 50 \ fs^{-1}$ and for barostat $\omega_b = 2000 \times \Delta t = 1.0 \ ps^{-1}$.
\floatsetup[figure]{style=plain,subcapbesideposition=top}
\begin{figure}[h]
\includegraphics[width=0.45\linewidth, height=5cm]{figs-si/conv-ekin.png}
\label{fig:ekin}
\includegraphics[width=0.45\linewidth, height=5cm]{figs-si/conv-etot.png}
\label{fig:etot}
\caption{\textbf{The convergence of NPT PIMD-B simulation with respect to the inverse of the number of beads $1/P$.} The convergence of (a) the kinetic energy and (b) the total energy of NPT PIMD-B simulation as the function of $1/P$ at $T=5$ K and $P=800$ GPa. The fitting line (red dashed line) indicates the extrapolated kinetic and total energy at the limit of $P \to \infty$.}
\label{fig:conv-pimd}
\end{figure}
The potential energy, kinetic energy and total energy of Bosonic PIMD NPT at $T = 5$ K and $p=800$ GPa are measured with respect to the number of beads $P =$ 4, 8, 16, 64, 80, 100, 130, 160, 180, 200, 240, 280 (Supplemental Fig. \ref{fig:conv-pimd}). At $P \sim 250$, 80 \% of kinetic energy (Supplemental Fig. \ref{fig:conv-pimd}a) and 0.1 \% of the total energy (Supplemental Fig. \ref{fig:conv-pimd}b) are converged.
\floatsetup[figure]{style=plain,subcapbesideposition=top}
\begin{figure}[h]
\includegraphics[height=6cm]{figs-si/rho_s_conv.png}
\caption{\textbf{The convergence of superfluid fraction with respect to the inverse of the number of beads $\mathbf{1/P}$ at $\mathbf{p=800}$ GPa and $\mathbf{T=0.4}$ K.} The convergence of superfluid fraction with respect to the inverse of number of beads (1/P) with error bars (cyan). The red dashed line indicates the average of superfluid fraction density at $P=256$.}
\label{fig:rho_s_conv}
\end{figure}
In addition, we test the convergence of superfluid fraction (Supplemental Fig. \ref{fig:rho_s_conv}) which will be discussed in detail in Section \ref{winding}. We note that the superfluid density measured at $p=800$ GPa and $T=0.4$ K converges well against $P=256$. Therefore, we used $P=256$ beads for PIMD-B simulation in this work.
\section{Bosonic path integral molecular dynamics}
In principle, one should account for the permutation of spin coordinates in addition to the spatial coordinates for deuterium atoms which are spin 1 Bosons. Including the spin variables is in principle possible but extremely difficult\cite{Lyubartsev93}, and often ignored such that the system is considered as a spin-polarized system. Ceperley pointed out that factoring the wavefunction to a spin-polarized one would complicate the analysis of rotational symmetry but is not known to cause problems in extended many-body systems\cite{Ceperley91}. \ctext[RGB]{255,251,204}{Following this argument, we neglect the permutation of spin coordinates and only permutes the positions. However, we note that this assumption might lead to a slight overestimation of the superfluid transition temperature}.
\subsection{Calculation of structural properties}
The Debye structure factors $S_D(q)$ of MD, PIMD and PIMD-B simulations (Supplemental Fig. \ref{fig:Sqdebye}) reveal that the peaks of $S_D(q)$ match for all three levels of theories except for the short range ($q > 22 \AA^{-1}$). This result provides that the crystalline long-range order of solid is maintained with dominant exchange of nuclei in PIMD-B simulation. The structure factor was calculated based on the analytic atomic scattering factor\cite{Colliex2004} with parameterized coefficients\cite{Peng96}.
\floatsetup[figure]{style=plain,subcapbesideposition=top}
\begin{figure}[h]
\includegraphics[height=6cm]{figs-si/Sq_debye.png}
\caption{\textbf{Debye structure factors at various of levels of theories.} The Debye structure factor $S_D(q)$ of the system in MD (yellow line), PIMD (blue line) and PIMD-B (red dashed line) simulations.}
\label{fig:Sqdebye}
\end{figure}
\begin{figure*}[]
\centering
\includegraphics[width=16cm]{./figs-si/dw-factor-trj.png}
\caption{\textbf{An ensemble of $\mathbf{P\times N}$ configurations in PIMD-B simulation.} (a-d) Sampled $(P\times N)$ configurations at different time steps in which dynamic exchange occurs. A blue sphere in the frame represents each $(P\times N)$ bead. (e-h) The relative structure factors $S_{rel}(q)$ measured at each time step (red line) referenced to the MD simulation (green dashed line). The amplitude of omitted $S_{rel}(q)$ peak (*) in the MD simulation is 14.9. (i-l) Instantaneous permutation probability of the system at a given time step.}
\label{fig:relSq}
\end{figure*}
In PIMD-B, at each time step, all permutations contribute to the force on each atom\cite{Hirshberg2019}. Therefore, the method can provide insight on how the sampled configurations of $P$ beads of $N$ deuterium atoms evolve in time ($P=256$ and $N=128$) (Supplemental Fig. \ref{fig:relSq}a-d). Along with the sampled configuration, we show a relative structure factor, $S_{rel}(q) =\frac{1}{PN}\sum^{P}_{\tau,\tau^{\prime}}\sum^{N}_{j,k}e^{-iq(R^{(\tau)}_j-R^{(\tau^{\prime})}_k)}$ (Supplemental Fig. \ref{fig:relSq}e-h). Although this is not the physical quantum structure factor, this rather describes the properties of the ensemble of $(P \times N)$ configurations sampled at any given time step. In spite of dynamic change in $(P\times N)$ distribution, the overall long-range order of $(P\times N)$ configuration is maintained (Supplemental Fig. \ref{fig:relSq}e-h). The corresponding instantaneous permutation probability also fluctuate in time while maintaining its overall shape of the average permutation probability (Supplemental Fig. \ref{fig:relSq}i-l).
\subsection{Particle indices shuffle in PIMD-B simulation}
\begin{figure}[]
\centering
\includegraphics[width=16cm]{./figs-si/shuffle_effect_Etot_T_P.png}
\caption{\textbf{Indices shuffling effect on thermodynamic quantities} (a) The total energy (eV/particle), (b) temperature and (c) pressure of the system at T = 0.4 K and $p=800$ GPa (blue line). Particle indices shuffling occurs every $N_s=100$ steps (50 $fs$), which is indicated by red vertical dashed lines.}
\label{fig:shuffle}
\end{figure}
Although particle permutation in PIMD-B simulation converges for enough simulation time, we develop a particle indices shuffling scheme for better permutation sampling given limited simulation time. Compared to the default recursive summation approach, the order of particle indices in the summation (equation 1 of main text) is randomly shuffled at every $N_s$ step. In our simulation, we used $N_s=100$ in which thermodynamic quantities are sufficiently equilibrized. Upon the indices shuffling, the thermodynamic observables, such as total energy, temperature and pressure, are not affected ensuring that the system is not pushed out of equilibrium (Supplemental Fig. \ref{fig:shuffle}).
\subsection{Winding number analysis in PIMD-B simulation} \label{winding}
\IncMargin{1em}
\begin{algorithm}
\SetKwData{Connectivity}{connectivity}\SetKwData{NNlist}{nnlist}\SetKwData{Winding}{W}\SetKwData{This}{this}\SetKwData{Up}{up}\SetKwData{Left}{left}
\SetKwFunction{Union}{Union}\SetKwFunction{CompareDistance}{CompareDistance}\SetKwFunction{FindCompress}{FindCompress}\SetKwFunction{Update}{Update}\SetKwFunction{Connect}{Connect}\SetKwFunction{CalculateWinding}{CalculateWinding}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{PIMD-B $(P\times N)$ trajectories}
\Output{Average Winding number}
\BlankLine
\emph{Iterate over the sampled time frame}\;
\For{$t\leftarrow 1$ \KwTo $t_{max}$}{
\emph{Iterate over the number of particle $N$}\;
\For{$i\leftarrow 1$ \KwTo $N$}{\label{forins}
\Connectivity$\leftarrow$ \CompareDistance{$(r^{(P-1)}_i-r^{(0)}_i), (r^{(P-1)}_i-r^{(0)}_j)$}\;
\If(\tcp*[h]{connect particle $i$ and $j$}){$i$ is not $j$}{\label{lt}
(\Connectivity,\NNlist)$\leftarrow$ \Update()
}
\Else(\tcp*[h]{particle $i$ is closed}){(\Connectivity,\NNlist)$\leftarrow$ \Update()}
\If(\tcp*[h]{revert the exchange}){$j$ is in \NNlist}{\label{lt}
(\Connectivity,\NNlist)$\leftarrow$ \Update()}
}
\Connectivity$\leftarrow$\Connect() \tcp*[h]{close any open rings}\;
\Winding$\leftarrow$\CalculateWinding() \tcp*[h]{calculate Winding number}\;
}
\caption{Winding number calculation in PIMD-B simulation}
\label{algo_winding}
\end{algorithm}\DecMargin{1em}
In a periodic system, the winding number \(\mathbf{W}\) analysis provides facile calculation of superfluid fraction $\rho_s/\rho$ in the path integral method\cite{Pollock1987}.
\begin{eqnarray}
\mathbf{W}\mathbf{L}= \sum^N_i (\mathbf{r}^{(P-1)}_i-\mathbf{r}^{(0)}_i)
\end{eqnarray} where N is the number of particles, P is the number of beads, $\mathbf{W}$ is the winding number and $\mathbf{L}$ is the unit-cell. However, the PIMD-B simulation calculates the force only, and it is not possible to obtain the exact permutation configuration directly from the trajectories, unlike the PIMC simulation.
To circumvent this problem, we attempt to identify the permutation configuration by comparing the distances between $(\mathbf{r}^{(P-1)}_i-\mathbf{r}^{(0)}_i)$ and $(\mathbf{r}^{(P-1)}_i-\mathbf{r}^{(0)}_j)$ (equation (\ref{eq:min_fn})),
\begin{eqnarray}
min\Big[|\mathbf{r}^{(P-1)}_i-\mathbf{r}^{(0)}_i|, |\mathbf{r}^{(P-1)}_i-\mathbf{r}^{(0)}_j|\Big].
\label{eq:min_fn}
\end{eqnarray}
If the nearest neighbour of the last bead of a particle $i$, is the first bead of the same particle $i$, it is reasonable to assume that no exchange occurs. On the other hands, if the nearest neighbour of the last bead of a particle $i$ is the first bead of the other particles $j$, $\mathbf{r}^{(0)}_j$, then we assume that the particle exchange occurs between the particle $i$ and $j$. The estimation is reasonable given that the ring-polymer potential is a harmonic function $ \propto (r^{(\tau+1)}_i-r^{(\tau)}_i)^2$. Algorithm \ref{algo_winding} ensure that the permutation configurations form closed loops. The superfluid fraction $\rho_s/\rho$ is given as
\begin{eqnarray}
\frac{\rho_s}{\rho} = 2\pi \sum_{\alpha=x,y,z} \Big[ \Big( \frac{L_\alpha}{\lambda_D} \Big)^2 \frac{<W_\alpha^2>}{3N} \Big].
\end{eqnarray}
\floatsetup[figure]{style=plain,subcapbesideposition=top}
\begin{figure}[h]
\includegraphics[height=6cm]{figs-si/4HeEx-rhos.png}
\caption{\textbf{A comparison of the superfluid fractions of $^4\text{He}$ liquid superfluid between the PIMC and PIMD-B simulations.} The superfluid fractions $\rho_s/\rho$ as the function of temperatures ranging from 1.0 to 3.0 K calculated by the PIMC (cyan)\cite{Pollock1987} and PIMD-B (red) simulations.}
\label{fig:rhos_pimc_pimd}
\end{figure}
\begin{figure}[h]
\includegraphics[width=15.5cm]{figs-si/he_solid.png}
\caption{\textbf{Benchmark PIMD-B simulation result of hcp He solid.} (a) The superfluid fractions $\rho_s/\rho$ as the function of temperatures ranging from 0.1 to 0.3 K calculated by the PIMD-B (red) simulations. (b) The permutation probabilities of length $l$ of $hcp$ He solid at $p = 5.5$ MPa and T = 0.1 K.}
\label{fig:rhos_he_solid}
\end{figure}
To validate our approach, we performed a benchmark on the superfluid liquid $^4\text{He}$ system using the \texttt{HFDHE2} potential\cite{Aziz1977} (Supplemental Fig. \ref{fig:rhos_pimc_pimd}). \ctext[RGB]{255,251,204}{We also did a benchmark on the $hcp$ $^4\text{He}$ solid at $p$ = 5.5 MPa between T = 0.1 K and T = 0.3 K using $P = 64$ beads. To eliminate any finite size effects in PIMD-B simulation, we set a sufficiently large unit cell with 216 atoms}\cite{Ceperley04}. \ctext[RGB]{255,251,204}{Although the experiment reported a superfluid transition around T $\sim$ 0.2 K}\cite{Kim2004}, \ctext[RGB]{255,251,204}{the PIMC simulation observed no sign of superfluid transition}\cite{Ceperley04}. \ctext[RGB]{255,251,204}{As expected, our PIMD-B simulation also do not observe the superfluid transition below T $\sim$ 0.2 K (Supplemental Fig.} \ref{fig:rhos_he_solid} \ctext[RGB]{255,251,204}{a). The permutation probability decays exponentially even below the transition temperature (T = 0.1 K), allowing only local permutations (Supplemental Fig.} \ref{fig:rhos_he_solid}\ctext[RGB]{255,251,204}{b)}\cite{Ceperley04}.
\subsection{Permutation probability in PIMD-B simulation}
\begin{figure*}[]
\centering
\includegraphics[width=16.5cm]{./figs-si/permutation_prob.png}
\caption{\textbf{Normalised permutation probability of high-pressure deuterium.} Normalised permutation probabilities of length $l$ of high-pressure deuterium at $p=800$ GPa are plotted in cyan bar as a function of temperature (a) T = 1.0 K, (b) T = 0.8 K, (c) T = 0.6 K, (d) T = 0.4 K, (e) T = 0.2 K and (f) T = 0.1 K. The permutation probability is normalised by $1/N$. The red dashed lines indicate the equal permutation probability of any $l$ permutation, which is 1.0 in the normalised probability.}
\label{fig:permutation_prob}
\end{figure*}
\floatsetup[figure]{style=plain,subcapbesideposition=top}
\begin{figure}[h]
\includegraphics[height=9cm]{figs-si/pnas-benchmark.png}
\caption{\textbf{Permutation probability of trapped Bosons in a 2D potential.} Permutation probability distributions of $l$ Bosons $p_l$ are plotted as cyan bars with respect to the temperatures (a) $\beta \hbar \omega_0=0.18$, (b) $\beta \hbar \omega_0=0.375$, (c) $\beta \hbar \omega_0=0.75$ and (d) $\beta \hbar \omega_0=6.0$. The red dashed line at each panel is $1/N$ where $N=32$.}
\label{fig:2dboson}
\end{figure}
Following a procedure already established in ref. \cite{Hirshberg2019}, we measure the probability of observing a permutation involving $l$ particle that can be directly extracted from equation (\ref{eq:pl}) (Supplemental Fig. \ref{fig:permutation_prob} and \ref{fig:2dboson}),
\begin{eqnarray}
\label{eq:pl}
p_l = \frac{e^{-\beta (E^{(l)}_N+V_N^{(N-l)})}}{\sum^N_{k=1} e^{-\beta (E^{(k)}_N+V_N^{(N-k)})}},
\end{eqnarray}
where $V^{(N-l)}_N$ is the bosonic potential of $(N-l)$ particles and $E^{(l)}_N$ is the spring energy of all the beads of $l$ particles. It has been shown that in the limit of a perfect superfluid, this probability is constant as a function of $l$ and equal to $1/N$. Thus a uniform probability can be understood as a sign of superfluidity\cite{Ceperley1995,Krauth96}.
The equation (\ref{eq:pl}) measures the probability of $l$ permutation occurrence, $p_l$. The most dominant term in the numerator $e^{-\beta(E^{(l)}_N+V_N^{(N-l)})}$ is the potential of the configuration composed of a ring of exchanged $l$ particles and $(N-l)$ independent rings. When normalised by $\sum^N_{k=1} e^{-\beta ( E^{(k)}_N+V_N^{(N-k)})}$, the probability of the remaining configurations are vanishingly small. To this effect, we are able to measure the permutation probability of $l$ bosonic particles.
As shown in Supplemental Fig. \ref{fig:permutation_prob}, as the temperature of solid deuterium is lowered from 1.0 K to 0.1 K at $p=800$ GPa, the probability of observing long paths increases significantly. At the low temperature limit, the permutation probability approaches to the Bose-Einstein condensation (Supplemental Fig. \ref{fig:permutation_prob}f).
To validate our approach, we perform the benchmark calculations of trapped bosonic particles in a 2D harmonic potential\cite{Hirshberg2019}. We observe the permutation probability of 32 non-interacting bosonic particles at various temperatures $\beta \hbar \omega_0=0.18, \ 0.375, \ 0.75, \ 6.0$, where the 2D trap frequency is $\hbar \omega_0=3$ meV and $\beta = 1/k_BT$ (Supplemental Fig. \ref{fig:2dboson}). It is well-known that the density matrix elements of any permutations $l$ are only dependent on the ground state in Bose-Einstein condensation\cite{Ceperley1995,Krauth96}. At the zero temperature limit $\beta \hbar \omega_0 \to \infty$ (Supplemental Fig. \ref{fig:2dboson}d), the permutation probability $p_l$ becomes equally probable $1/N$ where $N=32$. Also, the permutation probability recovers the distinguishable particle behaviour at the high temperature limit $\beta \hbar \omega_0 \to 0$ (Supplemental Fig. \ref{fig:2dboson}a).
\subsection{Density of states and inverse participation ratio of a supersolid phase}
We calculate the density of states of a supersolid phase by averaging over $P=256$ imaginary time slices. Since the density of states at given real time steps are similar, we sampled $\sim 10$ configurations at every $1 ps$ and averaged them all. The result (Supplemental Fig. 2d of main text) indicates that the deuterium supersolid is metallic under significant exchange effects.
To quantify the localisation properties of electronic states of supersolid phase, we calculate the inverse participation ratio (IPR) $p^{-1}_n$ of a given electronic eigenstate $u_n$ defined as
\begin{eqnarray}
\nonumber
p^{-1}_n = & \frac{\sum^N_i |\phi_{n,i}|^4}{\{ \sum^N_i |\phi_{n,i}|^2 \}^2}\\
= & \sum^N_i |q_{n,i}|^2
\end{eqnarray}
where $\phi_{n,i}$ is a projected atomic wavefunction of eigenstate $u_n$ to the atomic site of atom $i$, $q_{n,i}$ is a projected Löwdin charge population and $N$ is the number of atoms. IPR is a useful measure of localisation of any quantum states. If any states are localised at a particular atomic site $i$, $u_n \sim \delta(\mathbf{r}-\mathbf{r}_i)$, $p^{-1}_n$ becomes unity. On the other hands, $p^{-1}_n$ is $1/N$ for a perfectly delocalised quantum state.
\subsection{PIMD-B two-phase simulation}
\floatsetup[figure]{style=plain,subcapbesideposition=top}
\begin{figure}[h]
\includegraphics[width=16cm]{figs-si/2ph.png}
\caption{\textbf{Two-phase simulation and melting-line of high-pressure deuterium.} (a) A snapshot taken from two-phase PIMD-B simulation of solid-liquid deuterium interface. (b) P-T phase diagram of high-pressure deuterium at the ranges of $800$ GPa $<p<1200$ GPa and at $50$ K $<T<150$ K.}
\label{fig:2ph}
\end{figure}
The melting points of high-pressure deuterium are estimated at the pressure range of $800-1200$ GPa using ML potential and PIMD-B simulation. We prepared an initial configuration of the interface between the $I4_1/amd$ solid (128 atoms) and liquid (128 atoms) phases, following the previous DFT PIMD simulations\cite{chen13} (Supplemental Fig. \ref{fig:2ph}a). The initial configuration becomes either solid or liquid depending on the target temperature of PIMD-B simulation. We used $P=64$ beads at which previous DFT PIMD simulation converged\cite{chen13}. Although the DFT PIMD two-phase simulation of hydrogen suggested the existence of liquid metallic ground state from the negative slope of melting curve ($dP/dT<0$), we observe only a slight decrease of melting curve for deuterium.
\clearpage
|
1,314,259,996,487 | arxiv | \section{Introduction}
Barycenters are principled summaries (averages) of probability measures \citep{journals/siamma/AguehC11}, defined with respect to a similarity metric on the space of measures. They have been used in computer vision \citep{gramfort:hal-01135198}, economics \citep{carlier:hal-00987292}, Bayesian inference \citep{Srivastava2015}, physics \citep{pmlr-v48-peyre16}, and machine learning \citep{dognin2018wasserstein}.
Computing barycenters has been extensively studied by \citet{journals/siamma/AguehC11,pmlr-v32-cuturi14,benamou:hal-01096124,NIPS2019_9130}. It is extremely challenging, due to the need to optimize over spaces of measures. Current approaches typically use compactly-supported basis functions, in particular Diracs, to parametrize barycenters and optimize their weights and\slash or locations \citep{pmlr-v32-cuturi14,NIPS2019_9130}. The strictly local property of these functions requires an exponentially increasing number of basis functions as the dimensionality of their domain increases. As a result of this `curse of dimensionality', these methods are typically restricted to low-dimensional problems ($\mathbb{R}^{\leq 3}$). From a theoretical standpoint, \citet{altschuler} indeed highlights the NP-hardness of computing Wasserstein barycenters of measures, and hence the dimensionality curse. As a result, algorithms that do not incorporate structure (and leverage the low-dimensional structure in high dimensions) are doomed in high dimensions. Concurrent work by \citet{Shen2020SinkhornBV} takes a global approach to computing Sinkhorn barycenters and exploits a form of functional gradient descent to scale better with respect to dimensions than local methods. This approach, however, is limited to averaging under the Sinkhorn geometry, and was only used in synthetic settings.
In this paper, we introduce a practical algorithm for estimating barycenters that can be applied to high-dimensional settings. The key idea is to use a different parametrization of the barycenter by means of a generative model, turning the optimization over measures into a more tractable optimization over parameters of the generative model.
For instance, when learning a barycenter of measures on image space, we parametrize a CNN generating images, instead of parametrizing individual images constituting the barycenter.
Importantly, our approach allows to enforce a \emph{global structure} by treating the barycenter as a parametric model instead of a collection of point masses. It also introduces inductive biases to the model that can reach accurate solutions faster. The combination of global structure and inductive biases in the generator allows us to apply our algorithm to barycentric problems at unprecedented scales in terms of dimensions and support (e.g., in image space $\mathbb{R}^{\text{width}\times \text{height}\times \text{channels}}$). We also demonstrate that our approach leverages the problem structure to obtain additional speedups by incorporating inductive biases.
We also study convergence properties of our proposed algorithm to stationary points for general choices of discrepancies. In particular, we show that local convergence holds for all discrepancies that are either Lipschitz smooth or weakly-convex and Lipschitz continuous, which includes Sinkhorn as proved in \citet{10.5555/3327757.3327812} and MMD (with deep kernel) as proved in this paper.
We apply our algorithm to both traditional low-dimensional experiments (e.g., nested ellipses in $\mathbb{R}^2$ \citep{pmlr-v32-cuturi14}), and previously untackled high-dimensional experiments (e.g., on image datasets in $\mathbb{R}^{>10,000}$) for different choices of discrepancies, namely MMD, optimized MMD, and Sinkhorn. To the best of our knowledge, this is the first approach for estimating barycenters that is applied to non-toy, non-synthetic data in high dimensions.
\section{Barycenters of Measures}
\label{sec:background}
We consider the problem of computing barycenters of probability measures defined on a subset $\mathcal{X}$ of $\mathbb{R}^d$. We denote by $\mathcal{M}_{1}^{+}(\mathcal{X})$ the set of such measures on $\mathcal{X}$ and define the probability simplex $\Delta_P := \{\v{\beta} \in \mathbb{R}^{P} : \sum_{p=1}^{P}\beta_p=1, \beta_p\geq 0\}$.
Following \cite{journals/siamma/AguehC11}, the barycenter of $P$ probability measures $\mu_{1},\ldots,\mu_{P}\in \mathcal{M}_{1}^{+}(\mathcal{X})$ weighted by a vector $\v{\beta} \in \Delta_P$ can be expressed as the measure $\mu^{\star}$ solving
\begin{align}
\mu^{\star}=\argmin_{\mu \in \mathcal{M}_{1}^{+}(\mathcal{X})} \sum_{p=1}^{P}\beta_{p}D(\mu, \mu_{p}),
\label{eq:bary}
\end{align}
where $D: \mathcal{M}_{1}^{+}(\mathcal{X}) \times \mathcal{M}_{1}^{+}(\mathcal{X})\to \mathbb{R}^+$ is a discrepancy between measures . Depending on the choice of $D$, barycenters have significantly different properties. We discuss two families of barycenters obtained when using the Wasserstein distance and the maximum mean discrepancy (MMD) as discrepancy $D$ mainly based on the works of \citet{DBLP:conf/birthday/BottouALO17, journals/siamma/AguehC11, article_anderes}.
The characterization of barycentric properties will be useful to interpret results in the experiments section.
\subsection{Wasserstein Barycenters}
The \emph{$k$-Wasserstein distance} between two measures $\mu_{x}, \mu_{y}\in\mathcal{M}_{1}^{+}(\mathcal{X})$ is defined as \citep{villani}
\begin{align}
&\hspace{-2mm}\mathcal{W}_{k}(\mu_{x},\mu_{y})= \min_{\pi \in U(\mu_{x},\mu_{y})} \Big(\int_{\c{X}\times \c{X}}d^{k}(\v{x},\v{y})d\pi(\v{x},\v{y})\Big)^{\frac{1}{k}},\label{eq:wass}
\end{align}
where $d:\mathcal{X}\times \mathcal{X}\to \mathbb{R}$ is a distance representing the cost of transporting a unit of mass from $\v{x} \in \mathcal{X}$ to $\v{y} \in \mathcal{X}$, and $U(\mu_{x},\mu_{y})$ is the set of joint distributions with marginals $\mu_x,\mu_y$. Intuitively, $\mathcal{W}_k$ in~\eqref{eq:wass} corresponds to the minimal expected cost of transporting mass from $\mu_{x}$ to $\mu_{y}$ according to an optimal \textit{plan} $\pi\in U(\mu_{x},\mu_{y})$.
In general, computing the Wasserstein barycenter requires evaluating ~\eqref{eq:wass} several times, which is computationally challenging. Recent advances provide algorithms to solve~\eqref{eq:wass} approximately with a lower computational cost. \citet{Cuturi:2013:SDL:2999792.2999868} proposed to solve a regularized version of~\eqref{eq:wass} by adding a small (relative) \emph{entropic} term for regularization purposes, leading to a smooth convex objective
\begin{align}
\mathcal{W}^{k}_{k,\epsilon}(\mu_{x},\mu_{y})\!=\!\min_{\substack{\pi\in U}} \int\! d^{k}(\v{x},\v{y})d\pi(\v{x},\v{y})\!+\!\epsilon \f{KL}(\pi||\mu_x, \mu_y)
\label{eq:entwass}
\end{align}
for which optimization scales considerably better. Here, $\epsilon\geq 0$ controls the regularization. For simplicity, we refer to $\mathcal{W}_{\epsilon}$ as the entropic-regularized Wasserstein.
The objective in \eqref{eq:entwass} is biased as in general $\c{W}_{\epsilon}(\mu,\mu)\neq 0$
\citep{pmlr-v84-genevay18a}. Thus, \eqref{eq:entwass} does not define a distance. Also, the bias can lead to possibly wrong minima during optimization \citep{g.2018the}. To alleviate this issue, \citet{pmlr-v89-genevay19a} introduced the Sinkhorn divergence
\begin{align}
\c{S}\c{W}_{\epsilon}=2\c{W}_{\epsilon}(\mu_{x},\mu_{y})-\c{W}_{\epsilon}(\mu_{x},\mu_{x})-\c{W}_{\epsilon}(\mu_{y},\mu_{y}),
\label{eq:sink}
\end{align}
which removes that bias. Equation \eqref{eq:sink} is symmetric, non-negative, and unbiased while still approximating the Wasserstein distance for $\epsilon\to 0$. Hence, the Wasserstein barycenter can be in principle estimated using the Sinkhorn divergence instead of the less tractable Wasserstein distance~\citep{NIPS2019_9130}. We follow this approach in the paper.
\paragraph{Characterization of Wasserstein Barycenters}
It is well-known that Wasserstein barycenters of measures have interpolation properties.
We state formally the (known) result \citep{journals/siamma/AguehC11}, which will be useful in understanding the behavior of Wasserstein barycenters in high dimensions later. For completeness, we also provide a (new) proof in the Appendix of this known result.
\begin{proposition} \label{the:wass} (2-Wasserstein Barycenter): When the discrepancy between measures is $D=\c{W}_{2}^{2}$, and $d$ is the Euclidean $L^2$ norm, the barycenter $\mu^{\star}$ of measures $\mu_{1}, ...,\mu_{P}\in \mathcal{M}_{1}^{+}(\mathcal{X})$ with weights $\v{\beta} \in \Delta_P$ is
\begin{align}
\m{Y}\sim \mu^{\star} \iff \m{Y} = T(\m{X}), \quad \m{X}\sim \pi^\star,
\end{align}
where $\pi^\star$ is a multi-marginal transport plan (see Appendix for more details), and $T(\m{X}) = \frac{1}{P}\sum_{p=1}^P\v{x}_p$.
\end{proposition}
This means that a sample $\m{Y}$ from the barycenter distribution can be obtained by computing the Euclidean barycenter of samples $\m{X} = (\v{x}_1, ..., \v{x}_p)$ from a joint optimal coupling $\pi^\star$ of $\mu_1,\ldots,\mu_P$, i.e., $\m{Y} = T(\m{X})$.
As an illustration of Proposition \ref{the:wass}, Figure \ref{fig:wasse} shows that the 2-Wasserstein barycenter of four isotropic Gaussians located on the corners of a square indeed displaces the mass proportionally to the weights toward the mode with the highest weight (top left).
\begin{figure}
\centering
\subfigure[Wasserstein]{
\includegraphics[width=0.43\hsize]{images/sink_toy4gauss.pdf}
\label{fig:wasse}
}
\subfigure[MMD]{
\includegraphics[width=0.43\hsize]{images/mmd_toy4gauss.pdf}
\label{fig:mmd}
}
\caption{Barycenter (orange) of four Gaussians (black) with respect to \subref{fig:wasse} $\c{W}_\epsilon$; \subref{fig:mmd} MMD. Top-left Gaussian has three times the weight of the others: $\beta = [3/6,1/6,1/6,1/6]$.}
\label{fig:behaviors}
\end{figure}
\subsection{(Scaled) Maximum Mean Discrepancy Barycenters}
\label{sec:mmd}
The maximum mean discrepancy \citep{Gretton:2005:MSD:2101372.2101382}
\begin{align}
&\mathrm{MMD}(\mu_{x},\mu_{y})^2 :=\mathbb{E}_{\v{x},\v{x}'\sim \mu_x}[k(\v{x},\v{x}')]\nonumber \\&\quad + \mathbb{E}_{\v{y},\v{y}'\sim \mu_y}[k(\v{y},\v{y}')] - 2 \mathbb{E}_{\v{x}\sim \mu_x, \v{y}\sim\mu_y}[k(\v{x},\v{y})] \label{eq:mmd_explicit}
\end{align}
is a discrepancy between probability distributions and relies on a positive definite kernel $k:\mathcal{X}\times \mathcal{X}\rightarrow \mathbb{R}$
as a measure of similarity between pairwise samples.
The first two terms in \eqref{eq:mmd_explicit} compute the average similarity within each of $\mu_x$ and $\mu_y$ while the last term computes the average similarity between samples from $\mu_x$ and $\mu_y$. Unlike the Wasserstein, estimating the MMD using samples from $\mu_x$ and $\mu_y$ is straightforward \citep{Gretton:2005:MSD:2101372.2101382}.
\paragraph{Characterizing the MMD Barycenter}
\begin{proposition} \label{the:mmd} (MMD Barycenter): If $D=\text{MMD}^{2}$, the barycenter of measures $\mu_{1},..., \mu_{P}\in \mathcal{M}_{1}^{+}(\mathcal{X})$ with weights $\v{\beta} \in \Delta_P$ is the mixture of measures
\begin{align}
\mu^{\star}:=\sum_{p=1}^{P}\beta_{p}\mu_{p} \in \mathcal{M}_{1}^{+}(\mathcal{X}).
\end{align}
Proof in Appendix \ref{sec:proofmmd}.
\end{proposition}
Proposition \ref{the:mmd} can be seen as a direct extension of results describing the geodesic structure induced by the MMD (Th. 5.3 in \citet{DBLP:conf/birthday/BottouALO17}). It also suggests a basic generative process for sampling from MMD barycenters: (i) generate a draw $z \sim \text{Categorical}_{P}(\v{\beta})$; (ii) sample from measure $\mu_{z}$. Samples from the MMD barycenter (following this procedure) are shown in Figure \ref{fig:mmd}.
\paragraph{Scaled MMD}
Using MMD with a fixed kernel $k$ is ineffective, e.g., when training generative models on datasets of images, as the training signal may be small \citep{NIPS2017_6815}. To alleviate this, deep kernels $k_{f_\psi}(x,y)=k(f_\psi(x),f_\psi(y))$ can be used \citep{Calandra2016, Wilson2016}. When the feature $f_{\psi}$ is fixed, Proposition \ref{the:mmd} applies and the barycenter is still a mixture of measures. However, learning the feature along with the generator in an adversarial fashion has proven to be more effective \citep{NIPS2018_7904,binkowski2018demystifying,NIPS2017_6815}, allowing the gradient signal to increase at locations where measures differ. In this case, $D$ is of the form
\begin{align}
\text{SMMD}^2(\mathbb{P}_{\theta},\mathbb{P}) := \sup_{f_{\psi}\in \mathcal{E}}\lambda(\psi)\text{MMD}^2_{\psi}(\mathbb{P}_{\theta},\mathbb{P}),
\end{align}
where $\lambda(\psi)$ is a scaling function that acts as a regularizer. We assume that all $f_\psi \in \mathcal{E}$ are continuously parametrized by $\psi \in \Psi$ with $\Psi$ compact. Because the kernel changes during training, Proposition \ref{the:mmd} no longer applies, and as a result the barycenters of SMMD might have a different from. Nevertheless, in Section \ref{sec:experiments}, we provide empirical evidence that it retains properties of the MMD barycenter.
\subsection{Related Work on Barycentric Computations}
Most previous approaches to computing barycenters can be categorized into fixed \citep{pmlr-v32-cuturi14,NIPS2017_6858,NIPS2018_8274} and free \citep{pmlr-v32-cuturi14,pmlr-v80-claici18a,NIPS2019_9130} support. Fixed-support approaches choose a finite set of locations $\v{x}_1,...,\v{x}_N \in \c{X}$, parametrize the barycenter as a weighted sum of Diracs $\mu=\sum_{n=1}^{N}a_{n}\delta_{\v{x}_{n}}$, and optimize \eqref{eq:bary} with respect to weights $a_{n}$. Free-support approaches typically optimize both locations $\v x_{n}$ and weights $a_{n}$ by alternated optimization.
However, these methods hardly scale to high-dimensional problems due to the need to optimize locations $\v{x}_n$. The number of parameters to optimize scales exponentially with the dimensionality of the space, which makes them inapplicable to high-dimensional problems, such as considering datasets of images (where individual $\v{x}_n$ are images). Indeed, estimating barycenters without enforcing structure is doomed in high dimensions as demonstrated in various theoretical works. For instance, \citet{altschuler} show NP-hardness of the Wasserstein barycenter problem, which highlights the curse of dimensionality.
As a result of this computational challenge, previous approaches exclusively tackled problems in $\mathbb{R}^{\leq 3}$ \citep{pmlr-v32-cuturi14,benamou:hal-01096124,NIPS2018_8274,pmlr-v80-claici18a,NIPS2019_9130, bonneel:hal-00881872}.
Concurrent work by \citet{Shen2020SinkhornBV} tackles the problem of estimating Sinkhorn barycenters via functional gradient descent on the push-forward mapping (in a RKHS) of a base measure. The method is therefore tailored to Sinkhorn barycenters, while in this paper we propose a general method that can be used with other choices of $D$. Also, we propose, to the best of our knowledge, the first approach that is demonstrated to work in high-dimensions on non-synthetic data.
\section{Estimating Barycenters Using Generative Models}
In the following, we propose an algorithm for estimating barycenters between $P$ probability measures with discrepancies including MMD, MMD with optimized kernel \citep{NIPS2018_7904}, $\c{W}_\epsilon$ \citep{Cuturi:2013:SDL:2999792.2999868} and $\c{S}\c{W}_\epsilon$ \citep{pmlr-v84-genevay18a}.
The key idea behind our algorithm is to parametrize the barycenter using a generative model and thereby turn the optimization over the intractable space of measures into learning model parameters. With this we leverage the fact that high-dimensional data typically lies on significantly lower-dimensional manifolds. This also allows us to incorporate structural inductive biases (e.g., through a CNN), enabling our algorithm to scale to high dimensions. We also prove local convergence for common discrepancies.
\subsection{Algorithm}
\label{sec:baralg}
A generative model $\mathbb{P}_{\theta}$
is a probability measure in $\mathcal{M}_{1}^{+}(\mathcal{X})$, parametrized by a vector $\theta$. Generative models are typically defined as push-forwards of a latent measure $\rho \in \mathcal{M}_{1}^{+}(\mathcal{Z})$ on a lower-dimensional space through a generator function $G_{\theta}:\mathcal{Z}\rightarrow \mathcal{X}$. This means that a sample $\v{x}$ from $\mathbb{P}_{\theta}$ is obtained by first sampling $\v{z}$ from the latent $\rho$, then mapping it through $G_{\theta}$, i.e. $\v{x} = G_{\theta}(\v{z})$. More concisely, we simply write $\mathbb{P}_{\theta}=G_{\theta\#}\rho$.
In the context of estimating barycenters of measures, we propose to parametrize the barycenter using a generative model $\mathbb{P}_{\theta}$. This turns the problem of estimating the barycenter into finding optimal model parameters
\begin{align}\label{eq:approxbary}\theta^\star &= \argmin_{\theta} L(\theta), \\ L(\theta)& \coloneqq \sum_{p=1}^P\beta_p l_p(\theta),\qquad
l_p(\theta)\coloneqq D(G_{\theta \#}\rho, \mu_{p}),
\end{align}
where $D$ is a discrepancy between measures. Equation \eqref{eq:approxbary} (globally) parametrizes the barycentric problem \eqref{eq:bary} and can be solved by stochastic gradient descent as described in Algorithm \ref{algo:alg}.
In each training iteration, the algorithm receives a batch of data points from the individual measures as well as a batch of samples from the generator. Those are then used to compute stochastic gradients $g_p(\theta)$ of the distances between the generator and each of the measures $\mu_p$. The model parameters $\theta$ are then updated by running gradient descent steps using the stochastic barycentric gradient $\sum_{p=1}^P\beta_p g_p(\theta)$.
We note that the discrepancy $D$ needs to be well-defined for measures with discrete support as the barycenter is only accessible through its samples.
\begin{algorithm}[ht!]
\centering
\begin{algorithmic}
\REQUIRE Network $G_{\theta}$, measures $\{\mu_{p}\}_{p=1}^{P}$, weights $\{\beta_{p}\}_{p=1}^{P}$, base measure $\rho$, distances $\{D_{p}\}_{p=1}^{P}$, learning rate $\gamma$
\FOR{epoch in epochs}
\FOR{$p=1,..., P$}
\STATE Sample minibatches $\{x^{(p)}_{j} \sim \mu_{p} \}_{j=1}^{J}$
\STATE Sample $z_{j}\sim \rho$, $j=1,...,J$
\STATE Compute \begin{align*}g_p(\theta)=\nabla_\theta D_{p}(\sum\nolimits_{j=1}^{J}\delta_{\v x^{(p)}_{j}}, \sum\nolimits_{j=1}^{J}\delta_{G_{\theta}(\v z_{j})} )\end{align*}
\ENDFOR
\STATE Update $\theta = \theta - \gamma \sum_{p=1}^{P}\beta_{p} g_{p}(\theta) $
\ENDFOR
\end{algorithmic}
\caption{Algorithm for computing barycenters of arbitrary measures}
\label{algo:alg}
\end{algorithm}
\paragraph{Inductive Biases} We can incorporate prior knowledge on the form of the barycenter through the generator's structure (e.g., CNNs for barycenters of images) and leverage global basis functions (neural networks in particular). This enables scaling to high-dimensional settings, unlike Dirac-based approaches that suffer from the curse of dimensionality as they optimize locations of particles in a high-dimensional space. Note that the generator $G_\theta$ is not restricted to being a neural network, and domain knowledge can enable more efficient learning. For instance, if we know that the actual barycenter is Gaussian, we can set $\rho=\mathcal{N}(\m{0},\m{I})$, $G_{\theta}(\v{z}_n)=\m{S}^{\frac{1}{2}}\v{z}_n+\v{m}$ and optimize the mean $\v{m}$ and covariance $\m{S}$ using our algorithm as shown empirically in Section \ref{sec:experiments}.
\paragraph{Optimization}
As discussed in Section \ref{sec:mmd}, MMD with fixed kernels is not a sensible metric on high-dimensional spaces. MMD with deep kernels (SMMD) alleviates this issue by defining a metric between measures over learned features. In that case, the kernel and the generator $G_\theta$ are trained adversarially, similar to \cite{NIPS2018_7904}.
Analogous adversarial formulations of these discrepancies were advocated for $\c{W}_\epsilon$ and $\c{S}\c{W}_\epsilon$ \citep{pmlr-v84-genevay18a,bunne2019gwgan}. All these approaches require careful regularization of the critic (e.g., by penalizing its gradient \citep{NIPS2017_7159,binkowski2018demystifying,NIPS2018_7904} or weight clipping \citep{pmlr-v70-arjovsky17a}).
\paragraph{Special Cases}
The special case of computing the barycenter of a single measure ($P=1$) corresponds to the traditional implicit generative modeling objective. In that setting, different kinds of discrepancies $D$ have been considered, including MMD \citep{DBLP:conf/uai/DziugaiteRG15,NIPS2017_6815}, 1-Wasserstein \citep{pmlr-v70-arjovsky17a,NIPS2017_7159}, Sinkhorn divergence \citep{pmlr-v84-genevay18a}, and $\c{G}\c{W}_\epsilon$ \citep{bunne2019gwgan}.
From a purely computational perspective, \citet{ijcai2019-483} train a Wasserstein GAN on a single dataset by randomly splitting that dataset into $P$ subsets and minimizing the average 1-Wasserstein between samples from the GAN and from those subsets. This is a special case in which the individual measures are all equal to the same data distribution. This implies that the barycenter coincides with such data distribution leading to a significantly simpler problem.
In the case where all measures are Gaussians, \citet{rigolletbures} derive the gradients of the Wasserstein barycenter functional with respect to the mean and variance of the barycenter and use SGD to learn it.
\begin{remark}
In the MMD case, the barycenter computed using our algorithm targets the mixture of the datasets. Note that the generative MMD barycenter could thus be estimated by training a normal MMD GAN ($P=1$) on the mixture of the datasets. However, training with the barycentric objective allows for larger batches per mode as training scales as $O(PN^2)$, where $N$ is the number of samples per mode and $P$ is the number of modes, instead of $O(P^2N^2)$ for GANs.
\end{remark}
\subsection{Convergence Analysis}
\label{sec:barcon}
The non-convexity of the loss \eqref{eq:approxbary} with respect to model parameters $\theta$ makes it hard to guarantee global convergence. However, we study local convergence to stationary points, which is challenging on its own since the divergence $D$ often results from an optimization procedure. Recently, \citet{10.5555/3327757.3327812} provided related results for the regularized Wasserstein distance.
However, their approach cannot be applied to MMD and SMMD. We hence leverage different techniques to prove convergence for these discrepancies. More generally, we show that local convergence holds for all discrepancies that are either Lipschitz-smooth or weakly convex and Lipschitz-continuous, of which both entropic-regularized Wasserstein, Sinkhorn divergence MMD, and scaled MMD are special cases.
\subsubsection{Smoothness}
Typical local convergence results rely on notions of smoothness.
\textit{Lipschitz smoothness} is the most commonly-used notion to guarantee local convergence.
\begin{definition}\label{def:l_smooth}
A function $L:\Theta \mapsto \mathbb{R} $ is $M$-Lipschitz smooth if there exists an $M\geq 0$, such that
\begin{align}
\Vert\nabla L(\theta)- \nabla L(\theta')\Vert\leq M\Vert\theta-\theta'\Vert\ \ \forall \theta,\theta'\in \Theta.
\end{align}
\end{definition}
Lipschitz smoothness of a function $L$ requires that the gradient of $L$ exists and is Lipschitz continuous.
\citet{10.5555/3327757.3327812} showed that entropic-regularized Wasserstein GANs (\eqref{eq:approxbary} with $P=1$ and $D=\c{W}_\epsilon$) is $M$-Lipschitz smooth with respect to the generator parameters $\theta$. This is easily extended to $\c{SW}_{\epsilon}$, and to the barycentric case ($P \geq 1$):
\begin{proposition}
\label{prop:l_smoothness}
Let $\mathcal{X}$ and $\mathcal{Z}$ be compact and $G_\theta$ Lipschitz and Lipschitz-smooth. Then, the barycenter objective
\begin{align}
L(\theta) := \sum_{p=1}^P \beta_p (\c{S})\c{W}_{\epsilon}( G_{\theta\#}\rho,\mu_p)
\end{align}
is $M$-Lipschitz smooth for $M \in \mathbb{R}^+$.
Proof in Appendix \ref{sec:proofthm_l_smooth}.
\end{proposition}
In the case of the optimized MMD, the discriminator is also learned leading to a non-concave problem. Therefore, the approach by \citet{10.5555/3327757.3327812} cannot be applied to guarantee $M$-Lipschitz smoothness of the resulting objective with respect to generator parameters $\theta$. We use a different approach
that relies on the weaker notion of \emph{weak convexity}.
\begin{definition}\label{def:weak_convex}
A function $L:\Theta \to \mathbb{R} $ is $C$-weakly convex if there exists a positive constant $C$, such that $ L(\theta) + C\Vert\theta \Vert^2$ is convex.
\end{definition}
The next result shows that (optimized) MMD is Lipschitz continuous and weakly convex, which will turn out to be sufficient to guarantee local convergence:
\begin{proposition}
\label{prop:l_convex_mmd}
Assume the kernel $k$ is Lipschitz and Lipschitz-smooth and functions $f_\psi \in \mathcal{E}$ are Lipschitz, Lipschitz-smooth, and
absolutely continuous with respect to the parameters $\psi$ and inputs $\v{x}$. Further assume $G_\theta$ is Lipschitz and Lipschitz-smooth in $\theta$. Then,
\begin{align}
L(\theta) := \sum_{p=1}^P \beta_p \mathrm{(S)MMD}^2( G_{\theta\#}\rho,\mu_p)
\end{align}
is weakly convex and Lipschitz.
Proof in Appendix \ref{sec:proofthm_l_smooth}.
\end{proposition}
Proposition \ref{prop:l_convex_mmd} states that the optimized MMD (and hence the barycentric objective) is \textit{weakly convex} and Lipschitz provided that the discriminator satisfies additional smoothness constraints. This is also useful in the case $P=1$ as it proves that several instantiations of MMD GANs \citep{binkowski2018demystifying,NIPS2018_7904} are also weakly convex (guaranteeing convergence; see Section \ref{sec:localconv}). Next, we show that local convergence holds in both cases.
\subsubsection{Local Convergence}
\label{sec:localconv}
When Lipschitz smoothness holds (as in Proposition \ref{prop:l_smoothness}), standard arguments guarantee convergence to a local stationary value $\theta^{\star}$ for gradient descent or SGD. When only weak convexity and Lipschitz continuity hold (as in Proposition \ref{prop:l_convex_mmd}), it is still possible to guarantee local convergence as shown in \citet{Davis:2018}.
However, both cases require access to an unbiased estimate of the gradient of $L$.
In practice, this is not possible as $L$ is estimated by approximately solving an optimization problem.
Therefore, we propose to use a similar setting as in \citep{10.5555/3327757.3327812}, where we assume access to an unbiased estimate of a direction $g$ that approximates $ \nabla L(\theta) $ to a precision $\delta$. In other words, $g$ satisfies
$
\Vert\nabla L(\theta) - g\Vert^2\leq \delta^2,
$
and $\Tilde{g}$ is an unbiased stochastic estimator of $g$, i.e. $\mathbb{E}[\Tilde{g}]=g$, which we assume we have access to.
Such an estimate can be obtained by performing a few steps of gradient descent on the discriminator, in the case of the SMMD, and then evaluating the gradient of the resulting loss with respect to $\theta$ on new samples.
We further assume that the noise in $\Tilde{g}$ has a bounded variance, i.e.
$\mathbb{E}[\Vert g - \Tilde{g}\Vert^2]\leq \sigma^2$, and we define $\Delta := L(\theta_0)- \inf_{\theta} L(\theta)$ as the initial regret.
%
\begin{figure*}[ht!]
\centering
\subfigure[Nested ellipse]{
\includegraphics[width=0.23\hsize]{images/ellipses_dataset.pdf}
\label{fig:dataset_ell}
}%
\subfigure[MLP]{
\includegraphics[width=0.23\hsize]{images/nested_ellipses_mlp.pdf}
\label{fig:mlp}
}
\subfigure[Structural model]{
\includegraphics[width=0.23\hsize]{images/nested_ellipses_paramellipses.pdf}
\label{fig:elli}
}%
\subfigure[\citet{NIPS2019_9130}]{
\includegraphics[width=0.23\hsize]{images/giulia_ellips_1000.pdf}
\label{fig:giulia}
}%
\caption{Sinkhorn barycenter of 30 nested ellipses, of which a subset is displayed in \subref{fig:dataset_ell} using \subref{fig:mlp} the MLP parametrization, \subref{fig:elli} the nested ellipses parametrization, \subref{fig:giulia} \citet{NIPS2019_9130}.}
\label{fig:ellipses}
\end{figure*}
\begin{figure*}[ht!]
\centering
\subfigure[Sinkhorn barycenter]{
\includegraphics[width=0.30\hsize]{images/plot_sink_bary_gaussians_all_models_one_data.pdf}
\label{fig:sink_gauss}
}%
\hfill
\subfigure[Sinkhorn barycenter -- High Dim.]{
\includegraphics[width=0.31\hsize]{images/sink_highdim.pdf}
\label{fig:highdim}
}%
\hfill
\subfigure[MMD barycenter -- High Dim.]{
\includegraphics[width=0.30\hsize]{images/MMD_MLP_gaussians_highdim.pdf}
\label{fig:mmd_gauss}
}
\caption{\subref{fig:sink_gauss}:Convergence plot of the computation of the barycenter of 15 Gaussians ($d=2$) w.r.t. $\c{S}\c{W}_\epsilon$ using MLP/Gaussian parametrizations, and \citet{NIPS2019_9130}. We also consider higher dimensions ($d=5,20,50$) in \subref{fig:highdim} w.r.t $\c{SW}_\epsilon$ via MLP and Gaussian parametrizations, and in \subref{fig:mmd_gauss} w.r.t. MMD via a mixture parametrization.}
\label{fig:gaussians}
\end{figure*}
\begin{theorem}[\citep{10.5555/3327757.3327812}]
\label{thm:local_convergence}
Assume $\Vert\nabla L(\theta) - g\Vert^2\leq \delta^2$, $\mathbb{E}[\Vert g - \Tilde{g}\Vert^2]\leq \sigma^2$ and $\mathbb{E}[\Tilde{g}]= g$.
Also, if $L(\theta)$ is $M$-Lipschitz smooth (as in Proposition \ref{prop:l_smoothness}), then setting the learning rate to $\alpha:=\sqrt{\frac{2\Delta}{M\sigma^2}}$ yields \begin{equation}
\min_{0\leq t\leq T-1} \mathbb{E}[||\nabla L(\theta_{t})||^{2}] \leq\sqrt{\frac{8\Delta M\sigma^2}{T}}+\delta^2.
\label{eq:conv}
\end{equation}
\label{the:local}
\end{theorem}
Theorem \ref{thm:local_convergence} shows that stochastic gradient methods converge to a stationary point
when Proposition~\ref{prop:l_smoothness} holds.
If $L$ is only $C$-weakly convex as in Proposition \ref{prop:l_convex_mmd}, local convergence still holds \citep{Davis:2018}.
\section{Experiments}\label{sec:experiments}
We demonstrate that our approach can scale the computation of barycenters to high dimensions, while still recovering accurate barycenters. We provide extensive experimental details in the Appendix.
We emphasize that, while MMD barycenters are known in closed form (mixture of measures), and that potentially simpler optimization schemes targeting it exist (GAN on the mixture of the datasets), studying them empirically allows us to analyze the performance our algorithm. We also study barycenters for which a general closed form is not known, including Sinkhorn and SMMD barycenters. In such cases, a scalable algorithm is required, especially in high dimensions.
\subsection{Traditional Barycentric Problems}
We start with classical barycenter problems to demonstrate our approach yields sensible solutions to the barycentric problem \eqref{eq:bary}, and that leveraging structure can speed up computations.
\paragraph{Nested ellipses} We consider the computation of the $\c{S}\c{W}_\epsilon$ barycenter of $P=30$ nested ellipses, reproducing the example of \citet{pmlr-v32-cuturi14,NIPS2018_7827,NIPS2019_9130}. We compare to the algorithm proposed by \citet{NIPS2019_9130}. We consider two approaches to parametrizing the generator $G_\theta$, $(i)$ using a multi-layer perceptron (MLP) as $G_\theta$ and $(ii)$ exploiting inductive biases by parametrizing two ellipses ($\theta$: axis lengths and centers of both ellipses). Figure \ref{fig:ellipses} shows that both approaches recover the barycenter, and obtain a similar but more accurate solution than the approach proposed in \citep{NIPS2019_9130} (under a time budget). In particular, there is significantly more support on the ground truth barycenter due to the global nature of our algorithm.
\paragraph{Gaussians}
To illustrate the importance of the structural knowledge, we consider two different generative models for the barycenter: A model which contains the ground-truth barycenter (GT model) and a generic MLP network which doesn't explicitly encode structural knowledge about the barycenter. In the case of the MMD, the GT model is simply a mixture of Gaussians parametrized by their means and variances, while for $\c{S}\c{W}_\epsilon$, the GT model is given by a single Gaussian \citep{janati}.
Figure \ref{fig:sink_gauss} shows that (i) our algorithm converges to a stationary point (\cref{sec:localconv}) and the gradient bias is negligible; (ii) structural knowledge can lead to faster and more accurate approximations as the Gaussian parametrization converges to a better solution than the MLP; (iii) our algorithm is significantly faster than \citet{NIPS2019_9130} (runtimes/implementations discussion in Appendix).
\begin{figure}[tb]
\centering
\subfigure[Sinkhorn Barycenter]{\includegraphics[width=0.48\hsize]{response/images_012.png}\label{fig:sink01}}
\subfigure[MMD Barycenter]{\includegraphics[width=0.48\hsize]{images/images_01_mmd.png}\label{fig:mmd01}}
\caption{Samples of digits from barycenters on MNIST datasets of $0$s and $1$s with respect to Sinkhorn \subref{fig:sink01} and MMD \subref{fig:mmd01}. We observe the interpolation and mixture behaviors (See Propositions \ref{the:wass}, \ref{the:mmd}). We include barycenters of $0$s, $1$s, and $2$s in the Appendix.}
\label{fig:sinkmnist}
\end{figure}
Figures \ref{fig:highdim},\ref{fig:mmd_gauss} compare the GT model to the MLP model in higher dimensions for both $\c{SW}_\epsilon$ and MMD. In the case of $\c{SW}_\epsilon$ \subref{fig:highdim}, we observe that both GT model and MLP model recover accurate solutions of the barycentric problem even in higher dimensions where the algorithm from \citet{NIPS2019_9130} does not apply ($d>2$).
In the case of MMD (Figure \ref{fig:mmd_gauss}), the GT model outperforms the MLP model significantly, suggesting that an MLP is not necessarily a good model for mixtures of distributions.
This is consistent with the discussion in \citet[Section 6.2]{DBLP:conf/birthday/BottouALO17} which implies that implicit models families, such as MLPs, are better suited for parametrizing Wasserstein barycenters than MMD barycenters.
We thus conclude that enforcing sensible inductive biases is essential to scaling to high dimensions.
\subsection{Barycenters of Natural Images}
In the following, we demonstrate that the combination of structural knowledge and parametric models can scale barycentric computations to high dimensions. Previous papers considered problems in which measures are supported on low-dimensional spaces. Even in experiments with images, these were considered as densities on a 2D space \citep{pmlr-v32-cuturi14,NIPS2019_9130}. In the following, we consider a more challenging setting in which each measure consists of a dataset of $10^4$--$10^5$ images of dimension $10^3$--$10^5$.
{\bf MNIST} We define $\mu_m$ as the dataset of all $m^{th}$ MNIST digits (e.g. $\mu_0$ corresponds to the dataset of all MNIST $0s$). Each measure consists of approximately $5,000$ samples in a $32\times32$-dimensional space.
We compute the Sinkhorn barycenter of $\mu_0, \mu_1$ in Figure~\ref{fig:sinkmnist} (left) and of $\mu_0, \mu_1, \mu_2$ (Appendix). We use a moderate entropic coefficient; hence, barycentric properties should be close to those of Wasserstein barycenters described in Proposition \ref{the:wass}. Both figures show the expected interpolation behavior, i.e., each sample from the barycenter is the interpolation of a `similar' 0 and 1 (Figure \ref{fig:sinkmnist} (Left)), and of a `similar' 0, 1 and 2 (See in Appendix).
Behaviors for barycenters of measures on Euclidean spaces (Figure \ref{fig:ellipses}) and on image spaces (Figure~\ref{fig:sinkmnist}) may at first seem contradictory. However, this is due to the fact that in the former case, a single atom of a specific measure consists of a point on an ellipse, whilst in the latter case it consists of a single image. Hence, interpolation on these two spaces is different as in the former case the overall barycenter will result in a smoothed out ellipse, whilst in the latter case it will result in a collection of interpolated (similar) images from the different classes.
We also compute the MMD barycenter of $\mu_0$ and $\mu_1$ (without optimized features) using our algorithm, which is expected to be a mixture of the datasets (see Proposition \ref{the:mmd}). Figure~\ref{fig:sinkmnist} illustrates the expected mixture behavior, which is in stark contrast to the interpolation behavior of Wasserstein barycenters.
To continue, we compute the SMMD (MMD with optimized features) barycenter of 10 measures $\mu_0,...,\mu_{9}$ and emphasize that SMMD barycentric properties are not known in closed form. Figure~\ref{fig:mnist} shows that the SMMD barycenter generates meaningful samples from all classes. Its behavior is similar to the mixture behavior of MMD barycenters (see Proposition \ref{the:mmd}). In that case, barycenters average measures over features instead of over images themselves, which is in contrast to the MMD barycenter computed in Figure~\ref{fig:sinkmnist}.
\begin{figure}
\centering
\includegraphics[width=1.0\hsize]{images/mnistallrect.png}
\caption{Samples from our SMMD barycenter on MNIST datasets of different digits. We compute the barycenter of $\mu_0,.., \mu_9$ ($\mu_i$ is the dataset of all $i^{th}$ digit).} \label{fig:mnist}
\end{figure}
\paragraph{CelebA} We finally compute the SMMD barycenter of two measures, CelebA males and females, each having approximately $100,000$ locations (images). Images are re-scaled to $3\times 128\times 128$ pixels, so that each (males/females) lives in an approximately $50,000$-dimensional space. We use deep convolutional generators and critics to leverage the structural knowledge about the input locations (images).
Figure \ref{fig:celeba} illustrates that the SMMD barycenter generates meaningful high-quality samples from both measures. Overall, $(i)$ expected barycentric geometric properties are observed in high-dimensional problems; $(ii)$ using structural knowledge (here a CNN) enables (good) approximate solutions to barycentric problems at unprecedented scale.
\section{Discussion}
Our proposed approach relies on global parametric structured models and thus departs significantly from previous barycentric works with local unstructured models \citep{pmlr-v32-cuturi14,NIPS2017_6858,NIPS2018_8274,pmlr-v32-cuturi14,pmlr-v80-claici18a,NIPS2019_9130}. This allows us to scale to higher dimensions under the assumption that inductive biases on the optimal solution are known. Such biases can be enforced through the structure of the parametric model (e.g., CNNs for SMMD barycenters of measures over images).
Without enforcing structure, barycentric algorithms are doomed in high dimensions as studied by \citet{altschuler}.
However, we note that in low-dimensional problems, where absolutely no structure is known about the barycenter, more brute-force approaches that do not enforce inductive biases (e.g., \citep{NIPS2019_9130}) may be more appropriate.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{images/celeba1_crop.png}
\caption{Samples from our SMMD barycenter of all CelebA females and males (respectively $\mu_0$ and $\mu_1$).} \label{fig:celeba}
\end{figure}
Our approach also departs from classical GAN problems (recovering them in the case $P=1$). Indeed, we aim to find a model that achieves the best trade-off between multiple distribution according to some distance. Hence, the choice of the distance has a significant impact on the nature of the solution. This is unlike GANs where the goal is to approximate the data distribution and where the choice of the distance has little impact on the nature of the optimal solution \cite{lucic2017gans}.
Our work can hence be considered a generalization of their works in two orthogonal directions: i) the averaging direction (we consider $P>1$ measures), and ii) the distance direction as we consider general choices of discrepancies between measures.
Finally, we provided local convergence guarantees instead of global ones
due to the non-convexity of the objective. While \citet{NIPS2019_9130} provided global convergence guarantees, they only hold under the assumption that an inner non-convex problem is solved exactly. In general, this problem remains as challenging as ours.
\section{Conclusion}
We proposed an algorithm for estimating high-dimensional barycenters of probability measures with respect to general choices of discrepancies. The key idea is to leverage a different parametrization of the barycenter. This turns the barycentric problem into a problem of learning model parameters, thereby sidestepping the curse of dimensionality from which other algorithms for estimating barycenters suffer. Our approach also enables incorporating explicit structural inductive biases in the model (e.g., CNNs for measures over images). We proved local convergence of our algorithm to stationary points under mild smoothness assumptions on the discrepancy considered.
We applied our algorithm to problems at an unprecedented scale (for both Sinkhorn and SMMD discrepancies), which includes estimating barycenters of measures with more than $10^5$ locations in over $10^4$ dimensions.
\section*{Acknowledgments}
We are grateful to Giulia Luise for providing us code and data for experiments, and for providing feedback on the draft.
SC was supported by the Engineering and Physical Sciences Research Council (grant number EP/S021566/1).
\bibliographystyle{abbrvnat}
|
1,314,259,996,488 | arxiv | \section{Introduction}
\label{sec1}
Defects such as domain walls, vortices, and monopoles often host gapless boundstates when coupled to fermions \cite{Jackiw:1975fn}, the existence of which can be related to topology and anomalies \cite{atiyah1968index,fujikawa1979path}. However, their properties vary according to the dimension and symmetries of the bulk and defect theories. In some cases, the existence of gapless modes can be deduced from currents flowing onto or off of the defect, as with the Integer Quantum Hall Effect or lattice domain wall fermions \cite{Kaplan:1992bt,jansen1992chiral}, while in other examples there are no such currents. Furthermore, the topology that governs the gapless states is in momentum space, and so the number and nature of such states can be sensitive to how the theory is regulated at short distance \cite{Golterman:1992ub, thouless1982quantized, niu1985quantized}. That too can depend on the dimension of the bulk theory.
In this paper we demonstrate a generic framework which can be used to identify gapless defect modes in topological insulators and superconductors, associating all of them with a generalized Hall current with nonzero divergence flowing onto the defect, irrespective of whether the original theory contains any continuous internal symmetry, conserved currents, or chiral anomalies. This approach involves computing the index \cite{Callias:1977kg} for the Dirac operator in the Euclidean action, where one has added diagnostic background fields with nontrivial topology\footnote{The Callias index theorem has been discussed before in a different context, for determining time-independent solutions to the Dirac equation, see Refs.~ \cite{Weinberg:1981eu,Seiberg:2016rsg}. The Euclidian path integral has been used to investigate the role of global anomalies in topological materials \cite{witten2019anomaly}.}. The index is determined by computing a one-loop Feynman diagram, where the integration over loop momentum can be directly related to topological properties of the fermion dispersion relation. We describe the method here and briefly give three explicit examples involving Dirac and Majorana fermions in two, three, and four spacetime dimensions. A more detailed analysis, which includes consideration of the role of interactions, may be found in Ref.~\cite{longpaper}.
The connection between massless states in Minkowski spacetime and the index of the Euclidean Dirac operator is not direct. Consider the example of a Dirac fermion in two spacetime dimensions with ${\cal D} = \slashed{\partial} + m \epsilon(x)$, where a domain wall confines gapless states to the one-dimensional line $x=0$. In Minkowski spacetime this would correspond to a massless mode confined to the end of wire. In Euclidean spacetime, the only state annihilated by ${\cal D}$ will be one localized in the $x$ direction but constant in Euclidian time $\tau$; as this is not a normalizable state it will not contribute to the index of ${\cal D}$. However, we can now imagine adding a second domain wall defect as a function of $\tau$; if there exists a gapless mode in the first place, it will now be localized in two dimensions, is not a zeromode of ${\cal D}^\dagger$, and can contribute to the index of ${\cal D}$, no matter how weakly the fermion interacts with that second domain wall. If we remove the original domain wall at $x=0$ eliminating the massless edge state of interest, then the index vanishes. In this sense, the index of the modified theory reveals the gapless state in the original one. More generally, one can reveal edge states by considering fermions propagating in arbitrary background fields. We will show that when our heuristic example of crossed domain walls is replaced by an arbitrary, smoothly varying complex scalar field, and one finds that the index is proportional to the field's vorticity. In higher dimensions localizing the zeromodes requires additional fields, such as gauge fields.
Our approach then is to add extra diagnostic fields to the theory of interest and compute the index of Euclidian spacetime operator ${\cal D}$ in a derivative expansion, along the lines of Ref.~\cite{Goldstone:1981kk}.
We find that the index is proportional to a topological invariant of these fields in coordinate space, times a topological invariant constructed from the fermion dispersion relation in momentum space. A nonzero value for the product of these winding numbers is taken to indicate the existence of massless states in the Minkowski version of the original theory.
As discussed in Refs.~\cite{Callias:1977kg,Weinberg:1981eu}, the index of a non-Hermitian elliptic operator ${\cal D}$ can be defined as ${\cal I}(0)\equiv \lim_{M\to 0}\ {\cal I}(M)$, where
\beq
{\cal I}(M) &=&\text{Tr}\, \left(\frac{M^2}{{\cal D}^\dagger {\cal D} + M^2} - \frac{M^2}{{\cal D} {\cal D}^\dagger+M^2}\right)\cr
&=&\text{Tr}\, \Gamma_\chi \frac{M}{K+M}\ ,
\eqn{IM}\eeq
where
\beq
K = \begin{pmatrix} 0 & -{\cal D}^\dagger\\ {\cal D} & 0\end{pmatrix},\
\Gamma_\chi = \begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix},\ \{K,\Gamma_{\chi}\}=0\ .
\eqn{Kdef}
\eeq
Let us now imagine that $S=\int \bar\psi {\cal D} \psi$ is the Euclidean action for a system of interest with massless edge states in $(d+1)$-dimensional Minkowski spacetime, where we use the term ``edge states'' to refer massless fermions bound to a defect of any codimension. Then $1/(K+M)$ looks like the propagator in a new theory with Euclidean action
\beq
S =\int d^{d+1}x\, \mybar \Psi (K +M)\Psi\ ,
\eqn{act}
\eeq
where $\Psi$ is a complex fermion with its own continuous fermion number symmetry and, in the $M\rightarrow 0$ limit, a continuous axial symmetry, both of which are unrelated to any symmetries of the original theory. $\Psi$ also has twice as many components as the physical fermions. We can now express ${\cal I}(M)$ in terms of the new theory as
\beq
{\cal I}(M) =-M\int d^{d+1}x \,\vev{\mybar\Psi(x)\Gamma_\chi \Psi(x)}\ ,
\eqn{PS}
\eeq
where the quantum average is computed from a path integral over $\Psi$ and $\mybar\Psi$ with weight $e^{-S}$. We are not restricted to an action like $S=\int \bar\psi {\cal D} \psi$ which preserves fermion number; this analysis is relevant also for a Minkowski theory of real fermions with Euclidean action $\int \psi^T C {\cal D} \psi$, where $C$ is the charge conjugation operator.
In the cases we will examine, $K$ will be a linear differential operator of the form $K = \Gamma_\mu \partial_\mu + V$, where the $V$ is some spacetime dependent matrix. Then we can define the Noether current for the axial symmetry of \eq{act}, ${\cal J}_\mu = \mybar \Psi \Gamma_\mu \Gamma_\chi \Psi$ which obeys an anomalous Ward-Takahashi identity
\beq
\partial_\mu {\cal J}_\mu = 2 M \mybar\Psi \Gamma_\chi \Psi - {\cal A}\ ,
\eqn{divJ}
\eeq
where the first term on the right is due to the explicit chiral symmetry breaking by $M$, and ${\cal A}$ is the potential anomalous contribution due to the variance of the path integral measure \cite{fujikawa1979path}, with
\beq
\int d^{d+1}x\, {\cal A} =-2 \lim_{\Lambda\to \infty}\text{Tr}\,\Gamma_\chi e^{K^2/\Lambda^2} =-2\, {\cal I}(\infty)\ .
\eqn{fuji}
\eeq
It follows then from \Eq{PS} that
\beq
{\cal I}(M)= {\cal I}(\infty) -\tfrac{1}{2} \int d^{d+1}x\, \partial_\mu \vev{\mybar \Psi \Gamma_\mu \Gamma_\chi \Psi} \ .
\eqn{ind}
\eeq
Therefore, to compute the index ${\cal I}(0)$ one need only compute the two terms on the right in the massless limit. In all the cases we will consider, the anomaly ${\cal A}$ and hence ${\cal I}(\infty)$ trivially vanish, and one need only compute the axial current flowing in from infinity, which requires computing the one loop diagram for the chiral current $ \vev{\mybar \Psi \Gamma_\mu \Gamma_\chi \Psi}$ from the action in \Eq{act}. Note that in every case, a nontrivial index is associated with current inflow, independently from whether ${\cal D}$ has a continuous symmetry or anomalies. This chiral current exists for every Minkowski theory and is unrelated to any chiral symmetry the original Minkowski theory might have possessed; it behaves like a generalization of the familiar quantum Hall current and we will refer to it as such.
\section{Majorana fermion in $1+1$ dimensions}
Our first example is a massive Majorana fermion in $1+1$ dimensions. Our starting point is the Lagrangian in Minkowski spacetime,
\beq
{\cal L}_M = \tfrac{1}{2}\psi^T C \left(i\slashed{\partial}- m \right)\psi
\eeq
where $\psi$ is a real, two-component Grassmann spinor and we can take $\gamma^0= C = \sigma_2$, $\gamma^1 = -i\sigma_1$, $\gamma_\chi = \sigma_3$, where $\sigma_i$ are the Pauli matrices.
This is the model considered in Ref.~\cite{Fidkowski:2009dba}, although here we do not include interactions, and we consider the case of infinite dimensions with a domain wall mass $m=m_0\epsilon(x^1)$ rather than a finite wire with two ends. This theory has no continuous symmetries except Lorentz symmetry; it possesses the discrete spacetime symmetries $C$, $P$ and $T$ which play the roles as particle-hole, sublattice, and time reversal symmetries respectively in condensed matter systems. As is easy to see, a gapless fermion exists at $x^1=0$.
%
\begin{figure*}[t]
\centerline{\includegraphics[width=14 cm]{PRL_fig_loops.pdf}}
\caption{\it Loop diagrams for computing the generalized Hall current for the $1+1$ (left), $2+1$(center) and $3+1$ (right two) dimension examples. The black dot is an insertion of the chiral current $\Gamma_\mu\Gamma_\chi$, and the propagators are $K^{-1}$.}
\label{fig:1}
\end{figure*}
The Euclidean Lagrangian is obtained from the Minkowski Lagrangian in $d+1$ dimensions as ${\cal L}_E = -{\cal L}_M$ along with the replacement $\partial_0\to i\partial_{0}$ and a redefinition of the $\gamma$ matrices so that they obey the $SO(d+1)$ Clifford algebra (we use a mostly minus metric). In the present example with $d=1$ we have ${\cal L}_E =\tfrac{1}{2} \psi^T C {\cal D}\psi$, where ${\cal D} = \slashed{\partial} + m$ with $C = \gamma_0 = \sigma_2$, $\gamma_1 = -\sigma_1$, and $\gamma_\chi=\sigma_3$.
Following the discussion in the introduction we generalize the model by replacing the mass by a scalar field $\phi_1$ and add a pseudoscalar field $\phi_2$ so that our Euclidean Dirac operator becomes
\beq
{\cal D}= \left(\slashed{\partial}+\phi_1+i \phi_2 {\gamma_\chi}\right)\ .
\eqn{ddag}
\eeq
In order to compute the index of ${\cal D}$ we next construct the Euclidean theory ${\cal L} = \mybar\Psi (K+M)\Psi$ where $K$, is specified by \Eq{Kdef}. The operator $K$ can be written as
\beq
K &=& \sum_{\mu=1}^2 \Gamma_\mu \partial_\mu + i\phi_2\Gamma_2 + i \phi_1\Gamma_3
\eeq
where
\begin{equation}
\begin{aligned}
\Gamma_i &= \sigma_1\otimes \gamma_i\ ,\
&\Gamma_2 &= \sigma_1\otimes \gamma_\chi\ , \cr
\Gamma_3 &= -\sigma_2\otimes 1\ , \
&\Gamma_\chi &= \sigma_3\otimes 1\ ,
\end{aligned}
\end{equation}
with $i=0,1$.
The leading contribution to the current ${\cal J}_\mu$ in a derivative expansion of the scalar fields is shown in Fig.~\ref{fig:1} and is proportional to $\partial\theta$, where we write $\phi = \phi_1 + i\phi_2 = v e^{i\theta}$ and expand about $\theta=0$. To linear order the $\theta$ vertex is given by $-iv \Gamma_2$. We define $K_0=K(\theta=0)$. Taking $M\to 0$, the Feynman diagram yields
\beq
{\cal J}_\mu&=&v\, \frac{\partial\theta}{\partial x_\nu}\int \frac{d^2q}{(2\pi)^2}\,\text{Tr}\,\Biggl[
\Gamma_\mu\Gamma_\chi
\left(\frac{\partial \tilde K_0^{-1}}{\partial q_\nu} \right)\Gamma_2\tilde K_0^{-1} \Biggl]\cr
&=&
\epsilon_{\mu\nu} \partial_\nu \theta \int \frac{d^2q}{(2\pi)^2}\frac{4v^2}{(q^2+v^2)^2} \cr
&=&
\frac{1}{\pi} \epsilon_{\mu\nu} \partial_\nu \theta \ .
\eqn{index2}
\eeq
The index is then computed to be
\beq
\text{ind}({\cal D})={\cal I}(0) = -\tfrac{1}{2}\int d^2x \,\partial_\mu{\cal J}_\mu = \oint \frac{d\theta}{2\pi} = -\nu_\phi\ ,
\eqn{res2}\eeq
where $\nu_\phi$ is the winding number of the scalar field $\phi$ in the $x_0-x_1$ plane. This result is consistent with the heuristic example discussed in the introduction of the crossed domain wall configuration $\phi_1 = m\epsilon(x_1)$, $\phi_2=\mu\epsilon(x_0)$ with $m>0$, $\mu>0$, for which one finds $\nu_\phi=-1$. We see that the index of the Euclidean Dirac operator ${\cal D}$ reveals the existence of massless edge state in the presence of nontrivial spatial topology for the background $\phi$ field.
The above calculation does not fully reveal the topological nature of the edge state, however, and one might still ask as to why the Feynman integral over momentum results in an integer for the index rather than some arbitrary real number.
To address this we borrow the techniques of Ref.~\cite{Golterman:1992ub, ishikawa1987microscopic} to show that the Feynman integral is actually computing a winding number. The result in \eq{index2} is unchanged if we substitute for the skew diagonal blocks in $K_0$
\beq
{\cal D}_0\rightarrow \xi \equiv \frac{{\cal D}_0}{\sqrt{\det{\cal D}_0}} = \frac{v }{\sqrt{q^2+v^2}} +i \hat q_\mu\gamma_\mu\frac{q}{\sqrt{q^2+v^2}} \ ,
\eqn{xi}
\eeq
where ${\cal D}_0 = {\cal D}\vert_{\theta=0}$, while ${\cal D}_0^\dagger$ is replaced by $\xi^{\dagger}$. The matrix $\xi$ is unitary and the generalized Hall current in \eq{index2} may be written in terms of $\xi$ as
\beq
{\cal J}_\mu&=&\frac{i}{2}\epsilon_{\mu\nu}\partial_\nu \theta\,\epsilon_{\sigma\tau}
\cr
&&\times\int \frac{d^2q}{(2\pi)^2}\text{Tr}\,\gamma_\chi\biggl[\left(\xi^\dagger \partial_\sigma \xi\right)\left( \xi^\dagger\partial_\tau \xi \right)
+\left(\xi \partial_\sigma \xi^\dagger\right)\left(\xi \partial_\tau \xi^\dagger\right) \bigr]\ .\cr&&
\eqn{D3new}
\eeq
The momentum integral can be shown to be the winding number associated with the map $U(q) = \xi^2(q)$ from momentum space to $S^2$ (see Ref.~\cite{longpaper} for details), where
\beq
U = \frac{v^2-q^2}{v^2+q^2} + \frac{2 i v \slashed{q}}{v^2+q^2}\equiv a(q) + i \slashed{b}(q)\ .
\eeq
Since $a^2 + b_i b_i = 1$, $U$ describes a unit 3-vector, which lives on $S^2$. We see that all possible points on $S^2$ correspond to a unique value of the 2-momentum $q_i$, except that the limit of infinite momentum in any direction gets mapped to $U=-1$. So momentum space has itself been compactified to $S^2$, and $U$ describes a nontrivial map in the homotopy group $\pi_2(S^2) = {\mathbb Z}$. The winding number $\nu_q=1$ of this map is computed by the above integral, and we have the result for the index
\beq
\text{ind}({\cal D})=-\nu_\phi\nu_q\ .
\eeq
Now the full topological meaning of the index is manifest and it is evident how a Feynman diagram can produce an integer. In general the index can change only at values for the parameters in the theory when our substitution in \eq{xi} fails, namely where $\det{\cal D}_0$ vanishes for some momentum; such singularities are equivalent to the bulk gap vanishing, allowing the zeromode to delocalize, and can only happen in this case when $v=0$.
As a last comment on this model, the anomaly term ${\cal A}$ in \eq{fuji} vanishes because the trace over momenta gives a factor of $\Lambda^2$ in two dimensions, while a nonzero $\Gamma$ matrix trace requires two powers of $K^2/\Lambda^2$, since the $\Gamma$ matrices behave like those for $SO(4)$, and hence ${\cal A}$ vanishes in the limit $\Lambda\to \infty$.
\section{Topological insulator in $3+1$ dimensions}
We next compute the index for a topological insulator in $3+1$ dimensions, consisting of a Dirac fermion with a domain wall mass \cite{Seiberg:2016rsg}. The domain wall in this case is a $2+1$ dimensional defect hosting massless fermions. In order to construct the current-inflow picture we consider the following Euclidean Lagrangian
\beq
{\cal L}_E =\bar\psi {\cal D} \psi\ ,\qquad {\cal D} = \slashed{D}+ \phi_1 + i \phi_2\gamma_5 \ ,
\eeq
where $\phi_1$ takes the place of the Dirac mass, and $\phi_2$ is a diagnostic background scalar field needed to localize the edge mode as in the previous example.
More background fields are required in higher dimensions to localize a zeromode, and to that end we include an Abelian gauge field in addition to $\phi_{1,2}$. The operator $K$ in \eq{Kdef} is given by
\beq
K &=& \begin{pmatrix} 0 & -{\cal D}^\dagger\\ {\cal D} & 0 \end{pmatrix} = \sum_{a=0}^3 \Gamma_a D_a +i \phi_2 \Gamma_4 +i \phi_1 \Gamma_5\ ,
\eeq
our basis for the $8\times 8$ $\Gamma$-matrices being
\begin{equation}
\begin{aligned}
\Gamma_a &= \sigma_1\otimes \gamma_a\ ,\quad &a=1,\ldots,3\ ,\\
\Gamma_4 &= \sigma_1\otimes \gamma_5 & \Gamma_5 = \sigma_2\otimes 1\ ,
\end{aligned}
\end{equation}
with $\Gamma_\chi = \sigma_3\otimes 1$. We must now compute the divergence of the chiral current, $\vev{\mybar\Psi\Gamma_\mu\Gamma_\chi\Psi}$, where again the mismatch between the four spacetime dimensions, and $8\times 8$ $\Gamma$-matrices ensures that the anomaly ${\cal A}$ in \eq{fuji} vanishes. Computing in a derivative expansion requires evaluating the two diagrams to the left in \ref{fig:1}, with the $M=0$ result
\beq
{\cal J}_\mu=\vev{\mybar\Psi\Gamma_\mu\Gamma_\chi\Psi} = \frac{\nu_q}{2\pi^2} \epsilon_{\mu\alpha\beta\gamma} F_{\alpha\beta}\ \partial_\gamma\theta\ ,
\eqn{4dres} \eeq
with $\phi = \phi_1 + i \phi_2=\rho e^{i\theta}$. Similar to the previous example, $\nu_q=1$ is a winding number computed by the Feynman loop integral of the map from momentum space to to the space of spinor orientations defined by the bulk fermion propagator. In this case we find the map to be an element of $\pi_4(S^4)={\mathbb Z}$ \cite{longpaper}.
When we consider the case of the domain wall mass profile in the original theory, we have $\phi_1 = m_0 \epsilon(x_3)$; a suitable background field configuration for $\phi_2$ and ${\bf A}$ to localize a massless edge state is the monopole configuration discussed in Ref.~\cite{cheng2013fermion} with couplings set to $e=1$ and $g=2\pi$:
\beq
{\bf A }= -\frac{(1+\cos\theta){\bf e}_\varphi}{2 r\sin\theta}\ ,\qquad \phi_2 =\alpha -\frac{1}{2r}\ ,
\eqn{CF}
\eeq
where $\{r,\theta,\varphi\}$ are polar coordinates for the Euclidean space spanned by $\{x_0,x_1,x_2\}$. In this background we can compute the index $-\tfrac{1}{2}\int d^4x\,\partial_\mu {\cal J}_\mu $, finding
\beq
\int d^4x\,\partial_\mu {\cal J}_\mu =
\left(1+\frac{\alpha}{|\alpha|}\right)\ \Longrightarrow\ \text{ind}({\cal D}) =- \frac{\alpha}{|\alpha|}\ .
\eeq
This nontrivial index indicates the existence of gapless edge states: a transition in a topological quantity such as the gap is only possible when fields become delocalized, and so we see that happens at $\alpha=0$, indicating no other scale exists in the low energy spectrum of the theory. Our result \eq{4dres} can equally be applied to identify a massless fermion bound to a magnetic monopole, by adding an external scalar field in the form of a vortex. The index is negative when nonzero because the field configuration in \eq{CF} causes ${\cal D}^\dagger$ to have a zeromode when a massless edge state is present, instead of ${\cal D}$.
\section{Majorana fermions in $2+1$ dimensions}
For our third example we next consider a two component complex fermion $\psi$ in $2+1$ dimensions with both a constant Majorana mass term $\mu$ and a real Dirac mass term $m(x^1)$. This theory is known to describe chiral topological superconductors \cite{PhysRevB.82.184516, Lian10938, PhysRevB.61.10267}. Nonzero $\mu$ breaks the $U(1)$ fermion number symmetry of the Dirac theory to $Z_2$. When $m(x^1)$ has a domain wall profile there appear zero, one, or two massless Majorana-Weyl fermions on the defect, depending on the ratios $m_{\pm}/\mu$, where $\pm m_\pm$ are the asymptotic values of $m(x^1)$ at $x^1=\pm \infty$.
In Minkowski spacetime the Lagrangian for this system may be written as
\beq
{\cal L}_M = \bar\psi \left(i\slashed{\partial} - m\right)\psi + i\frac{\mu}{2}\psi^T C \psi+i\frac{\mu}{2}\bar \psi \,C\bar \psi^T \ ,
\eqn{majmod}
\eeq
where $M$ is real, $\mu$ is real and positive, and we can work in the explicit basis $\gamma^0 = \sigma_2$, $\gamma^1=-i\sigma_1$, $\gamma^2 = i\sigma_3$ and $C=\sigma_2$. After rotating to Euclidian spacetime, decomposing into its real and imaginary parts $\psi = \chi_1 + i \chi_2 $ with real 2-component spinors $\chi_i$, and then defining $\zeta_\pm = (\chi_1\pm\chi_2)/\sqrt{2}$, one can write the Euclidian Lagrangian as
\beq
{\cal L}_E &=& \tfrac{1}{2}\left[\zeta_+^T C {\cal D}_+ \zeta_+ +\zeta_-^T C {\cal D}_- \zeta_- \right]\ , \ {\cal D}_\pm = \slashed{\partial} + (m\pm \mu)\ ,\cr &&
\eqn{D2}
\eeq
where the gamma matrices are given by $C=\gamma_0=\sigma_2$, $\gamma_1 = -\sigma_1$, $\gamma_2 = \sigma_3$. The index will then be the sum of the indices of ${\cal D}_+$ and ${\cal D}_-$. To compute them we add a gauge field as the diagnostic field, construct $K_\pm$ from ${\cal D}_\pm$, and compute the generalized Hall current from the second Feynman diagram in Fig.~\ref{fig:1} to leading order in a derivative expansion. Consider the case $\mu=0$; then the current for either of the $\zeta_\pm$ may be written as \cite{longpaper}
\beq
{\cal J}_\alpha
&=&
-\epsilon_{\alpha \beta\gamma}\partial_\gamma A_\beta \left(\frac{1}{3}\epsilon_{ijk}\right)\cr &&\times
\int \frac{d^3q}{(2\pi)^3} \text{Tr}\, \left[ \left(\tilde {\cal D}_0^{-1} \partial_i\tilde{\cal D}_0\right) \left(\tilde {\cal D}_0^{-1} \partial_j \tilde {\cal D}_0\right) \left(\tilde {\cal D}_0^{-1} \partial_k \tilde {\cal D}_0\right)\right] \cr
&&\cr &&
\eqn{Dloop}\eeq
where for ${\cal D}_0$ we take ${\cal D}_\pm\vert_{{A_\mu}=0}$. To understand the underlying topology we can define
\beq
U({\mathbf q}) = \frac{\tilde {\cal D}_0({\mathbf q})}{\sqrt{\det \tilde{\cal D}_0({\mathbf q})}} \equiv \cos\frac{\theta}{2}+ i \hat{ \slashed{\theta}}\cdot{\boldsymbol\gamma}\sin\frac{\theta}{2}
\eqn{udef1}
\eeq
where ${\boldsymbol \theta}$ is a real 3-vector with $\theta= |{\boldsymbol \theta}| $ and
\beq
\cos\frac{\theta}{2}= \frac{m}{\sqrt{m^2+q^2}} \ ,\quad
\sin\frac{\theta}{2}=\frac{ q}{\sqrt{m^2+q^2}}\ ,\quad \hat {\theta}=\hat{\bf q}\ . \cr&&
\eqn{udef2}\eeq
which allows us to rewrite the expression for the current as
\beq
{\cal J}_\alpha &=&
-\frac{1 }{\pi} \epsilon_{\alpha \beta\gamma}\partial_\gamma A_\beta \left(\frac{1}{24\pi^2}\epsilon_{ijk}\right)
\cr &&
\times \int_{\theta\le \pi } d^3\theta\, \text{Tr}\, \left[ \left( U^\dagger \partial_i U \right)
\left( U^\dagger \partial_j U\right)
\left( U^\dagger \partial_k U\right)\right] \ , \cr&&
\eqn{U3}
\eeq
with $\partial_iU\equiv \partial U/\partial \theta_i$. The integral looks like the winding number of a map from momentum space compactified to $S^3$, to $SU(2)\cong S^3$, except that the integral is only over half of $S^3$. The problem can be seen in \eq{udef1}: $\cos\theta/2 \ge 0$ for all momenta, so $U$ cannot take on all values in $SU(2)$. The problem is solved when the theory is regulated. With a Pauli-Villars regulator one substitutes ${\cal D}(m) \to {\cal D}(m)/{\cal D}(\Lambda)$ and takes the $\Lambda\to\infty$ limit. The result is that $U\to U_\text{reg}$ in \eq{U3} where
\beq
U_\text{reg}({\mathbf q}) &=& \frac{ \tilde {\cal D}_\text{reg}({\mathbf q})}{\sqrt{\det \tilde {\cal D}_\text{reg}({\mathbf q}) }}
\equiv \cos\frac{\theta_\text{reg}}{2}+ i \hat {\boldsymbol \theta}_\text{reg}\cdot{\boldsymbol\gamma}\sin\frac{\theta_\text{reg}}{2} \cr&&
\eeq
where $ \hat {\boldsymbol \theta}_\text{reg}=\hat{\bf q}$ as before, and
\beq
\cos\frac{\theta_\text{reg}}{2}= \frac{\Lambda m+q^2}{\sqrt{\left(m^2+q^2\right)
\left(\Lambda ^2+q^2\right)}}\ .
\eeq
Now we have $U_\text{reg}({\mathbf q})\xrightarrow{q\to\infty} = 1$, while $U_\text{reg}(0) = \text{sgn}(m\Lambda)$. This regulated theory describes a well defined map $S^3\to S^3$ which is nontrivial if $m$ and $\Lambda$ have the opposite signs, and trivial if they don't. On replacing $m\to m(x) \pm\mu$ and integrating the divergence of the generalized Hall current over Euclidian spacetime, we arrive at the result for the index for the whole system
\beq
\text{ind}({\cal D}) = \nu_A \nu_q\ ,
\eeq
where
\beq
\nu_A &=& \frac{1}{2\pi} \oint {\bf A}\cdot d{\bf \ell}\ ,\cr
\nu_q &=& \theta(m_+ + |\mu|) + \theta(m_+ - |\mu|) \cr &&-\theta(m_- + |\mu|) - \theta(m_- - |\mu|)\ .
\eeq
where we have assumed that $\mu$ is spatially constant while $m(x)\xrightarrow{x\to\pm\infty} m_\pm$. Note that $\nu_q$ can take on the values $0,\pm1,\pm 2$ depending on the relative values of $m_\pm$ and $\mu$. Once can verify that this result agrees with explicit edge state solutions to the equations of motion \cite{longpaper}, and we see that the generalized Hall current is sensitive to topological phase transitions as one varies parameters in the theory.
\section{Discussion}
We have shown how gapless fermions modes bound to defects or solitons in various dimensions may be detected by computing the index of the Euclidean Dirac operator in the presence of additional background fields. The method involves determining the divergence of a generalized Hall current via a 1-loop Feynman integral, which calculates a topological winding number of the fermion propagator, the field theoretic generalization \cite{Jansen:1992tw,Golterman:1992ub} of the TKNN result \cite{thouless1982quantized}. Regularization is required to make topological sense of the result in odd spacetime dimensions. These currents can be computed for systems without chiral symmetries or anomalies, and generalize the concept of the Hall current. The examples considered here are in the $BDI$, $D$ and $DIII$ topological classes in one, two and three spatial dimensions respectively; each is known to have a topological invariant taking values in ${\mathbb Z}$, and so perhaps it is not surprising that in each case we find momentum space topology governed by the homotopy groups $\pi_n(S^n)={\mathbb Z}$. However, in Ref.~\cite{longpaper} we show this method correctly identifies the edge state spectrum for the $D$ class in one spatial dimension, with topological invariant ${\mathbb Z}_2$, yet surprisingly, even in that case the momentum space topology of the generalized Hall current is given by $\pi_n(S^n)$. It remains to be seen how comprehensive our approach is, whether it can be applied to theories with interactions, and whether this generalized Hall current has any experimental implications in Minkowski spacetime.\\
\section{acknowledgements}
We thank Biao Lian for introducing us to the literature on chiral topological superconductors.
DBK is supported in part by DOE Grant No. DE-FG02-00ER41132 and by the DOE QuantISED program through the
theory consortium ``Intersections of QIS and Theoretical Particle
Physics'' at Fermilab. SS acknowledges support from the U.S. Department of Energy
grant No. GR-024204-00001.
|
1,314,259,996,489 | arxiv | \section{Introduction}
The adiabatic algorithm\cite{farhi2001quantum} is a proposed quantum algorithm for optimization. In its simplest form, one considers a quantum Hamiltonian which is a sum of two terms, one term being diagonal and proportional to the objective function of some classical optimization problem and the other so-called ``driving term" being a ``transverse field" (some non-commuting additional term). One then adiabatically evolves the Hamiltonian from a large value of the transverse field (where the ground state is easy to prepare) to a small value of the transverse field, where the ground state encodes the desired solution of the optimization problem.
Unfortunately, there is tremendous theoretical evidence that gaps for random instances typically become super-exponentially small
\cite{altshuler2009adiabatic,altshuler2010anderson} so that the time required for adiabatic evolution to remain in the ground state is longer than the time required for even a classical brute force search\footnote{While some authors have disputed the perturbative calculation\cite{knysh2010relevance}, we believe that the general mechanism of localization on the hypercube will still apply for random instances with local driving terms. The short path algorithm exploiting a difference between $\ell_1$ and $\ell_2$ localization may however be able give a super-Grover speedup\cite{Hastings_2018,Hastings_2019}.}. Other authors have shown exponentially small gaps in simple problems\cite{laumann2012quantum} and some explicit simple examples show a super-exponentially small gap\cite{Wecker_2016}. Further, separate from any question about the scaling of the gap, numerical experiments have shown that classical algorithms which simulate the quantum dynamics can perform comparably to a quantum device\cite{ronnow2014defining}.
Nevertheless, it remains of some interest to ask about the computational power of adiabatic quantum computation. If we consider a Hamiltonian $H=sH_1+(1-s)H_0$ which is a linear combination of two {\it arbitrary} local Hamiltonians $H_0,H_1$, with some parameter $s$ controlling the dynamics, and require that the gap become only polynomially small so that the adiabatic evolution can be performed in polynomial time, then the problem is completely understood: this model is equivalent to standard quantum computation, and can solve any problem in BQP\cite{aharonov}.
However, if we restrict to the case that $H$ has no sign problem in the computational basis then the problem remains open. Here ``no sign problem" means that in the given basis, all off-diagonal terms in $H$ are negative; this is sometimes termed ``stoquastic".
This case of no sign problem includes the adiabatic optimization algorithm discussed at the start if $H_1$ is equal to an objective function and $H_0$ is equal to $-\sum_i X_i$ with $X_i$ being the Pauli $X$ matrix on the $i$-th qubit. It is important to emphasize that the arguments of \cite{altshuler2009adiabatic} apply even if the driving term has some sign problem; they depend rather on $H_1$ being an objective function and $H_0$ being chosen as some sum of local terms.
We remark that there is a problem of ``glued trees"\cite{Childs_2002} for which an exponential speedup over classical is known using a Hamiltonian with no sign problem that changes slowly in time\cite{Somma_2012}, but this problem is very different from the adiabatic annealing considered here. The gap becomes exponentially small and the evolving quantum state has a large overlap with excited states during the evolution.
More generally, if one allows dynamics in excited states, then it is possible to perform universal quantum computation using Hamiltonians with no sign problem\cite{Childs_2009}. Thus, Hamiltonians with no sign problem are universal (using excited states) and adiabatic evolution is universal (using Hamiltonians with a sign problem). The question we consider is what happens if we impose both these restrictions: no sign problem and adiabatic evolution in the ground state with a gap that is only polynomially small.
One piece of evidence that it may be hard to simulate this adiabatic evolution classically in general is the existence of topological obstructions to the equilibration of path integral Monte Carlo methods\cite{obs}. As explained later, these obstructions help motivate the construction here.
Further, these obstructions are perhaps the main reason one should be interested in the question: topological obstructions such as difficulty equilibrating between different winding numbers can have an important effect on practical simulations of quantum systems as is well-studied in the condensed matter community\cite{Henelius_1998}, and so it would be useful if there were a general classical method that could overcome all such obstructions.
\subsection{Problem Statement and Results}
In this paper, we address this question. Our results, in an oracle model defined below, will show a superpolynomial separation between the power of adiabatic computation with no sign problem and the power of classical computation. At the same time, our results give no reason to believe that adiabatic computation with no sign problem is capable of universal quantum computation.
We use a number $N$ to parameterize the problem size (for example, in the example above using qubits, the number of basis states in the computational basis is $2^N$), and all references to polynomial scaling will refer to this parameter $N$.
We will assume more generally that the number of computational basis states is $\leq 2^{p(N)}$ for some function $p$ which is polynomially bounded,
i.e., the computational basis states can be labelled using polynomial space.
We will define a path of Hamiltonians $H_s$, for $s$ in some interval to be {\it admissible} if it satisfies the following properties (when we refer to a parameter $s$ below, it is always assumed to be in the given interval).
First, for all $s$, $H_s$ must have no sign problem in the computational basis.
Second, for all $s$, $H_s$ must be polynomially sparse, meaning that for every computational basis element $|i\rangle$, there are at most
${\rm poly}(N)$ basis elements $|j\rangle$ such that $\langle j | H_s | i \rangle$ is nonzero.
Third, for all $s$, for every $i,j$, $|\langle j | H_s | i \rangle| \leq {\rm poly}(N)$.
Fourth, for all $s$, $\Vert \partial_s H_s \Vert \leq {\rm poly}(N)$.
Fifth, for all $s$, $H_s$ has a unique ground state and the spectral gap to the first excited state is $\Omega(1/{\rm poly}(N))$.
Sixth, the length of the interval is ${\rm poly}(N)$.
Note that the number of basis states $2^{p(N)}$ and the definition of an admissible path both depend upon many polynomials that we have left unspecified. The particular value of these polynomials is not important;
for example, for any such $p(\cdot)$ and $N$, we can define $N'=\lceil p(N) \rceil$, so that the number of basis states is $\leq 2^{N'}$. Then, if the number of queries needed to solve all instances with some given probability is superpolynomial in $N$, it is also superpolynomial in $N'$.
Similarly, if that gap is lower bounded by $1/{\rm poly}(N)$ for some polynomial, it is lower bounded by $1/N''$ for $N''={\rm poly}(N)$
and again the number of queries would be superpolynomial in $N''$.
Still these polynomials should be regarded as fixed in the main result: for some specific choice of polynomials, we show a superpolynomial number of queries.
For example, the construction we use gives
$p(N)=O(N^2 \log(N)^3)$; it can be tightened somewhat.
The reason it is convenient to leave these polynomials unspecified is that it simplifies some
accounting later:
we will often, given some problem, construct a new problem with a larger number of basis states (increasing $p(N)$ so that it is still polynomial) or different gap; any change in the polynomials from this construction is often not stated explicitly.
The interest in the conditions other than the no sign problem condition is that adiabatic evolution on admissible paths can be efficiently simulated on a quantum computer up to polynomially small error, at least so long as the Hamiltonian is ``row computable", meaning that give any row index one can efficiently compute the nonzero entries in that row. For us, we will consider an oracle problem where the oracle will give these nonzero entries. Then, evolution under a time-dependent Hamiltonian for a time that is ${\rm poly}(N)$ will give the desired approximation to adiabatic evolution.
We will say that a path $H_s$ for $s$ in some interval $[a,b]$ satisfies the {\it endpoint condition} if
for both $s=a$ and $s=b$, the ground state of $H_s$ is some computational basis state, i.e., for some $i$ (possibly different for $s=a$ and $s=b$), $|i\rangle$ is the ground state of $H_s$ (in a slight abuse of language, we will say that a vector is ``the" ground state of a Hamiltonian when of course the ground state is only defined up to phase).
We refer to $a$ as the start of the path and $b$ as the end of the path.
So, our interest will be in admissible paths of Hamiltonians which satisfy the endpoint condition because this gives a simple example of Hamiltonians for which it is easy to prepare the ground state of $H_a$ and for which one can measure in the computational basis\footnote{The reader might feel that the endpoint condition is a very strong restriction and wonder whether more general paths should be considered. Of course, such more general paths may be useful in practice but since we are able to prove a superpolynomial separation even with this restriction, it seems worth keeping the restriction. In particular, when the path satisfies the endpoint condition, it is very clear what it means to ``solve" the problem classically: one should be able to determine the computational basis state at the end of the path as in \cref{docl}.}
to determine the ground state of $H_b$.
We say that the path satisfies the condition at one endpoint if for one endpoint (either $s=a$ or $s=b$), the ground state is a computational basis state.
Also, we can easily concatenate admissible paths which satisfy the endpoint condition: given one such path $H_s$ for $s\in [a,b]$ with $|j\rangle$ being the ground state of $H_b$ and another admissible path $H'_s$ for $s\in [b,c]$ with $|j\rangle$ being the ground state of $H'_b$, we can concatenate the two paths to get a new admissible path if $H_b=H'_b$. Even if $H_b$ differs from $H'_b$, it is possible to interpolate between $H_b$ and $H'_b$ by a path which first makes the off-diagonal elements of the Hamiltonian tend to zero, then changes the diagonal entries until they agree with diagonal entries of $H'_b$, then increases the off-diagonal elements until they agree with $H'_b$; doing this in an obvious way (for example, linear interpolation) still gives an admissible path from $H_a$ to $H'_c$.
One might be slightly surprised in one respect at our endpoint condition, though, since in adiabatic evolution with a transverse field, at $s=0$ the ground state of the Hamiltonian is a uniform superposition of all computational basis states, which may be written as $|+\rangle^{\otimes N}$. However, for a system of $N$ qubits there is an obvious admissible path $H_s=-(1-s) \sum_i Z_i - s \sum_i (X_i)$ for $s\in [0,1]$ with the ground state of $H_0$ being a computational basis state and the ground state of $H_1$ being $|+\rangle^{\otimes N}$. So, given some admissible path which satisfies the endpoint condition at the end of the path and with the Hamiltonian at the start being $-\sum_i X_i$ (for example, interpolation between a transverse field Hamiltonian and some objective function for an optimization problem), we can concatenate with the path above to get an admissible path which satisfies the endpoint condition.
We will consider a version of the problem with an oracle in order to
give a superpolynomial lower bound on the ability of classical algorithms to solve this problem.
Our oracle for a Hamiltonian will be similar to those considered previously\cite{Aharonov_2003,childs2004quantum,Berry_2006}.
The Hamiltonians that we consider can be simulated efficiently on a quantum computer with quantum queries of the oracle\cite{Aharonov_2003}.
As a remark for readers not familiar with oracle separations: to prove such a separation in a problem without an oracle (for example, when the Hamiltonian is a sum of two-body terms)
would require proving that P is not equal to BQP, which is far beyond current techniques in computer science, although a separation between classical and quantum computation is known with respect to an oracle\cite{Simon}.
Also, an oracle separation suggests that if one wishes to have an efficient classical algorithm for the problem without an oracle, one should exploit some structure of the problem.
We define the oracle problem ${\rm AdNSP}$ as follows\footnote{The terminology ${\rm AdNSP}$ is an abbreviation of ``adiabatic no-sign problem".}.
\begin{definition}
\label{docl}
An instance of ${\rm AdNSP}$ is defined by some admissible path
$H_s$ which satisfies the endpoint condition.
For definiteness, we assume that $s$ lies in the interval $[0,1]$.
A query of the oracle consists of giving it any $i$ which labels some computational basis state $|i\rangle$ as well as giving it any $s\in [0,1]$, and the oracle will return the set of $j$ such that $\langle j | H_s | i \rangle$ is nonzero as well as returning the matrix elements $\langle j | H_s | i \rangle$ for those $j$ to precision $\exp(-{\rm poly}(N))$, i.e., returning the matrix elements to ${\rm poly}(N)$ bits accuracy for any desired polynomial. We will call those $j$ the {\it neighbors} of $i$, and we will say that we {\it query state} $i$ at the given $s$.
The oracle will also return the diagonal matrix element $\langle i | H_s | i \rangle$.
The problem is: given query access to the oracle, and given the computational basis state $|i\rangle$ which is the ground state of $H_0$, determine the computational basis state $|j\rangle$ which is the ground state of $H_1$.
We say a classical algorithm solves this problem for an instance with some given probability if it returns the correct $j$ with at least that probability.
(As remarked above, the definition of an admissible path implicitly depends on various polynomials; so implicitly the problem ${\rm AdNSP}$ also depends on various polynomials.)
\end{definition}
Note that given an unlimited number of queries to the oracle, it is possible to simulate the quantum evolution on a classical computer since one can determine the Hamiltonian to exponentially small error.
Remark: we have stated above that the oracle returns the matrix elements only to polynomially many bits. This restriction is unnecessary for all the lower bounds on queries later, which would still hold even if the oracle returned the matrix elements to infinite precision.
Our main result is:
\begin{theorem}
\label{mainth}
For some constant $c$, for some specific choice of polynomials $p(\cdot)$ and choice of polynomials defining an admissible path, there is no algorithm that solves every instance of ${\rm AdNSP}$ with probability greater than $\exp(-cN)$ using
fewer than $\exp(\Theta(\log(N)^2))$ classical queries.
\end{theorem}
Throughout, when we refer to an algorithm, the algorithm may be randomized and may take an arbitrary amount of time.
Remark: as is standard terminology, we refer to functions which are $O(\exp(\log(N)^\alpha)))$ for some fixed $\alpha$ as quasi-polynomial functions, and denote an arbitrary such function by ${\rm qpoly}(N)$. A function which is
$\exp(\Theta(\log(N)^2))$ is quasi-polynomial but is superpolynomial.
\subsection{Outline and Motivation for Proof}
The motivation for the proof of \cref{mainth} is, to some extent, an idea from \cite{obs}: path integral Monte Carlo (which is a very natural classical algorithm for simulating quantum systems with no sign problem) in many cases cannot distinguish between the dynamics of a quantum particle on some graph $G$ and the dynamics on its universal cover $\tilde G$.
The universal cover of a connected graph $G$ can be defined by picking an arbitrary vertex $v$ of $G$, and then vertices of the cover $\tilde G$ correspond to nonbacktracking paths starting at $v$, with an edge between vertices of $\tilde G$ if the corresponding paths on $G$ differ by adding exactly one step to the end of one of the paths.
However, at the same time, the largest eigenvalue of the adjacency matrix of $G$ (which will give us, up to a minus sign, the ground state energy of a Hamiltonian we define for that graph) may be much larger than it is on $\tilde G$ (to be precise, for an infinite graph we should not talk about ``the largest eigenvalue", but rather use spectral norm), so that a quantum algorithm can distinguish them. We emphasize that if $\tilde G$ is a {\it finite cover} of $G$, the largest eigenvalue of the adjacency matrix of $\tilde G$ is the same as that of $G$.
This difference on infinite graphs has a finitary analogue: there is a difference between the ground state energy on a complete graph and the ground state energy on a tree graph with the same degree as the complete graph, with the difference in energy persisting no matter how deep the tree is, assuming the degree is $\geq 3$. Here, the ground state energy of a graph is minus the largest eigenvalue of the adjacency matrix of that graph.
Our proof is based on the following main idea: we construct two different graphs which have different ground state energies, but for which we can give a superpolynomial lower bound on the number of classical queries to
distinguish those graphs; we quantify this ability to distinguish the graphs in terms of mutual information
between a random variable which is a random choice of the graphs and another random variable which is the query responses.
We will term these graphs $C$ and $D$; these actually refer to families of graphs depending on some parameters.
The proof has two main parts: first, using these graphs to construct a family of instances of ${\rm AdNSP}$ which cannot efficiently be solved with classical queries, and, second, proving the lower bound on the number of classical queries. Proving spectral properties of the quantum Hamiltonians of these graphs is an additional part of the proof, but is relatively simple.
The first part of the proof is in
\cref{smqm}, where we show that it suffices to prove \cref{mainth} in a different query model.
This part of the proof is perhaps less interesting than later parts of the proof, though it is important to understand the modified query model that we define.
In this query model, the oracle gives less information in response to queries, making it impossible to distinguish between a graph $G$ and some cover $\tilde G$.
Each vertex of the graph corresponds to a computational basis state so that neighbors of a vertex are also neighbors that one might receive in response to a query. On $G$, one might follow some cycle on the graph, but if $\tilde G$ is the universal cover this is not possible: in the modified query model, one will not be able to know that has followed a cycle.
Then, in \cref{distg}, we then
reduce the problem of proving \cref{mainth} to showing
two graphs $C,D$ satisfying certain properties exist. The needed properties of the graphs are summarized briefly in \cref{trq}. The main result is \cref{blemma}.
The rest of the paper is concerned with constructing $C,D$.
In \cref{toomany} we give a first attempt at a construction, taking $C$ to be a complete graph of $O(1)$ vertices and $D$ to be a bounded depth tree graph (an infinite tree graph would give an example of a cover of $C$).
The idea is that if one does not reach a leaf of $D$, then $D$ is indistinguishable from the cover of $C$, and by choosing the
height of the tree superpolynomially large, it may take superpolynomially many queries to reach a leaf.
Unfortunately, the construction of \cref{toomany} suffers from two serious defects. The first is that the it uses ``too many" states, i.e., number of vertices in the graph is not $\exp({\rm poly}(N))$ if we take the height of the tree superpolynomially large.
The second and more serious defect is that the
graphs come with a privileged vertex called the ``start vertex", and in the modified query model we will still be able to determine when we return to the start vertex; a random walk in $C$ will often return to the start vertex but in $D$ one will not so a classical algorithm can efficiently distinguish them.
Nevertheless, this construction is worthwhile as it introduces certain key ideas used later.
The second part of the proof, constructing $C$ and $D$ fulfilling all needed properties (including that they cannot be efficiently distinguished by any classical algorithm and that there are only $\exp({\rm poly}(N))$ vertices in the graph), starts in
\cref{decg}. Here, we introduce the notion of a ``decorated graph". The idea is to define some sequence of tree graphs for which it is ``hard" in some sense to reach certain vertices far from the start vertex because one tends (speaking very heuristically) to get ``lost" in other paths near the root.
This tendency to get lost will make it hard for classical algorithms to detect the difference between $C$ and $D$.
In that section, we also give bounds on the energy of the Hamiltonians corresponding to these graphs and prove some properties of the ground states.
In \cref{ch}, we give lower bounds on the number of classical queries needed to distinguish between $C$ and $D$.
\cref{distlemma} quantifies the difficulty of distinguishing $C$ and $D$ in terms of mutual information; \cref{distlemma} takes as input an assumption about difficulty of ``reaching" a certain set of vertices $\Delta$ in $D$ using queries starting from a given ``start vertex". Difficulty of reaching this set follows from an inductive \cref{inductivelemma}.
The results in \cref{decg} and \cref{ch} are given in terms of a number of parameters. In \cref{chs} we fix values for these parameters and prove \cref{mainth}.
In \cref{linear} we briefly discuss a case of linear interpolation rather than arbitrary paths.
\section{Modifications to Query Model}
\label{smqm}
This section consists of two different subsections, which allow us to consider a more restrictive query model that we call the modified query model.
\subsection{Related States}
We first show that
we may, in everything that follows, assume that every state queried is either
the initial state
$|i\rangle$ which is the ground state of $H_0$ or a
neighbor of some state queried in a previous query.
For example, it may query $i$ for some value of $s$, receiving neighbors $j_1,j_2,\ldots,$. It may then choose to query $j_1$ (possibly for some different $s$), receiving neighbors $k_1,k_2,\ldots$, at which point it may query any of $i, j_2,j_3,\ldots,k_1,k_2,\ldots$, but it will never query an ``unrelated state", meaning a state (other than $|i\rangle$) that it has not received in response to a previous query.
This result is essentially the same as Lemma 4 of the arXiv version of \cite{Childs_2002}. The basic idea of the proof is one that we will re-use in \cref{mqm}. At a high level, the idea is:
given any problem $H_s$ from an instance of ${\rm AdNSP}$, we will construct some new oracle which is ``weaker" in some sense than the original oracle; in this case, whenever a related state is queried it returns the same responses as the original oracle, but whenever an unrelated state is queried, it returns some fixed response (i.e., the same response for any $H_s$) which hence gives no information about $H_s$. Thus, queries of the weaker oracle can be simulated by queries of the original oracle (simply replace the response to a query of an unrelated state by this fixed response) but not vice versa.
We then construct (for any path $H_s$) some set of paths $H'_s$ which corresponds to some other instances of ${\rm AdNSP}$ such that the original oracle for $H'_s$ is almost equivalent to the weaker oracle for $H_s$. Here, ``almost equivalent" means that for a random choice of path $H'_s$ from this set, with probability close to $1$ the original oracle for $H'_s$ returns the same responses as does the weaker oracle for $H_s$.
Finally, if there is some algorithm $A$ that solves every instance of ${\rm AdNSP}$ with probability $p$, including in particular the paths $H'_s$,
we can define an algorithm $A'$ which is given by algorithm $A$ using queries of the weaker oracle for $H_s$. Then algorithm $A'$ will solve every instance of ${\rm AdNSP}$ with probability close to $p$ using only the weaker oracle.
This construction will imply some change in the polynomials defining ${\rm AdNSP}$; in particular, in this case the size of the Hilbert space will change.
Formally:
\begin{lemma}
\label{relstatelemma}
For any algorithm $A$ that solves every instance of ${\rm AdNSP}$ with probability $\geq p$ using only quasi-polynomially many queries and possibly using queries of unrelated states,
there is some algorithm $A'$ which only queries related states and solves every instance of ${\rm AdNSP}$ with probability $\geq p-{\rm qpoly}(N) 2^{-N}$ using at most as many queries as $A$ (albeit with some change in the polynomials defining ${\rm AdNSP}$).
Hence, if algorithm $A$ succeeds with probability large compared to ${\rm qpoly}(N)2^{-N}$, then algorithm $A'$ succeeds with a probability that is comparable to that of $A$.
\begin{proof}
Given any path $H_s$ with Hilbert space ${\cal H}$ of dimension ${\rm dim}({\cal H})=2^{p(N)}$, consider a new Hilbert space ${\cal H}'$ of dimension ${\rm dim}({\cal H}')=2^{p(N)+N}$ which is exponentially larger. Define
a path $H'_s$ by
\begin{equation}
\label{perminto}
H'_s=\Pi \begin{pmatrix} H_s & \\ & W I\end{pmatrix} \Pi^{-1},
\end{equation}
where the rows and columns correspond to computational basis states,
where the first block is of size $2^{p(N)}$ and the second block is of size $2^{p(N)+N}-2^{p(N)}$,
where $\Pi$ is a permutation matrix chosen uniformly at random,
and where $I$ is the identity matrix
and $W$ is a scalar. We choose $W$ larger than the largest eigenvalue of $H$ so that
the ground state of $\begin{pmatrix} H_s & \\ & W I\end{pmatrix}$ is completely supported in the first block and is given in the obvious way from the ground state of $H_s$, and hence the ground state of $H'_s$ is given by applying $\Pi$ to that state.
Suppose on the $q$-th query, the algorithm queries an unrelated state. The number of unrelated states whose image, under $\Pi^{-1}$, is in the second block is at least ${\rm dim}({\cal H}')-{\rm dim}({\cal H})-q$.
Since the permutation $\Pi$ is random, with probability at least $({\rm dim}({\cal H}')-{\rm dim}({\cal H})-q)/{\rm dim}({\cal H}')$ the response to the query will be that the state has no neighbors and that the diagonal matrix element of that state is $W$.
Call this ``response $R$".
If an algorithm makes only quasi-polynomially many queries, with probability $\geq 1-{\rm qpoly}(N)2^{-N}$ the response to
{\it all} queries of unrelated states will be $R$.
So, given some algorithm $A$ which may query unrelated states, and which makes only quasi-polynomially many queries, we may define a new algorithm $A'$ which modifies $A$ by assuming (without querying the oracle) that the response to any query of an unrelated state will be $R$. Remark: if $A'$ finds some inconsistency in this assumption, for example if it queries some unrelated state and assumes the response is $R$ and then later that state is returned as the neighbor of some previous query, algorithm $A'$ will terminate and return some arbitrary result.
Then, $A'$ queries only related states and, with probability $\geq 1-{\rm qpoly}(N) 2^{-N}$, algorithm $A$ returns the same result as algorithm $A'$ does. Here ``probability" refers to both random choice of $\Pi$ and randomness in $A$; if $A$ is randomized, of course we assume that $A'$ uses the same source of randomness.
Each instance of $H'_s$ is defined by an instance of $H_s$ and by a choice of $\Pi$.
If $A$ returns the correct result for every instance of $H'_s$ with at least probability $p$ for some $p$, then, trivially, for any $H_s$ the average over $\Pi$ of its probability of returning the correct result is at least $p$. Hence, for any $H_s$, the probability that $A'$ returns the correct result is at least $p-{\rm qpoly}(N) 2^{-N}$.
\end{proof}
\end{lemma}
\subsection{Modified Query Model}
\label{mqm}
We now introduce the modified query model
in contrast to the query model given previously (which we will refer to as the original query model).
We show that if \cref{mainth} holds using the modified query model, then it holds using the original query model.
Very briefly: the modified query model will be such that if the algorithm follows some nonbacktracking path of queries that forms a cycle (for example, querying $i$ to get some neighbor $j$, querying $j$ to get some neighbor $k$, querying $k$ to get $i$ which is a neighbor of $k$), then the query responses will make it impossible to determine that one has returned to the start of the cycle (in this case, $i$).
To explain the modified query model in more detail, we have an infinite set of {\it labels}. Each label will correspond to some computational basis state, but the correspondence is many-to-one; we describe this correspondence by some function $F(\cdot)$. The algorithm will initially be given some label $l$ that corresponds to the computational basis state $|i\rangle$ that is the ground state of $H_0$.
A query of the oracle consists of giving it any label $m$ that is either $l$ or is a label that the algorithm has received in response to some previous query, as well as giving it any $s\in [0,1]$, and the oracle will return
some set $S$ of labels such that $F(S)$ is
the set of $j$ such that $\langle j | H_s | F(m) \rangle$ is nonzero.
Distinct labels in $S$ will have different images under $F(\cdot)$ so that $|S|$ is equal to the number of neighbors.
The oracle will also return, for each label $n\in S$,
the matrix elements $\langle F(n) | H_s | F(m) \rangle$ to precision $\exp(-{\rm poly}(N))$, i.e., returning the matrix elements to ${\rm poly}(N)$ bits accuracy for any desired polynomial.
The oracle will also return the diagonal matrix element $\langle F(m) | H_s | F(m) \rangle$.
The labels in $S$ will be chosen as follows: if label $m$ was received in response to some previous query on a label $n$, so that
$\langle F(n) | H_s | F(m) \rangle$ is nonzero and hence $F(n)\in F(S)$, then label $n$ will be in $S$, i.e., we will ``continue to label $F(n)$ by label $n$", or equivalently ``on a backtracking path, one realizes that one has backtracked". However, for {\it all} other $j$ such that
$\langle j | H_s | F(m) \rangle$ is nonzero, we will choose a new label (distinct from all previous labels) to label the given vertex $j$, i.e., a new label $o$ such that $F(o)=j$.
Thus, after a sequence of queries by the algorithm, we can describe the queries by a tree, each vertex of which is some label, with neighboring vertices in the tree corresponding to computational basis states which are neighbors.
We use the same idea as in the proof of \cref{relstatelemma}. In this case, the weaker oracle is the oracle of the modified query model. This can clearly be simulated by the original oracle, since one can simply invent new labels for a state if the oracle gives one a label that one has seen previously, but not necessarily vice versa.
We define, for each Hamiltonian $H$,
a model which has a large but finite set of labels. The function
$F(\cdot)$ mapping labels to vertices will be $2^N$-to-one. If Hamiltonian $H$ acts on Hilbert space ${\cal H}$, then these labels $l$ will correspond one-to-one to computational basis states of some Hilbert space ${\cal H}'$ with dimension $2^N {\rm dim}({\cal H})$.
We will define a Hamiltonian $H'$ acting on ${\cal H}$ as follows.
Label computational basis states of ${\cal H}'$ by a pair $i,x$ where $i$ is a computational basis state of ${\cal H}$ and $x$ is a bit
string of length $N$.
For each $a\in \{0,1,\ldots,N-1\}$, let $X_{a}$ denote the operator that flips the $a$-th
bit of this bit string, i.e.,
$$
X_a=\sum_{i,x} |i,x\oplus 1_a\rangle \langle i,x|,$$
where $1_a$ is a binary vector with entry $1$ in the $a$-th position, and $0$ elsewhere, and $\oplus$ is the exclusive OR operator.
Thus, one may regard that $N$ bits of the bit string as additional qubits and $X_a$ as the Pauli $X$ operator on them.
For each pair $i,j$ of computational basis states of ${\cal H}$, choose randomly some permutation $\pi_{i,j}(\cdot)$ of the bit strings of length $N$.
Here we emphasize that this
permutation is chosen from a set of size $(2^N)!$, i.e. permutations of a set of size $2^N$, rather than the much smaller set of size $N!$ of permutations of individual bits in a string.
Choose these permutations uniformly and independently subject to the condition that $\pi_{j,i}$ is the inverse function of $\pi_{i,j}$ and subject to choosing $\pi_{i,i}$ to be the identity permutation for all $i$.
Define $H'$ by
\begin{equation}
\label{Hprimedef}
H'=-T \sum_a X_a +\sum_{i,j,x} |i,x\rangle \langle j,\pi_{i,j}(x)| \Bigl( \langle i | H | j \rangle \Bigr),
\end{equation}
where $T>0$ is a scalar and the second term, in words, means that for each pair $i,j$ of computational basis states of ${\cal H}$, for each $x$, if there is a matrix element of $H$ between $i$ and $j$, then there is a matrix element of $H'$ between $i,x$ and $j,\pi_{i,j}(x)$.
Note that since $\pi_{i,i}$ is the identity permutation, if $H$ has no sign problem then $H'$ has no sign problem.
Remark: we have written $H'$ using a particular choice of basis states. However, we can, as in \cref{perminto}, assume that $H'$ is conjugated by a further random permutation so that the algorithm has no information on the labels of the basis states.
In this case, if the algorithm receives some state $|i,x\rangle$ as a neighbor in response to some query, and some other state $|i,y\rangle$ in response to a query with $x\neq y$, the algorithm will be unable to know that in both cases the first index $i$ is the same.
The key idea of this construction, using this remark, is that we can ensure that the algorithm is exponentially unlikely to receive the same label twice in response to a query, except for some trivial situations.
Given a path of Hamiltonians $H_s$, we can define a path $H'_s$ in the obvious way.
Define an isometry ${\cal L}$ from ${\cal H}$ to ${\cal H}'$ by
$${\cal L}=2^{-N/2} \sum_{i,x} |i,x\rangle \langle i|.$$
Choosing $T>0$, the ground state subspace of $-T\sum_a X_a$ is the range of ${\cal L}$, and if $H_s$ has no sign problem then
for any $T>0$
the ground state of $H'_s$ is equal to
$$\psi'_s={\cal L} \psi_s,$$
where $\psi_s$ is the ground state of $H_s$.
To see this,
let us refer to the second term of \cref{Hprimedef}, i.e. $\sum_{i,j,x} |i,x\rangle \langle j,\pi_{i,j}(x)| \Bigl( \langle i | H | j \rangle \Bigr)$, as the {\it cover} of $H$.
By Perron-Frobenius, the ground state of the cover of $H_s$ has a nonzero projection onto the range of ${\cal L}$; further the cover of $H_s$ commutes with
the projector ${\cal L} {\cal L}^\dagger$, so the ground state of the cover of $H_s$ is in the range of ${\cal L}$.
So, the ground state of the cover of $H_s$ is ${\cal L} \psi_s$ as claimed. Hence, for $T>0$, the
ground state of $H'_s$ is as claimed.
Further,
if the path $H_s$ is admissible, so is the path $H'_s$, as long as we choose
$T$ at least inverse polynomially large so that the fifth condition on the spectral gap is satisfied.
The path $H'_s$ does not yet satisfy the endpoint condition. However, this is easy to resolve if the path $H_s$ obeys the endpoint condition.
Concatenate the path $H'_s$ with a final path along
which the term $-T \sum_a X_a$ is replaced by the more general term
$$K_{\theta}\equiv -T\Bigl( \cos(\theta) \sum_a X_a + \sin(\theta) \sum_a Z_a\Bigr),$$
with $Z_a$ being the Pauli $Z$ operator on the additional qubits, i.e., $Z_a=\sum_{i,x} (1-2x_a) |i,x\rangle \langle i,x|,$
where $x_a$ is the $a$-th entry of bit string $x$.
On this final path, vary from $\theta=0$ to $\theta=\pi/2$.
Similarly, also concatenate with an initial path along which we vary from $\theta=\pi/2$ to $\theta=0$.
Let $\tilde H_s$ denote the path of Hamiltonians given by $H'_s$ concatenated with these two additional paths.
Then, if the ground state of $H_0$ is $|i\rangle$, the ground state of the $\tilde H_s$ at the start of the path is given by $|i,0\rangle$, and similarly at the end of the paths.
Further, concatenation with these additional paths preserves the spectral gap.
Let the ground state of $H_0$ be $|i\rangle$. Define a $2^N$-dimensional subspace spanned by states of the form $|i,x\rangle$ over all $x$.
The Hamiltonian on the additional initial path does not couple this subspace to the orthogonal subspace. It is easy to compute the spectrum in this subspace,
since the Hamiltonian in this subspace is just a $\langle i | H_0 | i \rangle+K_\theta$. One may verify the gap in this subspace, and verify that the ground state energy in this subspace is
$\langle i | H_0 |i\rangle$ plus the minimum eigenvalue of $K_\theta$. Any state orthogonal to this subspace must have
energy at least equal to the minimum eigenvalue of
$K_\theta$ plus the smallest eigenvalue of
$H_0$ in the space orthogonal to $|i\rangle$ (i.e., at least the second smallest eigenvalue of $H_0$) so the gap follows.
A similar argument holds for the final path.
Thus, we have an admissible path of Hamiltonians $\tilde H_s$ satisfying the endpoint condition, with the ground state of $\tilde H_s$ trivially related to that of $H_s$ for $s=0,1$ (the polynomially small error in $\psi'_s={\cal L} \psi_s$ at intermediate steps of the path is unimportant for this).
Suppose now we give the algorithm some additional information in response to queries: if we query some state $|i,x\rangle$ and a neighbor is some other state $|i,y\rangle$, then the algorithm will be informed that the value of $i$ remains the same. This additional information can only help.
However, we claim that with this additional information, {\it up to exponentially small error, the queries of $\tilde H_s$ in the original query model can be described by queries of $H_s$ in the modified query model}.
More precisely, assume we know that $\tilde H_s$ is given by this construction. Then, the only information given by querying $\tilde H_s$ along the ``final" or ``initial" paths above where $\theta$ varies is that one may get multiple labels which are known to have the same first index, i.e., one may start with $|i,0\rangle$, and then get labels describing other states $|i,x\rangle$. Further, we claim that up to exponentially small error explained below, queries along the path $H'_s$ in the original query model can be described by queries of $H_s$ in the modified query model.
To show this,
consider the probability that the algorithm receives the same label twice in response to a query.
Suppose the
algorithm makes multiple queries in which the first label does not change and the algorithm knows it due to the additional information above. Thus, the algorithm will know that some set of labels will describe the same value of the first index.
After some number of queries, there will be several sets $S_1,S_2,\ldots$ where each set is a set of labels known to describe the same value of the first index. Formally, there is an equivalence relation on labels: two labels are equivalent if one label is received in response to a query on the other and it is known that the first index did not change, and we extend this equivalence transitively, and the sets $S_1,S_2,\ldots$ are equivalence classes under this relation.
Now, consider the probability that some query of some label $|i,x\rangle$ gives a label $|j,y\rangle$ that has been seen previously by the algorithm in response to a previous query, in the case that $j\neq i$ (so that this query does not simply increase the size of one of the equivalence classes, but actually yields new information about the Hamiltonian).
The second index $y$ equals $\pi_{j,i}(x)$ and $\pi_{j,i}(x)$ is chosen uniformly at random subject to the condition that $\pi_{j,i}$ is the inverse of $\pi_{i,j}$.
Hence, after only quasi-polynomially many queries (so that only quasi-polynomially values of $\pi_{j,i}$ have been fixed) it is exponentially unlikely that $\pi_{j,i}(x)$ will agree with any previously given value of the second index, unless it is the case that
we have previously queried $|j,\pi_{j,i}(x)\rangle$, i.e., unless $|i,x\rangle$ was received as a label of a neighbor of $|j,\pi_{j,i}(x)\rangle$, which is precisely the case in the modified query model that we receive the same label for a given value of the
state. Now consider a query in which the first index does not change;
suppose we queried a vertex in some equivalence class $S$, receiving some new label $|i,x\rangle$. It is exponentially unlikely that this label labels a state in some other equivalence class, though it may be only polynomially unlikely that it labels a state in the given class $S$. Hence, except for an exponentially small probability, a query in which the first index does not change will not collapse two different equivalence classes.
Hence we have:
\begin{lemma}
If \cref{mainth} holds in the modified query model, then it holds in the original query model.
\end{lemma}
\section{Distinguishing Graphs}
\label{distg}
Now, within the modified query model we show how to use two (families of) graphs $C,D$ to construct instances of ${\rm AdNSP}$ to prove \cref{mainth}. The main result is \cref{blemma}, which we give at the end of this section after developing the machinery of paths needed.
In this section we assume that several properties of $C,D$ hold. We summarize these in \cref{trq}. More detail is given below and these properties are proven in later sections of the paper
Both graphs $C,D$ will have a privileged vertex called the ``start vertex". For tree graphs, the start vertex will often be the root of the tree.
Given a graph $G$, we say that the Hamiltonian corresponding to that graph is equal to minus the adjacency matrix of that graph, where each vertex of the graph corresponds to a distinct computational basis state.
We will assume that the ground state energy of the Hamiltonian of $C$ is lower than the ground state energy of $D$ by a spectral gap that is $\Omega(1/{\rm poly}(N))$; indeed, the difference will be much larger than that in our construction.
We will also assume that the gap between the ground state of the Hamiltonian of $C$ and the first excited state of that Hamiltonian is also $\Omega(1/{\rm poly}(N))$; indeed, that difference is also much larger than that.
Further, we will assume that the amplitude of the ground state wavefunction of $C$ on the start vertex is also $\Omega(1/{\rm poly}(N))$.
At the same time, we will also assume a superpolynomial lower bound on the number $q$ of classical queries needed to distinguish $C$ from $D$ in the modified query model above, assuming that the first vertex queried is the start vertex.
The modified query model refers to querying a Hamiltonian; here the Hamiltonian will be the Hamiltonian of the given graph, so that computational basis states are neighbors if the corresponding vertices are neighbors.
The bound is given in terms of mutual information in \cref{distlemma}; if the algorithm is randomized, then the mutual information is conditioned on any randomness used by the algorithm.
Later choices of constants in \cref{chs} will make $q=\exp(\Theta(\log(N)^2))$.
\begin{center}
\begin{itemize}
\item[1.] The ground state energy of the Hamiltonian of $C$ lower than that of $D$ by $\Omega(1/{\rm poly}(N))$.
\item[2.] $\Omega(1/{\rm poly}(N))$ gap of Hamiltonian of $C$.
\item[3.] Amplitude of ground state of $C$ on start vertex is $\Omega(1/{\rm poly}(N))$.
\item[4.] Lower bound on number of classical queries to distinguish $C$ from $D$ in the modified query model. Precisely: if the graph is chosen randomly to be $C$ with probability $1/2$ and $D$ with probability $1/2$, then with fewer than $q$ classical queries the mutual information (in bits) between the query responses and the choice of graph is bounded for all sufficiently large $N$ by some quantity which is strictly less than $1$, for some $q$ which is superpolynomial.
\item[5.] Number of vertices is $O(2^{{\rm poly}(N)})$. This property is needed because each vertex will correspond to some computational basis state.
\end{itemize}
\captionof{table}{List of properties needed for graphs $C$ and $D$.} \label{trq}
\end{center}
Then, given these graphs, we now construct a path of Hamiltonians $H_s$. To describe this path, we start with a simplified case. Consider a problem where for every vertex of a graph $G$ there is a corresponding
computational basis state, and there is one additional computational basis state written $|0\rangle$ which is distinct from all the basis states corresponding to vertices of the graph. We will label the basis state corresponding to the start vertex of $G$ by $|s\rangle$ (hopefully no confusion will arise with the use of $s$ as a parameter in the path).
Consider the two parameter family of Hamiltonians
\begin{equation}
\label{HtU}
H(t,U)=-U|0\rangle\langle 0|+t\Bigl( |0\rangle\langle s| +h.c.\Bigr)+H(G),
\end{equation}
where $H(G)$ is the Hamiltonian corresponding to the graph $G$ and where $U,t$ are both scalars\footnote{The terminology $U$ is used as it is a commonly used notation in physics for such diagonal terms; hopefully no confusion arises with the use of $U$ for unitaries in quantum information.}.
We take $t<0$ so that the Hamiltonian has no sign problem.
Now consider a path of Hamiltonians starting at very negative $U$ and with $t=0$ (so that initially the ground state is $|0\rangle$ for both $C$ and $D$),
then making $t$ slightly negative and increasing $U$ to a value between the ground state energy of
$C$ and that of $D$, followed by returning $t$ to zero.
At the
end of this path, if $G=D$, the ground state of the Hamiltonian will still be $|0\rangle$ but if $G=C$, the ground state of the Hamiltonian will be the ground state of $H(C)$.
We make $t$ nonzero during the middle of the path to avoid a level crossing: if $t=0$ throughout the path, then the gap closes if $G=C$ when $U$ becomes equal to the ground state energy of $C$.
Indeed, we may choose the path dependence of parameters $U,t$
so that we get an admissible path assuming the properties of $C$ and $D$ above.
For both $C$ and $D$, the ground state energy of $H(G)$ is only ${\rm poly}(N)$ so we may take $U$ only polynomially large initially.
Giving the rest of the path in detail,
change $t$ to an amount $-\Omega(1/{\rm poly}(N))$ and then change $U$ so that it is $1/{\rm poly}(N)$ larger than the ground state energy of $H(C)$, but still much smaller than the ground state energy of $H(D)$ and than the first excited state energy of $H(C)$. Do this with $|t|$ much smaller than the spectral gap of $H(C)$ and much smaller than the difference between the ground state energy of
$H(C)$ and the ground state energy of $H(D)$; given the differences in ground state energies of $H(C)$ and $H(D)$ and the gap of $H(D)$, it is possible to do this with $|t|$ that is indeed $\Omega(1/{\rm poly}(N))$ so that the gap of the Hamiltonian $H(t,U)$ remains $\Omega(1/{\rm poly}(N))$. Finally return $t$ to $0$.
For use later, let us call the path defined in the above paragraph $P(G)$.
At first sight, this path $P(G)$, combined with the lower bound {\bf 4} of \cref{trq} might seem to solve the problem of the needed separation between problems in ${\rm AdNSP}$ and the power of classical algorithms: the classical algorithm cannot distinguish the two graphs but one can distinguish them with an admissible path of Hamiltonians. However, this is not true; for one, our path of Hamiltonians does not satisfy the endpoint condition as the ground state of the Hamiltonian at the end of the path is a superposition of basis states in the case that $G=C$. Further, the problem is to compute the basis vector at the end of the path, not to distinguish two graphs\footnote{Remark: we could have defined a different oracle problem which differs from ${\rm AdNSP}$ in two ways: first, we only require that the admissible path satisfy the endpoint condition at endpoint $s=0$ and second, we say that an algorithm ``solves" the problem if it computes the amplitude of the basis state $|i\rangle$ which is the ground state of $H_0$ in the wavefunction which is the ground state of $H_1$ to within error $1/{\rm poly}(N)$ with success probability $\geq 2/3$. Then, for graph $C$ this amplitude is $0$ and for $D$ this amplitude is $1$ but the classical algorithm cannot distinguish the two cases with probability much larger than $1/2$. If we considered this class, then the given path would prove the needed separation. However, this is not what we are considering.}.
We might try to solve this by concatenating the path of Hamiltonians above with an additional path that decreases $H(G)$ to zero while adding a term $V |s\rangle\langle s|$ and with $V$ decreasing from zero, so that the ground state of the Hamiltonian (in the case that $G=C$) evolves from being the ground state of $H(G)$ to being $|s\rangle$ and the path now satisfies the endpoint condition.
This still however does not solve the problem: for this path $H_s$, a classical algorithm can determine the ground state at the end of the path (which we will assume to occur at $s=1$) by querying the oracle three times, first querying $\langle 0 | H_1 | 0\rangle$, then querying the oracle to find the neighbors of $|0\rangle$ in the middle of the path (so that it can determine $s$), and
finally querying $\langle s | H_1 | s \rangle$.
So, to construct the path showing \cref{mainth}, we use an additional trick.
First, we consider $N$ different copies of the problem defined by Hamiltonian \cref{HtU} ``in parallel". Here, taking ``copies in parallel" means the following, given several Hamiltonians $H_1,H_2,\ldots,H_N$, with associated Hilbert spaces ${\cal H}_1,{\cal H}_2,\ldots,$, we define a Hamiltonian $H$ on Hilbert space ${\cal H}\equiv {\cal H}_1 \otimes {\cal H}_2 \otimes \ldots$ by
$H=H_1\otimes I \otimes I \ldots+ I\otimes H_2 \otimes I \otimes \ldots + \ldots$ where $I$ is the identity matrix.
There is an obvious choice of computational basis states for ${\cal H}$, given by tensor products of computational basis states for ${\cal H}_1,{\cal H}_2,\ldots$.
Similarly, given paths of Hamiltonians, we consider the paths in parallel in the obvious way.
If each path is admissible, then the path given by those paths in parallel is also admissible; note that the size of the Hilbert ${\cal H}$ is $2^{{\rm poly}(N)}$ if each ${\cal H}_i$ has dimension $2^{{\rm poly}(N)}$.
For each of these $N$ copies, we choose $G$ to be either $C$ or $D$ independently, so that there are $2^N$ possible instances.
Write $G_i$ to denote the graph chosen on the $i$-th copy.
For each copy $i$ let $P_i$ denote the path $P$ for that copy. Let $\tilde P$ denote the path given by taking all those paths $P_i$ in parallel. At the end of the path $\tilde P$, the ground state is a tensor product of $|0\rangle$ on some copies and the ground state of $H(C)$ on some other copies.
To give an intuitive explanation of the trick we use, it will be that we use this property of the ground state as a kind of key to find an entry in a database: there will be some projector ($\Pi$ below) which is diagonal in the computational basis and one must find entries of it which are nonzero, and the adiabatic evolution will use this property of the ground state to find it, but the classical will not be able to.
The trick is: concatenate that path $\tilde P$ with a further path $Q$. To define this path $Q$, add two additional terms to the Hamiltonian
$$V\sum_i |s\rangle_i\langle s| +W \Pi,$$
where $|s\rangle_i\langle s|$ denotes the projector onto $|s\rangle$ on the $i$-th copy tensored with the identity on the other copies, where $V,W$ are scalars, and where $\Pi$ is a projector which is diagonal in the computational basis.
The projector $\Pi$ will be equal to $1$ on a given computational basis state if and only if that computational basis state is a tensor product of basis state $|0\rangle_i$ on all copies for which $G_i=D$ and of basis states corresponding to vertices of $G_i$ for all copies on which $G_i=C$.
Thus, the ground state at the end of path $\tilde P$ is an eigenvector of $\Pi$ with eigenvalue $1$.
The path $Q$ is then to first decrease $W$ from zero so that it is large and negative (it suffices to take it polynomially large) while keeping $t=0$; then decrease the coefficient in front of $H(G)$ to zero while increasing $V$ to be $\Omega(1/{\rm poly}(N))$.
Making $W$ large and negative ensures that throughout $Q$,
the ground state of the Hamiltonian is in the eigenspace of $\Pi$ with unit eigenvalue.
This decrease in the coefficient in front of $H(G)$ combined with increase in $V$ ensures that the ground state at the end of the path is a computational basis state: it is a tensor product of
$|0\rangle_i$ on all copies for which $G_i=D$ and of states $|s\rangle_i$ for all copies on which $G_i=C$.
We choose $V$ to be $\Omega(1/{\rm poly}(N))$ so that the gap of the Hamiltonian will be $\Omega(1/{\rm poly}(N))$.
Let $\hat P$ be the concatenation of $\tilde P$ and $Q$. Note that there are $2^N$ possible instances of path $\hat P$, depending on different choices of $G_i$.
Now we bound the probability of a classical algorithm to determine the final basis state.
This lemma shows that if we construct graphs which satisfy {\bf 1-5} of \cref{trq}, then \cref{mainth} follows.
\begin{lemma}
\label{blemma}
If items {\bf 1-3,5} of \cref{trq} hold, then $\hat P$ is an admissible path.
Further, if item {\bf 4} holds,
no algorithm
which uses a number of classical queries which is quasi-polynomial and is smaller than $q$
can solve all instances with probability greater than $\exp(-cN)$.
\begin{proof}
By construction $\hat P$ is admissible.
Choose each $G_i$ independently, choosing it to be $C$ with probability $1/2$ and $D$ with probability $1/2$.
By item {\bf 4} of \cref{trq}, with fewer than $q$ queries of the oracle the mutual information\footnote{Here, the random variables considered are as follows. First, a single bit for each graph $G_i$ to determine whether the graph is $C$ or $D$. Then, additional randomness to determine the labels in the modified query model. Then, any randomness used by the algorithm to choose queries (if the algorithm is randomized we then consider the mutual information conditioned on that randomness, and similarly we consider entropies conditioned on that randomness later in the proof). Finally,
the query responses (which of course are determined by the other random variables).} in bits between each $G_i$ and the query responses is bounded by some quantity $S<1$.
Of course, if the algorithm makes less than $q$ queries, then the average number of queries per graph is even less (i.e., less than $q/N$), but certainly for any $i$, we make at most $q$ queries of graph $G_i$ and so the mutual information between any $G_i$ and the set of all query responses is at most $S<1$.
Hence, the average entropy of $G_i$ given the query responses is at least $1-S$,
as $G_i$ is a random variable with entropy $1$.
So, since the entropy of $G_i$ given the query responses is bounded by $1$, with probability at least $(1-S)/2/(1-(1-S)/2)$
the entropy of $G_i$ given the query responses is at least $(1-S)/2$.
Thus, with probability that is $\Omega(1)$, the entropy of $G_i$ is $\Omega(1)$.
To get oriented, assume that these events (the entropy of each $G_i$) are independent; that is, define $N$ additional binary random variables $S_i$ which quantify the entropy of $G_i$ being $\geq (1-S)/2$ or not, and assume that these are all independent.
Then, with probability $1-\exp(-\Omega(N))$, there are $\Theta(N)$ independent variables $G_i$ each of which\footnote{Indeed, Bayes' theorem implies that the $G_i$ are independent variables given the query responses. The prior distribution is to choose the $G_i$ uniformly and independently, so they are independent variables in the prior. The likelihood of getting a certain set of responses in response to a given set of queries is a product over graphs of a likelihood function for each graph (with that function determined just by the queries of that graph). So, they are independent variables in the posterior.}
have entropy $\Omega(1)$ and
so it is not possible to determine all $G_i$ with probability better than $\exp(-cN)$ using only quasi-polynomially many queries.
Finally, consider the possibility that the $S_i$ are not independent. For example, there is a rather silly algorithm that makes these $S_i$ dependent on each other: consider any algorithm $A$ that gives independent $S_i$ and define a new algorithm $A'$ that calls $A$ with probability $1/2$ and makes no queries with probability $1/2$ (in which case all $S_i=1$ since no information is known about any $G_i$). Then the variables $S_i$ for $A'$ are not independent. However, this ``silly algorithm" certainly does not help.
Still we must consider the possibility that there is some way of correlating the $S_i$ which would help.
Suppose there were an algorithm which gave correlated $S_i$, so that for some $i$
the mutual information between $G_i$ and query responses,
conditioned on some responses for $j\neq i$ and conditioned on the $G_j$ for $j\neq i$, was larger than $S$. However, we could then postselect this algorithm on the query responses to the set of $j \neq i$ to give an algorithm that just acted on copy $i$ but which gave mutual information greater than $S$.
\end{proof}
\end{lemma}
\section{A First Attempt}
\label{toomany}
In this section we give a first attempt at constructing two graphs, $C$ and $D$ which satisfy the properties of \cref{trq}.
Unfortunately, the example will not quite satisfy the fourth property (using the privileged start vertex it will be possible to efficiently distinguish them, but it will not be possible without that knowledge) or
for the fifth: the graph $D$ will have too many vertices.
The construction later will fix both of these defects.
Briefly, the graph $C$ is a complete graph on $4$ vertices, i.e., every one of the four vertices has degree $3$ so it connects to every other vertex.
The graph $D$ is a tree graph where every vertex except the leaves has degree $3$, i.e., $D$ is given by attaching three binary trees to some root vertex.
We choose all the leaves of $D$ to be at distance $h$ from the root for some $h$ so that $D$ has $1+3+2\cdot 3 + 2^2\cdot 3+\ldots+2^{h-1} \cdot 3$ vertices.
We choose any vertex of $C$ arbitrarily to be the start vertex. We choose the start vertex of $D$ to be the root. Then, it is trivial to verify item {\bf 1-3} of \cref{trq}.
Consider what it means to distinguish two graphs in the modified query model. A query of any vertex can return only the information of the degree of that vertex and whether or not that vertex is the start vertex. Since all matrix elements between the computational basis state of that vertex and its neighbors are the same, one cannot distinguish the different neighbors in any way from the response to the given query. As we have mentioned, though, one can determine if a queried vertex is the start vertex since the Hamiltonian will have an additional coupling
$t\Bigl( |0\rangle\langle s| +h.c.\Bigr)$.
Using the knowledge of which vertex is the start vertex it is not hard to distinguish the two graphs in $O(1)$ queries: query the start vertex $s$ to get some new vertex $v$, then query a neighbor of $v$ (other than $s$, i.e. nonbacktracking) to get some new vertex $w$, then finally query a neighbor of $w$, again without backtracking. For $C$, with probability $1/3$ that neighbor of $w$ will be the start vertex. On the other hand, for $D$, the neighbor will never be the start vertex.
Suppose however, that we use only the information about the degree of the vertex and not which vertex is the start vertex.
In this case, it is impossible to distinguish $C$ from $D$ using fewer than $h$ queries because, trivially, any vertex accessed with fewer than $h$ queries is not a leaf of $D$ and hence has the same degree (i.e., $3$) as every vertex in $C$, and hence the terms in the Hamiltonian coupling the corresponding computational basis state to its neighbors are the same.
So, if we choose $h$ superpolynomially large, then item {\bf 4} is ``almost satisfied", i.e., without information about the start vertex we cannot distinguish them with polynomially many queries.
However, if we choose $h$ superpolynomially large, then item {\bf 5} is not satisfied since the number of vertices is not $O(2^{{\rm poly}(N)})$.
If instead we choose $h$ only polynomially large,
there is a simple efficient classical algorithm to distinguish $C$ from $D$ even without using knowledge of the start vertex: since we can avoid backtracking in the given query model, we will arrive at a leaf in $h$ queries. Even if we perform a random walk on $D$, allowing backtracking, we will typically arrive at a leaf in $O(h)$ queries
\section{Decorated Graphs}
\label{decg}
In this section we define an operation called {\it decoration} and then define graphs $C,D$ in terms of this operation. Our decoration operation is very similar to (indeed, it is a special case of) the decoration
operation defined in \cite{Schenker_2000} (there are other uses of the term ``decoration" in the math literature, such as in set theory, which are unrelated to this).
These graphs $C,D$ will depend upon a large number of parameters; in \cref{chs} we will show that for appropriate choice of these parameters, all properties in \cref{trq} are fulfilled.
\cref{decdef} defines decoration.
The idea of decoration is to make it easy for a classical algorithm to ``get lost". Decoration will add additional vertices and edges to some graph, and a classical algorithm will tend to follow what one may call ``false leads" along these edges so that it is hard for it to determine properties of the graph before decoration because it takes a large number of queries to avoid these false leads.
\cref{applic} applies decoration to define $C,D$ and explains some motivation for this choice of $C,D$. \cref{spec} considers the spectrum of the adjacency matrices of $C,D$, as well as proving some properties of the eigenvector of the adjacency matrix of $C$ with largest eigenvalue. Here we, roughly speaking, bound the effect of decoration on the spectrum and leading eigenvector of the graph.
\subsection{Decoration}
\label{decdef}
In this subsection, we define an operation that we call {\it decoration} that maps one graph to another graph.
We first recall some graph theory definitions. For us, all graphs are undirected, so all edges are unordered pairs of vertices, and there are no multi-edges or self-edges so that the edge set is a set of unordered pairs of distinct vertices.
An $m$-ary tree is a rooted tree (i.e., one vertex is referred to as the root) in which each vertex has at most $m$ children. A full $m$-ary tree is a tree in which every vertex has either or $0$ or $m$ children (i.e., all vertices which are not leafs have $m$ children).
A perfect $m$-ary tree is a full $m$-ary tree with all leaves at the same distance from the root. A binary tree is an $m$-ary tree with $m=2$.
We will make an additional definition (this concept may already be defined in the literature but we do not know a term for it). We will say that a
graph is ``$d$-inner regular with terminal vertex set $T$" if $T$ is a set of vertices
such that every vertex not in $T$ has
degree $d$. The vertices in $T$ will be called terminal vertices and the vertices not in $T$ will be called inner vertices.
We define the height of a tree to be the length of the longest path from the root to a leaf, so that tree with just one vertex (the root) has height $0$ (sometimes it is defined this way in the literature, but other authors define it differing by one from our definition).
Then, the number of vertices in a perfect $m$-ary tree of height $h$ is
$1+m+m^2+\ldots+m^{h}=(m^{h+1}-1)/(m-1)$.
We now define decoration. If we $(c,m,h)$-decorate a graph $G$, the resulting graph is given by attaching
$c$ perfect $m$-ary trees of height $h-1$ to each vertex of $G$; that is, for each vertex $v$ of $G$, we add $c$ such trees, adding one edge from $v$ to the root of each tree, so that the degree of $v$ is increased by $c$.
Call the resulting graph $H$.
Then, any vertex of graph $G$ which has degree $d$ corresponds to a vertex of $H$ with degree $d+c$.
We may regard $G$ as an induced subgraph of $H$; if $G$ is a rooted tree, then the root of $G$ corresponds to some vertex in $H$ that we will regard as the root of $H$.
We will refer to those vertices of $H$ which correspond to some vertex of $G$ as the {\it original vertices} of $H$, i.e., the original vertices of $H$ are those in the subgraph $G$.
If $G$ is a full $n$-ary tree, and if we $(n,2n,h)$ decorate $G$, then $H$ is a full $(2n)$-ary tree.
Now we define a sequence of decorations. Consider some sequence of heights $h_1,h_2,\ldots,h_l$ for some $l$. Given a graph $G_0$ which is $(n+1)$-inner regular, we $(n,2n,h_1)$-decorate $G$, calling the result $G_1$.
The terminal set of $G_1$ will be the set of vertices which correspond to vertices in the terminal set of $G$, as well as additional leaves (vertices of degree $1$) added in decoration, so that $G_1$ is $(2n+1)$-inner regular.
We then $(2n,4n,h_2)$-decorate $G_1$, calling the result $G_2$, defining the terminal set of $G_2$ in the analogous way.
Proceeding in this fashion, we $(2^m\cdot n,2^{m+1}\cdot n,h_m)$-decorate $G_m$, giving $G_{m+1}$, until we have defined graph $G_l$. We say that $G_l$ is given by decorating $G$ with height sequence $h_1,\ldots,h_l$.
Note that each graph $G_m$ is inner regular.
We will call the {\it original vertices} of $G_l$ those vertices which correspond to some vertex of $G_0$ in the obvious way, i.e., $G_0$ is an induced subgraph of $G_k$ and the original vertices are the vertices in that subgraph.
Remark: the reason that we talk about $(n+1)$-inner regular, rather than $n$-inner regular is that if one avoids backtracking, this means that there are $n$, rather than $(n-1)$, choices of vertex to query from any given vertex.
Also, since we are decorating with trees, in this way $n$ refers to the arity of the tree rather than the degree of the tree.
This is just an unimportant choice of how we define things and other readers might find it more convenient to shift our value of $n$ by one.
For use later, let us define
$T_{n,h_0}$ where $T_{n,h_0}$ is constructed by attaching $n+1$ perfect $n$-ary trees to some given vertex, i.e., $T_{n,h_0}$ is $n$-inner regular with the terminal set being the leaves of the trees.
See Fig.~\ref{figdec} for an example of decoration of such a graph $T_{3,2}$.
\begin{figure}
\includegraphics[width=3in]{tree.pdf}
\caption{Example of decoration. Solid lines are edges of graph $G$ which is $3$-inner regular.
In the notation given, it is $T_{3,2}$. Dashed lines represent edges added after $(2,4,1)$-decorating $G$. Solid circles are inner vertices and terminal vertices are not shown.
Since $h_1=1$, the $4$-ary trees added are of height $h_1-1=0$, consisting just of the root; if we had taken $h_1=2$, then each of the dashed lines shown would have a solid circle and four additional dashed lines attached to them.}
\label{figdec}
\end{figure}
\subsection{Application of Decoration, and Universal Cover of Graph}
\label{applic}
We will apply this decoration to two different choices of $(n+1)$-inner regular graphs, each of which has some fixed vertex that we call the start vertex.
In the first case, we pick $G_0$ to have vertices labelled by a pair of integers $(i,j)$ with
$0\leq i <L$ for some $L>1$ and $0\leq j <m$ for $m= (n+1)/2$ for some odd $n$.
There is an edge between $(i,j)$ and $(k,l)$ if $i=k\pm 1 \mod L$; note that there is no constraint on $j,l$.
So, for $m=1$, the graph is a so-called ring graph.
The start vertex will have $i=\lfloor L/2\rfloor $ and $j=0$.
We denote this graph $R_{L,m}$ where $m=(n+1)/2$.
The second case is the same except that there is an edge between $(i,j)$ and $(k,l)$ if $i=k\pm 1$. Note that the ``$\mod L$" is missing in the definition of $D$.
Again the start vertex has $i=\lfloor L/2\rfloor $ and $j=0$.
So, for $m=1$, this graph is a so-called path graph or linear graph.
We denote this graph $P_{L,m}$.
See Fig.~\ref{figP}.
\begin{figure}
\includegraphics[width=3in]{rp.pdf}
\caption{Example of graph $P_{4,3}$. Solid circles are vertices and solid lines are edges. Note that the graph is $6$-inner regular so $n=5$. In the case of graph $R_{4,3}$, there are $9$ additional edges connecting each of the
three vertices at the left to each of the three vertices on the right.}
\label{figP}
\end{figure}
Define $C=G_l$ in the case that $G=R_{L,(n+1)/2}$ and define
$D=G_l$ in the case that $G=P_{L,(n+1)/2}$.
Note that $R_{L,(n+1)/2}$ is an $(n+1)$-regular graph and $P_{L,(n+1)/2}$ is an $(n+1)$-inner regular graph where the terminal vertices are those $(i,j)$ with $i=0$ or $i=L-1$.
Then, in both cases,
the graph $G_l$ is a $d$-inner regular graph with $d=2^{l+1} \cdot n$.
We will define a start vertex on $G_l$ in the obvious way: it is the original vertex in $G_l$ that corresponds to the start vertex of $G$.
The key then is that to distinguish $C$ and $D$, one must be able to go a long distance, of order $L$, on the graph. Decoration will make it hard for any classical algorithm to follow such a long path.
\subsection{Spectrum of Adjacency Matrix of Decorated Graph}
\label{spec}
We now provide bounds on the spectrum of the adjacency matrix of graph $G_l$ constructed from decorating $G$ with some height sequence. For any graph $H$, let $\lambda_{0}(H)$ denote the largest eigenvalue of the adjacency matrix of $H$. Since $G$ is a subgraph of $G_l$, it follows that $\lambda_0(G_l)\geq \lambda_0(G)$.
We have:
\begin{lemma}
\label{eigbounddec}
Assume $n/L^2=\omega(1)$.
For either choice of $G_0$, the largest eigenvalue of the adjacency matrix of $G_l$ is bounded by
$$\lambda_0(G)\leq \lambda_0(G_l)\leq \lambda_0(G)+2^{l/2+1} \cdot \sqrt{n}.$$
Further $\lambda_0(C)\geq n$ and
$\lambda_0(D)\leq n+2^{l/2+1} \cdot \sqrt{n}-\Theta(n/L^2)$.
The second largest eigenvalue of the adjacency matrix of $C$ is upper bounded by
$n+2^{l/2+1} \cdot \sqrt{n}-\Theta(n/L^2)$.
Finally, let $\psi$ be an eigenvector of the adjacency matrix of $C$ with largest eigenvalue (the eigenspace has dimension $1$ since $G_l$ is connected), with $|\psi|=1$.
Let $\Pi$ be a diagonal matrix which is $1$ on the original vertices of $C$ and $0$ on the other vertices.
Let $p=|\Pi \psi|^2$. Remark: heuristically, $p$ is the ``probability" that if one measures $\psi$ in a basis of the vertices, that the result will be one of the original vertices.
Then, if $n \geq 2^{l/2+2} \cdot \sqrt{n}$,
$p\geq 1/5$.
\begin{proof}
To show the first result,
let $A$ be the adjacency matrix of $G_l$. We decompose $A=A_0+A_1$ where $A_0=\Pi A \Pi$ so that $A_0$ is the adjacency matrix of the subgraph of $G_l$ obtained by deleting all edges except those which connect two original vertices. Then, $\Vert A_0 \Vert=\lambda_0(G)$ and so $\lambda_0(G_l)\leq \lambda_0(G)+\Vert A_1 \Vert$.
However, $A_1$ is equal to the adjacency matrix of the subgraph of $G_l$ obtained by deleting any edge connecting two original vertices of $G_l$. This subgraph is a forest, to use the terminology of graph theory: it consists of disconnected trees, one tree for each vertex in $G$.
Indeed each of these trees is an $m$-ary tree with $m=2^l\cdot n$ (recall that if $G$ has degree $n+1$, then $G_l$ has degree $2^{l}\cdot n+1$).
We now upper bound the largest eigenvalue of the adjacency matrix of an $m$-ary tree.
For a perfect $m$-ary tree of height $h$, it is possible to compute the largest eigenvalue: the tree has a symmetry under permuting the daughters of any given vertex and the largest eigenvector will be invariant under this symmetry. So, let $v_k$, for integer $k$ with $0\leq k \leq h$, denote the vector with norm $1$ which is an equal amplitude superposition of all vertices which are distance $k$ from the root, so that $v_0$ is $1$ on the root and $0$ elsewhere, $v_1$ has amplitude $1/\sqrt{m}$ on each of the daughters of the root, and so on. We can then write the adjacency matrix, restricted to the subspace spanned by $v_0,v_1,\ldots$, as
$\sqrt{m} \sum_{0\leq k \leq k+1} |v_k\rangle\langle v_{k+1}|+h.c.$ and so clearly the largest eigenvalue is bounded by $2\sqrt{m}$.
So, $\lambda_0(G_l)\leq \lambda_0(G)+2^{l/2+1} \cdot \sqrt{n},$ as claimed
The lower bound on $\lambda_0(C)\geq n$ follows from the fact that $\lambda_0(R_{L,(n+1)/2})=n+1>n$.
The upper bound on $\lambda_0(D)$ follows since $\lambda_0(P_{L,(n+1)/2})=n+1-\Theta(n/L^2)=n-\Theta(n/L^2)$ since $n/L^2=\omega(1)$.
To bound the second largest eigenvalue of the adjacency matrix of $C$, again decompose $A=A_0+A_1$.
By the Courant-Fischer-Weyl min-max principle, the second largest eigenvalue of $A$ is equal to the minimum, over all subspaces of codimension $1$, of the largest eigenvalue of the projection of $A$ into that subspace. Consider the subspace orthogonal to the eigenvector of $A_0$ of largest eigenvalue; the projection of $A_0$ into this subspace is bounded by $n-\Theta(n/L^2)$ and so
the second largest eigenvalue of $A$ is upper bounded by $n-\Theta(n/L^2)+\Vert A_1 \Vert$.
For the final claim, we have $\lambda_0(C) \leq p \Vert A_0 \Vert + 2 \sqrt{p(1-p)} \Vert A_1 \Vert + (1-p) \Vert A_1 \Vert$.
Since $\lambda_0(C)\geq n$ and $\Vert A_0 \Vert=n$, we have
$(1-p) n \leq \Bigl((1-p)+2\sqrt{p(1-p)} \Bigr) 2^{l/2+1} \cdot \sqrt{n+1}$.
So, for $n\geq 2 (2^{l/2+1} \cdot \sqrt{n})$, after a little algebra we find that $p\geq 1/5$.
\end{proof}
\end{lemma}
\section{Classical Hardness}
\label{ch}
We now show classical hardness. We will give a lower bound on the number of queries needed by a classical algorithm to distinguish $C$ from $D$, with the initial state of the classical algorithm being the start vertex. The
lower bound \cref{distlemma} will depend on the difficulty of reaching a certain set $\Delta$ defined below.
To show this difficulty,
the main result is in the inductive \cref{inductivelemma}; we then apply this lemma in \cref{chs}.
Given a $d$-inner regular graph, with some given choice of start vertex, and given some set $S$ which is a subset of the set of vertices, we say that it is $(p,q)$-hard to reach $S$ if no classical algorithm, starting from the start vertex, can reach $S$ with at most $q$ queries with probability greater than $p$. Here ``reach" means that at some point the classical algorithm queries a vertex in $S$.
Note that given a perfect $m$-ary tree of height $h$, it is clearly $(0,h+1)$-hard to reach the set of leaves of the tree if the root vertex is the start vertex: the first query will query the start vertex, the second query can query a vertex at most distance $1$ from the start vertex, and so on.
Our goal will be to show for graph $D$ that it is hard (for some choice of parameters $p,q$) to reach the set $\Delta$ of vertices of $D$ which correspond to terminal vertices of $G_0$, i.e., $\Delta$ is a subset of the terminal vertices of $G_l$, consisting only of those which correspond to terminal vertices of $G_0$ rather than leaves added in decoration.
This will then imply a bound on the ability of a classical algorithm to distinguish $C$ from $D$ in the modified query model:
\begin{lemma}
\label{distlemma}
Suppose it is $(p,q)$-hard to reach the set $\Delta$ above for graph $D$.
Suppose one chooses
some graph $G$
to be $C$ or $D$ with probability $1/2$ for each choice. Then, no classical algorithm using at most $q$ queries in the modified query model can correctly guess which graph $G$ is with probability greater than $1/2+p/2$.
Additionally, the mutual information between the random variable $G$ and the query responses is $\leq p$.
If the algorithm is randomized, the mutual information here is conditioned on the randomness used by the algorithm.
\begin{proof}
Consider the set of vertices in $C$ which correspond to an original vertex $(i,j)$ with $i=0$ or $i=L-1$. In an overload of notation, let us also call this set $\Delta$.
Consider some classical algorithm.
If we apply this algorithm to graph $D$ or graph $C$, in the modified query model the two algorithms have the same probability distribution of query responses conditioned on the case that the algorithm applied to $D$ does not reach $\Delta$.
Hence, it is $(p,q)$-hard to reach the set $\Delta$ in $C$.
So, for graph $D$, the probability distribution of query responses is $(1-p) \sigma+p \tau$ for some probability distribution $\tau$: the first term in the sum is the case conditioned on not reaching $\Delta$ and the second is the case when it reaches $\Delta$.
For $C$, the probability distribution of query responses is $(1-p)\sigma+p\mu$ for some probability distribution $\mu$.
The $\ell_1$ distance between the two probability distributions is at most $2p$. So, the probability that the algorithm guesses right is at most $1/2+p/2$ (to see this, let $P,P'$ be two probability distributions on some set of events; let $e$ label events; choose $P$ or $P'$ with probability $1/2$, then observe some event given the probability distribution; let $A_e=1$ if one choose $P$ on event $e$ and $A_e=0$ otherwise; then the probability of guessing right is $\sum_e A_e P_e/2 - \sum_e (1-A_e) P'_e/2=1/2+\sum_e (A_e-1/2) (P_e-P'_e)/2$ and $\sum_e (A_e-1/2) (P_e-P'_e)/2\leq |P-P'|_1/4$).
The claim about the mutual information follows because the optimal case for the mutual information is when the support of $\sigma,\tau,\mu$ are all disjoint.
\end{proof}
\end{lemma}
We now need a couple more definitions.
First,
\begin{definition}
We will assign a
number called the {\it level} to each vertex $v$ of any graph $G_k$ defined by some sequence of decorations of a graph as follows.
Decoration gives a sequence of graphs $G_0,G_1,\ldots,G_k$. For $j<k$, we may regard $G_j$ as an induced subgraph of $G_k$.
The level of a vertex $v\in G_k$ is equal to the smallest $j$ such that $v\in G_j$.
Hence, the original vertices of $G_k$ are those with level $0$.
\end{definition}
Second, consider the problem of tossing a coin which is heads with probability $1/2$ and tails with probability $1/2$.
Define $P_{coin}(n,N)$ to be the probability that one observes $n$ heads after at most $N$ tosses of the coin.
This is the same as the probability that, after tossing the coin $N$ times, one has observed at least $n$ heads.
Thus,
$$P_{coin}(n,N)=\sum_{m=n}^N 2^{-N} {N \choose m}.$$
Now we give a lemma with two parts. This first part is an inductive lemma that implies difficulty of reaching leaves of a graph obtained by decorating a tree $T_{n,h}$. The second part is used to prove difficulty of reaching $\Delta$; both parts have very similar proofs and in applications we will first apply the first part inductively for several height sequences and then use the result as an input to the second part of the lemma.
Throughout, we will use $S_{leaf}$ to refer to a set of leafs in a tree graph; this will be the set of terminal vertices.
Now we show:
\begin{lemma}
\label{inductivelemma}
Assume that for some $n,h_0,h_1,\ldots,h_k$, it is $(P,Q)$-hard to reach $S_{leaf}$ in graph $G$
given by decorating
$T_{2n,h_0}$ by height sequence $h_1,\ldots,h_k$.
Then:
\begin{itemize}
\item[{\bf 1.}]
For any integers $M,H\geq 0$, it is $(P',Q')$-hard
to reach $S_{leaf}$ in graph $G'$ given by decorating
$G=T_{n,H}$ with height sequence $h_0,h_1,\ldots,h_k$,
where
\begin{equation}
\label{precursion}
P'=P_{coin}(H-h_0,M)+Q'P,
\end{equation}
and
\begin{equation}
Q'=(M-(H-h_0)) Q.
\end{equation}
\item[{\bf 2.}] For any integers $M,L\geq 0$, it is $(P',Q')$-hard to reach $\Delta$ in graph $D$ given by decorating $P_{L,(n+1)/2}$ with height sequence
$h_0,h_1,\ldots,h_k$ where
\begin{equation}
\label{precursion2}
P'=P_{coin}(L/4-h_0-O(1),M)+Q'P,
\end{equation}
and
\begin{equation}
Q'=(M-(H-h_0)) Q.
\end{equation}
\end{itemize}
\begin{proof}
We prove the first claim first.
After $q$ queries by the algorithm, we can describe the queries by a tree $T(q)$ with $q$ edges, each vertex of which corresponds to some vertex in $G'$. We use letters $a,b,\ldots$ to denote vertices in this tree $T(q)$ and use $v,w,\ldots$ to denote vertices in $G'$.
The root of the tree $T(q)$ corresponds to the start vertex. If some vertex $a\in T(q)$ corresponds to some $v\in G'$, then the daughters of $a$ correspond to neighbors of $v$ obtained by querying $v$.
Given an $a\in T(q)$, we will say it has some given level if the corresponding vertex in $G'$ has that level.
For any vertex $a\in T(q)$ or any $v\in G'$, we say that the {\it subtree} of $a$ (respectively, $v$) is the tree consisting of $a$ (respectively, $v$) and all its descendants. ``Queries of a subtree" mean queries in the modified query model starting from the root of that subtree.
Define a subtree $S(q)$ of $T(q)$. $S(q)$ will be the induced subgraph whose vertices consist of the root of $T(q)$ and of all other vertices $a$ of $T(q)$ which have been queried at least $Q$ times and such that the parent of the given vertex $a$ is at level $0$.
Suppose after some number of queries $q$, some new vertex $a$ is added to $S(q)$. We will now consider the probability distribution of the level of $a$, conditioned on the level being $0$ or $1$.
Let $a$ correspond to vertex $v$ of $G'$.
Suppose that
$v$ has distance at most $H-h_0$ from the root of $G'$ so that $S_{leaf}$ in $G'$ is distance at least $h_0$ from $v$.
We will say that a vertex in $G'$ has property $(\dagger)$ if these three conditions hold:
it is distance $\leq H-h_0$ from the root of $G'$, and is level $0$ or $1$, and is the
the child of a vertex with level $0$.
For a $v$ with property $(\dagger)$, we will say that
event (*) occurs for that $v$ if we reach a vertex in the subtree of $v$ which is level $0$ or $1$ and distance $\geq h_0$ from $v$ with fewer than $Q$ queries in that subtree.
If $v$ has level $1$, then
the responses to queries of the subtree of $v$ are the same as in the
given query model on graph $G$ given by
decorating $T_{2n',h_0}$ by height sequence $h_1,\ldots,h_k$.
In that case, event (*) occurs iff we reach $S_{leaf}$ in $G$ in fewer than $Q$ queries; this probability
is bounded by $P$ by the inductive assumption.
If instead
$v$ has level $0$, then since by assumption $S_{leaf}$ in $G'$ is distance at least $h_0$ from $v$, the probability of
event (*) occurring for that $v$ is also bounded by $P$.
The key point here is that if $v$ has level $0$ and we consider the subtree of $v$, and then further consider the subgraph of that subtree consisting of vertices of distance $\leq h_0$ from $v$, this subgraph is isomorphic to $G$.
Now, let us condition on event (*) {\it not} occurring for {\it any} $v$. At the end of the proof of the lemma, we will upper bound the probability of event (*) occurring.
If event (*) does not occur, then the distribution of queries responses in the subtree of $v$ is the same whether $v$ has level $0$ or $1$.
So, when vertex $a$ is added to $S(q)$, it has probability $1/2$ of being $0$, conditioned on the level being $0$ or $1$, i.e.,
we toss an unbiased coin to determine the level of that vertex: ``heads" corresponds to level $0$ and ``tails" corresponds to level $1$. The level may also be $>1$, but including that possibility only increases the number of queries.
The number of queries $q$ is at least equal to $Q$ times the number of ``tails" that have occurred.
Remark:
``heads" also implies that there were at least $Q$ queries of the subtree of some vertex, but if we include those queries due to ``heads" in the total number, we must be careful to avoid overcounting
as those $Q$ queries of the subtree of some vertex $v_i$ will also give some number of queries (up to $Q-1$) of descendants of $v_i$.
So, for simplicity, we will use the lower bound that the number of queries is at least $Q$ times the number of tails.
Starting from the root of $G'$, to reach $S_{leaf}$ in $G'$ with at most $q$ queries, we must have one of these two possibilities: {\bf (1)} subtree $S(q)$ contains some vertex at level $0$ with distance $\geq H-h_0$ from the root;
or {\bf (2)} we reach $S_{leaf}$ in $G'$ in fewer than $Q$ queries starting with some vertex $v$ of level $0$ and distance $\leq H-h_0$ from the root, i.e., we reach $S_{leaf}$ in $G'$ in the subtree of such a vertex $v$ with fewer than $Q$ queries in that subtree.
Conditioned on event (*) not occurring,
the probability of {\bf 1} occurring in at most $(M-(H-h_0))Q$ queries s bounded by the probability of having at least $H-h_0$ heads out of $M$ coin tosses and so is bounded by
$P_{coin}(H-h_0,M)$.
If event {\bf 2} happens, then event (*) happens.
By a union bound, since there are only $Q'$ vertices that we query, event (*) happens with probability at most $Q' P$.
So, by a union bound, the probability of reaching $S_{leaf}$ in $G'$ with at most $QM$ queries is bounded by
$P_{coin}(H-h_0,M)+Q'P$.
Remark: likely the union bound in the previous paragraph on the probability of event (*) could be tightened.
If we query a vertex at distance $H-h_0$ from the root then indeed the probability of reaching $S_{leaf}$ is bounded by $P$ but for vertices of lower distance from the root the probability is less as would follow from a better inductive assumption. We will not need this tightening so we omit it.
Having proved the first claim, the proof of the second claim is almost identical.
We use two new ideas.
The set $\Delta$ is at distance $L/2-O(1)$ from the start vertex of $D$. So, to reach $\Delta$ one must at some time query some vertex $v$ which is at level $0$ and at distance $\lfloor L/4 \rfloor $ from the start vertex of $D$ such that in the subtree of $v$ one then queries a vertex in $\Delta$.
This introduction of vertex $v$ is the first new idea: by choosing such a vertex $v$ which is far from the start vertex, we will be able to ignore the possibility that the algorithm gets extra information about which vertex is the start vertex in response to queries by considering
only vertices near $v$.
Indeed, consider queries in the subtree of $v$.
Consider the set of vertices of $D$ including all vertices of level $\geq 1$ and all vertices of level $0$ which are
distance $L/4-O(1)$ so that this set does not intersect $\Delta$ and does not contain the start vertex.
Let $D'$ be the subgraph of $D$ induced by this set of vertices.
In the modified query model one cannot distinguish $D'$ from its universal cover $\tilde D'$, which is a tree graph; this is the second new idea.
Take the root of this tree graph to have its image under the covering map be $v$.
Then, responses to queries on this tree graph are the same as almost the same in queries on the graph given by decorating $T_{n,L/4-O(1)}$
$h_0,h_1,\ldots,h_k$. The only minor difference is that we may take the first query to be nonbacktracking, so that rather than decorating graph $T_{n,L/4-O(1)}$ we decorate an $n$-ary tree of depth $L/4-O(1)$, i.e., there are $n$ rather than $n+1$ new neighbors in response to the first query.
To reach $\Delta$, one must reach some vertex which is at level $0$ and at distance $L/4-O(1)$ from $v$.
Then, the rest of the proof is the same.
\end{proof}
\end{lemma}
\section{Choice of Height Sequence and Proof of Main Theorem}
\label{chs}
We now make specific choices of the height sequence to prove \cref{mainth}.
We pick $n=N^8$ in the construction of $C,D$. The value of $n$ does not matter for the classical lower bounds that follow from \cref{inductivelemma}. However, the choice of $n$ does affect the spectrum of the quantum Hamiltonian.
We pick $L=4(l+1) N+O(1)$.
Thus $n/L^2=\Theta(N^6/l^2)$.
We decorate with height sequence $h_1,\ldots,h_l$, choosing $l=\lfloor \log_2(N)\rfloor $.
Thus, the graph $D$ is $d$-inner regular, with $d=\Theta(N^9)$.
We pick $$h_k=N \cdot (l+1-k),$$
for that $h_{k-1}-h_k=N$ and $L/4-O(1)-h_1=N$.
We first quickly show items {\bf 1-3,5} of \cref{trq} before showing the harder result, item {\bf 4}. Then, \cref{mainth} follows from \cref{blemma}.
Note that $n/L^2\gg\sqrt{d}$. So,
from \cref{eigbounddec}, it follows that the
difference between the ground state energy of $C$ and $D$ is $\Theta(N/L^2))$ and so item {\bf 1} of \cref{trq} is satisfied.
Again since $n/L^2\gg\sqrt{d}$, from the bound on the second largest eigenvalue of the adjacency matrix of $C$ in \cref{eigbounddec}, we satisfy the condition on the spectral gap of the Hamiltonian of $C$, item {\bf 2} of \cref{trq}.
Item {\bf 3} of \cref{trq} is also satisfied by the last result in \cref{eigbounddec}, since there are only $O(N)$ original vertices and the amplitude of the ground state wavefunction is the same on all of them.
Item {\bf 5} of \cref{trq} trivially follows. The number of vertices in $G_0$ is $\leq {\rm poly}(N)$. Decoration by an $m$-ary tree of height $h$ multiplies the number of vertices in the graph by $O(m^h)$. All $h_k$ are $O(N \log(N))$, and the largest $m$ is $O({\rm poly}(N))$, and there are only $O(\log(N))$ steps of decoration, so the total number of vertices is $\exp(O(N \log(N)^3))$.
Remark: since the construction of an admissible path in \cref{distg} uses $N$ copies of the graph, the number of computational basis states needed is $\exp(O(N^2 \log(N^3))$. This number can be reduced somewhat since we could choose smaller values of $h_k$, such as $h_k=N^\alpha \cdot (l+1-k)$ for some smaller $\alpha<1$; we omit this.
To prove item {\bf 4}, we use \cref{inductivelemma}.
We first use the first part of the lemma.
Consider a sequence of graphs $J_l,J_{l-1},J_{l-2},\ldots,J_1$, where $J_k$ is given by decorating
$T_{n_k,h_k}$ by height sequence $h_{k+1},h_{k+2},\ldots,h_l$, for $n_k=n \cdot 2^k$.
Clearly, for $J_l$, it is $(0,N-1)$-hard to reach $S_{leaf}$, simply because the tree has height $N$.
We apply
\cref{inductivelemma} to use $(P_{k+1},Q_{k+1})$-hardness on $J_{k+1}$ to show $(P_{k},Q_{k})$-hardness on $J_{k}$.
For all $k$, we pick $M=(4/3)N$, so that
\begin{equation}
\label{Qk}
Q_k=(N/3)^{l-k} N.
\end{equation}
Then $P_{coin}=\exp(-\Omega(N))$ and
\cref{precursion}
gives that $P_{k-1}\leq \exp(-\Omega(N))+N P_k$.
Hence
\begin{eqnarray}
P_k &\leq & \exp(-\Omega(N))+Q_k P_{k+1} \\ \nonumber
&\leq & \exp(-\Omega(N))+Q_k \exp(-\Omega(N))+Q_k Q_{k+1} P_{k+2} \\ \nonumber
& \leq & \ldots \\ \nonumber
&\leq & \exp(-\Omega(N)) (1+Q_k+Q_k Q_{k+1}+\ldots) \\ \nonumber
&\leq & \exp(-\Omega(N)) O(Q_k^l) \\ \nonumber
& \leq & \exp(-\Omega(N)) \exp(O(\log(N)^3)) \\ \nonumber
& \leq & \exp(-\Omega(N)).
\end{eqnarray}
It is $(P,Q)$-hard to reach $S_{leaf}$ in $J_1$ with
\begin{equation}
P=\exp(-\Omega(N)),
\end{equation}
and
\begin{equation}
Q=\exp(\Theta(\log(N)^2)).
\end{equation}
Then, use the second part of the lemma, with the same $M$, to show
\begin{lemma}
\label{hardnesslemma}
It is $(P,Q)$-hard to reach $\Delta$ in $D$ with
\begin{equation}
P=\exp(-\Omega(N)),
\end{equation}
and
\begin{equation}
Q=\exp(\Theta(\log(N)^2)).
\end{equation}
\end{lemma}
This completes the proof of \cref{mainth}.
We make one final remark on the Hamiltonian on these decorated graphs. For the given parameters, the ground state wavefunction on $C$ has most of its probability (its $\ell_2$ norm) on the original vertices of $C$. However, the $\ell_1$ norm is concentrated near the terminal vertices. The distinction between $\ell_1$ and $\ell_2$ norm was used in \cite{obs} to ``pin" the worldline in the case of path integral Monte Carlo with open boundary conditions and was considered in \cite{Jarret_2016} as an obstruction for diffusion Monte Carlo methods.
The large $\ell_2$ norm of the ground state wavefunction on $C$ can be regarded as arising from all the short cycles on $C$; if we replace $C$ with a (finite) tree with the same degree, then ground state energy on $C$ shifts and the $\ell_2$ norm of the ground state becomes concentrated instead near the terminal vertices. Thus, one may say that the topological obstructions are related to the $\ell_1$ versus $\ell_2$ obstructions. The idea of our construction here is to make it so that no classical algorithm can (once one is far from the start vertex) efficiently distinguish original vertices from other vertices, since it cannot detect the short cycles between the original vertices and so it is unable to determine that the $\ell_2$ norm should be larger.
\section{Linear Paths}
\label{linear}
We have considered a path $H_s$ given by an oracle and we have implemented some complicated $s$-dependent terms so that certain terms increase and then later decrease over the path. One might be interested in the case of linear interpolation so that $H_s=(1-s) H_0+sH_1$ for some fixed and known $H_0$, such as a transverse field, with $H_1$ being a diagonal matrix given by an oracle. It seems likely that the construction here could be adapted to show hardness in that case too, using some perturbative gadgets. Since it is not too interesting a question, we only sketch how this might be done.
One might induce dynamics on an unknown graph (either $C$ or $D$) as follows.
Both graphs have the same vertex set but have different edge sets. Define a new graph $E$ which has one vertex for each vertex in $C$ and one vertex for each edge in $C$. Given a pair of vertices $v,w$ connected by an edge $e$ in $C$, the graph $E$ will have
an edge between the vertex corresponding to $v$ and the vertex corresponding to $e$, as well as an edge between the
vertex corresponding to $w$ and the vertex corresponding to $e$. We add also a diagonal term to the Hamiltonian on graph $E$ which assigns some positive energy to all vertices which correspond to edges of $C$. If all these diagonal terms are the same and are chosen appropriately, we have a perturbative gadget which gives us an effective Hamiltonian corresponding to the Hamiltonian of graph $C$, up to an overall multiplicative scalar. On the other hand, we could increase the diagonal term on vertices which correspond to edges of $C$ which are not edges of $D$ to effectively induce the Hamiltonian on $D$.
Also, in our construction of a path in \cref{distg} we used the ability to turn on and off a term $t|0\rangle\langle s| + h.c.$ in the Hamiltonian. Such a term it seems can be induced by some perturbative gadgets also. First consider a slightly more general case where $H_0=(1-s) H_0 + H^{diag}_s$ where $H_0$ is fixed, known term and $H^{diag}_s$ is a diagonal matrix depending on $s$ and given by an oracle. Then, we can effectively induce a $t$ which depends on $s$ by using hopping between an intermediate state, i.e., $-(|0\rangle\langle {\rm int}| + |{\rm int}\rangle\langle s|)+h.c.$ where $|{\rm int}\rangle$ is some intermediate state. Then adding an $s$-dependent diagonal term on $|{\rm int}\rangle$ can be used to turn on or off the effective hopping between $|0\rangle$ and $|s\rangle$.
To effectively induce this $s$-dependent diagonal term on $|{\rm int}\rangle$ we could use another trick: replace every basis state with some set of basis states of size ${\rm poly}(N)$ and adding hopping terms between every pair of basis states in each such set.
Then, each such set defines a subspace.
Further add some diagonal terms on some of the basis states in each set, so that, for example, one set might include $n_0$ states with energy $E_0$, some number $n_1\gg n_0$ of additional states with energy $E_1\gg E_0$, and so on. Then, if $s$ is close to $1$ so that $(1-s)H_0$ is small, almost of the amplitude will be in the states with energy $0$ and we can treat that set of states as a single state with energy $E_0-(1-s) (n_0-1)$; for slightly smaller $s$, we will start to occupy higher energy states and for some choices of sequences $n,E$ we can treat that set approximately as a single state with energy
$\approx E_1-(1-s) (n_0+n_1-1)$, and so on.
By adjusting the sequences of $n,E$ on each set, it seems that we can
effectively implement a problem with fairly complicated $s$-dependent diagonal terms.
\bibliographystyle{unsrturl}
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1,314,259,996,490 | arxiv | \part{\partial}
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\def\sgn{\text{\rm sgn}}
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\begin{document}
\pagestyle{plain}
\thispagestyle{plain}
\title[Chain level loop bracket and pseudo-holomorphic disks]
{Chain level loop bracket and pseudo-holomorphic disks}
\author[Kei Irie]{Kei Irie}
\address{Research Institute for Mathematical Sciences, Kyoto University,
Kyoto 606-8502, Japan
/
Simons Center for Geometry and Physics, State University of New York,
Stony Brook NY 11794-3636, USA (on visit)}
\email{iriek@kurims.kyoto-u.ac.jp}
\subjclass[2010]{53D12, 53D40, 55P50}
\date{\today}
\begin{abstract}
Let $L$ be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin.
Using virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks with boundaries on $L$,
one can define a Maurer-Cartan element of a Lie bracket operation in string topology (the loop bracket) defined at chain level.
This observation is due to Fukaya, who also pointed out its important consequences in symplectic topology.
The goal of this paper is to work out details of this observation.
Our argument is based on a string topology chain model previously introduced by the author,
and the theory of Kuranishi structures on moduli spaces of pseudo-holomorphic disks,
which has been developed by Fukaya-Oh-Ohta-Ono.
\end{abstract}
\maketitle
\section{Introduction}
The study of Lagrangian submanifolds is one of central topics in symplectic topology.
In the monumental paper \cite{Gromov_pseudoholomorphic},
Gromov proved that the first Betti number of
a closed Lagrangian submanifold in $\C^n$ (with the standard symplectic structure) is nonzero,
using moduli spaces of (perturbed) pseudo-holomorphic disks with boundaries on the Lagrangian submanifold.
On the other hand, string topology is the study of algebraic structures on (homology of) loop spaces,
introduced by Chas-Sullivan \cite{ChSu_99}.
In this paper we discuss an application of \textit{chain level} string topology operations to the pseudo-holomorphic curve theory in symplectic topology.
This idea is due to Fukaya \cite{Fukaya_06}, who also pointed out its important consequences,
including a proof of Audin's conjecture for (closed, oriented and spin) aspherical Lagrangian submanifolds in $\C^n$,
and a complete classification of
orientable, closed, prime three-manifolds admitting Lagrangian embeddings into $\C^3$.
Let us briefly sketch the key argument in \cite{Fukaya_06}.
Let $L$ be a closed, oriented and spin Lagrangian submanifold in $\C^n$, and
$\mca{L}L:= C^\infty(S^1, L)$ be the space of free loops on $L$.
Let $D:= \{ z \in \C \mid |z| \le 1\}$,
and let $\mca{M}$ denote the (compactified) moduli space of nonconstant holomorphic maps
$(D, \partial D) \to (\C^n, L)$ modulo $\text{Aut}(D, 1)$.
Then one can define a map $\ev_{\mca{M}}: \mca{M} \to \mca{L}L$ by $\ev_{\mca{M}}(u):= u|_{\partial D}$
(strictly speaking, this ``definition'' has an ambiguity up to parametrizations of loops, however we omit this issue for the moment).
Considering virtual fundamental chain of the moduli space $\mca{M}$,
the pair $(\mca{M}, \ev_{\mca{M}})$ defines a ``chain'' $x \in \mca{C}_*(\mca{L}L)$.
Here $\mca{C}_*(\mca{L}L)$ denotes the complex of ``chains'' on $\mca{L}L$,
on which the \textit{loop bracket} is defined and makes $\mca{C}_*(\mca{L}L)$ a dg Lie algebra.
Since the codimension $1$ boundary of $\mca{M}$ consists of configurations of two disks glued at a point, one sees that
the chain $x$ satisfies the \textit{Maurer-Cartan equation}
\begin{equation}\label{170627_4}
\partial x - \frac{1}{2} [x,x] = 0
\end{equation}
where $[\, , \,]$ denotes the loop bracket defined at \textit{chain level}.
Nextly, we take a time-dependent Hamiltonian $H$ on $\C^n$ which displaces $L$,
and define a moduli space $\mca{N}$ which consists of solutions of the Cauchy-Riemann equation perturbed by $H$.
Then, the associated chain $y:= (\mca{N}, \ev_{\mca{N}}) \in \mca{C}_*(\mca{L} L)$ satisfies
\begin{equation}\label{170627_5}
\partial y - [x, y] = z
\end{equation}
where $z$ is another chain whose ``symplectic area zero part''
is a cycle representing the fundamental class of $L$.
Once we obtain chains $x$, $y$, $z$ satisfying equations (\ref{170627_4}) and (\ref{170627_5}),
using the homotopy transfer theorem for $L_\infty$-algebras,
one can formulate an equivalent result on homology of the free loop space (Theorem \ref{161011_1}).
This result has the following remarkable consequences:
\begin{itemize}
\item[(i):]
If $L$ is aspherical, then
there exists $a \in H_1(L: \Z)$ with Maslov index $2$ and positive symplectic action
(Corollary \ref{170421_1}).
\item[(ii):]
If $n=3$ and $L$ is prime as a three-manifold, then $L$ is diffeomorphic to $S^1$ times a closed surface
(Corollary \ref{170622_1}).
\end{itemize}
(i) in particular confirms Audin's conjecture for Lagrangian tori in $\C^n$,
and (ii) gives
a complete classification of
orientable, closed, prime three-manifolds admitting Lagrangian embeddings into $\C^3$;
see Section 3 for details and previous related works.
In the above argument,
$\mca{C}_*(\mca{L}L)$
denotes the complex of ``chains'' on $\mca{L}L$,
on which the loop bracket is defined and makes it a dg Lie algebra.
It is a highly nontrivial technical problem to define such chain models of the free loop space, in particular those compatible with virtual techniques in the pseudo-holomorphic curve theory.
Partly due to this issue,
in spite of the importance of its consequences,
full details of the above argument have not been available so far.
In \cite{Irie_17}, the author developed foundations for part of chain level algebraic structures (specifically, Batalin-Vilkovisky structure) in string topology,
using de Rham chains on spaces of Moore loops with arbitrarily many marked points.
The goal of this paper is to combine techniques from \cite{Irie_17} with the theory of Kuranishi structures \cite{FOOO_Kuranishi}
to work out details of the argument sketched above.
Now we describe the structure of this paper.
The goal of the first part (Sections 2--6) is to state the main result and introduce our setup in string topology.
Section 2 explains some preliminaries on $L_\infty$-algebras, in particular the homotopy transfer theorem.
Section 3 states the main result (Theorem \ref{161011_1}) on homology of the free loop space.
We also recall a few applications in symplectic topology from \cite{Fukaya_06}.
In Sections 4--6, we introduce our setup in string topology, following \cite{Irie_17} with minor modifications,
and reduce Theorem \ref{161011_1} to a chain-level statement (Theorem \ref{161215_1}).
Further details will be explained in the last paragraph of Section 3.
The goal of the second part (Sections 7--10) is to prove Theorem \ref{161215_1}.
The plan of the proof will be explained at the beginning of Section 7.
Our proof uses the theory of Kuranishi structures on moduli spaces of (perturbed) pseudo-holomorphic disks.
In particular, our arguments heavily rely on \cite{FOOO_Kuranishi} by Fukaya-Oh-Ohta-Ono.
Section 10 very briefly explains some notions in the theory of Kuranishi structures, mainly to fix notations.
\textbf{Conventions.}
Throughout this paper all manifolds are assumed to be of $C^\infty$.
All vector spaces are over $\R$, unless otherwise specified.
\textbf{Acknowledgements.}
The author appreciates Kenji Fukaya for sharing his time and insights into virtual techniques in
the pseudo-holomorphic curve theory,
and his comments on an early version of this paper.
The author also appreciates the Simons Center for Geometry and Physics for a great work environment.
This work is supported by JSPS Postdoctoral Fellowship for Research Abroad.
\section {Preliminaries on $L_\infty$-algebras}
We briefly recall basics of $L_\infty$-algebras, partially following \cite{Latschev_15}.
\subsection{Bar construction}
Let $C = \bigoplus_{i \in \Z} C_i$ be a $\Z\,$-graded vector space.
For every integer $k \ge 1$,
let $\sym_k$ denote the $k$-th symmetric group,
and
let us define an $\sym_k$\,-action on $C^{\otimes k}$ by
\[
\rho \cdot (c_1 \otimes \cdots \otimes c_k) := \ep(\rho: c_1, \ldots, c_k) \cdot c_{\rho(1)} \otimes \cdots \otimes c_{\rho(k)}
\]
where $\ep(\rho: c_1, \ldots, c_k):= \prod_{\substack{i<j \\ \rho(i)>\rho(j)}} (-1)^{|c_i||c_j|}$.
Let $S^kC$ denote the quotient of $C^{\otimes k}$ by the $\sym_k$-action,
and set $SC:= \bigoplus_{1 \le k \le \infty} S^kC$.
We define a coproduct $\Delta: SC \to SC^{\otimes 2}$ by
\[
\Delta(c_1 \ldots c_k) := \sum_{\substack{k_1+k_2 = k \\ \rho \in \sym_k}} \frac{\ep(\rho:c_1,\ldots,c_k)}{k_1!k_2!} \cdot c_{\rho(1)} \cdots c_{\rho(k_1)} \otimes c_{\rho(k_1+1)} \cdots c_{\rho(k)}.
\]
Then $\Delta$ is coassociative, namely $(1 \otimes \Delta) \circ \Delta = (\Delta \otimes 1) \circ \Delta$.
We denote $\Delta$ by $\Delta_C$ when we need to specify $C$.
\subsection{$L_\infty$-algebras and $L_\infty$-homomorphisms}
For any $\Z$-graded vector space $C$ and $n \in \Z$,
we define a shifted complex $C[n]$ by
$C[n]_d:= C_{n+d} \,(\forall d \in \Z)$.
\begin{defn}
\begin{enumerate}
\item[(i):]
An $L_\infty$-algebra is a pair of a graded vector space $C$ and
a linear map $l: S(C[-1]) \to S(C[-1])$ such that
\begin{itemize}
\item $|l|=-1$.
\item $l$ is a coderivation; $\Delta \circ l = (l \otimes 1 + 1 \otimes l) \circ \Delta$.
\item $l^2=0$.
\end{itemize}
For each integer $k \ge 1$, we define $l_k: S^k(C[-1])\to C[-1]$ by
$l_k:= \pr_1 \circ l|_{S^k(C[-1])} $, where
$\pr_1: S(C[-1]) \to S^1(C[-1]) \cong C[-1]$
denotes the projection.
\item[(ii):]
Let $(C, l)$ and $(C', l')$ be $L_\infty$-algebras.
An $L_\infty$-homomorphism from $(C, l)$ to $(C', l')$ is a linear map
$f: S(C[-1]) \to S(C'[-1]) $ such that
\begin{itemize}
\item $|f|=0$.
\item $f$ is a coalgebra map; $\Delta_{C'[-1]} \circ f = (f \otimes f) \circ \Delta_{C[-1]}$.
\item $\l' \circ f = f \circ l$.
\end{itemize}
For each integer $k \ge 1$, we define $f_k: S^k(C[-1]) \to C'[-1]$ by
$f_k:= \pr_1 \circ f|_{S^k(C[-1]) }$.
\end{enumerate}
\end{defn}
Here is another definition of $L_\infty$-algebras and $L_\infty$-homomorphisms using exterior products.
For each integer $k \ge 1$, let $\Lambda^k C$ denote the quotient of $C^{\otimes k}$ by the $\sym_k$-action defined by
\[
\rho \cdot (c_1 \otimes \cdots \otimes c_k) : = \sgn(\rho) \cdot \ep(\rho: c_1, \ldots, c_k) \cdot c_{\rho(1)} \otimes \cdots \otimes c_{\rho(k)}.
\]
Then there exists a natural isomorphism
\[
\sigma_k: (\Lambda^k C)[-k] \to S^k (C[-1]); \quad c_1 \wedge \cdots \wedge c_k \mapsto (-1)^{ \sum_i (k-i) |c_i|} c_1 \cdots c_k.
\]
We define $\lambda_k: \Lambda^k C \to C$ by $\lambda_k:= \sigma_1^{-1} \circ l_k \circ \sigma_k$.
Then one can define an $L_\infty$-structure on $C$ as a sequence $(\lambda_k)_{k \ge 1}$ such that
each $\lambda_k: \Lambda^k C \to C$ is of degree $k-2$ and satisfies the equation
\[
\sum_{\substack{k_1+k_2 = k+1 \\ \rho \in \sym_k}} \pm \frac{1}{k_1 ! (k-k_1)!} \lambda_{k_2} ( \lambda_{k_1}(c_{\rho(1)} \wedge \cdots \wedge c_{\rho(k_1)}) \wedge c_{\rho(k_1+1)} \wedge \cdots \wedge c_{\rho(k)}) = 0
\]
where $\pm$ stands for appropriate signs.
Similarly, one can define an $L_\infty$-homomorphism from $(C, (\lambda_k)_k)$ to $(C', (\lambda'_k)_k)$ as a sequence $(\ph_k)_{k \ge 1}$ such that
each $\ph_k: \Lambda^k C \to C'$ is of degree $k-1$ and satisfies the equation
\begin{align*}
&\sum_{\substack{k_1+k_2=k+1 \\ \rho \in \sym_k}} \pm \frac{1}{k_2! (k_1-1)!} \cdot \ph_{k_1} (\lambda_{k_2} (c_{\rho(1)} \wedge \cdots \wedge c_{\rho(k_2)}) \wedge c_{\rho(k_2+1)} \wedge \cdots \wedge c_{\rho(k)}) \\
= &\sum_{\substack{k_1+\cdots +k_r = k \\ \rho \in \sym_k}} \pm \frac{1}{r! k_1! \cdots k_r!} \cdot \lambda'_r ( \ph_{k_1}(c_{\rho(1)} \wedge \cdots \wedge c_{\rho(k_1)}) \wedge \cdots \wedge \ph_{k_r}(c_{\rho(k-k_r+1)} \wedge \cdots \wedge c_{\rho(k)})) \\
\end{align*}
where $\pm$ stands for appropriate signs.
\begin{rem}\label{170829_1}
For later purposes we need to specify signs for dg Lie algebras.
A dg Lie algebra is an $L_\infty$-algebra such that $\lambda_k = 0$ for every $k \ge 3$.
Setting $\partial x := \lambda_1(x)$ and $[x,y]:= \lambda_2(x, y)$, there holds
\begin{align*}
& [x,y] + (-1)^{|x||y|} [y,x] = 0, \\
& \partial [x, y] = [\partial x, y] + (-1)^{|x|} [x, \partial y], \\
& [x, [y,z]] + (-1)^{|x|(|y|+|z|)} [y, [z,x]] + (-1)^{|z|(|x|+|y|)} [z, [x,y]] = 0.
\end{align*}
\end{rem}
We also introduce the following notions for later purposes.
\begin{defn}
\begin{enumerate}
\item[(i):] Let $V$ be a vector space and $A$ be a commutative semigroup.
A decomposition of $V$ over $A$ is a decomposition $V = \bigoplus_{a \in A} V(a)$,
where each $V(a)$ is a subspace of $V$,
and $V(a) + V(a') \subset V(a+a')$ for every $a, a' \in A$.
\item[(ii):] Let $V$ and $W$ be vector spaces with decompositions over $A$.
A linear map $f: V^{\otimes k} \to W$ respects these decompositions if
\[
f (V(a_1) \otimes \cdots \otimes V(a_k)) \subset W(a_1+ \cdots + a_k)
\]
for every $a_1, \ldots, a_k \in A$.
\item[(iii):] An $L_\infty$-algebra structure
$l = (l_k)_k$ on $V$ respects the decomposition of $V$
if $l_k$ respects the decomposition for every $k \ge 1$.
\end{enumerate}
\end{defn}
\subsection{Homotopy transfer theorem}
Finally we state the homotopy transfer theorem for $L_\infty$-algebras.
The proof is only sketched since it is now standard
(perhaps goes back to the paper by Kadeishvili \cite{Kadeishvili} on $A_\infty$-algebras).
\begin{thm}\label{170623_2}
Let $(C, l)$ be an $L_\infty$-algebra, and
$H(C) := \Ker l_1/ \Image l_1$.
Suppose that there are linear maps
\[
\iota: H_*(C) \to C_*, \quad
\pi: C_* \to H_*(C), \quad
\kappa: C_* \to C_{*+1}
\]
such that
\[
l_1 \circ \iota = 0, \quad
\pi \circ l_1 = 0, \quad
\pi \circ \iota = \id_{H(C)}, \quad
\id_C - \iota \circ \pi = l_1 \circ \kappa + \kappa \circ l_1.
\]
Then there exist
an $L_\infty$-algebra structure $l^H$ on $H(C)$
and an $L_\infty$-homomorphism $p: (C, l) \to (H(C), l^H)$,
such that $l^H_1 = 0$ and $p_1 = \pi$.
When $C$ has a decomposition over a commutative semigroup $A$
(thus $H(C)$ also has a decomposition over $A$),
and linear maps $\iota$, $\pi$ and $\kappa$ respect these decompositions,
then one can take $l^H$ and $p$ so that they respect the decompositions over $A$.
\end{thm}
\begin{proof}
It is sufficient to define sequences $(l^H_{\le k})_{k \ge 1}$ and $(p_{\le k})_{k \ge 1}$ satisfying the following conditions:
\begin{itemize}
\item $p_{\le k}$ is a coalgebra map of degree $0$ from $S(C[-1])$ to $S(H[-1])$ which respects decompositions over $A$.
\item $l^H_{\le k}$ is a coderivation of degree $-1$ from $S(H[-1])$ to $S(H[-1])$ which respects decompositions over $A$.
\item $l^H_{\le k} \circ l^H_{\le k} = 0$ on $S^{\le k+1} (H[-1]) := \bigoplus_{1 \le i \le k+1} S^i(H[-1])$.
\item $l^H_{\le k} \circ p_{\le k} - p_{\le k} \circ l = 0$ on $S^{\le k} (C[-1]) := \bigoplus_{1 \le i \le k} S^i(C[-1])$.
\item $p_{\le k} = p_{\le k+1}$ on $S^{\le k} (C[-1])$.
\item $l^H_{\le k} = l^H_{\le k+1}$ on $S^{\le k}(H[-1])$.
\item $p_{\le 1} = \pi$, $l^H_{\le 1} =0$.
\end{itemize}
Once we obtain these sequences, the limits
$p:= \lim_{k \to \infty} p_{\le k}$ and
$l^H:= \lim_{k \to \infty} l^H_{\le k}$
satisfy the conditions in the theorem.
We can define $(l^H_{\le k})_{k \ge 1}$ and $(p_{\le k})_{k \ge 1}$ by upward induction on $k$.
For $k=1$, $p_{\le 1}$ and $l^H_{\le 1}$ are defined by the last condition.
We assume that we have defined $p_{\le k}$, $l^H_{\le k}$ and
are going to define
$p_{\le k+1}$, $l^H_{\le k+1}$.
Let $\Hom_A(S^{k+1}C, H)$ denote the space of linear maps from $S^{k+1}C$ to $H$ preserving decompositions over $A$. Namely:
\begin{align*}
\Hom_A(S^{k+1}C, H)&:= \{ f \in \Hom(S^{k+1}C, H) \mid \\
&f(C(a_1) \cdots C(a_{k+1} )) \subset H(a_1+\cdots+a_{k+1} ) \, (a_1, \ldots, a_{k+1} \in A)\}.
\end{align*}
We define a boundary operator $\partial$ on $\Hom(S^{k+1}C, H)$ by $\partial f:= f \circ l_1$,
then the homology is isomorphic to $\Hom_A(S^{k+1}H, H)$.
Our induction assumption shows that $l^H_{\le k} \circ p_{\le k} - p_{\le k} \circ l$ is a cycle in $\Hom_A(S^{k+1}C, H)$, thus one can define
\[
l^H_{k+1}:= [p_{\le k} \circ l - l^H_{\le k} \circ p_{\le k}] \in \Hom_A(S^{k+1} H, H).
\]
Then
$ l^H_{k+1} \circ p_{\le 1} + l^H_{\le k} \circ p_{\le k} - p_{\le k} \circ l$
is a null-homologous cycle, thus there exists
$p_{k+1} \in \Hom_A(S^{k+1}C, H)$ such that
\[
p_{k+1} \circ l_1 = l^H_{k+1} \circ p_{\le 1} + l^H_{\le k} \circ p_{\le k} - p_{\le k} \circ l.
\]
Then we define a coderivation $l^H_{\le k+1}$ so that $l^H_{\le k+1} = l^H_{\le k}$ on $S^{\le k}H$, and
\[
l^H_{\le k+1}|_{S^iH \to H} = \begin{cases} l^H_{k+1} &(i=k+1) \\ 0 &(i>k+1). \end{cases}
\]
Similarly, we define a coalgebra map $p_{\le k+1}$ so that $p_{\le k+1} = p_{\le k}$ on $S^{\le k}C$, and
\[
p_{\le k+1}|_{S^i C \to H} = \begin{cases} p_{k+1} &(i=k+1) \\ 0 &(i>k+1). \end{cases}
\]
It is easy to check $l^H_{\le k+1} \circ l^H_{\le k+1} =0$ on $S^{\le k+2}H$.
\end{proof}
\section{Main result}
We state the main result (Theorem \ref{161011_1}) and
explain a few applications to symplectic topology of Lagrangian submanifolds.
Let $\omega_n$ denote the standard symplectic form on $\C^n$, namely $\omega_n:=\sum_{j=1}^n dx_j \wedge dy_j$,
and $L$ be a Lagrangian submanifold in $(\C^n, \omega_n)$.
We assume that $L$ is closed (compact and $\partial L = \emptyset$),
connected, oriented and spin.
Let $\mu \in H^1(L: \Z)$ denote the Maslov class.
Since $L$ is oriented, $\mu (H_1(L:\Z)) \subset 2\Z$.
\begin{rem}
Let us explicitly define the Maslov class $\mu$ as follows.
Let $\Lambda(n):= U(n)/O(n)$ denote the unoriented Lagrangian Grassmannian,
and consider maps
\[
\tau: L \to \Lambda(n); \, x \mapsto T_x L, \qquad
{\det}^2: \Lambda(n) = U(n)/O(n) \to U(1).
\]
Then we define $\mu: =(\text{det}^2 \circ \tau)^*[U(1)]$,
where $U(1)$ is identified with $\{ e^{\sqrt{-1} \theta}| \theta \in \R/2\pi \Z\}$,
and $[U(1)]$ is defined as $[U(1)] : = [d\theta]/2\pi$.
\end{rem}
Let $S^1:= \R/\Z$, and $\mca{L}L := C^\infty(S^1, L)$.
We will often abbreviate $\mca{L}L$ by $\mca{L}$.
For every $a \in H_1(L: \Z)$, we set
$\mca{L}(a):= \{ \gamma \in \mca{L} \mid [\gamma] = a\}$.
Obviously $\mca{L} = \bigsqcup_{a \in H_1(L: \Z)} \mca{L}(a)$.
For each $a \in H_1(L: \Z)$,
we consider the $C^\infty$-topology on $\mca{L}(a)$ and set
\[
H^{\mca{L}}(a)_*: = H^\sing_{*+n+\mu(a)-1}(\mca{L}(a): \R),
\]
where the RHS is the singular homology with respect to the $C^\infty$-topology on $\mca{L}(a)$
(in the following we often omit the superscript ``$\sing$'' ).
Now we consider the direct product
\[
H^\mca{L}_*:= \bigoplus_{a \in H_1(L: \Z)} H^\mca{L}(a)_*
\]
equipped with the energy filtration;
for each $E \in \R$, we set
\[
F^E H^\mca{L}_*: = \bigoplus_{\omega_n(\bar{a}) >E} H^\mca{L}(a)_*
\]
where $\bar{a}$ denotes the unique element in $H_2(\C^n, L)$ satisfying $\partial \bar{a}=a$.
Finally $\wh{H}^\mca{L}_*$ denotes the completion by the energy filtration:
\[
\wh{H}^\mca{L}_*:= \varprojlim_{E \to \infty} H^\mca{L}_* / F^E H^\mca{L}_*.
\]
Now let us state the main result of this paper:
\begin{thm}\label{161011_1}
Let $L$ be a Lagrangian submanifold in $(\C^n, \omega_n)$
which is closed, oriented and spin.
Then, there exist
an $L_\infty$-structure $(l^H_k)_{k \ge 1}$ on $H^\mca{L}$ and
$X \in \wh{H}^\mca{L}_{-1}$,
$Y \in \wh{H}^\mca{L}_2$,
satisfying the following conditions:
\begin{enumerate}
\item[(i):] $l^H_1=0$.
\item[(ii):] The $L_\infty$-structure $(l^H_k)_{k \ge 1}$ respects the decomposition of $H^\mca{L}$ over $H_1(L: \Z)$.
In particular, the $L_\infty$-structure extends to the completion $\wh{H}^\mca{L}$.
\item[(iii):] There exists $c>0$ such that $X \in F^c \wh{H}^\mca{L}_{-1}$.
\item[(iv):]
$X$ and $Y$ satisfy the following equations:
\begin{equation}\label{170827_1}
\sum_{k \ge 2} \frac{1}{k!} l^H_k(X, \ldots, X) = 0,
\end{equation}
\begin{equation}\label{170827_2}
\biggl( \sum_{k \ge 2} \frac{1} {(k-1)!} l^H_k(Y, X, \ldots, X) \biggr)_{a=0} = (-1)^{n+1} [L].
\end{equation}
Note that infinite sums in the LHS make sense by the condition (iii).
$[L]$ in the RHS of (\ref{170827_2}) denotes the image of the fundamental class $[L] \in H_n(L: \R)$
by the embedding map $H_*(L: \R) \to H_*(\mca{L}(0): \R)$
which is induced by
\[
L \to \mca{L}(0); \quad x \mapsto \text{constant loop at $x$}.
\]
\end{enumerate}
\end{thm}
\begin{rem}
It will be possible to show that $l^H_2$ coincides (up to sign) with the Chas-Sullivan loop bracket \cite{ChSu_99},
and the full $L_\infty$-algebra structure is homotopy equivalent to
the dg Lie algebra defined by the chain level loop bracket in \cite{Irie_17}.
However, we do not give complete proofs of these claims in this paper.
\end{rem}
\begin{rem}
If $l^H_k=0$ for $k \ge 3$, assuming that $l^H_2$ coincides with the loop bracket up to sign,
(\ref{170827_2}) implies $[Y(-a), X(a)] \ne 0$ for some $a \in H_1(L: \Z)$.
When $L$ is diffeomorphic to $S^1 \times S^2$, this equation implies nonvanishing of the Maslov class $\mu$,
contradicting a result by Ekholm-Eliashberg-Murphy-Smith (Corollary 1.6 in \cite{EEMS}).
Therefore, if $L$ is diffeomorphic to $S^1 \times S^2$, there exists at least one nonvanishing higher term in the LHS of (\ref{170827_2}).
\end{rem}
Let us quickly recall two applications of Theorem \ref{161011_1} from \cite{Fukaya_06}.
For further results, consult the original paper \cite{Fukaya_06}.
\begin{cor}\label{170421_1}
Suppose that $L$ is aspherical. Then there exists $a \in H_1(L: \Z)$ such that $\mu(a)=2$,
$\omega_n(\bar{a})>0$ and $H_n(\mca{L}(a): \R) \ne 0$.
\end{cor}
\begin{proof}
By the equation (\ref{170827_2}),
there exist $a_1, \ldots, a_{k-1} \in H_1(L: \Z)$ such that
\[
l^H_k ( Y(- (a_1+ \cdots + a_{k-1})), X(a_1), \ldots, X(a_{k-1})) \ne 0.
\]
On the other hand, the assumption that $L$ is aspherical implies that
$H_i(\mca{L}) \ne 0$ only if $0 \le i \le n $ (Lemma 12.11 in \cite{Fukaya_06}).
Therefore we obtain
\[
1 \le \mu(a_1 + \cdots + a_{k-1}) \le n+1, \qquad
2-n \le \mu(a_j) \le 2 \quad (1 \le \forall j \le k-1).
\]
Since $\mu(a_1+ \cdots + a_{k-1})>0$, there exists $j$ such that $\mu(a_j)>0$.
Since $\mu$ takes values in $2\Z$, we obtain $\mu(a_j)=2$.
Since $X(a_j) \ne 0$ we obtain $\omega_n(\bar{a_j})>0$ and $H_n(\mca{L}(a_j): \R) \ne 0$.
\end{proof}
Corollary \ref{170421_1} in particular confirms Audin's conjecture:
every Lagrangian torus in $\C^n$ bounds a disk with positive symplectic area and Maslov index $2$.
Note that Cieliebak-Mohnke \cite{Cieliebak_Mohnke}
proved Audin's conjecture
by an approach different from ours.
For other previous results on this conjecture see \cite{Cieliebak_Mohnke}.
Another important application is
a complete classification of
orientable, closed, prime three-manifolds admitting Lagrangian embeddings into $\C^3$:
\begin{cor}\label{170622_1}
A closed, connected, orientable and prime three-manifold $M$ admits a Lagrangian embedding into $(\C^3, \omega_3)$
if and only if $M$ is diffeomorphic to $S^1 \times \Sigma$
where $\Sigma$ is a closed orientable two-manifold.
\end{cor}
The ``if'' part in Corollary \ref{170622_1} is elementary and classically known.
The ``only if'' part follows from Corollary \ref{170421_1} and some classical results in three-dimensional topology.
See \cite{Fukaya_06} Section 11 or \cite{Latschev_15} Section 5 for details.
Note that Evans-K\c{e}dra \cite{Evans_Kedra} and Damian \cite{Damian} proved the same conclusion for \textit{monotone} Lagrangian submanifolds in $\C^3$ which are orientable but not necessarily prime.
The proof of Theorem \ref{161011_1} occupies the rest of this paper.
In Sections 4--6 we reduce Theorem \ref{161011_1} to Theorem \ref{161215_1} (see Section 6),
which will be proved in Sections 7--9 using the pseudo-holomorphic curve theory.
In Section 4 we introduce the space of Moore loops with marked points, and the notion of de Rham chains on these spaces,
following \cite{Irie_17} with minor modifications.
Then we define the chain complex of de Rham chains and study its basic properties.
In Section 5, we reduce Theorem \ref{161011_1} to Theorem \ref{161214_2}, which asserts the existence of a solution of (\ref{170827_1}), (\ref{170827_2}) at chain level.
In Section 6, we reduce Theorem \ref{161214_2} to Theorem \ref{161215_1},
which asserts the existence of a sequence of approximate solutions connected by ``gauge equivalences''.
\section{de Rham chains on the space of loops with marked points}
In Section 4.1, we introduce the space of Moore loops with $k+1$ marked points (where $k \in \Z_{\ge 0}$) which we denote by $\mca{L}_{k+1}$.
In Section 4.2 we fix our conventions on signs.
In Section 4.3, we define the chain complex $C^\dR_*(\mca{L}_{k+1})$ which consists of ``de Rham chains'' on $\mca{L}_{k+1}$.
In Section 4.4, we introduce a chain model of $[-1, 1] \times \mca{L}_{k+1}$.
In Section 4.5, we introduce a natural dg Lie algebra by taking direct products of de Rham chain complexes introduced in Sections 4.3 and 4.4.
\subsection{Space of Moore loops with marked points}
First we consider the space of Moore paths
\[
\Pi:= \{ (T, \gamma) \mid T \in \R_{>0}, \, \gamma \in C^\infty([0, T], L), \,
\partial_t^m \gamma(0) = \partial_t^m \gamma(T)=0 \, (\forall m \ge 1) \}.
\]
We define evaluation maps $\ev_0, \ev_1: \Pi \to L$ by
\[
\ev_0(T, \gamma):= \gamma(0), \qquad
\ev_1(T, \gamma):= \gamma(T)
\]
and a concatenation map
\[
\Pi \fbp{\ev_1}{\ev_0} \Pi \to \Pi ;\quad (\Gamma_0, \Gamma_1) \mapsto \Gamma_0 * \Gamma_1
\]
by
\[
(T_0, \gamma_0) * (T_1, \gamma_1) := (T_0 + T_1, \gamma_0 * \gamma_1)
\]
where
\[
(\gamma_0 * \gamma_1) (t) := \begin{cases} \gamma_0(t) &(0 \le t \le T_0), \\ \gamma_1(t-T_0) &(T_0 \le t \le T_0+T_1). \end{cases}
\]
Next we consider the space of Moore loops with marked points.
For every $k \in \Z_{\ge 0}$,
we define the space $\mca{L}_{k+1}$
which consists of $(T, \gamma, t_1, \ldots, t_k)$ such that
\begin{itemize}
\item $T>0$ and $\gamma \in C^\infty(\R/T \Z, L)$.
\item $0 < t_1 < \cdots < t_k < T$. We set $t_0:= 0 = T \in \R/T\Z$.
\item $\partial_t^m \gamma(t_j)=0$ for every $m \in \Z_{\ge 1}$ and $j \in \{0, \ldots, k\}$.
\end{itemize}
For every $j \in \{0, \ldots, k\}$, we define $\evl_j: \mca{L}_{k+1} \to L$ by
\[
\evl_j (T, \gamma, t_1, \ldots, t_k) := \gamma(t_j).
\]
$\evl_j$ will be abbreviated as $\ev_j$ when there is no risk of confusion.
For $k \in \Z_{\ge 1}$, $k' \in \Z_{\ge 0}$ and $j \in \{1, \ldots, k\}$,
we define a concatenation map
\[
\con_j: \mca{L}_{k+1} \fbp{\ev^{\mca{L}}_j}{\ev^{\mca{L}}_0} \mca{L}_{k'+1} \to \mca{L}_{k+k'}
\]
as follows.
Notice that one can identify $\mca{L}_{k+1}$ with
\[
\{ (\Gamma_0, \ldots, \Gamma_k) \in \Pi^{k+1} \mid \ev_1(\Gamma_i) = \ev_0(\Gamma_{i+1}) \,(0 \le i \le k-1), \, \ev_1(\Gamma_k) = \ev_0(\Gamma_0)\}.
\]
Then we define $\con_j $ by
\begin{align*}
&\con_j ((\Gamma_0, \ldots, \Gamma_k), (\Gamma'_0, \ldots, \Gamma'_{k'})) \\
&:= \begin{cases}
(\Gamma_0, \ldots, \Gamma_{j-2}, \Gamma_{j-1}*\Gamma'_0, \Gamma'_1, \ldots, \Gamma'_{k'-1}, \Gamma'_{k'}*\Gamma_j, \Gamma_{j+1}, \ldots, \Gamma_k) &(k' \ge 1) \\
(\Gamma_0, \ldots, \Gamma_{j-2}, \Gamma_{j-1}*\Gamma'_0*\Gamma_j, \Gamma_{j+1}, \ldots, \Gamma_k ) &(k' = 0).
\end{cases}
\end{align*}
For every $a \in H_1(L: \Z)$,
let $\mca{L}_{k+1}(a)$ denote the subset of $\mca{L}_{k+1}$
which consists of $(T, \gamma, t_1, \ldots, t_k)$ such that
$[\gamma]=a$.
Obviously $\mca{L}_{k+1} = \bigsqcup_{a \in H_1(L: \Z)} \mca{L}_{k+1}(a)$,
and the concatenation map $\con_j$ satisfies
\[
\con_j ( \mca{L}_{k+1}(a) \fbp{\ev^{\mca{L}}_j}{\ev^{\mca{L}}_0} \mca{L}_{k'+1}(a')) \subset \mca{L}_{k+k'}(a+a').
\]
\subsection{Signs}
Here we summarize some conventions on signs.
Our sign conventions for direct/fiber products follow \cite{FOOO_09} Section 8.2,
and for pushout of differntial forms we follow \cite{FOOO_Kuranishi} Section 7.1.
Note that these conventions are different from those in \cite{Irie_17}.
\textbf{Direct and fiber products of manifolds}
Let $X_1$ and $X_2$ be oriented manifolds.
Their direct product $X_1 \times X_2$ is oriented so that
\[
T (X_1 \times X_2) \cong TX_1 \oplus TX_2
\]
preserves orientations.
Next we consider fiber product.
Let $M$ be an oriented manifold and
$\pi_i: X_i \to M\,(i=1,2)$ be $C^\infty$-maps.
We assume that $\pi_2$ is a submersion.
$\ker d\pi_2$ is oriented so that the isomorphism
\[
T X_2 \cong TM \oplus \ker d \pi_2
\]
preserves orientations. Then we orient $X_1 \fbp{\pi_1}{\pi_2} X_2$ so that
\[
T (X_1 \fbp{\pi_1}{\pi_2} X_2) \cong T X_1 \oplus \ker d\pi_2
\]
preserves orientations.
\textbf{Direct and fiber products of K-spaces}
For later use we also fix sign conventions for direct and fiber products of K-spaces
(see Section 10 for basic notions in the theory of Kuranishi structures).
Let $X_1$, $X_2$ be topological spaces with K-structures,
and $\mca{U}_i = (U_i, \mca{E}_i, s_i, \psi_i)$ be a K-chart on $X_i$, for each $i=1, 2$.
Then the direct product $\mca{U}_1 \times \mca{U}_2$,
which is a K-chart of $X_1 \times X_2$,
is oriented by
\[
\mca{U}_1 \times \mca{U}_2 := (-1)^{\rk \mca{E}_2 (\dim U_1 - \rk \mca{E}_1)} (U_1 \times U_2, \mca{E}_1 \times \mca{E}_2, s_1 \times s_2, \psi_1 \times \psi_2),
\]
where $U_1 \times U_2$ is oriented as before, and $\mca{E}_1 \times \mca{E}_2$
is oriented so that the isomorphism
\[
(\mca{E}_1 \times \mca{E}_2)_{(x_1, x_2)} \cong (\mca{E}_1)_{x_1} \oplus (\mca{E}_2)_{x_2} \qquad (x_1 \in U_1, \, x_2 \in U_2)
\]
preserves orientations.
Next we consider the fiber product.
Let $M$ be an oriented $C^\infty$-manifold and
$\pi_i: X_i \to M\,(i=1,2)$ be strongly smooth maps.
We assume that $\pi_2$ is weakly submersive.
Then the fiber product
$\mca{U}_1 \fbp{\pi_1}{\pi_2} \mca{U}_2$,
which is a K-chart of
$X \fbp{\pi_1}{\pi_2} X_2$, is oriented by
\[
\mca{U}_1 \fbp{\pi_1}{\pi_2} \mca{U}_2:= (-1)^{\rk \mca{E}_2 (\dim U_1 - \dim M - \rk \mca{E}_1)}
(U_1 \fbp{\pi_1}{\pi_2} U_2, \mca{E}_1 \times \mca{E}_2, s_1 \times s_2, \psi_1 \times \psi_2),
\]
where $U_1 \fbp{\pi_1}{\pi_2} U_2$ is oriented as before.
\textbf{Pushout of differential forms}
For any manifold $X$ and $j \in \Z$,
let $\mca{A}^j(X)$ denote the space of degree $j$ differential forms on $X$, and
$\mca{A}^j_c(X)$ denote its subspace which consists of compactly supported differential forms.
We set $\mca{A}^j(X) = 0$ when $j<0$ or $j > \dim X$.
Suppose $X$ and $Y$ are oriented manifolds
and $\pi: X \to Y$ is a $C^\infty$-submersion.
We define a pushout (or integration along fibers)
\[
\pi_!: \mca{A}^*_c(X) \to \mca{A}^{*- \dim \pi}_c(Y)
\]
(here $\dim \pi:= \dim X - \dim Y$) so that the formula
\[
\int_Y \pi_! \omega \wedge \eta = \int_X \omega \wedge \pi^* \eta
\]
holds for any $\omega \in \mca{A}^*_c(X)$ and $\eta \in \mca{A}^*(Y)$.
Simple computations show
\begin{align*}
&d (\pi_! \omega) = (-1)^{\dim \pi} \pi_!(d\omega) \qquad\qquad ( \omega \in \mca{A}^*_c(X)), \\
&\pi_! (\omega \wedge \pi^* \omega') = \pi_! \omega \wedge \omega' \qquad\qquad\, ( \omega \in \mca{A}^*_c(X), \, \omega' \in \mca{A}^*(Y)).
\end{align*}
\subsection{de Rham chain complex of $\mca{L}_{k+1}$}
Let us define the ``de Rham chain complex'' of $\mca{L}_{k+1}(a)$.
First we need the following definition.
\begin{defn}\label{171205_1}
Let $U$ be a $C^\infty$-manifold and
$\ph: U \to \mca{L}_{k+1}$.
We set
\[
\ph(u)= (T(u), \gamma(u), t_1(u), \ldots, t_k(u)).
\]
We say that $\ph$ is of $C^\infty$, if the map
\[
U \to \R^{k+1}; \, u \mapsto (T(u), t_1(u), \ldots, t_k(u))
\]
is of $C^\infty$ and
\[
\{ (u,t) \mid u \in U, \, 0 \le t \le T(u) \} \to L; \, (u,t) \mapsto \gamma(u)(t)
\]
is of $C^\infty$, namely it extends to a $C^\infty$-map from an open neighborhood of the LHS in $U \times \R$ to $L$.
We say that $\ph$ is \textit{smooth}, if $\ph$ is of $C^\infty$ and
$\ev^{\mca{L}}_0 \circ \ph: U \to L$ is a submersion.
\end{defn}
For every $N \in \Z_{\ge 1}$, let $\mf{U}_N$ denote the set of oriented submanifolds in $\R^N$, and let $\mf{U}:= \bigsqcup_{N \ge 1} \mf{U}_N$.
Let $\mca{P}(\mca{L}_{k+1}(a))$ denote the set of pairs $(U, \ph)$ such that
$U \in \mf{U}$ and $\ph: U \to \mca{L}_{k+1}(a)$ is a smooth map.
For every $N \in \Z$, let us consider the vector space
\begin{equation}\label{170619_1}
\bigoplus_{(U,\ph) \in \mca{P}(\mca{L}_{k+1}(a))} \mca{A}^{\dim U -N}_c(U).
\end{equation}
For any $(U,\ph) \in \mca{P}(\mca{L}_{k+1}(a))$ and $\omega \in \mca{A}^{\dim U- N}_c(U)$,
let $(U, \ph, \omega)$ denote the vector in (\ref{170619_1})
such that its $(U,\ph)$-component is $\omega$ and the other components are $0$.
Let $Z_N$ denote the subspace of (\ref{170619_1}) which is generated by
\begin{align*}
&\{ (U, \ph, \pi_! \omega) - (U', \ph \circ \pi, \omega) \mid (U, \ph) \in \mca{P}(\mca{L}_{k+1}(a)), \quad U' \in \mf{U}, \\
& \quad \omega \in \mca{A}_c^{\dim U' -N} (U'), \quad \pi: U' \to U \, \text{is a $C^\infty$-submersion} \}.
\end{align*}
Then we define
\[
C^\dR_N(\mca{L}_{k+1}(a)) := \biggl( \bigoplus_{(U,\ph) \in \mca{P}(\mca{L}_{k+1}(a))} \mca{A}^{\dim U -N}_c(U) \biggr) / Z_N.
\]
We often abbreviate $[(U, \ph, \omega)] \in C^\dR_N(\mca{L}_{k+1}(a))$ by $(U, \ph, \omega)$.
We define a boundary operator
$\partial: C^\dR_*(\mca{L}_{k+1}(a)) \to C^\dR_{*-1}(\mca{L}_{k+1}(a))$ by
\begin{equation}\label{170828_1}
\partial (U, \ph, \omega):= (-1)^{|\omega|+1} (U, \ph, d\omega).
\end{equation}
It is easy to check that $\partial$ is well-defined and $\partial^2=0$.
We call this chain complex the \textit{de Rham chain complex} of $\mca{L}_{k+1}(a)$,
and denote its homology by $H^\dR_*(\mca{L}_{k+1}(a))$.
\begin{rem}
Here are slight differences between the presentation in this section and that in \cite{Irie_17}.
\begin{itemize}
\item
The sign for the boundary operator in (\ref{170828_1}) is different from that in \cite{Irie_17}.
\item
In the definition of $\mca{P}(\mca{L}_{k+1}(a))$ we only require that
$\ev^{\mca{L}}_0 \circ \ph: U \to L$ is a submersion.
On the other hand,
in Section 7.2 in \cite{Irie_17},
we consider maps $\ph: U \to \mca{L}_{k+1}$ such that
$\ev^{\mca{L}}_j \circ \ph: U \to L$ are submersions for all $j \in \{0, \ldots, k\}$.
However the resulting chain complexes are quasi-isomorphic.
\end{itemize}
\end{rem}
\begin{lem}\label{170619_2}
\begin{enumerate}
\item[(i):] The forgetting map
\[
\mca{L}_{k+1}(a) \to \mca{L}_1(a); \qquad (T, \gamma, t_1, \ldots, t_k) \mapsto (T, \gamma)
\]
induces an isomorphism $H^\dR_*(\mca{L}_{k+1}(a)) \cong H^\dR_*(\mca{L}_1(a))$.
In particular $H^\dR_*(\mca{L}_{k+1}(a))$ does not depend on $k$.
\item[(ii):] $H^\dR_*(\mca{L}_1(a) ) \cong H^\sing_*(\mca{L}(a): \R)$, where the RHS denotes the singular homology with respect to the $C^\infty$-topology on $\mca{L}(a)$.
\end{enumerate}
\end{lem}
\begin{proof}
For each $k \in \Z_{\ge 0}$, let $\Delta^k$ denote the $k$-dimensional simplex:
\[
\Delta^k:= \begin{cases} \R^0 &(k=0), \\ \{ (t_1,\ldots, t_k) \in \R^k \mid 0 \le t_1 \le \cdots \le t_k \le 1\} &(k \ge 1). \end{cases}
\]
Then we define $\mca{P}(\mca{L}(a) \times \Delta^k)$ to be the set
which consists of $(U, \ph)$ such that
$U \in \mf{U}$ and $\ph: U \to \mca{L}(a) \times \Delta^k$ is of $C^\infty$ (i.e. projections to each components are of $C^\infty$).
Then we can define a chain complex $C^\dR_*(\mca{L}(a) \times \Delta^k)$ in exactly the same manner as $C^\dR_*(\mca{L}_{k+1}(a))$.
Moreover, there exists a zig-zag of quasi-isomorphisms connecting $C^\dR_*(\mca{L}_{k+1}(a))$ and $C^\dR_*(\mca{L}(a) \times \Delta^k)$
(see the last part of Section 7.2 in \cite{Irie_17}, where $C^\dR_*(\mca{L}_{k+1}(a))$ is denoted by
$C^\dR_*(\bar{\mca{L}}^a_{k, \text{reg}})$).
Then, to prove (i) it is sufficient to show that the map
\begin{equation}\label{170828_2}
\mca{L}(a) \times \Delta^k \to \mca{L}(a) ; \qquad (\gamma, t_1, \ldots,t_k) \mapsto \gamma
\end{equation}
induces an isomorphism on $H^\dR_*$,
which follows from homotopy invariance of $H^\dR_*$ (see Proposition 4.7 in \cite{Irie_17}).
(ii) follows from $H^\dR_*(\mca{L}_1(a)) \cong H^\dR_*(\mca{L}(a))$
(apply the zig-zag mentioned above for $k=0$),
and $H^\dR_*(\mca{L}(a)) \cong H^\sing_*(\mca{L}(a): \R)$,
which follows from Theorem 6.1 in \cite{Irie_17}.
\end{proof}
Next we define the fiber product on de Rham chain complexes.
For every $k \in \Z_{\ge 1}$, $k' \in \Z_{\ge 0}$, $j \in \{1,\ldots, k\}$ and $a, a' \in H_1(L:\Z)$,
we define a linear map
\begin{equation}\label{170619_3}
\circ_j: C^\dR_{n+d}(\mca{L}_{k+1}(a)) \otimes C^\dR_{n+d'}(\mca{L}_{k'+1}(a')) \to C^\dR_{n+d+d'}(\mca{L}_{k+k'}(a+a')) ; \, x \otimes y \mapsto x \circ_j y
\end{equation}
in the following way.
Setting
\[
x := (U, \ph, \omega), \qquad
y:= (U', \ph', \omega')
\]
let
$\ph_j: = \evl_j \circ \ph$ and
$\ph'_0:= \evl_0 \circ \ph'$.
Then we define
\[
x \circ_j y:= (-1)^{(\dim U - |\omega|-n)|\omega'|} (U \fbp{\ph_j}{\ph'_0} U', \con_j \circ (\ph_j \times \ph'_0), \omega \times \omega'),
\]
where
the fiber product $U \fbp{\ph_j}{\ph'_0} U'$ is oriented as in Section 4.2, and
$\con_j$ denotes the concatenation map defined in Section 4.1.
It is straightforward to check that the fiber product (\ref{170619_3}) is a chain map,
i.e. it satisfies the Leibniz rule
\[
\partial (x \circ_j y) = \partial x \circ_j y + (-1)^d x \circ_j \partial y.
\]
It is also easy to check the associativity: given $x_i \in C^\dR_{n+d_i} (\mca{L}_{k_i+1}(a_i)) \,(i=1,2,3)$,
there holds
\begin{align*}
(x_1 \circ_{i_1} x_2) \circ_{k_2 + i_2 - 1} x_3 &= (-1)^{d_2d_3} (x_1 \circ_{i_2} x_3) \circ_{i_1} x_2 \quad\, ( 1 \le i_1 < i_2 \le k_1), \\
(x_1 \circ_{i_1} x_2) \circ_{i_1 + i_2 -1} x_3 &= x_1 \circ_{i_1} (x_2 \circ_{i_2} x_3) \qquad\qquad\qquad (1 \le i_1 \le k_1,\, 1 \le i_2 \le k_2).
\end{align*}
On homology level,
the fiber product corresponds to the Chas-Sullivan loop product,
which was originally defined in \cite{ChSu_99}:
\begin{lem}\label{170623_1}
The fiber product (\ref{170619_3}) induces a linear map
\[
H^\dR_{n+d} ( \mca{L}_{k+1}(a) ) \otimes H^\dR_{n+d'} (\mca{L}_{k'+1}(a') ) \to H^\dR_{n+d+d'}(\mca{L}_{k+k'} (a+a')).
\]
Via isomorphisms in Lemma \ref{170619_2}, this map corresponds to the loop product
\[
H_{n+d}(\mca{L}(a)) \otimes H_{n+d'} (\mca{L}(a')) \to H_{n+d+d'}(\mca{L}(a+a')).
\]
\end{lem}
\begin{proof}
By Lemma \ref{170619_2} (i), it is sufficient to prove the case $k=1$ and $k'=0$,
which follows from Proposition 8.7 in \cite{Irie_17}.
\end{proof}
\subsection{Chain model of $[-1, 1] \times \mca{L}_{k+1}$}
In this subsection, we define another chain complex $\bar{C}^\dR_*(\mca{L}_{k+1}(a))$.
Roughly speaking, it consists of chains on $[-1,1] \times \mca{L}_{k+1}(a)$ relative to $\{-1,1\} \times \mca{L}_{k+1}(a)$.
In Section 6, we use this chain complex to define ``gauge-equivalence'' of (approximate) solutions of the Maurer-Cartan equation of loop bracket.
Let $\bar{\mca{P}}$ denote the set consists of tuple $(U, \ph, \tau_+, \tau_-)$ such that the following conditions are satisfied:
\begin{itemize}
\item $U \in \mf{U}$ and $\ph: U \to \R \times \mca{L}_{k+1}(a)$. We denote $\ph:= (\ph_\R, \ph_{\mca{L}})$,
and for every interval $I \subset \R$ we denote $U_I:= (\ph_\R)^{-1}(I)$.
\item
$\ph_\R$ and $\ph_{\mca{L}}$ are of $C^\infty$. Moreover,
$U \to \R \times L; \, u \mapsto (\ph_\R(u), \ev_0 \circ \ph_{\mca{L}}(u))$ is a submersion.
\item $\tau_+: U_{\ge 1} \to \R_{\ge 1} \times U_1$ is a diffeomorphism ($U_{\ge 1}$ is an abbreviation of $U_{\R_{\ge 1}}$) such that
\[
\ph|_{U_{\ge 1}} = (i _{\ge 1} \times \ph_{\mca{L}}|_{U_1}) \circ \tau_+
\]
where $i_{\ge 1}: \R_{\ge 1} \to \R$ is the inclusion map.
\item $\tau_-: U_{\le -1} \to \R_{\le -1} \times U_{-1}$ is a diffeomorphism $(U_{\le -1}$ is an abbreviation of $U_{\R_{\le -1}}$) such that
\[
\ph|_{U_{\le -1}} = (i _{\le -1} \times \ph_{\mca{L}}|_{U_{-1}}) \circ \tau_-
\]
where $i_{\le -1}: \R_{\le -1} \to \R$ is the inclusion map.
\end{itemize}
\begin{rem}
$U_{\ge 1}$ and $U_{\le -1}$ may be the empty set.
\end{rem}
For any $(U, \ph, \tau_+, \tau_-) \in \bar{\mca{P}}$ and $N \in \Z$,
let $\mca{A}^N(U, \ph, \tau_+, \tau_-)$
denote the vector space which consists of $\omega \in \mca{A}^N(U)$ satisfying the following conditions:
\begin{itemize}
\item $\omega|_{U_{[-1, 1]}}$ is compactly supported.
\item $\omega|_{U_{\ge 1}} = (\tau_+)^* (1 \times \omega|_{U_1})$.
\item $\omega|_{U_{\le -1}} = (\tau_-)^* (1 \times \omega|_{U_{-1}})$.
\end{itemize}
For $N \in \Z$, let us define
\[
\bar{C}^\dR_N (\mca{L}_{k+1}(a)) := \biggl( \bigoplus_{(U,\ph, \tau_+, \tau_-) \in \bar{\mca{P}}} \mca{A}^{\dim U-N-1} (U, \ph, \tau_+, \tau_-) \biggr)/Z_N
\]
where $Z_N$ is a subspace generated by vectors
\[
(U, \ph, \tau_+, \tau_-, \omega) -
(U', \ph', \tau'_+, \tau'_-, \omega')
\]
such that there exists a submersion $\pi: U' \to U$ satisfying
\begin{align*}
\ph' &= \ph \circ \pi, \\
\omega &= \pi_! \omega', \\
\tau_+ \circ \pi|_{U'_{\ge 1}} &= (\id_{\R_{\ge 1}} \times \pi|_{U'_1}) \circ \tau'_+, \\
\tau_- \circ \pi|_{U'_{\le -1}} &= (\id_{\R_{\le -1}} \times \pi|_{U'_{-1}}) \circ \tau'_-.
\end{align*}
Let us define $\partial: \bar{C}^\dR_*(\mca{L}_{k+1}(a)) \to \bar{C}^\dR_{*-1}(\mca{L}_{k+1}(a))$
by
\[
\partial(U, \ph, \tau_+, \tau_-, \omega): = (-1)^{|\omega|+1} (U, \ph, \tau_+, \tau_-, d\omega).
\]
It is easy to check that $\partial$ is well-defined and $\partial^2=0$, thus we obtain a chain complex.
In the following argument of this subsection,
we abbreviate $\bar{C}^\dR_*(\mca{L}_{k+1}(a))$ and $C^\dR_*(\mca{L}_{k+1}(a))$ by
$\bar{C}_*$ and $C_*$, respectively.
Let us define $i: C_* \to \bar{C}_*$ by
\[
i( U, \ph, \omega) := (-1)^{\dim U} (\R \times U, \id_\R \times \ph, \tau_+, \tau_-, 1 \times \omega)
\]
where $\tau_+$ and $\tau_-$ are defined in the obvious way.
Also, we define
$e_+: \bar{C}_* \to C_*$ and $e_-: \bar{C}_* \to C_*$ by
\begin{align*}
e_+ (U, \ph, \tau_+, \tau_-, \omega) &:= (-1)^{\dim U - 1} (U_1, \ph|_{U_1}, \omega|_{U_1}), \\
e_- (U, \ph, \tau_+, \tau_-, \omega) &:= (-1)^{\dim U - 1}(U_{-1}, \ph|_{U_{-1}}, \omega|_{U_{-1}}),
\end{align*}
where $U_1$ (resp. $U_{-1}$) is oriented so that
$\tau_+: U_{\ge 1} \to \R_{\ge 1} \times U_1$
(resp. $\tau_-: U_{\le -1} \to \R_{\le -1} \times U_{-1}$)
is orientation-preserving, where
$\R_{\ge 1}$ (resp. $\R_{\le -1}$)
is oriented so that
$\partial/\partial t$
is of positive direction
($t$ denotes the standard coordinate on $\R$).
Then it is easy to see that $i$, $e_+$, $e_-$ are well-defined chain maps,
and there holds $e_+ \circ i = e_- \circ i = \id_C$.
\begin{lem}
$(e_+, e_-): \bar{C}_* \to C_* \oplus C_*$ is surjective.
\end{lem}
\begin{proof}
Let us take $\chi \in C^\infty(\R, [0, 1])$ so that
$\chi(t) = 1$ for every $t \ge 1$ and $\chi(t)=0$ for every $t \le -1$.
Given $x = (U, \ph, \omega) \in C_*$, let
\[
\bar{x} := (-1)^{\dim U} (\R \times U, \id_\R \times \ph, \tau_+, \tau_-, \chi \times \omega),
\]
where $\tau_+$ and $\tau_-$ are defined in the obvious manner.
Then $e_+(\bar{x}) = x$ and $e_-(\bar{x})=0$, thus we have proved
that the image of $(e_+, e_-)$ contains $C_* \oplus 0$.
A similar argument shows that the image contains $0 \oplus C_*$.
This completes the proof.
\end{proof}
\begin{lem}
$i \circ e_+$ and $i \circ e_-$ are chain homotopic to $\id_{\bar{C}}$.
\end{lem}
\begin{proof}
We only prove that $i \circ e_+$ is chain homotopic to $\id_{\bar{C}}$
by explicitly defining a linear map $K: \bar{C}_* \to \bar{C}_{*+1}$ which satisfies
\[
K \partial + \partial K = \id_{\bar{C}} - i \circ e_+.
\]
The proof for $e_-$ is completely parallel.
\textbf{Step 1.}
Let us take $C^\infty$-functions $\alpha: \R^2 \to \R$ and $\chi: \R^2 \to [0,1]$
so that the following conditions are satisfied:
\begin{itemize}
\item $x \le 0 \implies \alpha(x,y)=y$.
\item $x \ge 1, \, y \ge -1 \implies \alpha(x,y)=-x$.
\item $\nabla \alpha(x,y) \ne 0$ for every $(x,y) \in \R^2$.
\item $\nu_\alpha:= \nabla \alpha/|\nabla \alpha|$ is proper.
Namely, for any $p \in \R^2$, there exists $c: \R \to \R^2$ such that
$c(0)=p$ and $\dot{c}(t) = \nu_\alpha(c(t))$ for any $t \in \R$.
\item $\{ \alpha \ge 1\} \subset \{ y \ge 1\}$.
\item $\{ \alpha \le -1\} \subset \{y \le -1\} \cup \{ x \ge 1\}$.
\item $\chi \equiv 1$ on a neighborhood of $\{x=0\} \cup \{x \ge 0, y \le 1\}$.
\item $d\chi(\nabla \alpha)=0$ on $\{ \alpha \ge 1\} \cup \{ \alpha \le -1\}$.
\item $\supp \chi \cap \{ -1 \le \alpha \le 1\}$ is compact.
\end{itemize}
\begin{center}
\input{alpha.tpc}
\end{center}
\textbf{Step 2.}
We define a linear map $K: \bar{C}_* \to \bar{C}_{*+1}$ by
\[
K (U, \ph, \tau_+, \tau_-, \omega):= (-1)^{|\omega|+1} (\R \times U, \bar{\ph}, \bar{\tau}_+, \bar{\tau}_-, \bar{\omega})
\]
where $\bar{\ph}$, $\bar{\tau}_{\pm}$ and $\bar{\omega}$ are defined as follows.
$\bar{C}_*$ is defined by taking a quotient, however well-definedness of $K$ is easy to check.
\begin{itemize}
\item Let us denote $\ph: U \to \R \times \mca{L}_{k+1}(a)$ by $\ph = (\ph_\R, \ph_{\mca{L}})$.
We define
$\tilde{\alpha}: \R \times U \to \R$ and $\bar{\ph}: \R \times U \to \R \times \mca{L}_{k+1}(a)$ by
\[
\tilde{\alpha}(r,u):=\alpha(r, \ph_\R(u)), \qquad
\bar{\ph}(r,u):= (\tilde{\alpha}(r,u), \ph_{\mca{L}}(u)).
\]
We have to check that
\[
\R \times U \to \R \times L; \quad (r,u) \mapsto (\tilde{\alpha}(r,u) , \ev_0 \circ \ph_{\mca{L}}(u))
\]
is a submersion.
This is because
$U \to \R \times L; \, u \mapsto (\ph_\R(u), \ev_0 \circ \ph_{\mca{L}}(u))$ is a submersion,
and $d\alpha(x,y) \ne 0$ for every $(x,y) \in \R^2$.
\item To define $\bar{\tau}_+$ and $\bar{\tau}_-$,
we first define a vector field
\[
V \in \mf{X}(\R \times (U_{\ge 1} \cup U_{\le -1}) \cup \R_{\ge 1} \times U_{[-1,1]})
\]
as follows (recall $\nu_\alpha:= \nabla \alpha/|\nabla \alpha|$ from Step 1):
\begin{itemize}
\item On $\R \times U_{\ge 1} \cong \R \times \R_{\ge 1} \times U_1$, we set $V(x,y,u):= (\nu_\alpha(x,y), 0)$.
\item On $\R \times U_{\le -1} \cong \R \times \R_{\le -1} \times U_{-1}$, we set $V(x,y,u):= (\nu_\alpha(x,y), 0)$.
\item On $\R_{\ge 1} \times U_{[-1,1]}$, we set $V(x,u) := (-1,0)$.
\end{itemize}
In particular, $V$ is defined on $\{ \tilde{\alpha} \ge 1\} \cup \{ \tilde{\alpha} \le -1\}$, and
$V$ is forward (resp. backward) complete on $\{ \tilde{\alpha} \ge 1\}$ (resp. $\{ \tilde{\alpha} \le -1\}$).
Then we define a diffeomorphism $\bar{\tau}_+: \{ \tilde{\alpha} \ge 1\} \cong \R_{\ge 1} \times \{\tilde{\alpha} = 1\}$ by
$\bar{\tau}_+(p) := (\tilde{\alpha}(p), q)$, where $q$ is a unique point in $\{\tilde{\alpha}=1\}$ which is connected to $p$ by
an integral curve of $V$.
Similarly, we define a diffeomorphism $\bar{\tau}_-: \{ \tilde{\alpha} \le -1\} \cong \R_{\le -1} \times \{\tilde{\alpha} = -1\}$ by
$\bar{\tau}_-(p) := (\tilde{\alpha}(p), q)$, where $q$ is a unique point in $\{\tilde{\alpha}= -1\}$ which is connected to $p$ by
an integral curve of $V$.
\item We define $\bar{\omega}$ by
$\bar{\omega}(r,u): = \chi(r, \ph_\R(u)) \cdot (\text{pr}_U)^* \omega$.
\end{itemize}
\textbf{Step 3.}
We prove that $K \partial + \partial K = \id_{\bar{C}} - i \circ e_+$.
It is easy to see that
\[
(K \partial + \partial K) (U, \ph, \tau_+, \tau_-, \omega)
=
(\R \times U, \bar{\ph}, \bar{\tau}_+, \bar{\tau}_-, d \chi (r, \ph_\R(u)) \wedge (\pr_U)^*\omega).
\]
Let us denote $d\chi:= (d\chi)_+ + (d\chi)_-$ where
$(d\chi)_+$ is supported on $\{x>0, y>1\}$ and $(d\chi)_-$ is supported on $\{x<0\}$.
Then
\begin{equation}\label{170622_2}
(\R \times U, \bar{\ph}, \bar{\tau}_+, \bar{\tau}_-, (d\chi)_+ \wedge (\text{pr}_U)^* \omega) = - (i \circ e_+)( U, \ph, \tau_+, \tau_-, \omega).
\end{equation}
To check (\ref{170622_2}), let us consider a submersion
\[
\R_{>0} \times U_{>1} \cong \R_{>0} \times (\R_{>1} \times U_1) \to \R \times U_1
\]
where the first map (diffeomorphism) is $\id_{\R_{>0}} \times \tau_+|_{U_{>1}}$, and the second map is $(x,y,u) \mapsto (\alpha(x,y), u)$.
Then $\pi: \R_{>0} \times \R_{>1} \to \R; \, (x,y) \mapsto \alpha(x,y)$ satisfies $\pi_!((d\chi)_+) = -1$,
and this shows (\ref{170622_2}).
On the other hand
\[
(\R \times U, \bar{\ph}, \bar{\tau}_+, \bar{\tau}_-, (d\chi)_- \wedge \text{pr}_U^* \omega) = (U, \ph, \tau_+, \tau_-, \omega)
\]
can be proved by the projection $\R_{<0} \times U \to U$,
which pushes $(d\chi)_-$ to $1$.
\end{proof}
Let us define the fiber product on $\bar{C}_*$.
For every $k \in \Z_{\ge 1}$, $k' \in \Z_{\ge 0}$, $j \in \{1, \ldots, k\}$ and $a, a' \in H_1(L:\Z)$, we define
\begin{equation}\label{170928_1}
\circ_j: \bar{C}^\dR_{n+d}(\mca{L}_{k+1}(a)) \otimes \bar{C}^\dR_{n+d'}(\mca{L}_{k'+1}(a')) \to \bar{C}^\dR_{n+d+d'}(\mca{L}_{k+k'}(a+a')); \,
x \otimes y \mapsto x \circ_j y
\end{equation}
in the following way. Setting
\[
x = (U, \ph, \tau_+, \tau_-, \omega), \qquad
y = (U', \ph', \tau'_+, \tau'_-, \omega'),
\]
let us define $C^\infty$-maps
$\ph_j, \, \ph'_0: U \to \R \times L$ by
\[
\ph_j: = (\ph_\R, \ev^{\mca{L}}_j \circ \ph_{\mca{L}}), \qquad
\ph'_0:= (\ph'_\R, \ev^{\mca{L}}_0 \circ \ph'_{\mca{L}}).
\]
Note that $\ph'_0$ is a submersion, thus the fiber product
$U \fbp{\ph_j}{\ph'_0} U'$ is a $C^\infty$-manifold.
Then we define
\[
x \circ_j y:= (-1)^{(\dim U - |\omega| - n-1) |\omega'| + n}
(U \fbp{\ph_j}{\ph'_0} U', \ph'', \tau''_+, \tau''_-, \omega \times \omega')
\]
where $\ph''$ is defined by
\[
\ph''(u, u') := (\ph_\R(u), \con_j(\ph_{\mca{L}}(u), \ph'_{\mca{L}}(u'))),
\]
and $\tau''_+$, $\tau''_-$ are defined as follows:
\begin{align*}
\rho_+(u,u')&:= \pr_{\R_{\ge 1}} \circ \tau_+(u) = \pr_{\R_{\ge 1}} \circ \tau'_+(u'), \\
\tau''_+(u, u')&:= (\rho_+(u, u'), ( \pr_{U_1} \circ \tau_+(u), \pr_{U'_1} \circ \tau'_+(u'))), \\
\rho_-(u,u')&:=\pr_{\R_{\le -1}} \circ \tau_-(u) = \pr_{\R_{\le -1}} \circ \tau'_-(u'), \\
\tau''_-(u, u')&:= (\rho_-(u,u') , ( \pr_{U_{-1}} \circ \tau_-(u), \pr_{U'_{-1}} \circ \tau'_-(u'))).
\end{align*}
It is easy to see that the fiber product (\ref{170928_1}) is a chain map,
i.e. it satisfies the Leibniz rule:
\[
\partial (x \circ_j y) = \partial x \circ_j y + (-1)^d x \circ_j \partial y.
\]
Moreover, chain maps $i$, $e_+$, $e_-$ intertwine fiber products on $C_*$ and $\bar{C}_*$.
\subsection{dg Lie algebras $C^{\mca{L}}$, $\bar{C}^{\mca{L}}$ and their completions}
For every $a \in H_1(L:\Z)$ and $k \in \Z_{\ge 0}$, we define
\[
C^{\mca{L}}(a, k)_*:= C^\dR_{*+n+\mu(a)+k-1} (\mca{L}_{k+1}(a))
\]
and set
\[
C^{\mca{L}}_*: = \bigoplus_{\substack{a \in H_1(L: \Z) \\ k \in \Z_{\ge 0}}} C^{\mca{L}}(a, k)_*.
\]
To define the action filatration on $C^{\mca{L}}_*$,
for each $E \in \R$ we set
\[
F^E C^{\mca{L}}_* := \bigoplus_{\substack{\omega_n(\bar{a}) > E \\ k \in \Z_{\ge 0}}} C^{\mca{L}}(a, k)_*
\]
and take completion:
\[
\wh{C}^{\mca{L}}_*:= \varprojlim_{E \to \infty} C^{\mca{L}}_* / F^E C^{\mca{L}}_*.
\]
$C^{\mca{L}}_*$ has a dg Lie algebra structure
(see Remark \ref{170829_1} for our convention)
defined as follows:
\begin{align*}
(\partial x)(a,k) &:= \partial (x(a,k)), \\
(x \circ y)(a,k) &:= \sum_{\substack{k'+k'' = k+1 \\ 1 \le i \le k' \\ a'+a''=a}}(-1)^{(i-1)(k''-1)+(k'-1)(|y|+1+k'')}
x(a', k') \circ_i y(a'', k''), \\
[x, y]&:= x \circ y - (-1)^{|x||y|} y \circ x.
\end{align*}
For the sign in the RHS of the second formula, see Theorem 2.8 (ii) in \cite{Irie_17}.
The Jacobi identity follows from the associativity of the fiber product.
Since this dg Lie algebra structure respects the decomposition $C^{\mca{L}}_* \cong \bigoplus_{(a,k) \in H_1(L: \Z) \times \Z_{\ge 0} } C^{\mca{L}}(a,k)_*$,
it extends to the completion $\wh{C}^{\mca{L}}_*$.
We also consider $\bar{C}^{\mca{L}}_* := \bigoplus_{(a,k) \in H_1(L: \Z) \times \Z_{\ge 0}} \bar{C}^\dR_{*+n+\mu(a)+k-1} (\mca{L}_{k+1}(a))$
and its completion $\wh{\bar{C}}^{\mca{L}}_*$.
One can naturally define a dg Lie algebra structure on $\bar{C}^{\mca{L}}_*$,
$C^{\mca{L}}_*$, and it extends to the completion.
One can define morphisms of dg Lie algebras
\[
i: C ^{\mca{L}}_* \to \bar{C}^{\mca{L}}_*, \quad
e_+: \bar{C}^{\mca{L}}_* \to C^{\mca{L}}_*, \quad
e_-: \bar{C}^{\mca{L}}_* \to C^{\mca{L}}_*,
\]
so that the following properties hold:
\begin{itemize}
\item $e_+ \circ i = e_- \circ i = \id_{C^{\mca{L}}}$.
\item $i \circ e_+$ and $i \circ e_-$ are chain homotopic to $\id_{\bar{C}^{\mca{L}}}$.
One can take chain homotopies to respect decompositions over $H_1(L: \Z) \times \Z_{\ge 0}$.
\item $(e_+, e_-): \bar{C}^{\mca{L}}_* \to C^{\mca{L}}_* \oplus C^{\mca{L}}_*$ is surjective.
\end{itemize}
\section{Chain level statement}
The goal of this section is to reduce Theorem \ref{161011_1} to Theorem \ref{161214_2},
which is formulated on the dg Lie algebra $\wh{C}^{\mca{L}}$.
Throughout this section, $\R$-coefficient singular homology $H^\sing_*(\, \cdot \, : \R)$ is abbreviated as $H_*(\, \cdot \,)$.
\begin{thm}\label{161214_2}
There exist $x \in \wh{C}^{\mca{L}}_{-1}$, $y \in \wh{C}^{\mca{L}}_2$, $z \in \wh{C}^{\mca{L}}_1$
and $\ep>0$ such that:
\begin{enumerate}
\item[(i):] $\partial x - \frac{1}{2} [x,x] = 0$, namely $x$ is a Maurer-Cartan element of $\wh{C}^{\mca{L}}$.
\item[(ii):] $\partial y - [x,y] = z$.
\item[(iii):] $x(a,k) \ne 0$ only if $\omega(\bar{a}) \ge 2\ep$ or $a=0$, $k \ge 2$.
Moreover, $x(0, 2) \in C^\dR_n(\mca{L}_3(0) )$ is a cycle such that
$[x(0, 2) ] \in H^\dR_n(\mca{L}_3(0) ) \cong H_n(\mca{L}(0))$ corresponds to $(-1)^{n+1} [L]$.
\item[(iv):]
$z(a, k) \ne 0$ only if $\omega(\bar{a}) \ge 2\ep $ or $a=0$.
Moreover, $z(0, 0) \in C^\dR_n(\mca{L}_1(0) )$ is a cycle such that
$[z(0, 0) ] \in H^\dR_n(\mca{L}_1(0) ) \cong H_n(\mca{L}(0))$ corresponds to $(-1)^{n+1} [L]$.
\end{enumerate}
In (iii) and (iv), $[L] \in H_n(\mca{L}(0))$ is defined as in Theorem \ref{161011_1}.
\end{thm}
Conditions (i) and (iii) imply that
\[
x^0:= \sum_{k \ge 2} x(0, k) \in \wh{C}^{\mca{L}}_{-1}
\]
is a Maurer-Cartan element of $\wh{C}^{\mca{L}}$.
For every $a \in H_1(L: \Z)$, let
$C^{\mca{L}}(a)_*:= \prod_{k \ge 0} C^\mca{L}(a,k)_*$.
We define
$\partial_{x^0}: C^{\mca{L}}(a)_* \to C^{\mca{L}}(a)_{*-1}$ by
$\partial_{x^0}(u) := \partial u - [x^0 , u]$.
\begin{lem}\label{161225_1}
$H_*(C^{\mca{L}}(a)_*, \partial_{x^0}) \cong H_{*+n+\mu(a)-1}(\mca{L}(a))$.
\end{lem}
\begin{proof}
Let us consider the exact sequence
\[
0 \to \prod_{k \ge 1} C^{\mca{L}}(a,k)_* \to C^{\mca{L}}(a)_* \to C^{\mca{L}}(a,0)_* \to 0
\]
where the first map is inclusion and the second map is projection.
Since $H_*(C^{\mca{L}}(a,0)) \cong H_{*+n+\mu(a)-1}(\mca{L}(a))$ by Lemma \ref{170619_2},
it is sufficient to show that $\prod_{k \ge 1} C^{\mca{L}}(a,k)_*$ is acyclic
with the boundary operator $\partial_{x^0}$.
For every positive integer $N$,
the operator $\partial_{x^0}$ preserves $\prod_{k > 2N} C^{\mca{L}}(a,k)_*$.
Thus $\partial_{x^0}$ acts on $\prod_{1 \le k \le 2N} C^{\mca{L}}(a,k)_*$,
preserving the filtration
$\biggl( \prod_{i \le k \le 2N} C^{\mca{L}}(a,k)_* \biggr)_{1 \le i \le 2N}$.
Then the $E_1$-term is
\[
H_*(C^{\mca{L}}(a,k)) \cong H^\dR_{*+n+\mu(a)+k-1}(\mca{L}_{k+1}(a)) \cong H_{*+n+\mu(a)+k-1}(\mca{L}(a)).
\]
Let us compute $d_1: H_*(C^{\mca{L}}(a,k)) \to H_{*-1}(C^{\mca{L}}(a,k+1))$.
By direct computations
\begin{align*}
&[x^0,y](a, k+1) = \\
&\qquad (-1)^{|y|} \biggl (x(0,2) \circ_2 y(a,k) + \sum_{1 \le i \le k} (-1)^i y(a,k) \circ_i x(0,2) + (-1)^{k+1} x(0,2) \circ_1 y(a,k) \biggr).
\end{align*}
Since $[x(0,2)] \in H^\dR_n(\mca{L}_3(0))$ corresponds to $(-1)^{n+1} [L] \in H_n(\mca{L})$
and the fiber product on homology level corresponds to the loop product (Lemma \ref{170623_1}),
$d_1$ coincides with $\pm \sum_{0 \le i \le k+1} (-1)^i$.
Then all $E_2$-terms vanish, thus $\prod_{1 \le k \le 2N} C^{\mca{L}}(a,k)_*$ is acyclic.
Finally, by taking an inverse limit (Theorem 3.5.8 in \cite{Weibel_94}) we have shown that
$\prod_{k \ge 1} C^{\mca{L}}(a,k)_*$ is acyclic.
\end{proof}
\begin{lem}
$x^+:= x - x^0$ satisfies equations
\[
\partial_{x^0} x^+ - \frac{1}{2}[x^+, x^+] = 0, \qquad \partial_{x^0} y - [x^+, y] = z.
\]
\end{lem}
\begin{proof}
Straightforward from $\partial x - \frac{1}{2}[x,x]=0$, $\partial y - [x,y]=z$ and $x= x^0 + x^+$.
\end{proof}
Let $H^{\mca{L}}_* = \bigoplus_{a \in H_1(L: \Z)} H_{*+n+\mu(a)-1}(\mca{L}(a))$
as in Section 3. Then there exist linear maps
\[
\iota: H^{\mca{L}}_* \to C^{\mca{L}}_*, \quad
\pi: C^{\mca{L}}_* \to H^{\mca{L}}_*, \quad
\kappa: C^{\mca{L}}_* \to C^{\mca{L}}_{*+1}
\]
such that
\[
\partial_{x^0} \circ \iota = 0, \quad
\pi \circ \partial_{x^0} = 0, \quad
\pi \circ \iota = \id_{H^{\mca{L}}}, \quad
\kappa \circ \partial_{x^0} + \partial_{x^0} \circ \kappa = \id_{C^{\mca{L}}} - \iota \circ \pi.
\]
One can take these maps so that they preserve decompositions of $H^{\mca{L}}$ and $C^{\mca{L}}$
over $H_1(L: \Z)$, thus they extend to the completions.
Also, one can take $\pi$ so that it maps $\sum_{k \ge 0} z(0,k)$ to $(-1)^{n+1} [L] \in H_n(\mca{L}(0))$.
Using homotopy transfer theorem (Thoerem \ref{170623_2}),
one can define an $L_\infty$-structure $l^H$ on $H^{\mca{L}}_*$
and an $L_\infty$-homomorphism
$p$ from $(C^{\mca{L}}_*, \partial_{x^0}, [\, , \,])$ to
$(H^{\mca{L}}_*, l^H)$ so that $l^H_1=0$ and $p_1 = \pi$.
We can take $l^H$ and $p$ so that they respect decompositions over $H_1(L:\Z)$,
thus they extend to the completions.
Applying Proposition 4.9 in \cite{Latschev_15},
elements in $\wh{H}^{\mca{L}}$
\begin{align*}
X&:= \sum_{k=1}^\infty \frac{1}{k!} p_k(x^+, \ldots, x^+) \in \wh{H}^{\mca{L}}_{-1} \\
Y&:= \sum_{k=1}^\infty \frac{1}{(k-1)!} p_k(y, x^+, \ldots, x^+) \in \wh{H}^{\mca{L}}_2, \\
Z&:= \sum_{k=1}^\infty \frac{1}{(k-1)!} p_k(z, x^+, \ldots, x^+) \in \wh{H}^{\mca{L}}_1,
\end{align*}
satisfy
\begin{align*}
&\sum_{k=2}^\infty \frac{1}{k!} l^H_k(X, \ldots, X) = 0, \\
&\sum_{k=2}^\infty \frac{1}{(k-1)!} l^H_k(Y, X, \ldots, X) = Z.
\end{align*}
Notice that infinite sums in the definitions of $X$, $Y$, $Z$ make sense
since $x^+(a) \ne 0$ only if $\omega(\bar{a}) \ge 2\ep$ and
$p$ preserves decompositions over $H_1(L: \Z)$.
Moreover $X(a) \ne 0$ only if $\omega(\bar{a}) \ge 2\ep$,
thus Theorem \ref{161011_1} (iii) holds with $c:= 2\ep$.
To complete the proof of Theorem \ref{161011_1},
we need to show that $Z(0) = (-1)^{n+1} [L]$.
Since $p$ respects decompositions over $H_1(L: \Z)$
and $z(a,k) \ne 0$ only if $\omega(\bar{a}) \ge 2\ep$ or $a=0$, we obtain
\[
Z(0) = \pi \biggl( \sum_{k=0}^\infty z(0,k) \biggr) = (-1)^{n+1} [L].
\]
\section{Sequence of approximate solutions}
To prove Theorem \ref{161214_2}, we have to define chains $x$, $y$, $z$ satisfying the equations
\begin{equation}\label{170830_1}
\partial x - \frac{1}{2} [x,x] = 0, \qquad \partial y - [x,y] = z.
\end{equation}
These chains are defined from virtual fundamental chains of moduli spaces of pseudo-holomorphic disks with boundaries on $L$.
However, it is difficult to get such chains in \textit{one step} since it will involve simultaneous perturbations of Kuranishi maps of infinitely many moduli spaces.
Due to this difficulty, we first define a sequence $(x_i, y_i, z_i)_i$ such that the following conditions hold for every $i$:
\begin{itemize}
\item The tuple $(x_i, y_i, z_i)$ satisfies equations (\ref{170830_1}) up to certain energy level which goes to $\infty$ as $i \to \infty$.
\item Tuples $(x_i, y_i, z_i)$ and $(x_{i+1}, y_{i+1}, z_{i+1})$ are ``gauge equivalent'' up to certain energy level which goes to $\infty$ as $i \to \infty$.
\end{itemize}
By the second condition, limits $x:= \lim_{i \to \infty} x_i$, $y:= \lim_{i \to \infty} y_i$, $z:= \lim_{i \to \infty} z_i$ exist,
and by the first condition these limits satisfy (\ref{170830_1}).
This procedure is similar to the construction of the $A_\infty$-structure in Lagrangian Floer theory as a limit of $A_K$-structures;
see Section 7.2.3 in \cite{FOOO_09}, and Remark 22.27 in \cite{FOOO_Kuranishi}.
The goal of this section is to explain details of the algebraic procedure sketched above;
we reduce Theorem \ref{161214_2} to Theorem \ref{161215_1},
which asserts the existence of a sequence of approximate solutions connected by gauge equivalences.
The proof of Theorem \ref{161215_1} involves construction of virtual fundamental chains using the theory of Kuranishi structures,
which is carried out in Sections 7--9.
Let $J$ be the standard complex structure on $\C^n$,
and take $\ep>0$ so that
$2 \ep$ is less than the minimal symplectic area of nonconstant $J$-holomorphic disks with boundaries on $L$.
We fix such $\ep$ in the following arguments.
For each $m \in \Z$, we set
\[
F^m C^{\mca{L}}_*:= \bigoplus_{\substack{ a \in H_1(L:\Z) \\ k \in \Z_{\ge 0} \\ \omega_n(\bar{a}) \ge \ep(m+1-k)}} C^{\mca{L}}(a,k)_*.
\]
Obviously $\partial F^m \subset F^m$.
We can also show $[F^m, F^{m'}] \subset F^{m+m'}$ since
\[
\omega_n(\bar{a}) \ge \ep (m+1-k) , \,
\omega_n(\bar{a'}) \ge \ep (m'+1-k') \implies
\omega_n(\bar{a} + \bar{a'}) \ge \ep (m+ m' + 1 - (k+k'-1)).
\]
The filtration on $\bar{C}^{\mca{L}}$ (see Section 4.5)
is defined in a similar manner.
These filtrations extend to the completions.
In the following we abbreviate $C^{\mca{L}}$ and $\bar{C}^{\mca{L}}$ by $C$ and $\bar{C}$.
\begin{thm}\label{161215_1}
There exist integers $I, U \ge 2$
and a sequence $(x_i, y_i, z_i, \bar{x}_i, \bar{y}_i, \bar{z}_i)_{i \ge I}$
satisfying the following conditions for every $i \ge I$:
\[
x_i \in F^1 C_{-1}, \,
\bar{x}_i \in F^1 \bar{C}_{-1}, \,
y_i \in F^{-U} C_2, \,
\bar{y}_i \in F^{-U} \bar{C}_2, \,
z_i \in F^{-1} C_1, \,
\bar{z}_i \in F^{-1} \bar{C}_1.
\]
\[
x_i = e_-(\bar{x}_i), \quad
y_i = e_-(\bar{y}_i), \quad
z_i = e_-(\bar{z}_i).
\]
\[
\partial \bar{x}_i - \frac{1}{2} [\bar{x}_i, \bar{x}_i] \in F^i \bar{C}_{-2}, \quad
\partial \bar{y}_i - [\bar{x}_i, \bar{y}_i] - \bar{z}_i \in F^{i-U-1}\bar{C}_1, \quad
\partial \bar{z}_i - [\bar{x}_i, \bar{z}_i] \in F^{i-2} \bar{C}_0.
\]
\[
x_{i+1} - e_+(\bar{x}_i) \in F^iC_{-1}, \quad
y_{i+1} - e_+(\bar{y}_i) \in F^{i-U-1}C_2, \quad
z_{i+1} - e_+(\bar{z}_i) \in F^{i-2}C_1.
\]
\begin{itemize}
\item
$x_i(a,k) \ne 0$ only if $\omega(\bar{a}) \ge 2\ep$ or $a=0$, $k \ge 2$.
Moreover,
$x_i(0,2)$ is a cycle in $C^\dR_n(\mca{L}_3(0))$
such that $[x_i(0,2)]$ corresponds to $(-1)^{n+1} [L]$
via the isomorphism $H^\dR_n(\mca{L}_3(0)) \cong H_n(\mca{L})$.
\item
$z_i(a,k) \ne 0$ only if $\omega(\bar{a}) \ge 2\ep$ or $a=0$.
Moreover,
$z_i(0,0)$ is a cycle in $C^\dR_n(\mca{L}_1(0))$
such that $[z_i(0,0)]$ corresponds to $(-1)^{n+1} [L]$
via the isomorphism $H^\dR_n(\mca{L}_1(0)) \cong H_n(\mca{L})$.
\end{itemize}
\end{thm}
\begin{rem}
$\partial z - [x,z]=0$ follows from $\partial x - \frac{1}{2}[x,x]=0$ and $\partial y - [x,y]=z$,
thus the assumption $\partial \bar{z}_i - [\bar{x}_i, \bar{z}_i] \in F^{i-2}$
in Theorem \ref{161215_1} may seem redundant.
However, this assumption is necessary to carry out the induction argument.
\end{rem}
The rest of this section is devoted to
the proof of Theorem \ref{161214_2} assuming Theorem \ref{161215_1}.
We first need the following elementary lemma.
\begin{lem}\label{161221_1}
Let $V_*$, $W_*$ be chain complexes and $e: V_* \to W_*$ be a surjective quasi-isomorphism.
For any $x \in V_*$ and $y \in W_{*+1}$ such that
\[
\partial x = 0, \qquad e(x)= \partial y
\]
there exists $\bar{y} \in V_{*+1}$ such that $e(\bar{y}) = y$ and $\partial \bar{y}=x$.
\end{lem}
\begin{proof}
Since $e$ is surjective, there exists $z \in V_{*+1}$ such that $e(z)=y$.
Then $e(x - \partial z) = 0$ and $\partial (x- \partial z) = 0$.
Thus $x - \partial z$ is a cycle in $\Ker e$.
Since $\Ker e$ is acyclic, there exists $w \in \Ker e$ such that
$\partial w = x - \partial z$.
Then, $\bar{y}:= z+w$ satisfies the condition of the lemma.
\end{proof}
Key arguments are summarized in Lemma \ref{161216_1} below.
\begin{lem}\label{161216_1}
Let $I$, $U$ be as in Theorem \ref{161215_1}.
There exists a sequence
\[
(x_{i,j}, y_{i,j}, z_{i,j}, \bar{x}_{i,j}, \bar{y}_{i,j}, \bar{z}_{i,j})_{i \ge I, \, j \ge 0}
\]
satisfying the following conditions for every $i \ge I$ and $j \ge 0$:
\begin{equation}\label{170627_1}
x_{i,0} = x_i, \quad
y_{i,0} = y_i, \quad
z_{i,0} = z_i, \quad
\bar{x}_{i,0} = \bar{x}_i, \quad
\bar{y}_{i,0} = \bar{y}_i, \quad
\bar{z}_{i,0} = \bar{z}_i.
\end{equation}
\begin{eqnarray}
&x_{i,j} \in F^1 C_{-1}, \quad \bar{x}_{i,j} \in F^1 \bar{C}_{-1}, \quad y_{i,j} \in F^{-U} C_2, \\
&\bar{y}_{i,j} \in F^{-U} \bar{C}_2, \quad z_{i,j} \in F^{-1} C_1, \quad \bar{z}_{i,j} \in F^{-1} \bar{C}_1. \nonumber
\end{eqnarray}
\begin{equation}\label{170627_2}
x_{i,j} = e_-(\bar{x}_{i,j}), \quad
y_{i,j} = e_-(\bar{y}_{i,j}), \quad
z_{i,j} = e_-(\bar{z}_{i,j}).
\end{equation}
\begin{eqnarray}\label{170627_3}
&\partial \bar{x}_{i,j} - \frac{1}{2} [ \bar{x}_{i,j}, \bar{x}_{i,j} ] \in F^{i+j}\bar{C}_{-2}, \qquad \partial \bar{y}_{i,j} - [ \bar{x}_{i,j} , \bar{y}_{i,j} ] - \bar{z}_{i,j} \in F^{i+j-U-1}\bar{C}_1. \\
&\partial \bar{z}_{i,j} - [ \bar{x}_{i,j} , \bar{z}_{i,j}] \in F^{i+j-2}\bar{C}_0. \nonumber
\end{eqnarray}
\begin{eqnarray}
&x_{i+1, j} - e_+(\bar{x}_{i,j}) \in F^{i+j}C_{-1}, \qquad y_{i+1, j} - e_+(\bar{y}_{i,j}) \in F^{i+j-U-1}C_2, \\
&z_{i+1, j} - e_+(\bar{z}_{i,j}) \in F^{i+j-2}C_1. \nonumber
\end{eqnarray}
\begin{equation}
\bar{x}_{i, j+1} - \bar{x}_{i,j} \in F^{i+j}\bar{C}_{-1}, \quad
\bar{y}_{i, j+1} - \bar{y}_{i,j} \in F^{i+j-U-1}\bar{C}_2, \quad
\bar{z}_{i, j+1} - \bar{z}_{i,j} \in F^{i+j-2}\bar{C}_1.
\end{equation}
Moreover we require the following conditions:
\begin{itemize}
\item
$\bar{x}_{i,j}(a,k) \ne 0$ only if $\omega(\bar{a}) \ge 2\ep$ or $a=0$, $k \ge 2$.
Moreover, $\bar{x}_{i,j}(0,2)$ is a cycle in $C^\dR_n(\mca{L}_3(0))$
such that $[\bar{x}_{i,j}(0,2)]$ corresponds to $(-1)^{n+1} [L]$.
\item
$\bar{z}_{i,j}(a,k) \ne 0$ only if $\omega(\bar{a}) \ge 2\ep$ or $a=0$.
Moreover, $\bar{z}_{i,j}(0,0)$ is a cycle in $C^\dR_n(\mca{L}_1(0))$
such that $[\bar{z}_{i,j}(0,0)]$ corresponds to $(-1)^{n+1} [L]$.
\end{itemize}
\end{lem}
\begin{proof}
The proof is by induction on $j$.
Let us define $x_{i,0}, \ldots, \bar{z}_{i,0}$ by (\ref{170627_1}).
Assuming that we have defined a sequence $(x_{i,j}, \ldots, \bar{z}_{i,j})_{i \ge I}$
which satisfies the required conditions,
we are going to define a sequence $(x_{i,j+1}, \ldots, \bar{z}_{i, j+1})_{i \ge I}$.
Let us set
\begin{align*}
\Delta^i_x &:= x_{i+1, j} - e_+(\bar{x}_{i,j}) \in F^{i+j} C_{-1}, \\
\Delta^i_y &:= y_{i+1, j} - e_+(\bar{y}_{i,j}) \in F^{i+j-U-1} C_2, \\
\Delta^i_z &:= z_{i+1, j} - e_+(\bar{z}_{i,j}) \in F^{i+j-2} C_1.
\end{align*}
Since $e_-$ preserves $\partial$, $[\, , \,]$ and filtrations,
(\ref{170627_2}) and (\ref{170627_3}) show that
\begin{align*}
&\partial x_{i+1, j} - \frac{1}{2} [ x_{i+1, j}, x_{i+1, j} ] \in F^{i+j+1}C_{-2}, \\
&\partial y_{i+1, j} - [ x_{i+1, j}, y_{i+1, j}] - z_{i+1, j} \in F^{i+j-U}C_1, \\
&\partial z_{i+1, j} - [x_{i+1, j} , z_{i+1, j}] \in F^{i+j-1}C_{-2}.
\end{align*}
Then we obtain
\begin{align*}
&\partial \Delta^i_x + e_+(\partial \bar{x}_{i,j} - \frac{1}{2}[\bar{x}_{i,j} , \bar{x}_{i,j}] ) \in F^{i+j+1}C_{-2}, \\
&\partial \Delta^i_y + e_+(\partial \bar{y}_{i,j} - [\bar{x}_{i,j}, \bar{y}_{i,j}] - \bar{z}_{i,j}) \in F^{i+j-U}C_1, \\
&\partial \Delta^i_z + e_+(\partial \bar{z}_{i,j} - [\bar{x}_{i,j} , \bar{z}_{i,j}]) \in F^{i+j-1}C_0.
\end{align*}
On the other hand, we obtain
\begin{align*}
\partial \bigg( \partial \bar{x}_{i,j} - \frac{1}{2}[\bar{x}_{i,j}, \bar{x}_{i,j}] \bigg) &=
-\frac{1}{2} \bigg( \bigg[\partial \bar{x}_{i,j} - \frac{1}{2}[\bar{x}_{i,j}, \bar{x}_{i,j} ] , \bar{x}_{i,j} \bigg] - \bigg[ \bar{x}_{i,j}, \partial \bar{x}_{i,j} - \frac{1}{2}[\bar{x}_{i,j}, \bar{x}_{i,j} ] \bigg] \bigg) \in F^{i+j+1}\bar{C}_{-3}, \\
\partial (\partial \bar{y}_{i,j} - [ \bar{x}_{i,j} , \bar{y}_{i,j}] - \bar{z}_{i,j})
&= - \bigg[ \partial \bar{x}_{i,j} - \frac{1}{2}[\bar{x}_{i,j}, \bar{x}_{i,j}] , \bar{y}_{i,j} \bigg] + [\bar{x}_{i,j} , \partial \bar{y}_{i,j} - [\bar{x}_{i,j}, \bar{y}_{i,j}] - \bar{z}_{i,j} ] \\
&\qquad - (\partial \bar{z}_{i,j} - [\bar{x}_{i,j} , \bar{z}_{i,j}]) \in F^{i+j-U} \bar{C}_0, \\
\partial (\partial \bar{z}_{i,j} - [ \bar{x}_{i,j} , \bar{z}_{i,j}])
&= - \bigg[ \partial \bar{x}_{i,j} - \frac{1}{2}[\bar{x}_{i,j}, \bar{x}_{i,j}], \bar{z}_{i,j} \bigg]
+ [\bar{x}_{i,j}, \partial \bar{z}_{i,j} - [\bar{x}_{i,j}, \bar{z}_{i,j}]] \in F^{i+j-1}\bar{C}_{-1}.
\end{align*}
Applying Lemma \ref{161221_1} for
\[
e_+ : F^D \bar{C}/ F^{D+1} \bar{C} \to F^D C / F^{D+1} C
\]
where $D = i+j, i+j-U-1, i+j-2$
(recall that $e_+$ is a surjective quasi-isomorphism, as we have seen in Section 4.5),
we show that there exist
\[
\bar{\Delta}^i_x \in F^{i+j} \bar{C}_{-1}, \quad
\bar{\Delta}^i_y \in F^{i+j-U-1} \bar{C}_2, \quad
\bar{\Delta}^i_z \in F^{i+j-2} \bar{C}_1
\]
such that
\[
e_+(\bar{\Delta}^i_x) - \Delta^i_x \in F^{i+j+1}C_{-1}, \quad
e_+(\bar{\Delta}^i_y) - \Delta^i_y \in F^{i+j-U}C_2, \quad
e_+(\bar{\Delta}^i_z) - \Delta^i_z \in F^{i+j-1}C_1
\]
and
\begin{align*}
&\partial \bar{\Delta}^i_x + ( \partial \bar{x}_{i,j} - \frac{1}{2} [\bar{x}_{i,j}, \bar{x}_{i,j}] ) \in F^{i+j+1}\bar{C}_{-2}, \\
&\partial \bar{\Delta}^i_y + (\partial \bar{y}_{i,j} - [\bar{x}_{i,j}, \bar{y}_{i,j}] - \bar{z}_{i,j}) \in F^{i+j-U}\bar{C}_1, \\
&\partial \bar{\Delta}^i_z + ( \partial \bar{z}_{i,j} - [\bar{x}_{i,j}, \bar{z}_{i,j}]) \in F^{i+j-1}\bar{C}_0.
\end{align*}
\begin{rem}\label{170304_1}
\begin{itemize}
\item
The $(a,k)$-components of $\Delta^i_x$ and $\partial \bar{x}_{i,j} + \frac{1}{2}[\bar{x}_{i,j}, \bar{x}_{i,j}]$ are nonzero only if $\omega(\bar{a}) \ge 2\ep$ or $a=0$.
Then we may take $\bar{\Delta}^i_x$ so that its $(a,k)$-component is nonzero only if $\omega(\bar{a}) \ge 2 \ep$ or $a=0$.
\item
Similarly, we can take $\bar{\Delta}^i_z$ so that
$\bar{\Delta}^i_z(a,k) \ne 0$ only if $\omega(\bar{a}) \ge 2\ep$ or $a=0$.
\item
Since $\bar{\Delta}^i_x \in F^2 \bar{C}_{-1}$ and $\bar{\Delta}^i_z \in F^0 \bar{C}_1$,
it follows that
$\bar{\Delta}^i_x(0,k)=0$ if $k=0,1,2$ and
$\bar{\Delta}^i_z(0,k)=0$ if $k=0$.
\end{itemize}
\end{rem}
Finally, let us set
\[
\bar{x}_{i,j+1} := \bar{x}_{i,j} + \bar{\Delta}^i_x, \quad
\bar{y}_{i,j+1} := \bar{y}_{i,j} + \bar{\Delta}^i_y, \quad
\bar{z}_{i,j+1} := \bar{z}_{i,j} + \bar{\Delta}^i_z
\]
and
\[
x_{i,j+1}:= e_-(\bar{x}_{i,j+1}), \quad
y_{i,j+1}:= e_-(\bar{y}_{i,j+1}), \quad
z_{i,j+1}:= e_-(\bar{z}_{i,j+1}).
\]
Now we have to check that, for every $i \ge I$ there holds
\begin{align*}
&\partial \bar{x}_{i, j+1} - \frac{1}{2} [\bar{x}_{i, j+1} , \bar{x}_{i,j+1}] \in F^{i+j+1} \bar{C}_{-2}, \\
&\partial \bar{y}_{i, j+1} - [\bar{x}_{i, j+1}, \bar{y}_{i, j+1}] - \bar{z}_{i, j+1} \in F^{i+j-U} \bar{C}_1, \\
&\partial \bar{z}_{i, j+1} - [\bar{x}_{i, j+1}, \bar{z}_{i, j+1}] \in F^{i+j-1} \bar{C}_0, \\
&x_{i+1, j+1} - e_+(\bar{x}_{i, j+1})\in F^{i+j+1} C_{-1}, \\
&y_{i+1, j+1} - e_+(\bar{y}_{i, j+1}) \in F^{i+j-U}C_{-2}, \\
&z_{i+1, j+1} - e_+(\bar{z}_{i, j+1}) \in F^{i+j-1}C_1.
\end{align*}
The first formula holds since
\[
\partial \bar{x}_{i, j+1} - \frac{1}{2} [\bar{x}_{i, j+1} , \bar{x}_{i,j+1}]
= \biggl( \partial \bar{x}_{i,j} + \partial \bar{\Delta}^i_x - \frac{1}{2}[\bar{x}_{i,j}, \bar{x}_{i,j}] \biggr) - \frac{1}{2} [\bar{\Delta}^i_x, \bar{\Delta}^i_x] - [\bar{x}_{i,j} , \bar{\Delta}^i_x]
\]
and all three terms in the RHS are in $F^{i+j+1} \bar{C}_{-2}$.
Similarly, the second and the third formulas follow from
\begin{align*}
&\partial \bar{y}_{i, j+1} - [\bar{x}_{i, j+1}, \bar{y}_{i, j+1}] - \bar{z}_{i, j+1}
= \\
&\quad (\partial \bar{\Delta}^i_y + \partial \bar{y}_{i,j} - [\bar{x}_{i,j}, \bar{y}_{i,j}] - \bar{z}_{i,j}) - [\bar{\Delta}^i_x, \bar{y}_{i,j}] - [\bar{x}_{i,j}, \bar{\Delta}^i_y] - [\bar{\Delta}^i_x, \bar{\Delta}^i_y] - \bar{\Delta}^i_z, \\
&\partial \bar{z}_{i, j+1} - [\bar{x}_{i, j+1} , \bar{z}_{i, j+1}] = \\
&\quad (\partial \bar{z}_{i, j} + \partial \bar{\Delta}^i_z - [\bar{x}_{i, j} , \bar{z}_{i, j}])
- [\bar{\Delta}^i_x, \bar{z}_{i, j}] - [\bar{x}_{i, j} , \bar{\Delta}^i_z] - [\bar{\Delta}^i_x, \bar{\Delta}^i_z].
\end{align*}
The fourth formula holds since
\begin{align*}
x_{i+1, j+1} - e_+(\bar{x}_{i,j+1}) &= (x_{i+1,j+1} - x_{i+1,j}) + (x_{i+1,j} - e_+(\bar{x}_{i,j})) + e_+(\bar{x}_{i,j} - \bar{x}_{i, j+1}) \\
&= e_-(\bar{\Delta}^{i+1}_x) + (\Delta^i_x - e_+(\bar{\Delta}^i_x)).
\end{align*}
Similarly, the fifth and the sixth formulas follow from
\begin{align*}
y_{i+1, j+1} - e_+(\bar{y}_{i, j+1})
&=(y_{i+1, j+1} - y_{i+1, j}) + (y_{i+1, j} - e_+(\bar{y}_{i,j})) + e_+(\bar{y}_{i,j} - \bar{y}_{i, j+1}) \\
&=e_-(\bar{\Delta}^{i+1}_y) + \Delta^i_y - e_+(\bar{\Delta}^i_y), \\
z_{i+1, j+1} - e_+(\bar{z}_{i, j+1})
&=(z_{i+1, j+1} - z_{i+1, j}) + (z_{i+1, j} - e_+(\bar{z}_{i,j})) + e_+(\bar{z}_{i,j} - \bar{z}_{i, j+1}) \\
&=e_-(\bar{\Delta}^{i+1}_z) + \Delta^i_z - e_+(\bar{\Delta}^i_z).
\end{align*}
Finally, by the induction hypothesis, $\bar{x}_{i,j}$ and $\bar{z}_{i,j}$ satisfy the last two conditions in the statement.
Then Remark \ref{170304_1} shows that $\bar{x}_{i, j+1}$ and $\bar{z}_{i, j+1}$ also satisfy these conditions.
\end{proof}
\begin{proof}
[\textbf{Proof of Theorem \ref{161214_2} assuming Theorem \ref{161215_1}}]
Let us fix an integer $i \ge I$.
Then for every $j \ge 0$, there holds
\[
x_{i, j+1} - x_{i,j} \in F^{i+j} C_{-1}, \quad
y_{i, j+1} - y_{i,j} \in F^{i+j-U-1} C_2, \quad
z_{i, j+1} - z_{i,j} \in F^{i+j-2} C_1.
\]
Then, the limits
\[
x:= \lim_{j \to \infty} x_{i,j} \in \wh{C}_{-1}, \quad
y:= \lim_{j \to \infty} y_{i,j} \in \wh{C}_2, \quad
z:= \lim_{j \to \infty} z_{i,j} \in \wh{C}_1
\]
exist, and satisfy
\[
\partial x - \frac{1}{2} [x,x] = 0, \qquad \partial y - [x,y] = z.
\]
Conditions (iii), (iv) in Theorem \ref{161214_2} follow from the last two conditions in Lemma \ref{161216_1}.
\end{proof}
\section{Proof of Theorem \ref{161215_1} modulo technical results}
Our arguments in the rest of this paper are based on the theory of Kuranishi structures (abbreviated as K-structures),
in particular we heavily rely on \cite{FOOO_Kuranishi} by Fukaya-Oh-Ohta-Ono.
Section 10 very briefly explains some notions in the theory of Kuranishi structures.
The goal of this section is to prove Theorem \ref{161215_1} modulo technical results,
which are proved in Sections 8 and 9.
Our proof of Theorem \ref{161215_1} is based on the following principle:
given a compact K-space $(X, \wh{\mca{U}})$
with a CF-perturbation $\wh{\mf{S}} = (\wh{\mf{S}}^\ep)_{0 < \ep \le 1}$ and a strongly smooth map $\wh{f}: (X, \wh{\mca{U}}) \to \mca{L}_{k+1}$,
one can define a de Rham chain $\wh{f}_*(X, \wh{\mca{U}}, \wh{\mf{S}}^\ep) \in C^\dR_*(\mca{L}_{k+1})$ for sufficiently small $\ep$.
In Section 7.1, we state this principle and its variants in rigorous forms.
Proofs of these results will be carried out in Section 8.
In Sections 7.2--7.6, we apply this principle to prove Theorem \ref{161215_1}.
In Section 7.2, we introduce compactified moduli spaces of (perturbed) pseudo-holomorphic disks,
and equip K-structures (with boundaries and corners) on these spaces
so that the (normalized) boundary of each moduli space is naturally isomorphic to a disjoint union
of fiber products of lower (in terms of symplectic area) moduli spaces.
We call this relation ``boundary $\cong$ fiber product'' relation.
To apply the principle in Section 7.1 to moduli spaces defined in Section 7.2,
we have to define \textit{strongly smooth} maps from these moduli spaces to spaces of \textit{smooth} loops with marked points,
so that these maps are compatible with the ``boundary $\cong$ fiber product'' relation.
The idea is to assign the boundary loop to each pseudo-holomorphic disk, however we have to be careful to achieve smoothness,
which can be very subtle on boundary of moduli spaces.
In this paper, we achieve smoothness in the following two steps.
The first step is to define \textit{strongly continuous} maps from these moduli spaces to spaces of \textit{continuous} loops with marked points.
This is much easier and we explain details in Sections 7.3 and 7.4.
The second step is to approximate (with respect to the $C^0$-topology) these continuous maps by smooth maps.
In this step we use an abstract approximation result ($C^0$-approximation lemma) which we state in Section 7.5 and prove in Section 9.
In Section 7.5, we also introduce CF-perturbations of K-structures on these moduli spaces.
Finally in Section 7.6 we complete the proof assuming technical results presented in Sections 7.1 and 7.5,
which are proved in Sections 8 and 9, respectively.
\subsection{Strongly smooth map from a K-space with a CF-perturbation gives a de Rham chain}
Let $(X, \wh{\mca{U}})$ be a K-space equipped with a differential form $\wh{\omega}$ and a CF-perturbation $\wh{\mf{S}} = (\wh{\mf{S}}^\ep)_{0 < \ep \le 1}$.
(The notion of CF-perturbation is explained in Section 7 in \cite{FOOO_Kuranishi}).
Given a strongly smooth map (see Definition \ref{170902_2} below)
$\wh{f} : (X, \wh{\mca{U}}) \to \mca{L}_{k+1}$, we define
\[
\wh{f}_*(X, \wh{\mca{U}}, \wh{\omega}, \wh{\mf{S}}^\ep ) \in C^\dR_* (\mca{L}_{k+1})
\]
for sufficiently small $\ep>0$,
and state Stokes' formula and fiber product formula.
We also consider the version that $X$ is an admissible K-space with boundaries and corners.
The goal of this subsection is to state these results in a formal manner so that we can use them to complete the proof of Theorem \ref{161215_1}.
Proofs of these results are explained in Section 8.
\subsubsection{K-space without boundary}
Let us start from the following definition:
\begin{defn}\label{170902_2}
Let $(X, \wh{\mca{U}})$ be a K-space without boundary.
A \textit{strongly smooth map} from $(X, \wh{\mca{U}})$ to $\mca{L}_{k+1}$ is a family
$\wh{f} = (f_p)_{p \in X}$ such that the following conditions hold:
\begin{itemize}
\item $f_p$ is a smooth map (in the sense of Definition \ref{171205_1}) from $U_p$ to $\mca{L}_{k+1}$ for every $p \in X$.
\item For every $p \in X$ and $q \in \Image \psi_p$,
there holds $f_p \circ \ph_{pq} = f_q|_{U_{pq}}$.
\end{itemize}
The underlying set-theoretic map $X \to \mca{L}_{k+1}$ is denoted by $f$.
\end{defn}
\begin{thm}\label{170628_1}
Let $(X, \wh{\mca{U}})$ be a compact, oriented K-space of dimension $d$
with a
strongly smooth map $\wh{f} : (X, \wh{\mca{U}}) \to \mca{L}_{k+1}$,
a differential form $\wh{\omega}$,
and a CF-perturbation $\wh{\mf{S}} = (\wh{\mf{S}}^\ep)_{0<\ep \le 1}$.
We assume that $\wh{\mf{S}}$ is transversal to $0$,
and $\ev_0 \circ \wh{f}: (X, \wh{\mca{U}}) \to L$ is strongly submersive with respect to $\wh{\mf{S}}$ (see Definition 9.2 in \cite{FOOO_Kuranishi}).
Under these assumptions, one can define a de Rham chain
\begin{equation}\label{170918_2}
\wh{f}_* (X, \wh{\mca{U}}, \wh{\omega}, \wh{\mf{S}}^\ep) \in C^\dR_{d- |\hat{\omega}|} (\mca{L}_{k+1})
\end{equation}
for sufficiently small $\ep$,
so that Stokes' formula (Theorem \ref{170628_2}) and
the fiber product formula (Theorem \ref{170628_3}) hold.
\end{thm}
\begin{rem}\label{170918_3}
When $\wh{\omega} \equiv 1$, we abbreviate
the LHS of (\ref{170918_2})
as $\wh{f}_*(X, \wh{\mca{U}}, \wh{\mf{S}}^\ep)$.
\end{rem}
\begin{rem}
In the statement of Theorem \ref{170628_1},
``for sufficiently small $\ep$'' is used slightly loosely.
Strictly speaking, it means the following:
the definition of $\wh{f}_*(X, \wh{\mca{U}}, \wh{\omega}, \wh{\mf{S}}^\ep)$ involves
some auxiliary choices (good coordinate system and partition of unity),
however it is well-defined in the sense of $\spadesuit$ (see Definition \ref{180301_1}).
Namely, for any choices $c_1$ and $c_2$, there exists $\ep(c_1, c_2)>0$ such that,
the definition with $c_1$ coincides with the definition with $c_2$ when $\ep \in (0, \ep(c_1, c_2))$.
A similar remark applies to all places in Section 7.1 where we use
``for sufficiently small $\ep$''.
\end{rem}
Stokes' formula is easy to state:
\begin{thm}\label{170628_2}
For sufficiently small $\ep >0$, there holds
\[
\partial
(\wh{f}_* (X, \wh{\mca{U}}, \wh{\omega}, \wh{\mf{S}}^\ep)) =
(-1)^{|\wh{\omega}|+1} \wh{f}_*(X, \wh{\mca{U}}, d \wh{\omega}, \wh{\mf{S}}^\ep).
\]
\end{thm}
To state the fiber product formula,
we need some notations.
Suppose, for $i=1,2$, we have the following data:
\begin{itemize}
\item $(X_i, \wh{\mca{U}}_i)$: a compact oriented K-space of dimension $d_i$,
\item a strongly smooth map $\wh{f}_i: (X_i, \wh{\mca{U}_i}) \to \mca{L}_{k_i+1}$,
\item a differential form $\wh{\omega}_i$ on $(X_i, \wh{\mca{U}}_i )$,
\item a CF-perturbation $\wh{\mf{S}}_i $ on $(X_i, \wh{\mca{U}}_i )$ such that $\ev_0 \circ \wh{f}_i: (X_i, \wh{\mca{U}}_i) \to L$ is strongly submersive with respect to $\wh{\mf{S}}_i$.
\end{itemize}
Due to Theorem \ref{170628_1}, one can define
$(\wh{f}_i)_*(X_i, \wh{\mca{U}}_i, \wh{\omega}_i, \wh{\mf{S}}^\ep_i) \in C^\dR_{d_i - |\wh{\omega}_i|} (\mca{L}_{k_i+1})$
for sufficiently small $\ep$.
On the other hand, for every $j \in \{1, \ldots, k_1\}$,
one can take a fiber product of K-spaces and define
\[
(X_{12}, \wh{\mca{U}}_{12}) := (X_1, \wh{\mca{U}}_1) \fbp{\ev_j \circ \wh{f}_1}{\ev_0 \circ \wh{f}_2} (X_2, \wh{\mca{U}}_2).
\]
For the definition of fiber product of K-spaces, see Section 4.1 in \cite{FOOO_Kuranishi}.
Our sign convention for the fiber product is explained in Section 4.2.
One can also define fiber product of CF-perturbations $\wh{\mf{S}}_1 \times \wh{\mf{S}}_2$
on $(X_{12}, \wh{\mca{U}}_{12})$ (see Definition 10.13 in \cite{FOOO_Kuranishi}).
Finally we define a differential form $\wh{\omega}_{12}$ on $(X_{12}, \wh{\mca{U}}_{12})$ by
\[
\wh{\omega}_{12} := (-1)^{(d_1- |\hat{\omega}_1| - n)|\hat{\omega}_2|} \cdot \wh{\omega}_1 \times \wh{\omega}_2
\]
and a strongly smooth map $\wh{f}_{12}: (X_{12}, \wh{\mca{U}}_{12}) \to \mca{L}_{k_1+k_2}$ by
\begin{eqnarray}\label{171018_1}
&(f_{12})_{(p_1, p_2)}(x_1, x_2) := \con_j ( (f_1)_{p_1}(x_1) , (f_2)_{p_2}(x_2)), \\
&(x_1 \in U_{p_1}, \, x_2 \in U_{p_2}, \, \ev_j \circ f_{p_1}(x_1) = \ev_0 \circ f_{p_2}(x_2)). \nonumber
\end{eqnarray}
Then one can state the fiber product formula
(the proof will be sketched in Section 8):
\begin{thm}\label{170628_3}
In the situation described above, there holds
\[
(\wh{f}_{12})_*(X_{12}, \wh{\mca{U}}_{12}, \wh{\omega}_{12}, \wh{\mf{S}}_{12}^\ep) =
(\wh{f}_1)_*(X_1, \wh{\mca{U}}_1, \wh{\omega}_1, \wh{\mf{S}}_1^\ep) \circ_j
(\wh{f}_2)_*(X_2, \wh{\mca{U}}_2, \wh{\omega}_2, \wh{\mf{S}}_2^\ep)
\]
for sufficiently small $\ep>0$, where the RHS is the fiber product of de Rham chains (see Section 4.3).
\end{thm}
\subsubsection{Admissible K-space}
Next we consider the case that $X$ is an admissible K-space,
which roughly means that $X$ is a K-space with boundaries and corners,
and all coordinate change data decay exponentially near boundaries.
For rigorous definitions of
admissible manifolds, vector bundles, and K-structures etc.,
see Section 25 in \cite{FOOO_Kuranishi}.
\begin{defn}\label{170701_1}
\begin{enumerate}
\item[(i):] Let $U$ be an admissible manifold.
A $C^\infty$-map
\[
f: U \to \mca{L}_{k+1}; \quad u \mapsto (T(u), \gamma(u), t_1(u), \ldots, t_k(u))
\]
is called \textit{admissible} if the following conditions hold:
\begin{itemize}
\item $T, t_1, \ldots, t_k: U \to \R$ are admissible (see Definition 25.3 in \cite{FOOO_Kuranishi}).
\item $U \times S^1 \to L; \, (u, \theta) \mapsto \gamma(u)(T(u) \theta)$ is admissible (see Remark \ref{171205_2} below).
\item $\ev_0 \circ f: U \to L$ is strata-wise submersive
(i.e. the restriction of $\ev_0 \circ f$ to each open strata of $U$ is a submersion).
\end{itemize}
\item[(ii):] Let $(X, \wh{\mca{U}})$ be an admissible K-space.
An admissible map from $(X, \wh{\mca{U}})$ to $\mca{L}_{k+1}$ is a family $\wh{f} = (f_p)_{p \in X}$ such that
\begin{itemize}
\item $f_p$ is an admissible map from $U_p$ to $\mca{L}_{k+1}$ for every $p \in X$.
\item For every $p \in X$ and $q \in \Image \psi_p$, there holds $f_p \circ \ph_{pq} = f_q|_{U_{pq}}$.
\end{itemize}
The underlying set-theoretic map $X \to \mca{L}_{k+1}$
is denoted by $f$.
\end{enumerate}
\end{defn}
\begin{rem}\label{171205_2}
In the second bullet of the definition (i) above,
the admissible structure on $U \times S^1$ is defined as follows:
let $(x_1, \ldots, x_l, y_1, \ldots, y_k) \,(x_i \in \R, y_i \in \R_{\ge 0})$
be an admissible chart on $U$ near a point on the codimension $k$ corner,
and $z$ be any chart on $S^1$.
Then we define the admissible structure on $U \times S^1$ so that
$(x_1, \ldots, x_l, y_1, \ldots, y_k, z)$ is an admissible chart on $U \times S^1$.
\end{rem}
The next result is a version of Theorem \ref{170628_1}
for admissible K-spaces.
\begin{thm}\label{170628_4}
Let $(X, \wh{\mca{U}})$ be a compact, oriented, admissible K-space of dimension $d$,
and
$\wh{f}: (X, \wh{\mca{U}}) \to \mca{L}_{k+1}$ be an admissible map,
$\wh{\omega}$ be an admissible differential form on $(X, \wh{\mca{U}})$, and
$\wh{\mf{S}}$ be an admissible CF-perturbation of $(X, \wh{\mca{U}})$.
We assume that $\wh{\mf{S}}$ is transversal to $0$, and
$\ev_0 \circ \wh{f}: (X, \wh{\mca{U}}) \to L$ is strata-wise strongly submersive with respect to $\wh{\mf{S}}$.
Then one can define
\[
\wh{f}_*( X, \wh{\mca{U}}, \wh{\omega}, \wh{\mf{S}}^\ep) \in C^\dR_{d- |\hat{\omega}|} (\mca{L}_{k+1})
\]
for sufficiently small $\ep>0$, so that Stokes' formula (Theorem \ref{170628_5}) and
the fiber product formula (Theorem \ref{170628_6}) hold.
\end{thm}
\begin{rem}
To define the de Rham chain $\wh{f}_*( X, \wh{\mca{U}}, \wh{\omega}, \wh{\mf{S}}^\ep)$,
we put a collar on $X$ and take an auxiliary cut-off function on the collar.
It seems that the chain $\wh{f}_*( X, \wh{\mca{U}}, \wh{\omega}, \wh{\mf{S}}^\ep)$
depends on choice of the cut-off function.
See Section 8.1.2 for more details.
\end{rem}
Now we state Stokes' formula:
\begin{thm}\label{170628_5}
In the situation of Theorem \ref{170628_4}, for sufficiently small $\ep >0$ there holds
\[
\partial (\wh{f}_*(X, \wh{\mca{U}}, \wh{\omega}, \wh{\mf{S}}^\ep)) =
(-1)^{|\wh{\omega}|}
\wh{f}_*(\partial X , \wh{\mca{U}}|_{\partial X}, \wh{\omega}|_{\partial X}, \wh{\mf{S}}^\ep|_{\partial X}) + (-1)^{|\wh{\omega}|+1}\wh{f}_*(X, \wh{\mca{U}}, d \wh{\omega}, \wh{\mf{S}}^\ep),
\]
where $\partial X$ denotes the normalized boundary of $X$,
which is again an admissible K-space.
\end{thm}
Next we state the fiber product formula.
Suppose, for $i=1, 2$, we have the following data:
\begin{itemize}
\item An admissible K-space $(X_i, \wh{\mca{U}}_i)$.
\item An admissible map $\wh{f}_i: (X_i, \wh{\mca{U}}_i) \to \mca{L}_{k_i+1}$.
\item An admissible differential form $\wh{\omega}_i$ on $(X_i, \wh{\mca{U}}_i)$.
\item An admissible CF-perturbation $\wh{\mf{S}}_i$ on $(X_i, \wh{\mca{U}}_i)$ such that
$\ev_0 \circ \wh{f}_i: (X_i, \wh{\mca{U}}_i) \to L$ is strata-wise strongly submersive with respect to $\wh{\mf{S}}_i$.
\end{itemize}
Under these assumptions, for every $j \in \{1, \ldots, k_1 \}$,
the fiber product
\[
(X_{12}, \wh{\mca{U}}_{12}) := (X_1, \wh{\mca{U}}_1) \fbp{\ev_j \circ f_1}{\ev_0 \circ f_2} (X_2, \wh{\mca{U}}_2)
\]
has an admissible K-structure.
Moreover, both $\wh{\omega}_1 \times \wh{\omega}_2$ and $\wh{\mf{S}}_1 \times \wh{\mf{S}}_2$ are admissible.
Finally, a strongly smooth map $\wh{f}_{12}: (X_{12}, \wh{\mca{U}}_{12}) \to \mca{L}_{k_1+k_2}$
which is defined by the same formula as (\ref{171018_1}) is also admissible.
\begin{thm}\label{170628_6}
In the situation described above, there holds
\[
(\wh{f}_{12})_*(X_{12}, \wh{\mca{U}}_{12}, \wh{\omega}_{12}, \wh{\mf{S}}_{12}^\ep) =
(\wh{f}_1)_*(X_1, \wh{\mca{U}}_1, \wh{\omega}_1, \wh{\mf{S}}_1^\ep) \circ_j
(\wh{f}_2)_*(X_2, \wh{\mca{U}}_2, \wh{\omega}_2, \wh{\mf{S}}_2^\ep)
\]
for sufficiently small $\ep>0$.
\end{thm}
\subsubsection{Admissible K-space over an interval}
Finally we consider the case that $X$ is an admissible K-space and
the target of the map $f$ is $[a,b] \times \mca{L}_{k+1}$.
\begin{defn}\label{170701_2}
\begin{enumerate}
\item[(i):] Let $U$ be an admissible manifold.
A $C^\infty$-map $f: U \to [a, b] \times \mca{L}_{k+1}$
is called \textit{admissible} if the following conditions hold:
\begin{itemize}
\item $\pr_{\mca{L}_{k+1}} \circ f: U \to \mca{L}_{k+1}$ is admissible in the sense of Definition \ref{170701_1}.
\item Let us denote $\tau:= \pr_{[a,b ]} \circ f$, and suppose that $\tau(p)=a$ (resp. $\tau(p)=b$) for $p \in U$.
Then $p$ is on codimension $k \ge 1$
corner of $U$,
and there exists an admissible chart
$(y, t_1, \ldots, t_k) \,(t_i \in \R_{\ge 0})$ defined near $p$,
such that
$p$ corresponds to $(y_0, 0, \ldots, 0)$ and
$\tau(y, t_1, \ldots, t_k) = t_1 + a$ (resp. $b-t_1$).
\item $(\tau , \ev_0 \circ \pr_{\mca{L}_{k+1}} \circ f): U \to [a,b] \times L$ is a corner stratified submersion (see Definition 26.3 in \cite{FOOO_Kuranishi}).
\end{itemize}
\item[(ii):]
Let $(X, \wh{\mca{U}})$ be an admissible K-space.
An admissible map from $(X, \wh{\mca{U}})$ to $[a, b] \times \mca{L}_{k+1}$
is a family $\wh{f} = (f_p)_{p \in X}$ such that
\begin{itemize}
\item $f_p$ is an admissible map from $U_p$ to $[a, b] \times \mca{L}_{k+1}$ for every $p \in X$.
\item For any $p \in X$ and $q \in \Image \psi_p$, there holds $f_p \circ \ph_{pq} = f_q|_{U_{pq}}$.
\end{itemize}
The underlying set-theoretic map
$X \to [a,b] \times \mca{L}_{k+1}$
is denoted by $f$.
\end{enumerate}
\end{defn}
In the situation of Definition \ref{170701_2} (ii), the normalized boundary $\partial X$ is decomposed as
$\partial X = \partial_v X \sqcup \partial_h X$, where
$\partial_v X$ (resp. $\partial_h X$) denotes the \textit{vertical} (resp. \textit{horizontal}) boundary
(see Definition 26.10 in \cite{FOOO_Kuranishi}).
Moreover, $\partial_v X$ is decomposed into
$\part_- X:= f^{-1}( \{a\} \times \mca{L}_{k+1})$ and
$\part_+ X:= f^{-1}(\{b\} \times \mca{L}_{k+1})$.
\begin{thm}\label{170701_3}
Let $(X, \wh{\mca{U}})$ be a compact, oriented, admissible K-space of dimension $d$,
$\wh{f}$ be an admissible map (in the sense of Definition \ref{170701_2})
from $(X, \wh{\mca{U}})$ to $[a, b] \times \mca{L}_{k+1}$,
$\wh{\omega}$ be an admissible differential form on $(X, \wh{\mca{U}})$,
$\wh{\mf{S}}$ be an admissible CF-perturbation of $(X, \wh{\mca{U}})$.
Assume that $\wh{\mf{S}}$ is transversal to $0$, and
\[
(\pr_{[a,b]} \circ \wh{f} , \ev_0 \circ \pr_{\mca{L}_{k+1}} \circ \wh{f}): (X, \wh{\mca{U}}) \to [a,b] \times L
\]
is a stratified submersion with respect to $\wh{\mf{S}}$.
Then one can define
\[
\wh{f}_*(X, \wh{\mca{U}}, \wh{\omega}, \wh{\mf{S}}^\ep) \in \bar{C}^\dR_{d- |\wh{\omega}| - 1} (\mca{L}_{k+1})
\]
for sufficiently small $\ep>0$, so that there holds
\begin{equation}
e_+(\wh{f}_*(X, \wh{\mca{U}}, \wh{\omega}, \wh{\mf{S}}^\ep)) = (\wh{f}|_{\partial_+ X})_* (\partial_+ X, \wh{\mca{U}}|_{\partial_+ X}, \wh{\omega}|_{\partial_+ X}, \wh{\mf{S}}^\ep|_{\partial_+ X}),
\end{equation}
\begin{equation}\label{171206_1}
e_-(\wh{f}_*(X, \wh{\mca{U}}, \wh{\omega}, \wh{\mf{S}}^\ep)) = (\wh{f}|_{\partial_- X})_* (\partial_- X, \wh{\mca{U}}|_{\partial_- X}, \wh{\omega}|_{\partial_- X}, \wh{\mf{S}}^\ep|_{\partial_- X}).
\end{equation}
Moreover Stokes' formula and the fiber product formula hold.
\end{thm}
Let us state Stokes' formula:
\begin{prop}
For sufficiently small $\ep>0$, there holds
\[
\partial \wh{f}_*(X, \wh{\mca{U}}, \wh{\omega}, \wh{\mf{S}}^\ep) =
(-1)^{|\hat{\omega}|}
(\wh{f}|_{\part_h X})_*(\part_h X, \wh{\mca{U}}|_{\part_h X}, \wh{\omega}|_{\part_h X}, \wh{\mf{S}}^\ep|_{\part_h X})
+ (-1)^{|\hat{\omega}| + 1}
\wh{f}_*(X, \wh{\mca{U}}, d \wh{\omega}, \wh{\mf{S}}^\ep).
\]
\end{prop}
Finally we state the fiber product formula.
Suppose, for $i \in \{1, 2\}$, we have
$(X_i, \wh{\mca{U}}_i)$,
$\wh{\omega}_i$,
$\wh{\mf{S}}_i$,
$k_i \in \Z_{\ge 0}$ and
$\wh{f}_i: (X_i, \wh{\mca{U}}_i) \to [a,b] \times \mca{L}_{k_i+1}$ satisfying the assumptions
in Theorem \ref{170701_3}.
For every $j \in \{1, \ldots, k_1 \}$,
the fiber product
\[
(X_{12}, \wh{\mca{U}}_{12}) := (X_1, \wh{\mca{U}}_1) \fbp{\ev_j \circ \wh{f}_1}{\ev_0 \circ \wh{f}_2} (X_2, \wh{\mca{U}}_2)
\]
has an admissible K-structure.
Moreover, both $\wh{\omega}_1 \times \wh{\omega}_2$ and $\wh{\mf{S}}_1 \times \wh{\mf{S}}_2$ are admissible.
Finally, a smooth map $\wh{f}_{12}: (X_{12}, \wh{\mca{U}}_{12}) \to [a, b] \times \mca{L}_{k_1+k_2}$
which is defined by the same formula as (\ref{171018_1}) is also admissible.
\begin{thm}\label{171018_2}
In the situation described above, there holds
\[
(\wh{f}_{12})_*(X_{12}, \wh{\mca{U}}_{12}, \wh{\omega}_{12}, \wh{\mf{S}}_{12}^\ep) =
(\wh{f}_1)_*(X_1, \wh{\mca{U}}_1, \wh{\omega}_1, \wh{\mf{S}}_1^\ep) \circ_j
(\wh{f}_2)_*(X_2, \wh{\mca{U}}_2, \wh{\omega}_2, \wh{\mf{S}}_2^\ep)
\]
for sufficiently small $\ep>0$.
\end{thm}
\subsection{Moduli spaces of (perturbed) pseudo-holomorphic disks with boundary marked points}
\subsubsection{Uncompactified moduli spaces}
Let $D$ denote the unit holomorphic disk, namely $D:= \{ z \in \C \mid |z| \le 1\}$.
For every $k \in \Z_{\ge 0}$ and $\beta \in H_2(\C^n, L)$, we define a set $\mtrg{\mca{M}}_{k+1}(\beta)$
in the following way.
When $\beta=0$ and $k=0$ or $k=1$, we define
both
$\mtrg{\mca{M}}_1(0)$ and $\mtrg{\mca{M}}_2(0)$ to be empty set.
In the other cases, namely $\beta \ne 0$ or $k \ne 0, 1$, we define
\[
\mtrg{\mca{M}}_{k+1}(\beta):= \{(u,z_0, \ldots, z_k)\}/\Aut (D)
\]
where $u: (D, \partial D) \to (\C^n, L)$ satisfies $\bar{\partial} u = 0$, $[u] = \beta$, and $z_0, \ldots, z_k$ are distinct points on $\partial D$ aligned in anti-clockwise order.
The right action of $\Aut (D)$ is defined so that
\[
(u, z_0, \ldots, z_k)^\rho:= (u \circ \rho, \rho^{-1}(z_0), \ldots, \rho^{-1}(z_k)).
\]
For each $j \in \{0, \ldots, k\}$, we define an evaluation map
$\ev_j: \mtrg{\mca{M}}_{k+1}(\beta) \to L$ by
\begin{equation}\label{170610_1}
\ev_j [(u, z_0, \ldots, z_k)] := u(z_j).
\end{equation}
Next we take a Hamiltonian $H \in C^\infty_c([0,1] \times \C^n)$ which displaces $L$.
For every $t \in [0,1]$, let us set $H_t(z):= H(t,z)$,
define a Hamiltonian vector field $X_{H_t}$ by
$\omega_n( \, \cdot \, , X_{H_t}) = dH_t( \, \cdot \,)$,
and define an isotopy $(\ph^t_H)_{t \in [0,1]}$ on $\C^n$ by
\[
\ph^0_H = \id_{\C^n}, \qquad
\partial_t \ph^t_H = X_{H_t}(\ph^t_H) \quad (\forall t \in [0,1]).
\]
Then ``$H$ displaces $L$'' means that $\ph^1_H(L) \cap L = \emptyset$.
We also assume that $H_t \equiv 0$ when $t \in [0,1/3] \cup [2/3,1]$.
Let us define a family of perturbed Cauchy-Riemann operators $(\bar{\partial}_r)_{r \ge 0}$ in the following way.
We first take a $C^\infty$-function $\chi: \R \to [0,1]$ such that
$\chi \equiv 0$ on $\R_{\le 0}$
and $\chi \equiv 1$ on $\R_{\ge 1}$.
For every $r \in \R_{\ge 0}$, we define
$\chi_r(s):= \chi(r+s)\chi(-r-s)$.
In particular $\chi_0 \equiv 0$.
Let us take a complex structure $J$ on $\R \times [0,1]$ so that
$J(\partial_s) = \partial_t$, where $s$ denotes the coordinate on $\R$ and $t$ denotes the coordinate on $[0,1]$.
Next we take a biholomorphic map from $D \setminus \{-1, 1\}$ to $\R \times [0,1]$.
This defines $C^\infty$-functions
$s: D \setminus \{-1, 1\} \to \R$
and $t: D \setminus \{-1, 1\} \to [0,1]$.
For every $r \in \R_{\ge 0}$
and $u: (D, \partial D) \to (\C^n, L)$, we define
\[
\bar{\partial}_r u:= (du - X_{\chi_r(s) H_t}(u) \otimes dt)^{0,1}.
\]
Obviously $\bar{\partial}_0 = \bar{\partial}$.
Now let us define
\[
\mtrg{\mca{N}}^r_{k+1}(\beta):= \{(u, z_0=1, z_1, \ldots, z_k)\}
\]
where $u: (D, \partial D) \to (\C^n, L)$ satisfies $\bar{\partial}_r u = 0$, $[u] = \beta$, and $1, z_1, \ldots, z_k$ are distinct points on $\partial D$ aligned in anti-clockwise order.
For each $j \in \{0,\ldots, k\}$,
the evaluation map $\ev_j: \mtrg{\mca{N}}^r_{k+1}(\beta) \to L$ is defined in the same formula as (\ref{170610_1}).
Finally we define
\[
\mtrg{\mca{N}}^{\ge 0}_{k+1}(\beta):= \bigcup_{r \ge 0} \mtrg{\mca{N}}^r_{k+1}(\beta).
\]
Let us summarize basic properties of these moduli spaces:
\begin{lem}\label{170625_1}
\begin{enumerate}
\item[(i):] If $\omega(\beta)<0$ or $\omega(\beta)=0$ and $\beta \ne 0$, then $\mtrg{\mca{M}}_{k+1}(\beta) = \mtrg{\mca{N}}^0_{k+1}(\beta)=\emptyset$.
\item[(ii):] Consider the case $\beta=0$:
\begin{itemize}
\item $\mtrg{\mca{M}}_{k+1}(0)$ consists of constant maps for every $k \ge 2$.
\item $\mtrg{\mca{N}}^0_{k+1}(0)$ consists of constant maps for every $k \ge 0$.
\end{itemize}
\item[(iii):] Let $\| \, \cdot \, \|$ denote the Hofer's norm, namely
\[
\| H \| := \int_0^1 \, (\max H_t - \min H_t) \, dt.
\]
If $\omega(\beta) + 2 \| H\| < 0$, then $\mtrg{\mca{N}}^{\ge 0} _{k+1}(\beta) = \emptyset$.
\item[(iv):] First notice that, for every $k \in \Z_{\ge 0}$ and $r \in \R_{\ge 0}$,
\[
\mtrg{\mca{M}}_{k+1}(\beta) = \emptyset \iff \mtrg{\mca{M}}_1(\beta) = \emptyset, \qquad
\mtrg{\mca{N}}^r_{k+1}(\beta) = \emptyset \iff \mtrg{\mca{N}}^r_1(\beta)=\emptyset.
\]
Now for every $c \in \R$, sets
\begin{align*}
&\{ \beta \in H_2(\C^n, L) \mid \mtrg{\mca{M}}_1(\beta) \ne \emptyset, \, \omega(\beta) < c\}, \\
&\{ \beta \in H_2(\C^n, L) \mid \mtrg{\mca{N}}^r_1(\beta) \ne \emptyset, \, \omega(\beta) < c\}
\end{align*}
are both finite.
\item[(v):]
For every $\beta \in H_2(\C^n, L)$,
there exists $r(\beta)>0$ such that
$\bigcup_{r \ge r(\beta)} \mtrg{\mca{N}}^r_1(\beta) = \emptyset$.
\end{enumerate}
\end{lem}
\begin{proof}
(i), (ii) are straightforward from definitions.
(iii) follows from standard computations.
(iv) follows from Gromov compactness theorem.
If (v) is not the case, there exists
$v: (\R \times [0,1], \R \times \{0,1\}) \to (\C^n, L)$ satisfying
\[
\int_{\R \times [0,1]} | \partial_s v|^2 \, dsdt < \infty, \qquad
\partial_s v - J( \partial_t v - X_{H_t}(v)) = 0,
\]
where $J$ denotes the standard complex structure on $\C^n$.
Then $\gamma(t):= \lim_{s \to \infty} v(s,t)$ satisfies
$\gamma(0), \gamma(1) \in L$ and $\partial_t \gamma(t)= X_{H_t}(\gamma(t))$,
contradicting the assumption that $H$ displaces $L$.
\end{proof}
\subsubsection{Compactified moduli spaces}
In this subsubsection, we define compactified moduli spaces
$\mca{M}_{k+1}(\beta)$, $\mca{N}^0_{k+1}(\beta)$ and $\mca{N}^{\ge 0}_{k+1}(\beta)$,
taking fiber products of uncompactified moduli spaces along decorated rooted ribbon trees
(see Definition \ref{170902_1} below).
K-structures on these spaces are defined in the next subsubsection.
The next definition is a modified version
of Definition 21.2 in \cite{FOOO_Kuranishi}.
\begin{defn}\label{170902_1}
A \textit{decorated rooted ribbon tree} is a pair $(T, B)$ such that:
\begin{itemize}
\item $T$ is a connected tree. Let $C_0(T)$ and $C_1(T)$ be the set of all vertices and edges of $T$, respectively.
\item For each $v \in C_0(T)$ we fix a cyclic order of the set of edges containing $v$ (ribbon structure).
\item $C_0(T)$ is divided into the set of \textit{exterior} vertices $C_{0,\exterior}(T)$ and the set of \textit{interior} vertices $C_{0, \interior}(T)$.
For every $v \in C_{0, \interior}(T)$, we define $k_v$ to be the valency of $v$ minus $1$.
\item We fix one element of $C_{0, \exterior}(T)$, which we call the \textit{root}.
\item The valency of every exterior vertex is $1$.
\item $B$ is a map from $C_{0, \interior}(T)$ to $H_2(\C^n, L)$.
For every vertex $v$, either $\omega_n(B(v))>0$ or $B(v)=0$ holds.
\item Every vertex $v$ with $B(v)=0$ has valency at least $3$.
\end{itemize}
For every $k \in \Z_{\ge 0}$ and $\beta \in H_2(\C^n, L)$,
we denote by $\mca{G}(k+1, \beta)$
the set of decorated rooted ribbon trees $(T, B)$ such that:
\begin{itemize}
\item[(I):] $\# C_{0, \exterior}(T) = k+1$.
\item[(II):] $\sum_{v \in C_{0, \interior}(T)} B(v)= \beta$.
\end{itemize}
An edge is called \textit{exterior} if it contains an exterior vertex.
Otherwise it is called \textit{interior}.
$C_{1, \exterior}(T)$ (resp. $C_{1,\interior}(T)$)
denotes the set of exterior (resp. interior) edges.
\end{defn}
We also define a natural notion of \textit{reduction} on $\mca{G}(k+1: \beta)$.
\begin{defn}\label{171014_2}
Let $(T, B) \in \mca{G}(k+1: \beta)$ and $e \in C_{1, \interior}(T)$, and $v_0$, $v_1$ be vertices of $e$.
By collapsing $e$ to a new vertex $v_{01}$, we get $(T', B') \in \mca{G}(k+1: \beta)$ such that
\begin{align*}
C_0(T') &:= (C_0(T) \setminus \{v_0, v_1\}) \cup \{v_{01}\}, \\
C_1(T') &:= C_1(T) \setminus \{e\}, \\
B'(v)&:= \begin{cases} B(v) &(v \ne v_{01}) \\ B(v_0) + B(v_1) &(v = v_{01}). \end{cases}
\end{align*}
An element of $\mca{G}(k+1: \beta)$ which can be obtained from $(T, B)$ by repeating
this process is called a \textit{reduction} of $(T, B)$.
\end{defn}
Now let us define moduli spaces $\mca{M}_{k+1}(\beta)$, $\mca{N}^0_{k+1}(\beta)$ and $\mca{N}^{\ge 0}_{k+1}(\beta)$.
For every $(T, B) \in \mca{G}(k+1, \beta)$,
one can define
\begin{equation}\label{170611_1}
\ev_{\interior}: \prod_{v \in C_{0, \interior}(T)} \mtrg{\mca{M}}_{k_v+1}(B(v)) \to \prod_{e \in C_{1,\interior}(T)} L^2
\end{equation}
in the same manner as the definition of the
map (21.2) in \cite{FOOO_Kuranishi}.
We also consider
\begin{equation}\label{170611_3}
\ev_{\exterior}: \prod_{v \in C_{0, \interior}(T)} \mtrg{\mca{M}}_{k_v+1}(B(v)) \to \prod_{e \in C_{1,\exterior}(T)} L \cong L^{k+1}
\end{equation}
where the isomorphism on the right is defined
by labeling exterior vertices by $\{0, \ldots, k\}$ in positive cyclic order so that
the root is labeled by $0$.
We also consider the diagonal map:
\begin{equation}\label{170611_2}
\Delta: \prod_{e \in C_{1,\interior}(T)} L \to \prod_{e \in C_{1,\interior}(T)} L^2; \quad (x_e)_e \mapsto (x_e, x_e)_e.
\end{equation}
Then we define $\mca{M}_{k+1}(\beta)$
by taking fiber products of (\ref{170611_1}) and (\ref{170611_2}).
Namely
\[
\mca{M}_{k+1}(\beta) := \bigsqcup_{(T, B) \in \mca{G}(k+1, \beta)}
\bigg(\prod_{e \in C_{1,\interior}(T)} L \bigg)
\fbp{\Delta}{\ev_{\interior}}
\bigg(\prod_{v \in C_{0, \interior}(T)} \mtrg{\mca{M}}_{k_v+1}(B(v)) \bigg).
\]
We define $\ev^{\mca{M}} = (\ev^{\mca{M}}_0, \ldots, \ev^{\mca{M}}_k): \mca{M}_{k+1}(\beta )\to L^{k+1}$
by restricting $\ev_\exterior$.
The definition of $\mca{N}^0_{k+1}(\beta)$ is similar.
For any $(T,B) \in \mca{G}(k+1, \beta)$
and $v_0 \in C_{0, \interior}(T)$,
we define
\begin{equation}\label{170702_1}
\ev_{\interior}:
\prod_{v \in C_{0, \interior}(T) \setminus \{v_0\}} \mtrg{\mca{M}}_{k_v+1}(B(v)) \times \mtrg{\mca{N}}^0_{k_{v_0}+1} (B(v_0)) \to \prod_{e \in C_{1, \interior}(T)} L^2
\end{equation}
and
\begin{equation}\label{170702_2}
\ev_{\exterior}:
\prod_{v \in C_{0, \interior}(T) \setminus \{v_0\}} \mtrg{\mca{M}}_{k_v+1}(B(v)) \times \mtrg{\mca{N}}^0_{k_{v_0}+1} (B(v_0)) \to L^{k+1}
\end{equation}
in manners similar to (\ref{170611_1}) and (\ref{170611_3}).
Then we define:
\begin{align*}
\mca{N}^0_{k+1}(\beta)&:= \bigsqcup_{\substack{(T, B) \in \mca{G}(k+1, \beta) \\ v_0 \in C_{0, \interior}(T)}} \\
&\bigg(\prod_{e \in C_{1,\interior}(T)} L \bigg)
\fbp{\Delta}{\ev_\interior}
\bigg(
\prod_{v \in C_{0, \interior}(T) \setminus \{v_0\}} \mtrg{\mca{M}}_{k_v+1}(B(v)) \times \mtrg{\mca{N}}^0_{k_{v_0}+1} (B(v_0))
\bigg).
\end{align*}
We define $\ev^{\mca{N}^0} = (\ev^{\mca{N}^0}_0, \ldots, \ev^{\mca{N}^0}_k): \mca{N}^0_{k+1}(\beta) \to L^{k+1}$
by restring $\ev_\exterior$ defined in (\ref{170702_2}).
$\mca{N}^{\ge 0}_{k+1}(\beta)$
and
$\ev^{\mca{N}^{\ge 0}}: \mca{N}^{\ge 0}_{k+1}(\beta) \to L^{k+1}$
are defined in a similar way.
Now we have defined \textit{sets}
$\mca{M}_{k+1}(\beta)$,
$\mca{N}^0_{k+1}(\beta)$, and
$\mca{N}^{\ge 0}_{k+1}(\beta)$.
These sets have natural topologies:
the topology on $\mca{M}_{k+1}(\beta)$ is described in Section 7.1.4 \cite{FOOO_09}
(actually our situation is much simpler, since we have neither interior marked points nor sphere bubbles).
The topologies on
$\mca{N}^0_{k+1}(\beta)$ and
$\mca{N}^{\ge 0}_{k+1}(\beta)$ are defined in similar ways,
and details are omitted.
\subsubsection{K-structures on compactified moduli spaces}
In this subsubsection, we consider a system of K-structures on compactified moduli spaces defined in the previous subsubsection.
More accurately, we consider admissible K-structures (see Section 25 in \cite{FOOO_Kuranishi}),
and define a certain inductive system of such admissible K-spaces (this is a variant of ``inductive system of $A_\infty$ correspondence'' in Definition 21.17 in \cite{FOOO_Kuranishi}).
Our goal in this subsubsection is to spell out an accurate statement in Theorem \ref{170611_5}.
Recall that we took $\ep>0$ so that
every nonconstant holomorphic disk with boundary on $L$ has area at least $2\ep$.
We take $U \in \Z_{>0}$ so that
$\ep (U-1) \ge 2 \| H \|$.
\begin{thm}\label{170611_5}
For every $k \in \Z_{\ge 0}$, $m \in \Z_{\ge 0}$ and $P \in \{ \{m\}, [m, m+1]\}$,
there exist the following data:
\begin{enumerate}
\item[(i):] \textbf{(Moduli spaces)}
Compact, oriented, admissible K-spaces
\begin{align*}
&\mca{M}_{k+1}(\beta: P) \qquad (\beta \in H_2(\C^n, L), \, \omega_n(\beta) < \ep(m+1-k)), \\
&\mca{N}^0_{k+1}(\beta: P) \qquad (\beta \in H_2(\C^n, L), \, \omega_n(\beta) <\ep(m-1-k)), \\
&\mca{N}^{\ge 0}_{k+1}(\beta: P) \qquad (\beta \in H_2(\C^n, L), \, \omega_n(\beta) < \ep(m-k-U)),
\end{align*}
whose underlying topological spaces are
\[
P \times \mca{M}_{k+1}(\beta), \quad
P \times \mca{N}^0_{k+1}(\beta), \quad
P \times \mca{N}^{\ge 0}_{k+1}(\beta),
\]
respectively.
Dimensions of these K-spaces are given by
\begin{align*}
&\dim \mca{M}_{k+1}(\beta: P) = \mu(\beta) + n+k-2 + \dim P, \\
&\dim \mca{N}^0_{k+1}(\beta: P) = \mu(\beta) + n+k + \dim P, \\
&\dim \mca{N}^{\ge 0}_{k+1}(\beta: P) = \mu(\beta) + n+k+1 + \dim P.
\end{align*}
\item[(ii):]
\textbf{(Evaluation maps)}
Corner stratified strongly smooth maps
(from K-spaces to manifolds with corners, see Definition 26.6 (1) in \cite{FOOO_Kuranishi})
\begin{align*}
\ev^{\mca{M}, P}&: \mca{M}_{k+1}(\beta: P) \to P \times L^{k+1}, \\
\ev^{\mca{N}^0, P}&: \mca{N}^0_{k+1}(\beta: P) \to P \times L^{k+1}, \\
\ev^{\mca{N}^{\ge 0}, P}&:\mca{N}^{\ge 0}_{k+1}(\beta: P) \to P \times L^{k+1},
\end{align*}
such that their underlying set-theoretic maps are:
\[
\id_P \times \ev^{\mca{M}}, \quad
\id_P \times \ev^{\mca{N}^0}, \quad
\id_P \times \ev^{\mca{N}^{\ge 0}}.
\]
We require that the following maps to $P \times L$
\[
(\id_P \times \pr_0) \circ \ev^{\mca{M}, P}, \quad
(\id_P \times \pr_0) \circ \ev^{\mca{N}^0, P}, \quad
(\id_P \times \pr_0) \circ \ev^{\mca{N}^{\ge 0}, P}
\]
are corner stratified weak submersions (see Definition 26.6 (2) in \cite{FOOO_Kuranishi}).
Here $\pr_0: L^{k+1} \to L$ is defined by $(p_0, \ldots, p_k) \mapsto p_0$.
\item[(iii):] \textbf{(Energy zero part)}
An isomorphism of admissible K-structures
\[
\mca{M}_{k+1}(0: P) \cong P \times L \times D^{k-2}
\]
for every $k \ge 2$,
so that $\ev^{\mca{M}, P}: \mca{M}_{k+1}(0: P) \to P \times L^{k+1}$
coincides with $\pr_P \times (\pr_L)^{k+1}$.
Here $D^{k-2}$ in the RHS is identified with the Stasheff cell; see \cite{Fukaya_Oh} Section 10.
The coordinate near boundary is
$1/\log T$,
where $T$ denotes the length of neck region;
see Remark 25.45 in \cite{FOOO_Kuranishi}.
\item[(iv):] \textbf{(Compatibility at boundaries)}
The following isomorphisms of admissible K-spaces preserving orientations
($\partial$ in the LHS denotes normalized boundaries,
and the fiber product
$\fbp{\ev_i}{\ev_0}$ is abbreviated as $\fbp{i}{0}$):
\begin{equation}\label{170903_1}
\partial \mca{M}_{k+1}(\beta: m) \cong \bigsqcup_{\substack{k_1+k_2=k+1 \\ 1 \le i \le k_1 \\ \beta_1 + \beta_2 = \beta}} (-1)^{\ep_0}
\mca{M}_{k_1+1}(\beta_1: m) \fbp{i}{0} \mca{M}_{k_2+1}(\beta_2: m),
\end{equation}
\begin{eqnarray}\label{170903_2}
&\partial \mca{N}^0_{k+1}(\beta: m) \cong \\
&\bigsqcup_{\substack{k_1+k_2=k+1 \\ 1 \le i \le k_1 \\ \beta_1 + \beta_2 = \beta}}
\biggl( (-1)^{\ep_1} \mca{N}^0_{k_1+1}(\beta_1: m) \fbp{i}{0} \mca{M}_{k_2+1}(\beta_2: m) \nonumber \\
&\qquad\qquad \sqcup (-1)^{\ep_2} \mca{M}_{k_1+1}(\beta_1: m) \fbp{i}{0} \mca{N}^0_{k_2+1}(\beta_2: m) \biggr), \nonumber
\end{eqnarray}
\begin{eqnarray}\label{170903_3}
&\partial \mca{N}^{\ge 0} _{k+1}(\beta: m ) \cong \mca{N}^0_{k+1}(\beta:m) \sqcup \\
&\bigsqcup_{\substack{k_1+k_2=k+1 \\ 1 \le i \le k_1 \\ \beta_1 + \beta_2 = \beta}}
\biggl( (-1)^{\ep_3} \mca{N}^{\ge 0}_{k_1+1}(\beta_1: m) \fbp{i}{0} \mca{M}_{k_2+1}(\beta_2: m) \nonumber \\
&\qquad\qquad \sqcup (-1)^{\ep_4} \mca{M}_{k_1+1}(\beta_1: m) \fbp{i}{0} \mca{N}^{\ge 0}_{k_2+1}(\beta_2: m) \biggr), \nonumber
\end{eqnarray}
\begin{eqnarray}\label{170903_4}
&\partial \mca{M}_{k+1}(\beta: [m,m+1]) \cong (-1)^{\ep_5} \mca{M}_{k+1}(\beta:m) \sqcup (-1)^{\ep_6} \mca{M}_{k+1}(\beta: m+1) \, \sqcup \\
&\bigsqcup_{\substack{k_1+k_2=k+1 \\ 1 \le i \le k_1 \\ \beta_1 + \beta_2 = \beta}} (-1)^{\ep_7}
\mca{M}_{k_1+1}(\beta_1: [m, m+1]) \fbp{i}{0} \mca{M}_{k_2+1}(\beta_2: [m, m+1]), \nonumber
\end{eqnarray}
\begin{eqnarray}\label{170903_5}
&\partial \mca{N}^0_{k+1}(\beta: [m,m+1] ) \cong (-1)^{\ep_8} \mca{N}^0_{k+1}(\beta: m) \sqcup (-1)^{\ep_9} \mca{N}^0_{k+1}(\beta: m+1) \, \sqcup \\
&\bigsqcup_{\substack{k_1+k_2=k+1 \\ 1 \le i \le k_1 \\ \beta_1 + \beta_2 = \beta}}
\biggl( (-1)^{\ep_{10}} \mca{N}^0_{k_1+1}(\beta_1: [m, m+1]) \fbp{i}{0} \mca{M}_{k_2+1}(\beta_2: [m,m+1]) \nonumber \\
&\qquad\qquad \sqcup (-1)^{\ep_{11}} \mca{M}_{k_1+1}(\beta_1: [m, m+1]) \fbp{i}{0} \mca{N}^0_{k_2+1}(\beta_2: [m, m+1]) \biggr), \nonumber
\end{eqnarray}
\begin{eqnarray}\label{170903_6}
&\partial \mca{N}^{\ge 0} _{k+1}(\beta: [m,m+1]) \cong
(-1)^{\ep_{12}} \mca{N}^{\ge 0}_{k+1}(\beta: m) \sqcup (-1)^{\ep_{13}} \mca{N}^{\ge 0}_{k+1}(\beta: m+1) \\
&\sqcup (-1)^{\ep_{14}} \mca{N}^0_{k+1}(\beta: [m, m+1]) \sqcup \nonumber \\
&\bigsqcup_{\substack{k_1+k_2=k+1 \\ 1 \le i \le k_1 \\ \beta_1 + \beta_2 = \beta}}
\biggl( (-1)^{\ep_{15}} \mca{N}^{\ge 0}_{k_1+1}(\beta_1: [m,m+1] ) \fbp{i}{0} \mca{M}_{k_2+1}(\beta_2: [m,m+1]) \nonumber \\
&\qquad\qquad \sqcup (-1)^{\ep_{16}} \mca{M}_{k_1+1}(\beta_1: [m, m+1] ) \fbp{i}{0} \mca{N}^{\ge 0}_{k_2+1}(\beta_2: [m, m+1]) \biggr). \nonumber
\end{eqnarray}
The signs $\ep_0, \ldots, \ep_{16}$ are given as follows:
\begin{align*}
\ep_0 &= (k_1-i)(k_2-1) + n+ k_1, \\
\ep_1 &= \ep_0 +k+ k_1, \, \ep_2 = \ep_0 + k + k_2, \\
\ep_3 &= \ep_1 + 1, \, \ep_4 = \ep_2 +k_1 + 1, \\
\ep_5 &= 1,\, \ep_6 = 0, \, \ep_7 = \ep_0 + 1, \\
\ep_8 &= 1, \, \ep_9 = 0, \, \ep_{10} = \ep_1 + 1, \, \ep_{11} = \ep_2 + 1, \\
\ep_{12}&= 1, \, \ep_{13} = 0, \, \ep_{14}=1, \, \ep_{15} = \ep_3 + 1, \, \ep_{16} = \ep_4 + 1.
\end{align*}
\item[(v):]\textbf{(Compatibility at corners I)}
First we introduce the following notations:
\begin{itemize}
\item
For any admissible K-space $X$ and $l \in \Z_{\ge 1}$,
let $\wh{S}_l X$ denote the normalized codimension $l$ corner of $X$;
see \cite{FOOO_Kuranishi} Definition 24.17.
\item
For every nonnegative integers $d$ and $m$, we denote
\[
\wh{S}_d \{m\} := \begin{cases} \{m \} &(d=0) \\ \emptyset &(d \ge 1), \end{cases} \qquad
\wh{S}_d [m, m+1] := \begin{cases} [m, m+1] &(d=0) \\ \{m, m+1\} &(d=1) \\ \emptyset &(d \ge 2). \end{cases}
\]
\end{itemize}
Then, there are the following isomorphisms of admissible K-spaces
(\ref{170614_1}), (\ref{170614_2}), and (\ref{170614_3}):
here we do not consider orientations of moduli spaces.
\begin{align}\label{170614_1}
&\wh{S}_l \mca{M}_{k+1}(\beta: P) \cong \\
&\qquad \bigsqcup_{\substack{(T, B) \in \mca{G}(k+1,\beta) \\ \# C_{1, \interior}(T) + d = l}}
\bigg( \prod_{e \in C_{1, \interior}(T)} \wh{S}_d P \times L \bigg) \fbp{\Delta}{\ev_\interior}
\bigg( \prod_{v \in C_{0, \interior}(T)} \mca{M}_{k_v+1}(B(v): \wh{S}_d P) \bigg) \nonumber
\end{align}
where the fiber product in the RHS is taken over $\prod_{e \in C_{1, \interior}(T)} (\wh{S}_d P \times L)^2$.
Notice that the fiber product makes sense, since
\[
(\id_P \times \pr_0) \circ \ev^{\mca{M}, P}: \mca{M}_{k+1}(\beta: P) \to P \times L
\]
is a corner stratified weak submersion, as we assumed in (ii).
\begin{align}\label{170614_2}
&\wh{S}_l \mca{N}^0_{k+1}(\beta: P) \cong
\bigsqcup_{\substack{(T, B) \in \mca{G}(k+1,\beta) \\ \# C_{1, \interior}(T) + d = l \\ v_0 \in C_{0, \interior}(T)}}
\bigg( \prod_{e \in C_{1, \interior}(T)} \wh{S}_d P \times L \bigg) \fbp{\Delta}{\ev_\interior} \\
&\qquad \bigg( \prod_{v \in C_{0, \interior}(T)\setminus \{v_0\}} \mca{M}_{k_v+1}(B(v): \wh{S}_d P) \times \mca{N}_{k_{v_0}+1}^0(B(v_0): \wh{S}_d P) \bigg). \nonumber
\end{align}
\begin{align}\label{170614_3}
&\wh{S}_l \mca{N}^{\ge 0}_{k+1}(\beta: P) \cong
\bigsqcup_{\substack{(T, B) \in \mca{G}(k+1,\beta) \\ \# C_{1, \interior}(T) + d = l \\ v_0 \in C_{0, \interior}(T)}}
\bigg( \prod_{e \in C_{1, \interior}(T)} \wh{S}_d P \times L \bigg) \fbp{\Delta}{\ev_\interior} \\
&\qquad \bigg( \prod_{v \in C_{0, \interior}(T)\setminus \{v_0\}} \mca{M}_{k_v+1}(B(v): \wh{S}_d P) \times \mca{N}_{k_{v_0}+1}^{\ge 0}(B(v_0): \wh{S}_d P) \bigg) \nonumber \\
&\qquad \sqcup \, \bigsqcup_{\substack{(T, B) \in \mca{G}(k+1,\beta) \\ \# C_{1, \interior}(T)+d = l-1 \\ v_0 \in C_{0, \interior}(T)}}
\bigg( \prod_{e \in C_{1, \interior}(T)} \wh{S}_d P \times L \bigg) \fbp{\Delta}{\ev_\interior} \nonumber \\
&\qquad \bigg( \prod_{v \in C_{0, \interior}(T)\setminus \{v_0\}} \mca{M}_{k_v+1}(B(v): \wh{S}_d P) \times \mca{N}_{k_{v_0}+1}^0(B(v_0): \wh{S}_d P) \bigg). \nonumber
\end{align}
\item[(vi):]\textbf{(Compatibility at corners II)}
Let $X$ be either $\mca{M}_{k+1}(\beta: P)$, $\mca{N}^0_{k+1}(\beta: P)$ or $\mca{N}^{\ge 0}_{k+1}(\beta: P)$.
Then, for every $l, l' \in \Z_{\ge 1}$,
the canonical covering map
$\pi_{l', l}: \wh{S}_{l'}(\wh{S}_l X) \to \wh{S}_{l+l'} X$
(see Proposition 24.16 in \cite{FOOO_Kuranishi})
coincides with the map defined from the fiber product presentation in (v).
\end{enumerate}
\end{thm}
Construction of these K-structures are mostly the same as the
construction of K-structure on $\mca{M}_{k+1}(\beta)$,
which is explained in \cite{FOOO_09} Section 7.1.
Evaluation maps in (ii)
and isomorphisms in (iv), (v) naturally follow from the construction of these K-structures.
Admissibility of this K-structure is a consequence of the exponential decay estimate in \cite{FOOO_Gluing_2016}; see Section 25.5 in \cite{FOOO_Kuranishi}.
We do not spell out a detailed proof of Theorem \ref{170611_5}.
Nevertheless, in the rest of this subsubsection, we explain:
\begin{itemize}
\item Explicit description of Kuranishi charts of $\mca{M}_{k+1}(\beta: P)$, which we use in Section 7.4.
\item Total order on the set of moduli spaces, which we need to carry out inductive argument.
\item Orientations: how signs $\ep_0, \ldots, \ep_{16}$ are computed.
\end{itemize}
\textbf{Explicit description of Kuranishi charts of $\mca{M}_{k+1}(\beta: P)$.}
Let $k \in \Z_{\ge 0}$ and $\beta \in H_2(\C^n, L)$.
Let $\mca{MM}_{k+1}(\beta)$ denote the set consists of tuples $(u, z_0, z_1, \ldots, z_k)$ such that
$u: (D, \partial D) \to (\C^n, L)$ is a $C^\infty$-map such that $\bar{\partial} u = 0$ on a \textit{neighborhood of $\partial D$}
and $[u] = \beta$, and $z_0, z_1, \ldots, z_k$ are distinct points on $\partial D$ aligned in the anti-clockwise order.
\begin{rem}
To define K-structure on $\mca{M}_{k+1}(\beta)$ we consider an obstruction bundle $E$ and a perturbed Cauchy-Riemann equation $\bar{\partial}u \in E$.
One can take $E$ so that every section of $E$ is supported on a compact set of $\text{int} D$,
thus these perturbed holomorphic maps (with boundary marked points) are elements in $\mca{MM}_{k+1}(\beta)$.
\end{rem}
Now we can state the explicit description of Kuranishi chart in Lemma \ref{171014_1} below.
\begin{lem}\label{171014_1}
Let $p \in \mca{M}_{k+1}(\beta: P)$ and $\mca{U}_p = (U_p, \mca{E}_p, s_p, \psi_p)$ be a K-chart at $p$.
Let $(T, B)$ be an element in $\mca{G}(k+1: \beta)$ such that
\[
p \in P \times \bigg(\prod_{e \in C_{1,\interior}(T)} L \bigg)
\fbp{\Delta}{\ev_{\interior}}
\bigg(\prod_{v \in C_{0, \interior}(T)} \mtrg{\mca{M}}_{k_v+1}(B(v)) \bigg).
\]
Then, $U_p$ can be (set theoretically) embedded into
\[
\bigsqcup_{(T', B')}
P \times \bigg(\prod_{e \in C_{1,\interior}(T')} L \bigg)
\fbp{\Delta}{\ev_{\interior}}
\bigg(\prod_{v \in C_{0, \interior}(T')} \mca{MM}_{k_v+1}(B'(v)) \bigg)
\]
where $(T', B')$ runs over all reductions (see Definition \ref{171014_2}) of $(T, B)$.
\end{lem}
\textbf{Total order on the set of moduli spaces.}
To achieve compatibility conditions (iv), (v) and (vi) in Theorem \ref{170611_5},
we need a total order on the set of moduli spaces,
so that the following property is satisfied:
if a moduli space $X$ is a part of the normalized boundary $\partial Y$ of
another moduli space $Y$, then $X<Y$.
To achieve this property we take a total order
which is described as follows:
\begin{itemize}
\item If the $P$-part of a moduli space $X$ is $\{m\}$ and the $P$-part of a moduli space $Y$ is $[m', m'+1]$, then $X<Y$
($m$ and $m'$ are arbitrary elements in $\Z_{\ge 0}$).
\item If the $P$-part of $X$ is $\{m\}$ and the $P$-part of $Y$ is $\{m+1\}$, then $X<Y$.
\item If the $P$-part of $X$ is $[m, m+1]$ and the $P$-part of $Y$ is $[m+1, m+2]$, then $X<Y$.
\item For each $m \in \Z_{\ge 0}$, we take a total order on moduli spaces with $P$-parts equal to $\{m\}$, in the following manner:
\begin{itemize}
\item For any $k, k', k'' \in \Z_{\ge 0}$ and $\beta, \beta', \beta'' \in H_2(\C^n, L)$,
\[
\mca{M}_{k+1}(\beta: m) < \mca{N}^0_{k'+1}(\beta': m) < \mca{N}^{\ge 0}_{k''+1}(\beta'': m).
\]
\item If $k, k' \in \Z_{\ge 0}$ and $\beta, \beta' \in H_2(\C^n, L)$ satisfy
$\omega_n(\beta) + \ep(k-1) < \omega_n(\beta') + \ep(k'-1)$, then
$\mca{M}_{k+1}(\beta: m) < \mca{M}_{k'+1}(\beta': m)$,
$\mca{N}^0_{k+1}(\beta: m) < \mca{N}^0_{k'+1}(\beta': m)$, and
$\mca{N}^{\ge 0}_{k+1}(\beta: m) < \mca{N}^{\ge 0}_{k'+1}(\beta': m)$.
\item For each $c \in \R$,
we choose arbitrary total orders on sets
\begin{align*}
&\{ \mca{M}_{k+1}(\beta: m) \mid \omega_n(\beta) + \ep(k-1) = c\}, \\
&\{ \mca{N}^0_{k+1}(\beta: m) \mid \omega_n(\beta) + \ep(k-1) = c\}, \\
&\{ \mca{N}^{\ge 0}_{k+1}(\beta: m) \mid \omega_n(\beta) + \ep(k-1) = c\}. \\
\end{align*}
\end{itemize}
\item For each $m \in \Z_{\ge 0}$, we take a total order on moduli spaces with $P$-parts equal to $[m, m+1] $,
in a manner similar to above.
\end{itemize}
\begin{rem}\label{171206_2}
For each moduli space $X$,
there are only finitely many moduli spaces which are smaller than $X$.
Therefore, we can assign $\tau(X) \in (1/2, 1)$ for each moduli space $X$,
so that $Y<X \implies \tau(Y) > \tau(X)$.
We use these numbers in Section 7.6.
\end{rem}
\textbf{Orientations.}
We explain how signs $\ep_0, \ldots, \ep_{16}$ are computed.
$\ep_0$: First we orient $\mca{M}_{k+1}(\beta)$ following Section 8.3 in \cite{FOOO_09}.
It is sufficient to orient its interior $\mtrg{\mca{M}}_{k+1}(\beta) = \{(u, z_0, \ldots, z_k)\}/\Aut(D)$.
We consider the set $\{(u, z_0, \ldots, z_k)\}$ as an open subset of
$\mtrg{\mca{M}}(\beta) \times (\partial D)^{k+1}$, where
\[
\mtrg{\mca{M}}(\beta):= \{ u: (D, \partial D) \to (\C^n, L) \mid \bar{\partial} u=0, \, [u]=\beta\}
\]
is canonically oriented by Theorem 8.1.1 in \cite{FOOO_09} and
$\partial D = \{ e^{i \theta} \mid \theta \in \R/2\pi \Z\}$ is oriented so that $\partial_\theta$ is of positive direction
(anti-clockwise orientation).
On the other hand,
following Convention 8.3.1 in \cite{FOOO_09},
$\Aut (D)$ is oriented so that the diffeomorphism
\[
\Aut (D) \to (\partial D)^3; \, \rho \mapsto (\rho^{-1}(1), \rho^{-1}(e^{2\pi i/3}) , \rho^{-1}(e^{4 \pi i/3}))
\]
is orientation preserving.
Finally, the quotient $\mtrg{\mca{M}}_{k+1}(\beta)$ is oriented so that a natural isomorphism (determined uniquely up to homotopy)
\[
T(\mca{M}_{k+1}(\beta)) \oplus T(\Aut(D)) \cong T(\mca{M}(\beta)) \oplus T(\partial D)^{\oplus k+1}
\]
preserves orientations (see page 692 in \cite{FOOO_09}).
Now let us compute $\ep_0$.
When $i=1$, Proposition 8.3.3 in \cite{FOOO_09} shows $\ep_0 = (k_1-1)(k_2-1)+n+k_1$.
For arbitrary $i \in \{1, \ldots, k_1\}$,
we can compute $\ep_0$ as
\[
\ep_0 = (k_1-1)(k_2-1) + n+ k_1 + (i-1) + (i-1) k_2 = (k_1-i)(k_2-1) + n+ k_1
\]
where the term $(i-1)$ comes from exchanging input boundary marked points $1, \ldots, i-1$ and $i$,
and the term $(i-1)k_2$ comes from exchanging output boundary marked points $1, \ldots, i-1$ and $i, \ldots, i+k_2-1$.
$\ep_1$ and $\ep_2$:
We orient $\mtrg{\mca{N}}^0_{k+1}(\beta) = \{ (u, z_0=1, z_1, \ldots, z_k)\}$
as an open subset of $\mtrg{\mca{M}}(\beta) \times (\partial D)^k$.
Then $\ep_1 = \ep_0 + k+k_1$ and $\ep_2 = \ep_0 +k+ k_2$ follow from the next lemma.
Following \cite{FOOO_09} page 694,
we say that an isomorphism of K-spaces is $(-1)$-oriented (resp. $(+1)$-oriented)
if the isomorphism reverses (resp. preserves) orientations.
\begin{lem}
Let us orient $\Aut(D, 1)$ so that the embedding
\[
\Aut(D, 1) \to (\partial D)^2 ; \, \rho \mapsto (\rho^{-1}(e^{2\pi i/3}), \rho^{-1}(e^{4\pi i/3}))
\]
preserves orientations.
Then, the isomorphism of K-spaces
\[
\mtrg{\mca{M}}_{k+1}(\beta) \cong \mtrg{\mca{N}}^0_{k+1}(\beta)/\Aut (D,1)
\]
is $(-1)^k$-oriented.
\end{lem}
\begin{proof}
Recall that $\mtrg{\mca{M}}_{k+1}(\beta) =\{(u, z_0, \ldots, z_k)\}/\Aut (D)$.
The natural diffeomorphism $\Aut(D) \cong \partial D \times \Aut(D, 1)$ preserves orientations,
and the first factor $\partial D$ acts on the $0$-th marked point $z_0$,
thus the factor $(-1)^k$ comes from exchanging $z_0$ and $z_1, \ldots, z_k$.
\end{proof}
$\ep_3$ and $\ep_4$:
$\mca{N}^{\ge 0}_{k+1}(\beta)$ can be identified (on a neighborhood of $\mca{N}^0_{k+1}(\beta)$) with
$[0, 1) \times \mca{N}^0_{k+1}(\beta)$.
Following Convention 8.2.1 in \cite{FOOO_09},
we orient $[0,1)$ so that $\partial/\partial t$ is of \textit{negative} direction,
where $t$ denotes the canonical coordinate on $\R$.
Then
\[
\partial ([0, 1) \times \mca{N}^0) = \{0\} \times \mca{N}^0 \sqcup (-1) \cdot [0,1) \times \partial \mca{N}^0
\]
and $\ep_3 = \ep_1 + 1$ immediately follows.
$\ep_4 = \ep_2 + k_1 +1$ follows from
\[
[0,1) \times (\mca{M}_{k_1+1}(\beta_1) \fbp{i}{0} \mca{N}^0_{k_2+1}(\beta_2))
= (-1)^\delta \mca{M}_{k_1+1}(\beta_1) \fbp{i}{0} ( [0,1) \times \mca{N}^0_{k_2+1}(\beta_2) )
\]
where $\delta = \dim \mca{M}_{k_1+1}(\beta_1) - n \equiv k_1 \,(\text{mod}\, 2)$.
$\ep_5, \ldots, \ep_{16}$:
$\mca{M}_{k+1}(\beta:[m, m+1])$ and $[m, m+1] \times \mca{M}_{k+1}(\beta)$
are different K-spaces with a common underlying topological space.
One can take a common open substructure
(see Definition 3.20 in \cite{FOOO_Kuranishi})
of these two K-spaces.
Thus, to fix orientations, one can identify these two K-spaces.
The same remark applies to $\mca{N}^0_{k+1}(\beta: [m, m+1])$ and $\mca{N}^{\ge 0}_{k+1}(\beta: [m, m+1])$.
Now let us orient $[m, m+1]$ so that $\partial/\partial t$ is of \textit{positive} direction,
where $t$ denotes the canonical coordinate on $\R$.
Then
\[
\partial ([m, m+1]) = (+1) \cdot \{ m+1\} \sqcup (-1) \cdot \{m\}.
\]
Then $\ep_5, \ldots, \ep_{16}$ can be determined from
$\ep_0, \ldots, \ep_4$ using the following formulas ($X$ and $Y$ denote arbitrary K-spaces):
\begin{align*}
\partial ([m, m+1] \times X)&= \{m+1\} \times X \sqcup (-1) \cdot \{m\} \times X \sqcup (-1) \cdot [m, m+1] \times \partial X, \\
[m, m+1] \times ( X \times_L Y) &= ([m, m+1] \times X) \times_{[m, m+1] \times L} ([m, m+1] \times Y).
\end{align*}
\subsection{Spaces of continuous paths and loops}
Let $\Pi^\con$ denote the set of continuous Moore paths on $L$, namely:
\[
\Pi^{\con}:= \{ (T, \gamma) \mid T \in \R_{\ge 0}, \, \gamma \in C^0([0,T], L) \}.
\]
For each $q \in L$, let $c_q$ denote the constant path at $q$ with length $0$.
Namely, $c_q = (0, \gamma_q)$
where $\gamma_q$ denotes the unique map from $\{0\}$ to $q$.
To define a metric $d_\Pi$ on $\Pi^\con$,
we fix (throughout this paper) an auxiliary Riemannian metric on $L$,
and $d_L$ denotes the associated metric on $L$.
For later use we fix a constant $\rho_L \in \R_{>0}$
which is smaller than the injectivity radius of $L$
such that for any $r \in (0, \rho_L]$ and $x \in L$
the ball with center $x$ and radius $r$ is geodesically convex.
Now let us define a metric $d_\Pi$ on $\Pi^\con$ by
\[
d_\Pi((T, \gamma), (T', \gamma'))) := \max \big\{ |T-T'|, \, \max_{0 \le s \le 1} d_L(\gamma(sT), \gamma'(sT')) \big\}.
\]
We define $\ev_0, \ev_1: \Pi^\con \to L$ by
\[
\ev_0 (T, \gamma):= \gamma(0), \qquad \ev_1(T, \gamma):= \gamma(T).
\]
Then we can define a concatenation map
\[
\Pi^\con \fbp{\ev_1}{\ev_0} \Pi^\con \to \Pi^\con; \qquad (\Gamma_0, \Gamma_1) \mapsto \Gamma_0*\Gamma_1
\]
by $(T, \gamma) * (T', \gamma'):= (T+T', \gamma*\gamma')$,
where
$\gamma*\gamma': [0,T+T'] \to L$ is defined by
\[
(\gamma*\gamma')(t):= \begin{cases} \gamma(t) &(0 \le t \le T), \\ \gamma'(t-T) &(T \le t \le T+T'). \end{cases}
\]
\begin{lem}\label{170611_4}
The concatenation map defined as above is continuous with respect to $d_\Pi$.
\end{lem}
\begin{proof}
We have to show that, if sequences
$(T_j, \gamma_j)_{j \ge 1}$ and
$(T'_j, \gamma'_j)_{j \ge 1}$ satisfy
\[
\lim_{j \to \infty} d_\Pi( (T_j, \gamma_j), (T, \gamma)) = \lim_{j \to \infty} d_\Pi((T'_j, \gamma'_j), (T', \gamma')) = 0
\]
then
\begin{equation}\label{170414_1}
\lim_{j \to \infty} d_\Pi( (T_j + T'_j, \gamma_j * \gamma'_j), (T+T', \gamma*\gamma')) = 0.
\end{equation}
If (\ref{170414_1}) is not the case, there exists a sequence $(s_j)_{j \ge 1}$ in $[0,1]$ and $\ep>0$ such that
\begin{equation}\label{170414_2}
d_L((\gamma_j*\gamma'_j)(s_j(T_j+T'_j)), (\gamma*\gamma')(s_j(T+T'))) \ge \ep
\end{equation}
for every $j$.
Then $T_j, T'_j >0$ for sufficiently large $j$, thus we may assume that
$T_j, T'_j>0$ holds for every $j \ge 1$.
By taking a subsequence,
we may also assume that either $s_j \le T_j/(T_j+T'_j)$ or $s_j \ge T_j/(T_j+T'_j)$ holds for all $j$.
In the following we only consider the case $s_j \le T_j/(T_j+T'_j)$, since the proof in the other case is parallel.
Now we obtain
\[
(\gamma_j*\gamma'_j) ( s_j (T_j+T'_j)) = \gamma_j \biggl( s_j \cdot \frac{T_j+T'_j}{T_j} \cdot T_j \biggr).
\]
We set $\tau_j:= s_j (T_j+T'_j)/T_j \in [0,1]$. Then the LHS of (\ref{170414_2}) is bounded from above by
\begin{equation}\label{170422_1}
d_L(\gamma_j(\tau_j T_j), \gamma(\tau_j T)) + d_L(\gamma(\tau_j T), (\gamma*\gamma')(s_j (T+T'))).
\end{equation}
Then the first term of (\ref{170422_1}) goes to $0$ as $j \to \infty$, since
\[
d_L(\gamma_j(\tau_j T), \gamma(\tau_j T)) \le d_\Pi ((T_j, \gamma_j), (T, \gamma)).
\]
Since $\gamma*\gamma'$ is (uniformly) continuous,
to show that
the second term of (\ref{170422_1}) goes to $0$ as $j \to \infty$,
it is sufficient to check:
\begin{equation}\label{170422_2}
\lim_{j \to \infty} \tau_j T - s_j (T+T') = 0.
\end{equation}
Using the obvious identity
\begin{equation}\label{170613_1}
\tau_j T - s_j(T+T') = s_j (T_j+T'_j) (T/T_j) - s_j (T+T'),
\end{equation}
we can check (\ref{170422_2}) by considering the following three cases:
\begin{itemize}
\item $T>0$: this case $\lim_{j \to \infty} T/T_j = 1$, thus the RHS of (\ref{170613_1}) goes to $0$.
\item $T=0$ and $T'>0$: this case the first term is $0$, and the second term goes to $0$ because
$0 \le s_j \le T_j/(T_j+T'_j)$ implies $\lim_{j \to \infty} s_j = 0$.
\item $T,T'=0$: this case both the first and second terms are $0$.
\end{itemize}
This completes the proof.
\end{proof}
Next we consider the space of continuous Moore loops with marked points.
For every $k \in \Z_{\ge 0}$,
let $\mca{L}_{k+1}^\con$ denote the set consists of
$(T, \gamma, t_0, \ldots, t_k)$ such that
\begin{itemize}
\item $T \in \R_{\ge 0}$ and $\gamma \in C^0([0, T], L)$ such that $\gamma(0)=\gamma(T)$.
\item $0 = t_0 \le t_1 \le \ldots \le t_k \le T$.
\end{itemize}
For each $a \in H_1(L: \Z)$,
let $\mca{L}_{k+1}^\con(a)$ denote the subspace of
$\mca{L}_{k+1}^\con$ which consists of
$(T, \gamma, t_0, \ldots, t_k)$ such that $[\gamma]=a$.
The set $\mca{L}_{k+1}^\con$ can be identified with the set
\[
\{ (\Gamma_0, \ldots, \Gamma_k) \in (\Pi^\con)^{k+1} \mid \ev_1(\Gamma_i) = \ev_0(\Gamma_{i+1}) \,(0 \le i \le k-1), \, \ev_1(\Gamma_k) = \ev_0(\Gamma_0)\}.
\]
Using this identification, a metric $d_{\mca{L}_{k+1}}$ on $\mca{L}_{k+1}^\con$ is defined by
\[
d_{\mca{L}_{k+1}}((\Gamma_0, \ldots, \Gamma_k), (\Gamma'_0, \ldots, \Gamma'_k)) := \max_{0 \le i \le k} d_{\Pi} (\Gamma_i, \Gamma'_i).
\]
We consider the topology on $\mca{L}_{k+1}^\con$ induced from $d_{\mca{L}_{k+1}}$.
The evaluation map
\[
\evl_j: \mca{L}_{k+1}^\con \to L \quad (j \in \{0, \ldots, k\})
\]
and the concatenation map
\[
\con_j:
\mca{L}_{k+1}^\con \fbp{\evl_j}{\evl_0} \mca{L}_{k'+1}^\con \to \mca{L}_{k+k'}^\con \quad (j \in \{1, \ldots, k\})
\]
are defined in the same way as in the case of smooth loops (see Section 4.1).
The evaluation map is obviously continuous,
and the concatenation map is continuous by Lemma \ref{170611_4}.
The evaluation map $\evl_j$ will be abbreviated as $\ev_j$ when there is no risk of confusion.
\subsection{Strongly continuous maps to $\mca{L}_{k+1}^\con$}
The goal of this subsection is to define
strongly continuous maps
from moduli spaces of (perturbed) pseudo-holomorphic disks (with boundary marked points)
to spaces of continuous loops (with marked points),
so that natural compatibility conditions (spelled out in Proposition \ref{170613_2}) are satisfied.
Throughout this subsection, the space $\mca{L}_{k+1}^\con$ is equipped with the topology defined from the metric $d_{\mca{L}_{k+1}}$,
which is defined in Section 7.3.
To state Proposition \ref{170613_2} we have to introduce the following notations.
For any $\beta \in H_2(\C^n, L)$ and $(T, B) \in \mca{G}(k+1: \beta)$,
one can define a continuous map
\begin{equation}
\ev_{\interior}: \prod_{v \in C_{0, \interior}(T)} P \times \mca{L}_{k_v+1}^\con (\partial B(v)) \to \prod_{e \in C_{1, \interior}(T)} (P \times L)^2
\end{equation}
in a way similar to (\ref{170611_1}).
Also let us recall the diagonal map (\ref{170611_2}).
Then one can define a continuous map
\begin{equation}\label{170615_1}
\biggl(\prod_{e \in C_{1, \interior}(T)} P \times L \biggr) \fbp{\Delta}{\ev_{\interior}} \biggl( \prod_{v \in C_{0, \interior}(T)} P \times \mca{L}_{k_v+1}^\con (\partial B(v)) \biggr)
\to P \times \mca{L}_{k+1}^\con(\partial \beta)
\end{equation}
by taking concatenations of loops,
where the fiber product in the LHS is taken over $\prod_{e \in C_{1,\interior}(T)} (P \times L)^2$.
\begin{prop}\label{170613_2}
For every $k \in \Z_{\ge 0}$, $m \in \Z_{\ge 0}$, and $P \in \{ \{m\}, [m, m+1]\}$,
one can define strongly continuous maps
\begin{equation}\label{170705_1}
\ev^{\mca{M}}: \mca{M}_{k+1}(\beta : P) \to P \times \mca{L}_{k+1}^\con(\partial \beta) \qquad (\omega_n(\beta) < \ep (m+1-k)),
\end{equation}
\begin{equation}\label{170705_2}
\ev^{\mca{N}^0}: \mca{N}^0_{k+1}(\beta : P) \to P \times \mca{L}_{k+1}^\con(\partial \beta) \qquad(\omega_n(\beta) < \ep(m-1-k)),
\end{equation}
\begin{equation}\label{170705_3}
\ev^{\mca{N}^{\ge 0}}: \mca{N}^{\ge 0}_{k+1}(\beta: P) \to P \times \mca{L}_{k+1}^\con(\partial \beta) \qquad(\omega_n(\beta) < \ep(m-k-U)),
\end{equation}
so that the following diagrams commute
for every $(T, B) \in \mca{G}(k+1, \beta)$:
\begin{itemize}
\item
\[
\xymatrix{
\biggl(\prod_e P \times L \biggr) \fbp{\Delta}{\ev_\interior} \biggl( \prod_v \mca{M}_{k_v+1}(B(v): P) \biggr)
\ar[r]\ar[d] & \mca{M}_{k+1}(\beta: P)
\ar[d] \\
\biggl(\prod_e P \times L \biggr) \fbp{\Delta}{\ev_\interior}\biggl( \prod_v P \times \mca{L}_{k_v+1}^\con(\partial B(v)) \biggr)
\ar[r]_-{(\ref{170615_1})}& P \times \mca{L}_{k+1}^\con(\partial \beta)
\\
}
\]
where the first horizontal map is defined from (\ref{170614_1}) by setting $d=0$, and
vertical maps are defined by (\ref{170705_1}).
\item
\[
\xymatrix{
\biggl(\prod_e P \times L \biggr) \fbp{\Delta}{\ev_\interior} (\star)
\ar[r] \ar[d] & \mca{N}^0_{k+1}(\beta: P)
\ar[d] \\
\biggl(\prod_e P \times L \biggr) \fbp{\Delta}{\ev_\interior}\biggl( \prod_v P \times \mca{L}_{k_v+1}^\con(\partial B(v)) \biggr)
\ar[r]_-{ (\ref{170615_1})} & P \times \mca{L}_{k+1}^\con(\partial \beta)
\\
}
\]
where $(\star):= \prod_{v \ne v_0} \mca{M}_{k_v+1} (B(v): P) \times \mca{N}^0_{k_{v_0}+1}(B(v_0): P)$ and
the first horizontal map is defined from (\ref{170614_2}) by setting $d=0$, and
vertical maps are defined by (\ref{170705_2}).
\item
\[
\xymatrix{
\biggl(\prod_e P \times L \biggr) \fbp{\Delta}{\ev_\interior} (\star)
\ar[r] \ar[d] & \mca{N}^{\ge 0}_{k+1}(\beta: P)
\ar[d] \\
\biggl(\prod_e P \times L \biggr) \fbp{\Delta}{\ev_\interior}\biggl( \prod_v P \times \mca{L}_{k_v+1}^\con(\partial B(v)) \biggr)
\ar[r]_-{(\ref{170615_1})} & P \times \mca{L}_{k+1}^\con(\partial \beta)
\\
}
\]
where $(\star) =\prod_{v \ne v_0} \mca{M}_{k_v+1} (B(v): P) \times \mca{N}^{\ge 0}_{k_{v_0}+1}(B(v_0): P) $ and
the first horizontal map is defined from (\ref{170614_3}) by setting $d=0$, and
vertical maps are defined by (\ref{170705_3}).
\end{itemize}
\end{prop}
In the rest of this subsection,
we explain the definition of
the strongly continous map (\ref{170705_1}).
Definitions of
(\ref{170705_2}) and (\ref{170705_3}) are similar.
For each $p \in \mca{M}_{k+1}(\beta: P)$,
let $\mca{U}_p = (U_p, \mca{E}_p, s_p, \psi_p)$
be a K-chart at $p$.
To define a strongly continuous map (\ref{170705_1}),
it is sufficient to define a continuous map
$\ev^{\mca{M}}_p: U_p \to P \times \mca{L}_{k+1}^\con$
for each $p$,
so that compatibility conditions with coordinate changes are satisfied.
By Lemma \ref{171014_1}, every $x \in U_p$ is represented by an element of
\[
P \times \bigg(\prod_{e \in C_{1,\interior}(T')} L \bigg)
\fbp{\Delta}{\ev_{\interior}}
\bigg(\prod_{v \in C_{0, \interior}(T')} \mca{MM}_{k_v+1}(B'(v)) \bigg)
\]
where $(T', B')$ is a \textit{reduction} of $(T, B)$.
We denote the representative by
\[
(\pi , (u^v, z^v_0, \ldots, z^v_{k_v})_v).
\]
Now we define
\[
\ev^{\mca{M}}_p(x):= (\pi, \ev( (u^v, z^v_0, \ldots, z^v_{k_v})_v)) \in P \times \mca{L}_{k+1}^\con(\partial \beta),
\]
where $\ev ((u^v, z^v_0, \ldots, z^v_{k_v})_v)$ is the concatenation (defined by (\ref{170615_1}))
of $\ev(u^v, z^v_0, \ldots, z^v_{k_v}) \in \mca{L}_{k_v+1}^\con(\partial B'(v))$ (which is defined below) for all $v \in C_{0, \interior}(T')$.
To define $\ev(u^v, z^v_0, \ldots, z^v_{k_v})$,
we distinguish the case $u^v$ is constant and the case $u^v$ is nonconstant.
\begin{itemize}
\item
The case $u^v$ is a constant map to $q \in L$: we define
\[
\ev(u^v, z^v_0, \ldots, z^v_{k_v}) := (c_q, \ldots, c_q)
\]
where $c_q$ denotes the constant path at $q$ with length $0$.
\item
The case $u^v$ is nonconstant: we use the length parametrization.
For every $j \in \{0,\ldots, k_v\}$ we define $\theta_j \in [0, 2\pi)$ by $z^v_j / z^v_0= e^{\sqrt{-1} \theta_j}$, thus $0=\theta_0 < \theta_1 < \cdots < \theta_{k_v} < 2\pi$.
Then we define
\begin{align*}
T^{u^v} _j &:= \int_{\theta_j}^{\theta_{j+1}} \bigg\lvert \frac{d}{d\theta} u^v(z^v_0 \cdot e^{\sqrt{-1} \theta}) \bigg\rvert \, d \theta, \\
\lambda^{u^v} _j &: [\theta_j, \theta_{j+1}] \to [0, T^{u^v} _j] ;\quad \Theta \mapsto \int_{\theta_j}^\Theta \bigg\lvert \frac{d}{d\theta} u^v(z^v_0 \cdot e^{\sqrt{-1} \theta}) \bigg\rvert \, d \theta, \\
\gamma^{u^v} _j &: [0, T^{u^v} _j] \to L; \quad t \mapsto u^v (e^{\sqrt{-1} (\lambda^{u^v} _j)^{-1}(t)}).
\end{align*}
$\frac{d}{d \theta} u^v (e^{\sqrt{-1}\theta})$ vanishes at only finitely many $\theta$
since $u^v$ is nonconstant,
thus $T^{u^v} _j$ is positive and $\lambda^{u^v} _j$ is strictly increasing.
Hence $(\lambda^{u^v} _j)^{-1}: [0, T^{u^v} _j] \to [\theta_j, \theta_{j+1}]$ is a well-defined continuous map.
Then we define
\[
\ev(u^v, z^v_0, \ldots, z^v_{k_v} ):= (T^{u^v}_j, \gamma^{u^v}_j)_{0 \le j \le k_v}.
\]
This completes the definition of $\ev^{\mca{M}}_p(x)$.
\end{itemize}
\begin{rem}\label{171014_3}
For later purpose (proof of Lemma \ref{170615_2}) we define
$\Lambda^{u^v}_j: [0,1] \to [0, T^{u^v}_j]$ by
$\Lambda^{u^v}_j(s) := \lambda^{u^v}_j((1-s)\theta_j + s \theta_{j+1})$.
\end{rem}
The above definition of $\ev^{\mca{M}}_p(x)$ does not depend on choices of representatives of $x$.
In particular, the family of maps $(\ev^{\mca{M}}_p)_p$ is compatible with coordinate changes of K-charts.
To show that this family defines a strongly continuous map,
we have to check that $\ev^{\mca{M}}_p$ is continuous for each $p$.
\begin{lem}\label{170615_2}
$\ev^{\mca{M}}_p: U_p \to P \times \mca{L}_{k+1}^\con(\partial \beta)$ is continuous for every $p \in \mca{M}_{k+1}(\beta: P)$.
\end{lem}
\begin{proof}
\textbf{Step 1.}
Let $(x_l)_{l \ge 1}$ be a sequence in $U_p$ which converges to $x_\infty \in U_p$.
For simplicity, we work under the following assumptions:
\begin{itemize}
\item $k=0$.
\item Each $x_l$ is represented by a single holomorphic map; we denote the representative by $(\pi_l, u_l, z_{l,0})$.
\item $x_\infty$ is represented by an element of
\[
P \times \bigg(\prod_{e \in C_{1,\interior}(T_\infty)} L \bigg)
\fbp{\Delta}{\ev_{\interior}}
\bigg(\prod_{v \in C_{0, \interior}(T_\infty)} \mca{MM}_{k_v+1}(B_\infty(v)) \bigg)
\]
for some $(T_\infty, B_\infty) \in \mca{G}(k+1, \beta)$; see Lemma \ref{171014_1}.
We denote the representative by
$(\pi_\infty, (u^v_\infty , z^v_{\infty, 0}, \ldots, z^v_{\infty, k_v})_{v \in C_{0, \interior}(T_\infty)})$.
\end{itemize}
Since $\lim_{l \to \infty} \pi_l = \pi_\infty$ is obvious, it is sufficient to show that
\begin{equation}\label{171013_1}
\lim_{l \to \infty} d_{\mca{L}_1}(\ev(u_l, z_{l,0}), \ev( (u^v_\infty , z^v_{\infty, 0}, \ldots, z^v_{\infty, k_v})_v)) = 0.
\end{equation}
\textbf{Step 2.}
We first consider the case $\sharp C_{0, \interior}(T_\infty)=1$,
and abbreviate $u^v_\infty$ as $u_\infty$.
We may assume that
$\lim_{l \to \infty} u_l = u_\infty$ in the $C^\infty$-topology, and
$z_{l,0} = 1$ for every $l$ (including $l= \infty$).
When $u_\infty$ is constant, then $u_l|_{\partial D}$ converges to a constant map in $C^\infty$ (in particular $C^0$) topology,
then (\ref{171013_1}) follows.
When $u_\infty$ is nonconstant,
then $u_l$ is nonconstant for sufficiently large $l$.
Moreover
$\lim_{l \to \infty} T^{u_l}_0 = T^{u_\infty}_0$
and
$\lim_{l \to \infty} \Lambda^{u_l}_0 = \Lambda^{u_\infty}_0$ in the $C^0$-topology.
Then (\ref{171013_1}) follows from the next lemma,
the proof of which is completely elementary and omitted.
\begin{lem}
Let $(\tau_l)_l$ be a sequence in $C^0([0,1], [0,1])$ such that
\begin{itemize}
\item $(\tau_l)_l$ has a $C^0$-limit $\tau_\infty: [0,1] \to [0,1]$.
\item For every $l$ (including $\infty$), $\tau_l$ is strictly increasing and $\tau_l (0)=0$, $\tau_l(1)=1$.
\end{itemize}
Then $(\tau_l)^{-1}$ converges to $(\tau_\infty)^{-1}$ in the $C^0$-topology.
\end{lem}
\textbf{Step 3.}
Now we consider the case $\sharp C_{0, \interior}(T_\infty) >1$.
We fix $\ep>0$, which can be arbitrarily small.
For sufficiently large every $l$,
there exists a decomposition (depends on $\ep$ but we drop it from the following notations)
\[
D= V_l \sqcup \bigsqcup_{v \in C_{0, \interior}(T_\infty) } U^v_l
\]
such that the following conditions are satisfied (see \cite{FOOO_09} Section 7.1.4):
\begin{itemize}
\item $V_l$ is a compact set
with the number of connected components is $\sharp C_{0, \interior}(T_\infty)$,
and $U^v_l$ is a connected open set for every $v \in C_{0, \interior}(T_\infty)$.
\item $V_l \cap \partial D$ is a disjoint union of $2\sharp C_{0, \interior}(T_\infty) - 1$ closed intervals.
\item $\int_{V_l \cap \partial D} \bigg\lvert \frac{d}{d\theta} u_l(e^{\sqrt{-1} \theta}) \bigg\rvert < \ep$.
\item For every $v \in C_{0, \interior}(T_\infty)$,
$\overline{U^v_l} \cap \partial D$ is a disjoint union of $k_v+1$ closed intervals.
We denote them as $I^v_l(0), \ldots, I^v_l(k_v)$ aligned in anti-clockwise order.
\item There exists a sequence $(\rho^v_l)_l$ in $\Aut(D)$ such that $(\rho^v_l)^{-1}(U^v_l)$ is independent on $l$ (denoted by $U^v$),
and $u_l \circ \rho^v_l$ converges to $u^v_\infty$ (with respect to the $C^\infty$-topology) on $\overline{U^v}$.
\item $\overline{U^v} \cap \partial D$ is a disjoint union of $k_v+1$ closed intervals,
which we denote by $I^v_{\infty}(0), \ldots, I^v_{\infty}(k_v)$ in anti-clockwise order.
\end{itemize}
\begin{center}
\input{glueddisks.tpc}
\end{center}
For each $v \in C_{0, \interior}(T_\infty)$ and $j \in \{0, \ldots, k_v\}$,
our assumption $u_l \circ \rho^v_l \to u^v_\infty$
implies that
\[
\lim_{l \to \infty} d_\Pi (u_l|_{I^v_l(j)} , u^v_\infty|_{I^v_\infty(j)}) = 0.
\]
The proof is same as our argument in Step 2.
On the other hand, for every $l$, the total lengths of $u_l|_{V_l \cap \partial D}$ is at most $\ep$.
Since $\ev(u_l, z_{l,0}) $ is obtained as the concatenation of these paths,
it is not difficult to see that
\[
\limsup_{l \to \infty} \, d_{\mca{L}_1}(\ev(u_l, z_{l,0}), \ev( (u^v_\infty , z^v_{\infty, 0}, \ldots, z^v_{\infty, k_v})_v)) \le 10 \ep.
\]
Since $\ep$ can be arbitrarily small, this implies
the convergence (\ref{171013_1}).
\end{proof}
\subsection{$C^0$-approximation lemma and CF-perturbation}
We first introduce the notion of ``$\ep$-closeness''
for strongly continuous maps from a K-space to a metric space:
\begin{defn}
Let $(X, \wh{\mca{U}})$ be a K-space,
$(Y, d)$ be a metric space,
and $\wh{f}, \wh{g}: (X,\wh{\mca{U}}) \to Y$ be strongly continuous maps.
For any $\ep>0$, we say that
$\wh{f}$ and $\wh{g}$ are $\ep$-close, if
$d(f_p(x), g_p(x)) < \ep$ for every $p \in X$ and $x \in U_p$.
\end{defn}
Let us state a key technical result in this subsection, which we call $C^0$-approximation lemma.
The notion of ``open substrcture'' of a given K-structure is defined in Definition 3.20 in \cite{FOOO_Kuranishi}.
The constant $\rho_L$ in the statement was introduced in the second paragraph of Section 7.3.
\begin{thm}\label{170430_2}
Let $(X, \wh{\mca{U}})$ be a compact K-space and
$\wh{f}: (X, \wh{\mca{U}}) \to \mca{L}_{k+1}^\con$ be a strongly continuous map such that
$\ev^{\mca{L}}_j \circ \wh{f}: (X, \wh{\mca{U}}) \to L$ is strongly smooth for every $j \in \{0, \ldots, k\}$.
Let $Z$ be a closed subset of $X$ and $\wh{g}: (Z, \wh{\mca{U}}|_Z) \to \mca{L}_{k+1}$
be a strongly smooth map such that:
\begin{itemize}
\item $\ev^{\mca{L}}_j \circ \wh{g} = \ev^{\mca{L}}_j \circ \wh{f}|_Z$ for every $j \in \{0, \ldots, k\}$.
\item $\wh{g}$ is $\ep$-close to $\wh{f}|_Z$ with respect to $d_{\mca{L}_{k+1}}$.
\end{itemize}
If $\ep < \rho_L$,
there exists an open substructure $\wh{\mca{U}_0}$ of $\wh{\mca{U}}$
and a strongly smooth map $\wh{g'}: (X, \wh{\mca{U}_0}) \to \mca{L}_{k+1}$
such that the following conditions hold:
\begin{itemize}
\item
$\wh{g'}$ is $\ep$-close to $\wh{f}|_{\wh{\mca{U}_0}}$.
\item
$\evl_j \circ \wh{g'} = \evl_j \circ \wh{f}|_{\wh{\mca{U}_0}}$ for every $j \in \{0, \ldots, k\}$.
\item
$\wh{g'} = \wh{g}$ on $\wh{\mca{U}_0}|_Z$.
\end{itemize}
\end{thm}
The proof of Theorem \ref{170430_2} is carried out in Section 9.
Combining Theorem \ref{170430_2} with results from \cite{FOOO_Kuranishi},
we obtain Theorem \ref{170430_1} below.
In the statement of Theorem \ref{170430_1} we use the following notions from \cite{FOOO_Kuranishi}
without repeating their definitions:
\begin{itemize}
\item Thickening of K-structures: see Section 5.2 in \cite{FOOO_Kuranishi}.
\item Collared versions of K-structures, strongly smooth maps and CF-perturbations: see Section 17.5 in \cite{FOOO_Kuranishi}.
\end{itemize}
The assumptions in Theorem \ref{170430_1}
on K-structures and CF-perturbations are
similar to Situations 17.43 and 17.57 in \cite{FOOO_Kuranishi}, respectively.
\begin{thm}\label{170430_1}
Suppose that we are given the following data:
\begin{itemize}
\item $k \in \Z_{\ge 0}$, $\tau \in (0, 1)$ and $\ep \in (0, \rho_L)$.
\item A $\tau$-collared K-space $(X, \wh{\mca{U}})$.
\item A $\tau$-collared strongly continuous map $\wh{f}: (X, \wh{\mca{U}}) \to \mca{L}_{k+1}^\con$
such that $\ev^{\mca{L}}_j \circ \wh{f} : (X, \wh{\mca{U}}) \to L$ is admissible for every $j \in \{0, \ldots, k\}$,
and $\ev^{\mca{L}}_0 \circ \wh{f}$ is strata-wise weakly submersive.
\item For every $l \in \Z_{\ge 1}$,
a $\tau$-collared K-structure $\wh{\mca{U}^+_l}$ on $\wh{S}_l(X)$
which is a thickening of $\wh{\mca{U}}|_{\wh{S}_l(X)}$.
\item For every $l_1, l_2 \in \Z_{\ge 1}$,
$(l_1+l_2)!/(l_1)!(l_2)!$-fold covering of $\tau$-collared $K$-spaces
\[
\wh{S}_{l_1}(\wh{S}_{l_2}(X), \wh{\mca{U}^+_{l_2}}) \to
(\wh{S}_{l_1+l_2}(X), \wh{\mca{U}^+_{l_1+l_2}})
\]
such that the following diagrams commute for every $l_1, l_2, l_3 \in \Z_{\ge 1}$:
\[
\xymatrix{
\wh{S}_{l_1}(\wh{S}_{l_2}(\wh{S}_{l_3}(X), \wh{\mca{U}^+_{l_3}})) \ar[r] \ar[d] & \wh{S}_{l_1+l_2}(\wh{S}_{l_3}(X), \wh{\mca{U}^+_{l_3}}) \ar[d] \\
\wh{S}_{l_1}(\wh{S}_{l_2+l_3}(X), \wh{\mca{U}^+_{l_2+l_3}}) \ar[r] & (\wh{S}_{l_1+l_2+l_3}(X), \wh{\mca{U}^+_{l_1+l_2+l_3}}), \\
}
\]
\[
\xymatrix{
\wh{S}_{l_1}(\wh{S}_{l_2}(X, \wh{\mca{U}})) \ar[r] \ar[d] & \wh{S}_{l_1}(\wh{S}_{l_2}(X), \wh{\mca{U}^+_{l_2}}) \ar[d] \\
\wh{S}_{l_1+l_2}(X, \wh{\mca{U}}) \ar[r] & (\wh{S}_{l_1+l_2}(X), \wh{\mca{U}^+_{l_1+l_2}}). \\
}
\]
\item A $\tau$-collared CF-perturbation $\wh{\mf{S}^+_l}$ of $(\wh{S}_l(X), \wh{\mca{U}^+_l})$ for every $l \in \Z_{\ge 1}$,
such that the pullback of $\wh{\mf{S}^+_{l_1+l_2}}$ by
$\wh{S}_{l_1}(\wh{S}_{l_2}(X), \wh{\mca{U}^+_{l_2}}) \to (\wh{S}_{l_1+l_2}(X), \wh{\mca{U}^+_{l_1+l_2}})$
coincides with the restriction of $\wh{\mf{S}^+_{l_2}}$
for every $l_1, l_2 \in \Z_{\ge 1}$.
\item A $\tau$-collared admissible map $\wh{f^+_l}: (\wh{S}_l(X) , \wh{\mca{U}^+_l}) \to \mca{L}_{k+1}$
for every $l \in \Z_{\ge 1}$ such that:
\begin{itemize}
\item The pullback of $\wh{f^+_{l_1+l_2}}$ by $\wh{S}_{l_1}(\wh{S}_{l_2}(X), \wh{\mca{U}^+_{l_2}}) \to (\wh{S}_{l_1+l_2}(X), \wh{\mca{U}^+_{l_1+l_2}})$
coincides with the restriction of $\wh{f^+_{l_2}}$
for every $l_1, l_2 \in \Z_{\ge 1}$.
\item For every $j \in \{0, \ldots, k\}$,
$\ev^{\mca{L}}_j \circ \wh{f_l^+}: (\wh{S}_l(X), \wh{\mca{U}^+_l}) \to L$ coincides with
$\ev^{\mca{L}}_j \circ \wh{f} |_{\wh{S}_l(X)}: (\wh{S}_l(X), \wh{\mca{U}}|_{\wh{S}_l(X)}) \to L$
via the KK-embedding $\wh{\mca{U}}|_{\wh{S}_l(X)} \to \wh{\mca{U}^+_l}$.
\item $\ev^{\mca{L}}_0 \circ \wh{f_l^+}: (\wh{S}_l(X), \wh{\mca{U}^+_l}) \to L$
is strata-wise strongly submersive with respect to $\wh{\mf{S}^+_l}$.
\item $\wh{f^+_l}$ is $\ep$-close to $\wh{f}|_{\wh{S}_l(X)}$.
\end{itemize}
\end{itemize}
Then, for any $\tau' \in (0, \tau)$, there exist the following data:
\begin{itemize}
\item A $\tau'$-collared K-structure $\wh{\mca{U}^+}$ on $X$, which is a thickening of $\wh{\mca{U}}$.
\item An isomorphism of $\tau'$-collared K-structures $\wh{\mca{U}^+}|_{\wh{S}_l(X)} \cong \wh{\mca{U}^+_l}$ for every $l \in \Z_{\ge 1}$.
\item A $\tau'$-collared CF-perturbation $\wh{\mf{S}^+}$ of $(X, \wh{\mca{U}^+})$ such that
$\wh{\mf{S}^+}|_{\wh{S}_l(X)}$ coincides with $\wh{\mf{S}^+_l}$
via the isomorphism of K-spaces $\wh{\mca{U}^+}|_{\wh{S}_l(X)} \cong \wh{\mca{U}^+_l}$.
\item A $\tau'$-collared admissible map $\wh{f^+}: (X, \wh{\mca{U}^+}) \to \mca{L}_{k+1}$ such that:
\begin{itemize}
\item $\wh{f^+}$ is $\ep$-close to $\wh{f}$.
\item For every $j \in \{0, \ldots, k\}$, $\ev^{\mca{L}}_j \circ \wh{f^+}$ coincides with $\ev^{\mca{L}}_j \circ \wh{f}$ with respect to the KK-embedding $\wh{\mca{U}} \to \wh{\mca{U}^+}$.
\item $\ev^{\mca{L}}_0 \circ \wh{f^+} : (X, \wh{\mca{U}^+}) \to L$ is strata-wise strongly submersive with respect to $\wh{\mf{S}^+}$.
\end{itemize}
\end{itemize}
\end{thm}
\begin{proof}
The $K$-structure $\wh{\mca{U}^+}$ and the CF-perturbation $\wh{\mf{S}^+}$ are defined by
Propositions 17.62 and 17.65 in \cite{FOOO_Kuranishi}, respectively;
here we apply Proposition 17.65 (2) to $\ev^{\mca{L}}_0 \circ \wh{f}: (X, \wh{\mca{U}}) \to L$.
Moreover $\wh{f^+_1}: \wh{S}_1(X) \to \mca{L}_{k+1}$
extends to the $\tau'$-neighborhood of $\partial X$ (denoted by $N(\tau')$).
Then we can apply Theorem \ref{170430_2} to conclude the proof,
taking $\mca{N}(\tau')$ as $Z$ and the extension of $\wh{f^+_1}$ as $\wh{g}$.
\end{proof}
\begin{rem}\label{180308_1}
Nextly, we should state and prove a version of Theorem \ref{170430_1}
such that $(X, \wh{\mca{U}})$ is a $\tau$-collared K-space, and
\[
\wh{f}: (X, \wh{\mca{U}}) \to [a, b]^{\boxplus \tau} \times \mca{L}_{k+1}^\con
\]
is a $\tau$-collared strongly continuous map,
where $a<b$ are real numbers and $[a,b]^{\boxplus \tau}: = [a-\tau, b+\tau]$.
However, the statement is similar to Theorem \ref{170430_1}
and involves further notations,
thus here we choose not to write it down in detail.
\end{rem}
\subsection{Wrap-up of the proof}
Now we can complete the proof of Theorem \ref{161215_1} assuming results
presented in Sections 7.1 and 7.5.
Let $X$ be one of moduli spaces considered in Theorem \ref{170611_5} (i).
In Section 7.4, we defined a strongly continuous map from $X$ to the space of continuous loops (with marked points).
This map naturally extends to a $1$-collared strongly continuous map from $X^{\boxplus 1}$,
see Lemma-Definition 17.35 and Lemma 17.37 (3) in \cite{FOOO_Kuranishi}.
Taking $\tau(X) \in (1/2, 1)$ as in Remark \ref{171206_2}
and successively applying Theorem \ref{170430_1} and Remark \ref{180308_1},
$\bar{X}:= X^{\boxplus 1/2}$ is equipped with an admissible CF-perturbation and
an admissible strongly smooth map to the space of smooth loops (with marked points).
In conclusion, we obtain the following data
for every $k \in \Z_{\ge 0}$, $m \in \Z_{\ge 0}$ and $P \in \{ \{m\}, [m, m+1]\}$.
\begin{enumerate}
\item[(i):] Compact admissible K-spaces
\begin{align*}
&\bar{\mca{M}}_{k+1}(\beta: P) \qquad ( \omega_n(\beta) < \ep(m+1-k)), \\
&\bar{\mca{N}}^0_{k+1}(\beta: P) \qquad (\omega_n(\beta) <\ep(m-1-k)), \\
&\bar{\mca{N}}^{\ge 0}_{k+1}(\beta: P) \qquad (\omega_n(\beta) < \ep(m-k-U)),
\end{align*}
and admissible CF-perturbations on these K-spaces.
\item[(ii):] Admissible maps
\begin{align}
\ev^{\bar{\mca{M}}_{k+1}}: \bar{\mca{M}}_{k+1}(\beta: P) & \to P \times \mca{L}_{k+1}(\partial \beta) \\
\ev^{\bar{\mca{N}}^0_{k+1}}: \bar{\mca{N}}^0_{k+1}(\beta: P) & \to P \times \mca{L}_{k+1}(\partial \beta) \\
\ev^{\bar{\mca{N}}^{\ge 0}_{k+1}}: \bar{\mca{N}}^{\ge 0}_{k+1}(\beta: P) &\to P \times \mca{L}_{k+1}(\partial \beta)
\end{align}
such that compositions with $\id_P \times \ev_0$
(which are admissible maps to $P \times L$)
are strata-wise strongly submersive with respect to the CF-perturbations in (i).
\item[(iii):] Isomorphisms of admissible K-spaces,
which are obtained from isomorphisms (\ref{170903_1})--(\ref{170903_6})
by replacing each moduli space $X$ with $\bar{X}$.
For example, we obtain
\begin{equation}\label{171019_1}
\partial \bar{\mca{M}} _{k+1}(\beta: m) \cong \bigsqcup_{\substack{k_1+k_2=k+1 \\ 1 \le i \le k_1 \\ \beta_1 + \beta_2 = \beta}} (-1)^{\ep_0}
\bar{\mca{M}}_{k_1+1}(\beta_1: m) \fbp{i} {0} \bar{\mca{M}}_{k_2+1}(\beta_2: m)
\end{equation}
from (\ref{170903_1}).
We require that these isomorphisms are compatible with
CF-perturbations in (i) and evaluation maps (ii).
\end{enumerate}
Applying results in Section 7.1, one can define
\begin{align}
x_m(a,k)&:= \sum_{\omega_n(\bar{a}) < \ep(m+1-k)} (-1)^{n+1} \ev_*(\bar{\mca{M}}_{k+1}(\bar{a}, \{m\})) \\
\bar{x}_m(a,k)&:= \sum_{\omega_n(\bar{a}) < \ep(m+1-k)} (-1)^{n+1} \ev_*(\bar{\mca{M}}_{k+1}(\bar{a}, [m, m+1])), \\
y_m(a,k)&:= \sum_{\omega_n(\bar{a}) < \ep(m-U-k)} (-1)^{n+k+1} \ev_*(\bar{\mca{N}}_{k+1}^{\ge 0}(\bar{a}, \{m\})), \\
\bar{y}_m(a,k)&:= \sum_{\omega_n(\bar{a}) < \ep(m-U-k)} (-1)^{n+k+1} \ev_*(\bar{\mca{N}}_{k+1}^{\ge 0} (\bar{a}, [m, m+1])), \\
z_m(a,k)&:= \sum_{\omega_n(\bar{a}) < \ep(m-1-k)} (-1)^{n+k+1} \ev_*(\bar{\mca{N}}_{k+1}^0 (\bar{a}, \{m\})), \\
\bar{z}_m(a,k)&:= \sum_{\omega_n(\bar{a}) < \ep(m-1-k)} (-1)^{n+k+1} \ev_*(\bar{\mca{N}}_{k+1}^0 (\bar{a}, [m, m+1])).
\end{align}
\begin{rem}
In the above formulas we abbreviate differential form $1$ (see Remark \ref{170918_3}) and
CF-perturbations. We also abbreviate superscripts of $\ev$ for simplicity.
\end{rem}
\begin{rem}
Here we explain one issue which can be overlooked by abbreviating CF-perturbations in the above formulas.
As is clear from Section 7.1, to define a de Rham chain
one has to fix a parameter of CF-perturbation $e \in (0, 1]$
(here we use a letter $e$, not to be confused with $\ep$ used in the above formulas).
Hence we take a strictly decreasing sequence of positive real numbers $(e_m)_{m \ge 1}$ satisfying
$\lim_{m \to \infty} e_m = 0$,
and define $x_m(a,k)$ by
\[
x_m(a,k) := \sum_{\omega_n(\bar{a}) < \ep(m+1-k)} (-1)^{n+1} \ev_*(\bar{\mca{M}}_{k+1}(\bar{a}, \{m\}), \wh{\mf{S}}^{e_m}).
\]
$y_m(a,k)$ and $z_m(a,k)$ are defined by similar formulas, using $e_m$.
Then
$\bar{x}_m(a,k)$ is defined by interpolating $e_m$ and $e_{m+1}$ on the moduli space $\bar{\mca{M}}_{k+1}(\bar{a}, [m, m+1])$.
$\bar{y}_m(a,k)$ and $\bar{z}_m(a,k)$ are defined in a similar way.
\end{rem}
Let us check that the requirements in Theorem \ref{161215_1} are satisfied.
$x_m= e_-(\bar{x}_m)$, $y_m = e_-(\bar{y}_m)$, $z_m = e_-(\bar{z}_m)$ follow from the above definition
and (\ref{171206_1}).
Moreover
\[
(x_{m+1} - e_+(\bar{x}_m)) (a,k) \ne 0 \implies \omega_n(\bar{a}) \ge \ep(m+1-k)
\]
show $x_{m+1} - e_+(\bar{x}_m) \in F^m$.
By similar arguments one can show
$y_{m+1} - e_+(\bar{y}_m) \in F^{m-U-1}$ and
$z_{m+1} - e_+(\bar{z}_m) \in F^{m-2}$.
The isomorphism (\ref{171019_1}) shows
\[
\partial \bar{x}_m(a,k) = \sum_{\substack{k_1+k_2=k+1 \\ a_1+a_2 = a \\ 1 \le i \le k_1}} (-1)^{(k_1-m)(k_2-1) + (k_1-1)} \bar{x}_m (a_1, k_1) \circ_i \bar{x}_m (a_2, k_2)
\]
for every $(a,k)$ such that $\omega_n(\bar{a}) < \ep(m+1-k)$,
thus $\part \bar{x}_m - \frac{1}{2} [ \bar{x}_m, \bar{x}_m] \in F^m$.
By similar arguments one can show
$\part \bar{y}_m - [\bar{x}_m, \bar{y}_m] - \bar{z}_m \in F^{m-U-1}$ and
$\part \bar{z}_m - [\bar{x}_m, \bar{z}_m] \in F^{m-2}$.
Finally $x_m(\bar{a}, k) \ne 0 \implies \mca{M}_{k+1}(\bar{a}) \ne \emptyset$,
thus $\omega_n(\bar{a}) \ge 2\ep$ or $a=0$, $k \ge 2$.
Moreover
$[x_m(0,2)] = (-1)^{n+1} [\mca{M}_3(0)] = (-1)^{n+1}[L]$.
Similarly,
$z_m(\bar{a}, k) \ne 0 \implies \mca{N}^0_{k+1}(\bar{a}) \ne \emptyset$,
thus $\omega_n(\bar{a}) \ge 2\ep$ or $a=0$.
Moreover
$[z_m(0,0)] = (-1)^{n+1}[\mca{N}^0_1(0)] = (-1)^{n+1}[L]$.
\qed
\section{Strongly smooth map from a K-space with a CF-perturbation gives a de Rham chain}
The goal of this section is to explain proofs of results presented in Section 7.1.
Namely, given a strongly smooth map from a K-space (equipped with a differential form and a CF-perturbation) to $\mca{L}_{k+1}$ (the space of loops with $k+1$ marked points),
we define a de Rham chain on $\mca{L}_{k+1}$ and prove Stokes' formula and fiber product formula.
Here we imitate arguments in \cite{FOOO_Kuranishi} Sections 7 and 9 in our setting.
In Section 8.1 we consider smooth maps from single K-charts.
In Section 8.2 we consider strongly smooth maps from spaces with good coordinate system (GCS), and in Section 8.3 strongly smooth maps from K-spaces.
In Sections 8.2 and 8.3, we only consider spaces without boundaries (and corners),
since generalizations to spaces with boundaries are straightforward.
Throughout this section, $X$ denotes a separable, metrizable topolgoical space.
\subsection{Single K-chart}
In this subsection,
given a smooth map from a K-chart (equipped with a CF-perturbation and a differential form)
to $\mca{L}_{k+1}$,
we define a de Rham chain on $\mca{L}_{k+1}$.
We also prove Stokes's formula and fiber product formula.
We first consider K-charts without boundary, and then proceed to the case of K-charts with boundaries.
\subsubsection{K-chart without boundary}
Suppose we are given the following data:
\begin{itemize}
\item $\mca{U}=(U, \mca{E}, s, \psi)$ is a K-chart of $X$.
\item $f: U \to \mca{L}_{k+1}$ is a smooth map in the sense of Definition \ref{171205_1}.
\item $\omega \in \mca{A}^*_c(U)$.
\item $\mf{S} = (\mf{S}^\ep)_{0 < \ep \le 1}$ is a CF-perturbation of $\supp \omega$ such that $\ev_0 \circ f: U \to L$ is strongly submersive with respect to $\mf{S}$
(see Definition 7.24 and Definition-Lemma 7.25 in \cite{FOOO_Kuranishi}).
\end{itemize}
We are going to define $f_*(\mca{U}, \omega, \mf{S}^\ep) \in C^\dR_*(\mca{L}_{k+1})$ for every $\ep \in (0,1]$.
Let $( \mf{V}_{\mf{r}}, \mca{S}_{\mf{r}})_{\mf{r} \in \mf{R}}$ be a representative of $\mf{S}$
(see Definitions 7.15, 7.16 and 7.19 in \cite{FOOO_Kuranishi}), such that
\begin{itemize}
\item $\mf{V}_{\mf{r}} = (V_{\mf{r}}, E_{\mf{r}}, \phi_{\mf{r}}, \wh{\phi}_{\mf{r}})$ is a manifold chart
(we do not consider group actions: see Remark \ref{170912_1}) of $(U, \mca{E})$ such that
$(\phi_{\mf{r}}(V_{\mf{r}}))_{\mf{r} \in \mf{R}}$ covers $U$.
Let $s_{\mf{r}}: V_{\mf{r}} \to E_{\mf{r}}$ denote the pull back of $s$ by $\phi_{\mf{r}}$.
\item $\mca{S}_{\mf{r}} = (W_{\mf{r}}, \eta_{\mf{r}}, \{ \mf{s}^\ep_{\mf{r}}\}_{\ep})$ is a CF-perturbation of
$\mca{U}$ on $\mf{V}_{\mf{r}}$. Namely
\begin{itemize}
\item $W_{\mf{r}}$ is an open neighborhood of $0$ of a finite-dimensional oriented vector space $\wh{W_{\mf{r}}}$.
\item $\mf{s}^\ep_{\mf{r}}: V_{\mf{r}} \times W_{\mf{r}} \to E_{\mf{r}}$ is a $C^\infty$ map transversal to $0$ for every $\ep \in (0,1]$.
\item $\lim_{\ep \to 0} \mf{s}^\ep_{\mf{r}} (y, \xi) = s_{\mf{r}}(y)$ in compact $C^1$ topology on $V_{\mf{r}} \times W_{\mf{r}}$.
\item $\eta_{\mf{r}} \in \mca{A}^{\dim W_{\mf{r}}} _c(W_{\mf{r}})$ such that $\int_{W_{\mf{r}}} \eta_{\mf{r}} = 1$.
\end{itemize}
\item $\ev_0 \circ f \circ \phi_{\mf{r}} \circ \pr_{V_\mf{r}}: (\mf{s}^\ep_{\mf{r}})^{-1}(0) \to L $ is a submersion for every $\mf{r} \in \mf{R}$ and $\ep \in (0, 1]$.
\end{itemize}
$V_{\mf{r}}$ and $E_{\mf{r}}$ are oriented so that $\phi_{\mf{r}}$ and $\wh{\phi}_{\mf{r}}$ preserve orientations.
$(\mf{s}^\ep_{\mf{r}})^{-1}(0)$ is oriented so that the isomorphism
\[
E_{\mf{r}} \oplus T (\mf{s}^\ep_{\mf{r}})^{-1}(0) \cong T V_{\mf{r}} \oplus TW_{\mf{r}}
\]
preserves orientations, following Convention 8.2.1 in \cite{FOOO_09}.
We take a partition of unity $\{ \chi_{\mf{r}}\}_{\mf{r} \in \mf{R}}$ subordinate to $(\phi_{\mf{r}}(V_{\mf{r}}))_{\mf{r} \in \mf{R}}$,
i.e.,
$\chi_\mf{r} \in C^\infty_c(U, [0,1])$ and $\supp \chi_\mf{r} \subset \phi_\tau(V_\mf{r})$
for every $\mf{r} \in \mf{R}$, and
$\sum_{\mf{r} \in \mf{R}} \chi_{\mf{r}} \equiv 1$ on $\supp \omega$.
Then, for each $\ep \in (0, 1]$, we define
\begin{equation}
f_*(\mca{U}, \omega, \mf{S}^\ep):= \sum_{\mf{r} \in \mf{R}} f_*(\mca{U}, \chi_{\mf{r}} \omega, \mf{V}_{\mf{r}}, \mca{S}^\ep_{\mf{r}}),
\end{equation}
where the RHS is defined as
\begin{equation}\label{180112_1}
f_*(\mca{U}, \chi_{\mf{r}} \omega, \mf{V}_{\mf{r}}, \mca{S}^\ep_{\mf{r}}):= (-1)^\dagger ( (\mf{s}^\ep_{\mf{r}})^{-1}(0), f \circ \phi_{\mf{r}} \circ \pr_{V_{\mf{r}}} , (\phi_{\mf{r}} \circ \pr_{V_{\mf{r}}})^* (\chi_{\mf{r}} \omega) \wedge (\pr_{W_{\mf{r}}})^*(\eta_{\mf{r}})),
\end{equation}
\begin{equation}\label{171105_2}
\dagger:= \dim W_{\mf{r}} \cdot ( \rk \mca{E} + |\omega|).
\end{equation}
$(\mf{s}^\ep_{\mf{r}})^{-1}(0)$ is an oriented $C^\infty$-manifold,
and $\ev_0 \circ f \circ \phi_{\mf{r}} \circ \pr_{V_{\mf{r}}}$
is a submersion, thus
the RHS in (\ref{180112_1}) makes sense.
\begin{rem}
Strictly speaking, one has to fix an embedding of $(\mf{s}^\ep_{\mf{r}})^{-1}(0)$ to Euclidean space to define a de Rham chain.
However it is easy to check that the de Rham chain does not depend on choice of embedding.
\end{rem}
A priori, the above definition may depend on choices of representative and partition of unity.
In Lemma \ref{170323_1} below we prove well-definedness and Stokes' formula.
\begin{lem}\label{170323_1}
\begin{enumerate}
\item[(i):] For any $\omega_1, \omega_2 \in \mca{A}^*_c(U)$, $a_1, a_2 \in \R$ and $\ep \in (0, 1]$,
\[
f_* (\mca{U}, a_1 \omega_1 + a_2 \omega_2 , \mf{S}^\ep) = a_1 f_*(\mca{U}, \omega_1, \mf{S}^\ep) + a_2 f_*(\mca{U}, \omega_2, \mf{S}^\ep).
\]
\item[(ii):] The above definition does not depend on choices of representative of the CF-perturbation $\mf{S}$ and partition of unity.
\item[(iii):] $\partial f_*(\mca{U}, \omega, \mf{S}^\ep) = (-1)^{|\omega|+1} f_*(\mca{U}, d\omega, \mf{S}^\ep)$ for every $\ep \in (0, 1]$.
\end{enumerate}
\end{lem}
\begin{proof}
(i) is obvious as far as we fix a representative of $\mf{S}$ and a partition of unity.
Thus to prove (ii), it is sufficient to prove the following claim:
\begin{quote}
Let $\mca{S}_i = (W_i, \eta_i, \{\mf{s}^\ep_i\}_\ep) \, (i=1,2)$ be two equivalent CF-perturbations
of $\mca{U}$ on a manifold chart $(V, E, \phi, \wh{\phi})$ and
$\omega \in \mca{A}^*_c(U)$ such that $\supp \omega \subset \phi(V)$.
Then
\begin{align*}
&( (\mf{s}^\ep_1)^{-1}(0), f \circ \phi \circ \pr_V, (\phi \circ \pr_V)^*\omega \wedge (\pr_{W_1})^* \eta_1) \\
&=(-1)^\dagger ( (\mf{s}^\ep_2)^{-1}(0), f \circ \phi \circ \pr_V, (\phi \circ \pr_V)^*\omega \wedge (\pr_{W_2})^* \eta_2), \\
&\dagger:= (\dim W_1 - \dim W_2)(\rk \mca{E} + |\omega|)
\end{align*}
\end{quote}
By definition of equivalence (see Definition 7.5 in \cite{FOOO_Kuranishi}),
we may assume that there exists a linear projection
$\Pi: \wh{W}_2 \to \wh{W}_1$ such that $(\Pi)_! (\eta_2) = \eta_1$ and
$\mf{s}^\ep_2 = (\id_V \times \Pi)^*(\mf{s}^\ep_1)$.
Then, pushout of
$\id_V \times \Pi: (\mf{s}^\ep_2)^{-1}(0) \to (\mf{s}^\ep_1)^{-1}(0)$
sends
$(\phi \circ \pr_V)^*\omega \wedge (\pr_{W_2})^* \eta_2$
to $(-1)^\dagger (\phi \circ \pr_V)^*\omega \wedge (\pr_{W_1})^* \eta_1$,
which completes the proof.
To prove (iii), by (i) and (ii) we may assume that $\supp \omega$ is sufficiently small and $\chi_{\mf{r}} \equiv 1$ on $\supp \omega$ for some $\mf{r} \in \mf{R}$.
In this case (iii) is obvious except for signs, which can be checked by simple computations.
\end{proof}
Now we can state the fiber product formula.
Suppose, for each $i \in \{1, 2\}$, we have
$X_i$, $\mca{U}_i = (U_i, \mca{E}_i, s_i, \psi_i)$, $f_i: U_i \to \mca{L}_{k_i+1}$,
$\omega_i$, $\mf{S}_i$ as before.
Then, for every $\ep \in (0, 1]$ one can define
\[
(f_i)_*(\mca{U}_i, \omega_i, \mf{S}_i^\ep) \in C^\dR_* (\mca{L}_{k_i+1})
\]
for each $i \in \{1, 2\}$.
On the other hand, for each $j \in \{1, \ldots, k_1\}$,
one can take a fiber product of K-charts
\[
\mca{U}_{12} = \mca{U}_1 \fbp{\ev_j \circ f_1}{\ev_0 \circ f_2} \mca{U}_2.
\]
One can also define a fiber product of CF-perturbations $\mf{S}_1 \times \mf{S}_2$ on $\mca{U}_{12}$.
Finally we define a differential form $\omega_{12}$ on $\mca{U}_{12}$ by
\[
\omega_{12} := (-1)^{(\dim U_1 - \rk \mca{E}_1- |\omega_1| - n)|\omega_2|} \cdot \omega_1 \times \omega_2,
\]
and a smooth map
\[
f_{12}: U_1 \fbp{\ev_j \circ f_1}{\ev_0 \circ f_2} U_2 \to \mca{L}_{k_1+k_2}; \quad
(x_1, x_2) \mapsto \con_j (f_1(x_1), f_2(x_2)).
\]
Then one can state the fiber product formula as follows.
The proof is obvious except for signs,
which can be checked by direct computations.
\begin{lem}\label{171110_3}
In the situation described above, there holds
\[
(f_{12})_* (\mca{U}_{12}, \omega_{12}, \mf{S}^\ep_{12}) =
(f_1)_* (\mca{U}_1, \omega_1, \mf{S}^\ep_1) \circ_j
(f_2)_* (\mca{U}_2, \omega_2, \mf{S}^\ep_2)
\]
for every $\ep \in (0, 1]$.
\end{lem}
\subsubsection{K-chart with boundary}
Suppose we are given the following data:
\begin{itemize}
\item $\mca{U} = (U, \mca{E}, s, \psi)$ is an admissible K-chart on $X$,
i.e. $U$ is an admissible manifold with boundaries (and corners),
the vector bundle $\mca{E}$ and the section $s$ are also admissible.
\item $f: U \to \mca{L}_{k+1}$ is an admissible map in the sense of Definition \ref{170701_1} (i).
\item $\omega \in \mca{A}^*_c(U)$ is admissible.
\item $\mf{S} = ( \mf{S}^\ep)_{0<\ep \le 1}$ is an admissible CF-perturbation of $\supp \omega$
such that $\ev_0 \circ f: U \to L$ is strata-wise strongly submersive with respect to $\mf{S}$.
\end{itemize}
Under these assumptions,
our goal is to define $f_*(\mca{U}, \omega, \mf{S}^\ep) \in C^\dR_*(\mca{L}_{k+1})$
for every $\ep \in (0, 1]$.
Let $(\mf{V}_{\mf{r}}, \mca{S}_{\mf{r}})_{\mf{r} \in \mf{R}}$ be a representative of $\mf{S}$,
and $(\chi_{\mf{r}})_{\mf{r} \in \mf{R}}$ be a partition of unity subordinate to
$(\phi_{\mf{r}}(V_{\mf{r}}))_{\mf{r} \in \mf{R}}$ such that
$\sum_{\mf{r}} \chi_{\mf{r}} \equiv 1$
on $\supp \omega$.
Then we define
\begin{equation}\label{171105_1}
f_*( \mca{U}, \omega, \mf{S}^\ep) := \sum_{\mf{r} \in \mf{R}} f_*(\mca{U}, \chi_{\mf{r}} \omega, \mf{V}_{\mf{r}}, \mca{S}^\ep_{\mf{r}})
\end{equation}
where each term in the RHS is defined below.
The proof of well-definedness (the RHS does not depend on choices of representatives of $\mf{S}$ and partition of unity)
is straightforward and omitted.
Let $D:= \dim U$.
We may assume that $V_{\mf{r}}$ is an open neighborhood of $(t_1,\ldots, t_D) \in (\R_{\ge 0})^D$.
We define
\[
\mca{R}: \R^D \to (\R_{\ge 0})^D ; \,(t_1,\ldots, t_D) \mapsto (t'_1, \ldots, t'_D)
\]
by $t'_i := \begin{cases} t_i &(t_i \ge 0) \\ 0 &(t_i < 0) \\ \end{cases}$ for every $1 \le i \le D$.
We take $\kappa \in C^\infty(\R, [0,1])$
such that
$\kappa \equiv 1$ on a neighborhood of $\R_{\ge 0}$,
and $\kappa \equiv 0$ on a neighborhood of $\R_{\le -1}$.
Now let us define the following data:
\begin{itemize}
\item $\bar{V}_{\mf{r}}:= \mca{R}^{-1}(V_{\mf{r}}) $, $\bar{E}_{\mf{r}} := \mca{R}^*E_{\mf{r}}$.
\item $\bar{\mf{s}}^\ep_{\mf{r}}: = ( \mca{R}|_{\bar{V}_{\mf{r}}} \times \id_{W_{\mf{r}}} )^*(\mf{s}^\ep_{\mf{r}})$.
\item $\bar{f}_{\mf{r}} := f \circ \phi_{\mf{r}} \circ \mca{R}|_{\bar{V}_{\mf{r}}}$.
\item $\overline{\chi_{\mf{r}} \omega}(t_1, \ldots, t_D):= \kappa(t_1) \cdots \kappa(t_D) \cdot (\phi_{\mf{r}} \circ \mca{R}|_{\bar{V}_{\mf{r}}} )^*(\chi_{\mf{r}} \omega)$.
\end{itemize}
Finally, we define
\begin{equation}\label{170914_1}
f_*(\mca{U}, \chi_{\mf{r}} \omega, \mf{V}_{\mf{r}}, \mca{S}^\ep_{\mf{r}})
:= (-1)^\dagger ( (\bar{\mf{s}}^\ep_{\mf{r}})^{-1}(0), \bar{f}_{\mf{r}} \circ \pr_{\bar{V}_{\mf{r}}},
\pr^*_{\bar{V}_{\mf{r}}} ( \overline{\chi_{\mf{r}} \omega}) \wedge \pr^*_{W_{\mf{r}}} (\eta_{\mf{r}})),
\end{equation}
where the sign $\dagger$ is defined by (\ref{171105_2}).
Note that $\ev_0 \circ \bar{f}_{\mf{r}} \circ \pr_{\bar{V}_{\mf{r}}}: (\bar{\mf{s}}^\ep_{\mf{r}})^{-1}(0) \to L$
is a submersion, thus the map
$\bar{f}_{\mf{r}} \circ \pr_{\bar{V}_{\mf{r}}}: (\bar{\mf{s}}^\ep_{\mf{r}})^{-1}(0) \to L$
is smooth.
\begin{rem}
The RHS of (\ref{170914_1}) may depend on the choice of the cutoff function $\kappa$.
Here we fix such $\kappa$ and drop it from our notation in the following arguments.
\end{rem}
The fiber product formula holds in the obvious manner and omitted.
Stokes' formula is formulated as follows.
\begin{prop}
\[
\partial (f_*(\mca{U}, \omega, \mf{S}^\ep)) =
(-1)^{|\omega|} (f|_{\partial \mca{U}})_* (\partial \mca{U}, \omega|_{\partial \mca{U}}, \mf{S}^\ep|_{\partial \mca{U}}) +
(-1)^{|\omega|+1} f_*(\mca{U}, d \omega, \mf{S}^\ep).
\]
\end{prop}
\begin{proof}
This follows from
\begin{align*}
d(\overline{\chi_{\mf{r}}\omega}) (t_1,\ldots, t_D) &= \kappa(t_1) \cdots \kappa(t_D) (\phi_{\mf{r}} \circ \mca{R}|_{\bar{V}_{\mf{r}}})^* (d(\chi_{\mf{r}}\omega)) \\
&+ \sum_{i=1}^D \kappa(t_1) \cdots d\kappa(t_i) \cdots \kappa(t_D) ( \phi_{\mf{r}} \circ \mca{R}|_{\bar{V}_{\mf{r}}})^*(\chi_{\mf{r}} \omega)
\end{align*}
and $\partial U$ is oriented so that $TU \cong \R_{\text{out}} \oplus T(\partial U)$ preserves orientations.
\end{proof}
\subsubsection{K-chart with boundary over an interval}
Suppose we are given the following data:
\begin{itemize}
\item $\mca{U} = (U, \mca{E}, s, \psi)$ is an admissible K-chart on $X$.
\item $f: U \to [a, b] \times \mca{L}_{k+1}$ is an admissible map in the sense of Definition \ref{170701_2} (i).
\item $\omega \in \mca{A}^*_c(U)$ is admissible.
\item $\mf{S} = ( \mf{S}^\ep)_{0<\ep \le 1}$ is an admissible CF-perturbation of $\supp \omega$
such that $\ev_0 \circ f: U \to [a,b] \times L$ is strata-wise strongly submersive with respect to $\mf{S}$.
\end{itemize}
Under these assumptions,
our goal is to define
$f_*(\mca{U}, \omega, \mf{S}^\ep) \in \bar{C}^\dR_*(\mca{L}_{k+1})$ for every $\ep \in (0,1]$.
For simplicity of notations, in the following we assume that $a=-1$, $b=1$.
Let $(\mf{V}_{\mf{r}}, \mca{S}_{\mf{r}})_{\mf{r} \in \mf{R}}$ be a representative of $\mf{S}$,
and $(\chi_{\mf{r}})_{\mf{r} \in \mf{R}}$ be a partition of unity subordinate to
$(\phi_{\mf{r}}(V_{\mf{r}}))_{\mf{r} \in \mf{R}}$ such that
$\sum_{\mf{r}} \chi_{\mf{r}} \equiv 1$
on $\supp \omega$.
Then we define
\begin{equation}\label{170918_1}
f_*( \mca{U}, \omega, \mf{S}^\ep) := \sum_{\mf{r} \in \mf{R}} f_*(\mca{U}, \chi_{\mf{r}} \omega, \mf{V}_{\mf{r}}, \mca{S}^\ep_{\mf{r}})
\end{equation}
where each term in the RHS is defined below.
The proof of well-definedness is omitted.
To define
$f_*(\mca{U}, \chi_{\mf{r}} \omega, \mf{V}_{\mf{r}}, \mca{S}^\ep_{\mf{r}})$,
it is sufficient to consider the following three cases:
\begin{enumerate}
\item[(i):] $f \circ \phi_{\mf{r}} (V_{\mf{r}})$ is contained in $(-1, 1) \times \mca{L}_{k+1}$.
\item[(ii):] $f \circ \phi_{\mf{r}} (V_{\mf{r}})$ is contained in $[-1, 1) \times \mca{L}_{k+1}$ and intersects $\{-1\} \times \mca{L}_{k+1}$.
\item[(iii):] $f \circ \phi_{\mf{r}} (V_{\mf{r}})$ is contained in $(-1, 1] \times \mca{L}_{k+1}$ and intersects $\{1\} \times \mca{L}_{k+1}$.
\end{enumerate}
The case (i) is similar to the case in the previous subsubsection and omitted.
In the following we only consider the case (ii), since the case (iii) is completely parallel.
Let $D:= \dim U$.
We may assume that $V_{\mf{r}}$ is an open neighborhood of
$(0, t_2, \ldots, t_D)$ in $(\R_{\ge 0})^D$ and
$f_\R \circ \ph_{\mf{r}}(t_1, \ldots, t_D) = t_1-1$ .
Here $f_\R$ denotes $\pr_{\R} \circ f$. Similarly, we set $f_{\mca{L}} := \pr_{\mca{L}_{k+1}} \circ f$.
We define $\bar{V}_{\mf{r}}$, $\bar{E}_{\mf{r}}$ and $\bar{\mf{s}}^\ep_{\mf{r}}$ in the same way as in the previous subsubsection.
We also define
$\bar{f}_{\mf{r}}$ and $\overline{\chi_{\mf{r}}\omega}$ as follows:
\begin{itemize}
\item $\bar{f}_{\mf{r}}: \bar{V}_{\mf{r}} \to \R \times \mca{L}_{k+1}$ is defined by
\[
\pr_{\mca{L}_{k+1}} \circ \bar{f}_{\mf{r}} := f_{\mca{L}} \circ \phi_{\mf{r}} \circ \mca{R}|_{\bar{V}_{\mf{r}}}, \qquad
\pr_{\R} \circ \bar{f}_{\mf{r}} (t_1, \ldots, t_D): = t_1 - 1.
\]
\item $\overline{\chi_{\mf{r}} \omega}(t_1, \ldots, t_D):= \kappa(t_2) \cdots \kappa(t_D) \cdot (\phi_{\mf{r}} \circ \mca{R}|_{\bar{V}_{\mf{r}}})^*(\chi_{\mf{r}} \omega)$.
\end{itemize}
Here $\kappa \in C^\infty(\R, [0,1])$ is taken in the previous subsubsection.
Then we define
\[
f_*(\mca{U}, \chi_{\mf{r}} \omega, \mf{V}_{\mf{r}}, \mca{S}^\ep_{\mf{r}}):=
(-1)^\dagger
((\bar{\mf{s}}^\ep_{\mf{r}})^{-1}(0), \bar{f}_{\mf{r}} \circ \pr_{\bar{V}_{\mf{r}}} , \tau_+, \tau_-,
\pr^*_{\bar{V}_{\mf{r}}} ( \overline{\chi_{\mf{r}} \omega}) \wedge \pr^*_{W_{\mf{r}}} (\eta_{\mf{r}})),
\]
where the sign $\dagger$ is defined as before and
$\tau_-$, $\tau_+$ are defined as follows.
$\tau_+$ is defined in the obvious way, since $f \circ \phi_{\mf{r}}(V_{\mf{r}})$ does not intersect $\{1\} \times \mca{L}_{k+1}$.
On the other hand, $\tau_-$ is defined as a restriction of
\[
(\bar{V}_{\mf{r}} \cap \{t_1 \le 0\}) \times W_{\mf{r}} \to \R_{\le -1} \times ( (V_{\mf{r}} \cap \{t_1=0\}) \times W_{\mf{r}} );
\,
(t_1,\ldots, t_D, w) \mapsto (t_1-1, t_2, \ldots, t_D, w).
\]
This completes the definition of
$f_*(\mca{U}, \chi_{\mf{r}} \omega, \mf{V}_{\mf{r}}, \mca{S}^\ep_{\mf{r}})$,
thus the definition of
$f_*( \mca{U}, \omega, \mf{S}^\ep)$.
The fiber product formula is stated and proved in the obvious way.
Stokes' formula
\[
\partial f_*(\mca{U}, \omega, \mf{S}^\ep) = (-1)^{|\omega|} (f|_{\partial_h \mca{U}})*(\partial_h \mca{U}, \omega, \mf{S}^\ep) + (-1)^{|\omega|+1} f_*(\mca{U}, d \omega, \mf{S}^\ep),
\]
where $\partial_h \mca{U}$ is the restriction of $\mca{U}$ to $\partial_h U$,
and
\begin{align*}
&e_+(f_*(\mca{U}, \omega, \mf{S}^\ep)) = (f_{\mca{L}}|_{U_1})_*(\mca{U}|_{U_1}, \omega|_{U_1}, \mf{S}^\ep|_{U_1}), \\
&e_-(f_*(\mca{U}, \omega, \mf{S}^\ep)) = (f_{\mca{L}}|_{U_{-1}})_*(\mca{U}|_{U_{-1}}, \omega|_{U_{-1}}, \mf{S}^\ep|_{U_{-1}})
\end{align*}
can be checked directly and proofs are omitted.
\subsection{Space with GCS}
In this subsection, we assume that $X$ is compact.
Let $\wt{\mca{U}} = ( \{ \mca{U}_{\mf{p}} \}_{\mf{p} \in \mf{P}} , \{ \Phi_{\mf{p}\mf{q}}\}_{\mf{p} \ge \mf{q}})$
be a GCS (good coordinate system) without boundary on $X$ (see Section 10).
We start from the following definition.
\begin{defn}\label{171108_1}
A strongly smooth map $\wt{f}$ from $(X, \wt{\mca{U}})$ to $\mca{L}_{k+1}$ is a family
$(f_{\mf{p}})_{\mf{p} \in \mf{P}}$
which satisfies the following conditions:
\begin{itemize}
\item For every $\mf{p} \in \mf{P}$, $f_{\mf{p}}$ is a smooth map from $U_{\mf{p}}$ to $\mca{L}_{k+1}$.
\item For every $\mf{q} \le \mf{p}$, there holds $f_\mf{p} \circ \ph_{\mf{p}\mf{q}} = f_{\mf{q}}|_{U_{\mf{p}\mf{q}}}$.
\end{itemize}
\end{defn}
Let $\mca{K}$ be a support system of $\wt{\mca{U}}$
and $\wt{\mf{S}}$ be a CF-perturbation of $(\wt{\mca{U}}, \mca{K})$
(see Definitions 5.6 and 7.47 in \cite{FOOO_Kuranishi}).
Here we recall the definition of support system
(in the case $Z=X$):
\begin{defn}
A support system of $\wt{\mca{U}}$ is $\mca{K} = (\mca{K}_{\mf{p}}) _{\mf{p} \in \mf{P}}$
where $\mca{K}_{\mf{p}}$ is a compact set of $U_{\mf{p}}$ for each $\mf{p} \in \mf{P}$
which is a closure of an open subset $\mtrg{\mca{K}}_{\mf{p}}$,
and $\bigcup_{\mf{p} \in \mf{P}} \psi_{\mf{p}}(\mtrg{\mca{K}}_{\mf{p}} \cap s^{-1}_{\mf{p}}(0)) =X$.
\end{defn}
We assume that $\wt{\mf{S}}$ is transversal to $0$,
and $\ev_0 \circ \wt{f}: (X, \wt{\mca{U}}) \to L$ is strongly submersive with respect to $\wt{\mf{S}}$.
Also, let $\wt{\omega} = (\omega_{\mf{p}})_{\mf{p} \in \mf{P}}$ be a differential form on $(X, \wt{\mca{U}})$.
Given these data, we are going to define
\begin{equation}\label{171108_2}
\wt{f} _*(X, \wt{\mca{U}}, \wt{\omega}, \wt{\mf{S}}^\ep) \in C^\dR_*(\mca{L}_{k+1})
\end{equation}
for sufficiently small $\ep>0$.
Note that the support system $\mca{K}$ is a part of the data to define (\ref{171108_2}),
though it is implicit in the above formula.
\begin{rem}
In contrast to the case of single K-charts, where de Rham chain is defined for all $\ep \in (0, 1]$,
the de Rham chain (\ref{171108_2}) is defined only when $\ep>0$ is sufficiently small.
\end{rem}
In Section 8.2.1 we state and prove some technical lemmas.
In Section 8.2.2 we define (\ref{171108_2}) and check its well-definedness, invariance under GG-embedding, and Stokes' formula.
We only consider spaces with GCS \textit{without} boundaries, since generalization to spaces with boundaries (and corners) is straightforward.
The arguments of this subsection partly follow arguments in Sections 7.5--7.7 in \cite{FOOO_Kuranishi} with some modifications.
\subsubsection{Technical lemmas}
Recall that $|\mca{K}| := \bigg( \bigsqcup_{\mf{p} \in \mf{P}} \mca{K}_{\mf{p}} \bigg) / \sim$ equipped with the
topology from $|\wt{\mca{U}}|$ is metrizable; see Definition 5.6 (3) in \cite{FOOO_Kuranishi}.
We fix a metric $d$ on $|\mca{K}|$ which is compatible with this topology.
For any $S \subset |\mca{K}|$ and $\delta>0$ we set
\[
B_\delta(S) := \{ x \in |\mca{K}| \mid d(S, x) < \delta\}, \qquad
\bar{B}_\delta(S):= \{ x \in |\mca{K}| \mid d(S, x) \le \delta \}.
\]
Recall the notion of ``support set'' of CF-perturbations from Definition 7.72 in \cite{FOOO_Kuranishi}:
\begin{quote}
For each $\mf{p} \in \mf{P}$, let $\{ (\mf{V}_{\mf{r}}, \mca{S}_{\mf{r}}) \mid \mf{r} \in \mf{R}\}$ be a representative of the CF-perturbation $\mf{S}_{\mf{p}}$ of $\mca{U}_{\mf{p}}$.
Then for each $\ep \in (0,1]$, we define $\Pi ( (\mf{S}^\ep_{\mf{p}})^{-1}(0)))$ to be the set consisting of all $x \in U_{\mf{p}}$ such that
there exists $\mf{r} \in \mf{R}$, $y \in V_{\mf{r}}$, $\xi \in W_{\mf{r}}$ such that
\[
\phi_{\mf{r}} (y) = x, \qquad s^\ep_{\mf{r}}(y, \xi)=0, \qquad \xi \in \supp \eta_{\mf{r}}.
\]
This definition is independent of the choice of representative.
Then we define
\[
\Pi ((\wt{\mf{S}}^\ep)^{-1}(0)):= \bigcup_{\mf{p} \in \mf{P}} (\mca{K}_{\mf{p}} \cap \Pi ( (\mf{S}^\ep_{\mf{p}})^{-1}(0))) \subset |\mca{K}|
\]
and call it the \textit{support set} of $\wt{\mf{S}}^\ep$.
\end{quote}
\begin{lem}\label{170329_1}
For any neighborhood $U$ of $\bigcup_{\mf{p} \in \mf{P}} (\mca{K}_{\mf{p}} \cap s_{\mf{p}}^{-1}(0))$ in $|\mca{K}|$,
there exists $\ep_0>0$ such that
$0 < \ep < \ep_0 \implies \Pi ((\wt{\mf{S}}^\ep)^{-1}(0)) \subset U$.
\end{lem}
\begin{proof}
If this is not the case, there exists a sequence $(\ep_m)_{m \ge 1}$ of positive real numbers converging to $0$,
and a sequence $(x_m)_{m \ge 1}$ such that $x_m \in \Pi ( (\wt{\mf{S}}^{\ep_m})^{-1}(0)) \setminus U$ for every $m \ge 1$.
Up to subsequence $(x_m)_m$ has a limit in $|\mca{K}|$ which we denote by $x$.
There exists $\mf{q} \in \mf{P}$ such that $x_m \in \mca{K}_{\mf{q}}$ for infinitely many $m$, thus we may assume that $x_m \in \mca{K}_{\mf{q}}$ for all $m$,
then $x \in \mca{K}_{\mf{q}}$. It is sufficient to show that $s_{\mf{q}}(x)=0$,
which implies $x_m \in U$ for sufficiently large $m$, a contradiction.
Take a manifold chart $(V, E, \phi, \wh{\phi})$ of $(U_{\mf{q}}, \mca{E}_{\mf{q}})$ such that $x \in \phi(V)$
and $\mf{S}_{\mf{q}}$ is locally represented by $(W, \eta, (\mf{s}^\ep)_\ep)$
where $\mf{s}^\ep: V \times W \to E$ and $\eta$ is a compactly supported form on $W$.
Let $s: V \to E$ denote the pullback of $s_{\mf{q}}$ by $\phi$.
Then $\mf{s}^\ep(y, \xi) \to s(y) \,(y \in V, \, \xi \in W)$
as $\ep \to 0$ in the compact $C^1$ topology on $V \times W$.
Take $y \in V$ so that $\phi(y)=x$ and
$y_m \in V$ for sufficiently large $m$ so that $\phi(y_m)=x_m$.
Then $y_m \to y$ and
there exists $\xi_m \in \supp \eta$ such that $\mf{s}^{\ep_m}(y_m, \xi_m)=0$.
Since $\supp \eta$ is compact, this shows that $s(y) = 0$,
which implies $s_{\mf{q}}(x)=0$.
\end{proof}
\begin{lem}\label{170330_1}
For any support system $\mca{K'} < \mca{K}$
(see Definition 5.6 (2) in \cite{FOOO_Kuranishi}),
there exist $\delta>0$ and $\ep_0>0$ such that
\[
0 < \ep < \ep_0 \implies
B_\delta ( \mca{K}'_{\mf{p}}) \cap \Pi ( (\wt{\mf{S}}^\ep)^{-1}(0)) \subset \mca{K}_{\mf{p}} \, (\forall \mf{p} \in \mf{P}).
\]
\end{lem}
\begin{proof}
We take a support system $\mca{K}''$ such that
$\mca{K}' < \mca{K}'' < \mca{K}$.
\textbf{Step 1.} We first show that there exists $\delta>0$ such that
\[
\bar{B}_\delta ( \mca{K}'_{\mf{p}}) \cap \bigcup_{\mf{q} \in \mf{P}} (\mca{K}_{\mf{q}} \cap s_{\mf{q}}^{-1}(0)) \subset \mca{K}''_{\mf{p}} \,(\forall \mf{p} \in \mf{P}).
\]
Suppose that this is not the case.
Then there exist $\mf{p} \in \mf{P}$ and a sequence $(x_m)_{m \ge 1}$ on $\bigcup_{\mf{q} \in \mf{P}} (\mca{K}_{\mf{q}} \cap s_{\mf{q}}^{-1}(0)) \setminus \mca{K}''_{\mf{p}}$
such that $d(x_m, \mca{K}'_{\mf{p}}) \to 0$ as $m \to \infty$.
Since $\mf{P}$ is a finite set, there exists $\mf{q} \in \mf{P}$ such that $x_m \in \mca{K}_{\mf{q}} \cap s_{\mf{q}}^{-1}(0)$ for infinitely many $m$.
We may assume that $x_m \in \mca{K}_{\mf{q}} \cap s_{\mf{q}}^{-1}(0)$ for all $m$, and
there exists $x \in \mca{K}_{\mf{q}} \cap \mca{K}'_{\mf{p}}$ such that $d(x_m, x) \to 0$.
Then $s_{\mf{q}}(x) = 0$, since $x_m \to x$ in $\mca{K}_{\mf{q}}$ and $s_{\mf{q}}(x_m) = 0$ for all $m$.
Since $\mca{K}_{\mf{q}} \cap \mca{K}_{\mf{p}} \ne \emptyset$ in $|\wt{\mca{U}}|$,
at least one of the following three cases holds:
\begin{itemize}
\item $\mf{p} \ge \mf{q}$: This case $\mca{K}_{\mf{q}} \setminus U_{\mf{p}}$ is compact, thus $d(\mca{K}_{\mf{q}} \setminus U_{\mf{p}}, \mca{K}'_{\mf{p}})>0$.
Since $\lim_{m \to \infty} x_m = x$ and $x \in \mca{K}'_{\mf{p}}$, we obtain $x_m \in U_{\mf{p}}$ for sufficiently large $m$.
Since $\mca{K}'_{\mf{p}} \subset \mtrg{\mca{K}}''_{\mf{p}}$ (this follows from $\mca{K}' < \mca{K}''$; see Definition 5.6 (2) in \cite{FOOO_Kuranishi}),
we obtain $x_m \in \mca{K}''_{\mf{p}}$ for sufficiently large $m$, contradicting our assumption.
\item $\mf{p} \le \mf{q}$ and $\dim U_{\mf{p}} = \dim U_{\mf{q}}$: This case $\mca{K}_{\mf{q}} \setminus U_{\mf{p}}$ is compact
since $\Phi_{\mf{q}\mf{p}}$ is an open embedding, and we obtain a contradiction as in the previous case.
\item $\mf{p} < \mf{q}$ and $\dim U_{\mf{p}} < \dim U_{\mf{q}}$:
Since $\lim_{m \to \infty} x_m = x \in \mca{K}'_{\mf{p}} \subset \mtrg{\mca{K}}''_{\mf{p}}$ and $x_m \notin \mca{K}''_{\mf{p}}$ for every $m$,
we obtain $x_m \notin \ph_{\mf{q}\mf{p}}(U_{\mf{q}\mf{p}})$ for sufficiently large $m$.
On the other hand $x = \lim_{m \to \infty} x_m$ satisfies $s_{\mf{q}}(x)=0$ and
$D_x s_{\mf{q}} : T_xU_{\mf{q}}/ T_x U_{\mf{p}} \to (\mca{E}_{\mf{q}})_x/(\mca{E}_{\mf{p}})_x$
is an isomorphism,
which implies that $s_{\mf{q}}(x_m) \ne 0$
for sufficiently large every $m$,
contradicting our assumption.
\end{itemize}
\textbf{Step 2.}
Taking $\delta$ as in Step 1, we prove that there exists $\ep_0>0$ such that
\[
0 < \ep < \ep_0 \implies B_\delta(\mca{K}'_{\mf{p}}) \cap \Pi ( (\wt{\mf{S}}^\ep)^{-1}(0)) \subset \mca{K}_{\mf{p}} \quad(\forall \mf{p} \in \mf{P}).
\]
If this is not the case, there exist
a sequence $(\ep_m)_{m \ge 1} $ of positive real numbers converging to $0$,
$\mf{p} \in \mf{P}$, and
a sequence $(x_m)_{m \ge 1}$ such that
\[
x_ m \in (B_\delta(\mca{K}'_{\mf{p}}) \cap \Pi ( (\wt{\mf{S}}^{\ep_m})^{-1}(0)) ) \setminus \mca{K}_{\mf{p}}
\]
for every $m$.
Up to subsequence $(x_m)_m$ has a limit $x$ in $|\mca{K}|$.
By Lemma \ref{170329_1}, $x \in \bigcup_{\mf{q} \in \mf{P}} (\mca{K}_{\mf{q}} \cap s_{\mf{q}}^{-1}(0))$.
Also $x \in \bar{B}_\delta(\mca{K}'_{\mf{p}})$, thus by Step 1 we get $x \in \mca{K}''_{\mf{p}}$.
There exists $\mf{q} \in \mf{P}$ such that $x_m \in \mca{K}_{\mf{q}}$ for infinitely many $m$,
then we may assume that $x_m \in \mca{K}_{\mf{q}}$ for all $m$, in particular $x \in \mca{K}_{\mf{q}}$.
Then $\mca{K}_{\mf{q}} \cap \mca{K}''_{\mf{p}} \ne \emptyset$,
thus at least one of the following three cases holds:
\begin{itemize}
\item $\mf{p} \ge \mf{q}$:
This case $\mca{K}_{\mf{q}} \setminus U_{\mf{p}}$ is compact, thus $d(\mca{K}_{\mf{q}} \setminus U_{\mf{p}}, \mca{K}''_{\mf{p}})>0$.
Since $\lim_{m \to \infty} x_m = x \in \mca{K}''_{\mf{p}}$,
we obtain $x_m \in U_{\mf{p}}$ for sufficiently large $m$.
By $\mca{K}''_{\mf{p}} \subset \mtrg{\mca{K}}_{\mf{p}}$,
we obtain $x_m \in \mca{K}_{\mf{p}}$ for sufficiently large $m$, contradicting our assumption.
\item $\mf{p} \le \mf{q}$ and $\dim U_{\mf{p}} = \dim U_{\mf{q}}$:
Again $\mca{K}_{\mf{q}} \setminus U_{\mf{p}}$ is compact, and we obtain a contradiction as in the previous case.
\item $\mf{p} < \mf{q}$ and $\dim U_{\mf{p}} < \dim U_{\mf{q}}$:
Since $\lim_{m \to \infty} x_m = x \in \mtrg{\mca{K}_{\mf{p}}}$ and
$x_m \notin \mca{K}_{\mf{p}}$ for every $m \ge 1$,
we obtain $x_m \notin \ph_{\mf{q}\mf{p}}(U_{\mf{q}\mf{p}})$ for sufficiently large $m$.
Since the CF-perturbation $\mf{S}^{\ep_m}_{\mf{q}}$ converges to $s_{\mf{q}}$ in compact $C^1$-topology,
$s_{\mf{q}}(x)=0$ thus
$D_x s_{\mf{q}} : T_xU_{\mf{q}}/ T_x U_{\mf{p}} \to (\mca{E}_{\mf{q}})_x/(\mca{E}_{\mf{p}})_x$ is an isomorphism,
we obtain $x_m \notin \Pi ((\mf{S}_{\mf{q}}^{\ep_m})^{-1}(0))$ for sufficiently large $m$,
contradicting our assumption.
\end{itemize}
\end{proof}
\begin{lem}
For sufficiently small $\delta>0$ there holds
\[
\mf{q} < \mf{p} \implies B_\delta(\mca{K}'_{\mf{p}}) \cap \mca{K}_{\mf{q}} \subset U_{\mf{p}\mf{q}}.
\]
\end{lem}
\begin{proof}
Suppose that this is not the case.
Then there exist $\mf{p}$, $\mf{q}$ such that
$\mf{q} < \mf{p}$ and a sequence $(\delta_m)_m$ converging to $0$
and a sequence $(x_m)_m$ such that $x_m \in (B_{\delta_m}(\mca{K}'_{\mf{p}}) \cap \mca{K}_{\mf{q}}) \setminus U_{\mf{p}\mf{q}}$
for every $m$.
Then the sequence $(x_m)_m$ has a limit point $x \in \mca{K}_{\mf{q}} \setminus U_{\mf{p}\mf{q}}$,
then $d(x, \mca{K}'_{\mf{p}})>0$,
contradicting $x_m \in B_{\delta_m}(\mca{K}'_{\mf{p}})\,(\forall m)$.
\end{proof}
\subsubsection{Definition and well-definedness}
We start from data $X$, $\wt{\mca{U}}$, $\wt{\omega}$, $\mca{K}$, $\wt{\mf{S}}$ and $\wt{f}$.
We take the following choices
(our definition of partition of unity is slightly different from that of \cite{FOOO_Kuranishi}, Definition 7.64).
\begin{itemize}
\item A support system $\mca{K}' < \mca{K}$.
\item A positive real number $\delta$ such that
\begin{itemize}
\item There exists $\ep>0$ such that
\[
0 < \ep' < \ep \implies
B_{\delta} ( \mca{K}'_{\mf{p}}) \cap \Pi ( (\wt{\mf{S}}^{\ep'})^{-1}(0)) \subset \mca{K}_{\mf{p}} \quad (\forall \mf{p} \in \mf{P}).
\]
\item $2 \delta < d(\mca{K}'_{\mf{p}}, \mca{K}'_{\mf{q}})$ for any $\mf{p}, \mf{q} \in \mf{P}$ such that $U_{\mf{p}} \cap U_{\mf{q}} = \emptyset$.
\item $\mf{q} < \mf{p} \implies B_{\delta}(\mca{K}'_{\mf{p}}) \cap \mca{K}_{\mf{q}} \subset U_{\mf{p}\mf{q}}$.
\item $\mca{K}'_{\mf{p}}(\delta):= \{ x \in \mca{K}_{\mf{p}} \mid d(\mca{K}'_{\mf{p}}, x) \le \delta \} \subset \mtrg{\mca{K}_{\mf{p}}}$ for every $\mf{p} \in \mf{P}$.
\end{itemize}
\item
Partition of unity $\chi = (\chi_{\mf{p}})_{\mf{p} \in \mf{P}}$ of $(X, \wt{\mca{U}}, \mca{K}', \delta)$.
Namely, the following conditions are satisfied:
\begin{itemize}
\item $\chi_{\mf{p}} : |\mca{K}| \to [0,1]$ is a strongly smooth function for every $\mf{p} \in \mf{P}$.
Namely, for every $\mf{q} \in \mf{P}$,
$\chi_{\mf{p}}|_{\mca{K}_{\mf{q}}}$ can be extended to a $C^\infty$-function defined on a neighborhood of $\mca{K}_{\mf{q}}$ in $U_{\mf{q}}$.
\item $\supp \chi_{\mf{p}} \subset B_{\delta}(\mca{K}'_{\mf{p}})$ for every $\mf{p} \in \mf{P}$.
\item $\sum_{\mf{p} \in \mf{P}} \chi_{\mf{p}} = 1$ on a neighborhood of $\bigcup_{\mf{p} \in \mf{P}} s_{\mf{p}}^{-1}(0) \subset |\mca{K}|$.
\end{itemize}
\end{itemize}
Then we define
\[
\wt{f}_*(X, \wt{\mca{U}}, \wt{\omega}, \wt{\mf{S}}^\ep):= \sum_{\mf{p} \in \mf{P}} (f_{\mf{p}})_*( \mca{U}_{\mf{p}}, \chi_{\mf{p}} \omega_{\mf{p}} , \mf{S}^\ep_{\mf{p}}).
\]
$ (f_{\mf{p}})_*( \mca{U}_{\mf{p}}, \chi_{\mf{p}} \omega_{\mf{p}} , \mf{S}^\ep_{\mf{p}})$
in the RHS makes sense, since for every $\mf{p} \in \mf{P}$ there holds
\[
\supp \chi_{\mf{p}} \cap U_{\mf{p}} \subset
B_\delta(\mca{K}'_{\mf{p}}) \cap U_{\mf{p}} \subset
\mca{K}'_{\mf{p}} (\delta) \subset \mtrg{\mca{K}}_{\mf{p}}
\]
and $\mf{S}_{\mf{p}} \in \mca{S}(\mca{K}_{\mf{p}})$.
We need to recall the following definition (Definition 7.79 in \cite{FOOO_Kuranishi}):
\begin{defn}\label{180301_1}
Let $\mca{A}$ and $\mca{X}$ be sets, and
$(F_a)_{a \in \mca{A}}$ be a family of maps such that
$F_a: (0, \ep_a) \to \mca{X}$ for each $a \in \mca{A}$.
We say that $F_a$ is independent on choices of $a$ in the sense of $\spadesuit$ if the following holds:
\begin{quote}
$\spadesuit$: For any $a_1, a_2 \in \mca{A}$, there exists $0 < \ep_0 < \min \{\ep_{a_1}, \ep_{a_2}\}$ such that
$F_{a_1}(\ep) = F_{a_2}(\ep)$ for every $\ep \in (0, \ep_0)$.
\end{quote}
\end{defn}
\begin{lem}\label{170330_1.5}
The above definition of
$\wt{f}_*(X, \wt{\mca{U}}, \wt{\omega}, \wt{\mf{S}}^\ep)$
is independent on choices of $\mca{K}'$, $\delta$, $\chi$
in the sense of $\spadesuit$.
\end{lem}
\begin{proof}
Let us take two choices $((\mca{K}')^i, \delta^i, \chi^i )_{i=1,2}$.
We are going to prove
\[
\sum_{\mf{p} \in \mf{P}} (f_{\mf{p}})_*( \mca{U}_{\mf{p}}, \chi^1_{\mf{p}} \omega_{\mf{p}}, \mf{S}^\ep_{\mf{p}}) =
\sum_{\mf{p} \in \mf{P}} (f_{\mf{p}})_*( \mca{U}_{\mf{p}}, \chi^2_{\mf{p}} \omega_{\mf{p}}, \mf{S}^\ep_{\mf{p}})
\]
when $\ep$ is sufficiently small.
By our assumption, $\sum_{\mf{p} \in \mf{P}} \chi^1_{\mf{p}} = \sum_{\mf{p} \in \mf{P}} \chi^2_{\mf{p}} = 1$ on a neighborhood of $\bigcup_{\mf{p} \in \mf{P}} s^{-1}_{\mf{p}}(0)$.
Thus, Lemma \ref{170329_1} implies that,
for sufficiently small $\ep>0$ and every $\mf{p} \in \mf{P}$,
there holds $\sum_{\mf{p} \in \mf{P}} \chi^1_{\mf{p}} = \sum_{\mf{p} \in \mf{P}} \chi^2_{\mf{p}} = 1$
on $\mca{K}_{\mf{p}} \cap \Pi ( (\mf{S}^\ep_{\mf{p}})^{-1}(0))$.
Then we obtain
\begin{align*}
\sum_{\mf{p} \in \mf{P}} (f_{\mf{p}})_*( \mca{U}_{\mf{p}}, \chi^1_{\mf{p}} \omega_{\mf{p}}, \mf{S}^\ep_{\mf{p}})&= \sum_{\mf{p}_1, \mf{p}_2} (f_{\mf{p}_1})_* ( \mca{U}_{\mf{p}_1}, \chi^1_{\mf{p}_1} \chi^2_{\mf{p}_2} \omega_{\mf{p}_1} , \mf{S}^\ep_{\mf{p}_1}) \\
\sum_{\mf{p} \in \mf{P}} (f_{\mf{p}})_*( \mca{U}_{\mf{p}}, \chi^2_{\mf{p}} \omega_{\mf{p}}, \mf{S}^\ep_{\mf{p}})&=
\sum_{\mf{p}_1, \mf{p}_2} (f_{\mf{p}_2})_* ( \mca{U}_{\mf{p}_2}, \chi^1_{\mf{p}_1} \chi^2_{\mf{p}_2} \omega_{\mf{p}_2} , \mf{S}^\ep_{\mf{p}_2}).
\end{align*}
Then it is sufficient to show
\[
(f_{\mf{p}_1})_* ( \mca{U}_{\mf{p}_1}, \chi^1_{\mf{p}_1} \chi^2_{\mf{p}_2} \omega_{\mf{p}_1} , \mf{S}^\ep_{\mf{p}_1}) =
(f_{\mf{p}_2})_* ( \mca{U}_{\mf{p}_2}, \chi^1_{\mf{p}_1} \chi^2_{\mf{p}_2} \omega_{\mf{p}_2} , \mf{S}^\ep_{\mf{p}_2})
\]
for any $\mf{p}_1, \mf{p}_2 \in \mf{P}$.
We may assume either $\mf{p}_1 \le \mf{p}_2$ or $\mf{p}_1 \ge \mf{p}_2$,
since otherwise $U_{\mf{p}_1} \cap U_{\mf{p}_2} = \emptyset$, thus
$\chi^1_{\mf{p}_1} \chi^2_{\mf{p}_2} \equiv 0$ by $2\delta < d(\mca{K}'_{\mf{p}_1}, \mca{K}'_{\mf{p}_2})$.
Also this equality is clear when $\mf{p}_1 = \mf{p}_2$.
Thus we may assume that $\mf{p}_1 < \mf{p}_2$.
In the argument below,
we abbreviate
$\mf{p}_i$ by $i$. For example $\mca{U}_{\mf{p}_i}$ is abbreviated by $\mca{U}_i$.
Let us introduce the following notations:
\begin{itemize}
\item $\mca{U}_{21}$ denotes $\mca{U}_1|_{U_{21}} = (\ph_{21})^* \mca{U}_2$.
\item $f_{21}$ denotes $f_1|_{U_{21}} = f_2 \circ \ph_{21}$.
\item $\omega_{21}$ denotes $\omega_1|_{U_{21}} = (\ph_{21})^* \omega_2$.
\item $\mf{S}_{21}$ denotes $\mf{S}_1|_{U_{21}} = (\ph_{21})^* \mf{S}_2$.
\end{itemize}
Then there holds
\[
(f_1)_* (\mca{U}_1, \chi^1_1 \chi^2_2 \omega_1 , \mf{S}^\ep_1) = (f_{21})_* (\mca{U}_{21}, \chi^1_1 \chi^2_2 \omega_{21}, \mf{S}^\ep_{21})
\]
since $\supp \chi^2_{\mf{p}_2} \subset B_\delta(\mca{K}'_{\mf{p}_2})$ and
$B_\delta(\mca{K}'_{\mf{p}_2}) \cap \mca{K}_{\mf{p}_1} \subset U_{\mf{p}_2\mf{p}_1}$.
On the other hand
\[
(f_2)_* (\mca{U}_2, \chi^1_1 \chi^2_2 \omega_2 , \mf{S}^\ep_2) = (f_{21})_* (\mca{U}_{21}, \chi^1_1 \chi^2_2 \omega_{21}, \mf{S}^\ep_{21})
\]
when $\ep$ is sufficiently small,
since $\supp \chi^1_{\mf{p}_1} \subset B_\delta(\mca{K}'_{\mf{p}_1})$ and
$B_\delta(\mca{K}'_{\mf{p}_1}) \cap \Pi ((\wt{\mf{S}}^\ep)^{-1}(0)) \subset \mca{K}_{\mf{p}_1}$
by Lemma \ref{170330_1}.
\end{proof}
Next we prove the invariance by GG-embedding
(for the definition of GG-embedding, see Definition 3.24 in \cite{FOOO_Kuranishi}).
Lemma \ref{170330_3} below is an analogue of Proposition 9.16 in \cite{FOOO_Kuranishi}.
\begin{lem}\label{170330_3}
Let us consider $(\wt{\mca{U}}_i, \mca{K}_i, \wt{\mf{S}}_i, \wt{f}_i, \wt{\omega}_i)_{i=1,2}$
and $\wt{\Phi}$ such that
\begin{itemize}
\item For each $i \in \{1,2\}$, the tuple $(\wt{\mca{U}}_i, \mca{K}_i, \wt{\mf{S}}_i, \wt{f}_i, \wt{\omega}_i)$
satisfies the conditions to define $(\wt{f}_i)_*(X, \wt{\mca{U}}_i, \wt{\omega}_i, \wt{\mf{S}}^\ep_i)$ for sufficiently small $\ep>0$.
\item $\wt{\Phi}: \wt{\mca{U}_1} \to \wt{\mca{U}_2}$ is a GG-embedding.
Namely, $\wt{\Phi} = (\mf{i}, (\Phi_{\mf{p}})_{\mf{p} \in \mf{P}_1})$ where $\mf{i}: \mf{P}_1 \to \mf{P}_2$ is an order preserving map,
and $\Phi_{\mf{p}} = (\ph_{\mf{p}}, \wh{\ph}_{\mf{p}}): (\mca{U}_1)_{\mf{p}} \to (\mca{U}_2)_{\mf{i}(\mf{p})}$ is an embedding of K-charts for each $\mf{p} \in \mf{P}_1$,
such that compatibilities in Definition 3.24 in \cite{FOOO_Kuranishi} are satisfied.
\item $\mca{K}_1$, $\mca{K}_2$ and $\wt{\mf{S}}_1$, $\wt{\mf{S}}_2$ are compatible with $\wt{\Phi}$ (see Definition 9.3 (3), (4) in \cite{FOOO_Kuranishi}).
Moreover $\wt{\omega}_1 = \wt{\Phi}^* \wt{\omega}_2$, $\wt{f}_1 = \wt{f}_2 \circ \wt{\Phi}$.
\end{itemize}
Then, for sufficiently small $\ep>0$, there holds
\[
(\wt{f}_1)_*(X, \wt{\mca{U}}_1, \wt{\omega}_1, \wt{\mf{S}}^\ep_1) =
(\wt{f}_2)_*(X, \wt{\mca{U}}_2, \wt{\omega}_2, \wt{\mf{S}}^\ep_2).
\]
\end{lem}
\begin{proof}
Let us take a support system $\mca{K}'_i$ of $\wt{\mca{U}_i}$ for $i=1,2$ such that:
\begin{itemize}
\item $\mca{K}'_i < \mca{K}_i$ for $i=1, 2$.
\item $\mca{K}'_1$ and $\mca{K}'_2$ are compatible with $\wt{\Phi}$.
Namely, $\ph_{\mf{p}}((\mca{K}'_1)_{\mf{p}}) \subset \mtrg{(\mca{K}'_2)}_{\mf{i}(\mf{p})}$ for every $\mf{p} \in \mf{P}_1$.
\end{itemize}
First we need Lemma \ref{171208_2} below,
whose proof is almost the same as the proof of Proposition 7.67 in \cite{FOOO_Kuranishi}.
In the statement, we take a metric on $|\mca{K}_2|$ and
its pullback to $|\mca{K}_1|$ via a natural embedding map
$|\mca{K}_1| \to |\mca{K}_2|$.
This embedding map exists since
$\mca{K}_1$ and $\mca{K}_2$ are compatible with $\wt{\Phi}$.
\begin{lem}\label{171208_2}
For any $\delta>0$, there exists
$\chi^+_1 = ((\chi^+_1)_{\mf{p}}) _{\mf{p} \in \mf{P}_1}$
which satisfies the following conditions:
\begin{itemize}
\item $(\chi^+_1)_{\mf{p}}$ is a strongly smooth map from $|\mca{K}_2|$ to $[0,1]$ for every $\mf{p} \in \mf{P}_1$.
\item $\supp (\chi^+_1)_{\mf{p}} \subset B_\delta( \ph_{\mf{p}} ( (\mca{K}'_1)_{\mf{p}}))$ for every $\mf{p} \in \mf{P}_1$.
Here $B_\delta$ denotes the $\delta$-neighborhood in $|\mca{K}_2|$.
\item $\sum_{\mf{p} \in \mf{P}_1} (\chi^+_1)_{\mf{p}} = 1$ on a neighborhood of $\bigcup_{\mf{q} \in \mf{P}_2} s_{\mf{q}}^{-1}(0) \subset |\mca{K}_2|$.
\end{itemize}
\end{lem}
\begin{proof}
There exists a natural homeomorphism $\bigcup_{\mf{q} \in \mf{P}_2} s_{\mf{q}}^{-1}(0) \to X$.
In the following argument we identify these two spaces, and in particular consider $X$ as a subspace of $|\mca{K}_2|$.
First we need the following fact, which is essentially the same as Lemma 7.66 in \cite{FOOO_Kuranishi}:
\begin{quote}
($\star$): For any open set $W$ of $|\mca{K}_2|$ containing a compact subset $K$ of $X$,
there exists a strongly smooth function $g: |\mca{K}_2| \to [0,1]$ that has a compact support
contained in $W$ and $1$ on a neighborhood of $K$.
\end{quote}
Applying the claim ($\star$) to $K: =\ph_{\mf{p}}((\mca{K}'_1)_{\mf{p}})$ and $W:= B_{\delta}(\ph_{\mf{p}}((\mca{K}'_1)_{\mf{p}}))$,
we obtain $f_{\mf{p}}: |\mca{K}_2| \to [0, 1]$.
Applying the claim ($\star$) to $K:= X$ and
$W:= \bigg\{ x \in |\mca{K}_2| \, \bigg{|} \sum_{\mf{q} \in \mf{P}_1} f_{\mf{q}}(x)>1/2 \bigg\}$,
we obtain $g: |\mca{K}_2| \to [0,1]$.
Finally, for each $\mf{p} \in \mf{P}_1$
we define $(\chi^+_1)_{\mf{p}}: |\mca{K}_2| \to [0,1]$ by
\[
(\chi^+_1)_{\mf{p}}(x):=
\begin{cases}
g(x) f_{\mf{p}}(x) \biggl( \sum_{\mf{q} \in \mf{P}_1} f_{\mf{q}}(x) \biggr)^{-1} &(x \in W) \\
0 &(x \notin W).
\end{cases}
\]
\end{proof}
For each $\mf{p} \in \mf{P}_1$, we define $(\chi_1)_{\mf{p}}: |\mca{K}_1| \to [0,1]$ by
\[
(\chi_1)_{\mf{p}} |_{(\mca{K}_1)_{\mf{q}}} := (\chi^+_1)_{\mf{i}(\mf{p})} \circ \ph_{\mf{q}}|_{(\mca{K}_1)_{\mf{q}}}.
\]
Then $\chi_1 = ((\chi_1)_{\mf{p}})_{\mf{p} \in \mf{P}_1}$ is a partition of unity of
$(X, \wt{\mca{U}}_1 , \mca{K}'_1, \delta)$.
We also take a partition of unitiy of
$(X, \wt{\mca{U}}_2, \mca{K}'_2, \delta)$, which we denote by $\chi_2 = ((\chi_2)_{\mf{q}})_{\mf{q} \in \mf{P}_2}$.
Now we can complete the proof as follows:
\begin{align*}
&(\wt{f}_1)_*(X, \wt{\mca{U}}_1, \wt{\omega}_1, \wt{\mf{S}}^\ep_1 ) \\
&\quad = \sum_{\mf{p} \in \mf{P}_1} ( (f_1)_{\mf{p}})_* ((\mca{U}_1)_{\mf{p}}, (\chi_1)_{\mf{p}} (\omega_1)_{\mf{p}}, (\mf{S}^\ep_1)_{\mf{p}}) \\
&\quad = \sum_{\mf{p} \in \mf{P}_1}
((f_2)_{\mf{i}(\mf{p})})_*( (\mca{U}_2)_{\mf{i}(\mf{p})}, (\chi^+_1)_{\mf{i}(\mf{p})} (\omega_2)_{\mf{i}(\mf{p})} , (\mf{S}^\ep_2)_{\mf{i}(\mf{p})}) \\
&\quad = \sum_{\mf{q} \in \mf{P}_2} ((f_2)_{\mf{q}} )_*((\mca{U}_2)_{\mf{q}}, (\chi_2)_{\mf{q}} (\omega_2)_{\mf{q}} , (\mf{S}^\ep_2)_{\mf{q}}) \\
&\quad = (\wt{f}_2)_*(X, \wt{\mca{U}}_2, \wt{\omega}_2, \wt{\mf{S}}^\ep_2).
\end{align*}
The first and fourth equality follows from the definition.
The second equality holds since
$(\omega_1)_{\mf{p}} = (\ph_{\mf{p}})^*((\omega_2)_{\mf{i}(\mf{p})})\, (\forall \mf{p} \in \mf{P}_1)$ and
\[
\supp (\chi^+_1)_{\mf{p}} \cap \Pi ((\mf{S}^\ep_2)^{-1}_{\mf{i}(\mf{p})}(0)) \subset
B_\delta(\ph_{\mf{p}}((\mca{K}'_1)_{\mf{p}})) \cap \Pi ( (\mf{S}^\ep_2)^{-1}_{\mf{i}(\mf{p})} (0))
\subset \ph_{\mf{p}} ((\mca{K}_1)_{\mf{p}})
\]
when $\delta$ and $\ep$ are sufficiently small
(this can be proved by arguments similar to the proof of Lemma \ref{170330_1}).
Proof of the third equality is similar to the proof of Lemma \ref{170330_1.5}.
This completes the proof of Lemma \ref{170330_3}.
\end{proof}
Let us state and prove Stokes' formula.
\begin{prop}\label{171109_1}
For sufficiently small $\ep>0$, there holds
\[
\partial (\wt{f}_*(X, \wt{\mca{U}}, \wt{\omega}, \wt{\mf{S}}^\ep) )
= (-1)^{|\omega|+1} \wt{f}_*(X, \wt{\mca{U}}, d\wt{\omega}, \wt{\mf{S}}^\ep).
\]
\end{prop}
\begin{proof}
Take $\mca{K}' < \mca{K}$, $\delta>0$ and let $(\chi_{\mf{p}})_{\mf{p} \in \mf{P}}$ be a partition of unity of $(X, \wt{\mca{U}}, \mca{K}', \delta)$.
When $\ep>0$ is sufficiently small,
\begin{align*}
&\partial (\wt{f}_*(X, \wt{\mca{U}}, \wt{\omega}, \wt{\mf{S}}^\ep) )
- (-1)^{|\omega|+1} \wt{f}_*(X, \wt{\mca{U}}, d\wt{\omega}, \wt{\mf{S}}^\ep) \\
&\quad =
(-1)^{|\omega|+1} \sum_{\mf{p} \in \mf{P} } (f_{\mf{p}})_* (\mca{U}_{\mf{p}}, d \chi_{\mf{p}} \omega_{\mf{p}} , \mf{S}^\ep_{\mf{p}}) \\
&\quad =
(-1)^{|\omega|+1} \sum_{\mf{p}, \mf{q} \in \mf{P}} (f_{\mf{p}})_* (\mca{U}_{\mf{p}}, \chi_{\mf{q}} d \chi_{\mf{p}} \omega_{\mf{p}}, \mf{S}^\ep_{\mf{p}}) \\
&\quad =
(-1)^{|\omega|+1}
\sum_{\mf{p}, \mf{q} \in \mf{P}} (f_{\mf{q}})_* (\mca{U}_{\mf{q}}, \chi_{\mf{q}} d\chi_{\mf{p}} \omega_{\mf{q}}, \mf{S}^\ep_{\mf{q}}) \\
&\quad =
(-1)^{|\omega|+1}
\sum_{\mf{q} \in \mf{P}} (f_{\mf{q}})_*(\mca{U}_{\mf{q}}, \chi_{\mf{q}} (\sum_{\mf{p} \in \mf{P}} d\chi_{\mf{p}}) \omega_{\mf{q}}, \mf{S}^\ep_{\mf{q}}) = 0.
\end{align*}
The first equality follows from Stokes' formula for K-charts:
Lemma \ref{170323_1} (iii).
The second and fifth equality holds since $\sum_{\mf{p} \in \mf{P}} \chi_{\mf{p}} = 1$ on
$\Pi( (\wt{\mf{S}}^\ep)^{-1}(0))$ when $\ep$ is sufficiently small.
The third equality holds by the argument similar to the proof of Lemma \ref{170330_1.5}.
The fourth equality is obvious.
\end{proof}
\subsection{K-space}
Let $(X, \wh{\mca{U}})$ be a compact, oriented K-space with a strongly smooth map
$\wh{f}: (X, \wh{\mca{U}}) \to \mca{L}_{k+1}$,
a differential form $\wh{\omega}$ on $(X, \wh{\mca{U}})$ and a CF-perturbation
$\wh{\mf{S}} = (\wh{\mf{S}}^\ep)_{\ep \in (0,1]}$ on $(X, \wh{\mca{U}})$
which is transversal to $0$ and $\ev_0 \circ \wh{f}: (X, \wh{\mca{U}}) \to L$ is strongly submersive with respect to $\wh{\mf{S}}$.
Under these assumptions we define a de Rham chain
\[
\wh{f}_* (X, \wh{\mca{U}}, \wh{\omega}, \wh{\mf{S}}^\ep) \in C^\dR_*(\mca{L}_{k+1})
\]
for sufficiently small $\ep>0$, and check Stokes' formula and fiber product formula.
We only consider K-spaces \textit{without} boundaries,
since generalization to admissible K-spaces with boundaries (and corners)
will be straightforward.
By Lemma 9.10 in \cite{FOOO_Kuranishi},
there exist
\begin{itemize}
\item $\wt{\mca{U}}$: GCS of $X$,
\item $\wt{\omega}$: differential form on $\wt{\mca{U}}$,
\item $\wt{f}$: strongly smooth map from $(X, \wt{\mca{U}})$ to $\mca{L}_{k+1}$,
\item $\mca{K}$: support system of $\wt{\mca{U}}$,
\item $\wt{\mf{S}}$: CF-perturbation of $(\wt{\mca{U}}, \mca{K})$ which is transversal to $0$, and
$\ev_0 \circ \wt{f}: (X, \wt{\mca{U}}) \to L$ is strongly submersive with respect to $\wt{\mf{S}}$,
\item strict KG-embedding $\wh{\Phi}: \wh{\mca{U}}_0 \to \wt{\mca{U}}$, where $\wh{\mca{U}_0}$ is an open substructure of $\wh{\mca{U}}$,
\end{itemize}
satisfying the following compatibilities:
\begin{itemize}
\item $\wh{\omega}|_{\wh{\mca{U}_0}} = \wh{\Phi}^*(\wt{\omega})$,
\item $\wh{f}|_{\wh{\mca{U}_0}} = \wt{f} \circ \wh{\Phi}$,
\item $\wh{\mf{S}}|_{\wh{\mca{U}_0}} = \wh{\Phi}^*(\wt{\mf{S}})$.
\end{itemize}
Then we define
\begin{equation}\label{171208_1}
\wh{f}_*(X, \wh{\mca{U}}, \wh{\omega}, \wh{\mf{S}^\ep}):= \wt{f}_*(X, \wt{\mca{U}}, \wt{\omega}, \wt{\mf{S}^\ep})
\end{equation}
for sufficiently small $\ep>0$.
Well-definedness
(i.e. the RHS of (\ref{171208_1}) does not depend on choices of $\wt{\mca{U}}$, $\wt{\omega}$, etc. in the sense of $\spadesuit$)
follows from invariance by GG-embedding (Lemma \ref{170330_3})
and arguments in \cite{FOOO_Kuranishi} Section 9.2
(proof of Proposition 9.16 $\implies$ Theorem 9.14).
Stokes' formula in this setting (Theorem \ref{170628_2})
follows from Stokes' formula for GCS (Theorem \ref{171109_1}).
Finally we give a sketch of the proof
of the fiber product formula (Theorem \ref{170628_3}),
imitating the proof of Proposition 10.23 in \cite{FOOO_Kuranishi}.
Namely, we prove
\begin{equation}\label{171110_1}
(\wh{f}_{12})_*(X_{12}, \wh{\mca{U}}_{12}, \wh{\omega}_{12}, \wh{\mf{S}}_{12}^\ep) =
(\wh{f}_1)_*(X_1, \wh{\mca{U}}_1, \wh{\omega}_1, \wh{\mf{S}}_1^\ep) \circ_j
(\wh{f}_2)_*(X_2, \wh{\mca{U}}_2, \wh{\omega}_2, \wh{\mf{S}}_2^\ep)
\end{equation}
for sufficiently small $\ep>0$, where $(X_{12}, \wh{\mca{U}}_{12})$ denotes the fiber product of
$(X_1, \wh{\mca{U}}_1)$ and $(X_2, \wh{\mca{U}}_2)$.
For each $i \in \{1, 2\}$, there exist a GCS $\wt{\mca{U}}_i$ on $X_i$ and a KG-embedding
$\wh{\mca{U}}_i \to \wt{\mca{U}}_i$,
namely a strict KG-embedding
$(\wh{\mca{U}}_i)_0 \to \wt{\mca{U}}_i$,
where $(\wh{\mca{U}}_i)_0$ is an open substructure of $\wh{\mca{U}}_i$.
Let $(\wh{\mca{U}}_{12})_0$ denote the fiber product of
$(\wh{\mca{U}}_1)_0$ and $(\wh{\mca{U}}_2)_0$.
One may assume that, for each $i \in \{1, 2\}$
there exist $\wt{\omega}_i$, $\wt{f}_i$, $\wt{\mf{S}}_i$ satisfying compatibilities.
Let $\chi^i = (\chi^i_{\mf{p}_i})_{\mf{p}_i \in \mf{P}_i}$ be a partition of unity of $(X_i, \wt{\mca{U}}_i)$.
For each $\mf{p}_i \in \mf{P}_i$,
$\chi^i_{\mf{p}_i}$ induces a strongly smooth map
$\wh{\chi}^i_{\mf{p}_i}: (X_i, \wh{\mca{U}}_i) \to [0, 1]$,
and there holds
$\sum_{\mf{p}_i \in \mf{P}_i} \wh{\chi}^i_{\mf{p}_i} = 1$.
Then (\ref{171110_1}) reduces to proving
\begin{align}\label{171110_2}
&(\wh{f}_{12})_*(X_{12}, (\wh{\mca{U}}_{12})_0, \wh{\chi}^1_{\mf{p}_1} \wh{\chi}^2_{\mf{p}_2} \wh{\omega}_{12}, \wh{\mf{S}}^\ep_{12}) =\\
&\qquad\qquad (\wh{f}_1)_*(X_1, (\wh{\mca{U}}_1)_0, \wh{\chi}^1_{\mf{p}_1} \wh{\omega}_1, \wh{\mf{S}}^\ep_1) \circ_j
(\wh{f}_2)_*(X_2, (\wh{\mca{U}}_2)_0, \wh{\chi}^2_{\mf{p}_2} \wh{\omega}_2, \wh{\mf{S}}^\ep_2) \nonumber
\end{align}
for every $\mf{p}_1 \in \mf{P}_1$ and $\mf{p}_2 \in \mf{P}_2$.
Now there holds
\[
(\wh{f}_1)_*(X_1, (\wh{\mca{U}}_1)_0, \wh{\chi}^1_{\mf{p}_1} \wh{\omega}_1, \wh{\mf{S}}^\ep_1) =
((f_1)_{\mf{p}_1})_*( (\mca{U}_1)_{\mf{p}_1}, (\chi_1)_{\mf{p}_1} (\omega_1)_{\mf{p}_1} , (\mf{S}_1)^\ep_{\mf{p}_1}),
\]
and similar formulas hold for $(\wh{f}_{12})_*(X_{12}, (\wh{\mca{U}}_{12})_0, \wh{\chi}^1_{\mf{p}_1} \wh{\chi}^2_{\mf{p}_2} \wh{\omega}_{12}, \wh{\mf{S}}^\ep_{12})$
and $(\wh{f}_2)_*(X_2, (\wh{\mca{U}}_2)_0, \wh{\chi}^2_{\mf{p}_2} \wh{\omega}_2, \wh{\mf{S}}^\ep_2)$.
Then (\ref{171110_2}) follows from the fiber product formula for single K-charts (Lemma \ref{171110_3}).
This completes a sketch of the proof of Theorem \ref{170628_3}.
\section{Proof of $C^0$-approximation lemma (Theorem \ref{170430_2})}
Recall that $\mca{L}_{k+1}$ is a subspace of $\Pi^{k+1}$,
which consists of $(\Gamma_0, \ldots, \Gamma_k) \in \Pi^{k+1}$ such that $\ev_1(\Gamma_i) =\ev_0(\Gamma_{i+1})\,(0 \le i \le k-1)$ and $\ev_1(\Gamma_k) = \ev_0(\Gamma_0)$.
Then, Theorem \ref{170430_2} is reduced to Lemma \ref{170430_3} below.
We first give a proof of Lemma \ref{170430_3} assuming
Lemma \ref{171114_1} (which is stated in the proof),
and proceed to a proof of Lemma \ref{171114_1}.
\begin{lem}\label{170430_3}
Let $(X, \wh{\mca{U}})$ be a compact K-space and
$\wh{f}: (X, \wh{\mca{U}}) \to \Pi^\con$ be a strongly continuous map such that
$\ev_j \circ \wh{f}: (X, \wh{\mca{U}}) \to L$ is strongly smooth for $j=0, 1$.
Let $Z$ be a closed subset of $X$ and $\wh{g}: (Z, \wh{\mca{U}}|_Z) \to \Pi$
be a strongly smooth map
(the notion of smooth map from a K-space to $\Pi$ is defined in the same way as Definition \ref{170902_2})
such that
$\ev_j \circ \wh{g} = \ev_j \circ \wh{f}|_Z$ for $j=0, 1$ and
$\wh{g}$ is $\ep$-close to $\wh{f}|_Z$ (with respect to $d_\Pi$).
If $\ep<\rho_L$, there exist an open substructure $\wh{\mca{U}_0}$ of $\wh{\mca{U}}$
and a strongly smooth map $\wh{g'}: (X, \wh{\mca{U}_0}) \to \Pi$
such that the following conditions hold:
\begin{itemize}
\item $\wh{g'}$ is $\ep$-close to $\wh{f}|_{\wh{\mca{U}_0}}$.
\item $\ev_j \circ \wh{g'} = \ev_j \circ \wh{f} |_{\wh{\mca{U}_0}}$ for $j=0, 1$.
\item $\wh{g'} = \wh{g}$ on $\wh{\mca{U}_0}|_Z$.
\end{itemize}
\end{lem}
\begin{proof}
\textbf{Step 1.}
There exist
\begin{itemize}
\item a GCS $\wt{\mca{U}_Z}$ of $Z$,
\item a KG-embedding from $\wh{\mca{U}} |_Z$ to $\wt{\mca{U}}_Z$,
namely an open substructure $\wh{\mca{U}_{Z,0}}$ of $\wh{\mca{U}}|_Z$ and a strict KG-embedding
$\Phi_1: \wh{\mca{U}_{Z,0}} \to \wt{\mca{U}_Z}$,
\item a strongly continuous map $\wt{f_Z}: (Z, \wt{\mca{U}}_Z) \to \Pi^\con$,
\item a strongly smooth map $\wt{g}: (Z, \wt{\mca{U}}_Z) \to \Pi$,
\end{itemize}
such that
\begin{itemize}
\item $\Phi_1^* \wt{f}_Z = \wh{f} |_{\wh{\mca{U}_{Z,0}}}$ and $\Phi_1^* \wt{g} = \wh{g}|_{\wh{\mca{U}_{Z,0}}}$,
\item $\wt{g}$ is $\ep$-close to $\wt{f}_Z$,
\item $\ev_j \circ \wt{g} = \ev_j \circ \wt{f}_Z$ for $j=0, 1$.
\end{itemize}
The GCS $\wt{\mca{U}_Z}$ exists by Theorem 3.30 in \cite{FOOO_Kuranishi}.
By inspecting its proof (Section 11.1 in \cite{FOOO_Kuranishi}),
one can take $\wt{\mca{U}_Z}$ so that each chart of
$\wt{\mca{U}_Z}$ is an open subchart of a certain K-chart of $\wh{\mca{U}_Z}$,
thus one can define $\wt{g}$ and $\wt{f}_Z$ by pulling back $\wh{f}$ and $\wh{g}$.
\textbf{Step 2.}
By Proposition 7.52 and Lemma 7.53 in \cite{FOOO_Kuranishi}, there exist
\begin{itemize}
\item a GCS $\wt{\mca{U}}$ of $X$.
\item a KG-embedding from $\wh{\mca{U}}$ to $\wt{\mca{U}}$,
namely and open substructure $\wh{\mca{U}_{0, +}}$ of $\wh{\mca{U}}$
and a strict KG-embedding $\Phi_2: \wh{\mca{U}_{0, +}} \to \wt{\mca{U}}$,
\item an extension from $\wt{\mca{U}_Z}$ to $\wt{\mca{U}}$ (see Definition 7.50 of \cite{FOOO_Kuranishi}),
namely an open substructure $\wt{\mca{U}_{Z,0}}$ of $\wt{\mca{U}_Z}$
and a strict extension $\Phi_3: \wt{\mca{U}_{Z,0}} \to \wt{\mca{U}}$,
\item a strongly continuous map $\wt{f}: (X, \wt{\mca{U}}) \to \Pi^\con$
such that $\Phi_2^* \wt{f} = \wh{f} |_{\wh{\mca{U}_{0, +}}}$
and $\Phi_3^* \wt{f} = \wt{f_Z}|_{\wt{\mca{U}_{Z,0}}}$.
\end{itemize}
Now we can state a $C^0$-approximation result for GCS
in Lemma \ref{171114_1} below,
which we assume for the moment.
\begin{lem}\label{171114_1}
There exist
\begin{itemize}
\item $\wt{\mca{U}_{Z,00}}$: an open substructure of $\wt{\mca{U}_{Z, 0}}$,
\item $\wt{\mca{U}_0}$: an open substructure of $\wt{\mca{U}}$,
\item a strict extension from $\wt{\mca{U}_{Z,00}}$ to $\wt{\mca{U}_0}$,
\item a strongly smooth map $\wt{g'}: (X, \wt{\mca{U}_0}) \to \Pi$ such that
\begin{itemize}
\item $\wt{g'}$ is $\ep$-close to $\wt{f}|_{\wt{\mca{U}_0}}$,
\item $\ev_j \circ \wt{g'} = \ev_j \circ \wt{f}\,(j=0,1)$ on $\wt{\mca{U}_0}$,
\item $\wt{g'} = \wt{g}$ on $\wt{\mca{U}_{Z,00}}$.
\end{itemize}
\end{itemize}
\end{lem}
\textbf{Step 3.}
Take an open substructure $\wh{\mca{U}_0}$ of $\wh{\mca{U}}$ with a strict KG-embedding $\wh{\mca{U}_0} \to \wt{\mca{U}_0}$
such that $\wh{\mca{U}_0}|_Z \to \wt{\mca{U}_0}$ factors through a strict KG-embedding $\wh{\mca{U}_0}|_Z \to \wt{\mca{U}_{Z,00}}$.
Finally, we can define $\wh{g'}: (X, \wh{\mca{U}_0}) \to \Pi$ by pulling back $\wt{g'}: (X, \wt{\mca{U}_0}) \to \Pi$
by $\wh{\mca{U}_0} \to \wt{\mca{U}_0}$.
\end{proof}
The rest of this section is devoted to the proof of Lemma \ref{171114_1}.
First we need to prove Lemmas \ref{171114_2}, \ref{171114_3} and \ref{170513_1}.
\begin{lem}\label{171114_2}
For any nonempty finite set $I$, there exists a $C^\infty$-map
\[
G: \{ (x_i, t_i )_{i \in I} \mid x_i \in L, \, t_i \in [0,1], \, \max_{i, j \in I} d_L(x_i, x_j) < \rho_L, \, \sum_{i \in I} t_i = 1\} \to L
\]
such that the following properties hold:
\begin{enumerate}
\item[(i):] If $t_i = \begin{cases} 1 &(i= i_0) \\ 0 &(i \ne i_0) \end{cases}$ for some $i_0 \in I$, then $G (x_i, t_i)_i = x_{i_0}$.
\item[(ii):] If there exists $y \in L$ and $r \in (0, \rho_L]$ such that $d_L(y, x_i) < r\,(\forall i \in I)$, then
$d_L(y, G(x_i, t_i)_i) < r$.
\end{enumerate}
\end{lem}
\begin{proof}
We fix an arbitrary total order on $I$.
For each $t = (t_i)_{i \in I}$, let $i_0:= \max \{ i \mid t_i \ne 0\}$,
and for every $\theta \in [0,1]$ let
\[
(t^\theta)_i: = \begin{cases}
t_i (1-\theta t_{i_0})/(1-t_{i_0}) &(i < i_0) \\
\theta t_{i_0} &(i = i_0) \\
0 &(i>i_0).
\end{cases}
\]
Now we define $G$ so that it satisfies (i),
and for any $x = (x_i)_i$, the map
\[
[0, 1] \to L; \, \theta \mapsto G(x, t^\theta)
\]
is the shortest geodesic connecting the end points.
Now $G$ satisfies (ii) since
any geodesic ball with radius $r \in (0, \rho_L]$ is geodesically convex,
by the definition of $\rho_L$ (see Section 7.3).
\end{proof}
\begin{lem}\label{171114_3}
For any $(T, \gamma) \in \Pi^\con$ and $\ep>0$,
there exists $(T', \gamma') \in \Pi$ such that
$d_\Pi((T,\gamma), (T', \gamma')) < \ep$.
\end{lem}
\begin{proof}
Take $T':= T+\ep/2$, and
$\gamma': [0,T'] \to L$ so that $d_L(\gamma(sT), \gamma'(sT'))<\ep/2$ for any $s \in [0,1]$,
which is possible since $C^\infty([0,1], L)$ is dense in $C^0([0,1], L)$ with respect to the $C^0$-topology.
\end{proof}
\begin{lem}\label{170513_1}
Let $U$ be a $C^\infty$-manifold,
$f: U \to \Pi^\con$ be a continuous map,
such that $\ev_j \circ f: U \to L$ is of $C^\infty$ for $j=0, 1$.
Let $V$ be a submanifold of $U$,
$g: V \to \Pi$ be a smooth map such that
$\ev_j \circ g = \ev_j \circ f|_V\,(j=0, 1)$ and
$g$ is $\ep$-close to $f|_V$.
Then for any $x \in V$, there exists an open neighborhood $W$ of $x$ in $U$
and a smooth map $g': W \to \Pi$ such that
$g' = g$ on $W \cap V$,
$\ev_j \circ g' = \ev_j \circ f|_W\,(j=0,1 )$,
and $g'$ is $\ep$-close to $f|_W$.
\end{lem}
\begin{proof}
The last condition ``$g'$ is $\ep$-close to $f|_W$'' can be achieved by taking $W$ sufficiently small,
since $f$ is continuous and $\ep$-closeness is an open condition.
Thus it is sufficient to define $W$ and a smooth map $g': W \to \Pi$ such that
$g' = g$ on $W \cap V$ and $\ev_j \circ g' =\ev_j \circ f|_W\,(j=0, 1)$.
Let $W$ be a sufficiently small neighborhood of $x$
, so that there exists a $C^\infty$-map $r: W \to W \cap V$ satisfying $r|_{W \cap V} = \id_{W \cap V}$, and
\[
d_L(\ev_j (f(y)), \ev_j (g \circ r(y))) < r_{\text{inj}}(L)
\]
for every $y \in W$ and $j \in \{0,1\}$, where $r_{\text{inj}}(L)$ denotes the injectivity radius of $L$.
We define $\xi_j(y) \in T_{\ev_j(g \circ r(y))}L$ by
$\exp (\xi_j(y)) = \ev_j (f(y))$.
We set $g(z):= (T(z), \gamma(z))$ for any $z \in W \cap V$.
For every $y \in W$ and $j \in \{0, 1\}$,
we define $\xi^y_j(\theta) \in T_{\gamma(r(y))(\theta)} L \, (0 \le \theta \le T(r(y)))$ so that
\[
\xi^y_0(0) = \xi_0(y), \quad \nabla_\theta \xi^y_0 \equiv 0, \quad
\xi^y_1(T(r(y)) = \xi_1(y), \quad \nabla_\theta \xi^y_1 \equiv 0.
\]
Taking a $C^\infty$-function $\chi: [0,1] \to [0,1]$ such that
$\chi \equiv 0$ near $0$ and
$\chi \equiv 1$ near $1$,
we define
$\xi'(y) \in C^\infty( \gamma(r(y))^*TL)$ by
\[
\xi'(y) (\theta):= \chi(\theta/T(r(y))) \cdot \xi^y_1(\theta ) + (1 - \chi (\theta/T(r(y)))) \cdot \xi^y_0(\theta)
\qquad (0 \le \theta \le T(r(y))).
\]
Finally, we define
$\gamma'(y): [0, T(r(y))] \to L$ by
\[
\gamma'(y)(\theta):= \exp (\gamma(r(y))(\theta), \xi'(y)(\theta))
\]
and
$g'(y):= (T(r(y)), \gamma'(y))$.
It is easy to check that this map $g'$ is smooth and satisfies required conditions.
\end{proof}
Now let us start the proof of Lemma \ref{171114_1}.
Let $\mf{P}$ denote the index set of $\wt{\mca{U}}$,
and $\mf{P}_Z \subset \mf{P}$ denote the index set of $\wt{\mca{U}_Z}$.
Let $\{\mca{U}_{\mf{p}}\}_{\mf{p} \in \mf{P}}$ be K-charts of $\wt{\mca{U}}$, and we denote $\mca{U}_{\mf{p}} = (U_{\mf{p}}, \ldots )$ for each $\mf{p} \in \mf{P}$.
Let $\{\mca{U}^Z_{\mf{p}}\}_{\mf{p} \in \mf{P}_Z}$ be K-charts of $\wt{\mca{U}_{Z,0}}$ and we denote $\mca{U}^Z_{\mf{p}}=(U^Z_{\mf{p}}, \ldots )$ for each $\mf{p} \in \mf{P}_Z$.
We also take support systems $\{ \mca{K}_{\mf{p}}\}_{\mf{p} \in \mf{P}}$ of $\wt{\mca{U}}$
and $\{\mca{K}^Z_{\mf{p}}\}_{\mf{p} \in \mf{P}_Z}$ of $\wt{\mca{U}_{Z,0}}$,
such that
$(\ph_3)_{\mf{p}} (\mca{K}^Z_{\mf{p}}) \subset \mtrg{\mca{K}}_{\mf{p}}$
for every $\mf{p} \in \mf{P}_Z$,
where $\Phi_3 = (\ph_3, \wh{\ph}_3)$ is a strict extension from $\wt{\mca{U}_{Z,0}}$ to $\wt{\mca{U}}$.
A subset $\mf{F} \subset \mf{P}$ is called a \textit{filter}
if $\mf{p}, \mf{q} \in \mf{P}$, $\mf{p} \ge \mf{q}$, $\mf{p} \in \mf{F}$ imply $\mf{q} \in \mf{F}$.
In particular the empty set is a filter (see Section 12.3 in \cite{FOOO_Kuranishi}).
Now Lemma \ref{171114_1} is proved by applying Lemma \ref{170507_1} below for $\mf{F}=\mf{P}$,
and setting
\[
(\mca{U}_0)_{\mf{p}}: = \mca{U}_{\mf{p}}|_{V_{\mf{p}}} \,(\forall \mf{p} \in \mf{P}), \quad
(\mca{U}_{Z,00})_{\mf{p}}: = (\mca{U}_{Z,0})_{\mf{p}}|_{W_{\mf{p}}} \,(\forall \mf{p} \in \mf{P}_Z).
\]
\begin{lem}\label{170507_1}
For any filter $\mf{F}$ of $\mf{P}$,
there exist
$(V_{\mf{p}}, g'_{\mf{p}})_{\mf{p} \in \mf{F}}$ and $(W_{\mf{p}})_{\mf{p} \in \mf{P}_Z}$
such that the following conditions are satisfied:
\begin{itemize}
\item $V_{\mf{p}}$ is an open neighborhood of $\mca{K}_{\mf{p}}$ in $U_{\mf{p}}$ for every $\mf{p} \in \mf{F}$.
\item $g'_{\mf{p}}: V_{\mf{p}} \to \Pi$ is a smooth map.
Moreover, $g'_{\mf{p}}$ is $\ep$-close to $f_{\mf{p}}|_{V_{\mf{p}}}$ and $\ev_j \circ g'_{\mf{p}} = \ev_j \circ f_{\mf{p}}|_{V_{\mf{p}}}$ for $j=0,1$.
\item $(g'_{\mf{p}})_{\mf{p} \in \mf{F}}$ is compatible with coordinate changes.
Specifically, for any $\mf{p}, \mf{p}' \in \mf{F}$ such that $\mf{p} \ge \mf{p}'$,
there holds $g_{\mf{p}'} = g_{\mf{p}} \circ \ph_{\mf{p}\mf{p}'}$ on $V_{\mf{p}'} \cap \ph_{\mf{p}\mf{p}'}^{-1}(V_{\mf{p}})$.
\item $W_{\mf{p}}$ is an open neighborhood of $\mca{K}^Z_{\mf{p}}$ in $U^Z_{\mf{p}}$ for every $\mf{p} \in \mf{P}_Z$.
\item $(g'_{\mf{p}})_{\mf{p} \in \mf{F}}$ is compatible with $\wt{g}$. Specifically:
\begin{itemize}
\item For any $\mf{p} \in \mf{P}_Z \cap \mf{F}$, there holds $(\ph_3)_{\mf{p}}(W_{\mf{p}}) \subset V_{\mf{p}}$ and $g'_{\mf{p}} = g_{\mf{p}} \circ (\ph_3)_{\mf{p}}$ on $W_{\mf{p}}$.
\item For any $\mf{q} \in \mf{P}_Z \setminus \mf{F}$ and $\mf{p} \in \mf{F}$ satisfying $\mf{p} < \mf{q}$, there holds
$g_{\mf{q}} \circ \ph_{\mf{q}\mf{p}} = g'_{\mf{p}}$ on $V_{\mf{p}} \cap (\ph_{\mf{q}\mf{p}})^{-1}(W_{\mf{q}})$.
\end{itemize}
\end{itemize}
\end{lem}
\begin{proof}
The proof is by induction on the cardinality of $\mf{F}$.
There is nothing to prove when $\mf{F} = \emptyset$.
To discuss the induction step, let $\mf{F}$ be a filter, $\mf{p}_0$ be its maximal element
(namely $\mf{p} \in \mf{F}, \mf{p} \ge \mf{p}_0 \implies \mf{p} = \mf{p}_0$),
and suppose that there exist $(V_{\mf{p}}, g'_{\mf{p}})_{\mf{p} \in \mf{F} \setminus \{\mf{p}_0\}}$ and $(W_{\mf{q}})_{\mf{q} \in \mf{P}_Z}$
satisfying the conditions in the lemma.
For each $x \in \mca{K}_{\mf{p}_0}$, we take an open neighborhood $W_x$ of $x$ in $U_{\mf{p}_0}$
and a smooth map $g'_x: W_x \to \Pi$ in the way described below.
We consider three cases (here we consider $x$ as a point in $|\wt{\mca{U}}| = \bigsqcup_{\mf{p} \in \mf{P}} U_{\mf{p}}/\sim$, and $\mca{K}^Z_{\mf{p}}$, $\mca{K}_{\mf{p}}$ as subspaces of $|\wt{\mca{U}}|$).
\begin{itemize}
\item[(i):] There exists $\mf{q} \in \mf{P}_Z \setminus (\mf{F} \setminus \{\mf{p}_0\})$ such that $x \in \mca{K}^Z_{\mf{q}}$.
\item [(ii):] $x \notin \mca{K}^Z_{\mf{q}}$ for any $\mf{q} \in \mf{P}_Z \setminus (\mf{F} \setminus \{\mf{p}_0\})$, but there exists $\mf{q} \in \mf{F} \setminus \{\mf{p}_0\}$
such that $x \in \mca{K}_{\mf{q}}$.
\item[(iii):] $x \notin \mca{K}^Z_{\mf{q}}$ for any $\mf{q} \in \mf{P}_Z \setminus (\mf{F} \setminus \{\mf{p}_0\})$, and $x \notin \mca{K}_{\mf{q}}$ for any $\mf{q} \in \mf{F} \setminus \{\mf{p}_0\}$.
\end{itemize}
In case (i), take maximal $\mf{q} \in \mf{P}_Z \setminus (\mf{F} \setminus \{\mf{p}_0\})$ such that $x \in \mca{K}^Z_{\mf{q}}$.
This condition implies $\mf{q} > \mf{p}_0$ since $U_{\mf{q}} \cap U_{\mf{p}_0} \ne \emptyset$ and $\mf{q} \in \mf{F}$.
Then take $W_x$ such that:
\begin{itemize}
\item
$\ol{W_x} \cap \mca{K}^Z_{\mf{q}'} = \emptyset$ for any $\mf{q}' \in \mf{P}_Z \setminus (\mf{F} \setminus \{\mf{p}_0\})$ which does \textit{not} satisfy $\mf{q}' \le \mf{q}$
(note this condition implies that $\mca{K}^Z_{\mf{q}} \cap \mca{K}^Z_{\mf{q}'} = \emptyset$),
\item
$W_x \subset (\ph_{\mf{q}\mf{p}_0})^{-1}(W_{\mf{q}})$.
\end{itemize}
Then we define $g'_x: W_x \to \Pi$ by
$g'_x: = g_{\mf{q}} \circ \ph_{\mf{q}\mf{p}_0}|_{W_x}$.
In case (ii), take maximal $\mf{q} \in \mf{F} \setminus \{\mf{p}_0\}$ such that $x \in \mca{K}_{\mf{q}}$.
This condition implies $\mf{q} < \mf{p}_0$.
Then take $W_x$ such that:
\begin{itemize}
\item $\ol{W_x} \cap \mca{K}_{\mf{q}'} = \emptyset$ for any $\mf{q}' \in \mf{F} \setminus \{\mf{p}_0\}$ which does \textit{not} satisfy $\mf{q}' \le \mf{q}$
(note this condition implies that $\mca{K}_{\mf{q}} \cap \mca{K}_{\mf{q}'} = \emptyset$),
\item $\ol{W_x} \cap \mca{K}^Z_{\mf{q}'} = \emptyset$ for any $\mf{q}' \in \mf{P}_Z \setminus (\mf{F} \setminus \{\mf{p}_0\})$,
\item $(\ph_{\mf{p}_0\mf{q}})^{-1}(W_x) \subset V_{\mf{q}}$.
\end{itemize}
When $W_x$ is sufficiently small, Lemma \ref{170513_1} shows that there exists a smooth map $g'_x: W_x \to \Pi$ such that
\begin{itemize}
\item $g'_x$ is $\ep$-close to $f_{\mf{p}_0}|_{W_x}$.
\item $\ev_j \circ g'_x = \ev_j \circ f_{\mf{p}_0}|_{W_x}$ for $j=0, 1$.
\item $g'_x \circ \ph_{\mf{p}_0\mf{q}} = g'_{\mf{q}}$ on $(\ph_{\mf{p}_0\mf{q}})^{-1}(W_x)$.
\end{itemize}
In case (iii), take $W_x$ so that
$\ol{W_x} \cap \mca{K}_{\mf{q}} = \emptyset$ for every $\mf{q} \in \mf{F} \setminus \{\mf{p}_0\}$, and
$\ol{W_x} \cap \mca{K}^Z_{\mf{q}} = \emptyset$ for every $\mf{q} \in \mf{P}_Z \setminus (\mf{F} \setminus \{\mf{p}_0\})$.
When $W_x$ is sufficiently small,
Lemmas \ref{171114_3} and \ref{170513_1} (applied to $V= \{x\}$)
show that there exists a smooth map $g'_x: W_x \to \Pi$ such that
\begin{itemize}
\item $g'_x$ is $\ep$-close to $f_{\mf{p}_0}|_{W_x}$.
\item $\ev_j \circ g'_x = \ev_j \circ f_{\mf{p}}|_{W_x}$ for $j=0, 1$.
\end{itemize}
Since $\mca{K}_{\mf{p}_0}$ is compact,
one can take finitely many points $\{x_i\}_{i \in I}$ such that
$\{W_{x_i}\}_{i \in I}$ covers $\mca{K}_{\mf{p}_0}$.
For each $i \in I$, we take
$g'_{x_i}: W_{x_i} \to \Pi$ and denote it as
$g'_{x_i} = (\gamma_i, T_i)$.
Let us take a $C^\infty$-function $\chi_i: U_{\mf{p}_0} \to \R_{\ge 0}$ for each $i \in I$,
such that $\supp \chi_i \subset W_{x_i}$
and $\sum_{i \in I} \chi_i = 1$ on a neighborhood of $\mca{K}_{\mf{p}_0}$, which we denote by $V_{\mf{p}_0}$.
Then we define $g'_{\mf{p}_0}: V_{\mf{p}_0} \to \Pi$ by
$g'_{\mf{p}_0} := (T, \gamma)$, such that
$T: = \sum_{i \in I} \chi_i T_i$,
and for every $y \in V_{\mf{p}_0}$
\[
\gamma(y): [0, T(y)] \to L ; \quad
\theta \mapsto
G (\gamma_i(y)(T_i(y)\theta/T(y)), \chi_i(y))_{i \in I}
\]
where $G$ is defined in Lemma \ref{171114_2}.
If $i \in I$ satisfies $y \in V_{x_i}$,
then
$g'_{x_i}(y)$ is $\ep$-close to $f_{\mf{p}_0}(y)$,
and $\ep < \rho_L$, thus
$g'_{\mf{p}_0}(y)$ is $\ep$-close to $f_{\mf{p}_0}(y)$.
Now we can finish the proof by replacing $V_{\mf{p}}$ with a smaller neighborhood of $\mca{K}_{\mf{p}}$ for each $\mf{p} \in \mf{F} \setminus \{\mf{p}_0\}$,
and $W_{\mf{q}}$ with a smaller neighborhood of $\mca{K}^Z_{\mf{q}}$ for each $\mf{q} \in \mf{P}_Z$.
\end{proof}
\section{Some basic notions in the theory of Kuranishi structures}
Here we recall some basic notions
in the theory of Kuranishi structures (abbreviated as K-structures),
mainly to fix notations used throughout this paper.
When we use notions which are not recalled here,
we directly refer to \cite{FOOO_Kuranishi}.
Throughout this section,
$X$ denotes a separable, metrizable topological space.
\textbf{Kuranishi chart (K-chart)}
A \textit{K-chart} of $X$ is a tuple
$\mca{U}=(U, \mca{E}, s, \psi)$ such that:
\begin{itemize}
\item $U$ is a $C^\infty$-manifold,
\item $\mca{E}$ is a $C^\infty$-vector bundle on $U$,
\item $s$ is a $C^\infty$-section of $\mca{E}$,
\item $\psi: s^{-1}(0) \to X$ is a homeomorphism onto an open set of $X$.
\end{itemize}
$\dim \mca{U}: = \dim U - \rk \mca{E}$ is called the dimension of $\mca{U}$.
An orientation of $\mca{U}$ is a pair of orientations of $U$ and $\mca{E}$.
A K-chart at $p \in X$ is a K-chart $\mca{U}=(U, \mca{E}, s, \psi)$ such that
$p \in \Image \psi$. We denote $o_p:= \psi^{-1}(p) \in s^{-1}(0)$.
\begin{rem}\label{170912_1}
In the standard definition (see Definition 3.1 in \cite{FOOO_Kuranishi}),
one assumes that
$U$ is an orbifold and $\mca{E}$ is an orbibundle.
However,
in the present paper we are working with pseudo-holomorphic disks \textit{without} sphere bubbles,
thus we do not need to take quotients by finite group actions on moduli spaces,
hence it is sufficient to work with vector bundles on manifolds.
\end{rem}
\textbf{Embedding of K-charts}
Let $\mca{U}_i = (U_i, \mca{E}_i, s_i, \psi_i)\,(i=1, 2)$ be K-charts of $X$.
An \textit{embedding} of K-charts $\Phi: \mca{U}_1 \to \mca{U}_2$
is a pair $\Phi = (\ph, \wh{\ph})$ such that:
\begin{itemize}
\item $\ph: U_1 \to U_2$ is an embedding of $C^\infty$-manifolds,
\item $\wh{\ph}: \mca{E}_1 \to \mca{E}_2$ is an embedding of $C^\infty$-vector bundles over $\ph$,
\item $\wh{\ph} \circ s_1 = s_2 \circ \ph$,
\item $\psi_2 \circ \ph = \psi_1$ on $s_1^{-1}(0)$,
\item for any $x \in s_1^{-1}(0)$, the covariant derivative
\begin{equation}\label{171119_1}
D_{\ph(x)} s_2:
\frac{ T_{\ph(x)} U_2}{(D_x\ph)(T_x U_1)} \to \frac{ (\mca{E}_2)_{\ph(x)}}{\wh{\ph}((\mca{E}_1)_x)}
\end{equation}
is an isomorphism.
\end{itemize}
When K-charts $\mca{U}_1$ and $\mca{U}_2$ are oriented,
we say that $\Phi = (\ph, \wh{\ph})$ is orientation preserving,
if an isomorphism
\[
\det TU_2 \otimes (\det TU_1)^\vee \cong
\det \mca{E}_2 \otimes (\det \mca{E}_1)^\vee
\]
induced by (\ref{171119_1}) preserves orientations.
\textbf{Coordinate changes}
Let $\mca{U}_i = (U_i, \mca{E}_i, s_i, \psi_i) \,(i=1, 2)$ be K-charts of $X$.
A \textit{coordinate change} in weak sense (resp. strong sense)
from $\mca{U}_1$ to $\mca{U}_2$ is a triple
$\Phi_{21} = (U_{21}, \ph_{21}, \wh{\ph}_{21})$
satisfying (i) and (ii) (resp. (i), (ii) and (iii)):
\begin{itemize}
\item[(i):] $U_{21}$ is an open subset of $U_1$,
\item[(ii):] $(\ph_{21}, \wh{\ph}_{21})$ is an embedding of K-charts $\mca{U}_1|_{U_{21}} \to \mca{U}_2$.
\item[(iii):] $\psi_1(s_1^{-1}(0) \cap U_{21}) = \Image \psi_1 \cap \Image \psi_2$.
\end{itemize}
\textbf{Kuranishi structure (K-structure)}
A \textit{K-structure} $\wh{\mca{U}}$ of $X$ (of dimension $d$) consists of
\begin{itemize}
\item a K-chart (of dimension $d$) $\mca{U}_p = (U_p, \mca{E}_p, s_p, \psi_p)$ at $p$ for every $p \in X$,
\item a coordinate change in weak sense
$\Phi_{pq} = (U_{pq}, \ph_{pq}, \wh{\ph}_{pq}): \mca{U}_q \to \mca{U}_p$
for every $p \in X$ and $q \in \Image (\psi_p)$,
\end{itemize}
such that
\begin{itemize}
\item $o_q \in U_{pq}$ for every $q \in \Image \psi_p$,
\item for every $p \in X$, $q \in \Image \psi_p$ and $r \in \psi_q(s_q^{-1}(0) \cap U_{pq})$,
there holds
$\Phi_{pr}|_{U_{pqr}} = \Phi_{pq} \circ \Phi_{qr}|_{U_{pqr}}$
where $U_{pqr} := \ph^{-1}_{qr}(U_{pq}) \cap U_{pr}$.
\end{itemize}
The pair $(X, \wh{\mca{U}})$ is called a space with Kuranishi structure
(abbreviated as a \textit{K-space})
of dimension $d$.
We say $\wh{\mca{U}}$ is oriented if each K-chart $\mca{U}_p$ is oriented and
$\ph_{pq}$ preserves orientations for every $p \in X$ and $q \in \Image \psi_p$.
\begin{rem}
In Definition 3.11 \cite{FOOO_Kuranishi} the notion of a relative K-space $(X, Z; \wh{\mca{U}})$ is introduced,
however in this paper we only need the absolute case ($X=Z$).
\end{rem}
\textbf{Strongly continuous/smooth maps from K-spaces}
Let $(X, \wh{\mca{U}})$ be a K-space and $Y$ be a topological space.
A \textit{strongly continuous map}
$\wh{f}: (X, \wh{\mca{U}}) \to Y$
assigns a continuous map
$f_p: U_p \to Y$ for every $p \in X$ such that
$f_p \circ \ph_{pq} = f_q$ on $U_{pq}$ for every $p \in X$ and $q \in \Image \psi_p$.
When $Y$ has a structure of a $C^\infty$-manifold,
$\wh{f}$ is called
\textit{strongly smooth}
if $f_p: U_p \to Y$ is of $C^\infty$ for every $p \in X$.
Moreover,
$\wh{f}$ is called
\textit{weakly submersive},
if $f_p$ is a submersion for every $p \in X$.
\textbf{Good coordinate system (GCS)}
Finally we recall the definition of a
\textit{good coordinate system} (GCS).
A GCS $\wt{\mca{U}}$ of $X$ consists of
\[
( (\mf{P}, \le) , \{ \mca{U}_{\mf{p}}\}_{\mf{p} \in \mf{P}} , \{ \Phi_{\mf{p}\mf{q}}\}_{\mf{q} \le \mf{p}})
\]
such that:
\begin{itemize}
\item $(\mf{P}, \le)$ is a finite partially ordered set.
\item $\mca{U}_{\mf{p}} = (U_{\mf{p}}, \mca{E}_{\mf{p}}, s_{\mf{p}}, \psi_{\mf{p}})$ is a K-chart for each $\mf{p} \in \mf{P}$, and $\bigcup_{\mf{p} \in \mf{P}} \Image \psi_{\mf{p}} = X$.
\item $\Phi_{\mf{p}\mf{q}}$ is a coordinate change $\mca{U}_{\mf{q}} \to \mca{U}_{\mf{p}}$ in strong sense.
\item If $\mf{r} \le \mf{q} \le \mf{p}$,
there holds $\Phi_{\mf{p}\mf{r}}|_{U_{\mf{p}\mf{q}\mf{r}}} = \Phi_{\mf{p}\mf{q}} \circ \Phi_{\mf{q}\mf{r}}|_{U_{\mf{p}\mf{q}\mf{r}}}$,
where $U_{\mf{p}\mf{q}\mf{r}}: = \ph^{-1}_{\mf{q}\mf{r}}(U_{\mf{p}\mf{q}}) \cap U_{\mf{p}\mf{r}}$.
\item If $\Image \psi_{\mf{p}} \cap \Image \psi_{\mf{q}} \ne \emptyset$, then either $\mf{p} \le \mf{q}$ or $\mf{q} \le \mf{p}$ holds.
\item Let us define a relation $\sim$ on $\bigsqcup_{\mf{p} \in \mf{P}} U_{\mf{p}}$ as follows:
$x \sim y$ if and only if one of the following holds:
\begin{itemize}
\item $\mf{p} = \mf{q}$ and $x=y$.
\item $\mf{p} \le \mf{q}$ and $y = \ph_{\mf{q}\mf{p}}(x)$.
\item $\mf{q} \le \mf{p}$ and $x = \ph_{\mf{p}\mf{q}}(y)$.
\end{itemize}
Then the relation $\sim$ is an equivalence relation, and the quotient
$\bigg(\bigsqcup_{\mf{p} \in \mf{P}} U_{\mf{p}}\bigg)/\sim$,
equipped with the quotient topology, is Hausdorff.
\end{itemize}
The definitions of strongly continuous/smooth maps naturally extend to spaces with GCS.
Finally, there exists a natural notion of embeddings from a K-structure to a GCS (KG-embedding; see Definition 3.29 in \cite{FOOO_Kuranishi}).
For any K-structure $\wh{\mca{U}}$ on a compact space $X$,
there exist a GCS $\wt{\mca{U}}$ and a KG-embedding $\wh{\mca{U}} \to \wt{\mca{U}}$;
see Theorem 3.30 in \cite{FOOO_Kuranishi}.
|
1,314,259,996,491 | arxiv | \section*{Abstract}
One-dimensional systems comprising $s$-wave superconductivity with meticulously tuned magnetism and spin-orbit coupling can realize topologically gapped superconductors hosting Majorana edge modes whose stability is determined by the gap's size. The ongoing quest for larger topological gaps evolved into a material science issue. However, for atomic spin chains on superconductor surfaces, the effect of the substrate's spin-orbit coupling on the system's topological gap size is largely unexplored.
Here, we introduce an atomic layer of the heavy metal Au on Nb(110) which combines strong spin-orbit coupling and a large superconducting gap with a high crystallographic quality enabling the assembly of defect-free Fe chains using a scanning tunneling microscope tip. Scanning tunneling spectroscopy experiments and density functional theory calculations
reveal ferromagnetic coupling and ungapped YSR bands in the chain despite of the heavy substrate. By artificially imposing a spin spiral state our calculations indicate a minigap opening and zero-energy edge state formation. The presented methodology paves the way towards a material screening of heavy metal layers on elemental superconductors for ideal systems hosting Majorana edge modes protected by large topological gaps.
\end{abstract}
\section*{Main}
Inducing spin-orbit coupling (SOC) in nanostructures has recently been of great interest in a variety of disciplines related to surface science\cite{Soumyanarayanan2016} due to its close ties with the existence of non-collinear magnetic states\cite{Dzyaloshinsky1958, Moriya1960,Bode2007}, spin-split surface states\cite{Lashell1996,Ast2007}, topological surface states\cite{Fu2007, Hasan2010} and topological superconductivity\cite{Fu2008, Lutchyn2010, Potter2012} which can be accompanied by Majorana bound states (MBS)\cite{Mourik2012, Kjaergaard2012, Pientka2013, NadjPerge2013, NadjPerge2014, Ruby2015, Kim2018}. Since the latter is a promising candidate as a building block of
topological quantum computation\cite{Freedman2002}, systems which may potentially host MBS have attracted a lot of interest. MBS may be realized in chains of magnetic atoms, also called atomic spin chains, on $s$-wave superconductors\cite{Pientka2013, Kim2018, Schneider2020, Pientka2015}, artificially fabricated by atom manipulation with the tip of a scanning tunneling microscope (STM)\cite{Eigler1990}. Although first experimental realizations, e.g. Fe chains on Re(0001), show signatures of MBS in scanning tunneling spectroscopy (STS)\cite{Kim2018}, the system suffers from the small energy gap $\varDelta_{\textrm{s}}$ of the superconducting rhenium substrate, making a clear allocation of in-gap features difficult\cite{Schneider2020}. More recent results of such atomic spin chains on Nb(110)\cite{Schneider2021a, Schneider2021b, Kuster2021a} circumvent this, but presumably at the price of lower SOC, which manifests itself in the dominance of collinear magnetic ground states due to weak Dzyaloshinskii--Moriya interaction (DMI) terms\cite{Beck2021}, and the hybridization of the spatially extended precursors of MBS in experimentally accessible chain lengths\cite{Schneider2021b}.
\newline
Aiming at maintaining the largest $\varDelta_{\textrm{s}}=\SI{1.50}{\milli \electronvolt}$ of all elemental superconductors combined with the possibility of atom manipulation which the Nb substrate offer, there are two apparent approaches to induce a higher SOC in the system.
On the one hand, one may think of using atoms with larger atomic SOC, e.g. rare earth metals, for the formation of the atomic spin chain.
On the other hand, one can try to couple the chain to a heavy metal substrate with potentially larger SOC, which is grown on Nb(110) and therefore becomes superconducting by proximity. First attempts combining both approaches using Gd atoms on bismuth thin films grown on Nb(110) show hybridization of the Yu--Shiba--Rusinov (YSR) states of small ensembles of 3-4 Gd atoms\cite{Ding2021}. However, the assembly of longer chains and a formation of bands from the hybridizing YSR states, which are both prerequisites for the emergence of topological superconductivity and MBSs, were not possible for that system. Moreover, it is even unclear, whether a larger SOC in the constituents, i.e. chain, substrate, or both of them, will lead to a larger SOC in the YSR bands of the hybrid chain system which is finally relevant for the emergence of topological superconductivity with a large topological gap and well-defined MBSs isolated at the ends of the chain.
\newline
Here, we investigate these questions pursuing the second approach by constructing Fe chains on ultrathin Au films grown on Nb(110) used as a substrate. Au is well known to exhibit large SOC\cite{Lashell1996, Henk2004} and the proximity to Au has been demonstrated to enhance SOC-induced effects in light elements, including the scattering rate\cite{Bergman1982}, the Rashba spin-splitting\cite{BIHLMAYER2006,Marchenko2012}, and also the magnetocrystalline anisotropy energy\cite{Szunyogh1995}. Moreover, twisted spin textures were predicted to occur around single Fe atoms\cite{Lounis2012} and in Mn chains\cite{Cardias2016} on Au(111). Since previous LEED studies hint at the possibility to grow pseudomorphic thin films of Au on Nb(110)\cite{Ruckman1988}, it is a natural candidate to be used as a proximitized superconducting heavy metal layer on Nb(110).
Combining experimental STS and density functional theory (DFT) calculations by solving the fully relativistic Dirac--Bogoliubov--de Gennes (DBdG) equations of the single Fe adatom, dimers and chains, we study (i) whether there is indeed a topological gap opening in the YSR bands due to the large SOC in the Au substrate, and (ii) the effect of a different spin structure in the chain on the topological gap width and the localization of MBSs.
\subsection*{Monolayer Au on Nb(110)}
We first describe the growth and the superconducting properties of the heavy metal layer. We aimed at the preparation of ultrathin Au films, maintaining the surface structure of Nb(110) as it offers multiple distinct building directions for artificial chains, which enables some tuning of the hybridization of YSR states\cite{Beck2021, Kuester2021} and, therefore, of the in-gap band structure of such chains\cite{Schneider2021a, Schneider2021b}. An overview STM image of the Au/Nb(110) sample after the low-temperature deposition of Fe atoms is shown in \autoref{fig1}a (preparation details in the Methods section). The Nb(110) surface is almost completely covered by one monolayer (ML) of Au (see sketch in bottom panel of \autoref{fig1}a), with only a few remaining holes. Pseudomorphic growth, as predicted by LEED studies\cite{Ruckman1988}, is confirmed by manipulated atom STM images\cite{Stroscio2004} (Supplementary Note 1 and Supplementary Figure 1) of the first ML. Partially, the second ML Au has started to grow at step edges and in the form of a few free-standing islands. In this ultrathin limit, we find that the energy gap of the superconducting Nb is fully preserved in the ML Au due to the proximity effect, as demonstrated by the deconvoluted \didu~spectrum (see Methods and Supplementary Note 2 for the deconvolution procedure) taken on the bare ML Au on Nb(110) far from any Fe atom in \autoref{fig1}c (gray curve). The energy gap on the ML Au is of equal size as that of bare Nb(110) at our measurement temperature and the in-gap \didu~signal is zero (see Supplementary Figure 2). This behavior is crucial for our experiments, but might be altered for thicker films as indicated by recent theoretical\cite{Csire2016, Csire2016a} and experimental\cite{Gupta2004, Tomanic2016} studies.
\subsection*{YSR states of single Fe atoms}
We continue with the investigation of the YSR states induced by single Fe atoms, i.e. the building blocks of chains. According to our DFT calculations the Fe adatom has a magnetic moment of $3.57 \mu_\textrm{B}$ with a preferred orientation perpendicular to the Au surface (Supplementary Note 5). A close-up STM image of the statistically distributed single Fe adatoms is shown in \autoref{fig1}b, where they appear as shallower protrusions on the first ML and as brighter spheres on the second ML. We restrict ourselves to the first ML since only this layer has large enough terraces to construct artificial chains. The Fe adatoms are adsorbed on the fourfold coordinated hollow sites on this ML Au (Supplementary Note 1 and Supplementary Figure 1). Fe adatoms which are adsorbed far from other adatoms or defects show similar \didu~spectra as shown in the top panel of \autoref{fig1}c (red curve). This reproducibility is required for a well-defined band formation in bottom-up fabricated nanostructures made from such individual adatoms. On the adatoms we find two pairs of YSR states induced in the gap of the Au ML which are marked by black arrows and greek letters\cite{Yazdani1997,Ruby2016, Choi2017}. One of them is energetically located close to the Au ML gap edge ($\pm \alpha$, $\pm \SI{1.23}{\milli \electronvolt}$) while the other one is located close to the Fermi energy $E_\mathrm{F}$ ($\pm \beta$, $\pm \SI{0.27}{\milli \electronvolt}$). We use constant-contour maps (see Methods) to resolve the spatial distribution of both YSR states\cite{Ruby2016, Choi2017}, as shown in \autoref{fig1}d.
The $\pm \alpha$ state has a spatial distribution resembling that of a $d_{yz}$ orbital with two lobes pointing along the $[1\overline{1}0]$ direction. On the other hand, a spatial distribution resembling that of a $d_{xz}$ orbital extended along the $[001]$ direction is observed for the $\pm \beta$ state. Note that the shapes of the $+\beta$ and the $-\beta$ state are somewhat different since the former has additional faint lobes along the $[1\overline{1}0]$ direction probably indicating contributions from a $d_{x^2-y^2}$-like YSR state.
The spatial distributions are explainable by the orbital origin of YSR states\cite{Ruby2016, Choi2017,Beck2021} and the point group $C_{2\textrm{v}}$ for the system. However, note that the energetic order of the states is interchanged with respect to the case of bare Nb(110)\cite{Beck2021}, and that there are no obvious indications for the other two possible, $d_{xy}$- and $d_{z^2}$-like, YSR states, which might indicate an overlap of these peaks in energy with the $d_{xz}$-, $d_{yz}$-, or $d_{x^2-y^2}$-like YSR states, or that they are hidden in the coherence peaks of the substrate.
\newline
In order to further clarify these experimental results we calculated the local density of states (LDOS) for a single Fe atom on the Au/Nb(110) film as shown in the top panel of \autoref{fig2}a and \autoref{fig2}b (see Methods for the calculation details). There are three very close-by, almost overlapping YSR states in the vicinity of the substrate's coherence peaks (see also Supplementary Figure 5a). They correspond to the $d_{z^2}$, $d_{xy}$ and $d_{yz}$ orbitals (see the $+1.32$ meV map in \autoref{fig2}b) and the spatial distributions of the LDOS around these peaks are very sensitive to the exact energy (Supplementary Figure 5b). Due to their energetic location and orbital symmetries, we assign them to the experimental $\pm\alpha$ YSR state (c.f. \autoref{fig1}c and d) which appear as a single peak in the \didu~spectrum due to the finite-temperature smearing. Additionally, there are two most intense peaks in the calculations which stem from two energetically close-by YSR states near $E_\mathrm{F}$ where the one closest to $E_\mathrm{F}$ corresponds to the $d_{x^2-y^2}$ orbital (see the $+0.38$ meV map in \autoref{fig2}b), and the less intense one further apart from $E_\mathrm{F}$ to the $d_{xz}$ orbital (see the $-0.58$ meV map in \autoref{fig2}b). While the latter resembles the experimentally observed $-\beta$ state, the former has more similarities to the $+\beta$ state (c.f. \autoref{fig1}d). These theoretical results indicate that the different spatial experimental distributions of the $+\beta$ and $-\beta$ states in \autoref{fig1}d can be explained by supposing that they correspond to two different YSR states of $d_{x^2-y^2}$ and $d_{xz}$ orbital character which overlap within the experimental energy resolution, rather than to a single YSR state. The two YSR peaks also overlap in the theoretical calculations if a larger imaginary part is chosen for the energy, corresponding to a higher effective temperature. Finally, note that the electron-hole asymmetries in the intensities of the calculated peaks appear to be inverted compared to the experiment. With the exception of the $d_{xz}$-like YSR state, each pair of peaks has a larger electron contribution above $E_\mathrm{F}$ (Supplementary Figure 5a). The $d_{xz}$-like YSR state has a higher electron contribution below $E_\mathrm{F}$ which implies that this state has the strongest coupling to the substrate.
\subsection*{YSR states of Fe dimers}
Before we continue with the investigation of Fe dimers, we consider some intuitive ideas about the most promising orientations of chains built from individual Fe atoms towards the goal of topologically gapped YSR bands. As found in previous works\cite{Schneider2021a, Schneider2021b, Kuster2021a,Liebhaber2022}, enabling a sufficient hybridization of a YSR state which is already close to $E_\mathrm{F}$, while, at the same time, minimizing the hybridizations of all the other YSR states far from $E_\mathrm{F}$ may lead to a single YSR band overlapping with $E_\mathrm{F}$. Together with SOC, this can be a sufficient condition for the opening of a topologically non-trivial gap in the lowest-energy band. Starting from the experimentally detected shapes and energies of the $\alpha$ and $\beta$ YSR states (\autoref{fig1}d) we thus regard chains along the $[001]$ direction as most promising. For this orientation, we expect weak and strong hybridizations, respectively, for the $\alpha$ and $\beta$ YSR states which are far and close to $E_\mathrm{F}$. While a manipulation of close-packed dimers along $[001]$ turned out to be impossible, we were able to tune the system into the above conditions using dimers with a distance of $2a$ along $[001]$ (see STM image in \autoref{fig1}e). A \didu~spectrum measured above the center of the dimer as well as constant-contour maps of the spatial distributions of the three evident states are displayed in the bottom panel (orange curve) of \autoref{fig1}c and \autoref{fig1}e, respectively. In this configuration, the $\pm \alpha$ YSR states of the two atoms do not overlap significantly such that they do not split into hybridized states, but only slightly shift in energy. In contrast, the $\pm \beta$ YSR states of the two atoms strongly overlap, and split into an energetically higher one with a clear nodal line in the center between both impurities ($\pm \beta_\textrm{a}$) and another energetically lower one with an increased intensity in the center ($\pm \beta_\textrm{s}$).
\newline
These experimental conclusions are corroborated by our calculations (bottom panel of \autoref{fig2}a and \autoref{fig2}c). Apparently, all five pairs of single-atom YSR states are split, as expected from previous experimental and theoretical studies\cite{Beck2021,Nyari2021}. Although based on the orbital decomposition it is possible to separate all of the ten pairs (Supplementary Figure 6), the splitting of the three YSR states contributing to the $\alpha$ YSR state is particularly small, in accordance with the experiment, which makes it hard to resolve them in the total LDOS. In \autoref{fig2}c we plot the LDOS maps of the six most relevant peaks in \autoref{fig2}a. We find that the very weakly splitted $d_{yz}$ YSR states at $+1.32$ meV and $+1.30$ meV which are assigned to the experimental $\alpha$ YSR state appear with an almost identical shape as the single atom $d_{yz}$ YSR state (c.f. \autoref{fig2}b). In contrast, the $d_{x^2-y^2}$ and $d_{xz}$ YSR states strongly split into states with larger (at $+0.79$ meV and $-0.53$ meV) and smaller (at $+0.04$ meV and $-0.66$ meV) intensities in the center between both impurities and are thus associated with the experimental $\pm\beta_{s}$ and $\pm\beta_{a}$ YSR states, respectively. We thus conclude, that while the $\alpha$ YSR states hybridize only very weakly, the $\beta$ YSR states hybridize strongly and split into states which resemble anti-symmetric and symmetric linear combinations of the single atom YSR states\cite{Ruby2018, Flatte2000, Morr2003}, which can be seen as a prerequisite for band formation from the hybridizing $\beta$ YSR states.
\newline
\subsection*{Gapless YSR band in ferromagnetic Fe chains on Au monolayer}
Having identified a promising orientation and interatomic spacing from the investigation of the single atom and the dimer above, we move on to study artificial chains with the same interatomic separation, called $2a-[001]$ chains in the following. A sketch illustrating this geometry and an STM image of a nine Fe atoms long $\mathrm{Fe}_9$ $2a-[001]$ chain are shown in the top panels of \autoref{fig3}a and b. Spin-polarized measurements of a $\mathrm{Fe}_{19}$ $2a-[001]$ chain indicate that the atoms in this chain configuration prefer ferromagnetic alignment (Supplementary Note 3). This is also supported by our DFT calculations (Supplementary Note 5). We found that the DMI is around 10\% of the Heisenberg exchange interaction in the dimer. Although this is not particularly weak, the SOC in the Au layer additionally induces a very strong out-of-plane on-site anisotropy, which prevents the formation of spin-spirals and stabilizes a normal-to-plane ferromagnetic spin structure.
\newline
A \didu~line profile (see Methods) was measured in the center of such a chain along its main axis and is plotted in \autoref{fig3}a (bottom panel) alongside the acquired stabilization height profile (middle panel). The first apparent characteristic of this measurement is the modulation of every feature with the interatomic spacing of $2a$ in these chains, which is also visible in the height profile. It should be emphasized that this is not a feature of the chains' in-gap band structure but is just due to the lattice-periodic part of the wave function. However, we find additional states with different well-defined numbers of maxima at increasing energy and also very close to $E_\mathrm{F}$ as indicated by the labels $n_\beta$ ($n_\beta-1$) for the numbers of maxima (nodes). Note that all these states have particle-hole partners occurring on the other side of $E_\mathrm{F}$ with the same energetic distance to $E_\mathrm{F}$ and equal numbers of maxima and nodes. However, they mostly have much smaller intensities such that they are barely visible. These pairs of states can thus be assigned to confined Bogoliubov-de-Gennes (BdG) quasiparticles residing in a YSR band induced by the finite magnetic chain in the superconductor\cite{Schneider2021a}.
To determine the orbital origin of these states, we show \didu~maps (see Methods) of the $\mathrm{Fe}_9$ $2a-[001]$ chain in \autoref{fig3}b. We find that the confined BdG states identified before in \autoref{fig3}a are localized inside the spatial extent of the chain deduced from the STM image (dashed red elliptical circumference). We assign those states to a band formed by the strong hybridization of the $\pm\beta$ YSR states of the single adatoms as they are expected to be largely localized along the longitudinal axis of the chain. Additionally, there is a state at a similar energy as the single adatom and dimer $\pm\alpha$ YSR states around $\pm \SI{1.09}{\milli \electronvolt}$. This state has exactly as many maxima as there are atoms in the chain, namely 9, which are spatially localized along both sides of the chain with a similar distance to the chain axis as the lobes of the single adatom and dimer's $\pm\alpha$ YSR states (c.f. \autoref{fig1}d and e). Therefore, we assign this state to the very weakly hybridizing $\pm\alpha$ YSR states of the single atom. The state is not observed in the \didu~line profile of \autoref{fig3}a due to its nodal line along the longitudinal chain axis.
\newline
In order to measure the dispersion of the confined BdG states from the $\beta$ YSR band, we collect similar \didu~line profiles as the one in \autoref{fig3}a of defect-free chains for lengths ranging from $N = 7$ to $N = 14$ atoms ($\mathrm{Fe}_7 - \mathrm{Fe}_{14}$, see Supplementary Note 4 and Supplementary Figure 4). It can be observed that the confined BdG quasiparticle states shift in energy as a function of the length $L = N \cdot d = N \cdot 2\mathrm{a}$ of the chain, as expected from the length-dependent interference condition
\begin{equation}
\label{eq:interference_condition}
q = |\textbf{q}| =\pm \frac{2 \pi n}{L}
\end{equation}
where $n$ is an integer and $|\textbf{q}|$ is the length of the BdG quasiparticle scattering vector\cite{Schneider2021a}. For particular chain lengths, the confined BdG quasiparticle states can be located very close to $E_\mathrm{F}$ (c.f. $\mathrm{Fe}_8$ and $\mathrm{Fe}_{10}$ in Supplementary Figure 4).
We perform one-dimensional fast Fourier transforms (1D-FFT) of the columns of the \didu~line profiles at fixed energy $E$ averaging all data sets taken for chains of multiple lengths, and thereby obtain the dispersion of the scattering vectors $E(q)$ (\autoref{fig3}c). This dispersion is closely linked to the $\beta$ YSR band structure. We find that this band has an approximately parabolic dispersion ranging from $\SI{-0.9}{\milli \electronvolt}$ at $q/2=0$ to $+\SI{0.5}{\milli \electronvolt}$ at $q/2=\pi / d$. Note that, as already discussed for the \didu~line profiles above, the particle-hole partner of this band has a much lower intensity. It is only visible around the Brillouin zone center ($q/2=0$) in our measurements. Most importantly, the $\beta$ YSR band smoothly crosses $E_\mathrm{F}$ without any indications of a minigap opening.
\newline
An overall similar behaviour is found using our \textit{ab-initio} framework. We performed calculations for $2a-[001]$ chains of lengths ranging from 9 to 19 Fe atoms with ferromagnetic spin alignment (Supplementary Note 7 and Supplementary Figure 7). Exemplarily, the calculated LDOS along a Fe$_9$ chain is shown in \autoref{fig4}a and can be directly compared to the measured line profile in \autoref{fig3}a. The band formation of the YSR states can clearly be observed in a wide range of the substrate gap in the form of LDOS lines with a well-defined number of maxima along the chain as indicated in the figure. In \autoref{fig4}b we present the corresponding spatial distributions of the LDOS of the Fe$_9$ chain in the form of two-dimensional maps for a selection of confined BdG states with the indicated dominant orbital characters and numbers of maxima (see the Methods section for calculation details). The states closest to the substrate's coherence peaks with $n_{yz}=2$ (and admixed $n_{yz}=4$) and $n_{z^2}=3$ maxima have $d_{yz}$ and $d_{z^2}$ orbital characters, respectively. They reside in a very narrow band formed by the weakly interacting $\alpha$ YSR states of the Fe atoms (\autoref{fig2}), which explains the low dispersion of this band. On the contrary, wide bands are formed by the strong hybridization of the $\beta$ YSR states, i.e. a $d_{x^2-y^2}$ YSR band (between -0.2 meV and +1.1 meV) having high intensities on both sides along the longitudinal axis of the chain and a $d_{xz}$ YSR band (between -0.8 meV and 0 meV) characterized by high intensities between the atoms of the chain (\autoref{fig4}a,b). In order to deduce the dispersions of these YSR bands from the theoretical calculations we applied the same 1D-FFT method as in the experiment (see also Ref. \citeonline{Schneider2021a}), and averaged over chains containing 9, 11, 13, 14, 17 and 19 Fe atoms (Supplementary Figure 7). The result is plotted in \autoref{fig4}c and can be compared to the experimental dispersion in \autoref{fig3}c. The most characteristic, broad bands are the $d_{x^2-y^2}$ and $d_{xz}$ YSR bands between -0.2 meV and 1.1 meV and between -0.8 and 0 meV, respectively. While the energy range of the $d_{xz}$ YSR band agrees reasonably well with that of the experimental $\beta$ band, the $d_{x^2-y^2}$ YSR band is probably not detected significantly in the experimental data (\autoref{fig3}c). The latter might be explained by the small intensity of the experimental $+\beta$ state (\autoref{fig1}d) which is not reproduced by the calculations (\autoref{fig2}a). Most importantly, the theoretical study confirms the lack of a detectable minigap at $E_\mathrm{F}$ in the YSR bands.
\newline
\subsection*{Minigap and end states in spin spiral Fe chains on Au monolayer}
At first sight, the missing minigap seems surprising. For similar ferromagnetic chains on the lighter substrates Nb(110) and Ta(110) there are already clear indications for the openings of topological minigaps \cite{Beck2022b, Schneider2021a}. It is widely accepted that topological minigaps hosting MBSs can open in the quasiparticle spectrum of one-dimensional helical spin systems being proximity-coupled to a conventional $s$-wave superconductor \cite{NadjPerge2013,Pientka2013,Klinovaja2013,Kim2018}. For ferromagnetic chains, this phenomenon has been attributed to a Rashba-type SOC induced by the substrate\cite{Li2014}, which is equivalent to a spin spiral structure without SOC in a single-band tight-binding model\cite{Braunecker2010}. As outlined in the introduction above, the heavier material Au is well known to exhibit large SOC\cite{Lashell1996, Henk2004}. However, as our experiments and calculations show, it obviously does not induce a spin-spiral state in the Fe chain, and likewise does not induce a SOC of sufficient strength in the YSR bands of the ferromagnetic chain to open a detectable minigap. In order to trace whether we can still force the system into a state with a large topological gap $\varDelta_\textrm{ind}$ just by artificially imposing a suitable non-collinear spin state onto the chain, we performed calculations for the same chains as before, but now imposing a helical spin spiral state (\autoref{fig5}, Supplementary Note 8, and Supplementary Figure 8). The configuration of the spin spiral was such that the first Fe site had its spin pointing along the positive $z$ direction and then each spin is rotated by 90$^\circ$ around the chain axis when moving along the chain (\autoref{fig5}a). Indeed, there are two significant features which emerge in the LDOS of the spin-spiral chain with 19 iron atoms (\autoref{fig5}b), which were absent in the LDOS of the ferromagnetic chain (\autoref{fig4}a). First, a minigap at $E_\mathrm{F}$ opens up between $-0.22$ meV and $+0.22$ meV. Second, inside this minigap a single state can be observed at $E_\mathrm{F}$ with a pronounced intensity localized at the ends of the chain. This state has an electron-hole ratio of 1 and is robust against the variation of the chain length from 9 to 19 atoms as illustrated in \autoref{fig5}b and Supplementary Figure 8. The strongly different spatial LDOS distribution of the zero-energy state compared to that of some exemplary higher-energy states is further illustrated in \autoref{fig5}c. The former is localized over a few atoms at the two ends of the chain, while the latter states outside the minigap are extended along the whole chain. It should be mentioned that all these states, both the zero-energy one as well as those outside of the minigap show the same orbital character, indicating that the minigap emerges from the $d_{x^2-y^2}$ YSR states of the ferromagnetic chain. The induced minigap of $2\varDelta_\textrm{ind}=\SI{0.44}{\milli \electronvolt}$ width and the narrow spectral weight around $E_\mathrm{F}$ stemming from the zero-energy end states are also clearly visible in the dispersion of the scattering wave vectors deduced from the averaged 1D-FFTs of the LDOSs of chains of different lengths (\autoref{fig5}d). Thus, the calculations show evidence for the formation of a topological, most probably $p$-wave-like, minigap which hosts a MBS, if the Fe chain on Au(111) is forced into a helical spin spiral state, indicating that the absence of the non-collinear ground state is the limiting factor of this experimental system.
\subsection*{Conclusions and outlook}
In summary, our combined experimental and theoretical investigation shows that in contrast to what might be suggested by simplified tight-binding models\cite{Li2014, Braunecker2010}, a strong substrate SOC alone generally is not a sufficient condition for the opening of a topological minigap in a ferromagnetic chain in contact to an $s$-wave superconductor, since the SOC has to exist in the lowest-energy YSR band.
In fact, first-principles calculations of the magnetic interaction parameters in ultrathin film systems have demonstrated that also the connection between the formation of a spin-spiral state and SOC is considerably more complicated. In particular, the DMI preferring a non-collinear spin alignment is typically weak when a $3d$ transition metal is deposited on a Au surface compared to other $5d$ substrates\cite{Simon2014,Belabbes2016,Simon2018}, which may be tentatively attributed to the fully occupied $5d$ band of Au having a reduced effect on the DMI. Proximity to a Au layer is known to give rise to strong Heisenberg exchange interactions and anisotropy\cite{Szunyogh1995} in the magnetic layer instead, both of which prefer a collinear spin alignment and the latter being induced by the SOC. Our results indicate that, similarly to the competition between DMI and anisotropy terms in the formation of non-collinear spin structures, the role of SOC may be more complex for inducing topological superconductivity in the YSR bands of ferromagnetic spin chains.
\newline
Our study proves that it is experimentally possible to grow proximitized ultrathin heavy metal layers on a superconductor with a large $T_{\textrm{c}}$ that can be used as a substrate for the deposition of transition metal atoms and to construct defect-free one-dimensional structures with an excellent quality, enabling the tailoring of YSR bands. Further, we presented an \textit{ab-initio} method that accurately reproduces the main LDOS features observed in the experiments. Our work thus demonstrates the theoretical feasibility of an \textit{ab-initio} screening of other combinations of transition metal chains on heavy metal thin films on bulk superconductors in order to find the optimal conditions for the opening of a large topological minigap.
\begin{methods}
\subsection*{STM and STS measurements}
The experiments were performed in a custom home-built ultra-high vacuum system, equipped with an STM setup, which was operated at a temperature of $\SI{320}{\milli \kelvin}$\cite{Wiebe2004}.
STM images were obtained by applying a bias voltage $V_{\mathrm{ bias}}$ to the sample upon which the tip-sample distance is controlled by a feedback loop such that a constant current $I$ is achieved.
\didu~spectra were obtained in open feedback mode after stabilizing the tip at $V_{\mathrm{stab}}=\SI{6}{\milli \volt}$ and $I_{\mathrm{stab}}=\SI{1}{\nano \ampere}$ using a standard lock-in technique with an AC voltage $V_{\mathrm{mod}}=\SI{20}{\micro \volt}$ (rms value) of frequency $f_{\mathrm{mod}}=\SI{4142}{\hertz}$ added to the ramped $V_{\mathrm{bias}}$. If other stabilization parameters were used for a particular measurement, it is indicated in the respective figure caption.
\didu~maps were obtained by measuring \didu~spectra on a predefined spatial grid, which was positioned over the structure of interest, and selecting a slice at a given voltage. Typical measurement parameters are the same as for individual \didu~spectra.
\didu~line profiles are measured similarly to \didu~maps, with the exception that the spatial grid is one-dimensional.
Constant-contour maps were obtained by repeated scanning of individual lines of STM images. First, each line is measured as it would be the case in a regular STM image. The $z$-signal of this sweep is saved. Then, the feedback is turned off, the bias voltage $V_{\mathrm{bias}}$ is set to a predefined value of interest, and for the next sweep on the same scan line the \didu signal is measured while restoring the previously recorded $z$-signal.
\newline
A mechanically sharpened and in-situ flashed (\SI{50}{\watt}) bulk Nb tip was used for all measurements. While the usage of a superconducting tip is a crucial factor for obtaining a very good energy resolution, it has the downside that the \didu~spectra are convolutions of the tip and sample DOSs. However, we can determine the superconducting gaps of the tip and the sample, and deconvolute the \didu~spectra. This process is described in Supplementary Note 2 and is performed for every spectrum in the main manuscript.
\subsection*{Sample preparation}
A Nb(110) single crystal with a purity of $\SI{99.99}{\percent}$ was transferred into the ultra-high vacuum chamber. The sample was cleaned by cycles of Ar ion sputtering and flashes up to $\SI{2400}{ \celsius}$, which results in a clean surface with only few oxygen impurities remaining\cite{Odobesko2019}. We established flashing parameters which clean the surface of oxygen, and checked the results by STM. Once this cleaning procedure was reproducible with the given parameters, we evaporated Au from an e-beam evaporator (EFM3 by FOCUS GmbH) equipped with a Au rod ($\SI{99.99}{\percent}$ purity). Following this procedure, we achieved flat and spatially extended films (\autoref{fig1}a).
\newline
Fe was evaporated onto the surface from a carefully outgassed Fe rod using a second e-beam evaporator while keeping the sample temperature below $T=\SI{10}{\kelvin}$ to avoid clustering and diffusion and thus achieve a random distribution of single Fe adatoms (\autoref{fig1}b). From Supplementary Note 1, we conclude that the Au film grows pseudomorphically and that the Fe atoms are adsorbed in the fourfold coordinated hollow sites in the center of four Au atoms. This is further supported by the similarity of the spatial distributions of the YSR states of the Fe/Au/Nb(110) system compared to the Mn/Nb(110) system\cite{Beck2021}.
\newline
STM tip-induced atom manipulation\cite{Stroscio2004, Eigler1990} is used to position individual Fe atoms and construct artificial structures, such as dimers and chains. The structures built in this study have sufficient interatomic spacing to unambiguously identify the positions of the individual atoms forming the structure using STM images. We restrict the investigations here to Fe atoms positioned on the first ML of Au. Fe atoms on the first and second ML can easily be distinguished by their apparent height. In the top part of \autoref{fig1}b one can see that an Fe atom on the second ML appears as a bright spherical protrusion, while an Fe atom on the first ML is more shallow and has a relatively irregular shape. Thus, we can be sure that all of the experiments were carried out on the first ML.
\subsection*{First-principles calculations} The calculations were performed in terms of the Screened Korringa-Kohn-Rostoker method (SKKR), based on a fully relativistic Green's function formalism by solving the Dirac equation for the normal state\cite{Szunyogh1995} and the Dirac-Bogoliubov-de Gennes (DBdG) equation for the superconducting state within multiple scattering theory (MST)\cite{csire2018relativistic, Nyari2021}. The impurities are included within an embedding scheme\cite{Lazarovits2002}, being an efficient method to address the electronic and magnetic properties or the in-gap spectra of real-space atomic structures without introducing a supercell. The system consists of seven atomic layers of Nb, a single atomic layer of Au and four atomic layers of vacuum between semi-infinite bulk Nb and semi-infinite vacuum. The Fe adatoms are placed in the hollow position in the vacuum above the Au layer and relaxed towards the surface by 21\%, while the top Au layer is also relaxed inwards by 2\%.
The relaxations are obtained from total-energy minimization in a VASP\cite{Kresse1996,Kresse1996a,Hafner2008} calculation for a single Fe adatom and are used for the dimer and all the chains. For the potentials we employ the atomic sphere approximation (ASA), the normal state is calculated self-consistently in the local density approximation (LDA) as parametrized by Vosko \textit{et al.}\cite{Vosko1980} The partial waves within MST are treated with an angular momentum cutoff of $\ell_\mathrm{max}=2$. In the self-consistent normal state calculations we used a Brillouin zone (BZ) integration with 253 $\mathbf{k}$ points in the irreducible wedge of the BZ and a semicircular energy contour on the upper complex half plane with 16 points for energy integration. In order to take into account charge relaxation around the magnetic sites, the single impurity and the $2a-[001]$ dimer are calculated by including a neighborhood containing 48 and 84 atomic sites, respectively, corresponding to a spherical radius of $r=1.66~a$. The atomic chains are calculated with a somewhat smaller neighborhood corresponding to 2 atomic shells or a spherical radius of $r=1.01~a$ around the Fe atoms. This way our largest atomic cluster in the calculation with 19 Fe chain atoms contained 339 atomic sites.
After having obtained the self-consistent potentials in the normal state, the superconducting state is simulated within single-shot calculations by solving the DBdG equation with the experimental band gap used as the pairing potential in the Nb layers\cite{Nyari2021}. In the case of the single impurity and the dimer, the BZ integration for the host Green's function is performed by using the same $\mathbf{k}$ mesh as for the normal state, but in order to achieve convergence for the chains we had to increase the number of $\mathbf{k}$ points up to 1891 in the irreducible wedge of the BZ. A sufficient energy resolution of the LDOS in the superconducting gap is acquired by considering 301 energy points between $\pm 1.95$~meV with an imaginary part of 13.6~$\mu$eV related to the smearing of the resulting LDOS. Both the electron and the hole components of the LDOS are calculated, but in this paper we present only the electron part leading to the asymmetry of the spectrum as also seen in the experiments. Due to the ASA used in our method we obtain the LDOS for each atomic site of the cluster averaged inside the atomic spheres. The orbital resolution of the YSR states can be determined based on the orbital-resolved LDOS of the Fe atoms. Since the canonical $d$ orbitals hybridize due to the symmetry of the cluster and due to SOC, we assign the labels based on the orbital which has the largest contribution to the given peak in the LDOS. In addition, in order to mimic the constant-contour maps in the experiments, we evaluate the spatial distribution of the LDOS. These LDOS maps are taken from the first vacuum layer above the surface in which the magnetic atoms are embedded, reflecting the orbital characteristics obtained from the resolution of the LDOS of the Fe atoms.
In order to better reproduce the experimental constant-contour maps taken from the vacuum region, the LDOS of the magnetic sites are replaced by the average LDOS over the two vacuum sites (empty spheres) closest to them in the layer above. To get a continuous picture for the LDOS maps we applied an interpolation\cite{gouraud1971} scheme on the data calculated as described above.
\subsection*{Data availability}
The authors declare that all relevant data are included in the paper and its Supplementary Information files.
\end{methods}
|
1,314,259,996,492 | arxiv | \section{Introduction}\label{sec:intro}
White dwarfs are the degenerate remnants of stars with masses lower than $8-10$ $M_{\odot}$ and are the endpoint of stellar evolution for more than 97\% of Galactic stars \citep[e.g.,][and references therein]{ibeling13,doherty15}. White dwarfs are an important tool for studying the star formation history of the Milky Way, the late phases of stellar evolution, and can be used to improve our understanding of the physics of condensed matter.
Photometric white dwarf catalogs have so far mainly been based on the search for objects showing an ultraviolet excess, such as the Palomar-Green catalog \citep[PG,][]{PGS}, the Kiso survey \citep[KUV,][]{KUV1,KUV2}, the Kitt Peak-Downes survey \citep[KPD,][]{KPD}, or the Sloan Digital Sky Survey \citep[SDSS,][]{sdss_dr16}; and using reduced proper motions (e.g., \citealt{NLTT,harris06,rowell11,GF15,munn17}). Subsequent spectroscopic follow-ups led to $\sim 35\,000$ white dwarfs with spectroscopic information \citep[e.g.,][]{eggen65,mccook99,eisenstein06,kepler19}. The main limitation of these photometric and spectroscopic catalogs is their nontrivial selection functions, a situation that has been improved thanks to the {\it Gaia} mission \citep{gaia}. {\it Gaia} provides a unique source of astrometric and photometric information that can be used to define the largest and most secure white dwarf catalog to date, with $360\,000$ sources so far \citep[][GF21 hereafter]{GF21}, and permits the definition of high-confidence volume-limited white dwarf samples \citep[e.g.,][]{hollands18,jimenezesteban18,gentilefusillo19,kilic20,mccleery20,gaia_edr3_nearby}.
Spectroscopic analysis of the early white dwarf catalogs demonstrated their spectral diversity \citep{sion83}. Those white dwarfs with hydrogen lines in their spectra are classified as DA type, while those presenting helium absorption can be DO (\ion{He}{II}) or DB (\ion{He}{I}). Featureless spectra in the optical define the continuum DC class, while the presence of heavy metals polluting the white dwarf atmosphere lead to the DZ and DQ (carbon) classifications. In addition to these general classes, hybrid types have also been reported (DAB, DBA, DZA, etc.). The dominant atmospheric composition of white dwarfs, defined as H-dominated or He-dominated depending on the most common component in their outer atmosphere, changes with cooling age (i.e., with decreasing effective temperature $T_{\rm eff}$). The DOs dominate at high temperatures ($T_{\rm eff} \gtrsim 80\,000$~K), with a steady decline in the fraction of He-dominated atmospheres down to $T_{\rm eff} \sim 40\,000$~K, where it reaches a minimum of 5\%-10\%. The fraction of He-dominated white dwarfs increases again at $T_{\rm eff} \sim 20\,000$~K towards lower temperatures, with a transition between DBs to DCs at $T_{\rm eff} \sim 11\,000$~K, where the \ion{He}{I} lines are no longer visible. This is also the case for H-dominated white dwarfs at $T_{\rm eff} \lesssim 5\,000$~K, where both H- and He-dominated white dwarfs are classified as DCs in the absence of spectral lines from polluting metals. This general picture is based on extensive observational work \citep{sion84,fleming86,greenstein86,fontaine87,eisenstein06db,tremblay08,giammichele12,limoges15,GB19,ourique19,blouin19,bedard20,cunningham20,mccleery20}.
The change in the fraction of He-dominated atmospheres in white dwarfs at $T_{\rm eff} \lesssim 25\,000$~K, referred to as spectral evolution hereafter, has been interpreted as the effect of convective dilution and convective mixing processes \citep[e.g.,][]{rolland18,cunningham20}, where the outer hydrogen layer and the underlying helium layer in white dwarfs are mixed because of the appearance of convection zones. In the dilution scenario, convection in the helium layer reaches the bottom of the thin hydrogen shell and erodes it, while in the mixing scenario the convection in the outermost hydrogen layer reaches the inner helium zone. In both cases, the net effect is to mix the hydrogen with the more abundant helium, producing the spectral change from a H-dominated to a He-dominated atmosphere for hydrogen mass below $\log M_{\rm H}/M \sim -6$. Moreover, the size of the convection zone increases with decreasing temperature, providing an indirect estimation of the mass of the hydrogen layer \citep[e.g.,][]{tremblay08,cunningham20} that can be compared with results from asteroseismology \citep{romero17}. Therefore, detailed knowledge of the spectral evolution of white dwarfs provides clues about the physical processes acting in these objects as they cool over time.
Several of the above-mentioned studies of white dwarf spectral evolution at $T_{\rm eff} \lesssim 25\,000$~K are based on SDSS spectroscopy, which is affected by complicated selection effects \citep{GF15}. Unfortunately, current photometric data from the {\it Gaia} early data release three (EDR3) do not permit a spectral classification of the white dwarfs, preventing the study of their spectral evolution over the full sky. To mitigate this limitation, \citet{cunningham20} supplemented the {\it Gaia} catalog with optical and ultraviolet photometry, extending the white dwarf photometric classification down to $T_{\rm eff} = 9\,000$ K and obtaining a spectral evolution in agreement with the spectroscopic results.
In the present study, we used the 12 optical bands of the Javalambre Photometric Local Universe Survey (J-PLUS; \citealt{cenarro19}) second data release (DR2; \citealt{jplus_dr2}) over $2\,176$ deg$^2$ to supplement the GF21 catalog, which is based on {\it Gaia} EDR3, and to provide observational constraints to the white dwarf spectral evolution in the range $5\,000 < T_{\rm eff} < 30\,000$~K. The J-PLUS photometric system (Table~\ref{tab:JPLUS_filters}) is composed of five SDSS-like ($ugriz$) and seven medium-band filters located in key stellar features, such as the $4\,000~\AA$ break ($J0378$, $J0395$, $J0410$, and $J0430$), the Mg $b$ triplet ($J0515$), H$\alpha$ at rest frame ($J0660$), and the calcium triplet ($J0861$). We used this extra photometric information, coupled with a Bayesian analysis of the data, to disentangle the white dwarf spectral type over a wide range of effective temperatures.
This paper is organized as follows. In Sect.~\ref{sec:data} we detail the J-PLUS photometric data and the reference white dwarf catalog from {\it Gaia} EDR3 used in our analysis. The Bayesian fitting process is described in Sect.~\ref{sec:methods}. The final selection of the sample and the derived atmospheric parameters are presented in Sect.~\ref{sec:teff_logg}. The white dwarf spectral evolution from J-PLUS is reported in Sect.~\ref{sec:fnonda_teff}. Finally, a summary and the conclusions of our work are presented in Sect.~\ref{sec:conclusions}. All magnitudes are expressed in the AB system \citep{oke83}.
\begin{table}
\caption{J-PLUS passbands, including filter transmission, CCD efficiency, telescope optics, and atmosphere.}
\label{tab:JPLUS_filters}
\centering
\begin{tabular}{l c c}
\hline\hline\rule{0pt}{3ex}
Passband & Effective wavelength & Rectangular width \\
& [nm] & [nm]\\
\hline\rule{0pt}{2ex}
$u$ &353.6 & 34.3 \\
$J0378$ &378.2 & 13.8 \\
$J0395$ &393.9 & 9.9 \\
$J0410$ &410.8 & 19.4 \\
$J0430$ &430.3 & 19.6 \\
$g$ &481.0 & 129.5 \\
$J0515$ &514.1 & 20.5 \\
$r$ &627.2 & 143.4 \\
$J0660$ &660.4 & 14.6 \\
$i$ &766.9 & 139.7 \\
$J0861$ &861.1 & 40.2 \\
$z$ &898.0 & 124.9 \\
\hline
\end{tabular}
\end{table}
\begin{figure*}[t]
\centering
\resizebox{0.49\hsize}{!}{\includegraphics{figures/73822_16823_12p_hd.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/73315_2320_12p_hd.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/68023_9306_12p_hd.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/70019_8895_12p_hd.pdf}}
\caption{Spectral energy distributions of four white dwarfs analyzed as part of this work, selected to highlight the difference between H- and He-dominated atmospheres in the J-PLUS passbands. The colored points in all the panels are the 3 arcsec diameter photometry corrected to total magnitudes from J-PLUS (squares for broad bands, $ugriz$; circles for medium bands, $J0378$, $J0395$, $J0410$, $J0430$, $J0515$, $J0660$, and $J0861$). The gray diamonds connected with a solid line show the best-fitting solution, with the derived parameters and their uncertainties labeled in the panels: effective temperature ($T_{\rm eff}$), surface gravity ($\log {\rm g}$), parallax ($\varpi$), and probability of having a H-dominated atmosphere ($p_{\rm H}$). The unique J-PLUS identification, composed by the \texttt{TILE\_ID} of the reference $r-$band image and the \texttt{NUMBER} assigned by SExtractor to the source, is also reported in the panels for reference. Sources with $T_{\rm eff} \sim 16\,000$ K ({\it top panels}) and $T_{\rm eff} \sim 10\,000$ K ({\it bottom panels}) are presented. {\it Left panels}: Sources with H-dominated atmospheres, $p_{\rm H} = 1$. {\it Right panels}: Sources with He-dominated atmospheres, $p_{\rm He} = 0$.}
\label{fig:examples}
\end{figure*}
\section{Data}\label{sec:data}
\subsection{J-PLUS photometric data}\label{sec:jplus}
J-PLUS\footnote{\url{www.j-plus.es}} is being conducted from the Observatorio Astrof\'{\i}sico de Javalambre (OAJ, Teruel, Spain; \citealt{oaj}) using the 83\,cm Javalambre Auxiliary Survey Telescope (JAST80) and T80Cam, a panoramic camera of 9.2k $\times$ 9.2k pixels that provides a $2\deg^2$ field of view (FoV) with a pixel scale of 0.55 arsec pix$^{-1}$ \citep{t80cam}. The J-PLUS filter system is composed of 12 passbands (Table~\ref{tab:JPLUS_filters}). The J-PLUS observational strategy, image reduction, and main scientific goals are presented in \citet{cenarro19}.
The J-PLUS DR2 comprises $1\,088$ pointings ($2\,176$ deg$^2$) observed, reduced, and calibrated in all survey bands \citep{jplus_dr2, clsj21zsl}. The limiting magnitudes (5$\sigma$, 3 arsec aperture) of the DR2 are $\sim 22$ mag in $g$ and $r$ passbands, and $\sim 21$ mag in the another ten bands. The median point spread function (PSF) full width at half maximum (FWHM) in the DR2 $r$-band images is 1.1 arcsec. Source detection was done in the $r$ band using \texttt{SExtractor} \citep{sextractor}, and the flux measurement in the 12 J-PLUS bands was performed at the position of the detected sources using the aperture defined in the $r$-band image. Objects near the borders of the images, close to bright stars, or affected by optical artefacts were masked from the initial $2\,176$ deg$^2$, providing a high-quality area of $1\,941$ deg$^2$. The DR2 is publicly available on the J-PLUS web site\footnote{\url{www.j-plus.es/datareleases/data_release_dr2}}.
We used aperture photometry of 3 arcsec in diameter to analyze the white dwarf population. The observed fluxes were stored in the vector $\vec{f} = \{ f_j \}$, and their errors were stored in the vector $\sigma_{\vec{f}} = \{\sigma_j\}$, where the index $j$ runs the J-PLUS passbands. The error vector includes the uncertainties from photon counting, sky background, and photometric calibration \citep{clsj21zsl}.
\subsection{{\it Gaia} white dwarf catalog}\label{sec:gf21}
We used the {\it Gaia}-based catalog of white dwarfs presented in GF21 as a reference, and we did not attempt to derive a white dwarf catalog from J-PLUS photometry. The main reason for this choice is that the parallax information from {\it Gaia} EDR3 permits the estimation of the white dwarf surface gravity, which is poorly constrained from photometry alone, and the definition of volume-limited samples. In combination with the 12-band J-PLUS photometry in the present work, the atmospheric parameters of the common white dwarfs, including the atmospheric composition, can be computed (Sect.~\ref{sec:methods}). As a drawback, the selection effects of the GF21 catalog will be inherited by our common sample.
As a summary of the selection process performed by GF21, $1\,280\,266$ objects are selected using several quality flags and their location in the Hertzsprung-Russell diagram of {\it Gaia} EDR3. This initial selection represents a compromise between removing the majority of sources with non-optimal {\it Gaia} measurements and preserving all the stars in the white dwarf locus.
A total of $22\,998$ spectroscopically confirmed single white dwarfs and $7\,124$ contaminant objects obtained from SDSS DR16 \citep{sdss_dr16} were then used to map their distribution in the absolute $G$-band magnitude versus color space and to assign a white dwarf probability, $P_{\rm WD}$. Following the prescriptions in GF21, we selected the $259\,073$ sources with $P_{\rm WD} > 0.75$, of which $25\,632$ have SDSS spectroscopy. This spectroscopic sample comprises $91$\% confirmed white dwarfs, $1$\% contaminant objects, $3$\% white dwarf--main sequence binaries or cataclysmic variables, and the rest have unreliable classification. When comparing with confirmed SDSS spectroscopic white dwarfs, GF21 also find no significant color bias in the selection. We refer the reader to GF21 for a detailed description of the selection criteria and the properties of the reference sample.
We cross-matched the $259\,073$ sources with $P_{\rm WD} > 0.75$ in the GF21 catalog with the J-PLUS DR2 dataset using a $1.5$ arcsec radius, finding $11\,182$ sources with $r \leq 20.3$ mag in common. The final white dwarf sample, which ensures a well-defined volume and magnitude selection, is detailed in Sect.~\ref{sec:teff_logg}.
\section{Estimation of white dwarf atmospheric parameters and composition}\label{sec:methods}
We aim to estimate the following probability density function (PDF) for each white dwarf in the sample,
\begin{equation}
{\rm PDF}\,(t,\theta\,|\,\vec{f}, \vec{\sigma}_{\vec{f}}) \propto \mathcal{L}\,(\,\vec{f}\,|\,t,\theta,\sigma_{\vec{f}}) \times P\,(\theta) \times P\,(t),
\end{equation}
where $\theta$ are the parameters in the fitting, $t$ are the different atmospheric compositions considered in our analysis, $\mathcal{L}$ is the likelihood of the data for a given set of parameters and composition, and $P$ are the prior probabilities. The final PDF is normalized to one by definition:
\begin{equation}
\sum_t \int {\rm PDF}\,(t,\theta)\,{\rm d}\theta = 1.
\end{equation}
The parameters in the fitting were $\theta = \{T_{\rm eff}, \log {\rm g}, \varpi \}$, corresponding to effective temperature, surface gravity, and parallax. We explored two atmospheric compositions, corresponding to hydrogen- and helium-dominated atmospheres. We assumed that H-dominated atmospheres correspond to DA spectral type, while He-dominated atmospheres to DB and DC spectral types. The latter assumption imposes a lower limit on effective temperature in our analysis of $T_{\rm eff} = 5\,000$ K, because at lower temperatures the hydrogen lines are no longer visible in H-dominated atmospheres and they are also classified as DCs \citep[e.g.,][]{greenstein88}. For simplicity in the notation, hybrid spectral types such as DABs and DBAs were considered as their main composition type. We therefore used $t = \{ {\rm H}, {\rm He} \}$ in our analysis.
The probability of being H-dominated is therefore defined as
\begin{equation}
p_{\rm H} = \int {\rm PDF}\,({\rm H},\theta)\,{\rm d}\theta,
\end{equation}
and the probability of being He-dominated is $p_{\rm He} = 1 - p_{\rm H}$.
In the following sections, we describe the likelihood and the priors used in the analysis of J-PLUS + {\it Gaia} white dwarfs. We present four representative examples in Fig.~\ref{fig:examples}.
\subsection{Likelihood}\label{sec:likelihood}
We defined the likelihood of the data given a set of parameters as
\begin{equation}
\mathcal{L}\,(\,\vec{f}\,|\,t,\theta,{\boldmath \sigma}_{\vec{f}}) = \prod_{j = 1}^{12} P_{\rm G}\,(f_j\,|\,f_{j}^{\rm mod}, \sigma_{j}),
\end{equation}
where the index $j$ runs over the 12 J-PLUS passbands, the function $P_{\rm G}$ defines a Gaussian probability distribution,
\begin{equation}
P_{\rm G}\,(x\,|\,\mu,\sigma) = \frac{1}{\sqrt{2\pi}\,\sigma}\ {\rm exp}\Big[-\frac{(x - \mu)^2}{2\sigma^2}\Big] = \frac{1}{\sqrt{2\pi}\,\sigma}\ {\rm exp}\Big[\frac{- \chi^2}{2}\Big],
\end{equation}
and the model flux was estimated as
\begin{equation}
f_{j}^{\rm mod}\,(t,\theta)= \bigg( \frac{\varpi}{100} \bigg)^2\,F_{t,j}\,(T_{\rm eff},\log {\rm g})\,10^{-0.4\,k_j\,E(B-V)}\,10^{0.4\,C^{\rm aper}_j},
\end{equation}
where $E(B-V)$ is the assumed color excess of the white dwarf, $k_j$ is the extinction coefficient, $C^{\rm aper}_j$ is the aperture correction needed to translate the observed $3$ arcsec magnitudes to total magnitudes, and $F_{t,j}$ is the theoretical absolute flux emitted by a white dwarf of type $t$ located at 10 pc distance, that is, with a parallax of $100$ mas.
We assumed pure-H models to describe H-dominated atmospheres ($t = {\rm H}$, \citealt{tremblay11,tremblay13}), while mixed models with H/He = $10^{-5}$ at $T_{\rm eff} > 6\,500$ K and pure-He models at $T_{\rm eff} < 6\,500$ K were used to describe He-dominated atmospheres ($t = {\rm He}$, \citealt{cukanovaite18, cukanovaite19}). The mass--radius relation of \citet{fontaine01} for thick (H-atmospheres) and thin (He-atmospheres) hydrogen layers was assumed in the modeling. An extensive discussion about these choices and extra details of the models are presented in \citet{bergeron19}, \citet{GF20}, \citet{mccleery20}, and GF21.
The likelihood was estimated in a dense grid of models with $\Delta \log T_{\rm eff} = 0.005$ dex, $\Delta \log {\rm g} = 0.005$ dex, and $\Delta \varpi = 0.05$ mas. The fluxes for those parameters not included in the initial grid of theoretical models were estimated by linear interpolation.
The color excess was estimated using the $E(B-V)$ at infinity from \citet{sfd98}, properly scaled at a distance of $d = 1/\varpi$ with the Milky Way dust model presented in \citet{li18}. The uncertainty in $E(B-V)$ was fixed to $0.012$ mag. This error was estimated from the dispersion in the comparison between the color excess directly measured from the star-pair method \citep{yuan13} with the assumed $E(B-V)$. We note that this extinction scheme was used in the photometric calibration of J-PLUS DR2 \citep{clsj21zsl}, so we decided to also follow it here for consistency. The extinction coefficients for J-PLUS passbands are also reported in \citet{clsj21zsl}.
The aperture correction was defined as
\begin{equation}
C^{\rm aper} = C^{\rm tot}_{6} + C^{6}_{3},
\end{equation}
where the first term is the correction from $6$ arcsec magnitudes to total magnitudes, and the second term is the needed correction to go from $3$ arcsec to $6$ arcsec photometry. The 6 arcsec to total correction was estimated from the growth curve of bright, nonsaturated stars for each passband and pointing. For each star, increasingly large circular apertures were measured until convergence within errors. This defined the aperture size that provides the total magnitude of the sources in the pointing that is then compared with the magnitude at $6$ arcsec aperture to provide $C_{6}^{\rm tot}$.
The signal-to-noise ratio (S/N) from 3 arcsec photometry ($m_{3}$) is 30\% to 50\% larger than from 6 arcsec photometry ($m_{6}$) at $r = 19.5$ mag, the limiting magnitude of the present study; however, it is affected by PSF variation among passbands and along the FoV. To overcome these limitations, we estimated the 3 arcsec to 6 arcsec aperture correction using the measurements in the J-PLUS catalog. For each tile and passband, we derived the $c_{63}= m_{6} - m_{3}$ magnitude for objects with a stellarity parameter\footnote{We used the variable \texttt{sglc\_prob\_star} in the J-PLUS DR2 database as morphological classifier \citep{clsj19psmor}} larger than 0.9. We used the 250 brightest stars in the tile to estimate the median $c_{63}$ in a $5 \times 5$ grid in $(X,Y)$ to enhance the signal and ensure a smooth variation along the FoV. We parameterized the $(X,Y)$ variation with a combination of Chebysev polynomial up to second order. The resulting smooth function in $(X,Y)$ provides the aperture correction $C^{\rm 6}_{3}$ to transform 3 arcsec photometry to 6 arcsec photometry. The uncertainty in this correction, estimated from the dispersion of the $c_{63}$ measurements with respect to the best-fitting model after accounting for the observational errors, is typically below $5$ mmag. Because of its limited impact in the final error budget, the total aperture correction $C^{\rm aper}$ was assumed with no uncertainty in the fitting process.
We applied both the extinction and the aperture correction to the model fluxes instead of attempting to correct the observations. This is motivated by the fact that application of such factors to negative fluxes produces ill-defined values. The attenuation of model fluxes is always positive and well-defined, and permits the proper statistical treatment of those observations close to the detection limit of the survey.
\subsection{Priors}
The application of priors in the estimation of white dwarf parameters helps to break degeneracies and to avoid nonphysical or unlikely solutions \citep[e.g.,][]{mortlock09,omalley13}. We applied a prior in the parallax and the atmospheric composition, as detailed in the following sections.
\subsubsection{Parallax prior}\label{sec:prior_px}
The parallax prior was
\begin{equation}
P\,(\varpi) = P_{\rm G}\,(\varpi\,|\,\varpi_{\rm EDR3}, \sigma_{\varpi}),
\end{equation}
where $\varpi_{\rm EDR3}$ and $\sigma_{\varpi}$ are the parallax and its error obtained from ${\it Gaia}$ EDR3 \citep{gaiaedr3,lindegren21a}. The published values of the parallax were corrected by the {\it Gaia} zero-point offset following the prescription by \citet{lindegren21b}, already validated by several independent studies to a few $\mu$as level \citep[e.g.,][]{huang21par,ren21par,maiz21par}.
We note that the assumption of a mass--radius relation in the estimation of the theoretical fluxes coupled the parallax and the surface gravity variables. In this framework, the prior in $\varpi$ also imposes strong conditions on $\log {\rm g}$, which is poorly constrained by photometry alone. This issue is explored in Sect.~\ref{sec:chidist}.
\subsubsection{Atmospheric composition prior}\label{sec:prior_da}
The used prior in the atmospheric composition was
\begin{equation}
P\,({\rm H}\,|\,T_{\rm eff}) = 1 - f_{\rm He}\,(T_{\rm eff}),\label{eq:fdbprior}
\end{equation}
where $f_{\rm He}$ is the fraction of helium-dominated white dwarfs,
\begin{equation}
f_{\rm He} = \frac{N_{\rm He}}{N_{\rm H} + N_{\rm He}},
\end{equation}
$N_{\rm H}$ is the number of H-dominated white dwarfs, and $N_{\rm He}$ is the number of He-dominated white dwarfs at a given effective temperature. We note that this prior distribution is equivalent to the white dwarf spectral evolution with $T_{\rm eff}$, the elucidation of which is the main goal of our work.
Here, we present the technical details in the computation of $f_{\rm He}$, while the obtained results are detailed in Sect.~\ref{sec:fnonda_teff}. We can obtain the fraction of He-dominated white dwarfs from the final PDFs of the population as
\begin{equation}
f_{\rm He}\,(T_{\rm eff}) = \frac{\sum_i V^{-1}_i\,({\rm He})\times{\rm PDF}_i\,({\rm He}, T_{\rm eff})}{\sum_i \sum_t V_i^{-1}\,(t) \times {\rm PDF}_i\,(t,T_{\rm eff})},\label{eq:fdbpdf}
\end{equation}
where the index $i$ refers to the white dwarfs in the sample, the full PDFs were marginalized over the nonexplicit variables, and $V\,(t)$ is the maximum effective volume probed by each white dwarf as computed in Sect.~\ref{sec:selec}. This defines a continuum variable in effective temperature and a histogram can be created by integrating over $T_{\rm eff}$ bins. In the latter case, the errors in $f_{\rm He}$ were estimated by bootstrapping.
We note that $f_{\rm He}\,(T_{\rm eff})$ is present both in Eq.~(\ref{eq:fdbprior}) and Eq.~(\ref{eq:fdbpdf}). In other words, the input prior used to estimate the PDFs should be similar to the output fraction derived from the posteriors. Instead of imposing an externally computed prior, which would compromise the interpretation and significance of our results, we used the above argument to derive a prior from J-PLUS data in a self-consistent way and to estimate $f_{\rm He}\,(T_{\rm eff})$. Formally, this is a Bayesian hierarchical analysis where the parameters of the prior function are derived from the data. The assumed function in $T_{\rm eff}$ for the He-dominated fraction and the parameters that describe the white dwarf spectral evolution are presented and justified in Sect.~\ref{sec:fnonda_teff}.
To compute the self-consistent prior, we estimated the aggregated $\chi^2$ between the output PDF-based histogram and the binned input prior using the same $T_{\rm eff}$ ranges in both cases. We included the covariance terms between different temperature bins in the process, which were estimated from bootstrapping. We minimized the $\chi^2$ by exploring the parameters that define $f_{\rm He}\,(T_{\rm eff})$ with the \texttt{emcee} code \citep{emcee}, a \texttt{Python} implementation of the affine-invariant ensemble sampler for the Markov chain Monte Carlo (MCMC) technique proposed by \citet{goodman10}. The \texttt{emcee} code provides a collection of solutions in the parameter space, with the density of solutions being proportional to the posterior probability of the parameters. We obtained the central values of the parameters and their uncertainties from a Gaussian fit to the distribution of the solutions.
\subsection{Selection probability and derived quantities}\label{sec:selec}
In addition to the fundamental parameters obtained in the fitting process, other properties of the analyzed white dwarfs can be derived, such as the mass or the luminosity.
The mass PDF for a given type $t$ was estimated as
\begin{equation}
{\rm PDF}\,(M\,|\,t) = \int {\rm PDF}\,(t,\theta) \times \delta[M - \mathcal{M}\,(t,\theta)]\,{\rm d}\theta,
\end{equation}
where $\delta$ is the Dirac delta function, and $\mathcal{M}$ is the mass predicted for each model. The same procedure was used to obtain the PDF in $\hat{r}$, the total de-reddened $r$-band apparent magnitude, as
\begin{equation}
{\rm PDF}\,(\hat{r}\,|\,t) = \int {\rm PDF}\,(t,\theta) \times \delta[\hat{r} - \mathcal{R}\,(t,\theta)]\,{\rm d}\theta,
\end{equation}
where
\begin{equation}
\mathcal{R}\,(t,\theta) = -2.5\log_{10} [ f_r\,(t,\theta) ] + 46.8,\end{equation}
and
\begin{equation}
f_r\,(t,\theta) = \bigg( \frac{\varpi}{100} \bigg)^2\,F_{t,r}\,(T_{\rm eff},\log {\rm g}).
\end{equation}
We note that the variable $\hat{r}$ can be used to define a well-controlled magnitude-limited sample. We defined the selection probability as
\begin{equation}
p_{\rm sel}\,(r_{\lim}) = \sum_t \int_{-\infty}^{r_{\rm lim}}\,{\rm PDF}\,(\hat{r}\,|\,t)\,{\rm d}\hat{r},
\end{equation}
where the limiting magnitude $r_{\rm lim}$ defines the selection of the sample. In addition, we imposed a minimum of 10 pc and a maximum of $1$ kpc in the posterior analysis. In other words, the estimation of the PDF was performed in the full distance range, but we only kept those solutions with parallax $\varpi \in [1,100]$ mas. This selection helps to avoid extreme solutions and provides well-defined volumes. The final selection of the white dwarf sample and its distribution in $\hat{r}$ are presented in Sect.~\ref{sec:counts}.
Finally, we estimated the effective volume probed by each white dwarf in the sample. This quantity is needed to account for the different luminosities (i.e., probed volumes) of H- and He-dominated white dwarfs for a given effective temperature and surface gravity. We defined the effective volume as
\begin{equation}
V\,(t) = \frac{\int {\rm PDF}\,(t,\theta) \times \mathcal{V}\,(t,\theta)\,{\rm d}\theta}{\int {\rm PDF}\,(t,\theta)\,{\rm d}\theta}\ \ \ \ \rm{[kpc^{3}]},
\end{equation}
where
\begin{equation}
\mathcal{V}\,(t,\theta) = \frac{4}{3}\pi f_{\Omega}\,(\varpi_{\rm min}^{-3} - \varpi_{\rm max}^{-3})\ \ \ \ \rm{[kpc^{3}]},
\end{equation}
the fraction of the sphere subtended by the unmasked J-PLUS DR2 is $f_{\Omega} = 0.047$ ($1\,941$ deg$^2$), the maximum parallax was set to $\varpi_{\rm max} = 100$ mas, the minimum parallax was defined as
\begin{equation}
\varpi_{\rm min}\,(t,\theta) = {\rm max}[1, \varpi_{\rm lim}],
\end{equation}
and $\varpi_{\rm lim}$ is the parallax associated to the $r_{\rm lim}$ magnitude for a given set of parameters (i.e, the maximum distance at which a white dwarf can be observed given the selection magnitude $r_{\rm lim}$). The effective volume is used in the estimation of the spectral evolution of white dwarfs, as detailed in Sect.~\ref{sec:prior_da}.
\subsection{Summary statistics}
We obtained the summary statistics for each parameter by marginalizing the PDFs over the other parameters at a fixed atmospheric composition and performing a Gaussian fit to the resulting distribution. The retrieved parameters and their uncertainties are the median and the dispersion of the best-fitting Gaussian. We find that this process provides a proper description of the posteriors and allows us to gather the relevant information table format.
The reported values of $\hat{r}$ were estimated before the final selection in both magnitude and distance (Sect.~\ref{sec:selec}), while the remaining variables were computed from the remaining solutions after applying the selection to the posterior PDFs. The summary statistics are publicly available in the J-PLUS database and the Centre de Donn\'ees astronomiques de Strasbourg\footnote{\url{https://cds.u-strasbg.fr}} (CDS). A description of and links to the data are presented in Appendix~\ref{app:data}.
We also provide $p_{\rm H}$ in the table. In those cases where the probability of having a H-dominated atmosphere is different from both zero and one, we included the atmospheric parameters for both H- and He-dominated atmospheres. These values should be weighted with the corresponding $p_{\rm H}$ to provide meaningful results, as illustrated in Sect.~\ref{sec:massdist}.
\section{White dwarf sample and atmospheric parameters}\label{sec:teff_logg}
We present in this section the atmospheric parameters obtained in the analysis of the J-PLUS DR2 + ${\it Gaia}$ EDR3 catalog. The final selection of the sample is described in Sect.~\ref{sec:counts} and we analyze the quality of the fitting process in Sect.~\ref{sec:chidist}. The obtained effective temperatures and surface gravities are studied in Sect.~\ref{sec:tlogg}.
\begin{figure}[t]
\centering
\resizebox{\hsize}{!}{\includegraphics{figures/rdist.pdf}}
\caption{White dwarf number counts in J-PLUS DR2 as a function of the total and de-reddened $r$-band apparent magnitude, noted $\hat{r}$ (red histogram). The black line is the function that better describes the distribution, as labeled in the panel. The white area shows the magnitudes used to define the white dwarf sample, $\hat{r} \leq 19.5$ mag.}
\label{fig:rdist}
\end{figure}
\begin{figure}[t]
\centering
\resizebox{\hsize}{!}{\includegraphics{figures/snr_px.pdf}}
\caption{Signal-to-noise ratio in the {\it Gaia} EDR3 parallax (${\rm S/N}_{\varpi_{\rm EDR3}}$) as a function of the de-reddened $r$-band apparent magnitude ($\hat{r}$) for the final white dwarf sample. The black dots show individual measurements. The red areas from lighter to darker enclose 90\%, 50\%, and 10\% of the sources, respectively. The gray dashed line marks ${\rm S/N}_{\varpi_{\rm EDR3}} = 3$.}
\label{fig:snrpx}
\end{figure}
\subsection{Final selection and number density}\label{sec:counts}
We estimated the atmospheric parameters and composition for the $11\,182$ sources in common between the GF21 catalog and J-PLUS DR2 at $r < 20.3$ mag (Sect.~\ref{sec:gf21}), and only kept those with a selection probability $p_{\rm sel} > 0.01$ for $\varpi \in [1,100]$ mas and $\hat{r} \leq 19.5$ mag. We refer the reader to Sect.~\ref{sec:selec} for further details about the definition of the selection probability. This selection ensures enough S/N in the J-PLUS photometry to perform a meaningful statistical analysis of the sources and provides well-defined volumes.
The final sample comprises $5\,926$ white dwarfs, which translates to a number density of $2.8$ deg$^{-2}$. We present the white dwarf number counts in Fig.~\ref{fig:rdist}, obtained as the $p_{\rm sel}$-weighted histogram in the dust de-reddened $\hat{r}$ magnitude normalized by the J-PLUS DR2 surveyed area and the magnitude bin size. We note that the used $\hat{r}$ magnitudes refer to the full range of solutions, that is, before applying the selection constraints. Hence, sources with $\hat{r} > 19.5$ and $p_{\rm sel} < 1$ are present in the sample. The number counts are well described as
\begin{equation}
\log C_{\rm WD} = 0.5\,\hat{r} - 9.25\ \ \ \ {\rm [deg^{-2} mag^{-1}]}.
\end{equation}
We find no evidence of departure from linearity in log scale over the six magnitudes covered by our study, $13.5 \lesssim \hat{r} \leq 19.5$ mag.
We present the S/N in the {\it Gaia} EDR3 parallax as a function of $\hat{r}$ for the final sample in Fig.~\ref{fig:snrpx}. As expected, there is a trend towards lower signal at fainter magnitudes. The median value for the sample is ${\rm S/N} = 20.3$, with more than 99\% of the sources having ${\rm S/N} > 3$.
Finally, we estimated the purity of our final sample using the $42\,007$ spectral classifications based on SDSS DR16 \citep{sdss_dr16} also presented in the GF21 catalog. We cross-matched the spectroscopic catalog with our final sample, discarding duplicate entries and sources with unreliable classification. This provided a total of $1\,835$ sources with spectral classification, with $1\,805$ ($98.3$\%) white dwarfs, $25$ ($1.4$\%) white dwarf--main sequence binaries or cataclysmic variables, and $5$ ($0.3$\%) contaminants or objects with unknown classification. We conclude that our final sample of $5\,926$ white dwarfs presents a well-defined magnitude and volume selection with a purity above $98$\%.
\begin{figure}[t]
\centering
\resizebox{\hsize}{!}{\includegraphics{figures/chi2dist_pxprior.pdf}}
\caption{Distribution of $\chi^2_{\rm min}$ for the white dwarf sample. The open black histogram represents the solutions when the ${\it Gaia}$ EDR3 parallax was used as prior, and the blue solid histogram shows when no prior in parallax was used in the fitting. The red solid and blue dotted lines mark the expected $\chi^2$ distribution for 10 and 9 degrees of freedom, respectively.}
\label{fig:chi2dist}
\end{figure}
\begin{figure}[t]
\centering
\resizebox{\hsize}{!}{\includegraphics{figures/chidist_nfil12_case2.pdf}}\caption{Distribution of the observed flux minus the best-fitting flux normalized by the photometric error. The squares (broad bands) and circles (medium bands) show the median and the dispersion of the distribution, depicted with the violin plots. The black solid line marks a zero difference, and the gray area shows the $\pm 0.91$ value expected for the dispersion.}
\label{fig:chidist}
\end{figure}
\begin{figure*}[t]
\centering
\resizebox{0.49\hsize}{!}{\includegraphics{figures/teff_jvsgf.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/Dteff_jvsgf.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/Dteff_jvsgf_hist.pdf}}\resizebox{0.49\hsize}{!}{\includegraphics{figures/dteff_jvsgf_hist.pdf}}\caption{Comparison between the effective temperature derived from J-PLUS ($T_{\rm eff}^{\rm J{\textrm -}PLUS}$) and {\it Gaia} EDR3 ($T_{\rm eff}^{\rm GF21}$) photometry. {\it Top left panel}: Individual measurements (red dots) with gray error bars. The median in ten temperature intervals is marked with the white dots. Dashed line marks the one-to-one relation. {\it Top right panel}: Relative difference between J-PLUS and {\it Gaia} temperatures as a function of {\it Gaia} temperature. White dots show the median difference in ten temperature intervals. Dashed line depicts identity. {\it Bottom left panel}: Histogram of the relative difference. {\it Bottom right panel}: Histogram of the error-normalised difference between J-PLUS and {\it Gaia} temperatures. In both {\it bottom panels}, the red line shows the best Gaussian fit to the distribution, with parameters labeled in the panel.
}
\label{fig:teffgf}
\end{figure*}
\subsection{Comparison between photometry and models}\label{sec:chidist}
Before exploring the derived white dwarf parameters, we studied the comparison between the observed photometry and the predicted fluxes from the best-fitting model. We defined the minimum $\chi^2$ of each source as
\begin{equation}
\chi^2_{\rm min} = \sum_{j= 1}^{12}\frac{(f_j - f_j^{\rm best})^2}{\sigma^2_j} = \sum_{j= 1}^{12} (\Delta f_j)^2,
\end{equation}
where $f_j^{\rm best}$ represents the expected flux from the model with the largest probability for each analyzed white dwarf.
The histogram of the $\chi^2_{\rm min}$ for the white dwarf sample is presented in Fig.~\ref{fig:chi2dist}. We find that it is described by a $\chi^2$ distribution with 10 degrees of freedom (dof). We have 12 photometric points, implying that our modeling procedure has only two effective variables. This is a consequence of two issues: first, the surface gravity and the parallax are highly correlated and should be considered a unique effective variable. Second, the parallax prior from {\it Gaia} EDR3 (Sect.~\ref{sec:prior_px}) tightly constrains the parallax variable, and therefore the surface gravity. As a consequence, only the effective temperature and the atmospheric composition have freedom to vary in the fitting process.
The argument above was tested by repeating the Bayesian analysis but assuming a flat prior in parallax. Without the constraint from {\it Gaia} EDR3, the parallax and the surface gravity can vary in the fitting process and we expect an improvement in the minimum $\chi^2$ of the sources. We find that the distribution indeed improves (Fig.~\ref{fig:chi2dist}) and is described by a $\chi^2$ distribution with 9 dof. This means that we have three effective parameters, as anticipated.
In addition to the aggregated $\chi^2$, we can analyze the distribution of the individual passbands with respect to the best-fitting model, noted $\Delta f_j$. We expect this variable to follow a Gaussian distribution with median $\mu = 0$ and dispersion $\sigma = 0.91$, the square root of the ratio between the dof and the number of passbands. The measured distribution in $\Delta f_j$ is presented in Fig.~\ref{fig:chidist}. We find that, as desired, the median of the distributions is close to zero, with median $\mu \sim \pm 0.05$, and that the dispersion is close to expectations, with a median value of $\sigma = 0.94$ and fluctuations of only 0.1.
The median of close to zero suggests a proper match between the photometry and the models. The color calibration of J-PLUS DR2 was performed using the locus of 639 white dwarfs \citep{clsj19jcal,clsj21zsl}, and the match found was therefore expected. The measured dispersion implies that the photometric errors are properly estimated for each passband, as they account for the observed dispersion between the photometry and the models. We conclude that the J-PLUS photometry and the estimated observational errors are reliable, providing a proper data set with which to explore the spectral evolution of the white dwarf population.
\begin{figure*}[t]
\centering
\resizebox{0.49\hsize}{!}{\includegraphics{figures/logg_jvsgf.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/Dlogg_jvsgf.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/logg_jvsgf_hist.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/Dlogg_jvsgf_hist.pdf}}\caption{Comparison between the surface gravity derived from J-PLUS ($\log {\rm g}^{\rm J{\textrm -}PLUS}$) and {\it Gaia} EDR3 ($\log {\rm g}^{\rm GF21}$) photometry. {\it Top left panel}: Individual measurements (red dots) with gray error bars. The median in ten gravity intervals is marked with the white dots. Dashed line marks the one-to-one relation. {\it Top right panel}: Difference between J-PLUS and {\it Gaia} surface gravity as a function of {\it Gaia} temperature. White dots mark the median difference in ten temperature intervals. Dashed line depicts identity. {\it Bottom left panel}: Histogram of the difference. {\it Bottom right panel}: Histogram of the error-normalized difference between J-PLUS and {\it Gaia} surface gravity. In both {\it bottom panels}, the red line shows the best Gaussian fit to the distribution, with parameters labeled in the panel.}
\label{fig:logggf}
\end{figure*}
\subsection{Effective temperature and surface gravity}\label{sec:tlogg}
We compare the effective temperature and the surface gravity of the $5\,926$ white dwarfs in our sample derived from J-PLUS photometry against the estimations from GF21 using {\it Gaia} EDR3 data. In the comparison, the solutions from H-dominated models in both studies were used for sources with $p_{\rm H} \geq 0.5$ and the He-dominated solutions were used otherwise. The comparison between the {\it Gaia} estimations and the values derived from spectroscopy are presented elsewhere \citep[e.g.,][GF21]{tremblay19,cukanovaite21}, and so we do not duplicate such a comparison in this work.
The comparison between J-PLUS and {\it Gaia} effective temperature scales is presented in Fig.~\ref{fig:teffgf}. We find an excellent one-to-one correlation between both measurements in the temperature range $5\,000 \lesssim T_{\rm eff} \lesssim 100\,000$ K. Moreover, the fractional difference as a function of $T_{\rm eff}$ presents no trend with temperature. The histogram of the fractional difference resembles a Lorentzian profile, with a compact core and extended wings. This is the usual distribution when measurements with different uncertainties are combined. The Gaussian approach to this distribution provides a 0.6\% difference between the two values, that is, both photometric systems provide the same effective temperature scale. The dispersion of the Gaussian distribution is $7$\%. As in Sect.~\ref{sec:chidist}, we tested the $T_{\rm eff}$ uncertainties by normalizing the difference in effective temperatures with $\sigma_{T}$, the combined error from the individual measurements. We find that the obtained distribution is well described by a Gaussian with median $\mu = 0.08$ and dispersion $\sigma = 0.84$. The median reflects the reported offset in units of the uncertainty, and the dispersion below the expected unity implies that the temperature uncertainties in both {\it Gaia} and J-PLUS could be overestimated by just $\sim 10$ \%.
The surface gravity is compared in Fig.~\ref{fig:logggf}. Both measurements are again in excellent agreement, with an apparently larger $\log {\rm g}$ in J-PLUS at $T_{\rm eff} \gtrsim 30\,000$ K. The direct comparison provides no bias and a dispersion of 0.13 dex. The error-normalized distribution resembles a Gaussian with median $\mu = 0.003$ and dispersion $\sigma = 0.60$. In this case, the dispersion is clearly below unity. We interpret this as a reflection of the correlation between the measurements, both of which used the {\it Gaia} EDR3 parallax in the analysis. The assumption of a mass--radius relation largely couples the parallax and the surface gravity parameters, which are degenerated and poorly constrained from photometry alone. Thus, the main information used to estimate $\log {\rm g}$ is shared in both studies. When accounted for in the error budget, this covariance translates to a smaller $\sigma_{\log {\rm g}}$, the combined error, and thus to a larger dispersion than in the case of having truly independent measurements. A covariance of $\rho = 0.7$ is needed to obtain a unity dispersion.
We repeated the $T_{\rm eff}$ and $\log {\rm g}$ comparison for white dwarfs with $p_{\rm H} > 0.9$ (H-dominated, $4\,011$ sources) and $p_{\rm H} < 0.1$ (He-dominated, $685$ sources). On the one hand, the results for H-dominated white dwarfs are similar to the global case, as they dominate the statistics. On the other hand, the obtained figures for the He-dominated sample are also compatible with the general case, and we only find a trend at $T_{\rm eff}^{\rm GF21} \gtrsim 20\,000$ K, with J-PLUS effective temperatures being typically lower than those based on {\it Gaia} EDR3 photometry.
The comparison between J-PLUS and {\it Gaia} values reveals satisfactory results. Unfortunately, the only net improvement is in the $T_{\rm eff}$ errors, which decrease from 10\% in {\it Gaia} to 5\% in J-PLUS, but the effective temperature and surface gravity scales are similar for both data sets. We conclude that increasing the photometric optical information from three filters in {\it Gaia} EDR3 to 12 filters in J-PLUS is not critical for $T_{\rm eff}$ and $\log {\rm g}$ estimation. As we demonstrate in the following section, the great advantage of J-PLUS with respect to {\it Gaia} EDR3 is its capability to provide the atmospheric composition of the analyzed white dwarfs.
\section{White dwarf spectral evolution with temperature}\label{sec:fnonda_teff}
The main result of the present paper, that is, the spectral evolution in the fraction of He-dominated white dwarfs with effective temperature, is presented in Sect.~\ref{sec:fheteff}. We compare the J-PLUS results with the literature in Sect.~\ref{sec:fhe_lit}, and test the reliability of the $p_{\rm H}$ probabilities used in the estimation of the spectral evolution in Sect.~\ref{sec:pdatest}.
\subsection{Spectral evolution from J-PLUS photometry}\label{sec:fheteff}
The technical details about the reported results are presented in Sect.~\ref{sec:prior_da}. As a brief summary, the fraction of He-dominated white dwarfs ($f_{\rm He}$) is parameterized as an effective temperature function that is used as prior in the atmospheric composition and compared against the resulting $f_{\rm He}$ estimated from the posterior probabilities of the J-PLUS + {\it Gaia} white dwarf sample. The parameters of the $f_{\rm He}$ function were explored searching for self-consistency in the prior and posterior values of the He-dominated fraction.
The results from the literature (Sect.~\ref{sec:intro}) suggest that a linear function in $T_{\rm eff}$ is a proper proxy for the spectral evolution at $T_{ \rm eff} \lesssim 20\,000$ K, with a minimum fraction at the so-called DB minimum at $20\,000 \lesssim T_{\rm eff} \lesssim 45\,000$ K \citep{eisenstein06db,GB19,bedard20}. Thus, we described the spectral evolution as
\begin{eqnarray}
f_{\rm He} = b - a\times \bigg( \frac{T_{\rm eff}}{10^4\ {\rm K}} - 1 \bigg),
\end{eqnarray}
imposing a maximum value of $1$ and a minimum value of $f_{\rm He}^{\rm min}$. We defined $a$ with a negative sign to provide the evolution rate with cooling time. The three parameters that we aimed to estimate are $a$, $b$, and the minimum fraction of He-dominated white dwarfs.
We performed the spectral analysis in the range $5\,000 < T_{\rm eff} < 30\,000$~K. As hot white dwarfs emit most of their light in the ultraviolet, optical photometry samples the Rayleigh-Jeans tail of their spectral energy distribution and is therefore weakly sensitive to their effective temperature. Our upper limit in temperature ensures a precise estimation of $T_{\rm eff}$ with J-PLUS photometry, which provides $\sigma_{\log T_{\rm eff}} \simeq 0.02$ dex at $T_{\rm eff} \lesssim 30\,000$ K. At higher temperatures, the uncertainty starts to increase, reaching $\sigma_{\log T_{\rm eff}} \simeq 0.10$ dex at $T_{\rm eff} \sim 50\,000$ K. Moreover, the maximum effective temperature in our He-dominated models is $T_{\rm eff} = 40\,000$ K, and so we also avoid undesired border effects in the solutions. The lower limit in temperature ensures that Balmer lines are visible for H-dominated white dwarfs.
We used the full PDF in the estimation of the spectral evolution (Sect.~\ref{sec:prior_da}), and so individual sources can be spread over different temperature bins. To minimize the covariance between adjacent bins and to maximize the independent information in the calculation, we set the bin size to $\Delta \log T_{\rm eff} = 0.06 \approx 3 \times \sigma_{\log T_{\rm eff}}$. Therefore, 13 effective temperature bins were available for the spectral evolution analysis.
Those sources with mass $M \leq 0.45\ M_{\odot}$ were discarded to avoid unresolved double degenerates and low-mass white dwarfs from binary evolution, leaving $4\,962$ white dwarfs. The He-dominated fraction obtained from the posteriors with and without spectral type prior are presented in Fig.~\ref{fig:fnonda_jplus} and Table~\ref{tab:fnonda}. We find that the He-dominated fraction obtained without prior has a minimum of $f_{\rm He} \simeq 0.15$ at $T_{\rm eff} \gtrsim 17\,000$ K, and then increases at lower temperatures reaching $f_{\rm He} \simeq 0.40$ at $T_{\rm eff} \sim 5\,000$ K. In other words, the J-PLUS photometry suggests a spectral evolution. However, the increase in $f_{\rm He}$ with decreasing temperature could simply be a reflection of our lower capacity to distinguish between white dwarf types. The Bayesian analysis discards such a possibility and provides a solid statistical significance to the spectral evolution.
The application of the self-consistent prior, which also provides the best measurement of the spectral evolution with temperature using J-PLUS information, yields a minimum fraction of $f_{\rm He}^{\rm min} = 0.08 \pm 0.02$, a He-dominated fraction at $T_{\rm eff} = 10\,000$~K of $b = 0.24 \pm 0.01$, and a positive slope of $a = 0.14 \pm 0.02$. The parameter $a$, which reflects the rate in the spectral evolution with cooling time, is different from zero at 7$\sigma$ level.
\begin{figure}[t]
\centering
\resizebox{\hsize}{!}{\includegraphics{figures/fdb_vs_teff_prior0_lit0.pdf}}
\caption{Fraction of He-dominated white dwarfs ($f_{\rm He}$) as a function of the effective temperature ($T_{\rm eff}$). The red solid line is the best spectral evolution prior estimated from J-PLUS photometry. The red area encloses 68\% of the solutions. The green and red dots show the values obtained from the posteriors estimated without and with the spectral evolution self-consistent prior applied, respectively.}
\label{fig:fnonda_jplus}
\end{figure}
The differences obtained in the spectral evolution with and without the spectral type prior illustrate that the capability of the J-PLUS photometry to disentangle between H- and He-dominated atmospheres degrades at higher and lower temperatures, where the spectral differences between white dwarfs with different atmospheric composition are diluted. The larger difference between the observed fraction of He-dominated white dwarfs with and without prior occurs at $T_{\rm eff} \gtrsim 17\,000$ K and $T_{\rm eff} \lesssim 9\,000$ K, where the posterior fraction decreases by 50\%. The difference is small at $9\,000 \leq T_{\rm eff} \leq 17\,000$ K, where the Balmer lines are more prominent in DAs and the contrast with respect to DBs and DCs in the J-PLUS photometry is maximum.
The final spectral evolution from J-PLUS DR2 provides a $21 \pm 3$\% increase in the fraction of He-dominated white dwarfs from $T_{\rm eff} = 20\,000$ K to $T_{\rm eff} = 5\,000$ K. We recall that the prior and the posterior fractions as a function of $T_{\rm eff}$ are self-consistent and were obtained using only J-PLUS photometric data. We demonstrate the reliability of the final $p_{\rm H}$ in Sect.~\ref{sec:pdatest}.
\begin{table}
\caption{Spectral evolution of white dwarfs with effective temperature by PDF analysis.}
\label{tab:fnonda}
\centering
\begin{tabular}{c c c}
\hline\hline\rule{0pt}{3ex}
$T_{\rm eff}$ & $f_{\rm He}$ & $f_{\rm He}$ \\
${\rm [kK]}$ & without prior & with prior \\
\hline\rule{0pt}{3ex}
\!$5.34$ & $0.43 \pm 0.03$ & $0.27 \pm 0.02$ \\
$6.13$ & $0.40 \pm 0.02$ & $0.25 \pm 0.02$ \\
$7.04$ & $0.35 \pm 0.04$ & $0.25 \pm 0.03$ \\
$8.08$ & $0.35 \pm 0.03$ & $0.27 \pm 0.03$ \\
$9.28$ & $0.26 \pm 0.03$ & $0.24 \pm 0.02$ \\
$10.65$ & $0.27 \pm 0.02$ & $0.27 \pm 0.02$ \\
$12.23$ & $0.23 \pm 0.02$ & $0.22 \pm 0.02$ \\
$14.04$ & $0.18 \pm 0.02$ & $0.17 \pm 0.02$ \\
$16.13$ & $0.20 \pm 0.02$ & $0.18 \pm 0.02$ \\
$18.51$ & $0.14 \pm 0.02$ & $0.11 \pm 0.01$ \\
$21.26$ & $0.14 \pm 0.03$ & $0.08 \pm 0.02$ \\
$24.41$ & $0.14 \pm 0.03$ & $0.07 \pm 0.02$ \\
$28.02$ & $0.25 \pm 0.04$ & $0.12 \pm 0.03$ \\
\hline
\end{tabular}
\end{table}
\begin{figure*}[t]
\centering
\resizebox{0.49\hsize}{!}{\includegraphics{figures/fdb_vs_teff_prior1_lit1.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/fdb_vs_teff_prior1_lit4.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/fdb_vs_teff_prior1_lit2.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/fdb_vs_teff_prior1_lit3.pdf}}
\caption{Fraction of He-dominated white dwarfs ($f_{\rm He}$) as a function of effective temperature ($T_{\rm eff}$) from the literature and J-PLUS. We split the comparison in four panels to improve visualization. In all panels, the red solid line is the best-fitting spectral evolution estimated from J-PLUS photometry. The red area encloses 68\% of the solutions. The red dots show the J-PLUS values obtained from the posterior estimated with the spectral evolution prior applied. The black symbols labeled in the panels show results from the literature.}
\label{fig:fnonda_lit}
\end{figure*}
\subsection{Comparison with the literature}\label{sec:fhe_lit}
We now compare our final spectral evolution with previous results in the literature, as illustrated in Fig.~\ref{fig:fnonda_lit}. There is a general agreement with the trends and values derived from spectroscopy \citep{tremblay08,GB19,ourique19,blouin19,bedard20,mccleery20} and by NUV--optical photometry \citep{cunningham20}. We recall that the J-PLUS result is based only on optical photometry.
Regarding the high-temperature end, $T_{\rm eff} \geq 20\,000$ K, the results from \citet{GB19}, \citet{ourique19}, and \citet{bedard20} suggest a lower limit of $f_{\rm He} \sim 0.05-0.10$, compatible with our derived value of $f_{\rm He}^{\rm min} = 0.08 \pm 0.02$.
At lower temperatures, previous studies found an increase in the He-dominated fraction that is compatible with the J-PLUS trend. The quantitative agreement between the J-PLUS values and the findings of \citet{tremblay08} and \citet{ourique19}
is remarkable. We find discrepancies with \citet{ourique19} at $T_{\rm eff} \lesssim 15\,000$ K. This could be due to the nontrivial SDSS spectroscopic selection function affecting the sample used by these latter authors.
The agreement with \citet{cunningham20} is excellent over the entire temperature range, except at $T_{\rm eff} \sim 10\,000$ K. The classification used by \citet{cunningham20} is based on the $(NUV-g)$ versus $(g-r)$ color--color diagram, where H- and He-dominated white dwarfs present different loci. The separation between these loci decreases at lower $T_{\rm eff}$, making it more difficult to classify cool white dwarfs. We note that \citet{cunningham20} do not apply a spectral type prior to their classification. This suggests that the discrepancies are just a reflection of the noisier classification at their lower temperatures and highlights the importance of spectral priors on photometric studies.
The values from \citet{mccleery20} are based on the spectroscopic follow-up of a volume-limited 40 pc sample selected from {\it Gaia} DR2 \citep{tremblay20}. The most interesting feature is the nice agreement at low temperatures, $T_{\rm eff}~<~10\,000$~K. The comparison with \citet{blouin19} at this temperature range is also satisfactory within uncertainties, but their measurements are lower than J-PLUS values in certain temperature ranges. These could be real fluctuations in the He-dominated fraction, but the smooth functional form assumed to describe the J-PLUS data is not sensitive to these possible variations, and only the general trend with $T_{\rm eff}$ can be explored. With this limitation in mind, the results from \citet{blouin19} are compatible with the J-PLUS findings.
In summary, we find good agreement with recent studies regarding the spectral evolution of white dwarfs at $T_{\rm eff} < 40\,000$~K, supporting our analysis and the unique capabilities of the J-PLUS photometric data.
\subsection{Testing the probability of having a H-dominated atmosphere}\label{sec:pdatest}
The spectral evolution presented in the previous sections is based on the H- and He-dominated posteriors estimated from J-PLUS DR2 photometry. In this section, we present three tests to check the reliability of $p_{\rm H}$: we compare the J-PLUS spectral probability with the classification from spectroscopy (Sect.~\ref{sec:spec_pda}), analyze the usual $(u-r)$ versus $(g-i)$ color--color diagram (Sect.~\ref{sec:color}), and estimate the mass distribution of H- and He-dominated white dwarfs at $d < 100$ pc (Sect.~\ref{sec:massdist}).
\subsubsection{Spectroscopic classification and significance of $p_{\rm H}$}\label{sec:spec_pda}
The atmospheric composition prior scheme developed in Sect.~\ref{sec:prior_da} can be tested by comparing $p_{\rm H}$ with the classification from spectra. We again used the spectral labels available in the GF21 catalog, estimated from SDSS DR16 spectroscopy. We only used the classification from spectra with ${\rm S/N} \geq 10$. Those white dwarfs with a dominant presence of metals in their atmosphere according to the spectroscopic classification (DZ, DZA, DZB, etc.) were included in the He-dominated class. In addition, spectral subtypes (DAB, DBA, DAZ, etc.) were assigned to their main atmospheric composition. We also restricted the sample to our temperature and mass ranges of interest, $5\,000<T_{\rm eff}<30\,000$~K and $M > 0.45$~$M_{\sun}$. We found $929$ H-dominated (DA) and $289$ He-dominated (DB/DC/DZ) white dwarfs with spectral classification.
We first studied the $p_{\rm H}$ distribution for DA and DB/DC/DZ sources, as presented in Fig.~\ref{fig:pdahist}. We found that the DA distribution peaks at $p_{\rm H} = 1$ and the DB/DC/DZ distribution at $p_{\rm H} = 0$, as desired. Furthermore, 83\% of the DA sample have $p_{\rm H} \geq 0.95$, and 64\% of the DB/DC/DZ sample have $p_{\rm H} \leq 0.05$, demonstrating the capability of J-PLUS photometry to differentiate between different white dwarf types.
The performance of a categorical classification based on a $p_{\rm H}$ threshold can be estimated with the completeness, the purity, and other summary statistics that compare spectroscopic and photometric types. However, from a statistical point of view, we should use the measured $p_{\rm H}$ to weight the sources. In this case, all the white dwarfs are always used in the analysis and they are properly weighted with our best knowledge about their atmospheric composition. We therefore have to demonstrate that the estimated $p_{\rm H}$ is indeed the probability of having a H-dominated atmosphere.
The $p_{\rm H}$ reliability can be tested by comparing the fraction of spectroscopic DAs at a given $p_{\rm H}$ range with the median $p_{\rm H}$ in that range. A properly derived $p_{\rm H}$ must produce a one-to-one relation, that is, the fraction of true DAs is proportional to $p_{\rm H}$. The obtained values are presented in Fig.~\ref{fig:pdafda} for the $p_{\rm H}$ obtained with and without atmospheric composition prior. The uncertainties were estimated from bootstrapping. We find that the relation clearly departs from the one-to-one line when the prior is not applied, with the $p_{\rm H}$ being underestimated (i.e., the fraction of true DAs is larger than predicted). The values obtained with the self-consistent prior are compatible with the desired one-to-one line. We recall that the prior was computed using J-PLUS data, and that no spectroscopic label was used in the process. This confirms the reliability of the Bayesian analysis and strengthens the spectral evolution results obtained in Sect.~\ref{sec:fheteff}.
\begin{figure}[t]
\centering
\resizebox{\hsize}{!}{\includegraphics{figures/pda_hist_12p_aper3.pdf}}
\caption{Normalized histogram of the $p_{\rm H}$ probability for the sample of $929$ DAs (orange) and $289$ DB/DC/DZs (purple) with spectroscopic classification.
}
\label{fig:pdahist}
\end{figure}
\begin{figure}[t]
\centering
\resizebox{\hsize}{!}{\includegraphics{figures/pda_nspec_12p_prior1.pdf}}\caption{Fraction of spectroscopic DAs as a function of $p_{\rm H}$. The black and red dots show the results without and with the spectral type prior applied, respectively. The dashed line marks the expected one-to-one relation.}
\label{fig:pdafda}
\end{figure}
\begin{figure*}[t]
\centering
\resizebox{0.49\hsize}{!}{\includegraphics{figures/gi_ur_wddot_da1_snj3.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/gi_ur_wddot_da0_snj3.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/j0660r_j0378j0515_wddot_da1_snj3.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/j0660r_j0378j0515_wddot_da0_snj3.pdf}}
\caption{The $(u-r)_0$ versus $(g-i)_0$ ({\it top panels}) and $(J0378-J0515)_0$ versus $(J0660-r)_0$ ({\it bottom panels}) color--color diagrams for white dwarfs with ${\rm S/N} > 3$ in J-PLUS photometry ($5\,379$ sources). In both panels, the orange and purple lines show the theoretical loci for H-dominated and He-dominated atmospheres, respectively. The black contour enclose 50\% of the sources in each panel. The square and the diamond mark the colors for a H- and a He-dominated white dwarf, respectively, with $T_{\rm eff} = 8\,500$ K and $\log {\rm g} = 8$ dex. The color scale shows the probability of having a H-dominated atmosphere, $p_{\rm H}$. {\it Left panels}: Color--color diagrams for the $4\,470$ sources with $p_{\rm H} \geq 0.5$. {\it Right panels}: Color--color diagrams for the $909$ sources with $p_{\rm H} < 0.5$.
}
\label{fig:urgi}
\end{figure*}
As an extra test, we computed the number of H-dominated white dwarfs in the spectroscopic sample as
\begin{equation}
N_{\rm H}^{\rm phot} = \sum_i p_{\rm H}^i,
\end{equation}
and the number of He-dominated white dwarfs as the sum of the $(1-p_{\rm H})$ probabilities. The uncertainties were again estimated by bootstrapping. We obtained $N_{\rm H}^{\rm phot} = 936 \pm 7$ and $N_{\rm He}^{\rm phot} = 282 \pm 7$ in the prior case, and $N_{\rm H}^{\rm phot} = 876 \pm 7$ and $N_{\rm He}^{\rm phot} = 342 \pm 7$ when the prior was not used. These results must be compared with the spectroscopic values $N_{\rm H}^{\rm spec} = 929$ and $N_{\rm He}^{\rm spec} = 289$. We find that the photometric and the spectroscopic numbers are compatible at 1$\sigma$ when the self-consistent prior is applied, but the discrepancies are at $7\sigma$ when the prior is neglected.
Finally, we tested the J-PLUS performance with hybrid types. There are $9$ sources classified as DAB and $26$ as DBA. We analyzed these $35$ hybrid types using the J-PLUS probabilities, obtaining $N_{\rm H}^{\rm phot} = 5 \pm 2$ and $N_{\rm He}^{\rm phot} = 30 \pm 2$ for H- and He-dominated atmospheres, respectively. The photometric values are compatible at $2\sigma$ with the spectroscopic ones. We are therefore able to obtain the main composition of the hybrid types and their presence in the white dwarf sample does not impact the classification obtained from J-PLUS data.
We conclude that the $p_{\rm H}$ derived from J-PLUS photometry is reliable and that the self-consistent prior in atmospheric composition is needed to properly recover the number of spectroscopic types. We can therefore use $p_{\rm H}$ to study the properties of H- and He-dominated white dwarfs, as illustrated in Section~\ref{sec:massdist} with the mass distribution.
\subsubsection{Color--color diagrams}\label{sec:color}
The H-dominated and He-dominated white dwarfs present two separate loci in broad-band color--color diagrams that contain the $u$ passband \citep{greenstein88}. For comparison with previous studies and to illustrate the performance of J-PLUS spectral classification, in this section we study the interstellar dust de-reddened $(u-r)_0$ versus $(g-i)_0$ color--color diagram as a function of $p_{\rm H}$ (Fig.~\ref{fig:urgi}). These plots show sources with ${\rm S/N} \geq 3$ in all J-PLUS passbands ($5\,379$ white dwarfs or 91\% of the total sample).
We find that sources with $p_{\rm H} \geq 0.5$ are clustered in the redder $(u - r)_0$ branch of the white dwarf locus at $(g-i)_0 < 0$, which corresponds to $T_{\rm eff} \gtrsim 8\,500$ K. In this temperature range, sources with $p_{\rm H} < 0.5$ are located in the bluer $(u - r)_0$ branch of the locus, as expected. At lower temperatures, the theoretical and observational colors of H- and He-dominated white dwarfs converge, making it impossible to disentangle the nature of the observed white dwarf only using broad-band optical colors. The addition of the seven J-PLUS medium-bands and the application of the self-consistent spectral type prior improves the classification below $T_{\rm eff} \sim 9\,000$~K.
We complement this analysis with the J-PLUS color--color diagram $(J0378-J0515)_0$ versus $(J0660 - r)_0$ in the {\it bottom panels} of Fig.~\ref{fig:urgi}. This diagram includes three J-PLUS medium bands and shows part of the additional information provided by the J-PLUS filter system, which helps to better discriminate between different spectral types. On the one hand, the $(J0660 - r)$ color is sensitive to the presence of H$\alpha$ absorption. On the other hand, the combination of a J-PLUS passband with $\lambda < 4\,000\ \AA$ and the $J0515$ passband enhance the contrasts between the two types of white dwarf. An additional analysis of the white dwarf population in the J-PLUS color--color diagrams can be found in \citet{clsj19jcal}.
\begin{figure*}[t]
\centering
\resizebox{0.49\hsize}{!}{\includegraphics{figures/mass_hist_cum0_lit0_da.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/mass_hist_cum0_lit0_db.pdf}}
\caption{White dwarf mass distribution at $d \leq 100$ pc for sources with $M > 0.45\, M_{\odot}$ and $T_{\rm eff} > 6\,000$ K. {\it Left panel}: Normalized histogram weighted by $p_{\rm H}$. {\it Right panel}: Normalized histogram weighted by $(1-p_{\rm H})$. The dotted line in both panels marks a mass of $M = 0.6\, M_{\odot}$ for reference.}
\label{fig:mfwhite dwarf}
\end{figure*}
\begin{figure*}[t]
\centering
\resizebox{0.49\hsize}{!}{\includegraphics{figures/mass_hist_cum0_lit1_da.pdf}}
\resizebox{0.49\hsize}{!}{\includegraphics{figures/mass_hist_cum0_lit2_da.pdf}}
\caption{Comparison between the H-dominated white dwarf mass distribution at $d \leq 100$ pc, $T_{\rm eff} > 6\,000$ K, and $M > 0.45\,M_{\odot}$ from J-PLUS (dashed histograms), that of \citet[][black solid line in the {\it left panel}]{kilic20}, and that of \citet[][red solid line in the {\it right panel}]{jimenezesteban18}.
}
\label{fig:mfwhite dwarflit}
\end{figure*}
These results illustrate the performance of J-PLUS photometry in a common color--color diagram, highlighting the range of colors, $(g-i)_0 \gtrsim 0$, at which the J-PLUS data and the PDF analysis provide an advantage over broad-band photometry.
\subsubsection{Mass distribution at $d \leq 100$ pc}\label{sec:massdist}
In this section, we estimate the stellar mass distribution of H- and He-dominated white dwarfs as a final control check for the quality of the $p_{\rm H}$ probabilities.
We restricted our sample to $d \leq 100$~pc, $T_{\rm eff} > 6\,000$~K, and $M > 0.45$~$M_{\odot}$. This selection allows us to directly compare the J-PLUS measurements with the results of \citet{jimenezesteban18} and those of \citet{kilic20}. The restricted sample in this section comprises 351 white dwarfs.
The mass distribution for H-dominated white dwarfs was estimated as the weighted histogram of the sample, where the weights were defined as
\begin{equation}
w = p_{\rm sel} \times p_{\rm H} \times V^{-1}_{\rm eff},
\end{equation}
and the effective volume $V_{\rm eff}$ refers to the $10 \leq d \leq 100$~pc range. The weights for the He-dominated distribution were similar, but were computed with $(1 - p_{\rm H}),$ using the corresponding effective volume for He-dominated sources. The obtained distributions for H-dominated and He-dominated types, normalized to one, are presented in Fig.~\ref{fig:mfwhite dwarf}. Adding the weights, we obtain $277$ H-dominated and $84$ He-dominated white dwarfs in the restricted sample.
We find a clear peak at $M = 0.59$~M$_{\odot}$ in the mass distribution of H-dominated white dwarfs, with a high-mass tail that peaks at $M \sim 0.8$~$M_{\odot}$. This high-mass excess has been reported in several studies \citep[e.g.,][]{liebert05,limoges15,rebassa15,tremblay16,tremblay19,jimenezesteban18,kilic20}. We compare the H-dominated mass distribution with the results from \citet{jimenezesteban18} and \citet{kilic20} in Fig.~\ref{fig:mfwhite dwarflit}. We find close agreement between our results and the distribution presented by \citet{kilic20}, including the location and the amplitude of their two suggested components. These latter authors have the spectroscopic type for the sources, and we obtained similar results using our photometric classification. This result further supports our Bayesian analysis and the reliability of the $p_{\rm H}$ probabilities.
The distribution from \citet{jimenezesteban18} presents an excess of sources at $M \sim 0.75$~$M_{\sun}$ with respect to the J-PLUS distribution. \citet{jimenezesteban18} use photometric information from the UV to the mid-infrared, but do not perform a spectral classification and assume that all the observed white dwarfs are H-dominated. They find that poor fittings are obtained for spectroscopically confirmed DBs and DCs, but this is only significant for the hotter systems, where broad bands can be use to discriminate between both types (see previous section). The contamination of cool ($T_{\rm eff} \lesssim 9\,000$~K) He-dominated white dwarfs that have typical masses of $M \sim 0.7-0.8$~$M_{\sun}$ when analyzed with pure-H models \citep{bergeron19} is a plausible explanation for the observed discrepancy.
We must mention the apparent excess of H-dominated white dwarfs with $M \gtrsim 1.2$~$M_{\odot}$ in J-PLUS. There are only six sources in the sample at this mass range, and four have $T_{\rm eff} < 10\,000$~K. Their high mass, coupled with a low effective temperature, produces a small effective volume that boosts their number density in Fig.~\ref{fig:mfwhite dwarf}. We note that our He-dominated models only reach $\log {\rm g} = 9$~dex, and so these massive white dwarfs can only be classified as H-dominated. We checked that the high mass was dictated by the parallax information, with three of our six sources being confirmed as high-mass white dwarfs by the spectroscopic follow up in \citet{kilic20}. An extra source is photometrically selected as a high-mass white dwarf candidate by \citet{kilic21}. Finally, five of our high-mass sources have $\log {\rm g} \geq 9$~dex in the \citet{jimenezesteban18} catalog.
We turn now to the mass distribution of He-dominated white dwarfs, as presented in the {\it right panel} of Fig.~\ref{fig:mfwhite dwarf}. We find a unique population located at $M = 0.62$~$M_{\odot}$. This is slightly more massive ($0.03$~$M_{\odot}$) than the primary peak for H-dominated white dwarfs, but we do not report this difference as representative. Our He-dominated models have a mixed composition with ${\rm H/He} = 10^{-5}$, as suggested by \citet{bergeron19} to reconcile the masses of DB/DCs with the masses of the DA population at $T_{\rm eff} < 11\,000$~K. We repeated the analysis with pure-He models, and the only difference that we see in the results is that the peak of the He-dominated distribution is translated to $M = 0.69$~$M_{\sun}$. That is, our results follow the discussion in \citet{bergeron19} and changes in the assumed He-dominated models will modify the location of the observed peak.
\begin{figure}[t]
\centering
\resizebox{\hsize}{!}{\includegraphics{figures/mass_hist_cum1_lit4_da.pdf}}
\caption{Cumulative mass distribution at $d \leq 100$ pc, $T_{\rm eff} > 6\,000$~K, and $0.45 < M < 1.2$~$M_{\odot}$ for H-dominated (orange histogram) and He-dominated (purple histogram) white dwarfs in J-PLUS. The colored areas show the 99\% confidence intervals in the measured distributions.}
\label{fig:mfwhite dwarfcum}
\end{figure}
Interestingly, it seems that the high-mass tail is absent in the distribution of He-dominated white dwarfs. To better illustrate this issue, the cumulative mass distributions for both types in the range $0.45 < M < 1.2$ $M_{\sun}$, including $99$\% confidence intervals estimated by bootstrapping, are presented in Fig.~\ref{fig:mfwhite dwarfcum}. The high-mass limit was imposed to avoid border effects in the comparison, as our He-dominated models were restricted to $\log {\rm g} \leq 9$~dex. There is a clear difference between both types at $M > 0.65$~$M_{\sun}$, where the excess of H-dominated white dwarfs with respect to the He-dominated distribution seems significant at a level of more than 99\%. We performed a two-sample Kolmogorov-Smirnov test, finding that the maximum difference between the cumulative curves, $D = 0.22$, corresponds to a 0.4\% probability that both distributions were extracted from the same parent population. That is equivalent to a $3\sigma$ significance in the observed difference. The lack of a high-mass tail in the He-dominated distribution has previously been reported using spectroscopic classifications \citep[e.g.,][]{bergeron01,tremblay19}, and we reproduce this result here using only optical photometric data.
We conclude that the statistical type classification from J-PLUS photometry is able to provide reliable mass distributions for H- and He-dominated white dwarfs.
\section{Summary and conclusions}\label{sec:conclusions}
We analyzed a sample of $5\,926$ white dwarfs with $r \leq 19.5$ mag in common between the {\it Gaia} EDR3 catalog from \citet{GF21} and J-PLUS DR2. We estimated the effective temperature, surface gravity, parallax, and atmospheric composition (H-dominated or He-dominated) of the sources with a Bayesian analysis. We used the parallax from {\it Gaia} EDR3 as prior, and derived a self-consistent prior for the atmospheric composition as a function of $T_{\rm eff}$ using J-PLUS photometric data alone. A way to access to the derived parameters is described in Appendix~\ref{app:data}.
We find that the fraction of He-dominated white dwarfs ($f_{\rm He}$) increases by $21 \pm 3$\% from $T_{\rm eff} = 20\,000$~K to $T_{\rm eff} = 5\,000$~K. We describe the fraction of He-dominated white dwarfs with a linear function of the effective temperature at $5\,000 \leq T_{\rm eff} \leq 30\,000\ {\rm K}$. We find $f_{\rm He} = 0.24 \pm 0.01$ at $T_{\rm eff} = 10\,000$~K, a change rate along the cooling sequence of $0.14 \pm 0.02$ per $10$~kK, and a minimum He-dominated fraction of $f_{\rm He}^{\rm min} = 0.08 \pm 0.02$ at the high-temperature end. The derived values of the He-dominated fraction are in agreement with previous results in the literature, where the observed spectral evolution is interpreted as the effect of convective mixing and convective dilution.
We tested the estimated probabilities of being H-dominated by comparison with the classification from spectroscopy. We find that the derived $p_{\rm H}$ provides the true probability of being H-dominated, so it can be used to obtain reliable probability-weighted distributions of the white dwarf population. We highlighted the last point by estimating the mass distribution at $d \leq 100$ pc and $T_{\rm eff} > 6\,000$ K for H- and He-dominated white dwarfs. Our findings for the H-dominated distribution resemble those of previous work, with a dominant $M = 0.59$~$M_{\odot}$ peak and the presence of a high-mass tail at $M \sim 0.8$~$M_{\odot}$. This high-mass excess is absent in the He-dominated distribution, which presents a single peak at $M \simeq 0.6$~$M_{\odot}$.
This work also provides hints about the capabilities of low-spectral-resolution data ($R \sim 50$) in the study of the white dwarf population. The future spectro-photometry from {\it Gaia} DR3 and the photo-spectra from the Javalambre Physics of the accelerating Universe Astrophysical Survey (J-PAS, 56 narrow-bands of 14 nm width in the optical over thousands of square degrees up to $m \sim 22.5$ mag; \citealt{jpas, minijpas}) coupled with the massive spectroscopic follow up from the William Herschel Telescope Enhanced Area Velocity Explorer (WEAVE; \citealt{weave}) will expand our knowledge of white dwarf spectral evolution with large and homogeneous data sets.
\begin{acknowledgements}
We dedicate this paper to the memory of our six IAC colleagues and friends who met with a fatal accident in Piedra de los Cochinos, Tenerife, in February 2007, with special thanks to Maurizio Panniello, whose teachings of \texttt{python} were so important for this paper.
We thank the discussions with the members of the J-PLUS collaboration, especially to A.~J.~Dom\'{\i}nguez-Fern\'andez. We thank the anonymous referee for useful comments and suggestions.
Based on observations made with the JAST80 telescope at the Observatorio Astrof\'{\i}sico de Javalambre (OAJ), in Teruel, owned, managed, and operated by the Centro de Estudios de F\'{\i}sica del Cosmos de Arag\'on. We acknowledge the OAJ Data Processing and Archiving Unit (UPAD, \citealt{upad}) for reducing and calibrating the OAJ data used in this work.
Funding for the J-PLUS Project has been provided by the Governments of Spain and Arag\'on through the Fondo de Inversiones de Teruel; the Aragonese Government through the Reseach Groups E96, E103, and E16\_17R; the Spanish Ministry of Science, Innovation and Universities (MCIU/AEI/FEDER, UE) with grants PGC2018-097585-B-C21 and PGC2018-097585-B-C22; the Spanish Ministry of Economy and Competitiveness (MINECO) under AYA2015-66211-C2-1-P, AYA2015-66211-C2-2, AYA2012-30789, and ICTS-2009-14; and European FEDER funding (FCDD10-4E-867, FCDD13-4E-2685). The Brazilian agencies FINEP, FAPESP, and the National Observatory of Brazil have also contributed to this project.
P.~-E.~T. has received funding from the European Research Council under the European Union's Horizon 2020 research and innovation programme n. 677706 (white dwarf3D).
J.~M.~C. acknowledge financial support by the Spanish Ministry of Science, Innovation and University (MICIU/FEDER, UE) through grant RTI2018-095076-B-C21, and the Institute of Cosmos Sciences University of Barcelona (ICCUB, Unidad de Excelencia ’Mar\'{\i}a de Maeztu’) through grant CEX2019-000918-M.
J.~V. acknowledges the technical members of the UPAD for their invaluable work: Juan Castillo, Javier Hern\'andez, \'Angel L\'opez, Alberto Moreno, and David Muniesa.
B.~T.~G. was supported by STFC grantST/T000406/1 and by a Leverhulme
Research Fellowship.
F.~J.~E. acknowledges financial support from the Spanish MINECO/FEDER
through the grant AYA2017-84089 and MDM-2017-0737 at Centro de
Astrobiolog\'{\i}a (CSIC-INTA), Unidad de Excelencia Mar\'{\i}a de Maeztu, and
from the European Union’s Horizon 2020 research and innovation programme
under Grant Agreement no. 824064 through the ESCAPE - The European
Science Cluster of Astronomy \& Particle Physics ESFRI Research
Infrastructures project.
R.~A.~D. acknowledges support from the CNPq through BP grant 308105/2018-4.
This work has made use of data from the European Space Agency (ESA) mission
{\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement.
We also used the data from the SVO archive of White Dwarfs from {\it Gaia} (\url{http://svo2.cab.inta-csic.es/vocats/v2/wdw/}) at CAB (INTA-CSIC).
Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is \url{http://www.sdss3.org/}.
SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High Performance Computing at the University of Utah. The SDSS website is \url{www.sdss.org}. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard \& Smithsonian, the Chilean Participation Group, the French Participation Group, Instituto de Astrof\'isica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut f\"ur Astrophysik Potsdam (AIP), Max-Planck-Institut f\"ur Astronomie (MPIA Heidelberg), Max-Planck-Institut f\"ur Astrophysik (MPA Garching), Max-Planck-Institut f\"ur Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observat\'ario Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Aut\'onoma de M\'exico, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.
This research made use of \texttt{Astropy}, a community-developed core \texttt{Python} package for Astronomy \citep{astropy}, and \texttt{Matplotlib}, a 2D graphics package used for \texttt{Python} for publication-quality image generation across user interfaces and operating systems \citep{pylab}.
\end{acknowledgements}
\bibliographystyle{aa}
|
1,314,259,996,493 | arxiv | \section{Introduction}
More than a decade after the prediction of the $\eta$-mesic nuclei
\cite{hailiu}, their existence remains a matter of interesting debate
among nuclear physicists. The root of the debate lies
in the uncertainty in the knowledge of the elementary eta-nucleon
($\eta N$) interaction. Though once again, the first prediction of an
attractive $\eta N$ interaction \cite{bhale} (arising basically due
to the proximity of the $\eta N$ threshold to the $N^* (1535)$ resonance)
was made back in 1985, an agreement on the magnitude of the attraction
has not been reached.
Though most of the
models are constrained by the same sets of data on $\pi N$ elastic scattering
and the $\pi N \rightarrow \eta N$ cross sections, the $\eta N$ scattering
length predicted by different models is very different
due to the unavailability of direct
experimental information on $\eta N$ elastic scattering. Consequently, the
predictions of possible resonances or unstable bound states of $\eta$ mesons
and nuclei within these models also vary a lot.
In the present article we refer to $\eta$-nucleus states with a negative
binding energy but a finite lifetime (width) as ``unstable bound states".
These states are sometimes also called ``quasibound" states in literature.
Even if the present knowledge of the $\eta N$ interaction is somewhat poor,
the growing experimental efforts in the past few years could improve the
understanding in a not so distant future.
The photoproduction of $\eta$-mesic $^3$He investigated by the TAPS
collaboration \cite{pfieff}, has indeed proved to be a step forward in this
direction. The total inclusive cross section for the $\gamma \,^3$He
$\rightarrow \eta X$ reaction was measured at the Mainz Microtron accelerator
facility using the TAPS calorimeter and an unstable bound state with a
binding energy of $-4.4 \pm 4.2$ MeV was reported.
Certain evidence for the existence of the $\eta$-mesic nucleus $^{11}_{\eta}C$,
was reported in \cite{sokol} and a claim for the light $\eta^4$He
quasi-bound state was made in \cite{willis} from the study of the cross
section and tensor analysing power in the $\vec{d} d \rightarrow \,
^4$He$\,\eta$ reaction. Indirect evidence of the strong $\eta$-nucleus
attraction was also obtained from data on $\eta$ production in the $p d
\rightarrow \,^3$He$\,\eta$ \cite{mayer} and $n p \rightarrow d \eta$
\cite{npdata} reactions which display large enhancements in the cross
sections near threshold. More information is expected to be available
from the ongoing program of the COSY-GEM collaboration \cite{cosygem}.
Since the $\eta^3$He states which we recently predicted
\cite{wejphysg} from time delay were found to be in agreement with
the TAPS data, we thought it worthwhile to extend the calculations
and search for eta-mesic states in two other light nuclei.
In contrast
to conventional methods of resonance extraction like Argand diagrams and
poles of the $S$-matrix in the complex energy plane, the time delay method
has a physical meaning which was noticed more than 50 years ago by
Eisenbud \cite{eisen} and Wigner \cite{wigner1}. The fact
that a large positive time delay in a scattering process is associated
with the formation of a resonance is now mentioned and elaborately
discussed in most standard text books \cite{books}. This time delay is
related to the energy derivative of scattering phase shifts and can be easily
calculated. However, strangely but not surprisingly, this well-documented
method had rarely been used in literature to extract information on
resonances from data until recently when the authors of the present work
put it to a test with hadron-hadron elastic scattering
\cite{we3KN,we3pi,me,we4}. Being encouraged by the fact that this method was
successful in characterizing known nucleonic resonances \cite{we4}, could find
some evidence for penta-quark baryons \cite{we3KN} and revealed all known
meson resonances \cite{we3pi}, we carried out a similar study \cite{wejphysg}
for the $\eta^3$He system which also proved quite fruitful.
Hence, in the present work we shall search for $\eta$d and $\eta^4$He
unstable states using the time delay method.
The transition matrix for $\eta$-nucleus scattering (which is
required as an input to the calculation of time delay) is
constructed using few body equations and accounts
for off-shell re-scattering and
nuclear binding energy effects.
This model was successful \cite{umebkj}
in showing that the enhancement in the cross sections
(or the strong energy dependence of the scattering amplitude near threshold)
of the $p d \rightarrow \,^3$He$\,\eta$ reaction is due to the
$\eta\,^3$He final state interaction.
Since the calculations are purely theoretical, we have extended
the concept of time delay to negative energies too. In the following
sections, we shall demonstrate the validity and usefulness of this
concept in characterizing bound and virtual states occurring at negative
energies.
There exist scores of papers in literature, with predictions of
unstable bound (sometimes called quasi-bound) states of
$\eta$-mesic nuclei as well as $\eta$-nucleus resonances.
One can find a detailed account of the
existing literature in a recent work \cite{hailiu2}, where a search for
unstable $\eta$-mesic nuclei ranging from $A=3$ to $208$ was made. Here, we
shall briefly survey the literature on light nuclei, since we restrict
ourselves to the study of $\eta$d and $\eta^4$He elastic
scattering in the present work. One of the early calculations \cite{ueda}
using the 3-body equation, predicted a resonance with a mass of
2430 MeV and a width of 10-20 MeV in the $\pi N N$-$\eta N N$ coupled system.
This state led to a remarkable enhancement of the $\eta d$ elastic
cross section. In some other Faddeev-type calculations
\cite{shev,shev2} of $\eta d$ scattering, a resonance
at low energies was predicted. The exact values of the
predicted mass and the width of course varied with the value of the
$\eta$-nucleon
scattering length, $a_{\eta N}$, which in turn depends on the model of
the $\eta$-nucleon interaction. The existence of resonances in the above
works was inferred from Argand diagrams. Some other recent
(also Faddeev-type) calculations, however, ruled out the possibility of a
resonance in the $\eta d$ system \cite{gpena,delof,fix}. Predictions
of resonances in the light $\eta$-nucleus systems, namely, $d$, $^3$He and
$^4$He, from the positions and movements of the poles of the amplitude, can
be found in \cite{rakit96}. Finally to mention some predictions of
`virtual' states, a virtual (anti-bound) s-wave $\eta ^3$H state, which led
to a large enhancement of the cross section for $\eta$ production from the
three-body nuclei was found in \cite{Fix3N}, whereas a narrow virtual state
in the $\eta d$ system was found \cite{wycech} to have a rather weak effect on
the $p n \rightarrow \eta d$ cross sections.
In this work, we shall infer the existence of unstable states in the $\eta$d
and $\eta ^4$He systems, from positive peaks in time delay and
compare our results with existing predictions in literature using other
methods. In the next section, we present the basic elements of the ``time
delay" method and the characteristics expected
from it in the case of resonances, bound, quasi-bound, virtual and
quasi-virtual states.
Since one can find thorough discussions of
time delay in literature \cite{alltdpapers,smith},
and also in standard text-books
on quantum mechanics \cite{books,brans}, we do not perform a review of this
method here. However, the efficacy of the inferences from this method
is established by applying it to the known case of neutron-proton bound
and virtual states and referring to our earlier application to the known
hadron resonances. Section 3 is devoted to a brief discussion of the
few body equations involving the finite rank approximation (FRA) which we
use to calculate the $\eta$-nucleus transition matrix. This $t$-matrix is
subsequently used to evaluate the time delay in $\eta$-light nucleus
scattering. Since the applicability of the FRA for the $\eta d$ case
is limited (it was shown in \cite{shev2} that FRA agrees with the
Faddeev calculations for real part
of the scattering lengths less than $0.5$ fm only), we use a
parameterization of the $\eta$ d $t$-matrix using
relativistic Faddeev equations and present the
results of this calculation in Sec. 4. The time delay results for the
$\eta ^4$He system are given in Sec. 5.
The FRA calculations are done using two different models of the $\eta N$
interaction (constructed within coupled channel formalisms) which fit the
same set of data on pion scattering and pion induced $\eta$ production on a
nucleon, but give different values of the $\eta N$ scattering length.
Finally, Sec. 6
summarizes the findings of the present work.
\section{Time delay plots of bound, virtual and decaying states}
The collision time or delay time in scattering processes, was quantified
by Eisenbud and Wigner in terms of the observable phase shift
which can be extracted from cross section
data. In \cite{wigner1}, Wigner pictured the resonance formation in
elastic scattering as the capture and retention of the incident particle
for some time by the scattering centre, introducing thereby a time delay
in the emergence of the outgoing particles.
He further showed that the energy derivative of
the phase shift, $\delta$, which is related to the time delay,
$\Delta t(E)$, as
\begin{equation}\label{1}
\Delta t(E) = 2 \hbar {d \delta \over dE}
\end{equation}
and is large and positive close to resonances, can also take negative values
which are however limited from causality constraints.
In the presence of inelasticities, a one
to one correspondence between time delay and the lifetime of a resonance
does not hold and a more useful definition, namely, the time delay matrix
(later discussed in terms of a lifetime matrix by Smith \cite{smith}) was
given by Eisenbud. An element of this matrix, $\Delta t_{ij}$, which is the
time delay in the emergence of a particle in the $j^{th}$ channel after
being injected in the $i^{th}$ channel is given by,
\begin{equation}\label{2}
\Delta t_{ij} = \Re e \biggl [ -i \hbar (S_{ij})^{-1} {dS_{ij} \over dE}
\biggr ] \, ,
\end{equation}
where $S_{ij}$ is an element of the corresponding $S$-matrix.
Writing the $S$-matrix in terms of the $T$-matrix as,
\begin{equation}\label{6}
{\bf S} = 1 + 2\,i\,{\bf T}\, ,
\end{equation}
one can evaluate time delay in terms of the $T$-matrix.
The time delay in elastic scattering, i.e. $\Delta t_{ii}$, is given in terms
of the $T$-matrix as,
\begin{equation}\label{7}
S^*_{ii} \,S_{ii}\, \Delta t_{ii}\, =\, 2 \,\hbar\,
\biggl[ \Re e \biggl({dT_{ii} \over dE}\biggr)\,+ \,2 \,\Re e T_{ii}\,\,
\Im m \biggl ({dT_{ii} \over dE}\biggr) \,-\, 2\, \Im m T_{ii}\,\,
\Re e\biggl( {dT_{ii} \over dE}\biggr)\,
\biggr],
\end{equation}
where {\bf T} is the complex $T$-matrix such that,
\begin{equation} \label{7a}
T_{kj} = \Re e T_{kj} \,+ \,i\,\Im m T_{kj}\,.
\end{equation}
The time delay evaluated from eqs (\ref{1}) and (\ref{7}) is just the same.
The above relations were put
to a test in \cite{we3KN,we3pi,me,we4} to characterize the hadron resonances
occurring in meson-nucleon and meson-meson elastic scattering. The energy
distribution of the time delay evaluated in these works, nicely displayed
the known $N$ and $\Delta$ baryons, meson resonances like the
$\rho$, the scalars ($f_0$) and strange $K^*$'s found in $K \pi$ scattering,
in addition to confirming
some old claims of exotic states. The peaks in time delay, $\Delta t(E)$,
agreed well with the known resonance masses. It was also shown that the
time delay peaks and the $T$-matrix poles essentially contain the same
information. A theoretical discussion on this issue can be found in
\cite{brans,peres}.
The time delay concept is not only useful to locate resonances, but can
also be used to locate the bound, virtual and unstable bound states
which have negative binding energies.
We illustrate this assertion with the well-known case of the $n-p$ system.
The $S$-matrix for the neutron-proton system, constructed from a square
well potential which produces the correct binding
energy of the deuteron is given as a function of $l$ as,
\begin{equation}
S_l = - {\alpha h_l^{(2)'}(\alpha) j_l(\beta) - \beta h_l^{(2)}(\alpha)
j_l{'}(\beta) \over
\alpha h_l^{(1)'}(\alpha) j_l(\beta) - \beta h_l^{(1)}(\alpha) j_l^{'}(\beta)}
\end{equation}
where $j_l$, $h_l^{(1)}$ and $h_l^{(2)}$ are the spherical Bessel and Hankel
functions of the first and second kind respectively.
\begin{equation}
\alpha = k R \,\,\,\,\,{\rm and}\,\,\,\,\,
\beta = (\alpha^2 - 2 \mu U R^2/\hbar^2)^{1/2}
\end{equation}
where the potential $U$ is given by
\begin{equation}
U = V + i W , \,\,\,\,\, U(r) = U \theta (R-r)
\end{equation}
and $R$ is the width of the potential well.
A similar square well potential was used by Morimatsu and Yazaki \cite{mori}
while locating the ``unstable bound states" of $\Sigma$-hypernuclei as
second quadrant poles and by J. Fraxedas and J. Sesma using the time
delay method \cite{frax}.
\begin{figure}[ht]
\centerline{\vbox{
\psfig{file=figur1.eps,height=9cm,width=6cm}}}
\caption{The theoretical delay time in $n p$
elastic scattering as a function of the energy $E = \sqrt s - m_n - m_p$,
where $\sqrt s$ is the total energy available in the $n p$ centre of
mass system.
}
\end{figure}
We evaluate the time delay in $n-p$ scattering at negative energies $E$,
where $E = \sqrt s - m_n - m_p$ with $\sqrt s$ being the energy available
in the $n p$ centre of mass system.
Using the above $S$-matrix with $\alpha = i k R$ (hence $E = -k^2/2\mu$),
$l=0$ and the appropriate parameters for an $n-p$
square well potential, namely, $V = 34.6 MeV$ and $R = 2.07$ fm, the time delay
plot (as in Fig. 1) shows a sharp spike (positive infinite time delay)
exactly at the
binding energy of the deuteron ($E=-2.224$ MeV). If we add a small
imaginary part to the potential, then of course there is a
Breit-Wigner kind of distribution centered around the binding energy
of the deuteron (a fictitious ``unstable bound state" of the $n-p$
system at $E = -2.224$ MeV).
The correlation between the potential
parameters and the position of the spike at the correct deuteron
binding energy is very definite. If we take the potential parameters
which do not give the correct binding energy, then the spike appears
at a wrong place in the time delay plot.
In Fig. 2, we demonstrate this sensitivity of time delay.
In the upper half of the figure, we plot time
delay calculated using a fixed well depth of $V = 34.535$ MeV and
different choices for the width of the square well.
It can be seen that the position of the spike is very sensitive to
the value of $R$. The spike at the right binding energy is produced
only with $R=2.07$ fm.
In the lower half of the plot, we perform a
similar calculation but this time with the well width fixed and
the well depth changing. Again, a small change in the well depth
parameter, changes the energy at which the positive infinite time
delay appears.
\begin{figure}[ht]
\centerline{\vbox{
\psfig{file=figur2a.eps,height=5cm,width=6.5cm}
\psfig{file=figur2b.eps,height=5cm,width=6.5cm}}}
\caption{\label{fig:epsart2} Sensitivity of time delay in $n p$
scattering to the square well parameters. $V$ is the depth of the
square well potential and $R$ its width. The solid lines
indicate the sharp positive infinite time delay at the correct deuteron binding
energy.}
\end{figure}
Beyond quasibound states, an $S$-matrix pole in the third quadrant
of the complex momentum plane corresponds to a quasivirtual state,
which translates to a pole on the unphysical sheet
of the complex energy plane of the type $-|E| + i |\Gamma/2|$.
This, in contrast to a resonance pole of $|E| - i |\Gamma/2|$ (which
leads to an exponentially decaying state with a decay law of
$e^{-\Gamma t}$), gives rise to an exponential
growth, namely $e^{+\Gamma t}$. One can then see that in contrast to
the time ``delay" that one observes for a resonance, one would observe
a time ``advancement" for a quasivirtual state. In other words, for a
quasivirtual state one observes a finite `negative' time delay.
Similarly, a virtual state, in contrast to a bound state would
show an infinite negative time delay. This is indeed seen in Fig. 1 for
the known $n-p$ virtual state at $100$ keV, where the time delay
calculated from the square well potential with parameters corresponding
to this virtual state (namely, $V = 23.6$ MeV and $R = 2$ fm) is
plotted.
\section{$T$-matrix for $\eta$-nucleus elastic scattering}
We evaluate the transition matrix for $\eta$-nucleus ($\eta A$)
elastic scattering, using few body equations for the $\eta(2N)$
and $\eta(4N)$ systems. The calculation is done within a Finite Rank
Approximation (FRA) approach, which means that in the intermediate state,
the nucleus in $\eta A$ elastic
scattering remains in its ground state. Since the $\eta$-mesic bound states
and resonances are basically low energy phenomena, it seems justified to use
the FRA for calculations of the present work.
In \cite{shev2}, the authors mention that though
the use of FRA for $\eta^3$He and $\eta^4$He systems seems justified, it is
questionable for the case of $\eta d$ scattering and investigate
the shortcoming due to the neglect of excitations of the nuclear ground state
in $\eta$-deuteron calculations. Within their model, they find that the
FRA results differ from those evaluated using the rigorous
Alt-Grassberger-Sandhas (AGS) equations in the case of the strong $\eta N$
interaction (Re $a_{\eta N} > 0.5$ fm), while for small values of
Re $a_{\eta N}$, the FRA is reasonably good.
Therefore, for the $\eta$-d case, we also include calculations using
the results from the recent relativistic Faddeev equation (RFE) calculations
for the $\eta N N$ system.
The target Hamiltonian $H_A$, in the FRA is written as \cite{bela},
\begin{equation}\label{hamilt}
H_A \approx \varepsilon |\psi_0> <\psi_0|
\end{equation}
where $\psi_0$ is the nuclear ground state wave function and
$\varepsilon$
the binding energy.
The $\eta A$ $T$-matrix in the FRA is given as \cite{rakit96,bela,rakit1},
\begin{eqnarray}\label{tfsi}
t_{\eta A}(\vec{k^\prime},\, \vec{k}\,; z) &=& <\, \vec {k^\prime}\, ; \,
\psi_0\,|\, t^0(z)
\, | \, \vec{k} \, ; \, \psi_0\,> \, +\, \\ \nonumber
&&\varepsilon\, \int {\vec{dk^{\prime\prime}}
\over (2\pi)^3} {<\,\vec{k^\prime}\, ; \, \psi_0 \,|\, t^0(z)\, | \,
\vec{k^{\prime\prime}}\,
; \, \psi_0\,> \over (z - {k^{\prime\prime\,2} \over 2\mu})(z -
\varepsilon
- {k^{\prime\prime\,2} \over 2\mu})}
t_{\eta A}(\vec{k^{\prime\prime}},\, \vec{k}\, ; \, z)
\end{eqnarray}
where $z = E - |\varepsilon| + i0$. $E$ is the energy associated with
$\eta A$ relative motion, $\varepsilon$ is the binding energy of the
nucleus and $\mu$ is the reduced mass of the
$\eta A$ system. Though the operator $t^0$ describes the
scattering of the $\eta$ meson from nucleons fixed in their space position
within the nucleus, it differs from the usual fixed center $t$-matrices.
Here, $t^0$ is taken off the energy shell and involves the
motion of the $\eta$ meson with respect to the center of mass of the
target. The present scheme should not be confused with a conventional
optical potential approach which involves the impulse approximation
and omits the re-scattering of the $\eta$ meson from the nucleons.
The matrix elements for $t^0$ are given as,
\begin{equation}\label{t0mat}
<\, \vec{k^\prime} \, ; \, \psi_0\,|\, t^0(z)\, |\, \vec {k} \, ; \,
\psi_0\,> =
\int d\vec{r}\, |\, \psi_0(\vec{r})\, |^2 \, t^0\, (\vec{k^\prime},\,
\vec{k}\,;
\vec{r}\,;z)
\end{equation}
where,
\begin{equation}\label{t0mat2}
t^0\,(\vec{k^\prime},\, \vec{k} \,;\vec{r} \,;z) = \sum_{i=1}^A \,
t_i^0\,
(\vec{k^\prime},\, \vec{k}\,;\vec{r_i}\,;z)
\end{equation}
$t_i^0$ is the t-matrix for the scattering of the $\eta$-meson from the
$i^{th}$ nucleon in the nucleus, with the re-scattering
from the other (A-1) nucleons included. It is given as,
\begin{equation}\label{t0mat3}
t_i^0\,(\vec{k^\prime},\, \vec{k}\,;\vec{r_i}\,;z) =
t_i^{\eta N}(\vec{k^\prime},\, \vec{k}\,;\vec{r_i}\,;z) + \int
{d\vec{k^{\prime\prime}}
\over (2\pi)^3}\,{t_i^{\eta N}(\vec{k^\prime},\,
\vec{k^{\prime\prime}}\,;\vec{r_i}\,;z)
\over z - {k^{\prime\prime\,2} \over 2\mu}} \sum_{j\neq i}
t_j^0(\vec{k^{\prime\prime}},\, \vec{k}\,;\vec{r_j}\,;z)
\end{equation}
The t-matrix for elementary $\eta$-nucleon scattering, $t_i^{\eta N}$,
is written in terms of the two body $\eta N$ matrix
$t_{\eta N \rightarrow \eta N}$ as,
\begin{equation}\label{tetan}
t_i^{\eta N}(\vec{k^\prime},\, \vec{k}\,;\vec{r_i}\,;z) =
t_{\eta\,N \rightarrow \eta N}(\vec{k^\prime},\,
\vec{k}\,;z)\, exp [\,i (\, \vec{k} -
\vec{k^\prime}\,)\cdot\,\vec{r_i}\,]
\end{equation}
The $^4$He nuclear wave function, required in the
calculation of the $T$-matrix is taken to be of the Gaussian form.
The deuteron wave function is written using a parametrization of
the wave function \cite{paris} obtained using the Paris potential. The results
using the Paris potential are also compared with a calculation using a
Gaussian form of the deuteron wave function.
As mentioned in the introduction, there exists a lot of uncertainty
in the knowledge of the $\eta$-nucleon interaction and hence,
we use different prescriptions of
the $\eta$-N t-matrix, t$_{\eta \, N \, \rightarrow \, \eta \, N}$,
leading to different values of the $\eta N$ scattering length. We give
a brief description of two of these models of
t$_{\eta \, N \, \rightarrow \, \eta \, N}$ which we use for the FRA
calculations below. In
\cite{fix} a coupled channel t-matrix including the $\pi$N and
$\eta$N channels with the S$_{11}$ - $\eta$N interaction playing a
dominant role was constructed. The t-matrix thus consisted of the
meson - N* vertices and the N* propagator as given below:
\begin{equation}\label{tfix}
t_{\eta \, N \, \rightarrow \, \eta \, N} (\, k^\prime, \, k; z) =
{ { \rm g}_{_{N^*}}\beta^2 \over (k^{\prime\,2} +
\beta^2)}\,\tau_{_{N^*}}(z)\,{ {\rm g}_{_{N^*}}\beta^2 \over (k^2 + \beta^2)}
\end{equation}
with,
\begin{equation}\label{tau}
\tau_{_{N^*}}(z) = ( \, z - M_0- \Sigma_\pi(z) -
\Sigma_\eta(z) + i\epsilon \, )^{-1}
\end{equation}
where $\Sigma_\alpha(z)$ $(\alpha = \pi, \eta)$ are the self energy
contributions from the $\pi N$ and $\eta N$ loops.
We choose the parameter set with ${\rm g}_{_{N^*}} = 2.13, \,\beta = 13 \,{\rm fm}^{-1},\,\,\,{\rm and}
\,\,\,M_0 = 1656 \,{\rm MeV} $ which leads to $a_{\eta N}$ = (0.88, 0.41) fm.
We also present results using one of the earliest calculations of
the $\eta$-N t-matrix \cite{bhale} which gives
a much smaller value of the scattering length,
namely, $a_{\eta N}$ = (0.28, 0.19) fm.
In this model,
the $\pi$N, $\eta$N and $\pi \Delta$ ($\pi \pi N$) channels were treated
in a coupled channel formalism (so that an additional self-energy
term appears in the propagator in Eq. (\ref{tau})).
The parameters of this model are,
$ { \rm g}_{_{N^*}} = 0.616, \,\beta = 2.36 \,{\rm fm}^{-1},\,\,\,{\rm and}
\,\,\,M_0 = 1608.1 \,{\rm MeV}.$
There also exists a recent model of the $\eta N$ interaction
\cite{green2}, which
predicts a scattering length of $a_{\eta N} = (0.91, 0.27)$ fm, from a
fit to the $\pi N \to \pi N$, $\pi N \to \eta N$, $\gamma N \to \pi N$
and $\gamma N \to \eta N$ data. However, we have not used it for our present
FRA calculations since the $T$-matrix which fits the data very well is
an on-shell $T$-matrix.
The off-shell separable form given by the authors \cite{green2}
agrees with their on-shell $T$-matrix (which fits data) but does
not include the intermediate off shell $\pi$ and $\eta$ loops.
The off shell nature appears only in the vertex form factors.
The $T$-matrix for $\eta A$ elastic scattering, $t_{\eta A}$, is related to
the $S$-matrix as,
\begin{equation}\label{smat1}
S\,=\, 1 \,-\, {\mu \,i\,k \over \pi}\,t_{\eta A}
\end{equation}
where $k$ is the momentum in the $\eta A$ centre of mass system
and hence, the dimensionless $T$-matrix required in the evaluation of
time delay as given in eq. (\ref{7}) is evaluated using the relation,
\begin{equation}\label{ttd}
T\,=\,-\,{\mu \,k \over \,2\,\pi} \,t_{\eta A}
\end{equation}
We shall present the time delay plots for $\eta$d and
$\eta\,^4$He elastic scattering in the next sections.
\section{The $\eta$ deuteron system}
We make an analytic continuation of the $T$ matrix for $\eta$-nucleus
elastic scattering on to the complex energy plane. Evaluating the
matrix elements of the $\eta$-nucleus $T$-matrix at
negative energies (corresponding
to purely imaginary momentum), i.e.
$t_{\eta A}(\vec{ik},\, i\vec{k}\,; z)$, we evaluate the
time delay in $\eta$-nucleus elastic scattering and
search for the ``unstable bound states".
The resonances at positive energies are of course determined from
the positive time delay peaks at positive energies and real momenta.
\begin{figure}[ht]
\centerline{\vbox{
\psfig{file=figur3c.eps,height=8cm,width=6cm}}}
\caption{The delay time in $\eta$-deuteron
elastic scattering as a function of the energy $E = \sqrt s - m_{\eta} - m_d$,
where $\sqrt s$ is the total energy available in the $\eta d$ centre of
mass system. The shaded curves are calculations using the Gaussian form
of the deuteron wave function and dashed lines are evaluated using the
Paris deuteron wave function. The vertical axis scale is broken in order
to display the structure in the time delay plot clearly.}
\end{figure}
In Fig. 3, we plot the time delay in $\eta$ d $\rightarrow \eta$ d
elastic scattering with two different inputs for the elementary
$\eta N$ interaction.
The $\eta N$ scattering lengths of
$a_{\eta N} = (0.88, 0.41)$ fm and $a_{\eta N} = (0.28, 0.19)$ fm
correspond to $\eta$d scattering lengths of
$a_{\eta d} = (1.52, 2.57)$ fm and $a_{\eta d} = (0.67, 0.42)$ fm
respectively.
In both cases, we see a large positive time delay
located near threshold. The solid lines (with shaded regions)
are the calculations using a Gaussian form and the dashed lines
with a Paris potential parametrization of the deuteron wave function.
The vertical axis scale in Fig. 3 is broken in order
to display the structure in the time delay plot clearly.
We do not find a big difference in the results with the change of
the wave function.
In the case of the weaker $\eta N$ interaction, i.e.
$a_{\eta N} = (0.28,0.19)$ fm, there appears a very broad bump
around $-15$ MeV which could be due to an unstable bound
state.
On the positive energy side, there is a sharp negative time delay when
the kinetic energy
of the $\eta$-d system equals the binding energy of the deuteron.
This behaviour is expected because of the connection of the
energy derivative of the phase shift and hence the time delay
(\ref{1}) to the density of states as given by the
Beth-Uhlenbeck formula \cite{uhlen}.
It was shown in \cite{me} that a maximum negative time delay occurs at
the opening of an inelastic threshold.
In the present case, the negative dip around $2.22$ MeV, corresponds to
the break up threshold of the deuteron.
We also see a resonance just near this inelastic threshold.
However, the above result near the inelastic threshold should be taken
with some caution since we have used the FRA which might not be a very
good approximation at energies where new thresholds open up.
\begin{figure}
\centerline{\vbox{
\psfig{file=figur3a.eps,height=7cm,width=7cm}}}
\caption{Quasivirtual states of the $\eta d$ system
evaluated from a Faddeev calculation for $\eta d$ scattering. The various
model numbers correspond to the different strengths of the $\eta N$
interaction as explained in \cite{garcil}.}
\end{figure}
\begin{figure}
\centerline{\vbox{
\psfig{file=figur3b.eps,height=7cm,width=7cm}}}
\caption{Quasivirtual and quasibound states
of the $\eta d$ system
evaluated from a Faddeev calculation for $\eta d$ scattering using
model 0 as mentioned in \cite{garcil}.}
\end{figure}
In order to check the validity of the above FRA approach
for the $\eta d$ case (where it is known to have limitations \cite{shev2}),
we evaluate time delay
using a model \cite{garcil} which obtains the $\eta d$ elastic
scattering amplitude using a relativistic version of the Faddeev
equations described in \cite{garcil2}. The $\eta d$ amplitude in
\cite{garcil} is parametrized using the effective range formula:
\begin{equation}
f^{-1}_{\eta d} = {1 \over A_{\eta d}} \, + \, {1 \over 2}
\, R_{\eta d} \, k^2 \, + \, S_{\eta d}\, k^4\, -i\,k
\end{equation}
where $k$ is the momentum in the $\eta d$ centre of mass system
and the parameters $A_{\eta d}$, $S_{\eta d}$ and $R_{\eta d}$ are
as given in Table II of \cite{garcil}.
In Figs 4 and 5, we show the results obtained using the above
amplitude. The time delay in Fig. 4 has been evaluated using
$k \to -ik$ and hence the negative dips in this figure correspond
to the quasivirtual states which appear as poles
in the third quadrant of the complex k-plane. The locations of
these dips (using different models as given in \cite{garcil})
are exactly at the energy pole values given in Table III
of \cite{garcil} as expected. As explained already in Sec. II,
such a negative time delay (or time advancement) is expected for
quasivirtual states which give an exponential rise rather than an
exponential decay law.
\begin{figure}
\centerline{\vbox{
\psfig{file=spol.eps,height=8cm,width=12cm}}}
\caption{Magnitude of the complex amplitude
$S= 1+ 2ikf$ in the complex momentum plane
evaluated from a Faddeev calculation for $\eta d$ scattering using
model 0 as mentioned in \cite{garcil}. The sharp pole corresponds to
the quasibound state at $-16$ MeV as seen in the time delay plot in Fig. 5.}
\end{figure}
In the upper half of Fig. 5, we plot the time delay evaluated using
$k \to -ik$ and the lower half shows the time delay plot
corresponding to $k \to +ik$ in the amplitude with Model 0.
Thus the negative
dip in the upper half is the quasivirtual state as also given in
Table II of \cite{garcil} and the lower half shows the positive
time delay corresponding to a quasibound or what we address as
an ``unstable bound" state in the present work. In order to
demonstrate once again the one-to-one correspondence between the
time delay peak and $S$-matrix poles, in Fig. 6 we plot the
magnitude of $S$ as a function of the real and imaginary parts
of momentum $k$. The pole occurs at ($-0.283 + i0.674$) fm
in the complex momentum plane. This corresponds to a peak value
of about $-17$ MeV and a width of $35$ MeV which is in good agreement
with the time delay plot in Fig. 5.
Such a pole has however not been mentioned by the author in
\cite{garcil} (probably due to the fact that the author in \cite{garcil}
has been mostly concerned about the effect of the near threshold
quasivirtual states on the $n p \rightarrow d \eta$ reaction).
We do not find any such positive peaks in any of the models shown
in Fig. 4. It is interesting to note that as expected from \cite{shev2},
the FRA has some agreement with the Faddeev calculation for small
$\eta N$ scattering lengths, $a_{\eta N}$,
and the discrepancy increases for large $a_{\eta N}$.
Finally, we wish to caution the reader regarding
the interpretation of time delay peaks in the case of $s$-wave scattering.
To see this, substituting the phase shift expression,
$S = exp(2i\delta)$ and comparing it with (\ref{smat1}), one
can write,
\begin{equation}
\delta = {1 \over 2i} \, {\rm ln}(1 - {i \mu k \over \pi} t_{\eta A}) \, = \,
{1 \over 2i} \, {\rm ln}(1 + 2 i k f)
\end{equation}
where $f$ is the scattering amplitude. For small $k$, $\delta \simeq k f$ and
the behaviour of $d\delta/dE$ (the real part of
which is essentially the time delay) is
determined by the simple pole at $k=0$ (or $E_{\eta A} = E_{threshold}$) and
the energy dependence of the scattering amplitude $f$. In the absence of
a resonance, as $k \to 0$, $\delta = k a$, where $a = a_R + i a_I$
is the complex scattering length. For positive energies,
$\Re e \delta = k a_R$, whereas for energies below zero, $ k \to ik$ and
$\Re e \delta = - k a_I$. In such a situation, $\Re e (d\delta/dE)$ exhibits
a sharp peak at $E_{\eta A} = E_{threshold}$, the sign of which is determined
by the sign of the scattering length.
On the other hand, if the scattering amplitude has a resonant behaviour
near threshold, one would see a superposition of the two behaviours.
Consequently, a state reasonably close
to threshold gets distorted in shape and one very close
manifests simply by broadening the threshold singularity. A state
far from threshold, however, remains completely unaffected.
\section{The $\eta \,^4$He system}
\begin{figure}[ht]
\centerline{\vbox{
\psfig{file=figur5.eps,height=9cm,width=7cm}}}
\caption{The delay time in $\eta \,^4$He
elastic scattering as a function of the energy $E = \sqrt s - m_{\eta} -
m_{\eta ^4He}$,
where $\sqrt s$ is the total energy available in the $\eta \,^4$He centre of
mass system.}
\end{figure}
The time delay plot in the upper half of Fig. 7,
shows once
again the near threshold peak and one more broad one centered
around $-2$ MeV. However, the plot with time delay evaluated using
the model which gives $a_{\eta N} = (0.88, 0.41)$ fm, shows a
large negative time delay near threshold. The negative time delay
can sometimes also arise due to a repulsive interaction \cite{smith}.
Intuitively, an attractive interaction is something that causes
resonance formation and ``delays" the scattering process. A repulsive
interaction on the other hand would speed up the process and the
time taken for the process in the absence of interaction would be larger
than that with interaction.
Before ending this section, we note that the $\eta^4$He
scattering lengths obtained within the FRA and
using two different models of the $\eta N$ interaction, namely,
$a_{\eta N} = (0.88, 0.41)$ fm and $a_{\eta N} = (0.28, 0.19)$ fm
are $a_{\eta \,^4He} = (-3.94, 5.575)$ fm and
$a_{\eta \,^4He} = (1.678, 1.524)$ fm respectively.
\section{Summary}
In conclusion, we summarize the present work as: \\
(i) we have made a search for the
unstable states of $\eta$-mesic deuteron and $^4$He
extending the approach of time delay
which was used recently for the first time in the eta-mesic
case \cite{wejphysg}.\\
(ii) The established time delay method for searching resonances has
been extended to negative energies, to search for bound, virtual and unstable
bound states. The validity of this method has been established by
applying it first to the known case of the $n p$ system and then
to the case of the $\eta$-d system within a parameterized
Faddeev calculation.\\
(iii) The calculations were performed
with different values of the $\eta N$ scattering length considered
as acceptable in literature.
Within the Faddeev equation parameterization, we find one unstable bound state
far from threshold ($\sim -16 $ MeV) for an $\eta N$
scattering length of $(0.42,0.34)$ fm.
Within the FRA calculation, we find such an $\eta d$ state
around $-12$ MeV for $a_{\eta N} = (0.28, 0.19)$ fm.
These results seem to indicate that though the FRA in general is not
recommendable for $\eta d$ elastic scattering, the results are close to
those from the Faddeev calculations for low values of the $\eta N$
scattering length.
\\
(iv) In the $\eta^4$He case, within the FRA calculations, we find
an unstable bound state close to threshold
for a small scattering length of $(0.28,0.19)$ fm.
|
1,314,259,996,494 | arxiv | \section{Introduction}
It is a general conviction that, in the search of a quantum gravity
theory, a black hole should play a role similar to that of the hydrogen atom in quantum
mechanics \cite{key-9}. It should be a ``theoretical laboratory''
where one discusses and tries to understand conceptual problems and
potential contradictions in the attempt to unify Einstein's general
theory of relativity with quantum mechanics. This analogy suggests
that black holes should be regular quantum systems with a discrete mass spectrum
\cite{key-9}. In this paper, the authors attempt to contribute
to the above by finding the Schr\"odinger equation and the wave
function of the Schwarzschild BH. The knowledge of such quantities
could, in principle, also play a role in the solution of the famous
BH information paradox \cite{key-10} because here black holes seem to be well defined quantum mechanical systems, having
ordered and discrete quantum spectra. This issue appears consistent with
the unitarity of the underlying quantum gravity theory and with the
idea that information should come out in BH evaporation.
A quantization approach proposed 25 years ago by the historical collaborator of Einstein,
Nathan Rosen \cite{key-5}, will be applied to the gravitational collapse in the simple case of a pressureless \textquotedblleft \emph{star
of dust}\textquotedblright. Therefore, the gravitational potential, the
Schr\"odinger equation and the solution for the collapse's energy levels
will be found. After that, the constraints for a BH will be applied
and this will permit to find the BH quantum gravitational potential, Schr\"odinger equation and energy spectrum. Such an energy spectrum, in its absolute value, is in agreement with both the one conjectured by J. Bekenstein in 1974 \cite{key-7} and that found by Maggiore's description of black hole in terms of quantum membranes \cite{key-8}. Rosen's approach also allows us to find an interesting quantum representation of the Schwarzschild BH ground state at the Planck scale.
It is well-known that the canonical quantization of general relativity leads to the Wheeler-DeWitt equation introducing the so-called Superspace, an infinite-dimensional space of all possible 3-metrics; Rosen, instead, preferred to start his work from the classical cosmological equations using a simplified quantization scheme, reducing, at least formally, the cosmological Einstein-Friedman equations of general relativity to a quantum mechanical system; the Friedman equations can be then recast as a Schr\"odinger equation and the cosmological solutions can be read as eigensolutions of such a \textquotedblleft cosmological Schr\"odinger equation. In this way Rosen found that, in the case of a Universe filled with pressureless matter, the equation is like that for the $s$ states of a hydrogen-like atom \cite{key-5}. It is important to recall that quantization of FLRW universe date back at least to DeWitt's famous 1967 paper \cite{key-42}, where one understands that a large number of particles is required in order to ensure the semiclassical behaviour.
Furthermore, we try to clarify two important issues such as Bekenstein area law and singularity resolution. With regard to the former, a result similar to that obtained by Bekenstein, but with a different coefficient, has been found. About the latter, it is shown that the traditional classical singularity in the core of the Schwarzschild BH is replaced,
in a full quantum treatment, by a two-particle system where the two components strongly interact with each other via a quantum gravitational potential. The two-particle system seems to be non-singular from the quantum point of view and is analogous to the hydrogen atom because it consists of a \textquotedblleft core\textquotedblright{} and an \textquotedblleft electron\textquotedblright{}.
Returning to the Rosen's quantization approach, it has also been recently applied to a cosmological framework by one of the authors
(FF) and collaborators in \cite{key-11} and to the famous Hartle-Hawking
initial state by both of the authors and I. Licata in \cite{key-26}.
For the sake of completeness, one stresses that this quantization
approach is only applicable to homogeneous space-times where the Weyl
tensor vanishes. As soon as one introduces inhomogeneities (for example in the
Lema\^itre-Tolman-Bondi (LTB) models \cite{key-23,key-24,key-25}), there exists open
sets of initial data where the collapse ends to a BH absolutely similar
to Oppenheimer-Snyder-Datt (OSD) collapse. Hence, the exterior space-time
remains the same whereas the interior is absolutely different. A further
generalization of the proposed approach, which could
be the object of future works, should include also a cosmological
term or other sources of dark energy.
One also underlines that the Oppenheimer and Snyder gravitational collapse is not physical and can be considered a toy model. But here the key point is that, as it is well known from the historical paper of Oppenheimer and Snyder \cite{key-2}, the final state of this simplified gravitational collapse is the SBH, which, instead, has a fundamental role in quantum gravity. It is indeed a general conviction, arising from an idea of Bekenstein \cite{key-9}, that, in the search of a quantum gravity theory, the SBH should play a role similar to the hydrogen atom in quantum mechanics. Thus, despite non-physical, the Oppenheimer and Snyder gravitational collapse must be here considered as a tool which allows to understand a fundamental physical system, that is the SBH. In fact, it will be shown that, by setting the constrains for the formation of the SBH in the quantized Oppenheimer and Snyder gravitational collapse, one arrives to quantize the SBH, and this will be a remarkable, important result in the quantum gravity's search.
\section{Application of Rosen's quantization approach to the gravitational
collapse}
Classically, the gravitational collapse in the simple case of a pressureless
\textquotedblleft \emph{star
of dust}\textquotedblright{} with uniform density
is well known \cite{key-1}. Historically, it
was originally analysed in the famous paper of Oppenheimer and Snyder
\cite{key-2}, while a different approach has been developed by
Beckerdoff and Misner \cite{key-3}. Furthermore, a non-linear electrodynamics
Lagrangian has been recently added in this collapse's framework by one of the
authors (CC) and Herman J. Mosquera Cuesta in \cite{key-4}. This different approach
allows to find a way to remove the black hole singularity at the classical
level. The traditional, classical framework of this
kind of gravitational collapse is well known \cite{key-1,key-2,key-3}. In the following we will follow \cite{key-1}. In regard to the
interior of the collapsing star, it is described by the well-known Friedmann-Lema\^itre-Robertson-Walker
(FLRW) line-element with comoving hyper-spherical coordinates $\chi,$
$\theta,$ $\varphi$. Therefore, one writes down
(hereafter Planck units will be used, i.e., $G=c=k_{B}=\hbar=\frac{1}{4\pi\epsilon_{0}}=1$)
\begin{equation}
ds^{2}=d\tau^{2}+a^2(\tau)[(-d\chi^{2}-\sin^{2}\chi(d\theta^{2}+\sin^{2}\theta d\varphi^{2})],\label{eq: metrica conformemente piatta}
\end{equation}
where the origin of coordinates is set at the centre of the star,
and the following relations hold:
\begin{equation}
\begin{array}{c}
a=\frac{1}{2}a_{m}\left(1+\cos\eta\right),\\
\\
\tau=\frac{1}{2}a_{m}\left(\eta+\sin\eta\right).
\end{array}\label{eq: cycloidal relation}
\end{equation}
The density is given by
\begin{equation}
\rho=\left(\frac{3a_{m}}{8\pi}\right)a^{-3}=\left(\frac{3}{8\pi a_{m}^{2}}\right)\left[\frac{1}{2}\left(1+\cos\eta\right)\right]^{-3}.\label{eq: density}
\end{equation}
Setting $\sin^{2}\chi$ one chooses the case of positive curvature,
which corresponds to a gas sphere whose dynamics begins at rest with
a finite radius and, in turn, it is the only one of interest.
Thus, the choice $k=1$ is made for dynamical reasons (the initial
rate of change of density is null, that means ``momentum of maximum
expansion''), but the dynamics also depends on the
field equations. As it has been stressed in the Introduction,
for isotropic models, a cosmological term - or other sources of dark
energy - can be in principle included in future works, in order to
obtain a more realistic physical framework for the collapse.
In order to discuss the simplest model of a \emph{``star of dust''},
that is the case of zero pressure, one sets the stress-energy tensor
as
\begin{equation}
T=\rho u\otimes u,\label{eq: stress energy}
\end{equation}
where $\rho$ is the density of the collapsing star and $u$ the four-vector
velocity of the matter. On the other hand, the external geometry is
given by the Schwartzschild line-element
\begin{equation}
ds^{2}=\left(1-\frac{2M}{r}\right)dt^{2}-r^{2}\left(\sin^{2}\theta d\varphi^{2}+d\theta^{2}\right)-\frac{dr^{2}}{1-\frac{2M}{r}},\label{eq: Hilbert}
\end{equation}
where $M$ is the total mass of the collapsing star. Since there are
no pressure gradients, which can deflect the particles motion, the
particles on the surface of any ball of dust move along radial geodesics
in the exterior Schwarzschild space-time. Considering
a ball which begins at rest with finite radius (in terms of the Schwarzschild
radial coordinate) $r=r_{i}$ at the (Schwarzschild) time $t=0,$
the geodesics motion of its surface is given by the following equations:
\begin{equation}
r=\frac{1}{2}r_{i}\left(1+\cos\eta\right),\label{eq: geodesics surface radius}
\end{equation}
\vspace{-0.8cm}
\begin{equation}
\begin{array}{c}
\hspace{0.55cm} t=2M\ln\left[\frac{\sqrt{\frac{r_{i}}{2M}-1}+\tan\left(\frac{\eta}{2}\right)}{\sqrt{\frac{r_{i}}{2M}-1}-\tan\left(\frac{\eta}{2}\right)}\right]\\
\\
\quad\quad\quad\quad\quad\;\:\,\hspace{1.5cm}+2M\sqrt{\frac{r_{i}}{2M}-1}\left[\eta+\left(\frac{r_{i}}{4M}\right)\left(\eta+\sin\eta\right)\right].
\end{array}\label{eq: geodesic surface time}
\end{equation}
The proper time measured by a clock put on the surface of the collapsing
star can be written as
\begin{equation}
\tau=\sqrt{\frac{r_{i}^{3}}{8M}}\left(\eta+\sin\eta\right).\label{eq: tau}
\end{equation}
The collapse begins at $r=r_{i},$ $\eta=\tau=t=0,$ and ends
at the singularity $r=0,$ $\eta=\pi$ after a duration of proper
time measured by the falling particles
\begin{equation}
\Delta\tau=\pi\sqrt{\frac{r_{i}^{3}}{8M}},\label{eq: delta tau}
\end{equation}
which coincidentally corresponds, as it is well known, to the interval
of Newtonian time for free-fall collapse in Newtonian theory. Different
from the cosmological case, where the solution is homogeneous and
isotropic everywhere, here the internal homogeneity and isotropy of
the FLRW line-element are broken at the star's surface, that is, at
some radius $\chi=\chi_{0}$. At that surface, which
is a 3-dimensional world tube enclosing the star's fluid,
the interior FLRW geometry must smoothly match the exterior Schwarzschild
geometry. One considers a range of $\chi$ given by
$0\leq\chi\leq\chi_{0}$, with $\chi_{0}<\frac{\pi}{2}$ during the
collapse. For the pressureless case the match is possible. The external Schwarzschild solution indeed predicts
a cycloidal relation for the star's circumference
\begin{equation}
\begin{array}{c}
C=2\pi r=2\pi\left[\frac{1}{2}r_{i}\left(1+\cos\eta\right)\right],\\
\\
\tau=\sqrt{\frac{r_{i}^{3}}{8M}}\left(\eta+\sin\eta\right).
\end{array}\label{eq: Circonferenza esterna}
\end{equation}
The interior FLRW predicts a similar cycloidal relation
\begin{equation}
\begin{array}{c}
C=2\pi r=2\pi a\sin\chi_{0}=\pi\sin\chi_{0}a_{m}\left(1+\cos\eta\right),\\
\\
\tau=\frac{1}{2}a_{m}\left(\eta+\sin\eta\right).
\end{array}\label{eq: circonferenza esterna}
\end{equation}
Therefore, the two predictions agree perfectly for all time if and
only if
\begin{equation}
\begin{array}{c}
r_{i}=a_{0}\sin\chi_{0},\\
\\
\hspace{0.25cm}M=\frac{1}{2}a_{0}\sin^{3}\chi_{0,}
\end{array}\label{eq: matching}
\end{equation}
where $r_{i}$ and $a_{0}$ represent the values of the Schwarzschild radial
coordinate in Eq. (\ref{eq: Hilbert}) and of the scale factor in
Eq. (\ref{eq: metrica conformemente piatta}) at the beginning of
the collapse, respectively. Thus, Eqs. (\ref{eq: matching}) represent
the requested match, while the Schwarzschild radial coordinate, in
the case of the matching between the internal and external geometries,
is
\begin{equation}
r=a\sin\chi_{0}.\label{eq: mach}
\end{equation}
The attentive reader notes that the initial conditions on the matching
are the simplest possible that could be relaxed, still having a continuous
matching without extra surface terms. In fact, taking the interior
solution to be homogeneous requires very fine tuned initial conditions
for the collapse and the dynamics of the edge. So, on one hand,
further analyses for a better characterization of the initial conditions
on the matching between the internal and external geometries could
be the object of future works. On the other hand, despite the analysis
of this paper is not as general as possible, one stresses that the BH quantization is one of the most important problems of modern theoretical physics which has not yet been solved.
Thus, in order to attempt to solve such a fundamental problem, one
must start from the simplest case rather than from more complicated
ones. This is in complete analogy with the history of general relativity. In fact, the first solution of Einstein field equations was the Schwartzschild
solution, but it was not a general, rotating solution which included
cosmological term or other sources of dark energy, as well the corresponding
gravitational collapse developed by Oppenheimer and Snyder did not
include a class of non-homogeneous models. Thus, as this is a new
approach to the BH quantization, here one starts from the simplest
conditions rather than from more complicated ones. So, the initial conditions
on the matching that are applied here are exactly the ones proposed
by Oppenheimer and Snyder in their paper on the gravitational
collapse \cite{key-2}. It is well known that the final result of
the gravitational collapse studied by Oppenheimer and Snyder is the
Schwarzschild BH \cite{key-1}.
In the following, the quantization approach derived by Rosen in \cite{key-5}
will be applied to the above case; some differences will be found,
because here one analyses the case of a collapsing star, while Rosen
analysed a closed homogeneous and isotropic universe.
Let us start by rewriting the FLRW line-element (\ref{eq: metrica conformemente piatta})
in spherical coordinates and comoving time as \cite{key-1,key-5}
\begin{equation}
ds^{2}=d\tau^{2}-a^{2}(\tau)\left(\frac{dr^{2}}{1-r^{2}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\varphi^{2}\right).\label{eq: FLRW}
\end{equation}
The Einstein field equations
\begin{equation}
G_{\mu\nu}=-8\pi T_{\mu\nu}\label{eq: Einstein field equation}
\end{equation}
gives the relations (we are assuming zero pressure)
\begin{equation}
\begin{array}{c}
\dot{a}^{2}=\frac{8}{3}\pi a^{2}\rho-1,\\
\\
\ddot{a}=-\frac{4}{3}\pi a\rho,
\end{array}\label{eq: evoluzione}
\end{equation}
with $\dot{a}=\frac{da}{d\tau}$. For consistency, one gets
\begin{equation}
\frac{d\rho}{da}=-\frac{3\rho}{a},\label{eq: consistenza}
\end{equation}
which, when integrated, gives
\begin{equation}
\rho=\frac{C}{a^{3}}.\label{eq: densit=0000E0}
\end{equation}
In the case of a collapse $C$ is determined by the initial conditions which predict a cycloidal relation for the star's circumference, see Eqs. (\ref{eq: circonferenza esterna}) and Section 32.4 in Ref. \cite{key-1}. One gets
\begin{equation}
C=\frac{3a_{0}}{8\pi}.\label{eq: C}
\end{equation}
This value is consistent with the one found by Rosen \cite{key-5}.
Thus, one rewrites Eq. (\ref{eq: densit=0000E0}) as
\begin{equation}
\rho=\frac{3a_{0}}{8\pi a^{3}}.\label{eq: densit=0000E0 2}
\end{equation}
Multiplying the first of (\ref{eq: evoluzione}) by $M/2$ one
gets
\begin{equation}
\frac{1}{2}M\dot{a}^{2}-\frac{4}{3}\pi Ma^{2}\rho=-\frac{M}{2},\label{eq: energy equation for a particle}
\end{equation}
which seems like the energy equation for a particle in one-dimensional
motion having coordinate $a$:
\begin{equation}
E=T+V,\label{eq: energia totale}
\end{equation}
where the kinetic energy is
\begin{equation}
T=\frac{1}{2}M\dot{a}^{2},\label{eq: energia cinetica}
\end{equation}
and the potential energy is
\begin{equation}
V(a)=-\frac{4}{3}\pi Ma^{2}\rho.\label{eq: energia potenziale}
\end{equation}
Thus, the total energy is
\begin{equation}
E=-\frac{M}{2}.\label{eq: energia totale 2}
\end{equation}
From the second of Eqs. (\ref{eq: evoluzione}), one gets the equation
of motion of this particle:
\begin{equation}
M\ddot{a}=-\frac{4}{3}M\pi a\rho.\label{eq: equation of motion}
\end{equation}
The momentum of the particle is
\begin{equation}
P=M\dot{a},\label{eq: momentum}
\end{equation}
with an associated Hamiltonian
\begin{equation}
\mathcal{H}=\frac{P^{2}}{2M}+V.\label{eq: Hamiltonian}
\end{equation}
Till now, the problem has been discussed from the classical point
of view. In order to discuss it from the quantum point of view, one
needs to define a wave-function as
\begin{equation}
\Psi\equiv\Psi\left(a,\tau\right).\label{eq: wave-function}
\end{equation}
Thus, in correspondence of the classical equation (\ref{eq: Hamiltonian}),
one gets the traditional Schr\"odinger equation
\begin{equation}
i\frac{\partial\Psi}{\partial\tau}=-\frac{1}{2M}\frac{\partial^{2}\Psi}{\partial a^{2}}+V\Psi.\label{eq: Schrodinger equation}
\end{equation}
For a stationary state with energy $E$ one obtains
\begin{equation}
\Psi=\Psi\left(a\right)\exp\left(-iE\tau\right),\label{eq: separazione}
\end{equation}
and Eq. (\ref{eq: wave-function}) becomes
\begin{equation}
-\frac{1}{2M}\frac{\partial^{2}\Psi}{\partial a^{2}}+V\Psi=E\Psi.\label{eq: Schrodinger equation 2}
\end{equation}
Inserting Eq. (\ref{eq: densit=0000E0 2}) into Eq. (\ref{eq: energia potenziale})
one obtains
\begin{equation}
V(a)=-\frac{Ma_{0}}{2a}.\label{eq: energia potenziale 2}
\end{equation}
Setting
\begin{equation}
\Psi=aX,\label{eq: X}
\end{equation}
Eq. (\ref{eq: Schrodinger equation 2}) becomes
\begin{equation}
-\frac{1}{2M}\left(\frac{\partial^{2}X}{\partial a^{2}}+\frac{2}{a}\frac{\partial X}{\partial a}\right)+VX=EX.\label{eq: Schrodinger equation 3}
\end{equation}
With $V$ given by Eq. (\ref{eq: energia potenziale 2}), (\ref{eq: Schrodinger equation 3})
is analogous to the Schr\"odinger equation in polar coordinates for
the $s$ states ($l=0$) of a hydrogen-like atom \cite{key-6}
in which the squared electron charge $e^{2}$ is replaced by $\frac{Ma_{0}}{2}$.
Thus, for the bound states ($E<0$) the energy spectrum is
\begin{equation}
E_{n}=-\frac{a_{0}^{2}M^{3}}{8n^{2}},\label{eq: spettro energia}
\end{equation}
where $n$ is the principal quantum number. At this point, one inserts (\ref{eq: energia totale 2}) into (\ref{eq: spettro energia}),
obtaining the mass spectrum as
\begin{equation}
M_{n}=\frac{a_{0}^{2}M_{n}^{3}}{4n^{2}}\Rightarrow M_{n}=\frac{2n}{a_{0}}.\label{eq: spettro massa}
\end{equation}
On the other hand, by using Eq. (\ref{eq: energia totale 2}) one
finds the energy levels of the collapsing star as
\begin{equation}
E_{n}=-\frac{n}{a_{0}}.\label{eq: energy levels}
\end{equation}
In fact, Eq. (\ref{eq: spettro massa}) represents the spectrum of
the total mass of the collapsing star, while Eq. (\ref{eq: energy levels})
represents the energy spectrum of the collapsing
star where the gravitational energy, which is given by Eq. (\ref{eq: energia potenziale 2}),
is included. The total energy of a quantum system with bound states
is indeed negative.
What is the meaning of Eq. (\ref{eq: energy levels}) and of its ground state? One sees that the Hamiltonian (\ref{eq: Hamiltonian}) governs the quantum mechanics of the gravitational collapse. Therefore, the square of the wave function (\ref{eq: wave-function}) must be interpreted as the probability density of a single particle in a finite volume. Thus, the integral over the entire volume must be normalized to unity as
\begin{equation}
\int dx^{3}\left|X\right|^{2}=1.\label{eq: Normalization}
\end{equation}
For stable quantum systems, this normalization must remain the same at all times of the collapse's evolution. As the wave function (\ref{eq: wave-function})
obeys the Schrodinger equation (\ref{eq: Schrodinger equation}),
this is assured if and only if the Hamiltonian operator (\ref{eq: Hamiltonian})
is Hermitian \cite{key-43}. In other words, the Hamiltonian operator
(\ref{eq: Hamiltonian}) must satisfy for arbitrary wave functions
$X_{1}$ and $X_{2}$ the equality \cite{key-43}
\begin{equation}
\int dx^{3}\left[HX_{2}\right]^{*}X_{1}=\int dx^{3}X_{2}^{*}HX_{1}.\label{eq: hermiticit=0000E0}
\end{equation}
One notes that both $\vec{p}$ and $a$ are Hermitian operators. Therrefore,
the Hamiltonian (\ref{eq: Hamiltonian}) is automatically a Hermitian
operator because it is a sum of a kinetic and a potential energy \cite{key-43},
\begin{equation}
H=T+V.\label{eq: energia totale}
\end{equation}
This is always the case for non-relativistic particles in Cartesian-like coordinates and works also for the gravitational collapse under consideration. In this framework, the ground state of Eq. (\ref{eq: energy levels}) represents the minimum energy which can collapse in an astrophysical scenario. We conclude that the gravitational collapse can be interpreted as a ``perfect" quantum system.
It is also important to clarify the issue concerning the
gravitational energy. It is well known that, in the framework of the general theory of
relativity, the gravitational energy cannot be localized \cite{key-1}.
This is a consequence of Einstein's equivalence principle (EEP) \cite{key-1},
which implies that one can always find in any given locality a reference's
frame (the local Lorentz reference's frame) in which all local gravitational
fields are null. No local gravitational fields means no local gravitational
energy-momentum and, in turn, no stress-energy tensor for the gravitational
field. In any case, this general situation admits an
important exception \cite{key-1}, given by the case of a spherical
star \cite{key-1}, which is exactly the case analysed in this paper.
In fact, in this case the gravitational energy is localized not by
mathematical conventions, but by the circumstance that transfer of
energy is detectable by local measures, see Box 23.1 of \cite{key-1}
for details. Therefore, one can surely consider Eq. (\ref{eq: energia potenziale 2})
as the gravitational potential energy of the collapsing star.
\section{Black hole energy spectrum, ground state and singularity resolution}
Thus, let us see what happens when the star is completely collapsed,
i.e. when the star is a BH. One sees that, inserting $r_{i}=2M=r_{g},$
where $r_{g}$ is the gravitational radius (the Schwarzschild radius),
in Eqs. (\ref{eq: matching}), one obtains $\sin^{2}\chi_{0}=1$. Therefore,
as the range $\chi>\frac{\pi}{2}$ must be discarded \cite{key-1},
one concludes that it is $\chi_{0}=\frac{\pi}{2}$, $r=a$ and $r_{i}=a_{0}=2M=r_{g}$
in Eqs. (\ref{eq: matching}) and (\ref{eq: mach}) for a BH.
Then, Eqs. from (\ref{eq: energia potenziale 2}) to (\ref{eq: energy levels})
become, respectively,
\begin{equation}
V(r)=-\frac{M^{2}}{r},\label{eq: energia potenziale BH}
\end{equation}
\begin{equation}
\Psi=rX,\label{eq: X BH}
\end{equation}
\begin{equation}
-\frac{1}{2M}\left(\frac{\partial^{2}X}{\partial r^{2}}+\frac{2}{r}\frac{\partial X}{\partial r}\right)+VX=EX,\label{eq: Schrodinger equation BH}
\end{equation}
\begin{equation}
E_{n}=-\frac{r_{g}^{2}M^{3}}{8n^{2}},\label{eq: spettro energia BH}
\end{equation}
\begin{equation}
M_{n}=\sqrt{n},\label{eq: spettro massa buco nero}
\end{equation}
\begin{equation}
E_{n}=-\sqrt{\frac{n}{4}}.\label{eq: spettro energia buco nero}
\end{equation}
Eqs. (\ref{eq: energia potenziale BH}), (\ref{eq: Schrodinger equation BH}),
(\ref{eq: spettro massa buco nero}) and (\ref{eq: spettro energia buco nero})
should be the exact gravitation potential energy\emph{,} Schr\"odinger
equation, mass spectrum and energy spectrum for the Schwarzschild
BH interpreted as ``gravitational hydrogen atom'', respectively.
Actually, a further final correction is needed. To clarify this point,
let us compare Eq. (\ref{eq: energia potenziale BH}) with the analogous
potential energy of an hydrogen atom, which is \cite{key-6}
\begin{equation}
V(r)=-\frac{e^{2}}{r}.\label{eq: energia potenziale atomo idrogeno}
\end{equation}
Eqs. (\ref{eq: energia potenziale BH}) and (\ref{eq: energia potenziale atomo idrogeno})
are formally identical, but there is an important physical difference. In the
case of Eq. (\ref{eq: energia potenziale atomo idrogeno}) the electron's
charge is constant for all the energy levels of the hydrogen atom.
Instead, in the case of Eq. (\ref{eq: energia potenziale BH}), based
on the emissions of Hawking quanta or on the absorptions of external
particles, the BH mass changes during the jumps from one energy level
to another. In fact, such a BH mass decreases for emissions and increases
for absorptions. Therefore, one must also consider this dynamical behavior. One way to take into account this dynamical behavior
is by introducing the \emph{BH effective state} (see \cite{key-13,key-14} for details).
Let us start from the emissions of Hawking quanta. If one neglects
the above mentioned BH dynamical behavior, the probability of emission of Hawking
quanta is the one originally found by Hawking, which represents a
strictly thermal spectrum \cite{key-16}
\begin{equation}
\Gamma\sim\exp\left(-\frac{\omega}{T_{H}}\right),\label{eq: hawking probability}
\end{equation}
where $\omega$ is the energy-frequency of the emitted particle
and $T_{H}\equiv\frac{1}{8\pi M}$ is the Hawking temperature. Taking
into account the BH dynamical behavior, i.e., the BH contraction allowing
a varying BH geometry, one gets the famous correction found by Parikh and
Wilczek \cite{key-17}:
\begin{equation}
\Gamma\sim\exp\left[-\frac{\omega}{T_{H}}\left(1-\frac{\omega}{2M}\right)\right]\quad\Longrightarrow\quad\Gamma=\alpha\exp\left[-\frac{\omega}{T_{H}}\left(1-\frac{\omega}{2M}\right)\right],\label{eq: Parikh Correction}
\end{equation}
where $\alpha\sim1$ and the additional term $\frac{\omega}{2M}\:$
is present. By introducing the \emph{effective temperature }\cite{key-13,key-14}
\begin{equation}
T_{E}(\omega)\equiv\frac{2M}{2M-\omega}T_{H}=\frac{1}{4\pi\left(2M-\omega\right)},\label{eq: Corda Temperature}
\end{equation}
Eq. (\ref{eq: Parikh Correction}) can be rewritten in a Boltzmann-like
form \cite{key-13,key-14}, namely
\begin{equation}
\Gamma=\alpha\exp[-\beta_{E}(\omega)\omega]=\alpha\exp\left(-\frac{\omega}{T_{E}(\omega)}\right),\label{eq: Corda Probability}
\end{equation}
where $\exp[-\beta_{E}(\omega)\omega]$ is the \emph{effective Boltzmann
factor,} with \cite{key-13,key-14}
\begin{equation}
\beta_{E}(\omega)\equiv\frac{1}{T_{E}(\omega)}.\label{eq: beta E}
\end{equation}
Therefore, the effective temperature replaces the Hawking temperature
in the equation of the probability of emission as dynamical quantity.
There are indeed various fields of science where one can take into
account the deviation from the thermal spectrum of an emitting body
by introducing an effective temperature which represents the temperature
of a black body that would emit the same total amount of radiation\emph{
}\cite{key-13,key-14}\emph{.} The effective temperature depends on
the energy-frequency of the emitted radiation and the ratio $\frac{T_{E}(\omega)}{T_{H}}=\frac{2M}{2M-\omega}$
represents the deviation of the BH radiation spectrum from the strictly
thermal feature due to the BH dynamical behavior \cite{key-13,key-14}.
Besides, one can introduce other
\emph{effective quantities}. In particular, if $M$ is the initial
BH mass \emph{before} the emission, and $M-\omega$ is the final BH
mass \emph{after} the emission, the \emph{BH} \emph{effective mass
}and the \emph{BH effective horizon }can be\emph{ }introduced as \cite{key-13,key-14}
\begin{equation}
M_{E}\equiv M-\frac{\omega}{2},\mbox{ }r_{E}\equiv2M_{E}.\label{eq: effective quantities}
\end{equation}
They represent the BH mass and horizon\emph{ during} the BH
contraction, i.e. \emph{during} the emission of the particle \cite{key-13,key-14}, respectively.
These are average quantities. The variable \emph{$r_{E}$
}is indeed the average of the initial and final horizons while \emph{$M_{E}$
}is the average of the initial and final masses \cite{key-13,key-14}.
In regard to the effective temperature, it is the inverse of the average value of the inverses of the initial and final Hawking temperatures; \emph{before}
the emission we have $T_{H}^{i}=\frac{1}{8\pi M}$, \emph{after}
the emission $T_{H}^{f}=\frac{1}{8\pi(M-\omega)}$ \cite{key-13,key-14}.
To show that the effective mass is indeed the correct
quantity which characterizes the BH dynamical behavior, one can rely on
Hawking's periodicity argument \cite{key-16,key-17,key-18}. One rewrites Eq. (\ref{eq: beta E})
as \cite{key-20}
\begin{equation}
\beta_{E}(\omega)\equiv\frac{1}{T_{E}(\omega)}=\beta_{H}\left(1-\frac{\omega}{2M}\right),\label{eq: beta E-1}
\end{equation}
where $\beta_{H}\equiv\frac{1}{T_{H}}$. Following Hawking' s arguments
\cite{key-16,key-17,key-18}, the Euclidean form of the metric is given by \cite{key-20}
\begin{equation}
ds_{E}^{2}=x^{2}\left[\frac{d\tau}{4M\left(1-\frac{\omega}{2M}\right)}\right]^{2}+\left(\frac{r}{r_{E}}\right)^{2}dx^{2}+r^{2}(\sin^{2}\theta d\varphi^{2}+d\theta^{2}).\label{eq: euclidean form}
\end{equation}
This equation is regular at $x=0$ and $r=r_{E}$. One also treats
$\tau$ as an angular variable with period $\beta_{E}(\omega)$ \cite{key-13,key-14,key-16}. Following \cite{key-20}, one replaces the quantity $\sum_{i}\beta_{i}\frac{\hslash^{i}}{M^{2i}}$
in \cite{key-18} with $-\frac{\omega}{2M}.$ Then, following the analysis presented in \cite{key-18}, one
obtains \cite{key-20}
\begin{equation}
ds_{E}^{2}\equiv\left(1-\frac{2M_{E}}{r}\right)dt^{2}-\frac{dr^{2}}{1-\frac{2M_{E}}{r}}-r^{2}\left(\sin^{2}\theta d\varphi^{2}+d\theta^{2}\right).\label{eq: Hilbert effective}
\end{equation}
One can also show that $r_{E}$ in Eq. (\ref{eq: euclidean form})
is the same as in Eq. (\ref{eq: effective quantities}).
Despite the above analysis has been realized for emissions of particles,
one immediately argues by symmetry that the same analysis works also
in the case of absorptions of external particles, which can be considered
as emissions having opposite sign. Thus, the effective quantities
(\ref{eq: effective quantities}) become
\begin{equation}
M_{E}\equiv M+\frac{\omega}{2},\mbox{ }r_{E}\equiv2M_{E}.\label{eq: effective quantities absorption}
\end{equation}
Now they represents the BH mass and horizon\emph{ during}
the BH expansion, i.e., \emph{during} the absorption of the particle, respectively.
Hence, Eq. (\ref{eq: Hilbert effective}) implies that, in order to
take the BH dynamical behavior into due account, one must replace
the BH mass $M$ with the BH effective mass $M_{E}$ in Eqs. (\ref{eq: energia potenziale BH}),
(\ref{eq: Schrodinger equation BH}), (\ref{eq: spettro energia BH}), (\ref{eq: energia totale 2}), obtaining
\begin{equation}
V(r)=-\frac{M_{E}^{2}}{r},\label{eq: energia potenziale BH effettiva}
\end{equation}
\begin{equation}
-\frac{1}{2M_{E}}\left(\frac{\partial^{2}X}{\partial r^{2}}+\frac{2}{r}\frac{\partial X}{\partial r}\right)+VX=EX,\label{eq: Schrodinger equation BH effettiva}
\end{equation}
\begin{equation}
E_{n}=-\frac{r_{E}^{2}M_{E}^{3}}{8n^{2}},\label{eq: spettro energia BH effettivo}
\end{equation}
\begin{equation}
E=-\frac{M_{E}}{2}.\label{eq: energia totale effettiva}
\end{equation}
From the quantum point of view, we want to obtain the energy
eigenvalues as being absorptions starting from the BH formation, that
is from the BH having null mass. This implies that we must replace
$M\rightarrow0$ and $\omega\rightarrow M$ in Eq. (\ref{eq: effective quantities absorption}).
Thus, we obtain
\begin{equation}
M_{E}\equiv\frac{M}{2},\mbox{ }r_{E}\equiv2M_{E}=M.\label{eq: effective quantities absorption finali}
\end{equation}
Following again \cite{key-5}, one inserts Eqs. (\ref{eq: energia totale effettiva})
and (\ref{eq: effective quantities absorption finali}) into Eq. (\ref{eq: spettro energia BH effettivo}),
obtaining the BH mass spectrum as
\begin{equation}
M_{n}=2\sqrt{n},\label{eq: spettro massa BH finale}
\end{equation}
and by using Eq. (\ref{eq: energia totale effettiva}) one finds the
BH energy levels as
\begin{equation}
E_{n}=-\frac{1}{2}\sqrt{n}.\label{eq: BH energy levels finale.}
\end{equation}
Remarkably, in its absolute value, this final result is consistent with the
BH energy spectrum which was conjectured by Bekenstein in 1974 \cite{key-7}. Bekenstein indeed obtained $E_{n}\sim \sqrt{n}$ by using the
Bohr-Sommerfeld quantization condition because he argued that the
Schwarzschild BH behaves as an adiabatic invariant. Besides, Maggiore \cite{key-8}
conjectured a quantum description of BH in terms of quantum membranes.
He obtained the energy spectrum
\begin{equation}
E_{n}=\sqrt{\frac{A_{0}n}{16\pi}}.\label{eq: spettro massa buco nero membrane}
\end{equation}
One sees that, in its absolute value, the
result of Eq. (\ref{eq: BH energy levels finale.}) is consistent
also with Maggiore's result. On the other hand, it should be noted that both Bekenstein and
Maggiore used heuristic analyses, approximations and/or conjectures.
Instead, Eq. (\ref{eq: spettro massa buco nero}) has been obtained
through an exact quantization process. In addition, neither Bekenstein
nor Maggiore realized that the BH energy spectrum must have negative
eigenvalues because the ``gravitational hydrogen atom'' is a quantum
system composed by bound states.
Let us again consider the analogy between the potential energy of
a hydrogen atom, given by Eq. (\ref{eq: energia potenziale atomo idrogeno}),
and the effective potential energy of the ``gravitational hydrogen
atom'' given by Eq. (\ref{eq: energia potenziale BH effettiva}).
Eq. (\ref{eq: energia potenziale atomo idrogeno}) represents
the interaction between the nucleus of the hydrogen atom, having a
charge $e$ and the electron, having a charge $-e.$ Eq. (\ref{eq: energia potenziale BH effettiva})
represents the interaction between the nucleus of the ``gravitational
hydrogen atom'', i.e. the BH, having an effective, dynamical mass
$M_{E}$, and another, mysterious, particle, i.e., the ``electron''
of the ``gravitational hydrogen atom'' having again an effective,
dynamical mass $M_{E}$. Therefore, let us ask: what is the ``electron''
of the BH? An intriguing answer to this question has been given by
one of the authors (CC), who recently developed a semi-classical Bohr-like
approach to BH quantum physics where, for large values of the principal
quantum number $n,$ the BH quasi-normal modes (QNMs), \textquotedblleft triggered\textquotedblright{}
by emissions (Hawking radiation) and absorption of external particles,
represent the \textquotedblleft electron\textquotedblright{} which
jumps from a level to another one; the absolute values of the
QNMs frequencies represent the energy \textquotedblleft shells\textquotedblright{}
of the \textquotedblleft gravitational hydrogen atom\textquotedblright.
In this context, the QNM jumping from a level to another one has been
indeed interpreted in terms of a particle quantized on a circle \cite{key-13,key-14},
which is analogous to the electron travelling in circular orbits around
the hydrogen nucleus, similar in structure to the solar system, of
Bohr's semi-classical model of the hydrogen atom \cite{key-21,key-22}.
Therefore, the results in the present paper seem consistent with the above mentioned works \cite{key-13,key-14}.
For the BH ground state ($n=1$), from Eq. (\ref{eq: spettro massa BH finale})
one gets the mass as
\begin{equation}
M_{1}=2,\label{eq: massa minima}
\end{equation}
in Planck units. Thus, in standard units one gets $M_{1}=2m_{P},$
where $m_{P}$ is the Planck mass, $m_{P}=2,17645\text{\texttimes}10^{-8}\mathrm{~Kg}.$
To this mass is associated a total negative energy arising from Eq.
(\ref{eq: BH energy levels finale.}), which is
\begin{equation}
E_{1}=-\frac{1}{2},\label{eq: energia minima}
\end{equation}
and a Schwarzschild radius
\begin{equation}
r_{g1} = 4.
\end{equation}
Hence, this is the state having minimum mass and minimum energy (the
energy of this state is minimum in absolute value; in its real value,
being negative, it is maximum). In other words, this ground state
represents the smallest possible BH. In the case of Bohr's semi-classical
model of hydrogen atom, the Bohr radius, which represents the classical
radius of the electron at the ground state, is \cite{key-6}
\begin{equation}
\text{Bohr radius}=b_1=\frac{1}{m_{e}e^{2}},\label{eq: Bohr radius}
\end{equation}
where $m_{e}$ is the electron mass. To obtain the correspondent
``Bohr radius'' for the ``gravitational hydrogen atom'', one needs
to replace both the electron mass $m_{e}$ and the charge $e$ in Eq. (\ref{eq: Bohr radius})
with the effective mass of the BH ground state, which is $\frac{M_{1}}{2}=1.$
Thus, now the ``Bohr radius'' becomes
\begin{equation}
b_{1}=1,\label{eq: Bohr radius-1}
\end{equation}
which in standard units reads $b_{1}=l_{P},$ where $l_{P}=1,61625\text{\texttimes}10^{-35}\mathrm{~m}$
is the Planck length. Hence, we have found that the ``Bohr radius''
associated to the smallest possible black hole is equal to the Planck length. Following \cite{key-5}, the wave-function associated to the BH ground state is
\begin{equation}
\Psi_{1}=2b_{1}^{-\frac{3}{2}}r\exp\left(-\frac{r}{b_{1}}\right)=2r\exp\left(-r\right),\label{eq: wave-function 1 BH}
\end{equation}
where $\Psi_{1}$ is normalized as
\begin{equation}
\int_{0}^{\infty}\Psi_{1}^{2}dr=1.\label{eq: normalizzazione BH}
\end{equation}
The size of this BH is of the order of
\begin{equation}
\bar{r}_{1}=\int_{0}^{\infty}\Psi_{1}^{2}rdr=\frac{3}{2}b_{1}=\frac{3}{2}.\label{eq: size BH 1}
\end{equation}
The issue that the size of the BH ground state is, on average, shorter
than the gravitational radius could appear surprising, but one recalls
again that one interprets the ``BH electron states'' in terms of
BH QNMs \cite{key-13,key-14}. Thus, the BH size which is, on average,
shorter than the gravitational radius, seems consistent with the issue
that the BH horizon oscillates with damped oscillations when the BH
energy state jumps from a quantum level to another one through emissions
of Hawking quanta and/or absorption of external particles.
This seems an interesting quantum representation of the Schwarzschild
BH ground state at the Planck scale. This Schwarzschild BH ground
state represents the BH minimum energy level which is compatible with
the generalized uncertainty principle (GUP) \cite{key-12}. The GUP
indeed prevents a BH from its total evaporation by stopping Hawking's
evaporation process in exactly the same way that the usual uncertainty
principle prevents the hydrogen atom from total collapse \cite{key-12}.
Now, let us discuss a fundamental issue. Can one say that the quantum
BH expressed by the system of equations from (\ref{eq: energia potenziale BH effettiva}) to (\ref{eq: energia totale effettiva}) is non-singular? It seems
that the correct answer is yes. It is well known that, in the classical
general relativistic framework, in the internal geometry all time-like
radial geodesics of the collapsing star terminate after a lapse of
finite proper time in the termination point $r=0$ and it is impossible
to extend the internal space-time manifold beyond that termination
point \cite{key-1}. Thus, the point $r=0$ represents a singularity
based on the rigorous definition by Schmidt \cite{key-37}. But what
happens in the quantum framework that has been analysed in this paper
is completely different. By inserting the constraints for a Schwarzschild
BH in Rosen's quantization process applied to the gravitational collapse,
it has been shown that the completely collapsed object has been split
in a two-particle system where the two components strongly interact
with each other through a quantum gravitational interaction. In concrete
terms, the system that has been analysed is indeed formally equal
to the well known system of two quantum particles having finite distance
with the mutual attraction of the form $1/r$ \cite{key-6}. These
two particles are the ``nucleus'' and the ``electron'' of the
``gravitational hydrogen atom''. Thus, the key point is the meaning
of the word ``particle'' in a quantum framework. Quantum particles
remain in an uncertain, non-deterministic, smeared, probabilistic
wave-particle orbital state \cite{key-6}. Then, they cannot be localized
in a particular ``termination point where all time-like radial geodesics
terminate''. As it is well known, such a localization is also in
contrast with the Heisenberg uncertainty principle (HUP). The HUP says indeed that either the location
or the momentum of a quantum particle such as the BH ``electron''
can be known as precisely as desired, but as one of these quantities
is specified more precisely, the value of the other becomes increasingly
indeterminate. This is not simply a matter of observational difficulty,
but rather a fundamental property of nature. This means that,
within the tiny confines of the ``gravitational atom'', the ``electron''
cannot really be regarded as a ``point-like particle''
having a definite energy and location. Thus, it is somewhat misleading
to talk about the BH ``electron'' ``falling into''
the BH ``nucleus''. In other words, the Schwarzschild radial coordinate
cannot become equal to
zero. The GUP makes even stronger this last statement: as we can notice from its general expression \cite{key-38}
\begin{equation}
\Delta x\Delta p\geq\frac{1}{2}\left[1+\eta\left(\Delta p\right)^{2}+\ldots \right],\label{eq: GUP}
\end{equation}
it implies a non-zero lower bound on the minimum value of the uncertainty on the particle's position $\left(\Delta x\right)$ which is of order of the Planck
length \cite{key-38}. In other words, the GUP implies the existence
of a minimal length in quantum gravity.
One notes also another important difference between the hydrogen
atom of quantum mechanics \cite{key-6} and the ``gravitational hydrogen
atom'' discussed in this paper. In the standard hydrogen atom the
nucleus and the electron are different particles. In the quantum BH analysed here they are equal particles instead, as one easily checks
in the system of equations from (\ref{eq: energia potenziale BH effettiva})
to (\ref{eq: energia totale effettiva}). Thus, the ``nucleus''
and the ``electron'' can be mutually exchanged without varying the
physical properties of the system. Hence, the quantum state of the
two particles seems even more uncertain, more non-deterministic, more
smeared and more probabilistic than the corresponding quantum states
of the particles of the hydrogen atom. These quantum argumentations
seem to be strong argumentations in favour of the non-singular behavior
of the Schwarzschild BH in a quantum framework. Notice that the results
in this paper are also in agreement with the general conviction that
quantum gravity effects become fundamental in the presence of strong
gravitational fields. In a certain sense, the results in this paper
permit to ``see into'' the Schwartzschild BH. The authors hope to
further deepen these fundamental issues in future works.
\section{Area quantization}
Bekenstein proposed that the area
of the BH horizon is quantized in units of the Planck length in quantum
gravity (let us remember that the Planck length
is equal to one in Planck units) \cite{key-7}. His result was that the Schwarzschild
BH area quantum is $\Delta A=8\pi$ \cite{key-7}.
In the Schwarzschild
BH the \emph{horizon area} $A$ is related to the
mass by the relation $A=16\pi M^{2}.$ Thus, a variation $\Delta M\,$ of the mass implies a variation
\begin{equation}
\Delta A=32\pi M\Delta M\label{eq: variazione area}
\end{equation}
of the area. Let us consider a BH which is excited at the level $n$. The corresponding BH mass is given by Eq. (\ref{eq: spettro massa BH finale}),
that is
\begin{equation}
M_{n}=2\sqrt{n}.\label{eq: spettro massa BH finale n}
\end{equation}
Now, let us assume that a neighboring particle is captured by the
BH causing a transition from $n$ to $n+1.$ Then, the variation of the BH mass is
\begin{equation}
M_{n+1}-M_{n}=\Delta M_{n\rightarrow n+1},\label{eq: absorbed}
\end{equation}
where
\begin{equation}
M_{n+1}=2\sqrt{n+1}.\label{eq: spettro massa BH finale n+1}
\end{equation}
Therefore, using Eqs. (\ref{eq: variazione area}) and (\ref{eq: absorbed})
one gets
\begin{equation}
\Delta A_{n}\equiv32\pi M_{n}\Delta M_{n\rightarrow n+1}.\label{eq: area quantum a}
\end{equation}
Eq. (\ref{eq: area quantum a}) should give the area quantum
of an excited BH when one considers an absorption from the level $n$
to the level $n+1$ in function of the principal quantum number $n$.
But, let us consider the following problem. An emission from the level
$n+1$ to the level $n$ is now possible due to the potential emission
of a Hawking quantum. Then, the correspondent mass lost by the BH
will be
\begin{equation}
M_{n+1}-M_{n}=-\Delta M_{n\rightarrow n+1}\equiv\Delta M_{n+1\rightarrow n}.\label{eq: emitted}
\end{equation}
Hence, the area quantum for the transition (\ref{eq: emitted}) should
be
\begin{equation}
\Delta A_{n}\equiv32\pi M_{n+1}\Delta M_{n+1\rightarrow n},\label{eq: area quantum e}
\end{equation}
and one gets the strange result that the absolute value of the area
quantum for an emission from the level $n+1$ to the level $n\;$
is different from the absolute value of the area quantum for an absorption
from the level $n$ to the level $n+1$ because it is $M_{n+1}\neq M_{n}$.
One expects the area spectrum to be the same for absorption and emission
instead. In order to resolve this inconsistency, one considers the \emph{effective
mass,} which has
been introduced in Section 3, corresponding to the transitions between the two levels
$n$ and $n+1$. In fact, the effective mass is the same for emission
and absorption
\begin{equation}
M_{E(n,\;n+1)}\equiv\frac{1}{2}\left(M_{n}+M_{n+1}\right) = \sqrt{n}+\sqrt{n+1}.
\label{eq: massa effettiva n}
\end{equation}
By replacing $M_{n+1}$ with $M_{E(n,\;n+1)}$ in equation (\ref{eq: area quantum e})
and $M_{n}$ with $M_{E(n,\;n+1)}$ in Eq. (\ref{eq: area quantum a})
one obtains
\begin{equation}
\begin{array}{c}
\Delta A_{n+1}\equiv32\pi M_{E(n,\;n+1)}\,\Delta M_{n+1\rightarrow n}\qquad \text{emission}\\
\\
\hspace{0.75cm}\Delta A_{n}\equiv32\pi M_{E(n,\;n+1)}\,\Delta M_{n\rightarrow n+1}\qquad \text{absorption}
\end{array}\label{eq: expects}
\end{equation}
and now it is $|\Delta A_{n}|=|\Delta A_{n-1}|.$ By using Eqs.
(\ref{eq: absorbed}) and (\ref{eq: massa effettiva n}) one finds
\begin{equation}
|\Delta A_{n}|=|\Delta A_{n+1}|=64\pi,\label{eq: 8 pi planck}
\end{equation}
which is similar to the original result found by Bekenstein in 1974 \cite{key-7},
but with a different coefficient. This is not surprising because
there is no general consensus on the area quantum. In fact, in \cite{key-27,key-28}
Hod considered the black hole QNMs like quantum levels
for absorption of particles, obtaining a different numerical
coefficient. On the other hand, the expression found by Hod
\begin{equation}
\Delta A=4\ln3
\end{equation}
is actually a special case of the one suggested by Mukhanov in \cite{key-29}, who proposed
\begin{equation}
\Delta A=4\ln k, \quad k=2,3,\ldots
\end{equation}
This can be found in \cite{key-9,key-30}.
Thus, the approach in this paper seems consistent with the
Bekenstein area law.
\section{Discussion and conclusion remarks}
Rosen's quantization approach has been applied to the gravitational
collapse in the simple case of a pressureless \textquotedblleft \emph{star
of dust}\textquotedblright. In this way, the gravitational potential,
the Schr\"odinger equation and the solution for the collapse's energy
levels have been found. After that, by applying the constraints for
a BH and by using the concept of BH\emph{ }effective state \cite{key-13,key-14},
the analogous results and the energy spectrum have been
found for the Schwarzschild BH. Remarkably, such an energy spectrum
is consistent (in its absolute value) with both the one which was found
by J. Bekenstein in 1974 \cite{key-7} and that found
by Maggiore in \cite{key-8}.
The discussed approach also allowed to find an interesting quantum
representation of the Schwarzschild BH ground state at the Planck
scale; in other words, the smallest BH has been found, by also showing
that it has a mass of two Planck masses and a ``Bohr radius'' equal to the
Planck length. Furthermore, two fundamental issues such as Bekenstein area law and singularity resolution have been discussed. Thus, despite the gravitational collapse analysed in this paper is the simplest possible, the analysis that it has been performed permitted to obtain important results in BH quantum physics.
Finally, for the sake of completeness, it is necessary to discuss
how the results found in this paper are related to those appearing in the
literature. The results of Bekenstein and Maggiore
concerning the BH energy spectrum have been previously cited. In general, such an energy spectrum has been discussed and derived
in many different ways, see for example \cite{key-31,key-32}. In the so-called reduced phase space quantization method
\cite{key-31,key-32}, the BH energy spectrum gets augmented by an
additional zero-point energy; this becomes important if one attempts
to address the ultimate fate of BH evaporation, but, otherwise, it
can safely be ignored for macroscopic BHs for which the principal
quantum number $n$ will be extremely large. It is important to stress
that the discreteness of the energy spectrum needs a drastic departure
from the thermal behavior of the Hawking radiation spectrum. A popular way to realize this is through
the popular tunnelling framework arising from the paper of Parikh
and Wilczek \cite{key-17}. In that case, the energy conservation
forces the BH to contract during the emission of the particle; the horizon recedes from its original radius and becomes smaller at the end of the emission process \cite{key-17}.
Therefore, BHs do not exactly emit like perfect black bodies,
see also the discussion on this issue in Section 3. Moreover, Loop Quantum
Gravity predicts a discrete energy spectrum which indicates a physical
Planck scale cutoff of the Hawking temperature law \cite{key-33}.
In the framework of String-Theory one can identify microscopic BHs with long chains
living on the worldvolume of two dual Euclidean brane pairs \cite{key-34}.
This leads to a discrete Bekenstein-like energy spectrum for
the Schwarzschild black hole \cite{key-35}. The Bekenstein energy spectrum
is present also in canonical quantization schemes \cite{key-36}. This approach
yields a BH picture that is shown to be equivalent to a collection
of oscillators whose density of levels is corresponding to that of the statistical bootstrap model \cite{key-36}.
In a series of interesting papers \cite{key-37,key-38,key-39}, Stojkovic
and collaborators wrote down the Schr\"odinger equation for a collapsing
object and showed by explicit calculations that quantum mechanics
is perhaps able to remove the singularity at the BH center (in various
space-time slicings). This is consistent with our analysis. In \cite{key-37,key-38,key-39} it is indeed proved (among other things)
that the wave function of the collapsing object is non-singular at the center
even when the radius of the collapsing object (classically) reaches zero.
Moreover, in \cite{key-39}, they also considered charged BHs.
Another interesting approach to the area quantization, based on graph
theory, has been proposed by Davidson in [41]. In such a paper, the Bekenstein-Hawking
area entropy formula is obtained, being automatically accompanied
by a proper logarithmic term (a subleading correction), and the size
of the horizon unit area is fixed \cite{key-40}. Curiously, Davidson
also found a hydrogen-like spectrum in a totally different contest \cite{key-41}.
Thus, it seems that the results of this paper are consistent
with the previous literature on BH quantum physics.
\section*{Acknowledgements }
The authors thank Dennis Durairaj for useful suggestions. The authors are also grateful to the anonymous referee for very useful comments.
\providecommand{\href}[2]{#2}
|
1,314,259,996,495 | arxiv | \section{Introduction}
Research on Vietnamese NLP has been actively explored in the last decade, boosted by the successes of the 4-year KC01.01/2006-2010 national project on Vietnamese language and speech processing (VLSP). Over the last 5 years, standard benchmark datasets for key Vietnamese NLP tasks are publicly available: datasets for word segmentation and POS tagging were released for the first VLSP evaluation campaign in 2013; a dependency treebank was published in 2014 \cite{Nguyen2014NLDB}; and an NER dataset was released for the second VLSP campaign in 2016. So there is a need for building an NLP pipeline, such as the Stanford CoreNLP toolkit \cite{manning-EtAl:2014:P14-5}, for those key tasks to assist users and to support researchers and tool developers of downstream tasks.
\newcite{NguyenPN2010} and \newcite{Le:2013:VOS} built Vietnamese NLP pipelines by wrapping existing word segmenters and POS taggers including: JVnSegmenter \cite{Y06-1028}, vnTokenizer \cite{Le2008}, JVnTagger \cite{NguyenPN2010} and vnTagger \cite{lehong00526139}. However,
these word segmenters and POS taggers are no longer considered
SOTA models for Vietnamese \cite{NguyenL2016,JCSCE}. \newcite{PhamPNP2017b} built the NNVLP toolkit for Vietnamese sequence labeling tasks by applying a BiLSTM-CNN-CRF model \cite{ma-hovy:2016:P16-1}. However, \newcite{PhamPNP2017b} did not make a comparison to SOTA traditional feature-based models. In addition, NNVLP is slow with a processing speed at about 300 words per second, which is not practical for real-world application such as dealing with large-scale data.
\setlength{\abovecaptionskip}{5pt plus 2pt minus 1pt}
\begin{figure}[!t]
\centering
\includegraphics[width=7.5cm]{VnCoreNLP_Architecture.pdf}
\caption{In pipeline architecture of VnCoreNLP, annotations are performed on an {\tt Annotation} object.}
\label{fig:diagram}
\end{figure}
In this paper, we present a Java NLP toolkit for Vietnamese, namely VnCoreNLP, which aims to facilitate Vietnamese NLP research by providing rich linguistic annotations through key NLP components of word segmentation, POS tagging, NER and dependency parsing. Figure \ref{fig:diagram} describes the overall system architecture. The
following items highlight typical characteristics of VnCoreNLP:
\begin{itemize}
\setlength{\itemsep}{5pt}
\setlength{\parskip}{0pt}
\setlength{\parsep}{0pt}
\item \textbf{Easy-to-use} -- All VnCoreNLP components are wrapped into a single .jar file, so users do not have to install external dependencies. Users can run processing pipelines from either the command-line or the Java API.
\item \textbf{Fast} -- VnCoreNLP is fast, so it can be used for dealing with large-scale data. Also it benefits users suffering from limited computation resources (e.g. users from Vietnam).
\item \textbf{Accurate} -- VnCoreNLP components obtain higher results than all previous published results on the same benchmark datasets.
\end{itemize}
\section{Basic usages}
Our design goal is to make VnCoreNLP simple to setup and run from either the command-line or the Java API. Performing linguistic annotations for a given file can be done by using a simple command as in Figure \ref{fig:command}.
\begin{figure}[ht]
{\footnotesize\ttfamily \$ java -Xmx2g -jar VnCoreNLP.jar -fin input.txt -fout output.txt}
\caption{Minimal command to run VnCoreNLP.}
\label{fig:command}
\end{figure}
Suppose that the file {\ttfamily input.txt} in Figure \ref{fig:command} contains a sentence ``Ông Nguyễn Khắc Chúc đang làm việc tại Đại học Quốc gia Hà Nội.'' (Mr\textsubscript{Ông} Nguyen Khac Chuc is\textsubscript{đang} working\textsubscript{làm\_việc} at\textsubscript{tại} Vietnam National\textsubscript{quốc\_gia} University\textsubscript{đại\_học} Hanoi\textsubscript{Hà\_Nội}). Table \ref{tab:expoutput} shows the output for this sentence in plain text form.
\begin{table}[ht]
\centering
\resizebox{8cm}{!}{
\begin{tabular}{l l l l l l}
1 & Ông & Nc & O & 4 & sub \\
2 & Nguyễn\_Khắc\_Chúc & Np & B-PER & 1 & nmod\\
3 & đang & R & O & 4 & adv\\
4 & làm\_việc & V & O & 0 & root\\
5 & tại & E & O & 4 & loc\\
6 & Đại\_học & N & B-ORG & 5 & pob\\
7 & Quốc\_gia & N & I-ORG & 6 & nmod\\
8 & Hà\_Nội & Np & I-ORG & 6 & nmod\\
9 & . & CH & O & 4 & punct\\
\end{tabular}
}
\caption{The output in file {\ttfamily output.txt} for the sentence `Ông Nguyễn Khắc Chúc đang làm việc tại Đại học Quốc gia Hà Nội.'' from file {\ttfamily input.txt} in Figure \ref{fig:command}. The output is in a 6-column format representing word index, word form, POS tag, NER label, head index of the current word, and dependency relation type.}
\label{tab:expoutput}
\end{table}
Similarly, we can also get the same output by using the API as easy as in Listing \ref{lst1}.
\begin{lstlisting}[label=lst1,caption= {Minimal code for an analysis pipeline.}]
VnCoreNLP pipeline = new VnCoreNLP() ;
Annotation annotation = new Annotation(
pipeline.annotate(annotation);
String annotatedStr = annotation.toString();
\end{lstlisting}
In addition,
Listing \ref{lst2} provides a more realistic and complete example code, presenting key components of the toolkit.
Here an annotation pipeline can be used for any text rather than just a single sentence, e.g. for a paragraph or entire news story.
\section{Components}
This section briefly describes each component of VnCoreNLP. Note that our goal is not to develop new approach or model for each component task. Here we focus on incorporating existing models into a single pipeline. In particular, except a new model we develop for the language-dependent component of word segmentation, we apply traditional feature-based models which obtain SOTA results for English POS tagging, NER and dependency parsing to Vietnamese. The reason is based on a well-established belief in the literature that for a less-resourced language such as Vietnamese, we should consider using feature-based models to obtain fast and accurate performances, rather than using neural network-based models \cite{King2015}.
\setlength{\textfloatsep}{1pt plus 1.0pt minus 1.0pt}
\begin{lstlisting}[float=tp,label=lst2,caption= {A simple and complete example code.}]
import vn.pipeline.*;
import java.io.*;
public class VnCoreNLPExample {
public static void main(String[] args) throws IOException {
// "wseg", "pos", "ner", and "parse" refer to as word segmentation, POS tagging, NER and dependency parsing, respectively.
String[] annotators = {"wseg", "pos", "ner", "parse"};
VnCoreNLP pipeline = new VnCoreNLP(annotators);
// Mr Nguyen Khac Chuc is working at Vietnam National University, Hanoi. Mrs Lan, Mr Chuc's wife, is also working at this university.
String str =
Annotation annotation = new Annotation(str);
pipeline.annotate(annotation);
PrintStream outputPrinter = new PrintStream("output.txt");
pipeline.printToFile(annotation, outputPrinter);
// Users can get a single sentence to analyze individually
Sentence firstSentence = annotation.getSentences().get(0);
}
}
\end{lstlisting}
\begin{itemize}
\setlength{\itemsep}{5pt}
\setlength{\parskip}{0pt}
\setlength{\parsep}{0pt}
\item \textbf{wseg} -- Unlike English where white space is a strong indicator of word boundaries, when written in Vietnamese white space is also used to separate syllables that constitute words. So word segmentation is referred to as the key first step in Vietnamese NLP{.}\ {W}e have proposed a transformation rule-based learning model for Vietnamese word segmentation, which obtains better segmentation accuracy and speed than all previous word segmenters. See details in \newcite{NguyenNVDJ2018}.
\item \textbf{pos} -- To label words with their POS tag, we apply MarMoT which is a generic
CRF framework and a SOTA POS and morphological
tagger \cite{mueller-schmid-schutze:2013:EMNLP}.\footnote{\url{http://cistern.cis.lmu.de/marmot/}}
\item \textbf{ner} -- To recognize named entities, we apply a dynamic feature induction model that automatically optimizes feature combinations \cite{choi:2016:N16-1}.\footnote{\url{https://emorynlp.github.io/nlp4j/components/named-entity-recognition.html}}
\item \textbf{parse} -- To perform dependency parsing, we apply the greedy version of a transition-based parsing model with selectional branching \cite{choi2015ACL}.\footnote{\url{https://emorynlp.github.io/nlp4j/components/dependency-parsing.html}}
\end{itemize}
\section{Evaluation}
We detail experimental results of the word segmentation (\textbf{wseg}) and POS tagging (\textbf{pos}) components of VnCoreNLP in \newcite{NguyenNVDJ2018} and \newcite{NguyenVNDJ-ALTA-2017}, respectively. In particular, our word segmentation component gets the highest results in terms of both segmentation F1 score at 97.90\% and speed at 62K words per second.\footnote{All speeds reported in this paper are computed on a personal computer of Intel Core i7 2.2 GHz.} Our POS tagging component also obtains the highest accuracy to date at 95.88\% with a fast tagging speed at 25K words per second, and outperforms BiLSTM-CRF-based models.
Following subsections present evaluations for the NER (\textbf{ner}) and dependency parsing (\textbf{parse}) components.
\subsection{Named entity recognition}\label{ssec:ner}
We make a comparison between SOTA feature-based and neural network-based models, which, to the best of our knowledge, has not been done in any prior work on Vietnamese NER.
\paragraph{Dataset:} The NER shared task at the 2016 VLSP workshop provides a set of 16,861 manually annotated sentences for training and development, and a set of 2,831 manually annotated sentences for test, with four NER labels PER, LOC, ORG and MISC. Note that in both datasets, words are also supplied with gold POS tags. In addition, each word representing a full personal name are separated into syllables that constitute the word.
So this annotation scheme results in an unrealistic scenario for a pipeline evaluation because: (\textbf{i}) gold POS tags are not available in a real-world application, and (\textbf{ii}) in the standard annotation (and benchmark datasets) for Vietnamese word segmentation and POS tagging \cite{nguyen-EtAl:2009:LAW-III}, each full name is referred to as a word token (i.e., all word segmenters have been trained to output a full name as a word and all POS taggers have been trained to assign a label to the entire full-name).
For a more realistic scenario, we merge those contiguous syllables constituting a full name to form a word.\footnote{Based on the gold label PER, contiguous syllables such as ``Nguyễn/B-PER'', ``Khắc/I-PER'' and ``Chúc/I-PER'' are merged to form a word as ``Nguyễn\_Khắc\_Chúc/B-PER.''} Then we replace the gold POS tags by automatic tags predicted by our POS tagging component. From the set of 16,861 sentences, we sample 2,000 sentences for development and using the remaining 14,861 sentences for training.
\paragraph{Models:} We make an empirical comparison between the VnCoreNLP's NER component and the following neural network-based models:
\begin{itemize}
\setlength{\itemsep}{5pt}
\setlength{\parskip}{0pt}
\setlength{\parsep}{0pt}
\item {BiLSTM-CRF} \cite{HuangXY15} is a sequence labeling model which extends the BiLSTM model with a CRF layer.
\item {BiLSTM-CRF + CNN-char}, i.e. {BiLSTM-CNN-CRF}, is an extension of {BiLSTM-CRF}, using CNN to derive character-based word representations \cite{ma-hovy:2016:P16-1}
\item {BiLSTM-CRF + LSTM-char} is an extension of {BiLSTM-CRF}, using BiLSTM to derive the character-based word representations \cite{lample-EtAl:2016:N16-1}.
\item BiLSTM-CRF\textsubscript{+POS} is another extension to BiLSTM-CRF, incorporating embeddings of automatically predicted POS tags \cite{reimers-gurevych:2017:EMNLP2017}.
\end{itemize}
We use a well-known implementation which is optimized for performance of all BiLSTM-CRF-based models from \newcite{reimers-gurevych:2017:EMNLP2017}.\footnote{\url{https://github.com/UKPLab/emnlp2017-bilstm-cnn-crf}} We then follow \newcite[Section 3.4]{NguyenVNDJ-ALTA-2017} to perform hyper-parameter tuning.\footnote{We employ pre-trained Vietnamese word vectors from \url{https://github.com/sonvx/word2vecVN}.}
\setlength{\textfloatsep}{20.0pt plus 2.0pt minus 4.0pt}
\begin{table}[!t]
\centering
\begin{tabular}{l|c|l}
\hline
\textbf{Model} & \textbf{F1} & \textbf{Speed} \\
\hline
VnCoreNLP & \textbf{88.55} & \textbf{18K} \\
BiLSTM-CRF & 86.48 & 2.8K \\
\ \ \ \ \ + CNN-char & {88.28} & 1.8K \\
\ \ \ \ \ + LSTM-char & 87.71 & 1.3K \\
BiLSTM-CRF\textsubscript{+POS} & 86.12 & \_ \\
\ \ \ \ \ + CNN-char & 88.06 & \_ \\
\ \ \ \ \ + LSTM-char & 87.43 & \_ \\
\hline
\end{tabular}
\caption{F1 scores (in \%) on the test set w.r.t. gold word-segmentation. ``\textbf{Speed}'' denotes the processing speed of the number of words per second (for VnCoreNLP, we include the time POS tagging takes in the speed).}
\label{tab:ner}
\end{table}
\paragraph{Main results:} Table \ref{tab:ner} presents F1 score and speed of each model on the test set, where VnCoreNLP obtains the highest score at 88.55\% with a fast speed at 18K words per second. In particular, VnCoreNLP obtains 10 times faster speed than the second most accurate model BiLSTM-CRF + CNN-char.
It is initially surprising that for such an isolated language as Vietnamese where all words are not inflected, using character-based representations helps producing 1+\% improvements to the BiLSTM-CRF model. We find that the improvements to BiLSTM-CRF are mostly accounted for by the PER label. The reason turns out to be simple: about 50\% of named entities are labeled with tag PER, so character-based representations are in fact able to capture common family, middle or given name syllables in `unknown' full-name words. Furthermore, we also find that BiLSTM-CRF-based models do not benefit from additional predicted POS tags. It is probably because BiLSTM can take word order into account, while without word inflection, all grammatical information in Vietnamese is conveyed through its fixed word order, thus explicit predicted POS tags with noisy grammatical information are not helpful.
\subsection{Dependency parsing}\label{ssec:dep}
\paragraph{Experimental setup:} We use the Vietnamese dependency treebank VnDT \cite{Nguyen2014NLDB} consisting of 10,200 sentences in our experiments. Following \newcite{NguyenALTA2016}, we use the last 1020 sentences of VnDT for test while the remaining sentences are used for training. Evaluation metrics are the labeled attachment score (LAS) and unlabeled attachment score (UAS).
\paragraph{Main results:} Table \ref{tab:dep} compares the dependency parsing results of VnCoreNLP with results reported in prior work, using the same experimental setup. The first six rows present the scores with gold POS tags. The next two rows show scores of VnCoreNLP with automatic POS tags which are produced by our POS tagging component. The last row presents scores of the joint POS tagging and dependency parsing model jPTDP \protect\cite{NguyenCoNLL2017}. Table \ref{tab:dep} shows that compared to previously published results, VnCoreNLP produces the highest LAS score. Note that previous results for other systems are reported without using additional information of automatically predicted NER labels. In this case, the LAS score for VnCoreNLP without automatic NER features (i.e. VnCoreNLP\textsubscript{--NER} in Table \ref{tab:dep}) is still higher than previous ones. Notably, we also obtain a fast parsing speed at 8K words per second.
\begin{table}[!t]
\centering
\setlength{\tabcolsep}{0.5em}
\def1.1{1.1}
\begin{tabular}{l|l|c|c|l }
\hline
\multicolumn{2}{c|}{\textbf{Model}} & \textbf{LAS} & \textbf{UAS} & \textbf{Speed} \\
\hline
\multirow{6}{*}{\rotatebox[origin=c]{90}{Gold POS}}
& VnCoreNLP & \textbf{73.39} & 79.02 & \_ \\
& VnCoreNLP\textsubscript{--NER} & 73.21 & 78.91 & \_ \\
& BIST-bmstparser & 73.17 & \textbf{79.39} & \_ \\
& BIST-barchybrid & 72.53 & 79.33 & \_ \\
& MSTParser & 70.29 & 76.47 & \_\\
& MaltParser & 69.10 & 74.91 & \_\\
\hline
\multirow{3}{*}{\rotatebox[origin=c]{90}{Auto POS}}
& VnCoreNLP & \textbf{70.23} & 76.93 & 8K \\
& VnCoreNLP\textsubscript{--NER} & 70.10 & 76.85 & \textbf{9K} \\
& jPTDP & 69.49 & \textbf{77.68} & 700 \\
\hline
\end{tabular}
\caption{LAS and UAS scores (in \%) computed on all tokens (i.e. including punctuation) on the test set w.r.t. gold word-segmentation. ``\textbf{Speed}'' is defined as in Table \ref{tab:ner}. The subscript ``--NER'' denotes the model without using automatically predicted NER labels as features. The results of the MSTParser \protect\cite{McDonald2005OLT}, MaltParser \protect\cite{Nivre2007}, and BiLSTM-based parsing models BIST-bmstparser and BIST-barchybrid \protect\cite{TACL885} are reported in \protect\newcite{NguyenALTA2016}. The result of the jPTDP model for Vietnamese is mentioned in \protect\newcite{NguyenVNDJ-ALTA-2017}.
\label{tab:dep}
\end{table}
\section{Conclusion}
In this paper, we have presented the VnCoreNLP toolkit---an easy-to-use, fast and accurate processing pipeline for Vietnamese NLP. VnCoreNLP provides core NLP steps including word segmentation, POS tagging, NER and dependency parsing. Current version of VnCoreNLP has been trained without any linguistic optimization, i.e. we only employ existing pre-defined features in the traditional feature-based models for POS tagging, NER and dependency parsing. So future work will focus on incorporating Vietnamese linguistic features into these feature-based models.
VnCoreNLP is released for research and educational purposes, and available at: \url{https://github.com/vncorenlp/VnCoreNLP}.
|
1,314,259,996,496 | arxiv |
\section*{Acknowledgement}
Special thanks to my advisor Professor Damin Wu, who has provided me incrediblly great instruction and support. The author would like to thank Professor Shing-Tung Yau, Professor Lizhen Ji and Professor Vladimir Markovic for helpful comments. Last but not least, the auother would like to thank the anonymous referees for their careful reviews and suggestions.
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\section{Introduction}\label{sec:introduction}
The main object in this paper is the punctured Riemann sphere $\punctured$, where $\mathbb{CP}^1$ is the projective space over $\mathbb{C}$ of dimension one, and $\{a_1,\ldots,a_n\}$ are $n$ different points that are omitting from $\mathbb{CP}^1$. It is well-known that there exists a unique complete K\"{a}hler-Einstein metric on $\mathbb{CP}^1\setminus\{a_1,\ldots,a_n\}$, $n\ge 3$, with negative constant Gauss curvature. However, an explicit asymptotic expansion of the metric near $a_j$, $j=1,\ldots,n$, remains unknown. More precisely, the coefficients in the expansion are difficult to determine. In this article, we derive a precise asymptotic formula for the K\"{a}hler-Einstein metric whose coefficients are polynomials on the first two parameters, which are determined by the punctures $\{a_1,\ldots,a_n\}$. Furhtermore, all coefficient polynomials can be explicitly written down when $n=3,4,6,12$.
The asymptotic expansion of the complete K\"ahler-Einstein metric on the quasi-projective manifold $M = \overline{M} \setminus D$ was proposed by Yau~\cite[p. 377]{YauReviewOfGeometry}, where $D$ is a normal crossing divisor in the project manifold $\overline{M}$ such that $K_{\overline{M}} + D > 0$. The leading order term has been known to him since the late 1970s (see \cite{Yau1978}). Several people have worked on the asymptotic expansion, see for example \cite{Schumacher1998MathAnn, DaminWu2006, RochonZhang2012}, using techniques from partial differential equations. Another important class of the quasi-projective manifolds is that $M$ is the quotient of Siegel space $\mathcal{S}_g/\Gamma$ by an arithmetic subgroup $\Gamma$, $g \ge 2$, and $\overline{M}$ is Mumford's toroidal compactification. In this case $K_{\overline{M}} + D$ is big and nef. The complete K\"ahler-Einstein volume form on $M$ has been written down in \cite{WangSmoothSiegel1993,YauZhang2014}.
The open Riemann surface $\mathbb{C}\mathbb{P}^1 \setminus \{a_1, \ldots, a_n\}$, $n\ge 3$, is only a one-dimensional example of $\overline{M} \setminus D$ with $K_{\overline{M}} + [D] > 0$. It is nevertheless the building block of a general complete quasi-projective manifold with negative holomorphic sectional curvature. Indeed, given a projective manifold $X$, for any point $x \in X$, there exists a Zariski neighborhood $U = X \setminus Z$ (where $Z$ is an algebraic subvariety of $X$) of $x$ such that $U$ can be embedded into the product
$$ S_1 \times \cdots \times S_N $$
as a closed algebraic submanifold, in which $N$ is some positive integer, and each $S_j$ is of the form $\mathbb{C}\mathbb{P}^1 \setminus \{a_1, \ldots, a_n\}$ for $n \ge 3$ (see \cite[p. 25, Lemma 2.3]{GriffithsZariski1971}). Take the complete K\"ahler-Einstein metric on each $S_j$. Then, the product metric restricted to $U$ gives a complete K\"ahler metric $\omega$ on $U$ of finite volume and negatively pinched holomorphic sectional curvature. Furthermore, by a recent result \cite[Theorem 3]{DaminYau2017}, the quasi-projective manifold $U$ possesses a complete K\"ahler-Einstein metric which is uniformly equivalent to the metric $\omega$; see also \cite[Example 5.4]{DaminYau2018}.
To derive the expansion of K\"ahler-Einstein metric on $\mathbb{C}\mathbb{P}^1 \setminus \{a_1, \ldots, a_n\}$, we
make use of the modular forms in number theory. This technique enables us to obtain the precise expansion with explicit coefficients, which were obscure in the literature by the abstract series expansion or the kernel of local linear operators. Recall the well-known uniformization theorem (see \cite{HubbardTeichmuller}) which indicates that the punctured Riemann sphere has the unit disk $\mathbb{D}$ as its universal covering space, or equivalently, the upper half plane $\mathbb{H}$. The idea is to find this covering map, then induce the Poincar\'{e} metric from $\mathbb{H}$ to our object.
F. Klein and R. Fricke had included classic theories regard automorphic function in their lecture notes (see \cite{KleinBook}). Later, the Teichm\"{u}ller space was introduced by O. Teichm\"{u}ller and developed by L. Ahlfors and L. Bers (see \cite{BersTechmuller, BersUniModuliKlein, AhlforsTeichmuller}). One of the methods that turns out very useful is the application of Schwarzian derivative. Let $f$ be a covering map for the universal covering space $\mathbb{H}\rightarrow\punctured$, then $f$ uniquely determines its Schwarzian derivative
$$\{f,\tau\}=2(f_{\tau\tau}/f_{\tau})_{\tau}-(f_{\tau\tau}/f_{\tau})^2,\quad\mbox{where }f=f(\tau),\,\tau\in\mathbb{H}.$$
On the other hand, the Schwarzian derivative of the inverse of $f$ is well-defined and uniquely determined by the following equation
\begin{equation}\label{eq:intro-schwarzian-expansion}
\{\tau,f\}=\sum_{j=1}^{n}\left(\frac{1}{(f-a_j)^2}+\frac{2\beta_j}{\alpha_j}\frac{1}{f-a_j}\right),
\end{equation}
where all $a_j$ are the omitting points from the punctured Riemann sphere, and $\beta_j, \alpha_j$ are some constants, $j=1,\ldots,n$, which will be discussed in section \ref{section:schwarzian-derivative}. By an argument from its geometric property, there are $(n-3)$ ratios from the set $\{\frac{\beta_1}{\alpha_1},\ldots,\frac{\beta_n}{\alpha_n}\}$ that are independent to each other, these constants are often called the accessory parameters. Determining the accessory parameters is a way to figure out the automorphic function. This approach has been studied by I. Kra in \cite{IrwinKraAccessoryPara} and A. B. Venkov in \cite{VenkovRussianPaper}. However, despite the results from automorphic function and Teichm\"{u}ller theory, such covering map $f:\mathbb{H}\rightarrow\punctured$ is still quite difficult to write out.
On the other hand, the congruence subgroup in $\mbox{SL}_2(\mathbb{Z})$ is a great collection of Fuchsian groups with nice properties. J. McKay and A. Sebbar discovered a connection between modular forms and Schwarzian derivative of automorphic functions in \cite{McKay2000}, which will be mentioned in section \ref{section:modular-forms}. Furthermore, there are several interesting connections between modular curve, Schwarzian derivative and graph theory that are mentioned by A. Sebbar in \cite{SebbarTorsionFree, SebbarModularCurve}.
The main result of this article is given by the following theorem.
\begin{Theorem}\label{thm:into-main-1}
A universal covering map $f:\mathbb{H}\rightarrow\mathbb{CP}^1\setminus\{a_1=0,a_2,\ldots,a_n\}$ which vanishes at infinity can be given by the following expansion
\begin{equation}
f=f(\tau)=A\left(\qk+\frac{B}{A}\qk^2+\C_3(A,\frac{B}{A})\qk^3+\sum_{m=4}^{\infty}\C_m(A,\frac{B}{A})\qk^m\right)\notag
\end{equation}
in $\qk=\exp\{\frac{2\pi i}{k}\tau\}$, $\tau\in\mathbb{H}$, for some real number $k$, where $A,B$ are constants depending only on $a_2,\ldots,a_n$, and the coefficient term $\C_m(A,\frac{B}{A})$ is a polynomial on $A$ and $\frac{B}{A}$ for $m\ge 3$. In particular, $\C_3(A,\frac{B}{A})=\frac{1}{16}\left[19\frac{B^2}{A^2}-A^2\sum_{j=2}^{n}\left(\frac{1}{a^2_j}-\frac{1}{a_j}\frac{2\beta_j}{\alpha_j}\right)\right]$, where $\alpha_j,\beta_j$ are the constants in equation \eqref{eq:intro-schwarzian-expansion}. Consequently, the complete K\"{a}hler-Einstein metric can be given by the following asymptotic expansion
\begin{align}
|ds|=\frac{1}{|f|\log|\frac{f}{A}|}\left|1-\left(\frac{B}{A}\frac{f}{A}-\frac{\Re(\frac{B}{A}\frac{f}{A})}{\log|\frac{f}{A}|}\right)+\sum_{m=2}^{\infty}R_m(A,\frac{B}{A},\frac{f}{A},\frac{f^s\overline{f^{m-s}}}{A^s\overline{A^{m-s}}\log^j|\frac{f}{A}|})\right||df|
\end{align}
at the cusp $0$, where $f\in\mathbb{CP}^1\setminus\{a_1=0,a_2,\ldots,a_n\}$ and $R_m(A,\frac{B}{A},\frac{f}{A},\frac{f^s\overline{f^{m-s}}}{A^s\overline{A^{m-s}}\log^j|\frac{f}{A}|})$ is a polynomial in $A,\frac{B}{A},\frac{f}{A},\frac{f^s\overline{f^{m-s}}}{A^s\overline{A^{m-s}}\log^j|\frac{f}{A}|}$, $s,j=0,1,\ldots,m$, with constant coefficients for $m\ge 2$.
\end{Theorem}
To illustrate the second result, we define the following values for $n(N)$,
\begin{equation}\label{eq:cusp-number-Gamma-N}
n(2)=3,\quad n(3)=4,\quad n(4)=6,\quad n(5)=12.
\end{equation}
The second result is given by the following theorem and corollary.
\begin{Theorem}\label{thm:intro-thm-2}
Let $n(N)$ be the values in equation \eqref{eq:cusp-number-Gamma-N}, and let $f_N:\mathbb{H}\rightarrow\Npunctured$ be a universal covering space with deck transformation group $\Aut(f_N)=\Gamma(N)$-the principal congruence subgroup-which vanishes at infinity, $N=2,3,4,5$. Then the map $f_N$ can be given as the following expansion
\begin{equation}\label{eq:intro-thm-2}
f_N=f_N(\tau)=A\left( \q+\frac{B}{A}\q^2+\sum_{m=3}^{\infty}\C_{m}(\frac{B}{A})\q^m\right)
\end{equation}
in $\q=\exp\{\frac{2\pi}{N}i\tau\},\tau\in\mathbb{H}$, where the constants $A,B\in\mathbb{C}$ are uniquely determined by the set of values of the punctured points $\{a_1=0,a_2,\ldots,a_{n(N)}\}$, and the coefficient term $\C_m(\frac{B}{A})$ is a polynomial in $\frac{B}{A}$ with constant coefficients for $m\ge 3$.
\end{Theorem}
\begin{remark}
Explicit formulas of $\C_m(A,\frac{B}{A})$ are given in \cite{ChoQianKobayashi} in terms of Bell polynomials.
\end{remark}
In particular, when $n=2$, $n(2)=3$, it is a triple punctured Riemann sphere, the covering map $\mathbb{H}\rightarrow\mathbb{CP}^1\setminus\{a_1=0,a_2,a_3\}$ for arbitrary $a_2,a_3$ is given in the following example.
\begin{example}\label{example:intro-thm-2}
The covering map $f_2:\mathbb{H}\rightarrow\mathbb{CP}^1\setminus\{a_1=0,a_2,a_3\}$ which vanishes at infinity has the following expansion
\begin{equation*}
f_2(\tau)=\frac{16a_2a_3}{a_3-a_2}\left[\qsec-\left(8+\frac{16a_2a_3}{a_3-a_2}\right)\qsec^2+\sum_{m=3}^{\infty}\C_m\left(8+\frac{16a_2a_3}{a_3-a_2}\right)\qsec^m\right],
\end{equation*}
in $\qsec=\exp\{\pi i\tau\}$, $\tau\in\mathbb{H}$, where $\C_m\left(-8-\frac{16a_2a_3}{a_3-a_2}\right)$ is a polynomial in the term $\left(-8-\frac{16a_2a_3}{a_3-a_2}\right)$ for $m\ge 3$. In particular, $\C_3=\left(-8-\frac{16a_2a_3}{a_3-a_2}\right)^2-20$. In this case, the constants $A,B$ in Theorem \ref{thm:intro-thm-2} are given by the following
\begin{equation*}
A=\frac{16a_2a_3}{a_3-a_2},\qquad B=\left(\frac{16a_2a_3}{a_3-a_2}\right)\cdot\left[-\left(8+\frac{16a_2a_3}{a_3-a_2}\right)\right].
\end{equation*}
\end{example}
\begin{Coro}\label{coro:intro}
As a consequence of Theorem \ref{thm:intro-thm-2}, the complete K\"{a}hler-Einstein metric on $\Npunctured$ has the following asymptotic expansion
\begin{equation}
|ds|=\frac{1}{|f|\log|\frac{f}{A}|}\left|1-\left(\frac{B}{A}\frac{f}{A}-\frac{\Re(\frac{B}{A}\frac{f}{A})}{\log|\frac{f}{A}|}\right)+\sum_{m=2}^{\infty}R_m(\frac{B}{A},\frac{f}{A},\frac{f^s\overline{f^{m-s}}}{A^s\overline{A^{m-s}}\log^j|\frac{f}{A}|})\right||df|
\end{equation}
at the cusp $a_1=0$, where $R_m(\frac{B}{A},\frac{f}{A},\frac{f^s\overline{f^{m-s}}}{A^s\overline{A^{m-s}}\log^j|\frac{f}{A}|})$ is a polynomial in $\frac{B}{A},\frac{f}{A},\frac{f^s\overline{f^{m-s}}}{A^s\overline{A^{m-s}}\log^j|\frac{f}{A}|}$, $s,j=0,1,\ldots,m$, with constant coefficients for $m\ge 2$.
\end{Coro}
As an application of our result, several examples are given in the section \ref{section:example}. For example, we give the complete K\"{a}hler-Einstein metric of $\mathbb{CP}^1\setminus\{0,1,\infty\}$ in the following example.
\begin{example}\label{example:intro}
The covering map $f_2:\mathbb{H}\rightarrow\mathbb{CP}^1\setminus\{0,1,\infty\}$ has the following expansion
\begin{equation}
f=f_2(\tau)=16\qsec-128\qsec^2+704\qsec^3-3072\qsec^4+11488\qsec^5+O(\qsec^6)
\end{equation}
in $\qsec=\exp\{\pi i\tau\}$, $\tau\in\mathbb{H}$, and we write $f=f_2$ for convenience. In this case, the constants $A,B$ in Theorem \ref{thm:intro-thm-2} take values $A=16, B=-128$. Therefore the coefficients of $f=f_2$ in equation \eqref{eq:intro-thm-2} are givien by $B/A=\frac{-128}{16}=-8$, $\C_3=(B/A)^2-20=(-8)^2-20=44$. From Corollary \ref{coro:intro}, the complete K\"{a}hler-Einstein metric has the following explicit expansion
\begin{align*}
|ds|&=\frac{1}{|f|\log|f/16|}\left|1+\frac{1}{2}\left(f-\frac{\Re f}{\log|f/16|}\right)\right.\\
&\qquad\left.-\left[\frac{51}{32}f^2-\frac{1}{4}\frac{f\Re f}{\log|f/16|}+\frac{51}{64}\frac{\Re(f^2)}{\log|f/16|}+\frac{1}{4}\frac{(\Re f)^2}{\log^2|f/16|}\right]+O(f^3)\right||df|
\end{align*}
at the cusp $0$, where $f\in\mathbb{CP}^1\setminus\{0,1,\infty\}$.
\end{example}
\begin{remark}
The K\"ahler-Einstein metric on $\mathbb{CP}^1\backslash\{0,1,\infty\}$ was known by S. Agard. He gave a globle explicit formula in terms of a double integral (see equation (2.6) in \cite{AgardDist}).
\end{remark}
\begin{remark}
The argument of this article can derive a general case of $a_1,a_2,a_3$, please see Corollaries \ref{coro:section-8-1} and \ref{coro:section-8-2}.
\end{remark}
In this article, we start with studying the fundamental group $\pi_1$ of the punctured Riemann sphere, its topology indicates that every element from $\pi_1(\punctured)$ corresponds to an automorphism of its universal cover, more precisely an element in $\mbox{SL}_2(\mathbb{R})$. Two similar arguments from its monodromy action imply Theorem \ref{thm:generator-are-parabolic}, which indicates that each generator is of \textit{parabolic} type. This fact allows us to interpret the covering map $f$ as an expansion at the corresponding cusp, so as the complete K\"{a}hler-Einstein metric. In the next section, section \ref{section:schwarzian-derivative}, we introduce and discuss the Schwarzian derivative in the analytic sense, and derive Theorem \ref{thm:coefficient-depend-on-A-B} at the end, which plays an important role to our main result. In section \ref{section:ramification-point}, we will focus on the action of a discrete group on $\mathbb{H}$. Couple propositions are introduced in section \ref{section:modular-group} in order to target the congruence subgroups that are of genus \textit{zero} without \textit{elliptic} point. In section \ref{section:modular-forms}, we introduce the modular forms and discuss its relation with the Schwarzian derivative. The main results and couple examples are provided in the last section
\section{Construction of Universal Covering Space}\label{section:construction of universal covering space}
We will sketch the process of the universal covering from $\mathbb{H}$ to $\punctured$ in the graphic way. The general procedure of constructing the universal covering space is mentioned in \cite[p. 8-12, section 2]{NevaROLF}. I will describe the most straightforward case in this article, which will be sufficient and helpful for readers to understand the topic. To cover the punctured Riemann sphere $\punctured$ from the unit disk $\mathbb{D}$, or equivalently, the upper half plane $\mathbb{H}$, we first connect the $n$ punctures such that it is a polygon without self-crossing, rename the vertexes as $a_1,a_2,\ldots,a_n$ counter-clockwisely. Then, let us pick $n$ points randomly on the unit circle, and name them $c_1,\ldots,c_n$ counter-clockwisely which correspond to $a_1,\ldots,a_n$ respectively, and let $c_1=1, c_n=-1$. The collection of the $n$ points all lie on the upper half unit circle. Connecting the two points that are right next to each other by arcs. In particular, $c_1$ and $c_n$ are connected by the diameter, and we denote this circular polygon as $P_1$. Reflect $P_1$ with respect to the side $c_1c_n$, it will result in $(n-2)$ points from the reflections of $\{c_2,\ldots,c_{n-1}\}$ on the lower half circle, we mark those points counter-clockwisely by $c_{n+1},\ldots,c_{2n-2}$, and mark the reflected polygon by $P_2$. Then we call the combined polygon $P=P_1+P_2$ a fundamental polygon of this covering (see Figure \ref{fig:PolygoOnDisk}).
\begin{figure}
\input{picdiskcovergeodesic}
\caption{Polygon on Disk}
\label{fig:PolygoOnDisk}
\end{figure}
Next we construct the reproduction of the covering process. Let $V$ be the set of vertexes with either only odd index or even index. For a vertex $c_m\in V$, reflect $P$ with respect to an adjacent side $c_{m-1}c_m$, it results in a new polygon $\tilde{P}=\tilde{P}_1+\tilde{P}_2$, where $\tilde{P}_i$ is the reflecting image of $P_i$, $i=1,2$. Notice that $\tilde{P}_1$ is the image of $P_1$ from reflections of odd times. $\tilde{P}_2$ is the image of $P_1$ from reflections of even times since $\tilde{P}_2$ is the reflection of $P_2$ and $P_2$ is the reflection of $P_1$. Next we reflect $P$ with respect to $c_mc_{m+1}$, which is the other adjacent side of $c_m$. We replicate such reflections on both sides of each vertex in the set $V$, it will result in a new polygon $P^{(1)}$ on $\mathbb{D}$, where $P^{(1)}$ is composed by $2\times(n-1)+1=2n-1$ copies of $P^{(0)}=P$, and it has $2\times(2n-3)\times(n-1)$ sides and vertexes. Keep repeating this process on every $P^{(i)}$, it will cover the unit disk $\mathbb{D}$ as $i$ approaches $\infty$.
If we change the universal covering space to the upper half plane $\mathbb{H}$, the construction will be the same through the Cayley transformation,
\begin{equation}\label{eq:cayley-transformation}
t:\mathbb{H}\rightarrow\mathbb{D},\qquad z\mapsto t(z)=\frac{z-i}{z+i}.
\end{equation}
For example, Figure \ref{fig:PolygonOnH} shows the corresponding fundamental polygon on $\mathbb{H}$.
\begin{figure}
\input{picdisktoH}
\caption{Polygon on $\mathbb{H}$}
\label{fig:PolygonOnH}
\end{figure}
\section{Deck Transformation Group and Expansion in $q_{_k}$}
\subsection{Deck Transformation Group and Generators}\label{subsection:deck-group-generators}
Let $x\in\punctured$ be a base point and $\gamma:[0,1]\rightarrow\punctured$ be a non-trivial loop with $\gamma(0)=\gamma(1)=x$. Assume $f:\mathbb{H}\rightarrow\punctured$ is a universal covering map. For an arbitrary pre-image $\tau_1\in f^{-1}(x)\subseteq\mathbb{H}$, the end point $\tau_2$ of the lift $\tilde{\gamma}$ with initial point $\tau_1$ is uniquely determined by $\gamma$ and the choice of $\tau_1$. The uniqueness implies that the non-trivial loop $\gamma$ induces a permutation on points of $f^{-1}(x)$, and such permutation induces an isomorphism of its covering space $\mathbb{H}$.
\begin{definition}
We use $\Aut(f)$ to denote the deck transformation group of the universal covering $f:\mathbb{H}\rightarrow\punctured$. Equivalently, we have $f\circ \gamma(\tau)=f(\tau)$ for every element $\gamma\in\Aut(f)\subseteq\Aut(\mathbb{H})$ and every point $\tau\in\mathbb{H}$.
\end{definition}
There is a one-to-one correspondence between the deck transformation group $\Aut(f)$ and its fundamental group:
$$\Aut(f)\,\leftrightarrow\,\pi_1(\punctured), \qquad h\,\leftrightarrow\,\mbox{loop}.$$
Notice that the fundamental group $\pi_1(\punctured)$ is a free group generated by the $(n-1)$ loops, each of which circles around only one punctured point $a\in\{a_1,\cdots,a_n\}$. Therefore the corresponding relation implies that $\Aut(f)$ is generated by $(n-1)$ elements from $\Aut(\mathbb{H})$. We state the conclusion as the following statement.
\begin{theorem}\label{thm:fundalmental-group-first-thm}
The deck transformation group $\Aut(f)$ of the universal covering $f:\mathbb{H}\rightarrow\punctured$ is a free group generated by $(n-1)$ elements from $\Aut(\mathbb{H})$, where each generator corresponds to a single loop in $\pi_1(\punctured)$.
\end{theorem}
Next we will discuss the properties of $\Aut(f)$. Let us recall some elementary definitions.
\begin{definition}\label{def:matrix-acts-on-upper}
An element $\gamma=\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\in\mbox{GL}_2(\mathbb{C})$ acts on $[z_0:z_1]\in\mathbb{CP}^1$ in the following sense
\begin{equation}
\gamma([z_0:z_1])=\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\left(\begin{array}{c}
z_0\\
z_1
\end{array}\right)=\left(\begin{array}{c}
az_0+b\\
cz_1+d
\end{array}\right)=[az_0+b:cz_1+d].\notag
\end{equation}
\end{definition}
\begin{remark}
Let $\gamma,\gamma_1,\gamma_2\in\mbox{GL}_2(\mathbb{C})$, and $[z_0:z_1]\in\mathbb{CP}^1$.
\begin{enumerate}
\item $\gamma_2(\gamma_1([z_0:z_1]))=(\gamma_2\cdot\gamma_1)([z_0:z_1])$ because of the associativity of matrix multiplication.
\item If we consider $z=\frac{z_0}{z_1}$ when $z_1\neq 0$, and $z=\infty$ when $z_1=0$, then we have $\gamma(z)=\frac{az+b}{cz+d}$ and $\gamma(\infty)=\frac{a}{c}$.
\end{enumerate}
\end{remark}
\begin{definition}
Let $z\in\mathbb{CP}^1$ and an element $\gamma\in\mbox{GL}_2(\mathbb{C})$ such that $\gamma(z)=z$, we say that $z$ is fixed under the action of $\gamma$, or, equivalently, $z$ is invariant under $\gamma$.
\end{definition}
Elements in $\Aut(\mathbb{H})$ are the ones that we are interested in. On one side, we know that $\Aut(\mathbb{CP}^1)\subseteq\mbox{SL}_2(\mathbb{C})$, therefore we have $\Aut(\mathbb{H})\subseteq\mbox{SL}_2(\mathbb{C})$. On the other side, the isomorphism of $\mathbb{H}$ keeps its boundary $\mathbb{R}$ invariant, this implies that $\Aut(\mathbb{H})=\mbox{SL}_2(\mathbb{R})$. We will focus on the group $\mbox{SL}_2(\mathbb{R})$ from now on.
\begin{prop}\label{prop:invariant-pt-infty-0}
Let $\omega$ be a linear transformation in $\mbox{SL}_2(\mathbb{R})$, and $\tau\in\mathbb{H}$.
\begin{enumerate}
\item If $0,\infty$ are invariant under the action of $\omega$, i.e., $\omega(0)=0$ and $\omega(\infty)=\infty$, then $\omega(\tau)=\lambda \tau$ for some $0<\lambda\in\mathbb{R}$.
\item If $\omega$ has only one invariant point $\infty$, i.e., $\omega(\infty)=\infty$, then $\omega(\tau)=\tau+k$ for some $k\in\mathbb{R}$.
\item If $\omega$ has only one invariant point $i=\sqrt{-1}$, i.e., $\omega(i)=i$, then $\frac{\omega(\tau)-i}{\omega(\tau)+i}=e^{i\theta}\frac{\tau-i}{\tau+i}$ for some $\theta\in\mathbb{R}$.
\end{enumerate}
\end{prop}
\begin{proof}
Assume $\omega=\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\in\mbox{SL}_2(\mathbb{R})$. The condition $\omega(\infty)=\infty$ implies that $\frac{a}{c}=\infty$, so we get $c=0$ and $a\neq 0$ since $ad-bc=1$. And $\omega(0)=0$ implies $\frac{b}{d}=0$, so $b=0$ and $d\neq 0$.
\begin{enumerate}
\item If both of $0$ and $\infty$ are invariant under $\omega$, we have $\omega=\left(\begin{array}{cc}
a & 0\\
0 & d
\end{array}\right)$ and $\omega(z)=\frac{a}{d}z$, specially $\lambda=\frac{a}{d}>0$ since $ad=1>0$.\\
\item If $\infty$ is the only invariant point under the action of $\omega$, we have $\omega=\left(\begin{array}{cc}
a & b\\
0 & d
\end{array}\right)$ where $b\neq 0$. Any point other than $\infty$ are not invariant means that the equation $\omega(z)=\frac{a}{d}z+\frac{b}{d}=z$ has no solution, so $\frac{a}{d}=1$. Let $\frac{b}{d}=k$, we conclude $\omega(z)=z+k$.\\
\item For the last case, the composition $t\circ\omega\circ t^{-1}$ is an automorphism of the unit disk $\mathbb{D}$ with $0$ being fixed, where $t$ denotes the Cayley transformation \eqref{eq:cayley-transformation}. Therefore we have relation $t\circ\omega\circ t^{-1}(\tilde{z})=e^{i\theta}\tilde{z}$ for $\tilde{z}\in\mathbb{D}$. Let $\tilde{z}=t(z)$, the following equation
$$t\circ\omega\circ t^{-1}(\tilde{z})=t\circ\omega\circ t^{-1}\circ t(z)=t\circ\omega(z)=e^{t\theta}t(z)$$
holds, where the last equality implies $\frac{\omega(\tau)-i}{\omega(\tau)+i}=e^{i\theta}\frac{\tau-i}{\tau+i}$.
\end{enumerate}
\end{proof}
\begin{prop}\label{prop:Classification-SL2}
For an element $\omega\in\mbox{SL}_2(\mathbb{C})$, it fits in one of the following three situations.
\begin{enumerate}
\item If $\omega$ has two different invariant points on the boundary $\mathbb{R}\cup\{\infty\}$, it is called of \textit{hyperbolic} type. Furthermore, if $r_1<r_2\in\mathbb{R}$ are the two invariant points, then the transformation has expression $$\frac{\omega(z)-r_2}{\omega(z)-r_1}=\lambda\frac{z-r_2}{z-r_1},\quad\mbox{for some }0<\lambda\in\mathbb{R}.$$
\item If $\omega$ has only one invariant point on the boundary $\mathbb{R}\cup\{\infty\}$, it is called of \textit{parabolic} type. Furthermore, when the invariant point $r\neq\infty$, it has expression
$$-\frac{1}{\omega(z)-r}=-\frac{1}{z-r}+k,\quad\mbox{for some } k\in\mathbb{R}.$$
\item If $\omega$ has only one invariant point $r\in\mathbb{H}$ in the interior of $\mathbb{H}$, it is called of \textit{elliptic} type. It has expression $$\frac{\omega(z)-r}{\omega(z)-\overline{r}}=e^{i\theta}\frac{z-r}{z-\overline{r}},\quad \mbox{for some }\theta\in\mathbb{R}.$$
\end{enumerate}
\end{prop}
\begin{proof}
The proof is elementary. It directly follows from Proposition \ref{prop:invariant-pt-infty-0}. Also see \cite[p. 7, Proposition 1.13]{ShimuraIntroToArith} and \cite[p. 8, section 1.6]{NevaROLF}.
\end{proof}
\begin{comment}
\begin{proof}
Assume $\omega=\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\in\mbox{SL}_2(\mathbb{R})$, if $z\in\mathbb{C}$ is invariant under $\omega$, i.e., $\omega(z)=z$, we have the following equation:
\begin{equation}\notag
z=\frac{az+b}{cz+d}\quad\mbox{implies}\quad cz^2+(d-a)z-b=0.
\end{equation}
Its determinant $\Delta=(d-a)^2+4bc$ shows that when $\Delta>0$, there are two real solutions, it is of \textit{hyperbolic} type; when $\Delta=0$, it has one and only real solution, it is of \textit{parabolic} type; and when $\Delta<0$, it has two non-real solutions, we say it is of \textit{elliptic} type. We consider them separately.
\begin{enumerate}
\item\textit{Hyperbolic} type.
$\Delta>0$, we assume the two different real solutions are $r_1$ and $r_2$, without loss of generality, let us also assume $r_1< r_2\in\mathbb{R}$. Consider the map $\left(\begin{array}{cc}
1 & -r_1\\
1 & -r_2
\end{array}\right)\in\mbox{GL}_2(\mathbb{R})\subseteq\Aut(\mathbb{H})$ which maps the following way
\begin{equation*}
z\mapsto\frac{z-r_2}{z-r_1},\qquad r_2\mapsto 0,\qquad r_1\mapsto\infty,
\end{equation*}
it shows that the transformation between $\frac{\omega(z)-r_2}{\omega(z)-r_1}$ and $\frac{z-r_2}{z-r_1}$ is an automorphism of $\mathbb{H}$ that keeps $0$ and $\infty$ invariant. From proposition \ref{prop:invariant-pt-infty-0}, this automorphism has the form $\bullet\mapsto\frac{a}{d}\bullet=\lambda\cdot\bullet$ for some $\lambda\in\mathbb{R}^+$. Therefore, if $\omega$ fixes two points $r_1<r_2$ on the real line $\mathbb{R}$, there is a positive real number $\lambda$ such that should have the following relation:
\begin{equation}\label{HCase1}
\frac{\omega(z)-r_2}{\omega(z)-r_1}=\lambda\frac{z-r_2}{z-r_1},\qquad\mbox{for some } \lambda\in\mathbb{R}^+.
\end{equation}
\item \textit{Parabolic} type.
$\Delta=0$, we assume $r\in\mathbb{R}$ is the only invariant point under $\omega$. Consider the map $z\mapsto -\frac{1}{z-r}$ in $\Aut(\mathbb{H})$ which sends $r$ to $\infty$. Similarly, the transformation between $-\frac{1}{\omega(z)-r}$ and $-\frac{1}{z-r}$ is an automorphism of $\mathbb{H}$ that only fixes $\infty$. Proposition \ref{prop:invariant-pt-infty-0} implies that such transformation has the form $\bullet\mapsto\bullet+\frac{b}{d}=\bullet+k$ for some $k\in\mathbb{R}$. We conclude the following relation:
\begin{equation}\label{HCase2}
-\frac{1}{\omega(z)-r}=-\frac{1}{z-r}+k, \qquad\mbox{for some } k\in\mathbb{R}.
\end{equation}
\item\textit{Elliptic type.} If $\Delta<0$, the equation $cz^2+(d-a)z-b=0$ has two non-real solutions that are conjugate to each other, assume they are $r\in\mathbb{H}$ and $\overline{r}$. The transformation $\frac{z-r}{z-\overline{r}}$ maps $\mathbb{H}\rightarrow\mathbb{D}$ with $r\mapsto 0$. By a similar argument, proposition \ref{prop:invariant-pt-infty-0} implies the relation
\begin{equation}\label{HCase3}
\frac{\omega(z)-r}{\omega(z)-\overline{r}}=e^{i\theta}\frac{z-r}{z-\overline{r}},\qquad\mbox{for some }\theta\in\mathbb{R}.
\end{equation}
\end{enumerate}
\end{proof}
\end{comment}
Next, we will show that each generator in Theorem \ref{thm:fundalmental-group-first-thm} has to be of \textit{parabolic} type. For a universal covering space $f:\mathbb{H}\rightarrow\punctured$, it is equivalent to say that $f$ is biholomorphic. Suppose an element $\gamma\in\Aut(f)$ corresponds to a generating loop $l$ in $\pi_1(\punctured)$, where $l:[0,1]\rightarrow\punctured$ and $l$ isolates only one puncture $a\in\{a_1,\ldots,a_n\}$. Fix a point $\tau_0$ in the fiber $f^{-1}(l(0))$, the lift $\tilde{l}(t)$ of $l(t)$ with initial point $\tau_0$ is uniquely determined and so as its terminal point $\tilde{l}(1)=\tau_1$. Similarly,
the lift $\widetilde{l^n}$ of $l^n=l\ast\cdots\ast l$, is uniquely determined by the fixed initial point $\tau_0$, where $\ast$ means $(l\ast l)(t)=\left\{\begin{array}{ll}
l(2t), & 0\le t<\frac{1}{2},\\
l(2t-1), & \frac{1}{2}\le t\le 1.
\end{array}\right.$
Denote the terminal point $\tau_n=\widetilde{l^n}(1)$, meanwhile $\tau_n\in f^{-1}(l(0))$. By the corresponding relation between $l$ and $\gamma$, we have $\tau_n=(l\ast\cdots\ast l)^{\sim}(1)=\gamma\circ\cdots\circ \gamma(\tau_0)=\gamma^n(\tau_0)$. We will prove that this $\gamma$ can not be of \textit{hyperbolic} type by analyzing its monodormy behavior. Similarly, we can conclude that $\gamma$ can not be of \textit{elliptic} type either. The proofs of the following propositions are elementary, but it is helpful to understand the uniformizing function at singularities. For this purpose, I will include one of the proofs.
\begin{prop}\label{prop:generator-not-hyper}
The generator is not of \textit{hyperbolic} type.
\end{prop}
\begin{proof}
Suppose $l$ is a loop in $\punctured$ which isolates a puncture $a\in\{a_1,\ldots,a_n\}$. $\gamma$ is the corresponding element in $\Aut(f)$, where $f$ is the universal covering map. If $\gamma$ is of \textit{hyperbolic} type, then there exists $r_1<r_2\in\mathbb{R}$ which are the two invariant points of $\gamma$, and let $\lambda\in\mathbb{R}^+$ be the constant in Proposition \ref{prop:Classification-SL2} such that the relation
$$\frac{\gamma(\tau)-r_2}{\gamma(\tau)-r_1}=\lambda\frac{\tau-r_2}{\tau-r_1},\qquad\tau\in\mathbb{H}$$
is satisfied. Therefore the following equality
$$
\frac{\tau_{n+1}-r_2}{\tau_{n+1}-r_1}=\frac{\gamma(\tau_n)-r_2}{\gamma(\tau_n)-r_1}=\lambda\frac{\tau_n-r_2}{\tau_n-r_1}$$
implies relation
$$\frac{\widetilde{l^{n+1}}(1)-r_2}{\widetilde{l^{n+1}}(1)-r_1}=\lambda\frac{\widetilde{l^n}(1)-r_2}{\widetilde{l^n}(1)-r_1}=\lambda^n\frac{\widetilde{l}(1)-r_2}{\widetilde{l}(1)-r_1},$$
which leads to the following equation
\begin{equation}\label{eq:generator-not-hyper}
\frac{\tau_n-r_2}{\tau_n-r_1}=\lambda^n\frac{\tau_0-r_2}{\tau_0-r_1}.
\end{equation}
On the branch $0<\arg\left(\frac{\tau-r_2}{\tau-r_1}\right)<\pi$, the following calculation
\begin{align}\label{eq:not-hyperbolic-1}
\exp\{\frac{2\pi i}{\log\lambda}\cdot\log\frac{\tau_n-r_2}{\tau_n-r_1}\}
&=\exp\{\frac{2\pi i}{\log\lambda}\cdot\log\left(\lambda^n\cdot\frac{\tau_0-r_2}{\tau_0-r_1}\right)\},\notag\\
&=\exp\{\frac{2\pi i}{\log\lambda}\cdot n\log\lambda\}\cdot\exp\{\frac{2\pi i}{\log\lambda}\cdot\log\frac{\tau_0-r_2}{\tau_0-r_1}\},\notag\\
&=\exp\{\frac{2\pi i}{\log\lambda}\cdot\log\frac{\tau_0-r_2}{\tau_0-r_1}\}
\end{align}
holds since $\lambda\in\mathbb{R}^+$. Define a function
$$\Phi(x)=\exp\left\{\frac{2\pi i}{\log\lambda}\cdot\log\left(\frac{f^{-1}(x)-r_2}{f^{-1}(x)-r_1}\right)\right\},\quad U_l-\{a\}\rightarrow\mathbb{C},$$
where $U_{l}$ is the open set bounded by $l$ with $a$ inside. Notice that the value defined by the expression $\exp\left\{\frac{2\pi i}{\log\lambda}\cdot\log\left(\frac{f^{-1}\circ l(0)-r_2}{f^{-1}\circ l(0)-r_1}\right)\right\}$ is single valued over the fiber of $l(0)$ due to equation \eqref{eq:not-hyperbolic-1}. Applying the same argument on every point $x\in U_{l}-\{a\}$, we conclude that $\Phi(x)$ is a well-defined single valued function on $U_{l}-\{a\}$ over the branch $0<\arg \frac{f^{-1}(x)-r_2}{f^{-1}(x)-r_1}<\pi$. We estimate $|\Phi(x)|$ in $U_{l}-\{a\}$ as the following,
\begin{align}
|\Phi(x)| &=\left|\exp\left\{\frac{2\pi i}{\log\lambda}\cdot\log\left(\frac{f^{-1}(x)-r_2}{f^{-1}(x)-r_1}\right)\right\}\right|,\notag\\
&=\left|\exp\left\{\frac{2\pi i}{\log\lambda}\right\}\cdot\log\left|\frac{f^{-1}(x)-r_2}{f^{-1}(x)-r_2}\right|\right|\cdot\exp\left\{-\frac{2\pi}{\log\lambda}\cdot\arg\frac{f^{-1}(x)-r_2}{f^{-1}(x)-r_1}\right\},\notag\\
&=\exp\left\{-\frac{2\pi}{\log\lambda}\cdot\arg\frac{f^{-1}(x)-r_2}{f^{-1}(x)-r_1}\right\}.\label{eq:not-hyperbolic-2}
\end{align}
Over the branch $0<\arg \frac{f^{-1}(x)-r_2}{f^{-1}(x)-r_1}<\pi$, equation \eqref{eq:not-hyperbolic-2} implies that $|\Phi(x)|$ is bounded between two positive values $\exp\left\{-\frac{2\pi^2}{\log\lambda}\right\}$ and $1=\exp\{-\frac{2\pi}{\log\lambda}\cdot 0\}$. By Cauchy-Riemann theorem, $a$ is a removable singularity and $\Phi(a)\neq 0$, thus the exponential of $\Phi(x)$ will be regular at $x=a$, as well as $f^{-1}(x)$. This contradicts with the fact that $a$ is an essential singularity.
\end{proof}
\begin{prop}\label{prop:generator-not-elliptic}
The generator is not of \textit{elliptic} type.
\end{prop}
\begin{proof}
It is similar to the proof of Proposition \ref{prop:generator-not-hyper} (see \cite[p. 16-17, section 3.2]{NevaROLF}).
\end{proof}
From Propositions \ref{prop:generator-not-hyper} and \ref{prop:generator-not-elliptic}, we have the following conclusion.
\begin{theorem}\label{thm:generator-are-parabolic}
The deck transformation group $\Aut(f)$ of a universal covering space $f:\mathbb{H}\rightarrow\punctured$ is generated by $(n-1)$ \textit{parabolic} elements in $\mbox{SL}_2(\mathbb{R})$.
\end{theorem
\subsection{The Expansion and Metric Formula}\label{section:qk-expansion-metric-formula}
Suppose $\gamma_1,\ldots,\gamma_{n-1}$ are the $(n-1)$ \textit{parabolic} generators corresponding to the $(n-1)$ generating loops of $\pi_1(\punctured)$. Theorem \ref{thm:generator-are-parabolic} shows that each of them satisfies the following relation
\begin{equation*}
-\frac{1}{\gamma_j(\tau)-r_j}=-\frac{1}{\tau-r_j}+k_j,\qquad \mbox{when $r_j\neq\infty$},
\end{equation*}
or
\begin{equation*}
\gamma_j(\tau)=\tau+k_j,\qquad\mbox{when $r_j=\infty$,}
\end{equation*}
where $r_j$ and $k_j$ are the constants corresponding to $\gamma_j$, $j=1,\ldots,n-1$, in Proposition \ref{prop:Classification-SL2}. Let $a_j$ be the singularity that corresponds to $\gamma_j$, the function
\begin{equation*}
\Phi_j(x)=\left\{\begin{array}{ll}
e^{-\frac{2\pi i}{k_j}\frac{1}{f^{-1}(x)-r_j}}, & \mbox{when }r_j\neq\infty,\\
e^{\frac{2\pi i}{k_j}f^{-1}(x)}, & \mbox{when }r_j=\infty,
\end{array}\right.\qquad U_l-\{a_j\}\rightarrow\punctured
\end{equation*}
is biholomorphic, where $U_l-\{a_j\}$ is a punctured neighborhood that is defined similar to the one in Proposition \ref{prop:generator-not-hyper}. Such function $\Phi_j(x)$ is single valued on the neighborhood $U_l-\{a_j\}$ around the singularity $a_j$. We call this function a uniformizing function at $a_j$.
Without loss of generality, we will focus on the situation $\infty\in\{r_1,\ldots,r_{n-1}\}$ for convenience. Let $f:\mathbb{H}\rightarrow\punctured$ be a universal covering map, assume $r_1=\infty$, and let $\gamma_1$ be the corresponding generator and $a_1$ be the corresponding singularity. The uniformizing function $\Phi_1(x)$ is defined as the following
\begin{equation*}
\Phi_1(x)=e^{\frac{2\pi i}{k_1}f^{-1}(x)}, \qquad U_l-\{a_1\}\rightarrow\punctured.
\end{equation*}
If $\gamma_1\in\mbox{SL}_2(\mathbb{R})$ is a generator of $\Aut(f)$ that corresponds to $r_1$, so $\gamma_1$ fixes $\infty$, we have the following periodic property,
\begin{equation*}
f(x)=f\circ\gamma_1(x)=f(x+k_1).
\end{equation*}
Fundamental Fourier analysis implies the following proposition.
\begin{prop}\label{prop:coveringmap-global-expression}
Let $f$ be a covering map $f:\mathbb{H}\rightarrow\punctured$ such that a generator of $\Aut(f)$ fixes $\infty$. Then the map $f$ has the following global expansion
\begin{equation}\label{eq:f-qk-expansion}
f(\tau)=f(\qk)=c_0+A\qk+B\qk^2+\sum_{m=3}^{\infty}c_m\qk^m
\end{equation}
in $\qk=\qk(\tau)=\exp\{\frac{2\pi i}{k}\tau\}$ for any $\tau\in\mathbb{H}$, where $k\in\mathbb{R}$ is the constant that corresponds to the generator in Proposition \ref{prop:Classification-SL2}, and the constant $A\neq 0$.
\end{prop}
\begin{proof}
If $\infty$ is invariant under a generator $\gamma\in\Aut(f)$, then there is a constant $k\in\mathbb{R}$ such that $\gamma(\tau)=\tau+k$, which implies that $f$ has the following periodic property
$$f(\tau+k)=f(\tau),\qquad \mbox{for any }\tau\in\mathbb{H}.$$
Consider $f(\tau)=f(x+iy)=f_y(x)$ as a function of $x$ on $\mathbb{R}$. Its Fourier series
\begin{equation}\label{eq:FourierExpansion1}
f(x+iy)=f_y(x)=\sum_{n=-\infty}^{\infty}c_n(y)e^{\frac{2\pi}{k}inx},
\end{equation}
converges uniformly on $x$ due to Corollaries 2.3 and 2.4 in \cite[p. 41-42]{SteinFourier}, where the coefficients are given by the following equation
\begin{equation}
c_n(y)=\frac{1}{k}\int_{0}^{k}f_y(x)e^{-\frac{2\pi}{k}inx}dx\notag.
\end{equation}
The Cauchy-Riemann equation implies the following equation
\begin{equation}
i\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}
\end{equation}
holds since $f(\tau)$ is holomorphic. Calculating the left hand side, due to the uniform convergence on $x$, we have equation
\begin{align}
i\frac{\partial f}{\partial x}
&=i\frac{\partial}{\partial x}\left(\sum_{n=-\infty}^{\infty}c_n(y)e^{\frac{2\pi}{k}inx}\right)=i\left(\sum_{n=-\infty}^{\infty}c_n(y)e^{\frac{2\pi}{k_1}inx}\cdot\frac{2\pi}{k}in\right),\notag\\
&=\sum_{n=-\infty}^{\infty}-\frac{2n\pi}{k}c_n(y)e^{\frac{2\pi}{k}inx}.\notag
\end{align}
Also calculating the right hand side, we have equation
\begin{align*}
\frac{\partial f}{\partial y}=\sum_{n=-\infty}^{\infty}c_n'(y)e^{\frac{2\pi}{k}inx}.
\end{align*}
Matching the coefficients, the differential equation
\begin{equation*}
\frac{2n\pi}{k}c_n(y)+c_n'(y)=0
\end{equation*}
holds, which has solution
\begin{equation*}
c_n(y)=C_ne^{-\frac{2\pi}{k}ny},\quad\mbox{for some constant $C_n\in\mathbb{C}$.}
\end{equation*}
Substituting the solution set in equation \eqref{eq:FourierExpansion1}, we have equation
\begin{equation*}
f(\tau)=f(x+iy)=\sum_{n=-\infty}^{+\infty}C_ne^{-\frac{2\pi}{k}ny}e^{\frac{2\pi}{k}inx}=\sum_{n=-\infty}^{+\infty}C_ne^{\frac{2\pi}{k}in\tau},
\end{equation*}
the last equality holds because $inx-ny=in(x+iy)=in\tau$. On the other hand, the uniformizing function $\qk=e^{\frac{2\pi i}{k}\tau}$ is a local coordinate on a neighborhood of the corresponding singularity $a\in\{a_1,\ldots,a_n\}$ on $\punctured$, so $f$ has a Taylor expansion
\begin{equation}\label{eq:Taylor-Fourier}
f(\tau)=c_0+c_1\qk+c_2\qk^2+\sum_{m=3}^{\infty}c_m\qk^m
\end{equation}
in $\qk$ on a neighborhood of $a$, which will coincide with its Fourier series, thus equation \eqref{eq:Taylor-Fourier} holds globally. For convenience in later sections, we write $A=c_1, B=c_2$,
\begin{equation*}
f(\tau)=c_0+A\qk+B\qk^2+\sum_{j=3}^{\infty}c_j\qk^j.
\end{equation*}
To see $A\neq 0$, remember that $f(\tau)$ is a covering map without ramification point, this implies that $f_{\qk}|_{\qk=0}\neq 0$, so $A\neq 0$.
\end{proof}
The condition $A\neq0$ in equation \eqref{eq:f-qk-expansion} allows us to find an inversion series $\qk(f)$ in $f$ such that $\qk(f(\qk))=\qk$. We assume the inversion series $\qk(f)$ has expression
\begin{equation}\label{eq:qk-f-inversion}
\qk(f)=\tilde{c}_0+\tilde{A}f+\tilde{B}f^2+\sum_{j=3}^{\infty}\tilde{c}_jf^j.
\end{equation}
\begin{theorem}\label{thm:metric-expression}
Let $f:\mathbb{H}\rightarrow\punctured$ is a covering map for the universal covering space, and $\Aut(f)$ has a generator $\gamma$ which fixes infinity, then there exists a positive constant $k\in\mathbb{R}$ such that $f$ can be given as the following expansion
\begin{equation*}
f(\tau)=f(\qk)=c_0+A\qk+B\qk^2+c_3\qk^3+\sum_{m=4}^{\infty}c_m\qk^m
\end{equation*}
in $\qk=\exp\{\frac{2\pi i}{k}\tau\}, \tau\in\mathbb{H}$ and $A\neq 0$. Furthermore, such covering map induces a complete K\"{a}hler-Einstein metric on $\punctured$ from $\mathbb{H}$, it can be given by the following equation
\begin{equation}
ds^2=\frac{|\qk'(f)|^2}{|\qk(f)|^2\log^2|\qk(f)|}|df|^2,\quad f\in\punctured,
\end{equation}
where $\qk(f)$ is the inversion series \eqref{eq:qk-f-inversion}.
\end{theorem}
\begin{proof}
The first statement is directly from Proposition \ref{prop:coveringmap-global-expression}, we only need to show the second part. For any point $x_0\in\punctured$, there is an evenly covered neighborhood $U$ of $x_0$ that is simply connected. Let $\tilde{U}$ and $\tilde{U}'$ are two different pre-images of $U$, i.e., $\tilde{U}\neq\tilde{U}'\in f^{-1}(U)$. Notice that $\tilde{U}$ and $\tilde{U}'$ are both in $\mathbb{H}$, we can define logarithms of $\exp\{\frac{2\pi i}{k}\tau\}=\qk=\qk(f)$ on different branches as the following
\begin{align}
\tau=\tau(f)=\frac{k}{2\pi i}\log\qk(f),\qquad\mbox{on }\tilde{U},\notag\\
\tau'=\tau'(f)=\frac{k}{2\pi i}\log\qk(f),\qquad\mbox{on }\tilde{U}'.\notag
\end{align}
Recall that $\tilde{U}$ and $\tilde{U}'$ are homeomorphic through a cover transformation $h\in\Aut(f)\subseteq\mbox{SL}_2(\mathbb{R})$, i.e., $h\circ \tau(f)=\tau'(f)$. Since the Poincar\'{e} metric is invariant under actions of $\mbox{GL}_2(\mathbb{R})$, therefore we have equality
\begin{equation}\label{eq:invariance}
ds^2=\frac{-4}{(\tau-\overline{\tau})^2}|d\tau|^2=\frac{-4}{(\tau'-\overline{\tau'})^2}|d\tau'|^2,
\end{equation}
Thus the general logarithm property
$$\frac{d}{df}(\log \qk(f))=\qk'(f)/\qk(f)df$$
holds since $\log \qk(f)$ is biholomorphic on the branch $\tilde{U}$, . The metric can be induced on $\tilde{U}$ by the following calculation
\begin{align}
ds^2 & =\frac{-4}{(\tau-\overline{\tau})^2}|d\tau|^2,\notag\\
& =\frac{-4}{((\frac{k}{2\pi i}\log \qk(f)-\overline{\frac{k}{2\pi i}\log \qk(f)})^2}|d(\frac{k}{2\pi i}\log \qk(f))|^2,\notag\\
& =\frac{-4\cdot (2\pi i)^2}{k^2(\log \qk(f)+\overline{\log \qk(f)})^2}(\frac{k}{2\pi})^2\left|\frac{\qk'(f)}{\qk(f)}df\right|^2,\notag\\
& =\frac{|\qk'(f)|^2}{|\qk(f)|^2\log^2|\qk(f)|}|df|^2.\label{eq:metricexpression}
\end{align}
Furthermore, the invariant property in equation \eqref{eq:invariance} implies that the expression \eqref{eq:metricexpression} is independent of the choice of branches. Therefore the complete K\"{a}hler-Einstein metric $ds^2$ can be defined globally by equation \eqref{eq:metricexpression} on $\punctured$.
\end{proof}
One thing that is worth to mention is that Theorems \ref{thm:metric-expression} and \ref{thm:coefficient-depend-on-A-B}, which will be included in the later section, together imply the main result Theorem \ref{thm:into-main-1}
\section{Schwarzian Derivative}\label{section:schwarzian-derivative}
\subsection{Introduction}
The Schwarzian derivative is invariant under linear transformations, let $\tau=\tau(f)=f^{-1}$ be the inverse of the covering map $f:\mathbb{H}\rightarrow\punctured$, the Schwarzian derivative $\{\tau,f\}$ is actually well-defined due to the invariance. This fact builds a connection between $\{f,\tau\}$ and $\{\tau,f\}$, which will lead to the main result of this article.
\begin{definition}\label{def:schwarzian-deirvative}
If a function $f$ is locally biholomorphically defined on the complex $z-$plane, the Schwarzian derivative of $f$ with respect to $z$ is defined as the following:
\begin{equation*}
\{f,z\}=2\left(\frac{f_{zz}}{f_z}\right)_z-\left(\frac{f_{zz}}{f_z}\right)^2,
\end{equation*}
where we write $f_z=df/dz$ for convenience.
\end{definition}
In Definition \ref{def:schwarzian-deirvative}, we require $f$ being locally biholomorphic so that we are able to talk about the Schwarzian derivative of its inverse function $z=f^{-1}$. We simply write $\{z,f\}$ to denote the Schwarzian derivative of $z=z(f)$ with respect to $f$. If $w$ is also a locally biholomorphic function defined on the complex plane, then it allows us to consider the Schwarzian derivative of the composition $f\circ w(z)$. Since the functions $f,w$ are biholomorphic, we will simply write the inverse $f^{-1}$ as $z(f)$ and the inverse $w^{-1}=z(w)$.
\begin{prop}\label{prop:schwarzian-basic-properties}
Assume $f,w$ are two locally biholomorphic functions defined on the complex complex plane, then we have the following properties:
\begin{enumerate}
\item $\{\frac{af+b}{cf+d},z\}=\{f,z\}$ for any $\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\in\mbox{SL}_2(\mathbb{C})$,\\
\item $\{f,z\}=\{f,w\}w_z^2+\{w,z\}$ if $f=f\circ w(z)$. In particular, $\{z,f\}=-\{f,z\}z_f^2$.
\end{enumerate}
\end{prop}
\begin{proof}
The proof is obvious.
\end{proof}
Recall Theorem \ref{thm:metric-expression} in section \ref{section:qk-expansion-metric-formula}, the covering map $f$ can be expressed as an expansion in $\qk$, let us denote this expansion as $f(\qk)$. Notice that the three functions $f(\tau)$ and $f(\qk)$ and $\qk(\tau)$ are all locally biholomorphic, it makes sense to consider the relation of the Schwarzian derivatives among the three of them.
\begin{prop}\label{prop:schwarzian-expression-qk-tau-relation}
Let $f:\mathbb{H}\rightarrow\punctured, \tau\mapsto f(\tau)$ be a covering map and $\qk(\tau)=\exp\{\frac{2\pi i}{k}\tau\}$, then we have the following relation
\begin{equation}\label{eq:schwarzian-expression-qk-tau-relation}
\{f,\tau\}=\frac{4\pi^2}{k^2}(1-\qk^2\{f,\qk\}).
\end{equation}
\end{prop}
\begin{proof}
Assume $w=\qk=e^{\frac{2\pi}{k}i\tau}$ in Proposition \ref{prop:schwarzian-basic-properties}, we have equalities
$$(\qk)_{\tau}=\frac{2\pi i}{k}\qk,\qquad(\qk)_{\tau\tau}=-(\frac{2\pi}{k})^2\qk,\qquad (\qk)_{\tau}^2=-(\frac{2\pi}{k})^2\qk^2,$$
and
$$\{\qk,\tau\}=0-(\frac{2\pi i}{k})^2=\frac{4\pi^2}{k^2}.$$
Direct calculation leads to the conclusion of this proposition,
\begin{equation}
\{f,\tau\}=-(\frac{2\pi}{k})^2\qk^2\{f,\qk\}+(\frac{2\pi}{k})^2=\frac{4\pi^2}{k^2}(1-\qk^2\{f,\qk\}).\notag
\end{equation}
\end{proof}
It is beneficial to have a closer look at the expansion of $\{f,\qk\}$ in $\qk$.
\begin{prop}\label{prop:schwarzian-f-qk-expansion}
If a covering map $f:\mathbb{H}\rightarrow\punctured$ has the following expansion
\begin{equation}\label{eq:f-qk-expanion-with-0}
f(\tau)=f(\qk)=A\qk+B\qk^2+c_3\qk^3+c_4\qk^4+\sum_{m=5}^{\infty}c_m\qk^m
\end{equation}
in $\qk=\exp\{\frac{2\pi i}{k}\tau\}$, where $\tau\in\mathbb{H}$. Then the Schwarzian derivative of $f(\qk)$ with respect to $\qk$ has the following expansion
\begin{equation}\label{eq:schwarzian-f-qk-expansion-2}
\{f,\qk\}=P_0(\B,\C_3)+\sum_{m=1}^{\infty}P_m(\B,\C_3,\ldots,\C_{m+3})\qk^m
\end{equation}
in $\qk$, where $\B=\frac{B}{A}$, $\C_{m+3}=\frac{c_{m+3}}{A}$ and the coefficient term $P_m(\B,\C_3,\ldots,\C_{m+3})$ is a polynomial in $\B,\C_3,\ldots,\C_{m+3}$ with degree $1$ in $\C_{m+3}$, and degree $m+2$ in $\B$, $m\ge 0$. Specially $P_0(\B,\C_3)=12(\C_3-\B^2)$.
\end{prop}
\begin{proof}
We will write $f'(\qk)=\frac{\d f(\qk)}{\d\qk}$ and $f''(\qk)=\frac{\d}{\d\qk}\left(\frac{\d f}{\d\qk}\right)$ for convenience. Direct calculation gives the following equations
\begin{align}
f'(\qk)&=A+2B\qk+3c_3\qk^2+\sum_{m=4}^{\infty}mc_m\qk^{m-1}=A+A\cdot\left(2\B\qk+\sum_{m=2}^{\infty}Q^{(1)}_m(\C_{m+1})\qk^m\right),\notag\\
f''(\qk)&=2B+6c_3\qk+\sum_{m=4}^{\infty}m(m-1)c_m\qk^{m-2}=A\left(2\B+\sum_{m=1}^{\infty}Q^{(2)}_m(\C_{m+2})\qk^m\right),\notag
\end{align}
where $Q^{(1)}_m(\C_{m+1})$ is a polynomial in $\C_{m+1}$ with degree $1$, and $Q^{(2)}_m(\C_{m+2})$ is a polynomial in $\C_{m+2}$ with degree $1$ as well. We have the following calculation
\begin{align}
1/f'(\qk)&=\frac{1}{A}\cdot\frac{1}{1+\left(2\B\qk+\sum_{m=2}^{\infty}Q^{(1)}_m(\C_{m+1})\qk^m\right)},\notag\\
&=\frac{1}{A}\cdot\left[1-2\B\qk+\left(4\B^2-3\C_3\right)\qk^2+\sum_{m=3}^{\infty}Q^{(3)}_m(\B,\C_3,\ldots,\C_{m+1})\qk^m\right],\label{eq:schwarzian-1/fqk}
\end{align}
where the coefficients $Q^{(3)}_m(\B,\C_3,\ldots,\C_{m+1})$ are polynomials in $\B,\C_3,\ldots,\C_{m+1}$, which has degree $1$ in $\C_{m+1}$ with constant coefficient.
We applied the expansion $\frac{1}{1-z}=1+z+z^2+\sum_{m=3}^{\infty}z^m$ in the above calculation. The series \eqref{eq:schwarzian-1/fqk} converges since that $f'(\qk)$ is still analytic on the unit disk $\mathbb{D}$ (see \cite[p. 179-182]{AhlforsComplex}). Then we have the following equation
\begin{align}
f''(\qk)/f'(\qk)=2\B+(6\C_3-4\B^2)\qk+\sum_{m=2}^{\infty}Q^{(4)}_m(\B,\C_3,\ldots,\C_{m+2})\qk^2,\notag
\end{align}
where $Q^{(4)}_m$ are polynomials in $\B,\C_3,\ldots,\C_{m+2}$ with degree $1$ in $\C_{m+2}$ with constant coefficient.
Continue computing the terms in $\{f,\qk\}=2\left(\frac{f''(\qk)}{f'(\qk)}\right)'-\left(\frac{f''(\qk)}{f'(\qk)}\right)^2$, the Schwarzian derivative can be given by the following expansion
\begin{align*}
\{f,\qk\}&=\sum_{m=0}^{\infty}Q^{(5)}_m(\B,\C_3,\ldots,\C_{m+3})\qk^m-\sum_{m=0}^{\infty}Q^{(6)}_m(\B,\C_3,\ldots,\C_{m+2})\qk^m,\notag\\
&=12(\C_3-\B^2)+\sum_{m=1}^{\infty}P_m(\B,\C_3,\ldots,\C_{m+3})\qk^m
\end{align*}
in $\qk$, where $Q^{(5)}_m$ is a polynomial in $\B,\C_3,\ldots,\C_{m+3}$ which has degree $1$ in $\C_{m+3}$ with constant coefficient;
$Q^{(6)}_m$ is a polynomial in $\B,\C_3,\ldots,\C_{m+2}$ that has degree $1$ in $\C_{m+2}$ but with coefficient in terms of $\B$.
Therefore the coefficient $P_m(\B,\C_3,\ldots,\C_{m+2})=Q^{(5)}_m-Q^{(6)}_m$ of $\qk^m$ is a polynomial in $\B,\C_3,\ldots,\C_{m+2}$, which has degree $1$ in $\C_{m+3}$ with constant coefficient.
\end{proof}
\subsection{Differential Equation}
Recall section \ref{subsection:deck-group-generators}, the branches of the inverse $\tau(f)$ are related by linear transformations, Proposition \ref{prop:schwarzian-basic-properties} implies that $\{\tau(f),f\}=\{\tau,f\}$ is well-defined. In order to discuss its analytic property, we consider the following uniformizing function near each singularity $a\in\{a_1,\ldots,a_n\}$,
\begin{equation}
\Phi(f)=\exp\{-\frac{2\pi i}{k}\frac{1}{\tau(f)-r}\},\quad\mbox{when }r\neq\infty\label{eq:schwarzian-uniformize-1},
\end{equation}
or
\begin{equation}
\Phi(f)=\exp\{\frac{2\pi i}{k}\tau(f)\},\quad \mbox{when }r=\infty,\label{eq:schwarzian-uniformize-2}
\end{equation}
where $r$ and $k$ are the constants with respect to the parabolic transformation $\gamma$ in Propositions \ref{prop:invariant-pt-infty-0} and \ref{prop:Classification-SL2}, and $\gamma$ is the generator corresponding to the singularity $a$. On the other side, from section \ref{subsection:deck-group-generators}, the uniformizing functions are single valued in a neighborhood of $a$. So the uniformizing function $\Phi(f)$ has the following expansion near $a$,
\begin{equation}
\Phi(f)=\alpha_a(f-a)+\beta_a(f-a)^2+(f-a)^2\phi(f),\quad a\neq\infty,\alpha_a\neq 0,\label{eq:unif-func-singularity-expansion-finite}\\
\end{equation}
\begin{equation}
\Phi(f)=\alpha_{\infty} f+\beta_{\infty}+\varphi(\frac{1}{f}),\quad a=\infty,\alpha_{\infty}\neq 0,\label{eq:unif-func-singularity-expansion-infty}
\end{equation}
where $\phi(f)$ and $\varphi(\frac{1}{f})$ are regular at $a$ and satisfying
\begin{align*}
\phi(a)=0,\qquad\mbox{when }a\neq\infty,\\
\varphi(\frac{1}{a})=\varphi(0)=0,\qquad\mbox{when }a=\infty.
\end{align*}
Assume one of the singularities is $0$, and the parabolic transformation $\gamma$ that associates to this singularity $a=0$ fixes $r=\infty$, and $k$ is the constnat in Proposition \ref{prop:Classification-SL2}, then we have the following analytic expansion of the uniformizing function $\Phi(f)$:
\begin{equation}\label{eq:unif-func-singularity-expansion-calculation}
\exp\{\frac{2\pi i}{k}\tau(f)\}=\alpha_{(a=0)} f+\beta_{(a=0)} f^2+f^2\phi(f).
\end{equation}
We use equation \eqref{eq:unif-func-singularity-expansion-calculation} to calculate its Schwarzian derivative $\{\tau,f\}$. Notice that differentiation will make the branch problem no longer an issue if we take the derivative of its logarithm, we write $\alpha=\alpha_{(a=0)}$ and $\beta=\beta_{(a=0)}$ for convenience,
\begin{align}
\frac{2\pi i}{k}\tau(f)&=\log\alpha+\log f+\log(1+\frac{\beta}{\alpha}f+\frac{1}{\alpha}f\phi(f)),\notag\\
\frac{2\pi i}{k}\tau_{f}&=\frac{1}{f}+\frac{\frac{\beta}{\alpha}+\frac{1}{\alpha}(\phi(f)+f\phi'(f))}{1+\frac{\beta}{\alpha}f+\frac{1}{\alpha}f\phi(f)}=\frac{1}{f}+\frac{\beta}{\alpha}+\phi_1(f),\notag\\
\log{\frac{2\pi i}{k}}+\log\tau_f&=\log\frac{1}{f}+\log\left(1+\frac{\beta}{\alpha}f+f\phi_1(f)\right),\notag\\
\frac{\tau_{ff}}{\tau_f}&=-\frac{1}{f}+\frac{\frac{\beta}{\alpha}+\phi_2(f)}{1+\frac{\beta}{\alpha}f+f\phi_1(f)}=-\frac{1}{f}+\frac{\beta}{\alpha}+\phi_3(f)\notag,
\end{align}
where each $\phi_l(f)$, $l=1,2,3$, is regular at $a=0$, and $\phi_l(0)=0$. Therefore the Schwarzian derivative has expansion around $a=0$ as the following equation
\begin{align}
\{\tau,f\}&=2\left(\frac{\tau_{ff}}{\tau_f}\right)_f-\left(\frac{\tau_{ff}}{\tau_f}\right)^2,\notag\\
&=\frac{1}{f^2}+\frac{2\beta}{\alpha}\frac{1}{f}+\phi_4(f),\label{eq:tau-f-schwarzian-expansion-special-case}
\end{align}
where $\phi_4(f)$ is regular at $a=0$. Recall that $\{\tau,f\}$ is invariant under any linear transformation on $\tau$, so we have the following equality
\begin{equation*}
\{\tau(f),f\}=\{\frac{1}{\tau(f)-r},f\}.
\end{equation*}
For the special case $a=\infty$, equation \eqref{eq:unif-func-singularity-expansion-infty} implies that $\{\tau,f\}$ does not have a singularity at $a=\infty$. Therefore, due to Liouville's theorem, we have the following conclusion from equation \eqref{eq:tau-f-schwarzian-expansion-special-case} (see \cite[p. 201]{NehariConformalMap}),
\begin{equation}
\{\tau,f\}=\left\{
\begin{aligned}
&\sum_{j=1}^{n}\frac{1}{(f-a_j)^2}+\frac{2\beta_j/\alpha_j}{f-a_j}, \qquad\mbox{if }\infty\notin\{a_1,\ldots,a_n\},\\
&\sum_{j=1}^{n-1}\frac{1}{(f-a_j)^2}+\frac{2\beta_j/\alpha_j}{f-a_j}, \qquad\mbox{if }a_n=\infty.
\end{aligned}\right.
\end{equation}
\begin{comment}
The holomorphic function must be reduced to a constant due to Liouville's theorem and the fact that the upper half plane and unit disk are identified.
\end{comment}
From now on, we will fix one of the singularities to be $0$. Assume $f:\mathbb{H}\rightarrow\mathbb{CP}^1\setminus\{a_1=0,a_2,\ldots,a_n\}, \tau\mapsto f(\tau)$ is a covering map, then we have the following equation
\begin{equation}\label{eq:right-hand-side-of-relation}
\{\tau,f\}=\frac{1}{f^2}+\frac{2\beta}{\alpha}\frac{1}{f}+\sum_{j=2}^{n}\left[\frac{1}{(f-a_j)^2}+\frac{2\beta_j}{\alpha_j}\frac{1}{f-a_j}\right],
\end{equation}
where $\alpha=\alpha_{(a_1=0)},\beta=\beta_{(a_1=0)}$, and $\alpha_j=\alpha_{a_j},\beta_j=\beta_{a_j}$, $j=2,\ldots,n$, are the coefficients of the first and second order leading terms in the analytic expansion \eqref{eq:schwarzian-uniformize-1} of the unifomizing function at the singularity $a_j$. It is worth to mention that $\alpha_j,\beta_j$ are uniquely determined by the given covering map $f$ due to its analytic property.
Let us recall Propositions \ref{prop:schwarzian-basic-properties} and \ref{prop:schwarzian-expression-qk-tau-relation}, we have the following equation
\begin{equation}
\{\tau,f\}=-\frac{4\pi^2}{k^2}(1-\qk^2\{f,\qk\})\tau_f^2\notag
\end{equation}
and equation
\begin{equation}
\{\tau,f\}=\frac{1}{f^2_{\qk}}(\frac{1}{\qk^2}-\{f,\qk\})
\end{equation}
due to $\tau_f=\frac{k}{2\pi i}\qk^{-1}f_{\qk}^{-1}$ from chain rule. Therefore, we have the following relation that is connected by $\{\tau,f\}$,
\begin{equation}\label{eq:schwarzian-qk-relation}
\frac{1}{f^2_{\qk}}(\frac{1}{\qk^2}-\{f,\qk\})=\frac{1}{f^2}+\frac{2\beta}{\alpha}\frac{1}{f}+\sum_{j=2}^{n}\left[\frac{1}{(f-a_j)^2}+\frac{2\beta_j}{\alpha_j}\frac{1}{f-a_j}\right].
\end{equation}
From equations \eqref{eq:schwarzian-f-qk-expansion-2} and \eqref{eq:schwarzian-1/fqk}, we have the following expansion of the left hand side of equation \eqref{eq:schwarzian-qk-relation},
\begin{equation}
\frac{1}{A^{2}}\left[\frac{1}{q_{k}^{2}}-4\B\frac{1}{q_{k}}+(24\B^2-18\C_3)+\sum_{m=1}^{\infty}\tilde{P}_m(\B,\C_3,\ldots,\C_{m+3})\qk^m\right],\label{eq:schwarzian-tau-f-qk-expansion}
\end{equation}
where the coefficient term $\tilde{P}_m(\B,\C_3,\ldots,\C_{m+3})$ is a polynomial in $\B,\C_3,\ldots,\C_{m+3}$ which has degree $1$ in $\C_{m+3}$ for $m\ge1$.
For convenience, we denote the constant term $\tilde{P}_0(\B,\C_3)=(24\B^2-18\C_3)$.
On the punctured sphere $\mathbb{CP}^1\setminus\{a_1=0,a_2,\ldots,a_n\}$, the singularities are discrete since it is a set of finite points. Take a neighborhood $U$ which contains the one and only singularity $a_1=0$, then we have the following expansion in $U$,
\begin{equation}
\frac{1}{f}=\frac{1}{A}\frac{1}{\qk}[1-\B\qk+\sum_{m=2}^{\infty}Q^{(7)}_m(\B,\C_3,\ldots,\C_{m+1})\qk^m],
\end{equation}
where $Q^{(7)}_m$ is a polynomial in $\B,\C_3,\ldots,\C_{m+1}$ which has degree $1$ in $\C_{m+1}$ with constant coefficient.
And for other terms $\frac{1}{f-a_j}$, $j=2,\ldots,n$, it has expansion
\begin{align}
\frac{1}{f-a_{j}}=-\frac{1}{a_{j}}\cdot\left[1+\frac{A}{a_{j}}\qk+\sum_{m=2}^{\infty}Q^{(8)}_m(\frac{A}{a_j},\frac{c_2}{a_j},\ldots,\frac{c_{m}}{a_j})\qk^m\right],
\end{align}
where $c_2=B$, $Q^{(8)}_m$ is a polynomial in $\frac{A}{a_j},\frac{c_2}{a_j},\ldots,\frac{c_{m}}{a_j}$ which has degree $1$ in $\frac{c_m}{a_j}$ with constant coefficient.
Therefore equation \eqref{eq:right-hand-side-of-relation} on the right hand side of equation \eqref{eq:schwarzian-qk-relation} has expansion in $\qk$ as the following
\begin{equation}\label{eq:tau-f-right-hand-side-expansion-specific}
\begin{aligned}
\frac{1}{A^2}&\left\{\frac{1}{\qk^2}-4\B\frac{1}{\qk}+\left[5\B^2-2\C_3+A^2\sum_{j=2}^{n}\left(\frac{1}{a^2_j}-\frac{1}{a_j}\frac{2\beta_j}{\alpha_j}\right)\right]\right.\\
&\qquad+\left.\sum_{m=1}^{\infty}\left[Q^{(9)}_m(\B,\C_3,\ldots,\C_{m+3})+\sum_{j=2}^{n}Q^{(10)}_{j_m}(\frac{1}{a_j},c_1,\ldots,c_m)\right]\qk^m\right\},
\end{aligned}
\end{equation}
where $c_1=A,c_2=B$, and $Q^{(9)}_m(\B,\C_3,\ldots,\C_{m+3})$ is a polynomial in $\B,\C_3,\ldots,\C_{m+3}$ that has degree $1$ in $\C_{m+3}$ with constant coefficient;
$Q^{(10)}_{j_m}(\frac{1}{a_j},c_1,\ldots,c_m)$ is a polynomial in $\frac{1}{a_j},c_1,\ldots,c_m$ which has degree $1$ in $c_m$ with constant coefficient.
If we consider each $c_m=\C_m\cdot A$ for $m\ge 1$, then $Q^{(10)}_{j_m}$ is a polynomial $Q^{(10)'}_{j_m}$ in $A,\B,\C_3,\ldots,\C_m$. We identify equation \eqref{eq:tau-f-right-hand-side-expansion-specific} with equation \eqref{eq:schwarzian-tau-f-qk-expansion} to get a set of equations
\begin{equation}\label{eq:schwarzian-analytic-equation-set}
\tilde{P}_m(\B,\C_3,\ldots,\C_{m+3})=Q^{(9)}_m(\B,\C_3,\ldots,\C_{m+3})+\sum_{j=2}^{n}Q^{(10)'}_{j_m}(A,\B,\C_3,\ldots,\C_m),
\end{equation}
where $m\ge 1$. And the constant term gives equation
\begin{equation}
24\B^2-18\C_3=5\B^2-2\C_3+A^2\sum_{j=2}^{n}\left(\frac{1}{a^2_j}-\frac{1}{a_j}\frac{2\beta_j}{\alpha_j}\right),
\end{equation}
thus the solution for $\C_3$ is given by the following equation
\begin{equation}
\C_3=\C_3(A,\B)=\frac{1}{16}\left[19\B^2-A^2\sum_{j=2}^{n}\left(\frac{1}{a^2_j}-\frac{1}{a_j}\frac{2\beta_j}{\alpha_j}\right)\right].
\end{equation}
Notice that the left hand side of equation \eqref{eq:schwarzian-analytic-equation-set} is a polynomial in $\B,\C_3,\ldots,\C_{m+3}$ that has degree $1$ in $\C_{m+3}$ with constant coefficient, and the right hand side of equation \eqref{eq:schwarzian-analytic-equation-set} is a polynomial in $A,\B,\C_3,\ldots,\C_{m+3}$ that also has degree $1$ in $\C_{m+3}$ with constant coefficient. Let us start with $m=1$, equation \eqref{eq:schwarzian-analytic-equation-set} has one and the only one unknown term $\C_4$, we can solve for $\C_4$ and the solution is a polynomial in $\C_3,A$ and $\B$. The solution $\C_4(\C_3,A,\B)$ can be expressed as a polynomial $\C_4(A,\B)$ in $A,\B$ with constant coefficients since $\C_3=\C_3(A,\B)$. By induction, it is easy to conclude that $\C_m$ can be solved as a polynomial $\C_m(A,\B)$ in $A,\B$ with constant coefficients for $m\ge 3$, i.e.,
\begin{equation}
\C_m=\C_m(A,\B,a_2,\ldots,a_n)=\C_m(A,\B),
\end{equation}
where $a_2,\ldots,a_n$ are the singularities. Therefore we can conclude the following theorem.
\begin{theorem}\label{thm:coefficient-depend-on-A-B}
Let $f:\mathbb{H}\rightarrow\mathbb{CP}^1\setminus\{a_1=0,a_2,\ldots,a_n\}$ be a covering map, and the parabolic generator corresponding to $a_1=0$ fixes infinity, then the map $f$ can be uniquely determined up to the coefficients $A,B$ of the first two order leading terms of its expansion \eqref{eq:f-qk-expanion-with-0}, i.e.,
\begin{equation}\label{eq:coefficient-depend-on-A-B}
f/A=\qk+\B\qk^2+\C_3(A,\B)\qk^3+\sum_{m=4}^{\infty}\C_m(A,\B)\qk^m,
\end{equation}
where $\qk=\exp\{\frac{2\pi i}{k}\tau\}$, and $\C_m(A,\B)$ are polynomials in $A,\B=\frac{B}{A}$ with constant coefficients for $m\ge 3$.
\end{theorem}
Recall Theorem \ref{thm:metric-expression} in section \ref{section:qk-expansion-metric-formula} and Theorem \ref{thm:coefficient-depend-on-A-B}, they together imply our main result Theorem \ref{thm:into-main-1}, which will be concluded in the last section
\section{Ramification Point}\label{section:ramification-point}
\subsection{Introduction}
In section \ref{subsection:deck-group-generators}, we concluded that the deck transformation group $\Aut(f)$ is generated by $(n-1)$ \textit{parabolic} transformations from $\Aut(\mathbb{H})$. On the other hand, we will discover properties of such subgroups of $\mbox{GL}_2(\mathbb{R})$. Notice that $\gamma(\tau)=(\frac{\gamma}{\det{\gamma}})(\tau)$, so we will only consider $\mbox{SL}_2(\mathbb{R})$ from now. Consider $\mbox{SL}_2(\mathbb{R})$ with the normal $\mathbb{R}^4$ topology. The general matrix multiplication
$$\mbox{SL}_2(\mathbb{R})\times\mbox{SL}_2(\mathbb{R})\rightarrow\mbox{SL}_2(\mathbb{R}),\quad (\gamma_1,\gamma_2)\mapsto \gamma_1\gamma_2$$
and inversion
$$\mbox{SL}_2(\mathbb{R})\rightarrow\mbox{SL}_2(\mathbb{R}),\quad \gamma\mapsto\gamma^{-1}$$
are continuous maps.
\begin{definition}
We say that $\mbox{SL}_2(\mathbb{R})$ acts on $\mathbb{H}$ continuously if the following map
\begin{equation}
\mbox{SL}_2(\mathbb{R})\times\mathbb{H}\rightarrow \mathbb{H},\qquad (\gamma,\tau)\mapsto\gamma(\tau)\notag
\end{equation}
is continuous.
\end{definition}
In general, if $G$ is a subgroup of $\mbox{SL}_2(\mathbb{R})$. An orbit of a point $\tau\in\mathbb{H}$ is the set of images of $\tau$ under actions in $G$, i.e.,
$$\mbox{Orbit}_G(\tau)=\{g(\tau)\,|\,g\in G\}.$$
The stabilizer of a point $\tau\in\mathbb{H}$ are the elements in $G$ that fixes $\tau$, i.e.,
$$\mbox{Stab}_G(\tau)=\{g\,|\,g(\tau)=\tau, g\in G\}.$$
We say that two points $\tau,\tau'\in\mathbb{H}$ are equivalent under the action of $G$ if $\tau'\in\mbox{Orbit}_G(\tau)\tau$, i.e.,
$$\tau\sim\tau'\qquad\mbox{if and only if}\qquad\tau'=g(\tau),\,\mbox{for some } g\in G.$$
The set $\mbox{Orbit}_G(\tau)$ is the set of points that are equivalent to $\tau$ under the action of $G$.
\begin{definition}
If $G$ is a discrete subgroup of $\mbox{SL}_2(\mathbb{R})$, define the quotient space $\mathbb{H}/G$ to be the space of equivalent classes of $\mathbb{H}$ under the action of $G$, i.e.,
$$\mathbb{H}/G=\mathbb{H}/\sim,$$
equipped the induced topology through the quotient map.
\end{definition}
Let $\mbox{SL}_2(\mathbb{R})/G$ to be the left cosets of $G$ in $\mbox{SL}_2(\mathbb{R})$, i.e.,
$$\mbox{SL}_2(\mathbb{R})/G=\bigcup_{h\in H}hG,$$
where $H$ is the left coset representation of $G$ in $\mbox{SL}_2(\mathbb{R})$. The topology on $\mbox{SL}_2(\mathbb{R})/G$ is induced from the left group action.
\begin{definition}\label{def:elliptic-point}
A point $P\in\mathbb{H}$ is \textit{elliptic} of a discrete subgroup $G\subset\mbox{SL}_2(\mathbb{R})$ if it is invariant under the action of an \textit{elliptic} element in $G$.
\end{definition}
The goal for this section is to show that a point $P\in\mathbb{H}/G$ is \textit{elliptic} if and only if the point $P\in\mathbb{H}/G$ is a ramification point.
\subsection{Discrete Group, Quotient Space and Ramification Point}
All the conclusions can be found in \cite[p. 2-20, Chapter 1]{ShimuraIntroToArith}. For the completion of this article, I still state them.
\begin{lemma}\label{lemma:fact1-SO2-homeomorphism}
$\mbox{SO}_2(\mathbb{R})$ is compact in $\mbox{SL}_2(\mathbb{R})$. Furthermore, the map
\begin{equation}
\tilde{\alpha}:\mbox{SL}_2(\mathbb{R})/\mbox{SO}_2(\mathbb{R})\rightarrow\mathbb{H},\qquad [\gamma]\mapsto \tilde{\alpha}([\gamma])=\gamma(i)
\end{equation}
is a homeomorphism.
\end{lemma}
\begin{proof}
See \cite[p. 2, Theorem 1.1]{ShimuraIntroToArith} or \cite[p. 25, Proposition 2.1 (d)]{JSMilne}
\end{proof}
\begin{comment}
\begin{proof}
To show the compactness, define a map
$$\begin{array}{cccc}
\varphi: & S^1 & \rightarrow & \mbox{SL}_2(\mathbb{R}),\\
& e^{i\theta} & \mapsto & \varphi(e^{i\theta})=\left(\begin{array}{cc}
\cos\theta & \sin\theta\\
-\sin\theta & \cos\theta
\end{array}\right).
\end{array}$$
Notice that $\varphi$ is continuous and $S^1$ is compact, so its image $\varphi(S^1)=\mbox{SO}_2(\mathbb{R})$ is also compact.
To show $\tilde{\alpha}$ is a homeomorphism, it is equivalent to show that it is bijective and continuous and open. We define a map $\alpha$ on $\mbox{SL}_2(\mathbb{R})$ as the following,
$$\alpha:\mbox{SL}_2(\mathbb{R})\rightarrow\mathbb{H},\qquad \gamma\mapsto \alpha(\gamma)=\gamma(i).$$
Since $\mbox{SL}_2(\mathbb{R})$ acts transitively on $\mathbb{H}$, i.e., $\mbox{Orbit}_{\mbox{SL}_2(\mathbb{R})}(\tau)=\mathbb{H}$ for any $\tau\in\mathbb{H}$, so the map $\alpha$ is onto, which implies that $\tilde{\alpha}$ is surjective. For any $\gamma_1,\gamma_2\in\mbox{SL}_2(\mathbb{R})$, the condition $\tilde{\alpha}([\gamma_1])=\tilde{\alpha}([\gamma_2])$ holds if and only if the equation $\gamma_1(i)=\gamma_2(i)$ holds. If $\gamma_1(i)=\gamma_2(i)$ is true, then we have equality $$\gamma_2^{-1}(\gamma_1(i))=(\gamma_2^{-1}\gamma_1)(i)=i,$$
which implies the following relation
$$(\gamma_2^{-1}\gamma_1)\in\mbox{Stab}_{\mbox{SL}_1(\mathbb{R})}(i)=\mbox{SO}_2(\mathbb{R}),\quad\mbox{equivalently}\quad[\gamma_1]=[\gamma_2],$$
thus $\alpha$ is injective. Therefore $\tilde{\alpha}$ is a bijection. The continuity of $\tilde{\alpha}$ follows from the continuity of $\alpha$.\\
Now we only need to show it is open. Observe that if $x+yi\in\mathbb{H}$ such that $\alpha(\gamma)=x+yi$ for some $\gamma=\left(\begin{array}{cc}a & b\\
c & d
\end{array}\right)\in\mbox{SL}_2(\mathbb{R})$, then we have the following conclusion
$$ad-bc=1,\quad\frac{ai+b}{ci+d}=x+yi,\qquad\mbox{implies}\qquad (x^2+y^2)d-bx-ay=0,$$
since $y\neq 0, c=\frac{dx-b}{y}$. Suppose $\Omega\subseteq\mbox{SL}_2(\mathbb{R})/\mbox{SO}_2(\mathbb{R})$ is open, we will show that any point $x_0+y_0i\in\tilde{\alpha}(\Omega)$ is an interior point. Notice that $x_0+y_0i$ is in the image $\tilde{\alpha}(\Omega)$, so there exists an element $\gamma_0=\left(\begin{array}{cc}
a_0 & b_0\\
c_0 & d_0
\end{array}\right)\in\mbox{SL}_2(\mathbb{R})$ such that $[\gamma_0]\in\Omega$ and $\gamma_0(i)=x_0+y_0i$, we have the following equation
$$(x_0^2+y_0^2)d_0-b_0x_0-a_0y_0=0.$$
Define a function
$$\begin{array}{cccl}
s: & \mathbb{R}^4\times\mathbb{R} & \rightarrow & \mathbb{R},\\
& (x,y,a,b,d) & \mapsto & (x^2+y^2)d-bx-ay.
\end{array}$$
Notice that $\partial s/\partial d=x^2+y^2>0$ since $y>0$. By implicit function theorem, there is a continuous differentiable function $g(x,y,a,b)=d$ on a neighborhood $\Omega_0$ of $(x_0,y_0,a_0,b_0)$ in $\mathbb{R}^4$ with $s(x,y,a,b,d)=0$. Notice that $d=g(x_0,y_0,a,b_0)=\frac{y_0}{x_0^2+y_0^2}a+\frac{x_0}{x_0^2+y_0^2}b_0$ is linearly dependent on $a$, thus there is $\{x_0\}\times\{y_0\}\times(a_0-\varepsilon,a_0+\varepsilon)\times\{b_0\}\subseteq\Omega_0$ for some $\varepsilon>0$ such that $(d_0-\varepsilon',d_0+\varepsilon')=U_{d_0}\subseteq g(\Omega_0)$ for some $\varepsilon'>0$. Recall the fact that $\Omega\subseteq\mbox{SL}_2(\mathbb{R})/\mbox{SO}_2(\mathbb{R})$ is open, without loss of generality, we assume $\{a_0\}\times\{b_0\}\times\{c_0\}\times U_{d_0}\subseteq\Omega$. Therefore the solution set $g^{-1}(U_{d_0})\subseteq\mathbb{R}^4$ is open, so there is an neighborhood $U_{x_0}\times U_{y_0}\times\{a_0\}\times\{b_0\}\subseteq g^{-1}(U_{d_0})$ such that $U_{x_0}\times U_{y_0}$ is an open neighborhood of $(x_0,y_0)$, thus it is an interior point. We can conclude that $\alpha$ is a homeomorphism.
\end{proof}
\end{comment}
\begin{lemma}\label{lemma:fact2-discrete-compact-finite}
Assume $\Gamma$ is a discrete subgroup of $\mbox{SL}_2(\mathbb{R})$, if $V_1,V_2$ are any compact subsets in $\mathbb{H}$, then the set $\{\gamma\in\Gamma|\gamma(V_1)\cap V_2\neq\emptyset\}$ is finite.
\end{lemma}
\begin{comment}
\begin{proof}
Assume $V_1\subseteq\mathbb{H}$ is compact, we will show that $\alpha^{-1}(V_1)$ is also compact in $\mbox{SL}_2(\mathbb{R})$, where $\alpha$ is the same map defined in lemma \ref{lemma:fact1-SO2-homeomorphism}. Since $\mbox{SL}_2(\mathbb{R})$ has the normal topology in $\mathbb{R}^4$, we can consider the collection of balls $B_i=\{(a,b,c,d)\,|\,|(a,b,c,d)|<i\}\subseteq\mbox{SL}_2(\mathbb{R})$ which forms an open cover of $\mathbb{R}^4$, we have the following conclusion
$$\mbox{SL}_2(\mathbb{R})\subseteq\cup_{i=1}^{\infty}B_i\quad\mbox{implies}\quad V_1\subseteq\cup_{i=1}^{\infty}\alpha(B_i).$$
Thus there exists a finite number $m\in\mathbb{Z}$ such that $V_1\subseteq\alpha(B_m)$ since $V_1$ is compact. Therefore the image of the closure $\overline{B_m}$ of $B_m$ covers $V_1$, i.e., $V_1\subseteq\alpha(\overline{B_m})\subseteq\mathbb{H}$. Consider the composition with the inverse of $\alpha$,
$$\alpha^{-1}(V_1)\subseteq\alpha^{-1}(\alpha(\overline{B_m})),$$
we conclude that $\alpha^{-1}(V_1)$ is closed because $\alpha$ is continuous and $V_1$ is compact so it is closed. Consider the right hand side $\alpha^{-1}(\alpha(\overline{B_m}))=\tilde{\alpha}^{-1}(\alpha(\overline{B_m}))\times\mbox{SO}_2(\mathbb{R})$, $\alpha(\overline{B_m})$ is compact since $\overline{B_m}$ is compact, and $\tilde{\alpha}^{-1}(\alpha(\overline{B_m}))$ is compact since $\tilde{\alpha}$ is a homeomorphism. Therefore the set $\tilde{\alpha}^{-1}(\alpha(\overline{B_m}))\times\mbox{SO}_2(\mathbb{R})$ is compact since $\mbox{SO}_2(\mathbb{R})$ is compact. The pre-image $\alpha^{-1}(V_1)$ of $V_1$ is a closed subset in a compact subset, so it is also compact. Similarly, $\alpha^{-1}(V_2)$ is also compact. The fact that the intersection of a discrete set with a compact set $\Gamma\cap(\alpha^{-1}(V_1)\cup\alpha^{-1}(V_2))$ is finite, so there is only finitely $\gamma$ such that $\gamma(V_1)\cap V_2\neq\emptyset$.
\end{proof}
\end{comment}
\begin{proof}
See \cite[p. 3, Proposition 1.6]{ShimuraIntroToArith} or \cite[p. 26, Proposition 2.4]{JSMilne}.
\end{proof}
\begin{prop}\label{prop:discreteneighborhood}
If $\Gamma\in\mbox{SL}_2(\mathbb{R})$ is a discrete subgroup, then for any $\tau\in\mathbb{H}$, there is a neighborhood $U$ of $\tau$ such that if $\gamma\in\Gamma$ and $U\cap\gamma(U)\neq\emptyset$, then $\gamma(\tau)=\tau$.
\end{prop}
\begin{proof}
See \cite[p. 3, Proposition 1.7]{ShimuraIntroToArith} or \cite[p. 27, Proposition 2.5 (b)]{JSMilne}
\end{proof}
\begin{comment}
\begin{proof}
For any $\tau\in\mathbb{H}$, let $V$ be a compact neighborhood of $\tau$ in $\mathbb{H}$. By lemma \ref{lemma:fact2-discrete-compact-finite}, we assume there is a finite subset $\{\gamma_1,\ldots,\gamma_l\}\subseteq\Gamma$ such that $\gamma(V)\cap V\neq\emptyset$. Suppose $\{\gamma_1,\ldots,\gamma_s\}$ are the ones that fixes $\tau$, i.e., $\gamma_i(\tau)=\tau$ for $1\le i\le s$. And for any element $\gamma_i$ which is from the rest of the set $\{\gamma_{s+1},\ldots,\gamma_l\}$ does not fix $\tau$, i.e., $\gamma_i(\tau)\neq\tau(s+1\le i\le l)$. Take some disjoint neighborhoods $U_i^{(1)}$ of $\tau$ and $U_i^{(2)}$ of $\gamma_i(\tau)$ in $V$, consider the intersection
$$U=\cap_{i=s+1}^l (U_i^1\cap\gamma_i^{-1}(U_i^2)),$$
then $U$ is the neighborhood we are looking for.
\end{proof}
\end{comment}
Next, we are able to show that if $\Gamma$ is a discrete subgroup of $\mbox{SL}_2(\mathbb{R})$, a point $[\tau]=[\tau_0]\in\mathbb{H}/\Gamma$ is ramified if and only if following condition
\begin{equation}\label{eq:elliptic-point-equivalent-condition}
\mbox{Stab}_{\Gamma}(\tau)-\{I\}\neq \emptyset
\end{equation}
holds, for any $\tau\in[\tau_0]\subset\mathbb{H}$. Recall Definition \ref{def:elliptic-point}, a point $P$ satisfying condition \eqref{eq:elliptic-point-equivalent-condition} if and only if $P$ is an \textit{elliptic} point. The following proposition can be concluded from \cite[p. 8, Proposition 1.18]{ShimuraIntroToArith}, we will present proof here since it is not obvious.
\begin{prop}\label{proposition-5.7}
Let $\Gamma$ be a discrete subgroup of $\mbox{SL}_2(\mathbb{R})$. Then on the quotient space $\mathbb{H}/\Gamma$, a point $\tau\in\mathbb{H}$ is ramified if and only if it is an \textit{elliptic} point of $\Gamma$.
\end{prop}
\begin{proof}
If $\tau\in\mathbb{H}$ is not an elliptic point, then by Proposition \ref{prop:discreteneighborhood}, there is a neighborhood $U$ of $\tau$ such that $\gamma(U)\cap U=\emptyset$, for any $\gamma\in\Gamma$. It implies that $\tau$ is not ramified. If $\tau\in\mathbb{H}$ is an elliptic point of $\Gamma$, assume there is an elliptic element $\gamma\in\Gamma$ such that $\gamma(\tau)=\tau$. Let $\sigma\in\mbox{SL}_2(\mathbb{R})$ such that $\sigma(i)=\tau$, then the composition $\sigma^{-1}\circ\gamma\circ\sigma\in\mbox{Stab}(i)=\mbox{SO}_2(\mathbb{R})$, so $<\sigma^{-1}\circ\gamma\circ\sigma>=\sigma^{-1}\circ<\gamma>\circ\sigma$ is a subgroup of the conjugate $\sigma^{-1}\Gamma\sigma$, which is still a discrete subgroup of $\mbox{SL}_2(\mathbb{R})$. Also we have the following correspondence relation
\begin{equation*}
<\gamma>\cong \sigma^{-1}<\gamma>\sigma=\mbox{SO}_2(\mathbb{R})\bigcap\sigma^{-1}\Gamma\sigma,
\end{equation*}
the right hand side is an intersection of a compact set and a discrete set, so it has to be finite. Therefore if $m=\#\mbox{Stab}_{\Gamma}(\tau)$, $\tau$ is a ramification point of index $m$.
\end{proof
\section{Modular Group}\label{section:modular-group}
\subsection{The Full Modular Group $\Gamma(1)$}
In this section, we will start with introducing the full modular group.
\begin{definition}
The full modular group $\Gamma(1)$ is defined to be the image of $\mbox{SL}_2(\mathbb{Z})$ by identifying $+\gamma$ and $-\gamma$ for any element $\gamma\in\mbox{SL}_2(\mathbb{R})$. Equivalently, it is the same as the following definition:
\begin{equation}
\Gamma(1)=\mbox{PSL}_2(\mathbb{Z})=\mbox{SL}_2(\mathbb{Z})/\pm I.\notag
\end{equation}
\end{definition}
In this article, we focus on the principal congruence subgroups of $\Gamma(1)$.
\begin{definition}\label{def:principal-congruence-subgroup}
The principle congruence subgroup of level $N$ is defined as the following:
\begin{equation}
\Gamma(N)=\left\{\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\in\mbox{SL}_2(\mathbb{Z})\left|\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\equiv\pm\left(\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right)\,(\mbox{mod}\,N)\right\}\right. /\pm I.\notag
\end{equation}
\end{definition}
The goal is to determine all the principal congruence subgroups $\Gamma(N)$ such that they are candidates for $\Aut(f)$, i.e., $\mathbb{H}/\Aut(f)=\mathbb{H}/\Gamma(N)\cong\punctured$ for some $n\ge 3$. Notice that $\punctured$ does not have any ramification point, and the compactification of $\punctured$ is the Riemann sphere, which is of genus $zero$. We will use these two properties to determine all suitable $\Gamma(N)$. First, we will use the \textit{fundamental domain} of $\Gamma(1)$ to locate all ramification points of $\mathbb{H}/\Gamma(1)$.
\begin{definition}\label{def:fundamental domain}
The \textit{fundamental domain} for a discrete subgroup $\Gamma\subseteq\mbox{SL}_2(\mathbb{R})$ is a connected open subset $D$ of $\mathbb{H}$ such that every pair of points in $D$ are inequivalent under $\Gamma$, and meanwhile $\mathbb{H}\subseteq\bigcup_{\gamma\in\Gamma}[\gamma(\overline{D})]$.
\end{definition}
These conditions are equivalent to $D\rightarrow\mathbb{H}/\Gamma$ is injective and $\overline{D}\rightarrow\mathbb{H}/\Gamma$ is surjective. Now we recall the \textit{fundamental domain} of $\Gamma(1)$ (see \cite[p. 16]{ShimuraIntroToArith}, \cite[p. 19, Proposition 1.2.2]{BumpAuto} or \cite[p. 78-79]{SerreJP}).
\begin{theorem}\label{fact:fundamental-domain-Gamma-1}
The fundamental domain of $\Gamma(1)$ is the set
\begin{equation}
D=\{z=x+yi\,|\,|z|>1,|x|<\frac{1}{2},y>0\}.
\end{equation}
\end{theorem}
\begin{comment}
\begin{remark}\label{rmk:fundamental-domain-Gamma-1}
The half closure of $D$:
$$\{z=x+yi\,|\,-\frac{1}{2}\le x<\frac{1}{2},|z|>1,y>0\}\cup\{z=x+yi\,|\,-\frac{1}{2}\le x\le 0,|z|=1, y>0\}$$
is a set of representatives of the quotient space $\mathbb{H}/\Gamma(1)$. Since $\Gamma(1)$ acts transitively on $\mathbb{Q}\cup\{\infty\}$, we will write
$$\mathbb{H}^*=\mathbb{H}\cup\mathbb{Q}\cup\{\infty\}.$$
In fact, $\mathbb{H}^*/\Gamma\cong\mathbb{CP}^1$, where $\mathbb{CP}^1$ is the compactification of $\punctured$.
\end{remark}
\end{comment}
\begin{prop}
Let $p$ be the quotient map
$$p:\mathbb{H}\rightarrow\mathbb{H}/\Gamma(1),\qquad \tau\mapsto [p(\tau)].$$
The \textit{elliptic} points in $\mathbb{H}/\Gamma(1)$ are either $[i]$ or $[\rho]$, i.e.,
$$\{\mbox{ramification points of }\mathbb{H}\rightarrow\mathbb{H}/\Gamma(1)\}=p^{-1}(i)\bigcup p^{-1}(\rho),$$
where $\rho=e^{\frac{\pi}{3}i}=\frac{1+\sqrt{3}i}{2}$, the root of $1$ of order $6$.
\end{prop}
\begin{proof}
See \cite[p. 14-15, section 1.4]{ShimuraIntroToArith}.
\end{proof}
\begin{comment}
\begin{proof}
Assume $\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)$ is an \textit{elliptic} element in $\Gamma(1)$, so $a+d=0$ or $\pm 1$. By solving a quadratic equation, an \textit{elliptic} point $z_0$ satisfies the following equation
\begin{equation*}
z_0=\frac{a-d}{2c}+\frac{\sqrt{(a+d)^2-4}}{2c},\qquad\mbox{we assume $c>0$ from now on.}
\end{equation*}
Therefore $z_0$ takes values from the following choices
\begin{enumerate}
\item $z_0=\frac{a}{c}+\frac{i}{c},\quad\mbox{or}$\label{enumerate:case-1}
\item $z_0=\frac{2a\pm 1}{2c}+\frac{1}{c}\frac{\sqrt{3}i}{2},$\label{enumerate:case-2}
\end{enumerate}
where $c\in\mathbb{Z}^+, a\in\mathbb{Z}$. On the other side, consider the \textit{fundamental domain} $D$ of $\Gamma(1)$ in theorem \ref{fact:fundamental-domain-Gamma-1} and property in remark \ref{rmk:fundamental-domain-Gamma-1}.
It implies that $\overline{D}$ contains all representative points of equivalent classes of \textit{elliptic} points, so we only need to identity the \textit{elliptic} points in $\overline{D}$. If it is case (\ref{enumerate:case-1}), $\mbox{Im}(z_0)=\frac{i}{c}\in\{i,\frac{i}{2},\frac{i}{3},\cdots\}$, the only choice for $z_0\in\overline{D}$ is $\mbox{Im}(z_0)=i,\,c=1$, and accordingly $a=0$. Take $b=-1, d=0$, the point $z_0=i$ is invariant under the \textit{elliptic} matrix $\left(
\begin{array}{cc}
0 & -1\\
1 & 0
\end{array}\right)$. For case (\ref{enumerate:case-2}), $\mbox{Im}(z_0)=\{\frac{\sqrt{3}i}{2},\frac{\sqrt{3}i}{4},\frac{\sqrt{3}i}{6},\cdots\}$, the only choice is $\mbox{Im}(z_0)=\frac{\sqrt{3}i}{2}$ when $c=1$, then $a=0$, $z_0=\rho$ or $\rho^2$. $\rho=\frac{1+\sqrt{3}i}{2}$ is invariant under $\left(\begin{array}{cc}
0 & -1\\
1 & -1
\end{array}\right)$, and $\rho^2=\frac{-1+\sqrt{3}i}{2}$ is invariant under $\left(\begin{array}{cc}
0 & -1\\
1 & 1
\end{array}\right)$. Notice that $\rho$ and $\rho^2$ are equivalent in $\Gamma(1)$ by $\left(\begin{array}{cc}
1 & 1\\
0 & 1
\end{array}\right)$. Therefore we can conclude that every \textit{elliptic} point of $\Gamma(1)$ is either equivalent to $i$ or $\rho$.\\
\end{proof}
\end{comment}
\subsection{Genus Formula for Subgroups of Modular Group}
We consider a subgroup $\Gamma$ of $\Gamma(1)$ with finite index, then the quotient space $\mathbb{H}/\Gamma$ is composed by copies of $\mathbb{H}/\Gamma(1)$. The genus of $\mathbb{H}/\Gamma$ can be computed by applying the Riemann-Roch Theorem and the Riemann-Hurwitz Formula.
\begin{prop}\label{prop:genus-formula-subgroup-of-Gamma1}
If $\Gamma$ is a subgroup of $\Gamma(1)$ with finite index $m$, then the genus $g$ of $\mathbb{H}/\Gamma$ is given by the following formula
\begin{equation}\label{eq:genus-formula-subgroup-of-Gamma1}
g=1+m/12-\nu_2/4-\nu_3/3-\nu_{\infty}/2,
\end{equation}
where $\nu_2$ is the number of ramification points of order $2$, $\nu_3$ is the number of ramification points of order $3$, $\nu_{\infty}$ is the number of cusps and $m=[\Gamma(1):\Gamma]$.
\end{prop}
\begin{proof}
See \cite[p. 23, Proposition 1.40]{ShimuraIntroToArith} or \cite[p. 37-38, Theorem 2.22]{JSMilne}.
\end{proof}
\begin{comment}
we need the following conclusions. (see \cite{milneMF})
\begin{theorem}[Riemann-Roch Theorem]\label{thm: R-R thm}
If $X$ is a compact Riemann surface, $K$ is a canonical divisor, define $g=$genus$(X)$. The following equality
\begin{equation}
l(D)=1-g+\mbox{deg}(D)+l(K-D),
\end{equation}
holds for any divisor $D$ on $X$,
\end{theorem}
\begin{coro}
If $K$ is a canonical divisor on $X$, then the following equalities
\begin{enumerate}
\item $l(K)=g$,\label{enumerate:R-R-thm-1}
\item $\mbox{deg}(K)=2g-2$,\label{enumerate:R-R-thm-2}
\end{enumerate}
are true.
\end{coro}
\begin{proof}
For case \ref{enumerate:R-R-thm-1}, let $D=0$. then $l(D)=0$, $\mbox{deg}(D)=0$ and $l(K-D)=l(K)$, by theorem \ref{thm: R-R thm} we have equation
$$l(K)=g-1+l(D)+\mbox{deg}(D)=g.$$
To prove case \ref{enumerate:R-R-thm-2}, let $D=K$, apply theorem \ref{thm: R-R thm} again, we have $l(K)=g$ and $l(K-D)=l(0)=1$, therefore $l(K)=1-g+\mbox{deg}(K)+l(0)$ holds, we have conclusion
$$\mbox{deg}(K)=2g-2.$$
\end{proof}
\begin{theorem}[Riemann-Hurwitz Formula]\label{thm: R-H formula}
Let $X,Y$ be Riemann surfaces, assume $s:Y\rightarrow X$ is a biholomorphic covering map of degree $m$ with finite ramification points. Let $R$ be the set of ramification points on $Y$, $e_P$ be the ramification index at $P\in R$, then the following formula
\begin{equation}
2g(Y)-2=m(2g(X)-2)+\sum_{P\in R}(e_P-1),
\end{equation}
holds.
\end{theorem}
\begin{proof}
Let $\omega$ be a canonical divisor on $X$ such that it does not have zeros or poles at any ramification point $P\in R$. On $X$, we have equation
$$\mbox{deg}(\omega)=2g(X)-2.$$
Consider the pull-back $1-$form $s^*\omega$ on $Y$, we will calculate its degree. On un-ramified points, the degree satisfies relation
$$m\cdot\mbox{deg}(\omega)=m\cdot(2g(X)-2)$$
since it is $m:1$ on un-ramified points. On a ramified point $P\in R$, let $e_P$ be the multiplicity of $P$ in the fiber of $s$, then locally we have coordinate
$$X\rightarrow Y,\quad z\mapsto z^{e_P}=z'.$$
so we have $dz'=d(z^{e_P})=e_Pz^{e_P-1}dz$, therefore the total degree on $R$ is given by the summation
$$\Sigma_{P\in R}(e_P-1).$$
Therefore we conclude the formula
$$\mbox{deg}(s^*\omega)=2g(Y)-2=m(2g(X)-2)+\sum_{P\in R}(e_P-1).$$
\end{proof}
Now, we are able to prove proposition \ref{prop:genus-formula-subgroup-of-Gamma1}.
\begin{proof}[Proof of Proposition \ref{prop:genus-formula-subgroup-of-Gamma1}]
Let $Y=\mathbb{H}^*/\Gamma$ and $X=\mathbb{H}^*/\Gamma(1)$ in theorem \ref{thm: R-H formula}. Since $g(X)=g(\mathbb{H}^*/\Gamma(1))=0$, we have following equation
\begin{equation}
g(\mathbb{H}^*/\Gamma)=g(Y)=\frac{1}{2}m(0-2)+\frac{1}{2}\sum_{P\in R}(e_P-1)=1-m+\frac{1}{2}\sum_{P\in R}(e_P-1).\notag
\end{equation}
We will count $e^P$ for each $P\in R$. Write the maps $s,s_0,f$ in the following chart
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes, row sep=3em, column sep=4em, minimum width=2em]
{
\mathbb{H}^* & \mathbb{H}^*/\Gamma \supseteq R\\
& \mathbb{H}^*/\Gamma(1)\\
};
\path[->]
(m-1-1) edge node [above] {$f$} (m-1-2)
edge node [below] {$s_{_0}$} (m-2-2)
(m-1-2) edge node [right] {$s$} (m-2-2);
\matrix (m1) [matrix of math nodes, right=2cm of m, row sep=3em, column sep=4em, minimum width=2em]
{
Q & P \\
& \left[Q\right]\\
};
\path[|->]
(m1-1-1) edge node [above] {$f$} (m1-1-2)
edge node [below] {$s_{_0}$} (m1-2-2)
(m1-1-2) edge node [right] {$s$} (m1-2-2);
\end{tikzpicture}
\end{center}
Let $e_{s}(P/[Q])$ be the ramification index of $P$ in $s$, and let $e_{s_0}(Q/[Q])$ and $e_{f}(Q/P)$ be the index of $Q$ in $s_0$ and $f$ respectively.
Since above diagram commute, we have relation
\begin{equation}
e_{s_0}(Q/[Q])=e_f(Q/P)\cdot e_s(P/[Q]).\notag
\end{equation}
Since $[i],[\rho]$ are the only ramification points of $s_0$, so $R\subseteq s^{-1}([i])\cup s^{-1}([\rho])$. For any $P\in R$, $s(P)$ is either $[i]$, $[\rho]$ or $[\infty]$, consider each case separately
\begin{enumerate}
\item $s(P)=[i]$, $e_f(Q/P)\cdot e_s(P/[i])=e_{s_0}(Q/[i])=2$, so $e_s(P/[i])=1$ or $2$.
\begin{enumerate}
\item $e_s(P/[i])=1$, $e_f(Q/P)=2$. $P$ is not ramified in $s$, but $P$ is ramified in $f$, let $\nu_2$ be the number of this kind points in $\mathbb{H}^*/\Gamma$.
\item $e_s(P/[i])=2$, $e_f(Q/P)=1$. $P$ is ramified with index $2$ in $s$, but $P$ is not ramified in $f$. $\#(s^{-1}([i]))=m$ since $s$ is $(m:1)$, so there are a number of $\frac{1}{2}(m-\nu_2)$ for this type of points.
\end{enumerate}
\item $s(P)=[\rho]$, $e_f(Q/P)\cdot e_s(P/[\rho])=e_{s_0}(Q/[\rho])=3$, thus $e_s(P/[\rho])=1$ or $3$.
\begin{enumerate}
\item $e_s(P/[\rho])=1$, $e_f(Q/P)=3$. $P$ is not ramified in $s$, but $P$ is ramified in $f$, let $\nu_3$ be the number of this kind points in $\mathbb{H}^*/\Gamma$.
\item $e_s(P/[\rho])=3$, $e_f(Q/P)=1$. $P$ is ramified with index $3$ in $s$, there are $\frac{1}{3}(m-\nu_3)$ points of this kind.
\end{enumerate}
\item $s(P)=[\infty]$, $e_f(Q/P)=e_{s_0}(Q/[\infty])=\infty$, if $P$ is ramified, then by definition we have the following equation
$$\sum_{P\in s^{-1}([\infty])}e_s(P/[\infty])=m.$$
Let $\nu_{\infty}$ be the number of cusps of $\mathbb{H}^*/\Gamma$, i.e., the number of points $P$ such that $e_s(P/[\infty])>1$.
\end{enumerate}
Therefore the sum is given by the following calculation
\begin{align}
\sum_{P\in R}(e_s(P)-1)&=\sum_{e_s(P/[i])=2}(2-1)+\sum_{e_s(P)/[\rho]}(3-1)+\sum_{e_s(P/[\infty])=m}(e_s(P/[\infty])-1), \notag\\
&=\frac{1}{2}(m-\nu_2)+\frac{2}{3}(m-\nu_3)+(m-\nu_{\infty}),\notag\\
&=\frac{11}{6}m-\frac{1}{2}\nu_2-\frac{2}{3}\nu_3-\nu_{\infty}.\notag
\end{align}
The genus formula is completed as the following
\begin{align}
g(Y)&=1-m+\frac{1}{2}[\frac{11}{6}m-\frac{1}{2}\nu_2-\frac{2}{3}\nu_3-\nu_{\infty}],\notag\\
&=1+\frac{1}{12}m-\frac{1}{4}\nu_2-\frac{1}{3}\nu_3-\frac{1}{2}\nu_{\infty}.\notag
\end{align}
\end{proof}
\end{comment}
\subsection{Principle Congruence Subgroups}
Recall that our subject is the finitely punctured Riemann sphere, if we consider it as a quotient space $\mathbb{H}/\Gamma(N)$ for some suitable $N\in\mathbb{Z}$, the fact that the punctured Riemann sphere has genus $\it{zero}$ implies $g(\Gamma(N))=0<1$. The following theorem will show that $N=2,3,4,5$ are the only values such that $g(\Gamma(N))=0$.
\begin{theorem}\label{thm:genus-formula-Gamma-N}
The genus of $\mathbb{H}/\Gamma(N)$ is given by the following formula:
\begin{equation}\label{eq:genus-formula-Gamma-N}
g(\Gamma(N))=\left\{\begin{array}{lc}
0, & N=2,\\
1+\frac{N-6}{24}\cdot N^2\prod_{p|N}(1-p^{-2}), & N\ge 3.
\end{array}\right.
\end{equation}
\end{theorem}
Recall that if $N>1$, $\Gamma(N)$ does not have \textit{elliptic} element, which is equivalent to the fact that $\mathbb{H}/\Gamma(N)$ does not contain any elliptic point, so it does not have any ramification point as well. Therefore $\nu_2=\nu_3=0$ in equation \eqref{eq:genus-formula-subgroup-of-Gamma1}, the genus is given by the following simplified formula
\begin{equation}
g(\Gamma(N))=1+m/12-\nu_{\infty}/2,\qquad n\ge 2.\label{eq:genus-Gamma-N-formula-1}
\end{equation}
By the properties of elements in $\Gamma(N)$, the index $m$ of $\Gamma(N)$ in $\Gamma(1)$ is given by the following proposition.
\begin{prop}\label{prop:principle-congruence-index}
The index $m$ of a principle congruence group $\Gamma(N)$ in the full modular group is given by the formula
\begin{equation}\label{eq:principle-congruence-index-1}
m=(\Gamma(1):\Gamma(N))=\left\{\begin{array}{lc}
6, & N=2,\\
\frac{1}{2}\#(\mbox{SL}_2(\mathbb{Z}/N\mathbb{Z}))=\frac{1}{2}N^3\prod_{p|N}(1-p^{-2}), & N\ge 3,
\end{array}\right.
\end{equation}
where $p$ are the prime divisors of $N$.
\end{prop}
\begin{proof}
See \cite[p. 22, Equation (1.62)]{ShimuraIntroToArith} (where $\tilde{\Gamma}(N)$ is in our notation $\Gamma(N)$).
\end{proof}
Consider the stabilizer of $\infty$ in $\Gamma(1)$ and the stabilizer of $\infty$ in $\Gamma(N)$, we have the following proposition.
\begin{prop}\label{prop:number-cusps-principal-subgroup}
The number of cusps of $\Gamma(N)$ is $\nu_{\infty}=m/N$, where $N\ge 2$.
\end{prop}
\begin{proof}
See \cite[p. 22-23]{ShimuraIntroToArith}.
\end{proof}
Therefore equation \eqref{eq:genus-Gamma-N-formula-1} becomes
\begin{equation}
g(\Gamma(N))=1+\frac{m}{12N}(N-6),\label{eq:genus-Gamma-N-formula-2}
\end{equation}
equation \eqref{eq:genus-Gamma-N-formula-2} and \eqref{eq:principle-congruence-index-1} imply equation \eqref{eq:genus-formula-Gamma-N}. Therefore the number of cusps of $\Gamma(N)$ is given by the formula:
$$\nu_{\infty}(\Gamma(N))=m/N=\left\{\begin{array}{lc}
3, & N=2,\\
\frac{1}{2}N^2\Pi_{p|N}(1-p^{-2}), & N\ge 3.
\end{array}\right.$$
We list the number of cusps for each $\Gamma(N)$, $N=2,3,4,5$,
\begin{align}
&\nu_{\infty}(\Gamma(2))=\frac{m(\Gamma(2))}{2}=\frac{6}{2}=3, &\nu_{\infty}(\Gamma(3))&=\frac{1}{2}\cdot 9\cdot\frac{8}{9}=4,\notag\\
&\nu_{\infty}(\Gamma(4))=\frac{1}{2}\cdot 16\cdot\frac{3}{4}=6, &\nu_{\infty}(\Gamma(5))&=\frac{1}{2}\cdot 25\cdot\frac{24}{25}=12.\notag
\end{align}
From now on, we will use $n(N)$ to denote the number of cusps of $\Gamma(N)$. More precisely,
\begin{equation}
n(2)=3,\quad n(3)=4,\quad n(4)=6,\quad n(5)=12.
\end{equation}
Let us conclude the following theorem.
\begin{theorem}\label{thm:Gamma-Aut-sec-6}
Let us define $n(N)$, $N=2,3,4,5,$ as above. If we have a covering space
$$f:\mathbb{H}\rightarrow\mathbb{CP}^1\backslash\{a_1,a_2,\ldots,a_{_n(N)}\}=\mathbb{H}/\Gamma(N)$$
for approporiate choices of $\{a_1,a_2,\ldots,a_{_{n(N)}}\}\subset\mathbb{CP}^1$, then $\Aut(f)=\Gamma(N)$.
\end{theorem}
\begin{proof}
The definition of $\Gamma(N)$ implies that $\Gamma(N)$ does not contain elliptic elements. From Propositions \ref{prop:discreteneighborhood} and \ref{proposition-5.7}, for an arbitrary point $x\in\mathbb{H}$, there exists a neighborhood $x\in U$ such that $\gamma_1(U)\cap\gamma_2(U)=\emptyset$ for any $\gamma_1\neq\gamma_2\in\Gamma(N)$. The conclusion directly follows from \cite[p. 72, Proposition 1.40 (b)]{AllenHatcher}.
\end{proof}
\begin{comment}
\begin{prop}\label{prop:number-cusps-principal-subgroup}
The number of cusps of $\Gamma(N)$ is $\nu_{\infty}=m/N$ $(N\ge 2)$.
\end{prop}
\begin{proof}
Let $\mathbb{Q}^*=\mathbb{Q}\cup\{\infty\}$, and $\mathbb{Q}^*/\Gamma(N)$ is the quotient space of $\mathbb{Q}^*$ under the action of $\Gamma(N)$. The quotient space $\Gamma(1)/\Gamma(N)$ is well defined since $\Gamma(N)$ is normal in $\Gamma(1)$. We define the following short exact sequence,
\begin{center}
\begin{tikzpicture}[text depth=0.5ex]
\node (s1) at (0,1.5) {$0$};
\node (s2) at (3,1.5) {$\mbox{Stab}_{\Gamma(1)}(\infty)/\mbox{Stab}_{\Gamma(N)}(\infty)$};
\node (s3) at (7,1.5) {$\Gamma(1)/\Gamma(N)$};
\node (s4) at (10,1.5) {$\mathbb{Q}^*/\Gamma(N)$};
\node (s5) at (12,1.5) {$0$};
\draw [->]
(s1) edge node[above] {$\sigma_1$} (s2)
(s2) edge node[above] {$\sigma_2$} (s3)
(s3) edge node[above] {$\sigma_3$} (s4)
(s4) edge node[above] {$\sigma_4$} (s5);
\node (s1-2) at (3,0.75) {$[\gamma^{\infty}]$};
\node (s1-3) at (7,0.75) {$[\gamma^{\infty}]$};
\draw [|->]
(s1-2) edge node[above] {$\sigma_2$} (s1-3);
\node (s2-3) at (7,0) {$[\gamma]$};
\node (s2-4) at (10,0) {$[\gamma(\infty)]$};
\draw [|->]
(s2-3) edge node[above] {$\sigma_3$} (s2-4);
\end{tikzpicture}
\end{center}
which satisfies the following properties and facts
\begin{enumerate}
\item $\sigma_1,\sigma_4$ are trivial maps;
\item $\sigma_2$ is well-defined. If $[\gamma_1^{\infty}]=[\gamma_2^{\infty}]$ in $\mbox{Stab}_{\Gamma(1)}(\infty)/\mbox{Stab}_{\Gamma(N)}(\infty)$ for some $\gamma_1^{\infty}\neq\gamma_2^{\infty}\in\mbox{Stab}_{\Gamma(1)}(\infty)$, then $(\gamma_2^{\infty})\circ\gamma_1^{\infty}\in\mbox{Stab}_{\Gamma(N)}(\infty)\subseteq\Gamma(N)$. It implies that $\sigma_2([\gamma_1^{\infty}])=\sigma_2([\gamma_2^{\infty}])$ in $\Gamma(1)/\Gamma(N)$;
\item $(\mathbb{Q}^*/\Gamma(N),\odot)$ is a group with identity $[\infty]$. The fact that $\Gamma(1)$ acts transitively on $\mathbb{Q}^*$ allows us to find $\gamma_i\in\Gamma(1)$ for any $b_i\in\mathbb{Q}^*(i=1,2)$ such that $b_i=\gamma_i(\infty)$. For $[b_1],[b_2]\in\mathbb{Q}^*/\Gamma(N)$ and $\gamma_2\in\Gamma(1)$ such that $b_2=\gamma_2(\infty)$. We define operation $\odot$:
\begin{equation}
[b_1]\odot[b_2]=[\gamma_1\circ\gamma_2(\infty)] \qquad\mbox{on }\mathbb{Q}^*/\Gamma(N).\notag
\end{equation}
If $[b_1]=[b_1']$, there is $\gamma_N\in\Gamma(N)$ such that $\gamma_N(b_1')=b_1$, and an element $\gamma_1'\in\Gamma(1)$ such that $\gamma_1'(\infty)=b_1'$, so $b_1=\gamma_N\circ\gamma_1'(\infty)=\gamma_1(\infty)$, then the following equation
$$[b_1']\odot[b_2]=[\gamma_1'\circ\gamma_2(\infty)]=[\gamma_N\circ\gamma_1'\circ\gamma_2(\infty)]=[b_1]\cdot[b_2],$$
holds. If $[b_2]=[b_2']$, there is $\gamma_N\in\Gamma(N)$ such that $\gamma_N(b_2')=b_2$, and $\gamma_2'\in\Gamma(1)$ such that $\gamma_2'(\infty)=b_2'$, we have $\gamma_2(\infty)=\gamma_N\circ\gamma_2'(\infty)$. On the other hand $\Gamma(N)$ is normal in $\Gamma(1)$, there is $\gamma_N'\in\Gamma(N)$ such that $\gamma_N'\circ\gamma_1=\gamma_1\circ\gamma_N$, therefore the following equation
\begin{align}
[b_1]\odot[b_2']&=[\gamma_1\circ\gamma_2'(\infty)]=[\gamma_1\circ\gamma_N\circ\gamma_2(\infty)]\notag\\
&=[\gamma_N'\circ\gamma_1\circ\gamma_2(\infty)]=[b_1]\odot[b_2],\notag
\end{align}
holds. The associativity, identity and inverse conditions are trivial.
\item $\sigma_2,\sigma_3$ are group homomorphisms;\\
\item $\mbox{im}\sigma_2=\ker\sigma_3$. For $[\gamma]\in\Gamma(1)/\Gamma(N)$, we have the following equivalent relations
\begin{align}
&\sigma_3([\gamma])=[\gamma(\infty)]=[\infty]\in\mathbb{Q}^*/\Gamma(N),\notag\\
\mbox{if and only if \qquad}&\gamma(\infty)=\gamma_N(\infty),\notag\\
\mbox{if and only if \qquad}&\gamma_N^{-1}\circ\gamma(\infty)=\infty,\quad\mbox{exists }\gamma_N\in\Gamma(N),\notag
\end{align}
the last equation implies relation
$$\gamma_N^{-1}\circ\gamma\in\mbox{Stab}_{\Gamma(1)}(\infty)\subseteq\Gamma(1),$$
which is equivalent to $[\gamma_N^{-1}\circ\gamma]\in\mbox{Stab}_{\Gamma(1)}(\infty)/\mbox{Stab}_{\Gamma(N)}(\infty)$, thus $\mbox{im}\sigma_2=\ker\sigma_3$.
\end{enumerate}
This exact short sequence implies relation
\begin{align}
\mathbb{Q}^*/\Gamma(N)\cong(\Gamma(1)/\Gamma(N))/\ker\sigma_3=&(\Gamma(1)/\Gamma(N))/\mbox{im}\sigma_2,\notag\\
=&(\Gamma(1)/\Gamma(N))/(\mbox{Stab}_{\Gamma(1)}(\infty)/\mbox{Stab}_{\Gamma(N)}(\infty)).\notag
\end{align}
Counting their cardinalities,
\begin{align}
\nu_{\infty}(\Gamma(N))=\#(\mathbb{Q}^*/\Gamma(N))=&\#\left((\Gamma(1)/\Gamma(N))/(\mbox{Stab}_{\Gamma(1)}(\infty)/\mbox{Stab}_{\Gamma(N)}(\infty))\right)\notag\\
=&\#(\Gamma(1)/\Gamma(N))/\#(\mbox{Stab}_{\Gamma(1)}(\infty)/\mbox{Stab}_{\Gamma(N)}(\infty)),\notag
\end{align}
where $(\Gamma(1):\Gamma(N))=\#(\Gamma(1)/\Gamma(N))=m$. Direct calculation shows the following facts
$$\mbox{Stab}_{\Gamma(1)}(\infty)=<\left(\begin{array}{cc}
1 & 1\\
0 & 1
\end{array}\right)>,\qquad \mbox{Stab}_{\Gamma(N)}(\infty)=<\left(\begin{array}{cc}
1 & N\\
0 & 1
\end{array}\right)>,$$
therefore $\#(\mbox{Stab}_{\Gamma(1)}(\infty)/\mbox{Stab}_{\Gamma(N)}(\infty))=N$. We proved the conclusion
$$\nu_{\infty}(\Gamma(N))=m/N.$$
\end{proof}
\end{comment}
\begin{comment}
\begin{prop}\label{prop:principle-congruence-index}
The index $m$ of principle congruence group in the full modular group is given by the formula
\begin{equation}\label{eq:principle-congruence-index-1}
m=(\Gamma(1):\Gamma(N))=\left\{\begin{array}{lc}
6, & N=2,\\
\frac{1}{2}\#(\mbox{SL}_2(\mathbb{Z}/N\mathbb{Z}))=\frac{1}{2}N^3\prod_{p|N}(1-p^{-2}), & N\ge 3,
\end{array}\right.
\end{equation}
where $p$ are the prime divisors of $N$.
\end{prop}
\begin{proof}
First we show the equation on the left in equation \eqref{eq:principle-congruence-index-1}. Consider the following group homomorphism
$$\begin{array}{crcl}
\phi:&\mbox{SL}_2(\mathbb{Z}) & \rightarrow & \mbox{SL}_2(\mathbb{Z}/N\mathbb{Z}),\\
&\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right) & \mapsto & \left(\begin{array}{cc}
\overline{a} & \overline{b}\\
\overline{c} & \overline{d}
\end{array}\right),
\end{array}$$
where $\overline{a}$ is the image of $a$ in $\mathbb{Z}/N\mathbb{Z}$, so as $\overline{b},\overline{c}$ and $\overline{d}$. The identity in $\mbox{SL}_2(\mathbb{Z}/N\mathbb{Z})$ is $\left(\begin{array}{cc}
\overline{1} & \overline{0}\\
\overline{0} & \overline{1}
\end{array}\right)$, recall the fact
$$\Gamma(N)\cong\phi^{-1}\left(\begin{array}{cc}
\overline{1} & \overline{0}\\
\overline{0} & \overline{1}
\end{array}\right)=\ker\phi.$$
On the other hand,
$$\mbox{SL}_2(\mathbb{Z})/\ker\phi\cong\mbox{SL}_2(\mathbb{Z}/N\mathbb{Z}),$$
therefore the following relation
$$\Gamma(1)/\Gamma(N)\cong(\mbox{SL}_2(\mathbb{Z})/\pm I)/\ker\phi\cong\mbox{SL}_2(\mathbb{Z}/N\mathbb{Z})/\pm I,$$
holds, which implies equation
$$(\Gamma(1):\Gamma(N))=\#(\mbox{PSL}_2(\mathbb{Z}))/\ker\phi=\frac{1}{2}\#(\mbox{SL}_2(\mathbb{Z}/N\mathbb{Z})),\qquad N\ge 3,$$
and equation
$$(\Gamma(1):\Gamma(2))=\#(\mbox{PSL}_2(\mathbb{Z}))/\ker\phi=\#(\mbox{SL}_2(\mathbb{Z}/2\mathbb{Z}),$$
since $\overline{I}=\overline{-I}$ in $\mbox{SL}_2(\mathbb{Z}/2\mathbb{Z})$.
\begin{claim}\label{claim:Gamma-N-index-order}
We have the following
$$\#\mbox{SL}_2(\mathbb{Z}/N\mathbb{Z})=\Pi_{i=1}^mp_i^3(1-p_r^{-2})=N^3\Pi_{p|N}(1-p^{-2}).$$
\end{claim}
With the help of Claim \ref{claim:Gamma-N-index-order}, we conclude \ref{eq:principle-congruence-index-1}.
\end{proof}
\begin{proof}[Proof of Claim \ref{claim:Gamma-N-index-order}]
Write the integer $N$ as the following
$$N=p_1^{r_1}\ldots p_m^{r_m},$$
where $p_i(i=1,\ldots,m)$ are different prime numbers. By Chinese remainder theorem, we have the following isomorphism
\begin{equation*}
\mbox{GL}_2(\mathbb{Z}/N\mathbb{Z})\cong \mbox{GL}_2(\mathbb{Z}/p_1^{r_1}\mathbb{Z})\times\ldots\times\mbox{GL}_2((\mathbb{Z}/p_m^{r_m}\mathbb{Z})),
\end{equation*}
and isomorphism
\begin{equation*}
\mbox{SL}_2(\mathbb{Z}/N\mathbb{Z})\cong \mbox{SL}_2(\mathbb{Z}/p_1^{r_1}\mathbb{Z})\times\ldots\times\mbox{SL}_2((\mathbb{Z}/p_m^{r_m}\mathbb{Z})).
\end{equation*}
The first one holds because of the following equivalent relations
\begin{align*}
& \left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\in\mbox{GL}_2(\mathbb{Z}/N\mathbb{Z})\\
\mbox{ if and only if }&
\det \left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\in(\mathbb{Z}/N\mathbb{Z})^{\times}\\
\mbox{ if and only if }&
\det \left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\in(\mathbb{Z}/p_i^{r_i}\mathbb{Z})^{\times}\\
\mbox{ if and only if }&
\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\in\mbox{GL}_2(\mathbb{Z}/p_i^{r_i}\mathbb{Z}),\quad i=1,\ldots,m.
\end{align*}
The second one holds because of the equivalence relation
\begin{equation*}
a\equiv \overline{1}\in(\mathbb{Z}/N\mathbb{Z})^{\times}\quad\mbox{ if and only if }\quad a\equiv \overline{1}\in(\mathbb{Z}/p_i^{r_i}\mathbb{Z})^{\times}\quad i=1,\ldots,m.
\end{equation*}
It will be enough to calculate $\#(\mbox{SL}_2(\mathbb{Z}/p_i^{r_i}\mathbb{Z}))$. Claim
\begin{equation}
\#(\mbox{SL}_2(\mathbb{Z}/p_i^{r_i}\mathbb{Z}))=\#(\mbox{GL}_2(\mathbb{Z}/p_i^{r_i}\mathbb{Z}))/\#(\mathbb{Z}/p_i^{r_i}\mathbb{Z})^{\times}.
\end{equation}
To show the claim, consider the homomorphism map $\det$:
\begin{align*}
\det:\mbox{GL}_2(\mathbb{Z}/p_i^{r_i}\mathbb{Z})&\rightarrow(\mathbb{Z}/p_i^{r_i}\mathbb{Z})^{\times},\\
\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)&\mapsto\det\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)=\overline{ad-bc}.
\end{align*}
The kernel is $\det^{-1}(\overline{1})=\mbox{SL}_2(\mathbb{Z}/p_i^{r_i}\mathbb{Z})$, it induces an isomorphism
$$\mbox{GL}_2(\mathbb{Z}/p_i^{r_i}\mathbb{Z})/\mbox{SL}_2(\mathbb{Z}/p_i^{r_i}\mathbb{Z})\cong(\mathbb{Z}/p_i^{r_i}\mathbb{Z})^{\times},$$
which implies the claim. We have $\#(\mathbb{Z}/p_i^{r_i}\mathbb{Z})^{\times}=\varphi(p_i^{r_i})=p_i^{r_i}-p_i^{r_i-1}$. Now let us compute the order of $\mbox{GL}_2(\mathbb{Z}/p_i^{r_i}\mathbb{Z})$. Consider the natural homomorphism
$$h: \mbox{GL}_2(\mathbb{Z}/p_i^{r_i}\mathbb{Z})\rightarrow\mbox{GL}_2(\mathbb{Z}/p_i\mathbb{Z}).$$
It is an onto map, so $\#\mbox{GL}_2(\mathbb{Z}/p_i^{r_i}\mathbb{Z})=\#\mbox{GL}_2(\mathbb{Z}/p_i\mathbb{Z})\cdot\#\ker h$. Claim the following equation $$\#\mbox{GL}_2(\mathbb{Z}/p_i\mathbb{Z})=(p_i^2-1)(p_i^2-p_i).$$
Because $p_i$ is prime, we only need the first and second row to be linearly independent. There are $p_i^2-1$ choices for first row, and then $p_i^2-p_i$ choices for the second row, so the claim holds. For $\ker h$, the pre-image of $\left(\begin{array}{cc}
\overline{1} & \overline{0}\\
\overline{0} & \overline{1}
\end{array}\right)\in\mbox{GL}_2(\mathbb{Z}/p_i\mathbb{Z})$ has the following form
$$\left(\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right)+p_i\left(\begin{array}{cc}
\ast & \ast\\
\ast & \ast
\end{array}\right),$$
where $\left(\begin{array}{cc}
\ast & \ast\\
\ast & \ast
\end{array}\right)$ is any matrix with elements in $\mathbb{Z}/p_i^{r_i-1}\mathbb{Z}$. The cardinality is $(p_i^{r_i-1})^4=\#\ker h$. Therefore we have the following calculations
$$\#\mbox{GL}_2(\mathbb{Z}/p_i^{r_i}\mathbb{Z})=(p_i^2-1)(p_i^2-p_i)(p_i^{r_i-1})^4,$$
\begin{align*}
\#\mbox{SL}_2(\mathbb{Z}/p_i^{r_i}\mathbb{Z})&=(p_i^2-1)(p_i^2-p_i)(p_i^{r_i-1})^4/(p_i^{r_i}-p_i^{r_i-1}),\\
&=(p_i^2-1)p_i^{3r_i-2}=p_i^3(1-p_r^{-2}),
\end{align*}
and we conclude the claim
\begin{equation*}
\#\mbox{SL}_2(\mathbb{Z}/N\mathbb{Z})=\Pi_{i=1}^mp_i^3(1-p_r^{-2})=N^3\Pi_{p|N}(1-p^{-2}).
\end{equation*}
\end{proof}
\end{comment}
\section{Modular Forms and Modular Functions}\label{section:modular-forms}
\begin{definition}
A function $f:\mathbb{H}\rightarrow\mathbb{C}$ is called a modular function of weight $k$ if $f$ satisfies the following properties:
\begin{enumerate}
\item The function $f$ is meromorphic on $\mathbb{H}$;
\item The equation $f\circ\gamma(\tau)=(c\tau+d)^{k}f(\tau)$ holds for any $\tau\in\mathbb{H}$ and any $\gamma=\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\in\Gamma(1)$, where $\Gamma(1)$ is the modular group;\\
\item The function $f$ is meromorphic at infinity.
\end{enumerate}
Furthermore, if $f$ is a modular function of weight $k$ and holomorphic on $\mathbb{H}\cup\{\infty\}$, we call $f$ a modular form of weight $k$.
\end{definition}
As a matter of fact, the space of weight $4$ modular form for $\Gamma(1)$ is one dimensional and generated by the Eisenstein series $E_4(\tau)$. This is a classic result, see \cite[p. 88, Theorem 4 (ii) and p. 93, Examples]{SerreJP} (where $E_2$ is in our notation $E_4$).
\begin{theorem}\label{thm:weight-4-modular-form-E-4}
The weight $4$ modular form is a dimension one vector space generated by the Eisenstein series $E_4(\tau)$. $E_4(\tau)$ has the following expansion
\begin{equation}\label{eq:weight-4-modular-form-e-4}
E_4(\tau)=1+240\sum_{m=1}^{\infty}\sigma_3(m)q^m
\end{equation}
in $q=\exp\{2\pi i\tau\}$, $\tau\in\mathbb{H}$, where $\sigma_3(m)=\sum_{d|m}d^3$-the sum of the cubes of all positive divisors of $m$.
\end{theorem}
\subsection{Schwarzian Derivative and Modular Function and Modular Form}
For convenience, we introduce the following definition.
\begin{definition}
We say that a function $f:\mathbb{H}\rightarrow\mathbb{C}$ is an automorphic function for a discrete group $\Gamma'\subseteq\mbox{SL}_2(\mathbb{R})$ if $f$ is meromorphic on $\mathbb{H}\cup\{\infty\}$ and satisfies property
\begin{equation*}
f\circ\gamma(\tau)=f(\tau),\qquad\mbox{for all } \tau\in\mathbb{H} \mbox{ and }\gamma\in\Gamma'.
\end{equation*}
\end{definition}
\begin{remark}
Modular functions and modular forms are automorphic functions and automorphic forms of weight $zero$ for the modular group $\Gamma(1)$, respectively. Notice that the covering map $f:\mathbb{H}\rightarrow\punctured$ is an automorphic function, it is a generator of the function field over $\mathbb{H}/\Aut(f)$ which has a traditional name \textit{Hauptmodul}. We shall refer the covering map $f:\mathbb{H}\rightarrow \punctured$ as a Hauptmodul for the group $\Aut(f)$ in the future.
\end{remark}
The following lemma is mentioned in \cite[Proposition 3.2]{McKay2000}, for the completion of this article, I will restate and proof it.
\begin{lemma}\label{lemma:auto-func-dimension-one-fraction}
Let $f$ be a Hauptmodul for a genus zero discrete group $\Aut(f)\subset \mbox{SL}_2(\mathbb{R})$. For every $\gamma$ that normalizes $\Aut(f)$ in $\mbox{SL}_2(\mathbb{R})$, there exists a corresponding matrix $\eta=\left(\begin{array}{cc}
\eta_1 & \eta_2\\
\eta_3 & \eta_4
\end{array}\right)\in\mbox{SL}_2(\mathbb{C})$ such that the following equation
\begin{equation*}
f\circ\gamma(\tau)=\eta\circ f(\tau)
\end{equation*}
is true for any $\tau\in\mathbb{H}$.
\end{lemma}
\begin{proof}
Notice that $\gamma$ normalizes $Aut(f)$ in $\mbox{SL}_2(\mathbb{R})$, i.e., $\gamma\in\mathbb{H}=\mbox{SL}_2(\mathbb{R})$. Therefore the composition $f\circ\gamma(\tau)$ is also an automorphic function for $\Aut(f)$. The compactification of $\punctured\cong\mathbb{H}/\Aut(f)$ is the Riemann sphere $\mathbb{CP}^1$ which has genus zero, which implies that $\mathbb{H}^*/\Aut(f)$ has transcendental degree zero. Thus $f\circ\gamma(\tau)$ and $f(\tau)$ are related by an automorphism of $\mathbb{CP}^1$, i.e., an element $\eta$ in $\mbox{SL}_2(\mathbb{C})$. More precisely, assume $\gamma=\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\in\mbox{SL}_2(\mathbb{R})$ and $\eta=\left(\begin{array}{cc}
\eta_1 & \eta_2\\
\eta_3 & \eta_4
\end{array}\right)\in\mbox{SL}_2(\mathbb{C})$, we have the conclusion
\begin{equation*}
f\left(\frac{a\tau+b}{c\tau+d}\right)=\frac{\eta_1 f(\tau)+\eta_2}{\eta_3f(\tau)+\eta_4}
\end{equation*}
holds for every $\tau\in\mathbb{H}\cup\{\infty\}$.
\end{proof}
The following theorem plays an important role in determining the automorphic function for $\Gamma(N)$, $N=2,3,4,5$, it is also mentioned in \cite[Proposition 3.1]{McKay2000}, I restate and proof it here for the completion of this article. We will apply Proposition \ref{prop:schwarzian-basic-properties} and Lemma \ref{lemma:auto-func-dimension-one-fraction} to prove the following theorem.
\begin{theorem}\label{thm:schwarzian-f-tau-weight-4}
Let $f$ be a Hauptmodul for a genus zero discrete group $\Gamma'\subseteq\mbox{SL}_2(\mathbb{R})$, then $\{f,\tau\}$ is a weight $4$ automorphic form for $\Gamma'$.
\end{theorem}
\begin{proof}
First we show that $\{f,\tau\}$ is holomorphic on $\mathbb{H}$ and also holomorphic at infinity. From the assumption that $f(\tau)$ is a covering map, $f(\tau)$ is locally biholomorphic at any point $\tau_0\in\mathbb{H}$ with $f'(\tau_0)\neq0$. The definition of Schwarzian derivate
$$\{f,\tau\}=2\left(\frac{f_{\tau\tau}}{f_{\tau}}\right)_{\tau}-\left(\frac{f_{\tau\tau}}{f_{\tau}}\right)^2$$
shows the analyticity of $\{f,\tau\}$ at $\tau_0$. Recall equations \eqref{eq:schwarzian-expression-qk-tau-relation} and \eqref{eq:schwarzian-f-qk-expansion-2}, the following equation
\begin{align}
\{f,\tau\}&=\frac{4\pi^2}{k^2}(1-\qk^2\{f,\qk\}),\notag\\
&=\frac{4\pi^2}{k^2}(1-\sum_{m=0}^{\infty}P_m(\B,\C_3,\ldots,\C_{m+3})\qk^{m+2}),\notag
\end{align}
implies that $\{f,\tau\}$ is analytic at $\tau=\infty$. Therefore, $\{f,\tau\}$ is holomorphic on $\mathbb{H}\cup\{\infty\}$. To show $\{f,\tau\}$ is a weight $4$ modular form for $\Gamma'$ we only need to show equation
\begin{equation}\label{eq:schwarzian-weight-4-form-1}
\{f(\gamma(\tau)),\gamma(\tau)\}=(c\tau+d)^4\{f(\tau),\tau\}
\end{equation}
holds for every element $\gamma=\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\in\Gamma'$. On one hand, we know that $f\circ\gamma(\tau)$ and $f(\tau)$ are related by a linear transformation from Lemma \ref{lemma:auto-func-dimension-one-fraction}, and Proposition \ref{prop:schwarzian-basic-properties} implies equation
\begin{align}
\{f(\tau),\tau\}&=\{f(\tau),\gamma(\tau)\}(\gamma_{\tau})^2+\{\gamma(\tau),\tau\}=\{f(\tau),\gamma(\tau)\}(\gamma_{\tau})^2+\{\tau,\tau\},\notag\\
&=\{f(\tau),\gamma(\tau)\}\cdot(c\tau+d)^4,\label{eq:schwarzian-weight-4-form-3}
\end{align}
where $\{\tau,\tau\}=0$ since $\tau_{\tau\tau}=(1)_{\tau}=0$. Therefore equation \eqref{eq:schwarzian-weight-4-form-1} holds, i.e., $\{f,\tau\}$ is a weight $4$ modular form for $\Gamma'$.
\end{proof}
Next we have the following conclusion as a corollary of Theorem \ref{thm:schwarzian-f-tau-weight-4}.
\begin{coro}\label{coro:schwarzian-f-tau-weight-4-normalizer}
ILet $f$ be a Hauptmodul for a genus zero discrete group $\Gamma'\subseteq\mbox{SL}_2(\mathbb{R})$. Then $\{f,\tau\}$ is a weight $4$ automorphic form for the normalizer of $\Gamma'$ in $\mbox{SL}_2(\mathbb{R})$.
\end{coro}
\begin{proof}
Assume $\mu\in\mbox{SL}_2(\mathbb{R})$ normalizes $\Gamma'$. By Lemma \ref{lemma:auto-func-dimension-one-fraction}, there exists an matrix $\eta=\left(\begin{array}{cc}
\eta_1 & \eta_2\\
\eta_3 & \eta_4
\end{array}\right)\in\mbox{SL}_2(\mathbb{C})$ such that $f(\tau)=\eta\circ f(\tau)$.
Therefore we have the following equalities
\begin{align}
\{f(\mu(\tau)),\mu(\tau)\}&=\{\frac{\eta_1f(\tau)+\eta_2}{\eta_3f(\tau)+\eta_4},\mu(\tau)\}=\{f(\tau),\mu(\tau)\},\notag\\
&=(c\tau+d)^4\{f,\tau\},\notag
\end{align}
where $\mu=\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\in\Aut(f)$, and the last equation comes from direct calculation. This shows that $\{f,\tau\}$ is a weight $4$ auromorphic form for the normalizer of $\Gamma'$ in $\mbox{SL}_2(\mathbb{R})$.
\end{proof}
\begin{lemma}\label{lemma:gamma-N-gamma-1-normolizer}
$\Gamma(1)$ normalizes $\Gamma(N)$ in $\mbox{SL}_2(\mathbb{R})$. Consequently, $\Gamma(N)$ is normal in $\Gamma(1)$.
\end{lemma}
\begin{proof}
The proof is elementary.
\end{proof}
\subsection{Examples}\label{section:example}
We will see some examples by applying the main theorems.
\begin{example}[The covering space $\mathbb{CP}^1\setminus\{a_1=0,a_2,\ldots,a_{12}\}\cong\mathbb{H}/\Gamma(5)$]
In this case, $N=5$, $q_{_5}^5=q$, so the coefficients of $q_{_5}^2,q_{_5}^3,q_{_5}^4$ are all $0$. We have the following,
$$
\left\{\begin{array}{l}
12(\C_3-\B^2)=0,\\
48(\C_4-2\C_3\B+\B^3)=0,\\
24(5\C_5-10\C_4\B+17\C_3\B^2-6\C_3^2-6\B^4)=0,\\
\ldots,
\end{array}\right.\quad\mbox{implies}\quad
\left\{\begin{array}{c}
\C_3=\B^2,\\
\C_4=\B^3,\\
\C_5=\B^4,\\
\ldots.
\end{array}\right.
$$
Therefore the covering map $f_5:\mathbb{H}\rightarrow\mathbb{CP}^1\setminus\{a_1=0,a_2,\ldots,a_{12}\}$ with deck transformation group $\Gamma(5)$ can be given by the following expansion
\begin{equation}
\f_5=\frac{f_5(\tau)}{A}=q_{_5}+\B q^2_{_5}+\B^2q^3_{_5}+\B^3q^4_{_5}+\B^4q^5_{_5}+\sum_{m=6}^{\infty}P_m(\B)q^m_{_5},
\end{equation}
where the value of $\B$ is up to the collection of punctures $\{a_1=0,a_2,\ldots,a_{12}\}$. Therefore the complete K\"{a}hler-Einstein metric at the cusp $a_1=0$ is given by the following equation
\begin{align*}
|ds|=\frac{1}{|A||\f|\log|\f|}\left|1-\left(\B\f-\frac{\Re(\B\f)}{\log|\f|}\right)+\sum_{m=2}^{\infty}R_m(\B,\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})\right||df|,
\end{align*}
where $\f=\f_5=\frac{f_5}{A}$ for convenience, $f_5\in\mathbb{CP}^1\setminus\{a_1=0,a_2,\ldots,a_{12}\}$.
\end{example}
The following example is the case of the triple punctured Riemann sphere, which was mentioned in Example \ref{example:intro-thm-2}.
\begin{example}[The covering space of $\mathbb{H}/\Gamma(2)\cong\mathbb{CP}^1\setminus\{a_1=0,a_2,a_3\}$]\label{example:3-points-arbitrary}
In this example, $N=2$, $q^2_{_2}=q$, we have the following,
\begin{equation*}
\left\{\begin{array}{l}
12(\C_3-\B^2)=-240,\\
48(\C_4-2\C_3\B+\B^3)=0,\\
24(5\C_5-10\C_4\B+17\C_3\B^2-6\C_3^2-6\B^4)=-2160,\\
\ldots,
\end{array}\right.\quad\mbox{implies}\quad\left\{\begin{array}{l}
\C_3=\B^2-20,\\
\C_4=\B^3-40\B,\\
\C_5=\B^4-60\B^2+462,\\
\ldots.
\end{array}\right.
\end{equation*}
Consequently, the covering map $f_2:\mathbb{H}\rightarrow\mathbb{CP}^1\setminus\{a_1=0,a_2,a_3\}$ with deck transformation group $\Gamma(2)$ has expression
\begin{equation}\label{eq:f-2-general-q-2-expansion}
\f_2=\frac{f_2(\tau)}{A}=q_{_2}+\B q^2_{_2}+(\B^2-20)q^3_{_2}+\sum_{m=4}^{\infty}P_m(\B)q^m_{_2}.
\end{equation}
The metric equation \eqref{eq:ds-final-expansion-in-auto-coefficients} have the following expression
\begin{align*}
|ds|=\frac{1}{|A||\f|\log|\f|}\left|1-\left(\B\f-\frac{\Re(\B\f)}{\log|\f|}\right)+\sum_{m=2}^{\infty}R_m(\B,\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})\right||df|,
\end{align*}
where $\f=\f_2=\frac{f_2}{A}$, $f_2\in\mathbb{CP}^1\setminus\{0,a_2,a_3\}$, for convenience, and $\B=\frac{B}{A}$ is the only free parameter which is uniquely determined by the two punctures $a_2$ and $a_3$.
\end{example}
Next example is a special case of the triple punctured Riemann sphere, which was mentioned as Example \ref{example:intro} in section \ref{sec:introduction}. The explicit metric formula is also given by S. Agard from a different approach in \cite{AgardDist}.
\begin{example}[The quotient space $\mathbb{H}/\Gamma(2)\cong\mathbb{CP}^1\setminus\{0,1,\infty\}$]\label{example:metric}
When $A=16, B=-128$, equation \eqref{eq:f-2-general-q-2-expansion} is the covering map given by the modular lambda function
\begin{equation}
f_2=f_2(\tau)=\lambda(\tau)=16\qsec-128\qsec^2+704\qsec^3-3072\qsec^4+O(\qsec^5),
\end{equation}
where $f_2=\lambda(\tau)$ is the covering map of $\mathbb{H}\rightarrow\mathbb{CP}^1\setminus\{0,\infty,1\}$ with values at the cusps as below
$$\infty\mapsto 0,\qquad0\mapsto 1,\qquad1\mapsto\infty.$$
In this case, $\B=\frac{-128}{16}=-8$, $\C_3=\frac{704}{16}=44=(-8)^2-20$, the metric is given by the following
\begin{align*}
|ds|&=\frac{1}{|16||\f|\log|\f|}\left|1+8\left(\f-\frac{\Re\f}{\log|\f|}\right)\right.\\
&\qquad\left.-\left[(2\cdot 44+5\cdot 64)\f^2-64\frac{\f\Re\f}{\log|\f|}+(44+\frac{5}{2}\cdot 64)\frac{\Re(\f^2)}{\log|\f|}+64\frac{(\Re\f)^2}{\log^2|\f|}\right]+O(\f^3)\right||df|,\\
&=\frac{1}{|f|\log|f/16|}\left|1+\frac{1}{2}\left(f-\frac{\Re f}{\log|f/16|}\right)\right.\\
&\qquad\left.-\left[\frac{51}{32}f^2-\frac{1}{4}\frac{f\Re f}{\log|f/16|}+\frac{51}{64}\frac{\Re(f^2)}{\log|f/16|}+\frac{1}{4}\frac{(\Re f)^2}{\log^2|f/16|}\right]+O(f^3)\right||df|,
\end{align*}
where $f=f_2$ and $\f=\f_2=\frac{f_2}{16}$, $f=f_2\in\mathbb{CP}^1\setminus\{0,1,\infty\}$.
\end{example}
\begin{example}[The punctured Riemann sphere $\mathbb{CP}^1\setminus\{a_1,a_2,a_3\}$ for arbitrary $a_1,a_2,a_3$]
Assume $a_1, a_2$ and $a_3$ are three different points on $\mathbb{CP}^1$, the M\"{o}bius transformation
\begin{equation}
\lambda\mapsto\frac{a_1(a_2-a_3)-a_3(a_2-a_1)\lambda}{(a_2-a_3)-(a_2-a_1)\lambda}=\tilde{\lambda}\notag
\end{equation}
maps $\{0,1,\infty\}$ to $\{a_2,a_3,a_1\}$ respectively. Direct calculation indicates the following conclusion.
\begin{coro}\label{coro:section-8-1}
A covering map $f:\mathbb{H}\rightarrow\mathbb{CP}^1\setminus\{a_1,a_2,a_3\}$ for any three different points $a_1,a_2,a_3$ can be uniquely determined by the following equation
\begin{equation}
f(\tau)=\frac{a_1(a_2-a_3)-a_3(a_2-a_1)\lambda(\tau)}{(a_2-a_3)-(a_2-a_1)\lambda(\tau)}.
\end{equation}
Furthermore, $f(\tau)=\tilde{\lambda}(\tau)$ can be given by the following expansion,
\begin{equation}
f(\tau)=a_1+16\frac{(a_1-a_3)(a_2-a_1)}{a_2-a_3}\qsec+128(a_1-a_3)\left[2\frac{(a_2-a_1)^2}{(a_2-a_3)^2}-\frac{a_2-a_1}{a_2-a_3}\right]\qsec^2+O(\qsec^3).\notag
\end{equation}
\end{coro}
Therefore the metric expansion at $a_1$ can be given by the following corollary.
\begin{coro}\label{coro:section-8-2}
The complete K\"{a}hler-Einstein
metric on $\mathbb{CP}^1\setminus\{a_1,a_2,a_3\}$ at cusp $a_1$ has the following asymptotic expansion
\begin{equation}
|ds|=\frac{|a_2-a_3|}{16|a_1-a_3||a_2-a_1|}\frac{1}{\mathfrak{f}\log|\mathfrak{f}|}\left\{1-8\left[(2\frac{a_2-a_1}{a_2-a_3}-1)\mathfrak{f}-\frac{\Re((2\frac{a_2-a_1}{a_2-a_3}-1)\mathfrak{f})}{\log|\mathfrak{f}|}\right]+O(\mathfrak{f}^2)\right\}|df|\notag
\end{equation}
in $\mathfrak{f}=\frac{(a_2-a_3)}{16(a_1-a_3)(a_2-a_1)}(f-a_1)$, $f\in\mathbb{CP}^1\setminus\{a_1,a_2,a_3\}$.
\end{coro}
\begin{proof}
Direct calculation gives the value of coefficients $A, B$,
$$A=16\frac{(a_1-a_3)(a_2-a_1)}{a_2-a_3},\,B=128(a_1-a_3)\left[2\frac{(a_2-a_1)^2}{(a_2-a_3)^2}-\frac{a_2-a_1}{a_2-a_3}\right].$$
It implies the following value
$$\B=\frac{B}{A}=8\left(2\frac{a_2-a_1}{a_2-a_3}-1\right).$$
Then the result follows directly from Corollary \ref{coro:gamma-N-metric-expansion}.
\end{proof}
\end{example}
\section{Main Result and Examples}
\subsection{Main Result}
In section \ref{section:qk-expansion-metric-formula}, we mentioned the inversion series \eqref{eq:qk-f-inversion}. Now let us consider the situation that one of the singularities is $a_1=0$, and the corresponding parabolic generator fixes infinity. It is equivalent to say that a covering map $f:\mathbb{H}\rightarrow\punctured$ has expansion
\begin{equation*}
f=A\qk+B\qk^2+c_3\qk^3+\sum_{m=4}^{\infty}c_m\qk^m
\end{equation*}
in $\qk=\exp\{\frac{2\pi i}{k}\tau\}$, $\tau\in\mathbb{H}$, for some real constant $k$ with $A\neq 0$. Let us denote $\f=\frac{f}{A}$, $f\in\punctured$, for convenience, recall equation \eqref{eq:coefficient-depend-on-A-B} in Theorem \ref{thm:coefficient-depend-on-A-B},
\begin{equation*}
\f=f/A=\qk+\B\qk^2+\C_3(A,\B)\qk^3+\sum_{m=4}^{\infty}\C_m(A,\B)\qk^m,
\end{equation*}
where $\B=\frac{B}{A}$, $\C_m=\frac{c_m}{A}$ for $m\ge 3$. It is not hard to see that $\qk$ has the following expansion
\begin{equation}\label{eq:qk-f-inversion-poly-expression}
\qk(\f)=\f+\tB(\B)\f^2+\tc_3(\B,\C_3)\f^3+\sum_{m=4}^{\infty}\tc_m(\B,\C_3,\ldots,\C_m)\f^m
\end{equation}
in $\f$, where $\tB(\B)=-\B$ and $\tc_m(\B,\C_3,\ldots,\C_m)$ are polynomials in $\B,\C_3,\ldots,\C_m$ which has degree $1$ in $\C_m$ with constant coefficient. Let us restate the main result, Theorem \ref{thm:into-main-1}, and proof it here.
\begin{theorem}\label{thm:final-metric-formula-expansion-1}
Let $f:\mathbb{H}\rightarrow\mathbb{CP}^1\setminus\{a_1=0,a_2,\ldots,a_n\}$ be a covering map with expression
\begin{equation}
f=f(\tau)=A\qk+B\qk^2+c_3\qk^3+\sum_{m=4}^{\infty}c_m\qk^m.\notag
\end{equation}
Then the complete K\"{a}hler-Einstein metric has asymptotic expansion
\begin{align}
|ds|=\frac{1}{|A||\f|\log|\f|}\left|1-\left(\B\f-\frac{\Re(\B\f)}{\log|\f|}\right)+\sum_{m=2}^{\infty}R_m(A,\B,\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})\right||df|\label{eq:ds-final-expansion-in-auto-coefficients}
\end{align}
at the cusp $0$, where $\f=\frac{f}{A}$ for $f\in\mathbb{CP}^1\setminus\{a_1=0,a_2,\ldots,a_n\}$, $\B=\frac{B}{A}$, and $R_m(A,\B,\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})$ is a polynomial in $A,\B,\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|}$, $s,j=0,1,\ldots,m$, with constant coefficients for $m\ge 2$.
\end{theorem}
\begin{proof}
For convenience, we will simply write $q=\qk$ in this proof. Recall the proof of Theorem \ref{thm:metric-expression}, we apply the substitution $\f=\frac{f}{A}$,
\begin{align}
ds^2&=\frac{-4}{\left(\frac{k}{2\pi i}\log q(\f)-\overline{\frac{k}{2\pi i}\log q(\f)}\right)^2}\left|d\left(\frac{k}{2\pi i}\right)\log q(\f)\right|^2,\notag\\
&=\frac{|q_{\f}(\f)|^2}{|q(\f)|^2\log^2|q(\f)|}\left|\frac{d\f}{df}\right||df|^2,\notag\\
&=\frac{|q_{\f}(\f)|^2}{A^2|q(\f)|^2\log^2|q(\f)|}|df|^2.\label{eq:metric-expasion-in-bf(f)}
\end{align}
We calculate the expansion of each terms in equation \eqref{eq:metric-expasion-in-bf(f)} by \eqref{eq:qk-f-inversion-poly-expression},
\begin{align}
\frac{q_{_{\f}}(\f)}{q(\f)}&=(\log|q(\f)|)_{\f}=(\log|\f|+\log|1+\tB\f+\sum_{m=2}^{\infty}\tc_{m+1}\f^m|)_{_{\f}},\notag\\
&=\frac{1}{\f}+\tB+\sum_{m=1}^{\infty}\tilde{Q}^{(11)}_m(\tB,\tc_3,\ldots,\tc_{m+2})\f^m,\label{eq:q'-over-q-expansion}
\end{align}
where $\tilde{Q}^{(11)}_m(\tB,\tc_3,\ldots,\tc_{m+2})$ is a polynomial in $\tB,\tc_3,\ldots,\tc_{m+2}$ which has degree $1$ in $\tc_{m+2}$ with constant coefficients for $m\ge 1$. Next we calculate the expansion of $\frac{1}{\log|q(\f)|}$ in $\f$ and $\log|\f|$,
\begin{align}
\frac{1}{\log|q(\f)|}
&=\frac{1}{\log|\f|}\left(1+\frac{\log|1+\tB\f+\sum_{m=2}^{\infty}\tc_{m+1}\f^m|}{\log|\f|}\right)^{-1},\notag\\
&=\frac{1}{\log|\f|}\left[1-\frac{\log|1+\tB\f+\sum_{m=2}^{\infty}\tc_{m+1}\f^m|}{\log|\f|}+\sum_{l=2}^{\infty}(-1)^l\left(\frac{\log|1+\tB\f+\sum_{m=2}^{\infty}\tc_{m+1}\f^m|}{\log|\f|}\right)^l\right].\label{eq:1/log|q|-expansion-part-1}
\end{align}
Let us use the expansion $\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+\sum_{n\ge 4}(-1)^{n+1}\frac{x^n}{n}$ and write $\tc_1=1$ and $\tc_2=\tB$ for convenience,
\begin{align}
\log|1+\tB\f+\sum_{m=2}^{\infty}\tc_{m+1}\f^m|
&=\frac{1}{2}\log|1+\tB\f+\tc_3\f^2+\sum_{m=2}^{\infty}\tc_{m+1}\f^m|^2,\notag\\
&=\frac{1}{2}\log\left[1+2\Re(\tB\f)+\sum_{m=2}^{\infty}\left(\sum_{s+j=m,s,j\ge 0}\tc_{s+1}\overline{\tc_{j+1}}\f^s\overline{\f^j}\right)\right],\notag\\
&=\frac{1}{2}\left\{\left[2\Re(\tB\f)+\sum_{m=2}^{\infty}\left(\sum_{s+j=m,s,j\ge 0}\tc_{s+1}\overline{\tc_{j+1}}\f^s\overline{\f^j}\right)\right]\right.\notag\\
&\qquad\qquad\left.+\sum_{l=2}^{\infty}(-1)^{l+1}\frac{1}{l}\left[2\Re(\tB\f)+\sum_{m=2}^{\infty}\left(\sum_{s+j=m,s,j\ge 0}\tc_{s+1}\overline{\tc_{j+1}}\f^s\overline{\f^j}\right)\right]^l\right\},\notag\\
&=\frac{1}{2}\left[2\Re(\tB\f)+\sum_{m=2}^{\infty}R^{(1)}_m(\f^m,\f^{m-1}\overline{\f},\ldots,\f\overline{\f^{m-1}},\overline{\f^m})\right],\notag
\end{align}
where $R^{(1)}_m(\f^m,\f^{m-1}\overline{\f},\ldots,\f\overline{\f^{m-1}},\overline{\f^m})$ is a polynomial in $\f^m,\f^{m-1}\overline{\f},\ldots,\f\overline{\f^{m-1}},\overline{\f^m}$ with coefficients in $\tB,\tc_3,\ldots,\tc_{m+1}$. Therefore equation \eqref{eq:1/log|q|-expansion-part-1} has expansion
\begin{align}
\frac{1}{\log|q(\f)|}
&=\frac{1}{\log|\f|}\left(1-\frac{1}{2}\frac{1}{\log|\f|}\left[2\Re(\tB\f)+\sum_{m=2}^{\infty}R^{(1)}_m(\f^m,\f^{m-1}\overline{\f},\ldots,\f\overline{\f^{m-1}},\overline{\f^m})\right]\right.\notag\\
&\qquad\qquad \left.+\sum_{l=2}^{\infty}\frac{(-1)^l}{2^l\log^l|\f|}\left[2\Re(\tB\f)+\sum_{m=2}^{\infty}R^{(1)}_m(\f^m,\f^{m-1}\overline{\f},\ldots,\f\overline{\f^{m-1}},\overline{\f^m})\right]^l\right),\notag\\
&=\frac{1}{\log|\f|}\left[1-\frac{\Re(\tB\f)}{\log|\f|}+\sum_{m=2}^{\infty}R^{(2)}_m(\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})\right],\label{eq:1/log|q|-expansion-part-final}
\end{align}
where $R^{(2)}_m(\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})$ is a polynomial in $\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|}$, $s=0,1,\ldots,m$ and $j=1,\ldots,m$, which has coefficients in $\tB,\tc_3,\ldots,\tc_{m+1}$ for $m\ge 2$. Therefore equation \eqref{eq:q'-over-q-expansion} and \eqref{eq:1/log|q|-expansion-part-final} implies that the metric $|ds|$ defined by equation \eqref{eq:metric-expasion-in-bf(f)} has expression
\begin{align}
|ds|&=\frac{1}{|A||\f|\log|\f|}\left|1+\tB\f+\sum_{m=1}^{\infty}\tilde{Q}^{(11)}(\tB,\tc_3,\ldots,\tc_{m+2})\f^{m+1}\right|\cdot\left[1-\frac{\Re(\tB\f)}{\log|\f|}+\sum_{m=2}^{\infty}R^{(2)}_m(\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})\right],\notag\\
&=\frac{1}{|A||\f|\log|\f|}\left|1+\left(\tB\f-\frac{\Re(\tB\f)}{\log|\f|}\right)+\sum_{m=2}^{\infty}\tilde{R}_m(\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})\right||df|,
\end{align}
where $\tilde{R}_m(\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})$ is a polynomial in the variable set $\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|}$, $s,j=0,1,\ldots,m$, with coefficients in $\tB,\tc_2,\ldots,\tc_{m+1}$ for $m\ge 2$. Due to equation \eqref{eq:qk-f-inversion-poly-expression} and Theorem \ref{thm:coefficient-depend-on-A-B}, each term $\tB,\tc_2,\ldots,\tc_{m+1}$ can be solved as a polynomial in $A,\B$, so the coefficients in $\tilde{R}_m(\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})$ are polynomials in $A,\B$. Let us write $\tilde{R}_m(\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})$ as $R_m(A,\B,\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})$, which denotes a polynomial in $A,\B,\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|}$ with constant coefficients. Recall that $\tB=-\B$, we have conclusion that the metric $|ds|$ is given by the following expression
\begin{equation}
|ds|=\frac{1}{|A||\f|\log|\f|}\left|1-\left(\B\f-\frac{\Re(\B\f)}{\log|\f|}\right)+\sum_{m=2}^{\infty}R_m(A,\B,\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})\right||df|,
\end{equation}
and, consequently, $|ds|$ is uniquely determined up to a choice of $A,\B$.
\end{proof}
Recall Theorem \ref{thm:Gamma-Aut-sec-6}, now let us focus on the case that the deck transformation group is $\Gamma(N)$, $N=2,3,4,5$.
\begin{theorem}\label{thm:2nd-main-thm}
Let $n(N)$ be the numbers that is defined by equation \eqref{eq:cusp-number-Gamma-N}, and let $f_N:\mathbb{H}\rightarrow\Npunctured$ be a universal covering with deck transformation group $\Aut(f_N)=\Gamma(N)$, $N=2,3,4,5$, satisfying that $f_N$ vanishes at infinity, i.e., $f_N(\infty)=0$. Then $f_N$ can be given by the following expansion
\begin{equation}\label{eq:2nd-main-theorem-equation}
f_N(\tau)=A\q+B\q^2+\sum_{m=3}^{\infty}A\cdot\C_{m}(\B)\q^m
\end{equation}
in $\q=\exp\{\frac{2\pi}{N}i\tau\},\tau\in\mathbb{H}$, where the constants $A,B\in\mathbb{C}$ are uniquely determined by the set of values of the punctured points $\{a_1=0,a_2,\ldots,a_{n(N)}\}$, and the term $\C_m(\B)$ in the coefficient is a polynomial in $\B$ with constant coefficients for $m\ge 3$.
\end{theorem}
\begin{proof}
By Proposition \ref{prop:schwarzian-expression-qk-tau-relation}, the following identity
\begin{equation}
\{f_N,\tau\}=\frac{4\pi^2}{N^2}(1-\q^2\{f_N,\q\})\notag
\end{equation}
holds. Corollary \ref{coro:schwarzian-f-tau-weight-4-normalizer} and Lemma \ref{lemma:gamma-N-gamma-1-normolizer} imply that $\{f_N,\tau\}$ is a weight 4 modular form for $\Gamma(1)$. Recall Theorem \ref{thm:weight-4-modular-form-E-4}, there is a constant $\kappa$ such that the following equation
\begin{equation}
E_4(\tau)=\kappa\{f_N,\tau\}=\kappa\frac{4\pi^2}{N^2}(1-\q^2\{f_N,\q\})
\end{equation}
holds. Equations \eqref{eq:weight-4-modular-form-e-4} and \eqref{eq:schwarzian-f-qk-expansion-2} imply the following identity
\begin{equation}\label{eq:main-theorem-schwarzian-qN-expansion}
1+240\sum_{m=1}^{\infty}\sigma_3(m)q^m=\frac{4\pi^2}{N^2}\kappa[1-\sum_{m=0}^{\infty}P_m(\B,\C_3,\ldots,\C_{m+3})\q^{m+2}].
\end{equation}
Matching the constant terms in equation \eqref{eq:main-theorem-schwarzian-qN-expansion},
\begin{equation*}
1=\frac{4\pi^2}{N^2}\kappa\quad\mbox{implies}\quad\kappa=\frac{N^2}{4\pi^2}.
\end{equation*}
Therefore we have the following identity
\begin{equation*}
1+240\sum_{m=1}^{\infty}\sigma_3(m)q^m=1-P_0(\B,\C_3)\q^2-\sum_{m=1}^{\infty}P_m(\B,\C_3,\ldots,\C_{m+3})\q^{m+2}
\end{equation*}
holds. Notice the identity $\q^N=q$ by their definitions, we match the coefficient of $\q^{lN}$ in equation \eqref{eq:main-theorem-schwarzian-qN-expansion} with the coefficient of $q^l$ in equation \eqref{eq:weight-4-modular-form-e-4}, and let other coefficients in equation \eqref{eq:main-theorem-schwarzian-qN-expansion} be $0$. We get the following set of equations,
\begin{equation}
P_m(\B,\C_3,\ldots,\C_{m+3})=\left\{
\begin{array}{ll}
240\sigma_3(l), & \mbox{if }m=lN-2\mbox{ for }l=1,2,\ldots,\\
0, & \mbox{otherwise},
\end{array}\right.
\end{equation}
where $m\ge 0$. Therefore $\C_3$ can be solved in terms of $\B$ when $m=0$, and notice that every coefficient $P_m(\B,\C_3,\ldots,\C_{m+3})$ is a polynomial of degree $1$ in $\C_{m+3}$ with constant coefficient. By induction, we can conclude that $\C_m$ can be solved as a polynomial in $\B$. Equation \eqref{eq:2nd-main-theorem-equation} holds since $c_m=A\cdot\C_m$, $m\ge 3$.
\end{proof}
\begin{coro}\label{coro:gamma-N-metric-expansion}
Under the same assumption in Theorem \ref{thm:2nd-main-thm}, the complete K\"{a}hler-Einstein metric has asymptotic expansion
\begin{equation}
|ds|=\frac{1}{|A||\f|\log|\f|}\left|1-\left(\B\f-\frac{\Re(\B\f)}{\log|\f|}\right)+\sum_{m=2}^{\infty}R_m(\B,\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})\right||df|
\end{equation}
at the cusp $0$, where $R_m(\B,\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|})$ is a polynomial in $\B,\f,\frac{\f^s\overline{\f^{m-s}}}{\log^j|\f|}$, $s,j=0,1,\ldots,m$, with constant coefficients for $m\ge 2$.
\end{coro}
\begin{proof}
It directly follows from Theorems \ref{thm:final-metric-formula-expansion-1} and \ref{thm:2nd-main-thm}.
\end{proof} |
1,314,259,996,497 | arxiv | \section{Introduction}
"A theory of everything" (TOE) is usually considered as a final goal. It means an existence of some universal lagrangian containing fields, symmetries and parameters (coupling constants) that should describe all phenomena in the Universe. But even if the goal was achieved main questions listed above remain unanswered:
\begin{itemize}
\item What is the origin of the fields? Why just these fields are realized in the Nature?
\item Why just these symmetries are realized but not others?
\item Why the parameters have specific values?
\item Why our space has specific number of dimensions (4, 10, 11, 26)?
\item The situation is complicated by the known problem: the physical parameters acquire their observable values which are extremely fine tuned.
\end{itemize}
One may conclude that a TOE can not be a real final theory due to its internal reasons.
The alternative approach - multiverse - is also known. In this case it is supposed that many theories (universes) could be realized in Nature and we accidentally live in some space domain (Universe) described by one of the theory. This way is partly declared by the string theory. Main objection to this approach can be expressed shortly in one phrase: "If everything is possible, it is not a science". Let me suggest some example as a counterargument. Indeed, it would be strange if somebody would explain the observed Earth mass just saying that a lot of planets with different masses do exist. Does it mean that any scientific researches in this direction are meaningless? Not at all. Instead we develop the theory of planet formation and some other scientific directions like the geology, the geophysics.
\section{Formation of low energy physics}
Our aim is to propose the approach that connects observable physical parameters and initial parameters of nonlinear gravity acting in $D-$ dimensional. It will be shown that a universe similar to ours can be formed for a wide range of initial parameters. In case of success, answers to the questions itemized above would be quite evident. The main tools are multidimensional gravity and quantum theory. The latter is needed to justify a quantum creation of manifolds with various metrics. For our purposes we need only the same fact of a nonzero probability of any metric formation in a small space region.
The essence of the approach is as follows. Suppose that a sufficient number of extra dimensions does exist. A metric of a whole space $M_0$ is chosen in the form
\begin{equation}\label{times0}
M_0 =M_4\times M.
\end{equation}
Here $M_4$ is our 4-dim space and $M$ is an extra space. It is supposed that every 4-dim lagrangian, e.g. the Standard Model lagrangian is deduced from a primary one that acts in the whole space $M_0$. Let this lagrangian contains initial parameters $a_{in}$. The observed values of the parameters $a_{obs}$ of the Standard Model are their functions
\begin{equation}\label{aobs}
a_{obs}=f(a_{in},g_M).
\end{equation}
Here $g_M$ is a metric tensor of the extra space.
A form of the function $f$ is a separate problem which was discussed partly in \cite{RuZin}. The internal metric $g_M$ is usually fixed by low energy physics what leads to strict connection between two sets $a_{in}$ and $a_{obs}$. Thus, in spite of many interesting applications have been performed, see e.g. \cite{Brokoru07, BOBRU}, the problem of origin of the same parameters $a_{in}$ remains. Let us make a next step and suppose that the geometry has the form
\begin{equation}\label{times1}
M_0 =M_4\times M\times K,
\end{equation}
where $K$ is an another extra space with a metric $g_K$. Formula (\ref{aobs}) is transformed into the two-step expression
\begin{equation}\label{aobs1}
a_{in}=f_1 (b_{in},g_K), \quad a_{obs}=f(a_{in},g_M),
\end{equation}
where $b_{in}$ is a new set of initial parameters. For the first sight nothing has been changed. In fact we have obtain additional freedom - an internal metric $g_K$ may be varied freely. The set $a_{in}$ can be obtained from a variety of sets $b_{in}$ by fitting the metric $g_K$. We will refer to the set $a_{in}$ as a secondary set and to the set $b_{in}$ as a primary one. Our nearest goal is to elaborate some production mechanism of various metrics $g_K$. It seems a trivial task because quantum fluctuations claimed produce any form of metric. The problem becomes much less simple if one takes into account a classical motion of an extra space metric $g_K$ just after its nucleation. As a result, a set of final metrics $g_K (t\rightarrow \infty)$ appears to be quite limited \cite{RuZin}. This strongly reduces a number of possibilities for the metric variation in (\ref{aobs1}).
As was shown in \cite{RuZin} the metrics could evolve classically to some stationary configuration $g_{stat}(y)=const$ that does not depend on an initial metric. This way gives rise a large but limited number of an initial sets $a_{in}$ of parameters that lead to the universe similar to ours. The problem of fine tuning would be hardly solved.
Another wide class of theories could be obtained if classical equations for the metric $g_K$ have finite-dimensional attractors. In this case a behavior of the metric at large times depends on an arbitrary initial metric and form a continuous set. Thus we come to infinite set of effective theories.
So far we have implicitly assumed that the structure of the space is
$T\times M_{D_0 -1}$, i.e. we do not distinguish an extra space and the observable 3-dim space of our Universe. In fact, the huge difference in their sizes has to be explained.
As was shown in \cite{BR06}, the reason of their difference is laid in initial conditions. If an initial metric satisfies the conditions
\begin{equation}\label{times2}
M_0 = M_1\times K_1 ;\quad R_M \ll R_K ,
\end{equation}
where $R_M , R_K$ are the Ricci scalar of the main space and the extra space correspondingly, their following evolution is different. Relaxation time in the extra space $K$ is much smaller so that an evolution of metric of the main space $M$ proceeds at (almost) stationary extra metric. Initial conditions dictate a shape of the extra space while the latter influences a dynamics of the main space.
\section{Explicit form of secondary parameters}
To illustrate the above, consider a toy model with the specific pure gravitational action in a $D_0-$dim space
\begin{equation}\label{act1}
S=\int d^{D_0} z\sqrt{G}F(R_{M_0},\{b\} ) ;\quad F(R,\{b\} )=\sum_{n=1}^N b_n R^n.
\end{equation}
Here $R_{M_0}$ is the Ricci scalar and $\{b\} = b_1, b_2 , ... b_N$ .
We follow only those geometries that represent a direct product
\begin{equation}\label{UMK}
U=M_4 \times M \times K
\end{equation}
and satisfy the inequality written in (\ref{times2}). A classical motion of the metric $g_K$ as a function in the extra space $M$ is of interest and we will omit any mentioning about our 3-dim space.
Metric
\begin{eqnarray}\label{interval}
ds^{2}&=&dt^2 - g_{ab}(t,y) dy^{a}dy^{b}-
e^{2\beta_1 (t)}\gamma_{1,ij}(z)dz_1 ^{i}dz_1 ^{j} - e^{2\beta_2 (t)}\gamma_{2,ij}(z)dz_2 ^{i}dz_2 ^{j}.
\end{eqnarray}
where $g_{ab}(t,y)$ is the metric in $M$,
$\gamma_{1,2;ij}(z)$ are positively defined internal metrics of the extra space $K=K_1 \times K_2$ and $e^{2\beta_{1,2}(t)}$ are scaling factors (see \cite{Carroll,Brokoru07}) of the spaces $K_{1,2}$. Also, $D=dim M, d=dim K$.
Evolution of the extra space metric is governed by classical equations for the functions $\beta_{1,2}(t)$. As was shown in \cite{BR06}, the effective action for the field $\beta$ has the form
\begin{eqnarray}\label{E-H_ActionD-1}
&&S = \frac{1}{2} V [d_1] \int d^{D}y\sqrt{ \left|G^{(D)}\right|}\, \{ \sign F'\cdot [R_4 + K] - 2V(\phi) \} \\
&&K_E =\frac{1}{d} \left( \partial\sigma
+ \frac{F''}{F'}\partial\phi \right)^{\! 2}
+ \left(\frac{F''}{F'}\right)^{\! 2} (\partial\phi)^2
+ \sum_i d_i (\partial\beta_i)^2, \label{KE}
\\
&&-2V_E (\phi_i) = \label{VE}
e^{-\sum_i \beta_i d_i} |F'|^{-d_0/d} F(\phi),
\end{eqnarray}
\begin{equation} \label{def-phi}
\phi_{1(2)}(t) := (d_{1(2)} - 1) e^{-2\beta_{1(2)}(t)}, \quad
\phi := d_1\phi_1 + d_2\phi_2 .
\end{equation}
As was shown in \cite{RuInfl} the lines $\phi_{1,2}^* =0$ are attractors of the classical equations for the fields $\phi_{1,2}$. We are interested in solutions of the form $\phi_{1} \rightarrow 0; 0<\phi_{2}<\infty$.
Under these assumptions the Ricci scalar
\begin{equation}\label{ricci}
R = R_{M} + \phi_1 (t) + \phi_2 (t)
\end{equation}
is easily obtained.
After some algebra with formulas (\ref{ricci}) and (\ref{act1}) the reduced action can be obtained
\begin{equation}\label{act2}
S=\int d^{D_M} y\sqrt{g_M}F(R_{M},\{a\} ),
\end{equation}
where the intermediate set of parameters ${a}$ is connected to a set of primary parameters ${b}$
\begin{equation}\label{newparam}
a_n (t) = \int d^{D_K}z \sqrt{g_K }\frac{F^{(n)}( \phi_1 (t) + \phi_2 (t),\{b\})}{n!}.
\end{equation}
$g_M , g_K$ are determinants of the metric tensors of spaces $M$ and $K$.
This formula may be considered as the connection between the primary set $b$ and the metric $g_K$ of extra space $K$. Variety of the sets ${b}$ can be obtained by continuous variation the metric $g_K$ at fixed intermediate parameters ${a}$. This illustrates the hypothesis:
\emph{In the framework of multidimensional gravity, the observable set of effective parameter values could be obtained by a continuous set of initial parameter values.}
Formula (\ref{newparam}) indicates that all physical parameters vary with time. Their modern values represent the asymptotes at time tends to infinity.
In this paper we discuss the idea of "inverse landscape". By this is meant that quantum fluctuations of extra space metric and their subsequent classical evolution could lead to observable values of physical parameters in wide range of initial parameter values of a primary lagrangian. In this case the fine tuning problem seems solvable.
The study was supported by The Ministry of education and science of Russian Federation, project 14.A18.21.0789.
|
1,314,259,996,498 | arxiv | \section{Introduction}
\subsection{Software in science}
The use of computer software in research has resulted in significant hardware and software developments
in computing science. Nowadays, the number of different scientific software packages is overwhelming, and it has become
progressively difficult for users (e.g. a scientist) to evaluate the relevance, usage and the performance of these packages.
Firstly, installing scientific software can be cumbersome, especially when the installation
and/or compilation is poorly designed. The software code, the library dependencies, the host platform and the compilers
may change over time, making it unclear how the original developer(s) intended to install and use the software. Secondly,
conflicting dependencies may arise when different software packages are built together, making it difficult to install them on
the same system. Thirdly, software packages have non-uniform interfaces as they have varying expectations of interaction
with a user or with other packages on the same system.
Kliko is a Docker-based encapsulating and chaining framework that purports to mitigate these issues
by creating a container of the software thereby solving the first and second issue above. The third issue can then
be solved by building a Docker container that has minimal extra requirements, i.e. the Kliko definition.
Kliko consists of two parts: i) a set of utilities for creating a container, including parsers to
check if all (meta) data is valid; and ii) a support library that can be used to schedule a Kliko
container and run it from a command line or from a web interface.
Kliko is not a pipeline construction tool itself, nor a web interface, but it can assist in making these.
\subsection{Software containerization with Docker}
Containerization is a method for building self-contained environments (called ``containers'') for applications. These
containers can then be distributed and used with minimal effort on a large variety of platforms.
Containerizing applications is not new. Similar techniques have been applied before, e.g. jail for FreeBSD
\footnote{\url{https://www.freebsd.org/doc/handbook/jails.html}}, zones for Solaris \cite{PriceSolaris} and chroot for
GNU/Linux \footnote{\url{http://man7.org/linux/man-pages/man2/chroot.2.html}}. However, their application was mostly
limited to enhancing security and to carry out clean builds of the UNIX system. The addition of operating system (OS)
level process isolation, named control groups or cgroups\cite{rosen2013resource}), to the popular Linux kernel
(since 3.8, 2008) accelerated the adoption of containerization for the usage of software distribution.
There are multiple software projects leveraging cgroups, for example
rkt\footnote{\url{https://github.com/coreos/rkt}}, Docker\cite{Boettiger14}\footnote{\url{https://www.docker.com}},
Singularity\cite{kurtzer2016}\footnote{\url{http://singularity.lbl.gov}} and
LXC\footnote{\url{https://linuxcontainers.org}}. Docker \cite{Merkel2014} is currently the most popular container
technology with the largest community of users and the most momentum for future development and support. Kliko aims to
be agnostic of the container technology, but since Docker has the biggest user community, we focus on this
implementation.
In Docker, an image is built using a initialization script (a ``Dockerfile'') which contains the recipe
to install or build the application. The Dockerfile is a series of commands applied to a basic and clean Docker image,
typically a headless Linux distribution. These base images are retrieved from an online database provided by Docker,
and stored locally. The Dockerfile, when
executed, will create an ``image'' which is a ``inactive'' snapshot of the virtualized application. An image becomes a
container when instantiated (e.g. the application runs). The difference between active and inactive is important, a
container is an image with an unwritten (dirty) state.
An application that is containerized is self-contained and can be seen as a complete OS without kernel. The container
could even only contain a statically compiled binary, but in practise it is useful to have the tools and package
manager of a Linux distribution available inside the container. Theoretically, to run a Docker container using Docker
on a host machine, the only requirement is to have the Docker daemon running on the host. Unfortunately, there are
some hardware specific edge cases like CPU register usage optimization and GPU acceleration. These cases will be
discussed in section \S\ref{sec:Limitations}.
A Docker container ``image'' is basically a file system snapshot of a minimal OS. The ``target'' application (i.e.
the one to host) and its library dependencies are installed inside this virtual isolated file system. When the
application is started, the container file system is exposed to the application as the working environment.
On a kernel level, cgroups and namespaces are used to create a new isolated environment for the application, limiting
access to other processes on the host and presenting the isolated environment as if is a separate host to the application.
Intuitively, this can be seen as similar technology as CPU level OS virtualization like VirtualBox, but in the case of
containers, the kernel is shared by the host and the guest.
A Docker container also gets a private IP address on an internal network range. This makes the container appear as a
separate networked machine to the host. By default, access to network ports are restricted and access needs to be granted per
port. One can also forward the port to an external interface where it will appear as the service is running on the host
itself.
All of the above might appear similar to simple virtualization, but containerization has some clear additional advantages.
Firstly, when using Docker, available physical resources do not need to be
partitioned between the host and the guest. While memory size allocated for a virtual machine is fixed or not easy to change,
running containers does not require the user to fix this memory size, although it still is possible to limit the amount of memory
allocatable by the process. Secondly, there is no CPU instruction emulation, as the process is directly executed on the
host kernel. Thirdly, there is minimal startup and shutdown overhead for starting containers as the containerized OS is
reduced to minimal consumption. Startup time is instantaneous (in the millisecond range) and loading time will only
become noticeable when high numbers of containers are spawned.
In addition to containerization, Docker also offers other features: it uses an ``union'' file system to join
multiple layers of file systems together. The intermediate result of each command in the Dockerfile is cached and stored
in layers. These layers can be reused by other containers, allowing data sharing between them, which reduces the size of
the storage requirements. These layers can also be stored in a central location, where they can be distributed and reused
in both a public or private way.
\section{The Kliko specification}
The Kliko specification is designed to extend containerization with an uniform interface resulting in simplified
interaction with the containerized application.
The Kliko specification describes how a Kliko container should look like and what a Kliko container should expect during
runtime. The relevant terminology is listed below:
\underline{Def~1:} The Kliko Image\\
A Docker image complying to the Kliko specification. An image is a read-only ordered collection of root file system
changes and the corresponding execution parameters for use within a container runtime.
\underline{Def~2:} The Kliko Container\\
A container in an active (or inactive if exited) stateful instantiation of a Kliko image.
\underline{Def~3:} The Kliko Runner\\
A process that can run a Kliko image to make a container. For example the Kliko-run command line tool, or RODRIGUES
(see \S \ref{rodrigues}).
\underline{Def~4:} The Kliko Parameters\\
A list of parameters that can influence the behavior of the software in the container. The list can be arbitrary in
size and consists of any combination of primitive types listed in Table~\ref{tab:klikotypes}.
\subsection{The Kliko Image}
A Kliko image should contain a \texttt{/kliko.yml} file in YAML\footnote{\url{http://www.yaml.org}}
syntax following the Kliko schema \ref{klikoformat}. YAML is a human-readable data
serialization language and stands for \textit{YAML Ain't Markup Language}.
The Kliko image should also contain a \texttt{/kliko} file which is called during runtime by the Kliko runner. This
Kliko script can be anything executable, but in most cases, it will be a Python script using the Kliko library to check
and parse all related Kliko tasks during runtime. Note that we have deliberately chosen not to use the ENTRYPOINT or CMD
statements supported by Docker. This way, Kliko is non-intrusive and can be easily added to existing containers that already
set an ENTRYPOINT or CMD.
\subsection{Expected runtime behavior}
During runtime, the Kliko runner will gather the parameters and expose them to the Kliko container. The content of the
variables is exposed by the Kliko runner in the \texttt{/parameter.json} file, which should contain a flat dictionary
in JSON syntax\footnote{\url{https://www.json.org}}. JSON and YAML are structurally very similar, but YAML is designed
to be more human-readable, hence our choice of YAML for the Kliko definition. Future versions of Kliko will support
both formats.
While reading this text, one might get confused by the context
of the file location (inside or outside the container). As a rule of thumb if a path in this text starts with a slash
(\texttt{/}) it is \textit{inside} the container.
If one or more of the parameters is a file, those will be exposed by the Kliko runner in the read-only
\texttt{/param\_files} folder during runtime. It is the responsibility
of the Kliko container to parse the \texttt{/parameters.json} file, perform potential the run-time housekeeping and
convert the parameter keys, values and/or files into an eventual command do be executed.
It is recommended to write logging to stdout and stderr. This makes it easier for the Kliko runner to visualize or
parse the output of a Kliko image.
\subsection{Flavors of Kliko Images}
\label{flavors}
We distinguish two flavors of Kliko containers, \textit{joined Input/Output} (read-write) and
\textit{split IO} (read-only). The style of container is specified in the \texttt{io} field in \texttt{/kliko.yml} file
inside the container, see \S \ref{klikoformat}.
The difference is the way the contained software interacts with the working data. In the case of \textit{split IO}
the Kliko runner exposes the input data to the container in the \texttt{/input} folder. This folder is read-only, to
prevent accidental manipulation of the data. The Kliko container is expected to write any output data into the
\texttt{/output} folder. The Kliko Runner will then handle this output data after the container reaches the end if its
lifetime. A \textit{split IO} Kliko container should always yield the same results for multiple independent runs
when presented with the same data and parameters (formally is called, ``having no side effects''). This is basically the
essence of the functional programming paradigm.
In the case of \textit{joint IO} there is only one point of interaction with the Kliko host, \texttt{/work} which is exposed
read/write. Basically, the input and output folders are combined into one that is mounted with read/write permissions.
Contrary to the \textit{split IO} flavor, this might be potentially dangerous for data processing as it can alter the
original data.
From a run-time parallelization perspective, the \textit{split IO} flavor is preferred. A container without side effects
enables the Kliko Runner to do graph-based logical inference of dependencies and execution scheduling, reuse results and
also run various containers in parallel, potentially resulting in faster execution. In practice, existing software
does not
always support this type of operation, or it is simply not feasible to create a copy of the data. In that case, the
\textit{joined IO} style has to be used.
\subsection{The \texttt{/kliko.yml} schema}
\label{klikoformat}
A \emph{kliko.yml} file is a YAML file and it \emph{should} contain the fields listed in Table~\ref{tab:klikofields}.
\begin{table*}[t]
\caption{Required Kliko fields}
\label{tab:klikofields}
\begin{tabular}{l | p{0.7\textwidth} }
\hline
field & description \\ \hline
\textbf{schema\_version} & The version of the Kliko specification, independent of the versioning of the Kliko library \\
\textbf{name} & Name of the Kliko image. For example ``radioastro/simulator'' for RODRIGUES. \\
\textbf{description} & A more detailed description of the image. \\
\textbf{url} & Website of project or repository where project is maintained \\
\textbf{io} & ``join'' or ``split''. See the two flavors of Kliko Containers in \S \ref{flavors} \\
\textbf{Sections} & a list of one or more sections, grouping fields together.
\end{tabular}
\end{table*}
Each section contains a list of fields. Each fields statement should contain a list of field elements. Each field
element has two mandatory keys, a name and a type. Name is a short reference to the field which needs to be unique.
This will be the name for internal reference. The type defines the type of the field, possible types are listed in
Table~\ref{tab:klikotypes}. Depending on the type there are optional extra fields, listed in Table~\ref{tab:klikotypefields}.
\begin{table*}[t]
\caption{Kliko variable types}
\label{tab:klikotypes}
\begin{tabular}{l | p{0.8\textwidth} }
\hline
type & description \\ \hline
\textbf{choice} & field with a predefined set of options, see the optional choices field below \\
\textbf{str} & string value \\
\textbf{float} & float value \\
\textbf{file} & A file path. This file will be exposed in \texttt{/param\_files} at runtime by the Kliko Runner \\
\textbf{bool} & A boolean value \\
\textbf{int} & An integer value
\end{tabular}
\end{table*}
\begin{table*}[t]
\caption{Kliko field types}
\label{tab:klikotypefields}
\begin{tabular}{l | p{0.8\textwidth} }
\hline
field & description \\ \hline
\textbf{initial} & supply a initial (default) value for a field \\
\textbf{max\_length} & define a maximum length in case of string type \\
\textbf{choices} & define a list of choices in case of a choice field. The choices should be a mapping \\
\textbf{label} & The label used for representing the field to the end user. If no label is given the name of the field is used \\
\textbf{required} & Indicates if the field is required or optional \\
\textbf{help\_text} & An optional help text that is presented to the end user next to the field \\
\end{tabular}
\end{table*}
The schema described above is defined in the Kwalify format. Kwalify is a parser and schema validator for YAML and
JSON\footnote{\url{http://www.kuwata-lab.com/kwalify}}.
The definition itself is also written in YAML. The Kliko library pykwalify\footnote{\url{https://github.com/Grokzen/pykwalify}}
is used to validate the YAML file against a schema. The full Kliko version 2 schema is listed Listing~\ref{code:kliko.yml}
in \ref{appendix:example}.
\subsection{The \texttt{/parameters.json} file}
When a container is started, the Kliko runner will mount a \texttt{/parameters.json} file into the container. This file
contains all parameters for the container in the JSON format. The \texttt{/kliko} script supplied by the container
author should read and parse the \texttt{/parameters.json file}. The Kliko library (\ref{sec:klikolibrary}) supports
helper functions and scripts to parse and validate this file.
Validation is done based on the \texttt{/kliko.yml} definition, which is useful for preventing or tracking down problems.
An example parameters file that could be generated based on the kliko.yml definition is shown in Listing
\ref{code:parameters.json}.
\begin{listing}[htbp!]
\begin{minted}[frame=single,fontsize=\scriptsize]{json}
{
"int": 10,
"file": "some-file",
"char": "gijs",
"float": 0.0,
"choice": "first"
}
\end{minted}
\caption{Example \texttt{parameters.json} file}
\label{code:parameters.json}
\end{listing}
Note that the sections are not supplied since they are only used for grouping and representation to the user.
\section{Running Kliko containers}
\subsection{Running a container manually}
As an example, we will describe an extremely simple Kliko container named ``fitsimagerescaler", available on our github
repository described in \S \ref{sec:Software}. This container takes a FITS image file which resides in the
\texttt{/input} directory, opens it, multiplies the pixel values by a parameterized value (2 by default) and exports
the result as a new FITS image in \texttt{/output}. The actual code that is run in \texttt{/kliko} is shown in
Listing \ref{code:fitsimagerescaler}.
Starting the Kliko container is nothing more than starting the container using Docker with some specific flags. If the
\texttt{parameters.json} file already exists, starting the container from the command line looks like this:
\begin{listing}[htbp!]
\begin{minted}[frame=single,fontsize=\scriptsize]{console}
$ docker run -t -i \
-v `pwd`/parameters.json:/parameters.json:ro \
-v `pwd`/input:/input:ro \
-v `pwd`/output:/output:rw \
kliko/fitsimagerescaler /kliko
\end{minted}
\caption{Command for running a Kliko container manually. The \texttt{/parameters.json} file is mounted as well as the
input and output directories, in the ``split'' mode. The \texttt{kliko.yml} is already inside the container.}
\label{code:manual_kliko}
\end{listing}
\texttt{`pwd`/input} and \texttt{`pwd`/input} are input and output folders in your current working directory
outside the container. \texttt{`pwd`} is required since the Docker engine can only work with absolute paths.
This command fires up the \texttt{fitsimagerescaler} container, mounts the \texttt{parameters.json} file as well as
input/output directories and runs the \texttt{/kliko} script located in the root directory. In our case, the FITS image
file has to be present in the local input directory for the script to run properly.
For the reader unfamiliar with Docker this command might look cumbersome and error-prone but the command constitutes the
fundamental principle of Kliko (in addition to specification and the extensive test suite). This can be used a base to create your
own Kliko runner in any language that has Docker bindings.
This way of implementing inputs, outputs and running generic scripts, demonstrates that it becomes fairly easy to
connect the input parameters and data (generated by scripting and/or a web form) to software living inside the
container. Kliko implements a set of tools to insure the robustness of this implementation.
\subsection{Inside the Kliko container}
\label{sec:klikolibrary}
The \texttt{/kliko} script is the first entry point into the specifics of the container. We can easily parse the
\texttt{/parameters.json} file using a JSON parser in python, by performing the commands in code listing
\ref{code:parse_params}.
\begin{listing}[htbp!]
\begin{minted}[frame=single,fontsize=\scriptsize]{python}
import json
parameters = json.load(open('/parameters.json', 'r'))
\end{minted}
\caption{Example of how to parse \texttt{parameters.json} file with standard python packages.}
\label{code:parse_params}
\end{listing}
However, at this point, the parameters file is not yet validated. We can be sure that the parameters file is actually
generated from our Kliko definition by installing the Kliko library inside the Kliko container and using it from our
\texttt{/kliko} script\ref{code:validate_params}. Validation helps reduce human or programming error.
\begin{listing}[htbp!]
\begin{minted}[frame=single,fontsize=\scriptsize]{python}
from kliko.validate import validate
parameters = validate()
\end{minted}
\caption{Example how to parse \texttt{parameters.json} using the Kliko library}
\label{code:validate_params}
\end{listing}
After the Kliko validation is performed, a dictionary is created and all values can be used freely inside the script
itself (by passing them to functions) or passed directly to the container OS as environment variables.
All this validation is intended to reduce human or programming error as early as possible.
\begin{listing}[htbp!]
\begin{minted}[frame=single,fontsize=\scriptsize]{python}
import kliko
from kliko.validate import validate
from astropy.io import fits
parameters = validate()
file = parameters['file'] # filename in /input
factor = parameters['factor'] # multiplying factor
print('welcome to fits multiply!')
print("
data = fits.getdata(file)
multiplied = data * factor
output = path.join(kliko.output_path, path.basename(file))
fits.writeto(output, multiplied, clobber=True)
\end{minted}
\caption{Example of \texttt{/kliko} file scale the values in a FITS image}
\label{code:fitsimagerescaler}
\end{listing}
\subsection{kliko-run}
Instead of calling Docker directly, Kliko is bundled with \texttt{kliko-run}, a command line utility that enables a user
to run a Kliko container in a seamless way. It also assists in exploring the parameters that a given Kliko container
supports. Code listing \ref{code:kliko-run} shows the docstring of the kliko-run command for a simple container
(available as a test container shipped with Kliko). The optional arguments are generated automatically from the YAML
file. It shows how any shipped application can be easily interfaced with the host system, in a way that part (or all)
of the variable names of the application can be modified directly from the command line.
This enables Kliko, with the help of Docker, to ship a complex software as an application that is equipped with a
simple interface. Kliko provides a simple way to implement this interface in a controlled and robust way while being
completely agnostic about the mechanics happening inside the container.
\begin{listing}[htbp!]
\begin{minted}[frame=single,fontsize=\scriptsize,breaklines]{console}
$ kliko-run kliko/fitsimagerescaler --help
usage: kliko-run [-h] [--target_folder TARGET_FOLDER] --choice {second,first}
--char CHAR [--float FLOAT] --file FILE --int INT
image_name
positional arguments:
image_name
optional arguments:
-h, --help show this help message and exit
--target_folder TARGET_FOLDER
specify output or work folder (default: ./output)
--choice {second,first}
choice field (default: second)
--char CHAR char field, maximum of 10 chars (default: empty)
--float FLOAT float field (default: 0.0)
--file FILE file field, this file will be put in /input in case
of split io, /work in case of join io
--int INT int field
\end{minted}
\caption{Output of the \texttt{kliko-run} command}
\label{code:kliko-run}
\end{listing}
\section{Chaining containers}
Kliko containers can also be chained. Chaining means that the output of a container is connected to the input
of a consequetively executed container. This enables the creation of workflows. Additionally, if the Kliko containers
are ``split IO", we can execute containers that do not depend on each other in parallel. Their intermediate results
can be cached, which can speed up execution time of future workflow runs and can help debugging problems with the
workflow by examining intermediate results.
There are various workflow creation frameworks and libraries available. We evaluated two popular Python-based workflow
management libraries, airflow\footnote{\url{https://airflow.apache.org}} and
Luigi\footnote{\url{https://github.com/spotify/luigi}}. Although Kliko is designed to be workflow management independent, Luigi is a
better fit. Airflow is intended to visualize automated repetitive tasks like cron jobs, while Luigi is more
oriented towards once-off batch processing. Luigi is an open source Python library that handles dependency resolution,
does workflow management, optionally visualizes data in a web interface and can handle and retry failures. At the
core of a Luigi workflow is the Task, which is a Python class that defines what to be executed, how to check if this
task has already been executed and optionally if it depends on the result of another task. This is a very simple
but powerful concept that integrates fluently with Kliko. The Kliko library contains a KlikoTask definition which
can be used to integrate Kliko in a Luigi pipeline.
\section{Example usage of Kliko}
\subsection{VerMeerKAT}
\begin{figure}[!htbp]
\centering
\includegraphics[width=3in]{vermeerkat_kliko.png}
\caption{Flow diagram of the VerMeerKAT data reduction pipeline.}
\label{fig:vermeerkat}
\end{figure}
To illustrate the mechanics of chaining container together we explain a real world application here, the VerMeerKAT
pipeline.
VerMeerKAT is a semi-automated data reduction pipeline for the first phase of deployment of the MeerKAT
telescope\footnote{http://www.ska.ac.za/gallery/meerkat/}\cite{booth2009meerkat}\cite{jonas2009meerkat}. All steps in this
pipeline are shown in
Figure \ref{fig:vermeerkat}. It is a closed source project used internally at SKA South Africa and is based on a set of
bash scripts. Using bash for this is not ideal; it is hard to make a portable pipeline, not trivial to recover and
continue from errors, reuse intermediate results. Parallelization is possible, but this needs to be manually and
explicitly defined in the scripts.
For this paper we made a Kliko version of this pipeline\footnote{https://github.com/gijzelaerr/vermeerkat-kliko}.
Using Kliko for composing this pipelines has some key advantages; i) easy installation and deployment of the software
ii) optional caching of intermediate data products; ii) implicit parallelization of tasks independent steps; and iii)
progress visualization and reporting using a tool like Luigi.
The VerMeerKAT pipeline, (see Figure \ref{fig:vermeerkat}), starts by querying the MeerKAT data archive for a given set of
observations (step 1), along with the meta-data for that observation. Next step is to convert the downloaded data from the hdf5
format to a MeasurementSet (MS)\cite{kemball2000measurementset} (step 2), since most radio astronomy tools only support this
format. Once the data is in the MS format, it is then taken through a series of
manual and automated tools that excise data points that are contaminated by radio frequency interference
\cite{Prasad2012,aoflagger} (step 3, 4 and 5). The data are then calibrated \cite{hamaker2006,oms2011b} and imaged \cite{white7} (step 6).
For the creation of the VerMeerKAT Kliko containers, we make use of the packages from KERN\cite{KERN}. KERN is a
bi-annually released set of radio astronomical software packages. This suite contains most of the tools that a
radio astronomer needs to process radio telescope data. These packages are precompiled binaries in the Debian format
and contain all the metadata required for installing the package like dependencies and conflicts. KERN is only
supported on Ubuntu 16.04 at the moment of writing, but that is no problem inside a Docker container.
Listing \ref{code:kern} is an example Dockerfile for the wsclean\cite{offringa2014wsclean} Kliko container in VerMeerKAT.
When this file is built it will install the KERN package of wsclean inside the container and bundle the container with
the Kliko definition and parser script. The Docker files for the other steps are very similar.
\begin{listing}[ht]
\begin{minted}[frame=single,fontsize=\scriptsize,breaklines]{console}
FROM kernsuite/base:1
RUN docker-apt-install wsclean
ADD kliko.yml /
ADD kliko /
\end{minted}
\caption{Dockerfile for a KERN package}
\label{code:kern}
\end{listing}
Listing \ref{code:klikotask} is an example Kliko task definition. This example will use the rfimasker Kliko
containers. It depends on the H5tomsTask Kliko task. When this task is invoked using Luigi, Luigi will do the dependency
resolution, check if the required tasks have run and if not, run them. The progress can be visualized with the Luigi
interface (Figure \ref{fig:luigi}). All other steps in the workflow are very similar to this example.
\begin{figure}[!htbp]
\centering
\includegraphics[width=3in]{luigi.png}
\caption{Screenshot of Luigi, running the vermeerkat pipeline }
\label{fig:luigi}
\end{figure}
\begin{listing}[ht]
\begin{minted}[frame=single,fontsize=\scriptsize,breaklines]{python}
from kliko.luigi_util import KlikoTask
class RfiMaskerTask(KlikoTask):
@classmethod
def image_name(cls):
return "vermeerkat/rfimasker:0.1"
def requires(self):
return H5tomsTask()
\end{minted}
\caption{An example \texttt{KlikoTask}}
\label{code:klikotask}
\end{listing}
\subsection{RODRIGUES}
\label{rodrigues}
Another project using Kliko is RODRIGUES (RATT Online Deconvolved Radio Image Generation Using Esoteric
Software)\footnote{\url{https://github.com/ska-sa/rodrigues}}.
RODRIGUES is a web-based Kliko job scheduling tool and it uses the Kliko as a required format for the job.
RODRIGUES acts as a ``kliko runner''. A user of RODRIGUES can log into RODRIGUES and add a new
Kliko container. RODRIGUES will open the Kliko container, parse the parameters and expose these parameters to
the user using a web form (Figure~\ref{fig:rodrigues_form}). The user can then fill in the parameters in this form
and submit the job into the RODRIGUES container queue. The container will be run on the system configured by the
RODRIGUES system administrator. Once the job is finished the results are presented to the user in the same web interface
(Figure~\ref{fig:rodrigues_result}).
RODRIGUES makes it much easier to schedule new jobs with varying parameters, enabling scientists with minimal
programming or computing knowledge to run experiments in a clean, visual, structured and reproducible way.
\begin{figure}[!htbp]
\includegraphics[width=0.9\columnwidth]{rodrigues3.png}
\caption{Screenshot of RODRIGUES, the result visualizer.}
\label{fig:rodrigues_result}
\end{figure}
\begin{figure}[!htbp]
\includegraphics[width=0.9\columnwidth]{rodrigues.png}
\caption{Screenshot of RODRIGUES, parameter form is generated from Kliko definition.}
\label{fig:rodrigues_form}
\end{figure}
\section{Software availability}
\label{sec:Software}
Kliko and its library are open source, licensed under the GNU Public License
2.0\footnote{\url{https://github.com/gijzelaerr/kliko}}. Kliko is bundled with an extensive test suite which covers
80\% of the source code as of the current release \texttt{0.6.1}. The Kliko library is written in Python and is
compatible with Python 2.7, all Python 3 versions and even PyPy. Development and distribution is done on Github and a
third party continuous integration service runs the full test suite on all supported platforms for every commit and every
Github pull request.
\section{Discussions and Prospects}
\subsection{Limitations}
\label{sec:Limitations}
While developing Kliko, we ran into various issues with Docker which might limit the applicability. It is up to the user
to decide if this affects the usefulness of Docker and or Kliko. These issues are listed hereafter for your consideration.
That said, the field of containerization is evolving very fast and hopefully most of these issues will be resolved soon
or can be worked around.
First of all, being able to run a Docker container on a system is very similar to giving the user administrative
access to the machine, that is the user can escalate easily to root privileges
\footnote{\url{https://github.com/docker/docker/issues/6324}} \cite{Bui2015Analysis}. The singularity containerization
technology has a more secure design and we are planning to add support for this framework in future versions of
Kliko.
Additionally, using GPU acceleration with NVidia hardware is not trivial, since the kernel driver version and library
version need to match up, breaking the independence between host and container. There is a workaround available, but
this requires a replacement of the Docker daemon with a custom one \footnote{\url{https://github.com/NVIDIA/nvidia-docker}}.
A similar issue arises with optimization flags. For example, SIMD instructions can greatly enhance the run-time speed,
but not all x86 processors support all SIMD optimization. This will result in crashes of the binary if it is compiled
with optimization not supported by the host. Again this breaks the platform independence assumption. A good strategy
is to be conservative and compile your binaries for the oldest architecture you intend to support. The good news is
that it is easier to support multiple target platforms in the same binary when using modern versions of GCC \footnote{\url{https://lwn.net/Articles/691932}}.
Another issue is that it is easy to inherit Docker definitions from other Docker definitions, but it is currently not
possible to combine Docker definitions together or to inherit from multiple Docker definitions at the same time (merge).
Following the Docker philosophy, Docker containers should have a single responsibility, so this should not be a problem,
it does not require mixing of Docker definitions. However, in practice,
this is not always possible: sometimes various big libraries need to communicate in the same memory space so a new
Docker container with all software needs to be created. The results are far away from minimal small single purpose
Docker containers.
For network intensive applications Docker may be less well suited, since the use of network address
translation\cite{Felter2015Updated}. Also here there is a workaround available by disabling the translation and using
the host network stack directly.
\subsection{Future Work}
During the development and usage of Kliko we became aware of CommonWL\cite{Amstutz2016}, a more generic approach to
describe applications input/output flow and parameters. We will investigate how we can incorporate CommonWL into Kliko
to extend the usability and user base.
At the moment Kliko is designed with other container solutions in mind.
Singularity is an alternative that looks to be gaining momentum within the scientific compute (HPC)
community since it is more aware and careful of the security implications that come by allowing running
containers on a shared infrastructure.
\subsubsection{Streaming Kliko}
Kliko was born in the field of radio astronomy. Most tools in this field operate on data living on disk.
Radio astronomy uses several file formats, for example, casacore Measurement Sets, FITS images and, in
some cases HDF5. Using the file system is an easy to understand and technically stable approach,
but problems arise when the size of the dataset grows. Emerging telescope arrays such as MeerKAT,
followed by SKA phase 1, will result in an exponential growth in data rates. Repeatedly reading and writing
data from to slowest medium in a computer -- the disk -- is not going to scale and a new strategy is needed.
Directly streaming data between processing tasks will be required. Although this is already being done \footnote{\url{https://github.com/ska-sa/spead2}} in some
pipelines, there is no field-wide accepted standard that fits all needs. Our plan of action is to investigate
existing solutions being used, investigate industry standards \footnote{\url{https://github.com/google/protobuf}},
optionally create a sub-specification and open source reference implementation with support libraries for the most
used public languages\footnote{\url{http://arxiv.org/pdf/1507.03989.pdf}}.
\section{Conclusions}
Kliko is a Docker-based container specification. It is used to create abstract descriptions of the input
and output of existing software resulting in Kliko containers. These Kliko containers can be used to encapsulate a single
job or can be chained together in a pipeline. In the future, we probably will adopt the CWL standard to extend the
interoperability with other existing workflow tools. Kliko is written in Python, open source and available free to use.
\section{Acknowledgments}
J. Girard acknowledges the financial support from the UnivEarthS Labex program of
Sorbonne Paris Cit\'e (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02). The research of
O. Smirnov is supported by the South African Research Chairs Initiative of the Department of Science and
Technology and National Research Foundation.
\section{References}
|
1,314,259,996,499 | arxiv | \section{Introduction}\label{intro}
Let $\mathcal{N}$ be a set with $n\in\mathbb{N}\coloneqq\cb{1,2,\ldots}$ distinct objects and fix $k\in\mathbb{N}$. We consider the problem of choosing $k$ subsets $A_1,\ldots,A_k$ of $\mathcal{N}$ in such a way that these sets cover $\mathcal{N}$, that is, $\bigcup_{i=1}^k A_i = \mathcal{N}$, and no two of these sets are subsets of each other, that is, $A_i \setminus A_j \neq \emptyset$ for all $i,j\in\cb{1,\ldots,k}$ with $i\neq j$. We refer to such a cover of $\mathcal{N}$ as a \emph{constructive $k$-cover}. Due to the second requirement in the preceding definition, it is necessary that $k\leq n$ for a constructive $k$-cover to exist. In this paper, we address the following question: \emph{How many ways are there to choose a constructive $k$-cover of $\mathcal{N}$?}
In the literature, a related problem was considered by T.~Hearne, C.~Wagner in 1973 and by R.~J.~Clarke in 1990. In both of \cite{hearnewagner} and \cite{clarke}, distinct subsets $A_1,\ldots,A_k$ of $\mathcal{N}$ are said to form a \emph{minimal $k$-cover} of $\mathcal{N}$ if these sets cover $\mathcal{N}$, and $\mathcal{N}$ cannot be a covered by a strict subcollection of these sets, that is, $\bigcup_{i\neq j}A_i\neq\mathcal{N}$ for all $j\in\cb{1,\ldots,k}$. It is easy to observe that a minimal $k$-cover is necessarily a constructive $k$-cover. As a counterexample for the converse, when $n=4$ and $k=3$, note that the sets $A_1=\cb{1,2}, A_2=\cb{2,3}, A_3=\cb{1,3,4}$ form a constructive $k$-cover but not a minimal $k$-cover. In \cite{hearnewagner} and \cite{clarke}, summation-type formulae are provided for the number of minimal $k$-covers of $\mathcal{N}$ (in \cite{clarke}, also for the case where the elements of $\mathcal{N}$ are not necessarily distinguishable). In the present paper, the requirement that no two sets be subsets of each other makes the problem more involved and it seems that the enumeration ideas in \cite{clarke} cannot be replicated here to solve the problem. The one-to-one correspondence between minimal covers and the so-called \emph{split graphs} on $n$ vertices is studied in \cite{royle}. The more recent work \cite{collinstrenk} provides a more detailed analysis of the connection between minimal covers, split graphs and bipartite posets. In a similar line of research, the so-called \emph{$m$-balanced covers} which consist of sets with cardinality $m$ are studied in \cite{burger} and a recursive relation is provided for the number of $m$-balanced covers that include the minimum possible number of sets.
The consideration of constructive $k$-covers is motivated by \emph{coherent systems} in reliability theory. In \cite{barlow}, a coherent system is described in terms of its set $\mathcal{N}$ of distinct components and the way these components are configured. The precise configuration of the components is encoded by the so-called \emph{minimal path sets} $P_1,\ldots,P_k\subseteq\mathcal{N}$, which are supposed to form a constructive $k$-cover of $\mathcal{N}$. For each $i\in\cb{1,\ldots,k}$, the components in $P_i$ are interpreted as components that are connected in series, and the sub-systems $P_1,\ldots,P_k$ are connected in parallel. The class of coherent systems is rich enough to cover many system configurations in reliability applications. For instance, the system with $\mathcal{N}=\cb{1,2,3}$ and $P_1=\cb{1,2}, P_2=\cb{1,3}, P_3=\cb{2,3}$ corresponds to a $2$-out-of-$3$ system which functions as long as at least two of the three components function. Our main result provides a formula to calculate the number of coherent systems with $n$ distinct components and $k$ minimal path sets. Alternatively, the configuration of the components in a coherent system can also be encoded by the so-called \emph{minimal cut sets} $C_1,\ldots,C_k$, which are supposed to form a constructive $k$-cover of $\mathcal{N}$ as in the case of minimal path sets. However, the interpretation of the sets is different in this case: for each $i\in\cb{1,\ldots,k}$, the components in $C_i$ are interpreted as parallel components and the sub-systems $C_1,\ldots,C_k$ are connected in series. Hence, our result also provides the number of coherent systems with $n$ components and $k$ minimal cut sets.
In the present paper, the analysis of the problem is based on a main formula in Section~\ref{problemdefn} (see Theorem~\ref{mainthm} and Corollary~\ref{compcor}) which partitions the set of constructive $k$-covers into certain Cartesian products of pairs of sets. In each pair, the cardinality of the first set is a variant of Stirling numbers of the second kind. This new variant, which we call as \emph{integrated Stirling number (ISN)}, is introduced in Section~\ref{prelim} separately, along with its basic properties. The second set in each Cartesian product gives rise to an auxilary problem which may also be of independent interest: suppose that we are about to form $k\in\mathbb{N}$ sets $A_1,\ldots,A_k$ which partition the world into $2^k$ (disjoint) regions. We consider all the regions except for the one denoting the intersection of all $k$ sets ($A_1\cap\ldots\cap A_k$) and the one denoting the complement of the union of all $k$ sets ($\bar{A}_1\cap\ldots\cap \bar{A}_k$). Given a number $\ell\in\cb{1,\ldots,2^k-2}$, how many ways are there to label these $2^k-2$ regions as ``non-empty" and ``empty" so that there are exactly $\ell$ regions with label ``non-empty" and each set difference $A_i\setminus A_j$, where $i,j\in\cb{1,\ldots,k}$, $i\neq j$, contains at least one region with label ``non-empty"? Note that this new problem is free of $n$, the cardinality of $\mathcal{N}$, but it depends on $k$, the number of subsets in the constructive cover, as well as the auxiliary variable $\ell$ which keeps track of the regions that are labeled as ``non-empty." While calculating ISNs is an easy task, the above labeling problem is highly nontrivial and requires further analysis. As a result, when using our method, the level of difficulty in counting the number of constructive $k$-covers of $\mathcal{N}$ is largely determined by the value of $k$ rather than $n=|\mathcal{N}|$. In terms of the reliability theory application described above, this implies that our method of counting possible coherent systems is much more sensitive to the number of minimal path (or cut) sets than it is to the number of components.
We carry out a detailed analysis of the auxiliary labeling problem in Section~\ref{partitioningsec}. In particular, we exploit three types of symmetries: based on permutations (Section~\ref{permsym}), taking complements (Section~\ref{compsym}), and the so-called \emph{impact sets} (Section~\ref{impactsym}). These symmetries enable us to define an equivalence relation (Section~\ref{equiv1}) and prove finer results to calculate the answers to the auxiliary problem as well as the original problem. Finally, in Section~\ref{nogoodsec}, we describe the computational procedure based on solving feasibility problems iteratively using \emph{no-good cuts} for binary variables, namely, inequalities that ensure us, in each iteration, to find a solution that is different from the ones found in the former iterations. Some numerical results are also presented in Section~\ref{nogoodsec}. The connection between constructive covers and reliability theory, which was our initial motivation to study the subject, is explained in Section~\ref{reliability}. Section~\ref{conc} is the concluding section. The elementary proofs of the results in Section~\ref{prelim} are collected in Section~\ref{app}, the appendix.
\section{Preliminaries}\label{prelim}
We recall the definition of Stirling numbers of the second kind and introduce \emph{integrated Stirling numbers (ISN)}, a new variant of the former that will be used in the main results of Section~\ref{problemdefn}.
Let $\mathcal{N}$ be a set with $n\in\mathbb{N}\coloneqq\cb{1,2,\ldots}$ distinct elements. Let $\ell\in\mathbb{N}$. A finite (ordered) sequence $\of{B_1,\ldots,B_\ell}$ of disjoint nonempty subsets of $\mathcal{N}$ is said to be an \emph{ordered $\ell$-partition} of $\mathcal{N}$ if
\[
B_1\cup\ldots\cup B_\ell = \mathcal{N}.
\]
Let $S(\mathcal{N},\ell)$ be the set of all ordered $\ell$-partitions of $\mathcal{N}$.
For $n,\ell$ as above, the corresponding \emph{Stirling number (of the second kind)} is defined as
\begin{equation}\label{StirlingDefn}
s(n,\ell)\coloneqq \frac{1}{\ell!}\sum_{j=0}^\ell (-1)^{\ell-j}\binom{\ell}{j}j^n.
\end{equation}
Note that $\ell!s(n,\ell)$ gives the number of ordered $\ell$-partitions of $\mathcal{N}$, that is,
\[
|S(\mathcal{N},\ell)|=\ell!s(n,\ell).
\]
It can be checked by induction that
\begin{equation}\label{Stirlingzero}
s(n,\ell)=0
\end{equation}
for every $n<\ell$. Moreover, it is well-known that these numbers satisfy the recurrence relation
\[
s(n+1,\ell)=\ell s(n,\ell)+s(n,\ell-1),\quad n\in\mathbb{N}\!\setminus\!\cb{1},\ \ell\in\cb{2,\ldots,n}
\]
with the boundary conditions
\[
s(n,n)=1,\quad s(n,1)=1,\ n\in\mathbb{N}.
\]
For each $n\in\mathbb{N}$ and $\ell\in\mathbb{N}$, we define the ISN as
\[
\tilde{s}(n,\ell)\coloneqq \frac{1}{\ell!}\sum_{j=0}^\ell (-1)^{\ell-j}\binom{\ell}{j}(j+1)^n.
\]
The next proposition will be used to interpret $\tilde{s}(n,\ell)$ in terms of the number of ordered $\ell$-partitions of subsets of $\mathcal{N}$. Its proof is given in Section \ref{app}, the appendix.
\begin{proposition}\label{workhorse}
Let $n,\ell\in\mathbb{N}$. We have
\begin{equation}\label{stilderes}
\tilde{s}(n,\ell)=\sum_{i=1}^n\binom{n}{i}s(i,\ell).
\end{equation}
In particular, the following are valid.
\begin{enumerate}[(i)]
\item If $n<\ell$, then $\tilde{s}(n,\ell)=0$.
\item If $n\geq \ell$, then
\begin{equation}\label{stildeinterpret}
\tilde{s}(n,\ell)=\sum_{i=\ell}^n\binom{n}{i}s(i,\ell)=\sum_{i=0}^{n-\ell}\binom{n}{i}s(n-i,\ell).
\end{equation}
\item $\tilde{s}(n,n)=1$.
\item $\tilde{s}(n,1)=2^n-1$.
\end{enumerate}
\end{proposition}
\begin{remark}\label{interpretation}
Let $1\leq\ell\leq n$. The relation \eqref{stildeinterpret} in Proposition~\ref{workhorse} provides the following characterization of ISNs. For each $i\in\cb{\ell,\ldots,n}$, as mentioned above, $\ell! s(i,\ell)$ is the number of ordered $\ell$-partitions of a set of $i$ distinguishable objects; hence, $\binom ni \ell! s(i,\ell)$ gives the total number of ordered $\ell$-partitions of all subsets of $\mathcal{N}$ with size $i$. Let $\tilde{S}(\mathcal{N},\ell)$ be the set of all ordered $\ell$-partitions of all subsets of $\mathcal{N}$ (with at least size $\ell$), that is,
\begin{equation}\label{defnstilde}
\tilde{S}(\mathcal{N},\ell)\coloneqq \cb{S(\mathcal{I},\ell)\mid \mathcal{I}\subseteq\mathcal{N},|\mathcal{I}|\geq \ell}.
\end{equation}
Therefore,
\begin{equation}\label{stildecount}
|\tilde{S}(\mathcal{N},\ell)|=\ell!\tilde{s}(n,\ell).
\end{equation}
\end{remark}
\begin{proposition}\label{stilderec}
ISNs satisfy the recurrence relation
\[
\tilde{s}(n+1,\ell)=(\ell+1)\tilde{s}(n,\ell)+\tilde{s}(n,\ell-1),\quad n\in\mathbb{N}\!\setminus\!\cb{1}, \quad \ell\in\cb{2,\ldots,n}.
\]
with the boundary conditions
\[
\tilde{s}(n,n)=1,\ \tilde{s}(n,1)=2^n-1, \ n\in\mathbb{N}.
\]
\end{proposition}
Using the recurrence relation in \eqref{stilderec}, we calculate $\tilde{s}(n,\ell)$ for $n\in\cb{1,\ldots,10}$ and $\ell\in\cb{1,\ldots,n}$ as shown in Table~\ref{table2}. For completeness, we also provide the values of $s(n,\ell)$ for the same $(n,\ell)$ pairs in Table~\ref{table1}.
\begin{table}[h]
\centering
\begin{tabular}[htbp]{|c|*{10}{p{1cm}|}}\hline
\diagbox{$n$}{$\ell$} &$1$ & $ 2$ & $ 3$ & $ 4$ & $5$& $6$ & $ 7$ & $8$ & $9$ & $10$\\
\hline
$ 1$& $1$ \\ \cline{1-3}
$ 2$& $1$ & $1$ \\ \cline{1-4}
$3$& $ 1$ &$3$ &$1$ \\ \cline{1-5}
$4$& $1$ &$7$ & $6$& $1$ \\\cline{1-6}
$5$& $1$ &$15$&$25$ &$10$ &$1$ \\\cline{1-7}
$6$& $1$ &$31$&$90$ &$65$ &$15$ & $1$ \\\cline{1-8}
$7$& $1$ &$63$&$301$ &$350$ &$140$ & $21$ &$1$ \\ \cline{1-9}
$8$& $1$ &$127$&$966$ &$1701$ &$1050$ & $266$ & $28$ &$1$ \\ \cline{1-10}
$9$& $1$ &$255$&$3025$ &$7770$ &$6951$ & $2646$ & $462$ &$36$& $1$ \\ \cline{1-11}
$10$& $1$ &$511$&$9330$ &$34105$ &$42525$ & $22827$ & $5880$ &$750$ & $45$& $1$ \\ \hline
\end{tabular}
\caption{$s(n,\ell)$ values for $1\leq \ell\leq n\leq 10$}\label{table1}
\centering
\vspace{0.3cm}
\begin{tabular}{|c|*{10}{p{1cm}|}}\hline
\diagbox{$n$}{$\ell$} &$1$ & $ 2$ & $ 3$ & $ 4$ & $5$& $6$ & $ 7$ & $8$ & $9$ & $10$\\
\hline
$ 1$& $1$ \\ \cline{1-3}
$ 2$& $3$ & $1$ \\ \cline{1-4}
$3$& $ 7$ &$6$ &$1$ \\ \cline{1-5}
$4$& $15$ &$25$ & $10$& $1$ \\\cline{1-6}
$5$& $31$ &$90$&$65$ &$15$ &$1$ \\\cline{1-7}
$6$& $63$ &$301$&$350$ &$140$ &$21$ & $1$ \\\cline{1-8}
$7$& $127$ &$966$&$1701$ &$1050$ &$266$ & $28$ &$1$ \\ \cline{1-9}
$8$& $255$ &$3025$&$7770$ &$6951$ &$2646$ & $462$ & $36$ &$1$ \\ \cline{1-10}
$9$& $511$ &$9330$&$34105$ &$42525$ &$22827$ & $5880$ & $750$ &$45$& $1$ \\ \cline{1-11}
$10$& $1023$ &$28501$&$145750$ &$246730$ &$179487$ & $63987$ & $11880$ &$1155$ & $55$& $1$ \\ \hline
\end{tabular}
\caption{$\tilde{s}(n,\ell)$ values for $1\leq \ell\leq n\leq 10$}\label{table2}
\end{table}
\section{Constructive covers and sets of labelings}\label{problemdefn}
As in the previous section, we consider a set $\mathcal{N}$ of $n\in\mathbb{N}$ distinct elements. Let $k\in\cb{1,\ldots,n}$.
\begin{definition}\label{kcover}
A finite sequence $\mathscr{A}=\of{A_1,\ldots,A_k}$ of distinct subsets of $\mathcal{N}$ is said to be a constructive ordered $k$-cover of $\mathcal{N}$ if it satisfies the following conditions.
\begin{enumerate}[(i)]
\item The sets in $\mathscr{A}$ cover $\mathcal{N}$, that is, $\bigcup_{i=1}^k A_i= \mathcal{N}$.
\item Two distinct sets in $\mathscr{A}$ are not subsets of each other, that is, $A_i\!\setminus\! A_j\neq\emptyset$ for every $i,j\in\cb{1,\ldots,k}$ with $i\neq j$.
\end{enumerate}
\end{definition}
Let $C(\mathcal{N},k)$ be the set of all distinct constructive $k$-covers of $\mathcal{N}$. Our aim is to provide a characterization of the set $C(\mathcal{N},k)$ that also helps computing its cardinality $ |C(\mathcal{N},k)|$.
\begin{remark}\label{unordered}
Note that one can also define a constructive \emph{unordered} $k$-cover of $\mathcal{N}$ as a collection $\cb{A_1,\ldots,A_k}$ of distinct subsets of $\mathcal{N}$ satisfying the two conditions above. Clearly, the number of all distinct constructive unordered $k$-covers of $\mathcal{N}$ is $|C(\mathcal{N},k)|/k!$.
\end{remark}
Let $\mathscr{A}=\of{A_1,\ldots,A_k}$ be a constructive ordered $k$-cover of $\mathcal{N}$. Consider a binary index vector $t=(t_1,\ldots,t_k)\in\cb{e,c}^k$ ($e$ for ``excluded," $c$ for ``contained") and define the set
\[
B_t(\mathscr{A})\coloneqq \of{\bigcap_{i\colon t_i=c}A_i} \cap\of{\bigcap_{i\colon t_i=e}\bar{A}_i},
\]
where $\bar{A}$ denotes the complement of a subset $A$ of $\mathcal{N}$. Hence, $\mathscr{A}$ gives rise to $2^k$ disjoint (possibly empty) sets $B_t(\mathscr{A})$, $t\in\cb{e,c}^k$.
\begin{example}
To be more specific, let $k=5$ and $t=(e,e,c,e,c)$. Then,
\[
B_t(\mathscr{A})=\bar{A}_1\cap \bar{A}_2\cap A_3\cap \bar{A}_4\cap A_5.
\]
\end{example}
Two cases need special attention in the above construction. The special vector $\mathbf{e}\coloneqq(e,\ldots,e)$ corresponds to
\[
B_{\mathbf{e}}(\mathscr{A})=\bar{A}_1\cap\ldots \cap \bar{A}_k,
\]
which must be equal to the empty set since $\mathscr{A}$ is a constructive ordered $k$-cover. On the other hand, the special vector $\mathbf{c}\coloneqq(c,\ldots,c)$ corresponds to
\[
B_{\mathbf{c}}(\mathscr{A})=A_1\cap\ldots \cap A_k
\]
and the definition of constructive ordered $k$-cover does not impose any non-emptiness condition on this set. Next, we introduce the index sets
\begin{equation}\label{tk}
\mathbb{T}(k)\coloneqq \cb{e,c}^k \!\setminus\! \cb{\mathbf{e},\mathbf{c}},\quad \mathbb{T}^\ast(k) \coloneqq \cb{e,c}^k\!\setminus\! \cb{\mathbf{e}},
\end{equation}
and rewrite the conditions (i) and (ii) in Definition~\ref{kcover} as follows.
\begin{enumerate}[(i)]
\item $\bigcup_{i=1}^k A_i=\bigcup_{t\in\mathbb{T}^\ast(k)}B_t(\mathscr{A})=\mathcal{N}$.
\item For every $i,j\in\cb{1,\ldots,k}$ with $i\neq j$,
\[
A_i\!\setminus\! A_j = \bigcup_{t\in\mathbb{T}(k)\colon t_i=c, t_j=e}B_t(\mathscr{A})\neq \emptyset.
\]
\end{enumerate}
Note that condition (ii) depends on the sets $B_t(\mathscr{A}), t\in\mathbb{T}(k)$, only through the \emph{non-emptiness} of certain unions of these sets. Indeed, for each $t\in\mathbb{T}(k)$, let us introduce the binary number
\[
x_t(\mathscr{A})\coloneqq \begin{cases}1&\text{ if }B_t(\mathscr{A})\neq\emptyset,\\ 0&\text{ if }B_t(\mathscr{A})=\emptyset.\end{cases}
\]
Hence, condition (ii) is equivalent to having
\[
\sum_{t\in\mathbb{T}(k)\colon t_i=c, t_j=e}x_t(\mathscr{A})\geq 1
\]
for every $i,j\in\cb{1,\ldots,k}$ with $i\neq j$. Let us define
\begin{equation}\label{defng}
G(k)\coloneqq\cb{(x_t)_{t\in\mathbb{T}(k)}\mid \sum_{t\in\mathbb{T}(k)\colon t_i=c, t_j=e}x_t\geq 1\ \forall i\neq j,\ x_t\in\cb{0,1}\ \forall t\in\mathbb{T}(k)}.
\end{equation}
Since $\mathscr{A}$ is a constructive ordered $k$-cover of $\mathcal{N}$, we necessarily have $(x_t(\mathscr{A}))_{t\in\mathbb{T}(k)}\in G(k)$. Let us also define for each $\ell\in\cb{1,\ldots, 2^{k}-2}$ the set
\begin{equation}\label{DefnOfF}
F(k,\ell)\coloneqq\cb{(x_t)_{t\in\mathbb{T}(k)}\in G(k)\mid\sum_{t\in\mathbb{T}(k)}x_t=\ell},
\end{equation}
which is the set of all labelings in $G(k)$ where exactly $\ell$ sets are labeled as ``non-empty." In short, we refer to $F(k,\ell)$ as the set of all $(k,\ell)$-labelings. We have
\begin{equation}\label{disjointunion}
G(k)=\bigcup_{\ell=1}^{2^k-2}F(k,\ell),
\end{equation}
which is a disjoint union. Hence, we obtain
\[
|G(k)|=\sum_{\ell=1}^{2^k-2}|F(k,\ell)|.
\]
In the following example, we illustrate the notation and the structure of the sets defined above.
\begin{example}
Let us consider the case $k=3$. We have
\begin{equation}\label{tkorder}
\mathbb{T}(3)=\cb{(c,e,e),(e,c,e),(e,e,c),(c,c,e),(c,e,c),(e,c,c)}.
\end{equation}
Then, we can write the corresponding set $G(3)$ as
\begin{align*}
G(3)=\Big\{(x_t)_{t\in\mathbb{T}(3)}\mid \ & x_{(c,e,e)}+x_{(c,e,c)}\geq 1,\ x_{(e,c,e)}+x_{(e,c,c)}\geq 1,\ x_{(c,e,e)}+x_{(c,c,e)}\geq 1,\\
&x_{(e,e,c)}+x_{(e,c,c)}\geq 1,\ x_{(e,c,e)}+x_{(c,c,e)}\geq 1,\ x_{(e,e,c)}+x_{(c,e,c)}\geq 1,\\
& x_t\in\cb{0,1}\ \forall t\in\mathbb{T}(3)\Big\}.
\end{align*}
Hence, for each $\ell\in\cb{1,\ldots,6}$,
\begin{align*}
F(3,\ell)=\Big\{(x_t)_{t\in\mathbb{T}(3)}\mid \ & x_{(c,e,e)}+x_{(c,e,c)}\geq 1,\ x_{(e,c,e)}+x_{(e,c,c)}\geq 1,\ x_{(c,e,e)}+x_{(c,c,e)}\geq 1,\\
&x_{(e,e,c)}+x_{(e,c,c)}\geq 1,\ x_{(e,c,e)}+x_{(c,c,e)}\geq 1,\ x_{(e,e,c)}+x_{(c,e,c)}\geq 1,\\
&x_{(e,e,c)}+x_{(e,c,e)}+x_{(e,c,c)}+x_{(c,e,e)}+x_{(c,e,c)}+x_{(c,c,e)}=\ell
\Big\}.
\end{align*}
We observe that $F(3,1)=F(3,2)=\emptyset$. Further, we can explicitly express $F(3,3)$ as
\begin{align*}
&F(3,3)=\cb{(c,c,c,e,e,e),(e,e,e,c,c,c)},\\
&F(3,4)=\{(c,e,e,c,c,c),(e,c,e,c,c,c),(e,e,c,c,c,c),(c,c,c,c,e,e),(c,c,c,e,c,e),(c,c,c,e,e,c),\\
&\quad\quad\quad \quad \quad (c,c,e,e,c,c),(c,e,c,c,e,c),(e,c,c,c,c,e)\},\\
&F(3,5)=\cb{(c,c,c,c,c,e),(c,c,c,c,e,c),(c,c,c,e,c,c),(c,c,e,c,c,c),(c,c,e,c,c,c),(e,c,c,c,c,c)},\\
&F(3,6)=\cb{(c,c,c,c,c,c)},
\end{align*}
following the order in \eqref{tkorder}. Hence,
\[
G(3)=F(3,3)\cup F(3,4)\cup F(3,5)\cup F(3,6).
\]
\end{example}
Using \eqref{disjointunion}, we may write
\[
C(\mathcal{N},k)=\bigcup_{\ell=1}^{2^{k}-2}\cb{\mathscr{A}\in C(\mathcal{N},k)\mid (x_t(\mathscr{A}))_{t\in\mathbb{T}(k)}\in F(k,\ell)}
\]
as a disjoint union so that
\begin{equation}\label{subthm}
|C(\mathcal{N},k)|=\sum_{\ell=1}^{2^k-2}|\cb{\mathscr{A}\in C(\mathcal{N},k)\mid (x_t(\mathscr{A}))_{t\in\mathbb{T}(k)}\in F(k,\ell)}|.
\end{equation}
The next theorem is the main result of the paper. Up to isomorphisms, it characterizes $C(\mathcal{N},k)$ as a disjoint union of Cartesian products of basic sets. For two sets $E_1,E_2$, we write $E_1\cong E_2$ if there is a bijection $f\colon E_1\to E_2$.
\begin{theorem}\label{mainthm}
For each $\ell\in\cb{1,\ldots,2^k-2}$, it holds
\begin{equation}\label{congell}
\cb{\mathscr{A}\in C(\mathcal{N},k)\mid (x_t(\mathscr{A}))_{t\in\mathbb{T}(k)}\in F(k,\ell)}\cong\tilde{S}(n,\ell) \times F(k,\ell).
\end{equation}
In particular,
\begin{equation}\label{cong}
C(\mathcal{N},k)\cong\bigcup_{\ell=1}^{2^k-2}\tilde{S}(n,\ell) \times F(k,\ell).
\end{equation}
\end{theorem}
\begin{proof}
Let $\ell\in\cb{1,\ldots,2^k-2}$. Let $\mathscr{A}=(A_1,\ldots,A_k)\in C(\mathcal{N},k)$ with $(x_t(\mathscr{A}))_{t\in\mathbb{T}(k)}\in F(k,\ell)$. In other words, $\mathscr{A}$ is a constructive ordered $k$-cover of $\mathcal{N}$ for which exactly $\ell$ of the sets $B_t(\mathscr{A})$, $t\in\mathbb{T}(k)$, are nonempty. Denoting by $\prec$ the strict lexicographical ordering on $\mathbb{T}(k)$, let us order these $\ell$ indices as $t^1(\mathscr{A})\prec\ldots\prec t^\ell(\mathscr{A})$. Then, $\mathcal{B}(\mathscr{A})\coloneqq (B_{t^1(\mathscr{A})}(\mathscr{A}),\ldots,B_{t^\ell(\mathscr{A})}(\mathscr{A}))$ is an ordered $\ell$-partition of $\mathcal{N}\!\setminus\! B_{\mathbf{c}}(\mathscr{A})$. Hence, $\mathcal{B}(\mathscr{A})\in S(\mathcal{N}\!\setminus\! B_{\mathbf{c}}(\mathscr{A}),\ell)\subseteq \tilde{S}(\mathcal{N},\ell)$ by \eqref{defnstilde}.
The above construction establishes the mapping
\begin{equation}\label{mapping}
\mathscr{A}\mapsto (\mathcal{B}(\mathscr{A}),(x_t(\mathscr{A}))_{t\in\mathbb{T}(k)})
\end{equation}
from $\cb{\mathscr{A}\in C(\mathcal{N},k)\mid (x_t(\mathscr{A}))_{t\in\mathbb{T}(k)}\in F(k,\ell)}$ to $\tilde{S}(\mathcal{N},\ell)\times F(k,\ell)$. To check that this mapping is injective, let $\mathscr{A}^\prime=(A^\prime_1,\ldots,A^\prime_k)\in C(\mathcal{N},k)$ be another constructive ordered $k$-cover such that $(x_t(\mathscr{A}^\prime))_{t\in\mathbb{T}(k)}\in F(k,\ell)$. Suppose that
\[
(x_t(\mathscr{A}))_{t\in\mathbb{T}(k)}=(x_t(\mathscr{A}^\prime))_{t\in\mathbb{T}(k)},\quad \mathcal{B}(\mathscr{A})=\mathcal{B}(\mathscr{A}^\prime).
\]
The first supposition guarantees that $\mathscr{A}$ and $\mathscr{A}^\prime$ agree on the nonemptiness of their corresponding sets $B_t(\mathscr{A}), B_t(\mathscr{A}^\prime)$ for each $t\in\mathbb{T}(k)$. In other words,
\[
B_t(\mathscr{A})=\emptyset\quad\Leftrightarrow\quad B_t(\mathscr{A}^\prime)=\emptyset
\]
for each $t\in\mathbb{T}(k)$. Moreover, from the definition of lexicographical ordering, it follows that
\[
t^1\coloneqq t^1(\mathscr{A})=t^1(\mathscr{A}^\prime),\ldots, t^\ell\coloneqq t^\ell(\mathscr{A})=t^\ell(\mathscr{A}^\prime).
\]
Then, by the second supposition, we have
\[
B_{t^1}(\mathscr{A})=B_{t^1}(\mathscr{A}^\prime),\ldots, B_{t^\ell}(\mathscr{A})=B_{t^\ell}(\mathscr{A}^\prime).
\]
Hence,
\[
B_t(\mathscr{A})=B_t(\mathscr{A}^\prime)
\]
for every $t\in\mathbb{T}(k)$. Since $\bigcup_{i=1}^k A_i = \bigcup_{i=1}^k A^\prime_i=\mathcal{N}$, we also have
\[
B_{\mathbf{c}}(\mathscr{A})=B_{\mathbf{c}}(\mathscr{A}^\prime).
\]
Finally, we have
\[
A_i= \of{\bigcup_{t\in\mathbb{T}(k)\colon t_i=c}B_t(\mathscr{A})}\cup B_{\mathbf{c}}(\mathscr{A})= \of{\bigcup_{t\in\mathbb{T}(k)\colon t_i=c}B_t(\mathscr{A}^\prime)}\cup B_{\mathbf{c}}(\mathscr{A}^\prime)=A^\prime_i
\]
for every $i\in\cb{1,\ldots,k}$ so that $\mathscr{A}=\mathscr{A}^\prime$. This finishes the proof of injectivity.
Next, we show that the mapping in \eqref{mapping} is surjective. Let $\mathcal{B}=(\bar{B}_1,\ldots,\bar{B}_\ell)\in \tilde{S}(\mathcal{N},\ell)$ and $(x_t)_{t\in\mathbb{T}(k)}\in F(k,\ell)$. Hence, by \eqref{defnstilde} in Remark~\ref{interpretation}, there exists $\mathcal{I}\subseteq\mathcal{N}$ with $|\mathcal{I}|\geq \ell$ such that $\mathcal{B}\in S(\mathcal{I},\ell)$. Let us set
\[
B_{\mathbf{c}}\coloneqq \mathcal{N}\!\setminus\!\of{\bar{B}_1\cup\ldots\cup \bar{B}_{\ell}}=\mathcal{N}\!\setminus\!\mathcal{I}.
\]
On the other hand, consider the set of all $t\in\mathbb{T}(k)$ for which $x_t =1$. Since $(x_t)_{t\in\mathbb{T}(k)}\in F(k,\ell)$, there are $\ell$ such indices in $\mathbb{T}(k)$. As before, let us order them as $t^1\prec\ldots\prec t^\ell$ using the lexicographical ordering and set
\[
B_{t^1}\coloneqq \bar{B}_1,\ldots, B_{t^\ell}\coloneqq \bar{B}_\ell
\]
and
\[
B_t\coloneqq \emptyset
\]
for every $t\in\mathbb{T}(k)\!\setminus\!\cb{t^1,\ldots,t^\ell}$. Then, let
\[
A_i\coloneqq \of{\bigcup_{t\in\mathbb{T}(k)\colon t_i=c}B_t}\cup B_{\mathbf{c}}
\]
for each $i\in\cb{1,\ldots,k}$ and $\mathscr{A}\coloneqq (A_1,\ldots,A_k)$. It is clear that $\bigcup_{i=1}^k A_i=\mathcal{N}$. Moreover, the assumption that $(x_t)_{t\in\mathbb{T}(k)}\in F(k,\ell)\subseteq G(k)$ guarantees that $A_i\!\setminus\! A_j\neq\emptyset$ for every $i,j\in\cb{1,\ldots,k}$ with $i\neq j$. Finally, by construction of the mapping in \eqref{mapping}, we conclude that $\mathcal{B}(\mathscr{A})=\mathcal{B}$ and $(x_t(\mathscr{A}))_{t\in\mathbb{T}(k)}=(x_t)_{t\in\mathbb{T}(k)}$. This shows that every element of $\tilde{S}(\mathcal{N},\ell)\times F(k,\ell)$ is the value of the mapping in \eqref{mapping} for some $\mathscr{A}\in C(\mathcal{N},\ell)$ with $(x_t(\mathscr{A}))_{t\in\mathbb{T}(k)}\in F(k,\ell)$.
Therefore, \eqref{congell} follows. As an immediate consequence of disjointness, \eqref{cong} holds as well.
\end{proof}
\begin{corollary}\label{compcor}
It holds
\[
|C(\mathcal{N},k)|=\sum_{\ell=1}^{(2^k-2)\wedge n}\ell!\tilde{s}(n,\ell) |F(k,\ell)|.
\]
\end{corollary}
\begin{proof}
By \eqref{stildecount} in Remark~\ref{interpretation}, $|\tilde{S}(n,\ell)\times F(k,\ell)|=\ell!\tilde{s}(n,\ell) |F(k,\ell)|$ for each $\ell\in\cb{1,\ldots,2^k-2}$. Hence, the corollary follows from Theorem~\ref{mainthm}.
\end{proof}
Next, we aim to refine the result of Corollary~\ref{compcor} by showing that $F(k,\ell)$ is the empty set for small values of $\ell$. To that end, for a subset $T\subseteq\mathbb{T}(k)$ and $i,j\in\cb{1,\ldots,k}$ with $i\neq j$, let us define
\[
u_{ij}(T)\coloneqq\sum_{t\in T}1_{\cb{c}}(t_i)1_{\cb{e}}(t_j).
\]
\begin{proposition}\label{ell0result}
Let
\[
\ell_0(k)\coloneqq \min\cb{\abs{T}\mid u_{ij}(T)\geq 1\ \forall i\neq j,\ T\subseteq\mathbb{T}(k)}
\]
Then, for every $\ell\in\cb{1,\ldots,2^k-2}$,
\[
\ell \geq \ell_0(k)\quad\Leftrightarrow\quad F(k,\ell)\neq \emptyset.
\]
In particular,
\[
\ell_0(k)=\min\cb{\ell\mid F(k,\ell)\neq\emptyset}.
\]
\end{proposition}
\begin{proof}
Let $\ell\in\cb{1,\ldots,2^k-2}$. Suppose that $\ell\geq\ell_0(k)$. Let $T^\ast\subseteq\mathbb{T}(k)$ such that $|T^\ast|=\ell_0(k)$ and $u_{ij}(T^\ast)\geq 1$ for every $i,j\in\cb{1,\ldots,k}$ with $i\neq j$. By adding $\ell-\ell_0$ more elements to $T^\ast$ arbitrarily, one can find a set $T\subseteq\mathbb{T}(k)$ such that $T^\ast\subseteq T$ and $|T|=\ell$. For each $t\in\mathbb{T}(k)$, let us define a binary variable $x_t$ by $x_t=1$ if $t\in T$ and $x_t=0$ if $T\in\mathbb{T}(k)\!\setminus\! T$. The assumed properties of $T^\ast$ ensure that $(x_t)_{t\in\mathbb{T}(k)}\in F(k,\ell)$. Hence, $F(k,\ell)\neq \emptyset$. In particular, the case $\ell=\ell_0$ implies that $\ell_0(k)\geq \min\cb{\ell\mid F(k,\ell)\neq\emptyset}$.
For the converse, suppose that $F(k,\ell)\neq \emptyset$. Let $(x_t)_{t\in\mathbb{T}(k)}\in F(k,\ell)$ and $i,j\in\cb{1,\ldots,k}$ with $i\neq j$. By the definition of $F(k,\ell)$, there exists $t(i,j)\in \mathbb{T}(k)$ such that $x_{t(i,j)}=1$, $t(i,j)_i=c$, $t(i,j)_j=e$. Let
\[
T=\cb{t(i,j)\mid i\neq j}.
\]
Note that
\[
u_{ij}(T)=\sum_{t\in T}1_{\cb{c}}(t_i)1_{\cb{e}}(t_j)\geq 1_{\cb{c}}(t(i,j)_i)1_{\cb{e}}(t(i,j)_j)=1\cdot 1=1.
\]
Hence, $\ell_0(k)\leq |T|$. By the definition of $F(k,\ell)$ again,
\[
\ell_0(k)\leq |T|\leq \sum_{i\neq j}x_{t(i,j)}\leq \sum_{t\in\mathbb{T}(k)}x_t=\ell.
\]
Hence, $\ell_0(k)\leq \ell$. In particular, $\ell_0(k)\leq \min\cb{\ell\mid F(k,\ell)\neq\emptyset}$.
\end{proof}
\begin{corollary}\label{ell0forcounting}
It holds
\[
|C(\mathcal{N},k)|=\sum_{\ell=\ell_0(k)}^{(2^k-2)\wedge n}\ell!\tilde{s}(n,\ell) |F(k,\ell)|.
\]
\end{corollary}
\begin{proof}
This is an immediate consequence of Corollary~\ref{compcor} and Proposition~\ref{ell0result}.
\end{proof}
\section{Partitioning the sets of labelings}\label{partitioningsec}
By Theorem~\ref{mainthm} and Corollary~\ref{ell0forcounting} of Section~\ref{problemdefn}, we are able to calculate the cardinality of the set $C(\mathcal{N}, k)$ of constructive $k$-covers of $\mathcal{N}$ in terms of ISNs $\tilde{s}(n,\ell)$ as well as the cardinalities of the sets $F(k,\ell)$ of $(k,\ell)$-labelings for a range of $\ell$ values. While it is easy to numerically calculate ISNs by \eqref{stilderes} and {\eqref{StirlingDefn}}, the calculation of $|F(k,\ell)|$ by brute force enumeration could be quite difficult even for small values of $k,\ell$. In this section, we introduce three notions of symmetry for the sets $F(k,\ell)$, $\ell\in\cb{\ell_0(k),\ldots,2^k-2}$, which yield an equivalence relation. It turns out that the equivalence classes of this relation provide substantial reduction in the computational effort to find the cardinalities $|F(k,\cdot)|$.
Let $k\in\mathbb{N}$. Let us fix a nonempty subset $T$ of $\mathbb{T}(k)$. We call $T$ the \emph{branching set} of index vectors. Let us define
\[
\mathcal{Z}_T(k)\coloneqq \cb{y=(y_t)_{t\in T}\mid y_t\in\cb{0,1}\ \forall t\in T}.
\]
We also fix $\ell\in\mathbb{N}$ with $\ell_0(k)\leq \ell\leq 2^k-2$, where $\ell_0(k)$ is defined as in Proposition~\ref{ell0result}. In particular, $F(k,\ell)\neq\emptyset$. We may consider partitioning $F(k,\ell)$ with respect to the possible ways of assigning the binary variables $x_t$ associated to all $t\in T$. The next definition formalizes this idea.
\begin{definition}\label{DefnOfFm}
Let $y_t\in\cb{0,1}$ for each $t\in T$. Then, the set of all $(k,\ell)$-labelings with respect to $y= (y_t)_{t\in T}$ is defined as
\[
F_{y}(k,\ell)\coloneqq\cb{(x_t)_{t\in\mathbb{T}(k)}\in F(k,\ell)\mid x_t = y_t \ \forall t\in T}.
\]
\end{definition}
If $\sum_{t\in T}y_t >\ell$, then $F_{y}(k,\ell)=\emptyset$ obviously. Let us introduce the set
\[
\mathcal{Y}_T(k,\ell)\coloneqq \cb{y=(y_t)_{t\in T}\in\mathcal{Z}_T(k)\mid \sum_{t\in T}y_t \leq \ell}.
\]
\begin{remark}
Given $y\in \mathcal{Y}_T(k,\ell)$, depending on the structure of $T$ and $y$, the set $F_y(k,\ell)$ may still be empty. Nevertheless, such cases will be detected in the computational procedure presented later in this section and we do not need to distinguish them \emph{a priori} in the theoretical development.
\end{remark}
We begin with a simple result that provides a partitioning of $F(k,\ell)$ into smaller sets.
\begin{proposition}\label{PropBranch}
Let $y,z\in\mathcal{Y}_T(k,\ell)$.
\begin{enumerate}[(i)]
\item $F_{y}(k, \ell) \cap F_{z}(k, \ell) = \emptyset$ if $y\neq z$.
\item It holds
\[
F(k,\ell)=\bigcup_{y\in \mathcal{Y}_T(k,\ell)}F_{y}(k, \ell).
\]
\end{enumerate}
In particular,
\[
|F(k, \ell)| = \sum_{y\in \mathcal{Y}_T(k,\ell)} |F_y(k, \ell)|.
\]
\end{proposition}
\begin{proof}
\text{}
\begin{enumerate}[(i)]
\item This is an immediate consequence of Definition~\ref{DefnOfFm}.
\item The $\supseteq$ part of the equality is obvious since $F_{y}(k, \ell) \subseteq F(k, \ell)$ for each $y\in \mathcal{Y}_T(k,\ell)$. For the $\subseteq$ part, let $(x_t)_{t\in\mathbb{T}(k)}\in F(k, \ell)$. Let us define $y=(y_t)_{t\in T}$ by setting $y_t\coloneqq x_t$ for each $t\in T$. Then, $\sum_{t\in T}y_t= \sum_{t\in T}x_t\leq\sum_{t\in\mathbb{T}(k)}x_t=\ell$ so that $y\in\mathcal{Y}_T(k,\ell)$. Clearly, we also have $(x_t)_{t\in \mathbb{T}(k)}\in F_{y}(k,\ell)$. Hence, the $\subseteq$ part of the equality follows.
\end{enumerate}
The last statement follows directly from (i) and (ii).
\end{proof}
While Proposition~\ref{PropBranch} partitions $F(k,\ell)$ into the smaller sets $F_y(k,\ell)$, $y\in\mathcal{Y}_T(k,\ell)$, it may still be computationally expensive to calculate the cardinality of each of these sets by an enumerative method. By introducing three notions of symmetry below, we show that the cardinalities $|F_y(k,\ell)|$ are repeated for many $y\in\mathcal{Y}_T(k,\ell)$ so that we only need to calculate the distinct values of these cardinalities and the number of times each cardinality value is repeated.
\subsection{A symmetry based on permutations}\label{permsym}
The first notion of symmetry we introduce is based on permutations. To that end, let us denote by $\mathbb{S}_k$ the symmetric group of $\cb{1,\ldots,k}$, that is, the set of all permutations $\pi\colon\cb{1,\ldots,k}\to\cb{1,\ldots,k}$. Let $\pi\in\S_k$. Given $t=(t_1,\ldots,t_k)\in \cb{e,c}^k$, we may consider $t$ as a function $t\colon\cb{1,\ldots,k}\to \cb{e,c}$ and define the composition $t\circ \pi \colon\cb{1,\ldots, k}\to\cb{e,c}$ by
\[
(t\circ \pi)(i)=(t\circ\pi)_i\coloneqq t_{\pi(i)},\quad i\in\cb{1,\ldots,k},
\]
or, we may simply define $t\circ \pi$ as the vector
\[
t\circ \pi = ((t\circ \pi)_1,\ldots,(t\circ\pi)_k)\coloneqq (t_{\pi(1)},\ldots, t_{\pi(k)})\in\cb{e,c}^k.
\]
From the definition of $\mathbb{T}(k)$ (see \eqref{tk}), it is clear that $t\circ \pi \in \mathbb{T}(k)$ if and only if $t\in\mathbb{T}(k)$. Let us introduce the set
\[
T^\pi \coloneqq \cb{t\circ \pi \mid t\in T}.
\]
We call $T$ symmetric with respect to $\pi$ if $T=T^\pi$. In particular, it is always the case that $\mathbb{T}(k)$ is symmetric with respect to $\pi$. We denote by $\S_k^T$ the set of all permutations with respect to which $T$ is symmetric, that is,
\[
\S_k^T \coloneqq\cb{\pi\in\S_k\mid T=T^\pi}.
\]
Note that $\S_k^T\neq\emptyset$ as we always have $T=T^\pi$ when $\pi$ is the identity permutation.
The next proposition formulates how the number of $(k,\ell)$-labelings associated to a vector $y\in\mathcal{Y}_T(k,\ell)$ changes under the application of a permutation.
\begin{proposition}\label{equalityofcard}
Let $\pi\in\S_k^T$ and $y\in \mathcal{Y}_T(k,\ell)$. Define $y^\pi=(y^\pi_t)_{t\in T}$ by
\[
y^\pi_t \coloneqq y_{t\circ \pi},\quad t\in T.
\]
Then, $y^\pi \in \mathcal{Y}_T(k,\ell)$ and $|F_y(k,\ell)|=|F_{y^\pi}(k,\ell)|$.
\end{proposition}
\begin{proof}
Since $y\in\mathcal{Y}_T(k,\ell)$ and $T=T^\pi$, we have
\[
\sum_{t\in T}y^\pi_t = \sum_{t\in T}y_{t\circ\pi}=\sum_{t\in T}y_t \leq \ell.
\]
Hence, $y^\pi\in \mathcal{Y}_T(k,\ell)$.
To prove that $|F_y(k,\ell)|=|F_{y^\pi}(k,\ell)|$, it is sufficient to establish a bijection from $F_y(k,\ell)$ into $F_{y^\pi}(k,\ell)$. To that end, given $(x_t)_{t\in \mathbb{T}(k)}\in F_y(k,\ell)$, let us define $(x^\pi_t)_{t\in \mathbb{T}(k)}$ by
\[
x^\pi_t = x_{t\circ \pi},\quad t\in \mathbb{T}(k).
\]
We first show that $(x^\pi_t)_{t\in\mathbb{T}(k)}\in F_{y^\pi}(k,\ell)$. Let $i,j\in\cb{1,\ldots,k}$ with $i\neq j$. Denoting the inverse permuatation of $\pi$ by $\pi^{-1}$, we have
\begin{align*}
\sum_{t\in\mathbb{T}(k)\colon t_i=e,t_j=c}x^\pi_t
&= \sum_{t\in\mathbb{T}(k)\colon t_i=e,t_j=c}x_{t\circ \pi}\\
&=\sum_{t\in\mathbb{T}(k)\colon (t\circ\pi)_{\pi^{-1}(i)}=e,(t\circ\pi)_{\pi^{-1}(j)}=c}x_{t\circ \pi}\\
&=\sum_{t\in\mathbb{T}(k)\colon t_{\pi^{-1}(i)}=e,t_{\pi^{-1}(j)}=c}x_{t},
\end{align*}
where we make a change of variables using the fact that $\mathbb{T}(k)=(\mathbb{T}(k))^\pi$ in order to get the last equality. Since $i\neq j$, we have $\pi^{-1}(i)\neq \pi^{-1}(j)$. As we also have $(x_t)_{t\in\mathbb{T}(k)}\in G(k)$ (see \eqref{defng}), it follows that
\[
\sum_{t\in\mathbb{T}(k)\colon t_i=e,t_j=c}x^\pi_t=\sum_{t\in\mathbb{T}(k)\colon t_{\pi^{-1}(i)}=e,t_{\pi^{-1}(j)}=c}x_{t}\geq 1.
\]
Similarly, since $\mathbb{T}(k)=(\mathbb{T}(k))^\pi$,
\[
\sum_{t\in\mathbb{T}(k)}x^\pi_t = \sum_{t\in \mathbb{T}(k)}x_{t\circ\pi}=\sum_{t\in\mathbb{T}(k)}x_t = \ell.
\]
On the other hand, for each $t\in T$, we have $t\circ\pi\in T$ so that
\[
x^\pi_t = x_{t\circ \pi} = y_{t\circ \pi}=y^\pi_t.
\]
Therefore, $(x^\pi_t)_{t\in\mathbb{T}(k)}\in F_{y^\pi}(k,\ell)$.
It remains to check that the mapping $(x_t)_{t\in\mathbb{T}(k)}\mapsto (x_t^\pi)_{t\in\mathbb{T}(k)}$ is indeed a bijection from $F_y(k,\ell)$ into $F_{y^\pi}(k,\ell)$. Let $(x_t)_{t\in \mathbb{T}(k)},$ $(\bar{x}_t)_{t\in\mathbb{T}(k)}\in F_y(k,\ell)$ such that $x_t^\pi =\bar{x}_t^\pi$ for every $t\in\mathbb{T}(k)$, that is, $x_{t\circ \pi}=\bar{x}_{t\circ\pi}$ for every $t\in\mathbb{T}(k)$. Since $\mathbb{T}(k)=(\mathbb{T}(k))^\pi$, this is equivalent to having $x_t = \bar{x}_t$ for every $t\in\mathbb{T}(k)$. Hence, $(x_t)_{t\in\mathbb{T}(k)}\mapsto (x_t^\pi)_{t\in\mathbb{T}(k)}$ is injective. Next, let $(z_t)_{t\in\mathbb{T}(k)}\in F_{y^\pi}(k,\ell)$. Let us define $(x_t)_{t\in\mathbb{T}(k)}$ by
\[
x_t \coloneqq z^{\pi^{-1}}_t=z_{t\circ \pi^{-1}},\quad t\in\mathbb{T}(k).
\]
Let $i,j\in\cb{1,\ldots,k}$ such that $i\neq j$. Hence, we have $\pi(i)\neq\pi(j)$ so that
\[
\sum_{t\in\mathbb{T}(k)\colon t_i=e,\ t_j=c}x_t=\sum_{t\in\mathbb{T}(k)\colon t_i=e,\ t_j=c} z_{t\circ \pi^{-1}}=\sum_{t\in\mathbb{T}(k)\colon t_{\pi(i)}=e,\ t_{\pi(j)}=c} z_{t}\geq 1.
\]
Next, since $\mathbb{T}(k)=(\mathbb{T}(k))^{\pi^{-1}}$, we have
\[
\sum_{t\in\mathbb{T}(k)}x_t = \sum_{t\in\mathbb{T}(k)}z_{t\circ \pi^{-1}}=\sum_{t\in\mathbb{T}(k)}z_t = \ell.
\]
On the other hand, since $T=T^\pi$, we also have $T=T^{\pi^{-1}}$. Hence, for each $t\in T$, we have $t\circ\pi^{-1}\in T$ so that
\[
x_t = z_{t\circ\pi^{-1}}=y_{t\circ \pi^{-1}}^{\pi}=y_t.
\]
Therefore, $(x_t)_{t\in\mathbb{T}(k)}\in F_y(k,\ell)$, that is, $(x_t)_{t\in\mathbb{T}(k)}\mapsto (x_t^\pi)_{t\in\mathbb{T}(k)}$ is surjective as well.
\end{proof}
\subsection{A symmetry based on taking complements}\label{compsym}
In addition to the above notion of permutation-based symmetry, we introduce a second type of symmetry based on taking ``complements,'' that is, based on changing the roles of $e$ and $c$ in the index vectors. To be more precise, let us define two mappings $\alpha_1,\alpha_2\colon\cb{e,c}\to\cb{e,c}$ by
\[
\alpha_1(e)=\alpha_2(c)=e,\quad \alpha_1(c)=\alpha_2(e)=c.
\]
In other words, $\alpha_1$ is the identity mapping and $\alpha_2$ switches $e$ and $c$. Let us formally define the set $\mathbb{A}=\cb{\alpha_1,\alpha_2}$, which is indeed the symmetric group of $\cb{e,c}$. Let $\alpha\in\mathbb{A}$. Similar to what is done in Section \ref{permsym}, we may regard each $t=(t_1,\ldots,t_k)\in\cb{e,c}^k$ as a function $t\colon\cb{1,\ldots,k}\to\cb{e,c}$ and define the composition $\alpha\circ t\colon \cb{1,\ldots,k}\to\cb{e,c}$ by
\[
(\alpha\circ t)(i)=(\alpha\circ t)_i \coloneqq \alpha(t_i),\quad i\in\cb{1,\ldots,k},
\]
or, we define $\alpha\circ t$ as the vector
\[
\alpha\circ t = \of{(\alpha\circ t)_1,\ldots,(\alpha\circ t)_k}\coloneqq \of{\alpha(t_1),\ldots,\alpha(t_k)}\in \cb{e,c}^k.
\]
Clearly, $\alpha\circ t\in\mathbb{T}(k)$ if and only if $t\in \mathbb{T}(k)$. Let us also define
\[
T^\alpha \coloneqq \cb{\alpha\circ t\mid t\in T}
\]
and
\[
\mathbb{A}^T \coloneqq \cb{\alpha\in \mathbb{A}\mid T=T^\alpha}.
\]
Since $\alpha_1$ is the identity mapping, we always have $T=T^{\alpha_1}$ so that $\mathbb{A}^T \neq \emptyset$.
In the next proposition, we relate the numbers of $(k,\ell)$-labelings associated to a vector $y\in\mathcal{Y}_T(k,\ell)$ before and after taking complements.
\begin{proposition}\label{equalityofcard2}
Let $\alpha\in \mathbb{A}^T$ and $y\in \mathcal{Y}_T(k,\ell)$. Define ${}^\alpha y=({}^\alpha y_t)_{t\in T}$ by
\[
{}^\alpha y_t \coloneqq y_{\alpha\circ t},\quad t\in T.
\]
Then, ${}^\alpha y\in \mathcal{Y}_T(k,\ell)$ and $|F_y(k,\ell)|=|F_{{}^\alpha y}(k,\ell)|$.
\end{proposition}
\begin{proof}
The result is trivial for $\alpha=\alpha_1$. Let us assume that $\alpha=\alpha_2$. Since $y\in\mathcal{Y}_T(k,\ell)$ and $T=T^{\alpha}$, we have
\[
\sum_{t\in T}{}^\alpha y_t = \sum_{t\in T}y_{\alpha\circ t}=\sum_{t\in T}y_t \leq \ell.
\]
Hence, ${}^\alpha y\in \mathcal{Y}_T(k,\ell)$.
To prove that $|F_y(k,\ell)|=|F_{{}^\alpha y}(k,\ell)|$, we construct a bijection from $F_y(k,\ell)$ to $F_{{}^\alpha y}(k,\ell)$ as follows. Given $(x_t)_{t\in\mathbb{T}(k)}\in F_y(k,\ell)$, let us define $({}^\alpha x_t)_{t\in\mathbb{T}(k)}$ by
\[
{}^\alpha x_t =x_{\alpha \circ t},\quad t\in\mathbb{T}(k).
\]
Let $i,j\in\cb{1,\ldots,k}$ with $i\neq j$. Since $(x_t)_{t\in \mathbb{T}(k)}\in G(k)$, we have
\begin{align*}
\sum_{t\in\mathbb{T}(k)\colon t_i=e,t_j=c}{}^\alpha x_t &= \sum_{t\in\mathbb{T}(k)\colon t_i=e,t_j=c}x_{\alpha \circ t}\\
&= \sum_{t\in\mathbb{T}(k)\colon \alpha(t_i)=c,\alpha(t_j)=e}x_{\alpha \circ t}\\
&=\sum_{t\in\mathbb{T}(k)\colon t_i=c,t_j=e}x_t\geq 1.
\end{align*}
Similarly,
\[
\sum_{t\in\mathbb{T}(k)}{}^\alpha x_t= \sum_{t\in\mathbb{T}(k)}x_{\alpha \circ t}=\sum_{t\in\mathbb{T}(k)}x_t = \ell
\]
and
\[
{}^\alpha x_t = x_{\alpha \circ t}=y_{\alpha \circ t}={}^\alpha y_t,\quad t\in T.
\]
Hence, $({}^\alpha x_t)_{t\in\mathbb{T}(k)}\in F_{{}^\alpha y}(k,\ell)$. Using a similar argument as in the proof of Proposition \ref{equalityofcard}, it can be shown that the mapping $(x_t)_{t\in\mathbb{T}(k)}\mapsto ({}^\alpha x_t)_{t\in\mathbb{T}(k)}$ is a bijection. The details are omitted.
\end{proof}
\subsection{A symmetry based on impact sets}\label{impactsym}
In this subsection, we introduce a third notion of symmetry based on the idea that two binary vectors $y,z\in\mathcal{Z}_T(k)$ might impose the same set of constraints in the definition of $G(k)$, see \eqref{defng}, which we will refer to as the \emph{impact set} of these vectors.
Given $y=(y_t)_{t\in T}\in\mathcal{Z}_T(k)$, we define the \emph{impact set} $\mathcal{D}_T(y)$ of $y$ as
\[
\mathcal{D}_T(y)\coloneqq \cb{(i,j)\in\cb{1,\ldots,k}^2\mid \exists t\in T\colon (t_i=c\ \wedge\ \ t_j=e\ \wedge\ y_t=1)}.
\]
The next theorem provides a relationship for the values of $|F_y(k,\cdot)|$ and $|F_z(k,\cdot)|$ when $y,z\in\mathcal{Z}_T(k)$ have the same impact set.
\begin{theorem}\label{simeqprop}
Let $y,z\in \mathcal{Z}_T(k)$ be such that $\mathcal{D}_T(y)=\mathcal{D}_T(z)$. Let
\[
w \coloneqq \sum_{t\in T}y_t - \sum_{t\in T}z_t.
\]
Suppose that $w\geq 0$. The following results hold.
\begin{enumerate}[(i)]
\item $|F_y(k,\ell+w)|=|F_z(k,\ell)|$ for every $\ell\in\mathbb{N}$ such that $\ell_0(k)\leq \ell \leq \ell+w\leq 2^k-2$.
\item $|F_y(k,\ell+w)|=0$ for every $\ell\in\mathbb{N}$ such that $\ell< \ell_0(k)\leq \ell+w$.
\item $|F_z(k,\ell)|=0$ for every $\ell\in\mathbb{N}$ such that $\ell\leq 2^k-2 <\ell+w$.
\end{enumerate}
\end{theorem}
\begin{proof}
We consider the sets $F_y(k,\ell), F_z(k,\ell)$ defined by \eqref{DefnOfF} for every $\ell\in\mathbb{N}$. (Hence, we extend the definition in \eqref{DefnOfF} for $\ell\geq 2^k-2$). Let us fix $\ell\in\mathbb{N}$. We establish a bijection from $F_z(k,\ell)$ to $F_y(k,\ell+w)$. Given $x=(x_t)_{t\in\mathbb{T}(k)}\in F_z(k,\ell)$, let us define $\bar{x}=(\bar{x}_t)_{t\in\mathbb{T}(k)}$ by
\[
\bar{x}_t = \begin{cases}y_t&\text{if } t\in T,\\ x_t & \text{if }t\notin T.\end{cases}
\]
Let $i,j\in\cb{1,\ldots,k}$ with $i\neq j$. First, suppose that $(i,j)\in \mathcal{D}_T(y)=\mathcal{D}_T(z)$. Hence, there exists $t^1\in T$ such that $t^1_i=c$, $t^1_j=e$, $y_{t^1}=1$. So
\[
\sum_{t\in\mathbb{T}(k)\colon t_i=c,t_j=e}\bar{x}_t\geq \bar{x}_{t^1}=y_{t^1}=1.
\]
Next, suppose that $(i,j)\notin \mathcal{D}_T(y)=\mathcal{D}_T(z)$. Since $x\in F_z(k,\ell)$, there exists $t^2\in \mathbb{T}(k)\setminus T$ such that $t^2_i=c$, $t^2_j=e$, $x_{t^2}=1$. So
\[
\sum_{t\in\mathbb{T}(k)\colon t_i=c,t_j=e}\bar{x}_t\geq \bar{x}_{t^2}=x_{t^2}=1.
\]
Moreover,
\[
\sum_{t\in \mathbb{T}(k)}\bar{x}_t = \sum_{t\in T}y_t +\sum_{t\in\mathbb{T}(k)\setminus T}x_t =\sum_{t\in T}y_t +\ell - \sum_{t\in T}x_t=\sum_{t\in T}y_t +\ell - \sum_{t\in T}z_t=\ell+w.
\]
Hence, $\bar{x}\in F_y(k,\ell+w)$.
Next, we show that the mapping $x\mapsto \bar{x}$ is a bijection from $F_z(k,\ell)$ into $F_y(k,\ell+w)$. Let $x^1=(x^1_t)_{t\in\mathbb{T}(k)}, x^2=(x^2_t)_{t\in \mathbb{T}(k)}\in F_z(k,\ell)$ such that their images are equal, that is, $\bar{x}^1_t = \bar{x}^2_t$ for every $t\in\mathbb{T}(k)$. From the definition of the mapping, it is immediate that $x^1_t=x^2_t$ for every $t\in \mathbb{T}(k)\setminus T$. On the other hand, $x^1_t=x^2_t=z_t$ for every $t\in T$ since $x^1,x^2\in F_z(k,\ell)$. Therefore, $x^1=x^2$. This proves that the mapping is injective. Let $\tilde{x}=(\tilde{x}_t)_{t\in\mathbb{T}(k)}\in F_y(k,\ell+w)$. Define $x=(x_t)_{t\in\mathbb{T}(k)}$ by
\[
x_t=\begin{cases}z_t&\text{if }t\in T,\\ \tilde{x}_t&\text{if }t\notin T.\end{cases}
\]
It is not difficult to check that $x\in F_z(k,\ell)$ and $\bar{x}=\tilde{x}$, which shows that the mapping is surjective. Hence, thanks to the bijection, we conclude that $|F_y(k,\ell+w)|=|F_z(k,\ell)|$ for every $\ell\in\mathbb{N}$. If $\ell<\ell_0(k)$, then $F_z(k,\ell)=\emptyset$ so that $F_y(k,\ell+w)=\emptyset$ as well. Similarly, if $2^k-2<\ell+w $, then $F_y(k,\ell+w)=\emptyset$ so that $F_z(k,\ell)=\emptyset$ as well. Hence, all three results hold.
\end{proof}
\subsection{Equivalence relation for the symmetries}\label{equiv1}
Given $t\in \mathbb{T}(k)$, $x_t\in\cb{e,c}$, $\pi\in\S_k$ and $\alpha\in\mathbb{A}$, note that
\[
({}^\alpha x_t)^\pi = {}^\alpha (x^\pi_t)=x_{\alpha\circ t\circ \pi}.
\]
Hence, we simply write ${}^\alpha x^\pi_t\coloneqq x_{\alpha\circ t\circ \pi}$ for the common value. On $\mathcal{Z}_T(k)$, let us define the relation $\equiv$ by
\begin{equation}\label{equivalence}
y\equiv z\quad \Leftrightarrow\quad \exists(\pi,\alpha)\in\S_k^T\times\mathbb{A}^T\colon \mathcal{D}_T(z)=\mathcal{D}_T({}^\alpha y^\pi)
\end{equation}
for each $y=(y_t)_{t\in T},z=(z_t)_{t\in T}\in\mathcal{Z}_T(k)$.
\begin{proposition}\label{equivrel}
The relation $\equiv$ defined by \eqref{equivalence} is an equivalence relation on $\mathcal{Z}_T(k)$.
\end{proposition}
\begin{proof}
The reflexivity of $\equiv$ follows since the identity permutation on $\cb{1,\ldots,k}$ is always a member of $\S_k^T$. To show that $\equiv$ is symmetric, let $y,z\in \mathcal{Z}_T(k)$ such that $y\equiv z$. So $\mathcal{D}_T(z)=\mathcal{D}_T({}^\alpha y^\pi)$ for some $\pi\in\S_k^T$ and $\alpha\in\mathbb{A}^T$. Then,
\[
T^{\pi^{-1}}=\cb{t\circ \pi^{-1}\mid t\in T}=\cb{\bar{t}\circ\pi\circ\pi^{-1}\mid \bar{t}\in T}=T
\]
since $T=T^{\pi}$. So $\pi^{-1}\in\S_k^T$. On the other hand, it is easy to see that $\alpha^{-1}=\alpha$. We claim that $\mathcal{D}_T(y)=\mathcal{D}_T({}^\alpha z^{\pi^{-1}})$. To show this, first, let $(i,j)\in\mathcal{D}_T(y)$. So there exists $t\in T$ such that $t_i=c$, $t_j=e$, $y_t=1$. Letting $t^\prime\coloneqq \alpha\circ t\circ \pi^{-1}$, we may write $t=\alpha\circ t^\prime\circ \pi$. In particular, we have the following:
\begin{enumerate}[(i)]
\item $t^\prime_{\pi(i)}=\alpha(t_{\pi^{-1}(\pi(i))})=\alpha(t_i)=\alpha(c)$.
\item $t^{\prime}_{\pi(j)}=\alpha(t_{\pi^{-1}(\pi(j))})=\alpha(t_j)=\alpha(e)$.
\item ${}^\alpha y_{t^\prime}^{\pi}=y_{\alpha\circ t^{\prime}\circ \pi}=y_t=1$.
\end{enumerate}
It follows that $(\pi(i),\pi(j))\in\mathcal{D}_T({}^\alpha y^{\pi})$ if $\alpha=\alpha_1$, and $(\pi(j),\pi(i))\in\mathcal{D}_T({}^\alpha y^{\pi})$ if $\alpha=\alpha_2$. Let us consider the case $\alpha=\alpha_1$. Since $(\pi(i),\pi(j))\in\mathcal{D}_T({}^\alpha y^{\pi})=\mathcal{D}_T(z)$, there exists $t^{\prime\prime}\in T$ such that $t^{\prime\prime}_{\pi(i)}=c$, $t^{\prime\prime}_{\pi(j)}=e$, $z_{t^{\prime\prime}}=1$. Let us define $t^{\prime\prime\prime}\coloneqq \alpha\circ t^{\prime\prime}\circ \pi$ so that $t^{\prime\prime}=\alpha\circ t^{\prime\prime\prime}\circ \pi^{-1}$. In particular, we have the following:
\begin{enumerate}[(i)]
\item $t^{\prime\prime\prime}_{i}=\alpha(t^{\prime\prime}_{\pi(i)})=\alpha(c)=c$.
\item $t^{\prime\prime\prime}_{j}=\alpha(t^{\prime\prime}_{\pi(j)})=\alpha(e)=e$.
\item ${}^\alpha z_{t^{\prime\prime\prime}}^{\pi^{-1}}=z_{\alpha\circ t^{\prime\prime\prime}\circ \pi^{-1}}=z_{t^{\prime\prime}}=1$.
\end{enumerate}
Hence, $(i,j)\in \mathcal{D}_T({}^\alpha z^{\pi^{-1}})$. The case $\alpha=\alpha_2$ can be treated by similar arguments and we obtain $(i,j)\in \mathcal{D}_T({}^\alpha z^{\pi^{-1}})$ as well. Hence, $\mathcal{D}_T(y)\subseteq \mathcal{D}_T({}^\alpha z^{\pi^{-1}})$.
Conversely, let $(i,j)\in \mathcal{D}_T({}^\alpha z^{\pi^{-1}})$. So there exists $t\in T$ such that $t_i=c$, $t_j=e$, ${}^\alpha z^{\pi^{-1}}_t=1$. Let $t^\prime\coloneqq\alpha\circ t \circ \pi^{-1}$. In a similar way as above, we have $t^\prime_{\pi(i)}=\alpha(c)$, $t^{\prime}_{\pi(j)}=\alpha(e)$, $z_{t^\prime}=1$, that is, $(\pi(i),\pi(j))\in \mathcal{D}_T(z)$ if $\a=\a_1$, and $(\pi(j),\pi(i))\in \mathcal{D}_T(z)$ if $\a=\a_2$. Suppose that $\a=\a_1$. In this case, since $(\pi(i),\pi(j))\in \mathcal{D}_T(z)=\mathcal{D}_T({}^\alpha y^\pi)$, there exists $t^{\prime\prime}\in T$ such that $t^{\prime\prime}_{\pi(i)}=c$, $t^{\prime\prime}_{\pi(j)}=e$, ${}^\a y^\pi_{t^{\prime\prime}}=1$. Let $t^{\prime\prime\prime}\coloneqq \alpha\circ t^{\prime\prime}\circ \pi$. Then, we have $t^{\prime\prime\prime}_i=\a(c)=c$, $t^{\prime\prime\prime}_j=\a(e)=e$, $y_{t^{\prime\prime\prime}}=1$ so that $(i,j)\in \mathcal{D}_T(y)$. Similarly, we may show that $(i,j)\in \mathcal{D}_T(y)$ in the case $\a=\a_2$ as well. Hence, $\mathcal{D}_T({}^\alpha z^{\pi^{-1}})\subseteq\mathcal{D}_T(y)$. This completes the proof of $\mathcal{D}_T(y)=\mathcal{D}_T({}^\alpha z^{\pi^{-1}})$. Therefore, $z\equiv y$.
To show that $\equiv$ is transitive, let $y,z,v\in \mathcal{Z}_T(k)$ such that $y\equiv z$ and $z\equiv v$. So $\mathcal{D}_T(z)=\mathcal{D}_T({}^\alpha y^\pi)$ and $\mathcal{D}_T(v)=\mathcal{D}_T({}^\beta z^{\sigma})$ for some $\pi,\sigma\in\S_k^T$ and $\alpha,\beta\in\mathbb{A}^T$. Since $T^\pi=T^{\sigma}=T$, we also have $T^{\sigma\circ \pi}=\cb{t\circ\sigma\circ \pi\mid t\in T}=\cb{\bar{t}\circ \pi \mid \bar{t}\in T}=T$ so that $\sigma\circ\pi\in\S_k^T$. On the other hand, $\alpha\circ\beta $ is either equal to $\alpha_1$ or to $\alpha_2$ so that $\alpha\circ\beta\in\mathbb{A}^T$. We claim that $\mathcal{D}_T(v)=\mathcal{D}_T({}^{\alpha\circ\beta}y^{\sigma\circ\pi})$.
To prove the claim, let $(i,j)\in \mathcal{D}_T(v)$. Since $\mathcal{D}_T(v)=\mathcal{D}_T({}^\beta z^{\sigma})$, there exists $t\in T$ such that $t_i=c$, $t_j=e$, ${}^\beta z^\sigma_t=1$. Letting $t^\prime=\beta\circ t\circ\sigma$, we have $t^\prime_{\sigma^{-1}(i)}\coloneqq\beta(c)$, $t^\prime_{\sigma^{-1}(j)}=\beta(e)$, $z_{t^\prime}=1$. Hence, $(\sigma^{-1}(i),\sigma^{-1}(j))\in\mathcal{D}_T(z)$ if $\beta=\a_1$, and $(\sigma^{-1}(j),\sigma^{-1}(i))\in\mathcal{D}_T(z)$ if $\beta=\a_2$. Suppose that $\beta=\a_1$. Since $(\sigma^{-1}(i),\sigma^{-1}(j))\in\mathcal{D}_T(z)=\mathcal{D}_T({}^\alpha y^\pi)$, there exists $t^{\prime\prime}\in T$ such that $t^{\prime\prime}_{\sigma^{-1}(i)}=c$, $t^{\prime\prime}_{\sigma^{-1}(j)}=e$, ${}^\alpha y^\pi_{t^{\prime\prime}}=1$. Letting $t^{\prime\prime\prime}=\beta\circ t^{\prime\prime}\circ\sigma^{-1}$, we have $t^{\prime\prime\prime}_i=c$, $t^{\prime\prime\prime}_i=e$, ${}^{\alpha\circ\beta}y^{\sigma\circ\pi}_{t^{\prime\prime\prime}}={}^{\a}y^{\pi}_{t^{\prime\prime}}=1$. Hence, $(i,j)\in\mathcal{D}_T({}^{\alpha\circ\beta}y^{\sigma\circ\pi})$. Similarly, we can reach the same conclusion when $\b=\a_2$. So $\mathcal{D}_T(v)\subseteq \mathcal{D}_T({}^{\alpha\circ\beta}y^{\sigma\circ\pi})$. The proof of $ \mathcal{D}_T({}^{\alpha\circ\beta}y^{\sigma\circ\pi})\subseteq \mathcal{D}_T(v)$ is similar, hence we omit it. So we have $y\equiv v$. Therefore, $\equiv$ is an equivalence relation on $\mathcal{Z}_T(k)$.
\end{proof}
Next, we address the role of Proposition~\ref{equivrel} for computational purposes. Note that the relation $\equiv$ partitions $\mathcal{Z}_T(k)$ into equivalence classes; let us denote them by $\mathcal{Z}_{T,1}(k),\ldots,\mathcal{Z}_{T,A}(k)$, where $A\in\mathbb{N}$ is the number of distinct classes. Algorithm \ref{alg1} shows the precise steps of the main procedure in which these classes are calculated. The two subroutines of this procedure that are used to find the equivalent elements with respect to $\equiv$ are given in Algorithm \ref{alg2} and Algorithm \ref{alg3}. Let $a\in\cb{1,\ldots,A}$ and let $z^{T,a}$ be a fixed representative element of $\mathcal{Z}_{T,a}(k)$. For each $y\in\mathcal{Z}_{T,a}(k)$, the relationship between $|F_{y}(k,\cdot)|$ and $|F_{z^{T,a}}(k,\cdot)|$ is formulated by Theorem \ref{simeqprop}: indeed, letting
\[
w(y,a)\coloneqq \sum_{t\in T}y_{t}-\sum_{t\in T}z^{T,a}_t,
\]
we have
\begin{align}
|F_{y}(k,\ell)|=\begin{cases}
|F_{z^{T,a}}(k,\ell-w(y,a))|&\text{if }\ell,\ell-w(y,a)\in\cb{\ell_0(k),\ldots,2^{k}-2},\\
0 &\text{else}.\end{cases}\label{shift}
\end{align}
Therefore, the values $|F_y(k,\cdot)|$ for a given $y\in\mathcal{Z}_{T,a}(k)$ can be calculated by a ``shift" of the values $|F_{z^{T,a}}(k,\cdot)|$ according to \eqref{shift}. This will be illustrated further in Example \ref{k=4ex} and Example \ref{k=5ex} in the next section.
Let us introduce some additional notation that will make the presentation of the examples simpler. First, without loss of generality, we assume that $z^{T,a}$ achieves the minimum possible sum $\sum_{t\in T}y_t$ among all $y\in \mathcal{Z}_{T,a}(k)$; hence $w(y,a)\geq 0$ for each $y\in\mathcal{Z}_{T,a}(k)$ and $w(z^{T,a},a)=0$. Let $\mathcal{W}(a)$ be the set of all possible values of $w(y,a)$, that is,
\[
\mathcal{W}(a)\coloneqq\cb{w(y,a)\mid y\in\mathcal{Z}_{T,a}(k)},
\]
which is a finite subset of $\mathbb{Z}_+=\cb{0,1,2,\ldots}$. For each $w\in\mathcal{W}(a)$, let us define
\[
\mathcal{Z}_{T,a,w}(k)=\cb{y\in\mathcal{Z}_{T,a}(k)\mid w(y,a)=w}.
\]
Hence, we may partition $\mathcal{Z}_{T,a}(k)$ as
\[
\mathcal{Z}_{T,a}(k)=\bigcup_{w\in\mathcal{W}(a)}\mathcal{Z}_{T,a,w}(k).
\]
We fix one representative $z^{T,a,w}\in \mathcal{Z}_{T,a,w}(k)$ for each $w\in\mathcal{W}(a)$ such that $z^{T,a,0}=z^{T,a}$. Hence, in view of Proposition \ref{PropBranch}, we may write
\[
|F(k,\ell)|=\sum_{a=1}^A \sum_{w\in\mathcal{W}(a)}|F_{z^{T,a,w}}(k,\ell)|\cdot|\mathcal{Z}_{T,a,w}(k)|
\]
for each $\ell\in\cb{\ell_0(k),\ldots,2^k-2}$. We will use these notation and reformulations in the examples of the next section.
\algnewcommand{\NULL}{\textsc{null}}
\algnewcommand{\algvar}{\texttt}
\begin{algorithm}
\caption{Main Algorithm}\label{alg1}
\begin{algorithmic}[1]
\State $\bar{\mathcal{D}}_{T} \gets \emptyset $
\For{$y$ \textbf{in} $\mathcal{Z}_{T}(k)$}
\State $\bar{\mathcal{D}}_{T} \gets \bar{\mathcal{D}}_{T}\cup \mathcal{D}_{T}(y) $
\EndFor
\State $asymmetricMatrices \gets \emptyset$
\State $A \gets 0$
\For{$y$ \textbf{in} $\mathcal{Z}_{T}(k)$} \Comment{Finding Equivalence Classes}
\State $symmetryMatrix \gets \algvar{generateSymmetryMatrix}(\bar{\mathcal{D}}_{T}, y, T)$
\State $a \gets \algvar{findEquivalenceClass}(symmetryMatrix, asymmetricMatrices, \bar{\mathcal{D}}_{T} )$
\If{$a == \NULL$}
\State $A\gets A+1$
\State $\mathcal{Z}_{T,A}(k)\gets \{y\}$
\State $z^{T,A}\gets y$
\State $asymmetricMatrices \gets asymmetricMatrices \cup \cb{symmetryMatrix}$
\Else
\State $\mathcal{Z}_{T,a}(k)\gets \mathcal{Z}_{T,a}(k)\cup\{y\}$
\EndIf
\EndFor
\For{$a=1:A$}
\State $\mathcal{W}(a) \gets \emptyset$
\For{$y$ \textbf{in} $\mathcal{Z}_{T,a}(k)$}
\State $w \gets \sum_{t\in T}y_t-\sum_{t\in T}z^{T,a}_{t}$
\If{$w$ \textbf{in} $\mathcal{W}(a)$}
\State $\mathcal{Z}_{T, a, w} \gets \mathcal{Z}_{T, a, w} \cup \{y\}$
\Else
\State $\mathcal{W}(a) \gets \mathcal{W}(a) \cup \{ w \}$
\State $\mathcal{Z}_{T, a, w} \gets \{ y \}$
\State $z^{T,a,w} \gets y$
\EndIf
\EndFor
\EndFor
\For{$a=1:A$}
\For{$\ell=1:(2^k-2)$}
\State Calculate $|F_{z^{T,a}}(k,\ell)|$ by no-good cuts.\Comment{see text}
\EndFor
\EndFor
\State $|F(k,\ell)|\gets 0$
\For{$a=1:A$}\Comment{Total Number of Solutions}
\For{$w$ \textbf{in} $\mathcal{W}(a)$ }
\For{$\ell=\ell_0(k):(2^k-2)$}
\If{$\ell\in\{\ell_0(k),\ldots,2^{k}-2\}\textbf{ and }\ell-w\in\cb{\ell_0(k),\ldots,2^k-2}$}
\State $|F(k,\ell)|\gets|F(k,\ell)|+|F_{z^{T,a}}(k,\ell-w)|\cdot|\mathcal{Z}_{T,a,w}(k)|$
\EndIf
\EndFor
\EndFor
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{algorithm}
\caption{\algvar{generateSymmetryMatrix}($\bar{\mathcal{D}}_{T}, y, T$)}\label{alg2}
\begin{algorithmic}[1]
\State $symmetryMatrix \gets zeros(|\bar{\mathcal{D}}_{T}|, |\mathbb{S}^T_k|\cdot|\mathbb{A}^T|)$
\State $A \gets 1$
\For{$(i,j)$ \textbf{in} $\bar{\mathcal{D}}_{T}$}
\State $B \gets 1$
\For{$\pi$ \textbf{in} $\mathbb{S}^T_k$}
\For{$\alpha$ \textbf{in} $\mathbb{A}^T$}
\For{$t$ \textbf{in} $T$}
\If{$(\alpha\circ t\circ \pi)_i == c$ \textbf{and} $(\alpha\circ t\circ \pi)_j == e$ \textbf{and} ${}^\alpha y^\pi_t == 1$}
\State $symmetryMatrix(A,B) = 1$
\EndIf
\EndFor
\State $B \gets B + 1$
\EndFor
\EndFor
\State $A \gets A + 1$
\EndFor
\State \Return symmetryMatrix
\end{algorithmic}
\end{algorithm}
\begin{algorithm}
\caption{\algvar{findEquivalenceClass}($symmetryMatrix$, $asymmetricMatrices, \bar{\mathcal{D}}_{T}$)}\label{alg3}
\begin{algorithmic}[1]
\State $A \gets 1$
\For{$M$ \textbf{in} $asymmetricMatrices$}\Comment{\parbox[t]{.5\linewidth} {Checking $\mathcal{D}_T(z)=\mathcal{D}_T({}^\alpha y^\pi)$ for any $\alpha$ and $\pi$ by dot product of their corresponding vectors. All the column vectors inside $symmetryMatrix$ represent $\mathcal{D}_T({}^\alpha y^\pi)$ for different choices of $(\alpha,\pi)$.}}
\If{$sum(M) == sum(symmetryMatrix)$ \ldots\\
\quad \quad \quad \ldots\; \textbf{and} ($sum(M)/|\bar{\mathcal{D}}_{T}| $ \textbf{exists in} $(M^T * symmetryMatrix)$)}
\State \Return $A$
\EndIf
\State $A \gets A+1$
\EndFor
\State \Return \NULL
\end{algorithmic}
\end{algorithm}
\subsection{Computational procedure with no-good cuts}\label{nogoodsec}
With the equivalence relation $\equiv$ introduced in Section \ref{equiv1}, we calculate the cardinalities $|F(k,\cdot)|$ through the equivalence classes $\mathcal{Z}_{T,a}(k)$, $a\in\cb{1,\ldots,A}$, and the corresponding $|F_{z^{T,a}}(k,\cdot)|$ values for a fixed representative $z^{T,a}$ of each class.
It remains to calculate the number of solutions for each equivalence class (as needed in line 21 of Algorithm \ref{alg1}). To that end, let $a\in\cb{1,\ldots,A}$ and $z=z^{T,a}$. In order to calculate $|F_z(k,\ell)|$, we adopt an optimization approach based on the so-called \emph{no-good cuts}. Given a set $F^\ast\subseteq F_z(k,\ell)$, we consider the following integer-linear optimization problem:
\begin{align}
\text{maximize}& \qquad 0\tag{$\mathscr{P}(z,F^\ast)$} \\
\text{subject to}& \qquad \sum_{t\in\mathbb{T}(k)\colon t_i=c, t_j=e}x_t\geq 1\quad \forall i\neq j\ \label{constraint1} \\
\emph{}& \qquad \sum_{t\in\mathbb{T}(k)}x_t=\ell \label{constraint2} \\
\emph{}&\qquad x_t = y_t \quad \forall t\in T\label{constraint5}\\
\emph{}& \qquad \sum_{t\in\mathbb{T}(k)\colon x^*_t=1} (1-x_t) + \sum_{t\in\mathbb{T}(k)\colon x^*_t=0}x_t \quad \geq 1 \quad \forall (x^*_t)_{t\in\mathbb{T}(k)} \in F^* \label{constraint3} \\
\emph{}& \qquad x_t\in \{0,1\} \quad \forall t\in\mathbb{T}(k) \label{constraint4}
\end{align}
Note that $(\mathscr{P}(z,F^\ast))$ is basically a feasibility problem since we maximize a constant function. Here, constraints \eqref{constraint1}, \eqref{constraint2}, \eqref{constraint4} are the relations in the definition of $F(k,\ell)$; constraint \eqref{constraint5} is the additional requirement in Definition~\ref{DefnOfFm}. Constraint \eqref{constraint3} is the collection of no-good cuts; it makes sure that the new solution $(x_t)_{t\in\mathbb{T}(k)}$ to be found is different from each of the solutions $(x_t^\ast)_{t\in\mathbb{T}(k)}$ that are already stored in $F^\ast$. Indeed, it is not difficult to check that the inequality in \eqref{constraint3} is equivalent to having
\[
\exists t\in\mathbb{T}(k)\colon x_t\neq x^\ast_t
\]
for each $(x^\ast_t)_{t\in\mathbb{T}(k)}\in F^\ast$.
While more general formulations of no-good cuts for continuous variables are nonconvex inequalities, the formulation we use here is for binary variables due to the nature of our problem and it yields a linear inequality. The reader is referred to \cite{nogoodcut} for a detailed treatment of no-good cuts.
The computational procedure is initialized by solving $(\mathscr{P}(z,F^{(1)}))$ with $F^{(1)}\coloneqq \emptyset$. If $F_z(k,\ell)=\emptyset$, then this problem terminates by infeasibility. Otherwise, it yields a $(k,\ell)$-labeling $(x_t^{(1)})_{t\in\mathbb{T}(k)}$ in $F_z(k,\ell)$ as an optimal solution. For each $u\in\mathbb{N}$, in the $(u+1)^{\text{st}}$ iteration, we call $(\mathscr{P}(z,F^{(u+1)}))$ with $F^{(u+1)}\coloneqq F^{(u)}\cup\{(x_t^{(u)})_{t\in\mathbb{T}(k)}\}$. If $|F_z(k,\ell)|\leq u$, then the problem terminates by infeasibility. Otherwise, it yields a $(k,\ell)$-labeling in $F_z(k,\ell)$ that is different from each of the points in $F^{(u)}$. The procedure terminates by infeasibility when $u=|F_z(k,\ell)|$ in which case we have $F_z(k,\ell) = F^{(u)}\cup\{(x_t^{(u)})_{t\in\mathbb{T}(k)}\}$. Hence, we find the cardinality of $F_z(k,\ell)$ by finding all of its elements.
It should be noted that, as the iteration number increases, the number of no-good cuts in the optimization problem increases and it takes more time to compute a new solution. Hence, when deciding on the choice of $T$, it is desirable to control $|F_z(k,\ell)|$ by a reasonable upper bound that depends on $T$. For this purpose, we use the simple upper bound
\[
|F_z(k,\ell)|\leq \binom{2^k-2-\bar{z}}{\ell-\bar{z}},
\]
where $\bar{z}\coloneqq\sum_{t\in T}z_t$; however, better upper bounds can be obtained by exploiting the structure of $T$.
To illustrate the computational procedure, in the ‘‘Total" rows of Table \ref{k=4table1} and Table \ref{k=5table1}, we present the $|F(k,\cdot)|$ values for $k=4$ and $k=5$, respectively. From these tables, it is notable that for fixed $k$, the quantity $|F(k,\ell)|$ first increases with $\ell$, then makes a maximum around $\ell=2^{k-1}-1$ and then decreases with $\ell$. Detailed explanations on these tables are provided in the following two examples.
\begin{example}\label{k=4ex}
Suppose that $k=4$. In this case, we have $\ell_0(k)=4$ and $2^k-2=14$. Hence, we consider the values $\ell\in\{4,\ldots,14\}$. As the branching set of index vectors, we select $T=\{(c,e,e,e),(c,e,c,e),(c,e,e,c),(c,e,c,e)\}$, which corresponds to the regions $A_1 \cap \bar{A}_2 \cap \bar{A}_3 \cap \bar{A}_4$, $A_1 \cap \bar{A}_2 \cap A_3 \cap \bar{A}_4$, $A_1 \cap \bar{A}_2 \cap \bar{A}_3 \cap A_4$, $A_1 \cap \bar{A}_2 \cap A_3 \cap A_4$. Hence, $|\mathcal{Z}_T(k)|=16$. By Algorithm~\ref{alg1}, we find out that $\equiv$ has $A=6$ equivalence classes.
In the header column of Table~\ref{k=4table1}, we use the format $(a,w,s)$ to report the equivalence class index $a$, the value $w\in\mathcal{W}(a)$ that is fixed for the corresponding row, and the cardinality $s=|\mathcal{Z}_{T,a,w}(k)|$ that corresponds to $(a,w)$. For instance, the row of $(5,1,2)$ gives the $|F_{y}(4,\cdot)|$ values for each of the two members of $\mathcal{Z}_{T,5,1}(4)$. The entries in the ‘‘Total" row represent the values $|F(k,\ell)|=\sum_{a=1}^{6}\sum_{w\in\mathcal{W}(a)}|F_{z^{T,a,w}}(k,\ell)|\cdot|\mathcal{Z}_{T,a,w}(k)|$ for all $\ell\in\cb{\ell_0(k),\ldots,2^k-2}$. In Table~\ref{k=4table1}, the rows corresponding to the same equivalence class are placed consecutively and shaded with the same color; we alternate between two colors as the class index $a$ changes. The representatives $z^{T,a,w}$, $a\in\cb{1,\ldots,A}$, $w\in\mathcal{W}(a)$, are listed in Table~\ref{k=4table2}. For instance, the line for $(5,1)$ states that
\[
z^{T,5,1}_{(c,e,e,e)}=1,\quad z^{T,5,1}_{(c,e,c,e)}=1, \quad z^{T,5,1}_{(c,e,e,c)}=1,\quad z^{T,5,1}_{(c,e,c,e)}=0.
\]
\end{example}
\begin{table}[t]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}
{|c|*{12}{p{0.8cm}|}}
\hline
\rowcolor{lightgray}
\rule{-3.5pt}{3.5ex}
\diagbox{\small{$(a,w,s)$}}{\small{$\ell$}} &$4$ &$5$ &$6$ &$7$ &$8$ &$9$ &$10$ &$11$ &$12$ &$13$ &$14$ \\
\cline{1-12}
$\cellcolor{lightgray}(1,0,2)$ &\cellcolor{lightred}$4$ &\cellcolor{lightred}$40$ &\cellcolor{lightred}$115$ &\cellcolor{lightred}$146$ &\cellcolor{lightred}$103$ &\cellcolor{lightred}$43$ &\cellcolor{lightred}$10$ &\cellcolor{lightred}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \cline{1-12}
$\cellcolor{lightgray}(2,0,2)$ &\cellcolor{lightgreen}$6$ &\cellcolor{lightgreen}$47$ &\cellcolor{lightgreen}$120$ &\cellcolor{lightgreen}$147$ &\cellcolor{lightgreen}$103$ &\cellcolor{lightgreen}$43$ &\cellcolor{lightgreen}$10$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$0$ &\cellcolor{lightgreen}$0$ &\cellcolor{lightgreen}$0$ \\ \cline{1-12}
$\cellcolor{lightgray}(3,0,4)$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$14$ &\cellcolor{lightred}$77$ &\cellcolor{lightred}$159$ &\cellcolor{lightred}$172$ &\cellcolor{lightred}$111$ &\cellcolor{lightred}$44$ &\cellcolor{lightred}$10$ &\cellcolor{lightred}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \cline{1-12}
$\cellcolor{lightgray}(4,0,1)$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$20$ &\cellcolor{lightgreen}$94$ &\cellcolor{lightgreen}$184$ &\cellcolor{lightgreen}$191$ &\cellcolor{lightgreen}$118$ &\cellcolor{lightgreen}$45$ &\cellcolor{lightgreen}$10$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$0$ &\cellcolor{lightgreen}$0$ \\ \cline{1-12}
$\cellcolor{lightgray}(5,0,1)$ &\cellcolor{lightred}$4$ &\cellcolor{lightred}$44$ &\cellcolor{lightred}$141$ &\cellcolor{lightred}$222$ &\cellcolor{lightred}$205$ &\cellcolor{lightred}$120$ &\cellcolor{lightred}$45$ &\cellcolor{lightred}$10$ &\cellcolor{lightred}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \cline{1-12}
$\cellcolor{lightgray}(5,1,2)$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$4$ &\cellcolor{lightred}$44$ &\cellcolor{lightred}$141$ &\cellcolor{lightred}$222$ &\cellcolor{lightred}$205$ &\cellcolor{lightred}$120$ &\cellcolor{lightred}$45$ &\cellcolor{lightred}$10$ &\cellcolor{lightred}$1$ &\cellcolor{lightred}$0$ \\ \cline{1-12}
$\cellcolor{lightgray}(5,2,1)$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$4$ &\cellcolor{lightred}$44$ &\cellcolor{lightred}$141$ &\cellcolor{lightred}$222$ &\cellcolor{lightred}$205$ &\cellcolor{lightred}$120$ &\cellcolor{lightred}$45$ &\cellcolor{lightred}$10$ &\cellcolor{lightred}$1$ \\ \cline{1-12}
$\cellcolor{lightgray}(6,0,2)$ &\cellcolor{lightgreen}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$30$ &\cellcolor{lightgreen}$117$ &\cellcolor{lightgreen}$203$ &\cellcolor{lightgreen}$198$ &\cellcolor{lightgreen}$119$ &\cellcolor{lightgreen}$45$ &\cellcolor{lightgreen}$10$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$0$ \\ \cline{1-12}
\hline
\rowcolor{lightgray}
Total &$25$ &$304$ &$1165$ &$2188$ &$2487$ &$1882$ &$989$ &$364$ &$91$ &$14$ &$1$ \\ \cline{1-12}
\hline
\end{tabular}}
\caption{$|F_{z^{T,a,w}}(4, \ell)|$ values for $\ell_0(k) \leq \ell\leq 2^k-2 $ in Example \ref{k=4ex}} \label{k=4table1}
\end{table}
\begin{table}[t]
\resizebox{\textwidth}{!}{%
\begin{tabular}{|>{\columncolor{lightgray}}P{2.2cm}|P{3.2cm}|P{3.2cm}|P{3.2cm}|P{3.2cm}|}
\hline
\rowcolor{lightgray}
\rule{-4pt}{1ex}
\diagbox{\small{$(a,w)$}}{\small{$T$}} &$ A_1 \cap \bar{A}_2 \cap \bar{A}_3 \cap \bar{A}_4$
&$A_1 \cap \bar{A}_2 \cap A_3 \cap \bar{A}_4$
&$A_1 \cap \bar{A}_2 \cap \bar{A}_3 \cap A_4$
&$A_1 \cap \bar{A}_2 \cap A_3 \cap A_4$ \\
\hline
$(1,0)$ & \cellcolor{lightgreen}$1$ & \cellcolor{lightred}$0$ &\cellcolor{lightred} $0$ & \cellcolor{lightred}$0$ \\
\hline
$(2,0)$ & \cellcolor{lightred}$0$ &\cellcolor{lightgreen} $1$ &\cellcolor{lightred} $0$ & \cellcolor{lightred}$0$ \\
\hline
$(3,0)$ & \cellcolor{lightgreen}$1$ &\cellcolor{lightgreen} $1$ &\cellcolor{lightred} $0$ &\cellcolor{lightred} $0$ \\
\hline
$(4,0)$ & \cellcolor{lightgreen}$1$ & \cellcolor{lightred}$0$ & \cellcolor{lightred}$0$ &\cellcolor{lightgreen} $0$ \\
\hline
$(5,0)$ &\cellcolor{lightred} $1$ &\cellcolor{lightgreen} $0$ &\cellcolor{lightgreen} $0$ & \cellcolor{lightred}$0$ \\
\hline
$(5,1)$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen} $1$ & \cellcolor{lightgreen}$1$ & \cellcolor{lightred}$0$ \\
\hline
$(5,2)$ &\cellcolor{lightgreen} $1$ & \cellcolor{lightgreen}$1$ & \cellcolor{lightgreen}$1$ & \cellcolor{lightgreen}$1$ \\
\hline
$(6,0)$ &\cellcolor{lightgreen} $1$ &\cellcolor{lightgreen} $1$ & \cellcolor{lightred}$0$ & \cellcolor{lightgreen}$1$ \\
\hline
\end{tabular}
}
\caption{Definitions of the representatives $z^{T,a,w}$ in Example \ref{k=4ex}}\label{k=4table2}
\end{table}
\begin{example}\label{k=5ex}
Suppose that $k=5$. In this case, we have $\ell_0(k)=4$ and $2^k-2=30$. As the branching set of index vectors, we fix $T=\cb{t^1,\ldots,t^{15}}$ and the precise definitions of $t^1,\ldots,t^{15}$ are given in Table \ref{k=5table3}. For instance, $t^7=(c,e,c,e,e)$. We have $|\mathcal{Z}_T(k)|=2^{15}$ and there are $A=28$ equivalence classes of $\equiv$. Table \ref{k=5table1} provides the $|F_{z^{T,a,w}}(5,\ell)|$ values and it is presented in two pages. Its structure is the same as that of Table \ref{k=4table1}. The definitions of the representatives $z^{T,a,w}$, $a\in\cb{1,\ldots,28}$, $w\in\mathcal{W}(a)$, are given in Table \ref{k=5table2}.
Table \ref{k=5table1} illustrates the tremendous reduction in computational effort provided by the symmetries encoded in $\equiv$. For each value of $a$, we only compute the values $|F_{z^{T,a,w}}(k,\cdot)|$ for $w=0$, that is, the values in the first row for $a$; then, for $w>0$, the rows are obtained by simple shifts of the row $w=0$. The row of $(a,w)$ is repeated $s=|\mathcal{Z}_{T,a,w}(k)|$ times in the ultimate value of $|F(k,\ell)|$. For instance, in a brute-force calculation without using symmetries, the values in the row for $(27,4)$ would have been calculated $s=4055$ times, which is avoided due to the structure provided by $\equiv$.
\end{example}
\section{Application to reliability theory}\label{reliability}
In this section, we illustrate the use of constructive covers in reliability theory.
In the setting of \cite{barlow}, let us consider a \emph{multi-state coherent system} with $n\in\mathbb{N}$ components forming a set $\mathcal{N}$. Without loss of generality, let us write $\mathcal{N}=\cb{1,\ldots,n}$. Each component $p\in\mathcal{N}$ has a state variable $z_p$ taking values in the set $\mathcal{S}\coloneqq\cb{0,1,\ldots,s}$, where $s\in\mathbb{N}$ is fixed for all components. Then, the states of all components can be expressed as a vector $z=(z_1,\ldots,z_n)\in\mathcal{S}^n$. In classical reliability theory, the typical systems are binary, that is, one takes $\mathcal{S}=\cb{0,1}$. In such systems, the state $1$ corresponds to the ``functioning" state and $0$ corresponds to the ``failure" state. Hence, the general case with $s+1$ states can be used to model varying levels of perfection for how well the components operate.
The structure of the system is encoded by a function $\phi\colon\mathcal{S}^n\to\mathcal{S}$ such that $\phi(z)=\phi(z_1,\ldots,z_n)$ gives the state of the overall system when component $p$ is at state $z_p$ for each $p\in\mathcal{N}$. We call $\phi$ the \emph{structure function} of the system. For instance, a \emph{parallel} system is defined as a system whose structure function is given by $\phi(z)=\max\cb{z_1,\ldots,z_n}$ for each $z\in\mathcal{S}^n$. Similarly, a \emph{series} system has the structure function $\phi(z)=\min\cb{z_1,\ldots,z_n}$ for each $z\in\mathcal{S}^n$. In general, a \emph{coherent system} is a system which is built as a nested structure of parallel and series systems. More precisely, let $k\in\cb{1,\ldots,n}$ and consider distinct sets $P_1,\ldots,P_k\subseteq\mathcal{N}$ satisfying the following properties: the sets are not strict subsets of each other, and $\bigcup_{i=1}^k P_i=\mathcal{N}$. Then, the structure function of a coherent system described with these sets is defined by
\begin{equation}\label{phipath}
\phi(z)=\max_{i\in\cb{1,\ldots,k}}\min_{p\in P_i}z_p
\end{equation}
for each $z\in\mathcal{S}^n$. The sets $P_1,\ldots,P_k$ are called the \emph{minimal path sets} of the system.
Note that the above requirements for the minimal path sets of the system are precisely the conditions in the definition of constructive ordered $k$-cover (Definition~\ref{kcover}) except that the order of the sets is not important. Hence, in view of Remark~\ref{unordered}, every constructive unordered $k$-cover of $\mathcal{N}$ corresponds to a system design with $k$ minimal path sets and $n$ components, and vice versa. By Corollary~\ref{ell0forcounting}, the number of such system designs is given by
\[
\frac{|C(\mathcal{N},k)|}{k!}=\sum_{\ell=\ell_0}^{(2^k-2)\wedge n}\frac{\ell!}{k!}\tilde{s}(n,\ell) |F(k,\ell)|.
\]
Given a coherent system with minimal path sets $P_1,\ldots,P_k$, by following a certain procedure, one can construct the so-called \emph{minimal cut sets} $C_1,\ldots,C_r\subseteq\mathcal{N}$ with $r\in\cb{1,\ldots,n}$ satisfying the following properties: the sets are not strict subsets of each other, $\bigcup_{j=1}^r C_j=\mathcal{N}$ and each minimal cut set has a nonempty intersection with each minimal path set. It can be shown that \cite[Proposition~1.1]{barlow} the structure function of the system can also be computed by the formula
\begin{equation}\label{phicut}
\phi(z)=\min_{j\in\cb{1,\ldots,r}}\max_{p\in C_j}z_p
\end{equation}
for each $z\in\mathcal{S}^n$.
\begin{landscape}
\begin{table}[b]
\centering
\begin{adjustbox}{width=1.4\textwidth,totalheight=0.92\textheight-2\baselineskip}
\begin{tabular}{|c|*{28}{p{1cm}|}}\hline
\rowcolor{lightgray}
\diagbox{$(a,w,s)$}{$\ell$} &$4$ & $ 5$ & $ 6$ & $ 7$ & $8$& $9$ & $ 10$ & $11$ & $12$ & $13$
&$14$ & $ 15$ & $ 16$ & $ 17$ & $18$& $19$ & $ 20$ & $21$ & $22$ & $23$
&$24$ & $ 25$ & $ 26$ & $ 27$ & $28$& $29$ & $ 30$ \\
\hline
\cellcolor{lightgray} $(1,0,1)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $38$& \cellcolor{lightred} $615$& \cellcolor{lightred} $2245$& \cellcolor{lightred} $3900$& \cellcolor{lightred} $4055$& \cellcolor{lightred} $2798$& \cellcolor{lightred} $1345$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(2,0,5)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $11$& \cellcolor{lightgreen} $193$& \cellcolor{lightgreen} $1105$& \cellcolor{lightgreen} $2889$& \cellcolor{lightgreen} $4350$& \cellcolor{lightgreen} $4235$& \cellcolor{lightgreen} $2838$& \cellcolor{lightgreen} $1349$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(3,0,10)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $32$& \cellcolor{lightred} $324$& \cellcolor{lightred} $1397$& \cellcolor{lightred} $3229$& \cellcolor{lightred} $4578$& \cellcolor{lightred} $4325$& \cellcolor{lightred} $2858$& \cellcolor{lightred} $1351$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(4,0,10)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $48$& \cellcolor{lightgreen} $441$& \cellcolor{lightgreen} $1685$& \cellcolor{lightgreen} $3570$& \cellcolor{lightgreen} $4806$& \cellcolor{lightgreen} $4415$& \cellcolor{lightgreen} $2878$& \cellcolor{lightgreen} $1353$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(4,1,10)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $48$& \cellcolor{lightgreen} $441$& \cellcolor{lightgreen} $1685$& \cellcolor{lightgreen} $3570$& \cellcolor{lightgreen} $4806$& \cellcolor{lightgreen} $4415$& \cellcolor{lightgreen} $2878$& \cellcolor{lightgreen} $1353$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(5,0,20)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $39$& \cellcolor{lightred} $378$& \cellcolor{lightred} $1538$& \cellcolor{lightred} $3399$& \cellcolor{lightred} $4692$& \cellcolor{lightred} $4370$& \cellcolor{lightred} $2868$& \cellcolor{lightred} $1352$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(6,0,30)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $2$& \cellcolor{lightgreen} $85$& \cellcolor{lightgreen} $607$& \cellcolor{lightgreen} $2009$& \cellcolor{lightgreen} $3923$& \cellcolor{lightgreen} $5036$& \cellcolor{lightgreen} $4505$& \cellcolor{lightgreen} $2898$& \cellcolor{lightgreen} $1355$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(7,0,30)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $78$& \cellcolor{lightred} $596$& \cellcolor{lightred} $2003$& \cellcolor{lightred} $3922$& \cellcolor{lightred} $5036$& \cellcolor{lightred} $4505$& \cellcolor{lightred} $2898$& \cellcolor{lightred} $1355$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(7,1,30)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $78$& \cellcolor{lightred} $596$& \cellcolor{lightred} $2003$& \cellcolor{lightred} $3922$& \cellcolor{lightred} $5036$& \cellcolor{lightred} $4505$& \cellcolor{lightred} $2898$& \cellcolor{lightred} $1355$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(8,0,15)$& \cellcolor{lightgreen} $2$& \cellcolor{lightgreen} $34$& \cellcolor{lightgreen} $244$& \cellcolor{lightgreen} $990$& \cellcolor{lightgreen} $2549$& \cellcolor{lightgreen} $4405$& \cellcolor{lightgreen} $5313$& \cellcolor{lightgreen} $4605$& \cellcolor{lightgreen} $2919$& \cellcolor{lightgreen} $1357$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(9,0,20)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $3$& \cellcolor{lightred} $126$& \cellcolor{lightred} $788$& \cellcolor{lightred} $2355$& \cellcolor{lightred} $4288$& \cellcolor{lightred} $5268$& \cellcolor{lightred} $4595$& \cellcolor{lightred} $2918$& \cellcolor{lightred} $1357$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(9,1,90)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $3$& \cellcolor{lightred} $126$& \cellcolor{lightred} $788$& \cellcolor{lightred} $2355$& \cellcolor{lightred} $4288$& \cellcolor{lightred} $5268$& \cellcolor{lightred} $4595$& \cellcolor{lightred} $2918$& \cellcolor{lightred} $1357$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(9,2,60)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $3$& \cellcolor{lightred} $126$& \cellcolor{lightred} $788$& \cellcolor{lightred} $2355$& \cellcolor{lightred} $4288$& \cellcolor{lightred} $5268$& \cellcolor{lightred} $4595$& \cellcolor{lightred} $2918$& \cellcolor{lightred} $1357$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(9,3,10)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $3$& \cellcolor{lightred} $126$& \cellcolor{lightred} $788$& \cellcolor{lightred} $2355$& \cellcolor{lightred} $4288$& \cellcolor{lightred} $5268$& \cellcolor{lightred} $4595$& \cellcolor{lightred} $2918$& \cellcolor{lightred} $1357$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(10,0,120)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $2$& \cellcolor{lightgreen} $102$& \cellcolor{lightgreen} $692$& \cellcolor{lightgreen} $2179$& \cellcolor{lightgreen} $4105$& \cellcolor{lightgreen} $5152$& \cellcolor{lightgreen} $4550$& \cellcolor{lightgreen} $2908$& \cellcolor{lightgreen} $1356$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(10,1,60)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $2$& \cellcolor{lightgreen} $102$& \cellcolor{lightgreen} $692$& \cellcolor{lightgreen} $2179$& \cellcolor{lightgreen} $4105$& \cellcolor{lightgreen} $5152$& \cellcolor{lightgreen} $4550$& \cellcolor{lightgreen} $2908$& \cellcolor{lightgreen} $1356$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(11,0,30)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $2$& \cellcolor{lightred} $41$& \cellcolor{lightred} $303$& \cellcolor{lightred} $1191$& \cellcolor{lightred} $2905$& \cellcolor{lightred} $4772$& \cellcolor{lightred} $5545$& \cellcolor{lightred} $4695$& \cellcolor{lightred} $2939$& \cellcolor{lightred} $1359$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(11,1,30)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $2$& \cellcolor{lightred} $41$& \cellcolor{lightred} $303$& \cellcolor{lightred} $1191$& \cellcolor{lightred} $2905$& \cellcolor{lightred} $4772$& \cellcolor{lightred} $5545$& \cellcolor{lightred} $4695$& \cellcolor{lightred} $2939$& \cellcolor{lightred} $1359$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(12,0,60)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $2$& \cellcolor{lightgreen} $37$& \cellcolor{lightgreen} $270$& \cellcolor{lightgreen} $1085$& \cellcolor{lightgreen} $2724$& \cellcolor{lightgreen} $4588$& \cellcolor{lightgreen} $5429$& \cellcolor{lightgreen} $4650$& \cellcolor{lightgreen} $2929$& \cellcolor{lightgreen} $1358$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(13,0,60)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $35$& \cellcolor{lightred} $290$& \cellcolor{lightred} $1178$& \cellcolor{lightred} $2899$& \cellcolor{lightred} $4771$& \cellcolor{lightred} $5545$& \cellcolor{lightred} $4695$& \cellcolor{lightred} $2939$& \cellcolor{lightred} $1359$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(13,1,120)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $35$& \cellcolor{lightred} $290$& \cellcolor{lightred} $1178$& \cellcolor{lightred} $2899$& \cellcolor{lightred} $4771$& \cellcolor{lightred} $5545$& \cellcolor{lightred} $4695$& \cellcolor{lightred} $2939$& \cellcolor{lightred} $1359$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(13,2,60)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $35$& \cellcolor{lightred} $290$& \cellcolor{lightred} $1178$& \cellcolor{lightred} $2899$& \cellcolor{lightred} $4771$& \cellcolor{lightred} $5545$& \cellcolor{lightred} $4695$& \cellcolor{lightred} $2939$& \cellcolor{lightred} $1359$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(14,0,15)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $2$& \cellcolor{lightgreen} $50$& \cellcolor{lightgreen} $375$& \cellcolor{lightgreen} $1417$& \cellcolor{lightgreen} $3283$& \cellcolor{lightgreen} $5149$& \cellcolor{lightgreen} $5779$& \cellcolor{lightgreen} $4785$& \cellcolor{lightgreen} $2959$& \cellcolor{lightgreen} $1361$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(15,0,20)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $22$& \cellcolor{lightred} $230$& \cellcolor{lightred} $1033$& \cellcolor{lightred} $2688$& \cellcolor{lightred} $4575$& \cellcolor{lightred} $5427$& \cellcolor{lightred} $4650$& \cellcolor{lightred} $2929$& \cellcolor{lightred} $1358$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(15,1,20)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $22$& \cellcolor{lightred} $230$& \cellcolor{lightred} $1033$& \cellcolor{lightred} $2688$& \cellcolor{lightred} $4575$& \cellcolor{lightred} $5427$& \cellcolor{lightred} $4650$& \cellcolor{lightred} $2929$& \cellcolor{lightred} $1358$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(16,0,60)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $2$& \cellcolor{lightgreen} $41$& \cellcolor{lightgreen} $305$& \cellcolor{lightgreen} $1198$& \cellcolor{lightgreen} $2914$& \cellcolor{lightgreen} $4777$& \cellcolor{lightgreen} $5546$& \cellcolor{lightgreen} $4695$& \cellcolor{lightgreen} $2939$& \cellcolor{lightgreen} $1359$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(16,1,180)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $2$& \cellcolor{lightgreen} $41$& \cellcolor{lightgreen} $305$& \cellcolor{lightgreen} $1198$& \cellcolor{lightgreen} $2914$& \cellcolor{lightgreen} $4777$& \cellcolor{lightgreen} $5546$& \cellcolor{lightgreen} $4695$& \cellcolor{lightgreen} $2939$& \cellcolor{lightgreen} $1359$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(16,2,60)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $2$& \cellcolor{lightgreen} $41$& \cellcolor{lightgreen} $305$& \cellcolor{lightgreen} $1198$& \cellcolor{lightgreen} $2914$& \cellcolor{lightgreen} $4777$& \cellcolor{lightgreen} $5546$& \cellcolor{lightgreen} $4695$& \cellcolor{lightgreen} $2939$& \cellcolor{lightgreen} $1359$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(17,0,30)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $7$& \cellcolor{lightred} $88$& \cellcolor{lightred} $491$& \cellcolor{lightred} $1608$& \cellcolor{lightred} $3472$& \cellcolor{lightred} $5266$& \cellcolor{lightred} $5824$& \cellcolor{lightred} $4795$& \cellcolor{lightred} $2960$& \cellcolor{lightred} $1361$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(17,1,30)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $7$& \cellcolor{lightred} $88$& \cellcolor{lightred} $491$& \cellcolor{lightred} $1608$& \cellcolor{lightred} $3472$& \cellcolor{lightred} $5266$& \cellcolor{lightred} $5824$& \cellcolor{lightred} $4795$& \cellcolor{lightred} $2960$& \cellcolor{lightred} $1361$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(18,0,40)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $3$& \cellcolor{lightgreen} $61$& \cellcolor{lightgreen} $410$& \cellcolor{lightgreen} $1468$& \cellcolor{lightgreen} $3321$& \cellcolor{lightgreen} $5163$& \cellcolor{lightgreen} $5781$& \cellcolor{lightgreen} $4785$& \cellcolor{lightgreen} $2959$& \cellcolor{lightgreen} $1361$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(18,1,360)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $3$& \cellcolor{lightgreen} $61$& \cellcolor{lightgreen} $410$& \cellcolor{lightgreen} $1468$& \cellcolor{lightgreen} $3321$& \cellcolor{lightgreen} $5163$& \cellcolor{lightgreen} $5781$& \cellcolor{lightgreen} $4785$& \cellcolor{lightgreen} $2959$& \cellcolor{lightgreen} $1361$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(18,2,670)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $3$& \cellcolor{lightgreen} $61$& \cellcolor{lightgreen} $410$& \cellcolor{lightgreen} $1468$& \cellcolor{lightgreen} $3321$& \cellcolor{lightgreen} $5163$& \cellcolor{lightgreen} $5781$& \cellcolor{lightgreen} $4785$& \cellcolor{lightgreen} $2959$& \cellcolor{lightgreen} $1361$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(18,3,540)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $3$& \cellcolor{lightgreen} $61$& \cellcolor{lightgreen} $410$& \cellcolor{lightgreen} $1468$& \cellcolor{lightgreen} $3321$& \cellcolor{lightgreen} $5163$& \cellcolor{lightgreen} $5781$& \cellcolor{lightgreen} $4785$& \cellcolor{lightgreen} $2959$& \cellcolor{lightgreen} $1361$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(18,4,225)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $3$& \cellcolor{lightgreen} $61$& \cellcolor{lightgreen} $410$& \cellcolor{lightgreen} $1468$& \cellcolor{lightgreen} $3321$& \cellcolor{lightgreen} $5163$& \cellcolor{lightgreen} $5781$& \cellcolor{lightgreen} $4785$& \cellcolor{lightgreen} $2959$& \cellcolor{lightgreen} $1361$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(18,5,50)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $3$& \cellcolor{lightgreen} $61$& \cellcolor{lightgreen} $410$& \cellcolor{lightgreen} $1468$& \cellcolor{lightgreen} $3321$& \cellcolor{lightgreen} $5163$& \cellcolor{lightgreen} $5781$& \cellcolor{lightgreen} $4785$& \cellcolor{lightgreen} $2959$& \cellcolor{lightgreen} $1361$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(18,6,5)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $3$& \cellcolor{lightgreen} $61$& \cellcolor{lightgreen} $410$& \cellcolor{lightgreen} $1468$& \cellcolor{lightgreen} $3321$& \cellcolor{lightgreen} $5163$& \cellcolor{lightgreen} $5781$& \cellcolor{lightgreen} $4785$& \cellcolor{lightgreen} $2959$& \cellcolor{lightgreen} $1361$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(19,0,360)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $2$& \cellcolor{lightred} $48$& \cellcolor{lightred} $350$& \cellcolor{lightred} $1323$& \cellcolor{lightred} $3110$& \cellcolor{lightred} $4967$& \cellcolor{lightred} $5663$& \cellcolor{lightred} $4740$& \cellcolor{lightred} $2949$& \cellcolor{lightred} $1360$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(19,1,660)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $2$& \cellcolor{lightred} $48$& \cellcolor{lightred} $350$& \cellcolor{lightred} $1323$& \cellcolor{lightred} $3110$& \cellcolor{lightred} $4967$& \cellcolor{lightred} $5663$& \cellcolor{lightred} $4740$& \cellcolor{lightred} $2949$& \cellcolor{lightred} $1360$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(19,2,360)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $2$& \cellcolor{lightred} $48$& \cellcolor{lightred} $350$& \cellcolor{lightred} $1323$& \cellcolor{lightred} $3110$& \cellcolor{lightred} $4967$& \cellcolor{lightred} $5663$& \cellcolor{lightred} $4740$& \cellcolor{lightred} $2949$& \cellcolor{lightred} $1360$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(19,3,60)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $2$& \cellcolor{lightred} $48$& \cellcolor{lightred} $350$& \cellcolor{lightred} $1323$& \cellcolor{lightred} $3110$& \cellcolor{lightred} $4967$& \cellcolor{lightred} $5663$& \cellcolor{lightred} $4740$& \cellcolor{lightred} $2949$& \cellcolor{lightred} $1360$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(20,0,20)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $16$& \cellcolor{lightgreen} $160$& \cellcolor{lightgreen} $740$& \cellcolor{lightgreen} $2100$& \cellcolor{lightgreen} $4088$& \cellcolor{lightgreen} $5776$& \cellcolor{lightgreen} $6105$& \cellcolor{lightgreen} $4895$& \cellcolor{lightgreen} $2981$& \cellcolor{lightgreen} $1363$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline
\end{tabular}
\end{adjustbox}
\caption{$|F_{z^{T,a,w}}(5, \ell)|$ values for $\ell_0(k)\leq \ell\leq 2^k-2$ in Example \ref{k=5ex}}
\end{table}
\end{landscape}
\begin{landscape}
\begin{table}[b]
\centering
\begin{adjustbox}{width=1.4\textwidth,totalheight=0.92\textheight-2\baselineskip}
\begin{tabular}{|c|*{28}{p{1cm}|}}\hline
\rowcolor{lightgray}
\diagbox{$(a,w,s)$}{$\ell$} &$4$ & $ 5$ & $ 6$ & $ 7$ & $8$& $9$ & $ 10$ & $11$ & $12$ & $13$
&$14$ & $ 15$ & $ 16$ & $ 17$ & $18$& $19$ & $ 20$ & $21$ & $22$ & $23$
&$24$ & $ 25$ & $ 26$ & $ 27$ & $28$& $29$ & $ 30$ \\
\hline
\cellcolor{lightgray} $(20,1,90)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $16$& \cellcolor{lightgreen} $160$& \cellcolor{lightgreen} $740$& \cellcolor{lightgreen} $2100$& \cellcolor{lightgreen} $4088$& \cellcolor{lightgreen} $5776$& \cellcolor{lightgreen} $6105$& \cellcolor{lightgreen} $4895$& \cellcolor{lightgreen} $2981$& \cellcolor{lightgreen} $1363$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(20,2,60)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $16$& \cellcolor{lightgreen} $160$& \cellcolor{lightgreen} $740$& \cellcolor{lightgreen} $2100$& \cellcolor{lightgreen} $4088$& \cellcolor{lightgreen} $5776$& \cellcolor{lightgreen} $6105$& \cellcolor{lightgreen} $4895$& \cellcolor{lightgreen} $2981$& \cellcolor{lightgreen} $1363$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(20,3,10)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $16$& \cellcolor{lightgreen} $160$& \cellcolor{lightgreen} $740$& \cellcolor{lightgreen} $2100$& \cellcolor{lightgreen} $4088$& \cellcolor{lightgreen} $5776$& \cellcolor{lightgreen} $6105$& \cellcolor{lightgreen} $4895$& \cellcolor{lightgreen} $2981$& \cellcolor{lightgreen} $1363$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(21,0,120)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $7$& \cellcolor{lightred} $97$& \cellcolor{lightred} $545$& \cellcolor{lightred} $1749$& \cellcolor{lightred} $3682$& \cellcolor{lightred} $5462$& \cellcolor{lightred} $5942$& \cellcolor{lightred} $4840$& \cellcolor{lightred} $2970$& \cellcolor{lightred} $1362$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(21,1,60)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $7$& \cellcolor{lightred} $97$& \cellcolor{lightred} $545$& \cellcolor{lightred} $1749$& \cellcolor{lightred} $3682$& \cellcolor{lightred} $5462$& \cellcolor{lightred} $5942$& \cellcolor{lightred} $4840$& \cellcolor{lightred} $2970$& \cellcolor{lightred} $1362$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(22,0,30)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $3$& \cellcolor{lightgreen} $40$& \cellcolor{lightgreen} $245$& \cellcolor{lightgreen} $915$& \cellcolor{lightgreen} $2331$& \cellcolor{lightgreen} $4291$& \cellcolor{lightgreen} $5895$& \cellcolor{lightgreen} $6150$& \cellcolor{lightgreen} $4905$& \cellcolor{lightgreen} $2982$& \cellcolor{lightgreen} $1363$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(22,1,210)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $3$& \cellcolor{lightgreen} $40$& \cellcolor{lightgreen} $245$& \cellcolor{lightgreen} $915$& \cellcolor{lightgreen} $2331$& \cellcolor{lightgreen} $4291$& \cellcolor{lightgreen} $5895$& \cellcolor{lightgreen} $6150$& \cellcolor{lightgreen} $4905$& \cellcolor{lightgreen} $2982$& \cellcolor{lightgreen} $1363$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(22,2,330)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $3$& \cellcolor{lightgreen} $40$& \cellcolor{lightgreen} $245$& \cellcolor{lightgreen} $915$& \cellcolor{lightgreen} $2331$& \cellcolor{lightgreen} $4291$& \cellcolor{lightgreen} $5895$& \cellcolor{lightgreen} $6150$& \cellcolor{lightgreen} $4905$& \cellcolor{lightgreen} $2982$& \cellcolor{lightgreen} $1363$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(22,3,180)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $3$& \cellcolor{lightgreen} $40$& \cellcolor{lightgreen} $245$& \cellcolor{lightgreen} $915$& \cellcolor{lightgreen} $2331$& \cellcolor{lightgreen} $4291$& \cellcolor{lightgreen} $5895$& \cellcolor{lightgreen} $6150$& \cellcolor{lightgreen} $4905$& \cellcolor{lightgreen} $2982$& \cellcolor{lightgreen} $1363$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(22,4,30)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $3$& \cellcolor{lightgreen} $40$& \cellcolor{lightgreen} $245$& \cellcolor{lightgreen} $915$& \cellcolor{lightgreen} $2331$& \cellcolor{lightgreen} $4291$& \cellcolor{lightgreen} $5895$& \cellcolor{lightgreen} $6150$& \cellcolor{lightgreen} $4905$& \cellcolor{lightgreen} $2982$& \cellcolor{lightgreen} $1363$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(23,0,120)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $20$& \cellcolor{lightred} $157$& \cellcolor{lightred} $690$& \cellcolor{lightred} $1960$& \cellcolor{lightred} $3878$& \cellcolor{lightred} $5580$& \cellcolor{lightred} $5987$& \cellcolor{lightred} $4850$& \cellcolor{lightred} $2971$& \cellcolor{lightred} $1362$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(23,1,180)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $20$& \cellcolor{lightred} $157$& \cellcolor{lightred} $690$& \cellcolor{lightred} $1960$& \cellcolor{lightred} $3878$& \cellcolor{lightred} $5580$& \cellcolor{lightred} $5987$& \cellcolor{lightred} $4850$& \cellcolor{lightred} $2971$& \cellcolor{lightred} $1362$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(23,2,60)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $20$& \cellcolor{lightred} $157$& \cellcolor{lightred} $690$& \cellcolor{lightred} $1960$& \cellcolor{lightred} $3878$& \cellcolor{lightred} $5580$& \cellcolor{lightred} $5987$& \cellcolor{lightred} $4850$& \cellcolor{lightred} $2971$& \cellcolor{lightred} $1362$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(24,0,20)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $2$& \cellcolor{lightgreen} $30$& \cellcolor{lightgreen} $200$& \cellcolor{lightgreen} $795$& \cellcolor{lightgreen} $2121$& \cellcolor{lightgreen} $4039$& \cellcolor{lightgreen} $5685$& \cellcolor{lightgreen} $6030$& \cellcolor{lightgreen} $4860$& \cellcolor{lightgreen} $2972$& \cellcolor{lightgreen} $1362$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(24,1,40)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $2$& \cellcolor{lightgreen} $30$& \cellcolor{lightgreen} $200$& \cellcolor{lightgreen} $795$& \cellcolor{lightgreen} $2121$& \cellcolor{lightgreen} $4039$& \cellcolor{lightgreen} $5685$& \cellcolor{lightgreen} $6030$& \cellcolor{lightgreen} $4860$& \cellcolor{lightgreen} $2972$& \cellcolor{lightgreen} $1362$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(24,2,20)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $2$& \cellcolor{lightgreen} $30$& \cellcolor{lightgreen} $200$& \cellcolor{lightgreen} $795$& \cellcolor{lightgreen} $2121$& \cellcolor{lightgreen} $4039$& \cellcolor{lightgreen} $5685$& \cellcolor{lightgreen} $6030$& \cellcolor{lightgreen} $4860$& \cellcolor{lightgreen} $2972$& \cellcolor{lightgreen} $1362$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(25,0,120)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $23$& \cellcolor{lightred} $181$& \cellcolor{lightred} $775$& \cellcolor{lightred} $2135$& \cellcolor{lightred} $4109$& \cellcolor{lightred} $5783$& \cellcolor{lightred} $6106$& \cellcolor{lightred} $4895$& \cellcolor{lightred} $2981$& \cellcolor{lightred} $1363$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(25,1,660)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $23$& \cellcolor{lightred} $181$& \cellcolor{lightred} $775$& \cellcolor{lightred} $2135$& \cellcolor{lightred} $4109$& \cellcolor{lightred} $5783$& \cellcolor{lightred} $6106$& \cellcolor{lightred} $4895$& \cellcolor{lightred} $2981$& \cellcolor{lightred} $1363$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(25,2,780)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $23$& \cellcolor{lightred} $181$& \cellcolor{lightred} $775$& \cellcolor{lightred} $2135$& \cellcolor{lightred} $4109$& \cellcolor{lightred} $5783$& \cellcolor{lightred} $6106$& \cellcolor{lightred} $4895$& \cellcolor{lightred} $2981$& \cellcolor{lightred} $1363$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(25,3,360)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $23$& \cellcolor{lightred} $181$& \cellcolor{lightred} $775$& \cellcolor{lightred} $2135$& \cellcolor{lightred} $4109$& \cellcolor{lightred} $5783$& \cellcolor{lightred} $6106$& \cellcolor{lightred} $4895$& \cellcolor{lightred} $2981$& \cellcolor{lightred} $1363$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(25,4,60)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $23$& \cellcolor{lightred} $181$& \cellcolor{lightred} $775$& \cellcolor{lightred} $2135$& \cellcolor{lightred} $4109$& \cellcolor{lightred} $5783$& \cellcolor{lightred} $6106$& \cellcolor{lightred} $4895$& \cellcolor{lightred} $2981$& \cellcolor{lightred} $1363$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(26,0,5)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $20$& \cellcolor{lightgreen} $155$& \cellcolor{lightgreen} $675$& \cellcolor{lightgreen} $1911$& \cellcolor{lightgreen} $3787$& \cellcolor{lightgreen} $5475$& \cellcolor{lightgreen} $5910$& \cellcolor{lightgreen} $4815$& \cellcolor{lightgreen} $2962$& \cellcolor{lightgreen} $1361$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(26,1,5)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $20$& \cellcolor{lightgreen} $155$& \cellcolor{lightgreen} $675$& \cellcolor{lightgreen} $1911$& \cellcolor{lightgreen} $3787$& \cellcolor{lightgreen} $5475$& \cellcolor{lightgreen} $5910$& \cellcolor{lightgreen} $4815$& \cellcolor{lightgreen} $2962$& \cellcolor{lightgreen} $1361$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(27,0,38)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $15$& \cellcolor{lightred} $105$& \cellcolor{lightred} $455$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(27,1,615)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $15$& \cellcolor{lightred} $105$& \cellcolor{lightred} $455$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(27,2,2245)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $15$& \cellcolor{lightred} $105$& \cellcolor{lightred} $455$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(27,3,3900)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $15$& \cellcolor{lightred} $105$& \cellcolor{lightred} $455$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(27,4,4055)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $15$& \cellcolor{lightred} $105$& \cellcolor{lightred} $455$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(27,5,2798)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $15$& \cellcolor{lightred} $105$& \cellcolor{lightred} $455$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(27,6,1345)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $15$& \cellcolor{lightred} $105$& \cellcolor{lightred} $455$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(27,7,455)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $15$& \cellcolor{lightred} $105$& \cellcolor{lightred} $455$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(27,8,105)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $15$& \cellcolor{lightred} $105$& \cellcolor{lightred} $455$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(27,9,15)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $15$& \cellcolor{lightred} $105$& \cellcolor{lightred} $455$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$& \cellcolor{lightred} $0$ \\ \hline \cellcolor{lightgray} $(27,10,1)$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $0$& \cellcolor{lightred} $1$& \cellcolor{lightred} $15$& \cellcolor{lightred} $105$& \cellcolor{lightred} $455$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $6435$& \cellcolor{lightred} $5005$& \cellcolor{lightred} $3003$& \cellcolor{lightred} $1365$& \cellcolor{lightred} $455$& \cellcolor{lightred} $105$& \cellcolor{lightred} $15$& \cellcolor{lightred} $1$ \\ \hline \cellcolor{lightgray} $(28,0,520)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $4$& \cellcolor{lightgreen} $50$& \cellcolor{lightgreen} $290$& \cellcolor{lightgreen} $1035$& \cellcolor{lightgreen} $2541$& \cellcolor{lightgreen} $4543$& \cellcolor{lightgreen} $6105$& \cellcolor{lightgreen} $6270$& \cellcolor{lightgreen} $4950$& \cellcolor{lightgreen} $2992$& \cellcolor{lightgreen} $1364$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(28,1,2100)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $4$& \cellcolor{lightgreen} $50$& \cellcolor{lightgreen} $290$& \cellcolor{lightgreen} $1035$& \cellcolor{lightgreen} $2541$& \cellcolor{lightgreen} $4543$& \cellcolor{lightgreen} $6105$& \cellcolor{lightgreen} $6270$& \cellcolor{lightgreen} $4950$& \cellcolor{lightgreen} $2992$& \cellcolor{lightgreen} $1364$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(28,2,3040)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $4$& \cellcolor{lightgreen} $50$& \cellcolor{lightgreen} $290$& \cellcolor{lightgreen} $1035$& \cellcolor{lightgreen} $2541$& \cellcolor{lightgreen} $4543$& \cellcolor{lightgreen} $6105$& \cellcolor{lightgreen} $6270$& \cellcolor{lightgreen} $4950$& \cellcolor{lightgreen} $2992$& \cellcolor{lightgreen} $1364$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(28,3,2220)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $4$& \cellcolor{lightgreen} $50$& \cellcolor{lightgreen} $290$& \cellcolor{lightgreen} $1035$& \cellcolor{lightgreen} $2541$& \cellcolor{lightgreen} $4543$& \cellcolor{lightgreen} $6105$& \cellcolor{lightgreen} $6270$& \cellcolor{lightgreen} $4950$& \cellcolor{lightgreen} $2992$& \cellcolor{lightgreen} $1364$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(28,4,900)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $4$& \cellcolor{lightgreen} $50$& \cellcolor{lightgreen} $290$& \cellcolor{lightgreen} $1035$& \cellcolor{lightgreen} $2541$& \cellcolor{lightgreen} $4543$& \cellcolor{lightgreen} $6105$& \cellcolor{lightgreen} $6270$& \cellcolor{lightgreen} $4950$& \cellcolor{lightgreen} $2992$& \cellcolor{lightgreen} $1364$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(28,5,200)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $4$& \cellcolor{lightgreen} $50$& \cellcolor{lightgreen} $290$& \cellcolor{lightgreen} $1035$& \cellcolor{lightgreen} $2541$& \cellcolor{lightgreen} $4543$& \cellcolor{lightgreen} $6105$& \cellcolor{lightgreen} $6270$& \cellcolor{lightgreen} $4950$& \cellcolor{lightgreen} $2992$& \cellcolor{lightgreen} $1364$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\ \hline \cellcolor{lightgray} $(28,6,20)$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $4$& \cellcolor{lightgreen} $50$& \cellcolor{lightgreen} $290$& \cellcolor{lightgreen} $1035$& \cellcolor{lightgreen} $2541$& \cellcolor{lightgreen} $4543$& \cellcolor{lightgreen} $6105$& \cellcolor{lightgreen} $6270$& \cellcolor{lightgreen} $4950$& \cellcolor{lightgreen} $2992$& \cellcolor{lightgreen} $1364$& \cellcolor{lightgreen} $455$& \cellcolor{lightgreen} $105$& \cellcolor{lightgreen} $15$& \cellcolor{lightgreen} $1$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$& \cellcolor{lightgreen} $0$ \\
\hline
\rowcolor{lightgray}
\rule{0pt}{1ex}
\quad \footnotesize{ Total }\quad &
\tiny{ $30 $}& \tiny{$2026$}& \tiny{$ 41430$}&\tiny{ $ 376350$}& \tiny{$ 2003655$}& \tiny{$ 7286000$}&\tiny{$19794315$} &\tiny{$42481630$}&\tiny{ $ 74703675$}&\tiny{ $ 110336120$}& \tiny{$ 139213315 $}& \tiny{$151755930$} & \tiny{$143939615$}&\tiny{ $ 119234250$}&\tiny{ $ 86346985$}& \tiny{$ 54596500$}&\tiny{ $ 30040395$}&\tiny{ $ 14306710$}&\tiny{ $ 5852905$}&\tiny{ $ 2035800$}&\tiny{$ 593775$}& \tiny{$ 142506$}&\tiny{ $ 27405$}& \tiny{$ 4060$}&\tiny{ $ 435$}&\tiny{ $ 30$}& \tiny{$ 1$}\\ \hline
\end{tabular}
\end{adjustbox}
\ContinuedFloat
\caption{$|F_{z^{T,a,w}}(5, \ell)|$ values for $\ell_0(k)\leq \ell\leq 2^k-2$ in Example \ref{k=5ex} (continued)}\label{k=5table1}
\end{table}
\end{landscape}
\begin{table}
\scriptsize
\centering
\rule{0pt}{5ex}
\resizebox{\textwidth}{!}{%
\begin{tabular}{|c|*{16}{p{1cm}|}}
\hline
\rowcolor{lightgray}
\rule{0pt}{2ex}
\diagbox{\small{$(a,w)$}}{\small{$T$}} &$t^1$ & $t^2$ &$t^3$ & $t^4$ &$t^5$ & $t^6$ &$t^7$ & $t^8$ &$t^9$ & $t^{10}$ &$t^{11}$ & $t^{12}$ &$t^{13}$ & $t^{14}$ &$t^{15}$ \\
\hline
$\cellcolor{lightgray}(1,0)$&\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(2,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(3,0)$&\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(4,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(4,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(5,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(6,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(7,0)$&\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(7,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(8,0)$&\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(9,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(9,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(9,2)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(9,3)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(10,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(10,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(11,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(11,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(11,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(13,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(13,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(13,2)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(14,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ \\ \hline$\cellcolor{lightgray}(15,0)$&\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(15,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(16,0)$&\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(16,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(16,2)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(17,0)$&\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ \\ \hline$\cellcolor{lightgray}(17,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ \\ \hline$\cellcolor{lightgray}(18,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(18,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(18,2)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(18,3)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(18,4)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(18,5)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(18,6)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(19,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(19,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(19,2)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(19,3)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(20,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ \\ \hline$\cellcolor{lightgray}(20,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ \\ \hline$\cellcolor{lightgray}(20,2)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ \\ \hline$\cellcolor{lightgray}(20,3)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ \\ \hline$\cellcolor{lightgray}(21,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ \\ \hline$\cellcolor{lightgray}(21,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ \\ \hline$\cellcolor{lightgray}(22,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(22,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(22,2)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(22,3)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(22,4)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(23,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(23,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(23,2)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(24,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(24,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(24,2)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(25,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(25,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(25,2)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(25,3)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(25,4)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(26,0)$&\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(26,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(27,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(27,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(27,2)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(27,3)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(27,4)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(27,5)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(27,6)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(27,7)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(27,8)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(27,9)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(27,10)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ \\ \hline$\cellcolor{lightgray}(28,0)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(28,1)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(28,2)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(28,3)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(28,4)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(28,5)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline$\cellcolor{lightgray}(28,6)$&\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightgreen}$1$ &\cellcolor{lightred}$0$ &\cellcolor{lightred}$0$ \\ \hline
\end{tabular}}
\caption{Definitions of the representatives $z^{T,a,w}$ in Example \ref{k=5ex}}\label{k=5table2}
\end{table}
\begin{table}[htb]
\begin{tabular}{|>{\columncolor{lightgray}}P{1.2cm}|P{2.4cm}|P{2.4cm}|P{2.4cm}|P{2.4cm}|P{2.4cm}|}
\hline
\rowcolor{lightgray}
\diagbox{$T$}{$\mathscr{A}$} &$A_1$& $A_2$& $A_3$& $A_4$&$A_5$ \\
\hline
$t^1$&\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ \\ \hline$t^2$&\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ \\ \hline$t^3$&\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ \\ \hline$t^4$&\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ \\ \hline$t^5$&\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ \\ \hline$t^6$&\cellcolor{lightgreen}$c$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ \\ \hline$t^7$&\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ \\ \hline$t^8$&\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ \\ \hline$t^9$&\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ \\ \hline$t^{10}$&\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ \\ \hline$t^{11}$&\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ \\ \hline$t^{12}$&\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ \\ \hline$t^{13}$&\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ \\ \hline$t^{14}$&\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ \\ \hline$t^{15}$&\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightred}$e$ &\cellcolor{lightgreen}$c$ &\cellcolor{lightgreen}$c$ \\ \hline
\end{tabular}
\caption{Definition of the set $T$ in Example \ref{k=5ex}}\label{k=5table3}
\end{table}
Let us comment on the alternative ways of calculating the state of the system given in \eqref{phipath} and \eqref{phicut}. We observe that the system is in the failure state if there is at least one minimal cut set in which every component is in the failure state, and the system is in a functioning state if there is at least one minimal path set in which every component is in a functioning state. Indeed, let $z\in\mathcal{S}^n$ and $\bar{s}\in\mathcal{S}$. Thanks to \eqref{phipath}, we have $\phi(z)\leq \bar{s}$ if and only if, for every minimal path set $P_i$, there is at least one component $p\in P_i$ with $z_p \leq \bar{s}$. Conversely, thanks to \eqref{phicut}, we have $\phi(z)>\bar{s}$ if and only if, for every minimal cut set $C_j$, there is at least one component $p\in C_j$ with $z_p>\bar{s}$.
As mentioned above, the minimal cut sets corresponding to a collection of minimal path sets can be constructed by a certain procedure. To be able to describe this procedure, we review the notion of minimality next. Let $\tilde{\mathscr{C}}=\{\tilde{C}_1,\ldots,\tilde{C}_{\tilde{r}}\}$ be a collection of distinct subsets of $\mathcal{N}$ with $\tilde{r}\in\cb{1,\ldots,n}$. For $i\in\cb{1,\ldots,\tilde{r}}$, the set $\tilde{C}_i$ is called a \emph{dominated element} of $\tilde{\mathscr{C}}$ if there is $i^\prime\in\cb{1,\ldots,\tilde{r}}\!\setminus\!\cb{i}$ such that $\tilde{C}_{i^\prime}\subseteq \tilde{C}_i$; in this case, $\tilde{C}_i$ is also said to be \emph{dominated by} $\tilde{C}_{i^\prime}$. $\tilde{C}_k$ is called a \emph{minimal element} of $\tilde{\mathscr{C}}$ if it is not a dominated element. Since $\tilde{\mathscr{C}}$ is a finite collection, a minimal element of it always exists. Moreover, it is easy to observe that every dominated element is dominated by at least one minimal element. It is easy to see that $\tilde{\mathscr{C}}$ is an constructive unordered $\tilde{r}$-cover of $\mathcal{N}$ if and only if every set in $\tilde{\mathscr{C}}$ is a minimal element of $\tilde{\mathscr{C}}$ and $\bigcup_{i=1}^{\tilde{r}} \tilde{C}_i=\mathcal{N}$.
Let $\mathscr{P}=\cb{P_1,\ldots,P_k}$ be a constructive unordered $k$-cover of $\mathcal{N}$. A set $C\subseteq \cb{1,\ldots,n}$ is called an \emph{intersector} of $\mathscr{P}$ if there exists a surjective function $f\colon\cb{1,\ldots,k}\to C$ such that $f(i)\in P_i$ for every $i\in\cb{1,\ldots,k}.$ In other words, an intersector chooses one component from each set in $\mathscr{P}$ but the same component can be chosen from multiple sets. Consequently, for an intersector $C$ of $\mathscr{P}$, it holds $C\cap P_i\neq \emptyset$ for each $i\in\cb{1,\ldots,k}$ but the cardinality of such an intersection may exceed one, in general. Moreover, for every component $p\in\mathcal{N}$, there exists an intersector $C$ of $\mathscr{P}$ such that $p\in C$. Let us denote by $\mathscr{C}_0$ the collection of all intersectors of $\mathscr{P}$ and by $\mathscr{C}$ the collection of all minimal elements of $\mathscr{C}_0$. Let us write
\begin{equation}\label{minint}
\mathscr{C}=\cb{C_1,\ldots,C_r},
\end{equation}
where $r=|\mathscr{C}|$.
\begin{proposition}
The collection $\mathscr{C}$ given in \eqref{minint} is a constructive unordered $r$-cover of $\mathcal{N}$.
\end{proposition}
\begin{proof}
We first note that, by construction, every set in $\mathscr{C}$ is a minimal element of $\mathscr{C}_0$, hence it is a minimal element of $\mathscr{C}$ as well. It remains to show that $\bigcup_{j=1}^r C_j =\mathcal{N}$. The $\subseteq$ part is obvious since $C_1,\ldots,C_r$ are subsets of $\mathcal{N}$. To prove the $\supseteq$ part, let us fix $p\in\mathcal{N}$. Let us denote by $\mathscr{C}_0(p)$ the collection of all intersectors of $\mathscr{P}$ containing $p$ as an element and by $\mathscr{C}(p)$ the collection of all minimal elements of $\mathscr{C}_0(p)$. Let us write $\mathscr{C}(p)=\cb{K_1,\ldots,K_q}$ with $q\in\mathbb{N}$ as a collection of distinct sets. To conclude the proof, it suffices to show that $\mathscr{C}(p)\subseteq\mathscr{C}$. To that end, we let $\a\in\cb{1,\ldots,q}$ and show that $K_\a\in\mathscr{C}$. To get a contradiction, suppose that $K_\a\notin\mathscr{C}$. Note that $\mathscr{C}(p)\subseteq\mathscr{C}_0$. Hence, the supposition is equivalent to that, $K_\a$ is a dominated element of $\mathscr{C}_0$ so that there exists $j(\a)\in\cb{1,\ldots,r}$ such that $C_{j(\a)}\subseteq K_\a$. Moreover, such $C_{j(\a)}$ does not contain $p$ as an element as otherwise $C_{j(\a)}$ would be a set in $\mathscr{C}_0(p)$, which would contradict the minimality of $K_j$ in $\mathscr{C}_0(p)$. Since $C_{\a(j)}$ is an intersector of $\mathscr{P}$, there exists a surjective function $f\colon\cb{1,\ldots,k}\to C_{j(\a)}$ such that $f(i)\in P_i$ for every $i\in\cb{1,\ldots,k}$. Define a function $g\colon\cb{1,\ldots,k}\to \mathcal{N}$ by letting $g(i)=p$ if $p\in P_{i} $ and $g(i)=f(i)$ if $p\notin P_i$. Since $\mathscr{P}$ is constructive, there exists $i^\prime$ such that $p\in P_{i^\prime}$. Consequently, the image $g(\cb{1,\ldots,k})$ of $g$ is an intersector of $\mathscr{P}$ containing $p$ as an element, $C_{j(\a)}\!\setminus\! g(\cb{1,\ldots,m})\neq \emptyset$, and $g(\cb{1,\ldots,k})\subseteq C_{j(\a)}\cup\cb{p}\subseteq K_j$. Hence, $g(\cb{1,\ldots,k})\in\mathscr{C}_0(p)$ and the minimality of $K_j$ in $\mathscr{C}_0(p)$ implies $C_{j(\a)}\subseteq K_j=g(\cb{1,\ldots,k})$, which contradicts $C_{j(\a)}\!\setminus\! g(\cb{1,\ldots,k})\neq \emptyset$. Therefore, $K_j\in\mathscr{C}$.
\end{proof}
As an alternative construction, one can start with a constructive unordered $r$-cover $\{C_1,\ldots,C_r\}$ of $\mathcal{N}$ to be used as the collection of the minimal cut sets of the system, that is, the structure function is defined by \eqref{phicut}. Then, the corresponding minimal path sets can be constructed by the above procedure, which guarantees that each minimal path set has a nonempty intersection with each minimal cut set. The number of all system designs with $r$ minimal cut sets is given by
\[
\frac{|C(\mathcal{N},r)|}{r!}=\sum_{\ell=\ell_0(k)}^{(2^r-2)\wedge n}\frac{\ell!}{r!}\tilde{s}(n,\ell) |F(r,\ell)|.
\]
We finish this section by providing two exact calculations of the above quantity based on the earlier calculations.
\begin{example}
Using Example\ref{k=4ex}, we calculate the number of all system designs with $n=7$ components and $r=4$ minimal cut sets as
\begin{align*}
\frac{|C(\mathcal{N},r)|}{r!}&=\frac{1}{24}\sum_{\ell=4}^{7}\ell!\tilde{s}(7,\ell) |F(4,\ell)|\\
&=\frac{1}{24}\of{4!\tilde{s}(7,4)|F(4,4)|+5!\tilde{s}(7,5)|F(4,5)|+6!\tilde{s}(7,6)|F(4,6)|+7!\tilde{s}(7,7)|F(4,7)|}\\
&=\frac{1}{24}\of{4!\cdot 1050\cdot 25+5!\cdot 266\cdot 304 + 6!\cdot 28\cdot 1165+7!\cdot 1\cdot 2188}\\
&=1,868,650.
\end{align*}
\end{example}
\begin{example}
Using Example\ref{k=5ex}, we calculate the number of all system designs with $n=9$ components and $r=5$ minimal cut sets as
\begin{align*}
\frac{|C(\mathcal{N},r)|}{r!}&=\frac{1}{120}\sum_{\ell=4}^{9}\ell!\tilde{s}(9,\ell) |F(5,\ell)|\\
&=\frac{1}{120}(4!\tilde{s}(9,4)|F(5,4)|+5!\tilde{s}(9,5)|F(5,5)|+6!\tilde{s}(9,6)|F(5,6)|+7!\tilde{s}(9,7)|F(5,7)|\\
&\quad \quad\quad +8!\tilde{s}(9,8)|F(5,8)|+9!\tilde{s}(9,9)|F(5,9)|)\\
&=\frac{1}{120}(4!\cdot42525\cdot 30 + 5!\cdot 22827\cdot 2026+6!\cdot 5880\cdot 41430+7!\cdot 750 \cdot 376350\\
&\quad \quad\quad +8!\cdot 45 \cdot 2003655+9!\cdot 1 \cdot 7286000)\\
&=65,691,305,652.
\end{align*}
\end{example}
\section{Conclusion}\label{conc}
In this paper, we consider the problem of counting the number of ways that one can cover a finite set by a given number of subsets with the additional requirement that every pair of these subsets has a nonempty set difference. It turns out that this seemingly simple problem requires a deep enumeration argument which results in two auxiliary problems: calculating ISNs and counting certain labelings of the disjoint regions that are induced by a cover. While the calculation of ISNs can be handled by a simple recursive relation, we solve the labeling problem by a more sophisticated method that exploits certain symmetries available in the set of labelings and uses no-good cuts from optimization literature. As the numerical examples illustrate, even for small values of $k$, the number of subsets in the cover, one has to calculate very large cardinalities due to exponential growth. The enhancement of this method for larger values of $k$ as well as the asymptotic analysis of the overall problem are subjects of future research.
\section{Appendix: Proofs of the results in Section~\ref{prelim}}\label{app}
\begin{proof}[Proof of Proposition~\ref{workhorse}]
Using binomial expansion, we obtain
\begin{align*}
\tilde{s}(n,\ell)&=\frac{1}{\ell!}\sum_{j=0}^\ell (-1)^{\ell-j}\binom{\ell}{j}(j+1)^n=\frac{1}{\ell!}\sum_{j=0}^\ell (-1)^{\ell-j}\binom{\ell}{j}\sum_{i=0}^n\binom{n}{i}j^i\\
&=\frac{1}{\ell!}\sum_{i=0}^n\binom{n}{i}\sum_{j=0}^\ell (-1)^{\ell-j}\binom{\ell}{j}j^i=\frac{1}{\ell!}\sum_{i=1}^n\binom{n}{i}\sum_{j=0}^\ell (-1)^{\ell-j}\binom{\ell}{j}j^i=\sum_{i=1}^n\binom{n}{i}s(i,\ell).
\end{align*}
Here, to get the penultimate equality, we use the fact that, for $i=0$,
\[
\sum_{j=0}^\ell (-1)^{\ell-j}\binom{\ell}{j}j^i=\sum_{j=0}^\ell (-1)^{\ell-j}(+1)^j\binom{\ell}{j}=0.
\]
Hence, \eqref{stilderes} follows. If $n<\ell$, then \eqref{Stirlingzero} implies
\[
\tilde{s}(n,\ell)=\sum_{i=1}^n\binom{n}{i}s(i,\ell)=\sum_{i=1}^n\binom{n}{i}0=0.
\]
On the other hand, if $n\geq \ell$, then we have $s(i,\ell)=0$ for each $i\in\cb{1,\ldots,\ell-1}$ so that
\[
\tilde{s}(n,\ell)=\sum_{i=\ell}^n\binom{n}{i}s(i,\ell)=\sum_{i=0}^{n-\ell}\binom{n}{i}s(n-i,\ell).
\]
Hence, \eqref{stildeinterpret} follows. In particular, $\tilde{s}(n,n)=\binom{n}{0} s(n,n)=1$. Finally, by \eqref{stilderes} together with (i) and (iii), we have
\[
\tilde{s}(n,1)=\sum_{i=1}^n \binom{n}{i}s(i,1)=\sum_{i=1}^n \binom{n}{i}=2^n-1.
\]
\end{proof}
\begin{proof}[Proof of Proposition~\ref{stilderec}]
Let $n\in\mathbb{N}\!\setminus\!\cb{1}$ and $\ell\in\cb{2,\ldots,n}$. By elementary calculations, we obtain
\begin{align*}
&(\ell+1)\tilde{s}(n,\ell)+\tilde{s}(n,\ell-1)\\
&=\frac{\ell+1}{\ell!}\sum_{j=0}^\ell (-1)^{\ell-j}\binom{\ell}{j}(j+1)^n+\frac{1}{(\ell-1)!}\sum_{j=0}^{\ell-1} (-1)^{\ell-1-j}\binom{\ell-1}{j}(j+1)^n\\
&=\frac{\ell+1}{\ell!}\sum_{j=0}^{\ell-1} (-1)^{\ell-j}\binom{\ell}{j}(j+1)^n+\frac{\ell+1}{\ell!}(\ell+1)^n+\frac{1}{(\ell-1)!}\sum_{j=0}^{\ell-1} (-1)^{\ell-1-j}\binom{\ell-1}{j}(j+1)^n\\
&=\frac{\ell+1}{\ell}\frac{1}{(\ell-1)!}\sum_{j=0}^{\ell-1} (-1)^{\ell-j}\binom{\ell}{j}(j+1)^n+\frac{\ell+1}{\ell!}(\ell+1)^n+\frac{(-1)^{-1}}{(\ell-1)!}\sum_{j=0}^{\ell-1} (-1)^{\ell-j}\binom{\ell-1}{j}(j+1)^n\\
&=\frac{1}{(\ell-1)!}\sum_{j=0}^{\ell-1} (-1)^{\ell-j}(j+1)^n\of{\frac{\ell+1}{\ell}\binom{\ell}{j}-\binom{\ell-1}{j}}+\frac{\ell+1}{\ell!}(\ell+1)^n\\
&=\frac{1}{(\ell-1)!}\sum_{j=0}^{\ell-1} (-1)^{\ell-j}(j+1)^n\binom{\ell}{j}\of{\frac{\ell+1}{\ell}-\frac{\ell-j}{\ell}}+\frac{\ell+1}{\ell!}(\ell+1)^n\\
&=\frac{1}{\ell!}\sum_{j=0}^{\ell-1} (-1)^{\ell-j}\binom{\ell}{j}(j+1)^{n+1}+\frac{1}{\ell!}(\ell+1)^{n+1}=\frac{1}{\ell!}\sum_{j=0}^{\ell} (-1)^{\ell-j}\binom{\ell}{j}(j+1)^{n+1}=\tilde{s}(n+1,\ell),
\end{align*}
as desired. The boundary conditions are given by (iii), (iv) of Proposition~\ref{workhorse}.
\end{proof}
\bibliographystyle{named}
|
1,314,259,996,500 | arxiv | \section{Introduction}
Transverse momentum dependent (TMD) parton distribution functions (PDFs, TMDPDFs) are one of the important ingredients in nucleon tomography and a central focus of hadron physics research in recent years and especially at the future electron-ion collider~\cite{Accardi:2012qut,Boer:2011fh}. TMDPDFs can be experimentally extracted from hard processes in deep inelastic scattering (DIS) and lepton pair production in hadronic collisions~\cite{Collins:2011zzd,Collins:1981uk,Ji:2004wu,Bacchetta:2008xw}. The available
experimental data and global analysis have generated strong interest in the hadron physics community, see, e.g., recent efforts in Refs.~\cite{Cammarota:2020qcw,Bacchetta:2020gko}.
The first attempt to compute the moments of TMDPDFs from lattice QCD has been made in
Refs.~\cite{Musch:2010ka,Musch:2011er,Yoon:2017qzo}. Meanwhile, great progress has been made to compute
$x$-dependent parton physics on the lattice using large momentum effective theory (LaMET)~\cite{Ji:2013dva,Ji:2014gla}, see, some recent reviews on this topic~\cite{Cichy:2018mum,Ji:2020ect}. LaMET is based on the observation that parton physics defined in terms of lightcone correlations can be obtained from time-independent Euclidean correlations (called quasi-distributions) through well-defined effective field theory (EFT) expansion as well as matching and running. LaMET has been applied to compute various collinear PDFs and distribution amplitudes~\cite{Cichy:2018mum,Ji:2020ect}. In the last few years, an
important new development has been to apply LaMET to describe TMDPDFs and associated soft functions~\cite{Ji:2014hxa,Ji:2018hvs,Ebert:2018gzl,Ebert:2019okf,Ebert:2019tvc,Shanahan:2019zcq,Shanahan:2020zxr,Ebert:2020gxr,Ji:2019sxk,Ji:2019ewn,Vladimirov:2020ofp,Zhang:2020dbb}. In this paper, we study
single transverse-spin asymmetries in the region where the transverse momentum is on the order of $\Lambda_{\rm QCD}$, focusing on the non-perturbative calculation of the relevant TMDPDF---the quark Sivers function~\cite{Sivers:1989cc}---
in terms of a Euclidean-space quasi distribution.
The spin-dependent, $k_\perp$-even TMDPDFs have been studied in Ref.~\cite{Ebert:2020gxr}, where similar factorization and matching were found as for the unpolarized case. Because the quark Sivers function is a $k_\perp$-odd distribution, it has special features different from those of the $k_\perp$-even ones.
In particular, in the large $k_\perp$ or small transverse distance limit, the quark Sivers function can be expressed in terms of the collinear twist-three quark-gluon-quark correlation functions in the nucleon, whereas the $k_\perp$-even TMDPDFs depend on the leading twist collinear quark distribution functions. Therefore, the EFT matching calculation in the present case is more involved compared with that in Ref.~\cite{Ebert:2020gxr}.
In this paper, we will focus on computing the quark-Sivers function
in the leading-order expansion from large-momentum effective theory.
An extension to the gluon-Sivers function should be possible in a similar manner. The quark-Sivers function describes a nontrivial correlation between the quark's transverse momentum and the nucleon's transverse polarization vector. Therefore, it represents a spin asymmetry in the TMDPDF. The quark Sivers function is non-zero because the gauge link associated with the quark distribution contributes the phase needed to obtain a single spin asymmetry~\cite{Brodsky:2002cx,Collins:2002kn,Ji:2002aa,Belitsky:2002sm}.
The paper is organized as follows. In Sec.~II, we present the evolution equation of the quasi-Sivers function and its matching to the physical Sivers function. Resummation formulas for the quasi-Sivers function and the matching kernel are also given. A generic argument to demonstrate the matching between the quasi-Sivers function and the light-cone Sivers function will be presented based on the factorization of the hard, collinear and soft gluon radiation contributions for the TMDPDFs. Because the matching coefficient only concerns hard gluon radiation, it does not depend on the spin structure of the nucleon. This is consistent with the observation in Ref.~\cite{Ebert:2020gxr}.
In Sec.~III, we provide detailed derivations of the quasi-Sivers function in LaMET up to one-loop order at large transverse momentum. Our calculations is based on the collinear twist-three quark-gluon-quark correlation functions, and we compute the quasi-Sivers function in terms of the Qiu-Sterman matrix element~\cite{Efremov:1981sh,Efremov:1984ip,Qiu:1991pp,Qiu:1991wg,Qiu:1998ia} (defined below). This can be compared to the light-cone quark-Sivers function calculated in the same framework~\cite{Ji:2006ub,Ji:2006vf,Ji:2006br,Koike:2007dg,Kang:2011mr,Sun:2013hua,Scimemi:2019gge} and the associated matching coefficient can be obtained. In Sec.~IV, we show the application of the formalism to experimental and theoretical single spin asymmetries. Finally, We summarize our paper in Sec. V.
\section{LaMET Expansion of Sivers Function in Euclidean Quasi TMDPDF}
Let us start with the transverse-spin dependent quasi-TMDPDF for quarks in a proton moving along the $+\hat z$ direction~\cite{Ji:2018hvs,Ji:2019ewn}
\begin{align}\label{eq:quasi_tmd}
& \tilde q(x,k_\perp,S_\perp,\mu,\zeta_z)=\int d^2 b_{\perp} ~\int \frac{d\lambda}{2(2\pi)^3}e^{i\lambda x+i\vec{k}_\perp\cdot\vec{b}_\perp} \\
&\! \lim_{L \rightarrow \infty} \frac{\langle PS| \bar \psi\big(\frac{\lambda n_z }{2}\!+\!\vec{b}_\perp\big)\Gamma{\cal W}_{z}(\frac{\lambda n_z}{2}\!+\!\vec{b}_\perp;-L)\psi\big(\!-\!\frac{\lambda n_z}{2}\big) |PS\rangle}{\sqrt{Z_E(2L,b_\perp,\mu)}} \ , \nonumber
\end{align}
where $\overline{\rm MS}$ renormalization is implied, $b_\perp=|\vec b_\perp|$, and the staple-shaped gauge-link ${\cal W}_z$ is
\begin{align}\label{eq:staplez}
&{\cal W}_z(\xi;-L)=W^{\dagger}_{z}(\xi; -L)W_{\perp}W_{z}(-\xi^zn_z;-L) \ ,\\
&W_{z}(\xi;-L)= {\cal P}{\rm exp}\Big[-ig\int_{\xi^z}^{-L} ds\, n_z\cdot A(\vec{\xi}_\perp\!+\!n_z s)\Big] \ .
\end{align}
The spin dependence is introduced by the hadron state $|PS\rangle$. $x$ and $k_\perp$ are the longitudinal momentum fraction and the transverse momentum carried by the quark, and $\zeta_z=4x^2P_z^2$ is the rapidity or Collins-Soper scale. The direction vector of the gauge-link $n_z$ is defined as $n_z=(0,0,0,1)$ and all coordinates are 4-vectors, e.g. $\vec b_\perp=(0,b_1,b_2,0)$. In contrast, $L$ is just a number. $\mu$ is the ultra-violet (UV) renormalization scale. A transverse gauge link was included to make the gauge links connected. The spin-1/2 proton has momentum $P^z$
and is polarized transversely, with the polarization vector $\vec{S}_\perp$ being perpendicular to its momentum direction. The Dirac matrix $\Gamma$ can be chosen as $\Gamma=\gamma^t$ or $\Gamma=\gamma^z$. As we will show, to leading order in $1/P^z$ the two choices are equivalent. The subtraction factor $Z_E(2L,b_\perp,\mu)$ is the vacuum expectation value of a rectangular Wilson-loop that removes the pinch-pole singularity at large $L$~\cite{Ji:2018hvs,Ji:2019ewn}
\begin{align}\label{eq:Z_E}
Z_E(2L,b_\perp,\mu)=\frac{1}{N_c}{\rm Tr}\langle 0|W_{\perp}{\cal W}_z(\vec{b}_\perp;2L)|0\rangle \ .
\end{align}
As emphasized in~\cite{Ji:2018hvs,Ji:2019ewn}, The self-interactions of gauge links are subtracted using $\sqrt{Z_{E}}$ in order to remove the pinch-pole singularities~\cite{Ji:2019sxk} and to guarantee the existence of the large $L$ limit.
With the above definition, we can express the transverse-spin dependent quasi-TMDPDF in terms
of appropriate Lorentz structures,
\begin{align}\label{eq:sivermomen}
&\tilde q(x,k_\perp,S_\perp,\mu,\zeta_z)\nonumber \\ &=\tilde q(x,k_\perp,\mu,\zeta_z)+\frac{\tilde f_{1T}^{\perp}(x,k_\perp,\mu,\zeta_z)\epsilon^{\beta\alpha}S_{\perp\beta}k_{\perp\alpha}}{M_P} \ ,
\end{align}
where $M_P$ is the proton mass and $\epsilon^{12}=1$ in our convention. In the above equation,
the first term represents the spin-averaged, unpolarized quark distribution and the second term is the quark Sivers function in LaMET. It is also convenient to Fourier transform the $k_\perp$ distribution to get the $b_\perp$-space expression,
\begin{align}
\tilde q(x,b_\perp,S_\perp,\mu,\zeta_z)=\int d^2k_\perp e^{-i\vec{k}_\perp\cdot \vec{b}_\perp} \tilde q(x,k_\perp,S_\perp,\mu,\zeta_z) \ ,
\end{align}
which is convenient for factorization calculations.
We can similarly express the quasi-TMDPDF in $b_\perp$-space as,
\begin{align}\label{eq:decompo}
&\tilde q(x,b_\perp,S_\perp,\mu,\zeta_z)\nonumber \\ &=\tilde{q}(x,b_\perp,\mu,\zeta_z)+\epsilon^{\alpha\beta}S_\perp^\beta
\tilde{f}_{1T}^{\perp\alpha}(x,b_\perp,\mu,\zeta_z) \ .
\end{align}
We would like to point out that $\tilde{q}$ is the Fourier transform of the spin-average quark TMDPDF in momentum space, but $\tilde{f}^{\alpha}_{1T}(x,b_\perp,\mu,\zeta_z)$ is not a direct Fourier transform of $\tilde f_{1T}^{\perp}(x,k_\perp,\mu,\zeta_z)$ due to the presence of $k_{\perp\alpha}$ in Eq.~(\ref{eq:sivermomen}). Our focus in this paper is the large-momentum
factorization of $\tilde{f}_{1T}^{\perp\alpha}(x,b_\perp,\mu,\zeta_z)$.
\subsection{Evolution Equation}
We start with the renormalization property of quasi-TMDPDFs. Similar as for the unpolarized quasi-TMDPDFs, in the numerator of Eq.~(\ref{eq:quasi_tmd}), there are linear divergences associated to the self-energy of the staple-shaped gauge-link ${\cal W}_z$ and logarithmic divergences associated to the quark-link vertices. In addition, there are cusp-UV divergence associated to the junctions between longitudinal and transverse gauge-links at $z=-L$. After subtraction using $\sqrt{Z_E}$ in the denominator, the linear divergences and the cusp-UV divergences all cancel, and one is left with only the logarithmic divergences for quark-link vertices. The associated anomalous dimensions are all equal and are known to be equivalent to the anomalous dimension $\gamma_F$ for the heavy-light quark current~\cite{Falk:1990yz,Ji:1991pr}. It can also be derived as the anomalous dimension of the quark field in $A^z=0$ gauge. Thus, the spin-dependent quasi-TMDPDF, in particular the quasi-Sivers function satisfies the following renormalization group equation
\begin{align}
\mu^2\frac{d}{d\mu^2}\ln \tilde f_{1T}^{\perp\alpha}(x,b_\perp,\mu,\zeta_z)=\gamma_F \ .
\end{align}
At one-loop level one has $\gamma_F=\frac{3\alpha_sC_F}{4\pi}$ and high-order results can be found in references~\cite{Falk:1990yz,Ji:1991pr}.
We then come to the evolution equation of quasi-TMDPDFs with respect to $\zeta_z$, i.e. the momentum evolution equation~\cite{Ji:2020ect}. Similar to the case of quasi-PDFs or unpolarized quasi-TMDPDFs, at large $P^z$ there are large logarithms of $P^z$ that can be resummed by the corresponding momentum evolution equation. Using diagramatic methods developed in~\cite{Collins:1981uk,Collins:2011zzd}, it can be shown~\cite{Collins:1981uk,Ji:2014hxa} that $\tilde f_{1T}^{\perp\alpha}$ satisfies the evolution equation
\begin{align}
2\zeta_z\frac{d}{d\zeta_z}\ln \tilde f_{1T}^{\perp\alpha}(x,b_\perp,\mu,\zeta_z)=K(b_\perp,\mu)+G(\zeta_z,\mu)\ ,
\end{align}
where $K$ is the non-perturbative Collins-Soper kernel~\cite{Collins:1981uk} and the $G$ is a perturbative part of the evolution kernel. At one-loop level, one has~\cite{Collins:1981uk,Ji:2014hxa},
\begin{align}\label{eq:oneloopkg}
K^{(1)}(b_\perp,\mu)&=-\frac{\alpha_sC_F}{\pi}L_b\, ,\\
G^{(1)}(\zeta_z,\mu)&=\frac{\alpha_sC_F}{\pi}\left(1-L_z\right) \, .
\end{align}
Here $L_b=\ln \frac{\mu^2 b_\perp^2}{c_0^2}$ with $c_0=2e^{-\gamma_{E}}$ and $L_z=\ln \frac{\zeta_z}{\mu^2}$. These equations allows the quasi-Sivers function to be re-summed in the form in which $\mu_b=\frac{c_0}{b_\perp}$
\begin{align}
\tilde f_{1T}^{\perp\alpha}(x,b_\perp,\mu,\zeta_z)=\tilde f_{1T}^{\perp\alpha}(x,& b_\perp,\mu=\sqrt{\zeta_z}=\mu_b)\nonumber\\ \times \exp \bigg(\frac{1}{2}\ln\frac{\zeta_z}{\mu_b^2}K(b_\perp,\mu)&+\int_{\mu_b^2}^{\zeta_z} \frac{d\zeta'}{2\zeta'}G(\zeta',\mu)\nonumber \\
&+\int^{\mu}_{\mu_b}\frac{d\mu'^2}{\mu'^2}
\gamma_F(\alpha_s(\mu'))\bigg)
\end{align}
with $\mu_b=\frac{c_0}{b_\perp}$.
Using the renormalization group equation~\cite{Collins:1981uk,Ji:2019ewn} for $G$ and $K$, $\frac{d\ln G}{d\ln \mu} = -\frac{d\ln K}{d\ln \mu}=2\Gamma_{\rm cusp}$ where $\Gamma_{\rm cusp}$ is the light-like cusp-anomalous dimension, allows a more refined treatment of resummation for $K$ and $G$.
\subsection{LaMET Expansion for Sivers Function}
Similar to the unpolarized case~\cite{Ji:2019ewn}, the quasi-TMDPDFs can be used to calculate the physical TMDPDFs appearing in the factorizations of experimental cross sections. The LaMET expansion formula requires the off-light-cone reduced soft function $S_r(b_\perp,\mu)$, the definition and more properties of which can be found in~\cite{Ji:2019ewn,Ji:2019sxk,Ji:2020ect}.
In terms of the non-perturbative reduced soft function, the EFT expansion formula for spin-dependent quasi-TMDPDF reads:
\begin{align}\label{eq:factorization}
&f_{1T}^{\perp\alpha}(x,b_\perp,\mu,\zeta)\nonumber \\ &=\frac{ e^{-K(b_\perp,\mu)\ln (\frac{\zeta_z}{\zeta})}} {H\left(\frac{\zeta_z}{\mu^2}\right)}\sqrt{S_r(b_\perp,\mu)} \tilde f_{1T}^{\perp\alpha}(x,b_\perp,\mu,\zeta_z) + ...\ .
\end{align}
where $H$ is the perturbative kernel and the higher-order terms in $1/P^z$ expansoin
have been omitted.
Similar to the unpolarized case~\cite{Ji:2019ewn}, here we provide the sketch of a proof for the matching formula~(\ref{eq:factorization}). We also argue that $H$ is independent of the spin structure, as was recently argued in Ref.~\cite{Ebert:2020gxr}.
First of all, one can perform a standard leading region analysis~\cite{Collins:2011zzd} for all spin-structures with minor modifications to include the staple-shaped gauge-links of the quasi-TMDPDF as in~\cite{Ji:2019ewn}. The leading region or the reduced diagram for quasi-TMDPDFs is shown in Figure.~\ref{fig:quasireduce}. There are collinear and soft subdiagrams responsible for collinear and soft contributions. The collinear contributions are exactly the same as those for the light-cone TMDPDF defined with light-like gauge-links. The soft radiations between the fast moving color charges and the staple shaped gauge-links can be factorized by the off-light-cone soft function.
In addition to the collinear and soft subdiagrams, there are two hard subdiagrams around the vertices at $0$ and $\vec{b}_\perp$. The natural hard scale $\zeta_z$ for the hard diagram is formed by a Lorentz invariant combination of the parton momenta entering the hard subdiagram and the direction vector $n_z$ for the staple-shaped gauge links. At large $P^z$, small $k_\perp$ or large $b_\perp$, the hard contributions are confined within the vicinities of the quark-link vertices around $0$ and $b_\perp$, since any hard momenta flowing between $0$ and $b_\perp$ will cause additional power suppressions in $\frac{1}{P^z}$. In another words, there are two disconnected hard subdiagrams, one containing $0$ and another one containing $\vec{b}_\perp$. Therefore, the momentum fractions carried by the quasi-TMDPDF and the physical TMDPDF are the same and the matching formula contains no convolution.
Given the leading region of the quasi-TMDPDF, one can apply the standard Ward-identity argument of Ref.~\cite{Collins:2011zzd} to factorize the quasi-TMDPDF and obtain Eq.~(\ref{eq:factorization}). The reduced soft function, which is actually the inverse of the rapidity independent part of the off-light-cone soft function~\cite{Ji:2019sxk,Ji:2019ewn}, appears to compensate the differences of soft contributions for quasi-TMDPDFs and physical TMDPDFs. The exponential of the Collins-Soper kernel can be explained by the emergence of large logarithms for quasi-TMDPDFs in the form $K(b_\perp,\mu)\ln \frac{\zeta_z}{\mu^2}$ generated by momentum evolution. To match to the physical TMDPDF at rapidity scale $\zeta$, a factor $e^{-K(b_\perp,\mu)\ln (\frac{\zeta_z}{\zeta})}$ is therefore needed. Finally, the mismatch between $\tilde f$ and $f$ due to the hard contributions is captured by the hard kernel $H$ that depends on the hard scale $\zeta_z$ and the renormalization scale $\mu$. The above arguments are similar for the unpolarized case~\cite{Ji:2019ewn}.
Here we argue that the hard kernel $H$ is independent of the spin structure. As we already emphasized, the hard-cores around $0$ and $\vec{b}_\perp$ are disconnected. Any momentum that is allowed to flow between the vertices and sees the transverse separation is either soft or collinear. The hard momenta have essentially no effects on the other vertex. Therefore, in order to obtain the matching kernel, it is sufficient to consider only ``half'' of the quark quasi-TMDPDF, which one might want to call an ``amputated'' form factor containing only an incoming light-quark with momentum $p=xP$ and an ``outing going'' gauge-link along $n_z$ direction. This form factor is shown in Fig.~\ref{fig:formfactor}. For this form factor, the generic Lorentz structure can always be written as
\begin{align}
\Gamma\left(A +B \gamma \cdot n_z \gamma \cdot p +C\gamma \cdot p \gamma \cdot n_z \right) u(p,S)\ ,
\end{align}
where $\Gamma$ is a generic Dirac matrix at the quark-link vertex, unrelated to that in Eq.(1), $u(p,S)$ is the Dirac spinor for the incoming quark and $A$,$B$,$C$ are scalar functions of $p^2$, $n_z^2$ and $n_z \cdot p$ . Using the anti-commutation relation of Dirac matrices and the equation of motion $\gamma \cdot p u(p,S)=0$, the above equation can be rewritten as
\begin{align}
\Gamma\left(A +2Cn_z\cdot p\right) u(p,S) \ ,
\end{align}
which depends only on a universal scalar function $A +2Cn_z\cdot p$, independent of the spin $S$ and the Dirac matrix $\Gamma$. As a result, the matching kernel only depends on these scalar functions but not the spin $S$ and the Dirac matrix $\Gamma$.
\begin{figure}[t]
\includegraphics[width=0.7\columnwidth]{quasi_reduced.eps}
\caption{The leading regions of the quasi-TMDPDF where $C$ is the collinear subdiagram, $S$ is the soft subdiagram and $H$'s are hard subdiagrams. The two hard cores are not connected with each other (but their open Dirac indices are ontracted), and as a result, the momentum fraction of the quasi-TMDPDF receives only contributions from collinear modes and there is no convolution in the matching formula. }
\label{fig:quasireduce}
\end{figure}
\begin{figure}[t]
\includegraphics[width=0.6\columnwidth]{form_factor.eps}
\caption{The form factor shown here is sufficient for calculating the matching kernel, which
contains an incoming quark with momentum $p=xP$, spin $S$ and an outgoing gauge-link in $n_z$ direction. }
\label{fig:formfactor}
\end{figure}
The above general results can be verified at one-loop order when $b_\perp$ is small
and a perturbative QCD calculation is valid. The one-loop
calculation is more complicated compared with that in~\cite{Ebert:2020gxr}, because
it involves twist-three collinear factorization.
First of all, let us recall that the standard TMDPDF factorization at small $b_\perp$ follows the procedure in Refs.~\cite{Collins:2011zzd,Catani:2000vq,Catani:2013tia,Prokudin:2015ysa}. For the quark Sivers function, we have~\cite{Sun:2013hua,Scimemi:2019gge},
\begin{align}\label{eq:oneloopsiver}
&{f}_{1T}^{\perp\alpha}(x,b_\perp,\mu,\zeta)=\frac{ib_\perp^\alpha}{2}T_F(x,x)\nonumber \\ &+\frac{ib_\perp^\alpha}{2}\frac{\alpha_s}{2\pi}\left\{\left(-\frac{1}{\epsilon}
-L_b\right){\cal P}_{qg/qg}^T\otimes T_F(x,x)\right.\nonumber\\
&\left.+\int\frac{dx_q}{x_q}T_F(x_q,x_q)\left[-\frac{1}{2N_c}(1-\xi_x)+\delta(1-\xi_x)C_Fs^{(1)}\right]\right\}\ ,
\end{align}
where $\xi_x=\frac{x}{x_q}$, $T_F(x,x)$ is the twist-3 quark-gluon-quark correlation function (the Qiu-Sterman matrix element) defined below and ${\cal P}_{qg/qg}^T$ is the associated splitting kernel. For the part involved in the calculations of Sec.~III, we have~\cite{Braun:2009mi,Schafer:2012ra,Kang:2012em,Sun:2013hua,Scimemi:2019gge}~\footnote{Here and in the following calculations, we only keep the so-called soft gluon pole and hard gluon pole contributions in the twist-three formalism~\cite{Ji:2006ub,Ji:2006vf,Ji:2006br}. A complete kernel including soft-fermion pole contributions and other twist-3 matrix elements can be found in Refs.~\cite{Braun:2009mi,Scimemi:2019gge}.},
\begin{eqnarray}
&&{\cal P}_{qg/qg}^T\otimes T_F(x,x)\nonumber\\
&&~=\int\frac{dx_q}{x_q} \left\{T_F(x_q,x_q)\left[C_F\left(\frac{1+\xi_x^2}{1-\xi_x}\right)_+-C_A\delta(1-\xi_x)\right]\right.\nonumber\\
&&~~\left
+\frac{C_A}{2}\left(T_F(x_q,x)\frac{1+\xi_x}{1-\xi_x}-T_F(x_q,x_q)\frac{1+\xi_x^2}{1-\xi_x}\right)\right\} \ .\label{ptt}
\end{eqnarray}
The contribution $s^{(1)}$ reads
\begin{align}
s^{(1)}=-\frac{\pi^2}{12}+\frac{3}{2}L_b-\frac{1}{2}L_b^2-L_{\zeta}L_b \ ,
\end{align}
where $L_{\zeta}=\ln \frac{\zeta}{\mu^2}$. Our definition of the physical TMD-PDF follows the standard one in Ref.~\cite{Echevarria:2012js,Collins:2012uy}, although the numerical factors $\frac{\pi^2}{12}$ depends on the renormalization schemes due the presence of double $\frac{1}{\epsilon^2}$ poles, see section VI of Ref.~\cite{Collins:2012uy} for a discussion. Our results are in the standard $\overline {MS}$ scheme.
In the next section, we show that the quasi-Sivers function has a similar factorizaiton
at small $b_\perp$,
\begin{align}\label{eq:oneloopquasisiver}
&{\tilde f}_{1T}^{\perp\alpha}(x,b_\perp,\mu,\zeta_z)=\frac{ib_\perp^\alpha}{2}T_F(x,x)\nonumber \\ &+\frac{ib_\perp^\alpha}{2}\frac{\alpha_s}{2\pi}\left\{\left(-\frac{1}{\epsilon}
-L_b\right){\cal P}_{qg/qg}^T\otimes T_F(x,x)\right.\nonumber\\
&\left.+\int\frac{dx_q}{x_q}T_F(x_q,x_q)\left[-\frac{1}{2N_c}(1-\xi_x)+\delta(1-\xi_x)C_F\tilde s^{(1)}\right]\right\}\ .
\end{align}
where $\tilde s^{(1)}$ reads
\begin{align}
\tilde s^{(1)}=&-2+\frac{5}{2}L_b-\frac{1}{2}L_b^2-L_zL_b-\frac{1}{2}L_z^2+L_z \ .
\end{align}
Furthermore, the one-loop reduced soft function reads~\cite{Ji:2020ect}:
\begin{align}
S_r(b_\perp,\mu)=1-\frac{\alpha_sC_F}{\pi}L_b \ .
\end{align}
Combining all of the above and comparing it to the LaMET expansion in Eq. (\ref{eq:factorization}), one obtains the one-loop matching kernel
\begin{align}
H\left(\frac{\zeta_z}{\mu^2}\right)=1+\frac{\alpha_sC_F}{2\pi}\bigg(-2+\frac{\pi^2}{12}-\frac{1}{2}L_z^2+L_z\bigg) \ ,
\end{align}
which is exactly the answer we expected: The matching kernel is independent of spin-structure and is equal to that of the unpolarized case for $\Gamma=\gamma^z$ and $\Gamma=\gamma^t$.
It can be shown~\cite{Ji:2019ewn} that the matching kernel satisfies the renormalization group equation
\begin{align}
\mu\frac{d}{d\mu} \ln H\left(\frac{\zeta_z}{\mu^2}\right)=\Gamma_{\rm cusp} \ln \frac{\zeta_z}{\mu^2} + \gamma_C
\end{align}
where $\gamma_C$ can be found in Ref.~\cite{Ji:2019ewn}.
The general solution to the above equation reads
\begin{align}\label{eq:C_renormalization}
&H\left(\alpha_s(\mu),\frac{\zeta_z}{\mu^2}\right)=H\left(\alpha_s(\sqrt{\zeta_z}), 1\right)\\
&\times\exp\left\{\int_{\sqrt{\zeta_z}}^{\mu}\frac{d\mu'}{\mu'}\left[\Gamma_{\rm cusp}(\alpha_s(\mu')) \ln \frac{\zeta_z}{\mu^{'^2}}+\gamma_C\big(\alpha_s(\mu')\big)\right]\right\}\nonumber\,.
\end{align}
This equation allows the determination of the large logarithms for $H$ to all orders in perturbation theory, up to unknown constants related to the initial condition $H(\alpha_s,1)$.
\section{One-loop Calculation for Spin-Dependent quasi-TMPPDFs}
In this section we calculate the quasi-Sivers function at one-loop level. The idea and procedure is the same as for previous examples in the LaMET formalism~\cite{Ji:2014hxa,Ji:2018hvs,Ebert:2018gzl,Ebert:2019okf,Ebert:2019tvc,Ebert:2020gxr,Ji:2019sxk,Ji:2019ewn,Vladimirov:2020ofp}. An important difference is that we will not be able to formulate it in terms of a single quark target. Instead, we need to use the collinear twist-three quark-gluon-quark correlation description and compute the quark quasi-TMDPDF and Sivers asymmetry in these collinear
quark distributions at small $b_\perp\ll 1/\Lambda_{\rm QCD}$.
For the quasi-TMDPDFs, we follow the definition of Eq.~(\ref{eq:quasi_tmd})~\cite{Ji:2019ewn}, where a rectangular Wilson-loop was adopted to remove the pinch-pole singularities. To match to the physical TMDPDFs
at leading order in $1/P^z$, one needs the reduced soft function at small $b_\perp$, which can also be extracted from lattice simulations at any $b_\perp$~\cite{Zhang:2020dbb}.
The perturbative quasi-TMDPDFs at small $b_\perp\ll 1/\Lambda_{\rm QCD}$ can be expressed in terms of the collinear parton distribution and/or the twist-three quark-gluon-quark correlation functions. For the unpolarized quark distribution, the previous results of \cite{Ji:2018hvs} can be expressed as,
\begin{align}
&\tilde{q}(x,b_\perp,\mu,\zeta_z)\nonumber \\ &=f_q(x_,\mu)+\frac{\alpha_s}{2\pi}\left\{\left(-\frac{1}{\epsilon}
-L_b\right){\cal P}_{q/q}\otimes f_q(x)\right.\nonumber\\
&\left.+C_F\int\frac{dx_q}{x_q}f_q(x_q)\left[(1-\xi_x)+
\delta(1-\xi_x)\tilde s^{(1)}\right]\right\}\ ,
\label{oneloopun}
\end{align}
for the leading order plus next-to-leading order result in ${\vec b}_\perp$-space, where $\xi_x=x/x_q$, $\mu$ is the renormalization scale in the $\overline{\rm MS}$ scheme, ${\cal P}_{q/q}(\xi_x)=C_F\left(\frac{1+\xi_x^2}{1-\xi_x}\right)_+$ is the usual splitting kernel for the quark, and $f_q(x)$ represents the light-cone integrated quark distribution function. The one-loop coefficient in the subtraction scheme of Ref.~\cite{Ji:2018hvs} reads
\begin{align}
\tilde s^{(1)}=&\frac{3}{2}\ln\frac{b_\perp^2\mu^2}{c_0^2}+\ln\frac{\zeta_zL^2}{4c_0^2}
-\frac{1}{2}\left(\ln\frac{\zeta_zb_\perp^2}{c_0^2}\right)^2\nonumber \\ &+2{\cal K}(\xi_b)-{\cal K}(2\xi_b) \ ,
\end{align}
where the Collins-Soper scale $\zeta_z=4x_q^2P_z^2$ and the function ${\cal K}$ will be defined later on. At large $L$, all the $L$ dependencies cancel and we have:
\begin{align}
\tilde s^{(1)}=-2+\frac{5}{2}L_b-\frac{1}{2}L_b^2-L_zL_b-\frac{1}{2}L_z^2+L_z \ ,\label{h1}
\end{align}
where $L_b$ and $L_z$ are defined after Eq.~(\ref{eq:oneloopkg}).
After the renormalization of the integrated quark distribution $f_q(x)$ at $\mu=\mu_b$, we can write the quasi-TMD unpolarized quark distribution as
\begin{align}
\tilde{q}(x,b_\perp,\mu_b,\zeta_z)&\nonumber \\ =\int\frac{dx_q}{x_q}f_q(x_q,\mu_b)&\bigg(\delta(1-\xi_x)+\frac{\alpha_sC_F}{2\pi}\bigg[(1-\xi_x) \nonumber \\
&+\delta(1-\xi_x)\tilde s^{(1)}\bigg] \bigg)\ .
\end{align}
The goal of the following derivations in this section is to apply the collinear twist-three formalism and calculate the quasi-Sivers function in LaMET at leading order and next-to-leading order.
\subsection{Phase Contribution from the Gauge Link}
In Eq. (\ref{eq:quasi_tmd}), the quasi-TMDPDF contains a gauge link with a finite length, implying that the eikonal gauge link propagators will be modified. From previous works~\cite{Brodsky:2002cx,Collins:2002kn,Ji:2002aa,Belitsky:2002sm}, we know that the gauge link propagators contribute to the crucial phase which is necessary to generate a non-zero Sivers function. Therefore, we need to check that the finite length gauge link can still do so.
Because of the finite length of the gauge links, the eikonal propagator
in these diagrams will be modified according to
\begin{equation}
(-ig)\frac{i n^\mu }{n\cdot k\pm i\epsilon}\Longrightarrow
(-ig)\frac{i n^\mu}{n\cdot k}\left(1-e^{\pm in\cdot k L}\right) \ ,\label{eikonal}
\end{equation}
where $n^\mu$ represents the gauge link direction. In the current case $n^\mu=n_z^\mu$.
In perturbation calculations, we will make use of the large
length limit $|LP_z|\gg 1$. By doing so, many previous results can
be applied to our calculations.
For example, in the large $L$ limit, we have the
following identity:
\begin{equation}
\lim_{L\to \infty}\frac{1}{n\cdot k}e^{\pm i Ln\cdot k}=\pm i\pi \delta(n\cdot k) \ ,
\end{equation}
which will contribute to the phases needed for a non-zero quark Sivers function.
In the following calculations, we will take two limits whenever this is possible: The large $L$ limit and the large $P_z$ limit. In certain diagrams, we have to use finite $L$ and $P_z$ to regulate, for example, the pinch-pole singularity and/or the end-point singularity~\cite{Ji:2018hvs}. We will emphasize these important points when we carry out the detailed calculations.
\subsection{Leading Order}
We carry out the derivations in the twist-three collinear framework, where the quark Sivers function depends on the so-called twist-three quark-gluon-quark correlation function, aka, the Qiu-Sterman matrix elements~\cite{Efremov:1981sh,Efremov:1984ip,Qiu:1991pp,Qiu:1991wg,Qiu:1998ia}. It is defined as follows,
\begin{align}
&T_F(x_2,x_2') \equiv \int\frac{d\zeta^-d\eta^-}{4\pi} e^{i(x_2
P^+\eta^-+(x_2'-x_2)P_B^+\zeta^-)}\epsilon_\perp^{\beta\alpha}S_{\perp\beta}
\nonumber \\
&\times \,
\left\langle PS|\overline\psi(0){\cal L}(0,\zeta^-)\gamma^+
g{F_\alpha}^+ (\zeta^-) {\cal L}(\zeta^-,\eta^-)
\psi(\eta^-)|PS\right\rangle \ , \label{TF}
\end{align}
where $F^{\mu\nu}$ represent the gluon field strength tensor. From the leading order derivation~\cite{Boer:2003cm}, we have,
\begin{equation}
\frac{1}{M_P}\int d^2k_{\perp}\, k^2_\perp\, f_{1T}^{\perp(\rm SIDIS)}(x,k_\perp) = - T_F(x,x) \ ,\label{moment}
\end{equation}
where $f_{1T}^{\perp({\rm SIDIS})}$ represents the quark Sivers function for a SIDIS process with gauge link going to $+\infty$, corresponding to our choice of $-L$ in Eq.~(\ref{eq:quasi_tmd}).
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=7cm]{lo-sivers.eps}
\end{center}
\caption[*]{Leading order diagrams for quasi-Sivers function.}
\label{leadingorder}
\end{figure}
The method for calculating the single transverse-spin asymmetry in the twist-three formalism has been well developed~\cite{Qiu:1991pp,Qiu:1991wg,Qiu:1998ia,Ji:2006ub,Ji:2006vf,Ji:2006br,Kouvaris:2006zy,Eguchi:2006qz,Eguchi:2006mc,Koike:2006qv,Koike:2007rq,Koike:2007dg,Braun:2009mi,Kang:2008ey,Vogelsang:2009pj,Zhou:2008mz,Schafer:2012ra,Kang:2011mr,Sun:2013hua,Scimemi:2019gge}. There are different approaches to derive the final result, in the following, we follow the collinear $k_{g\perp}$-expansion method~\cite{Qiu:1991pp,Qiu:1991wg,Qiu:1998ia,Ji:2006ub,Ji:2006vf,Ji:2006br,Kouvaris:2006zy}. In this approach, the additional gluon from the polarized hadron is associated with a gauge potential $A^+$, assuming that the polarized nucleon is moving along the $+\hat z$ direction. Thus, the gluon will carry longitudinal polarization and its momentum is parameterized as $x_gP+k_{g\perp}$, where $x_g$ is the momentum fraction with respect to the polarized proton and $k_{g\perp}$ is the transverse momentum. The contribution to the single-transverse-spin asymmetry arises from terms linear in $k_{g\perp}$ in the expansion of the partonic amplitudes. When combined with $A^+$, these linear terms will yield $\partial^\perp A^+$, a part of the gauge field strength tensor $F^{\perp +}$ in Eq.~(\ref{TF}).
As shown in Fig.~\ref{leadingorder}, we have $\vec{k}_{g\perp}=\vec{k}_{q2\perp}-\vec{k}_{q1\perp}$. Therefore, the $k_{g\perp}$ expansion of the scattering amplitudes can be expresssed in terms of the transverse momenta $k_{q1\perp}$ and $k_{q2\perp}$. The associated quark momenta are parameterized as,
\begin{equation}
k_{q1}=x_{q1}P+k_{q1\perp},~~~k_{q2}=x_{q2}P+k_{q2\perp} \ . \label{e55}
\end{equation}
We compute the quasi-Sivers function defined in Eq.~(\ref{eq:quasi_tmd}) with the Gamma matrix $\Gamma=\gamma^t$ or $\gamma^z$. The results are the same in the leading power of $1/P_z$. The leading order diagrams of Fig.~\ref{leadingorder} can be calculated following the above general procedure. The method is similar to that for the standard quark Sivers function calculation in Ref.~\cite{Kang:2011mr,Sun:2013hua}. In particular, the phase comes from the gauge link propagator,
\begin{eqnarray}
\lim_{L\to \infty}\frac{1}{n_z\cdot k_g}e^{\pm iLn_z\cdot k_g}=\pm i\pi\frac{1}{n_z\cdot P}\delta(x_g) \ ,
\end{eqnarray}
which determines the kinematics for the twist-three Qiu-Sterman matrix element at $T_F(x,x)$. The plus/minus signs correspond to the left and right diagrams where the gluon attaches to the left and right sides of the cut-line, respectively. To calculate the Sivers function in $b_\perp$-space, we need to perform a Fourier transformation with respect to the probing quark transverse momentum $k_\perp$ in Fig.~\ref{leadingorder}. Because of momentum conservation, at leading order, $k_\perp=k_{q2\perp}$ for the left diagram and $k_\perp=k_{q1\perp}$ for the right diagram. As shown above, these two diagrams contribute with opposite sign to the Sivers function. Therefore, the total contribution is proportional to:
\begin{equation}
\left(e^{i\vec{k}_{q2\perp}\cdot \vec{b}_\perp}-e^{i\vec{k}_{q1\perp}\cdot \vec{b}_\perp}\right)
\to ib_\perp^\alpha(k_{q2\perp}^\alpha-k_{q1\perp}^\alpha) =ib_\perp^\alpha k_{g\perp}^\alpha\ ,
\end{equation}
in the collinear expansion. As a result, the leading order result for the quark Sivers function in LaMET reads
\begin{eqnarray}
\tilde{f}_{1T}^{\perp\alpha(0)}(x,b_\perp,\mu,\zeta_z)=\frac{ib_\perp^\alpha}{2}T_F(x,x) \ .
\end{eqnarray}
Here, the normalization is consistent with Eq.~(\ref{moment}).
\subsection{One-loop Order from Cut Diagrams}
It has been shown that the quark TMDPDFs in LaMET can be evaluated by the cut diagram approximation~\cite{Ji:2014hxa,Ji:2018hvs}. In particular, if we focus on the kinematic region $0<x<1$, the cut diagram approximation leads to the same results as the complete calculation. In the following, we will apply this approximation to simplify the derivation of the quark Sivers function in LaMET.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=4cm]{general-sivers.eps}
\end{center}
\caption[*]{Cut diagram approximation to calculate the Siver function in LaMET. A mirror diagram similar to that in Fig.~\ref{leadingorder} should be included as well. The longitudinal gluon from the polarized nucleon can attach to any lines associated with the blob.}
\label{general}
\end{figure}
In Fig.~\ref{general}, we show the generic diagrams to calculate the quark Sivers function in LaMET. The lower part represents the quark-gluon-quark correlation from the polarized nucleon. We follow the strategy of Ref.~\cite{Ji:2006vf} to evaluate these diagrams. The radiated gluon carries transverse momentum $k_{1\perp}$ equal in size but opposite to $k_\perp$. Similar as for the leading diagrams, we need to generate a phase from the gauge link propagators in these diagrams. This corresponds to the pole contributions to the single spin asymmetries in the twist-three formalism~\cite{Ji:2006ub,Ji:2006vf,Ji:2006br}. In the following calculations, we focus on the so-called soft-gluon pole and hard-gluon pole contributions. They are characterized by the longitudinal momentum fraction carried by the gluon attached to the hard partonic part from the polarized nucleon: $x_g=0$ corresponds to the soft-gluon-pole contribution, while $x_g\neq 0$ corresponds to the hard-gluon-pole contribution. It is straightforward to extend this treatment to other contributions such as the soft-fermion pole contribution, and those associated with the twist-three function $\tilde{G}_F$~\cite{Koike:2007dg}.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=8cm]{softpole-sivers.eps}
\end{center}
\caption[*]{Soft-gluonic pole contribution at one-loop order for the real gluon radiation.}
\label{softpole}
\end{figure}
We emphasize again that the quasi-Sivers function defined in Eq.~(\ref{eq:quasi_tmd}) can be computed with $\Gamma=\gamma^t$ or $\gamma^z$ and the results are the same in the TMD limit. The soft gluon pole diagrams are shown in Fig.~\ref{softpole}. The pole contribution is the same as that for the leading order diagrams, i.e., $\delta(n_z\cdot k_g)=\frac{1}{n_z\cdot P}\delta(x_g)$. An important step to obtain the final result is to perform the collinear expansion for the incoming parton momenta. Therefore, we will keep the transverse momenta for $k_{q1}$, $k_{q2}$, and $k_g$. Because of momentum conservation, we have $k_{g\perp}=k_{q2\perp}-k_{q1\perp}$. Therefore, there will be two independent transverse momenta in the expansion. One of the collinear expansion contributions comes from the on-shell condition of the radiated gluon in the cut-diagram approximation. This leads to the so-called derivative terms, which can be easily evaluated~\cite{Ji:2006vf}. The final result can be written as
\begin{align}
&\tilde f_{1T}^\perp(x,k_\perp,\mu,\zeta_z)|_{\partial}=-\frac{M_p}{(k_\perp^2)^2}\frac{\alpha_s}{2\pi^2}\frac{1}{2N_c}\nonumber \\ &\times \int\frac{dx_q}{x_q}\left(x_q\frac{\partial}{\partial x_q}T_F(x_q,x_q)\right)\left(1+\xi_x^2+(1-\xi_x)^2\frac{D-2}{2}\right) \ ,
\end{align}
where $\xi_x=x/x_q$ and $D$ represents the dimension for the transverse plane. In the following we will also use $\epsilon=(2-D)/2$. We have also applied the following relation between the momentum fractions along the $\hat z$ direction and those along the light-cone plus direction,
\begin{eqnarray}
(1-\xi)=(1-\xi_x)\frac{1+\sqrt{1+r_\perp^2}}{2} \ ,
\end{eqnarray}
where $r_\perp=|k_\perp|/(x_q(1-\xi_x)P_z)$. In the TMD limit away from the end-point of $\xi_x=1$, we will have $(1-\xi)\to (1-\xi_x)$. At the end-point, we will have to keep the full expression in order to derive the complete result. However, for the above derivative terms, we can simply substitute $(1-\xi)\to (1-\xi_x)$. We further notice that the derivative terms can be transformed into non-derivative terms by performing an integral by part,
\begin{align}
&\tilde f_{1T}^\perp(x,k_\perp,\mu,\zeta_z)|_{\partial}=-\frac{M_p}{(k_\perp^2)^2}\frac{\alpha_s}{2\pi^2}\frac{1}{2N_c}\int\frac{dx_q}{x_q}T_F(x_q,x_q)\nonumber \\ &\times\left[2\xi_x^2+2\epsilon\xi_x(1-\xi_x)+2\delta(1-\xi_x)\right] \ .
\end{align}
The last term in the square brackets comes from the boundary.
Now, we turn to the non-derivative terms. Fig.~\ref{softpole}(a) is easy to derive because it does not have end-point singularity, and we find that
\begin{align}
&\tilde f_{1T}^\perp(x,k_\perp,\mu,\zeta_z)|_{\rm fig.\ref{softpole}(a)}^{\rm ND}\nonumber \\ &=\frac{M_p}{(k_\perp^2)^2}\frac{\alpha_s}{2\pi^2}\frac{1}{2N_c}\int\frac{dx_q}{x_q}T_F(x_q,x_q)\nonumber\\
&\times \frac{(1-\xi)(1-\epsilon)}{(1-\xi_x)\sqrt{1+r_\perp^2}}\left[(1-2\xi)(1-\xi)+\frac{k_\perp^2}{P_z^2}\right] \ .
\end{align}
Taking the TMD limit, and adding the corresponding term from the non-derivative contribution, we obtain the final result for Fig.~\ref{softpole}(a)
\begin{align}
&\tilde f_{1T}^\perp(x,k_\perp,\mu,\zeta_z)|_{\rm fig.\ref{softpole}(a)}\nonumber \\ &=\frac{M_p}{(k_\perp^2)^2}\frac{\alpha_s}{2\pi^2}\frac{1}{2N_c}\int\frac{dx_q}{x_q}T_F(x_q,x_q) (1-\xi_x)(1-\epsilon) \ .
\end{align}
On the other hand, the diagrams $(b,c)$ of Fig.~\ref{softpole} contribute to the end-point singularities. The result can be written as
\begin{align}
&\tilde f_{1T}^\perp(x,k_\perp,\mu,\zeta_z)|_{\rm fig.\ref{softpole}(b,c)}^{\rm ND}\nonumber \\ &=\frac{M_p}{(k_\perp^2)^2}\frac{\alpha_s}{2\pi^2}\frac{1}{2N_c}\int\frac{dx_q}{x_q}T_F(x_q,x_q)\nonumber\\
&\times\frac{2\xi(1-\xi)^2}{(1-\xi_x)^3\sqrt{1+r_\perp^2}} \left[2-\xi+(1-\xi)r_\perp^2\right] \ .
\end{align}
Clearly, the last term in the bracket is power suppressed in the TMD limit. Furthermore, we can rewrite $[2-\xi]$ as two terms as $1+(1-\xi)$. The first term will have an end-point singularity, whereas the second term is regular. It is interesting to find that this regular term cancels the corresponding term from the derivative contribution derived above. Therefore, there are only end-point contributions from Fig.~\ref{softpole}(b,c),
\begin{align}
&\tilde f_{1T}^\perp(x,k_\perp,\mu,\zeta_z)|_{\rm fig.\ref{softpole}(b,c)}\nonumber \\&=\frac{M_p}{(k_\perp^2)^2}\frac{\alpha_s}{2\pi^2}\frac{1}{2N_c}\int\frac{dx_q}{x_q}T_F(x_q,x_q)\frac{2\xi(1-\xi)^2}{(1-\xi_x)^3\sqrt{1+r_\perp^2}}\ .
\end{align}
We further notice that $(1-\xi)^2$ can be simplified as $(1-\xi)^2= (1-\xi_x)^2(1+\sqrt{1+r_\perp^2})^2/4\approx (1-\xi_x)^2(1+\sqrt{1+r_\perp^2})/2$ in the TMD limit. With that, we obtain
\begin{align}
&\tilde f_{1T}^\perp(x,k_\perp,\mu,\zeta_z)|_{\rm fig.\ref{softpole}(b,c)} \nonumber\\ &=\frac{M_p}{(k_\perp^2)^2}\frac{\alpha_s}{2\pi^2}\frac{1}{2N_c}\int\frac{dx_q}{x_q}T_F(x_q,x_q)\frac{2\xi_x}{1-\xi_x}\frac{1+\sqrt{1+r_\perp^2}}{2\sqrt{1+r_\perp^2}}\nonumber\\
&=\frac{M_p}{(k_\perp^2)^2}\frac{\alpha_s}{2\pi^2}\frac{1}{2N_c}\int\frac{dx_q}{x_q}T_F(x_q,x_q)\nonumber \\ &\times \left[\frac{2\xi_x}{(1-\xi_x)_+}+\delta(1-\xi_x)\ln\frac{\zeta_z}{k_\perp^2}\right]\ ,
\end{align}
where $\zeta_z=4x^2P_z^2$. The last equation follows from a similar derivation for the unpolarized TMD quark calculation in Refs.~\cite{Ji:2018hvs} in the TMD limit.
Fig.~\ref{softpole}(d) is a little more involved, because it has the so-called pinch pole singularity if we take the limit $L\to \infty$ first,
\begin{align}
&\tilde f_{1T}^\perp(x,k_\perp,\mu,\zeta_z)|_{\rm fig.\ref{softpole}(d)}^{L\to \infty}=\nonumber \\ &\frac{M_p}{k_\perp^2}\frac{\alpha_s}{2\pi^2}\frac{1}{2N_c}\int\frac{dx_q}{x_q}T_F(x_q,x_q)\frac{1}{\sqrt{1+r_\perp^2}}\frac{1}{k_{1z}+i\epsilon}\frac{1}{k_{1z}-i\epsilon}\ ,
\end{align}
where $k_{1z}=x_q(1-\xi_x)P_z$ represents the longitudinal momentum carried by the radiated gluon crossing the cut line. Because of the pinch-pole singularity, the above contribution is not well defined around $x=x_q$ ($\xi_x=1$). A finite length of the gauge link will help to regulate the pinch pole singularity as shown for the unpolarized case. In addition, similar to the unpolarized case, the contribution is power suppressed when $x\neq x_q$. Therefore, it will only contribute to a Delta function at $\xi_x=1$. Following the same strategy as in Ref.~\cite{Ji:2018hvs}, we perform the Fourier transformation with respect to $k_\perp$ and carry out the $k_{1z}$ integral to derive the $b_\perp$-space expression.
Schematically, the quark Sivers function in $b_\perp$-space can be derived as follows,
\begin{align}
&\tilde{f}_{1T}^\perp(x,b_\perp,\mu,\zeta_z)|_{\rm fig.\ref{softpole}(d)}\nonumber \\ & =\int \frac{d^2k_\perp}{(2\pi)^2} e^{i\vec{k}_\perp\cdot \vec{b}_\perp}\left[{\cal M}_{\rm fig.\ref{softpole}(d)}-{\cal M}_{\rm fig.\ref{softpole}(d)}^{\rm mirror}\right] \ .
\end{align}
Here, the mirror diagram represents the amplitude with gluon attachment to the right side of the cut-line in Fig.~\ref{softpole}(d). We can further apply $\vec k_\perp=\vec k_{q2\perp}-\vec k_{1\perp}^L$ for Fig.~\ref{softpole}(d) and $\vec k_\perp=\vec k_{q1\perp}-\vec k_{1\perp}^R$ for its mirror graph. Note that $k_1^L$ and $k_1^R$ are different because they have different $k_{q1\perp}$ and $k_{q2\perp}$ dependences. We find that the Sivers function in $b_\perp$-space is proportional to
\begin{align}
&\tilde{f}_{1T}^\perp(x,b_\perp,\mu,\zeta_z)|_{\rm fig.\ref{softpole}(d)}\propto \nonumber \\ &e^{i\vec{k}_{q2\perp}\cdot \vec{b}_\perp}\int\frac{d^4k_1^L}{(2\pi)^2}e^{i\vec{k}_{1\perp}^L\cdot \vec{b}_\perp}\frac{1}{(k_z^L)^2}{\cal R}(k_{1z}^L)\delta((k_1^L)^2)\nonumber\\
&-e^{i\vec{k}_{q1\perp}\cdot \vec{b}_\perp}\int\frac{d^4k_1^R}{(2\pi)^2}e^{i\vec{k}_{1\perp}^R\cdot \vec{b}_\perp}\frac{1}{(k_z^R)^2}{\cal R}(k_{1z}^R)\delta((k_1^R)^2)\ ,
\end{align}
where ${\cal R}(k_z)=\left(1-e^{ ik_zL}\right)\left(1-e^{ -ik_zL}\right)$. Notice that although $k_1^L$ and $k_1^R$ are not identical, their contribution to the above equation is the same. So, we can combine the above two terms and obtain,
\begin{align}
&\tilde{f}_{1T}^\perp(x,b_\perp,\mu,\zeta_z)|_{\rm fig.\ref{softpole}(d)}\propto
\left[e^{i\vec{k}_{q2\perp}\cdot \vec{b}_\perp}-e^{i\vec{k}_{q1\perp}\cdot \vec{b}_\perp}\right]\nonumber \\ &\times\int\frac{dk_{1z}}{k_{1z}^2} \frac{d^2k_{1\perp}}{(2\pi)^2} e^{i\vec{k}_{1\perp}\cdot \vec{b}_\perp}\frac{1}{\sqrt{k_{1z}^2+k_{1\perp}^2}}{\cal R}(k_{1z}) \ .
\end{align}
It is interesting to observe that the first factor is just the leading order expression and the second factor represents the amplitude without gluon attachments and is a diagram similar to that for the unpolarized quark distribution at one-loop order. Applying the result from Ref.~\cite{Ji:2018hvs}, we have
\begin{align}
&\tilde{f}_{1T}^\perp(x,b_\perp,\mu,\zeta_z)|_{\rm fig.\ref{softpole}(d)}\nonumber \\ &=-\frac{ib_\perp^\alpha}{2}\frac{\alpha_s}{2\pi}\frac{1}{2N_c}T_F(x,x) \int\frac{dk_z}{k_z^2}\frac{d^2k_\perp}{(2\pi)^2}e^{i\vec{k}_\perp\cdot \vec{b}_\perp}\frac{{\cal R}(k_z)}{\sqrt{k_z^2+k_\perp^2}} \nonumber\\
&=-\frac{ib_\perp^\alpha}{2}\frac{\alpha_s}{2\pi}\frac{1}{2N_c}T_F(x,x) 2{\cal K}(\xi_b) \ ,
\end{align}
where $\xi_b=L/|{\vec b}_\perp|$ and the function ${\cal K}$ is defined as~\cite{Ji:2018hvs},
\begin{equation}
{\cal K}(\xi_b)=2\xi_b \tan^{-1}\xi_b-\ln(1+\xi_b^2) \ .
\end{equation}
At large $\xi_b$ the above ${\cal K}(\xi_b)$ becomes $\pi\xi_b-2\ln\xi_b$, while
at small $\xi_b$ it behaves as $\xi_b^2$.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=8cm]{hardpole-sivers.eps}
\end{center}
\caption[*]{Hard pole contributions at one-loop order for quasi-Sivers function.}
\label{hardpole}
\end{figure}
Now, let us move to hard-gluon-pole contributions, for which the diagrams are shown in Fig.~\ref{hardpole}. First, we notice that the phase contribution comes from the gauge link propagator,
\begin{equation}
\frac{1}{n_z\cdot (k_{q1}-k)}=\frac{1}{n_z\cdot P}\frac{1}{x_{q1}-x} \ .
\end{equation}
The pole contribution leads to $\delta(x_{q1}-x)$ which means that $x_g=(1-\xi_x)x_q$. Because the pole is situated at $x_g\neq 0$ this is a hard-gluon pole contribution. Similar as for the standard Sivers function, these diagrams
do not produce derivative terms. Again, we apply the collinear expansion of the incoming parton transverse momenta and for convenience we have chosen the physical polarization for the radiated gluon in these diagrams. The total contribution from Fig.~\ref{hardpole} can be separated into two terms: one contains the pinch-pole singularity and one is free of pinch-pole singularity.
We can derive the pinch-pole term following the above procedure, giving the contribution from the soft gluon pole diagrams. Again, we have to use a finite length to regulate the divergence and the result in $b_\perp$-space is the same as above but with a different color factor,
\begin{align}
&\tilde{f}_{1T}^\perp(x,b_\perp,\mu,\zeta_z)|_{\rm fig.\ref{hardpole}}^{\rm pinch~p.}\nonumber\\ &=\frac{ib_\perp^\alpha}{2}\frac{\alpha_s}{2\pi}\frac{C_A}{2}T_F(x,x)\int\frac{dk_z}{k_z^2}\frac{d^2k_\perp}{(2\pi)^2}e^{i\vec{k}_\perp\cdot \vec{b}_\perp}\frac{{\cal R}(k_z)}{\sqrt{k_z^2+k_\perp^2}} \nonumber\\
&=\frac{ib_\perp^\alpha}{2}\frac{\alpha_s}{2\pi}\frac{C_A}{2}T_F(x,x) 2{\cal K}(\xi_b) \ .
\end{align}
After subtracting the pinch-pole contribution, we derive the rest of the hard gluon pole contribution from Fig.~\ref{hardpole},
\begin{align}
&\tilde f_{1T}^\perp(x,k_\perp,\mu,\zeta_z)|_{\rm fig.\ref{hardpole}}^{\rm no-pinch~ p.}\nonumber \\ &=-\frac{M_p}{(k_\perp^2)^2}\frac{\alpha_s}{2\pi^2}\frac{C_A}{2}\int\frac{dx_q}{x_q}T_F(x,x_q)\frac{(1-\xi)^2(1+\xi_x)}{(1-\xi_x)^2\sqrt{1+r_\perp^2}}\ ,
\end{align}
where we have neglected power corrections in the TMD limit, and we have applied the symmetry property of the Qiu-Sterman matrix element $T_F(x,x_q)=T_F(x_q,x)$ to simplify the final result. In the TMD limit, the expression above can be further simplified to
\begin{align}
&\tilde f_{1T}^\perp(x,k_\perp,\mu,\zeta_z)|_{\rm fig.\ref{hardpole}}^{\rm no-pinch~ p.}\nonumber \\ &=-\frac{M_p}{(k_\perp^2)^2}\frac{\alpha_s}{2\pi^2}\frac{C_A}{2}\int\frac{dx_q}{x_q}T_F(x,x_q)\nonumber \\ &\times\left[\frac{1+\xi_x}{(1-\xi_x)_+}+\delta(1-\xi_x)\ln\frac{\zeta_z}{k_\perp^2}\right]\ .
\end{align}
Similar to the case for the standard quark Sivers function calculated in Ref.~\cite{Ji:2006vf}. We observe the following: There are cancellations between the hard gluon pole contributions and the soft gluon pole contributions. In particular, the hard-gluon-pole is proportional to the color factor $C_A/2$ while the soft-gluon-pole to $1/2N_c$. Their cancellation leads to the final result proportional to $C_F$ for the end-point contribution and the pinch-pole contributions. This is consistent with the soft gluon radiation contribution and the soft factor subtraction.
Since the soft factor and the subtraction is spin-independent, their contributions will be same as those calculated in Ref.~\cite{Ji:2018hvs}. Combining all these terms, we obtain the final result for the quark Sivers function at one-loop order in LaMET,
\begin{align}
&\tilde{f}_{1T}^{\perp\alpha(1)}(x,{\vec b}_\perp,\mu,\zeta_z)\nonumber \\ &=\frac{ib_\perp^\alpha}{2}\frac{\alpha_s}{2\pi}\left\{\left(-\frac{1}{\epsilon}
-L_b\right){\cal P}_{qg/qg}^T\otimes T_F(x,x)\right.\nonumber\\
&\left.+\int\frac{dx_q}{x_q}T_F(x_q,x_q)\left[-\frac{1}{2N_c}(1-\xi_x)+
\delta(1-\xi_x)C_F\tilde s^{(1)}\right]\right\}\ ,
\label{oneloop}
\end{align}
where ${\cal P}_{qg/qg}^T\otimes T_F(x,x)$ has been defined in Eq.~(\ref{ptt}) and $\tilde s^{(1)}$ is the same as for the unpolarized case of Eq.~(\ref{h1}). Similar to the unpolarized case, after renormalization, we can write the quasi Sivers function in terms of the collinear twist-three Qiu-Sterman matrix element,
\begin{align}
&\tilde{f}_{1T}^{\perp\alpha}(x,{\vec b}_\perp,\mu_b,\zeta_z)\nonumber \\ &=\frac{ib_\perp^\alpha}{2}\int\frac{dx_q}{x_q}T_F(x_q,x_q,\mu_b)\left\{\delta(1-\xi_x)\right.\nonumber\\
&\left.+\frac{\alpha_s}{2\pi}\left[-\frac{1}{2N_c}(1-\xi_x)+\delta(1-\xi_x)C_F\tilde s^{(1)}\right]\right\} \ ,\label{siversoneloop}
\end{align}
at one-loop order.
A couple of comments are in order before we close this section. First, the above result is obtained in the scheme of Ref.~\cite{Ji:2019ewn} in which $\sqrt{Z_E}$ was adopted to subtract out the pinch-pole singularity. The same strategy has already been adopted in Ref.~\cite{Ji:2018hvs} in which it is regarded as a soft factor subtraction.
Second, for the Sivers contribution, we focused our calculations for the soft-gluon and hard-gluon pole contributions from the Qiu-Sterman matrix element $T_F(x_1,x_2)$. Other contributions from the soft-fermion pole and those from $\tilde G_F(x_1,x_2)$ can be included as well. They contribute to both the evolution kernel ${\cal P}_{qg/qg}^T$ and the finite term of Eq.~(\ref{siversoneloop}), see, for example, the recent study for the light-cone Sivers function in Ref.~\cite{Scimemi:2019gge}.
\section{Single Transverse-Spin Asymmetry}
In this section, we discuss applications of the results obtained in previous sections. In particular, we consider the single spin asymmetry at large and small $b_\perp$. We also comment on previous lattice calculations
for moments of the relevant TMDPDFs.
\subsection{Single Spin Asymmetry}
One of the most important physical application of spin-dependent TMDPDFs are single-spin asymmetries defined by the ratio of physical cross sections. For example, in the Drell-Yan lepton pair production, we define,
\begin{align}
A_{DY}
= \frac{\frac{d^4\sigma_{+S_\perp}}{d^2Q_\perp dx_Adx_B}-\frac{d^4\sigma_{-S_\perp}}{d^2Q_\perp dx_Adx_B}}{\frac{d^4\sigma_{+S_\perp}}{d^2Q_\perp dx_Adx_B}+\frac{d^4\sigma_{-S_\perp}}{d^2Q_\perp dx_Adx_B}} \ ,
\end{align}
where $\frac{d^4\sigma_{\pm S_\perp}}{d^2Q_\perp dx_Adx_B}$ are the differential cross sections with the transverse spin for the polarized target being $\pm S_\perp$, $x_{A,B}$ denote the momentum fractions of the incoming hadrons carried by the quark and antiquark, $Q^2$ and $Q_\perp$ are the invariant mass and transverse momentum for the lepton pair.
The factorization formula~\cite{Collins:2011zzd} for the Drell-Yan or SIDIS process in terms of the physical TMDPDFs reads
\begin{align}
&\frac{d^4\sigma}{dQ_\perp^2dx_Adx_B}=\hat \sigma(\frac{Q^2}{\mu^2}) \times \nonumber \\ &\int d^2b_\perp e^{i\vec{b}_\perp \cdot \vec{Q}_\perp}q(x_A,b_\perp,S_\perp,\mu,\zeta_A)q(x_B,b_\perp,\mu,\zeta_B)
\end{align}
where
$x_A$ is the momentum fraction for the quark parton coming out of the polarized target and $x_B$ is for the unpolarized one. $\hat \sigma(\frac{Q^2}{\mu^2})$ is the hard-cross section with $Q^2=2x_Ax_BP^+P^-$ where $P^{\pm}$ are the largest light-front components of the hadron momenta. Given the factorization formula and utilizing a decomposition for the TMDPDF similar to Eq.~(\ref{eq:decompo}), one found that the single-spin asymmetry for the Drell-Yan process can be written in terms of an unpolarized TMDPDF and Sivers function:
\begin{align}
&A_{DY}=\nonumber \\ &\frac{\int d^2b_\perp e^{i\vec{Q}_\perp\cdot \vec{b}_\perp}\epsilon^{\beta\alpha}S_\perp^\beta f_{1T}^{\perp\alpha}(x_A,b_\perp,\mu,\zeta_A)q(x_B,b_\perp,\mu,\zeta_B)}{\int d^2b_\perp e^{i\vec{Q}_\perp\cdot \vec{b}_\perp}q(x_A,b_\perp,\mu,\zeta_A)q(x_B,b_\perp,\mu,\zeta_B)}
\end{align}
where the hard cross-section induced by unpolarized quark partons cancels between numerator and denominator. By using the matching relation Eq.~(\ref{eq:factorization}) and choosing $\zeta_{zA}=\zeta_A=2x_A^2(P^+)^2$, $\zeta_{zB}=\zeta_B=4x_B^2(P^-)^2$, one has
\begin{align} \label{eq:singleas}
&A_{DY}=\frac{D}{S} \ ,\\
&D=\int d^2b_\perp e^{i\vec{Q}_\perp\cdot \vec{b}_\perp}\epsilon^{\beta\alpha}S_\perp^\beta \tilde f_{1T}^{\perp\alpha}(x_A,b_\perp,\mu,\zeta_{zA})\nonumber \\ &\times \tilde q(x_B,b_\perp,\mu,\zeta_{zB})S_r(b_\perp,\mu) \ , \\
&S=\int d^2b_\perp e^{i\vec{Q}_\perp\cdot \vec{b}_\perp}\tilde q(x_A,b_\perp,\mu,\zeta_{zA})\tilde q(x_B,b_\perp,\mu,\zeta_{zB})\nonumber \\ &\times S_r(b_\perp,\mu) \ .
\end{align}
Notice that the matching kernel all cancel, but the reduced soft function $S_r$ does not cancel between $D$ and $S$. Since $\tilde q$, $\tilde f_{1T}^{\perp\alpha}$ and $S_r$ can all be extracted from lattice calculations, Eq.~(\ref{eq:singleas}) allows to predict the physical observable single-spin asymmetry from lattice data.
Because $A_{DY}$ is complicated due to the fact that the soft contribution fails to cancel, it is attractive to mimic $A_{DY}$ by using the following simplified version for the single-spin asymmetry ratio in Fourier transform $b_\perp$-space:
\begin{align}\label{eq:singlespin}
&{\cal R}_{S_\perp}(x,b_\perp)\nonumber \\ &=\frac{q(x,b_\perp,S_\perp,\mu,\zeta)-q(x,b_\perp,-S_\perp,\mu,\zeta)}{q(x,b_\perp,S_\perp,\mu,\zeta)+ q(x,b_\perp,-S_\perp,\mu,\zeta)} \nonumber \\ &=\epsilon^{\beta\alpha}S_\perp^\beta\frac{f_{1T}^{\perp\alpha}(x,b_\perp,\mu,\zeta)}{q(x,b_\perp,\mu,\zeta)}\ .
\end{align}
Instead of transforming to momentum space, in ${\cal R}_{S_\perp}$
we directly compare the asymmetry point by point in $b_\perp$ space. We emphasize that this asymmetry is not a physical observable, but a ratio between the quark-Sivers function and the unpolarized quark distribution.
An important feature is that the $\mu$, $\zeta$ dependencies cancel between numerator and denominator, thus the ${\cal R}_{S_\perp}$
is independent of the renormalization scale $\mu$ and the rapidity scale $\zeta$. This contribution is proportional to $\vec{S}_\perp\times \vec{b}_\perp$ and the coefficient defines the size of the single spin asymmetry in the quark distribution. The individual TMDPDFs will depend on the rapidity renormalization scheme. However, the ratio between the quark Sivers function and the unpolarized quark distribution does not depend on the scheme. In particular, the scheme dependent soft functions cancel in the ratio of Eq.~(\ref{eq:singlespin}).
With the relation between the quasi-TMDPDF and physical TMDPDF in Eq.~(\ref{eq:factorization}), we will be able to study the single spin asymmetry in the quark distribution. Using the fact that the matching kernel $H$ is independent of the spin-structure and is the same for both the Sivers function and the un-polarized quark TMDPDF, we found that by taking the ratio, the soft factor and matching kernel dependencies cancel:
\begin{align}\label{eq:singlespinquasi}
\widetilde{{\cal R}}_{S_\perp}
&=\epsilon^{\beta\alpha}S_\perp^\beta\frac{\tilde f_{1T}^{\perp\alpha}(x,b_\perp,\mu,\zeta_z)}{\tilde q(x,b_\perp,\mu,\zeta_z)}\nonumber \\
&=\epsilon^{\beta\alpha}S_\perp^\beta\frac{f_{1T}^{\perp\alpha}(x,b_\perp,\mu,\zeta)}{q(x,b_\perp,\mu,\zeta)}\equiv {\cal R}_{S_\perp}
(x,b_\perp) \ .
\end{align}
Therefore, the single-spin asymmetry ratio extracted from the quasi-Sivers function and the Sivers function are the same. This is one of the major results of this paper.
At small $b_\perp$, the single spin asymmetry ratio in the quark distribution at one-loop order can be extracted from the perturbative results of the Sivers function provided in the previous sections
\begin{align}
&{\cal R}_{S_\perp}
(x,b_\perp)|_{b_\perp\ll \frac{1}{\Lambda_{QCD}}}=\frac{i|b_\perp|\sin(\phi_b)}{2}\nonumber \\ &
\times \frac{T_F(x,x,\mu_b)-\frac{\alpha_s}{4\pi N_c} (1-x/x_q)\otimes T_F(x_q,x_q,\mu_b)}{f_q(x,\mu_b)+\frac{\alpha_sC_F}{2\pi} (1-x/x_q)\otimes f_q(x_q,\mu_b)} \ ,\label{tfan}
\end{align}
where $\phi_b$ is the azimuthal angle between $\vec{b}_\perp$ and $\vec{S}_\perp$ and we have performed the renormalization of the quark distribution at $\mu=\mu_b$. As one can see, the soft contribution in $s^{(1)}$ cancels.
The explicit derivations in the previous sections have confirmed that the single spin asymmetry extracted from quasi-Sivers function are the same for small $b_\perp$.
For the quark Sivers asymmetry, the $\alpha_s$ correction is very small. For example, numerically, the $\alpha_s$ corrections for the numerator and denominator are less than $1\%$ in most of kinematics of the valence quark distributions at $x\sim 0.2$. Therefore, we can safely neglect these corrections and interpret the asymmetry as the ratio between the Qiu-Sterman matrix element and the unpolarized quark distribution at the scale $\mu_b$.
\subsection{Asymmetry at Large $b_\perp$}
On the other hand, the spin asymmetry ratio at large-$b_\perp$ is determined by non-perturbative TMDPDFs, for which lattice calculations in terms of the formalism presented in Sec. II will be very important.
In previous phenomenology studies, a Gaussian distribution in the transverse momentum space has been assumed for both the unpolarized and Sivers quark distributions, e.g.,
\begin{eqnarray}
q(x,k_\perp,\mu=\zeta=\mu_0)&\propto & e^{-\frac{k_\perp^2}{Q_0^2}}\ , \\
f_{1T}^\perp(x,k_\perp,\mu=\zeta=\mu_0)&\propto & e^{-\frac{k_\perp^2}{Q_s^2}}\ ,
\end{eqnarray}
where $Q_0$ and $Q_s$ are parameters for the Gaussian distributions. Because the asymmetry ratio has to decrease at large transverse momentum, the Gaussian width for the quark Sivers function is smaller than that for the unpolarized quark distribution, i.e., $Q_s<Q_0$. If we translate this into $b_\perp$-space, it will generate a significantly increasing function for ${\cal R}_{S_\perp}
(x,b_\perp)$ at large $b_\perp$,
\begin{equation}
{\cal R}_{S_\perp}
(b_\perp)|_{\rm model}\propto b_\perp e^{\frac{(Q_0^2-Q_s^2)b_\perp^2}{4}} \ .
\end{equation}
For example, the parameterization in Ref.~\cite{Sun:2013hua} predicts a factor of $25$ increase from $b_\perp=0.2~\rm fm$ to $b_\perp=1~\rm fm$. Similar predictions exist for other parameterizations, see, some recent global analyses~\cite{Cammarota:2020qcw,Bacchetta:2020gko}. It will be crucially important to check this in lattice simulations.
\subsection{Relation to Previous Lattice Simulations}
In Refs.~\cite{Musch:2010ka,Musch:2011er,Yoon:2017qzo}, lattice computations for certain matrix
elements in a hadron state have been carried out. These matrix elements are defined through TMDPDF-like bi-local operators, which are separated by transverse distance $b_\perp$ perpendicular to the hadron's momentum direction.
It is easy to see that the matrix elements calculated there do not correspond exactly to the moments of the quasi-TMDPDF distribution. Therefore, it is hard to interpret them although interesting results were obtained.
Our results, obtained by using LaMET can help to improve these earlier results. For example, we can add the explicite $x$ dependence to the matrix elements calculated in Refs.~\cite{Musch:2010ka,Musch:2011er,Yoon:2017qzo}. This can help to resolve the difference between, e.g.,
\begin{align}
&\frac{\int dx x^n \tilde f^{\perp \alpha}_{1T}(x,b_\perp,\mu,\zeta_z=4x^2(P^z)^2)}{\int dx x^n \tilde q(x,b_\perp,\mu,\zeta_z=4x^2(P^z)^2)}\nonumber \\ \ne &\frac{\int dx x^n f^{\perp \alpha}_{1T}(x,b_\perp,\mu,\zeta)}{\int dx x^n q(x,b_\perp,\mu,\zeta)} \ .
\end{align}
The above point has also been observed in Ref.~\cite{Ebert:2020gxr}.
It will be useful to have lattice simulations in the LaMET framework to constrain the quark Sivers functions and compare to phenomenological studies~\cite{Sun:2013hua,Cammarota:2020qcw,Bacchetta:2020gko}.
\section{Conclusion}
In summary, we have investigated the quark Sivers function in LaMET. A number of important features have been found for these distribution functions. In our derivation, we adopted the definition of quasi-TMDPDF in Refs.~\cite{Ji:2018hvs,Ji:2019ewn}. We have shown that the quasi-Sivers function can be matched to the physical Sivers function using Eq.~(\ref{eq:factorization}). The matching kernel and the reduced soft function are the same as that of the unpolarized case. As a result, in the single spin asymmetry, the soft function and hard kernels cancel between the quark Sivers function and the unpolarized quark distribution and we can extract the physical single spin asymmetry knowing the lattice calculable quasi-TMDPDFs.
As a byproduct, Eq.~(\ref{tfan}) provides another useful method to compute twist-three quark-gluon-quark correlation function, in particular, for those directly connected to the leading order TMDPDFs.
We notice a recent study of twist-three parton distribution $g_T(x)$ in LaMET framework~\cite{Bhattacharya:2020xlt}. These studies demonstrate the powerful reach of the LaMET formalism and we hope that more studies of this type will become available in the future.
The methods discussed in this paper can be extended in various directions.
An immediate extension is the analysis of all other $k_\perp$-odd quark TMDPDFs. The large transverse momentum dependence for these distributions has been derived in Ref.~\cite{Zhou:2009jm}. In order to study them in LaMET, we need to translate these results into the LaMET formalism following the procedure of the current paper for the quark Sivers function. We plan to study this in a future publication. Together with a recent paper on $k_\perp$-even spin dependent quark TMDPDFs~\cite{Ebert:2020gxr}, this will complete all leading quark TMDPDFs in LaMET.
In addition, the method developed in this paper shpold be applied to other related parton distribution functions, especially for those relevant for the quantum phase space Wigner distributions. These distribution functions contain in principle the complete information needed for nucleon tomography and allow to unveil the origin of the parton orbital angular momentum in nucleons. We expect more developments on this subject soon.
\vspace{2em}
We thank Jianhui Zhang and Yong Zhao for discussion and suggestions. This material is based upon work partially supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract number DE-AC02-05CH11231 and DE-SC0020682, and Center for Nuclear Femtography, Southeastern Universities Research Associations in Washington DC, and within the framework of the TMD Topical Collaboration. AS acknowledges support by SFB/TRR-55.
\bibliographystyle{apsrev4-1}
|
1,314,259,996,501 | arxiv | \section{Introduction}\label{intro}
The creation of top quarks in $p\bar{p}$ collisions offers a unique test of pair-production in quantum chromodynamics (QCD) at very large momentum transfer as well as a promising potential avenue for the observation of new physical phenomena. Given the very large mass of the top quark, exotic processes may couple more strongly to top quarks than to the other known fundamental particles, and possible hints of new interactions could be first observed in top-quark production. In particular, asymmetries in $t\bar{t}$ production could provide the first evidence of new interactions, such as $t\bar{t}$ production via a heavy axial color octet or a flavor-changing $Z^{\prime}$ boson, that might not be easily observed as excesses in the top quark production rate or as resonances in the $t\bar{t}$ invariant mass distribution.
The CDF and D0 collaborations have previously reported on forward-backward asymmetries ($A_\mathrm{FB}$) in $p\bar{p}\rightarrow t\bar{t}$ production at $\sqrt{s} = 1.96$~TeV at the Fermilab Tevatron. In the standard model (SM), the $t\bar{t}$ production process is approximately symmetric in production angle, with a $\mathcal{O}$(7\%) charge asymmetry arising at next-to-leading order (NLO) and beyond~\cite{nlotheory}. Using a sample corresponding to $5.3~ {\rm fb}^{-1} $ of integrated luminosity, CDF measured a parton-level asymmetry $A_\mathrm{FB} = 0.158 \pm 0.074$~\cite{cdfafb} in the lepton+jets decay channel ($t\bar{t} \rightarrow (W^{+}b)(W^{-}b) \rightarrow (l^{+}\nu)(q\bar{q^{\prime}})b\bar{b}$~\cite{cc}), and very good agreement was found by the D0 measurement $A_\mathrm{FB} = 0.196\pm 0.065$~\cite{d0afb} in a lepton+jets sample corresponding to $5.4~ {\rm fb}^{-1} $. CDF and D0 have also performed simple differential measurements using two bins each in the top-antitop rapidity difference $|\Delta y|$ and the top-antitop invariant mass $M_{t\bar{t}}$. The two experiments agreed on a large $|\Delta y|$ dependence. CDF also saw a large $M_{t\bar{t}}$ dependence, and while that observed at D0 was smaller, the CDF and D0 results were statistically consistent. One of the aims of this paper is to clarify the $|\Delta y|$ and $M_{t\bar{t}}$ dependence of the asymmetry using the full CDF data set.
The $5~ {\rm fb}^{-1} $ results have stimulated new theoretical work, both within and outside the context of the SM. The SM calculation has been improved by calculations of electroweak processes that contribute to the asymmetry, studies of the choice of renormalization scale, and progress on a next-to-next-to-leading order (NNLO) calculation of the asymmetry~\cite{hollikpagani,kuhnrodrigo,manohartrott,brodsky,nnlo_xsec}. The new calculations result in a small increase in the expected asymmetry, but not enough to resolve the tension with observation. Other work has focused on the dependence of the asymmetry on the transverse momentum of the $t\bar{t}$ system~\cite{ttpt}, on which we report here.
A number of speculative papers invoke new interactions in the top sector~\cite{np} to explain the large asymmetry. In one class of models, $t\bar{t}$ pairs can be produced via new axial $s$-channel particles arising from extended gauge symmetries or extra dimensions. For these models, the asymmetry is caused by interference between the new $s$-channel mediator and the SM gluon. In other models, light $t$-channel particles with flavor-violating couplings create an asymmetry via a $u$, $d \to t$ flavor change into the forward Rutherford-scattering peak. All potential models of new interactions must accommodate the apparent consistency of the measured cross section and $M_{t\bar{t}}$ spectrum with the SM predictions. Tevatron and LHC searches for related phenomena, such as di-jet resonances, same-sign tops, and other exotic processes, can provide additional experimental limits on potential models. Measurements by the LHC experiments of the top-quark charge asymmetry $A_{\rm C}$, an observable that is distinct from $A_\mathrm{FB}$ but correlated with it, have found no significant disagreement with the SM~\cite{aclhc}; however, any observable effect at the LHC is expected to be small, and the nature of the relationship between $A_\mathrm{FB}$ and $A_{\rm C}$ is model-dependent~\cite{afbvsac}. A more precise measurement of the Tevatron forward-backward asymmetry and its mass and rapidity dependence may help untangle the potential new physics sources for $A_\mathrm{FB}$ from the standard model and from each other.
This paper reports on a study of the asymmetry in the lepton+jets topology, with several new features compared to the previous CDF analysis in this channel~\cite{cdfafb}. We use the complete Tevatron Run~II data set with an integrated luminosity of $9.4~ {\rm fb}^{-1} $. We additionally expand the event selection by including events triggered by large missing transverse energy and multiple hadronic jets, increasing the total data set by approximately $30\%$ beyond what is gained by the increase in luminosity. In total, the number of candidate events in this analysis is more than twice the number of events used in Ref.~\cite{cdfafb}. An improved NLO Monte Carlo generator is used to describe the predicted $t\bar{t}$ signal, and we also add small corrections reflecting new results on the electroweak contributions to the asymmetry~\cite{hollikpagani,kuhnrodrigo,manohartrott}. Finally, parton-level shape corrections utilize an improved algorithm which yields binned parton-level measurements of the rapidity and mass dependence of the asymmetry. We also study the dependence of the asymmetry on the $t\bar{t}$ transverse momentum, $\mbox{${p_T^{t\bar{t}}}$}$, showing that the modeling of this quanity is robust, and that the excess asymmetry above the SM prediction is consistent with being independent of $\mbox{${p_T^{t\bar{t}}}$}$.
\section{Expected asymmetries and Monte Carlo models}\label{sec:mc}
The asymmetry is measured using the difference of the $t$ and $\bar{t}$ rapidities, $\Delta y = y_{t} - y_{\tbar}$, where the rapidity $y$ is given by
\begin{equation} \label{eq:rap}
y = \frac{1}{2} \ln \left( \frac{E + p_z}{E - p_z} \right),
\end{equation}
\noindent with $E$ being the total top-quark energy and $p_z$ being the component of the top-quark momentum along the beam axis as measured in the detector rest frame. $\Delta y$ is invariant to boosts along the beamline, and in the limit where the transverse momentum of the $t\bar{t}$ system is small, the forward-backward asymmetry
\begin{equation} \label{eq:afb}
A_\mathrm{FB} = \frac{N(\Delta y > 0) - N(\Delta y< 0)}{N(\Delta y > 0) + N(\Delta y < 0)}
\end{equation}
\noindent is identical to the asymmetry in the top-quark production angle in the experimentally well-defined $t\bar{t}$ rest frame. The standard model predictions for the top-quark asymmetry referenced in this paper are based on the NLO event generator \textsc{powheg}\xspace~\cite{powheg}, using the {\sc cteq6.1M} set of parton-distribution functions (PDFs), validated by comparing \textsc{powheg}\xspace to the NLO generator \textsc{mc@nlo}\xspace~\cite{mcnlo} as well as the NLO calculation of \textsc{mcfm}\xspace~\cite{mcfm}. We find good consistency overall, as shown in Table~\ref{mc:tab:predictions}~\cite{nlo_v_lo}. Sources of asymmetry from electroweak processes in the standard model that are not included in the \textsc{powheg}\xspace calculations~\cite{hollikpagani,kuhnrodrigo,manohartrott} lead to an overall increase of the asymmetry by a factor of 26\% of the QCD expectation. This is included in all the predictions shown in Table~\ref{mc:tab:predictions} and in all predicted asymmetries and $\Delta y$ distributions in this paper. The electroweak asymmetry is assumed to have the same $M_{t\bar{t}}$ and $\Delta y$ dependence as the QCD asymmetry, and we apply a simple 26\% rescaling to the {\sc powheg} predictions there as well. Following Ref.~\cite{errors}, we include a $30\%$ uncertainty on all theoretical predictions for the SM asymmetry due to the choice of renormalization scale.
\begin{table*}[ht]
\caption{Parton-level asymmetry predictions of \textsc{powheg}\xspace, \textsc{mc@nlo}\xspace, and \textsc{mcfm}\xspace after applying electroweak corrections.\label{mc:tab:predictions}}
\begin{center}
\begin{tabular}{cccc}
\hline
\hline
& \textsc{mc@nlo}\xspace & \textsc{powheg}\xspace & \textsc{mcfm}\xspace \\ \hline
Inclusive & \phantom{0}$0.067 \pm 0.020$\phantom{0} & \phantom{0}$0.066 \pm 0.020$\phantom{0} & \phantom{0}$0.073 \pm 0.022$\phantom{0} \\ \hline
$\ensuremath{\left\lvert \Delta y \right\rvert}\xspace < 1$ & \phantom{0}$0.047 \pm 0.014$\phantom{0} & \phantom{0}$0.043 \pm 0.013$\phantom{0} & \phantom{0}$0.049 \pm 0.015$\phantom{0} \\
$\ensuremath{\left\lvert \Delta y \right\rvert}\xspace > 1$ & \phantom{0}$0.130 \pm 0.039$\phantom{0} & \phantom{0}$0.139 \pm 0.042$\phantom{0} & \phantom{0}$0.150 \pm 0.045$\phantom{0} \\ \hline
$M_{t\bar{t}} < 450~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$ & \phantom{0}$0.054 \pm 0.016$\phantom{0} & \phantom{0}$0.047 \pm 0.014$\phantom{0} & \phantom{0}$0.050 \pm 0.015$\phantom{0} \\
$M_{t\bar{t}} > 450~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$ & \phantom{0}$0.089 \pm 0.027$\phantom{0} & \phantom{0}$0.100 \pm 0.030$\phantom{0} & \phantom{0}$0.110 \pm 0.033$\phantom{0}\\
\hline
\hline
\end{tabular}
\end{center}
\end{table*}
To test the analysis methodology in the case of a large asymmetry, we study two models in which an asymmetry is generated by the interference of the gluon with massive axial color-octet particles. Each provides a reasonable approximation of the observed data in presenting a large, positive forward-backward asymmetry, while also being comparable to the Tevatron data in other important variables such as the $t\bar{t}$ invariant mass, $M_{t\bar{t}}$.
The first model, Octet A, contains an axigluon with a mass of 2~\ifmmode {\rm TeV}/c^2 \else TeV$/c^2$\fi. This hypothetical particle is massive enough that the pole is observed as only a small excess in the tail of the $M_{t\bar{t}}$ spectrum, but it creates an asymmetry via the interference between the off-shell axigluon and the SM gluon. The couplings are tuned ($g_V(q) = g_V(t) = 0$, $g_A(q) = 3$, $g_A(t) = -3$, where $q$ refers to light-quark couplings and $t$ to top-quark couplings) to produce a parton-level asymmetry consistent with the measurement in Ref.~\cite{cdfafb}. The second model, Octet B, contains an axigluon with the same couplings, but a smaller mass of 1.8~\ifmmode {\rm TeV}/c^2 \else TeV$/c^2$\fi. This model produces a larger excess in the tail of the $M_{t\bar{t}}$ spectrum and an even larger asymmetry than Octet A, allowing the measurement procedure to be tested in a regime with a very large asymmetry.
Both models are simulated using the leading order (LO) \textsc{madgraph}~\cite{madgraph,tait} Monte Carlo generator and are hadronized with \textsc{pythia}\xspace~\cite{pythia} before being passed to the CDF detector simulation and reconstruction software. We emphasize that these are not hypotheses - the physical applicability of these models is, in fact, quite constrained by $t\bar{t}$ resonance searches at the LHC~\cite{lhcresonances}. Rather, these models are used as controlled inputs to study the performance of the analysis in the presence of large asymmetries. Further information about these models can be found in Ref.~\cite{cdfafb}.
\section{Measurement strategy}\label{sec:strategy}
The analysis takes place in several steps. We first consider the asymmetry observed at the reconstruction level in all selected events. Next, to study the asymmetry for a pure sample of $t\bar{t}$ events as recorded in the detector, the calculated non-$t\bar{t}$ background contribution is subtracted and the appropriate systematic uncertainties related to the background prediction are applied. Finally, to study the asymmetry at the parton level, corrections are applied for the event reconstruction and detector acceptance, along with appropriate systematic uncertainties on the signal modeling. The reconstruction- and background-subtracted-level measurements have the advantage of fewer assumptions, while the parton-level measurement allows direct comparison to theory predictions.
After reviewing the event selection and reconstruction in Sec.~\ref{sec:data}, we describe the various steps of the correction procedure in detail and apply them to the $\Delta y$ distribution and the inclusive $A_\mathrm{FB}$ measurement in Sec.~\ref{sec:inclusive_afb}. In Sec.~\ref{sec:afb_v_dy} and Sec.~\ref{sec:afb_v_mtt}, we study the dependence of the asymmetry on $|\Delta y|$ and $M_{t\bar{t}}$, $A_{\rm FB}(|\Delta y|)$ and $A_{\rm FB}(M_{t\bar{t}})$ respectively, at all three stages of correction, and Sec.~\ref{sec:sig} discusses the significance of discrepancies observed in these dependencies between the data and the SM. Section~\ref{sec:afb_v_pt} discusses the dependence of the asymmetry on the $t\bar{t}$ transverse momentum.
\section{Detector, event selection, and reconstruction}\label{sec:data}
The data sample corresponds to an integrated luminosity of $9.4~ {\rm fb}^{-1} $ recorded with the CDF~II detector during $p\bar{p}$ collisions at $1.96$~TeV. CDF~II is a general purpose, azimuthally and forward-backward symmetric magnetic spectrometer with calorimeters and muon detectors~\cite{cdf}. Charged particle trajectories are measured with a silicon-microstrip detector surrounded by a large open-cell drift chamber, both within a 1.4 T solenoidal magnetic field. The solenoid is surrounded by pointing-tower-geometry electromagnetic and hadronic calorimeters for the measurement of particle energies and missing energy reconstruction. Surrounding the calorimeters, scintillators and proportional chambers provide muon identification. We use a cylindrical coordinate system with the origin at the center of the detector and the $z$-axis along the direction of the proton beam~\cite{coords}.
This measurement selects $t\bar{t}$ candidate events in the lepton+jets topology, where one top quark decays semileptonically ($t \rightarrow Wb \rightarrow l \nu b$) and the other hadronically ($t \rightarrow Wb \rightarrow q \bar{q}^{\prime} b$). We detect the lepton and hadronization-induced jets. The presence of missing transverse energy ($\mbox{$\protect \raisebox{.3ex}{$\not$}\et$}$)~\cite{coords} is used to infer the passage of a neutrino through the detector. Detector readout is initiated in one of two ways: either by indications of a high-momentum lepton (electron or muon) in the central portion of the detector or by events with indications of large $\mbox{$\protect \raisebox{.3ex}{$\not$}\et$}$ and at least two energetic jets. Events collected in the second manner, in which we require the presence of muon candidates reconstructed offline, make up the ``loose muon'' sample, a new addition compared to the previous version of this analysis. After offline event reconstruction, we require that all candidate events contain exactly one electron or muon with $\mbox{${E_T}$}(\mbox{${p_T}$}) >20$~GeV(GeV/$c$) and $|\eta| < 1.0$, as well as four or more hadronic jets with $E_T >20$~GeV and $|\eta| < 2.0$. Jets are reconstructed using a cone algorithm with $\Delta R = \sqrt{\Delta\phi^2+\Delta\eta^2} < 0.4$, and calorimeter signals are corrected for various detector and measurement effects as described in Ref.~\cite{jes}. We require $\mbox{$\protect \raisebox{.3ex}{$\not$}\et$} > 20$~GeV, consistent with the presence of an undetected neutrino. We finally require that $H_T$, the scalar sum of the transverse energy of the lepton, jets, and $\mbox{$\protect \raisebox{.3ex}{$\not$}\et$}$, be $H_T > 220$~GeV. This requirement reduces the backgrounds by $17\%$ while accepting $97\%$ of signal events. The {\sc secvtx} algorithm~\cite{secvtx} is used to identify $b$ jets by searching for displaced decay vertices within the jet cones, and at least one jet in each event must contain such a ``$b$ tag''. The coverage of the tracking detector limits the acceptance for jets with identified $b$ tags to $|\eta| < 1$.
The sample passing this selection, including the $b$-tag requirement, contains 2653 candidate events. The estimated non-$t\bar{t}$ background in the data sample is $530 \pm 124$ events. The predominant background source is QCD-induced $W$+multi-parton events containing either $b$-tagged heavy-flavor jets or erroneously tagged light-flavor jets. These events are modeled with the {\sc alpgen} Monte Carlo generator~\cite{alpgen}, with the normalizations determined by tagging efficiencies, mis-tagging rates, and other measurements in the data. QCD multi-jet (``Non-$W$'') events containing mis-measured $\mbox{$\protect \raisebox{.3ex}{$\not$}\et$}$ and jets that are mis-identified as leptons are modeled using real data events with lepton candidates that are rejected by the lepton identification requirements. This background, which is the most difficult to model properly, is also the one that is most efficiently suppressed by the $H_T$ requirement, which reduces it by approximately 30\%. Small backgrounds from electroweak processes ($WW$, $WZ$, single-top) are estimated using Monte Carlo generators. The expected background contributions from each source are given in Table~\ref{tab:method2}. We note that there are correlations among the various sources of uncertainty for the different background components, so that the total background uncertainty is not a simple sum in quadrature of the uncertainties on the individual background normalizations. Further information about the background modeling and event selection can be found in Ref.~\cite{tZxsec}.
\begin{table}[ht]
\caption{Expected contributions of the various background sources to the selected data.}\label{tab:method2}
\begin{center}
\begin{tabular}{c c}
\hline
\hline
Background source & Number of events \\
\hline
$W$+HF & \phantom{0}256 $\pm$ 83\phantom{0} \\
$W$+LF & \phantom{0}102 $\pm$ 32\phantom{0} \\
Non-$W$ & \phantom{00}97 $\pm$ 50\phantom{0} \\
Single top & \phantom{00}35 $\pm$ 3\phantom{00} \\
Diboson & \phantom{00}21 $\pm$ 3\phantom{00} \\
$Z$+jets & \phantom{00}19 $\pm$ 3\phantom{00} \\
\hline
Total background & \phantom{0}530 $\pm$ 124 \\
$t\bar{t}$ (7.4 pb) & \phantom{}2186 $\pm$ 314 \\
\hline
Total prediction & \phantom{}2716 $\pm$ 339 \\
\hline
Data & 2653 \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
The reconstruction of the $t\bar{t}$ kinematics employs the measured momenta of the lepton and the four leading jets in the event, along with the measured $\vec{\mbox{$\protect \raisebox{.3ex}{$\not$}\et$}}$. The calculation of the $t\bar{t}$ four-vectors uses a $\chi^2$-based fit of the lepton and jet kinematic properties to the $t\bar{t}$ hypothesis. Each of the possible jet-to-parton assignments is evaluated according to its consistency with resulting from the decay of a pair of top quarks. Two of the observed jets are required to be consistent with being decay products of a $W$ boson, while the lepton and $\mbox{$\protect \raisebox{.3ex}{$\not$}\et$}$ must be consistent with another $W$ boson. Each $W$ boson, when paired with one of the remaining ($b$) jets, is checked for consistency with having resulted from a top-quark decay. The lepton momentum, \mbox{$\protect \raisebox{.3ex}{$\not$}\et$}, and jet energies are allowed to float within their experimental uncertainties, and we apply the constraints that $M_W=80.4~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$, $M_t=172.5~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$, and any $b$-tagged jets must be associated with $b$ partons. The jet-to-parton assignment that best matches these requirements is chosen to define the parent top quarks in each event.
This algorithm has been studied and validated in many precision top-quark-property analyses, including mass measurements~\cite{reco}, which remove the top-quark mass constraint, and property measurements that do make use of the mass constraint~\cite{alice}. The top- and antitop-quark four-vectors determined from this procedure are used to find the rapidities of the quarks and the $\Delta y = y_{t} - y_{\bar{t}}$ variable used for the asymmetry analysis, with the charges of the reconstructed top quarks being fixed by the observed lepton charge. In the Appendix, we discuss a high-precision test of the lepton-charge determination in a large control sample with the goal of verifying that the lepton charge assigment is well-modeled by the detector simulation.
The validity of the analysis is checked at all stages by comparison to a standard model prediction created using the {\sc powheg} $t\bar{t}$ model, the lepton+jets background model described above, and a full simulation of the CDF~II detector. Figure~\ref{fig:yhad} shows the rapidity distribution for the hadronically-decaying top or antitop quark. In the measurement of the asymmetry, the observed lepton charge is used to determine whether each entry in this distribution corresponds to a top quark or an antitop quark, and this rapidity is combined with the rapidity of the leptonically decaying quark to calculate $\Delta y$ for each event. In Fig.~\ref{fig:yhad} and all that follow, the $t\bar{t}$ signal prediction is scaled such that the total signal normalization, when added to the background prediction in Table~\ref{tab:method2}, totals number of observed events.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig1}
\caption{{\small The rapidity of the hadronically-decaying top or antitop quark.} \label{fig:yhad}}
\end{center}
\end{figure}
Figure~\ref{fig:mtt} shows a comparison of the data to the prediction for the invariant mass of the $t\bar{t}$ system, $M_{t\bar{t}}$; there is good agreement. In the previous CDF analysis~\cite{cdfafb}, the forward-backward asymmetry was found to have a large dependence on this variable. In Sec.~\ref{sec:afb_v_mtt} we report a new measurement of this dependence.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig2}
\caption{{\small Reconstructed invariant mass of the $t\bar{t}$ system. The last bin contains overflow events.} \label{fig:mtt}}
\end{center}
\end{figure}
The transverse momentum of the $t\bar{t}$ system, $\mbox{${p_T^{t\bar{t}}}$}$, provides a sensitive test of the reconstruction and modeling, particularly at low momenta, where both the prediction and the reconstruction are challenged by the addition of soft gluon radiation external to the $t\bar{t}$ system. In Fig.~\ref{fig:ptsys_reso} we show the difference between the reconstructed and true values of the $x$-component of $\mbox{${p_T^{t\bar{t}}}$}$ in {\sc powheg}. The difference is centered on zero and well-fit by the sum of two gaussians with widths as shown. Most events fall in the central core with a resolution of $\sim 14~\ifmmode {\rm GeV}/c \else GeV$/c$\fi$. Doubling this in quadrature for the two transverse components gives an overall expected resolution $\delta\mbox{${p_T^{t\bar{t}}}$}\sim 20~\ifmmode {\rm GeV}/c \else GeV$/c$\fi$ for the bulk of the data. In Fig.~\ref{fig:ptsys} we show that the reconstructed data is in good agreement with the sum of the background prediction and the NLO $t\bar{t}$ model; the 10 GeV bin size here is chosen to be half the measured resolution. The $t\bar{t}$ forward-backward asymmetry can have a significant $\mbox{${p_T^{t\bar{t}}}$}$ dependence, and we discuss the expected and measured asymmetry as a function of this variable in Sec.~\ref{sec:afb_v_pt}.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig3}
\caption{{\small Resolution of the $x$- or $y$-component of the reconstructed $\mbox{${p_T^{t\bar{t}}}$}$ of the $t\bar{t}$ system as measured in {\sc powheg}.}\label{fig:ptsys_reso}}
\end{center}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig4}
\caption{{\small Reconstructed $\mbox{${p_T^{t\bar{t}}}$}$ of the $t\bar{t}$ system. The last bin contains overflow events.} \label{fig:ptsys}}
\end{center}
\end{figure}
\begin{figure*}[!htbp]
\begin{center}
\subfigure[]{
\includegraphics[width=0.45\textwidth, clip]{fig5a}\label{fig:njet}}
\subfigure[]{
\includegraphics[width=0.45\textwidth, clip]{fig5b}\label{fig:ntag}}
\caption{{\small \subref{fig:njet} The number of observed jets and \subref{fig:ntag} the number of jets with $b$ tags in the data compared to the signal plus background model. The last bin contains overflow events.} \label{fig:njet_ntag}}
\end{center}
\end{figure*}
We also consider a wide range of other variables, a selection of which are shown here, to validate the reconstruction algorithm and the modeling of the data set. In Fig.~\ref{fig:njet_ntag} we show the distributions of the number of jets and number of $b$ tags in events passing the selection requirements. Figure~\ref{fig:ntag} also includes events containing no $b$-tagged jets, which are not part of the final sample of candidate events but provide an important check on the modeling of the $b$-tagging algorithm. Figure~\ref{fig:jetet_leppt} shows the transverse energy of the most energetic jet and the transverse momentum of the lepton, while Fig.~\ref{fig:met_ht} shows the distribution of the reconstructed $\mbox{$\protect \raisebox{.3ex}{$\not$}\et$}$ and $H_T$. All distributions exhibit good agreement between the observed data and the model expectations.
\begin{figure*}[!htbp]
\begin{center}
\subfigure[]{
\includegraphics[width=0.45\textwidth, clip]{fig6a}\label{fig:jetet}}
\subfigure[]{
\includegraphics[width=0.45\textwidth, clip]{fig6b}\label{fig:leppt}}
\caption{{\small \subref{fig:jetet} The $E_T$ of the most energetic jet and \subref{fig:leppt} the transverse momentum of the lepton in the data compared to the signal plus background model. The last bin contains overflow events.} \label{fig:jetet_leppt}}
\end{center}
\end{figure*}
\begin{figure*}[!htbp]
\begin{center}
\subfigure[]{
\includegraphics[width=0.45\textwidth, clip]{fig7a}\label{fig:met}}
\subfigure[]{
\includegraphics[width=0.45\textwidth, clip]{fig7b}\label{fig:ht}}
\caption{{\small \subref{fig:met} The missing transverse energy and \subref{fig:ht} the scalar sum of the transverse energy of the lepton, jets, and $\mbox{$\protect \raisebox{.3ex}{$\not$}\et$}$ in the data compared to the signal plus background model. The last bin contains overflow events.} \label{fig:met_ht}}
\end{center}
\end{figure*}
\section{The inclusive asymmetry}\label{sec:inclusive_afb}
\subsection{$\Delta y$ in the reconstructed data}\label{sec:reco}
We first consider the reconstructed $\Delta y$ distribution and its asymmetry as defined in Eq.~(\ref{eq:afb}). The $\Delta y$ distribution is shown in Fig.~\ref{fig:qdely}, compared to prediction for the background plus the \textsc{powheg}\xspace $t\bar{t}$ model. Those bins with $\Delta y > 0$ contain data points that are consistently higher than the prediction, while in the bins with $\Delta y < 0$, the data is consistently below the prediction. This results in an inclusive reconstructed asymmetry of $A_\mathrm{FB} = 0.063 \pm 0.019$, compared to a prediction of $0.020 \pm 0.012$. The uncertainty on the data measurement is statistical only. Table~\ref{tab:inc_asyms_posneg} summarizes the reconstructed asymmetry values, with events split according to the charge of the identified lepton, and also reports the results of Ref.~\cite{cdfafb} for comparison. The uncertainties scale as expected from the previous analysis according to the increase in the number of candidate events. When the sample is separated according to the charge of the lepton, the asymmetries are equal within uncertainties, as would be expected from a {\it CP}-conserving effect.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig8}
\caption{{\small (top) The reconstructed $\Delta y$ distribution and the inclusive reconstruction-level asymmetry, compared to the prediction of the signal and background model. (bottom) The difference between the data and prediction divided by the prediction. N.B. the left-most and right-most bins are under- and over-flow bins, respectively.} \label{fig:qdely}}
\end{center}
\end{figure}
\begin{table}[!htb]
\caption{Measured reconstruction-level asymmetries in $\Delta y$ compared to the values measured in the previous CDF analysis~\cite{cdfafb}, as well as the predicted asymmetries for the signal and background contributions.}\label{tab:inc_asyms_posneg}
\begin{center}
\begin{tabular}{c c c}
\hline
\hline
& \multicolumn{2}{c}{Predicted $\phantom{0}A_\mathrm{FB}$} \\
\hline
SM $t\bar{t}$ & \multicolumn{2}{c}{$\phantom{-}0.033\pm 0.011\phantom{0}$} \\
Backgrounds & \multicolumn{2}{c}{$ -0.034\pm 0.013\phantom{0}$} \\
Total prediction & \multicolumn{2}{c}{$\phantom{-}0.020\pm 0.012\phantom{0}$} \\
\hline
& \multicolumn{2}{c}{ Observed $\phantom{0}A_\mathrm{FB} \pm $ stat $ $ } \\
& $\phantom{0}9.4~ {\rm fb}^{-1} $ & $\phantom{0}5.3~ {\rm fb}^{-1} $ \\
\hline
All data & $\phantom{-}0.063\pm 0.019\phantom{0}$ & $\phantom{-}0.057\pm 0.028\phantom{0}$ \\
Positive leptons & $\phantom{-}0.072\pm 0.028\phantom{0}$ & $\phantom{-}0.067\pm 0.040\phantom{0}$ \\
Negative leptons & $\phantom{-}0.055\pm 0.027\phantom{0}$ & $\phantom{-}0.048\pm 0.039\phantom{0}$ \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
\subsection{Subtracting the background contributions}\label{sec:bkg_sub}
Approximately 20\% of the selected data set is composed of events originating from various background sources. We remove the effect of these events by subtracting the predicted background contribution from each bin of the reconstructed distribution. This background-subtraction procedure introduces additional systematic uncertainty, which is added in quadrature to the statistical uncertainty for all background-subtracted results in this paper.
To derive this uncertainty, we start with a total prediction containing $n$ components ($n-1$ background sources and one signal), with each component $i$ having an asymmetry $A_i$ and contributing $N_i$ events. This leads to a total asymmetry for the prediction of
\begin{equation}
A_{\rm tot} = \frac{\sum\limits_{i=1}^{n} A_i N_i}{\sum\limits_{i=1}^{n} N_i} = \frac{\sum\limits_{i=1}^{n} A_i N_i}{N_{\rm tot}}.
\end{equation}
\noindent For the $i$'th component, we let $\sigma_{A_{i}}$ and $\sigma_{N_{i}}$ be the uncertainties on the asymmetry and the normalization respectively. For $\sigma_{N_{i}}$, we use the predicted uncertainty of each background component, as listed in Table~\ref{tab:method2}. The uncertainty due to the finite sample size of the model for a given background component is included as $\sigma_{A_{i}}$, though this is only appreciable for the non-$W$ component, which is taken from a statistically limited sideband in the data.
These uncertainties can be propagated in the usual way by calculating derivatives and adding in quadrature, leading to the term within the summation in Eq.~(\ref{eq:bkgsub_syst}). For the uncertainty due to background subtraction, the summation runs over the $n-1$ background components. We also include an overall uncertainty $\sigma_{A_{\rm bkg}}$ as the final term.
\begin{equation}\label{eq:bkgsub_syst}
\sigma^2_{\rm syst} = \sum\limits_{i=1}^{n-1}\left[\frac{N^2_i}{N^2_n}\sigma_{A_{i}}^2 + \frac{(A_i - A_{\rm tot})^2}{N^2_n}\sigma_{N_{i}}^2\right] + \sigma_{A_{\rm bkg}}^2.
\end{equation}
\noindent For the uncertainty $\sigma_{A_{\rm bkg}}$ on the overall background shape, we substitute an alternate model for the non-$W$ background component and determine the effect on the measured asymmetry, contributing an uncertainty of $0.002$ to the inclusive $A_\mathrm{FB}$ result. The summation term in Eq.~(\ref{eq:bkgsub_syst}) results in a total uncertainty of $0.008$. In total, the sum of the systematic contributions to the uncertainty is small compared to the statistical uncertainty.
The $\Delta y$ distribution after background subtraction is shown in Fig.~\ref{fig:qdely_signal}. Because the total background prediction is nearly symmetric, the removal of the backgrounds increases the asymmetry attributable to the $t\bar{t}$ signal. The resulting observed asymmetry in the background-subtracted sample is $0.087 \pm 0.026$ (stat$+$syst), compared to the \textsc{powheg}\xspace prediction of $0.033 \pm 0.011$.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig9}
\caption{{\small (top) The observed background-subtracted $\Delta y$ distribution compared to the SM prediction. Error bars include both statistical and background-related systematic uncertainties. (bottom) The difference between the data and prediction divided by the prediction.} \label{fig:qdely_signal}}
\end{center}
\end{figure}
\subsection{Correction to the parton level}\label{sec:parton}
The background-subtracted results provide a measurement of the asymmetry due to $t\bar{t}$ events. However, these results are not directly comparable to theoretical predictions because they include the effects of the limited acceptance and resolution of the detector. We correct for these effects so as to provide parton-level results, in the $t\bar{t}$ rest frame after radiation, that can be directly compared to theoretical predictions.
If the true parton-level binned distribution of a particular variable is given by $\vec{n}_{\text{parton}}$, then, after background subtraction, we will observe $\vec{n}_{\text{bkg.sub.}}=\mathbf{S}\mathbf{A}\vec{n}_{\text{parton}}$, where the diagonal matrix $\mathbf{A}$ encodes the effect of the detector acceptance and selection requirements, while the response matrix $\mathbf{S}$ describes the bin-to-bin migration that occurs in events passing the selection due to the limited resolution of the detector and $t\bar{t}$ reconstruction algorithm. To recover the parton-level distribution, the effects of $\mathbf{S}$ and $\mathbf{A}$ must be reversed.
The $5.3~ {\rm fb}^{-1} $ CDF analysis~\cite{cdfafb} used simple matrix inversion (``unfolding'') to perform the correction to the parton level. While effective, this technique was limited in its application because unfolding via matrix inversion tends to enhance statistical fluctuations (due to small eigenvalues in the migration matrix), which makes it reliable only in densely populated distributions. This limited the previous analysis to the extent that the determination the functional dependencies of the asymmetry could only use two bins of $|\Delta y|$ and $M_{t\bar{t}}$. In this paper, we employ a new algorithm, also based on matrix inversion but more sophisticated in application, to measure more finely-binned parton-level distributions, resulting in a more robust measurement of the functional dependence of $A_\mathrm{FB}$ on $|\Delta y|$ and $M_{t\bar{t}}$ at the parton level.
We first consider $\mathbf{S}$, correcting for the finite resolution of the detector using a regularized unfolding algorithm based on Singular Value Decomposition (SVD)~\cite{svd,roounfold}. We model the bin-to-bin migration caused by the detector and reconstruction using \textsc{powheg}\xspace. The matrix $\mathbf{S}$ in $\Delta y$ from \textsc{powheg}\xspace is represented graphically in Fig.~\ref{fig:delyresponse}. Along each row, the box area is proportional to the probability that each possible measured value $\Delta y_{\text{meas}}$ is observed in events with a given true rapidity difference $\Delta y_{\text{true}}$. The matrix population clusters along the diagonal where $\Delta y_{\text{meas}} = \Delta y_{\text{true}}$ and is approximately symmetric, showing no large biases in the $\Delta y$ reconstruction. Before inverting the matrix $\mathbf{S}$ and applying it to the background-subtracted data, a regularization term is introduced to prevent statistical fluctuations from dominating the correction procedure. It is this smoothing via regularization that allows an increase in the number of bins in the parton-level distributions compared to the previous analysis. Details regarding how the regularization term is included are given in Ref.~\cite{svd}, but in essence, a term $\sqrt{\tau}\mathbf{C}$, where $\mathbf{C}$ is the second-derivative matrix,
\begin{equation}\label{eq:secondderivatives}
\mathbf{C} = \left( \begin{array}{ccccc}
-1 & 1 & 0 & 0 & ... \\
1 & -2 & 1 & 0 & ... \\
0 & 1 & -2 & 1 & ... \\
... & ... & ... & ... & ... \\
... & 0 & 1 & -2 & 1 \\
... & 0 & 0 & 1 & -1
\end{array} \right),
\end{equation}
\noindent is added to the matrix equation relating $\vec{n}_{\text{parton}}$ to $\vec{n}_{\text{bkg.sub.}}$. This term imposes the {\it a priori} condition that the parton-level solution should be smooth (more precisely, the regularization assumes that the ratio between the data distribution after acceptance cuts and the model distribution after acceptance cuts is smooth, but given a smooth acceptance function and a model that is smooth at the parton level, this is equivalent to a condition that the data be smooth at parton level). The value of $\tau$ defines how strongly the regularization condition affects the result and is determined using the methods recommended in Ref.~\cite{svd}.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig10}
\caption{{\small Detector response in $\Delta y$ as modeled by \textsc{powheg}\xspace, showing the true value of $\Delta y$ as a function of the measured value for all events passing the selection criteria. The size of each rectangle is proportional to the number of entries in that bin.} \label{fig:delyresponse}}
\end{center}
\end{figure}
In the second step of the parton-level correction procedure, we account for events that are unobserved due to limited acceptance. The acceptance in each bin is derived from the \textsc{powheg}\xspace model, as shown in Fig.~\ref{fig:delyacceptance}, and these acceptances are applied to the data as an inverse-multiplicative correction to each bin. The acceptance is asymmetric in $\Delta y$, with backwards events passing the selection requirements more often than forward events. This effect is related to the $\mbox{${p_T^{t\bar{t}}}$}$ dependence of the asymmetry that is discussed in Sec.~\ref{sec:afb_v_pt}. Large $\mbox{${p_T^{t\bar{t}}}$}$ in a given event leads to $t\bar{t}$ decay products that also have large $\mbox{${p_T}$}$, and thus events with large $\mbox{${p_T^{t\bar{t}}}$}$ pass the selection requirements more often than events with small $\mbox{${p_T^{t\bar{t}}}$}$. As is shown in Sec.~\ref{sec:afb_v_pt}, high-$\mbox{${p_T^{t\bar{t}}}$}$ events are also predicted by \textsc{powheg}\xspace (and various other SM calculations) to have a negative asymmetry. The result is that events with a negative asymmetry are more likely to fulfill the selection requirements, leading to the asymmetric acceptance distribution in Fig.~\ref{fig:delyacceptance}.
The SVD unsmearing and bin-by-bin acceptance correction have similarly-sized impact on the final result. Both of the corrections lead to an increase in the asymmetry. The population of poorly reconstructed events tends to have zero asymmetry, and thus dilutes the true asymmetry. One effect of the unsmearing is to remove this dilution. The acceptance correction also increases $A_\mathrm{FB}$ because of the asymmetric acceptance shown in Fig.~\ref{fig:delyacceptance}.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig11}
\caption{{\small Acceptance as a function of $\Delta y$ as modeled by \textsc{powheg}\xspace.} \label{fig:delyacceptance}}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig12}
\caption{{\small Results from simulated parton-level $\Delta y$ measurements based on Octet A. The data points show the central values for the simulated results, with the error bars representing the $1\sigma$ spread of the results. } \label{fig:pe_octetA}}
\end{center}
\end{figure}
\begin{table*}[!htb]
\caption{Average parton-level asymmetry values in 10~000 simulated experiments with Octet A.}\label{tab:pe_octetA}
\begin{center}
\begin{tabular}{c c c c }
\hline
\hline
$|\Delta y|$ & \phantom{0}Average measured $A_\mathrm{FB}$\phantom{0} & \phantom{0} Average uncertainty \phantom{0} & \phantom{0}True $A_\mathrm{FB}$\phantom{0} \\
\hline
Inclusive & 0.162 & 0.039 & 0.156 \\
$0.0 \leq |\Delta y| < 0.5$ & 0.056 & 0.035 & 0.052 \\
$0.5 \leq |\Delta y| < 1.0$ & 0.180 & 0.055 & 0.158 \\
$1.0 \leq |\Delta y| < 1.5$ & 0.316 & 0.078 & 0.295 \\
$|\Delta y| \geq 1.5$ & 0.434 & 0.128 & 0.468 \\
\hline \hline
\end{tabular}
\end{center}
\end{table*}
\begin{table}[!htb]
\caption{Systematic uncertainties on the parton level $A_\mathrm{FB}$ measurement.}\label{tab:svd_systematics}
\begin{center}
\begin{tabular}{l c }
\hline
\hline
Source & Uncertainty \\
\hline
Background shape & 0.018 \\
Background normalization & 0.013 \\
Parton shower & 0.010 \\
Jet energy scale & 0.007 \\
Initial- and final-state radiation & 0.005 \\
Correction procedure & 0.004 \\
Color reconnection & 0.001 \\
Parton-distribution functions & 0.001 \\
\hline
Total systematic uncertainty & 0.026 \\
\hline
Statistical uncertainty & 0.039 \\
\hline
Total uncertainty & 0.047 \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
The combination of these two parts of the correction procedure allows the determination of the parton-level distribution of $\Delta y$, which is reported as a differential cross section. This algorithm is tested in various simulated $t\bar{t}$ samples, including standard model $\textsc{powheg}\xspace$ and the non-SM samples Octet A and Octet B. Analyzing these samples as if they are data, we measure the bias in the comparison of derived parton-level results to the true values in the generated samples. The {\sc powheg} results are self-consistent to better than 1\%, and, because the NLO standard model is assumed {\it a priori} to be the correct description of the underlying physics and is used to model the acceptance and detector response, any biases observed in this case are included as systematic uncertainties, as described below.
In the octet models, the derived distributions track the generator truth predictions well, but small biases (generally less than $3-4\%$) are observed in some of the differential asymmetry values. An example of the average corrected distribution across a set of 10~000 simulated experiments is shown in Fig.~\ref{fig:pe_octetA} for Octet A, with the asymmetry as a function of $|\Delta y|$ for these simulated experiments summarized in Table~\ref{tab:pe_octetA}. We do not attempt to correct the biases seen in the non-SM models or include them in the uncertainty because there is no reason to believe that these specific octet models actually represent the real underlying physics - these models exhibit small but significant discrepancies with the data in the $M_{t\bar{t}}$ spectrum, a variable that has a significant effect on the $t\bar{t}$ reconstruction, and thus the detector response matrix. In light of this model-dependence, we emphasize that the parton-level results need to be interpreted with some caution in relation to models that differ significantly from the NLO standard model.
Because the resolution corrections can cause migration of events across bins, the populations in the final parton-level distributions are correlated. In all binned parton-level distributions, the error bars on a given bin correspond to the uncertainty in the contents of that bin, but they are not independent of the uncertainties corresponding to other bins in the distribution. When we calculate derived quantities such as $A_\mathrm{FB}$, we use the covariance matrix associated with the unsmearing procedure to propagate the uncertainties correctly.
Several sources of systematic uncertainty must be accounted for when applying the correction procedure. In addition to uncertainties on the size and shape of the background prediction, there are also uncertainties related to the signal Monte Carlo sample used to model the acceptance and detector response. These signal uncertainties include the size of the jet energy scale corrections~\cite{jes}, the amount of initial- and final-state radiation, the underlying parton-distribution functions~\cite{IFSRPDFsyst}, the modeling of color reconnection~\cite{colorreconnection}, and the modeling of parton showering and color coherence. We evaluate these uncertainties by repeating the measurement after making reasonable variations to the assumptions that are used when modeling the detector response. For example, to estimate the effect on our measurement of uncertainty in parton shower and color coherence models, we compare two detector response models, one using the Lund string model~\cite{pythia} and one using the Catani-Seymour dipole model~\cite{herwig}. We also include a systematic uncertainty for the correction algorithm itself, taking the difference between the true value in \textsc{powheg}\xspace and the average result from the simulated experiments based on \textsc{powheg}\xspace described above as the uncertainty resulting from the correction procedure. The systematic uncertainties on the inclusive $A_\mathrm{FB}$ measurement are shown in Table~\ref{tab:svd_systematics}, and the total systematic uncertainty is found to be small compared to the statistical uncertainty. When adding the systematic uncertainties to the the covariance matrices that result from the unfolding procedure, the systematic uncertainties are assumed to be 100\% correlated across all bins.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig13}
\caption{{\small (top) The differential cross section $d\sigma/d(\Delta y)$ as measured in the data after correction to the parton level compared to the SM prediction. Uncertainties include both statistical and systematic contributions and are correlated between bins. (bottom) The difference between the data and prediction divided by the prediction.} \label{fig:dely_parton}}
\end{center}
\end{figure}
\begin{table}[!htb]
\caption{The measured differential cross section as a function of $\Delta y$. The total cross section is normalized to 7.4 pb. Errors include both statistical and systematic contributions, and are correlated across bins.}\label{tab:xsecmeas}
\begin{center}
\begin{tabular}{c c }
\hline
\hline
$\Delta y$ & $d\sigma/d(\Delta y)$ (pb) \\
\hline
$\le$ $-1.5$ & \phantom{0}0.13 $\pm$ 0.05\phantom{0} \\
$-1.5$ to $-1.0$ & \phantom{0}0.36 $\pm$ 0.07\phantom{0} \\
$-1.0$ to $-0.5$ & \phantom{0}0.95 $\pm$ 0.10\phantom{0} \\
$-0.5$ to $0.0$ & \phantom{0}1.66 $\pm$ 0.14\phantom{0} \\
$0.0$ to $0.5$ & \phantom{0}1.82 $\pm$ 0.13\phantom{0} \\
$0.5$ to $1.0$ & \phantom{0}1.37 $\pm$ 0.09\phantom{0} \\
$1.0$ to $1.5$ & \phantom{0}0.76 $\pm$ 0.09\phantom{0} \\
$\ge$ $1.5$ & \phantom{0}0.35 $\pm$ 0.07\phantom{0} \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
Applying the correction procedure to the data of Fig.~\ref{fig:qdely_signal} yields the distribution shown in Fig.~\ref{fig:dely_parton}, where the measured result is compared to the SM \textsc{powheg}\xspace prediction. Both the prediction and the observed data distributions are scaled to a total cross section of 7.4 pb, so that Fig.~\ref{fig:dely_parton} shows the differential cross section for $t\bar{t}$ production as a function of $\Delta y$. The measured values are summarized in Table~\ref{tab:xsecmeas}. We measure an inclusive parton-level asymmetry of $0.164 \pm 0.039(\rm stat)\pm 0.026(\rm syst) = 0.164 \pm 0.047$. At the parton level, the observed inclusive asymmetry is non-zero with a significance of $3.5\sigma$ and exceeds the NLO prediction of \textsc{powheg}\xspace by $1.9\sigma$, where we have included a $30\%$ uncertainty on the prediction.
\section{The dependence of the asymmetry on $|\Delta y|$}\label{sec:afb_v_dy}
The dependence of $A_\mathrm{FB}$ on the rapidity difference $|\Delta y|$ was studied in the $5~ {\rm fb}^{-1} $ analyses~\cite{cdfafb,d0afb}, but with only two bins of $|\Delta y|$. The CDF and D0 results were consistent and showed a rise of $A_\mathrm{FB}$ with increasing $|\Delta y|$. We perform a more detailed study of the rapidity dependence of $A_\mathrm{FB}$ using the full data set and improved analysis techniques.
The forward-backward asymmetry as a function of $|\Delta y|$ at the reconstruction level can be derived from the data shown in Fig.~\ref{fig:qdely} according to
\begin{equation}
A_\mathrm{FB}(|\Delta y|) = \frac{N_F(|\Delta y|) - N_B(|\Delta y|)}{N_F(|\Delta y|) + N_B(|\Delta y|)},
\label{afbvdy_data}
\end{equation}
\noindent where $N_F(|\Delta y|)$ is the number of events in a given $|\Delta y|$ bin with $\Delta y > 0$ and $N_B(|\Delta y|)$ is the number of events in the corresponding $|\Delta y|$ bin with $\Delta y < 0$. One important constraint on the $\Delta y$ dependance of the asymmetry may be anticipated: any theory that predicts a continuous and differentiable $\Delta y$ distribution must have $A_\mathrm{FB}(|\Delta y|=0)=0$, regardless of the size of the inclusive asymmetry.
Figure~\ref{fig:afb_v_dely} shows $A_\mathrm{FB}(|\Delta y|)$ in four bins of $|\Delta y|$, with the measured values and their uncertainties listed in Table~\ref{tab:afb_v_dy_reco}. To quantify the behavior in a simple way, we assume a linear relationship, which provides a good approximation of both the data and the \textsc{powheg}\xspace prediction (see also Ref.~\cite{almeida}). From the theoretical considerations described above, we make the assumption $A_\mathrm{FB}(|\Delta y|=0)=0$ and fit the slope only. The slope $\alpha_{\Delta y}$ of the line does not correspond to a specific parameter of any particular theory, but provides a quantitative comparison of the $|\Delta y|$ dependence of the asymmetry in the data and prediction. The measurements of $A_\mathrm{FB}(|\Delta y|)$ in data at the reconstruction level are well-fit by a line with a $\chi^2$ per degree of freedom of $1.7/3$ and a slope $\alpha_{\Delta y} = (11.4 \pm 2.5) \times 10^{-2}$, a rapidity dependence that is non-zero with significance in excess of $4 \sigma$. The predicted slope from \textsc{powheg}\xspace and the background model is $(3.6 \pm 0.9) \times 10^{-2}$.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig14}
\caption{{\small The reconstruction-level forward-backward asymmetry as a function of $|\Delta y|$ with a best-fit line superimposed. The errors on the data are statistical, and the shaded region represents the uncertainty on the slope of the prediction.} \label{fig:afb_v_dely}}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig15}
\caption{{\small The background-subtracted asymmetry as a function of $|\Delta y|$ with a best-fit line superimposed. Error bars include both statistical and background-related systematic uncertainties. The shaded region represents the theoretical uncertainty on the slope of the prediction.} \label{fig:afb_v_dely_signal}}
\end{center}
\end{figure}
\begin{table*}[!htb]
\caption{The asymmetry at the reconstructed level as measured in the data, compared to the SM $t\bar{t}$ plus background expectation, as a function of $|\Delta y|$.}\label{tab:afb_v_dy_reco}
\begin{center}
\begin{tabular}{c c c }
\hline
\hline
& Data & SM $t\bar{t}$ + Bkg. \\
$|\Delta y|$ & $A_\mathrm{FB}$ $\pm$ stat & $A_\mathrm{FB}$ \\
\hline
0.0 - 0.5 & \phantom{0}0.016 $\pm$ 0.028\phantom{0} & \phantom{0}0.001 $\pm$ 0.005\phantom{0} \\
0.5 - 1.0 & \phantom{0}0.055 $\pm$ 0.035\phantom{0} & \phantom{0}0.020 $\pm$ 0.012\phantom{0} \\
1.0 - 1.5 & \phantom{0}0.186 $\pm$ 0.049\phantom{0} & \phantom{0}0.050 $\pm$ 0.021\phantom{0} \\
$\ge$ 1.5 & \phantom{0}0.206 $\pm$ 0.085\phantom{0} & \phantom{0}0.109 $\pm$ 0.030\phantom{0} \\
\hline \hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}[!htb]
\caption{The asymmetry at the background-subtracted level as measured in the data, compared to the SM $t\bar{t}$ expectation, as a function of $|\Delta y|$.}\label{tab:afb_v_dy_signal}
\begin{center}
\begin{tabular}{c c c }
\hline
\hline
& Data & SM $t\bar{t}$ \\
$|\Delta y|$ & $A_\mathrm{FB}$ $\pm$ (stat$+$syst) & $A_\mathrm{FB}$ \\
\hline
0.0 - 0.5 & \phantom{0}0.027 $\pm$ 0.034\phantom{0} & \phantom{0}0.009 $\pm$ 0.005\phantom{0} \\
0.5 - 1.0 & \phantom{0}0.086 $\pm$ 0.045\phantom{0} & \phantom{0}0.040 $\pm$ 0.014\phantom{0} \\
1.0 - 1.5 & \phantom{0}0.246 $\pm$ 0.063\phantom{0} & \phantom{0}0.074 $\pm$ 0.026\phantom{0} \\
$\ge$ 1.5 & \phantom{0}0.254 $\pm$ 0.124\phantom{0} & \phantom{0}0.113 $\pm$ 0.039\phantom{0} \\
\hline \hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig16}
\caption{{\small The parton-level forward-backward asymmetry as a function of $|\Delta y|$ with a best-fit line superimposed. Uncertainties are correlated and include both statistical and systematic contributions. The shaded region represents the theoretical uncertainty on the slope of the prediction.} \label{fig:afb_v_dely_parton}}
\end{center}
\end{figure}
The behavior of the asymmetry as a function of $|\Delta y|$ is also measured after the removal of the background contribution as described previously. Figure~\ref{fig:afb_v_dely_signal} shows the distribution $A_\mathrm{FB}(|\Delta y|)$ for the background-subtracted data, with the measured values summarized in Table~\ref{tab:afb_v_dy_signal}. Systematic uncertainties on the background-subtraction procedure are included in the error bars. The data measurements and the predictions are well-fitted by the linear assumption, with an observed slope of $\alpha_{\Delta y} = (15.5 \pm 3.3) \times 10^{-2}$ that exceeds the prediction of $(5.3 \pm 1.0) \times 10^{-2}$ by approximately $3\sigma$. The observed slope is larger than at the reconstruction level owing to the removal of the background, with the significance of the difference relative to the standard model staying approximately the same.
The $|\Delta y|$ dependence of the asymmetry at the parton level can be derived from Fig.~\ref{fig:dely_parton} by comparing the forward and backward bins corresponding to a given value of $|\Delta y|$. This parton-level $A_\mathrm{FB}(|\Delta y|)$ distribution is shown in Fig.~\ref{fig:afb_v_dely_parton}, with the asymmetries in each bin also listed in Table~\ref{tab:afb_dely_parton}. A linear fit to the parton-level results yields a slope $\alpha_{\Delta y}$ = $(25.3 \pm 6.2) \times 10^{-2}$, compared to an expected slope of $(9.7 \pm 1.5) \times 10^{-2}$. We use the full covariance matrix (including both statistical and systematic contributions) for the corrected $A_\mathrm{FB}$ values when minimizing $\chi^{2}$ in order to account for the correlations between bins in the parton-level distribution.
\begin{table*}[!htb]
\caption{The asymmetry at the parton level as measured in the data, compared to the SM $t\bar{t}$ expectation, as a function of $|\Delta y|$.}\label{tab:afb_dely_parton}
\begin{center}
\begin{tabular}{c c c }
\hline
\hline
Parton level & Data & SM $t\bar{t}$ \\
$|\Delta y|$ & $A_\mathrm{FB}$ $\pm$ stat $\pm$ syst & $A_\mathrm{FB}$ \\
\hline
0.0 - 0.5 & \phantom{0}0.048 $\pm$ 0.034 $\pm$ 0.025\phantom{0} & \phantom{0}0.023 $\pm$ 0.007\phantom{0} \\
0.5 - 1.0 & \phantom{0}0.180 $\pm$ 0.057 $\pm$ 0.046\phantom{0} & \phantom{0}0.072 $\pm$ 0.022\phantom{0} \\
1.0 - 1.5 & \phantom{0}0.356 $\pm$ 0.080 $\pm$ 0.036\phantom{0} & \phantom{0}0.119 $\pm$ 0.036\phantom{0} \\
$\ge 1.5$ & \phantom{0}0.477 $\pm$ 0.132 $\pm$ 0.074\phantom{0} & \phantom{0}0.185 $\pm$ 0.056\phantom{0} \\
\hline
$< 1.0$ & \phantom{0}0.101 $\pm$ 0.040 $\pm$ 0.029\phantom{0} & \phantom{0}0.043 $\pm$ 0.013\phantom{0} \\
$\ge 1.0$ & \phantom{0}0.392 $\pm$ 0.093 $\pm$ 0.043\phantom{0} & \phantom{0}0.139 $\pm$ 0.042\phantom{0} \\
\hline \hline
\end{tabular}
\end{center}
\end{table*}
\section{Dependence of the asymmetry on $M_{t\bar{t}}$}\label{sec:afb_v_mtt}
The dependence of $A_\mathrm{FB}$ on the invariant mass of the $t\bar{t}$ system was also studied in the $5~ {\rm fb}^{-1} $ analyses~\cite{cdfafb,d0afb} with only two bins. $M_{t\bar{t}}$ is correlated with the rapidity difference $\Delta y$, but because $\Delta y$ depends on the top-quark production angle in addition to $M_{t\bar{t}}$, a measurement of the $M_{t\bar{t}}$ dependence can provide additional information about the underlying asymmetry relative to the $A_\mathrm{FB}(|\Delta y|)$ measurement. In the previous publications~\cite{cdfafb,d0afb}, the CDF and D0 measurements of $A_\mathrm{FB}$ at small and large $M_{t\bar{t}}$ were consistent within statistical uncertainties but had quite different central values, leading to an ambiguity in the comparison of the results and their interpretation. We use the full CDF data set and the new techniques introduced in this analysis to clarify the dependence of $A_\mathrm{FB}$ on $M_{t\bar{t}}$.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig17}
\caption{{\small The reconstruction-level forward-backward asymmetry as a function of $M_{t\bar{t}}$ with a best-fit line superimposed. The last bin contains overflow events. The errors on the data are statistical, and the shaded region represents the uncertainty on the slope of the prediction.} \label{fig:afb_v_mtt}}
\end{center}
\end{figure}
\begin{figure}[!htbp]
\begin{center}
\subfigure[]{
\includegraphics[width=0.45\textwidth, clip]{fig18a}\label{fig:mtt_forbac_sig}}
\vspace*{0.15in}
\subfigure[]{
\includegraphics[width=0.45\textwidth, clip]{fig18b}\label{fig:afb_v_mtt_sig}}
\caption{{\small \subref{fig:mtt_forbac_sig} $M_{t\bar{t}}$ after background subtraction in events with positive and negative $\Delta y$ and \subref{fig:afb_v_mtt_sig} background-subtracted $A_\mathrm{FB}$ as a function of $M_{t\bar{t}}$ with a best-fit line superimposed. The last bin contains overflow events. Error bars include both statistical and background-related systematic uncertainties. The shaded region in \subref{fig:afb_v_mtt_sig} represents the theoretical uncertainty on the slope of the prediction.} \label{fig:mtt_fb_signal}}
\end{center}
\end{figure}
\begin{figure*}[!htbp]
\begin{center}
\subfigure[]{
\includegraphics[width=0.45\textwidth, clip]{fig19a}\label{fig:dely_lo_sig}}
\subfigure[]{
\includegraphics[width=0.45\textwidth, clip]{fig19b}\label{fig:dely_hi_sig}}
\caption{{\small The background-subtracted $\Delta y$ distributions for events with \subref{fig:dely_lo_sig} $M_{t\bar{t}} < 450~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$ and \subref{fig:dely_hi_sig} $M_{t\bar{t}} \ge 450~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$. Error bars include both statistical and background-related systematic uncertainties.} \label{fig:dely_hilomass_signal}}
\end{center}
\end{figure*}
We start at the detector level, where we divide the data into several mass bins and determine the number of events with positive ($N_{F}$) and negative ($N_{B}$) $\Delta y$ in each bin, from which we calculate the asymmetry as a function of $M_{t\bar{t}}$ according to
\begin{equation}
A_\mathrm{FB}(M_{t\bar{t}}) = \frac{N_F(M_{t\bar{t}}) - N_B(M_{t\bar{t}})}{N_F(M_{t\bar{t}}) + N_B(M_{t\bar{t}})}.
\label{afbvm_data}
\end{equation}
\noindent The $M_{t\bar{t}}$-dependent asymmetry is compared to the NLO $t\bar{t}$ plus background prediction in Fig.~\ref{fig:afb_v_mtt} and Table~\ref{tab:afb_v_mtt_reco}. The $M_{t\bar{t}}$ spectrum is divided into intervals of $50~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$ below $600~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$ and $100~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$ intervals above $600~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$, with the final bin containing overflow events. The $M_{t\bar{t}}$ resolution across this range varies as a function of mass, being approximately $50~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$ at the lowest masses and increasing to near $100~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$ at very high mass. A linear fit of the observed data has $\chi^{2}/N_{dof} = 1.0/5$ and yields a slope of $\alpha_{M_{t\bar{t}}} = (8.9 \pm 2.3) \times 10^{-4}~(\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi)^{-1}$, which is non-zero with significance in excess of $3 \sigma$. The predicted slope at the reconstruction level is $(2.4 \pm 0.6) \times 10^{-4}~(\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi)^{-1}$.
\begin{table*}[!htb]
\caption{The asymmetry observed in the reconstructed data, compared to the SM $t\bar{t}$ plus background expectation, as a function of $M_{t\bar{t}}$.}\label{tab:afb_v_mtt_reco}
\begin{center}
\begin{tabular}{c c c }
\hline
\hline
& Data & SM $t\bar{t}$ + Bkg. \\
$M_{t\bar{t}}$ ($\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$) & $A_\mathrm{FB}$ $\pm$ stat & $A_\mathrm{FB}$ \\
\hline
$< 400$ & $-$0.005 $\pm$ 0.030\phantom{0} & \phantom{0}0.002 $\pm$ 0.006\phantom{0} \\
400 - 450 & \phantom{$-$}0.053 $\pm$ 0.039\phantom{0} & \phantom{0}0.017 $\pm$ 0.010\phantom{0} \\
450 - 500 & \phantom{$-$}0.118 $\pm$ 0.050\phantom{0} & \phantom{0}0.028 $\pm$ 0.012\phantom{0} \\
500 - 550 & \phantom{$-$}0.152 $\pm$ 0.067\phantom{0} & \phantom{0}0.040 $\pm$ 0.018\phantom{0} \\
550 - 600 & \phantom{$-$}0.128 $\pm$ 0.086\phantom{0} & \phantom{0}0.067 $\pm$ 0.025\phantom{0} \\
600 - 700 & \phantom{$-$}0.275 $\pm$ 0.101\phantom{0} & \phantom{0}0.054 $\pm$ 0.024\phantom{0} \\
$\ge 700$ & \phantom{$-$}0.294 $\pm$ 0.134\phantom{0} & \phantom{0}0.101 $\pm$ 0.042\phantom{0} \\
\hline \hline
\end{tabular}
\end{center}
\end{table*}
After removing the background contribution, Fig.~\ref{fig:mtt_forbac_sig} compares the observed $M_{t\bar{t}}$ distributions in forward and backward events, with an excess of forward events in many bins. These distributions are converted into asymmetries as a function of $M_{t\bar{t}}$, as shown in Fig.~\ref{fig:afb_v_mtt_sig} and summarized in Table~\ref{tab:afb_v_mtt_signal}. The linear fit to the background-subtracted asymmetries yields $\chi^{2}/N_{dof}= 1.1/5$ and a slope of $(10.9 \pm 2.8) \times 10^{-4}~(\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi)^{-1}$, with the predicted slope being $(3.0 \pm 0.7) \times 10^{-4}~(\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi)^{-1}$.
\begin{table*}[!htb]
\caption{The asymmetry at the background-subtracted level as measured in the data, compared to the SM $t\bar{t}$ expectation, as a function of $M_{t\bar{t}}$.}\label{tab:afb_v_mtt_signal}
\begin{center}
\begin{tabular}{c c c }
\hline
\hline
& Data & SM $t\bar{t}$ \\
$M_{t\bar{t}}$ ($\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$) & $A_\mathrm{FB}$ $\pm$ (stat$+$syst) & $A_\mathrm{FB}$ \\
\hline
$< 400$ & \phantom{$-$}0.003 $\pm$ 0.038\phantom{0} & \phantom{0}0.012 $\pm$ 0.006\phantom{0} \\
400 - 450 & \phantom{$-$}0.076 $\pm$ 0.049\phantom{0} & \phantom{0}0.031 $\pm$ 0.011\phantom{0} \\
450 - 500 & \phantom{$-$}0.149 $\pm$ 0.061\phantom{0} & \phantom{0}0.039 $\pm$ 0.015\phantom{0} \\
500 - 550 & \phantom{$-$}0.198 $\pm$ 0.083\phantom{0} & \phantom{0}0.060 $\pm$ 0.022\phantom{0} \\
550 - 600 & \phantom{$-$}0.156 $\pm$ 0.104\phantom{0} & \phantom{0}0.083 $\pm$ 0.030\phantom{0} \\
600 - 700 & \phantom{$-$}0.361 $\pm$ 0.128\phantom{0} & \phantom{0}0.077 $\pm$ 0.028\phantom{0} \\
$\ge 700$ & \phantom{$-$}0.369 $\pm$ 0.159\phantom{0} & \phantom{0}0.137 $\pm$ 0.049\phantom{0} \\
\hline \hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure}
\begin{center}
\subfigure[]{
\includegraphics[width=0.45\textwidth, clip]{fig20a}\label{fig:mtt_forbac_parton}}
\vspace*{0.15in}
\subfigure[]{
\includegraphics[width=0.45\textwidth, clip]{fig20b}\label{fig:afb_v_mtt_parton}}
\caption{{\small \subref{fig:mtt_forbac_parton} The parton-level $M_{t\bar{t}}$ distributions for events with positive and negative $\Delta y$ and \subref{fig:afb_v_mtt_parton} the parton-level forward-backward asymmetry as a function of $M_{t\bar{t}}$ with a best-fit line superimposed. The last bin contains overflow events. Uncertainties are correlated and include both statistical and systematic contributions. The shaded region in \subref{fig:afb_v_mtt_parton} represents the theoretical uncertainty on the slope of the prediction.} \label{fig:mtt_fb_parton}}
\end{center}
\end{figure}
At the background-subtracted level, we divide the data into two regions of $M_{t\bar{t}}$ (above and below $ 450~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$) for direct comparison to the $5.3~ {\rm fb}^{-1} $ CDF analysis~\cite{cdfafb}. The $\Delta y$ distributions at high and low mass are shown in Fig.~\ref{fig:dely_hilomass_signal}, yielding asymmetries of $0.030 \pm 0.031$ for $M_{t\bar{t}} < 450~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$ and $0.197 \pm 0.043$ for $M_{t\bar{t}}\geq 450~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$, where the uncertainties include statistical and background-related systematic contributions. These are in good agreement with the values from the $5.3~ {\rm fb}^{-1} $ analysis, which found background-subtracted asymmetries of $-0.022 \pm 0.043$ for $M_{t\bar{t}} < 450~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$ and $0.266 \pm 0.62$ for $M_{t\bar{t}}\geq 450~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$~\cite{cdfafb}. To check against potential systematic effects, the behavior of the background-subtracted asymmetry at high and low $M_{t\bar{t}}$ in various subsets of the data is summarized in Table~\ref{tab:afb_splits_signal}. The $M_{t\bar{t}}$ dependence is consistent across lepton charge and lepton type. It is consistent (within relatively large statistical uncertainties) across single- and double-$b$-tagged events. The asymmetry is larger in events with exactly four jets than it is in events with at least five jets, an effect that is discussed further in Sec.~\ref{sec:afb_v_pt}.
\begin{table*}[!htb]
\caption{Various measured asymmetries after background subtraction, inclusively and at small and large $M_{t\bar{t}}$.}\label{tab:afb_splits_signal}
\begin{center}
\begin{tabular}{l c c c}
\hline
\hline
& \multicolumn{3}{c}{$A_\mathrm{FB}$ $\pm$ (stat$+$syst)} \\
Sample & \phantom{0}Inclusive\phantom{0} & \phantom{0}$M_{t\bar{t}} < 450~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$\phantom{0} & \phantom{0}$M_{t\bar{t}} \ge 450~\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$\phantom{0} \\
\hline
All data & \phantom{0}0.087 $\pm$ 0.026\phantom{0} & \phantom{0$-$}0.030 $\pm$ 0.031\phantom{0} & \phantom{0}0.197 $\pm$ 0.043\phantom{0} \\
Positive leptons & \phantom{0}0.094 $\pm$ 0.036\phantom{0} & \phantom{0$-$}0.034 $\pm$ 0.044\phantom{0} & \phantom{0}0.207 $\pm$ 0.060\phantom{0} \\
Negative leptons & \phantom{0}0.080 $\pm$ 0.035\phantom{0} & \phantom{0$-$}0.027 $\pm$ 0.043\phantom{0} & \phantom{0}0.186 $\pm$ 0.057\phantom{0} \\
Exactly 1 $b$ tags &\phantom{0}0.100 $\pm$ 0.031\phantom{0} & \phantom{0$-$}0.047 $\pm$ 0.036\phantom{0} & \phantom{0}0.220 $\pm$ 0.049\phantom{0} \\
At least 2 $b$ tags & \phantom{0}0.037 $\pm$ 0.045\phantom{0} & \phantom{0}$-$0.018 $\pm$ 0.055\phantom{0} & \phantom{0}0.134 $\pm$ 0.073\phantom{0} \\
Electrons & \phantom{0}0.079 $\pm$ 0.039\phantom{0} & \phantom{0$-$}0.017 $\pm$ 0.047\phantom{0} & \phantom{0}0.195 $\pm$ 0.062\phantom{0} \\
Muons & \phantom{0}0.094 $\pm$ 0.033\phantom{0} & \phantom{0$-$}0.041 $\pm$ 0.040\phantom{0} & \phantom{0}0.197 $\pm$ 0.055\phantom{0} \\
Exactly 4 jets & \phantom{0}0.110 $\pm$ 0.031\phantom{0} & \phantom{0$-$}0.029 $\pm$ 0.037\phantom{0} & \phantom{0}0.256 $\pm$ 0.049\phantom{0} \\
At least 5 jets & \phantom{0}0.033 $\pm$ 0.044\phantom{0} & \phantom{0$-$}0.034 $\pm$ 0.053\phantom{0} & \phantom{0}0.033 $\pm$ 0.077\phantom{0} \\
\hline \hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}[!htb]
\caption{The asymmetry at the parton level as measured in the data, compared to the SM $t\bar{t}$ expectation, as a function of $M_{t\bar{t}}$.}\label{tab:afb_mtt_parton}
\begin{center}
\begin{tabular}{c c c }
\hline
\hline
Parton level & Data & SM $t\bar{t}$ \\
$M_{t\bar{t}}$ ($\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi$) & $A_\mathrm{FB}$ $\pm$ stat $\pm$ syst & $A_\mathrm{FB}$ \\
\hline
$< 450$ & \phantom{0}0.084 $\pm$ 0.046 $\pm$ 0.030\phantom{0} & \phantom{0}0.047 $\pm$ 0.014\phantom{0} \\
450 - 550 & \phantom{0}0.255 $\pm$ 0.062 $\pm$ 0.034\phantom{0} & \phantom{0}0.090 $\pm$ 0.027\phantom{0} \\
550 - 650 & \phantom{0}0.370 $\pm$ 0.084 $\pm$ 0.087\phantom{0} & \phantom{0}0.117 $\pm$ 0.035\phantom{0} \\
$\ge 650$ & \phantom{0}0.493 $\pm$ 0.158 $\pm$ 0.110\phantom{0} & \phantom{0}0.143 $\pm$ 0.043\phantom{0} \\
\hline
$< 450$ & \phantom{0}0.084 $\pm$ 0.046 $\pm$ 0.030\phantom{0} & \phantom{0}0.047 $\pm$ 0.014\phantom{0} \\
$\ge 450$ & \phantom{0}0.295 $\pm$ 0.058 $\pm$ 0.033\phantom{0} & \phantom{0}0.100 $\pm$ 0.030\phantom{0} \\
\hline \hline
\end{tabular}
\end{center}
\end{table*}
We determine the parton-level mass dependence of $A_\mathrm{FB}$ by correcting the $\Delta y$ and $M_{t\bar{t}}$ distributions simultaneously. To do so, we apply the unfolding procedure to a two-dimensional distribution consisting of two bins in $\Delta y$ (for forward and backward events) and four bins in $M_{t\bar{t}}$. Since regularization makes use of the second-derivative matrix, which is not well-defined for a two-bin distribution, the regularization constraint is applied only along the $M_{t\bar{t}}$ dimension. The resulting $M_{t\bar{t}}$ distributions for forward and backward events are shown in Fig.~\ref{fig:mtt_forbac_parton}. These distributions are combined to determine the differential asymmetry as a function of $M_{t\bar{t}}$ shown in Fig.~\ref{fig:afb_v_mtt_parton} and summarized in Table~\ref{tab:afb_mtt_parton}. The best-fit line to the measured data asymmetries at parton level has a slope $\alpha_{M_{t\bar{t}}}$ = $(15.5 \pm 4.8) \times 10^{-4}~(\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi)^{-1}$, compared to the \textsc{powheg}\xspace prediction of $(3.4 \pm 1.2) \times 10^{-4}~(\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi)^{-1}$.
\section{Determination of the significance of the dependence of the asymmetry on $|\Delta y|$ and $M_{t\bar{t}}$}\label{sec:sig}
The slopes of the linear dependencies of $A_\mathrm{FB}$ on $|\Delta y|$ and $M_{t\bar{t}}$ provide a measure of the consistency between the data and the SM prediction. We quantify this consistency in a more rigorous manner by repeating the measurement on large ensembles of simulated experiments generated according to the SM prediction and determining the probabilities, or p-values, for observing the actual data given the SM assumption. Each p-value is defined as the fraction of simulated experiments in which the measured slopes are at least as large as those found in the data, $\alpha_{\Delta y,M_{t\bar{t}}}^{\rm simulated} \ge \alpha_{\Delta y,M_{t\bar{t}}}^{\rm data}$.
We use the background-subtracted sample for measuring these p-values because it provides access to an asymmetry calculation that has been corrected for background but is still independent of the assumptions made when using a regularized unfolding procedure to extract parton-level information. We start from the predicted distribution at the reconstruction level, created from the standard model predictions of \textsc{powheg}\xspace and the various background contributions proportioned as in Table~\ref{tab:method2}. The population of each bin of this predicted distribution is fluctuated within its uncertainty, which includes the statistical uncertainty on the contents of that bin, the systematic uncertainties on the various background contributions, as described in Sec.~\ref{sec:bkg_sub} above, and the theoretical uncertainty on the \textsc{powheg}\xspace prediction.
Many systematic uncertainties may in principal simultaneously affect both the signal and background models. However, the theory uncertainty is the dominant uncertainty on the predicted asymmetry (0.011). As a point of comparison, the uncertainty due to jet energy scales is only 0.0008, and the effects of correlations between uncertainties on the $t\bar{t}$ prediction and backgrounds are negligible. For this reason we do not include the effect of correlations between uncertainties on the signal and background models.
For each simulated experiment, the nominal background prediction with the normalizations of Table~\ref{tab:method2} is subtracted, and the slopes of the remaining asymmetries as a function of $|\Delta y|$ and $M_{t\bar{t}}$ are fit. We find p-values of $2.2 \times 10^{-3}$ for $A_\mathrm{FB}(|\Delta y|)$ and $7.4 \times 10^{-3}$ for $A_\mathrm{FB}(M_{t\bar{t}})$, corresponding to $2.8\sigma$ and $2.4\sigma$ discrepancies respectively (based on a one-sided integration of the normal probability distribution).
\section{Dependence of the asymmetry on the transverse momentum of the $t\bar{t}$ system}\label{sec:afb_v_pt}
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig21}
\caption{{\small Interfering $\qqbar\rightarrowt\bar{t}$ (top) and $\qqbar\rightarrowt\bar{t} j$ (bottom) diagrams.} \label{fig:nlo}}
\end{center}
\end{figure}
The QCD asymmetry at NLO arises from the sum of two different effects~\cite{nlotheory}. The interference of the $2 \rightarrow 2$ LO tree-level diagrams (upper left of Fig.~\ref{fig:nlo}) and the NLO box diagrams (upper right) produces a positive asymmetry (``Born-box'' interference), while the interference of $2 \rightarrow 3$ tree-level diagrams with initial-state (lower left) and final-state radiation (lower right) produces a negative asymmetry (``ISR-FSR'' interference). In the latter final state, $t\bar{t}$ plus an additional jet, the $t\bar{t}$ system acquires a transverse momentum $\mbox{${p_T^{t\bar{t}}}$}$, while in the former case with an exclusive $t\bar{t}$ final state, all events have $\mbox{${p_T^{t\bar{t}}}$} = 0$. The resultant SM asymmetry at NLO is therefore the sum of two effects of different sign, with very different $\mbox{${p_T^{t\bar{t}}}$}$ dependence. The virtual effects from Born-box interference are larger, leading to a net positive asymmetry. Recent work has also emphasized that color coherence during the hadronization process can produce a significant $\mbox{${p_T^{t\bar{t}}}$}$ dependence for the asymmetry in Monte Carlo generators that include hadronization~\cite{d0afb}, with the degree of the $\mbox{${p_T^{t\bar{t}}}$}$ dependence varying greatly depending on the details of the implementation of color coherence~\cite{ttpt}. The verification of the $\mbox{${p_T^{t\bar{t}}}$}$ dependence of the asymmetry is therefore crucial to understanding the reliability of the SM predictions for $A_\mathrm{FB}$~\cite{d0afb}, as well as testing for possible new effects beyond the SM.
In this section, we first compare and discuss several predictions for $A_\mathrm{FB}(\mbox{${p_T^{t\bar{t}}}$})$. We then compare the data to two of these predictions (the NLO with hadronization prediction from \textsc{powheg}\xspace and the LO with hadronization prediction from \textsc{pythia}\xspace), showing that the asymmetry in the data displays the same trend with respect to $\mbox{${p_T^{t\bar{t}}}$}$ as observed in both \textsc{powheg}\xspace and \textsc{pythia}\xspace, and that the excess inclusive asymmetry in the data is consistent with a $\mbox{${p_T^{t\bar{t}}}$}$-independent component.
We define the $\mbox{${p_T^{t\bar{t}}}$}$ dependence of the asymmetry as
\begin{equation}
A_\mathrm{FB}(\mbox{${p_T^{t\bar{t}}}$}) = \frac{N_F(\mbox{${p_T^{t\bar{t}}}$}) - N_B(\mbox{${p_T^{t\bar{t}}}$})}{N_F(\mbox{${p_T^{t\bar{t}}}$}) + N_B(\mbox{${p_T^{t\bar{t}}}$})}.
\label{afbvpt_data}
\end{equation}
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig22}
\caption{{\small Expected $A_\mathrm{FB}$ as a function of the $\mbox{${p_T^{t\bar{t}}}$}$ of the $t\bar{t}$ system at the parton level from \textsc{mcfm}\xspace, \textsc{powheg}\xspace, and \textsc{pythia}\xspace, as well as a NLO prediction for events where the top-quark pair is produced in association with an extra energetic jet. } \label{fig:true_afb_vs_pt}}
\end{center}
\end{figure}
\noindent The expected SM parton-level asymmetry is shown for four predictions in Fig.~\ref{fig:true_afb_vs_pt}. The matrix elements for \textsc{pythia}\xspace are LO for $t\bar{t}$ production, with some higher-order effects approximated through hadronization. There is essentially no net inclusive asymmetry in \textsc{pythia}\xspace due to the underlying $2 \rightarrow 2$ matrix elements in the hard-scattering process; gluon emission during hadronization results in a negative asymmetry for non-zero $\mbox{${p_T^{t\bar{t}}}$}$ events, leaving a positive asymmetry in the low-$\mbox{${p_T^{t\bar{t}}}$}$ region.
The other three curves suggest a different behavior for the $\mbox{${p_T^{t\bar{t}}}$}$ dependence at NLO. The \textsc{mcfm}\xspace calculation uses NLO matrix elements for $t\bar{t}$ production, and includes both the Born-box and ISR-FSR interference terms, with the result being a parton-level output with two partons ($t\bar{t}$) or three partons ($t\bar{t}$ plus a gluon) in the final state. In \textsc{mcfm}\xspace, events produced by the virtual matrix elements with Born-box interference have $\mbox{${p_T^{t\bar{t}}}$}=0$ and a positive asymmetry, while events produced by the real matrix elements describing gluon radiation have non-zero $\mbox{${p_T^{t\bar{t}}}$}$ and a negative asymmetry. \textsc{powheg}\xspace has the same NLO matrix elements as \textsc{mcfm}\xspace, with additional higher-order effects approximated through hadronization performed by \textsc{pythia}\xspace. The additional radiation from the hadronization process results in a migration of events in $\mbox{${p_T^{t\bar{t}}}$}$ and thus a moderation of the otherwise bimodal $\mbox{${p_T^{t\bar{t}}}$}$ behavior observed in \textsc{mcfm}\xspace.
The \textsc{powheg}\xspace prediction with \textsc{pythia}\xspace hadronization can be partially checked against a recent NLO calculation for $t\bar{t}$ production in association with an extra energetic jet ($p_T^{\rm jet} > 20~\ifmmode {\rm GeV}/c \else GeV$/c$\fi$ and $|\eta_{\rm jet}| < 2.0$)~\cite{schulze}, shown as ``$t\bar{t}$+jet''. This calculation has a Born-level final state with three partons ($t\bar{t}$ plus a gluon), and thus it is most relevant for comparison to the other predictions at high $\mbox{${p_T^{t\bar{t}}}$}$. It contains virtual matrix elements for the $t\bar{t}$+jet final state as well as real corrections from final states with $t\bar{t}$ and two extra jets. The negative asymmetry observed in the tree-level prediction for $t\bar{t}$+jet (as shown in \textsc{mcfm}\xspace at high-$\mbox{${p_T^{t\bar{t}}}$}$) is reduced with the full NLO calculation of this final state. In the high-$\mbox{${p_T^{t\bar{t}}}$}$ region, we see that the \textsc{powheg}\xspace predictions are in good agreement with those from the NLO $t\bar{t}$+jet calculation.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig23}
\caption{{\small The background-subtracted forward-backward asymmetry in the data as a function of the transverse momentum of the $t\bar{t}$ system, compared to both \textsc{powheg}\xspace and \textsc{pythia}\xspace. Error bars include both statistical and background-related systematic uncertainties. The last bin contains overflow events.} \label{fig:afb_v_pt}}
\end{center}
\end{figure}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig24}
\caption{{\small The forward-backward asymmetry as a function of the transverse momentum of the $t\bar{t}$ system for three models at the background-subtracted level: Octet A, SM \textsc{pythia}\xspace, and SM \textsc{pythia}\xspace normalized by the addition of $\Delta A_{\rm Oct}$. The last bin contains overflow events.} \label{fig:afb_v_pt_pythia_v_octA}}
\end{center}
\end{figure}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.45\textwidth, clip]{fig25}
\caption{{\small The background-subtracted forward-backward asymmetry in the data as a function of the transverse momentum of the $t\bar{t}$ system, compared to both \textsc{powheg}\xspace and \textsc{pythia}\xspace. The model predictions have been normalized by the addition of $\Delta A_{\rm NLO}$ to \textsc{powheg}\xspace and $\Delta A_{\rm LO}$ to \textsc{pythia}\xspace as described in the text. Error bars include both statistical and background-related systematic uncertainties. The last bin contains overflow events.} \label{fig:afb_v_pt_scaled}}
\end{center}
\end{figure}
In Fig.~\ref{fig:ptsys} we show that the reconstructed $\mbox{${p_T^{t\bar{t}}}$}$ spectrum in the data is well-reproduced by the $t\bar{t}$ signal and background model simulations. Building on this, we study the $\mbox{${p_T^{t\bar{t}}}$}$ dependence of the asymmetry in the data. Figure~\ref{fig:afb_v_pt} shows $A_\mathrm{FB}(\mbox{${p_T^{t\bar{t}}}$})$ for the data after background subtraction compared to predictions from \textsc{powheg}\xspace (hadronized with \textsc{pythia}\xspace) and from \textsc{pythia}\xspace. The trends of the parton-level curves in Fig.~\ref{fig:true_afb_vs_pt} are reproduced: the LO prediction has a steady drop, while the NLO prediction tends to zero or slightly below. The data show a clear decrease with $\mbox{${p_T^{t\bar{t}}}$}$, but lie above the models. We investigate this using the ansatz that the data contain an additional source of asymmetry that is independent of $\mbox{${p_T^{t\bar{t}}}$}$. In this case, because independent asymmetries are additive, it should be possible to normalize the model predictions to the data by adding a constant offset $\Delta A$ that is equal to the excess observed inclusive asymmetry in the data.
We test this ansatz using the color-octet model Octet A (implemented in \textsc{madgraph} with hadronization performed by \textsc{pythia}\xspace) described at the end of Sec.~\ref{sec:mc}. In this LO model, the octet physics induces a $\mbox{${p_T^{t\bar{t}}}$}$-independent inclusive $t\bar{t}$ asymmetry 0.106 at the background-subtracted level (we neglect very small statistical uncertainties in these large Monte Carlo samples). We wish to compare the $\mbox{${p_T^{t\bar{t}}}$}$ dependence of this asymmetry to the LO \textsc{pythia}\xspace model, which has a background-subtracted asymmetry of $-0.021$. The inclusive difference is $\Delta A_{\rm Oct} = 0.127$. If the excess asymmetry in Octet A is independent of $\mbox{${p_T^{t\bar{t}}}$}$, we expect that $A_\mathrm{FB}^{\rm \textsc{pythia}\xspace}(\mbox{${p_T^{t\bar{t}}}$}) + \Delta A_{\rm Oct}$ reproduces satisfactorily $A_\mathrm{FB}^{\rm Octet~A}(\mbox{${p_T^{t\bar{t}}}$})$. Figure~\ref{fig:afb_v_pt_pythia_v_octA} shows this test in the simulated samples, with the $\mbox{${p_T^{t\bar{t}}}$}$-dependent behavior of Octet A being described well by the addition of the constant normalization factor $\Delta A_{\rm Oct}$ to $A_\mathrm{FB}^{\rm \textsc{pythia}\xspace}(\mbox{${p_T^{t\bar{t}}}$})$.
\begin{figure*}[!htbp]
\begin{center}
\subfigure[]{
\includegraphics[width=0.45\textwidth, clip]{fig26a}\label{fig:corr_steps_dy}}
\subfigure[]{
\includegraphics[width=0.45\textwidth, clip]{fig26b}\label{fig:corr_steps_mtt}}
\caption{{\small The measured asymmetries as a function of \subref{fig:corr_steps_dy} $|\Delta y|$ and \subref{fig:corr_steps_mtt} $M_{t\bar{t}}$ in the data at the three different levels of correction. The error bars include both statistical uncertainties and the appropriate systematic uncertainties for each correction level as described in the text. The last bin contains overflow events.} \label{fig:corr_steps}}
\end{center}
\end{figure*}
\begin{table*}[!htb]
\caption{The measured inclusive forward-backward asymmetry and the best-fit slopes for $A_\mathrm{FB}(|\Delta y|)$ and $A_\mathrm{FB}(M_{t\bar{t}})$ at the different levels of correction. The uncertainties include the statistical uncertainties and the appropriate systematic uncertainties for each correction level as discussed in the text.}\label{tab:corr_steps}\begin{center}
\begin{tabular}{c c c c}
\hline
\hline
& \phantom{0}Inclusive\phantom{0} & \phantom{0}Slope\phantom{0} & \phantom{0} Slope\phantom{0} \\
Correction level & \phantom{0}$A_\mathrm{FB}$\phantom{0} & \phantom{0}$\alpha_{\Delta y}$\phantom{0} & \phantom{0}$\alpha_{M_{t\bar{t}}}\phantom{0}$ \\
\hline
Reconstruction & \phantom{0}0.063 $\pm$ 0.019\phantom{0} & \phantom{0}$(11.4 \pm 2.5) \times 10^{-2}$\phantom{0} & \phantom{0} $(8.9 \pm 2.3) \times 10^{-4}~(\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi)^{-1}$\phantom{0} \\
Background-subtracted & \phantom{0}0.087 $\pm$ 0.026\phantom{0} & \phantom{0}$(15.5 \pm 3.3) \times 10^{-2}$\phantom{0} & \phantom{0}$(10.9 \pm 2.8) \times 10^{-4}~(\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi)^{-1}$\phantom{0} \\
Parton & \phantom{0}0.164 $\pm$ 0.047\phantom{0} & \phantom{0}$(25.3 \pm 6.2) \times 10^{-2}$\phantom{0} & \phantom{0}$(15.5 \pm 4.8) \times 10^{-4}~(\ifmmode {\rm GeV}/c^2 \else GeV$/c^2$\fi)^{-1}$\phantom{0} \\
\hline \hline
\end{tabular}
\end{center}
\end{table*}
We use this procedure to normalize the $A_\mathrm{FB}(\mbox{${p_T^{t\bar{t}}}$})$ models of \textsc{powheg}\xspace and \textsc{pythia}\xspace to the total inclusive asymmetry observed in the data. Since this artificial procedure adjusts the mean values such that they are exactly equal, we do not assign uncertainties to the offsets. The asymmetry after background subtraction is $0.087$ in the data, $0.033$ in NLO \textsc{powheg}\xspace (Table~\ref{tab:inc_asyms_posneg}), and $-0.021$ in LO \textsc{pythia}\xspace, resulting in offset terms $\Delta A_{\rm NLO} = 0.054$ and $\Delta A_{\rm LO} = 0.108$.
The normalized $A_\mathrm{FB}(\mbox{${p_T^{t\bar{t}}}$})$ models are compared to the data in Fig.~\ref{fig:afb_v_pt_scaled}. Within the experimental uncertainties, the $A_\mathrm{FB}(\mbox{${p_T^{t\bar{t}}}$})$ behavior of the data is described well by both models. We conclude that the excess asymmetry in the data is consistent with being independent of $\mbox{${p_T^{t\bar{t}}}$}$.
Finally we note the connection between $\mbox{${p_T^{t\bar{t}}}$}$ and jet multiplicity. In events with one or more extra energetic jets, we expect the $t\bar{t}$ system to have large $\mbox{${p_T^{t\bar{t}}}$}$ due to recoil against these additional jets. In Table~\ref{tab:afb_splits_signal} a difference was noted in the asymmetry measurements at the background-subtracted level between events with exactly four jets and at least five jets. Rephrasing this in terms of $\mbox{${p_T^{t\bar{t}}}$}$, we find that the mean $\mbox{${p_T^{t\bar{t}}}$}$ in five-jet events is $34.4 \pm 0.6~\ifmmode {\rm GeV}/c \else GeV$/c$\fi$ compared to $18.6 \pm 0.3~\ifmmode {\rm GeV}/c \else GeV$/c$\fi$ in events with only four jets. The smaller asymmetry in events with extra jets is seen to be consistent with the observed $A_\mathrm{FB}(\mbox{${p_T^{t\bar{t}}}$})$ behavior.
\section{Conclusions}
We study the forward-backward asymmetry $A_\mathrm{FB}$ in top-quark pair production using the full CDF Run~II data set. Using the reconstructed $t\bar{t}$ rapidity difference in the detector frame, after removal of backgrounds, we observe an inclusive asymmetry of $0.063 \pm 0.019$(stat) compared to $0.020 \pm 0.012$ expected from the NLO standard model (with both QCD and electroweak contributions). Looking differentially, the asymmetry is found to have approximately linear dependence on both $|\Delta y|$ and $M_{t\bar{t}}$, as expected for the NLO charge asymmetry, although with larger slopes then the NLO prediction. The probabilities to observe the measured values or larger for the detector-level dependencies are $2.8\sigma$ and $2.4\sigma$ for $|\Delta y|$ and $M_{t\bar{t}}$ respectively.
The results are corrected to the parton level to find the differential cross section $d\sigma/d(\Delta y)$, where we measure an inclusive parton-level asymmetry of $0.164 \pm 0.047$(stat$+$syst).The asymmetries and their functional dependencies at the three stages of the analysis procedure are summarized in Fig.~\ref{fig:corr_steps} and Table~\ref{tab:corr_steps}.
We also study the dependence of $A_\mathrm{FB}$ on the transverse momentum of the $t\bar{t}$ system. We find a significant momentum dependence that is consistent with either of the LO or NLO predictions, and evidence that the excess asymmetry is independent of the momentum.
This new measurement of the top quark production asymmetry serves as a means to better understand higher-order corrections to the standard model or potential effects from non-standard model processes.
\section*{Acknowledgments}
We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Culture, Sports, Science and Technology of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; the Swiss National Science Foundation; the A.P. Sloan Foundation; the Bundesministerium f\"ur Bildung und Forschung, Germany; the Korean World Class University Program, the National Research Foundation of Korea; the Science and Technology Facilities Council and the Royal Society, UK; the Russian Foundation for Basic Research; the Ministerio de Ciencia e Innovaci\'{o}n, and Programa Consolider-Ingenio 2010, Spain; the Slovak R\&D Agency; the Academy of Finland; and the Australian Research Council (ARC).
|
1,314,259,996,502 | arxiv | \section*{}
\vspace{-1cm}
\footnotetext{\textit{$^{a}$~Institut f\"ur Theoretische Physik, Westf\"alische Wilhelms-Universit\"at M\"unster, Wilhelm-Klemm-Str.\ 9, 48149 M\"unster, Germany.}}
\footnotetext{\textit{$^{b}$~Center for Nonlinear Science (CeNoS), Westf{\"a}lische Wilhelms-Universit\"at M\"unster, Corrensstr.\ 2, 48149 M\"unster, Germany}}
\footnotetext{$\ast$~E-mail:~daniel.greve@wwu.de}
\footnotetext{$\dag$~E-mail:~s.hartmann@wwu.de, ORCID:~0000-0002-3127-136X}
\footnotetext{$\ddag$~E-mail:~u.thiele@uni-muenster.de, ORCID:~0000-0001-7989-9271}
\section{Introduction}\label{sec:intro}
Many hydrodynamic processes of practical importance involve the motion of three-phase contact lines. In consequence, for many years phenomena involving static and dynamic contact lines have been of much interdisciplinary interest. This includes shapes of static and dynamic drops and bubbles at solid substrates and liquid bridges between solids as well as wetting and dewetting processes.\cite{Genn1985rmp,TeDS1988rpap,StVe2009jpm,BEIM2009rmp,CrMa2009rmp,SnAn2013arfm,EWGT2016prf}
An important class of problems that has gained much attention in recent years is the wetting of soft adaptive substrates,\cite{BiRR2018arfm,BBSV2018l,AnSn2020arfm} i.e., substrates with a proper dynamics that responds to the dynamics of the liquids on top of them.
To comprehend the intricacies of dynamic wetting processes on substrates like hydrogels and polymer brushes, first, a profound understanding of the statics of wetting is essential. For a sessile liquid drop on a substrate, two limiting cases are often considered: on the one hand, for a rigid solid substrate, the equilibrium contact angle $\theta_\mathrm{Y}$ at the three-phase contact line is given by a macroscopic horizontal force balance, namely, the Young law.\cite{Youn1805ptrs}
If, on the other hand, the substrate is liquid, also the vertical force balance has to be taken into account as the liquid-gas surface tension exerts a traction force that deforms the substrate. The two conditions form the Neumann law.\cite{DeGennesBrochard2004,MDSA2012prl} It determines the two independent angles between the three involved interfaces and is invariant under a rigid rotation of the three-phase contact region.
For other non-rigid substrates as soft elastic or otherwise adaptive substrates, normally, the Neumann law applies at least in the close vicinity of the contact line.\cite{PAKZ2020prx,AnSn2020arfm} At larger distances, the bulk influence of substrate elasticity becomes relevant and results in features like the wetting ridge and viscoelastic breaking.\cite{AnSn2020arfm}
Here, we are specifically concerned with liquid droplets and menisci on a rigid solid substrate covered by a polymer brush that can elastically deform, and also absorbs liquid due to mass transfer and imbibition processes.\cite{MaMu2005jpm,LeMu2011jcp,BBSV2018l,MeSB2019m,EDSD2021m} As a result, brush swelling and deformation interact with contact line motion, e.g., for spreading and sliding drops.\cite{MMHM2011jpm,ThHa2020epjt,WHNK2020l,LSSK2021l} In contrast to soft solid substrates, where, depending on substrate softness, large elastic deformations can be found even on the scale of macroscopic droplets,\cite{HeST2021sm} the comparatively small length (nanometer to micrometer) of the grafted polymer molecules restricts brush deformations to mesoscopic scales. We will show that, in consequence, the wetting behaviour of polymer brushes corresponds to an intermediate case, where characteristics of both, the macroscopic Young law and Neumann law are encountered.
Due to the involved small scales of the deformations, many of the features known from soft solid substrates, e.g., the occurrence of a wetting ridge at the contact line, have, to our knowledge, not yet been assessed in experiments involving three-phase contact lines on polymer brushes. However, they may be responsible for intricate observed macroscopic behaviour, namely, the emergence of stick-slip dynamics, i.e., stick-slip motion of the contact line. This occurs, e.g., in forced wetting experiments with droplets on polymer brushes.\cite{WMYT2010scc,SHNF2021acis} The length scale-bridging relation of stick-slip motion and the formation of wetting ridges and associated pinning effects is already more widely investigated for contact line motion on soft solid substrates.\cite{KDNR2013sm, KBRD2014sm, KDGP2015nc, PBDJ2017sm, GASK2018prl, AnSn2020arfm,MoAK2022el} However, even there, the bifurcations underlying transitions between stationary and stick-slip contact line motion are not yet well understood.
Note that related stick-slip phenomena occur well beyond systems involving soft and adaptive substrates. They are described in a wide range of (de)wetting, deposition and coating processes.\cite{HaLi2012acie,Thie2014acis} Detailed investigations exist for droplets spreading or sliding on rigid substrates with imposed regular wettability or topography patterns.\cite{CuFe2001el,ThKn2006njp,ZhMi2009l,BKHT2011pre,SaKa2013jfm,VFFP2013prl,VSFP2014l,SBAV2014pre} Rough rigid substrates often reveal a stochastic stick-slip motion due to a contact line pinning induced by a random topography.\cite{ScWo1998prl,CuFe2004jcis,TYYA2006l,SaKP2010prl} Furthermore, self-organised stick-slip motion is of key importance to deposition processes like in the evaporative dewetting or dip coating of a solution or suspension,\cite{BoDG2010l,HaLi2012acie,Thie2014acis,Lars2014aj,JEZT2018l} and the Langmuir-Blodgett transfer of a surfactant layer from a bath onto a moving plate.\cite{SpCR1994el,LKGF2012s} This is also closely related to the fine structure of coffee rings.\cite{Deeg2000pre} Beyond various technical applications,\cite{HaLi2012acie} similar time-periodic behaviour may as well be found in biology as cells can show a stick-slip motility.\cite{ZGLA2017jpdp,RMGV2020prr} Detailed nonlinear analyses of the rich dynamical behaviour for the examples of evaporative dewetting,\cite{FrAT2012sm} Langmuir-Blodgett\cite{LKGF2012s,KoTh2014n} and dip-coating transfer\cite{TWGT2019prf} are available. They show that the onset of stick-slip motion is often related to time-periodic states that appear at Hopf bifurcations (at large contact line speeds) and global bifurcations (at low contact line speeds). The emergence of the entire branch of such states has recently also been investigated.\cite{MiMT2021prf} Here, we aim at a detailed understanding of stick-slip motion of the contact line on adaptive brush-covered substrates.
Our work is structured as follows. In section~\ref{sec:model} we present a mesoscopic hydrodynamic model that allows us to study the coupled dynamics of the profiles of the brush-liquid and the liquid-gas interfaces under full consideration of absorption, swelling and imbibition processes.
In the subsequent section~\ref{sec:equilibrium} we consider steady sessile droplets on the swollen polymer brush and analytically obtain laws characterizing the macroscopic contact angle and the mesoscopic wetting ridge. Using numerical simulations we then analyze in section~\ref{sec:forced_wetting} the case of forced dynamic wetting, i.e., the inverse Landau-Levich case when a brush-covered plate is slowly pushed into a liquid bath. There, a particular interest lies on the dynamic behaviour of the wetting ridge close to the three-phase contact line. Then, section~\ref{sec:stickslip} focuses on the conditions for the emergence of stick-slip motion. Finally, section~\ref{sec:conclusions} provides a conclusion and outlook.
\section{Mesoscopic hydrodynamic model}\label{sec:model}
We develop a mesoscopic gradient dynamics model that allows for studies of situations involving the dynamics of a three-phase contact line on an adaptive substrate formed by a rigid smooth solid covered by a polymer brush. In particular, we extend a recently presented model by~\citet{ThHa2020epjt} to also capture the dependence of wettability on brush state. The additional incorporation of driving forces then allows us to study forced wetting.
\begin{figure}
\centering
\includegraphics{Fig1.pdf}
\caption{Shown is a sketch of the considered geometry close to a static three-phase contact line for the case of a liquid meniscus on a polymer brush. Such a meniscus may, e.g., be seen as the contact line region of a sessile liquid drop (inset). Indicated are the definitions of height profiles $h$ and $\zeta$, the equilibrium (Neumann) angles $\theta_\mathrm{LG}$, $\theta_\mathrm{BL}$ and $\theta_\mathrm{BG}$ defined at the tip of the wetting ridge. The horizontal dotted lines mark the dry brush thickness $H_\mathrm{dry}$, as well as the equilibrium brush height $\zeta_\mathrm{n}$ and the height of the wetting ridge $\zeta_\mathrm{wr}$, both above $H_\mathrm{dry}$. At positions $x_1$ and $x_4$ far away from the three-phase contact line region (inset) the brush is practically flat. Positions $x_2$ and $x_3$ mark the points of maximal and minimal slope of the brush-liquid interface, i.e., its inflection points. Far from the meniscus the liquid thickness approaches a mesoscopic adsorption layer height $h_\mathrm{p}$.}
\label{fig:Winkel}
\end{figure}
In particular, we consider a liquid drop, layer or meniscus of height profile $h(\mathbf x, t)$ on a polymer brush of height $H(\mathrm x, t) = H_\mathrm{dry} + \zeta(\mathrm x, t)$. Here, $\mathbf x=(x,y)^T$ are the substrate coordinates, $H_\mathrm{dry}$ denotes the reference height of a completely dry brush and $\zeta(\mathrm x, t)$ is the local increase in brush thickness due to swelling, i.e., it corresponds to an effective height of the liquid contained within the brush. The geometry in the case of a liquid meniscus is sketched in Fig.~\ref{fig:Winkel}.
The coupled dynamics of $h$ and $\zeta$ is then described within a gradient dynamics framework\cite{Thie2018csa} -- a common formulation of evolution equations for one-layer thin liquid films and shallow drops.\cite{Mitl1993jcis,Thie2010jpcm} This approach has been expanded to two-field systems as, for example, two-layer liquid films,\cite{PBMT2005jcp,BCJP2013epje} liquid drops/films covered by an insoluble surfactant,\cite{ThAP2016prf} drops on viscoelastic substrates,\cite{HeST2021sm} films of liquid mixtures,\cite{ThTL2013prl} and drops of volatile liquids in a vapor-filled gap.\cite{HDJT2022apa} It also forms a central building block for models for biofilms\cite{TrJT2018sm} and drops of active liquids.\cite{StJT2022sm}
To arrive at a gradient dynamics description for a meniscus on a brush-covered substrate we assume that the region of interest is sufficiently small to be able to neglect inertia, i.e., the dynamics is mainly driven by an underlying free energy functional $\mathcal{F}[h, \zeta]$ that depends on the drop profile and the brush state. Allowing for various occurring transport and transfer processes, the dynamical equations for the profiles $h$ and $\zeta$ have the form
\begin{equation}
\begin{aligned}
\partial_t h &= \nabla \cdot \left[\frac{h^3}{3\eta}\,\nabla\frac{\delta\mathcal{F}}{\delta h}\right] \ \, - \ M\left[\frac{\delta\mathcal{F}}{\delta h}-\frac{\delta\mathcal{F}}{\delta\zeta}\right] \ + \ U\partial_x h\\
\partial_t \zeta &=\nabla \cdot \left[D\zeta\,\nabla\frac{\delta\mathcal{F}}{\delta \zeta}\right] \ - \ M\left[\frac{\delta\mathcal{F}}{\delta \zeta}-\frac{\delta\mathcal{F}}{\delta h}\right] \ + \ U\partial_x \zeta.\label{eq:gradient_model}
\end{aligned}
\end{equation}
where $U$ is an imposed velocity of the substrate that is drawn out ($U>0$) or pushed into ($U<0$) the bath/drop. For $U=0$, Eqs.~\eqref{eq:gradient_model} have the same form as the ones in \citet{ThHa2020epjt}. The first term on the r.h.s.\ of the first equation describes advective transport within the liquid layer (with the dynamic viscosity $\eta$), driven by the pressure gradient $\nabla (\delta\mathcal{F} / \delta h)$, while the second term describes the loss/gain of liquid via transfer to/from the brush (with rate constant $M$). This absorption process is driven by the pressure difference between liquid and brush. Without transfer, the equation becomes the standard mesoscopic thin-film (lubrication, long-wave) equation for a nonvolatile liquid on a rigid solid substrate.\cite{Genn1985rmp,OrDB1997rmp,StVe2009jpm,BEIM2009rmp,CrMa2009rmp,Thie2010jpcm} The first term on the r.h.s.\ of the second equation describes diffusive transport of liquid within the brush (diffusive imbibition with the diffusion constant $D$) driven by the gradient in chemical potential\footnote{Note that conceptionally it is a chemical potential that drives diffusion, however, here it is literally a pressure as our considered field $\zeta$ is a height and not a particle number.} $\nabla (\delta\mathcal{F} / \delta \zeta)$ while the second term describes the loss [gain] of liquid via transfer to [from] the meniscus. As we assume a nonvolatile liquid, the nonconserved terms in the two equations exactly compensate, i.e., $h+\zeta$ follows a continuity equation.
Assuming a long-wave setting, i.e., small interface slopes,\cite{OrDB1997rmp,CrMa2009rmp} $h^3/3\eta$ and $D\zeta$ are the viscous mobility in the liquid and the diffusive mobility within the brush, respectively. Note that there is no dynamics cross-coupling between drop and brush as we neglect advective transport of liquid within the brush. The driving term has the form of a Galilean transformation into a reference frame moving with velocity $U$. It may represent a moving substrate similar, e.g., to dip coating, or the velocity of a drop imposed from the outside. Note that such a permanent external driving force can normally not be captured by a free energy functional $\mathcal{F}$. In other words, it represents a nonvariational influence that persistently keeps the system out of equilibrium making time-periodic behaviour possible.
The considered free energy functional is
\begin{equation}
\mathcal{F}[h,\zeta] = \int \left[ f_\mathrm{cap}(h,\zeta) + \xi_\zeta f_\mathrm{wet}(h, \zeta) + g_\mathrm{brush}(\zeta) \right]\mathrm{d}^2x.
\label{eq:free_energy}
\end{equation}
It incorporates contributions due to capillarity of the brush-liquid and the liquid-gas interfaces
\begin{equation}
f_\mathrm{cap}(h,\zeta)=\gamma_\mathrm{bl}(\zeta)\,\xi_\zeta + \gamma\,\xi_{h+\zeta},
\end{equation}
wettability $\xi_\zeta f_\mathrm{wet}(h, \zeta)$, and the local free energy of the (dry or swollen) brush $g_\mathrm{brush}(\zeta)$. Here, $\gamma$ and $\gamma_\mathrm{bl}(\zeta)$ denote the interface energies of the liquid-gas and brush-liquid interfaces, respectively. In general, both, wetting energy $f_\mathrm{wet}(h, \zeta)$ and brush-liquid interface energy $\gamma_\mathrm{bl}(\zeta)$ will depend on the brush state encoded in $\zeta$. This dependency is necessary to ensure that at very high liquid concentration within the brush, the brush-liquid interface energy tends to zero while the brush-air interface energy (as encoded in $f_\mathrm{wet}$, cf.~\citet{Genn1985rmp,BEIM2009rmp,TSTJ2018l}) tends to $\gamma$. Note that this generalizes the approach in earlier work \cite{ThHa2020epjt} where such $\zeta$-dependencies were not considered.
Further note that the metric factors of the two interfaces,
\begin{equation}
\xi_\zeta=\sqrt{1+(\nabla\zeta)^2}\quad
\text{and}\quad
\xi_{h+\zeta}=\sqrt{1+[\nabla(h+\zeta)]^2},
\end{equation}
enter the interface energies as well as the wetting energy. This ensures consistency with macroscopic relations (see below).
\begin{figure}
\centering
\includegraphics{Fig2.pdf}
\caption{Displayed is the typical qualitative behaviour of (a) the wetting energy $f_{\mathrm{wet}}(h,\zeta)$ as a function of $h$ at fixed $\zeta$ [Eq.~\eqref{eq:wetting_potential}], and (b) the brush energy $g_{\mathrm{brush}}(\zeta)$ as a function of $\zeta$ [Eq.~\eqref{eq:brush-en-area}].}
\label{fig:energies}
\end{figure}
The employed wetting energy we base on a simple standard form for partially wetting liquids that includes long-range attractive and short-range repulsive contributions, both as power laws\cite{Pism2001pre,Thie2010jpcm} (see \autoref{fig:energies}~(a))
\begin{equation}
f_\mathrm{wet}(h, \zeta) = A(\zeta)\,\left[\frac{h_\mathrm{p}^3}{5h^5}-\frac{1}{2h^2}\right].\label{eq:wetting_potential}
\end{equation}
Here, we employ a brush state-dependent Hamaker constant $A$ effectively letting the equilibrium contact angle adjust to the swelling state of the brush. For a simple ansatz that fulfils the aforementioned limiting cases, we assume that both, the Hamaker constant $A(\zeta)$ and brush-liquid surface energy $\gamma_\mathrm{bl}(\zeta)$, depend linearly on the polymer concentration in the brush $c$, i.e.,
\begin{equation}
A(\zeta) = A_0\,c(\zeta) \quad \text{and} \quad \gamma_\mathrm{bl}(\zeta) = \gamma_\mathrm{bl,0}\,c(\zeta),\label{eq:brush_dependencies}
\end{equation}
where the constants $A_0$ and $\gamma_\mathrm{bl,0}$ are the reference values obtained for a dry brush ($\zeta=0$). This in particular ensures that
\begin{equation}
f_\mathrm{wet}(h, \zeta\to\infty)=0 \quad \text{and} \quad \gamma_\mathrm{bl}(\zeta\to\infty) = 0,
\end{equation}
i.e. the brush-liquid interface turns into a liquid-liquid interface. The implications for the macroscopic brush-air interface are discussed below in section~\ref{sec:consistency}.
The volume fraction of polymer within the brush layer is defined as
\begin{equation}
c(\zeta) = \frac{H(\zeta=0)}{H(\zeta)} = \frac{H_\mathrm{dry}}{H_\mathrm{dry} + \zeta},
\end{equation}
i.e., it corresponds to the inverse of the local swelling ratio.
Finally, we employ the brush energy\cite{ThHa2020epjt}
\begin{equation}
g_\mathrm{brush}(\zeta) = \frac{H_\mathrm{dry} k_B T}{\ell_K^3}\left[\frac{\sigma^2}{2 c^2} +
(1/c - 1) \,\log\left(1-c\right) \right],
\label{eq:brush-en-area}
\end{equation}
where $T$ is the temperature, $\ell_K$ is the Kuhn length (or the length of a unit cell in the lattice model) and $\sigma$ is the relative grafting density, i.e., the number of grafted chains per unit area. Note that in a simple Alexander-de Gennes geometry of the brush\cite{Genn1991crasi,Alex1977jp,Somm2017m} the grafting density relates to the collapsed brush height via the degree of polymerization $N$ as $H_\mathrm{dry} = \sigma N \ell_K$.
Within the brush energy \eqref{eq:brush-en-area}, the first term accounts for the elastic energy due to the stretching of polymers, whereas the second one represents an entropic contribution as obtained from a Flory-Huggins lattice model for polymer-solvent mixtures.\cite{Flor1953} Independently of all parameters, the brush energy is convex and therefore has one global minimum, see \autoref{fig:energies}~(b). Note that our approach assumes that the liquid content of the brush is vertically homogeneous, i.e., the distribution of liquid in the brush does not depend on the height coordinate.
In summary, compared to~\citet{ThHa2020epjt}, we have generalized the theoretical description in four ways: First, we incorporate dependencies of the brush-liquid interface energy and of the wetting energy on the brush state. Second, for consistency with macroscopic laws (considered in section~\ref{sec:consistency}), the wetting energy is scaled with the metric factor of the brush-liquid interface. Third, we incorporate a permanent driving force to be able to investigate dip-coating processes. Fourth, we improve the energy functional by using the exact metric factors resulting in the exact curvature in the time-evolution equations.\cite{GaRa1988ces,Snoe2006pf} It has been shown that such improved representations of the underlying energy functional are more important for a correct description of the physical behaviour than the details of the dynamics even though the resulting model is not asymptotically exact\cite{BoTH2018jfm} (also see section~3 of the review by~\citet{Thie2018csa}). Naturally, for small inclinations $\nabla \zeta$ and $\nabla h$, the metric factors can be Taylor-expanded up to second order to arrive at a standard long-wave approximation of both, mobilities and energies \cite{OrDB1997rmp,CrMa2009rmp} as used by~\citet{ThHa2020epjt}.
\section{Equilibrium states}\label{sec:equilibrium}
\subsection{Grand potential and mechanical analogue}
We start by analysing the variational case ($U=0$), where the gradient dynamics structure of Eqs.~\eqref{eq:gradient_model} implies a continuous decrease of the free energy, $\mathrm{d} \mathcal{F} / \mathrm{d}t \leq 0$ (see Appendix~\ref{sec:dissipation_derivation}). With other words, $\mathcal{F}$ is a Lyapunov functional and the system always approaches a steady state corresponding to a minimum of $\mathcal{F}$ under the constraint of an imposed total liquid volume. Then, all equilibria correspond to minima of the grand potential
\begin{equation}
\mathcal{G}[h,\zeta] = \int \left[ f_\mathrm{cap} + \xi_\zeta f_\mathrm{wet} + g_\mathrm{brush} - P(h+\zeta) \right]\,\mathrm{d}^2x\label{eq:F_P}
\end{equation}
where $P$ is a Lagrange multiplier to ensure volume conservation, and equilibria fulfil the corresponding Euler-Lagrange equations
$\delta\mathcal{F}/\delta h=P$ and $\delta\mathcal{F}/\delta \zeta=P$.
As the integrand of $\mathcal{G}$ only depends on the fields $h$ and $\zeta$ and their first spatial derivatives, in the case of a one-dimensional substrate (1D),\footnote{With other words, we consider two-dimensional (2D) droplets, see inset of \autoref{fig:Winkel}, that may be seen as cross sections of liquid ridges in 3D that are translation-invariant in the transverse direction. Also in a dip coating geometry a 2D slice of a transversely translation-invariant configuration is considered.} there exists a strong analogue to classical Lagrangian and Hamiltonian mechanics: The functional $\mathcal{G}$ then corresponds to an action and its integrand to a Lagrangian $\mathcal{L}$, i.e., $\mathcal{G}=\int\mathcal{L}\,{\d}x$.
The fields $h(x)+\zeta(x)$ and $\zeta(x)$ are generalised coordinates, and the spatial coordinate $x$ becomes time. In consequence, the determination of steady interface profiles corresponds to solving the coupled generalised Newton equations
\begin{align}
\frac{\delta \mathcal{G}}{\delta h} = &-\gamma\frac{\partial_{xx}(h+\zeta)}{\xi_{h+\zeta}^{3}} + \xi_\zeta\partial_h f_\mathrm{wet}-P=0\label{eq:dFdh}\\
\frac{\delta \mathcal{G}}{\delta \zeta} = &-\gamma\frac{\partial_{xx}(h+\zeta)}{\xi_{h+\zeta}^{3}} - \partial_x \cdot \left[ (\gamma_\mathrm{bl} + f_\mathrm{wet})\frac{\partial_x\zeta}{\xi_\zeta}\right]\nonumber\\
&+ \xi_\zeta \partial_\zeta (\gamma_\mathrm{bl} + f_\mathrm{wet}) + \partial_\zeta g_\mathrm{brush}-P=0.\label{eq:dFdz}
\end{align}
Further employing the analogy, we define the appropriate generalised momenta $p_{h+\zeta}$ and $p_\zeta$
\begin{align}
p_{h+\zeta} &:= \frac{\partial \mathcal{L}}{\partial [\partial_{x} (h+\zeta)]} = \gamma \frac{\partial_{x} (h+\zeta)}{\xi_{h+\zeta}}\label{eq:ph}\\
p_\zeta &:= \frac{\partial \mathcal{L}}{\partial (\partial_{x} \zeta)} = (\gamma_\mathrm{bl}+f_\mathrm{wet})\frac{\partial_{x} \zeta}{\xi_\zeta}\label{eq:pzeta}
\end{align}
and determine the first integral
\begin{equation}
\begin{split}
\mathcal{H}&=p_{h+\zeta}\partial_{x} (h+\zeta) + p_\zeta \partial_{x}\zeta-\mathcal{L}\\&=-\frac{\gamma}{\xi_{h+\zeta}} -\frac{\gamma_\mathrm{bl} + f_\mathrm{wet}}{\xi_{\zeta}}-g_\mathrm{brush}+P(h+\zeta),\label{eq:Hamiltonian}
\end{split}
\end{equation}
that corresponds to the Hamiltonian. In consequence, $E=-\mathcal{H}$ is a constant energy density, which we can use to describe the balance of horizontal forces across the contact line.
\subsection{Limiting cases and consistency condition}\label{sec:consistency}
\subsubsection{Contact angles}
Next, we consider a 2D slice through a straight liquid meniscus as depicted in \autoref{fig:Winkel}. Then, the angles between local tangents to the brush-liquid and liquid-gas interface and the horizontal, $\theta_\mathrm{bl}$ and $\theta_\mathrm{lg}$, respectively, are directly related to the metric factors via
\begin{equation}\label{eq:thet-metric}
\begin{aligned}
\cos\theta_\mathrm{lg} &= \frac{1}{\xi_{h+\zeta}}, &
\sin\theta_\mathrm{lg} &= \frac{\partial_x (h+\zeta)}{\xi_{h+\zeta}},\\
\cos\theta_\mathrm{bl} &= \frac{1}{\xi_{\zeta}},\quad\text{and} &
\sin\theta_\mathrm{bl} &= \frac{\partial_x \zeta}{\xi_{\zeta}}.
\end{aligned}
\end{equation}
As there is no trivial definition of equilibrium contact angles in the mesocopic picture,\cite{TMTT2013jcp} here, we introduce them as extremal values of $\theta_{\mathrm{lg}}$ and $\theta_{\mathrm{bl}}$, i.e., as local steepest slopes. Namely, considering the geometry in the main panel of \autoref{fig:Winkel} we use
\begin{equation}\label{eq:thet-metric2}
\theta_{\mathrm{LG}}=\min(\theta_{\mathrm{lg}}), \quad\theta_{\mathrm{BL}}=\max(\theta_{\mathrm{bl}})\quad\text{and}\quad\theta_{\mathrm{BG}}=\min(\theta_{\mathrm{bl}}).
\end{equation}
Note that due to the identification with the metric factors, inclination and contact angles are signed values in the range $[-\pi/2,\pi/2]$
where a positive [negative] angle corresponds to a positive [negative] slope of the corresponding interface profile.
\subsubsection{Global Young law}\label{sec:young}
As the wetting energy~\eqref{eq:wetting_potential} and its derivatives approach zero at large film height $h$, for a macroscopic droplet far away from the contact line region (e.g., at position $x_1$ in Fig.~\ref{fig:Winkel}) Eq.~\eqref{eq:dFdh} becomes
\begin{equation}
P = -\gamma \kappa \quad \text{with the curvature}\quad \kappa = \frac{\partial_{xx} (h+\zeta)}{\xi_{h+\zeta}^3}.
\end{equation}
Therefore, macroscopically, $P$ corresponds to the Laplace pressure and the liquid-gas interface approaches a spherical cap. Next, we consider a very large drop, i.e., $\kappa\rightarrow 0$, at equilibrium on a brush. Aiming at a global picture that ignores the details of the configuration in the contact line region, we consider positions $x_1$ and $x_4$ far from the contact line (Fig.~\ref{fig:Winkel}). There, the brush approaches a flat state, i.e., all spatial derivatives of the brush profile approach zero.
In consequence, within the drop (at~$x_1$) Eqs.~\eqref{eq:dFdh} and \eqref{eq:dFdz} simplify to
\begin{equation}
P = 0
\quad \text{and} \quad
\partial_\zeta (\gamma_\mathrm{bl} + g_\mathrm{brush}) = 0,
\label{eq:zeta_n-x1}
\end{equation}
as the Derjaguin (or disjoining) pressure $- \partial_h f_\mathrm{wet}$ vanishes within the droplet due to large $h$ (as discussed above). The brush assumes a height $\zeta_\mathrm{d}$ given by the minimum of $\gamma_\mathrm{bl} + g_\mathrm{brush}$ with respect to $\zeta$ (Fig.~\ref{fig:energies}~(b)). In contrast, far away from the drop (at $x_4$), the Derjaguin pressure matters and Eqs.~\eqref{eq:dFdh} and \eqref{eq:dFdz} become
\begin{equation}
\partial_h f_{\mathrm{wet}} = 0
\quad \text{and} \quad
\partial_\zeta (\gamma_\mathrm{bl} + f_\mathrm{wet} + g_\mathrm{brush}) = 0,
\label{eq:zeta_n-x4}
\end{equation}
i.e., the minimum of the wetting energy w.r.t.\ $h$ gives the adsorption layer height $h_p$ (Fig.~\ref{fig:energies}~(a)) while the minimum of $\gamma_\mathrm{bl} + f_\mathrm{wet} + g_\mathrm{brush}$ w.r.t.\ $\zeta$ gives the brush height $\zeta_\mathrm{p}$ far away from the drop.
Using this result and Eq.~\eqref{eq:Hamiltonian} we evaluate the asymptotic energy density $E=-\mathcal{H}$ inside and outside of the drop
\begin{align}
E(x_1) &= \gamma\cos{\theta_{LG}}+\gamma_\mathrm{bl}(\zeta_\mathrm{d})+g_\mathrm{brush}({\zeta_\mathrm{d}})\label{eq:E(x1)}\\
E(x_4) &= \gamma+\gamma_\mathrm{bl}(\zeta_\mathrm{p})+f_\mathrm{wet}(h_\mathrm{p}, \zeta_\mathrm{p})+g_\mathrm{brush}({\zeta_\mathrm{p}})\label{eq:E(x4)}.
\end{align}
We set the two energies equal, identify the occurring contact angle $\theta_\mathrm{LG}$ as a macroscopic angle $\theta_\mathrm{Y}$ and obtain an equivalent to Young's law\cite{Youn1805ptrs}\begin{equation}
\begin{aligned}
\gamma\cos\theta_\mathrm{Y} = &\ \gamma+f_\mathrm{wet}(h_\mathrm{p}, \zeta_\mathrm{p})\\
&+ \gamma_\mathrm{bl}(\zeta_\mathrm{p}) + g_\mathrm{brush}(\zeta_\mathrm{p}) - \gamma_\mathrm{bl}(\zeta_\mathrm{d}) - g_\mathrm{brush}(\zeta_\mathrm{d})\label{eq:young}
\end{aligned}
\end{equation}
expressed in mesoscale quantities.
Note that the last four addends are solely due to the adaptive character of the brush and are not present in the classical result known from rigid non-adaptive substrates. However, as per the bulk equilibrium condition \eqref{eq:zeta_n-x1} the correction to Young's law is of second order in the height difference $\zeta_\mathrm{d}-\zeta_\mathrm{p}$ and therefore potentially small.
A similar consideration based solely on macroscopic quantities, see Appendix~\ref{sec:macroscopic}, gives an identically amended macroscopic Young's law
\begin{equation}
\begin{aligned}
\gamma\cos\theta_\mathrm{Y} = &\ \gamma_\mathrm{bg}(\zeta_p) - \gamma_\mathrm{bl}(\zeta_\mathrm{p})\\
&+ \gamma_\mathrm{bl}(\zeta_\mathrm{p}) + g_\mathrm{brush}(\zeta_\mathrm{p}) - \gamma_\mathrm{bl}(\zeta_\mathrm{d}) - g_\mathrm{brush}(\zeta_\mathrm{d}).\label{eq:young_macro}
\end{aligned}
\end{equation}
Note that here we added the terms $\gamma_\mathrm{bl}(\zeta_p)-\gamma_\mathrm{bl}(\zeta_p)=0$ to resemble the form of the mesoscopic Young law, namely Eq.~\eqref{eq:young}.
Comparing both versions of Young's law, we identify a consistency relation between the macroscopic and mesoscopic global picture
\begin{equation}
f_\mathrm{wet}(h_\mathrm{p}, \zeta_p)=\gamma_\mathrm{bg}(\zeta_\mathrm{p})-\gamma_\mathrm{bl}(\zeta_\mathrm{p})-\gamma.\label{eq:consistency}
\end{equation}
The relation determines how the dependencies of interface tensions and wetting potential on the brush state $\zeta$ are related.
Together with our assumptions on the brush-dependency of the mesoscopic wetting potential $f_\mathrm{wet}(h, \zeta)$ and the brush-liquid interface energy $\gamma_\mathrm{bl}(\zeta)$ in Eq.~\eqref{eq:brush_dependencies} this implies that the macroscopic brush-gas interface energy scales as
\begin{equation}
\gamma_\mathrm{bg}(\zeta) = (\gamma_\mathrm{bg,0} - \gamma) c(\zeta) + \gamma,
\end{equation}
i.e., it interpolates between the dry brush-gas interface energy $\gamma_\mathrm{bg,0}$, and the energy of a liquid-gas interface in case of a fully swollen brush, as expected.
\subsubsection{Local Neumann law}\label{sec:neumann}
Having established the global picture for a large drop on a brush within a mesoscale and a macroscale description, we next retain the mesoscopic view and consider the local picture of the contact line region. Equating the brush and film pressure (Eqs.~\eqref{eq:dFdh} and \eqref{eq:dFdz}), we obtain for the curvature of the brush-liquid interface
\begin{equation}
\partial_x \cdot (\gamma_\mathrm{bl} + f_\mathrm{wet})\frac{\partial_x\zeta}{\xi_\zeta^{3}} = \xi_\zeta \partial_\zeta (\gamma_\mathrm{bl} + f_\mathrm{wet}) + \partial_\zeta g_\mathrm{brush}-\xi_\zeta\partial_h f_\mathrm{wet}.\label{eq:KrBrush}
\end{equation}
Furthermore, Eq.~\eqref{eq:dFdh} indicates that the Derjaguin pressure term $-\xi_\zeta \partial_h f_\mathrm{wet}$ has to balance the (strong) curvature of the liquid profile in the contact line region. Thus, generally, the r.h.s.\ of Eq.~\eqref{eq:KrBrush} is non-zero, i.e., the curvature of the brush profile can not vanish. We expect that the brush forms some type of wetting ridge as known from elastic substrates.\cite{AnSn2020arfm}
To investigate the ridge we consider the inflection points $x_2$ and $x_3$ of the brush profile as indicated in Fig~\ref{fig:Winkel}. Close to three-phase contact, the Derjaguin pressure will dominate Eq.~\eqref{eq:KrBrush} over a small length scale of the order of the height $h_p$. Hence, the distance between $x_2$ and $x_3$ will scale with $h_p/\theta_{LG}$. Outside this small region, the wetting ridge will decay to the equilibrium brush height $\zeta_\mathrm{p}$ (or $\zeta_\mathrm{d}$) and its shape is governed by the differential equation~\eqref{eq:KrBrush}.
To further advance, we next focus on situations where the wetting ridge is large compared to the adsorption layer height and assume, motivated by the previous argument, that the brush heights at $x_2$ and $x_3$ well approximate the peak height $\zeta_\mathrm{wr}$. Also assuming that the inclination angle of the liquid-gas interface at $x_2$ equals $\theta_{LG}$, we again use Eq.~\eqref{eq:Hamiltonian} to evaluate the energy $E=-\mathcal{H}$, this time at $x_2$ and $x_3$. We obtain
\begin{align}
E(x_2) &=\gamma\cos{\theta_\mathrm{LG}}+\gamma_\mathrm{bl}\cos{\theta_\mathrm{BL}}+g_\mathrm{brush}({\zeta_\mathrm{wr}})\label{eq:E(x2)}\\
E(x_3) &=[\gamma+\gamma_\mathrm{bl}+f_\mathrm{wet}(h_\mathrm{p},\zeta_\mathrm{wr})]\cos\theta_\mathrm{BG}+g_\mathrm{brush}({\zeta_\mathrm{wr}}).\label{eq:E(x3)}
\end{align}
Equating the two expressions yields the horizontal component of the Neumann law, namely,
\begin{equation}
\gamma\cos\theta_\mathrm{LG}+\gamma_\mathrm{bl}\cos{\theta_\mathrm{BL}}=[\gamma+\gamma_\mathrm{bl}+f_\mathrm{wet}(h_\mathrm{p},\zeta_\mathrm{wr})]\cos\theta_\mathrm{BG}. \label{eq:Neumann_hor}
\end{equation}
To obtain the vertical component of the law, we consider conservation of the total generalised momentum $p_\zeta+p_{h+\zeta}$ (Eqs.~\eqref{eq:ph} and~\eqref{eq:pzeta}) across the contact line region, i.e., from $x_2$ to $x_3$.\footnote{In the mechanical analogue the approach of the brush-liquid and liquid-gas interfaces and subsequent fusion at the contact line into the brush-gas interface corresponds to a completely inelastic collision of two particles. The wetting energy takes the role of an internal energy like a spring that snaps in on close approach.}
Also using Eqs.~\eqref{eq:thet-metric} and~\eqref{eq:thet-metric2}, this directly gives
\begin{equation}
\gamma\sin\theta_\mathrm{LG}+\gamma_\mathrm{bl}\sin\theta_\mathrm{BL}=[\gamma+\gamma_\mathrm{bl}+f_\mathrm{wet}(h_p, \zeta_\mathrm{wr})]\sin\theta_\mathrm{BG}. \label{eq:Neumann_ver}
\end{equation}
Note that the brush energy does not enter as we have used $\zeta(x_2)\approx\zeta(x_3)\approx\zeta_\mathrm{wr}$.
The obtained mesoscopic Neumann law, Eqs.~\eqref{eq:Neumann_hor} and~\eqref{eq:Neumann_ver}, corresponds to the usual macroscopic form when taking the consistency condition~\eqref{eq:consistency} into account, see Appendix~\ref{sec:macroscopic}. We emphasise that the full agreement critically depends on the above discussed scaling of the wetting energy with the metric factor of the brush-liquid interface.
We therefore conclude that in the limit of a wetting ridge that is large compared to the adsorption layer height $h_\mathrm{p}$, its shape is governed by the Neumann law expressed in mesoscale quantities. This is in line with similar findings for elastic substrates, where~\citet{PAKZ2020prx} have shown that the classic macroscopic Neumann law applies at the wetting ridge as substrate elasticity is negligible when approaching the contact line sufficiently closely.\cite{AnSn2020arfm} However, here, the rotation invariance of the local Neumann law is broken by the request that also the global Young law is fulfilled as this fixes the angle of the liquid-gas interface at three-phase contact w.r.t.\ the horizontal.
\subsubsection{Height of the wetting ridge}
In a similar way one can equate the energy $E$ underneath the liquid far away from the wetting ridge \eqref{eq:E(x1)} and close to the peak of the wetting ridge \eqref{eq:E(x2)}. As a result we obtain the difference in brush energy at the peak and for the flat brush
\begin{equation}
g_\mathrm{brush}({\zeta_\mathrm{wr}})-g_\mathrm{brush}({\zeta_\mathrm{d}})=\gamma_\mathrm{bl}(\zeta_\mathrm{d})-\gamma_\mathrm{bl}(\zeta_\mathrm{wr})\cos{\theta_\mathrm{BL}}.\label{eq:ridge_height_def}
\end{equation}
This is a transcendent equation for the wetting ridge height $\zeta_\mathrm{wr}$ at the contact line of large drops at equilibrium. Next, we derive an explicit expression for $\Delta \zeta = \zeta_\mathrm{wr} - \zeta_\mathrm{d}$ by expanding Eq.~\eqref{eq:ridge_height_def} up to the second order in small values of $\Delta \zeta$ and $\theta_\mathrm{BL}$.
\begin{equation}
\begin{aligned}
&\partial_\zeta g_\mathrm{brush}(\zeta_\mathrm{d}) \Delta \zeta + \partial_{\zeta\zeta} g_\mathrm{brush}(\zeta_\mathrm{d}) \frac{(\Delta \zeta)^2}{2}\\&= -\partial_\zeta \gamma_\mathrm{bl}(\zeta_\mathrm{d})\Delta \zeta - \partial_{\zeta\zeta} \gamma_\mathrm{bl}(\zeta_\mathrm{d})\frac{(\Delta \zeta)^2}{2} + \gamma_\mathrm{bl}(\zeta_\mathrm{d})\frac{\theta_\mathrm{BL}^2}{2}.
\end{aligned}
\end{equation}
The first order contributions cancel due to Eq.~\eqref{eq:zeta_n-x1}. Hence, the height of the wetting ridge is approximately given by
\begin{equation}
\Delta \zeta \approx \theta_\mathrm{BL} \sqrt{\frac{\gamma_\mathrm{bl}(\zeta_\mathrm{d})}{\partial_{\zeta\zeta} [g_\mathrm{brush}(\zeta_\mathrm{d}) + \gamma_\mathrm{bl}(\zeta_\mathrm{d})]}}.\label{eq:ridge_height}
\end{equation}
The angle $\theta_\mathrm{BL}$ may be expressed via the Neumann relations given in the previous section.
Note that at no point of the above derivations the brush energy $g_\mathrm{brush}$ has to be specified, i.e., all results are generic for rather general adaptive substrates. For purely elastic substrates the square root in Eq.~\eqref{eq:ridge_height} corresponds to the elastocapillary length $\ell_\mathrm{ec}$.\cite{HeST2021sm}
\subsection{Approaching the limiting cases}\label{sec:NumerikGG}
Next we consider how the derived relations for large steady drops are approached when the volume of drops of finite size is increased. This is done numerically employing direct time simulations and pseudo-arclength path continuation using the C++ finite element library \textit{oomph-lib}.\cite{HeHa2006} In all calculations a nondimensional version of the dynamic model~\eqref{eq:gradient_model} is used. Note that we present results for both variants of the presented mesoscopic model, the full-curvature formulation and the long-wave approximation mentioned at the end of section~\ref{sec:model}. Details of the nondimensionalisation and the long-wave approximation are given in Appendix~\ref{sec:nondim_and_longwave}. For the sake of readability from here on we use the dimensionless formulation without the tildes.
As the boundary conditions (BC) for the numerical analysis of the steady states we employ homogeneous Neumann conditions
\begin{equation}
\partial_x h = \partial_x \zeta = \partial_x \frac{\delta \mathcal{F}}{\delta h} = \partial_x \frac{\delta \mathcal{F}}{\delta \zeta} = 0
\end{equation}
at both boundaries of the 1D domain. This ensures, in particular, that for $U = 0$ there is no liquid flux through the domain boundaries, i.e., the total volume is conserved. For the forced wetting case $U > 0$ we need to amend the BC to retain the global balance of liquid volume as discussed in Appendix~\ref{sec:numerics}. In order to reach the limit of very large droplets $P\to 0$, we exploit that it is sufficient to only simulate the vicinity of the contact line region. We therefore replace the homogeneous Neumann condition $\partial_x h(x=0) = 0$ with an imposed interface slope $\partial_x h(x=0) = -\theta_\mathrm{0}$ at the boundary associated with the drop. When we increase $\theta_0$ letting it approach the equilibrium angle $|\theta_Y|$, the curvature of the bulk part of the drop decreases and approaches zero, i.e., $P\to 0$, and an infinitely large drop is approached.
\begin{figure}[!htb]
\centering
\includegraphics{Fig3.pdf}
\caption{Steady state brush profiles for different values of (a-c) the dry brush height $H_\mathrm{dry}$ and (d-f) the drop size, demonstrating that the ridge scales with brush height but not drop size. The drop size is adjusted via the slope of the drop profile at $x=0$ that controls the Laplace pressure $P$. In particular, $P=0$ corresponds to an infinitely large drop. Note the small but significant difference $\zeta_\mathrm{d}-\zeta_\mathrm{p}$ between the brush heights inside and outside the drop. The remaining parameters are $T=0.05$, $\sigma=0.3$, and $\gamma_\mathrm{bl}=0.4$. The Laplace pressure in panels (a-c) is $P=\num{9e-4}$ and the dry brush height in panels (d-f) is $H_\mathrm{dry}=6$. A quantitative characterisation of the wetting ridge is provided in Figs.~\ref{fig:statics_variation_H_dry} and~\ref{fig:statics_variation_P}.}
\label{fig:steady_profiles}
\end{figure}
In Fig.~\ref{fig:steady_profiles} we depict equilibrium states obtained in simulations for different dry brush heights and drop volumes. In this way we obtain an overview of possible shapes of the wetting ridge. Panels (a)-(c) only differ in the dry height of the brush $ H_\mathrm{dry} = \sigma \ell$, whereas panels (d)-(f) have identical brushes with $H_\mathrm{dry}=6$ while drop size varies from (d) small to (f) the limit of an infinitely large drop. The dry brush height $H_\mathrm{dry}=\sigma N \ell_K$ is varied by controlling the polymer chain length $N \ell_K$ at fixed grafting density $\sigma=0.3$.
Careful inspection of Fig.~\ref{fig:steady_profiles} shows that there is always a significant difference $\zeta_\mathrm{d}-\zeta_\mathrm{p}$ of the brush swelling inside and outside the drop. Numerically, we find that in all panels the brush inside the drop is swollen to approximately $\SI{300}{\percent}$ of its dry height. Outside the drop, the brush is swollen only to approximately \SI{200}{\percent}. Here, we have used $1/c$ to calculate the swelling ratio.
\begin{figure}
\centering
\includegraphics{Fig4.pdf}
\caption{Shown are geometric measures of the equilibrium configuration in the contact line region in dependence of the dry brush height $H_\mathrm{dry}$. Given are the brush heights (a) below the drop $\zeta_\mathrm{d}$, (b) outside the drop $\zeta_\mathrm{d}$, (c) the wetting ridge height $\zeta_\mathrm{wr}$, and (d-f) the three Neumann angles $\theta_\mathrm{LG}$, $\theta_\mathrm{BG}$, and $\theta_\mathrm{BL}$. For the three angles we compare the full-curvature (dashed lines) and long-wave (solid lines) results. All other parameters are as in Fig.~\ref{fig:steady_profiles}~(a-c).
}
\label{fig:statics_variation_H_dry}
\end{figure}
This is further quantified in Fig.~\ref{fig:statics_variation_H_dry} that provides various geometric measures of the equilibrium configuration in the contact line region in dependence of the dry brush height $H_\mathrm{dry}$. There we find that the equilibrium (swollen) brush heights inside ($\zeta_\mathrm{d}$) and outside ($\zeta_\mathrm{p}$) the drop, and the wetting ridge height $\zeta_\mathrm{wr}$ all increase approximately linearly with $H_\mathrm{dry}$ while the Neumann angles strongly [weakly] increase at small [large] $H_\mathrm{dry}$.
\begin{figure}
\centering
\includegraphics{Fig5.pdf}
\caption{Shown are geometric measures of the equilibrium configuration in the contact line region in dependence of the drop volume as controlled by the Laplace pressure $P$. The limit $P=0$ corresponds to an infinitely large droplet. The shown quantities in panels (a-f) and linestyles are as in Fig.~\ref{fig:statics_variation_H_dry}. All other parameters are as in Fig.~\ref{fig:steady_profiles}~(d-f).}
\label{fig:statics_variation_P}
\end{figure}
In contrast to the brush height, the drop size has only a very small influence on the swelling state and the wetting ridge. Namely, in Fig.~\ref{fig:steady_profiles}~(d-f) one discerns nearly no change in $\zeta_\mathrm{d,p,wr}$ upon varying the Laplace pressure $P$. This visual impression is quantitatively supported by Fig.~\ref{fig:statics_variation_P}, where we display the geometric measures as obtained from simulated equilibrium drops at different $P$.
Note that the shape and size of the wetting ridge are also impacted by some of the parameters not discussed here. As Eq.~\eqref{eq:ridge_height} already indicates, the ridge height is governed by an interplay of the surface energy and the brush forces. Thus, the values of the surface energies as well as the scale of the brush energy, namely, the dimensionless temperature parameter $T$, are equally relevant. While the above considerations are entirely static, the situation may become more intricate when the system is taken outside of equilibrium. This will be addressed in the subsequent sections.
Besides demonstrating the influence of drop size and dry brush height on the wetting ridge, Figs.~\ref{fig:statics_variation_H_dry} and~\ref{fig:statics_variation_P} also compare the results obtained with the full-curvature and long-wave variants of the model (cf.~Section~\ref{sec:model}). In general, for the chosen values of the interface energies the differences are relatively small. The Neumann angles tend to have slightly lower values for the long-wave model. Differences are largest for thin brushes. The brush swelling is practically identical in the two cases. Moreover, Fig~\ref{fig:statics_variation_P} suggests that the effect is rather independent of the drop size.
In consequence, we only use the model in long-wave approximation for all remaining numerical calculations.
Finally, we assess how well the Young and Neumann laws are fulfilled by the simulation results by comparing the measured value of the liquid-gas contact angle $\theta_\mathrm{LG}$ to the predicted value $\theta_\mathrm{LG}^\mathrm{(theo)}$ that we obtain from either the Young or the Neumann laws, Eqs.~\eqref{eq:young},~\eqref{eq:Neumann_hor},~\&~\eqref{eq:Neumann_ver}, and the measured values of the brush angles. The relative deviation is shown in Fig.~\ref{fig:Neumann_deviation} for both the full-curvature and the long-wave version of the model. There, the results are shown in dependence of the parameters varied in Figs.~\ref{fig:statics_variation_H_dry}~\&~\ref{fig:statics_variation_P}, namely, $H_\mathrm{dry}$ and $P$.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{Fig6.pdf}
\caption{Relative deviation $(\theta_\mathrm{LG}-\theta_\mathrm{LG}^\mathrm{(theo)})/(\theta_\mathrm{LG}^\mathrm{(theo)})$ between the measured liquid-gas contact angle $\theta_\mathrm{LG}$ and the predicted contact angle $\theta_\mathrm{LG}^\mathrm{(theo)}$ using either the Young or the Neumann laws, namely, Eqs.~\eqref{eq:young},~\eqref{eq:Neumann_hor},~\&~\eqref{eq:Neumann_ver},. The result quantifies how well the relations are fulfilled in dependence of (a) the dry brush height $H_\mathrm{dry}$ and (b) the Laplace pressure $P$ (encoding the drop volume). Solid lines correspond to calculations based on the full-curvature version of the model and the long-wave results are shown as dashed lines. The simulation data corresponds to the one shown in Figs.~\ref{fig:statics_variation_H_dry}~\&~\ref{fig:statics_variation_P}. In particular panel (a) uses $P=\num{9e-4}$ and panel (b) uses $H_\mathrm{dry}=6$.}
\label{fig:Neumann_deviation}
\end{figure}
We find that in the full-curvature formulation the Young law is perfectly fulfilled in the limit of large droplets ($P=0$), whereas the long-wave approximation results in a slightly lower contact angle. For decreasing drop size (increasing Laplace pressure) the results for the Young law increasingly deviate. For the Neumann law a deviation is already notable in the full-curvature case for $P=0$: there, the horizontal [vertical] Neumann balance deviates by about \SI{4}{\percent} [more than \SI{10}{\percent}].
We suggest that the latter deviations are due to the mesoscopic modelling as they are connected to the intricacies hidden in the analytic derivation of the mesoscopic Neumann law. On the one hand, we have assumed that the wetting potential can be neglected at the inner inflection point $x_2$ of the wetting ridge, i.e., that it does not appear in Eq.~\eqref{eq:E(x2)}. Indeed the local film height is not sufficiently large, resulting in a contribution to the observed deviation in the horizontal Neumann balance. On the other hand, the conservation of the total generalised momentum $p_\zeta+p_{h+\zeta}$ is also only exact in the limit $h_p/\zeta_{\mathrm{wr}}\rightarrow 0$.\footnote{In the mechanical analogue the inelastic collision takes place in an external force field (corresponding to $\partial_\zeta g_\mathrm{brush}$), such that the total momentum is only approximately conserved during a relatively short collision time.} The finite height of the wetting ridge results in the observed deviation in the vertical Neumann balance.
Note that as the convergence towards the limit $h_p/\zeta_{\mathrm{wr}}\rightarrow 0$ is very slow, and reducing the adsorption layer height $h_p$ increases the numerical effort significantly, here, we do not go beyond the choice of parameters employed above.
\section{Forced wetting}\label{sec:forced_wetting}
\subsection{General remarks}
Having analysed the case of drops and menisci at equilibrium, we next consider the case of forced wetting of a brush-covered substrate. Namely, we activate the additional advection terms in Eqs.~\eqref{eq:gradient_model} by considering substrate velocities $U>0$, i.e., pushing the substrate covered by a nearly dry brush into the liquid. In other words, we invert the typical Landau-Levich setting where a plate is drawn from a bath.
As this corresponds to a persistent energy influx, thermodynamically, we keep the system permanently out of equilibrium. This implies that occurring steady interface profiles with $\partial_t h=\partial_t \zeta=0$ do not correspond to a quiescent state of the liquid but to an internal stationary flow profile. Neither do such steady profiles correspond to minima of the free energy $\mathcal{F}$, i.e., the discussion of equilibrium states in \autoref{sec:equilibrium} does only provide the behaviour in the static limiting case $U\rightarrow 0$ and can not be applied for $U\neq0$. Further, it is important to note that out of equilibrium other states are possible beside steady ones. In particular, we expect time-periodic stick-slip motion of the contact line to occur. First, however, we analyse the changes occurring in the steady profiles when the velocity $U$ is increased from zero.
As any analytic treatment becomes rather challenging, here, we entirely focus on numerical results. To facilitate the numerical computations and a clear discussion of the effect of the brush energy, from now on we only consider the dimensionless model in long-wave approximation. For simplification, we further neglect the dependency of capillarity and wettability on brush state, i.e., we use $\gamma_\mathrm{bl}$ and $f_\mathrm{wet}$ that do not depend on $\zeta$.
Also, from hereon the meniscus is modelled using the above mentioned inclined film boundary condition $\partial_x h(x=0) = -\theta_0$ in order to capture the scenario of a brush-covered plate being pushed into a large liquid reservoir rather than into a finite sized droplet.
\subsection{Small velocities -- Rotation of Neumann angles}\label{sec:rot_Neumann}
\begin{figure}[!htb]
\centering
\includegraphics[width=\hsize]{Fig7.pdf}
\caption{Characterization of the out-of-equilibrium configuration of the contact line region for (relatively small) plate velocities $U$. Shown are (a) the dynamic contact angles (as indicated in the legend) as a function of $U$. Numerical results (solid lines) are compared to predictions obtained with the Neumann law based on the measured dynamic liquid-gas contact angle $\theta_{\mathrm{LG}}$ (dot-dashed line). The horizontal line gives the value of the equilibrium Young angle $\theta_{\mathrm{Y}}$. Panel~(b) presents examples of steady interface profiles at different velocities $U$ as indicated by the colour code of the brush-liquid interface. Gray lines indicate the liquid-air interfaces. Note that the profiles are shifted such that the peak positions of the wetting ridges coincide. Panel~(c) indicates the decrease of the ridge height $\zeta_\mathrm{wr}$ with increasing $U$. The long-wave approximation is used with parameters $T=0.02$, $\sigma=0.3$, $\gamma_\mathrm{bl}=0.3$, $\ell=20$, $M=0.1$ and $D=\SI{4e-3}{}$.}
\label{fig:Braking}
\end{figure}
We start by considering stationary states that occur at relatively small advection velocities $U>0$. Fig.~\ref{fig:Braking} shows corresponding characteristics of the out-of-equilibrium configuration of the contact line region: As $U$ is increased from zero, all contact angles start to deviate from their equilibrium values (panel~(a)), also compare the typical profiles given in panel~(b). The wetting ridge height decreases with increasing $U$ (panel~(c)).
The solid lines in Fig.~\ref{fig:Braking}~(a) converge at very small values ($U\approx10^{-6}$) to the equilibrium values. Then, upon increasing $U$, the dynamic $|\theta_\mathrm{LG}|$ and $|\theta_\mathrm{BG}|$ increase while $\theta_\mathrm{BL}$ decreases. However, introducing the three dynamic angles into the Neumann law (Eqs.~\eqref{eq:Neumann_hor} and~\eqref{eq:Neumann_ver}) we find that it holds reasonably well at least up to $U\approx10^{-3}$. To show this we use the numerically determined $\theta_\mathrm{LG}$ (blue line) and determine the other two angles employing Eqs.~\eqref{eq:Neumann_hor} and~\eqref{eq:Neumann_ver} in long-wave approximation, i.e., Eqs.~\eqref{eq:app:neumann} in Appendix~\ref{app:lw}. The resulting theoretical values are given as dashed lines. At all $U$, the deviation is rather small for $\theta_\mathrm{BG}$ while for $\theta_\mathrm{LG}$ it remains constant at $\approx10\%$ as seen before in the static case in \autoref{sec:NumerikGG}. By checking the validity of the vertical and horizontal Neumann conditions for all three measured angles, we determine that any deviation is as in the static case mostly due to the vertical Neumann balance. Collecting all terms of Eqs.~\eqref{eq:Neumann_hor} and~\eqref{eq:Neumann_ver} on one side, we find that the expressions for $U=0$ deviate from zero by absolute errors of 0.003 (horizontal condition) and 0.174 (vertical condition).
The continued approximate validity of the Neumann law implies that the Neumann angles are merely rotated with increasing $U$ as known for moving contact lines on an elastic substrate.\cite{KDGP2015nc} The rotation is best seen in the dynamic Young angle that coincides with $\theta_\mathrm{LG}$ (blue line in Fig.~\ref{fig:Braking}~(a)). In the velocity range from \SI{e-5}{} to \SI{e-3}{} it undergoes a 75\% increase accompanied by a 60\% decline in the height of the wetting ridge above the dry brush height (from 25 to 10, see Fig.~\ref{fig:Braking}~(c)). This makes the continued validity of the Neumannn law particularly interesting.
Another remarkable observation is, that at $U\approx\SI{e-2}{}$ the three dynamic angles all pass an extremum, i.e., a further velocity increase results in a smaller rotation of the Neumann angles as compared to their equilibrium values. Indeed, there, the deviation from Neumann's law strongly increases, rendering its application invalid. As next discussed in section~\ref{sec:instability}, the qualitative change in behaviour is closely related to the onset of an instability that occurs at a velocity less than one order of magnitude larger. It gives rise to time-periodic stick-slip motion.
\subsection{Large velocities -- instability mechanism}\label{sec:instability}
\begin{figure}[!htb]
\centering
\includegraphics{Fig8.pdf}
\caption{The dynamic angles $|\theta_\mathrm{LG}|$ and $|\theta_\mathrm{BG}|$ as well as their difference are given for steady profiles for a large range of substrate velocities $U$. At small and large $U$, the profiles are linearly stable (solid lines), while they are unstable at intermediate velocities (dashed lines). There, time-periodic stick-slip behaviour occurs (green shading) with a hysteresis range beyond the lower instability threshold (indicated by vertical dotted lines). The remaining parameters are as in Fig.~\ref{fig:Braking}.}
\label{fig:var_Uhigh}
\end{figure}
Moving to larger velocities in the range \SI{e-2}{} to \SI{e+0}{} we restrict our attention to the angles $\theta_\mathrm{LG}$ and $\theta_\mathrm{BG}$. As the wetting ridge is very low, no well-defined $\theta_\mathrm{BL}$ can be measured. \autoref{fig:var_Uhigh} shows the remaining dynamic angles as a function of $U$. At small and large $U$ the angle $\theta_\mathrm{LG}$ increases with increasing $U$ while it decreases in the intermediate range \SI{5e-3}{}$<U<$\SI{8e-2}{}. In contrast, the angle $\theta_\mathrm{BG}$ first becomes larger and then, for $U>$\SI{3e-3}{}, continuously decreases. Asymptotically it approaches zero as the wetting ridge and the difference in the brush heights inside and outside the liquid both shrink. The latter results from the strong decrease in the advection time scale as compared to the time scale for mass transfer into the brush. This also illustrates that at higher velocities the deviation from the Neumann law must get large.
Evaluating the stability of the stationary states tracked in Fig.~\ref{fig:var_Uhigh} shows that they are linearly stable at small and large $U$ (solid lines), but unstable at intermediate $U$ (dashed lines). In the latter range, time simulations reveal time-periodic stick-slip behaviour. Furthermore, Fig.~\ref{fig:var_Uhigh} already hints at two important conditions for the instability to occur. First, the corresponding $U$-range nearly coincides with the range where $\theta_\mathrm{LG}$ decreases, i.e., where $\partial |\theta_\mathrm{LG}|/\partial U<0$ for the stationary state. In other words, the instability occurs when an increase in velocity results in a decrease of the global dynamic contact angle. This implies that a slowly increasing velocity would favour a decreasing angle, but to decrease the angle the contact line region has to advance even faster -- corresponding to a destabilising feedback loop.
\begin{figure}[!htb]
\centering
\includegraphics[width=.45\textwidth]{Fig9.pdf}
\caption{Sketch illustrating the Gibbs condition and the resulting depinning. In panel (a) the contact line is pinned at the ridge as the effective angle between liquid-gas interface and the substrate-gas interface in front of the contact line is lower than the Young angle, i.e., $|\theta_\mathrm{LG}|-|\theta_\mathrm{BG}| < |\theta_\mathrm{Y}|$. The liquid-gas interface steepens until equality (panel (b)). Any further steepening lets the contact line slip off the ridge (panel~(c)).}
\label{fig:Gibbs}
\end{figure}
Second, the instability occurs at an order of magnitude of $U$, where the difference $|\theta_\mathrm{LG}|-|\theta_\mathrm{BG}|$ starts to strongly increase as the Neumann law does not hold any more. Notably, the upper critical $U$ where the unstable range ends, occurs where the difference $|\theta_\mathrm{LG}|-|\theta_\mathrm{BG}|$, i.e., the effective liquid contact angle at the advancing side of the ridge, exceeds the equilibrium contact angle $|\theta_\mathrm{Y}|$. This is a consequence of the Gibbs condition\cite{Quer2008armr,GASK2018prl} which states that a contact line depins from a heterogeneity of a rigid solid substrate when the contact angle with respect to the local substrate slope exceeds the equilibrium contact angle, see \autoref{fig:Gibbs}. While for $|\theta_\mathrm{LG}|-|\theta_\mathrm{BG}|<|\theta_\mathrm{Y}|$ the pinning at the wetting ridge governs the contact line dynamics (or at least part of it during the stick-slip dynamics), above this critical angle the contact line is unable to pin at all and instead constantly slips. Consequently, the dynamics is purely governed by the moving contact line while the ridge, or swelling gradient, simply follows behind.
\section{Stick-slip dynamics}\label{sec:stickslip}
Having established the existence of a velocity range where unstable stationary profiles give rise to stick-slip motion, we next analyse in some detail this time-periodic state of the contact line region and its dependency on various parameters.
\subsection{Single stick-slip cycle}
\begin{figure}[!htb]
\centering
\includegraphics[width=.5\textwidth]{Fig10.pdf}
\caption{Shown are space-time plots of (a) the brush profile $\zeta$ and (b) the liquid height profile $h+\zeta$ in the contact line region during a single typical stick-slip cycle at $U=0.014$. The black dotted line indicates when the Gibbs condition is fulfilled and the ensuing depinning results in a slipping motion. The remaining parameters are as in Fig.~\ref{fig:Braking}. The period of the cycle is $\tau=4800$.}
\label{fig:spacetime}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics{Fig11.pdf}
\caption{The top row displays time sequences of snapshots from four different phases of the single cycle of stick-slip motion in Fig.~\ref{fig:spacetime}. Shown are profiles of the liquid-gas (blue) and brush-liquid (orange) interface in the contact line region. The final snapshot from each burst is shown in darker colours that the preceding ones. For a discussion see main text. The bottom row shows (left) a linearly stable stationary profile at identical parameters, and (right) the temporal change in contact angles over one period $\tau$ for the top row dynamics (solid lines) compared to the values for the stationary state (dashed lines). The vertical dotted line indicates when the Gibbs condition is fulfilled.}\label{fig:stickslip}
\end{figure}
\begin{figure*}[!htb]
\centering
\includegraphics{Fig12.pdf}
\caption{The top and bottom row present contributions to energy and dissipation, respectively,
for the single cycle of stick-slip motion shown in Figs.~\ref{fig:spacetime} and~\ref{fig:stickslip} (solid lines). In particular, the total free energy $F$ and its constituents $F_\mathrm{cap}$, $F_\mathrm{wet}$ and $G_\mathrm{brush}$ are given as differences w.r.t.\ to their values at equilibrium ($U=0$). The total energy dissipation $D_\mathrm{ges}$ is compared to contributions from the three dissipation channels $D_h$, $D_\zeta$, and $D_M$. The horizontal dashed lines give the corresponding values for the linearly stable stationary state that exists at identical parameters. All values are obtained by integrating over the spatial domain $x\in [0,420]$. In each panel the vertical dotted line marks the time when the Gibbs condition is fulfilled. The insets magnify the vicinity of this instant.}
\label{fig:Energiedissipation}
\end{figure*}
A single typical stick-slip cycle is displayed in different representations in Figs.~\ref{fig:spacetime} and~\ref{fig:stickslip}. The former gives space-time plots of the interface profiles while the latter presents detailed views of sequences of individual profiles close to crucial moments within the cycle as well as measurements of dynamic angles during the cycle. The accompanying \autoref{fig:Energiedissipation} presents an analysis of changes during the same cycle in overall free energy $F[h,\zeta]$ and its constituents as well as in total dissipation and the individual dissipation channels as defined in Appendix~\ref{sec:dissipation_derivation}.
We start with a discussion of the dynamics of the interfaces: During the phase where the contact line is pinned to the fully developed wetting ridge ($0.17\leq t/\tau\leq 0.72$), the ridge slowly recedes towards the left carrying the pinned contact line along. However, in parallel, the contact line slowly creeps onto the right flank of the ridge (Fig.~\ref{fig:stickslip}~(top, left)), i.e., it surfs the ridge (as described for elastic substrates by~\citet{KDGP2015nc,GASK2018prl}). Because the surfing contact line separates from the peak of the wetting ridge, the upwards traction at the peak decreases and the ridge becomes rounder. In this phase $|\theta_\mathrm{LG}|$ remains nearly constant while $|\theta_\mathrm{BG}|$ decreases more and more. This occurs, in particular, in the surfing phase when the difference $|\theta_\mathrm{LG}|-|\theta_\mathrm{BG}|$ sharply increases (Fig.~\ref{fig:stickslip}~(bottom, right)). In consequence, at $t/\tau\approx 0.10$ the difference $|\theta_\mathrm{LG}|-|\theta_\mathrm{BG}|$ passes the equilibrium angle thereby fulfilling the Gibbs condition. As a result, the contact line depins and suddenly moves to the right leaving the then very slowly shrinking ridge behind (Fig.~\ref{fig:stickslip}~(top, second from left)). In this short slipping phase $|\theta_\mathrm{LG}|$ sharply decreases while $|\theta_\mathrm{BG}|$ remains nearly constant but is irrelevant as it belongs to the ridge that is already detached from the contact line. When the contact line slows down a new wetting ridge starts to emerge and pins the contact line (Fig.~\ref{fig:stickslip} (top, second and first right)). In this long phase $|\theta_\mathrm{LG}|$ slowly increases as does $|\theta_\mathrm{BG}|$, now related to the new ridge (Fig.~\ref{fig:stickslip} (bottom, right)). Nevertheless, $|\theta_\mathrm{LG}|-|\theta_\mathrm{BG}|$ stays almost constant in this phase. Then the cycle starts again.
Focusing next on \autoref{fig:Energiedissipation} we note that the dominant contribution to the overall energy $F$ is in all phases the capillary energy $F_\mathrm{cap}$. This is even more pronounced when an entire droplet is considered instead of a domain limited to the contact line region. When the new ridge grows with the contact line sticking to it, the increasing $|\theta_\mathrm{LG}|$ results in an increase in capillary energy. In parallel, the wetting energy $F_\mathrm{wet}$ decreases as in an increasing part of the domain the brush is only covered by the energetically favourable adsorption layer. During the fast slipping process, $F_\mathrm{cap}$ and with it $F$ sharply decrease, relaxing to their lowest values within the cycle. $F_\mathrm{wet}$ behaves inversely and sharply increases. The contribution of the brush $G_\mathrm{brush}$ is comparatively small in the considered parameter range, and qualitatively similar to $F_\mathrm{cap}$.
Dissipation occurs through three channels: viscous dissipation $D_h$ within the liquid, mostly focused in the contact line region, dissipation $D_\zeta$ due to liquid diffusion within the brush, and dissipation $D_M$ due to mass transfer between liquid layer and brush. The total dissipation $D$ is dominated by $D_h$, followed by the about one magnitude weaker $D_M$, and the rather small $D_\zeta$. All contributions show a slow build-up in the stick-phase that develops into a sharp peak in the surf- and slip-phase with the maxima of the individual channels slightly shifted w.r.t.\ each other. The trough to peak contrast is about one magnitude for all channels. The total dissipation peaks after the Gibbs condition is reached, i.e., clearly in the slip-phase. In contrast, $D_M$ peaks earlier confirming that depinning is triggered by a mass transfer-induced deformation of the substrate. The very weak $D_\zeta$ seems to have a double-peak structure with the minimum between the two maxima coinciding with the fulfilment of the Gibbs condition.
Note, finally, that due to hysteresis there exists a stationary state at the same parameter values where the analyzed stick-slip cycle occurs. It is shown in Fig.~\ref{fig:stickslip}~(bottom, left) with the corresponding angles, energies and dissipation given as horizontal dashed lines in Fig.~\ref{fig:stickslip}~(bottom, right), Fig.~\ref{fig:Energiedissipation}~(top) and Fig.~\ref{fig:Energiedissipation}~(bottom), respectively. On average the stick-slip cycle has a lower total energy and shows also a lower mean dissipation than the stationary state.
After having discussed a single stick-slip cycle in detail for a particular set of parameters, in the following we explore how the behaviour depends on important system parameters. We focus on the driving substrate velocity in section~\ref{sec:para-velocity}, on the mass transfer rate between liquid layer and brush in section~\ref{sec:para-transfer}, and on interface and brush energies in section~\ref{sec:para-energy}.
\subsection{Dependence on substrate velocity}\label{sec:para-velocity}
It is our main interest to systematically establish in which parameter ranges stationary and time-periodic states dominate. To obtain these ranges we employ two numerical techniques: First, we use extensive time simulations of the liquid meniscus dynamics where we increase (decrease) the substrate velocity in small steps and make sure that the integration time of each step is long enough for transient behaviour to die out. This allows us to access stable dynamical states. Second, we employ path continuation to access the stationary state for arbitrary values of the substrate velocity and to obtain its linear stability. We proceed to combine the results of both approaches into one diagram.
As solution measures for the stationary states we use the height of the wetting ridge $\zeta_\mathrm{wr}$, and the dynamic contact angle $|\theta_\mathrm{LG}|$. For the time-periodic stick-slip motion we show the corresponding minimal and maximal values within a cycle. Thereby, for $\zeta_\mathrm{wr}$ we only consider the particular ridge located in the vicinity of the contact line (practically, we request $h<5$). This ensures that the newly growing ridge is analyzed and not the previous one that is slowly decaying. Additionally, we consider the frequency $\nu=1/\tau$ obtained as the inverse of the time period. Note that $\nu=0$ for the stationary states.
\begin{figure}[!htb]
\centering
\includegraphics{Fig13.pdf}
\caption{Stationary states and stick-slip cycles are analyzed in dependence of substrate velocity $U$. Shown are (a) frequency $\nu$, (b) the dynamic liquid-gas contact angle $|\theta_\mathrm{LG}|$, and (c) the wetting ridge peak height $\zeta_\mathrm{wr}$. Solid [dashed] orange lines indicate linearly stable [unstable] stationary states, while green lines give the corresponding range for linearly stable stick-slip cycles. Linear stability thresholds and existence ranges are indicated by vertical dotted lines, thereby marking the stick-slip and hysteresis regimes. The remaining parameters are $T=0.02$, $\sigma=0.3$, $\gamma_\mathrm{bl}=0.3$, $\ell=20$, $M=0.1$ and $D=\SI{4e-3}{}$.}
\label{fig:var_U}
\end{figure}
\autoref{fig:var_U} gives the described quantities in dependence of substrate velocity $U$. Inspecting the figure we see that there exists a regime of stick-slip behaviour in the range of decreasing angles $|\theta_\mathrm{LG}|$ as already established in section~\ref{sec:instability}. At large velocities, it ends at $U\approx0.068$ in a supercritical Hopf bifurcation: the amplitude of the temporal modulation of all quantities approaches zero while the frequency approaches a finite value. This is further confirmed by the fact that at the same velocity the stationary state changes its linear stability via a pair of complex conjugate eigenvalues that crosses the imaginary axis. The situation is more involved at small velocities where a range of hysteresis exists, i.e., where bistability occurs between stationary state and time-periodic stick-slip behaviour. The latter is first observed at $U\approx0.010$ while the stationary state looses stability in a (seemingly subcritical) Hopf bifurcation at $U\approx0.013$. This indicates that the time simulation stops to pick up the stick-slip motion close to a saddle-node bifurcation where the subcritical branch of cycles stabilises and turns back towards larger $U$. At this point the frequency is small but finite and the amplitude of the temporal modulation of $|\theta_\mathrm{LG}|$ and $\zeta_\mathrm{wr}$ reach their larges values. Both amplitudes monotonically decrease with increasing $U$. Note that we find no indication of more complicated behaviour like, e.g., period doubling.
\subsection{Dependence on transfer constant}\label{sec:para-transfer}
\begin{figure}[]
\centering
\includegraphics{Fig14.pdf}
\caption{For selected values of the transfer rate constant $M$ (stated above each panel) the dependencies of the dynamic angles $|\theta_\mathrm{LG}|$ and $|\theta_\mathrm{BG}|$ as well as of their difference on substrate velocity $U$ are given. As reference the equilibrium contact angle $|\theta_\mathrm{Y}|$ is indicated by a horizontal line. The green shading marks the range of stick-slip motion including the hysteresis regime. The remaining parameters and line styles are as in Fig.~\ref{fig:var_U}.}
\label{fig:var_M}
\end{figure}
\begin{figure}[]
\centering
\includegraphics{Fig15.pdf}
\caption{Illustration of the definitions of (a) the characteristic timescale $\tau_M$ (vertical lines) for ridge growth as the time when $\zeta_\mathrm{wr}-\zeta_\mathrm{n}$ reaches half its maximal equilibrium height, and (b) the characteristic equilibrium ridge width $L_0$ defined by a triangular approximation using the equilibrium values of $\theta_\mathrm{BL}$ and $\theta_\mathrm{BG}$ (see \autoref{eq:L0}). Panels (c-e) compare for selected values of the transfer rate $M$ the timescale $\tau_M$ and a characteristic timescale of substrate motion $\tau_U=L_0/U$. Their equality coincides with the lower border of the unstable $U$-range highlighted by the green shading.}\label{fig:scales}
\end{figure}
In section~\ref{sec:para-velocity} we have analysed how the system behaviour depends on substrate velocity for a fixed set of all other parameters. Thereby, we have identified a $U$-range where stick-slip behaviour occurs. Next, we explore how the stick-slip range changes when the transfer rate constant $M$ is varied. Accordingly, \autoref{fig:var_M} shows dependencies of dynamic angles $|\theta_\mathrm{LG}|$ and $|\theta_\mathrm{BG}|$ as well as of their difference $|\theta_\mathrm{LG}|-|\theta_\mathrm{BG}|$ on $U$ for six different values of $M$ between $M=10$ (very fast mass transfer between liquid meniscus and brush) and $M=10^{-4}$ (very slow transfer). The panel at intermediate $M=0.1$ reproduces Figure~\ref{fig:var_Uhigh}. Comparing the panels of Fig.~\ref{fig:var_M} we first note that for fast transfer $|\theta_\mathrm{LG}|$ is a monotonically increasing function of velocity $U$. There is no range of negative inclination and no stick-slip motion occurs. Both features only appear if $M$ is sufficiently small, here $M\ge0.1$. Further decreasing $M$ by three orders of magnitude widens the $U$-range of negative inclination (the dip get shallower though) as well as the range of stick-slip motion.
Both (necessary) criteria for the instability uncovered in section~\ref{sec:instability} still hold for the entire range of considered transfer constants $M$: On the one hand, the instability only occurs, when $|\theta_\mathrm{LG}|(U)$ has a negative slope. On the other hand, the maximal $U$ where the stationary state is unstable coincides with the fulfilment of the Gibbs condition, i.e., $|\theta_\mathrm{LG}|-|\theta_\mathrm{BG}|$ crosses $\theta_Y$. However, considering the sequence from $M=0.1$ to $M=10^{-4}$ it also becomes obvious that the negative slope criterion is not sufficient for the instability to occur. While at $M=0.1$ the instability threshold follows not too far behind the maximum of $|\theta_\mathrm{LG}|(U)$, the distance becomes larger for smaller $M$. For instance, for the slowest considered transfer, at $M=10^{-4}$, the unstable $U$-range only corresponds to the second half of the range with negative slope. Therefore we need to refine our criterion for the onset of instability.
To do so, we define a characteristic timescale $\tau_M$ for the growth of a wetting ridge. It is important as during the stick-slip motion periodically a new ridge is created (see \autoref{fig:var_U}). For simplicity, we assume that $U$ has only a minor influence on the growth and consider the relaxation from a flat brush state towards an equilibrium wetting ridge at $U=0$. The corresponding growth of $\zeta_\mathrm{wr}$ is shown for three values of $M$ in Fig.~\ref{fig:scales}~(a). Based on these simulations, we define $\tau_M$ as the time when the wetting ridge reaches half of its equilibrium height. This characteristic time (marked by vertical lines in Fig.~\ref{fig:scales}~(a)) increases by about one order of magnitude when $M$ is decreased from $0.1$ to $10^{-3}$.
Additionally, we define a typical horizontal length scale $L_0$ characterising a typical ridge width based on the triangular construction illustrated in Fig.~\ref{fig:scales}~(b), or, by formula
\begin{equation}
L_0=(\zeta_\mathrm{wr}-\zeta_\mathrm{n})\left[\frac{1}{|\theta_\mathrm{BL}|}+\frac{1}{|\theta_\mathrm{BG}|}\right] \label{eq:L0}
\end{equation}
as obtained in long-wave approximation. This length scale allows us to define a second characteristic time $\tau_U=\frac{L_0}{U}$ that characterises the forced motion of the substrate.
The values for the two defined timescales $\tau_M, \tau_U$ are plotted in dependence of the velocity $U$ in Figs.~\ref{fig:scales}~(c), (d) and (e) for $M=0.1$, $10^{-2}$, and $10^{-3}$, respectively. One immediately notices that the low-velocity border of the region of unstable stationary states very closely coincides with the velocity where the two time scales cross. In other words, there, the stationary state is linearly stable as long as the wetting ridge grows much faster than it is moved by the substrate (note Figs.~\ref{fig:scales}~(c-e) are log-log plots). One may say, the ridge is slaved to the liquid that moves with the substrate. In contrast, when the time scales become similar all influences compete on equal terms and an instability is possible.
\subsection{Dependence on energies}\label{sec:para-energy}
\begin{figure*}[!htb]
\centering
\includegraphics[width=\textwidth]{Fig16.pdf}
\caption{Influence of various parameters on stationary states and stick-slip cycles: (top row) Frequency $\nu$, (center row) contact angle $|\theta_\mathrm{LG}|$, and (bottom row) height of the wetting ridge $\zeta_\mathrm{wr}$ are given as function of (1st column) brush-liquid interface energy $\gamma_\mathrm{bl}$, (2nd column) effective temperature $T$, (3rd column) brush grafting density $\sigma$ and (4th column) brush Kuhn length $\ell_K$. Line styles and respective remaining parameters are as in Fig.~\ref{fig:var_U}.
}
\label{fig:dynVariation}
\end{figure*}
Finally, we briefly investigate the influence of a number of further parameters related to capillary and brush energies. An overview is given in Fig.~\ref{fig:dynVariation} where dependencies on brush-liquid interface energy $\gamma_\mathrm{bl}$, effective temperature $T$, brush grafting density $\sigma$ and the Kuhn length $\ell$ of the brush polymers are shown at fixed $U=\SI{0.014}{}$, i.e., at the velocity of our reference case in Fig.~\ref{fig:stickslip}. Both, stationary states and stick-slip cycles are included. Note that each of the chosen parameters allows us to approach the limiting case of a rigid, non-adaptive substrate, i.e., with vanishing wetting ridge. The respective limits correspond to $\gamma_\mathrm{bl}\to\infty$, $T\to\infty$, $\sigma\to\infty$ and $\ell\to0$. This implies that in each of these cases the stick-slip behaviour will eventually disappear when moving towards the corresponding limit.
Inspecting Fig.~\ref{fig:dynVariation} we note that hysteresis is observed for all parameter dependencies indicating that in each case a subcritical Hopf bifurcation occurs. It is remarkable that the frequency $\nu$ of the stick-slip cycle barely depends on the brush-related parameters and only shows a small decrease with decreasing interface energy $\gamma_\mathrm{bl}$. A similar observation holds for the amplitude of variations in the height of the wetting ridge over a stick-slip cycle. The abrupt stop of stick-slip motion with increasing $\gamma_\mathrm{bl}$, $T$, and $\sigma$ can in all cases be related to the occurrence of a saddle-node bifurcation of cycles, in particular, for $\gamma_\mathrm{bl}$ and $T$ curves become nearly vertical clearly indicating a fold. This is less pronounced for $\sigma$ and $\ell$.
Taken together these observations imply that there is a ``critical softness'' or ``critical adaptivity'' that must be reached to induce a stick-slip behaviour. The stick-slip motion itself only depends relatively weakly on the energetic parameters. This is in stark contrast to the much larger influence found in sections~\ref{sec:para-velocity} and \ref{sec:para-transfer}, respectively, for the dynamic parameters substrate velocity and transfer rate constant. Their direct correspondence to the timescales $\tau_U$ and $\tau_M$ reinforces our finding that their ratio controls the occurrence of stick-slip behaviour.
\section{Conclusion}\label{sec:conclusions}
We have presented a mesoscopic model for the dynamic wetting of a polymer brush-covered substrate. It accounts for the coupled dynamics of a liquid drop/meniscus on the brush and of the imbibed liquid within the brush as well as for the imbibition process itself. Due to its central gradient dynamics structure, the dynamics is driven by a single underlying free energy functional. The model extends the work by \citet{ThHa2020epjt} by introducing a refined energy functional that additionally accounts for the dependencies of the wetting and interface energies on the brush state. Moreover, now the energies may be employed in their full-curvature formulation as well as in their long-wave approximation.
The refined mesoscopic model has first allowed us to employ the full-curvature formulation to consider a static contact line region. There, we have analytically recovered both, the Young law for the global liquid-gas contact angle and the Neumann law that locally relates the contact angle to the opening angle of the occurring static wetting ridge. While our calculations have mostly been performed in the mesoscopic picture, the derivation of macroscale equivalents for both laws has also been sketched. This has allowed us to establish consistency conditions that relate mesoscale and macroscale quantities in a similar spirit as recently presented for surfactant-covered drops.\cite{TSTJ2018l} In passing, analytic considerations have provided an estimate for the shape and size of the wetting ridge. Due to its nanometric size, a quantification remains a challenge for experiments. However, observations of wetting ridges are reported for Molecular Dynamics simulations. \cite{LeMu2011jcp,MeSB2019m}
Second, we have employed numerical simulations not only to confirm our analytical findings but also to test how they change beyond the static equilibrium case when considering a more intricate forced wetting scenario. We have therefore investigated the case of an externally driven liquid meniscus moving over the brush substrate at an imposed velocity modelling, e.g., a brush-covered plate that is plunged into a liquid bath at constant relatively small velocity. In other words, the employed geometry resembles a dip coating process that is modelled by adding a simple advection term and adjusting the boundary conditions. This corresponds to an inverted Landau-Levich setting as usually one studies the transfer of liquid onto a plate that is drawn out of a bath.\cite{SADF2007jfm,MRRQ2011jcis,TWGT2019prf}
For the brush-covered substrate that is immersed into the bath, at low velocities we have observed a deformation of the interfaces in the contact line region. Specifically, the liquid contact angle as well as the inclination angles of the flanks of the wetting ridge all change under the additional stress that forces the ridge to advance along the substrate. However, we have found that at small velocities the three angles continue to satisfy the Neumann law even in this out-of-equilibrium scenario. In other words, the interfaces close to the contact line region undergo a nearly rigid rotation. This agrees with earlier observations for moving contact lines on viscoelastic substrates (i.e., without mass transfer into the substrate). \cite{KDGP2015nc,AnSn2020arfm} The motion of the wetting ridge we have observed here on the swelling polymer brush has been found to cause a similar dissipative braking of the motion, that counteracts the imposed movement of the liquid meniscus, and results in a steepening of the liquid contact angle.
At larger velocities, first, the deviation of the relation between the angles from the Neumann relations starts to increase such that the Neumann law does not hold anymore. Then, one enters the velocity range where the dynamic brush-gas and liquid-gas contact angles decrease with increasing plate velocity. Eventually, the growth of the ridge due to mass transfer is not able to keep up with the substrate motion, the stationary state becomes unstable. In the unstable regime, the liquid-gas interface depins from the wetting ridge and advances over the substrate in a rapid slipping motion. The contact line slows down and stops when the contact angle relaxes. Subsequently, the brush starts to form a new ridge via some liquid imbibition near the contact line, thereby pinning the front again. In consequence, the process starts over and a time-periodic stick-slip motion emerges. There exists a hysteresis between stationary state and stick-slip cycles in the vicinity of the stability threshold.
While a few experimental observations of stick-slip behaviour in the forced spreading of drops on polymer brushes exist,\cite{WMYT2010scc,SHNF2021acis} to our knowledge, the present work establishes the first theoretical description and analysis of the mechanism.
Overall, the phases of a stick-slip cycle are rather similar to the ones described for other systems where stick-slip motion occurs, e.g., for evaporation-induced moving contact lines of solutions and suspensions that periodically deposit material\cite{FrAT2012sm} and contact lines moving on viscoelastic substrates.\cite{PBDJ2017sm,KDGP2015nc,AlMo2021ijnme} Moreover, we have identified that the depinning of the contact line from the ridge within a stick-slip cycle coincides with the fulfilment of a Gibbs condition\cite{Quer2008armr} as it is known from the wetting of rough or otherwise topographically heterogeneous substrates. Our data suggest that this condition governs both the onset of slip within a cycle and the size of the stick-slip range of forcing velocities. Note that the relevance of Gibbs' condition is also reported for other out-of-equilibrium settings, namely, related to evaporation.\cite{TDGR2014l}
Similarly, our analysis has shown that the growth rate of the ridge that newly grows during a cycle is critical to the existence of a stick-slip motion. If the time scale of growth (here limited by the rate of imbibition) is much shorter than the time scale of contact line motion over a characteristic length, the wetting ridge is able to move along with the contact line without depinning. In other words, a rapid (de-)swelling of the polymer brush allows the contact line to surf the comoving ridge. If, however, the contact line is forced to advance at a much larger velocity than the swelling rate of the polymers, the wetting ridge is unable to form at all. It is only when the time scales of imbibition and motion are comparable that a periodic ridge formation, pinning and subsequent depinning are observed.
We acknowledge that the time scale of ridge growth is a theoretical measure that could be difficult to access in experiments. Using our findings on the stick-slip phenomenon in applied scenarios may therefore require a related characteristic quantity as a proxy, e.g., the swelling time of a homogeneous brush from vapour. The prevalence of stick-slip motion in numerous wetting problem renders this a very interesting subject for future investigations. We point out that some of our key findings should allow for a straightforward comparison with experimental works and theoretical results employing other methods. Prominently, our model predicts a hysteretic transition between the stationary regime at low velocities and the stick-slip regime. The recent theoretical work by \citet{MoAK2022el} on the forced wetting of soft elastic substrates gives comparable results. Namely, their investigation of the transitions in and out of the stick-slip regime reveals a similar, respectively sub- and supercritical behaviour although hysteresis is not explicitly mentioned.
We have concluded our investigation with an extensive study of the dependence of the dynamics on various control parameters thereby providing an overview of the stick-slip range in parameter space. This parameter study will help to understand and compare the role of the various parameters in different experimental settings and even in the analysis of stick-slip behaviour in related problems. For example, the presented results show that the periodic stick-slip motion can be suppressed by using a liquid of a low liquid-gas interface energy. Similarly, we predict that brushes of lower grafting density should be more prone to stick-slip behaviour.
Finally, we stress that the presented gradient dynamics model allows for refinements in numerous ways. While for the sake of efficiency all of our numerical results have been performed using the free energy in its long-wave approximation, we expect only subtle shifts in the results when computed using the full-curvature model. Moreover, an extended model may additionally account for miscibility effects (e.g., by using a Flory-Huggins energy with $\chi \neq 0$), substrate elasticity,\cite{HeST2021sm} or the effects of evaporation into an ambient vapour phase using, e.g., the approach suggested by \citet{HDJT2022apa}.
\section*{Author Contributions}
\textbf{Daniel Greve:} Writing -- Review \& Editing, Writing -- Original Draft, Investigation;
\textbf{Simon Hartmann:} Writing -- Review \& Editing, Writing -- Original Draft, Investigation, Software;
\textbf{Uwe Thiele:} Writing -- Review \& Editing, Supervision, Funding acquisition.
\section*{Conflicts of interest}
There are no conflicts to declare.
\section*{Acknowledgements}
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) within SPP~2171 by Grants No.\ TH781/12-1 and TH781/12-2. We acknowledge fruitful discussions on adaptive viscous and viscoelastic substrates with Jan Diekmann, Christopher Henkel and Jacco Snoeijer, as well discussions on the interaction of liquids and polymer brushes with the group of Sissi de Beer.
|
1,314,259,996,503 | arxiv | \section{Introduction}
During the last decades, topology has influenced many fields of physics through the renewed description of various phenomena. In condensed-matter physics, topological invariants | known as Chern numbers | have played an important role in the description of the integer quantum Hall effect (IQHE)~\cite{Klitzing1986}. Here, the quantized Hall conductivity of a two-dimensional (2D) electron system is expressed as a sum $\sigma_\text{H}=R_{\text{K}}^{-1} \sum_{E_n < E_\text{F}} N_{\text{ch}} (E_n)$
of Chern numbers $N_{\text{ch}} (E_n)$ that are integers associated with the energy band $E_n$~\cite{Thouless1982,Kohmoto1985}.
Here $R_{\text{K}}$ is von Klitzing's constant and $E_\text{F}$ denotes the Fermi energy assumed to lie inside a gap of the bulk energy spectrum. Furthermore, it has been proven that the sum of Chern numbers is expressing the number of gapless \emph{edge states} located inside the bulk energy gaps. These edge states carry the current in the IQHE~\cite{Hatsugai1993,Qi2006}.
The breaking of time-reversal symmetry (TRS) due to external magnetic fields plays a crucial role for the topological interpretation of the IQHE~\cite{Haldane1988}. Recently, the discovery of the so-called quantum spin-Hall effect (QSHE) has lightened a new path for the investigation of systems where TRS is preserved~\cite{Kane2005,Kane2005bis,Kane2006,Bernevig2006,Bernevig2006bis,Qi2008}. The QSHE manifests itself in insulating systems that show a non-trivial Z$_2$ index~\cite{Kane2005}.
These so-called \emph{topological insulators} are characterized by the presence of \emph{spin-filtered} edge states in the gaps of the bulk energy spectrum. Because of TRS invariance, the spin-up and spin-down states move in opposite directions along the edge of the system. As a consequence, the total charge current as well as the associated Chern numbers are zero~\cite{Avron1988,Qi2008}. Yet, a spin-Chern number has been introduced in order to measure the spin-transport~\cite{Sheng2006} and to distinguish the Z$_2$ class of the system~\cite{Fukui2007}.
The interplay between the lattice topology and the QSHE has been the focus of various recent investigations. In particular, the QSHE has been studied for the Kagome~\cite{Guo2009,Liu2010}, the Lieb and Perovskite~\cite{Franz:2010}, the honeycomb~\cite{Kane2005,Ruegg2010}, the square~\cite{Stanescu2010,goldman:2010}, the $\mathcal{T}_3$~\cite{bercioux:2010}, the checkerboard~\cite{Sun2009}, the pyrochlore~ \cite{Guo2009PRL}, diamond~\cite{Fu2007} and the square-octogon~\cite{Kargarian2010} lattices.
In this context, some lattice models are of particular interest as they show dispersionless energy bands. These \emph{flat bands} correspond to a macroscopic number of degenerate localized states.
Originally, flat bands played a fundamental role in magnetism, as they were shown to accompany the occurance of ferromagnetic ground states in multi-band Hubbard models \cite{Tasaki2008,Lieb1989,Mielke1991}. More recently, the existence and the robustness of these special bands have been extensively studied in a vast family of frustrated hopping models \cite{Bergman2008,Green2010} and for the case of electron localization due to magnetic fields and spin-orbit interactions~\cite{localization}. Interestingly, singular touchings between flat and dispersive bands have been shown to be topologically protected by real-space loops \cite{Bergman2008}. On the face-centered square lattice | also known as Lieb lattice | a flat band touches two linearly dispersing bands, \emph{i.e.}, the flat band intersects a single Dirac point, and the low-energy regime is described by a quasi-relativistic equation for spin-1 fermions~\cite{schen:2010,Dagotto}.
Nowadays, various lattices can be engineered using cold atoms trapped by electromagnetic fields \cite{Lewenstein2007,Bloch2008,schen:2010,apaja:2010,grynberg:1993,bercioux:2009}. In particular, the realization of topological states of matter with cold fermionic atoms appears to be a realistic and attractive goal from the experimental point of view \cite{Stanescu2010,goldman:2010,Liu:2010}. A significant advantage of these experiments is the full control of a wide range of system parameters as, \emph{e.g.}, lattice geometry, interaction and disorder. In these experiments, engineered gauge fields allow to mimic the effects of magnetic fields \cite{Lin2009,Spielman2009,Jaksch2003} or spin-orbit interactions (SOIs) \cite{Stanescu2010,goldman:2010,Stanescu2008,Spielman2010, Wang2010b,Ho2010,Juze2010}. These gauge fields can be generated by spatially-varying laser or magnetic fields which modify particle-hopping via non-trivial Berry's phases \cite{Juze2005,Dalibard2010}. Recent experiments have implemented light-induced external gauge fields and reproduced the physics of charges subjected to electric or magnetic fields \cite{Lin2009,Spielman2009,Lin2010}. Moreover, with such a setup one expects to observe several fundamental phenomena including the Hofstadter butterfly \cite{Jaksch2003,Hofstadter1976}, atomic analogues of the quantum Hall effects \cite{Goldman2009,Sorensen2004}, relativistic physics \cite{Merkl2008,Goldman2009bis}, and vortex structures \cite{Gunter2009,Lin2009,Lim2008}. Optical-lattice setups also allow to consider a generalization of the ongoing experiments, namely the implementation of non-Abelian gauge fields \cite{Osterloh2005, Ruseckas2005,Gerbier2009,Spielman2010}. In particular, non-Abelian gauge fields acting on multi-level atomic systems could mimic SOI \cite{Stanescu2008,Spielman2010, Wang2010,Ho2010,Juze2010}, paving the way to study the spin Hall \cite{Zhu2006} and quantum spin Hall effects \cite{goldman:2010,Liu2010}. Very recently, a concrete proposal of an optical Lieb lattice for cold atoms has been presented \cite{apaja:2010}. In the later work, Apaja \emph{et al.} have shown that a fermionic cloud expanding after the release of the harmonic trap should show clear signatures of the flat band's localized states. Finally, the existence of flat bands with non-trivial topological order has been demonstrated \cite{Tang2010}, contradicting the belief that non-dispersive bands were associated to vanishing Chern numbers \cite{Green2010}. \\
Motivated by the possibility to engineer an optical Lieb lattice for cold fermionic atoms, we investigate the emergence of topological properties for various configurations of synthesized gauge fields. We first provide an original analysis of a peculiar IQHE, in the case where a uniform magnetic field is present in the Lieb lattice. We then explore the effects of an intrinsic spin-orbit term \cite{Kane2005} and show how it leads to quantum spin Hall states. In this framework, we extend the seminal work of Ref.~[\onlinecite{Franz:2010}] and derive an effective Hamiltonian describing the low-energy regime. This Weyl-like Hamiltonian leads to a three-component quantum equation that resembles the relativistic equation for spin-1 particles. Besides, we obtain the Landau levels in the presence of an external magnetic field and spin-orbit interaction. Finally, we discuss the optical-lattice realization of this Lieb system and propose realistic methods for creating Abelian (magnetic) and non-Abelian (spin-orbit) gauge fields. We show that the Lieb lattice is particularly suited to reproduce the intrinsic spin-orbit term introduced by Kane and Mele \cite{Kane2005}. The later, which involves complex spin-dependent \emph{next}-nearest-neighbour hoppings, can be simply decomposed into nearest-neighbour hopping on a square sublattice. This elegant idea is a non-Abelian generalization of the method proposed in Ref.~[\onlinecite{Liu2010}] for generating the Abelian Haldane-type gauge field.
\section{The Lieb lattice and topological phases in external fields}
\begin{figure}
\centering
\includegraphics[width=0.7\columnwidth]{Figure1.pdf}
\caption{\label{fig:lattice} The face-centered square lattice or Lieb lattice. The Peierls phases $e^{i \theta(m)}$, where $\theta (m) = \pi \Phi m$, are associated to a uniform magnetic flux per plaquette $\Phi$ and are indicated by vectical black arrows. We set $x=2 m \ell_0$ and $y=2 n \ell_0$.}
\end{figure}
We consider the face-centered square (Lieb) lattice, which is shown in Fig. \ref{fig:lattice}. This lattice has a unit cell characterized by three lattice sites, hereafter referred to as H, A and B. Site H has four nearest-neighbors (NN), namely two A and two B sites. On the contrary the A and B sites have only two NN H sites.
The bulk properties of the Lieb lattice can be analyzed within a tight-binding (TB) approximation. In this limit, the Hamiltonian of the system can be written as $\mathcal{H}_0=t \sum_{\langle i,j \rangle\alpha} c^\dag_{i \alpha} c_{j\alpha}$ with spin independent NN hopping amplitude $t$. Here, $c_{j \alpha}^\dag(c_{j \alpha})$ is the creation (annihilation) operator for a particle with spin direction $\alpha$ on the lattice site $j$. In absence of external fields the problem can be diagonalized exactly and the spectrum reads
\begin{subequations}\label{eq:spectrum}
\begin{align}
\varepsilon_0 (\bm{k}) & = 0,\label{spec:zero}\\
\varepsilon_\pm (\bm{k}) & = \pm t \sqrt{4 + 2 \cos(\bm{v}_1\cdot \bm{k}) + 2 \cos(\bm{v}_2\cdot \bm{k})},
\end{align}
\end{subequations}
where $\bm{k}=(k_x,k_y)$ and $\bm{v}_{1/2}$ are the lattice vectors, c.f. Fig.~\ref{fig:lattice}.
The bulk energy spectrum is shown in Fig.~\ref{fig:bulk:spectrum}a | it depicts two identical, electron-hole symmetric branches $\varepsilon_\pm$. Moreover, the Lieb lattice presents a unique non-dispersive band at the charge neutrality point (CNP). This band is rooted in the lattice topology, which allows for insulating states with finite wave function amplitudes on the A and B sites and vanishing amplitudes on H sites. This property holds also when hopping to higher-order neighbors is allowed. Note that the three bands touch at the center of the first Brillouin zone, which we set for simplicity at $\Gamma=\pi/2\ell_0(1,1)$. The resulting properties of carriers in proximity of the $\Gamma$ point are investigated in Sec.~\ref{LWA}.
\begin{figure}
\centering
\includegraphics[width=.8\columnwidth]{Figure2.pdf}
\caption{\label{fig:bulk:spectrum} Energy spectrum of the Lieb lattice as a function of the momentum $\bm{k}=(k_x,k_y)$. The dashed lines delimit the first Brillouin zone. Panel (a): spectrum without external fields. Panel (b): energy spectrum in the presence of spin-orbit interaction $t_\text{SO}=0.1\ t$.}
\end{figure}
\subsection{Uniform magnetic field and quantum Hall phases}
We now study the effects of a uniform magnetic field $\bm{B}= B \hat{z}$ on the spectral and transport properties of the Lieb lattice. We consider the Landau gauge
\begin{equation}\label{gaugeu1}
\bm{A}= \left( 0, B x , 0 \right) = \left( 0, \frac{\pi \Phi m}{\ell_0} , 0 \right) ,
\end{equation}
where $\Phi= \Phi_0^{-1} \int_{\square} \bm{B} \cdot \text{d}\bm{S}$ is the number of magnetic flux quanta $\Phi_0$ per plaquette and $x=2 m \ell_0$. Hereafter we use the notation $(m,n, \zeta)$, with $\zeta=\{\text{A,B,H}\}$, to label the lattice sites.
The gauge field $\bm{A}$ modifies the hopping along the $y$ direction through $x$-dependent Peierls phases $t \to t \text{e}^{\text{i} \theta(m)}$, where the phase reads
\begin{equation}\label{peierls}
\theta (m) = \int_{(m, n,\text{H})}^{(m, n+1,\text{B})} \bm{A} \cdot \text{d}\bm{l}= \int_{(m, n,\text{H})}^{(m, n,\text{B})} \bm{A} \cdot \text{d}\bm{l}= \pi \Phi m ,
\end{equation}
as illustrated in Fig.~\ref{fig:lattice}. Here, the integrations are performed along the links connecting the neighboring B and H sites.
Setting $\Phi=p/q$, where $p$ and $q$ are mutually prime integers, the system becomes $q$-periodic along the $x$ direction. By considering periodic boundary conditions, it is possible to diagonalize the resulting $3q \times 3q$ spectral problem. This leads to the fractal energy spectrum shown in Fig.~\ref{fig:butterfly}. As a function of the flux $\Phi$, the allowed energies depict two Hofstadter butterflies separated by a flat band at $E=0$ \cite{footnote1}. This specific band is reminiscent of the flat band obtained at zero magnetic field.
The fractal energy spectrum of lattices subjected to uniform magnetic fields are intimately related to the IQHE~\cite{kohmoto:1989,hasegawa:1990}. When the Fermi energy $E_{\text{F}}$ is located in a spectral gap, the Hall transverse conductivity of the system is quantized. This relation is supported by a Diophantine equation~\cite{kohmoto:1992} which expresses the quantized Hall conductivity in terms of the magnetic flux and the position of the gap: In the $r$--th gap, the Hall conductivity is given by $\sigma_{xy}=(e^2/h) t_r$, where the integers $(t_r , s_r)$ satisfy
\begin{equation}
r = p t_r + q s_r . \comment{, \qquad \Phi=p/q .}
\end{equation}
In general, the solutions $(t_r , s_r)$ are not unique and additional criteria are needed in order to find quantized values of the Hall conductivity~\cite{kohmoto:1992}. The integer $t_r$ also has a topological interpretation, since it represents the sum of Chern numbers characterizing the bands below the Fermi energy $E_{\text{F}}$:
\begin{equation}
t_r = - \sum_{E_{\lambda} < E_{\text{F}}} N_\text{Ch} (E_{\lambda}) .
\end{equation}
In order to investigate the quantum Hall phases in the Lieb lattice, we have numerically evaluated the Chern numbers $N_\text{Ch} (E_{\lambda})$ using the method of Ref.~\cite{fukui:2005}. We have verified that the integer $t_r$ satisfies the Diophantine equation with the specific condition $\vert t_r \vert \le q/2$~\cite{kohmoto:1992}. The full phase diagram describing the integer quantum Hall effect for the Lieb lattice is drawn in Fig.~\ref{fig:butterfly}. It represents the infinitely many quantum Hall phases, characterized by the quantized transport coefficient $\sigma_{xy}=(e^2/h) t_r$, inside the spectral gaps. The different positive [resp. negative] values of the Hall conductivity are designated by cold [resp. warm] colors. \\
To identify the Hall plateaus stemming from the uniform magnetic field, we represent the Hall conductivity $\sigma_{xy}(E_{\text{F}})$ as a function of the Fermi energy in the low-flux regime $\Phi\ll1$ (cf. Fig.~\ref{fig:Hall:cond}). In this regime, the quantized conductivity evolves monotonically but suddenly changes sign around the van Hove singularities (VHS) ~\cite{hatsugai:2006}, located at $E = \pm 2\ t$ (see the alternation of cold and warm colors in Figs.~\ref{fig:butterfly} and \ref{fig:Hall:cond}). Note that the gaps surrounding the topological flat band at $E=0$, correspond to normal band insulators with vanishing conductivity $\sigma_{xy}=0$, for all values of the flux $\Phi$. This is a consequence of the flat band's vanishing Chern number \cite{Green2010}. Most importantly, we observe that the Hall sequence presented in Fig. \ref{fig:Hall:cond} shows steps of $\Delta \sigma_{xy}=(e^2 /h)$. It is interesting to compare the latter result with the Hall sequences obtained for the $\mathcal{T}_3$ and honeycomb lattices \cite{bercioux:2010,hatsugai:2006}, which are characterized by steps of $\Delta \sigma_{xy}=2 (e^2 /h)$ between the VHS. This major difference \cite{Lan2011} is due to the fact that the Lieb lattice is characterized by a \emph{single} Dirac-Weyl point, whereas the $\mathcal{T}_3$ and honeycomb lattices display \emph{two} Dirac cones.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.9\columnwidth]{Figure3.pdf}
\end{center}
\caption{\label{fig:butterfly} Spectrum $E=E(\Phi)$ and phase diagram for $\Phi ={p}/{q}$ with $q<47$. The eigenvalues are dark blue dots forming two successive butterflies. Gaps are filled with cold [resp. warm] colors according to the related positive [resp. negative] values of the quantized conductivity $\sigma_{xy}$. The white gaps located around $E=0$ correspond to $\sigma_{xy}=0$.}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.9\columnwidth]{Figure4.pdf}
\end{center}
\caption{\label{fig:Hall:cond} Hall conductivity $\sigma_{xy}(E_{\text{F}})$ as a function of the Fermi energy for $\Phi=1/11$. Cold [resp. warm] colors correspond to positive [resp. negative] values of the quantized conductivity, while the central plateau at $\sigma_{xy}=0$ is represented in magenta.}
\end{figure}
\subsection{Spin-orbit interaction and quantum spin Hall phases}
In this Section, we study the effects of spin-orbit interactions (SOIs) on the properties of the Lieb lattice. Specifically, we introduce an intrinsic SOI term in the TB Hamiltonian, in analogy with the model of Kane and Mele for graphene~\cite{Haldane1988,Kane2005,Kane2005bis, min:2006,bercioux:2010,Franz:2010}. This term is modeled via a spin-dependent next-nearest-neighbor (NNN) hopping term
\begin{equation}\label{eq:SO}
\mathcal{H}_\text{SO} = \text{i}\, t_\text{SO} \sum_{\alpha\beta}\!\!\sum_{\langle\langle k,l \rangle\rangle}
c_{k,\alpha}^\dag \left(\mathbf{d}_i\times\mathbf{d}_j\right)\cdot \bm{\sigma}_{\alpha\beta} \ c_{l,\beta}\, .
\end{equation}
The $\bm{\sigma}_{\alpha\beta}$ are matrix elements of the Pauli-matrices $\bm{\sigma}$ with respect to the final and initial spin states $\alpha$ and $\beta$ and $\mathbf{d}_{i/j}$ are the two displacement vectors of the NNN hopping process connecting sites $k$ and $l$. Since in 2D lattices hopping is naturally restricted to in-plane processes, the SOI is effectively proportional to $\sigma_z$. Because of the unequal connectivity of A/B and H sites, the term~(\ref{eq:SO}) effectively induces hopping between A and B sites only, \emph{i.e.}, other NNN-hopping processes cancel.
The spectrum is obtained by exact diagonalization~\cite{Franz:2010} and reads
\begin{subequations}
\begin{align}\label{spectrum:SOI}
\varepsilon^\text{(SO)}_0 (\bm{k})& = 0 \\
\varepsilon^\text{(SO)}_\pm(\bm{k}) & = 2 \left[ t^2 \left(\cos^2(\bm{v}_1\cdot\bm{k})+\cos^2(\bm{v}_2\cdot\bm{k})\right) \right. \\
& ~~~\left. +4 t_\text{SO} ^2 \sin^2(\bm{v}_1 \cdot \bm{k}) \sin^2(\bm{v}_2\cdot\bm{k})\right]^{\frac{1}{2}}\,.
\end{align}
\end{subequations}
The bulk energy spectrum is shown in Fig.~\ref{fig:bulk:spectrum}b for $t_\text{SO}=0.1\,t$. Due to its topological origin, the non-dispersive band at $E=0$ is not affected by SOI. However, this term opens two bulk energy gaps $\Delta_\text{gap}=4\ t_\text{SO}$ between the non-dispersive and the electron/hole branches, respectively.
The SOI has dramatic consequences on the transport properties of the Lieb lattice: as shown by Weeks and Franz, the gap $\Delta_\text{gap}$ allows for a topological insulating phase~\cite{Franz:2010}. The latter is characterized by a robust spin transport along the edges of the system. This quantum spin Hall phase, induced by the Haldane-type term~(\ref{eq:SO}), can thus be directly visualized when studying a finite piece of the lattice, \emph{i.e.}, by considering its edges.
A standard method consists in diagonalizing the TB Hamiltonian with periodic boundary conditions imposed along one of the spatial directions. This abstract cylinder contains two edges and already allows to demonstrate the existence of helical edge states induced by the SOI. The corresponding energy spectrum (c.f.~Fig.~\ref{fig:spectrum:cylinder}a) depicts several edge-state channels: for each energy value within the bulk energy-gap, there exists a single time-reversed (or Kramers) pair of eigenstates localized on each edge of the lattice. The conservation of TRS prevents the mixing of this couple of states by small external perturbations and scattering from disorder~\cite{Kane2005,Kane2005bis}.
The helical edge states characterizing the QSH phase are topologically protected against external perturbations. Their property can be quantified by looking at the $\text{Z}_2$-index $\nu$~\cite{Kane2005}. This topological invariant characterizes the eigenstates defined in the bulk | it is defined on a 2-torus, in direct analogy with the Chern numbers introduced in the quantum Hall effect. Following Ref.~\cite{Franz:2010}, we have calculated this Z$_2$-index $\nu$ using the inversion symmetry of the lattice~\cite{Fu2007}. We obtain that SOI opens a spectral gap characterized by the index $\nu=1$, therefore classifying the Lieb lattice as a quantum spin-Hall insulator.
In the absence of spin-mixing perturbations, the Z$_2$ index is related to the spin Chern number $n_\sigma$~\cite{Sheng2006,Fukui2007}, through the simple relation $\nu=n_\sigma\text{mod}\,2$, where $n_\sigma=(N_{\uparrow}-N_{\downarrow})/2$ and $N_{\uparrow, \downarrow}$ represent the Chern numbers associated to the individual spins. Using the numerical method of Ref.~\cite{fukui:2005}, we have obtained $n_\sigma=1$ in agreement with the above result.
It is interesting to extend the analysis above by considering the more realistic open geometry, i.e. a finite piece of Lieb lattice, thus characterized by a unique edge. We have solved the TB problem for a Lieb lattice of $39 \times 39$ sites with realistic straight edges. This yields a discrete energy spectrum extending in the range $E \in [-2.8 t , 2.8 t]$. In the vicinity of the CNP, within a range corresponding to $\Delta_\text{gap}=4\ t_\text{SO}$, the eigenvalues correspond to eigenstates that are localized at the edge of the system. This result is illustrated in Fig.~\ref{fig:spectrum:cylinder}b, where the amplitude $\vert \psi_{\uparrow} (x,y) \vert ^2$ is drawn for a particular edge-state at $E=0.5 t$. Note that this coincides with $\vert \psi_{\downarrow} (x,y) \vert ^2$. Using this geometry, one can verify the helical property of these edge-states by computing their associated current, which for spins $\sigma$ can be expressed as
\begin{align}
j_x^{\sigma}(m,n)&= -\text{i} \ell_0 \biggl ( \psi^*_{\sigma} (m+1, n) \, \mathcal{U}_x \, \psi_{\sigma} (m,n) \\
& + \sqrt{2} \psi^*_{\sigma} (m+1, n+1) \, \mathcal{D} \psi_{\sigma} (m,n) \, - \text{h.c} \biggr) , \notag \\
j_y^{\sigma}(m,n)&= -\text{i} \ell_0 \biggl ( \psi^*_{\sigma} (m, n+1) \, \mathcal{U}_y \, \psi_{\sigma} (m,n) \\
&+\sqrt{2} \psi^*_{\sigma} (m+1, n+1) \, \mathcal{D} \, \psi_{\sigma} (m,n) - \text{h.c} \biggr ). \notag
\end{align}
Here $\psi_{\sigma} (m,n)= ( \psi_{\sigma} (m,n, \text{H}), \psi_{\sigma} (m,n, \text{A}) , \psi_{\sigma} (m,n, \text{B}) )$ and
\begin{subequations}
\begin{align}
\mathcal{U}_x &=\begin{pmatrix}
0 &0 &0 \\
t \mathbb{I}_2 &0 &- \text{i} t_\text{SO} \sigma \\
0 &0 &0
\end{pmatrix}, \\
\mathcal{U}_y & =\begin{pmatrix}
0 &0 & t \mathbb{I}_2 \\
0 &0 & - \text{i} t_\text{SO} \sigma \\
0 &0 &0
\end{pmatrix},\\
\mathcal{D}& =\begin{pmatrix}
0 &0 & 0 \\
0 &0 & \text{i} t_\text{SO} \sigma \\
0 &0 &0
\end{pmatrix}.
\end{align}
\end{subequations}
We have verified that the currents $\boldsymbol{j}^{\uparrow}(m,n)$ and $\boldsymbol{j}^{\downarrow}(m,n)$ associated to the edge-states yield two vector fields circulating along the edge in opposite directions.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=\columnwidth]{Figure5.pdf}
\put(-147,110){$\vert$}
\put(-122,110){$\vert$}
\caption{\label{fig:spectrum:cylinder} Panel (a): Energy spectrum $E=E(k_y)$ in the cylinder geometry with $t_\text{SO}=0.2 t$. The bulk gap is traversed by gapless edge-states [i.e. helical states]. Panel (b): Edge-states amplitude $\vert \psi_{\uparrow} (x,y) \vert ^2$ for the open Lieb lattice with $39 \times 39$ sites, straight edges and $t_\text{SO}=0.5 t$. This localized eigenstate satisfies $H \psi_{\lambda} (x,y)= E_{\lambda} \psi_\lambda (x,y)$ and corresponds to the energy $E_\lambda=0.5 t$ which lies within the gap.}
\end{center}
\end{figure}
\section{Low-energy fermions: the quasirelativistic regime\label{LWA}}
In this Section, we focus on the the properties of the Lieb lattice for non-interacting fermions at low energy, i.e. close to the CNP. We perform a long wavelength approximation to the Schr\"odinger equation underlying the TB Hamiltonian
which consists in expressing the spatial part of the wave function as the product of a fast-varying part times a slow-varying part. Within this approximation, the wave function can be written as
\begin{equation}\label{eq:wf:lwa}
\Psi_\alpha(\bm{R}_\alpha) \propto \text{e}^{\text{i} \bm{k}\cdot\bm{R}_\alpha} F_\alpha(\bm{R}_\alpha)\,,
\end{equation}
where $\alpha\in\{\text{A,B,H}\}$ and $\bm{R}_\alpha$ is the lattice site coordinate. We substitute this wave function into the Schr\"odinger equation and expand the slow-varying part as
\begin{equation*}\label{lwa}
F_\alpha(\bm{R}_{\alpha'}\pm\bm{d}_j)\simeq F_\alpha(\bm{R}_{\alpha'}) \pm \bm{d}_j \cdot \bm{\nabla}_{\bm{r}} \left.F_\alpha(\bm{r})\right|_{\bm{r}=\bm{R}_{\alpha'}} + \mathcal{O}(|\bm{d}|^2)\,.
\end{equation*}
Collecting all terms, we are left with
\begin{equation}\label{lwaHam}
\tilde{\mathcal{H}}= v_\text{F} \bm{\Sigma}\cdot \bm{p} \,,
\end{equation}
where $v_\text{F}=2\ell_0t$ is the Fermi velocity, and $\bm{p}=(p_x,p_y,0)=-i\hbar(\partial_x,\partial_y,0)$. Here the pseudo-spin matrices are defined as
\begin{equation}\label{eq:gm}
\Sigma_x=
\begin{pmatrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix},\
\Sigma_y= \begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{pmatrix},\
\Sigma_z=\begin{pmatrix}
0 & 0 & -\text{i} \\
0 & 0 & 0 \\
\text{i} & 0 & 0
\end{pmatrix}\,.
\end{equation}
These matrices fulfill the algebra of the angular momentum $[\Sigma_i,\Sigma_j]=\text{i} \epsilon_{ijk}\Sigma_k$
and form a 3-dimensional representation of SU(2). However, contrary to the Pauli matrices, they do not form a Clifford algebra, \emph{i.e.}, \ $\{\Sigma_i,\Sigma_j\}\neq 2 \delta_{i,j} \mathbb{I}_3$. Therefore, while Eq.~\eqref{lwaHam} describes electrons with a linear energy spectrum, it does not represent a Dirac Hamiltonian.
By introducing a rotation operator around the $z$ axis defined by $\mathcal{D}_z(\phi) = \exp \left( -\text{i} \Sigma_z \phi \right)$, a generic state $\ket{\alpha}$ is transformed into itself by $\mathcal{D}_z(2\pi)\ket{\alpha}\to\ket{\alpha}$, implying that the pseudo-spin $\bm{\Sigma}$ describes an integer spin $S=1$.
\subsection{Spin-orbit interaction}
Within the long wavelength approximation we can also express the intrinsic SOI introduced in the previous Section. This term reads
\begin{equation}\label{hsoi:lwa}
\tilde{\mathcal{H}}_\text{SO} = \Delta_\text{SO} \Sigma_z \otimes \sigma_z ,
\end{equation}
where $\Delta_\text{SO}$ is the effective spin-orbit coupling strength and $\sigma_z$ is a Pauli matrix.
The energy spectrum can be computed in this regime and reads
\begin{subequations}\label{soi:lwa}
\begin{align}
\tilde{\varepsilon}_0^\text{(SO)} &= 0 \\
\tilde{\varepsilon}_\pm^\text{(SO)} &= \pm \sqrt{v_\text{F}^2 |\bm{k}|^2 + \Delta_\text{SO}^2}\,.
\end{align}
\end{subequations}
This is two-fold degenerate, with degeneracy corresponding to spin-up and spin-down. \comment{The eigenstates for the case of spin-up reads
\begin{subequations}\label{eigenvectors:soi}
\begin{align}
\bm{v}_0 & = \begin{pmatrix}
-t \cos(\bm{d}_\text{A}\cdot\bm{k}) \\ 2 \text{i} \Delta_\text{SO} \sin (\bm{d}_\text{A}\cdot\bm{k}) \sin
(\bm{d}_\text{B}\cdot\bm{k})\\ t \cos (\bm{d}_\text{A}\cdot\bm{k})
\end{pmatrix} , \\
\bm{v}_\pm & = \\
& \hspace{-0.6cm}\pm\!\!\nonumber\begin{pmatrix}
\varepsilon_\pm \cos (\bm{d}_\text{A}\!\cdot\!\bm{k})-4 \text{i} \Delta_\text{SO} \sin (\bm{d}_\text{A}\!\cdot\!\bm{k}) \sin
(\bm{d}_\text{B}\cdot\bm{k}) \cos (\bm{d}_\text{B}\cdot\bm{k})\\
t (\cos (\bm{d}_\text{A}\!\cdot\!\bm{k})+\cos (\bm{d}_\text{B}\!\cdot\!\bm{k})) \\
\varepsilon_\pm \cos (\bm{d}_\text{A}\!\cdot\!\bm{k})+4 \text{i} \Delta_\text{SO} \sin
(\bm{d}_\text{A}\!\cdot\!\bm{k}) \cos (\bm{d}_\text{A}\!\cdot\!\bm{k}) \sin (\bm{d}_\text{B}\!\cdot\!\bm{k})
\end{pmatrix}\!.
\end{align}
\end{subequations}
}
\subsection{Landau levels in a uniform magnetic field}
The long wavelength approximation also allows to compute the Landau levels that arise in the presence of the uniform magnetic field.
The system Hamiltonian reads
\begin{equation}\label{eq:ham:landau}
\tilde{\mathcal{H}}_B = v_\text{F}\,\bm{\Sigma}\cdot\left(\bm{p}-\frac{e}{c}\bm{A}\right)\,,
\end{equation}
where $\mathbf{A}$ is the
vector potential associated with the magnetic field
$\bm{B}=(\bm{\nabla}\times\bm{A})$
perpendicular to the lattice plane.
We solve the Schr\"odinger equation in the Landau gauge with $\mathbf{A}=(-By,0,0)$.
Further, we make the Ansatz $\Psi=\psi(y)\exp(\text{i} kx)$. Introducing a $k$--dependent shift
in the $y$--coordinate, $\sqrt{B}{\xi}=By+k$ we are left to solve a system
of coupled linear differential equations
\begin{subequations}
\begin{align}
{\xi}\psi_\text{H}({\xi})&=\tilde\varepsilon\psi_\text{A}({\xi})\\
{\xi}\psi_\text{A}({\xi})-\text{i}\psi_\text{B}'({\xi})&=\tilde\varepsilon\psi_\text{H}({\xi})\\
-\text{i}\psi_\text{H}'({\xi})&=\tilde\varepsilon\psi_\text{B}({\xi})
\end{align}
\end{subequations}
for the three components of $\psi({\xi})$. Here $\varepsilon=E/(\hbar v_\text{F}\sqrt{B})$ is the
rescaled eigenenergy.
The Landau levels at non-zero energy are given by
\begin{equation}\label{eq:LL_finiteE}
\psi_{\pm,\mathfrak{n}}({\xi}) =
\begin{pmatrix}
(\sqrt{\mathfrak{n}}\phi_{\mathfrak{n}-1}+\sqrt{\mathfrak{n}+1}\phi_{\mathfrak{n}+1})/\sqrt{2} \\
\pm\sqrt{2\mathfrak{n}+1}\phi_\mathfrak{n} \\
-i(\sqrt{\mathfrak{n}}\phi_{\mathfrak{n}-1}-\sqrt{\mathfrak{n}+1}\phi_{\mathfrak{n}+1})/\sqrt{2}
\end{pmatrix}
\end{equation}
with corresponding eigenvalues $\tilde\varepsilon_{\pm,\mathfrak{n}}=\pm\sqrt{2\mathfrak{n}+1}$ and $\mathfrak{n}$ integer.
Here the $\phi_\mathfrak{n}$ for $\mathfrak{n}\ge0$ are the eigenfunctions of the one-dimensional harmonic oscillator,
\begin{equation}
\phi_{\mathfrak{n}}({\xi}) = \frac{1}{\sqrt{2^\mathfrak{n}\pi^{1/2}\mathfrak{n}!}} h_{\mathfrak{n}}({\xi})e^{-{\xi}^2/2},
\end{equation}
where $h_{\mathfrak{n}}$ denotes the Hermite polynomial of order $\mathfrak{n}$, while we define $\phi_{-1}\equiv 0$.
In addition to the eigenfunctions (\ref{eq:LL_finiteE}) there are two different types of solutions at energy $\tilde\varepsilon_{0,\mathfrak{n}}=0$.
The first of these is related to the flat band at zero magnetic field (\ref{spec:zero}) and reads
\begin{equation}\label{eq:LL_zeroE}
\psi_{0,\mathfrak{n}}({\xi}) = \frac{1}{\sqrt{2}}
\begin{pmatrix}
\sqrt{\mathfrak{n}+1}\phi_{\mathfrak{n}-1}-\sqrt{\mathfrak{n}}\phi_{\mathfrak{n}+1} \\
0 \\
-\text{i}(\sqrt{\mathfrak{n}+1}\phi_{\mathfrak{n}-1}+\sqrt{\mathfrak{n}}\phi_{\mathfrak{n}+1})
\end{pmatrix},
\end{equation}
where $\mathfrak{n}>0$. The second type of zero-energy Landau level
is given by
\begin{equation}\label{eq:LL_zeroE_B}
\psi_{0}^{(0)}({\xi})=\frac{1}{\sqrt{2}}
\begin{pmatrix}
\phi_{0} \\
0 \\
\text{i}\phi_{0}
\end{pmatrix}
\end{equation}
Note that $\psi_0^{(0)}({\xi})$ is not a generalization of (\ref{eq:LL_zeroE}) to the case $\mathfrak{n}=0$.
\subsection{Spectrum with magnetic field and spin-orbit interaction}
Now we turn to the effects of finite SOI on the Landau levels obtained above. The energies $\tilde\varepsilon_{\alpha,\sigma,\mathfrak{n}}$ (with $\alpha=\{0,\pm\}$ and $\sigma=\{\uparrow,\downarrow\}$) of the Landau levels are the three solutions of
\begin{eqnarray}
\frac{\sqrt{\mathfrak{n}}}{\tilde\varepsilon_{\alpha,\sigma,\mathfrak{n}}-\sigma\Delta_\text{SO}}+\frac{\sqrt{\mathfrak{n}+1}}{\tilde\varepsilon_{\alpha,\sigma,\mathfrak{n}}+\sigma\Delta_\text{SO}}&=&\tilde\varepsilon_{\alpha,\sigma,\mathfrak{n}}\,.
\end{eqnarray}
The corresponding wave functions read
\begin{eqnarray}
\label{eq:LL_finiteSOI}
\psi_{\alpha,\sigma,\mathfrak{n}}&=&
\frac{\sqrt{\mathfrak{n}}}{\tilde\varepsilon_{\alpha,\sigma,\mathfrak{n}}-\sigma\Delta_\text{SO}}\phi_{\mathfrak{n}-1}
\begin{pmatrix}
1\\
0\\
-\text{i}
\end{pmatrix}
+\phi_\mathfrak{n}
\begin{pmatrix}
0\\
\sqrt{2}\\
0
\end{pmatrix}
\nonumber\\&&
+\frac{\sqrt{\mathfrak{n}+1}}{\tilde\varepsilon_{\alpha,\sigma,\mathfrak{n}}+\sigma\Delta_\text{SO}}\phi_{\mathfrak{n}+1}
\begin{pmatrix}
1\\
0\\
\text{i}
\end{pmatrix}.
\end{eqnarray}
In the case of weak SOI, i.e. $\Delta_\text{SO}\ll1$, the Landau levels are given by
\begin{eqnarray}
\tilde\varepsilon_{\pm,\sigma,\mathfrak{n}}&=&\pm\sqrt{2\mathfrak{n}+1}-\frac{\sigma\Delta_\text{SO}}{4\mathfrak{n}+2}+{\cal O}(\Delta_\text{SO}^2) \\
\varepsilon_{0,\sigma,\mathfrak{n}}&=&\frac{\sigma\Delta_\text{SO}}{2\mathfrak{n}+1}+{\cal O}(\Delta_\text{SO}^2)
\end{eqnarray}
Consequently, the main effect of finite SOI is to lift the spin degeneracy, with a
level separation that decreases with growing Landau level index $\mathfrak{n}$. Moreover, the former highly degenerate
zero energy levels, c.f. Eqs.~(\ref{eq:LL_zeroE}) and (\ref{eq:LL_zeroE_B}), are now split into a family of flat bands at energies $\varepsilon_{0,\sigma,\mathfrak{n}}$.\comment{\bcom{Missing plots of the Landau levels + Landau levels modified by SOI}}
\section{Optical lattice realization}
Experimentally, the Lieb lattice can be realized as an optical lattice created by six counter propagating
pairs of laser beams. Four pairs are aligned along the $x$ and $y$ directions, two with wavelength $\lambda =\ell_0$ and two with wavelength $\lambda =\ell_0/2$. Finally, it requires two other laser pairs with a direction of $\pm45^\circ$ with respect to the $x$-axis. A detailed procedure leading to this choice of laser configuration has been discussed in Refs.~\cite{schen:2010,apaja:2010}. The potential profile is given by the field
\begin{align}\label{v:ol}
V_\text{OL}(x,y) = & V_0 \left[ \sin^2(kx)+\sin^2(ky)\right] \\
& + V_1 \left[ \sin^2(2kx)+\sin^2(2ky)\right]\nonumber \\
& + V_2 \left[ \cos^2\left(k \frac{x+y}{2} \right)+\cos^2\left(k \frac{x-y}{2} \right)\right] , \nonumber
\end{align}
with $V_0=V_1=2V_2$ and $k=\pi/\lambda$, which is depicted in Fig.~\ref{fig:optical:lattice}.
It is apparent that the hopping probability between A and B
sites is exponentially small compared to the hopping probability
between neighboring H and A/B pairs.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{Figure6.pdf}
\caption{\label{fig:optical:lattice} (Color online) (a) Distribution of the
laser field intensity for generating a Lieb lattice. (b) Two cuts through (a) corresponding to the B and H sites (red line) and A sites (blue line)}
\end{figure}
\subsection{Simulation of the U(1) synthetic gauge field and quantum Hall phases}
Recently, synthetic U(1) magnetic fields for cold neutral atoms have been proposed~\cite{Jaksch2003,gerbier:2010,Dalibard2010} and experimentally realized~\cite{lin:2009}. In such setups, atoms reproduce the dynamics of charged particles subjected to a uniform magnetic field and can effectively show quantum Hall phases. Several methods can be used in order to simulate the
Hofstadter model~\cite{Hofstadter1976} with these systems. These methods are generally based on the fact that the Peierls phases can be engineered by external optical~\cite{Jaksch2003,gerbier:2010} or magnetic~\cite{goldman:2010} fields. These electromagnetic fields can indeed induce hopping between neighboring lattice sites, when the latter host atoms in different internal states, say $\vert g \rangle$ and $\vert e \rangle$. More precisely, the external fields trigger (Raman) couplings between these internal states, resulting in a NN-hopping amplitude
\begin{equation}
t_{g,e} e^{i \theta (\boldsymbol{x}_g)} \propto \int w^* (\boldsymbol{x}-\boldsymbol{x}_e) \Omega_{g , e} w (\boldsymbol{x}-\boldsymbol{x}_g) d^3 x,
\label{raman}
\end{equation}
where the Rabi frequency $\Omega_{g , e} $ typically includes space-dependent phase factors and where we suppose the states $\vert g \rangle$ and $\vert e \rangle$ to be trapped in neighboring sites ~\cite{Jaksch2003,gerbier:2010}. Here, the Peierls phase $\exp (i \theta (\boldsymbol{x}_g))$ is directly related to the coupling laser's wave vector.
The gauge field \eqref{gaugeu1}, which leads to the Peierls phase \eqref{peierls} can be readily engineered in an optical lattice experiment by exploiting these methods~\cite{Jaksch2003,gerbier:2010}. From Fig.~\ref{fig:lattice}, it is clear that the $x$-dependent assisted-hopping involves nearest-neighbors and occurs along the $y$ direction only, i.e. between B and H sites. In this sense, the phases $\theta(m)=\pi \Phi m$ can be realized on the optical Lieb lattice by extending the methods envisaged for the standard square lattice (cf. Ref.~\cite{Jaksch2003,gerbier:2010,Dalibard2010}): In the Lieb lattice case, one should trap two internal states alternatively along the $y$-direction: the B-sites (resp. H-sites) should host an atom in the internal state $\vert g \rangle$ (resp. $\vert e \rangle$). Coupling these states with external fields should then induce hoppings of the form \eqref{raman}, resulting in the space-dependent Peierls phase \eqref{peierls}. Note that for generating a magnetic flux $\Phi$ per plaquette, a double phase $\theta_{\square}=2 \theta=2 \pi \Phi m$ is required for the square lattice compared to the Lieb lattice.
In the previous sections, we have discussed the existence of quantum Hall phases in a fermionic Lieb lattice subjected to a uniform magnetic field. In order to produce these phases in a cold atom experiment, one should engineer a U(1) gauge field for \emph{fermionic} atoms. As already discussed in Ref.~\cite{goldman:2010}, most of the schemes generating gauge fields for bosons use Raman transitions to couple the internal states, and would lead to high spontaneous emission rates for fermionic atoms. Therefore realizing (integer) quantum Hall states would require alternative methods~\cite{Gerbier2009,mazza1,mazza2}. Such a proposal was introduced in Ref.~\cite{goldman:2010} and uses radio-frequency magnetic fields produced by a set of current-carrying wires. The latter are periodically spaced on an atom chip and drive transitions between several internal states of $^{6}$Li fermionic atoms. These effective ``Raman transitions" lead to assisted hopping \eqref{raman} and can be tuned in order to produce the desired Peierls phases. In order to engineer the U(1) gauge field \eqref{gaugeu1}, one can simplify the method initially proposed in Ref.~\cite{goldman:2010} (which leads to the creation of SU(2) gauge fields) and consider transitions between two internal states of $^{6}$Li, e.g. $\ket{g_1} = \ket{F\!=\!1/2,m_F\!=\!1/2}$ and $\ket{e_1} = \ket{3/2,1/2}$. We stress that this practical scheme can be directly generalized to the Lieb lattice in order to generate the phases $\theta(m)=\pi \Phi m$ accompanying the hopping between neighboring H and B sites.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.7\columnwidth]{Figure7.pdf}
\end{center}
\caption{\label{fig:Peierls:phases} The Lieb lattice with the intrinsic spin-orbit coupling. This coupling is equivalent to a non-Abelian gauge field $\bm{A}$ leading to spin-dependent Peierls phases, as indicated by red arrows.}
\end{figure}
\subsection{Simulation of the SU(2) gauge field with neutral atoms}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=1.2\columnwidth]{Figure8.pdf}
\end{center}
\caption{\label{fig:SU2} (a) The Lieb lattice with the intrinsic spin-orbit coupling. The Peierls phases (red arrows) are defined on the square sublattice formed by the A and B sites. (b) Laser assisted-hoppings (arrows), associated couplings and synthetic fluxes felt by the up-spin component. Four states $e_{1,2}$ and $g_{1,2}$ are trapped in state-dependent potentials and are represented by filled or empty orange or red disks (cf. figures below). (c)-(d) State-dependent potentials and couplings along the $\tilde{x}$ and $\tilde{y}$ directions.}
\end{figure}
Here, we describe a practical scheme to simulate the intrinsic SOI term in a fermionic Lieb lattice. This coupling is equivalent to a SU(2) gauge field and could therefore be engineered in a multi-component atomic system through state-dependent Peierls phases (cf. Fig.~\ref{fig:Peierls:phases}). Our proposal is based on the observation that the spin-dependent NNN-hoppings are equivalent to \emph{NN}-hoppings defined on the square sublattice formed by the A and B sites only~\cite{Liu:2010} (cf. Fig.~\ref{fig:SU2} (a)). Consequently, generating the SOI reduces to the simple problem of engineering the Peierls phases $\text{e}^{\text{i} \pi /2} \sigma_z$ on a \emph{rotated} square lattice, which we now label using the notations $\tilde{m}$ and $\tilde{n}$ (cf. Fig.~\ref{fig:SU2} (b)). Obviously, the subtlety relies in the orientation of these phases: the phases are positive for a particle hopping respectively clockwise and anti-clockwise in neighboring plaquettes (cf. Fig.~\ref{fig:SU2} (a)). Note that this important fact naturally leads to a staggered magnetic field, with fluxes $\pm \Phi$, for each spin component (cf. Fig.~\ref{fig:SU2}b). We note that, in order to reproduce such a staggered field, one can simply exploit the fact that the hopping induced by Raman transitions between the internal states $g$ and $e$ is such that $(t e^{i \theta (\boldsymbol{x}_g)})_{e,g}=(t e^{i \theta (\boldsymbol{x}_g)})_{g,e}^*$ \cite{Gerbier2009}. \\
Let us now describe a feasible and concrete scheme to synthesize the SOI term \eqref{eq:SO} in a cold-atom experiment. Our proposal requires four states $e_{1,2}$ and $g_{1,2}$ and external fields producing Raman transitions in both the $\tilde{x}$ and $\tilde{y}$ directions (cf. Figs.~\ref{fig:SU2} (a)-(d)). Such states can be chosen as being four internal states \mbox{$\ket{F,m_F}$} of $^{6}$Li, e.g. $\ket{g_1} =$\mbox{$\ket{1/2,1/2}$}, $\ket{g_2} =$\mbox{$\ket{3/2,-1/2}$}, $\ket{e_1} =$\mbox{$\ket{3/2,1/2}$}, and $\ket{e_2} =$\mbox{$\ket{1/2,-1/2}$}~\cite{goldman:2010}. First, one needs to trap these states in state-dependent lattices \cite{Osterloh2005,Gerbier2009,goldman:2010} along the $\tilde{x}$ and $\tilde{y}$ directions (cf. Figs.~\ref{fig:SU2}c-d). Then external fields should drive Raman transitions between these states, with the corresponding Rabi frequencies
\begin{align}
&\Omega_{g_1 , e_1}^{\tilde{x}}=\Omega_1^{\tilde{x}}, \qquad \Omega_{g_2 , e_2}^{\tilde{x}}=\Omega_2^{\tilde{x}} , \notag \\
&\Omega_{e_1 , g_1}^{\tilde{y}}=\Omega_1^{\tilde{y}}, \qquad \Omega_{e_2 , g_2}^{\tilde{y}}=\Omega_2^{\tilde{y}} ,
\label{omega}
\end{align}
as indicated by arrows in Figs.~\ref{fig:SU2}c-d. At this point, we emphasize that the Rabi frequencies are controlled by the coupling lasers and that they are chosen to be different for transitions driven along the $\tilde{x}$ and $\tilde{y}$ directions. Note that these Rabi frequencies typically contain phase factors depending on the coupling lasers wave vectors \cite{Jaksch2003}. Moreover the Rabi frequencies associated to the opposite transitions are simply given by $\Omega_{e_j , g_j}^{\tilde{\mu}}=(\Omega_{g_j , e_j}^{\tilde{\mu}})^*$, where $\tilde{\mu}=\tilde{x},\tilde{y}$. Therefore, the hopping amplitudes along a given direction are accompanied by Peierls phases with alternating signs \cite{Gerbier2009}. This leads to a staggered magnetic field for each spin component, with fluxes $\pm \Phi$, as illustrated in Fig.~\ref{fig:SU2}b.\\
Now, the SOI term \eqref{eq:SO} requires a specific configuration of these Rabi frequencies. The associated SU(2) gauge field is proportional to $\sigma_z$, which is simply achieved by imposing the constraint $\Omega_{1}^{\tilde{\mu}}=(\Omega_{2}^{\tilde{\mu}})^*$ (i.e. the coupling lasers should be characterized by opposite wave vectors). Besides, the desired gauge field is associated to constant Peierls phases (i.e. $\exp( \pm i \pi /2)$), which further requires that the Rabi frequencies do not depend on the variables $\tilde{x},\tilde{y}$ and also obey the relation $\Omega_{1,2}^{\tilde{x}}=\Omega_{1,2}^{\tilde{y}}$ (or equally $\Omega_{g_j , e_j}^{\tilde{x}}=(\Omega_{g_j , e_j}^{\tilde{y}})^*$, using the definitions \eqref{omega}). \\
We stress that this concrete scheme leads to the SOI studied in the previous sections and that it should open a QSH gap in an atomic setup. In order to observe the QSH phases induced by such a synthetic SOI in a cold atom experiment, these non-trivial Peierls phases need to be engineered in a fermionic lattice. Again, one can consider the atom-chip proposal of Ref.~\cite{goldman:2010}: different sets of wires, aligned along $\tilde{x}$ and $\tilde{y}$, should trap the states $e_j$ and $g_j$ alternatively in both directions, as illustrated in Fig.~\ref{fig:SU2}c-d. Additional ``Raman wires" should then trigger RF transitions and couple the states, producing the induced-hopping and associated phases described above. Another possibility would be to apply the superlattice methods of Refs.~\cite{Gerbier2009,mazza1,mazza2} to the Lieb lattice. Once the gauge field is synthesized, this setup needs to be superimposed with a state-independent Lieb lattice yielding the desired total Hamiltonian $\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_\text{SO}$. \\
Finally, we note that the scheme presented in this Section could be simplified in order to reproduce the Abelian Haldane model on the Lieb lattice~\cite{Haldane1988,Liu2010}. In this case, only two internal states $g$ and $e$ would be needed, instead of four. A realization of the Haldane model on the Lieb lattice would lead to integer quantum Hall states~\cite{Liu2010}.
\subsection{Detection}
The TB Hamiltonian and its long wavelength
approximation are valid for a Lieb lattice
populated by single-component fermionic atoms, \emph{e.g.}
$^{40}$K or $^6$Li. In this case the atomic
collisions are negligible at low
temperature~\cite{Bloch2008}. From the experimental point
of view, time-of-flight imaging via light
absorption~\cite{koehl:2005} can be used in order to detect the
presence of massless fermions. The harmonic trap potential
$V(\mathbf r)=m\omega^2\mathbf{r}^2/2$ confining the fermionic
cold atom gas is ramped down slowly enough for the atoms to stay
adiabatically in the lowest band while their quasi-momentum is
approximatively conserved. Under these conditions, free fermions
expand with
ballistic motion and, from the measured absorption images, it is
possible~\cite{ho:2000,Umucalilar:2008} to reconstruct the initial reciprocal-space density profile
of the trapped gas. Then, the local
density approximation is typically well satisfied and the
local chemical potential can be assumed to vary with the radial
coordinate as $\mu(\mathbf{r}) = \mu_0 -V(\mathbf{r})$, where $\mu_0$ is the
chemical potential at the center of the trap.
For a system of cold atoms at temperature $T$, the atomic density in the bulk is uniquely
determined by the chemical potential
\begin{equation}\label{eq:twoelve}
\rho(\mu) = \frac{1}{\mathcal{S}_0} \sum_{\alpha} \int f(\mathbf{k},\alpha, \mu) \, d \mathbf{k} \,.
\end{equation}
Here $\mathcal{S}_0$ is the area of the
first Brillouin zone of the Lieb lattice, and
$f(\mathbf{k},\alpha, \mu)=[\exp[(E_\alpha(\mathbf{k})-\mu)/k_\text{B}T]+1]^{-1}$
is the Fermi distribution function, where $E_\alpha(\mathbf{k})$
is the energy spectrum of the Lieb lattice,
cf.\ Eq.~(\ref{eq:spectrum}).
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{Figure9.pdf}
\caption{\label{fig:density} (Color online) Panel (a) Atomic density as a function of the chemical potential. Panel (b) Density of states as a function of the chemical potential. The full line corresponds to $\Delta_{\text{SO}}=0.25t$ and the dashed line to $\Delta_{\text{SO}}=0$.}
\end{figure}
Figure~\ref{fig:density}a shows the atomic density $\rho$ for the bulk
as a function of the chemical potential $\mu$.
The contribution from the highly degenerate topological band (\ref{spec:zero}) manifests itself at $\mu=0$ as a sharp jump in the atomic density. This feature is specific to the Lieb lattice.
For small finite $\mu$ we note that $\rho$ evolves proportional to $\mu^2$, which
reflects the linear dispersion of massless fermions near the band center as well as particle-hole symmetry.
On the contrary, for values of $\mu$ close to the maximum or minimum of
the energy band, \emph{i.e.}\ far away from the band center (where the
long wavelength approximation can no longer be applied),
$\rho$ varies proportional to $\mu$. When we consider a finite value of the SOI, a finite gap appears in the energy spectrum. This coincides with the horizontal segments in Fig.~\ref{fig:density}a. Moreover, we can observe that the atomic density in proximity of the SOI gap is not anymore a quadratic function of the chemical potential but is behaving linearly. This is a signature of the mass term introduced by the finite SOI. In Fig.~\ref{fig:density}b we report the density of states (DOS) for two values of $\Delta_{\text{SO}}$. Note the gap opening for $\Delta_{\text{SO}}\ne 0$ and the existence of two robust van Hove singularities at $\mu= \pm 2$. \\
Let us stress the great similarity between the density and DOS obtained here for the Lieb lattice and those reported in Ref.~\cite{bercioux:2009} concerning the $\mathcal{T}_3$~ lattice. However, these two lattices differ in their topological order since the SOI opens a \emph{trivial} insulating gap in the $\mathcal{T}_3$~ lattice~\cite{bercioux:2010}. This fundamental difference can be emphasized by computing the density profile $\rho(\mu)$ for the realistic finite-size system: in this case, gapless edge states contribute to the Lieb lattice atomic density and slightly tilt the SO plateaus. Note that this important effect, which is the direct signature of the topological phase, could only be observed for sufficiently small lattices (in which case the number of edge-states is not totally negligible compared to the number of bulk states). Another difficulty is that the edge-states could easily be destroyed by the harmonic potential: they should therefore be stabilized by sharp boundaries \cite{Stanescu2010} or by designed interfaces \cite{goldman:2010} . \\
Finally, the methods for detecting topological properties, such as quantized Hall conductivity \cite{Umucalilar:2008} and chiral edge-states \cite{Stanescu2010,Liu:2010,goldman:2010} have been discussed recently and could be easily generalized to our Lieb lattice setup.
\section{Summary}
We have investigated the fermionic properties of a face-centered square (Lieb) lattice. This peculiar system is characterized by the presence of a single Dirac cone at the center of the first Brillouin zone and a flat energy band at half filling. In particular we focused on the modification of this exquisite energy spectrum in the presence of an external magnetic field and a next-nearest-neighbor spin-orbit interaction. In the former case, we have shown the opening of multiple gaps due to the occurrence of Landau levels, which leads to the formation of two Hofstadter butterflies separated by the robust flat band at zero energy. We have characterized the topological nature of these gaps by investigating the IQHE. Inside the two Hofstadter butterflies, the Hall conductivity is quantized and each gap is characterized either by positive or negative values. Importantly, we find that the energy gaps separating the two butterflies have a trivial topological nature as they are characterized by zero Hall conductivity. This fact is a direct consequence of the flat band's trivial order. In the case of the spin-orbit interaction we have shown the opening of two symmetric gaps around the flat band. These are characterized by a non-trivial Z$_2$ topological phase leading to the quantum spin Hall effect. We have demonstrated the existence of helical edge states for an abstract cylindrical system, as well as for the more realistic finite lattice.
We have further investigated the Lieb lattice presenting a long wavelength approximation for the system Hamiltonian around the single Dirac cone. We have shown that the Hamiltonian can be expressed in a relativistic form -- similarly to the honeycomb lattice case -- which is characterized by a set of pseudo-spin matrices of size $3\times3$. These matrices fulfill the commutation relation of an angular momentum and describe a spin-1 particle. Within the same approximation we have demonstrated that the spin-orbit interaction simply results in a mass term within the quasirelativistic Hamiltonian. Furthermore, we have inspected the properties of the Landau levels. In addition to the dispersion relation ruled by the square root of the Landau level index -- as in the case of the honeycomb lattice -- we have shown that there are two competing Landau levels at zero energy, which are related to the lattice topology and to the symmetry class of the Hamiltonian operator.
This system and its associated properties could be engineered using fermionic cold atoms placed in an optical lattice resembling the Lieb lattice topology. We have proposed a method for implementing Abelian and non-Abelian synthetic gauge fields in order to simulate the presence of an external magnetic field and a next-nearest-neighbor spin-orbit interaction term. We have further shown that these synthetic fields trigger the opening of energy gaps while preserving the robust flat band at half-filling, properties which can be directly deduced from atomic density measurements. In particular, we emphasized that the Lieb lattice is very well suited to reproduce the intrinsic spin-orbit term introduced in Ref. \cite{Kane2005}, which in this case, can be simply decomposed into nearest-neighbour hoppings on a square sublattice.
\acknowledgments
NG thanks the F.R.S-F.N.R.S (Belgium) for financial support. DB is supported by the Excellence Initiative of the German
Federal and State Governments.
|
1,314,259,996,504 | arxiv | \section{Introduction}
In the context of AdS/CFT, entanglement entropy of the boundary quantum field theory can be calculated using the Ryu-Takayanagi \cite{Ryu:2006bv, chm} prescription and its generalizations to higher curvature gravity theories. The derivation of this holographic entanglement entropy for two derivative Einstein gravity was proposed by Lewkowycz and Maldacena in \cite{LM}. This derivation allows one to obtain a surface equation (for static situations) for the entangling surface which minimizes an entropy functional. There were attempts to extend this calculation to find the entangling surface equation in higher curvature theories in \cite{bss} and the corresponding entropy functionals were argued to be different from the Wald entropy \cite{Dong:2013qoa,Camps:2013zua,Miao}. In particular, for Lovelock theories the entropy functional is the so-called Jacobson-Myers (JM) \cite{Jacobson:1993xs,Hung} one while for general four derivative theories, it coincides with the Fursaev-Patrushev-Solodukhin (FPS) entropy functional \cite{Fursaev:2013fta}. However, there still exist problems in finding the entangling surface using the Lewkowycz-Maldacena method and there are potential ambiguities related to higher order extrinsic curvature terms in the entropy functionals. Moreover, it leads one to wonder if such entropy functionals could arise independent of AdS/CFT from different and perhaps more fundamental considerations.
Wald and collaborators \cite{Wald:1993nt, Iyer:1994ys} had established the first law for black hole mechanics for any diffeomorphism invariant theory of gravity and proposed that the entropy of a black hole is a Noether charge associated with the Killing isometry generating the horizon. The Wald entropy suffers from various ambiguities \cite{Jacobson:1995uq} and therefore does not provide a unique answer for the horizon entropy. Although, none of these ambiguities contribute to black hole entropy of a stationary Killing horizon with regular bifurcation surface, the study of the second law of black hole mechanics beyond general relativity (GR) shows that these ambiguities need to be carefully included in the expression of black hole entropy to obtain an increase theorem similar to Hawking area theorem in GR. In fact, for black holes in Lovelock gravity, it is the JM functional which leads to an increase theorem for linearized perturbations \cite{Sarkar:2013swa}.
Recently, it has been pointed out that the 2nd law for spherically symmetric black holes is satisfied at linear order in perturbations in general four derivative theories of gravity by the holographic entanglement entropy (HEE) functionals \cite{Bhattacharjee:2015yaa}. This has been also generalized beyond spherical symmetry and to generic perturbations up to linear order \cite{Wall:2015raa} and a detailed construction of the entropy functionals has been proposed for any metric theory of gravity such that the linearized second law holds true. For theories with higher curvature terms, the construction produces the HEE functionals. This is a remarkable result since it allows for a way independent of AdS/CFT of deriving entropy functionals--these same entropy functionals therefore find applications in diverse settings. However, what still needs to be addressed is if only the HEE functionals alone do this job. In this paper, we first show that at linear order in perturbations only the HEE functionals can satisfy the second law. We start with an expression for horizon entropy with all possible ambiguous terms relevant for linear order in perturbation. We fix these terms by demanding the validity of the linearized second law and the final entropy coincides with HEE functionals. To be clear, since our analysis is going to be perturbative, we will not be able to rule out higher order extrinsic curvature terms in the functionals (e.g. $O(K^4)$ in the FPS functional)--see section 5.
We will tackle the question keeping in mind two different motivations. First, we are interested in what happens in the case of asymptotically flat black holes. We will consider two different types of perturbations--the first kind will be due to radially symmetric, slowly falling matter and the second kind being a shear perturbation. We will find that for the Gauss-Bonnet (GB) theory, if we go to second order in perturbations, the second law is automatically satisfied for regular horizons if the GB coupling is positive. If the GB coupling is negative then a certain lower bound on the horizon radius ensures that the second law is satisfied; since this suggests that such black holes cannot be formed from collapsing matter, one can take this as an indication that the negative coupling is disfavoured. If we consider a Ricci-square theory then we find that similar extra conditions on the mass of the black hole may be needed for the second law to hold. Second, we are interested in what happens in the context of AdS/CFT. We will find that for the GB theory, the second order analysis leads to a bound on the Gauss-Bonnet coupling. For black branes, this bound coincides with the absence of sound channel instabilities at the black hole horizon. A similar analysis for the Ricci- square theory bounds the corresponding coupling constant. For GB topological black holes \cite{rgcai}, for the zero mass case we find that the second law bound coincides with the tensor channel causality constraint \cite{Brigante, holoGB}. This bound is what leads to the lower bound on the ratio of shear viscosity to entropy density in the dual plasma and agrees with the lower bound on the $a/c$ ratio in 4d CFTs \cite{hm} with $\mathcal N=1$ supersymmetry. We also investigate quadratic theories with $R_{ab}R^{ab}$ and $R^2$ terms. In these theories, the causality constraints following from \cite{hm} are trivial whereas the second law bound in certain examples we study is nontrivial.
Using the HEE entropy functionals and the Raychaudhuri equation, we then turn to the question of holographic c-theorems. We will argue that the holographic c-functions found in \cite{ctheorems} arise naturally from such a consideration provided the matter sector satisfies the null energy condition. In \cite{sahakian}, a derivation for the c-functions in Einstein gravity, found in \cite{cthor}, was given using the Raychaudhuri equation. In \cite{cremades} an attempt was made to extend this to higher curvature theories using the Iyer-Wald prescription. Unfortunately the resulting functions do not give the correct central charge (namely the A-type anomaly coefficient in even dimensions). In \cite{ctheorems} it was observed that in all sensible holographic models, which can be constructed by demanding the absence of higher derivative terms in the radial direction (to ameliorate the problem of ghosts) allow for a simple c-function which at the fixed points coincide with the A-type anomaly coefficient in even
dimensions. This c-function was monotonic under RG flow provided the matter sector satisfied the null energy condition. We will find that using the HEE entropy functionals and considering the Raychaudhuri equation naturally leads to the same c-functions as in \cite{ctheorems}.
The organization of the paper as follows: In the next section we briefly describe the ambiguities in Wald's Noether charge construction. In the section \ref{second-set} set up for proving second law has been described. Next, we determine the coefficients of the ambiguous terms in Wald's construction using linearized second law. In section \ref{beyond}, we go beyond linear order in perturbation to study the GB theory and determine bound on the coupling parameter using AdS black hole solution with different horizon topologies as background. We also consider various other examples like critical gravity theories and put bounds on the couplings in those theories. In section 6, we derive holographic c-functions using the HEE functionals and the Raychaudhuri equation. We end this paper by discussing several follow-up questions. We use $\{-,+,+,,\cdots\}$ signature and set $G=1$ throughout the paper.
\section{Ambiguities in Noether charge method for non-stationary horizons}\label{ambi-wald}
Let us start by describing the geometry of the horizon of a stationary black hole in $D$ dimensions. The event horizon is a null hyper-surface ${\cal H}$ parameterized by a non-affine parameter $t$. The vector field $k^a = (\partial_t)^a$ is tangent to the horizon and obeys non-affine geodesic equation $k^a\nabla_ak^b=\kappa~ k^b$ with $\kappa$ is the surface gravity of the horizon ($a,b,...$ are bulk indices). All $t=$ constant slices are co-dimension $2$ space-like surfaces and foliate the horizon. We construct another auxiliary null normal to the $t=$ constant slices $l^a$, and the inner product between $k^a$ and $l^a$ satisfies $k^a l_a = -1$. The induced metric on any $t =$ constant slice of the horizon is now constructed as $h_{ab} = g_{ab} + 2 k_{(a} l_{b)}$. The horizon binormals are then given by $\epsilon_{ab} = \left( k_a l_b - k_b l_a\right)$.
The change in the induced metric along these two null directions can be expressed as a sum of trace part and a trace-free symmetric part (assuming null congruences are hypersurface orthogonal therefore have zero twist). The trace part measures the rate of change of the area of the horizon cross section along the null generators which is known to be the expansion and the other part measures the shear of the null geodesic congruences. We denote $\theta_{k}$ and $\theta_{l}$ to be the two expansion parameters in these two null directions and similarly we have two shears $\sigma_{k}^{ab}$ and $\sigma_{l}^{ab}$. Note that $\theta_{k}$ and $\sigma_{k}^{ab}$ typically vanish on a stationary horizon but $\theta_l$ and $\sigma_l^{ab}$ in general do not vanish.
As mentioned in the introduction, the entropy of a stationary black hole in any general covariant theory is given by the Noether charge associated with the boost symmetry generating Killing vectors at the horizon \cite{Wald:1993nt,Iyer:1994ys}. The Wald entropy functional for any general covariant Lagrangian $\cal{L}$ in $D$ dimension takes the form:
\begin{equation}
S_W = - 2 \pi \int_{\mathcal C} \frac{ \partial {\cal L}}{\partial R_{abcd} }\epsilon_{ab} \epsilon_{cd} \sqrt{h}\,dA,\label{Wald_Exp}
\end{equation}
where ${\mathcal C}$ denotes any horizon slice and $\sqrt{h}$ is the area element. $R_{abcd}$ is the Riemann tensor of bulk geometry and $dA$ is the area of the $D-2$ dimensional cross-sections of horizon. This formula gives a unique expression for entropy so long one is confined to any stationary slice or to the bifurcation surface of the horizon. However, as pointed out in \cite{Jacobson:1995uq,Iyer:1994ys}, the entropy expression constructed via Noether charge approach has several ambiguities if one tries to apply it for nonstationary slices of the horizon. In fact it turns out, the Wald entropy formula is just one of several possible candidates for the entropy. In \cite{Jacobson:1995uq}, Jacobson, Kang and Myers (JKM) identified three different types of ambiguities which may alter the Wald construction. Among these, the only relevant one for our purpose will be the one which gives us the freedom to add to the Wald entropy a term of the form with arbitrary coefficients:
\begin{equation}
S_A^{(JKM)}=-2\pi \int_{\mathcal C} X.Y \sqrt{h}\,dA
\label{ambi}
\end{equation}
where the integrand is invariant under a Lorentz boost in a plane orthogonal to an arbitrary horizon slice ${\mathcal C}$ but the terms $X$ and $Y$ are not separately boost invariant \cite{Wall:2015raa,Sarkar:2010xp}. These ambiguities vanish for Killing horizons but do not necessarily vanish on a nonstationary slice of the horizon. As a result the entropy functional for a nonstationary horizon may be expressed as\footnote{In the context of entanglement entropy, one typically evaluates this formula on a codimension-2 surface with one timelike $n^{(1)a}$ and one spacelike $n^{(2) a}$ normal such that , $n^{(1)a}=\frac{k^a+l^a}{\sqrt{2}}$ and $n^{(2) a}=\frac{k^{a}-l^{a}}{\sqrt{2}}$ and the extrinsic curvature $\mathcal{K}_{ab}^{(i)}$ for that codimension-2 surface with a induced metric $h_{ab}$ defined as, $\mathcal{K}_{ab}^{(i)}=\frac{1}{2}\mathcal{L}_{n^{(i)}} g_{ab}.$ Trace of this is defined as $\mathcal{K}^{(i)}=\mathcal{K}_{ab}^{(i)}h^{ab}$. Also we have $\mathcal{K}_{i}\mathcal{K}^{i}=-2\theta_{k}\theta_{l}$ and $\mathcal{K}_{(i)ab}\mathcal{K}^{(i)ab}= -2\Big(\frac{\theta_{k}\theta_{l}}{D-2}+\sigma_{k}^{ab}\sigma_{l ab}\Big)$, where $D$ is the dimension of the bulk spacetime. }
\begin{equation}
{\mathcal S} = - 2\pi \int_{\mathcal C} \sqrt{h} \,dA\, \left[ \frac{\partial L}{\partial R_{abcd}} \epsilon_{ab} \epsilon_{cd}
-p\, \theta_{k}\theta_{l}- q\, \sigma_k.\sigma_l \right].
\label{genn1}\end{equation}
\vskip 0.5cm
Here we have used the notation $\sigma_{kab}\sigma^{ab}_l=\sigma_k\sigma_l$. As can be seen from (\ref{genn1}) that the ambiguous terms in the entropy formula involve equal number of $k$ and $l$ subscripts which follows from the fact that these are the only boost invariant combinations that can appear in the entropy functional. Also on a stationary slice the ambiguity terms vanish and then the expression coincides with the Wald formula (\ref{Wald_Exp}). The First Law of black hole mechanics \cite{Iyer:1994ys, Jacobson:1995uq} doesn't fix the coefficients $p$ and $q$ uniquely. So, to fix the coefficients of these terms one is thus forced to examine whether ${\mathcal S}$ obeys a local increase law.
Recently in \cite{Wall:2015raa}, the second law has been shown to hold for any arbitrary higher curvature theories if one allows only linear perturbations to a stationary black hole. This analysis also shows, the entropy functional proposed in the context of Holographic Entanglement Entropy (HEE) \cite{Dong:2013qoa,Camps:2013zua,Fursaev:2013fta} remarkably matches with the ${\mathcal S}$. However, it is not apparent from this analysis\footnote{The criterion in \cite{Wall:2015raa} is sufficient but may not be necessary.} how just one condition (entropy increases along $t$) fixes two coefficients in the entropy functional! In this paper, we will show explicitly how this fixing happens for curvature squared gravity theories. Note that the unknown coefficients are sitting in front of products of shear and
expansion terms. Since shear and expansion belong to different irreducible parts of a tensor, they will remain separated even when perturbation is turned on at the linearized level. Consequently we will obtain two independent equations when we demand that entropy increase law holds for every slice of the horizon. This fixes all the coefficients of the entropy functional uniquely for generic quadratic curvature gravity.
It is important to mention that we have only considered ambiguities which are quadratic order in expansion and shear. We may also have higher powers of such products added to the entropy functional. However, we cannot fix such terms using linearized second law but in curvature squared theories these terms do not enter at linear order \cite{Sarkar:2010xp}. Our analysis matches with the result obtained in \cite{Bhattacharjee:2015yaa} where the coefficients were fixed for Ricci$^2$ theory using a Vaidya like solution in the linearized increase law.
\section{Second law set up}\label{second-set}
Let us turn to the apparatus needed to verify the second law of black hole mechanics. The equation of motion for a generic higher curvature metric theory of gravity will be of the form,
\begin{equation}
G_{ab}\,+\,H_{ab}=8\pi T_{ab}\label{eqm} \,,
\end{equation}
where $G_{ab}$ is the Einstein tensor coming from the Einstein-Hilbert part of the action and $H_{ab}$ is the part coming from higher curvature terms--in theories with a cosmological constant there will also be an additional term proportional to the metric which we can absorb into $G_{ab}$. $T_{ab}$ is the energy momentum tensor which we will assume to obey the Null Energy condition (NEC): $T_{ab} k^a k^b>0$ for some null vector $k^a$.
We will use the Raychaudhuri equation for null geodesic congruence which describes the evolution of the expansion along the horizon generating parameter $t$. In nonaffine parametrization this looks like\footnote{Interested readers are referred to \cite{Kar, Pois} for discussions on the Raychaudhuri equation.}
\begin{equation}
\frac{d\theta_k}{dt}=\kappa \theta_k -\frac{\theta_k^2}{D-2}-\sigma_k^2-R_{kk}\,.\label{Ray1}
\end{equation}
Our notation is $A_{ab} k^a k^b = A_{kk}$ and $\sigma_{kab}\sigma^{ab}_k=\sigma^2$. Now, we define an entropy density for any generic higher curvature theory as:
\begin{equation}
{\mathcal S}=\frac{1}{4} \int_{\mathcal C} \left(1 + \rho \right)\, \sqrt{h}\,d^{D-2}x, \label{entropyG}
\end{equation}
where $\rho (t)$ contains contribution from the higher curvature terms including ambiguities. For general relativity, $\rho = 0$. In the stationary limit $\rho$ will coincide with the Wald expression (\ref{Wald_Exp}). Now from this expression the change in entropy per unit area gives us the following expression of generalized expansion $\Theta$,
\begin{equation}
\Theta = \frac{d \rho}{dt} + \theta_{k} \left(1 + \rho \right).
\label{Theta}\end{equation}
The evolution of $\Theta$ is governed by the following equation,
\begin{eqnarray}
\frac{d\Theta}{dt}- \kappa\Theta=&-&8\pi T_{kk}-\frac{\theta_k^2}{D-2}(1+\rho)-\sigma_{k}^2 (1+\rho)+\theta_k \frac{d\rho}{dt}\nonumber\\&+&H_{kk}+\frac{d^2\rho}{dt^2}-\rho R_{kk}-\kappa\frac{d\rho}{dt}\label{Ray2}\,,
\end{eqnarray}
where we have used the equation of motion (\ref{eqm}) and inserted eq. (\ref{Ray1}). We can now write eq, (\ref{Ray2}) in a convenient form
\begin{equation} \label{Ray3}
\frac{d\Theta}{dt}-\kappa \Theta=-8\pi T_{kk}+E_{kk}\,,
\end{equation}
where
\begin{equation} \label{eqn2}
E_{kk}= H_{kk}+\theta_k \frac{d\rho}{dt}-\rho R_{kk}+k^{a}k^{b}\nabla_{a}\nabla_{b} \rho-\left(\frac{\theta_k^2}{D-2} +\sigma_{k}^2\right)(1+\rho)\,.
\end{equation}
We have used $$\frac{d^2\rho}{dt^2}=k^{a}k^{b}\nabla_{a}\nabla_{b} \rho +\kappa \frac{d\rho}{dt}$$ and $$\frac{d}{dt}= k^{a}\nabla_{a}.$$
Next, consider a situation when a stationary black hole is perturbed by some matter flux obeying NEC. The perturbation can be parametrized by some dimensionless parameter $\epsilon$. Note that $T_{kk}$ is linear ($\cal{O}(\epsilon)$) in perturbation and so as $\theta_k$, $\sigma_k$ and $d \rho/dt$. Now, to establish a linearized second law we ignore higher order terms in eq. (\ref{Ray3}). Then, it is easy to see then eq. (\ref{eqn2}) reduces to,
\begin{equation}
E_{kk}\cong \nabla_k \nabla_k \rho - \rho R_{kk} + H_{kk}\label{ekkl}\,.
\end{equation}
We have already mentioned that $T_{kk}$ is of order $\epsilon$, so if the rest of the terms in (\ref{Ray3}) are also collectively of higher order, i.e.,$ E_{kk} \sim \mathcal{O}(\epsilon^2)$, then we obtain
\begin{equation}
\frac{d \Theta}{d t} - \kappa \Theta= - 8 \pi \,T_{kk}.
\end{equation}
The above equation implies $d \Theta / d t - \kappa \Theta < 0$, on every slice of the horizon. We assume that in the asymptotic future, the horizon again settles down to a stationary state, we must have $\Theta \to 0$ in the future.
This will imply that $\Theta$ must be positive on every slice prior to the future and as a result the entropy given by (\ref{entropyG}) obeys a local increase law.
Therefore, to establish the linearized second law, we only need to show that the linear order terms in $E_{kk}$ exactly cancel each other. Interestingly, this alone will be enough to obtain the values of both the coefficients $p$ and $q$ introduced in the entropy functional (\ref{gen1}). In the next section, we will demonstrate this explicitly for the curvature squared gravity theories.
\section{Linearized second law and fixing JKM ambiguities}\label{linear}
We start with the most general second order higher curvature theory of gravity in five dimensions. The action of such a theory can be expressed as,
\begin{equation} S = \frac{1}{16 \pi} \int d^5x \sqrt{-g}\left( R- 2\Lambda +\,\alpha\, R^2 +\,\beta \, R_{ab}R^{ab} +\, \gamma\,{\cal L_{GB}} \right) \label{Flag}
\end{equation}
where ${\cal L_{GB}} = R^2\,-\,4 R_{ab}^2\,+\,R_{abcd}^2$ is the Gauss- Bonnet (GB) combination and $\Lambda$ is the cosmological constant. It is reasonable to assume that such a theory admits a stationary black hole solution as in the case of GR. We also expect to have a non stationary black hole solution with in-falling matter by perturbing this solution. Such a spacetime will be the counterpart of the Vaidya solution in general relativity for spherically symmetric case and can be expressed as,
\begin{equation} \label{vd1}
ds^2=-f(r,v) dv^2+2 dv dr + r^2d\Sigma_{3}^2
\end{equation}
$\Sigma_{3}$ can be any three dimensional space with positive, negative or zero curvature. We want to use this solution to investigate the issue of second law of black hole mechanics. Note that, the location of the event horizon $r = r(v)$ for this solution can be obtained by by solving the following equation,
\begin{equation} \label{hor}
\dot r= \frac{dr(v)}{dv}=\frac{f(r,v)}{2}.
\end{equation}
with appropriate boundary condition. Note that, $(\,\dot{}\,)$ and $(\,'\,)$ denote respectively derivative with respect to $v$ and $r$. The null generator of the event horizon is given by $k^{a}=\{1,f(r,v)/2,0,0,0\}$ and the corresponding auxiliary null vector $l_{a}=\{-1,0,0,0,0\}$. The event horizon has nonzero expansion due to the perturbation caused by in falling matter. Next, we will write the entropy associated with the horizon as,
\begin{equation}
{\mathcal S} = - 2\pi \int_{\mathcal C} \sqrt{h} \,dA\, \left[ \frac{\partial L}{\partial R_{abcd}} \epsilon_{ab} \epsilon_{cd}
-\tilde p\, \theta_{k}\theta_{l}-\tilde q\, \sigma_k\sigma_l \right].
\label{gen1}\end{equation}
We consider several choices of the coefficients $\alpha$, $\beta$ and $\gamma$ etc and study the evolution of this horizon entropy. The aim is to fix the unknown coefficients $\tilde p$ and $\tilde q$ for different gravity theories by demanding the validity of the linearized second law.
\subsection{Gauss-Bonnet gravity}
We will first study the local increase law for Gauss-Bonnet (GB) case. This corresponds to the choice $\alpha=\beta=0$ in (\ref{Flag}). The action is:
\begin{equation}
S=\frac{1}{16\pi}\int d^{5}x\sqrt{-g} \Big[R-2\Lambda+\gamma \Big(R_{abcd}R^{abcd}-4R_{ab}R^{ab}+ R^2\Big)+\mathcal{L}_{m}\Big].
\end{equation}
where we have also introduced a matter sector obeying the NEC. $\gamma$ is a coupling constant of dimension $Mass^{-2}$. The equation of motion for this theory is given by,
\begin{equation} \label{eom}
G_{ab}+\Lambda g_{ab}+ H_{ab}=8\pi G\, T_{ab}.
\end{equation}
With,
\begin{eqnarray}
{H}_{ab}\equiv2\gamma\Bigl[RR_{ab}-2R_{ac}R^c_{~b}
-2R^{cd}R_{acbd} +R_{a}^{~cde}R_{bcde}\Bigr]
-\frac{1}{2}g_{ab} {\cal L}_{GB}.
\end{eqnarray}
Next we start with the expression of the entropy functional (\ref{entropyG}) and evaluate the entropy density for the EGB theory. This is given by sum of Wald entropy density plus the ambiguity terms,
\begin{eqnarray}
{\mathcal S}&=&\frac{1}{4} \int_{\mathcal C} \left(1 + \rho \right)\, \sqrt{h}\,d^3x, \nonumber \\
&=&\frac{1}{4} \int_{\mathcal C} \left(1 + 2\gamma(R + 4 R_{kl} - 2 R_{klkl}-p\, \theta_k\theta_l-q\, \sigma_k \sigma_l) \right)\, \sqrt{h}\,d^3x
\,,\label{eee}\end{eqnarray}
where $2(R + 4 R_{kl} - 2 R_{klkl})$ is the contribution from the Wald construction. Also $\tilde p= 2 \,\gamma\, p$ and $ \quad \tilde q=2\,\gamma\, q.$ The EGB theory belongs to the general Lovelock class of action functions which give rise to quasi linear equation of motions. It has already been shown in \cite{Sarkar:2013swa} that black holes in all Lovelock theories obey a linearized second law if one uses the JM entropy functional. For EGB gravity, the JM entropy functional is the intrinsic Ricci scalar $({\mathcal R})$ of the horizon slice and using the null Gauss-Codazzi equation we can cast ${\mathcal R}$ as,
\begin{equation}\label{JMGB}
{\mathcal R}=R + 4R_{kl} - 2 R_{klkl}-{4\over 3}\theta_k\theta_l+2\sigma_k\sigma_l.
\end{equation}
We will now explicitly show below that in EGB gravity the entropy functional which obeys the second law is indeed the JM entropy by fixing the coefficients $p$ and $q$ in (\ref{eee}).
First, we proceed to evaluate the {\it rhs} of eq. (\ref{Ray3}) to study the second law. Note that, if we use the metric (\ref{vd}) with the choice of $d\Sigma_{3}^2$ to be a flat metric then the shear term in (\ref{Ray3}) will vanish identically and $q$ will remain undetermined. This happens because isometries of the metric on a horizon slice essentially coincides with that of a sphere. So we will break the symmetry by adding a cross term and the metric on the horizon slice takes the form,
\begin{equation}
ds^2=-f(r,v) dv^2+2 dv dr + r^2( dx^2+ dy^2+ dz^2+ \epsilon_{1} h(r,v) dx dy ) .
\end{equation}
We will assume that this shear mode $h(r,v)$ will be balanced by some matter stress tensor still obeying the NEC. Also we do not require to find the explicit form of $h(r,v)$ for our analysis. We will calculate $E_{kk}$ order by order in $\epsilon_{1}$ and extract the coefficients of the linear order terms in $\epsilon$ from the evolution equation. Setting those terms to zero will satisfy the linearized second law and in the process $p$ and $q$ will be determined. Below we quote expressions for $\theta's$ and $\sigma's$ for this solution.
\begin{equation}\label{exp1}
\theta_{k}=\frac{3 f(r,v) }{2 r(v)}+\mathcal{O}(\epsilon_{1}^2)\,,
\end{equation}
\begin{equation}\label{shear1}
\theta_{l}=\frac{3}{ r(v)}+\mathcal{O}(\epsilon_{1}^2)\,
\end{equation}
and
\begin{equation}
\sigma_k\sigma_l=-\frac{\epsilon_1 ^2 \left(r(v)^2 h'(r,v)^2-3 r(v) h(r) h'(r,v)+2 h(r,v)^2\right) f(r,v)}{2 r(v)^6}+\mathcal{O}(\epsilon_{1}^4).
\end{equation}
Now evaluating $E_{kk}$ and extracting the zeroth order terms in $\epsilon_1$ (and linear order in $\epsilon$), we get the following equation,
\begin{equation}
\frac{\partial^2 f(r,v)}{\partial^2 v}(4-3 p)=0.
\end{equation}
From $\mathcal{O}(\epsilon_{1}^{2})$ terms (which is when $h(r,v)$ makes its first appearance) we get another equation,
\begin{equation}
2(-6+6p+q) h(r,v)^2+3(2-3p-q)h(r,v)r(v)h'(r,v)+(2+q)r(v)^2h'(r,v)^2=0.
\end{equation}
It is evident that, both of these equations are satisfied if $$p=\frac{4}{3}\,\quad q=-2.$$ This shows that linearized second law fixes all the quadratic ambiguities in entropy functional (\ref{gen1}) uniquely and it is the JM entropy which obeys the linearized second law for GB black holes! Although we have shown this calculation using the 5-dimensional lagrangian for concreteness, the same result holds in any dimension $D>4$.
\subsection{$R_{ab}^2$ theory}
Let us repeat the above analysis for other curvature squared theories. First we take $R_{ab}^2$ theory and we set $\alpha=\gamma=0$ and $\beta >0 $ in (\ref{Flag}). We will start with the following entropy functional,
\begin{equation}\label{e1}
{\mathcal S}=\frac{1}{4}\int d^{3} x \sqrt{h}\Big(1-2\beta(R_{kl}-p\theta_{k}\theta_{l}-q\sigma_k\sigma_l)\Big).
\end{equation}
When $p$ and $q$ are zero then this reduces to the corresponding Wald entropy for the stationary case. Again proceeding as before we will use the linearized second law to fix these coefficients. Like the Gauss-Bonnet case we get two independent equations and solving them we get, $$p=\frac{1}{2}\,\quad q=0.$$ {As argued in \cite{Bhattacharjee:2015yaa}, the shear part has not contributed to the entropy functional at linear order and the other coefficient of the entropy functional exactly matches with the one which has been shown to obey the linearized second law in \cite{Bhattacharjee:2015yaa}.} \par
\subsection{$R^2$ theory}
We now consider the case $\beta=\gamma=0$ and $\alpha >0 $ in (\ref{Flag}) and the entropy functional now reads,
\begin{equation}\label{e2}
{\mathcal S}=\frac{1}{4}\int d^{3} x \sqrt{h}\Big(1+2\alpha(R-p\theta_{k}\theta_{l}-q\sigma_k.\sigma_l\Big).
\end{equation}
It has already been argued in \cite{Jacobson:1995uq} that $f(R)$ theory obeys a linearized second law and the Wald entropy functional itself does the job. It is clear, in this case setting $p$ and $q$ equal to zero will correspond to the Wald entropy. If we again repeat the linearized second law analysis we find, $$p=0\,\quad q=0.$$ So this is exactly what is expected from the earlier analysis in \cite{Jacobson:1995uq}.
Therefore, considering all the individual terms in (\ref{Flag}) we have demonstrated that for any general curvature squared theory we can fix the entropy functional completely using only the linearized second law.
\section{Beyond linearized second law}\label{beyond}
In this section, we will consider local entropy increase law beyond linear order which means we are not allowed to neglect the terms in (\ref{Ray3}) which have been thrown away in the previous section. Now we will be considering all the relevant terms in equation (\ref{Ray3}) and will determine the criteria such that second law holds non-perturbatively in the coupling. For this, we will assume a spherically symmetric Vaidya like solution. This means, the metric in (\ref{vd1}) will now have the following form,
\begin{equation} \label{vd}
ds^2=-f(r,v) dv^2+2 dv dr + r^2d\Omega_{3}^2.
\end{equation}
In fact for EGB gravity one can obtain such a solution just setting the mass in the static spherically symmetric Boulware-Deser \cite{BD} solution to be a function of the advanced time $v$.
Beyond linearized level, one has to evaluate the full $E_{kk}$ in the evolution equation (\ref{Ray3}). Since the solution is spherically symmetric, shear is identically zero. In fact, for this particular case the evolution equation will be of the form,
\begin{equation}
\frac{d\Theta}{dt}-\kappa\Theta=-8\pi T_{kk}-\zeta \theta^{2}_k,
\end{equation}
Note that we must have $d\Theta / dt < 0$ to have a local entropy increase law. Also, we have $T_{kk} > 0$ by NEC. Now consider a situation where the stationary black hole is perturbed by some matter flux and we are examining the second law when the matter has already entered into the black hole. In that case, the above evolution equation does not have any contribution from matter stress energy tensor and the evolution will be driven solely by the $\theta_k^2$ term. So we will have a equation of the form,
\begin{equation}
\frac{d\Theta}{dt}-\kappa\Theta=-\zeta \theta^{2}_k\,.
\end{equation}
In such a situation, if we demand the entropy is increasing, we have to fix the sign of the coefficient of $\theta_k^2$ term. We evaluate the coefficient in the stationary background and impose the condition that overall sign in front of $\theta^{2}_k$ is negative. This will give us a bound on the parameters of the theory under consideration.
\subsection{Gauss-Bonnet Gravity}
We start with the EGB gravity. The event horizon is now a null surface whose equation is $r = r(v)$. Calculating the r.h.s. of (\ref{Ray3}) for the metric (\ref{vd}) we obtain,
\begin{equation}
\frac{d\Theta}{dt}-\kappa\Theta=-\zeta \frac{9f(r,v)^2}{4 r(v)^2},
\end{equation}
where we have identified $$\theta_k^2=\frac{9f(r,v)^2}{4 r(v)^2}$$ and introduced $\zeta$ as the coefficient of the $\theta_k^2$ terms. The expression of $\zeta$ is given by,
\begin{equation} \label{zeta}
\zeta=\frac{1}{3}\left[1+2\gamma\left(\, \mathcal{R}-\frac{2}{r(v)}\frac{\partial f(r,v)}{\partial r}\right)\right].\end{equation}\par
For $\gamma=0$, the coefficient reduces to $1/3$ which matches with GR. $\mathcal{R}$ is the ricci scalar evaluated on the horizon. To satisfy the entropy increase law we now set,
\begin{equation}\label{cond} \zeta>0.\end{equation}
As discussed earlier, we will evaluate $\zeta$ for different stationary backgrounds and determine bounds on the coefficient $\gamma$ imposing the condition (\ref{cond}).
\subsubsection{Asymptotically flat case} \label{BD-bnd}
Now we consider the Boulware-Deser (BD) \cite{BD} black hole as the background, for which,
\begin{equation}
f(r)=1+\frac{r^2}{4\gamma}\Big[1-\sqrt{1+\gamma \frac{8 M}{r^4}}\Big].
\end{equation}
Also,
\begin{equation} \label{eqn3}
r_{h}^{2}+2\gamma=M
\end{equation}
determines the location of the horizon. Existence of an event horizon demands $r_h^2>0$. In this case horizon topology is a sphere. Now, evaluating $\zeta$ for the above background at the horizon $r =r_{h}$, we get the following inequality
\begin{equation}
\frac{M^2+4 M \gamma +36 \gamma^2}{4 \gamma^2-M^2}<0\,, \label{ineq}
\end{equation}
which leads to
\begin{eqnarray}
&M>2 |\gamma|\,\quad if~ M>0\\
&M<-2|\gamma|\,\quad if~ M<0\,.
\end{eqnarray}
To understand this better, note that we require $M>2\gamma$ to avoid the naked singularity of the black hole solution for $\gamma>0$. Thus in this case for a spherically symmetric black hole $\zeta$ will be positive and hence second law will be automatically satisfied. The condition of the validity of the second law is same as that for having a regular event horizon.
Also, for $\gamma > 0$, it is possible to make $r_h$ as small as possible by tuning the mass $M$. But when $\gamma$ is negative (a situation that appears to be disfavoured by string theory, see \cite{BD}, \cite{bms} and references therein), $r_h$ cannot be made arbitrarily small and it would suggest that these black holes cannot be formed continuously from a zero temperature set up. Notice that we could have reached the conclusion without the second law if $M$ is considered to be positive--however, our current argument does not need to make this assumption. Due to this pathology, it would appear that the negative GB coupling case would be ruled out in a theory with no cosmological constant.
\subsubsection{AdS case}
Next we consider the 5-dimensional AdS black hole solution for EGB gravity as the background. The function $f(r)$ now becomes \cite{rgcai},
\begin{equation}
f(r)=k+\frac{r^2}{4\gamma}\Big(1-\sqrt{1-\frac{8 \gamma}{l^2}(1-\left(\frac{r_{0}}{r}\right)^{4}}\Big).
\end{equation}
Here $l$ is the length scale with the cosmological constant ($2\Lambda=-12/l^2$) and $k$ can take values $0,1$ and $-1$ corresponding to planar, spherical or hyperbolic horizons respectively. The intrinsic Ricci scalar on the horizon is $6k/r_h^2$ where, the horizon is at
\begin{equation}
r_h^2=\frac{l^2}{2}\left[-k+\left(\frac{4 r_0^4}{l^4}+k^2\left(1-\frac{8 \gamma}{l^2}\right) \right)^{1/2}\right]\,.
\end{equation}
First we will consider black brane solution for which $k=0$ and the horizon is planar. The intrinsic curvature of the codimension two slices of the event horizon vanishes. In this case the coefficient $\zeta$ reads,
\begin{equation} \label{betajm}
\zeta= \frac{1}{3}\left(1-16\frac{\gamma}{l^2}\right).
\end{equation}
Introducing a rescaled coupling $\lambda_{GB} l^2=2\gamma$ \cite{Brigante, Buchel} we get,
\begin{equation}
\zeta= \frac{1}{3}(1-8\lambda_{GB}).
\end{equation}
Again demanding positivity of $\zeta$ we get,
\begin{equation}
\lambda_{GB}< \frac{1}{8}\,,
\end{equation}
so to satisfy the second law using ${\mathcal S}$ at $\mathcal{O}(\epsilon^2)$ order we need to impose a bound on the GB coupling.
Incidentally the same has to be imposed to avoid instabilities in the sound channel analysis of the quasinormal mode for dual plasma. It was shown in \cite{Buchel} that when $\lambda_{GB} >\frac{1}{8}$ the Schroedinger potential develops a well which can support unstable quasinormal modes in the sound channel. It is interesting to see that the second law knows about this instability. \par
Next we consider $k=-1$ case. This will give us a black hole with a hyperbolic horizon. We will also set $r_{0}=0$, which corresponds to the zero mass solution. In this limit we find ,
\begin{equation}
\zeta=5\sqrt{1-4\lambda_{GB}}-4.
\end{equation}
Demanding $\zeta>0$ we get,
\begin{equation}
\lambda_{GB} < \frac{9}{100}.
\end{equation}
If we take the limit $r_0/l\rightarrow \infty$ in the expression of $\zeta$ with $k=\pm 1$ (i.e. the case when the horizon sections becomes planar), we recover the bound on GB coupling $\lambda_{GB}$ for the $k=0$ case. However, this is a weaker bound on $\lambda_{GB}$ and the strongest bound on $\lambda_{GB}$ comes from the hyperbolic black hole in the massless limit. This is illustrated in figures 1a and 1b. We have checked that for the $k=1$ case other thermodynamic considerations (e.g., positive entropy, black hole being the correct phase) do not lead to stronger bounds.
\begin{figure}[hb]
\begin{tabular}{cc}
\includegraphics[height=1.8in,width=3.5in ]{plot1.pdf} & \includegraphics[height=1.8in,width=3.5in ]{plot2.pdf} \\
(a) & (b)
\end{tabular}
\caption{(Colour online) (a) Plot of $\zeta$ for different $r_0$ for $k=-1$. (b) Plot of $\zeta$ for different $r_0$ for $k=1$. The black dot denotes $\lambda_{GB}=9/100$. We have set the $l=1$. Note that both the figures shows that $\zeta$ will be positive in all the cases provided $\lambda_ {GB} < 9/100.$}\label{strongbound}
\end{figure}
\par
Quite curiously we have recovered the tensor channel causality constraint \cite{hm, Brigante, holoGB}\footnote{We can repeat the same analysis for higher dimensions using the normalizations in \cite{holoGB}. Curiously, we find the same bound $\lambda_{GB} < \frac{9}{100}$ which is stronger than the causality bounds for $d>5$ in \cite{holoGB}. For flat case ($k=0$), the bound on GB coupling in $d$ dimensions can be expressed as $\lambda_{GB}<1/(2(d-1))$. This means for any dimension the strongest bound arises when the black holes have hyperbolic horizon ($k=-1$) with $r_0=0$. This would also suggest that in the large $d$ limit, $\eta/s\rightarrow 1/4\pi$ contrary to $1/8\pi$ obtained in \cite{holoGB}.}. The bound on $\lambda_{GB}$ in this channel leads to the lower bound on the ratio of shear viscosity ($\eta$) to entropy density ($s$), $4\pi \eta/s\geq 16/25$ in GB holography \cite{Brigante} . The vector and scalar channels also lead to bounds on the coupling from causality constraints. The strongest bound in GB arises from the tensor and scalar channels \cite{holoGB}. Our finding coincides with the tensor channel constraint. It will be interesting to see if the other channels can also be reproduced using the second law analysis.
We end this section with one last comment. Instead of putting a bound on the Gauss-Bonnet coupling, one can try to modify the entropy functional ${\mathcal S}$ itself so that all the $\mathcal{O}(\epsilon^2)$ terms will be canceled. One possible modification is to add a term like $\theta_{k}^2\theta_{l}^2$ which vanishes on the stationary horizon and boost invariant in the null direction. So this type of higher order ambiguous term can occur in the Wald derivation. We modify ${\mathcal S}$ as,
\begin{equation}
S_{modify}=\frac{1}{4}\int d^{3} x \sqrt{h}(1+2\gamma\mathcal{R}+ \chi\, \theta_{k}^2\theta_{l}^2).
\end{equation}
$\zeta$ in (\ref{betajm}) will be modified as,
\begin{equation} \label{modeq}
\zeta=\frac{1}{3}\left[1-\left(\frac{4 \gamma}{r(v)} \frac{\partial f(r,v)}{\partial r}+ \frac{54 \chi}{r(v)^2}\, \left(\frac{\partial f(r,v)}{\partial r}\right)^2\right)\right]
\end{equation}
So we can adjust $\chi$ such that the $\zeta$ vanishes and second law automatically holds up to $\mathcal{O}(\epsilon^2)$. One may also can add a term like $\nabla_{i}\theta_{k}\nabla^{i}\theta_{l}$ at this order where $\nabla_{i}$ is the covariant derivative with respect to the surface index. But these additional terms will generate further higher order terms starting from $\mathcal{O}(\epsilon^3).$ So one will need to add more terms to cancel them. It might possible to do this recursively.
However, the coefficients of these terms will also depend on the background data. So it will be prudent not to modify the entropy functional. We have put a bound on the coupling instead of modifying the entropy functional at $\mathcal{O}(\epsilon^2)$ order such that second law continues to hold up to this order. It will be interesting to investigate what happens to this bound on $\lambda_{GB}$ when one goes beyond $\mathcal{O}(\epsilon^2)$ order.
\subsection{$R_{ab}^2$ theory}
\subsubsection{Asymptotically flat space}
First consider the asymptotically flat black hole with spherically symmetric horizon in 5d. In this case,
\begin{equation}
\zeta={1\over 3}\Big(1+\frac{8 \beta}{ r(v)^2}+ \frac{15}{2} \beta f''(r,v)+\frac{3 }{2 r(v)}\beta f'(r,v)-\frac{25}{2 r(v)^2} \beta f(r,v)\Big)
\end{equation}
Now, to extract a bound, we need an explicit form of a spherically symmetric static black hole solution in $Ricci^2$ theory. We are not aware of any such solution except the 5d Schwarzschild solution of GR which is also a solution of $Ricci^2$ theory. So, as an example we take such a solution as the background and write,
\begin{equation}
f(r)=1-\frac{r_h^2}{r^2}.
\end{equation}
Then the validity of the second law gives us the condition,
\begin{equation}
\frac{r_h^2 }{34}> \beta.
\end{equation}
When $\beta<0$ this is automatically satisfied whenever a horizon exist whereas for $\beta>0$, a lower bound on the horizon size ensures the validity of the second law. \\
Repeating this same analysis for 4d with the metric function $f(r,v)=1- (r_h/ r)$ we get ,
\begin{equation}
\frac{r_h^2}{18}>\beta,
\end{equation} Therefore, in both cases, there is a minimum horizon radius for the second law to hold when $\beta>0$. Again as discussed in \ref{BD-bnd}, minimum $r_h$ scenario would appear pathological and may be taken as a reason to disfavour the $\beta>0$ sign.
\subsubsection{AdS case}
We repeat the same analysis of the previous section for $R_{ab}^2$ theory but with the background as an asymptotically AdS black brane in 5d. The metric function is then given by
\begin{equation}
f(r)=\frac{r^2}{l^2} f_{\infty}\Big(1-(r_{h}/{r})^4\Big )
\end{equation}
where $f_{\infty}=\big (\frac{l}{l_{AdS}}\big)^2$ and it satisfies the equation,
\begin{equation}
1-f_\infty+\frac{2}{3}\lambda_2 f_\infty^2=0\,.
\end{equation}
In this case we obtain,
\begin{equation}
\zeta=\Big(\frac{1}{3}+\frac{5}{2} \beta f''(r,v)+\frac{\beta f'(r,v)}{2 r(v)}-\frac{25 \beta f(r,v)}{6 r(v)^2}\Big)\,,
\end{equation}
Define $\lambda_{2} l^2=2\beta $ and after evaluating on the horizon with the above metric function as background, we get
\begin{equation}
\zeta=\sqrt{9-24 \lambda_{2}}-\frac{8}{3}.
\end{equation}
From this, the bound becomes,
\begin{equation}
\lambda_{2} <\frac{17}{216}.
\end{equation}
In this case if we repeat out analysis for a zero mass hyperbolic black hole we obtain a lower bound on $\lambda_{2}$. Combining them we get,
\begin{equation}
-\frac{29}{243} <\lambda_{2} <\frac{17}{216}.
\end{equation}
If we repeat the same analysis for 4d and obtain the following bound,
$
\lambda_{2} <\frac{2}{15}.
$
No analog of such a bound in the context of AdS/CFT is known in this case. This analysis also suggests that the second law bound arising from hyperbolic horizons is not necessarily connected to the causality constraints. The reason is that for these theories, in the analysis of \cite{hm}, we only get $c_T>0$.
\subsection{Critical gravity}
We will now consider bounds arising from the second law in quadratic curvature theories with a negative cosmological constant, which only involve terms like $R_{ab}R^{ab}$ and $R^2$. In particular, we will focus on the case of critical gravity \cite{pope,others}. For any theory with only Ricci$^2$ or $R^2$, the positive energy condition of Hofman-Maldacena \cite{hm} does not yield any constraint (see \cite{anoms} for general expressions) except for the positivity of the two point function of stress tensors which follows from unitarity. In all the examples below, the positivity of black hole entropy leads to the same condition arising from demanding $c_T>0$ where $c_T$ is the coefficient in the two point function of CFT stress tensors. The second law constraint will lead to further conditions as we will see.
\subsubsection{D=4}
The action for the 4d version of critical gravity is given by,
\begin{equation} S = \frac{1}{16\pi}\int d^4x \sqrt{-g}\left( R+\frac{6}{l^2} +\alpha l^2\left(R_{ab}R^{ab}-\frac{1}{3}R^2\right) \right) \label{clag}
\end{equation}
In general this theory posses a massive spin-2 mode in addition to the usual massless spin-2 degree of freedom. It has been shown in \cite{pope}, at $\alpha=-1/2$, which is called the critical point, the additional massive spin-2 mode becomes massless. Also, at the critical point the entropy of the Schwarzschild-AdS black hole becomes zero.
We will now use the second law to put a bound on the coupling $\alpha$. Proceeding as before at the second order we get,
\begin{equation}
\zeta=\Big(\frac{1}{2}-\frac{2\, \alpha \, l^2 \, f'(r,v)}{3\, r(v)}+\frac{2}{3} \alpha\, l^2 f''(r,v)-\frac{2}{3} \,\alpha\, l^2\, r(v) f'''(r,v)\Big).
\end{equation}
In this case, we take the background as,
\begin{equation}
f(r)=\frac{r^2}{l^2} \Big(1-\Big(\frac{r_{h}}{r}\Big)^3\Big ).
\end{equation}
From this we get the bound on the coupling as,
\begin{equation} \label{b1}
\alpha \leq \frac{1}{12}.
\end{equation}
Also, the black hole entropy has to be positive--if it was zero then it must remain zero at all times. From this we have the following condition,
\begin{equation}
1+2\,\alpha >0.
\end{equation}
So,
\begin{equation} \label{b2}
\alpha > -\frac{1}{2}.
\end{equation}
Combining (\ref{b1}) and (\ref{b2}) we get,
\begin{equation} \label{b3}
-\frac{1}{2} <\alpha \leq \frac{1}{12}.
\end{equation}
Now from the AdS/CFT perspective it is natural to assume the positivity of the coefficient $c_{T}$ of two point correlator of boundary stress tensor. That will give exactly the same condition as in (\ref{b2}) which is weaker than (\ref{b3}). Repeating the same analysis for a zero mass hyperbolic black hole we did not any further bounds.
\subsubsection{D=5}
Next we consider the critical gravity theory in 5d. The action is given by,
\begin{equation} S = \frac{1}{16\pi}\int d^5x \sqrt{-g}\left( R+\frac{12}{l^2} +\alpha\, l^2\left(R_{ab}R^{ab}-\frac{5}{16}R^2\right) \right) \label{clag}
\end{equation}
The critical point now corresponds to $\alpha=-5/27$ \cite{pope}. Proceeding as before at the second order we get,
\begin{equation}
\zeta=\Big(\frac{1}{3}+\frac{\alpha\,l^2 f'(r,v)}{2 r(v)}+\frac{5}{24} \alpha\,l^2 f''(r,v)-\frac{5}{12}\, \alpha \,l^2 r(v)\, f'''(r,v)\Big).
\end{equation}
In this case, the background solution is taken as,
\begin{equation}
f(r)=\frac{r^2}{l^2} f_{\infty}\Big(1-\Big(\frac{r_{h}}{r}\Big)^4\Big ),
\end{equation}
where the quantity $f_{\infty}$ satisfies the equation,
$
1-f_{\infty}-\frac{3}{4} \alpha f_{\infty}^2=0.
$
The second law thus leads to,
\begin{equation} \label{b4}
-\frac{1}{3}\leq \alpha \leq \frac{109}{2809}.
\end{equation}
Again demanding the entropy to be positive we get,
$
1+\frac{9}{2} f_{\infty}\alpha>0,
$
from which
$
\alpha > -\frac{5 }{27}.
$
Thus we have,
\begin{equation} \label{b6}
-\frac{5 }{27} < \alpha \leq \frac{109 }{2809}.
\end{equation}
If we repeat our analysis for a zero mass hyperbolic black hole, the resulting bound we obtain is weaker than this.
\subsubsection{D=3}
We end this section by exploring the case of NMG theory in 3d.
The action is given by,
\begin{equation} S = \frac{1}{16\pi}\int d^3x \sqrt{-g}\left( R+\frac{2}{l^2} +\alpha\, l^2\left(R_{ab}R^{ab}-\frac{3}{8}R^2\right) \right) \label{clag}
\end{equation}
The critical point is now at $\alpha=-3.$ For this case,
\begin{equation}
\zeta=\Big(1-\frac{7 \alpha\, l^2 f'(r,v)}{2 r(v)}+\frac{9}{4} \alpha\, l^2 f''(r,v)-\frac{3}{2} \alpha\, l^2 r(v) f'''(r,v)\Big).
\end{equation}
The background solution is taken as,
\begin{equation}
f(r)=\frac{r^2}{l^2} f_{\infty}\Big(1-\Big(\frac{r_{h}}{r}\Big)^2\Big ),
\end{equation}
where the quantity $f_{\infty}$ satisfies $
1-f_{\infty}+ \alpha f_{\infty}^2=0.
$
Then we get the following bound,
\begin{equation} \label{b7}
\alpha \leq \frac{9}{25}.
\end{equation}
Demanding the positivity of the black hole entropy we get,
$
1+\frac{ f_{\infty} \,\alpha}{2}>0.
$
This gives,
$
-3 < \alpha < 1.
$
Thus we have,
\begin{equation}
-3 < \alpha \leq \frac{9}{25}.
\end{equation}
In this case, even if we consider a zero mass hyperbolic black hole we obtain the same bound.
\section{Holographic $c$-functions from the entropy evolution equation}
In this section we will construct a holographic $c$-function using the FPS entropy functional for general curvature squared theories. Here we will again use the evolution equation for $\Theta$ (\ref{Ray3}) and in the process we will have a geometric derivation for the $c$-function. \par
For unitary Lorentz invariant theory one can define a function in the space of couplings which decreases monotonically from UV to IR, parametrizing the RG flow. It coincides with the central charges of the corresponding CFTs at the UV and IR fixed point. In $1+1$ and $3+1$ dimensions this has been established rigorously \cite{Zam,Kom}. Now from the holographic point view this $c$ function plays a pivotal role in understanding nature of RG flow. A proposal for the $c$ function for arbitrary theories of gravity has been given in \cite{ctheorems,ctheorems2, quasitop} for arbitrary dimensions. In this section we will use the Raychaudhuri equation as discussed in the previous sections and derive a $c$ function for a general curvature squared theory. For general higher curvature theories, we do not expect a c-theorem as there are bound to be problems with unitarity in such theories. Thus some condition on the couplings, which presumably only explore a subset of conditions leading to unitarty as in \cite{ctheorems}, is expected. This is what we will find as well. At the UV and IR fixed points this function will coincide with the A-type anomaly coefficient for the curvature squared theory and it will also have to be a monotonically decreasing quantity along the RG flow. To prove the monotonicity we will use Raychaudhuri equation. The basic setup is similar as before.
\subsection{Setup}
To derive a $c$ function purely in terms of geometric quantities we will first start with a specific example. We will consider domain wall type of geometry in $5$ dimensions, the metric for which is given below,
\begin{equation}
ds^2=dr^2+e^{2 A(r)}(-d\tau^2+dx^2+dy^2+ dz^2).
\end{equation}
The spacetime can be foliated by surfaces of codimension-2 at every constant time slice. The induced metric on such surfaces is given by,
\begin{equation}
ds^{2}_{surface}=e^{2 A(r)}(dx^2+dy^2+dz^2).
\end{equation}
Next we will consider a hypersurface orthogonal null congruence. The tangent vector for this congruences is given by,
\begin{equation}
k^{a}=(\partial_{\lambda})^a=\{-e^{-A(r)},e^{-2 A(r)},0,0,0\}.
\end{equation}
It satisfies $
k^{a}\nabla_{a}k^{b}=0$
and
$k_{a}k^{a}=0.
$
We can construct another auxiliary null vector $l^a$ as,
\begin{equation}
l^{a}=\{\frac{e^{A(r)}}{2},\frac{1}{2},0,0,0 \}
\end{equation}
such that, $k_{a}l^{a}=-1.$ Note that in this case $k^a$ and $l^a$ are not any kind of horizon null generators as compared to the previous sections. They simply correspond to a null congruence that converges along the light sheet projected out of the codimension-2 surface under consideration. Also in this section we will use affine parametrization.
\subsection{Holographic c-functions in four derivative gravity}
Next we will start with a guess for the $c$ function for curvature squared theory. The action for this theory is given by,
\begin{equation}
S=\frac{1}{16\pi }\int d^{4} x\Big[R+\alpha R^2+\beta R_{ab}R^{ab}+\gamma (R^2-4 R_{ab}R^{ab}+R_{abcd}R^{abcd})\Big].
\end{equation}
The corresponding entropy functional is the FPS functional.
\begin{equation}
S_{FPS}=\frac{1}{4}\int d^{3} x \sqrt{h}(1+\rho)= \frac{1}{4 } \int d^{3} x \sqrt{h} \Big(1+2\Big [\alpha R-\beta(R_{kl}-\frac{1}{2}\theta_{k}\theta_{l})+\gamma\mathcal{R}\Big]\Big).
\end{equation}
$\mathcal{R}$ is the 3 dimensional Ricci scalar intrinsic to the surface.
Now we will start with the following candidate,
\begin{equation}
c(\lambda)=\frac{\Theta}{\sqrt{h}\theta_k^4}
\end{equation}
where as usual, $\Theta= \theta_k(1+\rho)+\frac{d\rho}{d\lambda}$ and $\lambda$ is the affine parameter. This appears to be a natural guess since for the Einstein theory it goes over to the c-function in \cite{cthor}. Then we have to show two things,
\begin{enumerate}
\item $c(\lambda)$ is monotonically decreasing under the flow along the null geodesic congruences. That is we have to check the sign of $\frac{dc(\lambda)}{d\lambda}.$ \par
\item Further it has to coincide with the correct central charge (A-type anomaly in four dimensions) at the fixed points. \end{enumerate}
Now,
\begin{align}
\begin{split}
\frac{ d c(\lambda)}{d\lambda} &= \frac{1}{\sqrt{h}\theta_k^4}\frac{d \Theta}{d\lambda}-\frac{ \Theta }{(\sqrt{h}\theta_k^4)^2}\frac{d}{d\lambda}(\sqrt{h}\theta_k^4)\\ &=\frac{1}{\sqrt{h}\theta_k^4}\frac{d \Theta}{d\lambda}-\frac{\Theta }{\sqrt{h}\theta_k^5}(\theta_k^2+4 \frac{d\theta_k}{d\lambda})
\end{split}
\end{align}
using $\frac{d\sqrt{h}}{d\lambda}=\sqrt{h}\theta_k.$ Next we will replace $\frac{d\Theta}{d\lambda}$ by (\ref{Ray2}) with $\kappa=0$ and $\frac{d\theta_{k}}{d\lambda}$ by Raychaudhuri equation (\ref{Ray1}).
We obtain,
\begin{equation}
\frac{dc(\lambda)}{d\lambda}=\frac{24 \pi G T_{kk}}{\sqrt{h}\theta_k^4}+ E_{kk},
\end{equation}
where,
\begin{align}
\begin{split}
E_{kk}=&\frac{e^{-A(r)} (16 \alpha +5 \beta)^2}{243 A'(r)^5}\Big((2 A^{(3)}(r)+15 A'(r)^3+16 A'(r) A''(r)) (15 A'(r)^4+2 (A^{(4)}(r)+\\&8 A''(r)^2)+8 A^{(3)}(r) A'(r)+6 A'(r)^2 A''(r))\Big)+\frac{e^{-A(r)} (16 \alpha +5 \beta )}{486 A'(r)^4}\Big(-24 (A^{(4)}(r)\\&+8 A''(r)^2 )+60 A'(r)^4 (4 (10 \alpha +2 \beta +3 \gamma ) A''(r)-3)+A'(r)^2 A''(r)\\& (256 (10 \alpha +2 \beta +3 \gamma ) A''(r)-207 )+8 A^{(3)}(r) A'(r) (4 (10 \alpha +2 \beta +3 \gamma ) A''(r)-15)\Big)\\&-\frac{4 e^{-A(r)} (10 \alpha +2 \beta +3 \gamma ) A''(r)}{27 A'(r)^2}.
\end{split}
\end{align}
Also,
\begin{align}
\begin{split}\label{dcdt}
\frac{dc(\lambda)}{d\lambda}&=\frac{ e^{-A(r)} (16\alpha+5\beta)}{162 A'(r)^5}\Big(-2 A^{(4)}(r) A'(r)+8 A^{(3)}(r) A''(r)-16 A^{(3)}(r) A'(r)^2\\&+15 A'(r)^3 A''(r)+48 A'(r) A''(r)^2\Big)-\frac{e^{-A(r)} A''(r)}{9 A'(r)^4}
\end{split}
\end{align}
and
\begin{equation}
c(\lambda)=\frac{2 (16 \alpha +5 \beta ) A^{(3)}(r)+A'(r) \left((16 \alpha +5 \beta ) \left(16 A''(r)+15 A'(r)^2\right)-6\right)}{162 A'(r)^4}.
\end{equation}
Notice that in (\ref{dcdt}) the term proportional to $16\alpha+5\beta$ has no possibility of having a definite sign and it does not cancel with terms in $E_{kk}$ so it is natural to set the coefficient to zero.
Now $\sqrt{h}\theta_{k}^4$ is positive and $T_{kk}$ using the null energy condition is positive. With $ 16\alpha+5\beta=0$, we find
\begin{equation}
E_{kk}= \frac{d}{d\lambda}\Big(-\frac{4 (10 \alpha +2 \beta +3 \gamma )}{27 A'(r)}\Big).
\end{equation}
where we have used $\frac{d}{d\lambda}= k^{a}\nabla_{a}.$
So we will have now,
\begin{equation}
\frac{dc(\lambda)}{d\lambda}=\frac{24 \pi G T_{kk}}{\sqrt{h}\theta_k^4}+\frac{d}{d\lambda}\Big(-\frac{4 (10 \alpha +2 \beta +3 \gamma )}{27 A'(r)}\Big),
\end{equation}
Then we can define a effective quantity,
\begin{equation}
\bar c(\lambda)= c(\lambda)+\frac{4 (10 \alpha +2 \beta +3 \gamma )}{27 A'(r)}
\end{equation}
such that,
\begin{equation}
\frac{d\bar c(\lambda)}{d\lambda}=\frac{24 \pi G T_{kk}}{\sqrt{h}\theta_k^4} \geq 0,
\end{equation}
Next evaluating $\bar c(\lambda)$ and multiplying both side by a factor of $-\frac{1}{27}$ we get,
\begin{equation}
\frac{d}{d\lambda}\Big(\frac{1-4 (10 \alpha +2 \beta +3 \gamma ) A'(r)^2}{A'(r)^3}\Big) \leq 0.
\end{equation}
Now in terms of the radial coordinate $r$ this becomes,
\begin{equation}
(k^{r}\partial_{r} + k^{\tau}\partial_{\tau} )\Big(\frac{1-4 (10 \alpha +2 \beta +3 \gamma ) A'(r)^2}{A'(r)^3}\Big) \leq 0.
\end{equation}
From this,
\begin{equation}
e^{-2 A(r)} \frac{d}{dr } \Big(\frac{1-4 (10 \alpha +2 \beta +3 \gamma ) A'(r)^2}{A'(r)^3}\Big) \geq0.
\end{equation}
So , $$ \bar c(r)=\frac{1-4 (10 \alpha +2 \beta +3 \gamma ) A'(r)^2}{A'(r)^3}$$ is a monotonically decreasing quantity along the RG flow, where the RG scale is determined by the radial coordinate $r$. At the fixed point the geometry is AdS and hence $$A(r)\equiv \frac{r}{l_{ads}}.$$ The $\bar c(r)$ coincides (upto an inconsequential overall factor) with the A-type anomaly coefficient at UV and IR (see eg. \cite{anoms} for expressions in general higher curvature theories). In this construction, the starting point of the entropy functional played an important role in reaching the condition $16\alpha+5 \beta=0$. In \cite{ctheorems, ctheorems2}, this condition emerged by demanding the existence of a ``simple" c-function. Our analysis gives another way of looking at the same condition. One may wonder if it is possible to come up with a scenario where we did not need to absorb $E_{kk}$ into the c-function like what we have done. For example, instead of the HEE functional, perhaps one could try to construct a functional that cancels the $E_{kk}$ on the {\it rhs} of the Raychaudhuri equation. This does not appear to be the case--starting with the Wald entropy functional instead of the HEE functional does not naturally lead to the $16\alpha+5 \beta=0$ condition.
\section{Discussion}
In this paper, we considered four derivative gravity and subjected it to the condition that the second law of black hole thermodynamics is satisfied at linear and quadratic orders in perturbation. We showed that the entropy functional is uniquely fixed (upto second order in extrinsic curvatures) and coincides with the holographic entropy functionals. We also derived interesting bounds on the coupling which arises at next to leading order in perturbations.
We conclude here with a brief discussion on possible future directions.
\begin{itemize}
\item One should consider more generic perturbations to see what happens at linear and second order. This is important since the possibility exists that for generic perturbations the second law holds only for very specific cases like the Lovelock theories. In \cite{camanho}, it was argued that causality constraints would require adding an infinite set of higher spin massive modes in a GB theory for consistency. It will be interesting to see if the second law knows about this in the perturbative sense used in the current work.
\item By considering asymptotically flat black holes and the Gauss-Bonnet theory, we found that for the negative GB coupling we get a minimum horizon radius for the second law to hold. Taking this to be a pathology, this appears to disfavour this sign of the coupling.
We further saw that the second law appears to know about the sound channel instablity in the context of Gauss-Bonnet holography. It will be interesting to see if $\lambda>-1/8$ which corresponds to the scalar channel instability \cite{Brigante} can also be seen from the second law.
\item One should consider how our analysis will change if there is matter coupling to the higher curvature terms. In this case, it is not clear how to define the null energy condition. It seems natural that the matter couplings will get constrained by demanding the second law.
\item Our analysis can be extended to general theories such as the quasi-topological theories \cite{quasitop}. In order to get a sufficient number of conditions to fix all the extrinsic curvature terms, one will need to turn on generic perturbations.
\item In case of gravitational Chern-Simons terms, it will be interesting to see if and how the validity of second law can fix the entropy functionals \cite{loga}. Since the ${\it rhs}$ of the Raychaudhuri equation does not know about topological terms, the linearized analysis will not be able to capture the effect of such terms. One may need to consider topology changing processes like black hole mergers \cite{Sarkar:2010xp} to see the effect of such terms.
\item It will be interesting to compare our analysis with what arises from similar considerations in the fluid-gravity correspondence \cite{shiraz}. Demanding a positive entropy current for higher curvature theories will lead to constraints which are presumably going to be similar to what we have found.
\item The connection between the second law constraints and the positivity of relative entropy \cite{rel} in the context of holography can also be explored. We found that when we consider the second law for topological black holes with zero mass, we recover the bound from the tensor channel causality constraint. Why is this happening? A plausible explanation is as follows. The topological black holes with zero mass are just AdS spaces written in different co-ordinates and the finite Wald entropy of these black holes can be interpreted as an entanglement entropy across a sphere \cite{ctheorems, chm} in the dual field theory. Thus the second law in this context may be related to the positivity of relative entropy due to a time dependent perturbation corresponding to the infalling matter we have considered. Relative entropy is expected to be positive for a unitary field theory. Hence there appears to be an interesting interplay between bulk causality and field theory unitarity. It will be interesting to make this connection more precise. It will also be interesting to understand the bounds in the context of c-theorems using the second law for causal horizons as advocated recently in \cite{sb}.
\end{itemize}
\section*{Acknowledgments} We thank Shamik Banerjee, Sayantani Bhattacharyya, Menika Sharma and Aron Wall for discussions. AB thanks IIT-Gandhinagar for hospitality during the course of this work. The research of SS is partially supported by IIT Gandhinagar start up grant No: IP/IITGN/PHY/SS/\\201415-12. AS acknowledges support from a Swarnajayanti fellowship, Govt. of India.
|
1,314,259,996,505 | arxiv | \section*{Movie description}
The video is available here: \href{http://ecommons.library.cornell.edu/bitstream/1813/14112/2/ristenpart_gfm_2009_hires.mpg}{High resolution
version} or \href{http://ecommons.library.cornell.edu/bitstream/1813/14112/3/ristenpart_gfm_2009.mpg}{Low resolution version}. The video
contains six segments, all displayed at 25 frames per second, and are arranged in order as follows.
\begin{enumerate}
\item This movie shows immediate coalescence at a low field strength. A droplet of water (10 mM KCl, nominal diameter = 1.4 mm) moves through
silicone oil (PDMS, 1000 cS). The black rectangular object at top is the tip of the electrode; the long black arc at bottom is the oil/water
meniscus (cf. Figure 1 in main text). The applied potential was 0.5 kV, yielding an approximate field strength of 100 V/mm. The capture frame
rate was 1000 frames/sec; the time lapse covers a period of 150 ms total.
\item This movie shows non-coalescence (bouncing) at a higher field strength. The setup is identical to that shown in Movie 1,
except the applied potential was 1.0 kV (approximate field strength of 200 V/mm). The capture frame rate was 1000 frames/sec; the time lapse
covers a period of 230 ms total.
\item This movie shows vigorous bouncing at an even higher field strength. The setup is identical to that shown in Movies 1 and 2,
except the applied potential was 3.0 kV (approximate field strength of 600 V/mm). The capture frame rate was 125 frames/sec; the time lapse covers
a period of 4000 ms total.
\item This movie shows two droplets bouncing back and forth in a high field strength. The setup is identical to that shown in Movies
1-3, with the addition of a smaller second droplet (both drops 1 M KCl, with nominal diameters of 1.2 mm and 1 mm for top and bottom drops
respectively). The applied potential was 3.0 kV (approximate field strength of 600 V/mm). The capture frame rate was 500 frames/sec; the time
lapse covers a period of 2000 ms total.
\item This movie shows a zoomed-in view of the immediate vicinity at the point of contact for a bouncing drop. The dark regions near the top and
bottom are water (1 M KCl) and the intervening bright region is oil (PDMS 1000 cS). The total image width is 0.45 mm. The lighter regions within
the water are optical artifacts due to reflection. At the point of contact a clear meniscus bridge is observed; the corresponding still frame is
shown in Fig. 3a in the main text. The applied potential was 1.0 kV (approximate field strength of 200 V/mm). The capture frame rate was 25,000
frames/sec; the time lapse covers a period of 28 ms total.
\item This movie shows an example of partial coalescence, where an oppositely charged `daughter' droplet is emitted from the point of contact. The
setup is similar to that shown in Supplementary Movie 3, except the viscosity of the silicone oil is decreased (100 cS) and the water salt
concentration is increased (1 M KCl). The applied potential was 3.0 kV (approximate field strength of 600 V/mm). The object at top-left is
the pipette tip. The capture frame rate was 1000 frames/sec; the time lapse covers a period of 700 ms total.
\end{enumerate}
\end{document}
|
1,314,259,996,506 | arxiv | \section{Introduction}\label{s1}
Different \jgrv{methods} to solve Partial differential equations (PDEs) with
and without boundary condition \jgrv{are} proposed in the literature. PDEs
are very common in different fields of engineering such as
thermodynamic, chemical process, wave analysis, etc. In some recently
developed control methodologies, the process of controller design is
reduced to finding the solution of such equations. Interconnection
and damping assignment passivity-based control is one of these
methods that is based on the general solution of some
PDEs~\cite{franco2019ida}. Providing new methodologies to find the
required solution of PDEs opens new horizon to design novel
controllers for many applications.
Passivity-based control (PBC) is a well-known methodology which was
first introduced in~\cite{ortega1989adaptive} to define a controller
design for stabilization through passivity. In this method, the
control objective is to stabilize the desired equilibrium point of
the system which is the minimum of a preselected storage function.
This method, which clearly reminiscent of standard Lyapunov
procedure, is successfully applied to simple mechanical
systems that can be stabilized by shaping merely the potential
energy~\cite{ortega2001putting}. For applications that requires
kinetic energy shaping, PBC may be used \cite{ortega2013passivity},
but the structure of the system in closed--loop will be changed and
the storage function of the passive map does not have the
interpretation of total energy anymore, while an unnatural stable
invertibility requirement is imposed to the system
\cite{ortega2002interconnection}.
In order to conquer this drawback and expand the range of
applications of PBC, an extended version of this method is proposed. In this version, a storage function is not
fixed at first, but the desired structure of the closed-loop system
such as port-controlled Hamiltonian (PCH) or Lagrangian, is
selected. Then all assignable energy functions which are applicable
to this structure are obtained via the solution of partial
differential equations. The most popular examples of this method are the interconnection and damping assignment
(IDA)~\cite{ortega2002interconnection} and
the controlled Lagrangian~\cite{bloch2000controlled}.
IDA-PBC reform the
closed-loop system to a Hamiltonian structure with three matrices
containing interconnection between subsystems, damping term and
kernel of input matrix. One of the most important advantages of port
Hamiltonian modeling is that it is based on energy exchange and
dissipation of the system, thus passive structure of the system is
visible. Readers are referred to~\cite{ortega2004interconnection}
and references therein for examples and other features of IDA-PBC.
The most difficulty of IDA-PBC, which restricts the application of
this method, \jgrv{is solving a set of PDEs called matching equations.}
Especially, in the case of underactuated robots; which have fewer
actuators than the system degrees of freedom (DOF); a nonlinear PDE
arises for kinetic energy shaping. In order to obviate this
difficulty, some solutions are reported in the literature. As
representatives, consider~\cite{gomez2001stabilization} that focuses
on robots with one degree of underactuation, where the inertia
matrix depends only on unactuated configuration. In the proposed
method the PDEs are reduced to a simple set of nonlinear ODEs that
are only solvable for some 2~DOF robots. a Similar method for PDE of
kinetic energy is reported in~\cite{
ortega2002stabilization}.
In~\cite{acosta2005interconnection} a method for transforming PDEs
to ODEs is proposed. This method is based on some restrictive
assumptions that reduces its application to few simple cases. Shaping the energy of port Hamiltonian
systems without solving PDEs is proposed in~\cite{
donaire2015shaping} for a special case of systems. Based on the results of this paper, in \cite{romero2016energy,romero2018global} energy shaping via PID controller is designed. The major disadvantage of last three mentioned papers is that they are merely applicable to the systems which satisfy some restrictive assumptions.
A
method to simplify the PDEs associated with the potential energy for
a class of underactuated mechanical systems is developed
in~\cite{ryalat2016simplified}. In~\cite{viola2007total}
simplification of kinetic energy PDE via change of coordinate is
analyzed. By considering these articles, one may argue
that proposing a general method which solves the matching equations
of any system, is a prohibitive task. Hence, new techniques with a
large domain of applicability are required to simplify the matching
equations especially, the PDE of kinetic energy shaping.
In this paper, a method to solve PDE of kinetic energy of mechanical
systems with one degree of underactuation is proposed. The
interconnection matrix is used as a free parameter in IDA-PBC to
transform the PDE with respect to the desired inertia matrix to some algebraic equations and a PDE with respect
to the unactuated coordinate. For this purpose, a special structure
for the desired inertia matrix is proposed to simplify the resulting
matching equation. \jgrv{By this means, the inverse of desired inertia matrix is designed such that merely the diagonal and the row and column corresponding to unactuated joint are nonzero. Note that one of the reason of difficulty of this PDE is that the desired inertia matrix is multipling into its partial difference. Hence, only one of the elements of this matrix is state-dependent, results in simplification of PDE. After this, and by suitably defining the free sub-block of interconnection matrix, the closed form of algebraic equations together with single PDE is derived. Three different cases are considered for the resulting PDE in which in two cases the solution may be found easily. We may invoke \cite{harandi2020solution} to solve the PDE in the other case.}
A necessary condition is also proposed to ensure
suitably selection of the desired inertial matrix, since this matrix
plays a crucial role in the stabilization of an unstable equilibrium
point. In other words, an inertia matrix should be designed which
leads to a desired potential energy with a positive Hessian matrix
at the desired equilibrium point.
The paper is organized as follows. Section~\ref{s2} presents
background of IDA-PBC method for underactuated mechanical systems. A
necessary condition and an algorithm for simplifying the PDE of
kinetic energy are proposed in section~\ref{s3}. Three examples are
represented in Section~\ref{s4}. Finally, concluding remarks and
future works are summarized in Section~\ref{s5}.
\textbf{Notation}: $I_n$ denotes
$n\times n$ identity matrix, $0_{m\times n}$ is $m\times n$ zero matrix and $0_n$
is a $n$ dimensional column vector of zeros. $x^{(i)}$ and $\xi^{(ij)}$ with
$x\in\mathbb{R}^n,\xi\in\mathbb{R}^{m\times n}$ denote $i$-th and $(i,j)$-th element
of $x$ and $\xi$, respectively.
$e_i\in\mathbb{R}^n$ with $i\in\bar{n}$ is the Euclidean basis vector where $\bar{n}=\{1,..,n\}$. Gradient of a scalar function $f(x)$ with $x\in\mathtt{R}^n$ which is denoted by $\nabla f$ is a column vector as $\nabla f=[\frac{\partial f(x)}{\partial x^{(1)}},...,\frac{\partial f(x)}{\partial x^{(n)}}]^T$.
\section{Review of IDA-PBC methodology for simple mechanical systems}\label{s2}
In here, IDA-PBC method for underactuated mechanical systems is
reviewed. The readers are referred to
\cite{ortega2002stabilization, acosta2005interconnection} for more
details. If it is assumed that the system has no natural damping,
the equations of motion may be written in the PCH form as
\begin{equation}
\label{1}
\begin{bmatrix}
\dot{q} \\ \dot{p}
\end{bmatrix}
=\begin{bmatrix}
0_{n\times n} & I_n \\ -I_n & 0_{n\times n}
\end{bmatrix}
\begin{bmatrix}
\nabla_q H \\ \nabla_p H
\end{bmatrix}
+\begin{bmatrix}
0_{n\times m} \\ G(q)
\end{bmatrix}
u,
\end{equation}
where $H(q,p)=1/2p^TM^{-1}(q)p+V(q)$ is total energy of the system,
$q,p\in R^n$ are generalized position and momenta, respectively,
$M^T(q)=M(q)>0$ is the inertia matrix, $V(q)$ is the potential
energy and rank of $G(q)$ is equal to $m<n$. Suppose that the
desired structure for $H_d$ is given as follows
\begin{equation*}
H_d(q,p)=1/2p^TM_d^{-1}(q)p+V_d(q),
\end{equation*}
where $M_d(q)$ and $V_d(q)$ represent the desired inertia matrix and
potential energy function, respectively, and it is required that the
desired equilibrium point $q_*$ satisfies $q_*=\text{arg min}
V_d(q)$. The desired interconnection matrix is also given as follows
\begin{equation*}
J_d(q,p)=
\begin{bmatrix}
0_{n\times n} & M^{-1}(q)M_d(q) \\ -M_d(q)M^{-1}(q) & J_2(q,p)
\end{bmatrix}
\end{equation*}
in which the skew-symmetric matrix $J_2(q,p)$ is a free design
parameter. It is possible to split the control into
$u=u_{es}(q,p)+u_{di}(q,p)$, in which
\begin{equation}
\label{3}
\begin{split}
&u_{es}=(G^TG)^{-1}G^T\big(\nabla_q H-M_dM^{-1}\nabla_q H_d+J_2M_d^{-1}p\big)\\
&u_{di}=-K_vG^T\nabla_p H_d
\end{split}
\end{equation}
with $K_v>0$. This restricts the desired damping matrix to have the
form of
\begin{equation*}
R_d(q)=
\begin{bmatrix}
0_{n\times n} & 0_{n\times n} \\ 0_{n\times n} & GK_vG^T
\end{bmatrix}
\end{equation*}
The closed-loop system takes the Hamiltonian form
\begin{equation}
\label{5}
\begin{bmatrix}
\dot{q} \\ \dot{p}
\end{bmatrix}
=
\begin{bmatrix}
0_{n\times n} & M^{-1}M_d \\ -M_dM^{-1} & J_2-GK_vG^T
\end{bmatrix}
\begin{bmatrix}
\nabla_q H_d \\ \nabla_p H_d
\end{bmatrix}
\end{equation}
the matching equations of the IDA-PBC can be separated into the
terms that depend on the kinetic and the potential energies, i.e.
the terms depend on $p$ and terms which are independent of $p$,
respectively. This leads to
\begin{subequations} \label{67}
\begin{align}
&G^\bot (q)\{\nabla_q \big(p^TM^{-1}(q)p\big)-M_dM^{-1}(q)\nabla_q \big(p^TM_d^{-1}(q)p\big)+2J_{2}M_d^{-1}p\}=0_s,\label{6}\\
&G^\bot (q)\{\nabla_q V(q)-M_dM^{-1}\nabla_q V_d(q)\}=0_s,\label{7}
\end{align}
\end{subequations}
where $G^\bot\in\mathbb{R}^{s\times n}$ is left annihilator of $G$
and $s=n-m$. Equation (\ref{6}) is a nonlinear PDE respect to
positive definite desired inertia matrix. Given $M_d$, equation
(\ref{7}) is a linear PDE with respect to the desired potential
energy. Therefore, the main difficulty of these PDEs is finding
analytical solution for equation (\ref{6}).
In the sequel, we focus on the PDE of kinetic energy. Note that the
proposed method works for the system with one degree of
underactuation. The aim is to solve PDE (\ref{6}) or propose a
methodology to simplify it. Invoking~\cite{crasta2015matching}, and
by considering a special form for $M_d$ and utilizing $J_2(q,p)$,
this PDE is transformed to some algebraic equations and a single PDE
in which the unknown parameter is the unactuated diagonal parameter
of $M_d^{-1}$. Notice that $M_d$ has a critical role to ensure
$q_*=\text{arg min} V_d(q)$. Regardless of the most previous
researches such as \cite{viola2007total,donaire2016simultaneous} that just focus on solving
equation (\ref{6}) without directly considering PDE (\ref{7}), Here
a necessary condition is proposed to restrict selection of $M_d$ to
conduce a suitable $V_d$.
\section{Main results}\label{s3}
In this section a constructive method with respect to PDE of
kinetic energy is proposed. To accomplish that, \jgrv{let us introduce} a condition
on the selection of $M_d$ as stated in the following proposition.
\begin{proposition}\label{pr1}\normalfont
Consider PDE (\ref{7}) and assume that $n-m=1$. If Hessian matrix $ \left.\frac{\partial^2 V_d}{\partial q^2}\right|_{q=q^*}$ is positive definite, then the following inequality holds
\begin{equation}
\label{8}
\left(G^\perp M_dM^{-1}\frac{\partial (G^\perp \nabla V)}
{\partial q}\right)_{q=q^*}>0.
\end{equation}
\carrew
\end{proposition}
\vspace{2mm} \textbf{Proof}:
Differentiate both side of PDE
(\ref{7}) respect to $q$:
\begin{equation}
\label{9}
\frac{\partial (G^\perp \nabla V)}{\partial q}=\Big(G^\perp M_dM^{-1}\frac{\partial^2 V_d}{\partial q^2}\Big)^T+\frac{\partial G^\perp M_dM^{-1}}{\partial q}\nabla V_d.
\end{equation}
Note that $\left.\nabla V_d\right|_{q=q^*}=0_n$. Thus, (\ref{9}) at $q=q^*$ is
\begin{equation*}
\left.\frac{\partial (G^\perp \nabla V)}{\partial q}\right|_{q=q^*}=\left.\Big(G^\perp M_dM^{-1}\frac{\partial^2 V_d}{\partial q^2}\Big)^T\right|_{q=q^*}=\Big(\left.\frac{\partial^2 V_d}{\partial q^2}(G^\perp M_dM^{-1})^T\Big)\right|_{q=q^*}
\end{equation*}
To complete the proof, multiply both side of above equation from
left to $\left.\big(G^\perp M_dM^{-1}\big)\right|_{q=q^*}$ and
notice that arbitrary matrix $A$ is positive definite if
$\xi^TA\xi>0$ for any $\xi\neq 0$.
\begin{flushright} \rule{2mm}{2mm} \end{flushright}
As explained before, we suppose that $m=n-1$. Thus, with a minor
loss of generality, suppose that \jgrv{$G=P[I_m,0_{m\times n-m}]^T$ with $P$ a permutation matrix which results in $G^\bot=e_k^T,k\in\bar{n}$}. Simplify PDE
(\ref{6}) term by term as follows. The first term is:
\begin{equation*}
\label{11}
G^\bot (q)\nabla_q \big(p^TM^{-1}(q)p\big)=p^T\frac{\partial M^{-1}}{\partial q^{(k)}}p
\end{equation*}
In the sequel, the following notations are used
\begin{equation}
\label{12}
M^{-1}=\frac{1}{\det{M}} \mathfrak{M}(q) \quad \implies \quad \frac{\partial M^{-1}}{\partial q^{(k)}}=\frac{1}{(\det{M})^2}\mathbb{M}(q)
\end{equation}
where $\mathfrak{M}\in \mathbb{R}^{n\times n}$ is adjugate matrix of
$M$ and $\mathbb{M}\in \mathbb{R}^{n\times n}$ is matrix of
nominator elements of $\frac{\partial M^{-1}}{\partial q^{(k)}}$.
Note that in spite of most previously reported research on this
topic, it is not assumed that $M(q)$ merely depends on some
specified configuration variables. \jgrv{Regard to second term of (\ref{6}),} it is assumed that
$M_d^{-1}(q)$ has the following structure:
\begin{equation}
\label{13}
M_d^{-1}=
\begin{bmatrix}
a_1 & 0 & \dots & b_1 & 0 & \dots & 0 \\
0 & a_2 & \dots & b_2 & 0 & \dots & 0 \\
\vdots & \vdots & & \vdots & & & 0 \\
b_1 & b_2 & \dots & a(q) & b_k & \dots & b_{n-1} \\
0 & 0 & \dots & b_k & a_k & \dots & 0 \\
\vdots & \vdots & & \vdots & & \ddots & \vdots\\
0 & 0 & \dots & b_{n-1} & 0 & \dots & a_{n-1}
\end{bmatrix}
\end{equation}
in which, all the elements of it are zero except diagonal elements,
and the $k$-th row and column. Notice that $a(q)$ is the only
element which is state dependent. In other words, since our aim is
to solve PDE of kinetic energy as simple as possible or at least
simplify it, $a_i$s and $b_i$s are considered to be constant. Notice
that the most important property of this structure is that $k$-th
row of adjugate matrix $\mathfrak{M}$ is independent of
configuration variables and $b_i$s.
In order to simplify second term of (\ref{6}), $G^\bot M_dM^{-1}$
is represented as follows
\begin{equation}
\label{14}
G^\bot M_dM^{-1}=\frac{1}{\det{M}\det{M_d^{-1}}}\gamma
\end{equation}
where $\gamma\in \mathbb{R}^n$ is a row vector independent of
$a(q)$. This is another advantage of the selected form of
(\ref{13}). determinant of ${M_d^{-1}}$ is
\begin{equation*}
\det{M_d^{-1}}=\phi_1a(q)+\phi_2,
\end{equation*}
where $\phi_1,\phi_2$ are constant parameters depending on other
elements of $M_d^{-1}$. Finally, second term of (\ref{6}) may be
reduced to
\begin{equation}
\label{15}
G^\bot M_dM^{-1}\nabla_q \big(p^TM_d^{-1}(q)p\big)=\frac{p^T\displaystyle\sum_{i=1}^{n}\bigg(\gamma^{(i)}\frac{\partial M_d^{-1}}{\partial q^{(i)}}\bigg)\hspace{1mm}p}{\det{M}\det{M_d^{-1}}}
\end{equation}
in which all elements of $\frac{\partial M_d^{-1}}{\partial
q^{(i)}}$ are zero except the $(k,k)$ element. Notice that if $M_d$
was selected like what is reported in previous
works~\cite{acosta2005interconnection, gomez2001stabilization} as a
function of only $q^{(k)}$, then the above equation reduces to
\begin{equation*}
\frac{\gamma^{(k)}p^T\frac{\partial M_d^{-1}}{\partial
q^{(k)}}p}{\det{M}\det{M_d^{-1}}}.
\end{equation*}
In order to simplify the last term of (\ref{6}), as reported in
\cite{acosta2005interconnection}, $J_2$ is linear with
respect to $p$. Therefore, $J_2$ can be parameterized in the
following form
\begin{equation}
\label{17}
J_2(q,p)=
\frac{1}{\det{M}}\begin{bmatrix}
0 & {p}^T\alpha_1(q) & \dots & {p}^T\alpha_{n-1}(q) \\
{p}^T\alpha_n(q) & 0 & \dots & {p}^T\alpha_{2n-2}(q) \\
\vdots & \vdots & \ddots & \vdots \\
{p}^T\alpha_{n^2-2n+2}(q) & {p}^T\alpha_{n^2-2n+3}(q) & \dots & 0
\end{bmatrix}
\end{equation}
where $\alpha_i\in \mathbb{R}^n, i\in \overline{n(n-1)}$. Note that
this form of $J_2$ is not generally skew-symmetric. However, only a
row of this matrix will be determined, thus, the column
corresponding to this row will be selected in such a way that $J_2$
becomes skew-symmetric. Other elements of this matrix are free
design parameters. Invoking \cite{acosta2005interconnection}, $J_2$
may be rewritten as follows
\begin{equation*}
J_2=\frac{1}{\det{M}}\displaystyle\sum_{i=1}^{n_0} {p}^T\alpha_i W_i, \qquad n_0=n(n-1),
\end{equation*}
where $W_i$ are matrices which are set as follows
\begin{equation*}
\begin{split}
W_1=&W^{1,2}, W_2=W^{1,3}, \dots, W_{n-1}=W^{1,n}, W_n=W^{2,1}, \dots, W_{n_{0}}=W^{n,n-1},
\end{split}
\end{equation*}
in which $W^{i,j}$ is a matrix such that all of its elements are
zero except the $(i,j)$ element which is equal to 1. Hence,
$G^{\perp}J_2$ can be written as follows
\begin{equation*}
G^{\perp}(q)J_2(p,q)=\frac{1}{\det{M}}{p}^T\mathcal{J}(q)A,\quad \mathcal{J}=
\begin{bmatrix}
\alpha_1 & \dots & \alpha_{n_0}
\end{bmatrix}
\in R^{n \times n_0},\quad A=
\begin{bmatrix}
(G^{\perp}W_1)^T & \dots & (G^{\perp}W_{n_0})^T
\end{bmatrix}^T
\in R^{n_0 \times n}.
\end{equation*}
Thus, $\mathcal{J}A$ may be written as:
\begin{equation*}
\begin{split}
\mathcal{J}A&=[\alpha_{(k-1)n-k+2},\dots,\alpha_{(k-1)n},0_n,\alpha_{(k-1)n+1},\dots,\alpha_{kn-k}]\triangleq B(q)\in\mathbb{R}^{n\times n}.
\end{split}
\end{equation*}
Notice that just one of the rows of $J_2$ appears in this equation. Finally,
third term in PDE (\ref{6}) is reduced to
\begin{equation}
\label{23}
G^\bot J_2(q,p)M_d^{-1}(q)p=\frac{1}{\det{M}}p^TBM_d^{-1}p
\end{equation}
All of the terms in (\ref{6}) are quadratic with respect to $p$ and
should be symmetric. Replacing (\ref{12}), (\ref{15}) and (\ref{23})
in (\ref{6}) results in the following relation:
\begin{equation}
\label{24}
\frac{\mathbb{M}}{\det{M}}-\frac{\displaystyle\sum_{i=1}^{n}\gamma^{(i)}\frac{\partial M_d^{-1}}{\partial q^{(i)}}}{\det{M_d^{-1}}}+(BM_d^{-1}+M_d^{-1}B^T)=0
\end{equation}
There are $\frac{n(n+1)}{2}$ equations and $n(n-1)$ free
parameters in above relation. At first, it seems that for $n\geq 3$
there is no need to calculate
$\displaystyle\sum_{i=1}^{n}\gamma^{(i)}\frac{\partial
M_d^{-1}}{\partial q^{(i)}}$. However, invoking Lemma2 in
\cite{acosta2005interconnection}, it is easy to show that rank of
$(BM_d^{-1}+M_d^{-1}B^T)$ is always $n-1$. It is also shown in
\cite{crasta2015matching} that the number of PDEs which should be
solved is $\frac{1}{6}s(s+1)(s+2)$. Therefore, equality (\ref{24})
leads to $\frac{n(n+1)}{2}-1$ algebraic equations and one PDE with
respect to $a(q)$. Notice that base on the structure of $M_d^{-1}$,
all the elements of the second term are zero except the $(k,k)$
element. Therefore, the $(k,k)$ element of equation (\ref{24})
leads to a PDE and other elements results in simple algebraic
equations. These algebraic equations are derived by some
manipulation as follows:
\begin{align}
\label{25}
&\frac{1}{\det{M}}
[\mathbb{M}^{(11)}, \mathbb{M}^{(12)}, \dots, \mathbb{M}^{(1n)}, \mathbb{M}^{(22)},\dots, \mathbb{M}^{((k-1)k)}, \mathbb{M}^{(k(k+1))}, \dots, \mathbb{M}^{(nn)}]^T\nonumber\\&= -\Psi
[\alpha_{(k-1)n-k+2}^{(1)},\alpha_{(k-1)n-k+2}^{(2)} \dots, \alpha_{(k-1)n-k+2}^{(n)}, \alpha_{(k-1)n-k+3}^{(1)}, \dots,\alpha_{(k-1)n-k+3}^{(n)},\dots, \alpha_{kn-k}^{(1)}, \dots \alpha_{kn-k}^{(n)}]^T
\end{align}
where
\begin{equation*}
\Psi=\begin{bmatrix}
\psi_1 & \dots & \psi_{n(n-1)}
\end{bmatrix}
\in \mathbb{R}^{\frac{n(n+1)-2}{2}\times n(n-1)}
\end{equation*}
\begin{equation*}
\begin{split}
&\psi_1=[2a_1, 0_{k-2}^T, b_1, 0_{\frac{n(n+1)}{2}-k-1}^T]^T,\qquad
\psi_2=[0, a_1, 0_{n+k-4}^T, b_1, 0_{\frac{n(n+1)}{2}-n-k}^T]^T,\qquad \dots \\
&\psi_{n+1}=[0, a_2,0_{k-3}^T, b_1, 0_{\frac{n(n+1)}{2}-k-1}^T]^T, \quad\dots\quad
\psi_{n(n-1)}=[0_{\frac{k(2n-k-1)}{2}-1}^T, b_{n-1},0_{\frac{n(n+1)-k(2n-k-1)}{2}-1}^T]^T,
\end{split}
\end{equation*}
in which $\alpha_i^{(j)}$ is the $j$-th element of vector
$\alpha_i$. Matrix $\Psi$ is generally full rank; therefore,
equation (\ref{25}) has at least one solution. The remaining PDE is
given by
\begin{equation}
\label{27}
\frac{\displaystyle\sum_{i=1}^{n}\gamma^{(i)}\frac{\partial a(q)}{\partial q^{(i)}}}{\det{M_d^{-1}}}-\frac{\mathbb{M}^{(kk)}}{\det{M}}-2\displaystyle\sum_{i=1}^{n-1} b_i\alpha_{(k-1)n-k+1+i}^{(k)}=0.
\end{equation}
Note that $a_i$s and $b_i$s should be determined such that $\Psi$ is full rank, $M_d^{-1}$ is positive definite and proposition \ref{pr1} is satisfied.
The following statements can be verified for (\ref{24}):
\begin{itemize}
\item If the assumption of \cite{gomez2001stabilization} holds, i.e. $M$ is
only function of unactuated coordinate $q^{(k)}$, the second term of (\ref{24})
is reduced to $\frac{\gamma^{(k)}\frac{\partial M_d^{-1}}{\partial q^{(k)}}}
{\det M_d^{-1}}$ and PDE (\ref{27}) is replaced by the following ODE
\begin{equation}
\label{28}
\begin{split}
&\frac{\gamma^{(k)}\frac{d a(q^{(k)})}{d q^{(k)}}}{\phi_1a(q^{(k)})+\phi_2}=\frac{\mathbb{M}^{(kk)}}{\det{M}}+2\displaystyle\sum_{i=1}^{k-1} b_i\alpha_{(k-1)n-k+1+i}^{(k)}=f(q^{(k)})
\end{split}
\end{equation}
Analytic solution of this ODE is
\begin{equation}
\label{29}
\begin{split}
a(q^{(k)})=\frac{\lambda e^{\phi_1F(q^{(k)})}-\phi_2}{\phi_1}
\end{split}
\end{equation}
where $\lambda$ is a constant parameter and $F=\int
\frac{f}{\gamma^{(k)}} dq^{(k)}$. Note that in the method proposed
in \cite{gomez2001stabilization}, the obtained ODEs is generally a
set of ODE and has analytic solution if $n=2$.
\item \label{it2} If $\mathbb{M}^{(kk)}$ is equal to zero, then the second term
in (\ref{27}) is omitted. Hence, it is easy to select $a(q)$ with respect to free
parameters. Special cases of this condition is considered in
\cite{acosta2005interconnection} where an analytic solution is proposed in
which $a(q)$ is merely \jgrv{a function of one of the} $q^{(i)}$s. In the proposed solution,
since the aim is to derive a simple solution, $M_d$ will be considered to be a
constant matrix such that Proposition~\ref{pr1} is satisfied.
Note that if
$M$ is constant, we can choose a constant value for $M_d$ without considering
a special form for $G(q)$. The proposed IDA-PBC in \cite{d2006further} for acrobot is an example of this case.
\item In other cases, a PDE should be solved. One may invoke the methods
proposed in~\cite{acosta2009pdes,viola2007total} to simplify it.
Another powerful method is using Pfaffian differential equations
detailed in~\cite{harandi2020solution,sneddon2006elements}. Base on this method, the
corresponding Pfaffian equations to PDE (\ref{27}) are
\begin{equation*}
\frac{\det {M_d^{-1}}dq^{(1)}}{\gamma^{(1)}}=\dots=\frac{\det
{M_d^{-1}}dq^n}{\gamma^{(n)}}=\frac{d
a}{\frac{\mathbb{M}^{(kk)}}{\det{M}}+2\displaystyle\sum_{i=1}^{n-1}
b_i\alpha_{(k-1)n-k+1+i}^{(k)}}
\end{equation*}
In \cite{harandi2020solution} (see also
\cite[ch.2]{sneddon2006elements}) some tips are proposed to solve
Pfaffian differential equations.
\end{itemize}
In the next section, some illustrative case studies are examined to
show the applicability of proposed method. Note that similar to
\cite{viola2007total,donaire2016simultaneous} we only concentrate on the matching equation
related to kinetic energy.
\section{Case Studies}\label{s4}
In the following three case studies is proposed to verify the three
above mentioned statements. The first example is Pendubot in which
its matching equation is replaced by an ODE. The second example is
VTOL aircraft where the corresponding PDE is solved easily by a
constant $M_d$. The last example is 2D SpiderCrane in which its
matching equation is solved by Pffafian differential equations.
\subsection{Pendubot} \label{pen}
Pendubot is a 2R planar serial robot in which the first joint is
only actuated. In~\cite{sandoval2008interconnection} an IDA-PBC
controller is designed for this robot by suitably defining new
variables. In here the PDE of kinetic energy shaping is solved
systematically. Inertia matrix and potential energy of this robot
are given by:
\begin{equation}
\label{45}
M=
\begin{bmatrix}
c_1+c_2+2c_3\cos(q^{(2)}) & c_2+c_3\cos(q^{(2)}) \\ c_2+c_3\cos(q^{(2)}) & c_2
\end{bmatrix},\qquad
V=c_4g\cos(q^{(1)})+c_5g\cos(q^{(1)}+q^{(2)}),\qquad G=\begin{bmatrix}
1 \\ 0
\end{bmatrix},
\end{equation}
with $c_i$s defined in \cite{sandoval2008interconnection}. After
some manipulation, the following expressions are obtained
\begin{equation}
\label{46}
\begin{split}
&\det{M}=c_1c_2-c_3^2\cos^2(q^{(2)}), \qquad \mathfrak{M}=
\begin{bmatrix}
c_2 & -c_2-c_3\cos(q^{(2)}) \\ -c_2-c_3\cos(q^{(2)}) & c_1+c_2+2c_3\cos(q^{(2)})
\end{bmatrix},\\
&\mathbb{M}=
\begin{bmatrix}
-2c_3^2\sin(q^{(2)})\cos^2(q^{(2)}) & \begin{array}{c} c_1c_2c_3\sin(q^{(2)})+c_3^3\sin(q^{(2)})\cos^2(q^{(2)})\\+2c_2c_3^2\sin(q^{(2)})\cos(q^{(2)})\end{array} \\ \begin{array}{c} c_1c_2c_3\sin(q^{(2)})+c_3^3\sin(q^{(2)})\cos^2(q^{(2)})\\+2c_2c_3^2\sin(q^{(2)})\cos(q^{(2)})\end{array} & \begin{array}{c} -2c_1c_2c_3\sin(q^{(2)})-2c_3^3\sin(q^{(2)})\cos^2(q^{(2)})\\-2c_3^2(c_1+c_2)\sin(q^{(2)})\cos(q^{(2)}) \end{array}
\end{bmatrix},\\
&M_d^{-1}=
\begin{bmatrix}
a_1 & b_1 \\ b_1 & a(q^{(2)})
\end{bmatrix},\quad J_2=
\begin{bmatrix}
0 & p^T\alpha_1 \\ p^T\alpha_2 & 0
\end{bmatrix},\quad
\gamma^T=\begin{bmatrix}-c_2b_1-c_2a_1-c_3a_1\cos(q^{(2)}) \\ a_1c_1+a_1c_2+b_1c_2+b_1c_3\cos(q^{(2)})+2a_1c_3\cos(q^{(2)}) \end{bmatrix}
\end{split}
\end{equation}
Note that in the following $\alpha_2$ will be determined and $\alpha_1=-\alpha_2$ will be set.
Equation (\ref{24}) for this case is derived as follows
\begin{equation}
\label{47}
\frac{\gamma^{(2)}}{\det{M_d^{-1}}}
\begin{bmatrix}
0 & 0 \\ 0 & \frac{\partial a}{\partial q^{(2)}}
\end{bmatrix}
=\frac{1}{\det{M}}\mathbb{M}+
\begin{bmatrix}
2\alpha_{2}^{(1)}a_1 & \alpha_{2}^{(2)}a_1+\alpha_{2}^{(1)}b_1 \\ \alpha_{2}^{(2)}a_1+\alpha_{2}^{(1)}b_1 & 2\alpha_{2}^{(2)}b_1
\end{bmatrix}
\end{equation}
By solving two algebraic equations, $\alpha_2$ is obtained as follows
\begin{equation}
\label{48}
\begin{split}
&\alpha_{2}^{(1)}=-\frac{\mathbb{M}^{(11)} }{2a_1\det{M}}=\frac{c_3^2\sin(q^{(2)})\cos^2(q^{(2)})}{a_1\big(c_1c_2-c_3^2\cos^2(q^{(2)})\big)},\\
&\alpha_{2}^{(2)}=-\frac{\mathbb{M}^{(21)} }{a_1\det{M}}-\frac{\alpha_{2}^{(1)}b_1}{a_1}=-\frac{c_1c_2c_3\sin(q^{(2)})+c_3^3\sin(q^{(2)})\cos^2(q^{(2)})+2c_2c_3^2\sin(q^{(2)})\cos(q^{(2)})}{a_1\big(c_1c_2-c_3^2\cos^2(q^{(2)})\big)}\\&-\frac{b_1c_3^2\sin(q^{(2)})\cos^2(q^{(2)})}{a_1^2\big(c_1c_2-c_3^2\cos^2(q^{(2)})\big)}
\end{split}
\end{equation}
Finally, the following ODE should be solved
\begin{equation}
\label{49}
\frac{1}{a_1a(q^{(2)})-b_1^2}\frac{da}{dq^{(2)}}=\frac{1}{\gamma^{(2)}\det{M}}\bigg(\mathbb{M}^{(22)}-\frac{-2b_1\mathbb{M}^{(21)}}{a_1}+\frac{b_1^2\mathbb{M}^{(11)}}{a_1^2}\bigg)
\end{equation}
This ODE is in the form of (\ref{28}) and its solution is derived from (\ref{29}) with
\begin{equation*}
\phi_1=b_1,\quad \phi_2=-b_1^2,\quad F(q^{(2)})=\int \frac{1}{\gamma^{(2)}\det{M}}\bigg(\mathbb{M}^{(22)}-\frac{-2b_1\mathbb{M}^{(21)}}{a_1}+\frac{b_1^2\mathbb{M}^{(11)}}{a_1^2}\bigg) dq^{(2)}.
\end{equation*}
For example, assume that $c_1=4, c_2=1$ and $c_3=1.5 $. By some
manipulation, $a(q^{(2)})$ is obtained as follows
\begin{equation}
\label{52}
a(q^{(2)})=\cos(q^{(2)})^{-7/3}+(4-3\cos(q^{(2)}))^{49/6}-(4+3\cos(q^{(2)}))^{-7/2},
\end{equation}
where $a_1=1,b_1=-5,\lambda=1$ are chosen to simplify the ODE (\ref{49}) and also the necessary condition (\ref{8}) is satisfied. The solution of potential energy PDE is proposed in Appendix.
\subsection{VTOL Aircraft}
Dynamic model of VTOL in PCH form (\ref{1}) is given as follows
\begin{equation}
\label{26}
G(q)=
\begin{bmatrix}
-\sin(\theta) & \epsilon\cos(\theta) \\
\cos(\theta) & \epsilon\sin(\theta) \\
0 & 1
\end{bmatrix},\qquad\qquad
M=I, \qquad \qquad
V=gy,\qquad \qquad q=\begin{bmatrix}
x\\y\\ \theta
\end{bmatrix}
\end{equation}
where, $x$ and $y$ denote the position of center of mass, $\theta$
is the roll angle and $\epsilon$ models the effect of the slopped
wings. The desired equilibrium point of the system is
$[x_*,y_*,0]^T$. In \cite{acosta2005interconnection} a controller
with state-dependent $M_d$ by defining new inputs is derived. Since the
inertia matrix is constant, it is possible to solve the PDE of
kinetic energy with a constant $M_d$, represented by:
\begin{equation*}
M_d=
\begin{bmatrix}
a & d & e \\ d & b & f \\ e & f & c
\end{bmatrix}
\end{equation*}
Necessary condition (\ref{8}) in this case leads to following
inequality
\begin{equation*}
\left.\Big(g\cos(\theta)(\epsilon\cos(\theta)+f\sin(\theta)-c\epsilon)\Big)\right|_{\theta=0}>0.
\end{equation*}
A suitable choice for the matrix parameters is
\begin{equation*}
a=\kappa\epsilon^2,\quad b=1,\quad c=\kappa',\quad d=0,\quad e=\epsilon,\quad f=0,
\end{equation*}
where the constants $\kappa,\kappa'>0$ should be selected such that
$\kappa\kappa'>1$. Note that $M_d=I$ does not satisfy the necessary
condition (\ref{8}) which is in line with our prior knowledge that
it is not possible to stabilize the system with merely potential
energy shaping. Although solving the potential energy PDE (\ref{7})
is out of scope of this paper, but in this case its solution with
$\kappa=20$ and $\kappa'=0.1$ is derived as follows
\begin{equation*}
\begin{split}
&V_d=\Big(\epsilon(y-y^*)+\ln\big(\epsilon\cos(\theta)-0.1\epsilon\big)\Big)^2+\Big(\frac{1}{20\epsilon}(x-x^*)-(\theta-\theta^*)-0.1\text{arctanh}\big(1.1055\tan(\frac{\theta}{2})\big)\Big)^2\\&-2\epsilon\ln(0.9\epsilon)(y-y^*)-\frac{g-2\epsilon\ln(0.9\epsilon)}{g\epsilon}\ln\big(\epsilon\cos(\theta)-0.1\epsilon\big).
\end{split}
\end{equation*}
\begin{figure}[b]
\centering
\subfigure[The stabilization errors and control effots with proposed controller.]{
\includegraphics[width=.45\linewidth]{f1}
\label{p31}
}\hspace{1mm}
\subfigure[The motion of VTOL aircraft in plene.]{
\includegraphics[width=.45\linewidth]{f2}
}
\label{p32}
\caption{Simulation results of proposed controller on VTOL aircraft. The aircraft moves toward its desired position with smooth states and control law.}
\label{p3}
\end{figure}
\begin{wrapfigure}{r}{0.5\textwidth}
\vspace{-1pt}
\begin{center}\includegraphics[scale=0.4]{MRD}
\end{center}
\vspace{-15pt}
\caption{\label{fig:Sch} Schematic of 2D SpiderCrane system.}
\vspace{-8pt}
\end{wrapfigure}
To simulate the response, consider the initial condition as
$q(0)=[6,-5,-1]^T$ with zero velocity while the desired position is
$q_*=[0,0,0]^T$. Set, $\epsilon=0.3$ and $K_v=\mbox{diag}\{1,0.5\}$.
Simulation results are illustrated in Fig.~\ref{p3}. As shown in
Fig.~\ref{p31}, the errors converge to zero in about 20 second with
an acceptable control efforts amplitude. The motion of the robot in
$X-Y$ plane is depicted in Fig.~\ref{p31}. Due to coupling of
inputs, the aircraft first moves to farther position to correct its
orientation and then goes to the desired position. Note that the
advantage of the proposed controller in comparison with that
reported in \cite{acosta2005interconnection} is its simplicity.
\subsection{2D SpiderCrane}
This system consists of a load suspended from a ring which is
controlled by two cables. The schematic of this system is
depicted in Fig.~\ref{fig:Sch}. The position of the ring and the
mass are denoted by $(x_r,y_r)$ and $(x,y)$, respectively, and their
mass is denoted by $M$ and $m$, respectively. The length of the
controlled cables is denoted by $l_1$ and $l_2$, while $l_3$ denotes
the fixed length of the cable between ring and the mass. Dynamic
equation of the system is in the form (\ref{1}) with following
parameters
\begin{equation*}
\hspace{-2mm}
q=\begin{bmatrix}
x_r \\ y_r\\ \theta
\end{bmatrix},\quad
G=\begin{bmatrix}
1 & 0 \\ 0 & 1 \\ 0 & 0
\end{bmatrix}^T,\quad
V=(M+m)gy_r-mgl_3\cos(\theta), \quad
M(q)=\begin{bmatrix}
M+m & 0 & ml_3\cos(\theta) \\
0 & M+m & ml_3\sin(\theta) \\
ml_3\cos(\theta) & ml_3\sin(\theta) & ml_3^2
\end{bmatrix}.
\end{equation*}
Two IDA-PBC controller have been designed for SpiderCrane. In
\cite{kazi2008stabilization} merely the potential energy is shaped
while in \cite{sarras2010total} total energy shaping method proposed
in \cite{acosta2005interconnection} is used such that first a
partial feedback linearization is applied to the system and then a
desired inertia matrix which is merely a function of $\theta$ is
chosen. In here, the aim is to derive a more general solution such
that $M_d$ may be set as a function of $x_r$ and $y_y$. Consider
$M_d^{-1}$ in the form of (\ref{13}). One can easily check that
necessary condition (\ref{8}) is satisfied if $b_1ml_3+a_2(M+m)>0$.
In order to solve matching equation (\ref{6}), the following
parameters are derived
\begin{equation*}
\begin{split}
&\det M(q)=(M+m)^2ml_3^2-(M+m)m^2l_3^2,\quad \\&\mathfrak{M}=\begin{bmatrix}
(M+m)ml_3^2-m^2l_3^2\sin^2(\theta) & m^2l_3^2\sin(\theta)\cos(\theta) & -(m+M)ml_3\cos(\theta) \\ m^2l_3^2\sin(\theta)\cos(\theta) & (M+m)ml_3^2-m^2l_3^2\cos^2(\theta) & -(M+m)ml\sin(\theta) \\ -(m+M)ml_3\cos(\theta) & -(M+m)ml\sin(\theta) & (M+m)^2
\end{bmatrix},
\\&\mathbb{M}=\det M(q)\begin{bmatrix}
-2m^2l_3^2\sin(\theta)\cos(\theta) & m^2l_3^2\cos(2\theta) & (M+m)ml_3\sin(\theta) \\ m^2l_3^2\cos(2\theta) & m^2l^2\sin(2\theta) & -(M+m)ml_3\cos(\theta) \\ (M+m)ml_3\sin(\theta) & -(M+m)ml_3\cos(\theta) & 0
\end{bmatrix},\\&
\gamma^T=\begin{bmatrix}
-a_2b_1(M+m)ml_3^2+a_2b_1m^2l_3^2\sin^2(\theta)-a_1b_2m^2l_3^2\sin(\theta)\cos(\theta)-a_1a_2(M+m)ml_3\cos(\theta) \\ -a_2b_1m^2l_3^2\sin(\theta)\cos(\theta)-a_1b_2(M+m)ml_3^2+a_1b_2m^2l_3^2\cos^2(\theta)-a_1a_2(M+m)ml_3\sin(\theta) \\ a_2b_1(M+m)ml_3\cos(\theta)+a_1b_2(M+m)ml_3\sin(\theta)+a_1a_2(M+m)^2
\end{bmatrix}
\end{split}
\end{equation*}
Based on necessary condition and simplifying the corresponding
matching equation, we choose $b_1=b_2=0$. By this means, the matrix
$\Psi$ in equality (\ref{25}) is in the following form
\begin{equation*}
\Psi=\begin{bmatrix}
2a_1 & 0 & 0 & 0 & 0 & 0 \\
0 & a_1 & 0& a_2& 0& 0 \\
0 & 0 & a_1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 2a_2 & 0 \\
0 & 0 & 0 & 0 & 0 & a_2
\end{bmatrix}
\end{equation*}
This matrix is full rank, hence $[\alpha_5^T,\alpha_6^T]^T$ is
determined by right pseudo-inverse of $\Psi$. The Pfaffian
differential equations of PDE (\ref{27}) for this system is given as
follows
\begin{equation*}
\frac{dx}{-ml_3\cos(\theta)}=\frac{dy}{-ml_3\sin(\theta)}=\frac{d\theta}{M+m}=\frac{d a}{0}
\end{equation*}
The solutions to these equations are
\begin{equation*}
x+\frac{ml_3}{M+m}\sin(\theta)=c_1,\qquad\qquad y-\frac{ml_3}{M+m}\cos(\theta)=c_2,
\end{equation*}
with $c_1$ and $c_2$ as free parameters.
Invoking \cite{harandi2020solution} $a(q)$ is
\begin{equation*}
a(q)=\phi\Big(x+\frac{ml_3}{M+m}\sin(\theta),y-\frac{ml_3}{M+m}\cos(\theta)\Big),
\end{equation*}
where $\phi$ is an arbitrary function. General form of $V_d$ is proposed in Appendix.
\section{Conclusions and Future Prospects}\label{s5}
In this paper a systematic method to simplify the matching equation
related to kinetic energy shaping for underactuated robots with one
degree of underactuation was proposed. A special structure of
desired inertia matrix was considered in such a way that just one of
its elements depends on configuration variables. By this means, the
arisen PDE can be analytically solved for robots with \jgrv{some properties including
manipulators} with inertia matrix depending on just one variable. The
proposed method was successfully implemented on VTOL aircraft,
pendubot and 2D SpiderCrane. Extension of this method to robots with
more degrees of underactuation and also consideration of potential
energy PDE are currently being examined in our research group.
\section*{Appendix}
Here, the solution of potential energy PDE for pendubot and 2D SpiderCrane is proposed. PDE (\ref{7}) for pendubot based on $M_d$ designed in section~\ref{pen} is in the following form
\begin{equation}
\label{54}
-15\sin(q_1+q_2)\det{M}\det{M_d^{-1}}=(4-1.5\cos(q_2))\frac{\partial V_d}{\partial q_1}-4.5\cos(q_2)\frac{\partial V_d}{\partial q_2}
\end{equation}
Homogeneous solution of this PDE is derived as follows
\begin{equation*}
V_d=\phi\bigg(\int \frac{4-1.5\cos(q_2)}{-4.5\cos(q_2)}dq_2-q_1\bigg)=\phi\bigg(-\frac{8}{9}\ln(\frac{1+sin(q_2)}{cos(q_2)})+\frac{1}{3}q_2-q_1\bigg).
\end{equation*}
In order to derive non-homogeneous solution of (\ref{54}), we may consider $V_d$ in the following form
\begin{equation*}
V_d=g_1(\theta_2)\sin(\theta_1)+g_2(\theta_2)\cos(\theta_1).
\end{equation*}
Hence, PDE \ref{54} is equal to following set of ODEs
\begin{equation*}
\begin{split}
f_3(q_2)g_1(q_2)+f_4(q_2)\frac{d g_2(q_2)}{dq_2}=f_2(q_2),\qquad
f_4(q_2)\frac{d g_1(q_2)}{d q_2}-f_3(q_2)g_2(q_2)=f_1(q_2),
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
&f_1=-15\cos(q_2)\det{M}\det{M_d^{-1}}, \qquad f_2=-15\sin(q_2)\det{M}\det{M_d^{-1}}, \\
&f_3=4-1.5\cos(q_2), \qquad\qquad\qquad\hspace{.4cm} f_4=-4.5cos(q_2).
\end{split}
\end{equation*}
For the case of 2D SpiderCrane, one can easily shown that PDE (\ref{7}) is in the following form
$$a(q)\det M mgl_3\sin(\theta)=-(M+m)ml_3\cos(\theta)\frac{\partial V_d}{\partial x}-(M+m)ml_3\sin(\theta)\frac{\partial V_d}{\partial y}+(M+m)^2\frac{\partial V_d}{\partial \theta}.$$
For simplicity, consider $a(q)=y-\frac{ml_3}{M+m}\cos(\theta)$. Notice that $M_d$ will be positive definite by suitable defining the center of coordinate. By this means, the above PDE yields
$$a(q)\det M mgl_3\sin(\theta)=-(M+m)ml_3\sin(\theta)\frac{\partial V_d}{\partial y}+(M+m)^2\frac{\partial V_d}{\partial \theta}.$$
We rewrite it in the following compact form
$$\sin(\theta)x+\sin^2(\theta)=\beta_1\frac{\partial V_d}{\partial y}+\beta_2\frac{\partial V_d}{\partial \theta},$$
which has the following solution
$$V_d=\phi\big(\beta_2y-\beta_1\cos(\theta)\big)+\frac{y\cos(\theta)}{\beta_1}-\frac{(\beta_1+\beta_2)y^2}{2\beta_1^2}.$$
\bibliographystyle{model1-num-names}
|
1,314,259,996,507 | arxiv | \section{Introduction}\label{S:intro}
Let ${\mathbb D}$ be the open unit disk,
let $dA$ be normalized area measure on ${\mathbb D}$,
and let $\hol({\mathbb D})$ be the space of holomorphic functions on ${\mathbb D}$.
Given a non-negative function $\omega\in L^1({\mathbb D})$ and $f\in\hol({\mathbb D})$,
we define
\[
{\cal D}_\omega(f):=\int_{\mathbb D}|f'(z)|^2\,\omega(z)\,dA(z).
\]
The \emph{weighted Dirichlet space} ${\cal D}_\omega$
is the set of $f\in\hol({\mathbb D})$ with ${\cal D}_\omega(f)<\infty$.
Obviously, if $\omega\equiv1$,
then ${\cal D}_\omega$ is just the classical Dirichlet space ${\cal D}$.
In this article we shall be mainly interested in the case
where $\omega$ is a superharmonic weight.
This class of weights was introduced by Aleman \cite{Al93}.
For such $\omega$, we automatically have $\omega\in L^1({\mathbb D})$, and,
provided that $\omega\not\equiv0$,
we also have ${\cal D}_\omega\subset H^2$, the Hardy space.
It is customary to define
\[
\|f\|_{{\cal D}_\omega}^2:=\|f\|_{H^2}^2+{\cal D}_\omega(f)
\qquad(f\in{\cal D}_\omega),
\]
making ${\cal D}_\omega$ a Hilbert space.
The class of superharmonic weights includes two important subclasses:
\begin{itemize}
\item the power weights
$\omega(z):=(1-|z|^2)^\alpha ~(0\le\alpha\le 1)$,
which form a scale linking the classical Dirichlet space ($\alpha=0$)
to the Hardy space ($\alpha=1$).
\item the harmonic weights, introduced by Richter \cite{Ri91} in connection with
his analysis of shift-invariant subspaces of the classical Dirichlet space.
\end{itemize}
Superharmonic weights have the important property that
the dilates of $f_r(z):=f(rz)$ of each function $f\in{\cal D}_\omega$ satisfy
\begin{equation}\label{E:frineq}
{\cal D}_\omega(f_r)\le C{\cal D}_\omega(f)
\qquad(0\le r<1),
\end{equation}
where $C$ is an absolute constant.
From this, it is not hard to deduce that
$\|f_r-f\|_{{\cal D}_\omega}\to0$ as $r\to1^-$,
and that polynomials are dense in ${\cal D}_\omega$.
Inequality \eqref{E:frineq} was first proved by Richter and Sundberg \cite{RS91}
in the case where $\omega$ is harmonic, with $C=4$.
It was generalized to superharmonic weights by Aleman \cite{Al93},
with an improved constant $C=5/2$.
For harmonic weights,
the constant was further improved to $C=1$ by Sarason \cite{Sa97}.
The basis of Sarason's method was to show that,
if $\omega=P_\zeta$ (the Poisson kernel at a point $\zeta\in{\mathbb T}$),
then it is possible to identify ${\cal D}_\omega$ as
a de Branges--Rovnyak space ${\cal H}(b)$, the norms being identical
(we shall define ${\cal H}(b)$ later).
Working within ${\cal H}(b)$,
one can deduce that \eqref{E:frineq} holds with $C=1$,
at least for these special $\omega$.
Inequality \eqref{E:frineq} for general harmonic $\omega$
then follows easily by using an averaging argument.
It turns out that Sarason's construction has a sort of converse.
The only harmonic weights $\omega$ for which
${\cal D}_\omega$ can be identified as a de Branges--Rovnyak space
are multiples of $P_\zeta$, where $\zeta\in{\mathbb T}$.
This was proved in \cite{CGR10}.
(One can also study a weaker notion of `identified',
where the spaces ${\cal D}_\omega$ and ${\cal H}(b)$ are allowed to carry different norms.
This appears to be more subtle: see \cite{CR13}.)
Our purpose in this paper is twofold.
First, we shall extend Sarason's result
by exhibiting a new family of superharmonic weights $\omega$
for which ${\cal D}_\omega$ can be identified as a de Branges--Rovnyak space ${\cal H}(b)$.
We shall deduce from this that
the inequality \eqref{E:frineq} holds with $C=1$ for all superharmonic weights.
Second, we shall prove a converse result,
characterizing those superharmonic weights $\omega$
for which ${\cal D}_\omega$ is equal to an ${\cal H}(b)$.
We shall also consider what happens for more general weights.
\section{Superharmonic weights}
Let $\omega$ be a positive superharmonic function on ${\mathbb D}$.
By standard results from potential theory,
$\omega$ is locally integrable on ${\mathbb D}$,
and $(1/r^2)\int_{|z|\le r}\omega\,dA$ is
a decreasing function of $r$ for $0<r<1$.
It follows that $\omega\in L^1({\mathbb D})$,
so it is admissible as a Dirichlet weight.
Now suppose that $\omega\not\equiv0$.
We shall show that ${\cal D}_\omega\subset H^2$.
By the minimum principle $\omega(z)>0$ for all $z\in{\mathbb D}$
(possibly infinite at some points).
By lower semicontinuity,
$m:=\inf_{|z|\le1/e}\omega(z)$ is attained and therefore $m>0$.
By the minimum principle again,
$\omega(z)\ge m\log(1/|z|)$ for all $z$ with $1/e<|z|<1$.
Since the Dirichlet space with weight $\log(1/|z|)$ is just $H^2$,
it follows easily that ${\cal D}_\omega\subset H^2$, as claimed.
It thus makes sense to define the norm
$\|\cdot\|_{{\cal D}_\omega}$ on ${\cal D}_\omega$ by
\begin{equation}\label{E:Dnormdef}
\|f\|_{{\cal D}_\omega}^2:=\|f\|_{H^2}^2+{\cal D}_\omega(f)
\qquad(f\in{\cal D}_\omega).
\end{equation}
With respect to this norm, ${\cal D}_\omega$ is a Hilbert space.
We shall make extensive use of the following representation formula
for positive superharmonic functions $\omega$ on ${\mathbb D}$.
Given such an $\omega$,
there exists a unique positive finite Borel measure $\mu$
on $\overline{{\mathbb D}}$ such that, for all $z\in{\mathbb D}$,
\begin{equation}\label{E:murep}
\omega(z)
=\int_{\mathbb D} \log\Bigl|\frac{1-\overline{\zeta}z}{\zeta-z}\Bigr|\frac{2}{1-|\zeta|^2}\,d\mu(\zeta)
+\int_{\mathbb T}\frac{1-|z|^2}{|\zeta-z|^2}\,d\mu(\zeta).
\end{equation}
(This is the usual decomposition of $\omega$ as a potential plus a harmonic function.)
We then write ${\cal D}_\mu$ for ${\cal D}_\omega$,
and further ${\cal D}_\zeta$ for ${\cal D}_{\delta_\zeta}$.
Thus, for $f\in\hol({\mathbb D})$,
\[
{\cal D}_\zeta(f)=
\begin{cases}
\displaystyle\int_{\mathbb D} \log\Bigl|\frac{1-\overline{\zeta}z}{\zeta-z}\Bigr|\frac{2}{1-|\zeta|^2}|f'(z)|^2\,dA(z),
&\zeta\in{\mathbb D},\\
\displaystyle\int_{\mathbb D}\frac{1-|z|^2}{|\zeta-z|^2}|f'(z)|^2\,dA(z),
&\zeta\in{\mathbb T}.
\end{cases}
\]
For general $\mu$, we can recover ${\cal D}_\mu(f)$ from ${\cal D}_\zeta(f)$ via Fubini's theorem:
\begin{equation}\label{E:average}
{\cal D}_\mu(f)=\int_{\overline{{\mathbb D}}}{\cal D}_\zeta(f)\,d\mu(\zeta).
\end{equation}
Finally, we shall need the following Douglas-type formula for ${\cal D}_\zeta(f)$.
\begin{theorem}\label{T:Douglas}
Let $f\in H^2$ and let $\zeta\in\overline{{\mathbb D}}$.
If $\zeta\in{\mathbb D}$, then
\begin{equation}\label{E:Douglas}
{\cal D}_\zeta(f)=
\frac{1}{2\pi}\int_{\mathbb T} \Bigl|\frac{f(\lambda)-f(\zeta)}{\lambda-\zeta}\Bigr|^2\,|d\lambda|.
\end{equation}
If $\zeta\in{\mathbb T}$ and $f(\zeta):=\lim_{r\to1^-}f(r\zeta)$ exists,
then the same formula holds.
Otherwise ${\cal D}_\zeta(f)=\infty$.
\end{theorem}
For a proof, and more background on superharmonic weights, see \cite{Al93}.
\section{De Branges--Rovnyak spaces}\label{S:deBR}
The de Branges--Rovnyak spaces
are a family of (not necessarily closed) subspaces ${\cal H}(b)$ of $H^2$,
para\-metrized by elements $b$ of the closed unit ball of $H^\infty$.
They were introduced by de Branges and Rovnyak
in the appendix of \cite{dBR66a}
and further studied in \cite{dBR66b}.
For background information we refer to the books of
de Branges and Rovnyak \cite{dBR66b}, Sarason \cite{Sa94},
and the forthcoming monograph of Fricain and Mashreghi \cite{FM15}.
Given $\psi\in L^\infty({\mathbb T})$, we define
the Toeplitz operator $T_\psi:H^2\to H^2$ by
\[
T_\psi f:=P_+(\psi f) \qquad(f\in H^2),
\]
where $P_+:L^2({\mathbb T})\to H^2$ is
the orthogonal projection of $L^2({\mathbb T})$ onto $H^2$.
Clearly $T_\psi$ is a bounded operator on $H^2$,
and its adjoint is $T_{\overline{\psi}}$.
Given $b\in H^\infty$
with $\|b\|_{H^\infty}\le1$,
the \emph{de Branges--Rovnyak space}
${\cal H}(b)$ is defined as the image of $H^2$
under the operator $(I-T_ bT_{\overline b})^{1/2}$.
We put a norm on ${\cal H}(b)$ making $(I-T_bT_{\overline{b}})^{1/2}$
a partial isometry from $H^2$ onto ${\cal H}(b)$, namely
\[
\|(I-T_bT_{\overline{b}})^{1/2}f\|_{{\cal H}(b)}:=\|f\|_{H^2}
\qquad(f\in H^2\ominus \ker(I-T_bT_{\overline{b}})^{1/2})).
\]
We have defined ${\cal H}(b)$ as in Sarason's book \cite{Sa94}.
The original definition of de Branges and Rovnyak,
based on the notion of complementary space,
is different but equivalent.
An explanation of the equivalence can be found in \cite[pp.7--8]{Sa94}.
A third approach is to start from the positive kernel
\[
k_w^b(z):=\frac{1-\overline{b(w)}b(z)}{1-\overline{w}z}
\qquad(z,w\in{\mathbb D}),
\]
and to define ${\cal H}(b)$ as
the reproducing kernel Hilbert space associated with this kernel.
The theory of ${\cal H}(b)$-spaces is pervaded by a fundamental dichotomy,
namely whether $b$ is or is not an extreme point
of the closed unit ball of $H^\infty$.
This is illustrated by following result.
\begin{theorem}\label{T:nonextreme}
Let $b\in H^\infty$ with $\|b\|_{H^\infty}\le 1$.
The following are equivalent:
\begin{itemize}
\item[\rm(i)] $b$ is a non-extreme point of the closed unit ball of $H^\infty$;
\item[\rm(ii)] $\log (1-|b|^2)\in L^1({\mathbb T})$;
\item[\rm(iii)] ${\cal H}(b)$ contains
all functions holomorphic in a neighborhood of $\overline{{\mathbb D}}$.
\end{itemize}
Furthermore, if $b$ is non-extreme, then the polynomials are dense in ${\cal H}(b)$.
\end{theorem}
\begin{proof}
The equivalence between (i) and (ii) is proved in \cite[Theorem~7.9]{Du00}.
That (i) implies (iii) is proved in \cite[\S IV-6]{Sa94},
and that (iii) implies (i) follows from \cite[\S V-1]{Sa94}.
Finally, the density of polynomials when $b$ is non-extreme
is proved in \cite[\S IV-3]{Sa94};
a constructive proof of this result is given in \cite{EFKMR15}.
\end{proof}
Henceforth we shall simply say that $b$ is `extreme' or `non-extreme',
it being understood that this relative to the closed unit ball of $H^\infty$.
From the equivalence between (i) and (ii), it follows that,
if $b$ is non-extreme,
then there exists a unique outer function $a$
such that $a(0)>0$ and $|a|^2+|b|^2=1$ a.e.\ on ${\mathbb T}$
(see \cite[\S IV-1]{Sa94}).
We shall call $(b,a)$ a \emph{pair}.
The following result gives
a useful characterization of ${\cal H}(b)$ in this case.
\begin{theorem}[\protect{\cite[\S IV-1]{Sa94}}]\label{T:f+}
Let $b$ be non-extreme,
let $(b,a)$ be a pair and let $f\in H^2$.
Then $f\in{\cal H}(b)$ if and only if
$T_{\overline b}f\in T_{\overline a}(H^2)$.
In this case, there exists a unique function $f^+\in H^2$
such that $T_{\overline b}f=T_{\overline a}f^+$, and
\begin{equation}\label{E:f+}
\|f\|_{{\cal H}(b)}^2=\|f\|_{H^2}^2+\|f^+\|_{H^2}^2.
\end{equation}
\end{theorem}
Given a pair $(b,a)$,
the function $\phi:=b/a$ is the quotient of two functions in $H^\infty$,
the denominator being outer.
In other words, $\phi\in N^+$, the Smirnov class.
Conversely, given $\phi\in N^+$, we can write $\phi=b/a$,
where $a,b\in H^\infty$ and $a$ is outer.
Multiplying $a$ and $b$ by an appropriately chosen outer function,
we may further ensure that $|a|^2+|b|^2=1$ a.e.\ on ${\mathbb T}$
and that $a(0)>0$,
in other words, that $(b,a)$ is a pair.
Then $a$ and $b$ are uniquely determined.
There is thus a bijection $b\leftrightarrow \phi$
between non-extreme functions $b$
and elements $\phi$ of the Smirnov class.
Note that $b$ and $\phi$ have the same inner factor.
Also $\phi$ is bounded if and only if $\|b\|_{H^\infty}<1$.
In this case, $(I-T_bT_{\overline{b}})$ is an invertible operator on $H^2$,
and consequently ${\cal H}(b)=H^2$ as vector spaces,
although the norms may be different.
\section{Extension of Sarason's theorem}
Our goal in this section is to establish the following theorem
and examine one of its consequences.
\begin{theorem}\label{T:Dmu=Hb}
Let $\zeta\in\overline{{\mathbb D}}$,
and let $b\in H^\infty$ be the function
corresponding to $\phi(z):=z/(1-\overline{\zeta}z)$.
Then ${\cal D}_\zeta={\cal H}(b)$ with equality of norms.
\end{theorem}
\begin{remarks}
(1) This theorem extends Sarason's result \cite{Sa97},
which is the special case $\zeta\in{\mathbb T}$.
(2) A computation shows that,
with $\phi(z)=z/(1-\overline{\zeta}z)$, we have
\[
b(z)=\frac{z}{A-Bz}
\qquad\text{and}\qquad
a(z)=\frac{1-\overline{\zeta}z}{A-Bz},
\]
where
\[
A:=\Bigl(\frac{2+|\zeta|^2+\sqrt{4+|\zeta|^4}}{2}\Bigr)^{1/2}
\quad\text{and}\quad
B:=\Bigl(\frac{2+|\zeta|^2-\sqrt{4+|\zeta|^4}}{2}\Bigr)^{1/2}\frac{\overline{\zeta}}{|\zeta|}.
\]
However, we do not need the precise formulas for $b$ and $a$ in what follows.
\end{remarks}
\begin{proof}
As the result is already known for $\zeta\in{\mathbb T}$,
we shall concentrate on the case $\zeta\in{\mathbb D}$.
In this case, ${\cal D}_\zeta$ and ${\cal H}_b$ are both equal to $H^2$ as vector spaces,
with equivalent norms.
For ${\cal D}_\zeta$ this follows from Theorem~\ref{T:Douglas},
and for ${\cal H}(b)$ we already observed this to be true whenever $\phi$ is bounded.
The content of the theorem is that
the norms on ${\cal D}_\zeta$ and ${\cal H}(b)$ are in fact identical.
For $w\in{\mathbb D}$, let $k_w$ denote the Cauchy kernel, namely
\[
k_w(z):=\frac{1}{1-\overline{w}z}
\qquad(z\in{\mathbb D}).
\]
It is enough to prove that
$\langle f,g\rangle_{{\cal D}_\zeta}=\langle f,g\rangle_{{\cal H}(b)}$
when $f$ and $g$ are finite linear combinations of Cauchy kernels,
since such $f,g$ are dense in $H^2$.
By sesquilinearity, this reduces to checking that
$\langle k_{w_1},k_{w_2}\rangle_{{\cal D}_\zeta}=\langle k_{w_1},k_{w_2}\rangle_{{\cal H}(b)}$
for all $w_1,w_2\in{\mathbb D}$. As both sides of this last equation
are holomorphic in $w_2$ and antiholomorphic in $w_1$,
it is sufficient to prove it in the case when $w_1=w_2$.
Thus we need to show that
$\|k_w\|_{{\cal D}_\zeta}^2=\|k_w\|_{{\cal H}(b)}^2$ for all $w\in{\mathbb D}$.
By \eqref{E:Dnormdef} and \eqref{E:f+},
this amounts to verifying that
${\cal D}_\zeta(k_w)=\|k_w^+\|_{H^2}^2$ for all $w\in{\mathbb D}$,
which we now proceed to do.
On the one hand, by Theorem~\ref{T:Douglas}, we have
\[
{\cal D}_\zeta(k_w)
=\frac{1}{2\pi}\int_{\mathbb T} \Bigl|\frac{k_w(\lambda)-k_w(\zeta)}{\lambda-\zeta}\Bigr|^2\,|d\lambda|
=\Bigl\|\frac{\overline{w}}{1-\overline{w}\zeta}k_w\Bigr\|_{H^2}^2.
\]
The Cauchy kernel is the reproducing kernel for $H^2$,
so $\langle f,k_w\rangle_{H^2}=f(w)$ for all $f\in H^2$,
and in particular $\|k_w\|_{H^2}^2=k_w(w)=1/(1-|w|^2)$.
Hence
\[
{\cal D}_\zeta(k_w)=\frac{|w|^2}{|1-\overline{w}\zeta|^2(1-|w|^2)}.
\]
On the other hand,
by a standard property of adjoints of multiplication operators acting on reproducing kernels,
we have $T_{\overline{h}}k_w=\overline{h(w)}k_w$
for all $h\in H^\infty$ and $w\in{\mathbb D}$.
In particular this is true when $h=b$ and when $h=a$,
where $(b,a)$ is the pair with $b/a=\phi$.
In the notation of Theorem~\ref{T:f+},
it follows that $k_w^+=\overline{\phi(w)}k_w$, whence
\[
\|k_w^+\|_{H^2}^2=|\phi(w)|^2\|k_w\|_{H^2}^2
=\frac{|w|^2}{|1-\overline{w}\zeta|^2(1-|w|^2)}.
\]
Thus $\|k_w^+\|_{H^2}^2={\cal D}_\zeta(k_w)$ for all $w\in{\mathbb D}$, as desired.
\end{proof}
Using the same basic idea as in \cite{Sa97},
we can use this theorem to deduce that \eqref{E:frineq} holds
for all superharmonic weights $\omega$
with constant $C=1$.
In fact, just as in \cite{Sa97}, we have the following even stronger result.
\begin{theorem}
Let $\mu$ be a finite positive measure on $\overline{{\mathbb D}}$
and let $f\in{\cal D}_\mu$. Then
\[
{\cal D}_\mu(f_r)\le \frac{2r}{1+r}{\cal D}_\mu(f) \qquad(0\le r< 1).
\]
\end{theorem}
\begin{proof}
We prove the result for $\mu=\delta_\zeta~(\zeta\in\overline{{\mathbb D}})$.
The general case follows by integrating up and using \eqref{E:average}.
By Theorem~\ref{T:Dmu=Hb},
we have ${\cal D}_\zeta(f)=\|f^+\|_{{\cal H}_b}^2$,
where $(b,a)$ is the pair for which
$\phi(z):=b(z)/a(z)=z/(1-\overline{\zeta}z)$.
So we need to show that, with this choice of $\phi$, we have
\[
\|(f_r)^+\|_{H^2}^2\le \frac{2r}{1+r}\|f^+\|_{H^2}^2.
\]
Given $h\in H^2$, we have
\[
\langle (f_r)^+,ah\rangle_{H^2}
=\langle f_r,bh\rangle_{H^2}
=\langle f,b_rh_r\rangle_{H^2}
=\langle f^+,(\phi_r/\phi)a_rh_r\rangle_{H^2},
\]
and thus
\[
|\langle (f_r)^+,ah\rangle_{H^2}|
\le \|f^+\|_{H^2}\|\phi_r/\phi\|_{H^\infty}\|ah\|_{H^2}.
\]
As $a$ is an outer function, $aH^2$ is dense in $H^2$, and so
\[
\|(f_r)^+\|_{H^2}\le\|\phi_r/\phi\|_{H^\infty}\|f^+\|_{H^2}.
\]
Finally, an elementary computation shows that
\[
\|\phi_r/\phi\|_{H^\infty}=r(1+|\zeta|)/(1+r|\zeta|)\le 2r/(1+r),
\]
whence the result.
\end{proof}
\section{A converse result}
Theorem~\ref{T:Dmu=Hb} furnishes a list of couples $(\mu,b)$
for which ${\cal D}_\mu={\cal H}(b)$.
In this section we prove a converse, which shows that,
apart from scalar multiples taken in a natural sense,
these are the only such couples.
It generalizes a result of \cite{CGR10},
where it was proved in the case when
$\mu$ is a measure on ${\mathbb T}$.
Note that we may assume from the outset that $b$ is
a non-extreme point of the closed unit ball of $H^\infty$.
Indeed, if we are to have ${\cal D}_\mu={\cal H}(b)$, then,
since ${\cal D}_\mu$ contains all functions holomorphic
on a neighborhood of $\overline{{\mathbb D}}$,
the same must be true of ${\cal H}(b)$.
By Theorem~\ref{T:nonextreme}, this entails that $b$ is non-extreme.
\begin{theorem}\label{T:converse}
Let $\mu$ be a finite positive measure on $\overline{{\mathbb D}}$ with $\mu\not\equiv0$.
Let $(b,a)$ be pair and let $\phi:=b/a$.
Then ${\cal D}_\mu={\cal H}(b)$ with equality of norms
if and only if there exist
$\zeta\in\overline{{\mathbb D}}$ and $\alpha\in{\mathbb C}\setminus\{0\}$ such that
\[
\mu=|\alpha|^2\delta_\zeta
\qquad\text{and}\qquad
\phi(z)=\alpha z/(1-\overline{\zeta}z).
\]
\end{theorem}
\begin{proof}
The `if' part follows directly from Theorem~\ref{T:Dmu=Hb}.
For the `only if', we observe that,
if we have equality of norms,
then $\|f^+\|_{H^2}^2={\cal D}_\mu(f)$ for all $f$ in the space,
in particular for $f=k_w$, the Cauchy kernels.
Performing similar calculations
to those in the proof of Theorem~\ref{T:Dmu=Hb},
we obtain the relation
\begin{equation}\label{E:phieqn}
|\phi(w)|^2=|w|^2\int_{\overline{{\mathbb D}}} \frac{d\mu(z)}{|1-z\overline{w}|^2}
\qquad(w\in{\mathbb D}).
\end{equation}
In particular $\phi(0)=0$.
If we write $\phi(w)/w$ as a Taylor series,
substitute it into the formula above,
expand in powers of $w$ and $\overline{w}$,
and equate coefficients,
then we end up with the following relation between the moments of $\mu$:
\begin{equation}\label{E:moment}
\mu(\overline{{\mathbb D}})\int_{\overline{{\mathbb D}}}z^n \overline{z}^{m}\,d\mu(z)
=\int_{\overline{{\mathbb D}}}z^n \,d\mu(z)\int_{\overline{{\mathbb D}}} \overline{z}^{m}\,d\mu(z)
\qquad(m,n\ge0).
\end{equation}
In particular, we have
$\mu(\overline{{\mathbb D}})(\int|z|^2\,d\mu)=(\int z\,d\mu)(\int\overline{z}\,d\mu)$.
This can be re-written as
$\int|z-\zeta|^2\,d\mu(z)=0$,
where $\zeta:=(\int z\,d\mu)/\mu(\overline{{\mathbb D}})\in\overline{{\mathbb D}}$.
It follows that $\mu=c\delta_\zeta$ for some $c>0$.
Substituting this back into formula \eqref{E:phieqn}, we find that
$|\phi(w)|^2=c|w|^2/|1-\overline{\zeta}w|^2$,
and so $\phi(w)=\alpha w/(1-\overline{\zeta}w)$,
where $\alpha$ is a complex constant with $|\alpha|^2=c$.
This completes the proof.
\end{proof}
\section{More general weights}
We have identified those superharmonic weights $\omega$ for which
${\cal D}_\omega$ is equal to a de Branges--Rovnyak space ${\cal H}(b)$.
What about more general weights?
In this section we obtain some results in this direction,
leading to questions that we think are interesting in their own right.
We begin with a result about which functions $b$ can arise in this context.
\begin{theorem}\label{T:binner}
Let $b$ be in the closed unit ball of $H^\infty$.
If there exists a weight $\omega$
with $0<\|\omega\|_{L^1({\mathbb D})}<\infty$
such that ${\cal H}(b)={\cal D}_\omega$, with equality of norms,
then $b$ is non-extreme and its inner factor is exactly $z$.
Consequently the inner factor of the associated function $\phi$ is also $z$.
\end{theorem}
\begin{remark}
This result is in stark contrast with what happens
if we do not insist on equality of norms.
See \cite[Theorem~4.5]{CR13}.
\end{remark}
\begin{proof}
That $b$ is non-extreme is proved just as in
the remark preceding the proof of Theorem~\ref{T:converse}.
We then have $\|f^+\|_{H^2}^2={\cal D}_\omega(f)$ for all $f\in {\cal H}(b)$.
In particular $f^+=0$ if and only if ${\cal D}_\omega(f)=0$. Now,
\[
f^+=0 \iff T_{\overline{b}}f=0 \iff f\perp bH^2 \iff f\perp b_iH^2,
\]
where $b_i$ is the inner factor of $b$.
Also, since $\|\omega\|_{L^1({\mathbb D})}>0$,
we have
\[
{\cal D}_\omega(f)=0 \iff f \text{~is a constant~}\iff f\perp zH^2.
\]
Combining these remarks, we conclude that $b_iH^2=zH^2$,
whence $b_i=z$.
The final statement of the theorem is a consequence of the fact,
remarked in \S\ref{S:deBR},
that $b$ and $\phi$ have the same inner factor.
\end{proof}
We next characterize those weights $\omega$ for which
${\cal D}_\omega$ is equal to some de Branges--Rovnyak space ${\cal H}(b)$.
There is no harm in normalizing $\omega$ so that $\|\omega\|_{L^1({\mathbb D})}=1$.
To state our result we need to introduce some notation.
Given $\psi\in L^1({\mathbb D})$,
we denote by $Q\psi$ its \emph{Bergman projection}, given by
\[
Q\psi(z):=\int_{\mathbb D} \frac{\psi(w)}{(1-\overline{w}z)^2}\,dA(w)
\qquad(z\in{\mathbb D}),
\]
and by $B\psi$ its \emph{Berezin transform}, defined by
\[
B\psi(z):=\int_{\mathbb D} \frac{(1-|z|^2)^2}{|1-\overline{w}z|^4}\psi(w)\,dA(w)
\qquad(z\in{\mathbb D}).
\]
For further information about these, we refer to \cite{HKZ00}.
\begin{theorem}
Let $\omega$ be a weight with $\|\omega\|_{L^1({\mathbb D})}=1$.
Then there exists $b$ such that ${\cal D}_\omega={\cal H}(b)$,
with equality of norms,
if and only if:
\begin{itemize}
\item[\rm(i)] ${\cal D}_\omega\subset H^2$,
\item[\rm(ii)] polynomials are dense in ${\cal D}_\omega$, and
\item[\rm(iii)] $Q\omega$ is outer and satisfies
\begin{equation}\label{E:QBeqn}
(1-|z|^2)|Q\omega(z)|^2 =B\omega(z)
\qquad(z\in{\mathbb D}).
\end{equation}
\end{itemize}
In this case the associated $\phi$ is given by
$\phi(z)=cz(Q\omega)(z)$, where $c\in{\mathbb T}$.
\end{theorem}
\begin{proof}
First we prove the `only if'.
If ${\cal D}_\omega={\cal H}(b)$, then
clearly ${\cal D}_\omega\subset H^2$.
Also, as remarked in the previous theorem,
$b$ is non-extreme,
so by Theorem~\ref{T:nonextreme} polynomials are dense in ${\cal H}(b)$,
and thus also in ${\cal D}_\omega$.
This proves (i) and (ii).
For (iii), note that equality of norms implies
$\|k_z^+\|_{H^2}^2={\cal D}_\omega(k_z)$ for all $z\in{\mathbb D}$,
where once again $k_z$ denotes the Cauchy kernel.
This yields the identity
\begin{equation}\label{E:phiomega}
\frac{|\phi(z)|^2}{1-|z|^2}
=\int_{\mathbb D}\frac{|z|^2}{|1-\overline{z}w|^4}\omega(w)\,dA(w)
\qquad(z\in{\mathbb D}).
\end{equation}
By Theorem~\ref{T:binner},
we have $\phi(z)=z\phi_o(z)$,
where $\phi_o$ is outer.
It follows that
\[
|\phi_o(z)|^2=\int_{\mathbb D}\frac{1-|z|^2}{|1-\overline{z}w|^4}\omega(w)\,dA(w)
\qquad(z\in{\mathbb D}).
\]
In particular $|\phi_o(0)|^2=\|\omega\|_{L^1({\mathbb D})}=1$.
Also, polarizing, we obtain
\[
\phi_o(z_1)\overline{\phi_o(z_2)}
=\int_{\mathbb D}\frac{1-z_1\overline{z}_2}{(1-\overline{w}z_1)^2(1-\overline{z}_2w)^2}\omega(w)\,dA(w)
\qquad(z_1,z_2\in{\mathbb D}).
\]
Setting $z_2=0$, we find that $\phi_o=c(Q\omega)$, where $|c|=1$.
Hence $Q\omega$ is outer,
and substituting this back into \eqref{E:phiomega} gives \eqref{E:QBeqn}.
This establishes (iii),
and also shows that $\phi(z)=cz(Q\omega)(z)$.
Now we prove the `if'.
Suppose that $\omega$ satisfies (i), (ii) and (iii).
Define $\phi(z):=z(Q\omega)(z)$.
By (iii), the function $\phi$ belongs to the Smirnov class $N^+$,
so it can be written as $\phi=b/a$ for some pair $(b,a)$.
We claim that ${\cal D}_\omega={\cal H}(b)$ with equality of norms.
Property~(i) implies that the norm $\|\cdot\|_{{\cal D}_\omega}$ is well-defined.
Property~(ii) implies that the Cauchy kernels $k_z~(z\in{\mathbb D})$
span a dense subspace of ${\cal D}_\omega$
(as they do for ${\cal H}(b)$).
It thus suffices to establish equality of norms,
and by the same argument as in the proof of Theorem~\ref{T:Dmu=Hb},
it is enough to prove that
${\cal D}_\omega(k_z)=\|k_z^+\|_{H^2}^2$ for all $z\in{\mathbb D}$.
This boils down to showing that \eqref{E:phiomega} holds,
which, with our definition of $\phi$,
is equivalent to equation \eqref{E:QBeqn} of Property~(iii).
This establishes our claim and completes the proof.
\end{proof}
The last theorem obviously begs the question as to
which weights satisfy properties (i), (ii) and (iii).
In particular:
\begin{question}
Which weights $\omega$ obey the relation \eqref{E:QBeqn}?
\end{question}
\noindent
According to Theorem~\ref{T:Dmu=Hb},
equation \eqref{E:QBeqn} is satisfied by the weights
\[
\omega_\zeta(z):=
\begin{cases}
\displaystyle\frac{1-|z|^2}{|\zeta-z|^2},
&\zeta\in{\mathbb T},\\
\displaystyle\log\Bigl|\frac{1-\overline{\zeta}z}{\zeta-z}\Bigr|\frac{2}{1-|\zeta|^2},
&\zeta\in{\mathbb D},
\end{cases}
\]
as can also be verified by direct calculation.
Are there any other solutions?
A possible source of examples are weights that can be expressed as
the difference of two positive superharmonic functions.
Such weights are given by the formula \eqref{E:murep},
where now $\mu$ is a finite \emph{signed} measure on $\overline{{\mathbb D}}$.
Part of the argument of the proof of Theorem~\ref{T:converse} carries over to this case,
showing that $\mu$ must still satisfy the moment relation \eqref{E:moment}.
This raises another question:
\begin{question}
Which signed measures $\mu$ on $\overline{{\mathbb D}}$ satisfy the relation \eqref{E:moment}?
\end{question}
\noindent
Obviously this is the case if $\mu$ is a multiple of a Dirac measure. Are there any others?
\subsection*{Acknowledgment}
Part of this research was carried out
during a Research-in-Teams meeting
at the Banff International Research Station (BIRS).
We thank BIRS for its hospitality.
|
1,314,259,996,508 | arxiv | \section{Introduction}
Launched March 2009, the \textit{Kepler} mission continuously observed a field in the sky centered on the Cygnus-Lyra region with the primary goal of detecting (small) exoplanets, by photometrically measuring planetary transits to a high level of precision \citep{borucki2008}. Apart from a growing list of confirmed planets (currently 152), the \textit{Kepler} catalog contains 3548 planetary candidates \citep{batalha2013}. The order of magnitude difference between those numbers illustrates the intrinsic difficulty of exoplanet confirmation.
Stars showing transit-like features are termed Kepler Objects of Interest (KOIs). Here we study KOI-42 (KIC 8866102, HD 175289, subsequently refered to as Kepler-410), which shows transit-like features consistent with a small planet ($R_\textrm{p} \approx 2.6 $ R$_\oplus$) on a relatively long orbit \citep[17.83 d;][]{borucki2011}. Apart from the bright host star (\textit{Kepler} magnitude $K_\textrm{p}$ = 9.4) Kepler-410 also consists of a fainter blended object \citep[$K_\textrm{p}$ = 12.2, ][]{adams2012}. We refer to this object as Kepler-410B, while we use Kepler-410A for the bright host star. The brightness of the system would make it a prime target for follow-up studies, if it can be confirmed that the transits are indeed occurring around Kepler-410A. Unfortunately the added complexity due to the presence of Kepler-410B, and the presumably small mass of the planet candidate, has so far prevented the planetary candidate to be confirmed as planet, or shown to be a false positive.
In this paper we will show that the transit-like features are indeed caused by a planet orbiting Kepler-410A. For this we combine information from the well-determined transit shape with additional (ground-based) observations and \textit{Spitzer} measurements. We also take advantage of Kepler-410 being almost exclusively observed in \textit{Kepler}'s short-cadence mode \citep[sampling it every 58.8 s,][]{borucki2008}, which allows for the detection of solar-like oscillations. Analyzing the stellar pulsations aids the confirmation of Kepler-410A as planet host and leads to accurate determination of the stellar parameters. We further measure the stellar rotation and its inclination by analyzing the pulsation modes. Such an analysis was recently carried out for Kepler-50 and Kepler-65 by \citet{2013ApJ...766..101C}.
In \S~\ref{sec:asteroseismology}, we describe the asteroseismic modeling before we present the various arguments that validate Kepler-410A b as a planet in \S~\ref{sec:planetary_validation}. The planetary and orbital parameters are presented in \S~\ref{sec:planetary_parameters}. We discuss the characteristics of the the system in \S~\ref{sec:discussion} and our conclusions are presented in \S~\ref{sec:conclusion}.
\section{Stellar properties from asteroseismology}
\label{sec:asteroseismology}
Kepler-410 was observed in short-cadence mode for the entire duration of the \textit{Kepler} mission, except during the second quarter of observations (Q2) where the long cadence mode was used. The latter observations are not included in the asteroseismic analysis, and we use short-cadence simple aperture photometry (SAP) data from Q0-Q1 and Q3-Q13. Before using the data as input for asteroseismology, it is de-trended and normalized using a specifically designed median filter to remove all transit features from the time series. The resulting time series is then used to derive a power spectrum\footnote{The power spectrum was calculated using a sine-wave fitting method \citep[see, e.\,g.,][]{1992PhDT.......208K,1995A&A...301..123F} which is normalized according to the amplitude-scaled version of Parseval's theorem \citep[see, e.\,g.,][]{1992PASP..104..413K}, in which a sine wave of peak amplitude, A, will have a corresponding peak in the power spectrum of $\rm A^2$.}, which is shown in \fref{fig:power_spectrum}.
\begin{figure*}[ht]
\centering
\includegraphics[scale=0.48]{figure1.png}
\caption{ \emph{ \footnotesize Power spectrum of Kepler-410 (gray). Overlain are the model fits (\eqref{eq:limitspec}) obtained from the MCMC peak-bagging. The black curve gives the model when including modes from the range $1370-2630\, \rm \mu Hz$ - all mode frequencies in this range were included in the stellar modeling. The red curve gives the model obtained when excluding the five outermost modes obtained in the first fit (black curve) in each end of the frequency scale. From this fit we get the estimates of the stellar inclination and frequency splitting.}}
\label{fig:power_spectrum}
\end{figure*}
\subsection{Asteroseismic frequency analysis}
The extraction of mode parameters for the asteroseismic analysis was performed by \emph{Peak-bagging} the power spectrum \citep[see, e.\,g.,][]{2003Ap&SS.284..109A}. This was done by making a global optimization of the power spectrum using an \emph{Markov Chain Monte Carlo} (MCMC) routine\footnote{The program StellarMC was used, which was written and is maintained by Rasmus Handberg.}, including a parallel tempering scheme to better search the full parameter space \citep[see][]{2011A&A...527A..56H}. In the fit the following model was used for the power spectrum:
\begin{equation}
\centering
\mathcal{P}(\nu_j ; \mathbf \Theta) = \sum_{n=n_{a}}^{n_{b}}\sum_{\ell=0}^{2}\sum_{m=-\ell}^{\ell} \frac{\mathcal{E}_{\ell m}(i) \tilde{V}_{\ell}^2 \alpha_{n\ell}}{1+\frac{4}{\Gamma_{n\ell}^2}(\nu-\nu_{n\ell m})^2 } + B(\nu),
\label{eq:limitspec}
\end{equation}
here $n_a$ and $n_b$ represent respectively the first and last radial order included from the power spectrum. We include modes of degree $\ell=0-2$. Each mode is described by a Lorentzian profile \citep[see, e.\,g.,][]{1990ApJ...364..699A, 2003ApJ...589.1009G} due to the way in which the p-modes are excited, namely stochastically by the turbulent convection in the outer envelope upon which they are intrinsically damped \citep{1994ApJ...424..466G}. In this description $\nu_{n\ell m}$ is the frequency of the mode while $\Gamma_{n\ell}$ is a measure for the damping\footnote{The mode life time is given by $\tau=1/\pi\Gamma_{n\ell}$.} rate of the mode and equals the full width at half maximum of the Lorentzian.
$\mathcal{E}_{\ell m}(i)$ is a function that sets the relative heights between the azimuthal $m$-components in a split multiplet as a function of the stellar inclination \citep[see, e.\,g.,][]{1977AcA....27..203D, 2003ApJ...589.1009G}. The factor $\tilde{V}_{\ell}^2$ is the relative visibility (in power) of a mode relative to the radial and non-split $\ell=0$ modes. The factor $\alpha_{n\ell}$ represents an amplitude modulation which mainly depends on frequency and is generally well approximated by a Gaussian.
In this work we do not fix the relative visibilities, as recent studies \citep[see, e.\,g.,][]{2010A&A...515A..87D,2011A&A...528A..25S,lund_to_come} have suggested that the theoretical computed values \citep[see, e.\,g.,][]{2011A&A...531A.124B} are generally not in good agreement with observations. In line with this notion we find that the theoretically expected values for the relative visibilities from \citet[][]{2011A&A...531A.124B}, which, using the spectroscopic parameters for Kepler-410 (see Table~\ref{table:spectroscopic_parameters2}), are given by $\tilde{V}_{\ell=1}^2\approx 1.51$ and $\tilde{V}_{\ell=2}^2\approx 0.53$, do not conform with the values obtained from our optimization (see Table~\ref{table:final_parameters}).
We describe the granulation background signal given by $B(\nu)$ by a sum of powerlaws \citep{1985ESASP.235..199H}, specifically in the version proposed by \cite{KarPhD}:
\begin{equation}
\centering
B(\nu) = B_n +\sum_{i=1}^{2}{\frac{4\sigma_i^2 \tau_i}{1 + (2\pi \nu \tau_i)^2 + (2\pi \nu \tau_i)^4}}.
\label{eq:bg}
\end{equation}
In this equation $\sigma_i$ and $\tau_i$ gives, respectively, the rms variation in the time domain and the characteristic time scale for the granulation and the faculae components. The constant $B_n$ is a measure for the photon shot-noise.
The frequencies of the individual modes in the interval $1370-2630\, \rm \mu Hz$ found from this optimization are used in the stellar modeling, see \S~\ref{sec:stellar_param} and \fref{fig:power_spectrum}.
\subsubsection{Stellar inclination and rotational splitting}
\label{sec:splitting}
Asteroseismology can via a fit of \eqref{eq:limitspec} be used to infer parameters such as the stellar rotation period and inclination\footnote{Going from $i=0^{\circ}$ at a pole-on view to $i=90^{\circ}$ for equator-on view.}. The information on these properties are found from the rotationally induced splitting of a oscillation mode of degree $\ell$ into $2\ell+1$ azimuthal $m$-components with values going from $m=-\ell$ to $m=\ell$.
In the case of a slow stellar rotation the star is generally assumed to rotate as a rigid body and the modes will be split as \citep{1951ApJ...114..373L}:
\begin{equation}
\centering
\nu_{n\ell m} = \nu_{n\ell} + m\frac{\Omega}{2\pi}(1-C_{n\ell}) \approx \nu_{n\ell} + m\nu_s,
\label{eq:split}
\end{equation}
with $\nu_{n\ell m}$ being the frequency entering into \eqref{eq:limitspec}, while $\nu_{n\ell}$ gives the unperturbed resonance frequency. The azimuthal order of the mode is given by $m$, $\Omega$ is the angular rotation rate of the star and $C_{n\ell}$ is the so-called \emph{Ledoux constant}; a dimensionless quantity describing the effect of the Coriolis force. For high-order, low-degree solar oscillations, as the ones seen in Kepler-410A, this quantity is of the order $C_{n\ell} < 10^{-2}$ and is therefore neglected. The splittings can thereby be seen as being dominated by advection. In this way we see that the splitting due to rotation between adjacent components of a multiplet will approximately be $\nu_s = \Omega/2\pi$, which will be referred to as the rotational frequency splitting.
In the optimization we use for the inclination a flat prior in the range $0-180^{\circ}$ and then fold the results from the MCMC around $i=90^{\circ}$. The reason for this is to better sample the posterior of the inclination very close to $i=90^{\circ}$, which would not be possible with a fixed boundary for the inclination at $i=90^{\circ}$ \citep[see, e.\,g.,][]{2013ApJ...766..101C}. A correct sampling of this region mainly has an influence on the credible regions computed for the value of the inclination. For the splitting we use a flat prior in the range $0-5\, \rm \mu Hz$.
\begin{figure*}[ht]
\centering
\includegraphics[scale=0.5]{figure2.eps}
\caption{ \emph{ \footnotesize Rotational splitting and inclination angle for Kepler-410 from the MCMC peak-bagging. The bottom left panel shows the correlation map between the inclination and the rotational splitting, while the panels above (inclination) and to the right (splitting) give the marginal probability density (PDF) functions for these two parameters. Our estimates for the parameters are given by the median values of their respective PDFs, indicated by the solid lines. The $68\%$ credible regions (found as the highest posterior density credible regions) are indicated by the dark gray part of the PDFs (bounded by dash-dotted lines), while the light gray indicates the additional part of the PDFs covered in a $95\%$ credible regions (bounded by dashed lines). The PDF for the inclination was found after first having folded the part of the full distribution ($0-180^{\circ}$) in the range $90-180^{\circ}$ onto the part in the range $0-90^{\circ}$. In the correlation map we have indicated the splitting as a
function of inclination (dark gray), with associated uncertainty (light gray), corresponding to the values
of $v\sin(i)$ estimated by \citet{molenda2013} (bottom lines) and \citet[][]{2013ApJ...767..127H} (top lines) and the radius estimate from our analysis.}}
\label{fig:split}
\end{figure*}
For the estimation of the inclination and rotational splitting we did not include the entire range used in estimating frequencies for the modeling, see \fref{fig:power_spectrum}. The rationale for using a narrower range that excludes the modes at highest and lowest frequencies is that we want only the modes with the highest signal-to-noise ratio. Furthermore, for modes at high frequencies the mode width becomes problematic for a proper estimate of the splitting.
Figure~\ref{fig:split} shows the correlation map from the MCMC analysis for the stellar inclination ($i$) and rotational splitting ($\nu_s$), going from low (light) to high (dark). The adopted values for the inclination and splitting are found by the median of the marginalized distribution, and indicated in the figure by the intersection of the two solid lines, final values are given in Table~\ref{table:final_parameters}. The dark gray part of the PDFs (bounded by dash-dotted lines) show the $68\%$ highest posterior density credible regions for each parameter and serve as the error for our estimates. The light gray (dashed lines) indicate the additional part of the PDFs covered by the $95\%$ credible regions. We find a value of $i=82.5^{\circ +7.5}_{-2.5} $ for the stellar inclination of the star, indicating a nearly equator-on view.
\begin{figure}[ht]
\centering
\resizebox{\hsize}{!}{\includegraphics{figure3.eps}}
\caption{ \emph{ \footnotesize Power spectrum of Kepler-410 (light gray) for a single order, with the dark gray giving the $0.1\, \rm \mu Hz$ smoothed version. Overlain is the model fit obtained from the MCMC peak-bagging (dashed red), in addition to the limit spectrum obtained when using an inclination angle of $45^{\circ}$ (full green).}}
\label{fig:single_order}
\end{figure}
Figure~\ref{fig:single_order} shows, for a single order, the best fitting model in red (i.\,e.\ using $i=82.5^{\circ}$). In green the limit spectrum is given when instead using a value for the stellar inclination of $i=45^{\circ}$ and keeping all other parameters fixed to the best fit values. This shows the large effect of the stellar inclination on the appearance of the limit spectrum from the variation in the relative heights of different azimuthal components.
To compare our result with the literature, we can compute the value of the splitting from literature values of $v\sin(i)$ via:
\begin{equation}
\nu_s = \frac{ [ v\sin(i) ] }{2\pi R \sin(i)}.
\label{eq:vsini}
\end{equation}
Using the radius found from the asteroseismic modeling (see \S~\ref{sec:stellar_param}), we have in \fref{fig:split} illustrated the corresponding values for the splitting from the estimate of $v\sin(i)$ by \citet{molenda2013} of $11.0\pm0.8\,\rm kms^{-1}$, and \citet[][]{2013ApJ...767..127H} of $15.0 \pm 0.5\,\rm kms^{-1}$. From our asteroseismic modeling we get, as expected from \fref{fig:split}, a value in between these estimates of $v\sin(i) = 12.9 \pm 0.6\,\rm kms^{-1}$ (see Table~\ref{table:final_parameters}).
\subsection{Asteroseismic modeling}
\label{sec:stellar_param}
\begin{table*}[ht]
\caption{ \emph{ \footnotesize Stellar parameters from spectroscopy. \textit{a:} \citet[][]{2013ApJ...767..127H}. \textit{b:} \citet{molenda2013}.}}
\label{table:spectroscopic_parameters2}
\centering
\begin{tabular}{l l l l l l}
\hline\hline\\[-0.35cm]
Reference & $\rm T_{eff}$ (K) & $\log\,g$ & $\rm [Fe/H]$ (dex) & $ v\rm\sin(i)$ ($\rm km s^{-1}$) & Instrument \\[0.05cm]
\hline\\[-0.3cm]
\textit{a} & $6325 \pm 75$ & - & $+0.01\pm 0.10$ & $15.0\pm0.5$ & HiRES, McDonald \\
\textit{b} & $6195\pm 134$ & $3.95\pm 0.21$ & $-0.16 \pm 0.21$ & $11.0\pm 0.8$ & ESPaDOnS \\[0.05cm]
\hline
\end{tabular}
\end{table*}
The stellar parameters were determined based on grids of models constructed using the GARching STellar Evolution Code \citep[GARSTEC,][]{Weiss:2008jy}. The input physics consists of the NACRE compilation of nuclear reaction rates \citep{Angulo:1999kp}, the \citet{Grevesse:1998cy} solar mixture, OPAL opacities \citep{Iglesias:1996dp} for high temperatures complemented by low-temperature opacities from \citet{Ferguson:2005gn}, the 2005 version of the OPAL equation of state \citep{Rogers:1996iv}, and the mixing-length theory of convection as described in \citet{2013sse..book.....K}. One grid of models also included the effect of convective overshooting from the stellar core when present. This is implemented in GARSTEC as an exponential decay of the convective velocities in the radiative region, and the used efficiency of mixing is the one calibrated to reproduce the CMD of open clusters \citep[e.\,g.][]{magic2010}. Diffusion of helium and heavy elements was not considered.
Our grid of models spans a mass range between 1.10-1.40~M$_\odot$ in steps of 0.02~M$_\odot$, and comprises five different compositions for each mass value spanning the 1-$\sigma$ uncertainty in metallicity as found from spectroscopy by \citet[][]{2013ApJ...767..127H}, see Table~\ref{table:spectroscopic_parameters2}. We chose this set of atmospheric constraints for the host star since they were derived using an asteroseismic determination of the surface gravity to avoid degeneracies from the correlations between $T_{\rm eff}$, $\log\,g$, and [Fe/H] \citep[see][for a thorough discussion]{2012ApJ...757..161T}. While the relative abundance of heavy elements over hydrogen can be directly determined from the measured [Fe/H] value, the assumption of a galactic chemical evolution law of $\Delta Y/\Delta Z = 1.4$ \citep[e.\,g.,][]{Casagrande:2007ck} allows a complete determination of the chemical composition. For both grids of models we computed frequencies of oscillations using the Aarhus Adiabatic Oscillations
Package \citep[ADIPLS,][]{ChristensenDalsgaard:2008kr}, and determined the goodness of fit by calculating a $\chi^2$ fit to the spectroscopic data and frequency combinations sensitive to the interior as described in \citet{SilvaAguirre:2013in}. Final parameters and
uncertainties were obtained by a
weighted mean and standard deviation using the $\chi^2$ values
of the grid without overshooting, and we added in quadrature the difference between these central values and those from the grid with overshooting to encompass in our error bar determinations the systematics introduced by the different input physics.
By combining the \citet{Casagrande:2010hj} implementation of the InfraRed Flux Method (IRFM) with the asteroseismic determinations as described in \citet{SilvaAguirre:2011es,SilvaAguirre:2012du}, it is possible to obtain a distance to the host star which is in principle accurate to a level of ${\sim}5\%$. Since the photometry of the host star might be contaminated by the close companion, we carefully checked the 2MASS photometry used in the implementation of the IRFM for warnings in the quality flags. The effective temperature determined by this method of $T_{\rm eff}=6273\pm140$K is in excellent agreement with those given in Table~\ref{table:spectroscopic_parameters2}, giving us confidence that the distance to the host star is accurately determined.
The final parameters of the star, including this distance, are given in Table~\ref{table:final_parameters}. From the stellar model parameters obtained from the peak-bagged frequencies we can calculate the Keplerian (rotational) break-up frequency of the star as:
\begin{equation}
\frac{\Omega_{\rm K}}{2\pi} = \rm \frac{1}{2\pi}\sqrt{\frac{GM}{R^3}} \approx 70.0 \pm 1.4\, \mu \rm Hz,
\end{equation}
whereby the star rotates at a rate of ${\sim}3\%$ of break-up as $2\pi\nu_s \approx 0.03 \Omega_{\rm K}$.
With this splitting it is worth considering the effect of second-order perturbations, $\delta\nu_2$, on the rotational frequency splitting:
\begin{equation}
\nu_{n\ell m} = \nu_{n\ell} + m\nu_s + \delta\nu_2.
\end{equation}
As described in Appendix~\ref{append}, this effect produces a small offset in the frequencies of the pulsations that in turn affects the stellar parameters derived from asteroseismic modeling. For this reason we iterated the frequency extraction with the stellar properties until the value of the break-up frequency converged (obtained after only a few iterations). The final model parameters are given in Table.~\ref{table:final_parameters}. We note that the change in parameters from including second-order effects is quite negligible, generally less than one per mil, with the exception of the age which is changed by about ${\sim}1.5\%$.
In \fref{fig:echelle} the \'echelle diagram \citep[][]{1983SoPh...82...55G} is shown, with observations overlaid by the frequencies from the best stellar model after the above iteration. For the sake of the comparison in the \'echelle diagram a surface correction has been applied to the model frequencies following the procedure of \citet[][]{2008ApJ...683L.175K}\footnote{As reference frequency we use the mean value of the radial modes, while $b$ is set to the solar calibrated value of $4.823$ \citep{2012ApJ...749..152M}}. Note, that the surface correction is not needed for the model optimization as frequency ratios, unaffected by the surface layers, are used rather than the actual frequencies. The splitting of the $\ell=1$ modes is clearly visible, with mode power mainly contained in the sectoral $m =\pm 1$ azimuthal components around the zonal $m=0$ components found in the peak-bagging and the modeling. This distribution of power between the azimuthal components is a function of the stellar inclination
angle \citep[see, e.\,g.,][]{2003ApJ...589.1009G}, where we indeed for $i$ close to $90^{\circ}$ (as found for Kepler-410) should expect to see power mainly in the sectoral components of $\ell=1$.
\begin{figure}[ht]
\centering
\resizebox{\hsize}{!}{\includegraphics{figure4.eps}}
\caption{ \emph{ \footnotesize \'Echelle diagram showing, in gray scale, the power spectrum of Kepler-410. Overlaid are the frequencies estimated from the MCMC peak-bagging (circles), along with the frequencies from the best-fitting stellar model after a surface correction (triangles). The frequencies estimated from the peak-bagging are the $m=0$ components, while $|m|>0$ components are included in \eqref{eq:limitspec} by the splitting. For an inclination as found for Kepler-410 mode power will for $\ell=1$ modes mainly be contained in the $m=\pm 1$ components, whereby the estimated $m=0$ component needed for the asteroseismic modeling should as seen be found in-between the $m=\pm 1$ power concentrations (assuming a symmetric splitting). }}
\label{fig:echelle}
\end{figure}
\section{Planetary validation}\label{sec:planetary_validation}
In this Section, we investigate the possible scenarios causing the transit-like features in the \textit{Kepler} data for Kepler-410. In \S~\ref{sec:confirmation_constraints}, we describe the constraints, as provided by the \textit{Kepler} data themselves, \textit{Spitzer} data and additional observations from ground. In \S~\ref{sec:confirmation_scenarios}, we then use those constraints to assess the likelihood of various scenarios, to conclude that the transits are indeed caused by a planet in orbit around Kepler-410A.
\subsection{Constraints\label{sec:confirmation_constraints}}
\label{sec:constraints}
\subsubsection{Geometry of transit signal}
\label{sec:transit_geometry}
A first constraint on what could be causing the transit signal in the \textit{Kepler} data, comes from the \textit{geometry of the transit signal} itself. While the transit signal could be diluted by additional stellar flux (i.e. by Kepler-410B, or additional unseen blends), the shape of the transit, as governed by the four contact points, remains the same. We use $T_\mathrm{tot}$ for the total transit duration, $T_\mathrm{full}$ for the duration between contact points two and three (the transit duration minus ingress and egress), $b$ for the impact parameter, and find \citep[see e.\,g.\ ][]{2010arXiv1001.2010W}:
\begin{equation}
\label{equ:transit_shape}
\frac{\sin(\pi T_\mathrm{tot} / P)}{\sin(\pi T_\mathrm{full} / P)} = \frac{\sqrt{(1 + R_\mathrm{p}/R_\star)^2 - b^2}}{\sqrt{(1 - R_\mathrm{p}/R_\star)^2 - b^2}}\, .
\end{equation}
Here $R_\star$ and $R_{\rm p}$ indicate the stellar and planetary radii. This equation (which only strictly holds for a zero-eccentricity orbit) can be understood by considering the most extreme case, namely a binary with two stars of the same size and $b = 0$, causing the equation to go to infinity. The transit becomes fully V-shaped, half of the transit is in ingress, while the other half is in egress. As it turns out, the short-cadence data constrains the transit shape to be clearly different from a V-shape as can be seen in \fref{fig:transit_fit}. With the left-hand side of \eqref{equ:transit_shape} determined by the data and setting $b\equiv0$, an upper limit on $R_\mathrm{p}/R_\star$ can be determined. Given the observed transit depth this ratio can now be used to establish an upper limit on any light dilution.
We model the planetary transit for various degrees of dilution until the transit fits for the ingress and egress get significantly worse (3$\sigma$ on a $\chi^2$ distribution) and we thereby reject transits occurring at a star more than 3.5 magnitudes fainter than Kepler-410A, and therefore exclude this region of the parameter space. This region is shown as the \textit{geometric limit} in hatched-gray in \fref{fig:confirming_planet}.
\begin{figure}[ht]
\resizebox{\hsize}{!}{\includegraphics{figure5.eps}}
\caption{\emph{ \footnotesize Planetary transit using the phase-folded observations (see \S~\ref{sec:period}), which were binned for clarity. The best fit is shown with a red line, together with the residuals (offset).}}
\label{fig:transit_fit}
\end{figure}
\begin{figure}[ht]
\centering
\resizebox{\hsize}{!}{\includegraphics{figure6}}
\caption{\emph{ \footnotesize Magnitude relative to Kepler-410A plotted versus the distance to the star. Kepler-410B is observed at 1.6$\arcsec$ and $\Delta m$ = 2.7 \citep{adams2012}. A large magnitude difference is excluded because of the transit geometry (\S~\ref{sec:transit_geometry}), while a large angular separation can be ruled out by analyzing the centroid (\S~\ref{sec:centroid}). Finally, ground-based photometry using adaptive optics (A.O.) and Speckle imaging (\S~\ref{sec:ao_speckle}) rules out all but a very small area of the parameter space.}}
\label{fig:confirming_planet}
\end{figure}
\subsubsection{Centroid}
\label{sec:centroid}
\textit{Pixel analysis} during transits can also unmask blends. Transits occurring around a slightly offset blended star would lead to centroid shifts on the {\it Kepler} CCD between in transit and out of transit data. A non detection of such shifts can give an upper limit on the brightness of a potential blend as function of projected distance on the sky.
The \textit{Kepler} team runs elaborate vetting procedures to determine if planetary signatures are caused by blends and centroid shifts are part of this procedure. These procedures are described in detail by \citet{bryson2013}. Kepler-410A, however, is a highly saturated star, which invalidates centroid shift measurements that appear in the Data Validation Report\footnote{http://exoplanetarchive.ipac.caltech.edu/docs/deprecated/KeplerDV.html}. Visual inspection of the difference images \citep[][Section 5]{bryson2013} in the Kepler-410 Data Validation report gives no indication that the transit source is not on the same pixel ($3.98\arcsec$ by $3.98\arcsec$) as Kepler-410A. This analysis is qualitative, however, and does not rule out the companion star. We therefore rely on other evidence given in this paper that the transit occurs on Kepler-410A.
\subsubsection{Ground-based photometry}\label{sec:ao_speckle}
\citet{adams2012} and \citet{howell2011} independently observed a blended object (Kepler-410B) at a distance of $1.6\arcsec$. These observations can also be used to exclude further objects inside certain magnitude and separation limits. The limits from the adaptive optics (A.O.) observations by \cite{adams2012} are shown in \fref{fig:confirming_planet}. The inner spatial limit for detections is at $0.2\arcsec$, where unseen objects up to a contrast of $4.2$ magnitudes in the \textit{Kepler} bandpass are excluded. This increases to $11.5$ magnitudes at $6\arcsec$.
Speckle images of Kepler-410 at $562$ and $692$~nm by \cite{howell2011} provide even tighter spatial constraints (\fref{fig:confirming_planet}), achieving a magnitude contrast of $3.55$ magnitudes between $0.05\arcsec$ and $0.30\arcsec$, with an increasing contrast up to $1.9\arcsec$. The limits in \fref{fig:confirming_planet} are 3$\sigma$ limits for observations at $562$~nm, while those for $692$~nm are about $0.5$ magnitudes tighter for the closest separations.
We further note that the 562 nm detection of Kepler-410B estimates it to be 4.24 magnitudes fainter than Kepler-410A, which, if the same difference holds in the broader \textit{Kepler} band, would place it below our geometric limit of possible planet hosting stars. However, \citet{howell2011} note that at $1.6\arcsec$ separation, the magnitude estimation of detected targets might be underestimated and we choose to adopt the \textit{Kepler} magnitude value for Kepler-410B as claimed in \citet{adams2012}, placing it just above our geometric limit.
\subsubsection{Spectroscopy}
Spectroscopic observations of Kepler-410 were taken with the HIRES\footnote{High Resolution Echelle Spectrometer on the Keck observatory.} echelle spectrometer at the Keck I telescope and reduced following a procedure described in \cite{chubak2012}. The spectra have a spectral resolution of $R = 55 000$ and stellar lines in the the near-IR wavelength region 654 - 800 nm were used to calculate the Doppler shift. The wavelength scale was determined from thorium-argon lamp spectra taken in twilight before and after each observing night while the wavelength zero-point was determined using telluric lines (from the A and B absorption bands) present in the target spectra. Due to the relatively high $v \sin i$ of the star (see Table \ref{table:spectroscopic_parameters2}), the errors listed here are slightly higher than the typical value (0.1km/s) stated in \cite{chubak2012}. The data are listed in Table~\ref{tab:rv}. We will use these RVs later to constrain scenarios involving binary systems.
\begin{table}
\caption{\emph{ \footnotesize Radial velocity measurements of Kepler-410.}}
\label{tab:rv}
\centering
\begin{tabular}{c c}
\hline\hline\\[-0.35cm]
Date (JD) & Radial Velocity (km/s)\\[0.05cm]
\hline\\[-0.3cm]
2454988.979733 & -40.30 $\pm$ 0.4\\
2455318.048353 & -40.995 $\pm$ 0.3\\
2455726.094382 & -40.18 $\pm$ 0.6\\
\hline
\end{tabular}
\end{table}
\subsubsection{Spitzer observations}
\label{sec:spitzer}
Kepler-410 was observed on 11 July and 18 December 2010 in-transit with the \textit{Spitzer Space telescope} \citep{werner2004}. The first visit consists of full-frame images with a longer integration time and lower accuracy than the second visit, which used \textit{Spitzer}'s subarray mode. We only analyze the subarray data. They consist of 310 sets of 64 individual subarray images, obtained using IRAC's channel 2 \citep{fazio2004}, which is centered at 4.5 $\mu$m. The data are available for download from the \textit{Spitzer} Heritage Archive database\footnote{http://sha.ipac.caltech.edu/applications/Spitzer/SHA} as basic calibrated data (BCD) files. The first observations (which are often more noisy due to the telescope's ramp up) are often ignored \citep[see e.\,g.][]{knutson2008}, but we omit the first 55 observations to keep an equal amount of observations before and after the transit (62 observations on each wing, with 131 in-transit observations).
We analyzed the data following a procedure described by \cite{desert2009}. A square aperture ($11\times11$ pixels) is used to collect the stellar flux (where 64 images of each subarray observation are immediately combined) and the centroid position is calculated. Since a pixel spans 1.2\arcsec, the flux contains the combined light of Kepler-410A and Kepler-410B. Subsequently, a linear function in time is used to de-trend the data, in combination with a quadratic function of the $x$ and $y$ coordinates of the centroids, resulting in five free fitting parameters, \citep[see e.g.][]{knutson2008,desert2009,demory2011} to correct for the pixel-phase effect. We fit only the out-of-transit data, but correct the full dataset.
\begin{figure}[htb]
\centering
\resizebox{\hsize}{!}{\includegraphics{figure7.eps}}
\caption{\emph{ \footnotesize The reduced \textit{Spitzer} observations are shown, together with the best fitted transit model (black) and a 1$\sigma$ confidence interval. The blue dots show binned data points. The red dotted line indicates a lower limit for the expected transit depth if the transit occurs on Kepler-410A. The blue dotted line shows a minimum depth if the transit would occur on Kepler-410B.}}
\label{fig:spitzer_transit}
\end{figure}
Now we compare the average flux level of the in-transit data to the out-of-transit data finding a transit depth of $240 \pm 90$~ppm. The uncertainty is calculated by bootstrapping (we re-sample without replacement, treating the in-transit and out-of-transit data separately), which we find to result in a slightly higher error level compared to simply using the scatter on the data points. We adopt this value and show the result in \fref{fig:spitzer_transit}. A similar procedure, comparing median flux levels rather than mean flux levels, gives a transit depth of $260 \pm 90$~ppm.
\subsubsection{Asteroseismology}
\label{sec:asteroseismic_constraints}
Finally, the \textit{Kepler} data provide an asteroseismic constraint on additional objects, by looking at the (absence of) stellar pulsations in the power spectrum (see \fref{fig:power_spectrum}). We searched the power spectrum for excess power from stellar oscillations using the so-called \emph{MWPS} method \citep[see][]{2012MNRAS.427.1784L}. With this, only one set of (solar-like) pulsations was detected, which can be attributed to Kepler-410A because of their high amplitudes, and we can thereby rule out additional signal from bright, large stars to be present in the light curve. We exclude solar-like oscillations of main-sequence stars or red giants up to $K_\textrm{p} = 13$, the geometric exclusion limit (\fref{fig:confirming_planet}).
We can translate this magnitude limit on additional solar-like oscillations into limits on the surface gravity using the method developed by \citet[][]{2011ApJ...732...54C} \citep[see also][]{campante_to_come}.
We estimate a lower limit for the value of $\nu_{\rm max}$\footnote{The frequency at which the oscillations have the largest amplitude.} for a marginal detection of oscillations in the power spectrum. This lower limit on $\nu_{\rm max}$ can in turn be translated into a lower limit for the surface gravity ($g$) of the star (or $\log\,g$ as most often used) via the simple relation:
\begin{equation}
g \simeq g_{\sun} \left( \frac{\nu_{\rm max}}{\nu_{\rm max, \sun}} \right) \left(\frac{T_{\rm eff}}{T_{\rm eff,\sun}} \right)^{1/2}.
\end{equation}
The above relation builds on the proportionality between $\nu_{\rm max}$ and the acoustic cut-off frequency \citep[$\nu_{ac}$; see, e.\,g.,][]{1991ApJ...368..599B,2011A&A...530A.142B}. In addition, the procedure uses various scaling relations for e.\,g.\ the amplitudes of the oscillations and the stellar noise background - we refer the reader to \citet[][]{2011ApJ...732...54C} for further details.
For temperatures in the range $T_{\rm eff} = 5500 - 5777$~K we estimate that non-detection of oscillations in any second component (i.e. a star other than Kepler-410A) sets limiting (lower-limit) values for $\log\,g$ of ${\gtrsim} 4.51 \pm 0.05\,\rm dex$ (5500 K) and ${\gtrsim} 4.57 \pm 0.05\,\rm dex$ (5777 K).
For higher assumed values $T_{\rm eff}$, the limiting values for $\log\,g$ are inconsistent with allowed combinations for $\log\,g$ and $T_{\rm eff}$ from stellar evolutionary theory. From these limiting values for $\log\,g$ any potential second component must necessarily be a small dwarf star.
For Kepler-410 the asteroseismic constraint, together with the geometric constraint, is enough to establish the planetary nature of the transit signal. As shown in Section~\ref{sec:transit_geometry} the signal cannot occur on a star fainter than $K_\textrm{p} = 13$ (limiting the maximum true transit depth) and due to the asteroseismic constraint, any object brighter than this is necessarily small. Since the transit depth is given by the size of the transiting object relative to its host star, the two constraints together limit the size of the transiting object to be smaller than Jupiter. For both constraints, observations in a short-cadence sampling are crucial.
\subsection{Scenarios\label{sec:confirmation_scenarios}}
We now use the constraints established in the last section to evaluate three possible scenarios which could cause the transit signal; a chance alignment with a background system (\S~\ref{sec:chance_alignment}), an unseen companion to Kepler-410 (\S~\ref{sec:physical_companion}), and a planet in orbit around Kepler-410B (\S~\ref{sec:koi42b}). Given the available data we can rule them out and conclude that the transit signal occurs on Kepler-410A.
\subsubsection{Chance alignment}
\label{sec:chance_alignment}
The scenario of a background system, largely diluted by a much brighter foreground object (Kepler-410A), is disfavored by a combination of the geometric constraints and the additional observations described in \S~\ref{sec:confirmation_constraints}. With most of the parameter space ruled out, a relevant system would need to have a $K_\textrm{p}$ between $9.5$ and $13$ (see \S~\ref{sec:transit_geometry}) and a separation which is less than $0.02\arcsec$ from Kepler-410A (see \S~\ref{sec:ao_speckle}).
A detailed analysis on false positive scenarios can be found in \cite{fressin2013}. Following a similar approach we use the Besancon model of the galaxy \citep{robin2003} to simulate the stellar background around Kepler-410. This leads to the prediction of $319$ objects brighter than $13$th magnitude in the R-band (which is close to the \textit{Kepler} band\footnote{\textit{Kepler} magnitudes are nearly equivalent to R band magnitudes \citep[][]{2010ApJ...713L..79K}.}), in an area of one square degree. This places on average $6 \times 10^{-8}$ background stars of sufficient brightness in the confusion region of $0.05$\arcsec around Kepler-410A, the region which is not ruled out by any constraints (see \fref{fig:confirming_planet}). Even without further consideration of whether any background objects could be eclipsing binaries or hosting a transiting planet, we consider this number too small for such a scenario to be feasible. From here on we therefore assume that the transit signal is not caused by a chance
alignment of a background system.
\subsubsection{Physically associated system}
\label{sec:physical_companion}
We now consider the possibility that the transit occurs on a star physically associated to Kepler-410A but not Kepler-410A itself. According to \citet{fressin2013}, transiting planets on a physically associated star are the most likely source of false positives for small Neptunes. Prior to constraints, they estimate 4.7 $\pm$ 1.0\% of the small Neptune \textit{Kepler} candidates are misidentified in this way.
For Kepler-410 the spatial constraints from the ground-based photometry (see \S~\ref{sec:ao_speckle}) are far more strict than what was used by \citet{fressin2013}, who only use the \textit{Kepler} data itself to determine the region of confusion.
From the transit geometry stars fainter by $\Delta K_\textrm{p} = 3.5$ are already excluded as possible host stars. Since a physical companion would have the same age as Kepler-410A, we can use the mass-luminosity relation for main sequence stars to derive a lower mass limit. We find this to be about 0.5 M$_\odot$. Furthermore, the companion star cannot be more massive than Kepler-410A itself, otherwise it would be more luminous and thereby visible in the spectra and produce an asteroseismic signal.
We proceed with a simple calculation to quantify the chance that Kepler-410 has an unseen companion with a planet that causes the transit signal. As in \cite{fressin2013}, we assign a binary companion to Kepler-410A following the distribution of binary objects from \cite{raghavan2010}; a random mass ratio and eccentricity and a log-normal distribution for the orbital period. We calculate the semi-major axis using Kepler's third law and assign a random inclination angle, argument of periastron, and orbital phase to the system.
From the simulated companions, we reject those with a mass lower than 0.5 M$_\odot$. We calculate their angular separation (using the distance estimate from Table~\ref{table:final_parameters}) and reject those which would have been detected in the ground-based photometry. Finally, we compute the radial velocity (RV) signal the companion would produce at the times of the RV measurements (Table \ref{tab:rv}) and reject those objects inconsistent with the observations. For this, we calculate the $\chi^2$ value for each simulated companion, and assign a chance of rejection to each one based on the $\chi^2$ distribution.
We find that only 0.46\% of the simulated objects could pass these tests. The frequency of non-single stars is 44\% \citep{raghavan2010}, resulting in a chance of 0.2\% that an undetected star is associated with Kepler-410A. This limit would be even lower if we assume Kepler-410B is physically associated with Kepler-410A, since the probability of additional companions in a multiple system is lower than the value quoted above \citep[an estimated 11\% of all stars are triple system or more complex;][]{raghavan2010}. More elaborate simulations could also further reduce this statistical chance, as we have not taken into account the \textit{Spitzer} transit depth, visibility in spectra, or visibility of asteroseismic features, of this hypothetical companion.
\subsubsection{Kepler-410B}
\label{sec:koi42b}
While the nature of Kepler-410B is largely unknown, some information on the star ca2012ApJn be derived from the observations by \cite{adams2012}. Using their 2MASS $J$ and $Ks$ magnitude, we can convert the measured brightness difference into a temperature estimate, using color-temperature transformations as described by \cite{casagrande2010}. We find a temperature of around 4850 K, assuming a solar metallicity. This indicates a small (dwarf) star, which is consistent with the non-detection of an asteroseismic signal of the object in the blended \textit{Kepler} light (see \S~\ref{sec:asteroseismic_constraints}).
There is modulation signal present in the \textit{Kepler} data, which is presumably caused by the rotation of Kepler-410B. It indicates a brightness variation of the object of $\approx$ 2.5 \% \citep[assuming the brightness contrast by][see \S~\ref{sec:ao_speckle}]{adams2012}, over a rotation period of 20 days. In fact, the modulation signal has previously been mis-attributed to Kepler-410A \citep{2013arXiv1308.1845M}, resulting in a rotation period inconsistent with what we derive through asteroseismology ($5.25 \pm 0.16$ days, see \S~\ref{sec:splitting}).
The different colors of Kepler-410A and Kepler-410B can be used to rule out Kepler-410B as a host star, by comparing the transit depth measured in the \textit{Spitzer} IRAC band with the depth as measured by \textit{Kepler}. Kepler-410B is $2.7$ magnitudes fainter than Kepler-410A in the \text{Kepler} band \citep{adams2012}. The flux of Kepler-410B is ${\approx} 8$\% the flux of Kepler-410A. In 2MASS $Ks$ $(2.1 \mu$m) the magnitude difference reduces to 1.9 ($\approx 17$\% flux). We conservatively assume that in \textit{Spitzer}'s IRAC band (4.5 $\mu$m), $\Delta m$ $\leq$ 1.9. Using this assumption, a transit occurring on Kepler-410A would be blended somewhat more in the \textit{Spitzer} observations (depth $\leq 300$ ppm), while a transit occurring on Kepler-410B would only be diluted by less than
half the dilution in the \textit{Kepler} light (depth $\geq$ 600 ppm).
A measured \textit{Spitzer} transit depth of $240 \pm 90$ ppm distinctly (at a 4$\sigma$ level) rules out Kepler-410B as a potential host star to the transiting planet and is consistent with the planet occurring on Kepler-410A. From here on, we assume that the transits occur on Kepler-410A.
\section{Planetary analysis}\label{sec:planetary_parameters}
\subsection{Period and transit timing variations}
\label{sec:period}
For the planetary analysis, we start from the same dataset as for the asteroseismic analysis (see \S~\ref{sec:asteroseismology}), where we normalize the planetary transits by fitting a second-order polynomial to the transit wings. To determine the planetary parameters we first create a phase folded high signal-to-noise light curve out of the {\it Kepler} light curve. As transit timing variations (TTVs) are present (see below) we cannot simply co-add the light curve on a linear ephemeris but we use the following steps:
\begin{enumerate}
\renewcommand{\theenumi}{\roman{enumi}}
\item Estimate the planetary period and produce a phase-folded light curve;
\item Use the phase-folded light curve as an empirical model for the shape of the transit and use this model to determine individual transit times;
\item Repeat the first two steps until convergence is reached;
\item Determine TTVs and produce a phase-folded lightcurve which takes this into account;
\item Model the transit, taking into account the dilution caused by Kepler-410B.
\end{enumerate}
We find the usage of the phase-folded light curve as an empirical model for the transit quite efficient in determining the times of individual transits. The time for an individual transit event is determined by shifting the empirical model around the predicted transit time. The new time for the transit event is determined by comparing data points with the time-shifted empirical model and minimizing $\chi^2$. Based on the new transit times, a new period estimate can be made and the procedure can be repeated. Following this approach, we reached convergence after only two iterations.
After convergence is reached on determining transit times of individual transit events, the planetary period can be determined. Under the assumption of a perfectly Keplerian orbit, the planetary period is given by a linear interpolation of the transit times:
\begin{equation}
T (n) = T (0) + n \times \mathrm{Period} \label{eq:linear_period},
\end{equation}
where $T(n)$ and $T(0)$ refer to the $n$th and 0th transit times (taking into account possible data gaps). The period found in this way is $17.833648 \pm 0.000054$ days. Subsequently, we produce an O-C ($Observed-Calculated$) diagram in which for each transit the calculated transit time is subtracted from the observed transit time, and which we present in \fref{fig:oc_diagram}. Transit Timing Variations (TTVs) are clearly visible.
The interpretation of TTVs is difficult. Short-period trends can be caused by stellar variability (e.\,g.\ stellar spots causing an apparent TTV signal), while longer-period trends such as here are in most cases attributed to a third body (e.\,g. planet), whose gravitational influence causes the deviation from the strictly Keplarian orbit.
The signal can be highly degenerate, with bodies in or close to different resonance orbits resulting in very similar TTV signals. Attempts of interpretations have been made by performing three-body simulations, with unique solutions for non-transiting objects in only a limited number of cases \citep[see, e.\,g.,][]{nesvorny2013}. TTVs have been successfully used to characterize systems with multiple transiting exoplanets, by studying their mutual gravitational influence \citep[e.g.][]{carter2012}. We have made a visual inspection of the time series to look for additional transit signals, but found none.
Based on limited data, \citet{ford2011} reported a possible detection of TTVs in the orbit of Kepler-410, and a study of TTVs on the full sample of KOIs \citep{mazeh2013} resulted in an amplitude of $13.95 \pm 0.86$ minutes and a period of $990$ days (no error given) for Kepler-410, using a sinusoidal model. We find a peak-to-peak amplitude of $0.023$ days ($33$ minutes), and a period of $957$ days, not using a sinusoidal but a \textit{zigzag} model, as indicated in \fref{fig:oc_diagram} by the solid line. It is not immediately clear what is causing the seemingly non-sinusoidal shape of the TTVs \citep[see e.\,g. ][for a discussion]{nesvorny2009}. A similar shape is seen for Kepler-36 \citep{carter2012}, where discontinuities occur when the planets are at conjunction. We speculate that the eccentricity of Kepler-410A b could be influencing the shape (see \S~\ref{sec:eccentricity}).
\begin{figure}[ht]
\resizebox{\hsize}{!}{\includegraphics{figure8.eps}}
\caption{\emph{ \footnotesize O-C diagram showing the observed transit times minus the calculated transit times following a Keplerian orbit (\eqref{eq:linear_period}). The black points represent individual transit measurements (with their error bars), the green dots are a copy of the observed data points, offset by one full period. They are for illustration only, and were not included in the fit. A clear trend is visible, which is fitted by a model with discontinuities at the turning points.}}
\label{fig:oc_diagram}
\end{figure}
\subsection{Parameters}
\label{sec:parameters}
The period-folded data are then used to determine the planetary parameters. The blending from Kepler-410B (see \S~\ref{sec:ao_speckle}) needs to be taken into account before estimating the planetary parameters, so we subtract the estimated flux due to Kepler-410B (8\%) from the light curve before starting our analysis.
\begin{table}
\caption{\emph{ \footnotesize Stellar parameters are derived from asteroseismic modeling. Values are from the best fitting model without overshoot; the differences between these values and the ones from the best fitting model including overshoot are taken as a measure of the systematic error from differing input physics in the modeling; this difference is added in quadrature to the uncertainties from the grid optimization. Planetary values are derived from transit modeling combined with asteroseismic results.}}
\label{table:final_parameters}
\begin{center}
\begin{tabular}{l c}
\hline\\[-0.3cm]
Stellar parameters & {Kepler-410A} \\
\hline\\[-0.35cm]
Mass [M$_\odot$] & 1.214 $\pm$ 0.033\\
Radius $R_\star$ [R$_\odot$] & 1.352 $\pm$ 0.010\\
$\log\,g$ [cgs] & 4.261 $\pm$ 0.007\\
$\rho$ [g cm$^{-3}$] & 0.693 $\pm$ 0.009\\
Age [Gyr] & 2.76 $\pm$ 0.54\\
Luminosity [L$_\odot$] & 2.72 $\pm$ 0.18 \\
Distance [pc] & 132 $\pm$ 6.9 \\
Inclination $i_\star$ [$\rm ^{\circ}$] & $82.5^{+7.5}_{-2.5}$ \\
Rotation period$^*$, $P_{\rm rot}$ [days] & $5.25 \pm 0.16$ \\[0.05cm]
\hline\\[-0.3cm]
Model parameters & \\
\hline\\[-0.35cm]
Rotational splitting, $\nu_s$ [$\rm \mu Hz$] & $2.206^{+0.067}_{-0.065}$ \\
$v\sin(i_\star)^{\dagger}$ [$\rm km s^{-1}$] & $12.9 \pm 0.6$ \\
$(V_1/V_0)^2$ & $1.796^{+0.090}_{-0.085}$\\
$(V_2/V_0)^2$ & $0.861^{+0.073}_{-0.068}$\\[0.05cm]
\hline\hline\\[-0.3cm]
Planetary parameters & Kepler-410A b\\
\hline\\[-0.35cm]
Period [days] & 17.833648 $\pm$ 0.000054 \\
Radius $R_\mathrm{p}$ [R$_\oplus$] & 2.838 $\pm$ 0.054 \\
Semi-major axis $a$ [AU] & 0.1226 $\pm$ 0.0047 \\
Eccentricity $e$ & 0.17$^{+0.07}_{-0.06}$\\[0.05cm]
Inclination $i_p$ [$^\circ$] & 87.72 $^{+0.13}_{-0.15}$ \\[0.1cm]
\hline\\[-0.3cm]
Model parameters & \\
\hline\\[-0.35cm]
a/R$_*$ & 19.50 $^{+0.68}_{-0.77}$ \\[0.1cm]
R$_\mathrm{p}$/R$_*$ & 0.01923 $^{+0.00034}_{-0.00033}$ \\[0.1cm]
Linear LD & 0.57 $^{+0.22}_{-0.28}$ \\[0.1cm]
Quad LD & -0.04 $^{+0.26}_{-0.22}$ \\[0.1cm]
\hline
\end{tabular}
\end{center}
$^*${\footnotesize Found as $P_{\rm rot}=1/\nu_s$, and using the uncertainty (asymmetric uncertainties are added quadrature) on $\nu_s$ to find uncertainty for $P_{\rm rot}$.}\\
$^{\dagger}${\footnotesize Found via \eqref{eq:vsini} and using the uncertainties (asymmetric uncertainties are added quadrature) on the parameters $R$, $\nu_s$, and $i$.}
\end{table}
The transits are fitted using the Transit Analysis Package (TAP) which is freely available \citep{gazak2012}. An MCMC analysis is carried out, using the analytical model of \citet{mandel2002}. An orbital eccentricity of zero is assumed for the entire fitting procedure. Flat priors were imposed on the limb darkening coefficients, and they were simply treated as free parameters in our approach. The folded datasets were binned to improve the speed of the MCMC procedure. \fref{fig:transit_fit} shows the transit curve. A list of all parameters is provided in Table \ref{table:final_parameters}.
We finally note that the true errors are likely to be slightly larger than the formal errors reported in Table~\ref{table:final_parameters}. These are the result of the MCMC fitting procedure, and do not take into account systematics in the \textit{Kepler} data \citep{vaneylen2013}, or the uncertainty in the flux contribution by the blended light from Kepler-410B, both of which could affect the transit depth.
\subsection{Planetary eccentricity}
\label{sec:eccentricity}
We have access to two estimates of the stellar density. One value was obtained from the asteroseismic modeling of the stellar pulsations ($\rho_\textrm{asteros.}$) and one from modeling the planetary transit \citep[${\rho_\textrm{transit}}$;][]{seager2003,tingley2011},
\begin{equation}\label{eq:stellar_density}
\rho_\textrm{transit} = \frac{3 \pi}{G P^2} \left( \frac{a}{R_*}\right)^3 = 0.441 \pm 0.050 \mathrm{~g/cm}^3 \, ,
\end{equation}
where $G$ is the gravitational constant and all other parameters are listed in Table~\ref{table:final_parameters}. To obtain an estimate of $\rho_\textrm{transit}$ a particular orbital eccentricity ($e$) needs to be assumed, which in this equation was set to zero. Therefore calculating the ratio of the two density estimates leads to a lower limit on the orbital eccentricity.
Following the notation in \cite{dawson2012} we obtain
\begin{equation}
\label{eq:eccentricity_periastron}
\frac{\rho_\textrm{asteros.}}{\rho_\textrm{transit}} = \frac{(1-e^2)^{3/2}}{(1+e\sin \omega)^3} = 1.57 \pm 0.18 \, ,
\end{equation}
where $\omega$ is the argument of periastron and we took $\rho_\textrm{asteros.} = 0.693 \pm 0.009$ from Table~\ref{table:final_parameters}. As the value is not consistent with unity within error bars, a circular orbit for the planet is ruled out. The eccentricity is a function of $\omega$, as can be seen in \fref{fig:eccentricity}.
\begin{figure}[ht]
\centering
\resizebox{\hsize}{!}{\includegraphics{figure9.eps}}
\caption{\emph{ \footnotesize Change of perceived stellar density, compared to zero-eccentricity density, for different angles of periastron. The inner solid line represents an eccentricity of $e=0$, the outer depicts $e=0.9$. The dashed line gives the location of the density ratio given in \eqref{eq:eccentricity_periastron} for Kepler-410 (uncertainty of the ratio is given by the red band).}}
\label{fig:eccentricity}
\end{figure}
\fref{fig:eccentricity} and \eqref{eq:eccentricity_periastron} indicate that a lower limit on the system's eccentricity can be derived. For certain arguments of periastron (around $\omega \approx 210^\circ$ or $\omega \approx 320^\circ$), high eccentricities cannot be excluded. However, the range of periastron angles becomes increasingly narrow for increasing eccentricities. Taking a sample assuming random angles of periastron, and a Gaussian distribution for $\rho_\textrm{asteros.}/\rho_\textrm{transit}$ to take into account the uncertainty of \eqref{eq:eccentricity_periastron}, and using a correction factor for non-grazing transits as described in \cite{dawson2012}, we find that the mode of the eccentricity is $0.17$ and 68\% of the eccentricities are contained in the interval [0.11,0.24], as indicated on \fref{fig:eccentricity_pdf}.
\begin{figure}[ht]
\centering
\resizebox{\hsize}{!}{\includegraphics{figure10.eps}}
\caption{\emph{ \footnotesize Kernel density distribution of the eccentricity values. The mode (dotted red line) is seen at an eccentricity of 0.17 and the uncertainties (highest posterior density credible regions) are indicated in grey.}}
\label{fig:eccentricity_pdf}
\end{figure}
For the above analysis to deliver unbiased eccentricity results, it is important to remove TTVs (see \S~\ref{sec:period}) and third light from Kepler-410B (\S~\ref{sec:parameters}) from the light curve \citep[see e.\,g.][]{kipping2013}. We expect no additional light dilution because of the additional constraints presented in \S~\ref{sec:constraints}, and specifically the asteroseismic constraint in \S~\ref{sec:asteroseismic_constraints} which rules out bright companion stars.
\subsection{Stellar obliquity}
\label{sec:obliquity}
The stellar obliquity (the opening angle between the stellar rotation angle and the orbital angular momentum) can be constrained with our asteroseismic modeling and transit measurements. We measure similar values for the inclination of the stellar rotation axis ($i_\star=82.5^{+7.5}_{-2.5}$ [$^\circ$]) and the planetary orbital axis ($i_p=87.72\pm0.15^\circ$). We have no information on the other variable defining the stellar obliquity, its projection on the plane of the sky. However we can ask how likely it would be that we measure similar inclinations if the orientation of the stellar rotation axis is uncorrelated to the planetary orbit and randomly oriented. For this we look at a distribution which is flat in $\cos i_\star$, and which leads to a random orientation of the angular momentum axis on a sphere. This way we find that there there is a 17\% chance to find the stellar rotation axis inclined as close to 90$^\circ$ as is the case, assuming no correlation between the stellar rotation axis and the
planetary orbital inclination. Therefore our asteroseismic measurement of $i_\star$ suggests a low obliquity in the Kepler-410A system.
\section{Discussion}
\label{sec:discussion}
With the validation of Kepler-410b as a small Neptune-sized exoplanet ($2.838 \pm 0.054$~R$_\oplus$), it joins the current list of 167 confirmed {\it Kepler} exoplanets around 90 stars. Thanks to the sampling in short cadence and the brightness of the host star, the stellar (and therefore planetary) parameters are known to very high accuracy. The star is the third brightest of the current sample of confirmed \textit{Kepler} planet host stars, only preceded by Kepler-21 \citep{howell2012} and Kepler-3 \citep[HAT-P-11][]{bakos2010}, both of which have a period of only a few days.
Even outside the \textit{Kepler} field, only about 10-20 planets (which are typically not as well-characterised) are known to transit around stars which are brighter or of similar brightness. 55 Cnc e \citep{mcarthur2004} is the brightest and together with HD 97658 \citep{howard2011,dragomir2013} and Kepler-21b \citep{howell2012}, they are the only planets smaller than Kepler-410A b around stars brighter than Kepler-410A, and they all have a shorter orbital period. The only ones with longer orbital periods are the Jupiter-sized planets HD 17156b \citep{fischer2007} and HD 80606b \citep{naef2001}. Perhaps the most similar system is the bright star Kepler-37, which has three planets of sub- and super-Earth size on orbital periods of 13, 21 and 39 days \citep{barclay2013}.
That the host star can be well-studied has its implications on the planetary parameters, which are now also well-known. This makes Kepler-410 an interesting object for follow-up observations. High-quality radial velocity observations might be able to constrain the planetary mass and therefore also its density, and, if the latter is favorable, even transmission spectroscopy might be within reach of some instruments. In addition, such observations might shed more light on the observed transit timing variations, which we suspect are caused by one or more additional planets in the system. With a relatively high TTV amplitude ($\sim$ 30 minutes peak-to-peak), one might hope (an) additional planet(s) can be revealed with radial velocity observations. Full simulations of the observed TTVs were beyond the scope of this paper but might be fruitful due to the eccentricity of Kepler-410A b; our observed transit times are available upon request.
Our finding of a low obliquity in Kepler-410A can be compared to obliquity measurements in other exoplanet systems with multiple planets. The first multiple system for which the projected obliquities has been measured is the Kepler-30 system which harbors three transiting planets \citep{sanchisojeda2012}. The authors found a low projected obliquity by analyzing spot crossing events. \cite{2013ApJ...766..101C} found good alignment between the orbital and stellar inclinations for Kepler-50 and Kepler-65, analyzing the splitting of the rotational modes in a similar way as presented in this work. \cite{hirano2012} and \cite{albrecht2013} analyzed the KOI-94 and Kepler-25 systems and found low projected obliquity. For the multiple transiting planet system KOI-56, asteroseismic modeling revealed a high obliquity between the orbit of the two planets and the stellar rotation \citep{huber2013}. The authors suggest that a companion leads to a misalignmemt of one planet, which then influenced the orbital plane of the
other planet. With the measurements at hand, it appears as if the obliquity distribution for multiple planet systems is flatter than what is observed for systems with single close-in Jupiter-sized planets \citep{albrecht2012,albrecht2013}.
One way to learn more about the obliquity in Kepler-410A would be to also measure the projection of the stellar rotation axis via the Rossiter-McLaughlin (RM) effect \citep[][]{1924ApJ....60...15R,1924ApJ....60...22M}. The amplitude of the RM effect would be of the order of a few m\,s$^{-1}$, despite the small transit depth, as $v \sin i_\star$ is large.
Of particular interest in this regard is also the tight constraints derived on the planetary eccentricity, which is measured to be inconsistent with a circular orbit. While eccentricities routinely result from radial velocity observations of exoplanet hosts, this is one of the first stars for which the eccentricity is tightly constrained using only photometric measurements, which to our knowledge has only resulted in excluding circular orbits in the case of Kepler-63b \citep{sanchisojeda2013}.
\section{Conclusions}
\label{sec:conclusion}
Using a combination of high-quality \textit{Kepler} data and ground-based photometry and spectroscopy, we are able to validate the presence of a planet around Kepler-410A; a small Neptune in an orbit with a period of $17.8$ days. This makes Kepler-410A the third brightest \textit{Kepler} planet host star currently known. A detailed analysis of the solar-like oscillations allows for a characterization of the stellar mass to within 3\%, while the radius is known to less than a 1\% and the age is determined to within 20\%. The asteroseismic study also allowed a precise determination of the distance to the star.
Furthermore, we constrain the rotation rate and inclination angle of the host star and find the results to be consistent with low obliquity. This is a result similar to most obliquity measurements in multiple planet systems, which is in contrast to measurements of obliquities in Hot-Jupiter systems, where the obliquities are much more diverse. With an accurate determination of the stellar density through asteroseismology, we are able to photometrically constrain the planetary eccentricity to 0.17$^{+0.07}_{-0.06}$. We finally note that transit timing variations strongly suggest the presence of at least one additional (non-transiting) planet in the system.\\
\vspace*{1cm}
\small{\emph{Acknowledgements}. We thank Joanna Molenda-\.{Z}akowicz, Lars A. Buchhave and Christoffer Karoff for sharing stellar spectra. We thank Luca Casagrande for help with the InfraRed Flux Method to obtain the stellar distance and David Kipping for helpful comments in reviewing the manuscript. The referee's helpful comments and suggestions have led to significant improvements. MNL would like to thank Dennis Stello and his colleagues at the Sydney Institute for Astronomy (SIfA) for their hospitality during a stay where some of the presented work was done. WJC and TLC acknowledge the support of the UK Science and Technology Facilities Council (STFC). Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation (Grant agreement no.: DNRF106). The research is supported by the ASTERISK project (ASTERoseismic Investigations with SONG and Kepler) funded by the European Research Council (Grant agreement no.: 267864). Funding for the \textit{Kepler} Discovery mission is
provided by NASA’s Science Mission Directorate. The Spitzer Space Telescope is operated by the Jet Propulsion Laboratory, California Institute of
Technology under a contract with NASA. We thank the entire \textit{Kepler} team, without whom these results would not be possible. }
|
1,314,259,996,509 | arxiv | \section{Early-Type Galaxies}
For a long time, the conventional view of early-type galaxies was that they were simple, single-component objects which were either almost spherical and hardly rotating, or flattened systems whose shape could be simply accounted for by rotation around a single axis. As telescopes became larger, and instruments and detectors became more efficient, it was possible to obtain a more detailed picture of the stellar motions of these systems. It became clear that many of these objects rotated too slowly to account for their apparent flattening: extra pressure support, in the form of an anisotropic velocity ellipsoid, was required to explain their shape \citep{bertola75,illingworth77,binney78}. Overall, early-type galaxies seemed to span a broad range of dynamical states, from rotating disk-like systems to dynamically hot, pressure-supported spheroids \citep[and references therein]{binney82,dezeeuw91}.
Furthermore, some objects were found to exhibit peculiar stellar kinematics, rotating around both the apparent short- and long-axes, for example, or showing a reversal in the rotation direction close to the galaxy center. Such galaxies were said to contain kinematically decoupled components (KDCs), indicating that there existed a sub-component of the galaxy which did not share the same orbital distribution as the rest of the object. The formation of such a sub-component can occur either by the accretion of an external system \citep[e.g.][]{kormendy84, balcells90}, or by the formation of stars from accreted external material \citep[e.g.][]{hernquist91,weil93}. Both scenarios have different implications for our theoretical understanding of galaxy formation.
Our understanding of the morphology of early-type galaxies has undergone a similar evolution. In particular, elliptical galaxies, as their name implies, were once considered to have apparent shapes described by a simple ellipse. With the advent of CCD detectors, improved photometry showed that elliptical galaxy isophotes exhibit small but measurable deviations from a simple elliptical shape. In general, elliptical galaxies can be divided into `boxy' and `disky' objects, quantified by a Fourier expansion around the best-fitting ellipse \citep[e.g.][]{bender88}. This property was found to correlate with other quantities \citep[such as rotational support, and even X-ray luminosity:][]{bender89}, indicating a close link between the morphology and evolution of these objects. Deep imaging studies have also shown that at very large radii, early-type galaxies can show faint asymmetric or `shell'-like features \citep[e.g.][]{schweizer90}. Such `fine structure' features are generally interpreted as evidence for interactions of some kind, perhaps remnants of the galaxy's formation.
Colour gradients were also observed, suggesting that early-type galaxies were not composed of a single population of stars, but were in fact composite populations, albeit rather smoothly varying \citep[e.g.][]{peletier90}. With further improvement in observational data, it became possible to study the individual elemental absorption lines from these stars, allowing the effects of increasing age and metallicity (which have a degenerate influence on broad-band colours) to be decoupled. By comparing the strength of various absorption lines (via equivalent width measurements, known as line-strength indices) in galaxies and in stellar population models, it was found that early-type galaxies are generally old, evolved systems, but that a significant fraction still appear to contain young stellar populations.\looseness-2
The spread in the apparent (luminosity-weighted) age of early-type galaxies is approximately 2--15~Gyr \citep[e.g.][]{trager00}. The majority of studies on significant samples of objects have been limited to aperture measurements, thus ignoring information on the spatial distribution of different populations within a given object. Moreover, because young populations are generally brighter than old populations, a small fraction of young stars {\em by mass} can result in a dramatic reduction in the integrated {\em luminosity-weighted} age of the system. This spread in age is therefore indicative of a spread in recent star-formation histories between these galaxies. How are the young stars distributed in these galaxies? And what physical conditions lead to the presence of young stars in some galaxies and not others?\looseness-2
The importance of early-type galaxies in the context of galaxy formation and evolution is paramount. The evolved nature of these objects endow them with a wealth of so-called `fossil' evidence of the processes of galaxy evolution. In the nearby universe, we can study these objects in far greater detail than at higher redshifts, allowing us to uncover clues from which we can re-trace the steps along the evolutionary path of the galaxy. From the above empirical evidence, early-type galaxies in the nearby universe can show signs for relatively recent formation: disturbed morphologies, kinematic sub-components, and young stellar populations. However, studies at high redshift indicate the existence of a population of objects which have properties similar to those of `conventional' early-type galaxies: smooth, spheroidal morphologies, dynamically hot, with evolved populations and little or no ongoing star-formation \citep[e.g.][]{franx03,chapman04,treu05}. Consolidating the various observational evidence concerning the origin and evolution of early-type galaxies remains a key challenge for galaxy formation theories.\looseness-2
\newpage
\section{The {\texttt {SAURON}}\/ Survey}
The development of the study of early-type galaxies is illustrative of how many advancements in astronomy are linked to advancements in instrumentation (not just telescope aperture). Until recently, one of the limiting factors in our understanding of these extended objects was having information only along one or a few axes from a long-slit spectrograph. Integral-field spectroscopy provides the next leap forwards in our perspective on these galaxies. At the forefront of this development is the {\texttt {SAURON}}\/ survey. In response to the aforementioned scientific questions, we have undertaken a survey of 72 representative early-type galaxies (24 each of ellipticals (Es), lenticulars (S0s) and Sa spiral bulges) using our custom-built panoramic integral-field spectrograph, {\texttt {SAURON}}, mounted on the William Herschel Telescope (WHT), La Palma \citep[Paper I]{bacon01}. With this survey, we map both the stellar kinematics and line-strengths within the 4800-5300\AA\/ wavelength region, and extract properties of the ionised gas where present. A review of the project aims is given in \citet[Paper II]{dezeeuw02}, and maps of the stellar kinematics, ionised gas properties and absorption line-strengths of the 48 E and S0 galaxies are given in \citet[Paper III]{emsellem04}, \citet[Paper V]{sarzi05} and \citet[Paper VI]{kuntschner05} respectively (for a summary of Paper V, see the contribution of Falc\'on-Barroso et al. in these proceedings).
\subsection{Stellar Kinematics with {\texttt {SAURON}}}
\label{sec:sau_zoo}
\begin{figure}
\begin{center}
\includegraphics[width=14.5cm, angle=0]{Fig1_lr.eps}
\end{center}
\caption{Different velocity field morphologies observed with {\texttt {SAURON}}. (a) NGC\,821, (b) NGC\,4550, (c) NGC\,4660, (d) NGC\,4477, (e) NGC\,4486, (f) NGC\,5813. The maximum and minimum velocities (in $\mbox{km s}^{-1}$) plotted are indicated in the vertical tab. The data are spatially binned using the Voronoi tesselation method of \citet{cappellari03} to obtain a minimum signal-to-noise ratio of 60 per spectral resolution element, and isophotes are overlaid. (See Paper III for details)}
\label{fig:sauron_zoo}
\end{figure}
The stellar kinematics presented in Paper III exhibit a wealth of structure. Figure \ref{fig:sauron_zoo} shows examples of the kinds of different velocity field morphologies found in our sample. There are a large number of objects which show symmetric rotation around a single axis, consistent with a disk-like structure embedded in a hotter bulge-like component (e.g. NGC\,821, Figure \ref{fig:sauron_zoo}a). Figure \ref{fig:sauron_zoo}b shows a dramatic example of two counter-rotating disks with different scale heights in NGC\,4550, unambiguously determined from these data. Secondary central disks are also present in a number of objects (e.g. NGC\,4660, Figure \ref{fig:sauron_zoo}c), visible as a second peak in the velocity distribution near the center of the object. As well as aligned rotation, several objects show a rotation axis that is misaligned with the local photometric minor-axis, and which can vary in orientation across the field (e.g. NGC\,4477, Figure \ref{fig:sauron_zoo}d), indicating a non-axisymmetric system, such as a bar or triaxial figure. Some galaxies show negligible rotation (e.g. M87, Figure \ref{fig:sauron_zoo}e), of which most are giant elliptical galaxies. And several objects show KDCs, which clearly rotate around a different axis compared to the main body of the galaxy, and in some cases in the opposite direction (e.g. NGC \,5813, Figure \ref{fig:sauron_zoo}f).\looseness-2
Generalising these different kinematic morphologies, there are basically two types of behaviour found in the velocity maps of our sample galaxies: galaxies exhibiting regular and ordered rotation across the field, which is generally flat or increasing at the edge of our observations (typically 1~R$_e$) - these correspond to objects like those shown in Figure \ref{fig:sauron_zoo}a-d; and galaxies that show little rotation, except for a possible KDC, and which have decreasing or zero rotation at the edge of our field-of-view - these correspond to objects like those shown in Figure \ref{fig:sauron_zoo}e-f. We refer to these two types of object as `fast-rotators' and `slow-rotators' respectively. This separation of objects based on the morphology of their velocity fields does not correlate exactly with the classical division of `elliptical' and `lenticular'. More specifically, there are a number of fast-rotating objects classified as elliptical from their surface brightness morphology alone, even though dynamically they have more in common with lenticular objects \citep[see also][]{kormendy96}.
\subsection{Kinematically Decoupled Components with {\texttt {SAURON}}}
We focus here on the KDCs found in our {\texttt {SAURON}}\/ observations. As described in the introduction, the presence of a KDC is often taken as evidence for a major galaxy merger, in which the core of one galaxy becomes embedded in the center of the merger remnant. Due to the random orientation of the merger, this scenario can naturally account for the generally misaligned orientation of the embedded sub-component. The KDCs found with {\texttt {SAURON}}\/ tend to reside in slow-rotating galaxies, where the outer parts are not rotating significantly. Such a hot, pressure-supported outer body may also be the natural consequence of a major merger event. In the hierarchical paradigm, larger systems form more recently, and so such mergers should have happened quite recently.
The alternative scenario for the formation of a KDC is to `grow' the KDC via star-formation {\em in situ}. To explain the misalignment of the KDC, the material from which the stars form must itself be kinematically misaligned with the rest of the galaxy. It is difficult to explain how this could occur for material with an internal origin, accumulated from the ejecta of stellar evolution. Therefore the gas would most likely have an external origin.
In both of these scenarios, the KDC would most likely be composed of a different stellar population to that of the main galaxy. In general, studies using both ground- and space-based imaging and spectroscopy have found little evidence to suggest that KDCs have very different star-formation histories from their host galaxies \citep[e.g.][but see \citealt{bender92}]{forbes96,carollo97,davies01}. This suggests that, rather than having formed by the most recent merger, KDCs (or at least the stars from which they are made) were formed at early epochs.
Integral-field spectroscopy is the ideal tool for studying KDCs. Measuring the stellar kinematics and populations over a significant area of the galaxy provides an accurate picture of the KDC shape and orientation, and also allows structures to be directly associated by their spatial distribution. Figure \ref{fig:sau_kdcs} presents preliminary stellar population analyses of six galaxies from the {\texttt {SAURON}}\/ survey which exhibit clear and well-resolved KDCs. The velocity fields for these galaxies are given in the insert. The small symbols indicate the line-strength measurements obtained in every spatially-binned aperture of our data, typically involving a few hundred points. The large symbols show the same values averaged along isophotes. Lighter symbols imply smaller radii. Overplotted on the line-strengths is a grid of varying age (horizontal lines) and metallicity (vertical lines) from the stellar population models of \citet{vazdekis99}. This figure shows that, in all cases, the central KDC shows very little evidence of young stars. Indeed, these objects are predominantly old ($> 8$~Gyr), with remarkably small age gradients. All objects do show a smoothly increasing metallicity towards the center of the galaxy. Such metallicity gradients are common in early-type galaxies, however, and do not strongly distinguish these galaxies from non-KDC objects. We note that our parent sample of 48 objects is representative, not complete, and that examples of large KDCs with younger stellar populations may well exist, but are not found in our survey.
\begin{figure}
\begin{center}
\includegraphics[height=12cm, angle=0]{Fig2_lr.eps}
\end{center}
\caption{Line-strength indices of six galaxies from the representative sample of 48 E/S0s presented in Paper III which show clear KDCs. Small symbols show the values measured in each (spatially binned) {\texttt {SAURON}}\/ aperture. Large symbols show values averaged along isophotes, with lighter symbols indicating smaller radii. In all cases, the populations are relatively old, and show a steady metallicity increase towards the centre with very little change in age between the outer body and KDC region. The inset images show the {\texttt {SAURON}}\/ velocity fields for reference, where the spatial scale is in arcseconds.}
\label{fig:sau_kdcs}
\end{figure}
From this analysis, it appears that the KDCs in our sample are composed of old, evolved stellar populations, which, while exhibiting dramatically different kinematics from the rest of the galaxy, show little evidence to suggest a significantly different evolution to that of the main body of the galaxy. This is consistent with previous findings, and implies that these KDCs were formed early in the evolution of these galaxies. Although these objects appear kinematically peculiar, in a triaxial potential such orbital configurations may remain stable over a Hubble time. Over long timescales ($\sim 10$~Gyr), differences in stellar population age become less pronounced, making populations which differ in age by a few Gyr much more difficult to separate in older objects. All of this points to a picture where these KDCs are formed at early epochs, and then evolve quiescently until the current day. The specific formation mechanism, via either galaxy merging or `growth' from accreted material, is difficult to determine, as whatever differences exist between the star-formation histories of the KDC and its host galaxy are now diluted by the sands of time.
An alternative explanation may be that the {\em stars} in these systems formed at early epochs, but the KDC galaxies we observe have themselves {\em assembled} only recently via a dissipationless merger between two gas-free spheroidal systems. It has been shown that such `dry mergers' do occur, at least in cluster environments \citep{vdokkum99}, and that they may play a significant role in the latter stages of massive galaxy assembly \citep{khochfar03,bell05}. Distinguishing this scenario from an intrinsically early formation of KDC galaxies as we see them today is not trivial. An indicator of assembly at early epochs may be the presence of old, disk-like KDCs, since any pre-existing stellar disk would be significantly dynamically heated during a recent major merger. Determining the dynamical structure of the KDCs themselves is beyond the scope of this contribution, but will be considered in forthcoming papers of the {\texttt {SAURON}}\/ series. Also, in the dry-merger scenario, the remnant KDC is formed by the more compact and tightly-bound progenitor. It is difficult to reconcile why the progenitors, while strongly differing in density, should have such similar stellar populations.
\section{{\texttt {OASIS}}\/ Observations of Galaxy Centres}
The central regions of early-type galaxies are crucial to our understanding of galaxy formation and evolution. Since the advent of the Hubble Space Telescope (HST), high spatial-resolution studies of these galaxy nuclei have made key discoveries in their structure, which are apparently linked to their large-scale properties. Spectroscopic studies with HST have given the strongest evidence for the existence of super-massive black holes in the centres of galaxies other than our own \citep[e.g.][and references therein]{kormendy95}. These black holes are thought to be ubiquitous, at least amongst spheroidal systems, and have masses which are closely linked to the mass of the host system \citep[e.g.][]{ferrarese00,gebhardt00}. Galaxy formation theories suggest that the co-evolution of black holes and their host galaxies is a fundamental aspect of how galaxies form, where accretion of material onto the black hole controls star-formation efficiency via feedback mechanisms \citep[e.g.][]{silk98,granato04,binney04,springel05}.\looseness-2
In addition, the central light profiles of early-type galaxies show differing gradients at HST resolution, ranging from a continuously rising or steepening profile (`cusp'), to a flat profile (`core') with a distinct break in slope from the outer parts \citep{jaffe94,lauer95,lauer05}. These light profile types correlate with other global galaxy parameters, such as disky or boxy isophotes, and the degree of rotational support \citep{faber97}. The role of the central black hole is thought to be important in determining the shape of the central light profile. Cusp nuclei form through the build-up of material in the deep potential well of the black hole environment, which then forms stars giving a central peak in the light profile. Core nuclei, on the other hand, are thought to form through `scouring' of the central regions by a coalescing black hole binary. This binary forms as the result of a galaxy merger, where the central black holes of the merging galaxies quickly fall to the center of the potential. The binary hardens by losing angular momentum to the nearby stars, which are ejected from the core region. The result is a deficit of stars in the environment of the central black hole, giving a flat light profile \citep[e.g.][]{quinlan97,merritt01}.\looseness-2
For our survey, the spatial sampling of the {\texttt {SAURON}}\/ spectrograph was set to 0\farcsec94 $\times$ 0\farcsec94 per spatial element. This gives maximum field of view (33\arcsec\/ $\times$ 43\arcsec), but often undersamples the typical seeing at La Palma (0\farcsec7--0\farcsec8 FWHM). For the main body of the galaxies, this is not important, since the properties of early-type galaxies generally vary smoothly on scales larger than the seeing. In the central regions of these objects, however, the light profile can rise steeply, becoming unresolved even in the best seeing. In these regions, the coarse spatial sampling of {\texttt {SAURON}}\/ becomes a problem, resulting in poorly resolved measurements of the galaxy centres. For this reason, we are conducting a campaign of follow-up observations of the centres of the {\texttt {SAURON}}\/ survey E and S0 galaxies, using high spatial-resolution integral-field spectroscopy. The first steps in this project have been taken with the {\texttt {OASIS}}\/ spectrograph during its operation at the Canada-France-Hawaii Telescope (CFHT), on Hawaii. In July 2003, {\texttt {OASIS}}\/ was moved to the WHT to operate behind the NAOMI Adaptive Optics (AO) system, which is described by Benn et al. in these proceedings.\footnote{The capabilities of {\texttt {OASIS}}\/ at the WHT are described in \citet{mcdermid04b}}
In total, 28 of the 48 {\texttt {SAURON}}\/ E and S0 galaxies have been observed, using a wavelength region and spectral resolution comparable to that of {\texttt {SAURON}}. The observations were seeing-limited, due to a lack of suitable natural guide stars. However, the spatial sampling of {\texttt {OASIS}}\/ was set to 0\farcsec27 $\times$ 0\farcsec27 per spatial element, thus adequately sampling even the best conditions on Mauna Kea. The median spatial-resolution of the {\texttt {OASIS}}\/ observations is almost a factor two better than that of our {\texttt {SAURON}}\/ observations of the same objects. A full description of the sample, observations and data reduction, as well as measured parameter maps is given in \citet{ mcdermid05}.
\subsection{Comparing {\texttt {SAURON}}\/ and {\texttt {OASIS}}}
\label{sec:oas_sau_comp}
It is instructive to explore what additional information we obtain by observing the {\texttt {SAURON}}\/ objects with higher spatial-resolution by considering some illustrative examples. Figure \ref{fig:sau_oas_comp}a shows the stellar velocity field of NGC\,5982 measured with {\texttt {SAURON}}\/ and {\texttt {OASIS}}. The center appears misaligned in the {\texttt {SAURON}}\/ observations, indicating a probable decoupled component or kinematic twist. This is confirmed by {\texttt {OASIS}}, showing that the misaligned component actually rotates rapidly around the major axis of the galaxy, indicating a prolate core.\looseness-2
\begin{figure}
\begin{center}
\includegraphics[width=14cm, angle=0]{Fig3_lr.eps}
\end{center}
\caption{Comparison of {\texttt {SAURON}}\/ ({\em left}) and {\texttt {OASIS}}\/ ({\em right}) observations of six early-type galaxy centres. The field sizes are the same between the pairs of panels, as are the intensity scales. Isophotes from the integrated spectra are overlaid. The maximum and minimum levels are given in the inserted tab, where units are $\mbox{km s}^{-1}$\/ for velocity fields and \AA\/ for line-strength indices. (a) Stellar velocity field of NGC\,5982. (b) Stellar velocity field of NGC\,4621. (c) Stellar velocity field of NGC\,4382. (d) Stellar velocity field of NGC\,5845. (e) H$\beta$ absorption map of NGC\,3489. (f) Gas velocity field of NGC\,4486.}
\label{fig:sau_oas_comp}
\end{figure}
Figure \ref{fig:sau_oas_comp}b shows the stellar velocity field of NGC\,4621. The `kink' in the zero-velocity contour observed with {\texttt {SAURON}}\/ is resolved into a clearly counter-rotating core, which was previously found using {\texttt {OASIS}}\/ with AO \citep{wernli02}. Similarly, Figure \ref{fig:sau_oas_comp}c shows that NGC\,4382 contains a clear decoupled core when observed with {\texttt {OASIS}}. Figure \ref{fig:sau_oas_comp}d further shows that it is not only misaligned and counter-rotating central components which are measured with far greater accuracy using {\texttt {OASIS}}. NGC\,5845 harbours an edge-on, thin central disk, which is clearly visible in HST imaging. The {\texttt {SAURON}}\/ observations indicate the presence of a central rotating component, although the amplitude of rotation does not conclusively indicate a dynamically cold disc. The {\texttt {OASIS}}\/ observations, on the other hand, beautifully resolve this disk component, showing rapid rotation tightly confined to the major-axis of the galaxy.
Other spectral properties show improvement when measured with {\texttt {OASIS}}. Figure \ref{fig:sau_oas_comp}e compares the H$\beta$ absorption index observed with {\texttt {SAURON}}\/ and {\texttt {OASIS}}\/ for NGC\,3489, which shows strong H$\beta$ absorption in the center. The improved image-quality of the {\texttt {OASIS}}\/ observations shows that the strength of absorption in the center is increasing within the resolution limit of {\texttt {SAURON}}. This makes a significant difference to the luminosity-weighted age determined in the central lenslets. Likewise, the kinematics of ionised gas can show complex distributions around galaxy nuclei. Figure \ref{fig:sau_oas_comp}f shows the gas velocity field for NGC\,4486 (M87). {\texttt {SAURON}}\/ shows a stream of gas spiraling to the center of the galaxy, connecting smoothly to a component rotating around the center. The {\texttt {OASIS}}\/ observations show that this central component rotates much more rapidly than the surrounding gas, and in fact is probably related to the Keplerian disk of material orbiting the central super-massive black hole \citep{ford94}.\looseness-2
\subsection{Stellar Kinematics of Young Nuclei}
Our {\texttt {SAURON}}\/ observations show that in the majority of galaxies exhibiting young stellar populations, the young stars are strongly concentrated towards the galaxy center, rather than distributed evenly throughout the object. This is in contrast to the distribution of ionised gas, which can exist in regular, large-scale structures extending beyond an effective radius (Paper V, see also Falc\'on-Barroso et al. in this proceedings). It would seem that only in the central regions, deep in the galaxy potential well, are the conditions suitable to trigger the conversion of the gas reservoir into stars. What are the causes and implications of this star-formation? What is the relationship between the star-formation and the stellar potential? What is the significance of this star-formation to the global evolution of the system?
\begin{figure}
\begin{center}
\includegraphics[width=14cm, angle=0]{Fig4_lr.eps}
\end{center}
\caption{{\em Central panel:} Line-strength indices measured within an R$_e/8$ circular aperture on our {\texttt {SAURON}}\/ data from Paper VI, overplotted with model stellar population predictions from \citet{vazdekis99}. Large symbols show galaxies observed in our {\texttt {OASIS}}\/ sub-sample. Small points show the remaining objects in the full {\texttt {SAURON}}\/ E/S0 sample. Triangles and circles indicate E and S0 galaxies respectively. Open symbols show `field' objects, and filled symbols `cluster' objects, as described in Paper II. {\em Insert plots:} {\texttt {SAURON}}\/ stellar velocity fields of the indicated galaxies, with a secondary `zoom' on the central regions showing our high spatial-resolution {\texttt {OASIS}}\/ measurements, except in the case of NGC\,4150, where we present data from the {\texttt {GMOS}}\/ integral-field spectrograph.}
\label{fig:oas_kdcs}
\vspace{3cm}
\end{figure}
Our {\texttt {OASIS}}\/ observations are the ideal tool to investigate these questions. The central panel of Figure \ref{fig:oas_kdcs} shows the distribution of aperture line-strength values, measured within an R$_e/8$ circular aperture with our {\texttt {SAURON}}\/ data. The large symbols indicate objects also observed with {\texttt {OASIS}}. Of the {\texttt {OASIS}}\/ sub-sample, four objects show relatively strong H$\beta$ absorption, with luminosity-weighted age $<5$~Gyr. The surrounding panels show, for each of these four objects, the {\texttt {SAURON}}\/ stellar velocity field, with a central insert showing the stellar velocity field observed at higher spatial-resolution with either {\texttt {OASIS}}, or in the case of NGC\,4150, with {\texttt {GMOS}}\/ on Gemini-North.
We find that three of these four galaxies (NGC\,3032, NGC\,4150, and NGC\,4382) with strong H$\beta$ absorption also exhibit a clearly decoupled component in their stellar kinematics. The KDCs in NGC\,3032 and NGC\,4382 have been discovered from our {\texttt {OASIS}}\/ observations \citep{mcdermid05}. The fourth object, NGC\,3489, does not exhibit the clear counter-rotation that the other objects show. Closer inspection of the velocity field does, however, indicate that there is a central fast-rotating sub-component. The curve of zero velocity also shows some evidence of a (spatially resolved) mild twist across the galaxy center, indicating that there may be some departure from regular axisymmetric rotation in the central parts.
Figure \ref{fig:oasis_kdcs_pops} presents the stellar population parameters, comparable to Figure \ref{fig:sau_kdcs}, but with the addition of our high spatial-resolution values inside the central 3\arcsec, shown as open symbols. This demonstrates that the central regions, where the decoupled components are located, show a significant age gradient towards younger populations. {\texttt {SAURON}}\/ data points inside 3\arcsec\/ are excluded here, as they may be biased by PSF effects.
\begin{figure}
\begin{center}
\includegraphics[width=14cm, angle=0]{Fig5_lr.eps}
\end{center}
\caption{Similar to Figure \ref{fig:sau_kdcs}, but for the four galaxies with central luminosity-weighted age $<5$~Gyr highlighted in Figure \ref{fig:oas_kdcs}. {\texttt {OASIS}}\/ and {\texttt {GMOS}}\/ measurements are included (open symbols) in addition to the {\texttt {SAURON}}\/ data (filled symbols), of which the central 3\arcsec\/ are excluded to avoid PSF effects. All four objects show a sharp decrease in luminosity-weighted age within the central few arcseconds, coincident with the KDC component in the high spatial-resolution data.\looseness-2}
\label{fig:oasis_kdcs_pops}
\vspace{3cm}
\end{figure}
\subsection{Stellar Populations of Kinematically Decoupled Components}
Comparing Figures \ref{fig:oasis_kdcs_pops} and \ref{fig:sau_kdcs}, it seems that there are basically two types of KDC: apparently large KDCs, which have populations indistinguishable from their host galaxy, and which exist in galaxies that tend to show little ordered rotation; and small KDCs, which are composed of young populations, and are found in galaxies that exhibit significant rotation. This was found, however, by considering only the four youngest objects in our {\texttt {OASIS}}\/ sub-sample. Here we explore the relationship, if any, between the intrinsic size of KDCs found in our sample (either from {\texttt {SAURON}}\/ or {\texttt {OASIS}}\/ data) and the stellar population within the KDC. We only consider KDCs which are unambiguously distinct from the rest of the galaxy, showing significantly misaligned or counter-rotating stellar motions (for example, NGC \,3489 shown above is not included here). This results in a list of 13 objects. To the 9 KDCs we have already considered in the above discussion, we add four objects: NGC\,4621 and NGC \,5982 - these objects appear to have KDCs from our {\texttt {OASIS}}\/ observations (see section \ref{sec:oas_sau_comp}); and NGC\,7332 and NGC\,7457 - although not observed with {\texttt {OASIS}}, these galaxies show small (unresolved) but distinct central twists in their {\texttt {SAURON}}\/ velocity fields \citep[Paper III, see also][]{falcon04}.
First we must quantify the spatial size of the KDC. From the map of mean stellar velocity, it is possible to determine where the contribution of the KDC to the line-of-sight velocity distribution becomes comparable to that of the outer body of the galaxy. At this point, the mean net rotation shows a minimum, and in some cases reaches zero, isolating the KDC from the rest of the galaxy. This point can be used to make an estimate of the KDC size, although there may still be an appreciable fraction of the KDC beyond this radius. For KDCs larger than the seeing FWHM, this will generally underestimate the size. For KDCs that appear similar in size to the seeing FWHM, this estimate may be too large. With this in mind, we use the largest diameter across the center of the galaxy which connects the zero velocity curve on opposite sides of the KDC.\looseness-2
We convert this apparent diameter into an intrinsic diameter using distances from surface brightness fluctuation studies \citep{tonry01}. Figure \ref{fig:kdc_size_age} plots the estimated intrinsic diameter of each KDC against the luminosity-weighted age derived from line-indices measured within the central pixel of the highest spatial-resolution data available (either {\texttt {SAURON}}, {\texttt {OASIS}}\/ or {\texttt {GMOS}}). From this plot, it can be seen that the intrinsically smallest KDCs (less than 400pc in diameter) also tend to exhibit the youngest populations. KDCs of around 1~kpc or more in diameter are all older than 8~Gyr.
There are a number of implications to these findings. In the initial discussion, we mentioned that KDCs found in early-type galaxies have often been used as evidence of recent merging activity. We also described how the spread in apparent ages of early-type galaxies has supported the scenario where these objects are amongst the last to form, as a hierarchical sequence of galaxy mergers. However, inspection of the stellar populations of `classical' KDC galaxies show that the KDC populations are coeval with the rest of the galaxy, and exhibit no peculiar properties to distinguish KDC galaxies from ordinary galaxies. We therefore cannot directly associate the {\em dynamical} evidence for recent merging and the {\em population} evidence for such merging by considering these objects. From our high spatial-resolution integral field spectroscopy of the subset of {\texttt {SAURON}}\/ survey galaxies, however, we have found that the youngest galaxies do exhibit KDCs which can be directly associated with the young population.
\begin{figure}
\begin{center}
\includegraphics[width=12cm, angle=0]{Fig6_lr.eps}
\end{center}
\caption{Luminosity-weighted age of the central aperture plotted as a function of KDC intrinsic diameter. Ages are computed from the stellar population models of \protect\citet{thomas03}, and include element abundance ratios as a free parameter. Circlular symbols denote fast-rotating galaxies; squares denote slow-rotating galaxies. Arrows denote cases where the apparent diameter of the KDC is less than four times the seeing FWHM, implying that we only obtain an upper limit. Filled symbols have both {\texttt {SAURON}}\/ and {\texttt {OASIS}}\/ or {\texttt {GMOS}}\/ observations. Open symbols have only {\texttt {SAURON}}\/ data.}
\label{fig:kdc_size_age}
\end{figure}
A tempting, but naive interpretation of this would be that the case of the missing young KDCs has been solved, and that these sub-components show both stellar dynamical and population evidence for recent merging. However, comparing the properties of the host galaxies reveals that the 100~pc-scale KDCs are not directly comparable to their kpc-scale counterparts. The different plotting symbols in Figure \ref{fig:kdc_size_age} denote the separation of objects into fast-rotators and slow-rotators, as discussed in section \ref{sec:sau_zoo}. Here we find a strong connection between the large, old KDCs and non-rotating galaxies, and the small, generally young KDCs and fast-rotating galaxies. The global properties of these objects, therefore, are rather different.
The fraction of the galaxy's mass contained in the KDC is also rather different between the two types of object. The young KDCs generally extend to only $\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}} 0.1$R$_e$, whereas the older KDCs can extend up to $\sim 0.4$R$_e$. Although this is only a rough indication of the actual mass fraction of the KDC, we should remember that the small KDCs have also young populations with low M/L values, thus contributing significantly to the light, but relatively little in mass.\looseness-2
We therefore conclude that these two manifestations of KDCs are distinct, both in origin and destiny. On one hand, large, old KDCs are consistent with having formed through major merging (having a large mass fraction, and residing in dynamically hot triaxial systems), being either dissipative at early epochs, followed by largely quiescent evolution thereafter; or dissipationless at more recent times. On the other hand, compact, young KDCs appear to be consistent with more modest dissipational accretion events, residing in fast-rotating disk-like galaxies containing plenty of star-forming material. In time, this young population will evolve and diminish in brightness, and the KDC may reduce in apparent size as it becomes less distinct above the background of the rest of the galaxy. The precise connection between the dynamical decoupling of the stars and the recent or ongoing star-formation in these young KDCs requires further investigation, and it is not yet clear whether the decoupled stars are formed from seed material which is dynamically distinct from the rest of the galaxy; or whether pre-existing decoupled stars themselves trigger star-formation as material enters the KDC potential. The dynamical structure of the young KDCs is also uncertain, and difficult to determine given their embedded nature and the limited spatial resolution of our data. However, the centrally concentrated nature of the young stars suggests an efficient transfer of material to the galaxy center. Furthermore, although close to being aligned, most (if not all) of the young KDCs exhibit some offset in their rotation axis to that of the rest of the galaxy. This points towards the involvement of asymmetry in the central potential, and possibly the presence of a nuclear bar. In any case, these objects seem to be driving the large spread in age observed for early-type galaxies, and the existence of a small KDC seems closely linked to the associated star-formation.
\section{Conclusions}
The study of nearby early-type galaxies is crucial to our understanding of galaxy formation and evolution, and sets key constraints for complementary high-redshift studies. The advent of integral-field spectroscopy is heralding a new era in our observational understanding of these objects. In this contribution, we have reviewed some of the key observational constraints on how early-type galaxies form and evolve, looking specifically at the connection between the apparent spread in luminosity-weighted age of these galaxies and the existence of kinematically decoupled sub-components, in the context of galaxy formation through hierarchical merging. From our wide-field {\texttt {SAURON}}\/ integral-field spectroscopy, we have shown that the clearly detected KDCs in our sample exhibit old stellar populations, and are coeval with their host galaxies. This strongly suggests that these KDCs are not the product of recent merging, but instead were formed at early epochs (either through merging or from accreted material which subsequently formed stars), and that no such significant evolutionary event has occurred in these objects since. We note, however, that formation via recent `dry' merging cannot be strongly excluded.
We have introduced a series of follow-up observations on the central regions of the {\texttt {SAURON}}\/ E and S0 galaxies, obtained at higher spatial-resolution using the {\texttt {OASIS}}\/ integral-field spectrograph. These observations reveal a wealth of structure in the central regions of these galaxies which is poorly resolved with {\texttt {SAURON}}. In particular, we find that the youngest galaxies in our current {\texttt {OASIS}}\/ sub-sample show clear evidence of harbouring compact central KDCs, which were previously unresolved by {\texttt {SAURON}}\/ or other studies. The apparent size of these KDCs are consistent with the spatial extent of the young populations, implying that the two phenomena are directly linked.
Estimating the intrinsic size of the various KDCs found in our full sample, either with {\texttt {SAURON}}\/ or {\texttt {OASIS}}, we find that, whilst small (100~pc-scale) KDCs can exhibit a large range of ages, larger (kpc-scale) KDCs are generally old ($> 8$~Gyr). We find that there is a connection between these two types of KDC and their host galaxy, such that young, compact KDCs are found in fast-rotating objects, whereas old, large KDCs are found in slowly-rotating systems. We conclude that these objects have distinct formation histories, and that these young KDCs are not strong evidence for the formation of early-type galaxies via recent merging. Further investigation will reveal how the dynamical decoupling of the central stars is related to recent or ongoing star-formation in these objects.
We have extended this sub-sample of the 48 {\texttt {SAURON}}\/ E/S0 by obtaining similar quality data for the lowest velocity dispersion objects using the {\texttt {GMOS}}\/ spectrographs on Gemini North and South, and hope to observe the remaining objects in due course with {\texttt {OASIS}}\/ at the WHT. This collection of data will be used to complement ongoing studies based on the {\texttt {SAURON}}\/ project, including using them to constrain general dynamical models. With these high-spatial-resolution integral-field measurements, and including HST spectroscopy where available, we will place strong constraints on the central orbital structure, in order to probe the formation of central super-massive black holes (Cappellari \& McDermid, 2005). These seeing-limited observations will form the foundation of future studies that will make use of the enhanced spatial-resolution offered by adaptive optics facilities. Development of laser guide star systems, such as GLAS (see the contribution by Rutten in these proceedings) will greatly expand the sky coverage for AO-fed integral-field spectrographs, such as {\texttt {OASIS}}\/ (WHT), {\texttt {SINFONI}}\/ (VLT), {\texttt {OSIRIS}}\/ (Keck) and {\texttt {NIFS}}\/ (Gemini), allowing systematic study of large, representative samples of nearby nuclei at high spatial resolution.\looseness-2
\newpage
{\bf ACKNOWLEDGMENTS}
RMcD would like to thank R.~Bacon for presenting this talk on his behalf - Baby McDermid was born one week later. It is a pleasure to thank the ING staff, both for the organisation of a fruitful and timely workshop, and for their continuuing support of the {\texttt {SAURON}}\/ project and instrument.
{\small Based in part on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Particle Physics and Astronomy Research Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), CNPq (Brazil) and CONICET (Argentina).}
|
1,314,259,996,510 | arxiv | \section{Introduction}
The Laser Interferometer Space Antenna (LISA)~\cite{Audley:2017drz} is a space-based Gravitational Wave (GW) detector scheduled to be launched by ESA, with junior
partnership from NASA, around 2034. LISA is currently in Phase A (assessment of feasibility), and among the items that are being investigated is the
scientific return as function of the mission design. The LISA observatory will be comprised by three spacecraft trailing the Earth around the Sun (by 10--15 degrees),
on an equilateral configuration with arms of about 2.5 million km. Along these arms, laser beams will be exchanged to monitor changes in the proper distance
between the free-falling test masses carried by the spacecraft, in order to detect GW signals with frequencies in the mHz band.
The GW sky at such low frequencies is expected to be much more crowded than in the band $\gtrsim 10$--$100$ Hz accessible
from the ground~\cite{TheLIGOScientific:2014jea, Advanced-Virgo, Hild:2010id}. Indeed, LISA sources are intrinsically long-lived, i.e. they are expected to be active for the whole duration
of the mission (which is nominally of 4 yr), and they are foreseen to number in the tens of thousands~\cite{Audley:2017drz} (including resolved sources alone).
In more detail, the strongest sources, with Signal to Noise Ratios (SNRs) up to thousands, are expected to be the mergers of massive
black hole binaries (with masses $\sim 10^4$--$10^7 M_\odot$), which could be between a few and hundreds depending on the (currently unknown) astrophysical formation scenario~\cite{Sesana2004,Sesana2005,Barausse2012,Klein2016,2019MNRAS.486.4044B}. Extreme mass ratio inspirals comprised of a massive black hole (with mass again $\sim 10^4$--$10^7 M_\odot$) and a smaller satellite stellar-origin
black hole (with mass $\sim 10$--$50M_\odot$) or neutron star may also be present, with very uncertain rates ranging from a few up to hundreds per year~\cite{Babak2017}.
LISA is also guaranteed to be able to observe a handful of white dwarf binaries that have already been observed in the electromagnetic band. These are referred to
as ``verification binaries''~\cite{verification}, and are just the tip of the iceberg of a much broader population of Galactic and extragalactic white dwarf binaries that are potentially
observable with LISA. Indeed, models of this population in the Milky Way predict that LISA should detect the signal of tens of thousands of resolved
Galactic white dwarf binaries at low frequencies~\cite{WD1,WD2,WD3,WD4,WD5}. Moreover, even more such sources will not have sufficient SNR to be detectable individually, but
will nevertheless constitute a formidable stochastic background signal, which will be potentially strong enough to degrade the mission's
sensitivity at low frequencies, i.e. to act as a foreground for other sources~\cite{cornish1,Adams:2013qma}.
After the first LIGO detection of GW150914~\cite{Abbott:2016blz}, it
was realized that binaries of stellar-origin black holes, which merge in the band of existing or future ground based interferometers, will
also be detectable in LISA much earlier (by a time ranging from weeks to years) than their merger~\cite{Sesana:2017vsj} (see also \cite{Wong:2018uwb, Gerosa:2019dbe}). While the presence of
these sources is a blessing, since their long low frequency inspiral allows for measuring
their parameters to high accuracy, thus permitting precise tests of their formation scenario~\cite{Nishizawa1,Nishizawa2,Tamanini:2019usx}, of General Relativity~\cite{dipole,Carson:2019rda,Gnocchi:2019jzp}, and possibly identify
their electromagnetic counterparts (if any)~\cite{caputo_sberna}, many of them may not be detectable individually. The unresolved signal
from stellar-origin black hole binaries is indeed foreseen to be present in the LIGO/Virgo band too, and should be detected
when those detectors reach design sensitivity. Based on the current estimates of this
background~\cite{LIGOScientific:2019vic}, its SNR in the LISA band is expected to be
around 53\footnote{This value is obtained (via~\cref{eq:SNR_definition}) by extrapolating the current estimates of the background
from stellar origin black hole and neutron star binaries in the LIGO/Virgo band to the LISA band.}
While this is not enough to degrade
the detector's sensitivity to resolved sources (unlike the case of Galactic white dwarf binaries), this
unresolved signal is expected to be certainly detectable once all resolved sources have been subtracted, by looking at the
auto-correlation of the data residuals.
Unlike the stochastic background from Galactic binaries, however, the background from stellar-origin black holes is
expected to be overwhelmingly extra-galactic in origin. As a result, while the unresolved signal from Galactic binaries
can for the most part be subtracted by exploiting its yearly modulation (induced by the constellation's motion around the Sun)
and its anisotropy~\cite{Adams:2013qma},
the background from stellar-origin black holes will be (to a very good approximation) isotropic and stationary. This makes its subtraction difficult,
and may hamper the detection of other, weaker SGWBs that may be present in the data.
Indeed, one of the most important goals of the LISA mission is the possible detection of SGWBs of exotic origin. While
some of these, if present, may be very strong (e.g. stochastic signals from the superradiance driven spin down of the astrophysical black hole population, which may occur in models of fuzzy dark matter~\cite{Brito1,Brito2} or in the presence of exotic physics near the event horizon~\cite{Barausse:2018vdb}), most signals
of cosmological origin are expected to be rather weak. Indeed, it is possible to construct scenarios in which phase transitions in the early Universe~\cite{Caprini:2019egz}, networks of topological defects (e.g. cosmic strings)~\cite{Auclair:2019wcv} and even inflationary constructions~\cite{Bartolo:2016ami} may induce a sufficiently large SGWB to reach the sensitivity of LISA (for a review of these models see for example~\cite{Caprini:2018mtu} and reference therein). However, for these relatively weak signals, the background from stellar-origin black holes will act as
a foreground and may jeopardize their detection, unless it is successfully subtracted.
Here, we introduce a template free approach for the detection and reconstruction of the LISA SGWB. Our method is based on a Principal Component Analysis (PCA; or singular value decomposition), and we show that it allows for the simultaneous detection and characterization of the foreground from stellar-origin black holes and a signal of unknown origin and spectral shape. While similar in spirit to~\cite{Karnesis:2019mph, Caprini:2019pxz}, where template free approaches were also put forward, our technique has the advantage of being very fast, because the determination of the posteriors becomes equivalent to solving a linear problem, and allows for
detecting signals up to ten times weaker than the one from stellar-origin black holes.
This paper is organized as follows: in~\cref{likelihood_sec} we describe the procedure to generate our data and the technique on which our analysis hinges. In~\cref{sec:results} we show the results obtained by applying these techniques to a set of mock signals. Finally, in~\cref{sec:conclusions} we draw our conclusions. This work contains two appendices: in~\cref{sec:noise_fg} we describe the noise model used in our analyses and in~\cref{sec:appendixdownsampling} we discuss the procedure to downsample our data.
\section{A principal component analysis for the stochastic background}\label{likelihood_sec}
We start by making the standard simplifying assumption that once all resolved sources have been subtracted, the LISA
data (i.e. the residuals) in a given channel are described by a stationary Gaussian process~\footnote{Residual non-Gaussianities
and non-stationarities may be present due to instrumental glitches and the subtraction procedure of thee resolved sources~\cite{Ginat:2019aed}.
To account for this, it is
possible to include additional parameters in the model for the power spectral density.}.
If we deconvolve the LISA data with the response function of the detector, as described in \cref{sec:noise_fg},
the resulting data $d(t)$ will satisfy
\begin{equation}\label{psd}
\langle d(f) d^*(f')\rangle=S(f)\delta(f-f')\,,
\end{equation}
where \begin{equation}d(f)=\int_{-\infty}^{\infty} d(t) \exp(-2 \pi i f t)dt\end{equation}
is the (complex) Fourier transform of the time domain data $d(t)$,
$\langle\ldots\rangle$ denotes
an ensemble average (i.e. an average over many data realizations), and $S(f)$ is referred to as the (double sided)
spectral density of the data $d$~\cite{Maggiore:1900zz}.
Since LISA samples the data for a finite
observation time $T$, the Fourier transform is defined only at discrete frequencies $f_i$ (with $i$ the frequency index ranging e.g. from $1$ to $n$)
spaced by $\Delta f=1/T$, and \cref{psd} becomes
\begin{equation}\label{psd2}
\langle d_i d^*_j\rangle=\frac{1}{\Delta f} S(f_i)\delta_{ij}\,,
\end{equation}
where $d_i\equiv d(f_i)$.
Since the residuals are Gaussian, the real and imaginary parts of the $d_i$ obey a Gaussian distribution
with variance set by \cref{psd2}, i.e.
\begin{equation}
\label{eq:gaussian}
p({\rm Re}\,d_i,{\rm Im}\,d_i)= \frac{1}{\pi S_i}e^{-[({\rm Re}\,d_i)^2+({\rm Im}\,d_i)^2]/S_i}\,.
\end{equation}
By changing variables to the absolute value ($|d_i|$) and phase
of the data, one obtains that the phase is uniformly distributed, while $|d_i|$ is described
by $p(|d_i|)=2|d_i| e^{- |d_i|^2/S_i}/S_i$.
Since different frequencies are uncorrelated, as per \cref{psd}, the likelihood can be obtained
by multiplying the probability distributions of the various sampled frequencies $f_i$. In particular,
by working not with $|d_i|$ but with $D_i\equiv |d_i|^2$, one obtains
\begin{equation}\label{likelihood0}
p(D_{i=1,...,n}|S)= \prod_i^n\frac{1}{S_i}e^{-D_i/S_i}\,,
\end{equation}
for the data in each channel.
At a fixed frequency $f_i$, the mean value of $D_i$ is $\mu_i=S_i$ and its variance
$\sigma^2_{i}=\mu_i^2= S_i^2$.
We now consider $\bar{D}_i$, which we define as the average of $D_i$ (with $i$ fixed) over $N\gg 1$ chunks in which we divide the time series.
By the central limit theorem, we can approximate the probability distribution
function for $\bar{D}_i$ with a Gaussian centered in $\mu_i=S_i$ and with variance $\sigma^2_{i}/N=\mu_i^2/N=S_i^2/N$.
The likelihood for the averaged data $\bar{D}_i$ then becomes
\begin{equation}\label{likelihood}
p(\bar{D}_{i=1,...,n}|S)\approx \frac{N^{n/2}}{(2\pi)^{n/2}}\prod_{i=1}^n \frac{1}{S_i}e^{-\frac{N(\bar{D}_i-S_i)^2}{2S_i^2}}\,.
\end{equation}
We can simplify the likelihood (which eventually will allow us to make the problem linear) by further noting that $\bar{D}_i\approx S_i$ near the peak of the likelihood, which
allows for writing
\begin{equation}\label{likelihood2}
p(\bar{D}_{i=1,...,n}|S)\approx \frac{N^{n/2}}{(2\pi)^{n/2}}\prod_{i=1}^n \frac{1}{\bar{D}_i}e^{-\frac{N(\bar{D}_i-S_i)^2}{2\bar{D}_i^2}}\,.
\end{equation}
Let us assume that the power spectral density $S$ is the sum of three contributions, from the signal, instrumental noise
and astrophysical foregrounds, i.e.
\begin{equation}\label{tot}
S(f)=S_{\rm signal}(f)+S_{\rm noise}(f)+S_{\rm foreground}(f)\,.
\end{equation}
For the signal, let us now assume the
form
\begin{equation}
\label{eq:signal_model}
S_{\rm signal}(f)=\sum_{j=1}^{m} a_j \delta_w(f-f^a_j)\,,
\end{equation}
where the $a_j$, $j=1,...,m$ are parameters to be determined, $f^a_j$ are pivot frequencies associated with the $a_j$, and the functions $\delta_w(F)$ are defined by
\begin{equation}\label{deltaw}
\delta_w(F)=\frac{1}{\sqrt{2\pi} w} e^{-\frac12 F^2/w^2} \,.
\end{equation}
Clearly, by evaluating \cref{eq:signal_model} at the sampled frequencies $f_i$, one obtains
\begin{equation}\label{model}
S^{\rm signal}_i=S_{\rm signal}(f_i)=\sum_{j=1}^{m} a_j \delta_w(f_i-f^a_j)\,.
\end{equation}
Note that in the limit $w\rightarrow 0$, one has $\delta_w(F) \rightarrow \delta(F)$. Therefore, by choosing $w\rightarrow 0$,
$f^a_j = f_i$ and $m = n$, the parameters $a_j$ would simply be the reconstructed values of the signal at the sampled frequencies.
Nevertheless, a non-zero $w$ allows for encoding the fact that the signal is expected to be a smooth function of $f$.
Moreover, allowing for $f^a_j \neq f_i$ and $m$ potentially lower than $n$ allows for extra flexibility in the
technique and will prove useful in practice, as we show in Sec.~\ref{sec:results}. Indeed,
we will show that in some situations it is useful to
work with a ``reduced'' basis, i.e. set $m\ll n$, as this
turns out to enhance stability of the results and also decreases
the dimensionality of the problem.
Indeed, one may even consider the number of
coefficients to be used as a free parameter of the model, and estimate its optimal
value within a Bayesian framework (i.e. by analyzing its posterior distribution or by using other criteria, as we discuss in the following).
While we leave this possibility for future work, in Sec.~\ref{sec:results} we briefly
investigate how our results change when the dimension of the working basis is reduced.
In practice, instead of working with the parameters
$a_j$, we utilize the rescaled parameters $\alpha_j=a_j/K_j$, $j=1,...,m$,
where the arbitrary normalizations $K_j$ are chosen to make the parameters $\alpha_j$ dimensionless and (as much as possible) of order
unity.\footnote{A possible choice is to take $K_j\sim | \bar{D}(f^a_j) - S_{\rm noise,exp}(f^a_j) -S_{\rm foreground,exp}(f^a_j) |$,
with $\bar{D}(f^a_j)$ set to the data $\bar{D}_i$ with frequency $f_i$ closest to $f^a_j$,
and with $S_{\rm noise,\,exp}$ and $S_{\rm foreground,\,exp}$ the
values of the instrumental noise and LIGO/Virgo foregrounds expected based on (i.e. maximizing) the priors. These
are given by \cref{noise} and \cref{fg} below, with $A=O=L=1$. We stress, however, that other prescriptions for the $K_j$ are also possible and our results are robust with respect
to the specific choice made.} We find that this improves the numerical stability of the technique.
Clearly, this is obviously equivalent to multiplying ${\delta_w(f-f_j)}$ by $K_j$. We then assume uniform priors for the parameters $\alpha_j$.
For the instrumental noise, let us assume that we know the functional form
of the acceleration and optical metrology system contributions~\cite{Caprini:2019pxz}, save for two normalization coefficients $A$ and $O$~\footnote{The $A$ and $O$ parameters used in this work are respectively the square of $A$ and of $P/10$ of~\cite{Caprini:2019pxz}. Note that more complicated/different uncertainties in the spectral shape and amplitude of
the instrumental noise can also be incorporated in our technique (although care needs to be used to keep the problem linear
if computational cost is an issue). However, irrespective of the adopted extraction technique, the
detection of any SGWB with LISA crucially relies on informative priors on the instrumental noise, because
cross correlation techniques such as those used in LIGO and Virgo are not applicable to LISA
(since noise in the three arms is correlated).},
for which we assume Gaussian priors of about 20\% width~\cite{LISA_docs}:
\begin{align}\label{noise}
&S_{\rm noise}(f)=A \, S_{\rm acc}(f)+O \, S_{\rm OMS}(f)\,,\\
&p(A,O)\propto e^{-\frac12 [(A-1)^2/\sigma_A^2+(O-1)^2/\sigma_O^2]}\,,
\end{align}
with $\sigma_A=\sigma_O=0.2$, while the functional forms $S_{\rm acc}(f)$ and $S_{\rm OMS}(f)$ are given in \cref{sec:noise_fg}.
For the astrophysical foreground, we assume that the spectral shape of
the signal from binaries of stellar origin black holes and neutron stars is known (see \cref{sec:noise_fg} for details) up to a normalization coefficient $L$. We can thus write
\begin{align}\label{fg}
&S_{\rm foreground}(f)= L \, S_{\rm LV}(f) \,,\\
\label{eq:prior_fg}
&p(L)\propto e^{-\frac12 [ (L-1)^2/\sigma_L^2]}\,,
\end{align}
where we assume $\sigma_L=0.5$. Present~\cite{ligostoch, Martynov:2016fzi} and upcoming~\cite{Punturo:2010zz, Sathyaprakash:2012jk, Maggiore:2019uih} ground based detectors are actually expected to measure this foreground, providing more stringent constraints on $L$. However, in order to test the robustness of our method, we use a relatively large value for $\sigma_L$. Furthermore, as an additional test for robustness, we checked that less stringent choices for the prior leave the results shown in~\cref{sec:results} unaffected.
As mentioned previously, we neglect here the contribution of the
foreground from Galactic binaries, since its time dependence (caused by the motion
of the detector) and it anisotropy allow for measuring its power spectral density to high precision (and thus
for removing it)~\cite{Adams:2013qma}. In any case, a residual contribution of the
population of Galactic binaries could be accounted for simply by adding a suitable term to eq.~\eqref{fg},
and does not significantly affect the results.
With these assumptions, the posterior probability distribution
for the parameters $\boldsymbol{\theta}~=~(\{\alpha_j\}_{j=1,\ldots,n}, A, O, L)$
is given by Bayes' theorem and reads
\begin{align}
&\ln p(\alpha_{j=1,...,n_{\rm max}},A,O,L|\bar{D}_{i=1,...,n})= -\frac{\chi^2}{2} +{\rm const}\,,\\
&\chi^2=\sum_{i=1}^n {N\frac{(\bar{D}_i-S_i)^2}{\bar{D}_i^2}} +\frac{(A-1)^2}{\sigma_A^2}+\frac{(O-1)^2}{\sigma_O^2}
+\frac{(L-1)^2}{\sigma_L^2}
\,,\label{chi2}\\
&S_i=\sum_{j=1}^{n} \alpha_j K_j \delta_w(f_i-f^a_j)+A S_{\rm acc}(f_i)+O S_{\rm OMS}(f_i) + L S_{\rm LV}(f_i)
\end{align}
Since $S_i$ depends linearly on the parameters,
finding the maximum of the posterior distribution (or equivalently the minimum $\chi^2$) is a linear problem, i.e. one has to solve the linear system $\partial \chi^2/\partial \theta_j=0$ with $j=1,...,m+3$.
Note that for $m+3>n$, the problem becomes degenerate, since there are more
parameters than data. However,
even for $m+3\leq n$
the problem may still be degenerate in practice, especially for large $m$, since we do not expect to be able to extract all the parameters reliably due to the errors affecting the data.
This issue can be bypassed by performing a PCA of the Fisher matrix $F_{ij}$, as we will outline
in the following.
The Fisher matrix is defined by
\begin{equation}
F_{ij}= \frac12\frac{\partial^2 \chi^2}{\partial \theta_i \partial \theta_j}\,,
\end{equation}
with $i,j=1,\ldots,m+3$. This matrix, which is independent of the parameters since the problem is linear, encodes their errors and correlations.
Solving for the eigenvalues and eigenvectors of the Fisher matrix then allows for
identifying linear combinations of the parameters that are uncorrelated with one another,
as well as computing their errors. In more detail, one can
define the functions $\{\eta_{(i)}(f)\}_{i=1,...,m+3}$ with
\begin{align}\label{basis}
\eta_{(i)}(f) & =\eta^{\rm signal}_{(i)}(f)+\eta^{\rm noise}_{(i)}(f) +\eta^{\rm LV}_{(i)}(f)\,\\
\eta^{\rm signal}_{(i)}(f) & =\sum_{j=1}^{m} e^{(i)}_j K_j\delta_w(f-f_j^a)\,\\
\eta^{\rm noise}_{(i)}(f) & =e^{(i)}_{m+1}S_{\rm acc}(f)+e^{(i)}_{m+2}S_{\rm OMS}(f)\,\\
\eta^{\rm LV}_{(i)}(f) & =e^{(i)}_{m+3}S_{\rm LV}(f)
\end{align}
where the vectors $\{\vec{e}_{(i)}\}_{i=1,...,m+3}$ are $n$ \textit{orthonormal} eigenvectors of the Fisher matrix. Note that if $m+3 > n $, at least $m-n+3$ of these eigenvectors will correspond to vanishing eigenvalues, since the problem is degenerate.
However, even if $m+3 \leq n $ many eigenvalues may be very small. This is central for the PCA/singular value decomposition technique,
as will become clear below.
The total power spectral density given
by eq.~\eqref{tot} can then be re-written as a sum on the functions $\{\eta_{(i)}(f)\}_{i=1,...,m+3}$, i.e.
\begin{align}\label{model2}
S_i&=S(f_i)=\sum_{k=1}^{m+3} b_k \eta_{(k)}(f_i)\,,\\
b_k&=\sum_{j=1}^m \alpha_j e^{(k)}_j+ A e^{(k)}_{m+1}+ O e^{(k)}_{m+2}
+Le^{(k)}_{m+3}\,.
\end{align}
The coefficients $b_k$ are now uncorrelated Gaussian variables, and their errors are given by
$\lambda_{(k)}^{-1/2}$, where $\lambda_{(k)}$ is the eigenvalue corresponding to the eigenvector $\vec{e}_{(k)}$. In particular, by using eq.~\eqref{basis} we can write the reconstructed signal, noise spectral density and astrophysical foregrounds as
\begin{align}\label{model3}
S^{\rm signal}_i&=\sum_{k=1}^{m+3} b_k \eta^{\rm signal}_{(k)}(f_i)\,,\\
S^{\rm noise}_i&=\sum_{k=1}^{m+3} b_k \eta^{\rm noise}_{(k)}(f_i)\,,\label{model3bis}\\
S^{\rm LV}_i&=\sum_{k=1}^{m+3} b_k \eta^{\rm LV}_{(k)}(f_i)\,.\label{model3tris}
\end{align}
As already mentioned, we can then perform
a PCA to ``de-noise'' the reconstructed quantities,
i.e. one can rewrite eqs.~\eqref{model3}--\eqref{model3tris} by including only
the coefficients $b_k$ which are ``well determined'' (\emph{e.g.} one possibility is to only include coefficients
whose values are not compatible with zero at one $\sigma$, namely $|b_k| > \lambda_{(k)}^{-1/2}$). This yields
\begin{align}\label{model4}
S^{\rm signal}_i &=\sum_{|b_k| > \lambda_{(k)}^{-1/2}} b_k \eta^{\rm signal}_{(k)}(f_i)\,,\\
S^{\rm noise}_i &=\sum_{|b_k| > \lambda_{(k)}^{-1/2}} b_k \eta^{\rm noise}_{(k)}(f_i)\,,\label{model4bis}\\\
S^{\rm LV}_i &=\sum_{|b_k| > \lambda_{(k)}^{-1/2}} b_k \eta^{\rm LV}_{(k)}(f_i)\,.\label{model4tris}
\end{align}
Since the $b_k$ are uncorrelated Gaussian variables, the (Gaussian) errors on $S^{\rm signal}_i$, $S^{\rm noise}_i$ and $S^{\rm LV}_i$ can then be obtained
by summing in quadrature the errors of the $b_k$ (i.e. $\lambda_{(k)}^{-1/2}$), with coefficients given by these equations. These are
the errors for the reconstructed signal shown in Figs.~\ref{fig:large_flat}--\ref{fig:bump} below.
Finding explicitly the eigenvalues and eigenvectors of the Fisher matrix can be challenging, since
the matrix is singular or almost singular, and the dimensionality of the parameter space
is huge. Indeed, $m$ can be as large as the number of sampled frequencies $n$, which
for LISA could be of the order of $10^6$, because the data sampling rate $\Delta t$ will be of the order of a few seconds.\footnote{Note that even though the nominal duration of the LISA mission
will be 4 yr, our technique assumes that the data are divided in $N\gg 1$ chunks, in order to write the likelihood of \cref{likelihood}
and \cref{likelihood2}.} The first problem can be addressed in practice by
replacing $F_{ij}\to F_{ij}+\epsilon \delta_{ij}$. This make
the Fisher matrix formally non-singular, with eigenvalues $\gtrsim \epsilon$. As long
as $\epsilon$ is much smaller than the minimum $b_k$ contributing to the sum in eqs.~\eqref{model4}--\eqref{model4tris}, the reconstructed ``cleaned'' model is unaffected.
As for the high dimensionality of the problem, the computational cost may
be reduced by noting that the Fisher matrix becomes sparse in a suitable basis. In more detail,
one can write
\begin{align}
F_{ij}=& \sum_{k=1}^m\frac{N}{\bar{D}_k^2}\frac{\partial S_k}{\partial \theta_i} \frac{\partial S_k}{\partial \theta_j}+ \frac{1}{\sigma_A^2}\delta_{i}^{m+1}\delta_{j}^{m+1}
+\frac{1}{\sigma_O^2}\delta_{i}^{m+2}\delta_{j}^{m+2}
+\frac{1}{\sigma_L^2}\delta_{i}^{m+3}\delta_{j}^{m+3}
\,,
\end{align}
where
\begin{align}
\frac{\partial S_k}{\partial \theta_i}&=\sum_{l=1}^m K_l\delta_w(f_k-f_l^a) \delta^{l}_{i}+S_{\rm acc}(f_k)\delta^{m+1}_i +S_{\rm pos}(f_k)\delta^{m+2}_i
+S_{\rm LV}(f_k) \delta^{m+3}_i \, .
\end{align}
From this expression, it is clear that unless $w\Delta t\gg 1$, the Fisher matrix becomes sparse -- with non-zero elements only near the diagonal or in the
last four rows/columns -- which might allow for decreasing the burden of the computation
when real data are available.
Nevertheless, for the purpose of this work, where we are just concerned with applying
our technique to mock simulated data, we downsample the latter in the following manner.
We start by bundling the $n$ data $\bar{D}_i$ in groups of $M$
data adjacent in frequency. Since the variance of $\bar{D}_i$
is $S_i^2/N$ [c.f.~eq.~\eqref{likelihood}], it seems natural to replace each group by the weighted average
$\bar{\bar{D}}_k=(\sum_{i=1+kM}^{(k+1)M} \bar{D}_i/S_i^2)/(\sum_{i=1+kM}^{(k+1)M} 1/S_i^2)$,
and to assign this value to the frequency
$\bar{f}_k \equiv (\sum_{i=1+kM}^{(k+1)M} f_i/S_i^2)/(\sum_{i=1+kM}^{(k+1)M} 1/S_i^2)$.
Indeed, in~\cref{sec:appendixdownsampling} we show that the
best estimate of the (total) power spectral density $\bar{S}_k= S(\bar{f}_k)$ at the
frequency $\bar{f}_k$ is indeed given by the ``downsampled'' data $\bar{\bar{D}}_k$,
with a (Gaussian) variance approximately given by $\bar{S}_k^2/M\approx\bar{\bar{D}}_k/M$.
Therefore, eqs.~\eqref{likelihood}--\eqref{likelihood2} remain valid
for the ```downsampled'' data, modulo the replacement $N\to NM$, as one would intuitively expect. The rest of the analysis proceeds unchanged, again
with $N\to NM$. Note that to compute the weighted averages needed to
define the $\bar{\bar{D}}_k$ and the $\bar{f}_k$, one can replace $S_i\approx \bar{D}_i$,
as we did when going from
eq.~\eqref{likelihood} to eq.~\eqref{likelihood2}.
For all the analyses presented in the next section, we consider frequencies in the range $f \in [10^{-4} , 2 \times 10^{-2} ]\text{Hz}$, which in logarithmic scale corresponds to a region which is roughly symmetric around to the peak of the LISA sensitivity (which is around $2 \times 10^{-3} $Hz). We start with by generating $N=94$ Gaussian realizations\footnote{We assume that the data are divided into chunks of around 11 days each (with a total observation time of 4 years and $75\%$ duty cycle). Note that the number of chunks controls how well the central limit theorem
holds when going from \cref{likelihood0} to \cref{likelihood}. Since $N$ is large but finite, we expect some small bias to be
potentially present in our results. Further bias may be introduced when approximating \cref{likelihood} with \cref{likelihood2}, which allows for
making the problem linear, at the expense of neglecting the skewness of \cref{likelihood} (which disappears in
\cref{likelihood2}). We have verified that all these biases are reduced by choosing larger $N$.}
of $d(f)$ obeying~\cref{psd2} with an initial spacing of $10^{-6}$Hz, and we then set $M =10$ to downsample our data.
\section{Results}
\label{sec:results}
We can now proceed to demonstrate the robustness and effectiveness of the procedure described in the previous section. To this purpose, we apply our technique to a set of mock signals with various SNRs. Note that depending on the particular shape of the input signal, a different ``correlation length'' $w$, as
defined in \cref{deltaw}, will be required for capturing the features of the signal.
For the analysis shown in this section, we choose $w$ by ``trial and error'',
i.e. by choosing the value best suited for recovering the injected signal. In reality,
when the injected signal is unknown, different correlation lengths should be tested in order to determine the one best describing the shape of the unknown SGWB. This corresponds to finding the $w$ giving the largest posterior probability (\emph{i.e.} the smallest $\chi^2$), which
can be found simply by varying $w$ on a uniform grid ``by brute force''. A similar procedure can
be applied to ascertain the optimal dimensionality $m$ of the basis, as defined in the previous section.
That would entail varying $w$ and $m$ on a grid, and then applying our technique to each point, in order
to find which one gives the lowest $\chi^2$ (c.f. sec. 4.2.1 of \cite{DAbook} for a problem solved by a similar technique).
Alternatively, Markov Chain
Monte Carlo techniques can be used to sample the posterior distribution. Another possibility for model comparison, which can also keep track of the number of the degrees of freedom employed in the analysis, is the Akaike Information Criterion (AIC)~\cite{Akaike}.
These analyses are however beyond the scope of this work, in which we instead choose $w$ and $m$ by trial and error.
\begin{figure}[h!]
\includegraphics[width=0.99\textwidth]{plots/Large_flat}
\caption{PCA analysis for a flat input signal with amplitude $h^2 \Omega_{*}=3\times 10^{-12}$ and SNR $\simeq 156$. For this analysis we have fixed $m=10$ and $w = 1$Hz (see main text for details). \emph{Top left}: a plot of all inputs, the simulated data and the reconstructed data. \emph{Top right}: input signal, linear fit and PCA reconstructions with $2\sigma$ error bands. \emph{Bottom left}: input LIGO/Virgo foreground (see main text for details), linear fit and PCA reconstructions with $2\sigma$ error bands. \emph{Bottom right}: input LISA noise (see main text for details), linear fit and PCA reconstructions with $2\sigma$ error bands. \label{fig:large_flat}}
\end{figure}
As a first example, we consider in~\cref{fig:large_flat} a flat background with amplitude $h^2 \Omega_{*}=3 \times 10^{-12}$, corresponding to SNR $\simeq 156$. Note that in this case the injected signal is larger than the foreground from LIGO/Virgo binaries in most of the frequency range. Moreover, since this signal is constant in the LISA band, it is appropriate to choose a large correlation length $w$, but that causes a large degeneracy between $S_{LV}$ and $S_{\rm signal}$. This makes it
numerically difficult to disentangle the two components. An efficient way to resolve this issue is to reduce the dimensionality $m$ of the signal basis\footnote{Clearly, a lower $m$ must be compensated by a larger $w$ in order for the basis to remain sufficiently dense to accurately model the signal.} In the analysis presented in~\cref{fig:large_flat}, we choose a basis of $m = 10$ Gaussians (with a uniform logarithmic spacing for the pivot frequencies $f^a_i$) and $w=1$Hz.
As can be seen from~\cref{fig:large_flat}, which shows the PCA reconstructed signal, foreground and noise, this choice
very efficiently disentangles the different components hidden in the data.
The reconstructed value for the LIGO/Virgo foreground parameter is $L \simeq 1.044 \pm 0.114$. As for the LISA noise parameters, we obtain $A \simeq 0.980 \pm 0.005$ and $O \simeq 0.976 \pm 0.001$.\footnote{Note that the parameters $A$ and $O$ are significantly different from 1. As mentioned earlier, this bias comes about
because the central limit theorem, used to go from \cref{likelihood0} to \cref{likelihood}, only holds
in the limit in which the number of chunks $N$ diverges. We have indeed verified that by increasing $N$, $A$ and $O$
become compatible with $1$. The same applies to the cases shown in the figures below.} Concerning the reconstruction of the signal, we can see from the top right panel that the linear fit
(defined as the model minimizing the $\chi^2$ in \cref{chi2}, or equivalently by the sum given in eqs.~\eqref{model3}--\eqref{model3tris}, where all coefficient are included, irrespectively of their value)
correctly reconstructs the signal in most of the frequency range. However, at large frequencies, where the foreground is larger than the signal, it fails. On the other hand the PCA reconstruction, by dropping low information components, produces a fairly accurate reconstruction of the signal even in this range.
In~\cref{fig:low_flat}, we consider the opposite situation, \emph{i.e.} we consider an injected signal smaller than the LIGO/Virgo foreground in most of the frequency range. In more detail, we choose a flat signal with amplitude $h^2 \Omega_{*}=6\times10^{-13}$, corresponding to an SNR $\simeq 31$. Consistently with the analysis of~\cref{fig:large_flat}, we have used $w = 1$Hz and $m= 10$ (with the same uniform logarithmic spacing for the pivot frequencies). We can see that while in this case the reconstruction the LISA noise parameters is basically unaffected ($A \simeq 0.978 \pm 0.005$ and $O \simeq 0.975 \pm 0.001$),
the determination of LIGO/Virgo foreground parameter is slightly more accurate ($L \simeq 0.896 \pm 0.106$). As for the signal reconstruction, the top right panel of~\cref{fig:low_flat} clearly shows that, despite the signal being much smaller than the LIGO/Virgo foreground, our procedure still captures it in most of the frequency range. Once again the PCA reconstruction works better than a simple linear fit in the region where the signal is much weaker than the foreground.
\begin{figure}[h!]
\includegraphics[width=0.99\textwidth]{plots/Small_flat}
\caption{PCA analysis for a flat input signal with amplitude $h^2 \Omega_{*}=6 \times 10^{-13}$ and SNR $\simeq 31$. This analysis was performed with $m=10$ and $w = 1$Hz. Plot structure as in~\cref{fig:large_flat}. \label{fig:low_flat}}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=0.99\textwidth]{plots/Broken}
\caption{PCA analysis for an input broken power law signal (see~\cref{footnote:broken}) with amplitude $h^2 \Omega_{*}=9 \times 10^{-11}$, $n_{s1} = 5$, $n_{s2} = -6$ and $f_* = 3 \times 10^{-4}$Hz. The SNR of this signal is $\simeq 34$. This analysis was performed with $m=n$, $w=2\times10^{-5}$Hz. Plot structure as in~\cref{fig:large_flat}. \label{fig:broken}}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=0.99\textwidth]{plots/Broken_2}
\caption{PCA analysis for an input broken power law signal (see~\cref{footnote:broken}) with amplitude $h^2 \Omega_{*}=9 \times 2.7 \times 10^{-11}$, $n_{s1} = 10$, $n_{s2} = -12$ and $f_* = 10^{-2}$Hz. The SNR of this signal is $\simeq 34$. This analysis was performed with $m=100$ with uniform log spacing and $w = 5 \times 10^{-4}$Hz. Plot structure as in~\cref{fig:large_flat}.
\label{fig:bump} }
\end{figure}
We conclude this section by showing two cases where the input signal is chosen to be less degenerate with the foreground. In~\cref{fig:broken} we show the results obtained by applying our procedure ($m=n$, $w=2\times10^{-5}$Hz) to a broken power law signal~\footnote{\label{footnote:broken} The signal is given by
$h^2 \Omega_{GW} (f) =h^2 \Omega_* \, ({f}/{f_*})^{n_{s1}} /[1 +({f}/{f_*})^{n_{s1} -n_{s2}} ]$, where $ \Omega_* $ is the amplitude, $f_*$ is the pivot frequency, and $n_{s1}$, $n_{s2}$ are respectively the spectral tilts before and after the pivot. } placed at small frequencies
and with SNR $\simeq 34$. In~\cref{fig:bump} we show instead the results obtained with $m=100$ with uniform log spacing and $w=5 \times 10^{-4}$Hz on an injected
signal given by
a broken power law placed at high frequencies and with SNR $\simeq 34$. In both cases, the reconstruction of the LIGO/Virgo foreground is accurate ($L \simeq 0.948 \pm 0.034$ and $L \simeq 0.989 \pm 0.045$ respectively), as is the case for the LISA noise parameters ($A \simeq 1.000 \pm 0.004$, $A \simeq 0.983 \pm 0.009$ and $O \simeq 0.981 \pm 0.001$, $O \simeq 0.984 \pm 0.002$ respectively).
\section{Discussion}
\label{sec:conclusions}
In this work we have proposed a template-free PCA technique for the reconstruction of SGWBs with LISA. We have shown that our procedure is very efficient at disentangling an unknown input signal from the instrumental noise even in the presence of a foreground. Remarkably, the component separation obtained with the techniques presented in this work is effective even for signals with fairly small SNR $\simeq30$, compared to a power-law foreground from
LIGO/Virgo binaries with SNR$\simeq 53$.
Our approach essentially consists of expanding the signal onto a basis of Gaussians with fixed width $w$ and centered
on a set of pivot frequencies within the LISA frequency band. Generically, an unknown signal can thus be expressed as a linear combination of these functions through some unknown parameters to be determined. We model
uncertainties in the LISA noise budget and in the amplitude of
the LIGO/Virgo foreground with global normalization parameters also to be determined. The best fit for all
these parameters can be then be found by maximizing the posterior distribution given by Bayes' theorem.
By dividing the time series of the data in chunks and using the central limit theorem,
we manage to reduce this maximization procedure to a linear problem, which only requires
the inversion of the Fisher matrix of the parameters. This inversion
can be challenging because of the large dimensionality of the problem, which makes the matrix
singular or almost singular (i.e. most combinations of the parameters are essentially unconstrained by the data).
The PCA technique provides in fact a robust technique to tackle this issue, by essentially dropping
all the linear combinations of the parameters on which the data provide no information, and retaining
only the ``components'' that are informed by the data. This procedure leads to a ``de-noised''
agnostic reconstruction of the underlying signal, as well as of the LISA instrumental noise and astrophysical LIGO/Virgo foreground.
In conclusion, the method described in this work is efficient at disentangling the different components
of the SGWB that LISA will measure. Our results are of major relevance since they clearly show that foreground subtraction can be successfully performed. This would allow for the identification (and eventually for the characterization) of the possible cosmological signals that might be hidden behind the foreground. Moreover, since our technique is based on a template-free approach, it provides a robust tool applicable to any background signal. Indeed, while in this work we have restricted our analysis to LISA, our techniques can be easily extended to different detectors.
\begin{acknowledgments}
This paper is dedicated to the memory of Pierre Bin\'etruy in the third anniversary of his death. Among many more things, Pierre pushed
us both to think about stochastic backgrounds for LISA, and this work would not have seen the light of day without his example and inspiration. We thank Carlo Contaldi, Vincent Desjacques, Valerie Domcke, Nikolaos Karnesis, Antoine Petiteau and Angelo Ricciardone for useful comments and discussions. We acknowledge financial support provided under the European Union's H2020 ERC Consolidator Grant ``GRavity from Astrophysical to Microscopic Scales'' grant agreement no. GRAMS-815673. M.P. acknowledges the support of the Science and Technology Facilities Council consolidated grants ST/P000762/1.
\end{acknowledgments}
|
1,314,259,996,511 | arxiv | \section{Introduction}
\label{sec:intro}
We consider the Cauchy problem for the following Hartree equation
\begin{equation}
\label{eq:r3}
i{\partial}_t u + \Delta u = \(K\ast |u|^2\)u,\quad t\in \R, \
x\in\R^d;\quad u_{\mid t=0}=u_0,
\end{equation}
where $K$ denotes the Hartree kernel.
We first deal with the case of a homogeneous kernel,
\begin{equation}
\label{eq:noyau}
K(x)=\frac{\lambda}{|x|^\gamma},\quad \lambda\in \R,\ \gamma>0.
\end{equation}
In \cite{Mou12}, it was proved that if $1\leqslant d\leqslant 3$ and $\gamma<d$,
the Cauchy problem \eqref{eq:r3} is locally well-posed in
$L^2(\R^d)\cap W$, where $W$ stands for the Wiener algebra (also
called Fourier algebra, according to the context)
\begin{equation*}
W=\left\{f\in \Sch'(\R^d;\C),\quad \widehat f\in L^1(\R^d)\right\},
\end{equation*}
and the Fourier transform is defined, for $f\in L^1(\R^d)$, as
\begin{equation*}
\widehat f(\xi) = \frac{1}{(2\pi)^{d/2}}\int_{\R^d}e^{-ix\cdot \xi}f(x)\mathrm{d}x.
\end{equation*}
In this paper, we investigate the global well-posedness for
\eqref{eq:r3}: we prove that if $d\geqslant 1$ and $\gamma<\min (2,d/2)$, then
the solution to \eqref{eq:r3}
is global in time in $L^2\cap W$. In view of the classical result
according to which \eqref{eq:r3} is globally well-posed in $L^2(\R^d)$,
our result can be understood as a propagation of the Wiener
regularity. On the other hand, the mere Wiener regularity does not
suffice to ensure even local well-posedness for \eqref{eq:r3}.
An advantage of working in $W$ lies in the fact that $W\hookrightarrow
L^\infty(\R^d)$, and, contrary to e.g. $H^s(\R^d$), $s>d/2$, $W$
scales like $L^\infty(\R^d)$ (note also that if $s>d/2$,
$H^s(\R^d)\hookrightarrow W$). So in a way, $W$ is the largest space
included in $L^\infty(\R^d)$ on which the Schr\"odinger group
$e^{it\Delta}$ acts continuously --- see Remark~\ref{rem:amalgam}
though. Recall that $e^{it\Delta}$ does not
map $L^\infty(\R^d)$ to itself, as shown by the explicit formula
$e^{i\Delta}(e^{-i|x|^2/4}) = \delta_{x=0}$, and the parabolic scale
invariance.
\begin{theorem}\label{theo:main}
Let $d\geqslant 1$, $K$ given by \eqref{eq:noyau} with $\lambda\in \R$ and
$0<\gamma<\min (2,d/2)$. If $u_0\in L^2(\R^d)\cap W$, then \eqref{eq:r3}
has a unique, global solution $u\in C(\R; L^2\cap W)$. In addition,
its $L^2$-norm is conserved,
\begin{equation*}
\|u(t)\|_{L^2}=\|u_0\|_{L^2},\quad \forall t\in \R.
\end{equation*}
\end{theorem}
\begin{remark}\label{rem:amalgam}
One might be tempted to consider a space included in
$L^\infty$ which also
scales like $L^\infty$, and which is
larger than $W=\mathcal F L^1$, namely the amalgam space $W(\mathcal F L^1;L^\infty)$. This
space consists essentially of functions which are locally in $W$,
and globally in $L^\infty$ (see e.g. \cite{CoNi08,CoZu10} for a
precise definition). Strichartz estimates in amalgam spaces have been
established in \cite{CoNi08} (even though the case $W(\mathcal F L^1;L^\infty)$
can never be considered). However, since the map $x\mapsto e^{i|x|^2}$
belongs to
$W(\mathcal F L^1;L^\infty)$, we see that $e^{it\Delta}$ does not act
continuously on
$W(\mathcal F L^1;L^\infty)$.
\end{remark}
We next show that \eqref{eq:r3} is not well-posed
in the mere Wiener algebra. Precisely have the following theorem:
\begin{theorem}
\label{th:ill}
Let $d\geqslant 1$, $K$ given by \eqref{eq:noyau} with $0<\gamma<d$. The Cauchy problem \eqref{eq:r3} is
locally well-posed in $W\cap L^2$, but not in $W$:
for all ball $B$ of $W$, for all $T>0$ the solution map
$ \varphi \in B \mapsto \psi \in C([0,T]; W)$ is not uniformly continuous.
\end{theorem}
\begin{remark}
In the case of the nonlinear Schr\"odinger equation
\begin{equation}\label{eq:nls}
i\partial_{t}u+\Delta u = \lambda|u|^{2{\sigma}}u,
\end{equation}
where ${\sigma} $ is an integer and $\lambda \in \R$, the Cauchy problem is
locally well-posed in $W$ (see \cite{CDS10}). From the above result,
this is in sharp contrast with the case of the Hartree equation. On
the other hand, it is not clear that the Cauchy problem for
\eqref{eq:nls} is \emph{globally} well-posed in $L^2(\R^d)\cap W$,
even in the case $d={\sigma}=1$. We note that in \cite{HyTs12}, the authors
study \eqref{eq:nls} in the one-dimensional case $d=1$, with ${\sigma}<2$
not necessarily an integer. They prove local well-posedness in
\begin{equation*}
\hat L^p= \{f\ : \ \hat f\in L^{p'}\},
\end{equation*}
for $p$ in some open neighborhood of $2$. Global well-posedness
results for initial data in $\hat L^p$ are established, in spaces
based on dispersive estimates.
\end{remark}
The kernel $K$ given by
\eqref{eq:noyau} is such that its Fourier transform belongs to no
Lebesgue space, but to a weak Lebesgue space, from the following property (see
e.g. \cite[Proposition~1.29]{BCD11}):
\begin{proposition}\label{prop:hatK}
Let $d\geqslant 1$ and $0<\gamma<d$. There exists $C=C(\gamma,d)$ such
that the Fourier transform of $K$ defined by \eqref{eq:noyau} is
\begin{equation*}
\widehat K(\xi) = \frac{\lambda C}{|\xi|^{d-\gamma}}.
\end{equation*}
\end{proposition}
The final result of this paper is concerned with the case where the
kernel $K$ is such that its Fourier transform belongs to some Lebesgue space.
\begin{theorem}\label{theo:Lp}
Let $d\geqslant 1$.
\begin{itemize}
\item Let $p\in [1,\infty]$, and suppose that $K$ is such that $\widehat
K\in L^p(\R^d)$. If
$u_0\in L^2(\R^d)\cap W$, then there exists $T>0$ such that
\eqref{eq:r3} has a unique solution $u\in C([-T,T]; L^2\cap W)$.
In addition, its $L^2$-norm is conserved,
\begin{equation*}
\|u(t)\|_{L^2}=\|u_0\|_{L^2},\quad \forall t\in [-T,T].
\end{equation*}
If $K\in W$ ($p=1$), then the solution is global: $u\in C(\R; L^2\cap W)$.
\item Suppose that $K$ is such that $\widehat K\in L^\infty(\R^d)$. If
$u_0\in W$, then there exists $T>0$ such that
\eqref{eq:r3} has a unique solution $u\in C([-T,T]; W)$.
\item For any $p\in [1,\infty)$, one can find $K$ with $\widehat K\in
L^p(\R^d)\setminus L^\infty(\R^d)$, such that
\eqref{eq:r3} is locally well-posed in $L^2(\R^d)\cap W$, but not
in $W$: for all ball $B$ of
$W$, for all $T>0$ the solution map
$ \varphi \in B \mapsto \psi \in C([0,T]; W)$ is not uniformly continuous.
\end{itemize}
\end{theorem}
\section{Standard existence results and properties}
\label{sec:standard}
\subsection{Main properties of the Wiener algebra}
\label{sec:wiener}
The space $W$ enjoys the following elementary properties (see
\cite{CoLa09,CDS10}):
\begin{enumerate}
\item $W$ is a Banach space, continuously embedded into
$L^\infty(\R^d)$.
\item $W$ is an algebra, and the
mapping $(f,g)\mapsto fg$ is continuous from $W^2$ to
$W$, with
\begin{equation*}
\|fg\|_{W}\leqslant \|f\|_{W}\|g\|_{W},\quad \forall f,g\in W.
\end{equation*}
\item For all $t\in \R$, the free Schr\"odinger group
$e^{i t \Delta}$
is unitary on $W$.
\end{enumerate}
\subsection{Existence results based on Strichartz estimates}
\label{sec:stri}
For the sake of completeness, we recall standard definition and
results.
\begin{definition}\label{def:adm}
A pair $(p,q)\not =(2,\infty)$ is admissible if $p\geqslant 2$, $q\geqslant 2$,
and
$$\frac{2}{p}= d\left( \frac{1}{2}-\frac{1}{q}\right).$$
\end{definition}
\begin{proposition}[From \cite{GV85c,KT}]\label{prop:strichartz}
$(1)$ For any admissible pair $(p,q)$, there exists $C_{q}$ such that
$$
\|e^{it\Delta} \varphi\|_{L^{p}(\R;L^{q})} \leqslant C_q
\|\varphi \|_{L^2},\quad \forall \varphi\in L^2(\R^d).
$$
$(2)$ Denote
\begin{equation*}
D(F)(t,x) = \int_0^t e^{i(t-\tau)\Delta}F(\tau,x)\mathrm{d}\tau.
\end{equation*}
For all admissible pairs $(p_1,q_1)$ and~$
(p_2,q_2)$, there exists $C=C_{q_1,q_2}$
such that for all interval $I\ni 0$,
\begin{equation*}
\left\lVert D(F)
\right\rVert_{L^{p_1}(I;L^{q_1})}\leqslant C \left\lVert
F\right\rVert_{L^{p'_2}\(I;L^{q'_2}\)},\quad \forall F\in
L^{p'_2}(I;L^{q'_2}).
\end{equation*}
\end{proposition}
\begin{proposition}\label{prop:r3L2}
Let $d\geqslant 1$, $K$ given by
\eqref{eq:noyau} with $\lambda\in \R$ and $0<\gamma<\min(2,d)$. If
$u_0\in L^2(\R^d)$, then \eqref{eq:r3} has a
unique, global solution
\begin{equation*}
u\in C(\R;L^2)\cap L^{8/\gamma}_{\rm loc}(\R;L^{4d/(2d-\gamma)} ).
\end{equation*}
In addition,
its $L^2$-norm is conserved,
\begin{equation*}
\|u(t)\|_{L^2}=\|u_0\|_{L^2},\quad \forall t\in \R,
\end{equation*}
and for all admissible pair $(p,q)$, $u \in L^p_{\rm
loc}(\R;L^q(\R^d))$.
\end{proposition}
\begin{proof}
We give the main technical steps of the proof, and refer to \cite{CazCourant}
for details.
By Duhamel's formula, we write \eqref{eq:r3} as
\begin{equation*}
u(t)=e^{it\Delta}u_0-
i\int_0^{t}e^{i(t-\tau)\Delta}(K\ast |u|^{2}u)(\tau)\mathrm{d}\tau=:\Phi(u)(t).
\end{equation*}
Introduce the space
\begin{equation*}
\begin{aligned}
Y(T)&=\{\phi\in C([0,T]; L^{2}(\R^{d})):
\|\phi\|_{L^{\infty}([0,T];\ L^{2}(\R^{d}))}\leqslant
2\|u_0\|_{L^{2}(\R^{d})},\\
&\quad \|\phi\|_{L^{8/\gamma}([0,T];\
L^{4d/(2d-\gamma)}(\R^{d}))}\leqslant 2C(8/\gamma)\|u_0\|_{L^{2}(\R^{d})}\},
\end{aligned}
\end{equation*}
and the distance $$ d(\phi_{1},
\phi_{2})=\|\phi_{1}-\phi_{2}\|_{L^{8/\gamma}([0,T];\
L^{4d/(2d-\gamma)})},$$
where $C(8/\gamma)$ stems from Proposition~\ref{prop:strichartz}.
Then $(Y(T), d)$ is a complete metric space,
as remarked in \cite{Kato87} (see also \cite{CazCourant}).
Hereafter, we denote by
\begin{equation*}
q=\frac{8}{\gamma}, \quad r=\frac{4d}{2d-\gamma}, \quad
\theta=\frac{8}{4-\gamma},
\end{equation*}
and $\|\cdot\|_{_{L^{a}([0,T];\ L^{b}(\R^{d}))}}$ by
$\|\cdot\|_{L^{a}L^{b}}$ for simplicity. Notice that $\left(q, r\right)$ is
admissible and
\begin{equation*}
\frac{1}{q^{\prime}}=\frac{4-\gamma}{4}+\frac{1}{q}=\frac{1}{2}+\frac{1}{\theta}
\quad ; \quad
\frac{1}{r^{\prime}}=\frac{\gamma}{2d}+\frac{1}{r}\quad;\quad
\frac{1}{2}=\frac{1}{\theta}+\frac{1}{q}.
\end{equation*}
By using Strichartz estimates, H\"older inequality and
Hardy--Littlewood--Sobolev inequality, we have, for $(\underline
q,\underline r)\in \{(q,r),(\infty,2)\}$:
\begin{equation*}
\begin{aligned}
\|\Phi(u)\|_{L^{\underline q}L^{\underline r}}&\leqslant C(\underline q)
\|u_0\|_{L^{2}}+C(\underline q,q)\left\lVert (K\ast|u|^{2})u
\right\rVert_{L^{q'}L^{r'}}\\
&\leqslant C(\underline q)\|u_0\|_{L^{2}}+
C(\underline q,q)\left\lVert
K\ast |u|^{2}\right\rVert_{L^{4/(4-\gamma)}L^{2d/\gamma}}
\|u\|_{L^{q}L^{r}}\\
& \leqslant C(\underline q)
\|u_0\|_{L^{2}}+C\|u\|_{L^{\theta}L^{r}}^{2}\|u\|_{L^{q}L^{r}}\\
&\leqslant C(\underline q)\|u_0\|_{L^{2}}+CT^{1-\gamma/2}\|u\|_{L^{q}L^{r}}^{3},
\end{aligned}
\end{equation*}
for any $u\in Y(T)$, with $C(\infty)=1$ by the standard energy
estimate. To show the contraction property
of $\Phi$, for any $v, w\in
Y(T)$, we get
\begin{align*}
\|\Phi(v)-\Phi(w)\|_{L^{q}L^{r}}
&\lesssim\|K\ast |v|^{2}\|_{L^{4/(4-\gamma)}L^{2d/\gamma}}
\|v-w\|_{L^{q}L^{r}}\\
&\quad
+\left\lVert K\ast \left\lvert |v|^{2}-|w|^{2}\right\rvert
\right\rVert_{L^{2}L^{2d/\gamma}}\|w\|_{L^{\theta}L^{r}}\\
&\lesssim
\left(\|v\|_{L^{\theta}L^{r}}^{2}+\|w\|_{L^{\theta}L^{r}}^{2}\right)\|v-w\|_{L^{q}L^{r}}\\
& \leqslant
C T^{1-\gamma/2}(\|v\|_{L^{q}L^{r}}^{2}+\|w\|_{L^{q}L^{r}}^{2})\|v-w\|_{L^{q}L^{r}}.
\end{align*}
Thus $\Phi$ is a contraction from $Y(T)$ to $Y(T)$ provided
that $T$ is sufficiently small. Then there exists a unique $u\in
Y(T)$ solving \eqref{eq:r3}. The global existence of the
solution for \eqref{eq:r3} follows from the conservation
of the $L^{2}$-norm of $u$. The last property of the proposition then
follows from Strichartz estimates applied with an arbitrary admissible
pair on the left hand side, and the same pairs as above on the right
hand side.
\end{proof}
\section{Proof of Theorem~\ref{theo:main}}
\label{sec:inter}
Thoughout this section, we assume that the kernel $K$ is given by
\eqref{eq:noyau}.
\subsection{Uniqueness}
\label{sec:unique}
Uniqueness stems from the local well-posedness result established in
\cite{Mou12}, based on the following lemma, whose proof is recalled
for the sake of completeness.
\begin{lemma}\label{lem:estW}
Let $0<\gamma<d$. There exists $C$ such that for all $f,g\in
L^2(\R^d)\cap W$,
\begin{equation*}
\left\| \(K\ast |f|^2\)f -\(K\ast |g|^2\)g\right\|_{L^2\cap W}
\leqslant C\(\|f\|_{L^2\cap W}^2 + \|g\|_{L^2\cap W}^2\)\|f-g\|_{L^2\cap W}.
\end{equation*}
\end{lemma}
\begin{proof}
Let $\kappa_1= {\bf 1}_{\{|\xi|\leqslant 1\}}\widehat K$ and $\kappa_2=
{\bf 1}_{\{|\xi|>1\}}\widehat K$. In view of
Proposition~\ref{prop:hatK}, $\kappa_1\in L^p(\R^d)$ for all $p\in
[1,\frac{d}{d-\gamma})$ and $\kappa_2\in L^q(\R^d)$ for all $q\in
(\frac{d}{d-\gamma},\infty]$. For $h\in L^1(\R^d)\cap W$, we have
\begin{align*}
\|K\ast h\|_{W} & \lesssim \|\kappa_1 \widehat h\|_{L^1} + \|\kappa_2
\widehat h\|_{L^1} \lesssim \|\kappa_1\|_{L^1}\|\widehat
h\|_{L^\infty} +\|\kappa_2\|_{L^\infty}\|\widehat h\|_{L^1}\\
& \lesssim \|h\|_{L^1} + \|\widehat h\|_{L^1},
\end{align*}
where we have used Hausdorff--Young inequality. Writing
\begin{align*}
\(K\ast |f|^2\)f -\(K\ast |g|^2\)g = \(K\ast |f|^2\)(f-g) +\(K\ast
(|f|^2-|g|^2)\)g,
\end{align*}
the lemma follows, since $W$ is a Banach algebra embedded
into $L^\infty(\R^d)$.
\end{proof}
We infer uniqueness in $L^2(\R^d)\cap W$ for \eqref{eq:r3} as soon as
$0<\gamma<d$:
\begin{proposition}\label{prop:uniqueness}
Let $0<\gamma<d$, $T>0$, and $u,v\in C([0,T];L^2\cap W)$ solve \eqref{eq:r3},
with the same initial datum $u_0\in L^2(\R^d)\cap W$. Then $u\equiv v$.
\end{proposition}
\begin{proof}
Duhamel's
formula yields
\begin{equation*}
u(t)-v(t) = -i \int_0^t e^{i(t-\tau)\Delta}\(\(K\ast |u|^2\)u
-\(K\ast |v|^2\)v\)(\tau)\mathrm{d}\tau.
\end{equation*}
Since the Schr\"odinger group is unitary on $L^2$ and on $W$, Minkowski
inequality and Lemma~\ref{lem:estW} yield, for $t\geqslant 0$,
\begin{align*}
\|u(t)-v(t)&\|_{L^2\cap W} \lesssim \int_0^t \(\|u(\tau)\|_{L^2\cap
W}^2 +\|v(\tau)\|_{L^2\cap W}^2 \)\|u(\tau)-v(\tau)\|_{L^2\cap
W}\mathrm{d}\tau\\
&\lesssim \( \|u\|_{L^\infty([0,T];L^2\cap W)}^2
+\|v\|_{L^\infty([0,T];L^2\cap W)}^2\) \int_0^t \|u(\tau)-v(\tau)\|_{L^2\cap
W}\mathrm{d}\tau.
\end{align*}
Gronwall lemma implies $u\equiv v$.
\end{proof}
\subsection{Existence}
\label{sec:existence}
In view of Lemma~\ref{lem:estW}, the standard fixed point argument
yields:
\begin{proposition}\label{prop:local}
Let $d\geqslant 1$, $\lambda\in \R$, $0<\gamma<d$, and $K$ given by
\eqref{eq:noyau}. If $u_0\in L^2(\R^d)\cap W$, then there exists
$T>0$ depending only on $\lambda,\gamma,d$ and $\|u_0\|_{L^2\cap
W}$, and a unique $u\in C([0,T];L^2\cap W)$ to \eqref{eq:r3}.
\end{proposition}
Taking Proposition~\ref{prop:r3L2} into account, to establish
Theorem~\ref{theo:main}, it suffices to prove that the Wiener norm of
$u$ cannot become unbounded in finite time.
Resuming the decomposition of $\widehat K$ introduced in the proof of
Lemma~\ref{lem:estW}, we find
\begin{align*}
\|u(t)\|_{W} &\leqslant \|u_0\|_{W} + \int_0^t \left\| \(K\ast
|u(\tau)|^2\)u(\tau)\right\|_{W} \mathrm{d}\tau \\
&\leqslant \|u_0\|_{W} + \int_0^t \left\| K\ast
|u(\tau)|^2\right\|_{W}\|u(\tau)\|_{W} \mathrm{d}\tau \\
&\leqslant \|u_0\|_{W} + \int_0^t \(\| \kappa_1\|_{L^1}
\|u(\tau)\|_{L^2}^2+
\|\kappa_2\|_{L^p}\left\|\widehat{|u(\tau)|^2}\right\|_{L^{p'}}\)
\|u(\tau)\|_{W} \mathrm{d}\tau ,
\end{align*}
provided that $p>\frac{d}{d-\gamma}$. Using the conservation of the
$L^2$-norm of $u$ and Hausdorff--Young inequality, we infer, if $p\leqslant
2$:
\begin{align*}
\|u(t)\|_{W} \lesssim \|u_0\|_{W} + \int_0^t \(\| \kappa_1\|_{L^1}
\|u_0\|_{L^2}^2+
\|\kappa_2\|_{L^p}\left\||u(\tau)|^2\right\|_{L^{p}}\)
\|u(\tau)\|_{W} \mathrm{d}\tau.
\end{align*}
To summarize, for all $1<\frac{d}{d-\gamma}<p\leqslant 2$, there exists $C$
such that
\begin{equation*}
\|u(t)\|_{W} \leqslant \|u_0\|_{W} + C \int_0^t \|u(\tau)\|_{W} \mathrm{d}\tau +
C\int_0^t \left\|u(\tau)\right\|^2_{L^{2p}}
\|u(\tau)\|_{W} \mathrm{d}\tau.
\end{equation*}
The above requirement on $p$ can be fulfilled if and only if
$0<\gamma<d/2$.
To take advantage of Proposition~\ref{prop:r3L2}, introduce $\alpha>
1$
such that $(2\alpha,2p)$ is admissible. This is possible provided that
$2p<\frac{2d}{d-2}$ when $d\geqslant 3$: this condition is compatible with
the requirement $p>\frac{d}{d-\gamma}$ if and only if
$\gamma<2$. Using H\"older inequality for the
last integral, we have
\begin{equation*}
\|u(t)\|_{W} \leqslant \|u_0\|_{W} + C \int_0^t \|u(\tau)\|_{W} \mathrm{d}\tau +
C\left\|u\right\|^2_{L^{2\alpha}([0,t];L^{2p})}
\|u\|_{L^{\alpha'}([0,t];W)}.
\end{equation*}
Set
\begin{equation*}
\omega (t) =\sup_{0\leqslant \tau\leqslant t}\|u(\tau)\|_{W}.
\end{equation*}
For a given $T>0$, $\omega$ satisfies an
estimate of the form
\begin{equation*}
\omega(t) \leqslant \|u_0\|_{W} + C\int_0^t\omega(\tau)\mathrm{d}\tau +
C_0(T)\(\int_0^t \omega(\tau)^{\alpha'}\mathrm{d}\tau\)^{1/\alpha'},
\end{equation*}
provided that $0\leqslant t\leqslant T$, and where we have used the fact that
$\alpha'$ is finite.
Using H\"older inequality, we infer
\begin{equation*}
\omega(t) \leqslant \|u_0\|_{W} +
C_1(T)\(\int_0^t \omega(\tau)^{\alpha'}\mathrm{d}\tau\)^{1/\alpha'}.
\end{equation*}
Raising the above estimate to the power $\alpha'$, we find
\begin{equation*}
\omega(t)^{\alpha'} \lesssim 1 + \int_0^t \omega(\tau)^{\alpha'}\mathrm{d}\tau.
\end{equation*}
Gronwall lemma shows that $\omega \in L^\infty([0,T])$. Since $T>0$ is
arbitrary, $\omega\in L^\infty_{\rm loc}(\R)$, and the result
follows.
\section{Ill-posedness in the mere Wiener algebra}
\label{sec:ill}
In this section we still assume that $K$ is given by
\eqref{eq:noyau}. We show that the Cauchy problem \eqref{eq:r3} is ill-posed
in the mere Wiener algebra, i.e without including $ L^{2}$.
We recall the definition of well-posedness for the problem \eqref{eq:r3}.
\begin{definition}
\label{well}
Let $(S, \|\cdot\|_{S})$ be a Banach space of initial data, and
$(D,\|\cdot\|_{D})$ be a Banach space of space-time functions.
The Cauchy problem \eqref{eq:r3} is well posed from $D$ to $S$ if, for all bounded
subset $B\subset D$, there exist $T > 0$ and a Banach
space $X_{T} \hookrightarrow C([0,T]; S)$ such that:
\begin{enumerate}
\item For all $\varphi\in B$, \eqref{eq:r3} has a unique solution $ \psi\in X_{T}$
with $\psi|_{t=0}=\varphi$.
\item The mapping $B\ni\varphi\mapsto \psi\in C([0,T];S)$ is uniformly
continuous.
\end{enumerate}
\end{definition}
\begin{proof}[Proof of Theorem~\ref{th:ill}]
In view of \cite{BeTa06}, it suffices to prove that one term in the Picard
iterations of $\Phi$ defined
in Section~\ref{sec:standard} does not verify the Definition~\ref{well}.
We argue by contradiction and assume that \eqref{eq:r3} is well-posed from
$W$ to $W$. Then, let $T>0$ be the local time existence of the solution. We recall that for $0<\gamma<d$, \eqref{eq:r3} is
well-posed from $W\cap L^{2}$ to $W\cap L^{2}$ (see
\cite[Theorem~2.1]{Mou12}, recalled in Proposition~\ref{prop:local}).
We define the following operator associated to the second Picard iterate:
\[
D(f)(t,x)=-i\int_{0}^{t} e^{i(t-\tau)\Delta}(K*|\psi|^{2}\psi)(\tau,x)\; \mathrm{d}x,
\]
where $\psi$ is the solution of the free equation
\[
i\partial_{t}\psi + \Delta \psi = 0; \quad \psi |_{t=0}=f,
\]
that is $\psi(t) = e^{it\Delta}f$. We denote $e^{it\Delta}f=U(t)f$.
By \cite[Proposition 1]{BeTa06} the operator $D$ is continuous
from $W$ to $W$, that is,
\begin{equation}
\label{ill-control}
\|D(f)(t)\|_{W}\leqslant C\|f\|^{3}_{W},\quad \forall t\in [0,T],
\end{equation}
for some positive constant $C$. Let $f\in \Sch(\R^{d})$ be an element of
the Schwartz space. We define a family of functions indexed by $h >0$
by
\[
f^{h}(x)=f(hx).
\]
For all $h>0, \|f^{h}\|_{L^{2}}=\frac{1}{h^{d/2}}\|f\|_{L^{2}}$ so for $h>0$ close to $0$
the family $(f^{h})_{h>0}$ leaves any compact of $L^{2}$. Remark that
$\widehat{f^{h}}(\xi)=h^{-d}\widehat{f}(\frac{\xi}{h})$ and
$ \|f^{h}\|_{W}=\|f\|_{W} < \infty$, so \eqref{ill-control} yields
\begin{equation}
\label{fh_W}
\|D(f^{h})(t)\|_{W}\leqslant C\|f\|^{3}_{W}.
\end{equation}
We develop the expression of $\|D(f^{h})(t)\|_{W}$. We have
\[
\begin{aligned}
&\mathcal{F}(K*|U(\tau)f^{h}|^{2}U(\tau)f^{h})(\xi)=\phantom{\int_{\R^{d}}
\int_{\R^{d}}\widehat{K}(\xi-y) e^{i\tau|\xi-y-z|^{2}}\widehat{f^{h}}(\xi-y-z)
e^{i\tau|y|^{2}}\widehat{f^{h}}(y)\;\mathrm{d}y}\\
&=(2\pi)^{d/2}\mathcal{F}(K*|U(\tau)f^{h}|^{2})*\mathcal{F}(U(\tau)f^{h})(\xi)\\
&=(2\pi)^{d/2}\int_{\R^{d}} \mathcal{F}(K*|U(\tau)f^{h}|^{2})(\xi-y)
\mathcal{F}(U(\tau)f^{h})(y)\;\mathrm{d}y\\
&=\int_{\R^{d}} e^{i\tau|y|^{2}}\widehat{K}(\xi-y)\mathcal{F}
(|U(\tau)f^{h}|^{2})(\xi-y)\widehat{f^{h}}(y)\;\mathrm{d}y\\
&=(2\pi)^{d/2}\iint_{\R^{2d}} e^{i\tau|y|^{2}}
e^{i\tau|\xi-y-z|^{2}}e^{-i\tau|z|^{2}}\widehat{K}(\xi-y)
\widehat{f^{h}}(\xi-y-z)\widehat{\overline{f^{h}}}(z)
\widehat{f^{h}}(y)\;\mathrm{d}y\mathrm{d}z\\
&=\frac{(2\pi)^{d/2}}{h^{3d}}\iint_{\R^{2d}}
e^{i\tau|y|^{2}}
e^{i\tau|\xi-y-z|^{2}}e^{-i\tau|z|^{2}}\widehat{K}(\xi-y)
\widehat{f}\(\frac{\xi-y-z}{h}\)\widehat{\overline{f}}\(\frac{z}{h}\)
\widehat{f}\(\frac{y}{h}\)\;\mathrm{d}y\mathrm{d}z.\\
\end{aligned}
\]
Taking the $W$-norm gives
\[
\begin{aligned}
\|D(f^{h})(t)\|_{W}&=\|\widehat{D(f^{h})}(t)\|_{L^{1}}\\
&=\int_{\R^{d}}\left|\int_{0}^{t} \mathcal{F}(U(t-\tau)(K*|U(\tau)f^{h}|^{2}
U(\tau)f^{h})(\xi)\;\mathrm{d}\tau\right|\;\mathrm{d}\xi\\
&=\int_{\R^{d}}\left|\int_{0}^{t} e^{i(t-\tau)|\xi|^{2}}
\mathcal{F}(K*|U(\tau)f^{h}|^{2}U(\tau)f^{h})(\xi)\;\mathrm{d}\tau\right|
\;\mathrm{d}\xi.\\
\end{aligned}
\]
We replace $\mathcal{F}(K*|U(\tau)f^{h}|^{2}U(\tau)f^{h})(\xi)$ by
its integral
formula above and apply the following changes of variable:
$\xi'=\xi/h, y'=y/h, z'=z/h, \tau'=\tau h^{2}$. We obtain
\begin{equation}
\label{Dfh_W}
\|D(f^{h})(t)\|_{W}=\frac{1}{h^{d-\gamma+2}}\|D(f)(th^{2})\|_{W}.
\end{equation}
Let $s\in (0, T)$. We examine more closely the term $F(s):=\int_{0}^{s}
U(s-\tau)g(\tau)\mathrm{d}\tau$ where $g(s):= (K*|U(s)f|^{2})U(s)f.$
Taylor formula yields
\[
F(s)=F(0)+F'(0)s + \frac{s^{2}}{2}\int_{0}^{1} (1-\theta)F''(s\theta)\;\mathrm{d}\theta.
\]
We have $F(0)=0$, so for $s\in [0,1]$,
\[
\begin{aligned}
\|F(s)-F'(0)s\|_{W} &\leqslant s^{2}\left\|\int_{0}^{1}
(1-\theta)F''(s\theta)\;\mathrm{d}\theta \right\|_{W}\;\mathrm{d}\theta\\
& \leqslant s^{2} \int_{0}^{1} \|F''(s\theta)\|_{W} \;\mathrm{d}\theta
\leqslant s^{2} \|F''\|_{L^{\infty}([0,1];W)}.
\end{aligned}
\]
The first and second derivatives of $F$ are given by
\begin{align*}
F'(s)&=g(s)+\int_{0}^{s}U(s-\tau)i\Delta g(\tau)\;\mathrm{d}\tau,\\
F''(s)&=g'(s)+ i\Delta g(s) - \int_{0}^{s}
U(s-\tau)\Delta^{2}g(\tau)\;\mathrm{d}\tau,
\end{align*}
so $F'(0)=g(0)$ and
\[
\|F''\|_{L^{\infty}([0,1];W)} \leqslant \|g'\|_{L^{\infty}([0,1];W)} +
\|\Delta g\|_{L^{\infty}([0,1];W)}+
\|\Delta^{2}g\|_{L^{\infty}([0,1];W)}.
\]
From the formula of $g$, since $f \in \mathcal{S}$ we can see easily
that $F''\in L^{\infty}([0,1];W)$ (uniformly in $h\in (0,1]$).
We obtain
\[
\|F(s)-sg(0)\|_{W} \leqslant C s^{2},
\]
where $C$ is a positive constant independent of $s$ (it depends on $f$
and $\gamma$). Thus,
\[
\|F(s)\|_{W}= s\|g(0)\|_{W} + \mathcal O(s^{2}).
\]
In particular, for all $t, h>0$
\[
\|D(f)(th^{2})\|_{W}=\|F(th^{2})\|_{W}= th^{2}\|g(0)\|_{W} + \mathcal O\(t^{2}h^{4}\).
\]
This implies that
\[
\|D(f^{h})(t)\|_{W}=\frac{1}{h^{d-\gamma+2}}\|D(f)(th^{2})\|_{W}=
\frac{t}{h^{d-\gamma}}\|g(0)\|_{W}+\frac{1}{h^{d-\gamma}}\mathcal O\(t^{2}h^{2}\).
\]
Fix $t>0$:
\[
\lim\limits_{h\to 0^{+}}
\frac{t}{h^{d-\gamma}}\|g(0)\|_{W}+\frac{1}{h^{d-\gamma}}\mathcal O\(t^{2}h^{2}\)= +\infty.
\]
We deduce that for $h>0$ sufficiently close to $0$,
\[
\|D(f^{h})(t)\|_{W} > C\|f\|_{W}^{3}.
\]
This contradicts
\eqref{ill-control}, and Theorem \ref{th:ill} follows.
\end{proof}
\section{Proof of Theorem~\ref{theo:Lp}}
We decompose the proof of Theorem~\ref{theo:Lp}
according to the three cases considered.
\subsection{Well-posedness in $L^{2}\cap W$}
We assume that
$K$ is such that $\widehat{K}\in L^{p}$ for some $p\in [1,\infty]$, and
we consider an initial
data $u_{0}\in L^{2}\cap W$.
For $T > 0$ we define the following space
\[
E_{T}=\{u\in C([0,T];L^{2}\cap W),\quad \|u\|_{L^{\infty}([0,T];L^{2}\cap W)}\leqslant 2\|u_{0}\|_{L^{2}\cap W}\}.
\]
It is a complete space metric when equipped with the metric
\[
d(u,v)= \|u-v\|_{L^{\infty}([0,T];L^{2}\cap W)}.
\]
\begin{lemma}\label{estWL^2}
Let $p\in [1,\infty]$, and $K$ such that $\widehat K\in L^p$. There exists
$C$ such that for all $f, g\in L^{2}\cap W$,
\[
\|K*(fg)\|_{W} \leqslant C\|\widehat{K}\|_{L^{p}} \|f\|_{L^{2}\cap W}\|g\|_{L^{2}\cap W}.
\]
\end{lemma}
\begin{proof}
Let $ f, g\in L^{2}\cap W$. We denote by $p'$ the H\"older conjugate
exponent of $p$. We have
\[
\begin{aligned}
\|K*(fg)\|_{W} = (2\pi)^{d/2} \|\widehat{K}\; \widehat{fg}\|_{L^{1}}
&\leqslant C\|\widehat{K}\|_{L^{p}}\|\widehat{fg}\|_{L^{p'}}
\leqslant C\|\widehat{K}\|_{L^{p}} \|\widehat{f}*\widehat{g}\|_{L^{p'}}\\
&\leqslant C \|\widehat{K}\|_{L^{p}} \|\widehat{f}\|_{L^{q}}\|\widehat{g}\|_{L^{q}},
\end{aligned}
\]
where $1+\frac{1}{p'}= \frac{2}{q}$. Since $q\in [1,2]$ there exists
$\theta \in [0,1]$ such that $\frac{1}{2}+ \frac{\theta}{2}= \frac{1}{q}$
and
\[
\|\widehat{f}\|_{L^{q}} \leqslant \|\widehat{f}\|^{\theta}_{L^{1}}\|\widehat{f}\|^{1-\theta}_{L^{2}}
=\|f\|^{\theta}_{W}\|f\|^{1-\theta}_{L^{2}}
\leqslant \|f\|_{L^{2}\cap W}.
\]
The lemma follows.
\end{proof}
Let $\Phi$ as defined in Section~\ref{sec:standard}. In view of the
previous lemma we have, for $u\in E_{T}$ and $t\in [0,T]$,
\[
\begin{aligned}
\|\Phi(u)(t)\|_{L^{2}\cap W} &\leqslant \|u_{0}\|_{L^{2}\cap W} +
\int_{0}^{t} \|K*|u(\tau)|^{2}u(\tau)\|_{L^{2}\cap W}\;\mathrm{d}\tau\\
&\leqslant \|u_{0}\|_{L^{2}\cap W} +
\int_{0}^{t} \|K*|u(\tau)|^{2}\|_{W}\|u(\tau)\|_{L^{2}\cap W}\;\mathrm{d}\tau\\
&\leqslant \|u_{0}\|_{L^{2}\cap W} +
C\|\widehat{K}\|_{L^{p}}\int_{0}^{t} \|u(\tau)\|^{3}_{L^{2}\cap W} \;\mathrm{d}\tau.\\
\end{aligned}
\]
We obtain
\[
\|\Phi(u)(t)\|_{L^{\infty}([0,T];L^{2}\cap W)} \leqslant \|u_{0}\|_{L^{2}\cap W}
+ C\|\widehat{K}\|_{L^{p}}\|u_{0}\|^{3}_{L^{\infty}([0,T];L^{2}\cap W)}T.
\]
For $T$ sufficiently small (depending on $\|u_{0}\|_{L^{2}\cap W}$)
$\|\Phi(u)\|_{L^{\infty}_{T}L^{2}} \leqslant 2\|u_{0}\|_{L^{2}\cap W}$.
Let $u,v \in E_{T}$. From Lemma \ref{estWL^2} we have
\[
\begin{aligned}
&\|\Phi(u)(t)-\Phi(v)(t)\|_{L^{2}\cap W}
\\
&\leqslant \int_{0}^{t}
\|K*(|u(\tau)|^{2}-|v(\tau)|^{2})u(\tau)\|_{L^{2}\cap
W}\;\mathrm{d}\tau\\
&\quad+\int_{0}^{t}\|K*|v(\tau)|^{2}(u(\tau)-v(\tau)\|_{L^{2}\cap
W}\;\mathrm{d}\tau\\
&\leqslant \int_{0}^{t} \|K*(|u(\tau)|^{2}-|v(\tau)|^{2})\|_{W}\|u(\tau)\|_{L^{2}\cap W}\;\mathrm{d}\tau\\
&\quad+ \int_{0}^{t} \|K*|v(\tau)|^{2}\|_{W}\|u(\tau)-v(\tau)\|_{L^{2}\cap W}\;\mathrm{d}\tau\\
&\leqslant\int_{0}^{t} C\|\widehat{K}\|_{L^{p}} \(\|u(\tau)\|^{2}_{L^{2}\cap
W}+
\|v(\tau)\|_{L^{2}\cap W}^2\)\|u(\tau)-v(\tau)\|_{L^{2}\cap W}\;\mathrm{d}\tau.
\end{aligned}
\]
We deduce that
\[
\|\Phi(u)-\Phi(v)\|_{L^{\infty}([0,T];L^{2}\cap W)} \leqslant
C\|\widehat{K}\|_{L^{p}}\|u_{0}\|^{2}_{L^{2}\cap W}T\|u-v\|_{L^{\infty}([0,T];L^{2}\cap W)}.
\]
For $T$ possibly smaller (still depending on $\|u_{0}\|_{L^{2}\cap W}$) $\Phi$ is a contraction from $E_{T}$ to $E_{T}$,
so admits a unique fixed point in $E_{T}$, which is a solution for the Cauchy problem.
By resuming the same arguments as in Proposition \ref{prop:uniqueness}
we deduce that the fixed point obtained before is the unique solution
for the Cauchy problem.
\bigbreak
For $p=1$, the solution constructed before is global in time. In view
of the conservation of the $L^2$-norm, we have
\[
\begin{aligned}
\|u(t)\|_{W}&\leqslant \|u_{0}\|_{W} +
\int_{0}^{t} \|K*|u(\tau)|^{2}\|_{W}\|u(\tau)\|_{L^{2}\cap W}\;\mathrm{d}\tau\\
&\leqslant \|u_{0}\|_{W}+ \int_{0}^{t} C\|\widehat{K}\|_{L^{1}}\|u(\tau)\|_{L^{2}}^{2} \|u(\tau)\|_{W}\;\mathrm{d}\tau\\
&\leqslant \|u_{0}\|_{W} + C\|\widehat{K}\|_{L^{1}}\|u_{0}\|_{L^{2}}^{2}\int_{0}^{t} \|u(\tau)\|_{W}\;\mathrm{d}\tau,\\
\end{aligned}
\]
and by the Gronwall lemma, we conclude that $\|u(t)\|_{W}$ remains bounded
on finite time intervals. This completes the proof of the first point in
Theorem~\ref{theo:Lp}.
\subsection{Well-posedness in $W$}
Let $\widehat{K}\in L^{\infty}$ and consider the Cauchy problem \eqref{eq:r3}
with an initial data in $W$. For $T>0$ we define the following space
\[
Y_{T}=\{u\in C([0,T]; W), \|u\|_{L^{\infty}([0,T];W)} \leqslant 2\|u_{0}\|_{W}\}.
\]
This later is a complete metric space when equipped with the metric
\[
d(u,v)=\|u-v\|_{L^{\infty}([0,T];W)}.
\]
As previously, the local existence of a solution is easily shown by a fixed point argument,
since $\Phi$ is a contraction from $Y_{T}$ to $Y_{T}$, and we show that it is unique.
The proof relies on the following lemma:
\begin{lemma}\label{lem:2}
There exists $C$ such that for all $f,g \in W$,
\[
\|K*(fg)\|_{W}\leqslant C\|\widehat{K}\|_{L^{\infty}}\|f\|_{W}\|g\|_{W}.
\]
\end{lemma}
\begin{proof}
Let $f,g\in W$. We have
\[
\begin{aligned}
\|K*(fg)\|_{W} = (2\pi)^{d/2} \|\widehat{K}\;\widehat{fg}\|_{L^{1}}
&\leqslant (2\pi)^{d/2}\|\widehat{K}\|_{L^{\infty}}\|\widehat{f}*\widehat{g}\|_{L^{1}}\\
&\leqslant
(2\pi)^{d/2}\|\widehat{K}\|_{L^{\infty}}\|\widehat{f}\|_{L^{1}}\|\widehat{g}\|_{L^{1}}\\
&\leqslant
(2\pi)^{d/2}\|\widehat{K}\|_{L^{\infty}}\|f\|_{W}\|g\|_{W}.
\end{aligned}
\]
\end{proof}
It then suffices to reproduce the proof given in the previous
subsection in order to prove the second point of
Theorem~\ref{theo:Lp}, by replacing Lemma~\ref{estWL^2} with
Lemma~\ref{lem:2}.
\subsection{Ill-posedness in $W$}
For $\gamma \in (0,d)$ we consider the Cauchy problem \eqref{eq:r3}
with the kernel $K$ given by its
Fourier transform
\begin{equation}\label{eq:Khat}
\widehat{K}(\xi)= \frac{1}{|\xi|^{d-\gamma}}{\bf
1}_{\{|\xi|\leqslant 1\}}.
\end{equation}
Then, $\widehat{K}\in L^{p}$ for all $p \in [1,
\frac{d}{d-\gamma})$. Conversely, for $p\in [1,\infty)$, we can always
find $\gamma \in (0,d)$ such that $K$, defined by \eqref{eq:Khat},
satisfies $\widehat{K}\in L^{p}$.
From the first point of Theorem~\ref{theo:Lp}, the Cauchy problem
\eqref{eq:r3} is locally well-posed in $L^2\cap W$.
We suppose that \eqref{eq:r3} is well posed in $W$: let $T>0$ be the local time
existence of the solution. Arguing as in Section~\ref{sec:ill}, this
implies that there exists $C>0$ such that
for all $f\in W$ and for all $t\in [0, T]$,
\begin{equation}\label{eq:ill2}
\|D(f)(t)\|_{W} \leqslant C\|f\|_{W}^{3},
\end{equation}
where $D$ is the operator defined in Section~\ref{sec:ill}.
Let $f\in \mathcal{S}(\R^d)$. As in Section~\ref{sec:ill} we define the
family of functions
$(f^{h})_{0<h\leqslant 1}$ by
\[
f^{h}(x) = f(hx).
\]
From \eqref{eq:ill2} we also have for all $t\in [0,T]$
\begin{equation}
\|D(f^{h})(t)\|_{W}\leqslant C\|f^{h}\|^{3}_{W}=C \|f\|^{3}_{W}.
\end{equation}
We define $K_{h}$ by setting
$\widehat{K_{h}}(\xi)=\frac{1}{|\xi|^{d-\gamma}} {\bf
1}_{\{|\xi|>\frac{1}{h}\}}$. We use the following identity:
\[
\begin{aligned}
&\mathcal{F}(K*|U(\tau)f^{h}|^{2}U(\tau)f^{h})(\xi)=\phantom{\int_{\R^{d}}
\int_{\R^{d}}\widehat{K}(\xi-y) e^{i\tau|\xi-y-z|^{2}}\widehat{f^{h}}(\xi-y-z)
e^{i\tau|y|^{2}}\widehat{f^{h}}(y)\;\mathrm{d}y}\\
&=\frac{(2\pi)^{d/2}}{h^{3d}}\iint_{\R^{2d}}
e^{i\tau|\xi-y|^{2}}
e^{i\tau|y-z|^{2}}e^{-i\tau|z|^{2}}\widehat{K}(y)
\widehat{f}\(\frac{\xi-y}{h}\)\widehat{\overline{f}}\(\frac{z}{h}\)
\widehat{f}\(\frac{y-z}{h}\)\;\mathrm{d}y\mathrm{d}z\\
&=\frac{(2\pi)^{d/2}}{h^{3d}}\iint_{\R^{2d}}
e^{i\tau|\xi-y|^{2}}
e^{i\tau|y-z|^{2}}e^{-i\tau|z|^{2}}\frac{1}{|y|^{d-\gamma}}
\widehat{f}\(\frac{\xi-y}{h}\)\widehat{\overline{f}}\(\frac{z}{h}\)
\widehat{f}\(\frac{y-z}{h}\)\;\mathrm{d}y\mathrm{d}z\\
&-\frac{(2\pi)^{d/2}}{h^{3d}}\iint_{\R^{2d}}
e^{i\tau|\xi-y|^{2}}
e^{i\tau|y-z|^{2}}e^{-i\tau|z|^{2}}\widehat{K_{h}}(y)
\widehat{f}\(\frac{\xi-y}{h}\)\widehat{\overline{f}}\(\frac{z}{h}\)
\widehat{f}\(\frac{y-z}{h}\)\;\mathrm{d}y\mathrm{d}z.\\
\end{aligned}
\]
We inject the above formula into the expression of
$\|D(f^{h})(t)\|_{W}$ and perform the same changes of variables
as in Section~\ref{sec:ill}. We obtain
\begin{equation*}
\|D(f^{h})(t)\|_{W} \geqslant \frac{t}{h^{d-\gamma}}\(\|(K\ast|f|^2)f\|_{W} +
\mathcal O\(t h^{2}\)\)- \frac{1}{h^{d-\gamma+2}}X^{h},
\end{equation*}
where
\[
X^{h}= \left\lVert\int_{0}^{th^{2}}
U(t-\tau)(K_{h}*|U(\tau)f|^{2}U(\tau)f)\;\mathrm{d}\tau\right\rVert_W.
\]
For $q\in (\frac{d}{d-\gamma}, \infty)$,
\[
\begin{aligned}
X^{h} &\leqslant \int_{0}^{th^{2}}
\left\|\(K_{h}*|U(\tau)f|^{2}\)U(\tau)f\right\|_{W}\;\mathrm{d}\tau
\leqslant \int_{0}^{th^{2}} \|K_{h}*|U(\tau)f|^{2}\|_{W} \|U(\tau)f\|_{W}\;\mathrm{d}\tau\\
&\leqslant \|f\|_W \int_{0}^{th^{2}}
\|\widehat{K_{h}}\|_{L^{q}}
\left\|\mathcal F\(|U(\tau)f|^{2}\)\right\|_{L^{q'}}\;\mathrm{d}\tau
\leqslant C th^{2} \|\widehat{K_{h}}\|_{L^{q}} ,
\end{aligned}
\]
for some constant $C$ independent of $h\in (0,1]$ and $t\in [0,T]$
(recall that $f\in \Sch(\R^d)$).
Moreover,
\[
\|\widehat{K_{h}}\|_{L^{q}} = \bigg(\int_{\R^{d}}
\frac{1}{|y|^{(d-\gamma)q}}{\bf 1}_{\{|y|>\frac{1}{h}\}}\;\mathrm{d}y\bigg)^{1/q}
= C \bigg(\int_{1/h}^{+\infty}
\frac{r^{d-1}}{r^{(d-\gamma)q}}\;\mathrm{d}r\bigg)^{1/q},
\]
so, $\|\widehat{K_{h}}\|_{L^{p}}=C h^{d-\gamma-\frac{d}{q}}$ and
\[
\begin{aligned}
\|D(f^{h})(t)\|_{W} & \geqslant \frac{t}{h^{d-\gamma}}\(\|(K\ast|f|^2)f\|_{W} +
\mathcal O\(t h^{2}\)- C h^{d-\gamma-\frac{d}{q}}\).\\
\end{aligned}
\]
Fix $t>0$:
\[
\lim\limits_{h\to +\infty} \frac{t}{h^{d-\gamma}}\(\|(K\ast|f|^2)f\|_{W} +
\mathcal O\(t h^{2}\)- C h^{d-\gamma-\frac{d}{q}}\) = +\infty.
\]
So, for $h>0$ small enough we have
\[
\|D(f^{h})(t)\|_{W} > C\|f\|_{W}^{3},
\]
hence a contradiction.
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\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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|
1,314,259,996,512 | arxiv | \section{Introduction}
Let $T^2 = \mathbb{R}^2/\mathbb{Z}^2$ denote the usual torus and, using the standard basis, identify $T^2$ with the quotient $H_1(T^2, \mathbb{R})/H_1(T^2, \mathbb{Z})$. Equip $T^2$ with the Euclidean metric. The following result is well-known:
\begin{thm}
\label{thm:torusversion}
Suppose $f: T^2 \to T^2$ is a self-covering map of the torus.
Then $f$ is homotopic through covering maps to an expanding map $g: T^2 \to T^2$ if and only if the spectrum of $f_*: H_1(T^2, \mathbb{R}) \to H_1(T^2, \mathbb{R})$ lies outside the closed unit disk.
\end{thm}
In this setting, one may take $g$ to be the linear map on $T^2=H_1(T^2, \mathbb{R})/H_1(T^2, \mathbb{Z})$ induced by $f_*$. One may further arrange so that the map $g$ is a factor of $f$ via a map $\pi: T^2 \to T^2$ (that is, $\pi \circ f = g \circ \pi$) where $\pi$ is homotopic to the identity, is monotone, has fibers not disconnecting $T^2$, and, if $f$ is itself expanding, is a homeomorphism. The proof is now standard: one lifts the identity map $h_0: T^2 \to T^2$ under $f$ and $g$ to obtain $h_1: T^2 \to T^2$ and a homotopy $h_t, t \in [0,1]$ from $h_0$ to $h_1$. By induction, lifting and concatenating the homotopies, one obtains a family $h_t, t \in [0,\infty)$, such that $g\circ h_{t+1} = h_t \circ f$. Expansion of $g$ implies the family $h_t$ converges uniformly to the desired semiconjugacy. Similar and much more general results hold, cf. \cite[Theorem 3]{shub:endomporphisms}.
In this work, we formulate and prove an analog of this theorem for a certain class of continuous maps $f: S^2 \to S^2$ of the sphere to itself, called {\em Thurston maps}.
A Thurston map $f: S^2 \to S^2$ is an orientation-preserving branched covering of the two-sphere of degree $d \geq 2$ for which the necessarily nonempty {\em postcritical set}
$P_f= P = \union_{n>0}f^{\circ n}(\{\mbox{branch points}\})$ is finite. These were originally introduced by Thurston \cite{DH1} as combinatorial objects
with which to classify rational functions as dynamical systems on the Riemann sphere $\mbox{$\widehat{\C}$}$. Two Thurston maps $f, g$ are {\em combinatorially equivalent}
if there exist orientation-preserving homeomorphisms
$h_0, h_1: (S^2, P_f) \to (S^2, P_g)$ such that $h_0\circ f = g \circ h_1$ and $h_0, h_1$ are homotopic relative to $P_f$.
The relation of combinatorial equivalence can be thought of as conjugacy, up to isotopy relative to postcritical sets.
Recently, Thurston maps have been investigated in connection with the study of finite subdivision rules, the classification of quasisymmetry classes
of metrics on $S^2$, and Cannon's conjecture; see \cite{cfp:fsr}, \cite{bonk:meyer:subdivisions}, \cite{kmp:ph:cxci}.
A formulation of a variant of Theorem \ref{thm:torusversion} for Thurston maps requires the development of a notion of expansion for non-locally injective maps,
algebraic/ homotopy-theoretic invariants of a self-map of a simply-connected surface, and the synthetic construction of an expanding model map from the algebraic data.
\gap
\noindent{\bf Expansion.} Let $f$ be a Thurston map. Since $f$ has branch points, it can never be expanding with respect to a Riemannian metric on $S^2$, nor can it be positively expansive. We therefore reformulate the notion of expansion in terms of contraction of inverse branches. Let ${\mathcal U}_0$ be a finite cover of $S^2$ by connected open sets, and let ${\mathcal U}_n$ denote the covering whose elements are connected components of $f^{-n}(U), U \in {\mathcal U}_0$. We say that $f$ is {\em expanding} if the following condition holds.
\begin{defn}
\label{defn:expansion} The map $f$ is {\em expanding} if the mesh of the coverings ${\mathcal U}_n$ tends to zero as $n \to \infty$. That is, for any finite open cover ${\mathcal Y}$ of $S^2$ by open sets, there exists $N$ such that for all $n \geq N$ and all $U \in {\mathcal U}_n$, there exists $Y \in {\mathcal Y}$ with $U \subset Y$. Equivalently: the diameters of the elements of ${\mathcal U}_n$, with respect to some (any) metric, tend to zero as $n \to \infty$.
\end{defn}
Note that this is a purely topological condition. It is at present unknown whether every expanding Thurston map is topologically conjugate to a map which is smooth away from a finite set of points. Perhaps more surprisingly, it is unknown whether every expanding Thurston map is topologically conjugate to a map which preserves some complete length structure on $S^2$.
\gap
\noindent{\bf Algebraic invariants.} Nekrashevych \cite{nekrashevych:book:selfsimilar} introduced a collection of algebraic invariants associated to a very wide class
of dynamical settings, including that of a Thurston map $f$. These may be unfamiliar to most readers; see \S 3 for details. As in \cite{DH1}, associated to
$f$ is an orbifold structure ${\mathcal O}$ on $S^2$. In order to have needed finiteness properties, we assume $f$ has no periodic branch points, so that ${\mathcal O}$ is compact.
Once a basepoint is fixed, there is a corresponding orbifold fundamental group $\pi_1({\mathcal O})$, and a so-called {\em virtual endomorphism}
$\phi: \pi_1({\mathcal O}) \dasharrow \pi_1({\mathcal O})$, well-defined up to postcomposition by inner automorphisms.
Roughly speaking, $\phi$ is the homomorphism induced by the {\em inverse} of $f$. The property of $\phi$ being {\em contracting} plays
the role of the spectral condition on the induced map $f_*$ on homology in Theorem \ref{thm:torusversion}.
\gap
\noindent{\bf Synthetic model map.} When the virtual endomorphism $\phi$ is contracting, by choosing suitable defining data
(basis; generators) one may construct a negatively curved {\em self-similarity complex} $\Sigma$. The construction is similar to that of the boundary
of a hyperbolic group from a Cayley graph. The boundary at infinity $\mathcal{J}$ of $\Sigma$ inherits a self-map $f_\Sigma: {\mathcal J} \to {\mathcal J}$.
The dynamical system $f_\Sigma: {\mathcal J} \to {\mathcal J}$ plays the role of the induced linear map $g$ in Theorem \ref{thm:torusversion}.
\gap
Our main result is
\begin{thm}[Characterization of expanding Thurston maps]
\label{thm:characterization}
Suppose $f$ is a Thurston map without periodic critical points. Then $f$ is homotopic (through Thurston maps relative to its postcritical set) to an expanding map $g$
if and only if the virtual endomorphism $\phi: \pi_1({\mathcal O}) \dasharrow \pi_1({\mathcal O})$ is contracting. \end{thm}
By applying an observation of Rees \cite[\S\,1.4]{rees:degree2:part1}, we conclude that a Thurston map without periodic branch points is combinatorially equivalent to an expanding map if and only if its virtual endomorphism on the orbifold fundamental group is contracting.
The hypothesis of no periodic critical points cannot be dropped: the rational function $f(z)=z^2$ acting on the Riemann sphere has contracting virtual endomorphism, but is not homotopic to any expanding map: the nonnegative real axis is an invariant arc joining periodic points in its postcritical set, and by Proposition \ref{prop:obstructions_to_expansion} this is an obstruction to expansion.
If the defining data for $\Sigma$ and an open covering ${\mathcal U}_0$ of $S^2$ is chosen in a suitable way, one obtains a semiconjugacy $\pi: S^2 \to {\mathcal J}$ from
$f: S^2 \to S^2$ to $f_\Sigma: {\mathcal J} \to {\mathcal J}$ playing the role of the map $\pi$ introduced after Theorem \ref{thm:torusversion}. In the setting of Theorem \ref{thm:characterization},
${\mathcal J}$ is a sphere, and $g$ is a limit of conjugates of $f$ via a path of conjugacies $h_t$ (with now $t \in [0,1])$ which gradually collapse the fibers of $\pi$ to points.
Formally, $h_t$ is a {\em pseudo-isotopy}.
A {\em finite subdivision rule} ${\mathcal R}$ on the sphere (in the sense of \cite{cfp:fsr}) is, roughly, a Thurston map $f: S^2 \to S^2$ together with a cell structure
$S_{\mathcal R}$ on $S^2$ and a refinement ${\mathcal R}(S_{\mathcal R})$ of $S_{\mathcal R}$ such that $f: {\mathcal R}(S_{\mathcal R}) \to S_{\mathcal R}$ is cellular and a homeomorphism on each cell.
By taking preimages, one obtains a sequence of refinements ${\mathcal R}^n(S_{\mathcal R})$, $n \in \mathbb{N}$. One says that ${\mathcal R}$ has {\em mesh going to zero}
if the diameters of the cells at level $n$ tend to zero as $n$ tends to infinity. It has {\em mesh going to zero combinatorially} if for some $n$, (i) every edge of
$S_{\mathcal R}$ is properly subdivided in ${\mathcal R}^n(S_{\mathcal R})$, and (ii) given a pair of disjoint edges $e, e'$ of $S_{\mathcal R}$ in the boundary of a tile,
no tile $t$ of ${\mathcal R}^n(S_{\mathcal R})$ contains edges in both $e$ and in $e'$. Thus mesh going to zero is clearly an invariant of topological conjugacy,
while mesh going to zero combinatorially is a combinatorial property.
By appealing to recent work of Bonk and Meyer \cite{bonk:meyer:subdivisions}, we obtain
\begin{thm}[Equivalence of combinatorial expansion]
\label{thm:equivalence}
Suppose $f$ is a Thurston map without periodic branch points. The following conditions are equivalent.
\begin{enumerate}
\item The map $f$ is equivalent to an expanding map, $g$.
\item There exists $m$ such that $f^m$ is equivalent to a map $g$ which is the subdivision map of a subdivision rule ${\mathcal R}$ with mesh going to zero.
\item There exists $m$ such that $f^m$ is equivalent to a map $g$ which is the subdivision map of a subdivision rule ${\mathcal R}$ with mesh going to zero combinatorially.
\item The virtual endomorphism $\phi_f$ is contracting.
\end{enumerate}
\end{thm}
\noindent {\bf Proof: } That (1) implies (2) follows from \cite[Theorem 1.2]{bonk:meyer:subdivisions}; we remark that they do not require the absence of periodic branch points. That (2) implies (1) is a straightforward consequence of the definitions. The equivalence of (2) and (3) is \cite[Theorem 3.2]{cfp:fsr}. Now suppose (3) holds. By \cite[Theorem 1.1]{cfpp:fsr-cve}, the virtual endomorphism $\phi_g$ is contracting. This implies (4). That (4) implies (1) is Theorem \ref{thm:characterization}. \qed
In Theorem \ref{thm:equivalence}, the topological conjugacy class of the expanding map $g$ is necessarily unique, by e.g. \cite[Theorem 10.4]{bonk:meyer:subdivisions}. A corollary of Theorem \ref{thm:equivalence} is thus that the dynamics of an expanding Thurston map is determined by either of two pieces of combinatorial data: a subdivision rule with mesh going to zero combinatorially, or by an expanding virtual endomorphism. If a Thurston map $f$ is already presented in terms of a subdivision rule ${\mathcal R}$, there is a finite, easily checkable, necessary and sufficient condition for ${\mathcal R}$ to have mesh going to zero combinatorially \cite[Theorem 6.1]{cfpp:fsr-cve}. If $f$ is not so presented, one can try to verify the contraction of $\phi_f$ instead. Bartholdi has pointed out that this can be done algorithmically, e.g. by combining a normal forms algorithm for writing elements of the orbifold fundamental group with the FR package \cite{bartholdi:FR} in the computational algebra program GAP.
In the paragraphs below, we outline in detail the reduction of the proof of Theorem \ref{thm:characterization} to a few key steps.
\gap
\noindent{\bf Necessity.}
The implications (1) $\Rightarrow$ (2) $\Rightarrow$ (3) $\Rightarrow$ (4) have already been established in the above proof of Theorem \ref{thm:equivalence}. In \S 7, however, we provide a new, direct proof. The idea is that
the group $\pi_1({\mathcal O})$ acts on the universal cover $p:\tilde{{\mathcal O}}\to{\mathcal O}$ and that we may find a lift $\tilde{f}_{-}$
of ``$f^{-1}$'' i.e, a map satisfying $f\circ p\circ \tilde{f}_{-}=p$. The action of the virtual endomorphism on $\pi_1({\mathcal O})$
can be interpreted in terms of the action of $\tilde{f}_{-}$ on $\tilde{{\mathcal O}}$.
Assuming that $f$ is expanding yields a metric on ${\mathcal O}$
which can be lifted to $\tilde{{\mathcal O}}$ in such a way that (a) $\pi_1({\mathcal O})$ acts geometrically on $\tilde{{\mathcal O}}$
implying that $\pi_1({\mathcal O})$ is quasi-isometric to $\tilde{{\mathcal O}}$ and (b) the action of $\tilde{f}_{-}$ is contracting.
Bringing together these two facts leads to the contraction of the virtual endomorphism.
\gap
\noindent{\bf Sufficiency.} Suppose the virtual endomorphism $\phi$ is contracting and $f$ has no periodic branch points. Necessarily $\#P \geq 3$. For technical reasons, it is convenient later to assume $\#P > 3$. When $\#P=3$,
$f$ is equivalent to a rational function which is chaotic on the whole sphere, by Thurston's characterization \cite{DH1}, and such a map is expanding.
We first construct the semiconjugacy $\pi: S^2 \to {\mathcal J}$ from $f: S^2 \to S^2$ to the model map $f_\Sigma: {\mathcal J} \to {\mathcal J}$; recall that ${\mathcal J}$ arises as the boundary of the hyperbolic selfsimilarity complex, $\Sigma$. Let ${\mathcal U}_0$ be an open cover of $S^2$ by Jordan domains. By taking inverse images, there is another associated infinite one-complex $\Gamma$; see \cite{kmp:ph:cxci} and \S 5. Using the absence of periodic branch points, we show (Theorem \ref{thm:coincidence}) that for suitable choices of the covering ${\mathcal U}_0$ defining $\Gamma$ and the defining data for $\Sigma$, there is a natural quasi-isometry $\Phi: \Sigma \to \Gamma$. Thus $\Gamma$ is also hyperbolic. Extending $\Phi$ to the boundary by a map of the same name, the composition $\pi:=\Phi^{-1} \circ \pi_\Gamma$ is the desired semiconjugacy.
Next, we show that ${\mathcal J}$ is homeomorphic to $S^2$, and that the model map $f_\Sigma$ is a Thurston map. A key role is played by the fact, established in \cite{kmp:ph:cxcii}, that $f_\Sigma: {\mathcal J} \to {\mathcal J}$ is a branched covering that, as a topological dynamical system, is expanding and is of so-called {\em finite type}. In particular, the local degrees of iterates are uniformly bounded. We first develop (\S 4) some dynamical consequences of the hypothesis that $\phi$ is contracting, e.g. that there are no so-called Levy cycles. Next, after showing $\pi$ is monotone (Lemma \ref{lemma:monotone}), it suffices, by a classical theorem of Moore \cite[Theorem 25.1]{daverman:decompositions}, to show that the fibers of $\pi$ do not disconnect $S^2$. To see this, we use a case-by-case analysis of the possibilities for how such fibers separate and intersect $P$ and the absence of Levy cycles and other related obstructions to expansion. We conclude that ${\mathcal J}$ is a sphere, that $f_\Sigma: {\mathcal J} \to {\mathcal J}$ is an expanding Thurston map, and that $\pi|_P$ is injective.
It remains to construct an isotopy from $f$ to an expanding map $g$. By \cite[Theorems 13.4, 25.1]{daverman:decompositions}, the decomposition ${\mathcal G}$ of $S^2$ by the fibers of the semiconjugacy $\pi: S^2 \to {\mathcal J}$ has the property of being {\em strongly shrinkable}: there is a one-parameter family of continuous maps $h_t: S^2 \to S^2, t \in [0,1]$, called a {\em pseudoisotopy}, such that $h_0 = \id_{S^2}$, $h_t$ is a homeomorphism for $t \in [0,1)$, $h_t|_{P_f} = \id_{P_f}$ for all $t$, and the fibers of $h_1$ and those of $\pi$ coincide. The induced homeomorphism $S^2 \to {\mathcal J}$ conjugates $f_\Sigma$ to an expanding Thurston map $g: S^2 \to S^2$, and the family of maps $f_t$, $t \in [0,1]$ defined as the unique continuous solution of $h_t \circ f = f_t \circ h_t$, with $f_1=g$, gives an homotopy through Thurston maps. The desired isotopy follows.
\gap
In the remainder of this work, we assume we are given a degree $d \geq 2$ Thurston map $f: S^2 \to S^2$ with postcritical set $P$.
\gap
\noindent{\bf Outline.} In \S 2, we collect generalities on Thurston maps and related combinatorial objects,
e.g. the pullback relations on curves and on arcs. In \S 3, we discuss algebraic invariants and define
the selfsimilarity complex, $\Sigma$. In \S 4, we derive dynamical consequences from the contraction
of the virtual endomorphism. \S 5 introduces the complex $\Gamma$ associated to an open cover ${\mathcal U}_0$,
while \S 6 gives the proof that $\Sigma$ and $\Gamma$ can be chosen so that the natural map between them is a quasi-isometry.
\S\S 7 and 8 conclude the proof of Theorem \ref{thm:characterization}.
\gap
\noindent{\bf Remark.} Nekrashevych \cite[Cor. 5.13]{nekrashevych:expanding} has shown that in great generality, a pair of maps $\iota, f: {\mathcal M}_1 \to {\mathcal M}_0$ satisfying analogous algebraic contraction conditions is combinatorially equivalent to a pair $\iota', f': {\mathcal M}_1' \to {\mathcal M}_0'$ where the ${\mathcal M}_i'$ are simplicial complexes, the maps $\iota', f'$ are piecewise affine, and the pair is expanding. While the result applies to our setting, the dimensions of ${\mathcal M}_i'$ can be larger than two, and it seems difficult to arrange for the spaces ${\mathcal M}_i'$ to coincide.
\section{Thurston maps}
\subsection{Curves and arcs} Here, we introduce the pullback relations on curves and arcs defined by a Thurston map, and formulate obstructions to expansion in terms of these relations.
Denote by ${\mathcal C}$ the set of free homotopy classes of essential, {\em oriented}, simple, closed, nonperipheral (that is, not homotopic into arbitrarily small neighborhoods of elements of $P$) curves in $S^2-P$; we use the term {\em curve} for an element of ${\mathcal C}$.
Let $o$ denote the union of the homotopy classes in $S^2-P$ of curves which are either inessential or peripheral; we call such curves {\em trivial}. The {\em pullback relation} $\longleftarrow$ on ${\mathcal C}\union \{o\}$ is defined by setting $o \longleftarrow o$ and
$ \gamma_1 \longleftarrow \gamma_2$
if and only if $\gamma_2$ is homotopic in $S^2-P$ to a component of $f^{-1}(\gamma_1)$. Thus $\gamma \longleftarrow o$ if and only if some preimage of $\gamma$ is inessential or peripheral in $S^2-P$.
If $\gamma_1 \longleftarrow \gamma_2$ we write $\gamma_1 \hookleftarrow \gamma_2$ if there is a representative of $\gamma_2$ mapping injectively to a representative of $\gamma_1$. The relation defined by $\hookleftarrow $ we refer to as the {\em univalent pullback relation}.
We next define a similar pullback relation on certain arcs with endpoints in $P$. Let ${\mathcal A}$ denote the set of isotopy classes (fixing $P$) of embedded
arcs $\alpha \subset S^2$ whose interiors are contained in $S^{2}-P$
and whose endpoints lie in $P$. A {\em preimage} $\tilde{\alpha}$ of $\alpha$ is, by definition, the closure of a connected component of the inverse image of the interior of $\alpha$ under $f$. If the endpoints of $\tilde{\alpha}$ lie in $P$, then $\tilde{\alpha}$ represents an element of ${\mathcal A}$. Adjoining a symbol $o$ to the set ${\mathcal A}$ to stand for the collection of arcs with one or both endpoints outside $P$, we obtain similarly a pullback relation on the set ${\mathcal A}\union \{o\}$.
\gap
We recall that a {\em multicurve} is a finite subset of ${\mathcal C}$ represented by disjoint curves. A {\em Levy cycle} is a sequence $(\gamma_0, \gamma_1, \ldots, \gamma_{p-1})$ with $\gamma_i \hookleftarrow \gamma_{i+1 \ \bmod p}$ whose elements comprise a multicurve.
The result below is not needed for the proofs of the theorems, since it could be derived from Theorems \ref{thm:characterization} and Theorem \ref{thm:prevents_levy} below. We include the statement and proof since they are simple and help build intuition. It applies to maps which may have periodic branch points.
\begin{prop}
\label{prop:obstructions_to_expansion}
Suppose $f$ is expanding. Given any curve $C_0$ or arc $\alpha_0$, there exists an integer $N$ such that if $C_0 \hookleftarrow \ldots \hookleftarrow C_n$ or $\alpha_0 \longleftarrow \ldots \longleftarrow \alpha_n$ is an orbit with $C_n, \alpha_n$ nontrivial, then $n \leq N$. In particular, the univalent pullback relation on curves has no (Levy) cycles or wandering curves, and the pullback relation on arcs has no cycles and no wandering arcs.
\end{prop}
Abusing terminology somewhat, we summarize the conclusion by saying that the univalent pullback relation on curves and pullback relation on arcs have {\em no elements with arbitrarily long nontrivial iterates}.
\gap
\noindent {\bf Proof: } If $f$ were expanding with respect to a length structure, the conclusion would follow immediately,
since lengths would decrease exponentially as one pulls back.
Without this additional structure, we use instead coverings by small open balls.
Let ${\mathcal U}_0, {\mathcal U}_i, i\in \mathbb{N}$ be as in the definition of expanding.
Choose $m$ sufficiently large so that $C_0, \alpha_0$ are covered by some number, say $L$, of elements of ${\mathcal U}_m$ and the union of these elements are contained in a regular neighborhood of $C_0, \alpha_0$ which meets $P$ in the same manner as does $C_0, \alpha_0$ (that is, either not at all, or only at the endpoints of $\alpha_0$) and which is homotopic, relative to $P$, to $C_0, \alpha_0$. It follows that for each $n$, $C_n, \alpha_n$ are covered by the same number $L$ of elements of ${\mathcal U}_{m+n}$. Hence if the diameters of the elements of the ${\mathcal U}_i$ tend to zero with $i$, the integer $n$ cannot be arbitrarily large.
\qed
Since the pullback relations are natural with respect to combinatorial equivalence, Proposition \ref{prop:obstructions_to_expansion} gives a necessary condition for a Thurston map $f$ to be equivalent to an expanding map. It is not, however, a sufficient condition; see \S \ref{secn:consequences} for examples.
\subsection{Orbifolds} We review here the definition of the orbifolds associated to a Thurston map and their fundamental groups.
For $x, y \in S^2$ let $\nu_0(y)=\lcm\{ \deg(f^n, x) : f^n(x)=y\}$
and $\nu_1(x)=\nu_0(f(x))/\deg(f,x)$. For $i=0,1$ let $\Sigma_i=\{x \in S^2 : \nu_i(x)>1\}$, and let ${\mathcal O}_i$ be the orbifold
whose underlying topological space is $\{ x \in S^2: \nu_i(x)<\infty\}$ and whose weight function is $\nu_i$. The sets $\Sigma_i$ are called the {\em singular sets} of ${\mathcal O}_i$. Note that $\Sigma_0=P$. There are no singular points of infinite weight if and only if there are no periodic branch points of $f$. In this case, the orbifolds ${\mathcal O}_i$ are compact.
The singular sets satisfy $f^{-1}(\Sigma_0) \supset \Sigma_1$. Set $U_0 = S^2-\Sigma_0$ and $U_1=S^2-f^{-1}(\Sigma_0)$. Then $U_1 \subset U_0$ and $f: U_1 \to U_0$ is a covering
map.
Let $b_0 \in U_0$ be a basepoint and $b_1\in f^{-1}(b_0)$ be one of
its preimages.
For $i=0, 1$ let $N_i$ denote the normal subgroup of $\pi_1(U_i,
b_i)$ generated by the set of elements of the form $g^k$, where $g$
is represented by a simple closed peripheral loop $\gamma$
surrounding a puncture $x$ of $U_i$, and the exponent $k$ is the
weight $\nu_i(x)<\infty$; if the weight is infinite, we do not add
such a loop as a generator. The {\em orbifold fundamental groups}
$\pi_1({\mathcal O}_i, b_i)$ are by definition the quotient groups
$\pi_1(U_i, b_i)/N_i$.
In what follows, we put ${\mathcal O}={\mathcal O}_0$ and $G:=\pi_1({\mathcal O}, b_0)$.
As the (orbifold) universal covering deck group, $G$ acts properly discontinuously on the universal covering of ${\mathcal O}$.
Note that each element of ${\mathcal C}$ corresponds to a conjugacy class in $G$ whose elements are of infinite order.
\section{Algebraic constructions}
In this section, we briefly summarize constructions and results of Nekrashevych \cite{nekrashevych:book:selfsimilar}, specializing to the case of Thurston maps.
\subsection{The virtual endomorphism}
In the setup of \S 2.2, let $f_*: \pi_1(U_1, b_1) \to \pi_1(U_0, b_0)$ be the injective
homomorphism induced by the covering $f: U_1 \to U_0$. Since $f$
sends peripheral loops to peripheral loops, it follows from the
definitions of the weight functions $\nu_i$ that $f_*: N_1 \to N_0$
is an isomorphism. This observation and the ``Five Lemma" of
homological algebra imply that the homomorphism $f_*$ descends to a
well-defined and injective map $\cl{f}_*: \pi_1({\mathcal O}_1, b_1) \to
\pi_1({\mathcal O}_0, b_0)=G$. We denote the image group
$\cl{f}_*(\pi_1({\mathcal O}_1, b_1))$ by $H$; it has finite index in $G$.
Let $\alpha: [0,1] \to U_0$ be a path joining $b_1$ to $b_0$ and
$\alpha_*: \pi_1(U_0, b_1) \to \pi_1(U_0, b_0)$ the induced
isomorphism. Let $N_0'=\alpha_*^{-1}(N_0)$. Since $N_0$ is normal,
the subgroup $N_0'$ is normal and is independent of the choice of
path $\alpha$. Set $\pi_1({\mathcal O}_0, b_1)=\pi_1(U_0, b_1)/N_0'$.
Again, the map $\alpha_*$ descends to a well-defined isomorphism
$\cl{\alpha}_*: \pi_1({\mathcal O}_0, b_1) \to \pi_1({\mathcal O}_0, b_0)$.
Since the inclusion $\iota: U_1 \hookrightarrow U_0$ sends peripheral
loops to loops which are either peripheral or trivial, and since
$\nu_0(x)$ divides $\nu_1(x)$ for all $x$, the induced map $\iota_*:
\pi_1(U_1, b_1) \to \pi_1(U_0, b_1)$ is surjective and sends $N_1$ to
$N_0'$. It easily follows that the map $\iota_*$ also descends to a
surjective map $\cl{\iota}_*: \pi_1({\mathcal O}_1, b_1) \to \pi_1({\mathcal O}_0,
b_1)$.
\begin{defn}
\label{defn:virtual_endo}
The {\em virtual endomorphism induced by $f$} is the homomorphism
$\phi: H \to G$ defined by
\[ \phi= \cl{\alpha}_* \circ \cl{\iota}_* \circ (\cl{f}_*)^{-1}.\]
\end{defn}
By construction, the virtual endomorphism $\phi$ associated to $f$ is
surjective.
The virtual endomorphism depends on the choices of the basepoint $b_0$, the
preimage $b_1$, and the homotopy class of the path $\alpha$. Different
choices yield virtual endomorphisms which differ by pre- and/or
post-composition by inner automorphisms. Up to this ambiguity,
the virtual endomorphism is an invariant of the homotopy class of $f$ relative to $P$.
A combinatorial equivalence between Thurston maps conjugates, up to inner automorphisms, their virtual endomorphisms; the property of being contracting is preserved.
For $n \geq 2$, the {\em $n$th iterate} $\phi^n$ is the homomorphism whose domain is defined
inductively by
\[ \mbox{\rm dom}\phi = H; \;\;\; \mbox{\rm dom} \phi^n=\{ g \in H :
\phi(g) \in \mbox{\rm dom}\phi^{n-1}\}\]
and whose rule is given by iterating $\phi$ a total of $n$ times.
Suppose $S$ is a finite generating set for $G$. We denote by $||g||$
the word length of $g$ in the generators $S$.
\begin{defn}
\label{defn:contracting1}
The virtual endomorphism $\phi: H \to G$ is called {\em contracting}
if the {\em contraction ratio}
\[ \rho= \limsup_{n\to\infty}\left( \limsup_{||g||\to\infty}
\frac{||\phi^n(g)||}{||g||}\right)^{1/n} < 1.\]
\end{defn}
The contraction ratio of the virtual endomorphism $\phi$ is
independent of the choices used in its construction.
As an algebraic object, the virtual endomorphism is rather straightforward to describe.
The next subsection gives a less familiar, but more natural, construction.
\subsection{The biset $\frak{M}$}
Suppose $\ell_0, \ell_1: [0,1] \to S^2 - P$ are two arcs joining the basepoint $b_0$ to a common point in $S^2 - P$.
We say $\ell_0, \ell_1$ are {\em homotopic in ${\mathcal O}$} if the loop formed by traversing $\ell_0$ first and then the reverse of $\ell_1$ represents
the trivial element of $G$. We denote by $\frak{M}$ the set of all homotopy classes $[\ell]$ of arcs $\ell$ in ${\mathcal O}$ joining
$b_0$ to one of the $d$ elements of $f^{-1}(b_0)$. For $[\ell] \in \frak{M}$ we denote by $z_{[\ell]} \in f^{-1}(b_0)$ the corresponding common endpoint.
The group $G$ acts on $\frak{M}$ via two commuting actions as follows. On the right, it acts by pre-concatenation:
\[ \ell \cdot g := \ell * g \]
where first $g$, then $\ell$ is traversed. Note that the right action is free and has $d$ orbits. The left action is obtained by taking preimages of the loops under $f$:
\[ g \cdot \ell := f^{-1}(g)[z_{[\ell]}]* \ell \]
where first $\ell$, then the lift $f^{-1}(g)[z_{[\ell]}]$ of $g$ based at $z_{[\ell]}$, is traversed.
The set $\frak{M}$ equipped with these two commuting actions of $G$ defines the $G$-{\em biset} associated to $f$.
In the next subsection, we show that choosing certain representatives of elements of $\frak{M}$ and representatives of generators of $G$ gives rise to an abstract $1$-complex $\Sigma$ and a projection map $\pi_\Sigma: \Sigma \to S^2$. Along the way, we reformulate the notion of contraction in terms of an action of $G$ on the set of words in an alphabet $X$ of size $d$.
\subsection{The selfsimilarity complex $\Sigma$}
Choose a bijection $\Lambda: X \to f^{-1}(b_0)$, where $X$ is a finite set of cardinality $d$.
For each $x \in X$, let $\lambda_x$ be an arc in $S^2 - P_f$ joining $b_0$ to $\Lambda(x)$.
The collection $\{[\lambda_x]: x \in X\}$ of elements of $\frak{M}$ is called a {\em basis} of the biset $\frak{M}=\frak{M}_f$.
For $n \in \mathbb{N}$, consider now the $n$th iterate $f^n$ of $f$. The orbifolds ${\mathcal O}={\mathcal O}_f$ and ${\mathcal O}_{f^n}$ coincide, so $G=G_f = G_{f^n}$. By lifting the arcs $\lambda_x$ under $f^n$, we obtain an identification $\Lambda: X^n \to f^{-n}(b_0) \times \{n\}$ of the set $X^n$ of words of length $n$ in the alphabet $X$ with the fiber $f^{-n}(b_0) \times \{n\}$ and a corresponding basis for the $G$-biset $\frak{M}_{f^n}$ associated to the $n$th iterate of $f$. We denote by $X^* = \union_n X^n$, with $X_0 = \{\emptyset\}$ consisting of the empty word. We write $|w|=n$ if $w \in X^n$.
Suppose $S$ is a generating set for $G$. For each $s \in S$, let $\gamma_s$ be a loop in $S^2-P$ based at $b_0$ representing $s$. By lifting loops under iterates of $f$, we obtain an action of $G$ on $X^*$ preserving the lengths of words and acting transitively on each $X^n$.
The basepoint $b_0$, arcs $\lambda_x, x \in X$, and loops $\gamma_s, s \in S$, we refer to as {\em defining data}.
Since the right action is free, for each word $u \in X^*$, and each $g \in G$, there are a unique word $v \in X^n$ and a unique $h \in G$ with $g.u = v.h$; the element $h$ is called the {\em restriction}\footnote{sometimes the term ``section'' is used.} of $g$ to $u$, and is denoted $g|_u$. The interpretation in terms of defining data is as follows. Let $\lambda_u, \lambda_v$ denote the concatenations of lifts of arcs corresponding to $u$ and $v$, so that the endpoints of $\lambda_u, \lambda_v$ correspond to $u$ and $v$. Then $v$ is the endpoint of $f^{-n}(g)[u]$ and $h$ is represented by the loop which first traverses $\lambda_u$, then $f^{-n}(g)[u]$, then the reverse of $\lambda_v$.
By \cite[Prop. 2.11.11]{nekrashevych:book:selfsimilar}, contraction of the virtual endomorphism is equivalent to the following: {\em there is a finite set ${\mathcal N}$ such that for each $g \in G$, there is an integer $n$ such that for all $v \in X^*$ satisfying $|v| \geq n$, the restriction $g|_v \in {\mathcal N}$}. In this case, one says the biset $\frak{M}$ is {\em hyperbolic}. This is an asymptotic condition: $\frak{M}_f$ is hyperbolic if and only if $\frak{M}_{f^m}$ is hyperbolic for some $m \geq 1$, since the contraction coefficients for $\phi$ and $\phi^m$ are necessarily simultaneously less than $1$.
Suppose now that defining data has been chosen. The {\em selfsimilarity complex} $\Sigma$ is the infinite $1$-complex whose vertex set is $X^*$ and whose edges, which come in two types, {\em horizontal} and {\em vertical}, are defined as follows. Given $w \in X^*$, a horizontal edge, labelled $s$, joins $w$ to $s.w$; a vertical edge, labelled $x$, joins $w$ to $xw$. The selfsimilarity complex, with edges of length $1$, is a proper geodesic metric space, and the local valence at each nonempty vertex is the same if $S$ is symmetric.
The right shift $\sigma: X^* \to X^*$ determines a cellular self-map $f_\Sigma: \Sigma' \to \Sigma$ where $\Sigma'$ is the subcomplex determined by nonempty words. When $\frak{M}$ is hyperbolic, $\Sigma$ is hyperbolic in the sense of Gromov, and the boundary ${\mathcal J}$ is compact. Since the virtual endomorphism
$\phi$ is surjective, the action is recurrent, so ${\mathcal J}$ is also connected and locally connected \cite[Theorem 3.5.1]{nekrashevych:book:selfsimilar}. The map $f_\Sigma$ extends to a map of the boundary $f_\Sigma: {\mathcal J} \to {\mathcal J}$, yielding a dynamical system.
Although $\Sigma$ is an abstract $1$-complex, path-lifting of the defining data determines a map
$\pi_\Sigma: \Sigma \to S^2$; see \cite{kmp:gromov} and \cite[Theorem 5.5.3]{nekrashevych:book:selfsimilar}.
If $f$ is not expanding with respect to a length structure on $S^2$, however, $\pi_\Sigma$ need not
extend to a map ${\mathcal J} \to S^2$.
\section{Dynamical consequences of the hyperbolicity of $\frak{M}$}
\label{secn:consequences}
Suppose $f$ has no periodic branch points. Proposition \ref{prop:obstructions_to_expansion} gives necessary conditions for $f$ to be expanding. The main result of this section is to derive the same conclusion from the algebraic assumption that $\phi$ is contracting; equivalently, that $\frak{M}$ is hyperbolic.
\begin{thm}
\label{thm:prevents_levy}
Suppose $f$ has no periodic branch points. If $\frak{M}$ is hyperbolic, the univalent pullback relation on curves and the pullback relation on arcs have no elements with arbitrarily long nontrivial iterates.
\end{thm}
\noindent {\bf Proof: } The central ingredient in the proof is the following Lemma, which will be proved later.
\begin{lemma}
\label{lemma:topology_algebra}
Suppose $C_0\hookleftarrow C_1$. Given any $g_0 \in G$ whose conjugacy class corresponds to $C_0$, there exists $g_1 \in G$ whose conjugacy class corresponds to $C_1$ and $x \in X$ with $g_0 \cdot x = x \cdot g_1 \in \frak{M}$.
\end{lemma}
To prove the Theorem, suppose $C_0 \hookleftarrow C_1 \hookleftarrow \ldots \hookleftarrow C_n$. Induction and Lemma \ref{lemma:topology_algebra} implies that there exist $g_i \in G, x_i \in X$ such that
\[ g_0 \cdot x_0x_1\ldots x_i = x_0x_1\ldots x_i \cdot g_{i+1}, \;\;\; i=0, \ldots, n-1.\]
In particular, upon setting $w=x_0x_1 \ldots x_n \in X^{n+1}$, we have
\[ g_0 \cdot w = w \cdot g_n;\]
recall that by definition, $g_n = g_0|_w$.
Since $\frak{M}$ is hyperbolic, there is a finite set ${\mathcal N} \subset G$ such that $g_n =g_0|_w \in {\mathcal N}$ whenever $n=|w|$ is sufficiently large.
This immediately implies that the number of distinct elements $g_n$ of ${\mathcal C}$ arising in this way is bounded and hence that the collection of curves $\{C_n\}$ arising in this way is also bounded, independent of $n$.
To rule out cycles, we use topology to simplify the algebra. Consider first the case of a Levy cycle of period one, i.e. a curve $C$ for which $C\hookleftarrow C$. By varying $f$ within its homotopy class, and using the fact that $C$ is oriented, we may assume $f=\id$ on a neighborhood of $C$. Take the basepoint $b_0$ for $\frak{M}$ to lie on $C$; it becomes a fixed point of $f$. Let $g$ be represented by $C$, regarded as a loop based at $b_0$. By choosing a basis for $\frak{M}$ with one arc equal to the constant path at $b_0$ and corresponding to an element $x$ in the alphabet $X$, we find $g \cdot x = x \cdot g$. Induction shows then that for all $n \in \mathbb{N}$ and $m \in \mathbb{Z}$ we have that if $w=xx\ldots x \in X^n$ then
\[ g^m \cdot w = w \cdot g^m.\]
But this is impossible if $\frak{M}$ is hyperbolic: $g$ has infinite order, and the identity above shows that for all $n$, $g^m|_w = g^m$ which cannot lie in ${\mathcal N}$ if $m$ is sufficiently large.
The case of a Levy cycle of period larger than one can be ruled out by passing to an iterate.
Analyzing the pullback relation on arcs is reduced to that of the pullback relation on curves. An arc which does not eventually become trivial must have both endpoints in cycles of $P$. For such arcs $\alpha_0, \alpha_1$, since $f$ has no periodic branch points, whenever $\alpha_0 \longleftarrow \alpha_1$, the boundaries of regular neighborhoods of $\alpha_0$ and $\alpha_1$ yield curves $C_0$ and $C_1$ for which $C_0 \hookleftarrow C_1$.
If $\#P> 3$ then $C_0, C_1$ must be nontrivial, and the eventual triviality of curve $\hookleftarrow$-orbits, already established, implies eventual triviality of arc $\longleftarrow$-orbits. So assume $\#P = 3$. By Thurston's characterization \cite{DH1}, $f$ is equivalent to a rational function without periodic critical points. Such a map is uniformly expanding with respect to a complete length structure, so such arc orbits are always eventually trivial.
\qed
We now turn to the proof of the Lemma, which is just an exercise in definitions.
\gap
\noindent {\bf Proof: } Suppose defining data has been chosen. Choose a representative $\gamma_0$ for $g$, and let
$\widetilde{\gamma}_0$ be a preimage of $\gamma_0$ corresponding to $C_1$; it exists, by homotopy lifting and the fact that
$C_1$ maps to $C_0$ by degree one. Let $\tilde{b}$ be the unique preimage of $b$ lying on $\widetilde{\gamma}_0$,
and suppose $x$ corresponds to $\tilde{b}$, i.e. $\Lambda(x)=\tilde{b}$. The loop given by
\[ \gamma_1 := \cl{\lambda}_x*\widetilde{\gamma}_0 * \lambda_x\]
(where $\lambda_x$ is traversed first) represents an element $g_1 \in G$. Moreover, by the definition of the biset,
\[ x \cdot g_1 = [\lambda_x * \cl{\lambda}_x * \widetilde{\gamma}_0 * \lambda] = g_0 \cdot x.\]
\qedspecial{Lemma \ref{lemma:topology_algebra}}
Examples showing the conditions in Propositions \ref{prop:obstructions_to_expansion} and Theorem \ref{thm:prevents_levy} are necessary but not sufficient can be found among Thurston maps induced by torus endomorphisms that are hyperbolic but not expanding.
Let $T^2 = \mathbb{R}^2/\mathbb{Z}^2$, let $A$ be an integral matrix with determinant larger than one, and let $F: T^2 \to T^2$ be the noninvertible endomorphism on the torus induced by $A$. There is a corresponding pullback relation on curves, defined analogously as in \S 2. In this setting, all preimages of nontrivial curves are nontrivial, are homotopic to each other, and map by the same degree.
\begin{prop}
\label{prop:eval_condition}
If a curve $C$ has arbitrarily long nontrivial iterates under the univalent pullback relation, then either $C \hookleftarrow C$ or $C \hookleftarrow -C$, where $-C$ denotes the curve $C$ with the opposite orientation.
This occurs if and only if the matrix $A$ has an eigenvalue $\lambda$ equal to $+1$ or $-1$.
\end{prop}
\noindent {\bf Proof: } The sufficiency in the second statement is obvious. The proof will show both the first statement, and the necessity; it is a straightforward reinterpretation of the arguments of \cite[Prop. 2.9.2]{nekrashevych:book:selfsimilar}.
Use the standard basis to identify $\mathbb{Q}^2$ with $H_1(T^2, \mathbb{Q})$ and $\pi_1(T^2)$ with $H_1(T^2, \mathbb{Z})=\mathbb{Z}^2$, so that $A$ is the matrix for the induced map on rational and integral homology. Let $B=A^{-1}$. Observe that if $C=C_0 \hookleftarrow C_1$ then in fact all preimages of $C_0$ map by degree one and the monodromy corresponding to $C_0$ is trivial. Thus if $C_0 \hookleftarrow C_1 \hookleftarrow C_2 \hookleftarrow \ldots $ is an infinite sequence of related curves (either wandering, or cycling) then as an element of $\pi_1(T^2)$, the curve $C_0$ represents a nontrivial element of the kernel $U < \mathbb{Z}^2< \mathbb{Q}^2$ of the corresponding iterated monodromy group action. The group $U$ is $B$-invariant, so $U \otimes \mathbb{Q}$ is a $B$-invariant $\mathbb{Q}$-linear subspace of $\mathbb{Q}^2$; let $C=B|_{U \otimes \mathbb{Q}}$. Then the characteristic polynomial $q$ of $C$ is monic, has integer coefficients, and is a factor of the characteristic polynomial of $B$. If $1/\lambda$ is a root of $q$ then $\lambda$ is an eigenvalue of $A$. Both $\lambda$ and $1/\lambda$ are algebraic integers, hence the norm of $\lambda$ is equal to $1$. The condition $\det(A)>1$ implies the algebraic degree of $\lambda$ must be equal to $1$, so $\lambda = \pm 1$.
\qed
The endomorphism $F: T^2 \to T^2$ is {\em hyperbolic} if $A$ has no eigenvalues of unit modulus. Under the assumption $\det(A)>1$, the following is easily shown via straightforward computations.
\begin{prop}
\label{prop:hyperbolic_endo}
The endomorphism $F$ is hyperbolic but not expanding if and only if $\mbox{\rm tr}(A)-\det(A)> 1$.
\end{prop}
The endomorphism $F: T^2 \to T^2$ commutes with $\pm \id_{T^2}$ and descends to a Thurston map $f: S^2 \to S^2$; the corresponding orbifold ${\mathcal O}$ has signature $(2,2,2,2)$.
The singular points are the images of the points of order at most $2$.
\begin{thm}
\label{thm:not_expanding}
Suppose $\det(A)>1$ and $\mbox{\rm tr}(A)-\det(A)> 1$, and let $f: S^2 \to S^2$ be the induced Thurston map. Then
\begin{enumerate}
\item $f$ is not equivalent to an expanding map, but
\item the univalent pullback relation on curves and the pullback relation on arcs have no elements with arbitrarily long nontrivial iterates.
\end{enumerate}
\end{thm}
\noindent {\bf Proof: } Proposition, \ref{prop:hyperbolic_endo}, Proposition \ref{prop:eval_condition}, and Theorem \ref{thm:torusversion} imply both conclusion (1) and the first part of (2). Suppose $\alpha \subset S^2$ is an arc. The two preimages of $\alpha$ under the branched covering $T^2 \to S^2$, when concatenated, yield a nontrivial curve, $C$. If $\alpha_0 \longleftarrow \alpha_1$ then for the corresponding curves in $T^2$ we have $C_0 \hookleftarrow C_1$. Thus nontrivial arc orbits lift to nontrivial univalent curve orbits.
\qed
\section{The covering complex $\mathbf{\Gamma}$} \label{scn:covering}
In this section, we begin by summarizing constructions and results of \cite[Chapter 3]{kmp:ph:cxci},
specializing to the case of Thurston maps. We associate to $f$ and a suitable open covering ${\mathcal U}_0$ another infinite
one-complex, $\Gamma$, this time a graph. When it is hyperbolic, it has a natural boundary, and there is
a natural map $\pi_\Gamma: S^2 \to \bdry \Gamma$.
For convenience, equip $S^2$ with the usual Euclidean length metric.
Let ${\mathcal U}_0$ be a finite open cover of $S^2$ by small balls $B$ for which $\#B \intersect P \leq 1$
with equality only when the corresponding point of $P$ is in the center of $B$.
For $n \in \mathbb{N}$ let ${\mathcal U}_n$ be the open cover whose elements are connected components of sets of the form
$f^{-n}(U), U \in {\mathcal U}_0$. It is convenient to set ${\mathcal U}_{-1}:=\{S^2\}$.
If $U \in {\mathcal U}_n$ we write $|U|=n$. A key observation here is that if $f$ has no periodic branch points,
then there is an upper bound on the degrees of restrictions $f^n: {\widetilde{U}} \to U$, ${\widetilde{U}} \in {\mathcal U}_n, n \in \mathbb{N}$.
The vertex set of $\Gamma$ is defined as $\union_{n\geq -1}{\mathcal U}_n$, i.e. one vertex for each component.
We denote the vertex corresponding to the unique element of ${\mathcal U}_{-1}$ by $o$.
Horizontal edges join $U_1, U_2$ if $|U_1|=|U_2|$ and $U_1 \intersect U_2 \neq \emptyset$. Vertical edges join $U, V$ if $||U|-|V|| =1$ and $U \intersect V \neq \emptyset$. $\Gamma$ is a graph with root $o$. We equip $\Gamma$ with the graph path metric, so that each edge has length $1$.
Then $\Gamma$ is a proper, geodesic metric space; let $d$ denote the corresponding distance function.
If $f$ has no periodic branch points, then the valence of a vertex of $\Gamma$ is uniformly bounded,
independent of the vertex.
Let $\varepsilon>0$ be small, and let $\varrho_\varepsilon: \Gamma \to (0,\infty)$ be given by
$\varrho_\varepsilon(x)=\exp(-\varepsilon d(x, o))$. Upon redefining the lengths of curves via
\[ \ell_\varepsilon(\gamma):=\int_\gamma \varrho_\varepsilon \, ds\]
we obtain an incomplete length metric space. Its completion compactifies $\Gamma$
by adding a boundary at infinity $\partial_{\ep}\Gamma$.
Points in its boundary $\bdry_\ep \Gamma$ are represented by vertical geodesic rays from the root. Given $a \in S^2$, let $U_n(a), n=-1, 0, 1, 2, \ldots$ be a sequence of open sets for which $a \in U_n \in {\mathcal U}_n$. Then $\{U_n\}$ defines a geodesic ray and, therefore, a point of $\bdry_\ep \Gamma$.
We obtain a well-defined surjective continuous map $\pi_\Gamma: S^2 \to \bdry_\ep \Gamma$. In particular, $\bdry_\ep \Gamma$ is connected
and locally connected. When $\Gamma$ is hyperbolic and $\ep>0$ is chosen small enough,
then $\bdry_\ep\Gamma$ coincides with its visual boundary $\bdry\Gamma$. We will always choose $\ep$ as above when $\Gamma$ is hyperbolic.
\begin{lemma}
\label{lemma:monotone}
If $\Gamma$ is hyperbolic, the projection $\pi_\Gamma: S^2 \to \bdry\Gamma$ is monotone, i.e. its fibers are connected.
\end{lemma}
\noindent {\bf Proof: } For $z\in S^2$ and $n\ge -1$, pick $U_n(z)\in{\mathcal U}_n$ such that $z\in U_n(z)$. It follows that the sequence $(U_n(z))_n$ defines a ray in $\Gamma$.
Suppose $\pi_\Gamma(x)=\pi_\Gamma(y)=\xi \in \bdry \Gamma$. From the definition of $\pi$ and the hyperbolicity of $\Gamma$,
there is a positive integer $D$ such that for every $n \in \mathbb{N}$, there exist a ``chain''
$U_n^0, U_n^1, \ldots, U_n^D$ of elements of ${\mathcal U}_n$
such that $x\in U_n^0$, $y \in U_n^D$, and $U_n^{i-1}\intersect U_n^i \neq \emptyset$ for all $i=1, \ldots, D$.
For $n \in \mathbb{N}$ let $Y_n = U_n^0 \union \ldots\union U_n^D$. Then $Y_1, Y_2, Y_3, \ldots$ is a sequence of open connected sets containing both $x$ and $y$.
Set $$K=\cap_{k\ge 0} \overline{\cup_{n\ge k} Y_n}\,.$$ It follows that $K\subset S^2$ is a continuum containing both $x$ and $y$.
We will prove that $\pi_\Gamma(K)=\xi$ and, therefore, that $\pi_\Gamma^{-1}(\xi)$ is connected.
Fix $z \in K$. For any $k\ge 0$, $U_k(z)\cap K\ne\emptyset$, so we may find $n\ge k$ such that $Y_n\cap U_k(z)\ne\emptyset$.
This means that
\begin{eqnarray*}
d_{\ep}(\pi_\Gamma(z),\xi)& \le & d_{\ep}(\pi_{\Gamma}(z),U_k(z))+ d_{\ep}(U_k(z),\{U_n^j, 0\le j\le D\})\\
& & \quad +\diam_{\ep} \{U_n^j, 0\le j\le D\} + d_{\ep}(\xi,\{U_n^j, 0\le j\le D\})\\
& \lesssim & e^{-\ep k} + e^{-\ep k} + e^{-\ep n} + e^{-\ep n}\lesssim e^{-\ep k}\,.\end{eqnarray*}
Since $k$ is arbitrary, we obtain that $\pi_{\Gamma}(z)=\xi$.
\qed
From \cite[Theorem 5]{daverman:decompositions}, we have that the decomposition of $S^2$ by the fibers of $\pi_\Gamma$ is upper-semicontinuous, and the quotient space is homeomorphic, via the map induced by $\pi_\Gamma$, to $\bdry\Gamma$.
The map $f$ induces a cellular map $f_\Gamma: \Gamma' \to \Gamma$, where $\Gamma'$ is the subgraph on vertices at level $\geq 1$. Since it sends vertical rays to vertical rays, it induces a continuous map $f_\Gamma: \bdry \Gamma \to \bdry \Gamma$.
Since it sends sufficiently small round balls to round balls \cite[Prop. 3.2.2]{kmp:ph:cxci}, it is an open map; it is easily seen to be at most $d$-to-$1$, so it is closed as well. In particular, the local degree function
\[ \deg(f_\Gamma,\xi):=\inf_W\{\#{f_\Gamma}^{-1}(\zeta) \intersect W : \zeta \in f_\Gamma(W), \;\; \xi \in W \; \mbox{open} \}\]
is well-defined.
\section{Coincidence of quasi-isometry types}
\label{secn:coincidence}
Recall that if $X, Y$ are metric spaces, a function $\Phi: X \to Y$ is a {\em quasi-isometry} if there exist constants $\lambda \geq 1, c \geq 0, b \geq 0$ such that
\[ \frac{1}{\lambda}|x-x'| - c \leq |\Phi(x)-\Phi(x')| \leq \lambda|x-x'| + c \]
and such that $|y - \Phi(X)|\leq b$ for all $y \in Y$; here $|x-x'|=d_X(x,x')$, etc.. For geodesic metric spaces, hyperbolicity is preserved under quasi-isometries.
Recall that a choice of defining data defines a projection $\pi_\Sigma: \Sigma \to S^2$ and an identification map $\Lambda:X^{*}\to \cup_{n\ge 0} f^{-n}(\{b\})$. The main result of this section is
\begin{thm}
\label{thm:coincidence}
Let $f: S^2 \to S^2$ be a Thurston map without periodic critical points.
Then there exist defining data for the associated selfsimilarity complex $\Sigma$ and the covering graph $\Gamma$
and a level-preserving quasi-isometry $\Phi: \Sigma \to \Gamma$ such that,
for every vertex $v\in\Sigma^0$, we have $\Lambda(v)\in\Phi(v)$ and $\Phi \circ f_\Sigma = f_\Gamma \circ \Phi$.
\end{thm}
\begin{remark} If we admit periodic branch points, then the same proof will show that $\Sigma$
is quasi-isometric to a covering complex $\Gamma$ built from a cover of the complement in the sphere of a neighborhood of the set of periodic branch points.
\end{remark}
\noindent {\bf Proof: } We first choose the defining data and list some elementary properties.
For convenience, we endow $S^2$ with its ordinary Euclidean length metric.
\gap
\noindent{\bf Choosing the defining data.}
Choose the basepoint $b \in S^2-P$ to be a regular point. We may choose representative loops
$\gamma_s, s \in S$, so that each is simple, peripheral, and has the property that for some common $\epsilon>0$,
the closed $\epsilon$-neighborhood $N_\epsilon(\gamma_s)$ is contained in a closed annulus $A$ contained in $S^2-P$ surrounding on one side
at most one point of $P$. The peripheral condition and the assumption that there are no periodic critical points implies that
there exists $N_0$ such that for all $n$,
\[ \deg(f^n: \widetilde{A} \to A) \leq N_0\]
for all preimages $\widetilde{A}$ of $A$ under $f^{-n}$.
In this situation, if the above degree is say $\delta$, then in the graph $\Lambda$, for each vertex $v$ at level $n$ with $\Lambda(v) \in \widetilde{A}$, there is a closed horizontal edge-
path of length $\delta$ whose edges are labelled by $s$ corresponding and which corresponds to the monodromy at level $n$ induced by $\gamma_s$.
We may also choose representative arcs $\lambda_x \subset S^2-P, x \in X$,
so that each is simple and has the property that for some common $\epsilon>0$,
the closed $\epsilon$-neighborhood $N_\epsilon(\lambda_x)$ is a closed disk $D$ contained in $S^2-P$. Thus
\[ \deg(f^n:\widetilde{D} \to D) = 1\]
for all preimages $\widetilde{D}$ of $D$ under $f^{-n}$.
For $w\in X^*$, let $\lambda_{xw}$ be the lift of $\lambda_{x}$ under $f^{|w|}$ which starts at $\pi_{\Sigma}(w)$.
\gap
Since $f$ is a branched covering, given $\delta>0$, there is a constant $r(\delta)>0$ with $r(\delta) \to 0$ as $\delta \to 0$
such that for any ball $B(y,\delta) \subset S^2$, any preimage of $B$ under $f$ has diameter $<r(\delta)$.
It follows that we may choose the defining data ${\mathcal U}_0$ for the covering graph $\Gamma$ to consist of a finite collection
of small open balls satisfying the following properties.
Given a set $E \subset S^2$, we denote by
\[ {\mathcal U}_n(E) = \{ U \in {\mathcal U}_n : E \intersect U \neq \emptyset\}.\]
\begin{enumerate}
\item $\forall U \in {\mathcal U}_0$, $P \intersect \bdry U = \emptyset$, $\# P \intersect U \leq 1$, and if equality holds, then the postcritical point in $U$ is at the center of $U$;
\item $\forall U \in {\mathcal U}_0$, ${\mathcal U}_0(U) \intersect P \leq 1$;
\item $\forall x \in X$, $\cl{{\mathcal U}_0(\lambda_x)} \subset N_\epsilon(\lambda_x)$;
\item $\forall s \in S$, $\cl{{\mathcal U}_0(\gamma_s)} \subset N_\epsilon(\gamma_s)$;
\item $\forall U_0 \in {\mathcal U}_0$ and $U_1 \in {\mathcal U}_1$ with $U_0 \intersect U_1 \neq \emptyset$,
the union $U_0 \union U_1$ is contained in a ball $B$ such that the ball $2B$ of twice the radius meets at most one point of $P$;
\item $\forall p \in S^2$ there exists an arc $\alpha$ joining the basepoint $b$ to $p$ such that ${\mathcal U}_0(\alpha)$ is contained in a disk which meets $P$ in at most one point.
\item There is exactly one element of ${\mathcal U}_0$ containing the basepoint $b$, and this basepoint is the center of the ball.
\end{enumerate}
{\noindent\bf Basic properties.} \\
\gap
\noindent{\bf (BP1).} Condition 1 implies that every element $U\in{\mathcal U}_n$ has a unique preferred point $c_U$
(we call it the center of $U$) which is the unique preimage of the center $c$ of the ball $f^{\circ n}(U)$ under $f^n$.
\gap
{\noindent\bf (BP2).} Conditions 1 and 2 imply the following. Since there are no periodic branch points,
there is a uniform upper bound $q$ on the local degree of all iterates of $f$.
Suppose $f^n:{\widetilde{U}}_i \to U_i \in {\mathcal U}_0, i=0,1$. If ${\widetilde{U}}_0, {\widetilde{U}}_1 \in {\mathcal U}_n$ intersect,
then at most one of ${\widetilde{U}}_0, {\widetilde{U}}_1$ meets a branch point of $f^n$; say it is ${\widetilde{U}}_0$.
Then there are $q' \leq q-1$ elements ${\widetilde{U}}_2, \ldots, {\widetilde{U}}_{q'}$ of ${\mathcal U}_n$, each a preimage of $U_1$,
such that $f^{\circ n}: {\widetilde{U}}_0 \union {\widetilde{U}}_1 \union \ldots \union {\widetilde{U}}_{q'} \to U_0 \union U_1$ is proper and of degree $q'$.
\gap
{\noindent\bf (BP3).} From condition 6, we define $A_0$ to be the finite set of curves $\alpha_U:[0,1]\to S^2$ joining the center
$c_U =\alpha_U(0)$ of $U$
to $b=\alpha_U(1)$. Let also $A_n$ denote all the lifts of these curves under $f^n$: these are curves joining every center
of $U\in{\mathcal U}_n$ to a point of $f^{-n}(b)$. Since every $\alpha_U$ is an arc contained in a disk which intersects $P$
in at most one single point, there is some $L$ independent of $n$ and $U\in{\mathcal U}_n$
such that $\alpha_U$ is covered by at most $L$ elements of ${\mathcal U}_n$.
\gap
{\noindent\bf (BP4).} Condition 7 is merely to make the exposition in what follows more convenient,
for with this choice we will define $\Phi: \Sigma \to \Gamma$ in a simple way.
\gap
\noindent{\bf Strategy of the proof. } Define $\Phi:V(\Sigma)\to V(\Gamma)$ by setting $\Phi(w)$ to be the unique set
$U\in{\mathcal U}_{|w|}$ which
contains $\Lambda(w)$. We extend $\Phi$ over the edges to all of $\Sigma$ as follows.
Suppose $\gamma_s$ represents a horizontal edge at level $0$ (a loop) in $\Sigma$. In the sphere, as this edge is traversed,
it passes through a sequence of elements of ${\mathcal U}_0$; we pick such a sequence, and associate to this a horizontal loop $\Phi(\gamma_s)$ in $\Gamma$ at level $0$,
parameterized at constant speed. We define $\Phi$ to send the edge of $\Sigma$ corresponding to $\gamma_s$ to this loop $\Phi(\gamma_s)$.
Similarly, each horizontal edge $e=e_s$ at level $n$ in $\Sigma^1$ corresponds to
a lift under $f^n$ of a unique edge $\gamma_s$, $s\in S$, at level zero; hence by lifting the chosen covering for $\gamma_s$ we obtain again
a curve $\Phi(e)$ in $\Gamma$ joining the images under $\Phi$ of
their extremities by definition.
Similarly, for $x\in X$, we associate to the arc $\lambda_x$ an edge-path $\Phi(\lambda_x)$ in $\Gamma$ corresponding to a covering of the chosen arc $\alpha_{\Lambda(x)}$ by elements of ${\mathcal U}_0$
concatenated with the vertical edge of $\Gamma^1$ between the elements $U\in{\mathcal U}_0$ and $V\in{\mathcal U}_1 $ containing $\Lambda(x)$.
We again extend $\Phi$ to the other vertical edges
equivariantly by path-lifting.
We also define a coarse inverse $\Psi : V(\Gamma)\to V(\Sigma)$ by setting $\Psi(U)=w$, where $w\in V(\Sigma)$ is the vertex such that
$\Lambda(w)$ is the endpoint of $\alpha_U$. Note that $\Psi\circ\Phi$ is the identity on $V(\Sigma)$ since $\alpha_U$
is trivial for preimages
of $U_b\in{\mathcal U}_0$ containing $b$, and $\Phi\circ\Psi$ maps a vertex $U\in\Gamma^0$ to the endpoint of $\alpha_U$.
\gap
To prove that $\Phi$ is a quasi-isometry, we will show that both $\Phi$ and $\Psi$ are Lipschitz on the set of vertices
and that the image of $\Phi$ is cobounded. The conclusion will follow since $\Psi\circ\Phi$ is the identity on $\Sigma^0$.
To prove that each map is Lipschitz, we will show that the distance between the images
of the two extremities of an edge is uniformly bounded.
\gap
\noindent{\bf $\mathbf{\Phi} $ is Lipschitz.}
Let us consider the horizontal case first. Suppose $v, w \in V(\Sigma)$, $|v|=|w|=n$, and $v, w$ are joined by a horizontal
edge, so that $w=s.v$ for some $s \in S$. Equivalently, there is a lift $\widetilde{\gamma}_s$ of $\gamma_s$ under $f^n$ joining
$\pi_\Sigma(v)$ to $\pi_\Sigma(w)$. Note that $f^n:\widetilde{\gamma}_s\to \gamma_s$ is $1-1$ away from the extremities.
The curve $\gamma_s$ is covered by $\#{\mathcal U}_0(\gamma_s)$ elements of ${\mathcal U}_0$. The lift $\widetilde{\gamma}_s$ is covered by at most $\#{\mathcal U}_0(\gamma_s)+1$ elements of ${\mathcal U}_n$.
By the definition of the metric in $\Gamma$, this implies that $|\Phi(v)-\Phi(w)|_{hor} \leq (\#{\mathcal U}_0(\gamma_s) +1)$. Note that we may assume that $ \#{\mathcal U}_0(\gamma_s)\le
\#{\mathcal U}_0-1$.
Since $|\Phi(v)-\Phi(w)|_\Gamma \leq |\Phi(v)-\Phi(w)|_{hor}$, we conclude that
\[ |\Phi(v)-\Phi(w)|_\Gamma \leq C_{hor}:= \#{\mathcal U}_0. \]
Now let us consider the vertical case. Suppose $|v|=n+1$, $|w|=n$, and $v$ and $w$ are joined by a vertical edge in $\Sigma$. By definition, $w=xv$ for some $x \in X$, and the points $\Lambda(w)$ and $\Lambda(v)$ in $S^2$ are joined by a lift of the arc $\lambda_x$ under $f^{-n}$ starting at $\Lambda(v)$.
The arc $\lambda_{x}$ is covered by $\#{\mathcal U}_0(\lambda_x)$ elements of ${\mathcal U}_0$,
each contained in the neighborhood $N_\epsilon(\lambda_x)$, which is a disk in $S^2-P$.
It follows that $\widetilde{\lambda}_x$ is covered by the same number $\#{\mathcal U}_0(\lambda_x)$ of elements of ${\mathcal U}_n$.
By the definition of the metric in $\Gamma$, this implies that $|\Phi(v)-\Phi(w)|_\Gamma \leq \#{\mathcal U}_0(\lambda_x) +1$. Hence
\[ |\Phi(v)-\Phi(w)|_\Gamma \leq C_{ver}:=\max\{\#{\mathcal U}_0(\lambda_x) : x \in X\}+1.\]
We conclude that $\Phi$ is $C=\max\{C_{hor}, C_{ver}\}$-Lipschitz on $V(\Sigma)$.
\gap
\noindent{\bf The image of $\Phi$ is cobounded in $\Sigma$.} The proof of this follows along exactly the same lines.
For any $U\in {\mathcal U}_n$, the curve $\alpha_U\in A_n $ joins $c_U$ to $\Phi\circ\Psi(U)$, whose
distance is bounded by $L$ according to the basic property (BP3).
\gap
\noindent{\bf Realization of the covering complex $\Gamma$.} Proving that $\Psi$ is Lipschitz
will require some more data.
The covering complex $\Gamma$ comes equipped with a self-map $f_{\Gamma}:\Gamma\setminus B(X,1)\to\Gamma$.
We will find a continuous map $\Gamma \to S^2$ which conjugates $f_{\Gamma}$ and $f$ where defined.
First, if $e=(U_1,U_2)$ denotes an edge with $U_1,U_2\in {\mathcal U}_0$, we consider $\beta_e$ to be the geodesic
(in the round metric) joining their centers; recall that $U_j$ are spherical balls at level $0$, so that
$\beta_e\subset U_1\cup U_2$. Denote by $B_0$ this finite set of curves, and let $B_n$ denote the lifts of the elements
of $B_0$ under $f^n$.
If $e=(U_0,U_1)$ is a vertical edge of $\Gamma$ with $U_0\in{\mathcal U}_0$ and $U_1\in{\mathcal U}_1$, we may find an arc $\eta_e$
joining their centers inside $U_0\cup U_1$. Let $H_0$ denote the sets of arcs defined for every such edge, and for each $n \geq 0$, define
$H_n$ to be the set of their lifts under $f^n$.
\gap
\noindent{\bf $\mathbf{\Psi:=\Phi^{-1}}$ is Lipschitz.} Let us first consider an edge $e$ joining two sets
${\widetilde{U}}_1$, ${\widetilde{U}}_2$ at the same level $n\ge 0$.
Let $\widetilde{\gamma}=\alpha_{{\widetilde{U}}_1} * \beta_e *\overline{\alpha_{{\widetilde{U}}_2}}$.
The image $f^n(\widetilde{\gamma})$ is a closed curve based at $b$ which is homotopically nontrivial
in $S^2\setminus P$ since $\widetilde{\gamma}$ is not closed.
This curve is also the concatenation of three curves belonging to $A_0\cup B_0$ which is a finite set.
Let
\[ C_{hor}:=\max_{\gamma} ||g(\gamma)|| \]
be the maximum word length of the corresponding group elements $g(\gamma)$, where $\gamma$ varies over this finite collection
of loops just constructed.
We conclude that
\[|\Psi({\widetilde{U}}_1)-\Psi({\widetilde{U}}_2)|_\Sigma \leq C_{hor}.\]
The case of vertically adjacent vertices proceeds similarly in spirit.
Suppose\footnote{Here, it is convenient to index things using zero and one, instead of one and two as above.}
${\widetilde{U}}_1, {\widetilde{U}}_0$ are vertically adjacent in $\Gamma$, with $|{\widetilde{U}}_1| = |{\widetilde{U}}_0|+1=n+1$, and let $e=({\widetilde{U}}_1, {\widetilde{U}}_0)$.
The curves $\alpha_{{\widetilde{U}}_1}$
and $\alpha_{{\widetilde{U}}_2}$ join their centers to elements $\pi_{\Sigma}(w)$ and $\pi_{\Sigma}(v)$, $w,v\in V(\Sigma)$.
Set \[\widetilde{\gamma} = \lambda_w * \alpha_{{\widetilde{U}}_1} * \eta_e * \overline{\alpha}_{{\widetilde{U}}_0}\]
which is a curve joining to points from $\pi_{\Sigma}(V(\Sigma))$ at the same level $n$.
It follows that $\gamma= f^n(\widetilde{\gamma})$ is a loop based at $b$ of the form
\[ \gamma = \lambda_x * \alpha_1 * \eta * \overline{\alpha}_0.\]
Hence it belongs to a finite set of curves (see Figure 1).
\begin{figure}[h]
\label{fig:part3}
\includegraphics{qisom.ps}
\caption{Showing $|\Psi({\widetilde{U}}_1)-\Psi({\widetilde{U}}_0)|_\Sigma = O(1)$. The disk $f(U_1)$ is not shown.}
\end{figure}
We let
\[ C_{ver}:=\max_{\gamma} ||g(\gamma)|| \]
be the maximum word length of the corresponding group elements $g(\gamma)$, where $\gamma$ varies over the finite collection
of loops just constructed. Thus
\[ |\Psi({\widetilde{U}}_1) - \Psi({\widetilde{U}}_0)|_\Sigma \leq C_{ver}.\]
We conclude that $\Psi$ is $\max\{C_{hor}, C_{ver}\}$-Lipschitz, and the proof is complete.
\qed
Suppose now that $f$ has no periodic critical points, and that the virtual endomorphism $\phi$ is contracting.
Then $\Sigma$ is hyperbolic; Theorem \ref{thm:coincidence} implies that $\Gamma$ is also hyperbolic. Hence if $\epsilon$ is chosen sufficiently small, the Floyd boundary $\bdry\Gamma$ coincides with the Gromov boundary, and $\Phi$ induces a homeomorphism $\Phi: {\mathcal J} \to \bdry \Gamma$ which conjugates the shift $f_\Sigma: {\mathcal J} \to {\mathcal J}$ to the induced map $f_\Gamma: \bdry \Gamma \to \bdry \Gamma$. It follows that
\[ \pi:= \Phi^{-1} \circ \pi_\Gamma: S^2 \to {\mathcal J}\]
gives a semiconjugacy of branched coverings from $f: S^2 \to S^2$ to $f_\Sigma: {\mathcal J} \to {\mathcal J}$. The remainder of the proof will use finiteness properties of the dynamical system $f_\Sigma: {\mathcal J} \to {\mathcal J}$ to analyze the possibilities for the fibers of $\pi$.
We conclude this section with a technical result needed for the analysis in the next section.
By \cite[Thm. 6.13]{kmp:ph:cxcii}, $f_\Sigma: {\mathcal J} \to {\mathcal J}$ is a branched covering of degree $d$. Given $E \subset S^2$ a continuum and a component $\widetilde{E}$ of $f^{-1}(E)$, we define
\[ \deg(f:\widetilde{E} \to E) = \min\{ \deg(f: {\widetilde{U}} \to U): E \subset U\}\]
where $U$ is a connected open neighborhood of $E$.
\begin{lemma}
\label{lemma:degree}
Suppose $f_\Sigma(\tilde{\xi}) = \xi$, and set $E = \pi^{-1}(\xi), \widetilde{E} = \pi^{-1}(\tilde{\xi})$. Then $\deg(f:\widetilde{E} \to E) = \deg(f_\Sigma, \tilde{\xi})$. In particular, the inverse image under $f$ of a fiber of $\pi$ is a disjoint union of fibers of $\pi$, and the degrees of $f$ on these fibers sum to $d$.
\end{lemma}
\noindent {\bf Proof: } We first prove $\deg(f:\widetilde{E} \to E) \geq \deg(f_\Sigma, \tilde{\xi})$. Let $U$ be a small open connected neighborhood of $E$ and ${\widetilde{U}}$ the pullback containing $\widetilde{E}$. Since the decomposition of $X$ by the fibers of $\pi$ is upper-semicontinuous, and $f_\Sigma$ is a branched covering, we may assume that the images $V=\pi(U), \widetilde{V}=\pi({\widetilde{U}})$ are neighborhoods of $\xi, \tilde{\xi}$, respectively, that $U=\pi^{-1}(V), {\widetilde{U}}=\pi^{-1}(\widetilde{V})$, and that $\deg(f_\Sigma,\tilde{\xi}) = \deg(f_\Sigma: \widetilde{V} \to V)$.
Since $f_\Sigma: {\mathcal J} \to {\mathcal J}$ is a branched covering, the set of branch values is nowhere dense, so there exists $\zeta \in V$ which is not a branch value of $f_\Sigma$. Hence $\#f_\Sigma^{-1}(\zeta)\intersect \widetilde{V} = \deg(f_\Sigma,\tilde{\xi})$. It follows easily that $\deg(f_\Sigma, \tilde{\xi})$ is a lower bound for $\deg(f: {\widetilde{U}} \to U)$.
The proof of the inequality $\deg(f:\widetilde{E} \to E) \geq \deg(f_\Sigma, \tilde{\xi})$ is similar. Suppose $V, \widetilde{V}$ are respectively neighborhoods of $\xi, \tilde{\xi}$ and set $U = \pi^{-1}(V), {\widetilde{U}} = \pi^{-1}(\widetilde{V})$. Choose $\zeta \in V$ which is not a branch value of $f_\Sigma$. Then $\#f_\Sigma^{-1}(\zeta)=d$, and since $f$ is a branched covering of degree $d$, this implies that for each $z\in \pi^{-1}(\zeta)$, $\#f^{-1}(z)=d$ also. Hence no two elements of $f^{-1}(z)$ belong to the same fiber of $\pi$. We conclude $\#f_\Sigma^{-1}(\zeta)\intersect \widetilde{V} = \#f^{-1}(z)\intersect {\widetilde{U}}$ and so $\deg(f: {\widetilde{U}} \to U)$ is a lower bound for $\deg(f_\Sigma, \tilde{\xi})$.
\qed
\section{Proof of necessity}
\begin{prop} Let $f:S^2\to S^2$ be a postcritically finite topological cxc map of the sphere.
Then its virtual endomorphism is contracting.\end{prop}
Let ${\mathcal O}$ be the orbifold modeled on $(S^2, P)$, $\nu: {\mathcal O} \to \mathbb{N}$ its weight function, and as usual choose a basepoint $b\in S^2\setminus P$.
Taking an iterate if necessary, we may assume
that $f(b)=b$. The group $G=\pi_1({\mathcal O},b)$
acts naturally as the deck transformation group of a Galois (regular) ramified universal orbifold covering $p:\widetilde{{\mathcal O}}\to S^2$ such that
$\deg(p, \tilde{x}) = \nu(p(\tilde{x}))$ for all $\tilde{x}\in\widetilde{{\mathcal O}}$.
It follows that there exists a lift of ``$f^{-1}$'', that is, a map
${\widetilde{f}_{-}}:\widetilde{{\mathcal O}}\to\widetilde{{\mathcal O}}$ such that
$f\circ p \circ{\widetilde{f}_{-}}= p$.
Since $f(b)=b$, we may choose ${\widetilde{f}_{-}}$ so that a given $\tilde{b} \in p^{-1}(b)$ is fixed by ${\widetilde{f}_{-}}$:
\[
\begin{array}{ccc}
(\widetilde{{\mathcal O}}, \tilde{b}) & \stackrel{{\widetilde{f}_{-}}}{\longleftarrow} &(\widetilde{{\mathcal O}}, \tilde{b})
\\
p \downarrow & \; & \downarrow p \\
({\mathcal O}, b)& \stackrel{f}{\longrightarrow} &({\mathcal O}, b) \\
\end{array}
\]
To see this: at lower-left, replace ${\mathcal O}$ with ${\mathcal O}_1$, so that the bottom arrow becomes an orbifold covering.
Lift the orbifold structure on ${\mathcal O}_1$ at lower-left under $p$ to obtain a new orbifold structure $\widetilde{{\mathcal O}}_1$
on the underlying space of $\widetilde{{\mathcal O}}$ so that $p: \widetilde{{\mathcal O}}_1 \to {\mathcal O}_1$ is an orbifold covering.
The map $\widetilde{f}_{-}$ is then a universal orbifold covering of $\widetilde{{\mathcal O}}_1$.
In this paragraph, we define a metric $\tilde{d}$ on ${\mathcal O}$ with respect to which ${\widetilde{f}_{-}}$ becomes a uniform contraction. Let ${\mathcal U}_0$ be a covering as in \S\ref{scn:covering}, and
endow $S^2$ with the metric $d_{\ep}$ described in \S\,\ref{scn:covering}.
Choose $M>0$ so small that (i) any $d_{\ep}$-ball on $S^2$ of radius $2M$ is contained
in some element of ${\mathcal U}_0$; from the choice of ${\mathcal U}_0$ it follows that
(ii) any such ball meets $P$ in at most one point.
Let ${\mathcal D}$ denote the set of Jordan domains $U$ of diameter at most $M$.
Condition (ii) implies that if $U \in {\mathcal D}$, then the inverse image $p^{-1}(U)$
is a collection $\{{\widetilde{U}}_k\}$ of pairwise disjoint Jordan domains in $\widetilde{{\mathcal O}}$
invariant under the covering group action of $G$
(their closures might meet, but we don't care), and the restriction of $p$ to any domain
${\widetilde{U}}_k$ is proper.
Let $\tilde{U}$ be one such component and set, for $x,y\in \tilde{U}$,
$$q_{\tilde{U}}(x,y)= \inf \diam_\ep p(K)$$
where the infimum is over all continua $K\subset \overline{\tilde{U}}$ joining $x$ to $y$.
Now suppose $x,y$ in $\widetilde{{\mathcal O}}$. We define the metric $\tilde{d}$ on $\widetilde{{\mathcal O}}$ by
$$\tilde{d}(x,y)=\inf \sum_{j} q_{\tilde{U}_j} (x_j,x_{j+1})$$
where the infimum is taken over every chain $x=x_1, x_2, \ldots, x_\ell = y$ joining $x$ to $y$, with the property that for each $j$, there exists $U_j \in {\mathcal D}$ and a component ${\widetilde{U}}_j$ of $p^{-1}(U_j)$ such that $\{x_j, x_{j+1}\}\subset {\widetilde{U}}_j$.
We note that if $\tilde{d}(x,y)\le M$, then we may find $\tilde{U}\in{\mathcal D}$ such that $\tilde{d}(x,y)=q_{\tilde{U}}(x,y)$.
It follows that the infimum in the definition of $\tilde{d}$ is actually attained.
The function $\tilde{d}$ defines a metric and the group $G$ acts by isometries on $(\widetilde{{\mathcal O}},\tilde{d})$.
\begin{lemma}
\label{lemma:orbqi}
The map $\Phi:g\mapsto g(\tilde{b} )$ defines a quasi-isometry from $G$ to $(\widetilde{{\mathcal O}},\tilde{d})$.\end{lemma}
\noindent {\bf Proof: } We already know that $G$ is finitely generated and that it
acts on $\widetilde{{\mathcal O}}$ by isometries, properly discontinuously and cocompactly.
We mimic the proof of the Svarc-Milnor lemma.
Let $C\subset \widetilde{{\mathcal O}}$ be a compact set containing $\tilde{b}$ such that $G.C=\widetilde{{\mathcal O}}$
and fix $D\ge 8M$ such that $C\subset B(\tilde{b} , D/4)$ and let $S=\{g\in G, \tilde{d}(g(\tilde{b} ),\tilde{b})\le D\}$.
Then $\Phi(G)$ is cobounded, $S$ is a finite generating set and there are constants $\lambda$, $c>0$
such that $\tilde{d}(\Phi(g_1),\Phi(g_2))\le \lambda \|g_1g_2^{-1}\|_S + c$
for all $g_1,g_2\in G$, see \cite[Prop.\,8.19]{bridson:haefliger:book}. The only
point to prove is the reverse inequality.
Fix $g\in G$ and let us consider $x_0,\ldots, x_n$ with $x_0=\tilde{b} $ and $x_n=g(\tilde{b} )$ and
let $\tilde{U_1},\ldots, \tilde{U}_n$
be such that $$\tilde{d}(\tilde{b} ,g(\tilde{b} ))=\sum q_{\tilde{U}_j} (x_{j-1},x_j)\,.$$
Let $N$ be the integer part of $4\tilde{d}(\tilde{b} ,g(\tilde{b} ))/D$. For any $1\le k \le N$, let $y_k$ be the first $x_i$ such that
$$\sum_{j=1}^i q_{\tilde{U}_j} (x_{j-1},x_j) \ge k D/4\,.$$ It follows that
$\tilde{d}(y_k,y_{k+1})\le D/2$ since the points $(x_j)$ are at most $M$-separated and $M\le D/8$.
Choose now $g_k\in G$ such that $\tilde{d}(g_k(\tilde{b} ),y_k)\le D/4$; then $\tilde{d}(g_k(\tilde{b} ),g_{k+1}(\tilde{b} ))\le D$, so that
$g_k\circ g_{k+1}^{-1}\in S$ and $\|g\|_S\le N+1$. Therefore, $\|g\|_S\le (4/ D ) (\tilde{d}(\tilde{b} ,g(\tilde{b} )) +1)+1$.
The proof is complete.\qed
\gap
\noindent {\bf Proof: } (Proposition)
Let $x,y\in\widetilde{{\mathcal O}}$ be such that $\tilde{d}(x,y)\le M$, then we may find a continuum $K$ containing $\{x,y\}$
such that $\tilde{d}(x,y)=\diam_\ep p(K)$. Note that $p(K)$ is contained in an element of ${\mathcal U}_0$.
Therefore, for any $n\ge 0$, $(p \circ {\widetilde{f}_{-}^{n}})(K)$ is contained in an element of ${\mathcal U}_n$ and
so $\tilde{d}( {\widetilde{f}_{-}^n}(x),{\widetilde{f}_{-}^n}(y))\lesssim e^{-\ep n}$.
It follows easily that, for any pair of points $x,y$ without any restriction on their distance, we have
$$\tilde{d}( {\widetilde{f}_{-}^n}(x),{\widetilde{f}_{-}^n}(y))\lesssim e^{-\ep n} \tilde{d}(x,y)\,.$$
This implies that
$$\limsup_{n\to\infty}\left( \sup_{x,y\in\widetilde{{\mathcal O}}}
\frac{ \tilde{d}( {\widetilde{f}_{-}^n}(x),{\widetilde{f}_{-}^n}(y)) }{\tilde{d}(x,y)}\right)^{1/n} = e^{-\ep} < 1.$$
Now, since $f(b)=b$ is fixed, we may define the virtual endomorphism $\phi$ by choosing the connecting path $\al$
(as in the definition given in \S 3.1) to be the constant path.
Hence, $g\in \mbox{\rm dom} \phi^n$ if and only if ${\widetilde{f}_{-}^n}(g(\tilde{b} ))$ belongs to the orbit $G.\tilde{b} $,
and in this case, $\Phi(\phi^n(g))= {\widetilde{f}_{-}^n}(g(\tilde{b} ))$.
From the preceding Lemma \ref{lemma:orbqi}, the metrics $|| \cdot ||$ and $\tilde{d}$ are comparable, so we obtain
$$\limsup_{n\to\infty}\left( \limsup_{||g||\to\infty}
\frac{||\phi^n(g)||}{||g||}\right)^{1/n} \le \limsup_{n\to\infty}\left( \sup_{x,y\in\widetilde{{\mathcal O}}}
\frac{ \tilde{d}( {\widetilde{f}_{-}^n}(x),{\widetilde{f}_{-}^n}(y)) }{\tilde{d}(x,y)}\right)^{1/n} < 1.$$
\qed
\section{Proof of sufficiency}
From the introduction, recall that the key step is to establish
\gap
\begin{prop}\label{prop:fibers} Let $f:S^2\to S^2$ be a Thurston map with postcritical set $P$ and with hyperbolic biset $\frak{M}$.
The semiconjugacy $\pi: S^2 \to {\mathcal J}$ is injective on $P$, and no fiber of $\pi$ separates $S^2$.
\end{prop}
The main ingredients in the proof of the claim are \cite[Lemma 6.11, Theorem 6.15]{kmp:ph:cxcii}, which assert that the dynamical system $f_\Sigma: {\mathcal J} \to {\mathcal J}$ is expanding, irreducible, and of {\em topologically finite type}. Finite type implies that there is a positive integer $p$ such that for all $\xi \in {\mathcal J}$ and all $n > 0$, $\deg(f_\Sigma^n, \xi) \leq p$. Thus in particular $f_\Sigma$ has no periodic branch points. Irreducible in this context is equivalent to the fact that given any nonempty open set $W$, there is an integer $n$ with $f_\Sigma^n(W)={\mathcal J}$.
We remark that these properties imply that $f_\Sigma: {\mathcal J} \to {\mathcal J}$, as a topological dynamical system, is a so-called finite type topologically coarse expanding conformal (cxc) system; cf. \cite{kmp:ph:cxci}.
This forces strong constraints on the fibers of $\pi$. We prove the claim by a case-by-case analysis of the possibilities.
We remark that there is much overlap with the flavor of the arguments in \cite{kmp:tan:rmwdjs}.
\gap
A continuum $E \subset S^2$ we call
\begin{itemize}
\item {\em special} if $E \intersect P\neq \emptyset$;
\item {\em type III} if $S^2 - E$ has at least $ 3$ components that intersect $P$
\item {\em type II} if $S^2 - E$ has exactly $2$ components that intersect $P$
\item {\em type I} if $S^2 - E$ has exactly $1$ component that intersects $P$
\end{itemize}
The map $f$ sends fibers of $\pi$ onto fibers of $\pi$. Suppose $E=\pi^{-1}(\xi)$ for some $\xi\in {\mathcal J}$.
\begin{itemize}
\item[(i)] If $E$ is special, then $f^k(E)$ is special for all $k\ge 0$, and since $P$ is finite, $E$ is eventually periodic.
Moreover, there are only finitely many such fibers.
\item[(ii)] It is easily verified that if $f(E)$ is not special, then neither is $E$, and the type of $f(E)$ is at least the type of $E$.
\item[(iii)] If $E$ is type II and non-special then it determines a homotopy class of unoriented simple closed curve in $S^2-P$.
\item[(iv)] Since distinct fibers are disjoint, there can be only finitely many type III fibers, and only finitely many homotopy classes of curves arise from type II fibers.
\item[(v)] If $U$ is a component of $S^2\setminus E$, then $\pi(U)$ is a component of ${\mathcal J}\setminus\{\xi\}$ and is open. This follows
from the facts that $\pi$ is monotone, continuous and onto and that $S^2$ is locally connected.
\end{itemize}
The proof of the following technical lemma is straightforward.
\begin{lemma}
\label{lemma:thickening}
Given a continuum $E \subset S^2$, there is a ``thickening'' $\widehat{E}$ of $E$ given by an open regular neighborhood of $E$ with the following properties.
\begin{itemize}
\item the closure of $\widehat{E}$ is a compact surface with boundary.
\item $\widehat{E}$ has the same type as $E$.
\item $\widehat{E} \intersect P = E \intersect P$.
\item for each component $\widetilde{E}$ of $f^{-1}(E)$, we have $\deg(f: \widetilde{\widehat{E}} \to \widehat{E}) = \deg(f: \widetilde{E} \to E)$,
where $\widetilde{\widehat{E}}$ is the component of $f^{-1}(\widehat{E})$ containing $\widetilde{E}$.
\end{itemize}
\end{lemma}
In what follows, given a fiber $E=\pi^{-1}(\xi)$, the symbol $\widehat{E}$ denotes a thickening of $E$ satisfying the conditions of Lemma \ref{lemma:thickening}. The following paragraphs rule out certain types of fibers.
\gap
\begin{lemma}[no periodic separating fibers]\label{lma:periodicfibers}
No periodic $E$ separates $S^2$, and there are no type II fiber homotopic rel $P$ to one of its iterates. Moreover, if
$E$ is periodic and special, then $\# E \intersect P= 1$ and $E\cap P$ is periodic of the same period
as $E$.
\end{lemma}
\noindent {\bf Proof: } Taking an iterate if necessary, we may assume that $E$ is fixed.
Since $f_\Sigma$ has no periodic branch points, $\deg(f_\Gamma, \xi)=\deg(f: E \to E)=1$, by Lemma \ref{lemma:degree}.
Then $E$ is contained in a thickening $\widehat{E}$ for which $\deg(\widetilde{\widehat{E}} \to \widehat{E})=1$.
This implies $f|_E$ is a homeomorphism which extends to a homeomorphism on a neighborhood of $E$.
\begin{enumerate}
\item {\bf Subcase $E$ special, $\# E \intersect P\geq 2$.} Since $f:E\to E$ is a homeomorphism and $P$ is forward invariant and finite, we may assume that
$P\cap E$ is pointwise fixed. Note that $\hat{E}\cap f(\hat{E})$ is a neighborhood of $E$.
Let $\gamma\subset f(\hat{E})$ join two distinct point of $E\cap P$. Then $f^{-1}\circ\gamma$ joins the same points
and is homotopic rel. $P$ to a curve $\gamma_1\subset f(\hat{E})$
since $f|E$ is $1-1$.
This implies the existence of an arc with arbitrarily long nontrivial iterates
under the pullback relation, which is impossible by Theorem \ref{thm:prevents_levy}.
\item {\bf Subcase $E$ special and type II or type III.} Let us first establish that
the boundary $C$ of some complementary component $W$ of $\widehat{E}$ must be essential and nonperipheral in $S^2-P$.
If each such $W$ contains at most one point of $P$ then $f^{-1}(\widehat{E})$ is connected. Thus
$\deg(f: f^{-1}(\widehat{E}) \to \widehat{E})=d>1$ holds, and this contradicts the fact that $\deg(f: E \to E)=1$.
Hence at least one $W$ contains two points of $P$. If $E$ is type III or type II and special, then $C$ is essential and nonperipheral.
We first note that $f^k:C\to f^k(C)$ has degree $1$ up to homotopy rel. $P$ for all $k\ge 1$. Moreover, $f^k(C)$ is also essential and nonperipheral
for otherwise, $C$ would not be. Finally, the forward orbit of $C$ has to cycle eventually up to homotopy since there are only finitely homotopic
classes of fibers of type at least II. This
means that a Levy cycle can be extracted from $\{f^k(C)\}_{k\ge 0}$, contradicting Theorem \ref{thm:prevents_levy}.
\item {\bf Subcase $E$ non special and type II or type I.} Let us assume that $E$ separates $S^2$.
There ia a component $W$ of $S^2\setminus E$
which contains all of $P$ but at most one single point. Let
$\widehat{W}$ be the component of $S^2\setminus \widehat{E}$ in $W$. Then the complement
$\widehat{U}=S^2\setminus \widehat{W}$ is a Jordan domain containing $E$ and at most one critical value of $f$.
Therefore, $f^{-1}(\widehat{U})$ is a union of Jordan domains.
One of them contains $E$, hence $U= S^2\setminus W$. It follows that $f(U)\subset U$. But $U$ contains a component
of $S^2\setminus E$, so $\pi(U)$ has interior by (v) and $f_\Sigma(\pi(U))\subset \pi(U)$. which contradicts the irreducibility of $f_\Sigma$.
\item {\bf Subcase type II $E$ periodic up to homotopy.} We assume that $E$ is type II and $f^k(E)$ is homotopic
to $E$ rel. $P$. If $\deg(f^k, E)>1$ then $\deg(f^{nk},E) =\deg(f_\Sigma^{nk}, \xi) \to \infty$ as $n \to \infty$, violating $f_\Sigma$ being finite type.
So $f^k|_E$ is injective, and after thickening, we extract a Levy cycle as above.
\end{enumerate}\qed
\gap
{\noindent\bf No preperiodic separating fibers nor type II and type III fibers.}
Let $E$ be a preperiodic separating fiber. If all its iterates are nonspecial, then we would obtain
a periodic separating fiber by (ii) and (iv), which is impossible according to Lemma \ref{lma:periodicfibers}.
Therefore, there is some $n\ge 1$ such that $E'=f^n(E)$ is special and periodic by (i).
By Lemma \ref{lma:periodicfibers} again, $E'$ does not separate $S^2$ and intersects $P$ at a single point, so
the inclusion $U:=S^2-(P\union E')\hookrightarrow S^2-P$ is a homotopy equivalence.
Therefore $f^{-n}(U)$ is homotopy equivalent to the connected set
$S^2-f^{-1}(P)$. Hence $U$ is connected, $E$ is nonseparating, and $\#E \intersect P \leq 1$.
Let us now prove that any type II and type III fibers have to cycle up to homotopy, which will establish their nonexistence..
Since fibers cannot intersect, there can be only finitely many of them up to homotopy, cf. (iv).
Let $E$ be of type at least II. If its iterates are non special, then they remain of type at least II by (ii), and hence have to cycle up to homotopy by (iv).
Otherwise, one iterate becomes special, hence is preperiodic as well.
\gap
In order to complete the proof, we must prove that there are no type I fiber. We have already ruled out preperiodic ones.
\gap
\noindent{\bf No wandering separating type I fiber.} All the possible remaining separating type I fibers are nonspecial and hence form a totally invariant set of $S^2$.
We use the same notation as in Lemma \ref{lma:periodicfibers}: given such a fiber $E$, let $\widehat{W}_E\subset W_E$
contain $P$, $U_E=S^2\setminus W_E$ and
$\widehat{U}=S^2\setminus \widehat{W}$.
Therefore, $f^{-1}(\widehat{U}_{f(E)})$ is a union of Jordan domains disjoint from $P$.
One of them contains $E$, hence $U_E$. It follows that $f(U_E)\subset U_{f(E)}$.
This implies that no preimage of $P$ can enter the sets $U_E$: this contradicts again the irreducibility of $f_\Sigma$.
Therefore, no type I fiber separates the sphere.
\gap
\noindent {\bf Proof: } (Proposition \ref{prop:fibers}) It follows from the discussion above that
fibers of $\pi$ are connected, do not separate the sphere, and meet $P$ in at most one point. \qed
\noindent {\bf Proof: } (Theorem \ref{thm:characterization}) Suppose the virtual endomorphism $\phi$ is contracting and $f$ has no periodic branch points.
Necessarily $\#P \geq 3$. For technical reasons, it is convenient later to assume $\#P > 3$ (see the introduction for the case $\#P=3$).
According to Theorem \ref{thm:coincidence}, we may consider defining data for $\Sigma$ and $\Gamma$ so that both graphs are quasi-isometric.
Since $\phi$ is contracting, it follows that $\Gamma$ is hyperbolic and that the quasi-isometry defines a homeomorphism
$\Phi:{\mathcal J}\to\partial\Gamma$. The map $\pi=\Phi^{-1}\circ\pi_\Gamma:S^2\to {\mathcal J}$ semiconjugates $f$ to $f_\Sigma$.
Moreover, Lemma \ref{lemma:monotone} and Proposition \ref{prop:fibers} imply that $\pi$ is
monotone, injective on $P$ and that its fibers do not separate $S^2$.
By Moore's Theorem \cite[Theorem 25.1]{daverman:decompositions}, the quotient ${\mathcal J}$ is homeomorphic to the sphere.
The induced map $f_\Sigma: {\mathcal J} \to {\mathcal J}$ is thus an expanding Thurston map.
It remains to construct a homotopy from $f$ to an expanding map $g$. By \cite[Theorems 13.4, 25.1]{daverman:decompositions},
the decomposition ${\mathcal G}$ of $S^2$ by the fibers of the semiconjugacy $\pi: S^2 \to {\mathcal J}$ has the property of being {\em strongly shrinkable}:
there is a one-parameter family of continuous maps $h_t: S^2 \to S^2, t \in [0,1]$, called a {\em pseudoisotopy}, such that $h_0 = \id_{S^2}$,
$h_t$ is a homeomorphism for $t \in [0,1)$, $h_t|_{P_f} = \id_{P_f}$ for all $t$, and the fibers of $h_1$ and those of $\pi$ coincide.
The induced homeomorphism $S^2 \to {\mathcal J}$ conjugates $f_\Sigma$ to an expanding Thurston map $g: S^2 \to S^2$, and
the family of maps $f_t$, $t \in [0,1]$ defined as the unique continuous solution of $h_t \circ f = f_t \circ h_t$, with $f_1=g$, gives a homotopy
through Thurston maps.
The proof of the sufficiency in
Theorem \ref{thm:characterization} is complete. \qed
|
1,314,259,996,513 | arxiv | \section{Introduction}
In the recent years, quantum information processes have developed rapidly \cite{book}, among which the most important branches
are the long-distance quantum communication and quantum computation. Entanglement, which is a uniquely quantum mechanical
feature, is considered to be an essential resource for both the two branches. In practical applications, entanglement is usually produced locally and can be distributed to the remote parties. It not only can hold the power for the quantum nonlocality \cite{Einstein}, but also can provide wide applications in the quantum information processing (QIP) \cite{rmp}. For example, many popular research areas such as the quantum teleportation \cite{teleportation,cteleportation1,cteleportation2}, quantum denescoding \cite{densecoding}, quantum secret sharing\cite{QSS1,QSS2,QSS3}, quantum state
sharing\cite{QSTS1,QSTS2,QSTS3}, and quantum secure direct communication \cite{QSDC1,QSDC2,QSDC3}, all require entanglement to set up the quantum entanglement channels.
Among various entanglement forms, the multi-mode and multi-particle W states have quite important applications. The perfect entangled W states are the maximally entangled W state, which can be written as
\begin{eqnarray}
|W\rangle_{multi-mode}&=&\frac{1}{\sqrt{N}}(|100\cdots 0\rangle+|010\cdots 0\rangle+|001\cdots 0\rangle+\cdots+|000\cdots 01\rangle),\nonumber\\
|W\rangle_{muti-photon}&=&\frac{1}{\sqrt{N}}(|HVV\cdots V\rangle+|VHV\cdots V\rangle+|VVH\cdots V\rangle+\cdots+|VVV\cdots VH\rangle),
\end{eqnarray}
where the $|H\rangle$ and $|V\rangle$ represent the horizontal and vertical polarization of the photon state, while $|1\rangle$ and $|0\rangle$ represent one photon and no photon, respectively. It has been proved that the W states are highly robust against the loss of one or two qubits \cite{W,W1,W2}. There are many works have been done based on both multi-particle W state and single-photon multi-mode W state, such as the protocols of perfect teleportation and superdense coding with W states \cite{wteleportation},
the generation of the W state \cite{generation1,generation2,generation3,generation4,generation5,generation6,generation7,generation8,generation9,generation10,generation11}, entanglement transformation \cite{wtransformations},
distillation \cite{wdistill1,wdistill2,wdistill3,cao} and concentration \cite{zhanglihua,wanghf,shengPRA,gub,duff,zhoujosa,zhouqip2} of the W states. Interestingly, Gottesman \emph{et al.} proposed a protocol for building an interferometric telescope based on the single-photon multi-mode W state \cite{telescope}. The protocol has the potential to eliminate the baseline length limit, and allows in principle the interferometers with arbitrarily long baselines.
However, in practical applications, the signals will inevitably interact with the environment during the storage and transmission process. In this way, the perfect entangled W states also may be degraded to a mixed state or a pure partially entangled states because of the environmental noise. During the applications, such partially entangled state may further decrease and cannot ultimately set up the high quality quantum entanglement channel \cite{memory}. Therefore, we need to recover the mixed state or pure partially entangled W state into the maximally entangled W state.
Here, we focus on recovering the pure partially entangled W state into the maximally entangled W state. The entanglement concentration is a powerful method to distill the maximally entangled state from the pure partially entangled state \cite{C.H.Bennett2,swapping1,swapping2,zhao1,Yamamoto1,shengpra2,shengqic,cao,zhanglihua,wanghf,gub,duff,deng4,shengPRA,zhoujosa,shengqip,zhouqip2,zhouqip1,shengpra3,dengpra,wangc1,wangc2}. In 1996, Bennett \emph{et al.} proposed the first entanglement concentration protocol (ECP) which is known as the Schmidt projection method \cite{C.H.Bennett2}. Since then, various ECPs have been put forward successively, such as the ECP based on the entanglement swapping \cite{swapping1} and the ECP based on unitary transformation \cite{swapping2}. In 2001, Zhao \emph{ et al.} and Yamamoto \emph{et al.} proposed two similar concentration protocol independently with linear optical elements, and later realized them in experiments, respectively \cite{zhao1,Yamamoto1}. In 2008, Sheng \emph{et al.} developed their protocols with the help of the cross-Kerr nonlinearity \cite{shengpra2}. However, most ECPs described above
are focused on the two-particles entanglement, which can not be used to concentrate the pure partially entangled W state. In 2003, Cao and Yang
firstly proposed an ECP for W state with the joint unitary transformation \cite{cao}. In 2007, Zhang \emph{et al.} proposed an ECP for the W state with the help of the collective Bell-state measurement \cite{zhanglihua}. In 2010, Wang\emph{ et al.} proposed an ECP for
a special W state as $\alpha|HVV\rangle+\beta(|VHV\rangle+|VVH\rangle)$ with linear optics \cite{wanghf}. In 2012, Gu \emph{et al.} and Du \emph{et al.} improved the ECP for the special W state with the help of the cross-Kerr nonlinearity \cite{gub,duff}. Later, Ren \emph{et al.} proposed an ECP for multipartite electron-spin states with CNOT gates \cite{deng4}. The concentration protocols for both arbitrary multi-photon partially entangled W state and
single-photon multi-mode W state were proposed \cite{shengPRA,zhoujosa,shengqip,zhouqip2}. Unfortunately, all the previous ECPs for partially entangled
W state are not optimal. Some of the ECPs are focused on the special types of the W states, and some ECPs need the cross-Kerr nonlinearity medium
to complete the task, which cannot be realized in current experimental conditional. Moreover, Most ECPs cannot reach a high success probability.
In this paper, we will present two optimal ECPs for multi-mode single-photon W state and multi-photon polarization W state, respectively, inspired by the recent excellent concentration work for two-photon system proposed by the group of Deng \cite{dengarxiv}. Both of our two ECPs do not require any auxiliary photon, and only resort to the linear optical elements. Therefore, they can be easily realized under current experimental condition. Meanwhile, our ECPs only require local operations, which can simplify the operations largely. Moreover, our ECP only need to be operated for one time, and its success probability is higher than all the previous ECPs for W states \cite{shengPRA,zhoujosa,shengqip,zhouqip2}. Based on the features above, our ECPs may be useful in current quantum communications.
The paper is organized as follows. In Sec. 2, we first briefly explain the ECP for the single-photon multi-mode partially entangled W state. In Sec. 3, we explain the ECP for the multi-photon polarization partially entangled W state. In Sec. 4, we make a discussion and summary.
\section{The efficient ECP for the single-photon multi-mode W state}
\begin{figure}[!h
\begin{center}
\includegraphics[width=10cm,angle=0]{ECPnumber.eps}
\caption{The schematic drawing of the ECP for the single-photon multi-mode W state. The ECP can be divided into two steps. The two concentration steps are independent. Alice and Bob can operate the two steps alone, respectively. In each concentration step, a variable beam splitter (VBS) is used to adjust the entanglement coefficient.}
\end{center}
\end{figure}
Now we first start to explain our ECP for the single photon three-mode W state and then extend this method to the case of single-photon multi-mode partially entangled W state. The basic principle of our ECP is shown in Fig. 1. Suppose a single photon source S emits a single photon, and sends it to three parties, say Alice, Bob and Charlie. In this way, it can create a single photon multi-mode W state in the spatial mode a1, b1 and c1 as
\begin{eqnarray}
|\Phi_{1}\rangle_{a1b1c1}=\alpha|100\rangle_{a1b1c1}+\beta|010\rangle_{a1b1c1}+\gamma|001\rangle_{a1b1c1}. \label{initial}
\end{eqnarray}
Here, $\alpha,\beta$, and $\gamma$ are the initial entanglement coefficients and $|\alpha|^{2}+|\beta|^{2}+|\gamma|^{2}=1$. Meanwhile, we suppose $|\alpha|>|\beta|>|\gamma|$.
Our ECP can be divided into two steps. In the first step, Alice makes the photon in the a1 mode pass through a variable beam splitter (VBS1) with the transmittance of $t_{1}$. After VBS1, the photon state in the a1 mode evolves to
\begin{eqnarray}
|\phi\rangle_{a1}= \alpha\sqrt{t_{1}}|1\rangle_{a2}+\alpha\sqrt{1-t_{1}}|1\rangle_{a3}.
\end{eqnarray}
After passing through the VBS1, the initial state becomes
\begin{eqnarray}
|\Phi_{1}\rangle_{a1b1c1}&=&\alpha|100\rangle_{a1b1c1}+\beta|010\rangle_{a1b1c1}+\gamma|001\rangle_{a1b1c1}\nonumber\\
&\rightarrow&(\alpha\sqrt{t_{1}}|100\rangle_{a2b1c1}+\alpha\sqrt{1-t_{1}}|100\rangle_{a3b1c1})+\beta|010\rangle_{a2b1c1}+\gamma|001\rangle_{a2b1c1}.\label{initial2}
\end{eqnarray}
Then, Alice detects the photon in the a3 mode by the single photon detector D1. It is easily to found that D1 may detect one photon or no photon. Alice selects the case that D1 detects no photon. In this way, the single photon state in the three parties becomes
\begin{eqnarray}
|\Phi_{2}\rangle_{a2b1c1}=\alpha\sqrt{t_{1}}|100\rangle_{a2b1c1}+\beta|010\rangle_{a2b1c1}+\gamma|001\rangle_{a2b1c1},\label{vbs1}
\end{eqnarray}
with the success probability of $|\alpha|^{2}t_{1}+|\beta|^{2}+|\gamma|^{2}$.
It can be found that if Alice can find a suitable VBS1 with $t_{1}=\frac{|\gamma|^{2}}{|\alpha|^{2}}$, Eq. (\ref{vbs1}) can be rewritten as
\begin{eqnarray}
|\Phi_{2}\rangle_{a2b1c1}=\gamma|100\rangle_{a2b1c1}+\beta|010\rangle_{a2b1c1}+\gamma|001\rangle_{a2b1c1},\label{step1}
\end{eqnarray}
which only has two different entanglement coefficients $\gamma$ and $\beta$.
Until now, the first concentration step is completed. In the first step, by selecting the suitable VBS with $t_{1}=\frac{|\gamma|^{2}}{|\alpha|^{2}}$ and the case that the photon detector D1 detects no photon, Alice successfully convert Eq. (\ref{initial}) to Eq. (\ref{step1}) with the success probability of
\begin{eqnarray}
P_{1}=2|\gamma|^{2}+|\beta|^{2},
\end{eqnarray}
where the subscript "1" means in the first concentration step.
The second concentration step is operated by Bob and the whole operation process is quite similar with the first step. Firstly, Bob makes the photon in the b1 mode pass through the VBS2 with the transmittance of $t_{2}$. After the VBS2, the photon state in the b1 mode can evolve to
\begin{eqnarray}
|\phi'\rangle_{b1}=\beta\sqrt{t_{2}}|1\rangle_{b2}+\beta\sqrt{1-t_{2}}|1\rangle_{b3}.\label{vbs2}
\end{eqnarray}
Bob also detects the photon in the b3 mode by the single photon detector D2. When D2 detects no photon, the single photon state in the three parties can evolve to
\begin{eqnarray}
|\Phi_{3}\rangle_{a2b2c1}=\gamma|100\rangle_{a2b2c1}+\beta\sqrt{t_{2}}|010\rangle_{a2b2c1}+\gamma|001\rangle_{a2b2c1},\label{step2}
\end{eqnarray}
with the success probability of $\frac{2|\gamma|^{2}+|\beta|^{2}t_{2}}{2|\gamma|^{2}+|\beta|^{2}}$.
Similarly, if Bob can select a suitable VBS2 with $t_{2}=\frac{|\gamma|^{2}}{|\beta|^{2}}$, Eq. (\ref{step2}) can finally evolve to
\begin{eqnarray}
|\Phi\rangle_{a2b2c1}&=&\gamma|100\rangle_{a2b2c1}+\gamma|010\rangle_{a2b2c1}+\gamma|001\rangle_{a2b2c1}\nonumber\\
&\longrightarrow&\frac{1}{\sqrt{3}}(|100\rangle_{a2b2c1}+\gamma|010\rangle_{a2b2c1}+\gamma|001\rangle_{a2b2c1}),\label{max}
\end{eqnarray}
which is the maximally entangled single photon W state. When $t_{2}=\frac{|\gamma|^{2}}{|\beta|^{2}}$, the success probability of the second concentration step is
\begin{eqnarray}
P_{2}=\frac{3|\gamma|^{2}}{2|\gamma|^{2}+|\beta|^{2}},
\end{eqnarray}
where the subscript "2" means in the second concentration step.
So far, the whole ECP is completed and the three parties can finally share a maximally entangled W state from the partially entangled single photon W state. In the practical experiment, the two concentration steps are absolutely independent, which can be completed by Alice and Bob alone, respectively. The total success probability equals to the product of the success probability in each concentration step, which can be written as
\begin{eqnarray}
P_{total}=P_{1}P_{2}=(2|\gamma|^{2}+|\beta|^{2})\frac{3|\gamma|^{2}}{2|\gamma|^{2}+|\beta|^{2}}=3|\gamma|^{2}.\label{probability}
\end{eqnarray}
Similarly, it is obvious that our ECP can be extended to concentrate single photon N-mode partially entangled W state. Suppose the N-mode single photon W state is shared by N parties, which can be written as
\begin{eqnarray}
|\Phi_{N}\rangle=a_{1}|100\cdots 0\rangle+a_{2}|010\cdots 0\rangle+a_{3}|001\cdots 0\rangle+\cdots+a_{N}|000\cdots 01\rangle,\label{initial-N}
\end{eqnarray}
where $|a_{1}|^{2}+|a_{2}|^{2}+|a_{3}|^{2}+\cdots+|a_{N}|^{2}=1$, and $|a_{1}|>|a_{2}|>|a_{3}|>\cdots >|a_{N}|$. Under this case, N-1 parties need to perform the concentration step, respectively. In each concentration step, a suitable VBS with the transmittance of $t_{i}=\frac{|a_{n}|^{2}}{|a_{i}|^{2}}$ should be provided. After the N-1 concentration steps, Eq. (\ref{initial-N}) can be finally converted to the maximally entangled W state as
\begin{eqnarray}
|\Phi'_{N}\rangle=\frac{1}{\sqrt{N}}(|100\cdots 0\rangle+|010\cdots 0\rangle+|001\cdots 0\rangle+\cdots+|000\cdots 01\rangle),\label{max-N}
\end{eqnarray}
with the success probability of
\begin{eqnarray}
P_{N_{total}}=N|a_{N}|^{2}.\label{probability-N}
\end{eqnarray}
\section{The ECP for the multi-photon polarization W state}
\begin{figure}[!h
\begin{center}
\includegraphics[width=10cm,angle=0]{ECPpolarization.eps}
\caption{The schematic drawing of the ECP for the three-photon polarization W state. The ECP also can be divided into two independent steps, which only requires local operations from Alice and Bob, respectively. In each step, the polarization beam splitters (PBSs) are used to transmit the $|H\rangle$ polarization photon state and reflect the $|V\rangle$ polarization photon state. The VBSs are used to adjust the entanglement coefficients.}
\end{center}
\end{figure}
Interestingly, with the basic principle in Sec. 2, we can still propose an efficient ECP for concentrating the partially entangled multi-photon polarization state. We take the three-photon W state as an example. The basic principle of the ECP is shown in Fig. 2. Suppose a single photon source S emits three photons and sends them to Alice, Bob and Charlie, respectively, which creates a partially entangled three-photon W state in a1, b1 and c1 modes as
\begin{eqnarray}
|\Psi_{1}\rangle_{a1b1c1}=\alpha|HVV\rangle_{a1b1c1}+\beta|VHV\rangle_{a1b1c1}+\gamma|VVH\rangle_{a1b1c1}. \label{initial2}
\end{eqnarray}
Here, $|\alpha|^{2}+|\beta|^{2}+|\gamma|^{2}=1$ and we also suppose $|\alpha|>|\beta|>|\gamma|$.
The ECP also can be divided into two steps. In the first step, Alice firstly maks the photon in the a1 mode pass through a polarization beam splitter (PBS), here named PBS1, which can transfer a $|H\rangle$ polarization photon and reflet a $|V\rangle$ polarization photon. After PBS1, $|\Psi_{1}\rangle_{a1b1c1}$ can evolve to
\begin{eqnarray}
|\Psi_{2}\rangle=\alpha|HVV\rangle_{a2b1c1}+\beta|VHV\rangle_{a3b1c1}+\gamma|VVH\rangle_{a3b1c1}.\label{PBS1}
\end{eqnarray}
Then Alice makes the photon in the a2 mode pass through a variable beam splitter (VBS1) with the transmittance of $t_{1}$. In this way, Eq. (\ref{PBS1}) can evolve to
\begin{eqnarray}
|\Psi_{2}\rangle=\alpha\sqrt{t_{1}}|HVV\rangle_{a4b1c1}+\alpha\sqrt{1-t_{1}}|HVV\rangle_{a5b1c1}+
\beta|VHV\rangle_{a3b1c1}+\gamma|VVH\rangle_{a3b1c1}.\label{PBS1-1}
\end{eqnarray}
After that, Alice detects the photon in the a5 mode by the single photon detector D1. If D1 detects no photon, Eq. (\ref{PBS1-1}) will collapse to
\begin{eqnarray}
|\Psi_{2}\rangle=\alpha\sqrt{t_{1}}|HVV\rangle_{a4b1c1}+\beta|VHV\rangle_{a3b1c1}+\gamma|VVH\rangle_{a3b1c1},\label{PBS1-2}
\end{eqnarray}
with the possibility of $|\alpha|^{2}t_{1}+|\beta|^{2}+|\gamma|^{2}$. Similar with Sec. 2, if $t_{1}=\frac{|\gamma|^{2}}{|\alpha|^{2}}$, Eq. (\ref{PBS1-2}) can be written as
\begin{eqnarray}
|\Psi_{2}\rangle\rightarrow\gamma|HVV\rangle_{a4b1c1}+\beta|VHV\rangle_{a3b1c1}+\gamma|VVH\rangle_{a3b1c1},\label{PBS1-3}
\end{eqnarray}
which only has two different entanglement coefficients $\beta$ and $\gamma$.
Finally, Alice makes the photon in the a3 and a4 mode pass through another PBS, here named PBS2. After PBS2, Eq. (\ref{PBS1-3}) evolves to
\begin{eqnarray}
|\Psi_{3}\rangle_{a6b1c1}=\gamma|HVV\rangle_{a6b1c1}+\beta|VHV\rangle_{a6b1c1}+\gamma|VVH\rangle_{a6b1c1}. \label{2-step1}
\end{eqnarray}
Until now, the first concentration step is completed and we successfully obtain the three-photon W state with only two different entanglement coefficients, with the success probability of $P_{1}=|\beta|^{2}+2|\gamma|^{2}$.
The second concentration step is operated by Bob alone, which is quite similar with the first concentration step. As shown in Fig. 2, by making the photon in the b1 mode pass through PBS3 and the photon in b2 mode pass through the VBS2 with the transmittance of $t_{2}$, Eq. (\ref{2-step1}) can ultimately evolve to
\begin{eqnarray}
|\Psi_{4}\rangle=\gamma|HVV\rangle_{a6b3c1}+\beta\sqrt{t_{2}}|VHV\rangle_{a6b4c1}+\beta\sqrt{1-t_{2}}|VHV\rangle_{a6b5c1}
+\gamma|VVH\rangle_{a6b3c1}.\label{2-PBS1}
\end{eqnarray}
Then, the photon in the b5 mode is detected by the single photon detector D2. Under the case that D2 detects no photon, Eq. (\ref{2-PBS1}) will collapse to
\begin{eqnarray}
|\Psi_{4}\rangle=\gamma|HVV\rangle_{a6b3c1}+\beta\sqrt{t_{2}}|VHV\rangle_{a6b4c1}+\gamma|VVH\rangle_{a6b3c1},\label{2-PBS2}
\end{eqnarray}
with the probability of $t_{2}$.
If $t_{2}=\frac{|\gamma|^{2}}{|\beta|^{2}}$, Eq. (\ref{2-PBS2}) can finally be written as
\begin{eqnarray}
|\Psi_{5}\rangle=\frac{1}{\sqrt{3}}(|HVV\rangle_{a6b3c1}+|VHV\rangle_{a6b4c1}+|VVH\rangle_{a6b3c1}).
\end{eqnarray}
Finally, Bob makes the photon in the b3 and b4 modes pass through the PBS4. After the PBS4, the three parties can share a maximally entangled polarization W state as
\begin{eqnarray}
|\Psi_{6}\rangle=\frac{1}{\sqrt{3}}(|HVV\rangle_{a6b6c1}+|VHV\rangle_{a6b6c1}+|VVH\rangle_{a6b6c1}).
\end{eqnarray}
The total success probability of the ECP also equals the product of the success probability in each concentration round, which is the same as that in Eq. (\ref{probability})
Similarly, by performing N-1 concentration steps described above, our ECP can also be extended to concentrate the partially entangled N-photon polarization W state as
\begin{eqnarray}
|\Psi_{N}\rangle=a_{1}|HVV\cdots V\rangle+a_{2}|VHV\cdots V\rangle+a_{3}|VVH\cdots V\rangle+\cdots+a_{N}|VVV\cdots VH\rangle,\label{initial-N2}
\end{eqnarray}
where $|a_{1}|^{2}+|a_{2}|^{2}+|a_{3}|^{2}+\cdots +|a_{4}|^{2}=1$, and $|a_{1}|>|a_{2}|>|a_{3}|>\cdots >|a_{N}|$. With the help of the PBSs and suitable VBS in each concentration step, Eq. (\ref{initial-N2}) can be finally recovered to the maximally entangled N-photon polarization W state as
\begin{eqnarray}
|\Psi'_{N}\rangle=\frac{1}{\sqrt{N}}(|HVV\cdots V\rangle+|VHV\cdots V\rangle+|VVH\cdots V\rangle+\cdots+|VVV\cdots VH\rangle),\label{initial-N2}
\end{eqnarray}
with the same success probability in Eq. (\ref{probability-N}).
\section{Discussion and summary}
\begin{figure}[!h
\begin{center}
\includegraphics[width=8cm,angle=0]{probability.eps}
\caption{The total probability P for obtaining a maximally entangled three-photon polarization W state of the ECPs in Ref \cite{shengPRA,zhoujosa,shengqip,zhouqip2} (curve A, B, C) and our paper (curve D), which is altered with the initial coefficient $\alpha$. Here, we choose $\beta=\frac{1}{\sqrt{3}}$. In the current ECP, we suppose $|\alpha|>|\beta|>|\gamma|$, so that we make $\alpha\in(\sqrt{\frac{1}{3}},\sqrt{\frac{2}{3}})$. As the ECPs in Ref \cite{shengPRA,zhoujosa,shengqip,zhouqip2} can be used repeatedly to further concentrate the partially entangled W state, we make curve A represents both the two steps are operated for one time, curve B represents both two steps are operated for three times, and curve C represents both the two steps are operated for five times. Curve D represents the ECP in our current paper.}
\end{center}
\end{figure}
In the paper, we propose two efficient ECPs for partially entangled multi-mode single photon W state and multi-photon polarization W state. Our ECPs only require the linear optical elements, among which the VBS is the key elements. We require the VBSs with suitable transmittance to adjust the entanglement coefficients and finally obtain the maximally entangled W state. Actually, the VBS is a common linear optical element in current experiment conditions. Recently, Osorio \emph{et al.} reported their results about the photon amplification for quantum communication with the help of the VBSs \cite{amplification}.
In their paper, they increased the probability $\eta_{t}$ of the single photon $|1\rangle$ from a mixed state as $\eta_{t}|1\rangle\langle1|+(1-\eta_{t})|0\rangle\langle0|$ with the help of the VBSs. In their experiment, they successfully adjust the transmittance of the VBSs from 50:50 to 90:10 to increase the visibility from 46.7 $\pm$ 3.1\% to 96.3 $\pm$ 3.8\%. This result ensures our ECP can be realized under current experimental conditions. Meanwhile, in our ECPs, each concentration step only requires local operation, which can simplify the experimental operation largely. On the other hand, in linear optics, when the photon is detected by the detector, it will be destroyed, which is well known as the post selection principle. In our ECPs, as each party only selects the case that the photon detector measures no photon, the generated maximally entangled W state will not be destroyed, and can be used in other applications.
Moreover, although our current ECPs can not be recycled, their success probability is higher than the previous ECPs for the W state \cite{shengPRA,zhoujosa,shengqip,zhouqip2}. Now, we will compare the success probability of our current ECPs with our previous ECPs for W state. We just take the three-photon polarization W state in Eq. (\ref{initial2}) as an example. In our previous papers \cite{shengPRA}, the ECPs also contain two concentration steps. In each step, we require an auxiliary single photon. Meanwhile, the ECPs can be used repeatedly to further concentrate the partially entangled W state. The success probability of the two concentration steps can be written as
\begin{eqnarray}
P_{N}^{1}&=&\frac{|\alpha|^{2^{N}}(|\beta|^{2^{N}-2}|\gamma|^{2}+2|\beta|^{2^{N}})}{(|\alpha|^{2^{N}}+|\beta|^{2^{N}})(|\alpha|^{2^{N-1}}
+|\beta|^{2^{N-1}})\cdots(|\alpha|^{2}+|\beta|^{2})},\nonumber\\
P_{M}^{2}&=&\frac{3\beta^{2^{M}}\gamma^{2^{M}}}{(\gamma^{2^{M}}+\beta^{2^{M}})(\gamma^{2^{M-1}}+\beta^{2^{M-1}})\cdots(\gamma^{2}+\beta^{2})}
\cdot\frac{1}{(\gamma^{2}+2\beta^{2})},
\end{eqnarray}
where the superscript "1" and "2" mean in the first and second concentration step, respectively. The subscripts "N" and "M" mean in the Nth and Mth concentration round.
Therefore, by repeating both steps, the total success probability is
\begin{eqnarray}
P_{total}&=&P_{1}^{1}(P_{1}^{2}+P_{2}^{2}+\cdots+P_{M}^{2}
+P_{2}^{1}(P_{1}^{2}+P_{2}^{2}+\cdots+P_{M}^{2})\nonumber\\
&+&\cdots +P_{N}^{1}(P_{1}^{2}+P_{2}^{2}+\cdots+P_{M}^{2})\nonumber\\
&=&\sum_{N=1}^{\infty}P^{1}_{N}\sum_{M=1}^{\infty}P^{2}_{M}.\label{total}
\end{eqnarray}
Here, we calculate the total success probability of both our current ECPs in Eq. (\ref{probability}) and the previous ECP in Eq. (\ref{total}) in Fig. 3. Here, we choose $\beta=\frac{1}{\sqrt{3}}$. In the current paper, we suppose $|\alpha|>|\beta|>|\gamma|$, so that we make $\alpha\in(\sqrt{\frac{1}{3}},\sqrt{\frac{2}{3}})$. In Fig. 3, curves A, B, and C represent the total success probability of the ECP in Ref. \cite{shengPRA}. Curve A represents that both the two steps are operated for one time. Curve B represents that both two steps are operated for three times. Curve C represents that both the two steps are operated for five times. Curve D represents the ECP in the current paper. It can be found that in both two ECPs, the success probability is largely altered with the initial entanglement coefficient $\alpha$. The higher initial entanglement can obtain the higher success probability. Moreover, although the success probability of the ECP in Ref. \cite{shengPRA} increases with the cycle times, it is still lower than that of our current ECPs. Especially, when $\alpha=\frac{1}{\sqrt{3}}$, the success probability of our current ECP is 1, while that of the ECP in Ref. \cite{shengPRA} can only obtain about 0.93, when both two concentration steps are operated for five times. Certainly, we can further increase its success probability by increasing its cycle times. However, by mathematical calculation, we can get when the ECP in Ref. \cite{shengPRA} is repeated indefinitely, its success probability curve (which will not be presented in Fig. 3) will be coincided with curve D. During our ECPs, the total success probability essentially is decided by the smallest coefficient of the initial state.
Actually, in the early theoretical work of concentration of the two-particle Bell state, Lo
and Popescu showed that the maximum probability
with which a Bell state can be obtained by purifying a single
entangled pair is twice the modulus square
of the Schmidt coefficient of smaller magnitude \cite{lo}. The result of the recent work of Deng's group is consist with Lo
and Popescu \cite{dengarxiv}. It reveals that the total entanglement is a conserved quantity. Interestingly, our ECPs can be regarded as the
extension of the result from the previous work of two-particle case, which can be concluded as the maximum probability of concentrating a N-particle partially entangled W
state or single-photon N-mode partially entangled state is Nth the modulus square of the Schmidt coefficient of the smallest magnitude.
In summary, we proposed two optimal ECPs for concentrating the single-photon multi-mode W state and N-photon polarization W state. In both ECPs, we only require one pair of partially entangled W state, and do not consume any auxiliary photon. In each concentration step, we mainly require the VBS to adjust the entanglement coefficients. Our ECPs have some obvious advantages. First, they only require the linear optical elements, which makes them can be easily realized under current experimental condition. Second, the generated maximally entangled W state will not be destroyed, and can be used in other applications. Third, our ECPs only need to be operated for one time, but they can obtain higher success probability than previous ECPs. Based on the advantages above, our ECPs may be useful in current quantum communication fields.
\section{Acknowledgements}
The project is supported by the National Natural Science Foundation of China
(Grant No. 11104159), the Open Research Fund Program of National
Laboratory of Solid State Microstructures, Nanjing University (Grant No. M25022), the Open Research Fund Program of the State Key Laboratory of
Low-Dimensional Quantum Physics, Tsinghua University (Grant No. 20120904), and the open research fund of Key Lab of Broadband Wireless Communication and Sensor Network Technology, Nanjing University of Posts and Telecommunications, Ministry of Education (No. NYKL201303), and the Priority Academic Program Development of Jiangsu Higher Education Institutions.
|
1,314,259,996,514 | arxiv | \section{Introduction}
Spectropolarimetry has been used in many fields as a diagnostic tool for deriving properties of astrophysical fluids. There are many types of spectropolarimetric models and observed polarized line profile morphologies across the HR diagram. The mechanisms invoked to explain these profiles include scattering, emission, absorption, atomic population imbalances (pumping) and potential modification of all these by magnetic fields.
In the context of circumstellar material, there are a few scattering models that are commonly invoked. The depolarization morphology where unpolarized line emission dilutes a polarized stellar continuum was developed in the Be star context and requires an asymmetric circumstellar envelope for creating continuum polarization via electron scattering and dilution of the polarized continuum by unpolarized line emission. There are line polarization effects that can arise from doppler shifts induced when scattering off thin circumstellar disks. The asymmetric circumstellar envelope and electron-scattering concept has been used to model or constrain Be circumstellar environments by fitting spectropolarimetric observations and also comparing polarization measurements to interferrometric measurements. There are also codes to produce synthetic polarized line profiles and continuum polarization for a hot star winds.
There have been a number of linear spectropolarimetric studies performed to date at spectral resolutions of a few to several thousand. Depolarization effects have been used in interpreting spectropolarimetric line profiles in many stars including Herbig Ae/Be, Be, B[e] and O-supergiants. The disk-scattering model was used to discuss various types of line profiles in TTauri and Herbig Ae/Be stars, O-type stars, as well as hot-massive stars. Wolf-Rayet stars have been observed and interpreted using these scattering frameworks as have massive post-red supergiants.
\begin{figure*}
\begin{center}
\includegraphics[width=0.35\linewidth, angle=90]{esp-halpha-espadonsswap-psiper-rebin.eps}
\includegraphics[width=0.35\linewidth, angle=90]{esp-hbeta-espadonsswap-psiper-rebin.eps} \\
\includegraphics[width=0.35\linewidth, angle=90]{esp-hgamma-espadonsswap-psiper-rebin.eps}
\includegraphics[width=0.35\linewidth, angle=90]{esp-hdelta-espadonsswap-psiper-rebin.eps}
\caption{The ESPaDOnS spectropolarimetry for $\psi$ Per in the H$_\alpha$ H$_\beta$ H$_\gamma$, H$_\delta$ lines for multiple epochs. \label{hbalmer} }
\end{center}
\end{figure*}
A review of the large number of papers outlining the scattering models and observations is beyond the scope of this proceedings, but some general features can be mentioned. Most of the observations presented in the observations were taken at low to moderate spectral resolution with various resolutions and binning procedures, with typically less than 10 independent polarization measurements across a spectral line. In addition, most studies use a few hours to a few nights worth of observations. In a series of papers, we outlined an H$_\alpha$ survey of many stars at high spectral resolution we have performed on over 100 observing nights (\citealt{har07, har08, har09a, har09b}). These observations found several examples of more complex line profile morphologies. Many of the Herbig Ae/Be observations showed polarization line profiles where only the absorptive component of a P-Cygni profile was polarized differently than the adjacent stellar continuum. The emission line polarization matched the continuum, ruling out the depolarization or typical scattering mechanisms.
\citealt{kuh07} introduced optical pumping (atomic alignment) as a mechanism for interpreting H$_\alpha$ spectropolarimetric line profiles from circumstellar material. This and related mechanisms are known in the solar community as atomic alignment (cf. \citealt{tru97}, \citealt{tru02}, \citealt{cas02}, \citealt{tru07}, \citealt{bel09}). It has also been called Zero-field Dichroism (\citealt{man03}). \citealt{ase05} apply this concept to circumstellar masers. Atomic alignment effects have been used in interpreting solar H$_\alpha$ observations (cf. \citealt{lop05}). A related computational method for a similar effect was explored by \citealt{yan06, yan07, yan08}. There the alignment of atoms by magnetic-field-modified emission is explored while the optical pumping effect of \citealt{kuh07} is independent of magnetic fields.
\section{The HiVIS Survey \& ESPaDOnS follow-up}
The High-resolution Visible and Infrared Spectrograph (HiVIS) is a spectropolarimeter on the 3.67m Advanced Electro-Optical System (AEOS) telescope with resolutions between 10,000 and 50,000. HiVIS has both achromatic wave plates and liquid crystal variable retarder (LCVR) full-Stokes modes and a dedicated IDL data reduction package (Harrington et al. 2006, Harrington \& Kuhn 2008, 2010). We have performed a $>$100 night survey with HiVIS and have many detections of linear polarization signals associated with H$_\alpha$ line profile. With high spectral resolution (R$>$10000) there are comparatively large amplitude (0.2-2\%) signals in Herbig Ae/Be, Be, PostAGB and other emission-line stars presented in \citealt{kuh07}, \citealt{har07, har08, har09a, har09b}. The absorptive polarization effects are ubiquitous in the H$_\alpha$ line of Herbig Ae/Be stars with roughly 2/3 of the stars showing this linear polarization signature. The classical Be stars more typically showed a broad `depolarization' morphology (10/30). About half of these show additional antisymmetric signatures in the absorptive part of the line profile (4/10) and another sub-set (5/30) show complex linear polarization spectral morphologies. The presence of absorptive polarization effects in most of the obscured stars in this survey suggests that the phenomenon may be present in any obscured star.
The follow-up observations for this project were performed with the 3.6m Canada France Hawaii Telescope (CFHT) using the ESPaDOnS spectropolarimeter with a nominal average spectral resolution of $R=\frac{\lambda}{\delta\lambda}=68000$ (cf. \citealt{don97, don99}). The Canadian Astronomy Data Centre (CADC) also provides archival access to all ESPaDOnS data which was searched for additional observations. All observations were reduced with the dedicated script, Libre-ESPRIT (\citealt{don97}).
The ESPaDOnS targets were known detections from \citealt{har09a} and \citealt{har09b} as well as supplementary archive observations. Our present sample includes 18 HAeBe stars, 8 Be stars, 9 Post-AGB stars and 8 other emission-line stars. We commonly find spectropolarimetric signatures in 15 lines at or above the 0.1\% amplitude: H$_\alpha$, H$_\beta$, H$_\gamma$, H$_\delta$, Na D lines, Ca NIR Triplet as well as Fe, He and Ox lines. As an example, Figure \ref{hbalmer} shows the H$_\alpha$ - H$_\beta$ - H$_\gamma$ - H$_\delta$ sequence for the Be star $\psi$ Per. There is substantial change in the line profile morphology and amplitude for each of the lines. Figure \ref{caNIR} shows examples of the Ca near-infrared triplet lines at 850nm, 854nm and 866nm. The detections are lower amplitude but the detections still show some morphological variations. With high spectral resolution and high precision detections across multiple lines, much more quantitative constraints on stellar environments have become possible.
\begin{figure*}
\begin{center}
\includegraphics[width=0.24\linewidth, angle=90]{esp-ca850-espadonsswap-mwc480-rebin.eps}
\includegraphics[width=0.24\linewidth, angle=90]{esp-ca854-espadonsswap-mwc480-rebin.eps}
\includegraphics[width=0.24\linewidth, angle=90]{esp-ca866-espadonsswap-mwc480-rebin.eps} \\
\includegraphics[width=0.24\linewidth, angle=90]{esp-ca850-espadonsswap-psiper-rebin.eps}
\includegraphics[width=0.24\linewidth, angle=90]{esp-ca854-espadonsswap-psiper-rebin.eps}
\includegraphics[width=0.24\linewidth, angle=90]{esp-ca866-espadonsswap-psiper-rebin.eps} \\
\caption{The spectropolarimetry for MWC 480 and Psi Per in the Ca near-infrared triplet lines 850, 854 and 866nm. \label{caNIR}}
\end{center}
\end{figure*}
\section{New Charge-Shuffling Detector}
For many targets it is calibration of the telescope and instrument systematic errors that limit the polarimetric precision of an observation. Moving optical elements, detector effects such as non-uniform sensitivity or nonlinearity, unstable optical beams, atmospheric seeing and transparency variations can all induce systematic errors that can be much larger than the photon noise. Efforts to stabilize the instrument and telescope, use non-moving optics, calibrate and characterize the system are essential when observing small amplitude signatures.
\begin{figure*} [!h, !t, !b]
\begin{center}
\includegraphics[width=0.35\linewidth, angle=90]{fig3.eps}
\includegraphics[width=0.35\linewidth, angle=90]{csd-data-extraction.eps}
\caption{ \label{rawcsd} The left hand panel shows a first-light Charge-Shifting Detector image in 'weave' mode of HR 3980. The right hand panel shows a calibration frame illustrating the extracted orders. The polcal stage is feeing in pure +Q. The LCVR's are set to measure $\pm$Q. The image is linearly scaled from 0 to 32,000 DN to show the bright orders as orange and the dark orders as blue/black. }
\end{center}
\end{figure*}
High-resolution night time astronomical spectropolarimeters have many common design elements and are almost always dual-beam systems with rotating achromatic retarders. Detector errors must be minimized by differencing signals on identical pixels given night-time observing constraints. Rotating achromatic retarders are placed before a calcite analyzer such as a Wollaston prism or a Savart plate. ESPaDOnS and Narval use two half-wave and one quarter-wave Fresnel rhombs before a Wollaston prism. Two other dual-beam instruments, PEPSI on the 8.4m LBT at R$\sim$310,000 and HARPS on the ESO 3.6m telescope at R$\sim$115,000 are in various stages of construction (\citealt{str03, str08}, \citealt{sni08, sni10}). The HARPS polarimetric package required the analyzer to be a Foster prism with separate quarter-wave and half-wave super-achromatic plates for circular and linear polarization that cannot be used simultaneously and must change between observations. The PEPSI design is similar to ESPaDOnS in that a lens collimates the beam before the retarders and a Wollaston analyzer. Instead of Fresnel rhomb retarders, a super-achromatic quarter-wave plate is chosen and linear polarization sensitivity is achieved only by physically rotating the entire polarization package. With HiVIS, a Savart plate is used as the analyzer after either rotating achromatic wave plates or LCVRs.
HiVIS is somewhat unique in that there are many oblique reflections in the optical path before the analyzer. The reflections cause quite severe cross-talk but the induced polarization and depolarization are almost always less than 5\%. The detected spectropolarimetric signatures are thus difficult to disentangle, but both detections and non-detections are significant and the spectropolarimetric morphology is preserved across an individual spectral line. We have found that effectively the cross-talk can be approximated well as a series of rotations in quv space.
\begin{figure} [!h, !t, !b]
\begin{center}
\includegraphics[width=0.32\linewidth, angle=90]{fig2e.eps}
\includegraphics[width=0.32\linewidth, angle=90]{fig2f.eps} \\
\includegraphics[width=0.32\linewidth, angle=90]{fig2c.eps}
\includegraphics[width=0.32\linewidth, angle=90]{fig2d.eps} \\
\caption{ \label{csdmodes} This Figure shows the first-light sequences for the new detector in 'weave' and 'smear' modes. Left side panels show the point-spread function while right side panels show a small region of the detector. The first row shows the first-light 'weave' routine where the spectra are shuffled by half the Savart plate separation to allow spectral orders to inter-lace. The second row shows the 'smear' routine where a 6-position loop is run during integration to spread charge to adjacent pixels to increase the effective well-depth of the pixels. }
\end{center}
\end{figure}
In an effort to remove systematic errors, many instruments use rapid modulation of the incident beam polarization state synchronous with charge motion the detector to remove time-dependent systematic effects. In some instruments, the temporal modulation is used to record multiple individual Stokes parameters at the same time. Other instruments use the temporal modulation to build up charge recording different polarization states on identical pixels. If this temporal shifting exceeds a kilohertz, then seeing errors can be minimized. For instance, the various incarnations of the ZIMPOL I, II and III solar imaging polarimeter have used piezo-elastic modulators or ferro-electric liquid crystals in combination with charge shuffling on a masked CCD to remove seeing induced systematic errors (cf. \citealt{gan04}, \citealt{pov01}, \citealt{ste07}). The Advanced Stokes Polarimeter (ASP) and La Palma Stokes Polarimeter (LPSP) are other notable polarimeters (c.f. \citealt{elm92}, \citealt{lit96}). This technique has been adapted for night-time spectropolarimetric use at the Dominion Astrophysical Observatory using ferro-electric liquid crystals and a fast-shuffling unmasked CCD (Monin, Private Communication).
We have adapted the two CCID20 arrays of the HiVIS detector to now include bi-directional parallel clocking synchronized with LCVR phase changes. The Pan-STARRS group has developed a detector controller hardware and software package called STARGRASP that allows for user control of the charge motion (cf \citealt{ona08}, \citealt{bur07}, \citealt{ton97, ton08}). The detector is oriented so the charge motion along the parallel clock corresponds effectively to the spatial direction of the recorded spectra. The Savart plate produces sufficient beam displacement between orthogonally polarized beams that a dekked slit length of 1.5`` allows the detection of four polarized spectral orders on the CCD with clear separation between beams. The new observing mode allows us to accumulate charge from two different LCVR settings in four beams on the same two groups of pixels within a single exposure.
\begin{figure*} [!h, !t, !b]
\begin{center}
\includegraphics[width=0.32\linewidth, angle=90]{hivis-csd-0525-polarization.eps}
\includegraphics[width=0.32\linewidth, angle=90]{hivis-csd-0525-polarization-residuals-paper.eps}
\caption{ \label{noiseavg} The left hand panel shows the detected polarization across a single spectral order averaged for 400 individual exposures. The right hand panel shows the measured residual polarization when 200 exposures are calibrated and averaged. }
\end{center}
\end{figure*}
The basic sequence is to record light for some time interval with one setting of the LCVRs, shuffle the charge to the unilluminated region of the detector, switch the LCVR voltages and to record light for another time interval. This loop is repeated until enough charge has accumulated that a detector readout is desired. We term this the 'weave' mode since the spectral orders are interlaced. Figure \ref{rawcsd} illustrates this observing mode. The left hand panel shows a full frame recording a stellar spectrum. The right hand panel shows a small region of the detector during a calibration exposure. The actively illuminated pixels are represented by beams a and b. The charge accumulated from these beams is shuffled down to the region labeled c and d while the LCVRs are switched. In this illustration, pure linear polarization was input with the HiVIS polarization calibration optics so that beams a and d have $\sim$30,000 counts while beams b and c only have $\sim$1,000 counts.
First-light with this detector was achieved in March of 2008. Figure \ref{csdmodes} shows the spatial profile (effectively the point-spread function) and a small region of the detector during a 'weave' sequence in the top row. The bottom row of this Figure shows what we call a 'smear' sequence where only two spectra are recorded but the effective well-depth of the pixels is increased by shuffling charge around during an exposure. This increases the efficiency of observing on bright targets by utilizing the detector area more efficiently and reducing the time lost to reading out the devices.
Rotating optics as well as seeing and telescope pointing drift change the illumination pattern on the detector. This combined with imprecise calibrations can lead to systematic errors that limit the precision of any measurement. For instance, Figure \ref{noiseavg} shows on the left the polarization calibrated calculated with LCVRs at a single voltage setting. Systematic errors at the 0.1\% level are present even over small wavelength ranges. However, when the calibrations are applied to differences on identical pixels, we achieve 10$^{-4}$ precision in a single spectral pixel with no systematics seen. This is shown in the right panel of Figure \ref{noiseavg} along with the calculated intensity spectrum.
\section{Summary}
There are many obscured stars across the HR diagram that show 0.1\% to over 1\% linear polarization signatures in many spectral lines of multiple atomic species at spectral resolutions above 10,000. We have performed a large survey of many stars in H$_\alpha$ and have followed up on these targets with ESPaDOnS to obtain detections across many spectral lines at 68,000 spectral resolution. In an effort to increase our precision and push the detection limit for linear spectropolarimetric signatures, we have added a new detector and new LCVR retarders to HiVIS. The detector has bi-directional clocking synchronized with the LCVRs to allow for a two order of magnitude decrease in phase switching time compared to large night-time astronomical spectropolarimeters. Precision in excess of 10$^{-4}$ has been demonstrated in the lab and first light with the instrument has been achieved. With this new mode, we plan to extend our HiVIS survey to include a high-precision component.
|
1,314,259,996,515 | arxiv | \section{INTRODUCTION}\label{sec:intro}
\vspace{-0.1cm}
A reliable determination of the valence-band offset (VBO) at the
(0001) polar interface between wurtzite AlN and GaN is still missing.
The few experimental investigations
available \cite{Waldrop.APL,Martin.APL68}
are in mutual disagreement, and theoretical studies
refer either to zincblende interfaces \cite{Albanesi.TechB},
or artificially lattice-matched wurtzite interfaces \cite{Wei}.
The latter approximation leads to a less accurate determination
for the VBO, and cannot pick up the possible forward-backward asymmetry
characteristic of lattice-mismatched interfaces.
In the case of the AlN/GaN interface, lattice mismatch amounts to
2.5 \%, and may cause a very large asymmetry. This asymmetry has
not yet been clearly determined experimentally (it
was not even found in early experimental work
\cite{Martin.APL65}), being
hidden by the large uncertainties in the measured data.
Even the best experimental investigations available face two
kinds of problem: (i) the determination of the
core level alignment with the valence-band maximum
(VBM) is obtained indirectly using theoretical estimates of the
VBM position, which (as underlined by Vogel {\it et al.} \cite{Vogel}) is affected by large systematic errors;
(ii) the existence of strong polarization fields
in both the substrate and the overlayer tends to modify the apparent
value of the VBO deduced from the core-level shift measurements.
The present {\it ab-initio} investigation includes all strain and
relaxation-induced effects, and overcomes the difficulty
in VBO determination due to polarization fields by the use
of a novel charge-decomposition technique. An estimate
of the formation energy of the interfaces studied
is also given.
\vspace{-0.5cm}\section{BULK PROPERTIES}
\vspace{-0.1cm}
Valence-band offset calculations at lattice-mismatched interfaces
require the evaluation of the band structure energies for the
bulk crystals in equilibrium, and subjected to biaxial strain.
The calculations are done using density-functional theory in
the local-density approximation (LDA) to describe the exchange-correlation
energy, and ultrasoft pseudopotentials \cite{Vanderbilt.PRB41}
for the electron-ion interaction. Plane-wave basis sets up to 25 Ry,
and 24 special {\bf k}-points
are found to give fully converged values for the bulk properties.
Since the properties of GaN are affected by Ga 3{\it d}
states \cite{Fiorentini.PRB},
our Ga pseudopotential includes 3{\it d} electrons in the valence.
This yields very good structural bulk
parameters (see below).
In wurtzite crystals, the determination of the atomic structure
at a given lattice constant {\it a}
implies the calculation of the {\it c/a} and {\it u} parameters.
The equilibrium $c$ as been determined by fitting with a polynomial
the total energy computed for six different values of {\it c},
with {\it u} being determined for each value of {\it c/a} via
minimization of the Hellmann-Feynman forces, with a threshold of
10$^{-4}$ Hartree/bohr.
The calculated structural parameters are given in Table \ref{tab:struct}.
The structural parameters of
AlN and GaN behave similarly under strain [$\Delta
({\it c/a})/({\it c/a}) \sim 7 \%$,
$\Delta {\it u}/{\it u} \sim 2 \%$] with similar total
energy variations. Instead, the effect of strain on
the valence-band edge is very different.
A rationale for this difference is that the
AlN (GaN) band edge is a singlet (doublet)
formed by the hybridization along the $c$-axis (in the $a$-plane)
of N 2{\it s} orbitals with Al {\it p}$_z$ (Ga {\it p$_{xy}$})
states, so that biaxial compression pushes the edge upward in GaN and
downward in AlN.
\begin{table}[t]
\begin{center}
\caption{Predicted structural parameters and valence band maxima
for equilibrium and
strained AlN and GaN.}
\vspace{0.2cm}
\begin{tabular}{|l|c|c|c|c|}\hline
Material & AlN & AlN & GaN & GaN \\\hline
Substrate & --- & GaN & --- & AlN \\\hline
{\it a} & 5.814 & 6.04 & 6.04 & 5.814 \\\hline
{\it c/a} & 1.619 & 1.51 & 1.6336 & 1.73 \\\hline
{\it u} & 0.38 & 0.3927 & 0.3761 & 0.3653 \\\hline
$E_{\rm strain}$ (eV)
& & +0.179 & & +0.155 \\\hline
$E_{\rm VBM}$ (eV)
&--0.16 & 0.09 &--4.90 &--4.69 \\\hline
\hline
\end{tabular}
\label{tab:struct}
\end{center}
\vspace{-0.5cm}
\end{table}
\vspace{-0.5cm}\section{BAND OFFSET}
\vspace{-0.1cm}
As pointed out by Baldereschi {\it et al.} \cite{Baldereschi.PRL}, the
valence-band offset $\Delta$ E$_v$ may be split in two terms:
\[
\Delta E_v = \Delta E_{\rm VBM} + \Delta V_{el}.
\]
The first contribution $\Delta E_{\rm VBM}$
is the difference between the valence-band edge energy
in the two bulk materials, each edge being referred to
the average bulk electrostatic potential.
The second contribution, the potential lineup $\Delta V_{el}$,
is the drop of the macroscopic average of the
electrostatic potential across the interface.
The latter term requires a selfconsistent calculation of the
electronic density distribution for the real interface system.
Our interface has been modeled using a (GaN)$_4$/(AlN)$_4$(0001)
superlattice (see below), both
ideal and fully relaxed. The
material being grown epitaxially on the chosen substrate, has been
pre-strained to have the same $a$ lattice constant as the substrate.
The lineup term is customarily obtained by solving the Poisson equation
for the macroscopic average of the charge density, neutralized by a
suitable distribution of gaussian charges centered on the ion sites.
The potential drop across the interface is usually calculated
as the difference of potential values in bulk-like regions
inside the two interfaced materials.
This turns out to be non-trivial
for a system such as the present one, in which the existence of
polarization fields in the equilibrium bulk makes it
impossible to define asymptotic values for the electrostatic
potentials. The existence of such fields, moreover, limits the maximum
length of our slab.
Indeed, beyond a certain critical thickness the drop of the
potential inside each slab would make the system metallic,
with a related transfer of charge, which
would spoil the exact determination of the lineup term.
Our choice of a 16-atom supercell is a compromise between the
need to have an insulating system, and at the same time to avoid
a spurious coupling of the interfaces at the sides of the slabs.
Tests were performed in supercells of up to 40 atoms.
In Fig. \ref{fig:scf}, we show the macroscopic average
of the charge density and of the electrostatic potential.
The potential drop is inextricably linked to the
polarization fields within the AlN and GaN bulks.
\begin{figure}[th]
\unitlength=1cm
\begin{center}
\begin{picture}(9,9.3)
\put(-2.0,-1.0){\epsfysize=11cm
\epsffile{./scf.eps}}
\end{picture}
\end{center}
\caption{Supercell electron density and electrostatic potential.
Electron density has been compensated in the two bulks by
a distribution of gaussians placed at the atomic sites.}
\label{fig:scf}
\end{figure}
We have circumvented this problem by employing a new method.
The basic idea is that at the polar AlN/GaN interfaces,
the existence of polarization fields reveals itself by an accumulation
of charge in the form of a {\it monopole} distribution
whose density is proportional to the difference between the
polarizations inside the two interfaced bulks.
On top of this monopole term, we have the traditional
{\it dipole} term representing the local charge transfer across the
interface.
This dipole term is the quantity we are interested in,
as the band offset is by definition related to the dipolar part of
the potential drop. Since the monopole contributions are related to the
polarization fields,
they must be equal and opposite for the two (geometrically inequivalent)
interfaces in our AlN/GaN superlattice.
To filter out the monopole term we superimpose the two interface
distributions by folding them around a plane placed halfway between
the two junctions. We define the dipole term $\bar{\bar{n}}^{dip}$ as
the average of the superimposed charges,
\[
\bar{\bar{n}}^{dip}(z-z_0) = \frac{1}{2}
\left[\bar{\bar{n}}(z-z_0) +
\bar{\bar{n}}(z_0-z) \right],
\]
where $z$ is a coordinate along the $c$-axis, $z_0$ the plane position
and $\bar{\bar{n}}$ the macroscopic average for the charge density.
The monopole term $\bar{\bar{n}}^{mono}$
is just the difference between the
dipole term and the total macroscopic charge:
\[
\bar{\bar{n}}^{mono}(z) = \bar{\bar{n}}(z) - \bar{\bar{n}}^{dip}(z).
\]
Such a decomposition allows a determination
of the polarization charges and dipole terms which is nearly
independent of the position of the folding plane.
Fig. \ref{fig:decomp} shows the decomposition for the AlN/GaN
interface.
\begin{figure}[th]
\unitlength=1cm
\begin{center}
\begin{picture}(9,9.3)
\put(-2.0,-1.0){\epsfysize=11cm
\epsffile{./monopole.eps}}
\end{picture}
\end{center}
\caption{Decomposition of the macroscopic average of the electronic
density (dotted line) into monopole (solid) and dipole (dashed) terms.
Such a decomposition allows the determination of the lineup term
$\Delta$ V$_{el}$ from the solution of the Poisson equation (dot-dashed)
of the dipole term.}
\label{fig:decomp}
\end{figure}
The decomposition reveals the origin of the asymmetry in the total
charge distribution,
and at the same time it enables us to evaluate the lineup term.
In Table \ref{tab:vbo} we report the values for the
VBO obtained via this decomposition.
\begin{table}[th]
\caption{Valence-band offset $\Delta$ $E_v$, potential lineup $\Delta$ $V_{el}$,
relaxation energies $E_{rel}$, monopole charge densities $\sigma_{int}$ and
electric fields $\vec{E}$ in the ideal
and relaxed the AlN/GaN (0001) interface.}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}\hline
Interface & \multicolumn{2}{c|}{AlN/GaN} & \multicolumn{2}{c|}{GaN/AlN} & units \\ \hline
structure & ideal & relaxed & ideal & relaxed& \\ \hline
$\Delta$ $E_v$ & 0.29 & 0.20 & 1.00 & 0.85 & eV \\ \hline
$\Delta$ $V_{el}$ & 5.28 & 5.18 & 5.52 & 5.36 & eV \\ \hline
$\sigma_{int}$& 0.029 & 0.014 & 0.022 & 0.011 & C/m$^{2}$ \\ \hline
$\vec{E}$& 32.7 & 15.6 & 24.4 & 12.9 & 10$^8$ V/m \\\hline
\hline
\end{tabular}
\end{center}
\vspace{-0.5cm}
\label{tab:vbo}
\end{table}
There is a very large forward-backward asymmetry of 0.65 eV
between AlN/GaN and GaN/AlN interface VBOs.
This
is only marginally due to the lineup term (contributing 0.18 eV),
its main component being the band structure term (0.47 eV).
The relaxation is responsible for comparatively small deviations
of $\sim$ 0.1 eV from the ideal-interface values.
The relaxation pattern is characterized in both cases
by a contraction of the Al-N axial interface bond ($\sim$ --0.04 a.u.)
and an expansion of the axial Ga-N bond ($\sim$ +0.02 a.u.).
\vspace{-0.5cm}\section{POLARIZATION}
\vspace{-0.1cm}
Supercell calculations are not the only way to obtain the
interface charge density $\sigma_{int}$.
As shown in Ref. \cite{Vanderbilt.PRB48},
given the polarization $P_1$ and $P_2$ and
the dielectric constants $\vareps_1$ and $\vareps_2$
of the component materials,
$\sigma_{int}$ is given by
\begin{equation}
\sigma_{int} = \pm ~2\, (P_2 - P_1)/(\vareps_1+\vareps_2).
\label{eq:sigma}
\end{equation}
in periodic boundary conditions. We have calculated the macroscopic
polarization for equilibrium and strained GaN and AlN via the Berry
phase technique of Ref. \cite{King.PRB47}.
The (high-frequency) dielectric constants of AlN and GaN have been calculated
using the relation
\[
\Delta P_{\rm T} = \vareps_{\infty} \Delta P_{\rm L},
\]
where $\Delta P_{\rm T}$ is the (so-called transverse) polarization
change induced by a
small cation sublattice displacement in the bulk
in zero field, and $\Delta P_{\rm L}$ is the (so-called longitudinal)
polarization change due to a uniform displacement of few cation
planes in a periodic bulk supercell. In the latter, a depolarizing
field is present due to the periodic boundary conditions.
As a by-product of our calculations, we obtained the
Born effective charges for AlN and GaN which, as expected for highly polar
semiconductors, are quite close to the nominal ionicity.
The results are shown in Table \ref{tab:pol}.
\vspace{-0.3cm}
\begin{table}[bh]
\caption{Polarization in GaN and AlN:
electronic $P_{el}$ and total $P_{tot}$ polarization,
derivative of the latter with respect to $u$, Born effective
charge, and dielectric constant are shown.}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline
System & $a$ & $P_{el}$&$P_{tot}$& $\partial P
/ \partial u$ & $Z^*$ &$\varepsilon_{\infty}$ \\\hline
----- & bohr & C/m$^2$ &C/m$^2$ & C/m$^2$ & $e$ & -----\\\hline
AlN & 5.814 &--0.178 &--0.0812 & 10.51 & 2.69 & 4.59 \\\hline
AlN & 6.04 &--0.492 &--0.1712 & 9.95 & 2.74 & 4.64 \\\hline
GaN & 5.814 & +0.223 & +0.0343 & 10.69 & 2.74 & 5.27 \\\hline
GaN & 6.04 &--0.0511 &--0.0308 & 9.88 & 2.72 & 5.52 \\\hline
\hline
\end{tabular}
\end{center}
\vspace{-0.5cm}
\label{tab:pol}
\end{table}
Substituting $P_{tot}$ and $\vareps$ in Eq.\ref{eq:sigma} we
obtain for the AlN/GaN (GaN/AlN) interface a monopole density
of 0.028 (0.023) C/m$^2$ in embarrassing agreement with the
outcomes of the supercell calculations. This proves directly that the
sources of the internal fields in the interface system are the
charges accumulated at the interface by the
bulk polarization effects. Also, the latter finding provides an a
posteriori justification of our somewhat ad-hoc charge decomposition
procedure.
\vspace{-0.5cm}\section{FORMATION ENERGIES}
\vspace{-0.1cm}
An important issue for the present selfconsistent calculation is the
evaluation of the formation energy for the AlN/GaN interfaces.
Contrary to the case of non-polar interfaces, for the wurtzite
(0001) system it is impossible to build a superlattice with symmetric
interfaces. This means that only the
average value of the formation energy for the two interfaces
can be obtained from a total-energy calculation.
We define the average formation energy for the AlN/GaN interface as
\[
E_f = \frac{1}{2A} \left[ E^{\rm tot} - n^{\rm Ga}\mu^{\rm GaN}
- n^{\rm Al}\mu^{\rm AlN}\right],
\]
where $\mu^X$ are the total energies per pair of GaN and AlN,
$n^{X}$ the number of Ga and Al atoms, $E^{\rm tot}$ the supercell
total energy and $A$ its cross-sectional area.
A reliable determination of $E_f$ requires equivalent
k-point sampling for bulk and interface calculations.
This is easily accomplished if the interface is lattice-matched.
In the present case, the supercell length is not simply an integer
multiple of the bulk unit cell of either constituent material.
This means that an exact equivalence between k-point meshes cannot
be achieved.
A good approximation for $E_f$
can however be obtained by defining, for each component material,
an auxiliary bulk cell having the same
lattice constant {\it a}, and an axial length
${\it\bar{c}}$ being a sub-multiple
of the supercell length ${\it l}$.
This value in the present case is just the average of ${\it c_{\rm
AlN}}$ and
${\it c_{\it GaN}}$. It is then possible to downfold exactly the
supercell mesh into the auxiliary bulk cell.
The next step is to uniformly scale the k-points coordinates
to adapt the mesh to the real value of $c$.
We should point out that the accuracy of this procedure (compared with
an exact computation of the energy integral over the IBZ) increases
with the number of points in the mesh. It is therefore
possible to find a suitable mesh to accomplish any required accuracy.
\vspace{-0.5cm}
\begin{table}[h]
\caption{Average formation energy for the AlN/GaN (0001) interfaces.}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}\hline
Interface & \multicolumn{2}{c|}{AlN/GaN} & \multicolumn{2}{c|}{GaN/AlN} & units \\ \hline
& ideal & relaxed & ideal & relaxed & \\ \hline
$E_f$ &--3.4 &--16.4 & +11.7 & --6.2 &meV \\ \hline
\hline
\end{tabular}
\end{center}
\label{tab:eform}
\end{table}
\vspace{-0.6cm}
The results for the formation energies reported in
Table \ref{tab:eform} have been obtained using
a 6-point Chadi-Cohen mesh \cite{Chadi.Cohen}
in the supercell, which when
downfolded in the auxiliary cell produces 24 special points.
We estimate the k-point sampling error in the formation energies
to be $\sim$ 10 meV.
It should be noted that such low formation energies are not
surprising when compared with the results obtained by Chetty
{\it et al.} \cite{Chetty.PRB} for GaAs/AlAs(111) interfaces.
\vspace{-0.6cm}
|
1,314,259,996,516 | arxiv |
\section{S1. Verification of Sampler Using Synthetic Data}
The figure below shows a bivariate density plot for rates and excitation probabilities learned using the sampler described in the main manuscript for synthetically generated data. The six rates that were used are 0.01, 0.1, 0.5, 2.0, 4.0, and 10.0 ns$^{-1}$, and the corresponding excitation probabilities are 0.2, 0.3, 0.1, 0.2, 0.1, and 0.1. These ground truth values are shown as orange dots in the plot. Our Markov-Chain Monte-Carlo sampler accurately predicts most of these rates and probabilities, however, rates that are not well separated in magnitude have much larger spread in their distributions. For instance, the learned distribution around the rates corresponding to 2.0 and 4.0 ns$^{-1}$ is much wider in the plot below.
\begin{figure}[h!]
\centering
\includegraphics[width =0.7 \textwidth]{bivariate_six_state.png}
\caption{\textbf{Robustness Test.} Bivariate density plot for a system
with six photophysical states with escape rates 0.01, 0.1, 0.5, 2.0,
4.0, and 10.0 ns$^{-1}$, and the corresponding excitation probabilities
are 0.2, 0.3, 0.1, 0.2, 0.1, and 0.1. The ground truth is shown as
dots. }
\label{ }
\end{figure}
\section{S2. Results Using Metropolis-Hastings Algorithm}
\begin{figure*}[h!]
\includegraphics[width=0.8\textwidth]{bivariate_plots_new.pdf}]
\caption{\textbf{Experimental Data Analysis Using Metropolis-Hastings.}
In row (a), we have three $256 \times 256$ pixels raster-scanned images
of samples containing three cubes labeled with ATTO 647N dyes, each.
The images on the left and the right are for single-dye cubes and
fluorocubes illuminated with 80\% maximum laser power, respectively.
However, the center image for large-cubes was produced using only 30\%
laser power. The laser power is varied in order to obtain approximately
equal photon numbers per probe per experimental run. Histograms for
photon arrival times (microtimes) recorded from these images are shown
in row (b). In row (c), each panel shows a bivariate posterior for
escape rates $\lambda_i$ (log scale) and corresponding probabilities
$\pi_i$ as in Eq.~5 of the main manuscript. We smooth out the bivariate
plots using the kernel density estimation (KDE) tool available in
Julia. The left panel shows the distribution for single-dye cubes. The
distribution is concentrated about $\lambda_i \approx 0.25$ ns$^{-1}$
or a lifetime of around $1/\lambda_i$ $= 4.0$ ns. $\pi_b \approx 0.10$
and $\pi_0 \approx 0.49$ in this case. The plot for large cubes in the
center panel shows no change in lifetime, suggesting insignificant
interactions. $\pi_b \approx 0.08$ and $\pi_0 \approx 0.37$ for these
cubes. In the rightmost panel for fluorocubes, the peak moves towards a
significantly shorter lifetime of around 1.54 ns. The background
probability $\pi_b$ here is 0.12, slightly higher than that of
non-interacting cubes, and $\pi_0 \approx 0.59$. }
\label{distributions}
\end{figure*}
\newpage
\section{S3. Results For Cy3 Fluorocubes}
\begin{figure*}[h!]
\includegraphics[width=0.65\textwidth]{bivariate_plots_cy3_new.pdf}
\caption{\textbf{Experimental Data Analysis.} Here, we show raster
scanned images (a), microtime histograms (b), and learned posterior
distributions (c) for probes labeled with Cy3 dyes. Each panel in row
(c) shows a bivariate posterior for escape rates $\lambda_i$ (log
scale) and corresponding probabilities, learned from a single
experimental run with samples containing multiple fluorescent probes
of the same type (fluorocubes or large cubes) in the field of view. In
the left panel, we show the distribution for large cubes and
illuminated with 85\% of the maximum laser power. The distribution is
concentrated about $\lambda_i \approx 0.465$ ns$^{-1}$ or a lifetime of
around $1/\lambda_i$ $= 2.15$ ns. $\pi_b \approx 0.03$ and $\pi_0
\approx 0.09$ in this case. In the second panel on the right, for
fluorocubes labeled with six dyes illuminated with 85\% laser power,
the peak moves towards a significantly shorter lifetime of around 1.31
ns. The background probability is about 0.15 here and 27 \% of the
pulses do not lead to any photon detections.
}
\label{distributions}
\end{figure*}
\newpage
\section{S4. Experimental Details}
\begin{table}
\scalebox{0.6}{
\begin{tabular}{ |c|c|c|}
\hline
\textbf{Name} & \textbf{Sequence and modification} & Vendor \\
\hline
FC\_SC\_01\_Cy3 & /5Cy3/ATGAGGTGTATGTGTAGAGTGATGGATGTAGT/3Cy3/ & IDT \\
\hline
FC\_SC\_02\_Cy3 & /5Cy3/AGGATGAGTGAGAGTGAGATGAGAGTAGATGT/3Cy3/
& IDT \\
\hline
FC\_St\_02\_Cy3 & /5Cy3/CACTCTCACACCTCATACATCTACCATCACTC/3Cy3/ & IDT \\
\hline
FC\_SC\_01\_ATTO647N & /5ATTO647NN/ATGAGGTGTATGTGTAGAGTGATGGATGTAGT/3ATTO647NN/
& IDT \\
\hline
FC\_SC\_02\_ATTO647N & /5ATTO647NN/AGGATGAGTGAGAGTGAGATGAGAGTAGATGT/3ATTO647NN/ & IDT \\
\hline
FC\_St\_02\_ATTO647N & /5ATTO647NN/CACTCTCACACCTCATACATCTACCATCACTC/3ATTO647NN/ & IDT \\
\hline
SD\_SC\_01\_UN & TTATGAGGTGTATGTGTAGAGTGATGGATGTAGTTT & IDT \\
\hline
SD\_SC\_02\_UN & TTAGGATGAGTGAGAGTGAGATGAGAGTAGATGTTT & IDT \\
\hline
SD\_St\_02\_ATT0647N & /5ATTO647NN/CACTCTCACACCTCATACATCTACCATCACTCTT & IDT \\
\hline
SD\_St\_02\_Cy3 & /5Cy3/CACTCTCACACCTCATACATCTACCATCACTCTT & IDT \\
\hline
FC\_St\_01\_Bio & Biotin TACACATACTCATCCTACTACATCTCTCATCTTT & IDT \\
\hline
LC\_SC\_01 & TTGAAAATTATCTCGATAAGCAGAAGGACCTGTATAACTGGCAAGAGACAAGGCCGCTTCAGAA & IDT \\
\hline
LC\_SC\_02 & AGGATAGCCGGACCGTATTAATGCCGCGCCAACGGTTTCCCGGACCTAGTGTCTATCAAGTCTA & IDT \\
\hline
LC\_SC\_03 & TTCTATGAAACCATTCTCGGGTCGAGCGGGTCACTGTTGTGACCTACGAGAAGCGTATAGATGT & IDT \\
\hline
LC\_SC\_04 & TCCGCGCGAATAGCTCACAGGCGAACTACGTATGAATTGGTTTAAACGCTCCTCGGGAATTAAT & IDT \\
\hline
LC\_SC\_05 & ACGACAGGTGGCAAACCACCTCCGATGTCAGCGCCGCATACCCATTCACTGTGAATTTCCACAC & IDT \\
\hline
LC\_SC\_06 & CGAGGATTCGCAGGTCCATGGGATTCACCAAGCTCGTATACACCCTGATTCTCCATGGCAGCGC & IDT \\
\hline
LC\_SC\_07 & GTAAGTTGAAGTAGGAAGCTTTTTCTAGCCATAGCATCGACACTACGACCTGCTTTTCGACACA & IDT \\
\hline
LC\_SC\_08 & GGACTGCATTCTGGACAGTAACTGCATTAACTACGTGCTCCCAACATAAGTGACGTCCTCAGCA & IDT \\
\hline
LC\_St\_01 & TTTGCTGAGGTGGAAATTTT & IDT \\
\hline
LC\_St\_02 & TTCCGGGAAACCGTTGGCCCTTCTGCTCGCCTGTCGTAGGTCGGGT & IDT \\
\hline
LC\_St\_03\_ATTO647N & GGTTCTGCGAATCCTCGGTGACGTCACTTCCTACTTCAACTTACTT/ATTO647N & IDT \\
\hline
LC\_St\_03\_Cy3 & GGTTCTGCGAATCCTCGGTGACGTCACTTCCTACTTCAACTTACTT/Cy3 & IDT \\
\hline
LC\_St\_04 & GTATGCGGCGCTCAGTTACTTCGTAGTGTCGATGCTTT & IDT \\
\hline
LC\_St\_05\_Bio & /5Biosg/TTCCATGGACTCATAGAATT & IDT \\
\hline
LC\_St\_06 & TTGAGAATCAACAACAGTTT & IDT \\
\hline
LC\_St\_07\_ATTO647N & TTACATCTATCACTAGGTTT/ATTO647N & IDT \\
\hline
LC\_St\_07\_Cy3 & TTACATCTATCACTAGGTTT/Cy3 & IDT \\
\hline
LC\_St\_08 & TTTACGTAGTTTATCGAGTT & IDT \\
\hline
LC\_St\_09\_ATTO647N & ATTO647N/TTCCGGCTATCCTTTCTGAAGCGGCCCCGAGGAGGAAT & IDT \\
\hline
LC\_St\_09\_Cy3 & Cy3/TTCCGGCTATCCTTTCTGAAGCGGCCCCGAGGAGGAAT & IDT \\
\hline
LC\_St\_10 & GTATACGAGCTTAAAAAGCTTATGTTGGGAGCACGTTT & IDT \\
\hline
LC\_St\_11 & TTTGCCAGTTATACAGGTGCGGCATTCGACCCGACGTTTAAAATGG & IDT \\
\hline
LC\_St\_12 & TTAGGTGGTTCGCGCGGATT & IDT \\
\hline
LC\_St\_13 & TTAGTTAATGGACATCGGTT & IDT \\
\hline
LC\_St\_14\_ATTO647N & ATTO647N/TTATAATTTTCAATAGACTTGATAGAACGCTTCTGAGCTATTTGCC & IDT \\
\hline
LC\_St\_14\_Cy3 & Cy3/TTATAATTTTCAATAGACTTGATAGAACGCTTCTGAGCTATTTGCC & IDT \\
\hline
LC\_St\_15 & TTGACCCGCTAATACGGTTT & IDT \\
\hline
LC\_St\_16\_ATTO647N & ACCTGTCGTGCGAAAGCAGGGTCCAGAATGCAGTCCTT/ATTO647N & IDT \\
\hline
LC\_St\_16\_Cy3 & ACCTGTCGTGCGAAAGCAGGGTCCAGAATGCAGTCCTT/Cy3 & IDT \\
\hline
LC\_St\_17 & TTATGGCTAGGGTGAATCTT & IDT \\
\hline
LC\_St\_18 & TTCACAGTGACCAATTCATT & IDT \\
\hline
LC\_St\_19 & TTTGTGTCGACTGCCATGTT & IDT \\
\hline
LC\_St\_20\_ATTO647N & TTATTAATTCTTGTCTCTTT/ATTO647N & IDT \\
\hline
LC\_St\_20\_Cy3 & TTATTAATTCTTGTCTCTTT/Cy3 & IDT \\
\hline
\end{tabular}}
\caption{Sequences for Assembly of DNA FluoroCubes and Large Cubes.}
\end{table}
\end{document}
|
1,314,259,996,517 | arxiv | \section{Introduction}
During the last two decades observational cosmology has entered an era of unprecedented precision. Cosmic microwave background (CMB) measurements~\cite{Bennett:2012zja, Ade:2013zuv}, baryon acoustic oscillations (BAO)~\cite{Lampeitl:2009jq} and observations of Type Ia Supernovae~\cite{Riess:2009pu} have shown very good agreement with the predictions of the standard cosmological model ($\Lambda$CDM) consisting of dark energy in the form of a cosmological constant $\Lambda$ and cold dark matter (CDM). However this agreement is not perfect: the Planck CMB data~\cite{Ade:2013zuv} are in tension with low redshift data such as cluster counts~\cite{Ade:2013lmv}, redshift space distortions (RSD)~\cite{Samushia:2012iq, Macaulay:2013swa}, weak lensing data~\cite{Heymans:2012gg} and local measurements of the Hubble constant, $H_0$~\cite{Riess:2011yx,Riess:2016jrr}. More specifically, the low redshift probes point towards a lower rate of structure growth (equivalently, a lower $\sigma_8$) than the Planck results for the base $\Lambda$CDM would prefer.
The most significant tensions are the ones coming from the cluster and weak lensing data.
A possible explanation for these tensions is that there are systematic effects that have not been accounted for, such as systematics that affect the determination of the mass bias in the cluster case, and small scales effects in the weak lensing case. Another possibility is that $\Lambda$CDM is not the correct model describing the evolution of the Universe. Finding a different model which gives a better (or, at the very least, equally good) fit to the data has proven very difficult. However, the motivation for exploring alternatives is strong because of the fundamental problems that plague $\Lambda$CDM, namely the \emph{fine-tuning} and \emph{coincidence} problems. These problems are associated with the cosmological constant and they have led to a plethora of alternative scenarios. One popular example is quintessence~\cite{Wetterich88, Ratra:1987rm, Wetterich:1994bg}, in which an evolving scalar field plays the role of dark energy. A different viewpoint suggests that General Relativity is modified on cosmological scales and this modification is responsible for the accelerated expansion of the Universe (see~\cite{Copeland:2006wr, CliftonEtal2011} and references therein).
The nature of the two constituents of the dark sector is currently unknown and the fact that they are considered uncoupled in $\Lambda$CDM is just an assumption of the model. Let us now consider a non-gravitational coupling between cold dark matter (CDM) and dark energy (DE). The energy momentum tensors for CDM and DE are then no longer separately conserved but instead we find
\begin{equation}
\nabla_\mu T^{{\rm (CDM)}\mu}_{\phantom{{\rm (CDM)}\mu} \nu} = J_\nu = -\nabla_\mu T^{{\rm (DE)}\mu}_{\phantom{{\rm (DE)}\mu} \nu},
\end{equation} where the coupling current $J_\nu$ represents the energy and momentum exchange between dark energy and dark matter. Note that we assume that the standard model is not coupled to the dark sector --- an assumption which well justified by observations that strongly constrain such couplings~\cite{Carroll:1998zi}.
Traditionally DE is modelled by a quintessence field $\phi$, so we are going to set ${\rm{DE}}\equiv \phi$ from now on, and ${\rm{cdm}}\equiv c$.
In the background, this means that the energy conservation relations become
\begin{align} \nonumber
\dot{\ensuremath{\bar{\rho}}}_c+3{\cal H}\ensuremath{\bar{\rho}}_c &= Q \\
\dot{\ensuremath{\bar{\rho}}}_\phi+3{\cal H}\ensuremath{\bar{\rho}}_\phi (1+w_\phi) &= -Q,
\end{align} where ${\cal H}=\dot{a}/a$ is the conformal Hubble rate with scale factor $a$ and a dot denotes the derivative with respect to conformal time $\tau$. $\ensuremath{\bar{\rho}}_c, \ensuremath{\bar{\rho}}_\phi$ are the energy densities of cold dark matter and dark energy (with the bar denoting background quantities) and $Q \equiv \bar{J}_0$ is the background energy transfer.
The form of $Q$ is usually chosen phenomenologically, and one of the most widely used forms is
$
Q = \alpha_0 \dot{\ensuremath{\bar{\phi}}} \ensuremath{\bar{\rho}}_c
$~\cite{Amendola:1999er}, where $\alpha_0$ is a constant parameter which determines the strength of the interaction.
This case has been extensively studied in the literature, see e.g.~\cite{Amendola:1999er, Xia:2009zzb, Xia:2013nua,Pourtsidou:2013nha} and references therein. Other widely used forms take $Q$ to be proportional to the energy densities of cdm and/or DE, i.e. $Q \propto \Gamma \rho $ with $\Gamma$ a constant interaction rate, or $Q \propto {\cal H} \rho$, see e.g.~\cite{CalderaCabral:2008bx, CalderaCabral:2009ja,Valiviita:2009nu, Jackson:2009mz,Clemson:2011an,Gavela:2009cy,Yang:2014gza,Salvatelli:2013wra, yang:2014vza, Murgia:2016ccp} and references therein.
Interest in interacting dark energy models has grown as they are arguably better motivated than the uncoupled $\Lambda$CDM concordance model, and they can possibly lift the tensions present in the available data. For example, an interaction between vacuum energy and dark matter was shown to be able to remove some of the tensions~\cite{Salvatelli:2014zta, Wang:2015wga}.
Very recently there have also been important theoretical developments in the field.
In \cite{DAmico:2016aa}, the authors formulate a quantum field theory of interacting dark matter - dark energy and show that the quantum corrections coming from the usual assumption for the form of the interaction (i.e. that dark matter is made up of ``heavy" particles whose mass depend on the dark energy field value) are huge and severely constrain the allowed couplings and the nature of the dark sector (see also \cite{Marsh:2016aa}).
However, in the majority of cases there is no Lagrangian description of the model, and an ad-hoc expression for the coupling $Q$ is just added at the level of the equations. In~\cite{Pourtsidou:2013nha} the authors used the pull-back formalism for fluids to generalise the fluid action and involve couplings between the scalar field playing the role of dark energy, and dark matter. Their construction led to three distinct \emph{families} of coupled models. The first two (Types 1 and 2) involve both energy and momentum transfer between dark matter and dark energy. The commonly used coupled quintessence model~\cite{Amendola:1999er} was shown to be a sub-class of Type 1~\cite{Pourtsidou:2013nha, Skordis:2015yra}. The third family of models (Type 3) is a pure momentum transfer theory with $Q=0$.
In this work we investigate a specific Type 3 model that was first presented in~\cite{Pourtsidou:2013nha}. We implement this model in the Einstein-Boltzmann solver \textsc{class}{}~\cite{Blas:2011rf}, perform a global fitting of cosmological parameters using the Markov chain Monte Carlo (MCMC) code~\textsc{MontePython}{}~\cite{Audren:2012wb}, and compare our findings to $\Lambda$CDM. The structure of the paper is as follows: In Section~\ref{sec:T3model} we present the specific Type 3 model we are going to study and state the background and linear perturbation cosmological equations which were derived in detail in~\cite{Pourtsidou:2013nha}. We also demonstrate the coupling's effects on the CMB and linear matter power spectra using \textsc{class}{}. In Section~\ref{sec:results} we describe the datasets and priors we use and then present the results of our MCMC analysis. We concentrate on the comparison of the Type 3 model with $\Lambda$CDM and the effects on the $(\sigma_8, H_0)$ parameters. We present the best fit cosmological parameters for $\Lambda$CDM and Type 3 and compare their $\chi^2$ values. We conclude in Section~\ref{sec:conclusions}.
\section{The pure momentum transfer model}
\label{sec:T3model}
As we have already stated, the model we are going to investigate belongs to the Type 3 class of theories constructed in~\cite{Pourtsidou:2013nha}. The distinctive characteristic of this class is that there is only momentum transfer between the two components of the dark sector. No coupling appears at the background level, regarding the fluid equations --- they remain the same as in the uncoupled case. Furthermore, the energy-conservation equation remains uncoupled even at the linear level. Hence, the theory provides for a pure momentum-transfer coupling at the level of linear perturbations.
Type 3 theories are classified via the Lagrangian~\cite{Pourtsidou:2013nha}
\begin{equation}
L(n,Y,Z,\phi) = F(Y,Z,\phi) + f(n),
\end{equation} where $n$ is the fluid number density, $Y=\frac{1}{2}\nabla_\mu \phi \nabla^\mu \phi$ is used to construct a kinetic term for $\phi$, and $Z=u^\mu \nabla_\mu \phi$ plays the role of a direct coupling of the fluid velocity $u^\mu$ to the gradient of the scalar field \footnote{
Note that the fluid velocity $u^\mu$ is uniquely defined in terms of the fluid number density $n$ and the dual number density (see \cite{Pourtsidou:2013nha} for details).}.
Considering a coupled quintessence function of the form
\begin{equation}
F=Y+V(\phi)+h(Z),
\end{equation} with $V(\phi)$ the quintessence potential, we have the freedom to choose the coupling function $h(Z)$ in order to construct specific models belonging to the same class. Following~\cite{Pourtsidou:2013nha}, we are going to concentrate on the sub-case with
\begin{equation}
h(Z) = \beta Z^2,
\end{equation} where $\beta$, the coupling parameter, is taken to be constant.
Defining $\tilde{g}^{\mu \nu}=g^{\mu \nu}+2\beta u^\mu u^\nu$, we can write the action for the scalar field $\phi$ as \cite{Pourtsidou:2013nha}
\begin{eqnarray}
\nonumber
S_{\phi}&=&
-\int d^4x \sqrt{-g}\left[
\frac{1}{2}\tilde{g}^{\mu \nu} \phi_\mu \phi_\nu
+ V(\phi)
\right]
\\
&\rightarrow &
\int dt \, d^3x \, a^3 \left[
\frac{1}{2}(1 - 2 \beta) \dot{\phi}^2
- \frac{1}{2} |\ensuremath{\vec{\nabla}}\phi|^2
- V(\phi)
\right] \;\;\;
\label{eq:action}
\end{eqnarray}
where the arrow denotes working in a frame where the CDM $3$-velocity is zero.
Hence, the model is physically acceptable for $\beta < \frac{1}{2}$. For $\beta \rightarrow 1/2$
we have a strong coupling problem, while for $\beta>1/2$ there is a ghost in the theory since the kinetic term becomes negative~\cite{Pourtsidou:2013nha}.
\subsection{Background Evolution}
We assume a Universe described by a flat Friedmann-Lema\^itre-Robertson-Walker (FLRW) metric
\begin{equation}
ds^2=a^2(\tau)(-d\tau^2+ dx_idx^i),
\end{equation} where $a(\tau)$ satisfies the Friedmann equation
\begin{equation}
{\cal H}^2 =\frac{8\pi G}{3}a^2\ensuremath{\bar{\rho}}_{\rm tot}
= \frac{8\pi G}{3}a^2(\ensuremath{\bar{\rho}}_\gamma + \ensuremath{\bar{\rho}}_b + \ensuremath{\bar{\rho}}_c + \ensuremath{\bar{\rho}}_\phi).
\end{equation} Here $\ensuremath{\bar{\rho}}_{\rm tot}$ is the total background energy density of all species and the subscripts $\gamma, b$ denote radiation and baryons, respectively.
The background energy density and pressure for quintessence are~\cite{Pourtsidou:2013nha}
\begin{eqnarray}
\ensuremath{\bar{\rho}}_{\phi}&=&\left(\frac{1}{2}-\beta\right)\frac{\dot{\ensuremath{\bar{\phi}}}^2}{a^2}+V(\phi), \\
\ensuremath{\bar{P}}_{\phi}&=&\left(\frac{1}{2}-\beta\right)\frac{\dot{\ensuremath{\bar{\phi}}}^2}{a^2}-V(\phi),
\end{eqnarray}
and the energy conservation equations are the same as in uncoupled quintessence:
\begin{eqnarray}
\label{eq:drhophiType3}
\dot{\ensuremath{\bar{\rho}}}_\phi &+& 3{\cal H}(\ensuremath{\bar{\rho}}_\phi+\ensuremath{\bar{P}}_\phi)=0, \\
\dot{\ensuremath{\bar{\rho}}}_c &+& 3{\cal H}\ensuremath{\bar{\rho}}_c=0,
\end{eqnarray} i.e. $Q=0$ for these models.
Hence the cold dark matter density obeys the usual scaling relation:
\begin{equation}
\ensuremath{\bar{\rho}}_c = \rho_{c,0} a^{-3}.
\end{equation}
In contrast to the usual coupled dark energy models which exhibit background energy exchange, the evolution of the CDM and quintessence energy densities
\begin{equation}
\Omega_c \equiv \frac{\ensuremath{\bar{\rho}}_c}{\ensuremath{\bar{\rho}}_{\rm tot}}, \;\;\; \Omega_\phi \equiv \frac{\ensuremath{\bar{\rho}}_\phi}{\ensuremath{\bar{\rho}}_{\rm tot}}
\end{equation} are unchanged in Type 3 models. The background Klein-Gordon equation is given by
\begin{equation}
\ddot{\ensuremath{\bar{\phi}}}+2{\cal H}\dot{\ensuremath{\bar{\phi}}}+
\left(\frac{1}{1-2\beta}\right)a^2\frac{dV}{d\phi}=0,
\end{equation}
and the speed of sound is $c^2_s=\frac{1-2\beta}{1-2\beta}=1$~\cite{Pourtsidou:2013nha}.
\subsection{Linear Perturbations}
In order to study the observational effects of the coupled models on the Cosmic Microwave Background and Large Scale Structure (LSS), we need to consider linear perturbations around the FLRW background. Choosing the synchronous gauge the metric is
\begin{eqnarray}
ds^2 &=& -a^2d\tau^2 + a^2 \left[ (1 + \frac{1}{3}h)\gamma_{ij} + D_{ij}\nu\right] dx^i dx^j,
\ \
\label{perturbed_metric}
\end{eqnarray}
where $\tau$ is the conformal time, $\ensuremath{\vec{\nabla}}_k$ is the covariant derivative associated with $\gamma_{ij}$, i.e. $\ensuremath{\vec{\nabla}}_k \gamma_{ij} = 0$
and $D_{ij}$ is the traceless derivative operator $D_{ij} = \ensuremath{\vec{\nabla}}_i \ensuremath{\vec{\nabla}}_j -\frac{1}{3} \ensuremath{\vec{\nabla}}^2 \gamma_{ij}$.
The unit-timelike vector field $u^\mu$ is perturbed as
\begin{eqnarray}
u_\mu &=& a( 1, \ensuremath{\vec{\nabla}}_i \theta).
\label{u_mu_theta}
\end{eqnarray}
We denote the field perturbation $\delta \phi$ by $\varphi$, the $\phi$-derivative of the potential by $V_\phi \equiv dV/d\phi$ and we also have $\ensuremath{\bar{Z}} \equiv -\dot{\ensuremath{\bar{\phi}}}/a$. In this notation the perturbed scalar field energy density and pressure are given by~\cite{Pourtsidou:2013nha}
\begin{align}
\delta \ensuremath{\rho_{\phi}} = -\frac{1}{a}\ensuremath{\bar{Z}} (1 - 2\beta) \dot{\varphi}
+ V_{\phi} \varphi, \\
\delta P_\phi = - \frac{1}{a} \ensuremath{\bar{Z}} \left(1 - 2\beta \right)\dot{\varphi} - V_{\phi}\varphi,
\end{align}
while the velocity divergence of the scalar field is
\begin{equation}
\ensuremath{\theta_{\phi}} =\frac{1}{(2\beta -1)\ensuremath{\bar{Z}}}\left(\frac{1}{a}\varphi+ 2\beta \ensuremath{\bar{Z}} \theta_c\right).
\label{eq:thetaphiType3}
\end{equation}
This is one other significant difference from other types of coupling, namely that $\ensuremath{\theta_{\phi}}$ depends also on the CDM velocity divergence $\theta_c$.
The linearised scalar field equation is~\cite{Pourtsidou:2013nha}
\begin{eqnarray}
&&
(1 - 2\beta) (\ddot{\varphi} + 2 \ensuremath{{\cal H}} \dot{\varphi})
+ \left(k^2 + a^2 V_{\phi\phi} \right) \varphi
\nonumber
\\
&&
\ \ \ \
+ \frac{1}{2} \dot{\ensuremath{\bar{\phi}}} (1 - 2\beta) \dot{h}
- 2\beta \dot{\ensuremath{\bar{\phi}}}k^2 \theta_c
= 0,
\label{eq:pertKG}
\end{eqnarray}
while the density contrast $\delta_c \equiv \delta \rho_c / \ensuremath{\bar{\rho}}_c$ obeys the standard evolution equation
\begin{equation}
\label{eq:deltac}
\dot{\delta}_c = -k^2 \theta_c - \frac{1}{2}\dot{h}.
\end{equation}
The momentum-transfer equation depends on the coupling and is given by
\begin{equation}
\dot{\theta}_c = - \ensuremath{{\cal H}} \theta_c +
\frac{( 6 \ensuremath{{\cal H}} \beta \ensuremath{\bar{Z}} + 2\beta \dot{\ensuremath{\bar{Z}}} ) \varphi + 2\beta \ensuremath{\bar{Z}} \dot{\varphi}}{a \left(\ensuremath{\bar{\rho}}_c - 2 \beta \ensuremath{\bar{Z}}^2 \right)} .
\label{dP_type3_pert}
\end{equation}
We implemented the above equations in \textsc{class}{} in order to compute the CMB temperature and matter power spectra. At this stage we will fix our quintessence potential $V(\phi)$ to be the widely used single exponential form (1EXP)
\begin{equation}
V(\phi)=V_0 e^{-\lambda \phi}.
\end{equation} In order to demonstrate the effect of the coupling before we perform an MCMC analysis, we will first compare the uncoupled $\beta = 0$ case with
the coupled model under consideration keeping the parameter $\lambda$ of the potential fixed ($\lambda=1.22$), while the potential normalisation $V_0$ is varied automatically by \textsc{class}{} is order to match a fixed $\Omega_\phi$ today. Our initial conditions for the quintessence field are $\phi_i=10^{-4}, \dot{\phi}_i=0$. Note, however, that in this case and using the 1EXP potential, the same cosmological evolution is expected for a wide range of initial conditions.
Before we present our results, we should briefly discuss the range of $\beta$ values we are going to consider. As we have already mentioned, it is clear from the Lagrangian in Equation~(\ref{eq:action}) that we are not allowed to consider the case $\beta > 1/2$ due to the strong coupling and ghost pathologies. We are free to consider any negative value $\beta$, but the fact that $\beta$ is a dimensionless coupling parameter in the Lagrangian suggests that its magnitude should be small. We will therefore initially choose a prior for our negative $\beta$ values such that $-0.5 \leq \beta \leq 0$ and call this model T3. However, as we shall see later the data allow for the model to span a much wider range of negative $\beta$, so in this case we will use a log prior $-3 \leq {\rm Log}_{10}(-\beta) \leq 7$ and denote this (phenomenological) Type 3 model $\text{T3}_\text{ph}$.
\begin{figure*}[tb]
\includegraphics[width=0.495\linewidth]{CoupledQuintessenceCLTT.pdf}
\hfill
\includegraphics[width=0.495\linewidth]{CoupledQuintessencePk.pdf}
\caption{Comparison of the CMB TT power spectra (left panel) and the linear (total) matter power spectra $P(k)$ at $z=0$ (right panel). Black solid lines denote the uncoupled quintessence model, blue dotted lines denote the coupled model with positive coupling parameter $\beta=0.1$, and dashed red lines denote the coupled model with negative coupling parameter $\beta=-0.5$.}
\label{fig:ClTT}
\end{figure*}
In Fig.~\ref{fig:ClTT} we show the CMB temperature spectra and the matter power spectra for the uncoupled case $\beta=0$, for a positive coupling parameter $\beta=0.1$, and for $\beta=-0.5$. We see that the effect of the coupling in the CMB temperature power spectrum is very small. There is no visible effect on the small scale amplitude and no shift of the location of the peaks, like in usual coupled quintessence models (\cite{Amendola:1999er, Xia:2009zzb, Xia:2013nua}). The only visible effect is an integrated Sachs-Wolfe (ISW) effect on large scales, but it remains small in contrast to coupled quintessence models with background energy exchange. This is because the background CDM energy density remains uncoupled in our model.
However, the effects on the matter power spectrum $P(k)$ are significant. For the positive coupling case there is enhanced growth at small scales (and, consequently, a larger value of $\sigma_8$ is obtained), while for the negative coupling T3 case the growth is suppressed and the associated $\sigma_8$ is smaller. Since the positive coupling model results in enhanced growth, it aggravates the tension between the Planck CMB data and low redshift observations. We will therefore exclude the $0<\beta<1/2$ branch from our MCMC analysis.
We will now show the comparison between the $\text{T3}_\text{ph}$ model - which spans a wide range of negative $\beta$ values - and the uncoupled quintessence case. In order to highlight the effect of $\beta$ we fix the sound speed at recombination $\theta_s$ and the physical energy densities of CDM and baryons, $\omega_{c,b}=\Omega_{c,b}h^2$. In Fig.~\ref{fig:ClTT_T3ph} we plot the CMB TT power spectrum and the matter power spectrum divided by their uncoupled counterparts. Similar to the T3 model, we see that the effect of the coupling on the TT power spectrum only shows up on very large scales. This is due to the late-time ISW effect which changes because of the effect of the coupling on the evolution of the matter density perturbations. We also notice that the effect changes direction (from enhancement to suppression with respect to the uncoupled case) around $\beta \simeq-10^2$.
In the matter power spectrum we also see an interesting feature, namely that for $\beta<-10^2$ there is a range of $k$-values where the matter power spectrum is actually~\emph{enhanced} compared to the uncoupled case. This gives $\sigma_8(\beta)>\sigma_8(\beta=0)$ for $\beta\lesssim-10^4$. Note that we are comparing models that reproduce the same TT-power spectrum at small scales according to the left panel of Fig.~\ref{fig:ClTT_T3ph}, and these models have slightly different values of $H_0$.
\begin{figure*}[tb]
\centering
\includegraphics[width=0.495\linewidth]{cl_ratio_beta.pdf}
\hfill
\includegraphics[width=0.495\linewidth]{pk_ratio_beta.pdf}
\caption{Comparison of the CMB TT power spectra (left panel) and the linear (total) matter power spectra at $z=0$ (right panel) for a wide range of negative $\beta$-values. In both cases we show the ratio between the $\text{T3}_\text{ph}$ model and the predictions of the uncoupled quintessence model.}
\label{fig:ClTT_T3ph}
\end{figure*}
For completeness, we also show the evolution of the dark energy equation of state parameter $w_\phi = p_\phi /\rho_\phi$ in Fig.~\ref{fig:wbeta}. We see that as the magnitude of the $\beta$ parameter increases, $w_\phi$ gets closer to $-1$, i.e. the background dark energy evolution resembles that of a cosmological constant. That is because of the effects of the coupling on the evolution of $\ensuremath{\bar{\phi}}$: the $\beta \dot{\ensuremath{\bar{\phi}}}^2/a^2$ term becomes completely subdominant to $V(\phi)$ and $w_\phi \rightarrow -1$.
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{w_beta_log.pdf}
\caption{ The evolution of the dark energy equation of state parameter $w_\phi = \ensuremath{\bar{P}}_\phi /\ensuremath{\bar{\rho}}_\phi$ as a function of the coupling parameter $\beta$. A constant $w_\phi = -0.9$ is shown for comparison. }
\label{fig:wbeta}
\end{figure}
Before we move on to our full MCMC analysis, we should stress that the effect of growth suppression in coupled dark energy models is quite rare and difficult to achieve.
The same is true for modified gravity models, the vast majority of which exhibit growth increase \footnote{This is easy to understand as such models result to an enhanced effective gravitational constant $G_{\rm eff}$ with respect to Newton's constant. However, a recent study of theories beyond Hordenski showed that there is the possibility of constructing stable and ghost-free weak gravity scenarios~\cite{Tsujikawa:2015mga}.
} and make the tension worse as they favour a large $\sigma_8$ value~\cite{Hu:2015rva}.
In coupled dark energy models that exhibit both energy and momentum exchange, like the coupled quintessence model with $Q=\alpha_0 \dot{\bar{\phi}}\ensuremath{\bar{\rho}}_c$~\cite{Amendola:1999er}, the growth rate depends on two terms: a fifth-force contribution $\propto \alpha^2_0$, and a friction term $\propto \alpha_0 \dot{\ensuremath{\bar{\phi}}}$, whose sign is determined by the sign of the coupling parameter $\alpha_0$ and the quintessence potential through the $\dot{\ensuremath{\bar{\phi}}}$ dependence.
In general, getting suppression of growth is highly non-trivial, and often requires potentials with more than one free parameters in order to make the friction term dominate over the competing fifth-force term, which tends to give growth increase for positive and negative coupling values (see~\cite{Tarrant:2011qe} for details and further discussion).
On the contrary, our pure momentum transfer model has a straightforward behaviour for the simplest case of the 1EXP potential: Considering the positive coupling Type 3 model and the negative coupling T3 model we
get growth increase for the former case and suppression of growth for the latter.
This effect comes from the modified density contrast evolution due to the presence of the momentum transfer coupling, and there is no competing coupling term in Equation~(\ref{eq:deltac}).
For the phenomenological $\text{T3}_\text{ph}$ model we also get growth suppression for a very wide range of negative $\beta$, but due to the scale dependent feature demonstrated in Fig.~\ref{fig:ClTT_T3ph} we also have a turning point where growth and $\sigma_8$ start to increase. For these reasons we chose to perform our MCMC analysis on the T3 and $\text{T3}_\text{ph}$ models separately.
\section{Results}
\label{sec:results}
\subsection{Datasets and priors}
\begin{figure*}[tb]
\centering
\includegraphics[width=0.495\linewidth]{plot1G.pdf}
\hfill
\includegraphics[width=0.495\linewidth]{plot3G.pdf}
\caption{$1\sigma$ and $2\sigma$ constraints for $\Lambda$CDM and the T3 and $\text{T3}_\text{ph}$ models in the $(\sigma_8,\Omega_M)$-plane using TT data (left panel) and using all data (right panel). The $1\sigma$-band from SZ clusters and CFHTLenS is shown in dotted green and solid red, respectively. $\Lambda$CDM is in tension with the SZ clusters and CFHTLenS while the T3 and $\text{T3}_\text{ph}$ models are compatible. Even after including the SZ-data, the $2\sigma$ $\Lambda$CDM contour does not overlap with the $1\sigma$ SZ contour, which illustrates the tension. On the contrary, the T3 and $\text{T3}_\text{ph}$ contours overlap with the SZ contour.}
\label{fig:plot3}
\end{figure*}
In our analysis we use a variety of recent datasets, including CMB data, baryon acoustic oscillation measurements, supernovae, and cluster counts. The specific likelihoods we employ are:
\begin{description}[leftmargin=4em,style=nextline]
\item[TT:] $C_\ell^{TT}$-data from Planck 2015~\cite{Ade:2015xua}, including low-$\ell$ polarisation.
\item[CMB:] TT and the lensing reconstruction from Planck 2015 data~\cite{Ade:2013tyw}.
\item[B:] Baryon Acoustic Oscillation (BAO) data from BOSS~\cite{Anderson:2013zyy}.
\item[J:] Joint Light-curve Analysis (JLA)~\cite{Betoule:2014frx}.
\item[SZ:] Planck SZ cluster counts~\cite{Ade:2013lmv,Ade:2015fva}.
\end{description}
We did not include weak lensing data in the MCMC run --- however, we show the constraint $\sigma_8\left(\Omega_M/0.27 \right)^{0.46} = 0.774 \pm0.040$ derived from CFHTLenS~\cite{Heymans:2013fya} in Fig.~\ref{fig:plot3}. Note that this constraint has been inferred assuming the $\Lambda$CDM model.
We must also comment on the applicability of the cluster count likelihood we use~\cite{Ade:2013lmv}. This likelihood assumes a mass bias $(1-b)\simeq 0.8$ that agrees well with simulations, but the authors note that a significantly lower value would alleviate the CMB-SZ tension. It also uses the Tinker et al halo mass function~\cite{Tinker:2008ff}, which has been calibrated against N-body simulations assuming $\Lambda$CDM. This implies that including this likelihood is not entirely self-consistent, see for instance Ref.~\cite{Cataneo:2014kaa} for a discussion of the similar problem in $f(R)$ gravity. Making quantitative predictions for the non-linear effects of our model is not possible at this stage. However, a very recent paper on structure formation simulations with momentum exchange showed that the qualitatively similar dark scattering model~\cite{Simpson:2010vh} can alleviate the CMB-LSS tensions while keeping non-linear effects very mild~\cite{Baldi:2016zom}. Running a suite of N-body simulations for a coupled quintessence model with pure momentum exchange is the subject of future work.
We chose flat priors on the following set of cosmological parameters,
\begin{equation}
\left\{\omega_b, \omega_\text{cdm}, \theta_s, A_s, n_s, \tau_\text{reio}, \lambda\right\},
\end{equation}
and the collection of nuisance parameters required by the Planck and JLA likelihoods. The prior ranges of $\lambda$ and $\beta$ were chosen as
\begin{align}
\lambda &\in [0; 2.1], & \beta &\in [-0.5,0],
\intertext{for the T3 model with a flat prior on $\beta$ and }
\lambda &\in [0; 2.1], & {\rm log}_{10}(-\beta) &\in [-3,7],
\end{align}
for the $\text{T3}_\text{ph}$ model with a logarithmic prior on $\beta$ as indicated. As we have already mentioned, we exclude the $\beta>0$ branch from our analysis since we want to focus on the branch which can provide suppression of growth.
The initial conditions for the quintessence field were chosen as $(\phi_i, \dot{\phi}_i)=(10^{-4},0)$.
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{plot2bG.pdf}
\caption{$1\sigma$ and $2\sigma$ temperature (TT) constraints in the $(\sigma_8,H_0)$-plane for $\Lambda$CDM and the T3 and $\text{T3}_\text{ph}$ models. The two parameters are uncorrelated in $\Lambda$CDM and $\text{T3}_\text{ph}$ while they are correlated in the T3 model}.
\label{fig:plot2b}
\end{figure}
\subsection{Parameter inference}
We perform a Markov chain Monte Carlo (MCMC) analysis using the publicly available code~\textsc{MontePython}{}.
In the left panel of \fref{fig:plot3} we show the constraint from TT data alone in the $(\Omega_M,\sigma_8)$-plane for $\Lambda$CDM and the T3 and $\text{T3}_\text{ph}$ models. The $2\sigma$-contours of the T3 and $\text{T3}_\text{ph}$ models overlap with the $1\sigma$-constraint from SZ clusters and CFHTLenS whereas the $\Lambda$CDM model is in tension with both SZ and CFHTLenS data.
\begin{figure*}[tb]
\centering
\includegraphics[width=0.495\linewidth]{plot4.pdf}
\hfill
\includegraphics[width=0.495\linewidth]{plot4G.pdf}
\caption{One dimensional posterior distributions of the parameters $\{\sigma_8, H_0, \beta, \lambda\}$ excluding (solid black lines) and including (magenta dashed lines) the SZ cluster data along with the rest of our datasets. In the left panel we show the T3 model while the right panel shows the $\text{T3}_\text{ph}$ model.}
\label{fig:plot4}
\end{figure*}
When we combine all our datasets including the SZ cluster data we find quite different $(\Omega_M,\sigma_8)$-constraints as illustrated in the right panel of \fref{fig:plot3}. The T3 and $\text{T3}_\text{ph}$ models generally prefer lower $\sigma_8$ and larger $\Omega_M$-values compared to $\Lambda$CDM. As we shall see in \sref{sec:chisq} the T3 and $\text{T3}_\text{ph}$ models are significantly better fits to the data than $\Lambda$CDM. In \fref{fig:plot2b} we show the $(\sigma_8,H_0)$-constraints. In the T3 model, $\sigma_8$ and $H_0$ become strongly correlated, but in $\Lambda$CDM and $\text{T3}_\text{ph}$ no correlation exists. As we will see later, this difference between T3 and $\text{T3}_\text{ph}$ has important implications for the $H_0$ tension.
In \fref{fig:plot4} we show the effect of including the SZ cluster data or not on the parameters $\{\sigma_8, H_0, \beta, \lambda\}$ for the T3 model (left panel) and $\text{T3}_\text{ph}$ model (right panel). Let us first comment on the T3 case: Taken at face value, the cluster data rules out the non-interacting case, $\beta=0$. Looking at the posterior distribution for $\lambda$ reveals that the non-cluster datasets roughly prefer $\lambda < 1$ while we have $\lambda > 1$ after the inclusion of the cluster data. This suggests that another potential could give an even better fit to the data than the single exponential potential (but would probably rely on additional free parameters). The T3 posterior distribution for $\beta$ suggests that a better fit can possibly be found if we let $\beta < -1/2$. Moving on to the $\text{T3}_\text{ph}$ case we verify this as the posterior distribution peaks at $\beta=-10^4$.
We also note that the $H_0$ distributions for $\Lambda$CDM and $\text{T3}_\text{ph}$ are very similar. In $\Lambda$CDM it is well known that the inclusion of SZ cluster data drives $H_0$ to larger values. In the T3 model the opposite happens, the cluster data pushes $H_0$ to lower values. We illustrate these effects in \fref{fig:plot5}.
This does put the T3 model in mild tension with local measurements of the Hubble constant but not much more than pure $\Lambda$CDM. However, as we already saw in the $\text{T3}_\text{ph}$ model the $H_0$ posterior distribution is very similar to the $\Lambda$CDM one.
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{plot5.pdf}
\caption{One dimensional posterior distributions of the Hubble parameter. The inclusion of SZ clusters drives $H_0$ to larger values in $\Lambda$CDM and lower values in T3.}
\label{fig:plot5}
\end{figure}
In Table~I we show the mean values and $1\sigma$ confidence intervals on the cosmological parameters for various datasets combinations for the $\Lambda$CDM, T3, and $\text{T3}_\text{ph}$ models. It is important to note that $\beta$ is essentially not constrained unless we include the SZ cluster data in the analysis.
\begin{table*}
\caption{\label{tab:parameters}Cosmological parameters for $\Lambda$CDM and the T3-model for the 4 different data set combinations used in the plots. Note that $\beta$ is essentially unconstrained unless cluster data is added.}
\begin{ruledtabular}
\begin{tabular}{l|rrr|rrr|rrr|}
& \multicolumn{3}{c|}{CMB} & \multicolumn{3}{c|}{CMB+B+J} & \multicolumn{3}{c|}{CMB+B+SZ+J}\\
& $\Lambda$CDM & T3 & $\text{T3}_\text{ph}$ & $\Lambda$CDM & T3 & $\text{T3}_\text{ph}$ & $\Lambda$CDM & T3 & $\text{T3}_\text{ph}$\\ \hline
$100~\omega_{b }$ & $2.22_{-0.02}^{+0.02}$ & $2.22_{-0.02}^{+0.02}$ & $2.22_{-0.02}^{+0.02}$ & $2.23_{-0.02}^{+0.02}$ & $2.23_{-0.02}^{+0.02}$ & $2.23_{-0.02}^{+0.02}$ & $2.24_{-0.02}^{+0.02}$ & $2.25_{-0.02}^{+0.02}$ & $2.23_{-0.02}^{+0.02}$\\
$\omega_\text{cdm}$ & $0.119_{-0.002}^{+0.002}$ & $0.119_{-0.002}^{+0.002}$ & $0.119_{-0.002}^{+0.002}$ & $0.118_{-0.001}^{+0.001}$ & $0.117_{-0.001}^{+0.001}$ & $0.118_{-0.001}^{+0.001}$ & $0.116_{-0.001}^{+0.001}$ & $0.116_{-0.001}^{+0.001}$ & $0.118_{-0.001}^{+0.001}$\\
$10^4\theta_s$ & $104.20_{-0.04}^{+0.04}$ & $104.20_{-0.04}^{+0.04}$ & $104.20_{-0.04}^{+0.04}$ & $104.21_{-0.04}^{+0.04}$ & $104.21_{-0.04}^{+0.04}$ & $104.21_{-0.04}^{+0.04}$ & $104.20_{-0.04}^{+0.04}$ & $104.22_{-0.04}^{+0.04}$ & $104.21_{-0.04}^{+0.04}$\\
$10^9 A_s$ & $2.15_{-0.07}^{+0.06}$ & $2.16_{-0.07}^{+0.06}$ & $2.16_{-0.07}^{+0.06}$ & $2.18_{-0.06}^{+0.05}$ & $2.19_{-0.06}^{+0.05}$ & $2.18_{-0.06}^{+0.05}$ & $2.08_{-0.05}^{+0.05}$ & $2.18_{-0.06}^{+0.06}$ & $2.19_{-0.06}^{+0.05}$\\
$n_{s }$ & $0.967_{-0.006}^{+0.006}$ & $0.967_{-0.006}^{+0.006}$ & $0.967_{-0.006}^{+0.006}$ & $0.970_{-0.005}^{+0.005}$ & $0.970_{-0.005}^{+0.005}$ & $0.970_{-0.005}^{+0.005}$ & $0.971_{-0.005}^{+0.004}$ & $0.974_{-0.005}^{+0.005}$ & $0.970_{-0.005}^{+0.004}$\\
$\tau_\text{reio}$ & $0.07_{-0.02}^{+0.02}$ & $0.07_{-0.02}^{+0.02}$ & $0.07_{-0.02}^{+0.02}$ & $0.08_{-0.01}^{+0.01}$ & $0.08_{-0.01}^{+0.01}$ & $0.08_{-0.01}^{+0.01}$ & $0.06_{-0.01}^{+0.01}$ & $0.08_{-0.02}^{+0.02}$ & $0.08_{-0.01}^{+0.01}$\\
$\Omega_M$ & $0.31_{-0.01}^{+0.01}$ & $0.32_{-0.03}^{+0.01}$ & $0.31_{-0.02}^{+0.01}$ & $0.301_{-0.007}^{+0.007}$ & $0.304_{-0.009}^{+0.008}$ & $0.301_{-0.008}^{+0.007}$ & $0.291_{-0.007}^{+0.007}$ & $0.310_{-0.010}^{+0.010}$ & $0.300_{-0.008}^{+0.008}$\\
$\sigma_{8 }$ & $0.818_{-0.010}^{+0.010}$ & $0.79_{-0.01}^{+0.03}$ & $0.796_{-0.009}^{+0.034}$ & $0.819_{-0.009}^{+0.009}$ & $0.81_{-0.01}^{+0.02}$ & $0.801_{-0.006}^{+0.030}$ & $0.795_{-0.008}^{+0.008}$ & $0.76_{-0.01}^{+0.01}$ & $0.76_{-0.01}^{+0.01}$\\
$H_0$ & $67.8_{-1.0}^{+0.9}$ & $66.2_{-1.1}^{+2.4}$ & $67.1_{-0.8}^{+1.7}$ & $68.3_{-0.6}^{+0.6}$ & $67.8_{-0.6}^{+0.8}$ & $68.2_{-0.6}^{+0.6}$ & $69.0_{-0.6}^{+0.6}$ & $66.8_{-0.9}^{+0.9}$ & $68.2_{-0.6}^{+0.6}$\\
$\lambda$ & \multicolumn{1}{c}{---} & $0.8_{-0.8}^{+0.2}$ & $0.9_{-0.9}^{+0.3}$ & \multicolumn{1}{c}{---} & $0.5_{-0.5}^{+0.1}$ & $0.8_{-0.8}^{+0.3}$ & \multicolumn{1}{c}{---} & $1.2_{-0.2}^{+0.2}$ & $1.4_{-0.4}^{+0.2}$\\
$\beta$ & \multicolumn{1}{c}{---} & $-0.24_{-0.09}^{+0.24}$ & \multicolumn{1}{c|}{---} & \multicolumn{1}{c}{---} & $-0.26_{-0.24}^{+0.07}$ & \multicolumn{1}{c|}{---} & \multicolumn{1}{c}{---} & $-0.39_{-0.11}^{+0.03}$ & \multicolumn{1}{c|}{---}\\
$\log_{10}(-\beta)$ & \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} & $2.5_{-5.5}^{+4.5}$ & \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} & $3.1_{-1.1}^{+3.9}$ & \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} & $2.8_{-0.8}^{+1.7}$\\
\end{tabular}
\end{ruledtabular}
\end{table*}
\subsection{$\chi^2$-values}\label{sec:chisq}
In Table~II we show the $\chi^2$ values for the best-fitting $\Lambda$CDM, T3 and $\text{T3}_\text{ph}$ models. Our T3 and $\text{T3}_\text{ph}$ models can reconcile CMB, BAO and LSS data. As can be seen from the last two lines in Table~II, when the SZ cluster dataset is included the preference for the T3 and $\text{T3}_\text{ph}$ models is strong.
\begin{table}
\caption{\label{tab:chisq}$\chi^2$-values for $\Lambda$CDM, the T3-model and the T3ph-model for all tested datasets. The preference for T3 and $\text{T3}_\text{ph}$ is strong when cluster data is included.}
\begin{ruledtabular}
\begin{tabular}{l|rrrrr}
Dataset & $\chi^2_{\Lambda\text{CDM}}$ & $\chi^2_\text{T3}$ & $\chi^2_{\text{T3}_\text{ph}}$ &$\Delta \chi^2_\text{T3}$ & $\Delta \chi^2_{\text{T3}_\text{ph}}$\\
\hline
TT & $11261.80$ & $11265.12$ & $11265.20$ & $-3.32$ & $-3.40$\\
TT+J & $11946.80$ & $11949.54$ & $11950.34$ & $-2.74$ & $-3.54$\\
CMB & $11271.80$ & $11271.78$ & $11273.18$ & $0.02$ & $-1.38$\\
CMB+J & $11956.52$ & $11956.86$ & $11957.26$ & $-0.34$ & $-0.74$\\
CMB+B & $11274.46$ & $11275.58$ & $11274.88$ & $-1.12$ & $-0.42$\\
CMB+B+J & $11958.80$ & $11958.68$ & $11960.22$ & $0.12$ & $-1.42$\\
CMB+B+SZ & $11293.50$ & $11279.44$ & $11276.30$ & $14.06$ & $17.20$\\
CMB+B+SZ+J & $11978.38$ & $11965.84$ & $11961.90$ & $12.54$ & $16.48$\\
\end{tabular}
\end{ruledtabular}
\end{table}
\section{Conclusions}
\label{sec:conclusions}
We have identified an interacting dark energy model which can suppress structure growth and can reconcile CMB and LSS observations. It is a pure momentum transfer model and belongs to a class of theories constructed using the Lagrangian pull-back formalism for fluids --- the coupling function characterising the theory is not added at the level of the equations, but at the level of the action. In this way various pathologies and instabilities can be very easily identified; considering the ghost-free branch of the model, we investigated its observational signatures on the CMB and linear matter power spectra. For a constant coupling parameter $\beta$ and our specific choice of potential, the model exhibits structure growth for positive $\beta$ and growth suppression for negative $\beta$. Focusing on the latter case, we performed an MCMC analysis and found that using CMB and BAO data our model is as good as $\Lambda$CDM, while adding cluster data our model becomes strongly prefered. We note, however, that it still exhibits tension with local measurements of the Hubble constant. A full model selection analysis based on the Bayesian evidence is left for future work. However, we note that when the likelihood method shows preference for an extra parameter at a level of $3 \sigma$, so does the Bayesian analysis which is also quite sensitive to the priors used (see~\cite{Battye:2014qga} for a related analysis and discussion for the case of massive neutrinos).
Our work offers a promising alternative for resolving the CMB and LSS tension.
Another alternative is massive neutrinos, which have also been proposed to lift the discrepancy~\cite{Battye:2013xqa} but they increase the tension with the Hubble constant~\cite{Giusarma:2013pmn}. A recent interesting proposal was presented in~\cite{Lesgourgues:2015wza}, in which dark matter interacts with a new form of dark radiation and structure growth is damped via momentum transfer effects.
On the other hand, there is also the possibility that this tension is a result of poorly understood systematic effects. In the imminent future, a set of larger and better optical large scale structure surveys (the Dark Energy Survey, the Euclid satellite, the Large Synoptic Survey Telescope) as well as new probes with completely different methodology and systematics (e.g. 21cm intensity mapping with the Square Kilometre Array~\cite{Santos:2015gra}) will either resolve this tension or confirm the exciting prospect of new physics.
In order to take full advantage of current and future large scale structure datasets, understanding of the non-linear effects of exotic dark energy models is crucial. For example, in order to use the full range of the available data with confidence, one needs to correct the power spectrum on small (non-linear) scales. $N$-body simulations related to pure momentum transfer in the dark sector have been performed in~\cite{Baldi:2014ica, Baldi:2016zom}, based on the elastic scattering model presented in~\cite{Simpson:2010vh}. We plan to investigate non-linear effects for the negative coupling Type-3 models in future work.
To conclude, we may have discovered a whole family of models (Type-3-like models with pure momentum transfer) that can give suppression of growth and reconcile the tension between CMB and LSS. In this work we focused on a particular case, but there is a plethora of different choices that give models belonging to the same class. For example, we could use another coupling function $h(Z)$ and/or another form for the quintessence potential. However, this way the results are strongly model-dependent. Using the PPF approach developed in~\cite{Skordis:2015yra}, we can try to parametrise the free non-zero functions that define Type-3 models in a model-independent way and study their observational consequences; arguably, we should be able to constrain these free functions (or their combinations) such that they give late time growth suppression relative to $\Lambda$CDM --- which is what the available data currently prefer. This would be very important for phenomenological model building and for determining the constraining and discriminating power of future surveys.
\FloatBarrier
\section{Acknowledgments}
A.P. acknowledges support by STFC grant ST/H002774/1. T.T. acknowledges support by STFC grant ST/K00090X/1. Numerical computations were performed using the Sciama High Performance Compute (HPC) cluster which is supported by the ICG, SEPNet and the University of Portsmouth. We are indebted to Robert Crittenden for very useful comments and feedback.
We would also like to thank Ed Copeland, Kazuya Koyama, Jeremy Sakstein and Constantinos Skordis for useful discussions.
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\bibliographystyle{apsrev}
\section{Introduction}
During the last two decades observational cosmology has entered an era of unprecedented precision. Cosmic microwave background (CMB) measurements~\cite{Bennett:2012zja, Ade:2013zuv}, baryon acoustic oscillations (BAO)~\cite{Lampeitl:2009jq} and observations of Type Ia Supernovae~\cite{Riess:2009pu} have shown very good agreement with the predictions of the standard cosmological model ($\Lambda$CDM) consisting of dark energy in the form of a cosmological constant $\Lambda$ and cold dark matter (CDM). However this agreement is not perfect: the Planck CMB data~\cite{Ade:2013zuv} are in tension with low redshift data such as cluster counts~\cite{Ade:2013lmv}, redshift space distortions (RSD)~\cite{Samushia:2012iq, Macaulay:2013swa}, weak lensing data~\cite{Heymans:2012gg} and local measurements of the Hubble constant, $H_0$~\cite{Riess:2011yx,Riess:2016jrr}. More specifically, the low redshift probes point towards a lower rate of structure growth (equivalently, a lower $\sigma_8$) than the Planck results for the base $\Lambda$CDM would prefer.
The most significant tensions are the ones coming from the cluster and weak lensing data.
A possible explanation for these tensions is that there are systematic effects that have not been accounted for, such as systematics that affect the determination of the mass bias in the cluster case, and small scales effects in the weak lensing case. Another possibility is that $\Lambda$CDM is not the correct model describing the evolution of the Universe. Finding a different model which gives a better (or, at the very least, equally good) fit to the data has proven very difficult. However, the motivation for exploring alternatives is strong because of the fundamental problems that plague $\Lambda$CDM, namely the \emph{fine-tuning} and \emph{coincidence} problems. These problems are associated with the cosmological constant and they have led to a plethora of alternative scenarios. One popular example is quintessence~\cite{Wetterich88, Ratra:1987rm, Wetterich:1994bg}, in which an evolving scalar field plays the role of dark energy. A different viewpoint suggests that General Relativity is modified on cosmological scales and this modification is responsible for the accelerated expansion of the Universe (see~\cite{Copeland:2006wr, CliftonEtal2011} and references therein).
The nature of the two constituents of the dark sector is currently unknown and the fact that they are considered uncoupled in $\Lambda$CDM is just an assumption of the model. Let us now consider a non-gravitational coupling between cold dark matter (CDM) and dark energy (DE). The energy momentum tensors for CDM and DE are then no longer separately conserved but instead we find
\begin{equation}
\nabla_\mu T^{{\rm (CDM)}\mu}_{\phantom{{\rm (CDM)}\mu} \nu} = J_\nu = -\nabla_\mu T^{{\rm (DE)}\mu}_{\phantom{{\rm (DE)}\mu} \nu},
\end{equation} where the coupling current $J_\nu$ represents the energy and momentum exchange between dark energy and dark matter. Note that we assume that the standard model is not coupled to the dark sector --- an assumption which well justified by observations that strongly constrain such couplings~\cite{Carroll:1998zi}.
Traditionally DE is modelled by a quintessence field $\phi$, so we are going to set ${\rm{DE}}\equiv \phi$ from now on, and ${\rm{cdm}}\equiv c$.
In the background, this means that the energy conservation relations become
\begin{align} \nonumber
\dot{\ensuremath{\bar{\rho}}}_c+3{\cal H}\ensuremath{\bar{\rho}}_c &= Q \\
\dot{\ensuremath{\bar{\rho}}}_\phi+3{\cal H}\ensuremath{\bar{\rho}}_\phi (1+w_\phi) &= -Q,
\end{align} where ${\cal H}=\dot{a}/a$ is the conformal Hubble rate with scale factor $a$ and a dot denotes the derivative with respect to conformal time $\tau$. $\ensuremath{\bar{\rho}}_c, \ensuremath{\bar{\rho}}_\phi$ are the energy densities of cold dark matter and dark energy (with the bar denoting background quantities) and $Q \equiv \bar{J}_0$ is the background energy transfer.
The form of $Q$ is usually chosen phenomenologically, and one of the most widely used forms is
$
Q = \alpha_0 \dot{\ensuremath{\bar{\phi}}} \ensuremath{\bar{\rho}}_c
$~\cite{Amendola:1999er}, where $\alpha_0$ is a constant parameter which determines the strength of the interaction.
This case has been extensively studied in the literature, see e.g.~\cite{Amendola:1999er, Xia:2009zzb, Xia:2013nua,Pourtsidou:2013nha} and references therein. Other widely used forms take $Q$ to be proportional to the energy densities of cdm and/or DE, i.e. $Q \propto \Gamma \rho $ with $\Gamma$ a constant interaction rate, or $Q \propto {\cal H} \rho$, see e.g.~\cite{CalderaCabral:2008bx, CalderaCabral:2009ja,Valiviita:2009nu, Jackson:2009mz,Clemson:2011an,Gavela:2009cy,Yang:2014gza,Salvatelli:2013wra, yang:2014vza, Murgia:2016ccp} and references therein.
Interest in interacting dark energy models has grown as they are arguably better motivated than the uncoupled $\Lambda$CDM concordance model, and they can possibly lift the tensions present in the available data. For example, an interaction between vacuum energy and dark matter was shown to be able to remove some of the tensions~\cite{Salvatelli:2014zta, Wang:2015wga}.
Very recently there have also been important theoretical developments in the field.
In \cite{DAmico:2016aa}, the authors formulate a quantum field theory of interacting dark matter - dark energy and show that the quantum corrections coming from the usual assumption for the form of the interaction (i.e. that dark matter is made up of ``heavy" particles whose mass depend on the dark energy field value) are huge and severely constrain the allowed couplings and the nature of the dark sector (see also \cite{Marsh:2016aa}).
However, in the majority of cases there is no Lagrangian description of the model, and an ad-hoc expression for the coupling $Q$ is just added at the level of the equations. In~\cite{Pourtsidou:2013nha} the authors used the pull-back formalism for fluids to generalise the fluid action and involve couplings between the scalar field playing the role of dark energy, and dark matter. Their construction led to three distinct \emph{families} of coupled models. The first two (Types 1 and 2) involve both energy and momentum transfer between dark matter and dark energy. The commonly used coupled quintessence model~\cite{Amendola:1999er} was shown to be a sub-class of Type 1~\cite{Pourtsidou:2013nha, Skordis:2015yra}. The third family of models (Type 3) is a pure momentum transfer theory with $Q=0$.
In this work we investigate a specific Type 3 model that was first presented in~\cite{Pourtsidou:2013nha}. We implement this model in the Einstein-Boltzmann solver \textsc{class}{}~\cite{Blas:2011rf}, perform a global fitting of cosmological parameters using the Markov chain Monte Carlo (MCMC) code~\textsc{MontePython}{}~\cite{Audren:2012wb}, and compare our findings to $\Lambda$CDM. The structure of the paper is as follows: In Section~\ref{sec:T3model} we present the specific Type 3 model we are going to study and state the background and linear perturbation cosmological equations which were derived in detail in~\cite{Pourtsidou:2013nha}. We also demonstrate the coupling's effects on the CMB and linear matter power spectra using \textsc{class}{}. In Section~\ref{sec:results} we describe the datasets and priors we use and then present the results of our MCMC analysis. We concentrate on the comparison of the Type 3 model with $\Lambda$CDM and the effects on the $(\sigma_8, H_0)$ parameters. We present the best fit cosmological parameters for $\Lambda$CDM and Type 3 and compare their $\chi^2$ values. We conclude in Section~\ref{sec:conclusions}.
\section{The pure momentum transfer model}
\label{sec:T3model}
As we have already stated, the model we are going to investigate belongs to the Type 3 class of theories constructed in~\cite{Pourtsidou:2013nha}. The distinctive characteristic of this class is that there is only momentum transfer between the two components of the dark sector. No coupling appears at the background level, regarding the fluid equations --- they remain the same as in the uncoupled case. Furthermore, the energy-conservation equation remains uncoupled even at the linear level. Hence, the theory provides for a pure momentum-transfer coupling at the level of linear perturbations.
Type 3 theories are classified via the Lagrangian~\cite{Pourtsidou:2013nha}
\begin{equation}
L(n,Y,Z,\phi) = F(Y,Z,\phi) + f(n),
\end{equation} where $n$ is the fluid number density, $Y=\frac{1}{2}\nabla_\mu \phi \nabla^\mu \phi$ is used to construct a kinetic term for $\phi$, and $Z=u^\mu \nabla_\mu \phi$ plays the role of a direct coupling of the fluid velocity $u^\mu$ to the gradient of the scalar field \footnote{
Note that the fluid velocity $u^\mu$ is uniquely defined in terms of the fluid number density $n$ and the dual number density (see \cite{Pourtsidou:2013nha} for details).}.
Considering a coupled quintessence function of the form
\begin{equation}
F=Y+V(\phi)+h(Z),
\end{equation} with $V(\phi)$ the quintessence potential, we have the freedom to choose the coupling function $h(Z)$ in order to construct specific models belonging to the same class. Following~\cite{Pourtsidou:2013nha}, we are going to concentrate on the sub-case with
\begin{equation}
h(Z) = \beta Z^2,
\end{equation} where $\beta$, the coupling parameter, is taken to be constant.
Defining $\tilde{g}^{\mu \nu}=g^{\mu \nu}+2\beta u^\mu u^\nu$, we can write the action for the scalar field $\phi$ as \cite{Pourtsidou:2013nha}
\begin{eqnarray}
\nonumber
S_{\phi}&=&
-\int d^4x \sqrt{-g}\left[
\frac{1}{2}\tilde{g}^{\mu \nu} \phi_\mu \phi_\nu
+ V(\phi)
\right]
\\
&\rightarrow &
\int dt \, d^3x \, a^3 \left[
\frac{1}{2}(1 - 2 \beta) \dot{\phi}^2
- \frac{1}{2} |\ensuremath{\vec{\nabla}}\phi|^2
- V(\phi)
\right] \;\;\;
\label{eq:action}
\end{eqnarray}
where the arrow denotes working in a frame where the CDM $3$-velocity is zero.
Hence, the model is physically acceptable for $\beta < \frac{1}{2}$. For $\beta \rightarrow 1/2$
we have a strong coupling problem, while for $\beta>1/2$ there is a ghost in the theory since the kinetic term becomes negative~\cite{Pourtsidou:2013nha}.
\subsection{Background Evolution}
We assume a Universe described by a flat Friedmann-Lema\^itre-Robertson-Walker (FLRW) metric
\begin{equation}
ds^2=a^2(\tau)(-d\tau^2+ dx_idx^i),
\end{equation} where $a(\tau)$ satisfies the Friedmann equation
\begin{equation}
{\cal H}^2 =\frac{8\pi G}{3}a^2\ensuremath{\bar{\rho}}_{\rm tot}
= \frac{8\pi G}{3}a^2(\ensuremath{\bar{\rho}}_\gamma + \ensuremath{\bar{\rho}}_b + \ensuremath{\bar{\rho}}_c + \ensuremath{\bar{\rho}}_\phi).
\end{equation} Here $\ensuremath{\bar{\rho}}_{\rm tot}$ is the total background energy density of all species and the subscripts $\gamma, b$ denote radiation and baryons, respectively.
The background energy density and pressure for quintessence are~\cite{Pourtsidou:2013nha}
\begin{eqnarray}
\ensuremath{\bar{\rho}}_{\phi}&=&\left(\frac{1}{2}-\beta\right)\frac{\dot{\ensuremath{\bar{\phi}}}^2}{a^2}+V(\phi), \\
\ensuremath{\bar{P}}_{\phi}&=&\left(\frac{1}{2}-\beta\right)\frac{\dot{\ensuremath{\bar{\phi}}}^2}{a^2}-V(\phi),
\end{eqnarray}
and the energy conservation equations are the same as in uncoupled quintessence:
\begin{eqnarray}
\label{eq:drhophiType3}
\dot{\ensuremath{\bar{\rho}}}_\phi &+& 3{\cal H}(\ensuremath{\bar{\rho}}_\phi+\ensuremath{\bar{P}}_\phi)=0, \\
\dot{\ensuremath{\bar{\rho}}}_c &+& 3{\cal H}\ensuremath{\bar{\rho}}_c=0,
\end{eqnarray} i.e. $Q=0$ for these models.
Hence the cold dark matter density obeys the usual scaling relation:
\begin{equation}
\ensuremath{\bar{\rho}}_c = \rho_{c,0} a^{-3}.
\end{equation}
In contrast to the usual coupled dark energy models which exhibit background energy exchange, the evolution of the CDM and quintessence energy densities
\begin{equation}
\Omega_c \equiv \frac{\ensuremath{\bar{\rho}}_c}{\ensuremath{\bar{\rho}}_{\rm tot}}, \;\;\; \Omega_\phi \equiv \frac{\ensuremath{\bar{\rho}}_\phi}{\ensuremath{\bar{\rho}}_{\rm tot}}
\end{equation} are unchanged in Type 3 models. The background Klein-Gordon equation is given by
\begin{equation}
\ddot{\ensuremath{\bar{\phi}}}+2{\cal H}\dot{\ensuremath{\bar{\phi}}}+
\left(\frac{1}{1-2\beta}\right)a^2\frac{dV}{d\phi}=0,
\end{equation}
and the speed of sound is $c^2_s=\frac{1-2\beta}{1-2\beta}=1$~\cite{Pourtsidou:2013nha}.
\subsection{Linear Perturbations}
In order to study the observational effects of the coupled models on the Cosmic Microwave Background and Large Scale Structure (LSS), we need to consider linear perturbations around the FLRW background. Choosing the synchronous gauge the metric is
\begin{eqnarray}
ds^2 &=& -a^2d\tau^2 + a^2 \left[ (1 + \frac{1}{3}h)\gamma_{ij} + D_{ij}\nu\right] dx^i dx^j,
\ \
\label{perturbed_metric}
\end{eqnarray}
where $\tau$ is the conformal time, $\ensuremath{\vec{\nabla}}_k$ is the covariant derivative associated with $\gamma_{ij}$, i.e. $\ensuremath{\vec{\nabla}}_k \gamma_{ij} = 0$
and $D_{ij}$ is the traceless derivative operator $D_{ij} = \ensuremath{\vec{\nabla}}_i \ensuremath{\vec{\nabla}}_j -\frac{1}{3} \ensuremath{\vec{\nabla}}^2 \gamma_{ij}$.
The unit-timelike vector field $u^\mu$ is perturbed as
\begin{eqnarray}
u_\mu &=& a( 1, \ensuremath{\vec{\nabla}}_i \theta).
\label{u_mu_theta}
\end{eqnarray}
We denote the field perturbation $\delta \phi$ by $\varphi$, the $\phi$-derivative of the potential by $V_\phi \equiv dV/d\phi$ and we also have $\ensuremath{\bar{Z}} \equiv -\dot{\ensuremath{\bar{\phi}}}/a$. In this notation the perturbed scalar field energy density and pressure are given by~\cite{Pourtsidou:2013nha}
\begin{align}
\delta \ensuremath{\rho_{\phi}} = -\frac{1}{a}\ensuremath{\bar{Z}} (1 - 2\beta) \dot{\varphi}
+ V_{\phi} \varphi, \\
\delta P_\phi = - \frac{1}{a} \ensuremath{\bar{Z}} \left(1 - 2\beta \right)\dot{\varphi} - V_{\phi}\varphi,
\end{align}
while the velocity divergence of the scalar field is
\begin{equation}
\ensuremath{\theta_{\phi}} =\frac{1}{(2\beta -1)\ensuremath{\bar{Z}}}\left(\frac{1}{a}\varphi+ 2\beta \ensuremath{\bar{Z}} \theta_c\right).
\label{eq:thetaphiType3}
\end{equation}
This is one other significant difference from other types of coupling, namely that $\ensuremath{\theta_{\phi}}$ depends also on the CDM velocity divergence $\theta_c$.
The linearised scalar field equation is~\cite{Pourtsidou:2013nha}
\begin{eqnarray}
&&
(1 - 2\beta) (\ddot{\varphi} + 2 \ensuremath{{\cal H}} \dot{\varphi})
+ \left(k^2 + a^2 V_{\phi\phi} \right) \varphi
\nonumber
\\
&&
\ \ \ \
+ \frac{1}{2} \dot{\ensuremath{\bar{\phi}}} (1 - 2\beta) \dot{h}
- 2\beta \dot{\ensuremath{\bar{\phi}}}k^2 \theta_c
= 0,
\label{eq:pertKG}
\end{eqnarray}
while the density contrast $\delta_c \equiv \delta \rho_c / \ensuremath{\bar{\rho}}_c$ obeys the standard evolution equation
\begin{equation}
\label{eq:deltac}
\dot{\delta}_c = -k^2 \theta_c - \frac{1}{2}\dot{h}.
\end{equation}
The momentum-transfer equation depends on the coupling and is given by
\begin{equation}
\dot{\theta}_c = - \ensuremath{{\cal H}} \theta_c +
\frac{( 6 \ensuremath{{\cal H}} \beta \ensuremath{\bar{Z}} + 2\beta \dot{\ensuremath{\bar{Z}}} ) \varphi + 2\beta \ensuremath{\bar{Z}} \dot{\varphi}}{a \left(\ensuremath{\bar{\rho}}_c - 2 \beta \ensuremath{\bar{Z}}^2 \right)} .
\label{dP_type3_pert}
\end{equation}
We implemented the above equations in \textsc{class}{} in order to compute the CMB temperature and matter power spectra. At this stage we will fix our quintessence potential $V(\phi)$ to be the widely used single exponential form (1EXP)
\begin{equation}
V(\phi)=V_0 e^{-\lambda \phi}.
\end{equation} In order to demonstrate the effect of the coupling before we perform an MCMC analysis, we will first compare the uncoupled $\beta = 0$ case with
the coupled model under consideration keeping the parameter $\lambda$ of the potential fixed ($\lambda=1.22$), while the potential normalisation $V_0$ is varied automatically by \textsc{class}{} is order to match a fixed $\Omega_\phi$ today. Our initial conditions for the quintessence field are $\phi_i=10^{-4}, \dot{\phi}_i=0$. Note, however, that in this case and using the 1EXP potential, the same cosmological evolution is expected for a wide range of initial conditions.
Before we present our results, we should briefly discuss the range of $\beta$ values we are going to consider. As we have already mentioned, it is clear from the Lagrangian in Equation~(\ref{eq:action}) that we are not allowed to consider the case $\beta > 1/2$ due to the strong coupling and ghost pathologies. We are free to consider any negative value $\beta$, but the fact that $\beta$ is a dimensionless coupling parameter in the Lagrangian suggests that its magnitude should be small. We will therefore initially choose a prior for our negative $\beta$ values such that $-0.5 \leq \beta \leq 0$ and call this model T3. However, as we shall see later the data allow for the model to span a much wider range of negative $\beta$, so in this case we will use a log prior $-3 \leq {\rm Log}_{10}(-\beta) \leq 7$ and denote this (phenomenological) Type 3 model $\text{T3}_\text{ph}$.
\begin{figure*}[tb]
\includegraphics[width=0.495\linewidth]{CoupledQuintessenceCLTT.pdf}
\hfill
\includegraphics[width=0.495\linewidth]{CoupledQuintessencePk.pdf}
\caption{Comparison of the CMB TT power spectra (left panel) and the linear (total) matter power spectra $P(k)$ at $z=0$ (right panel). Black solid lines denote the uncoupled quintessence model, blue dotted lines denote the coupled model with positive coupling parameter $\beta=0.1$, and dashed red lines denote the coupled model with negative coupling parameter $\beta=-0.5$.}
\label{fig:ClTT}
\end{figure*}
In Fig.~\ref{fig:ClTT} we show the CMB temperature spectra and the matter power spectra for the uncoupled case $\beta=0$, for a positive coupling parameter $\beta=0.1$, and for $\beta=-0.5$. We see that the effect of the coupling in the CMB temperature power spectrum is very small. There is no visible effect on the small scale amplitude and no shift of the location of the peaks, like in usual coupled quintessence models (\cite{Amendola:1999er, Xia:2009zzb, Xia:2013nua}). The only visible effect is an integrated Sachs-Wolfe (ISW) effect on large scales, but it remains small in contrast to coupled quintessence models with background energy exchange. This is because the background CDM energy density remains uncoupled in our model.
However, the effects on the matter power spectrum $P(k)$ are significant. For the positive coupling case there is enhanced growth at small scales (and, consequently, a larger value of $\sigma_8$ is obtained), while for the negative coupling T3 case the growth is suppressed and the associated $\sigma_8$ is smaller. Since the positive coupling model results in enhanced growth, it aggravates the tension between the Planck CMB data and low redshift observations. We will therefore exclude the $0<\beta<1/2$ branch from our MCMC analysis.
We will now show the comparison between the $\text{T3}_\text{ph}$ model - which spans a wide range of negative $\beta$ values - and the uncoupled quintessence case. In order to highlight the effect of $\beta$ we fix the sound speed at recombination $\theta_s$ and the physical energy densities of CDM and baryons, $\omega_{c,b}=\Omega_{c,b}h^2$. In Fig.~\ref{fig:ClTT_T3ph} we plot the CMB TT power spectrum and the matter power spectrum divided by their uncoupled counterparts. Similar to the T3 model, we see that the effect of the coupling on the TT power spectrum only shows up on very large scales. This is due to the late-time ISW effect which changes because of the effect of the coupling on the evolution of the matter density perturbations. We also notice that the effect changes direction (from enhancement to suppression with respect to the uncoupled case) around $\beta \simeq-10^2$.
In the matter power spectrum we also see an interesting feature, namely that for $\beta<-10^2$ there is a range of $k$-values where the matter power spectrum is actually~\emph{enhanced} compared to the uncoupled case. This gives $\sigma_8(\beta)>\sigma_8(\beta=0)$ for $\beta\lesssim-10^4$. Note that we are comparing models that reproduce the same TT-power spectrum at small scales according to the left panel of Fig.~\ref{fig:ClTT_T3ph}, and these models have slightly different values of $H_0$.
\begin{figure*}[tb]
\centering
\includegraphics[width=0.495\linewidth]{cl_ratio_beta.pdf}
\hfill
\includegraphics[width=0.495\linewidth]{pk_ratio_beta.pdf}
\caption{Comparison of the CMB TT power spectra (left panel) and the linear (total) matter power spectra at $z=0$ (right panel) for a wide range of negative $\beta$-values. In both cases we show the ratio between the $\text{T3}_\text{ph}$ model and the predictions of the uncoupled quintessence model.}
\label{fig:ClTT_T3ph}
\end{figure*}
For completeness, we also show the evolution of the dark energy equation of state parameter $w_\phi = p_\phi /\rho_\phi$ in Fig.~\ref{fig:wbeta}. We see that as the magnitude of the $\beta$ parameter increases, $w_\phi$ gets closer to $-1$, i.e. the background dark energy evolution resembles that of a cosmological constant. That is because of the effects of the coupling on the evolution of $\ensuremath{\bar{\phi}}$: the $\beta \dot{\ensuremath{\bar{\phi}}}^2/a^2$ term becomes completely subdominant to $V(\phi)$ and $w_\phi \rightarrow -1$.
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{w_beta_log.pdf}
\caption{ The evolution of the dark energy equation of state parameter $w_\phi = \ensuremath{\bar{P}}_\phi /\ensuremath{\bar{\rho}}_\phi$ as a function of the coupling parameter $\beta$. A constant $w_\phi = -0.9$ is shown for comparison. }
\label{fig:wbeta}
\end{figure}
Before we move on to our full MCMC analysis, we should stress that the effect of growth suppression in coupled dark energy models is quite rare and difficult to achieve.
The same is true for modified gravity models, the vast majority of which exhibit growth increase \footnote{This is easy to understand as such models result to an enhanced effective gravitational constant $G_{\rm eff}$ with respect to Newton's constant. However, a recent study of theories beyond Hordenski showed that there is the possibility of constructing stable and ghost-free weak gravity scenarios~\cite{Tsujikawa:2015mga}.
} and make the tension worse as they favour a large $\sigma_8$ value~\cite{Hu:2015rva}.
In coupled dark energy models that exhibit both energy and momentum exchange, like the coupled quintessence model with $Q=\alpha_0 \dot{\bar{\phi}}\ensuremath{\bar{\rho}}_c$~\cite{Amendola:1999er}, the growth rate depends on two terms: a fifth-force contribution $\propto \alpha^2_0$, and a friction term $\propto \alpha_0 \dot{\ensuremath{\bar{\phi}}}$, whose sign is determined by the sign of the coupling parameter $\alpha_0$ and the quintessence potential through the $\dot{\ensuremath{\bar{\phi}}}$ dependence.
In general, getting suppression of growth is highly non-trivial, and often requires potentials with more than one free parameters in order to make the friction term dominate over the competing fifth-force term, which tends to give growth increase for positive and negative coupling values (see~\cite{Tarrant:2011qe} for details and further discussion).
On the contrary, our pure momentum transfer model has a straightforward behaviour for the simplest case of the 1EXP potential: Considering the positive coupling Type 3 model and the negative coupling T3 model we
get growth increase for the former case and suppression of growth for the latter.
This effect comes from the modified density contrast evolution due to the presence of the momentum transfer coupling, and there is no competing coupling term in Equation~(\ref{eq:deltac}).
For the phenomenological $\text{T3}_\text{ph}$ model we also get growth suppression for a very wide range of negative $\beta$, but due to the scale dependent feature demonstrated in Fig.~\ref{fig:ClTT_T3ph} we also have a turning point where growth and $\sigma_8$ start to increase. For these reasons we chose to perform our MCMC analysis on the T3 and $\text{T3}_\text{ph}$ models separately.
\section{Results}
\label{sec:results}
\subsection{Datasets and priors}
\begin{figure*}[tb]
\centering
\includegraphics[width=0.495\linewidth]{plot1G.pdf}
\hfill
\includegraphics[width=0.495\linewidth]{plot3G.pdf}
\caption{$1\sigma$ and $2\sigma$ constraints for $\Lambda$CDM and the T3 and $\text{T3}_\text{ph}$ models in the $(\sigma_8,\Omega_M)$-plane using TT data (left panel) and using all data (right panel). The $1\sigma$-band from SZ clusters and CFHTLenS is shown in dotted green and solid red, respectively. $\Lambda$CDM is in tension with the SZ clusters and CFHTLenS while the T3 and $\text{T3}_\text{ph}$ models are compatible. Even after including the SZ-data, the $2\sigma$ $\Lambda$CDM contour does not overlap with the $1\sigma$ SZ contour, which illustrates the tension. On the contrary, the T3 and $\text{T3}_\text{ph}$ contours overlap with the SZ contour.}
\label{fig:plot3}
\end{figure*}
In our analysis we use a variety of recent datasets, including CMB data, baryon acoustic oscillation measurements, supernovae, and cluster counts. The specific likelihoods we employ are:
\begin{description}[leftmargin=4em,style=nextline]
\item[TT:] $C_\ell^{TT}$-data from Planck 2015~\cite{Ade:2015xua}, including low-$\ell$ polarisation.
\item[CMB:] TT and the lensing reconstruction from Planck 2015 data~\cite{Ade:2013tyw}.
\item[B:] Baryon Acoustic Oscillation (BAO) data from BOSS~\cite{Anderson:2013zyy}.
\item[J:] Joint Light-curve Analysis (JLA)~\cite{Betoule:2014frx}.
\item[SZ:] Planck SZ cluster counts~\cite{Ade:2013lmv,Ade:2015fva}.
\end{description}
We did not include weak lensing data in the MCMC run --- however, we show the constraint $\sigma_8\left(\Omega_M/0.27 \right)^{0.46} = 0.774 \pm0.040$ derived from CFHTLenS~\cite{Heymans:2013fya} in Fig.~\ref{fig:plot3}. Note that this constraint has been inferred assuming the $\Lambda$CDM model.
We must also comment on the applicability of the cluster count likelihood we use~\cite{Ade:2013lmv}. This likelihood assumes a mass bias $(1-b)\simeq 0.8$ that agrees well with simulations, but the authors note that a significantly lower value would alleviate the CMB-SZ tension. It also uses the Tinker et al halo mass function~\cite{Tinker:2008ff}, which has been calibrated against N-body simulations assuming $\Lambda$CDM. This implies that including this likelihood is not entirely self-consistent, see for instance Ref.~\cite{Cataneo:2014kaa} for a discussion of the similar problem in $f(R)$ gravity. Making quantitative predictions for the non-linear effects of our model is not possible at this stage. However, a very recent paper on structure formation simulations with momentum exchange showed that the qualitatively similar dark scattering model~\cite{Simpson:2010vh} can alleviate the CMB-LSS tensions while keeping non-linear effects very mild~\cite{Baldi:2016zom}. Running a suite of N-body simulations for a coupled quintessence model with pure momentum exchange is the subject of future work.
We chose flat priors on the following set of cosmological parameters,
\begin{equation}
\left\{\omega_b, \omega_\text{cdm}, \theta_s, A_s, n_s, \tau_\text{reio}, \lambda\right\},
\end{equation}
and the collection of nuisance parameters required by the Planck and JLA likelihoods. The prior ranges of $\lambda$ and $\beta$ were chosen as
\begin{align}
\lambda &\in [0; 2.1], & \beta &\in [-0.5,0],
\intertext{for the T3 model with a flat prior on $\beta$ and }
\lambda &\in [0; 2.1], & {\rm log}_{10}(-\beta) &\in [-3,7],
\end{align}
for the $\text{T3}_\text{ph}$ model with a logarithmic prior on $\beta$ as indicated. As we have already mentioned, we exclude the $\beta>0$ branch from our analysis since we want to focus on the branch which can provide suppression of growth.
The initial conditions for the quintessence field were chosen as $(\phi_i, \dot{\phi}_i)=(10^{-4},0)$.
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{plot2bG.pdf}
\caption{$1\sigma$ and $2\sigma$ temperature (TT) constraints in the $(\sigma_8,H_0)$-plane for $\Lambda$CDM and the T3 and $\text{T3}_\text{ph}$ models. The two parameters are uncorrelated in $\Lambda$CDM and $\text{T3}_\text{ph}$ while they are correlated in the T3 model}.
\label{fig:plot2b}
\end{figure}
\subsection{Parameter inference}
We perform a Markov chain Monte Carlo (MCMC) analysis using the publicly available code~\textsc{MontePython}{}.
In the left panel of \fref{fig:plot3} we show the constraint from TT data alone in the $(\Omega_M,\sigma_8)$-plane for $\Lambda$CDM and the T3 and $\text{T3}_\text{ph}$ models. The $2\sigma$-contours of the T3 and $\text{T3}_\text{ph}$ models overlap with the $1\sigma$-constraint from SZ clusters and CFHTLenS whereas the $\Lambda$CDM model is in tension with both SZ and CFHTLenS data.
\begin{figure*}[tb]
\centering
\includegraphics[width=0.495\linewidth]{plot4.pdf}
\hfill
\includegraphics[width=0.495\linewidth]{plot4G.pdf}
\caption{One dimensional posterior distributions of the parameters $\{\sigma_8, H_0, \beta, \lambda\}$ excluding (solid black lines) and including (magenta dashed lines) the SZ cluster data along with the rest of our datasets. In the left panel we show the T3 model while the right panel shows the $\text{T3}_\text{ph}$ model.}
\label{fig:plot4}
\end{figure*}
When we combine all our datasets including the SZ cluster data we find quite different $(\Omega_M,\sigma_8)$-constraints as illustrated in the right panel of \fref{fig:plot3}. The T3 and $\text{T3}_\text{ph}$ models generally prefer lower $\sigma_8$ and larger $\Omega_M$-values compared to $\Lambda$CDM. As we shall see in \sref{sec:chisq} the T3 and $\text{T3}_\text{ph}$ models are significantly better fits to the data than $\Lambda$CDM. In \fref{fig:plot2b} we show the $(\sigma_8,H_0)$-constraints. In the T3 model, $\sigma_8$ and $H_0$ become strongly correlated, but in $\Lambda$CDM and $\text{T3}_\text{ph}$ no correlation exists. As we will see later, this difference between T3 and $\text{T3}_\text{ph}$ has important implications for the $H_0$ tension.
In \fref{fig:plot4} we show the effect of including the SZ cluster data or not on the parameters $\{\sigma_8, H_0, \beta, \lambda\}$ for the T3 model (left panel) and $\text{T3}_\text{ph}$ model (right panel). Let us first comment on the T3 case: Taken at face value, the cluster data rules out the non-interacting case, $\beta=0$. Looking at the posterior distribution for $\lambda$ reveals that the non-cluster datasets roughly prefer $\lambda < 1$ while we have $\lambda > 1$ after the inclusion of the cluster data. This suggests that another potential could give an even better fit to the data than the single exponential potential (but would probably rely on additional free parameters). The T3 posterior distribution for $\beta$ suggests that a better fit can possibly be found if we let $\beta < -1/2$. Moving on to the $\text{T3}_\text{ph}$ case we verify this as the posterior distribution peaks at $\beta=-10^4$.
We also note that the $H_0$ distributions for $\Lambda$CDM and $\text{T3}_\text{ph}$ are very similar. In $\Lambda$CDM it is well known that the inclusion of SZ cluster data drives $H_0$ to larger values. In the T3 model the opposite happens, the cluster data pushes $H_0$ to lower values. We illustrate these effects in \fref{fig:plot5}.
This does put the T3 model in mild tension with local measurements of the Hubble constant but not much more than pure $\Lambda$CDM. However, as we already saw in the $\text{T3}_\text{ph}$ model the $H_0$ posterior distribution is very similar to the $\Lambda$CDM one.
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{plot5.pdf}
\caption{One dimensional posterior distributions of the Hubble parameter. The inclusion of SZ clusters drives $H_0$ to larger values in $\Lambda$CDM and lower values in T3.}
\label{fig:plot5}
\end{figure}
In Table~I we show the mean values and $1\sigma$ confidence intervals on the cosmological parameters for various datasets combinations for the $\Lambda$CDM, T3, and $\text{T3}_\text{ph}$ models. It is important to note that $\beta$ is essentially not constrained unless we include the SZ cluster data in the analysis.
\begin{table*}
\caption{\label{tab:parameters}Cosmological parameters for $\Lambda$CDM and the T3-model for the 4 different data set combinations used in the plots. Note that $\beta$ is essentially unconstrained unless cluster data is added.}
\begin{ruledtabular}
\begin{tabular}{l|rrr|rrr|rrr|}
& \multicolumn{3}{c|}{CMB} & \multicolumn{3}{c|}{CMB+B+J} & \multicolumn{3}{c|}{CMB+B+SZ+J}\\
& $\Lambda$CDM & T3 & $\text{T3}_\text{ph}$ & $\Lambda$CDM & T3 & $\text{T3}_\text{ph}$ & $\Lambda$CDM & T3 & $\text{T3}_\text{ph}$\\ \hline
$100~\omega_{b }$ & $2.22_{-0.02}^{+0.02}$ & $2.22_{-0.02}^{+0.02}$ & $2.22_{-0.02}^{+0.02}$ & $2.23_{-0.02}^{+0.02}$ & $2.23_{-0.02}^{+0.02}$ & $2.23_{-0.02}^{+0.02}$ & $2.24_{-0.02}^{+0.02}$ & $2.25_{-0.02}^{+0.02}$ & $2.23_{-0.02}^{+0.02}$\\
$\omega_\text{cdm}$ & $0.119_{-0.002}^{+0.002}$ & $0.119_{-0.002}^{+0.002}$ & $0.119_{-0.002}^{+0.002}$ & $0.118_{-0.001}^{+0.001}$ & $0.117_{-0.001}^{+0.001}$ & $0.118_{-0.001}^{+0.001}$ & $0.116_{-0.001}^{+0.001}$ & $0.116_{-0.001}^{+0.001}$ & $0.118_{-0.001}^{+0.001}$\\
$10^4\theta_s$ & $104.20_{-0.04}^{+0.04}$ & $104.20_{-0.04}^{+0.04}$ & $104.20_{-0.04}^{+0.04}$ & $104.21_{-0.04}^{+0.04}$ & $104.21_{-0.04}^{+0.04}$ & $104.21_{-0.04}^{+0.04}$ & $104.20_{-0.04}^{+0.04}$ & $104.22_{-0.04}^{+0.04}$ & $104.21_{-0.04}^{+0.04}$\\
$10^9 A_s$ & $2.15_{-0.07}^{+0.06}$ & $2.16_{-0.07}^{+0.06}$ & $2.16_{-0.07}^{+0.06}$ & $2.18_{-0.06}^{+0.05}$ & $2.19_{-0.06}^{+0.05}$ & $2.18_{-0.06}^{+0.05}$ & $2.08_{-0.05}^{+0.05}$ & $2.18_{-0.06}^{+0.06}$ & $2.19_{-0.06}^{+0.05}$\\
$n_{s }$ & $0.967_{-0.006}^{+0.006}$ & $0.967_{-0.006}^{+0.006}$ & $0.967_{-0.006}^{+0.006}$ & $0.970_{-0.005}^{+0.005}$ & $0.970_{-0.005}^{+0.005}$ & $0.970_{-0.005}^{+0.005}$ & $0.971_{-0.005}^{+0.004}$ & $0.974_{-0.005}^{+0.005}$ & $0.970_{-0.005}^{+0.004}$\\
$\tau_\text{reio}$ & $0.07_{-0.02}^{+0.02}$ & $0.07_{-0.02}^{+0.02}$ & $0.07_{-0.02}^{+0.02}$ & $0.08_{-0.01}^{+0.01}$ & $0.08_{-0.01}^{+0.01}$ & $0.08_{-0.01}^{+0.01}$ & $0.06_{-0.01}^{+0.01}$ & $0.08_{-0.02}^{+0.02}$ & $0.08_{-0.01}^{+0.01}$\\
$\Omega_M$ & $0.31_{-0.01}^{+0.01}$ & $0.32_{-0.03}^{+0.01}$ & $0.31_{-0.02}^{+0.01}$ & $0.301_{-0.007}^{+0.007}$ & $0.304_{-0.009}^{+0.008}$ & $0.301_{-0.008}^{+0.007}$ & $0.291_{-0.007}^{+0.007}$ & $0.310_{-0.010}^{+0.010}$ & $0.300_{-0.008}^{+0.008}$\\
$\sigma_{8 }$ & $0.818_{-0.010}^{+0.010}$ & $0.79_{-0.01}^{+0.03}$ & $0.796_{-0.009}^{+0.034}$ & $0.819_{-0.009}^{+0.009}$ & $0.81_{-0.01}^{+0.02}$ & $0.801_{-0.006}^{+0.030}$ & $0.795_{-0.008}^{+0.008}$ & $0.76_{-0.01}^{+0.01}$ & $0.76_{-0.01}^{+0.01}$\\
$H_0$ & $67.8_{-1.0}^{+0.9}$ & $66.2_{-1.1}^{+2.4}$ & $67.1_{-0.8}^{+1.7}$ & $68.3_{-0.6}^{+0.6}$ & $67.8_{-0.6}^{+0.8}$ & $68.2_{-0.6}^{+0.6}$ & $69.0_{-0.6}^{+0.6}$ & $66.8_{-0.9}^{+0.9}$ & $68.2_{-0.6}^{+0.6}$\\
$\lambda$ & \multicolumn{1}{c}{---} & $0.8_{-0.8}^{+0.2}$ & $0.9_{-0.9}^{+0.3}$ & \multicolumn{1}{c}{---} & $0.5_{-0.5}^{+0.1}$ & $0.8_{-0.8}^{+0.3}$ & \multicolumn{1}{c}{---} & $1.2_{-0.2}^{+0.2}$ & $1.4_{-0.4}^{+0.2}$\\
$\beta$ & \multicolumn{1}{c}{---} & $-0.24_{-0.09}^{+0.24}$ & \multicolumn{1}{c|}{---} & \multicolumn{1}{c}{---} & $-0.26_{-0.24}^{+0.07}$ & \multicolumn{1}{c|}{---} & \multicolumn{1}{c}{---} & $-0.39_{-0.11}^{+0.03}$ & \multicolumn{1}{c|}{---}\\
$\log_{10}(-\beta)$ & \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} & $2.5_{-5.5}^{+4.5}$ & \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} & $3.1_{-1.1}^{+3.9}$ & \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} & $2.8_{-0.8}^{+1.7}$\\
\end{tabular}
\end{ruledtabular}
\end{table*}
\subsection{$\chi^2$-values}\label{sec:chisq}
In Table~II we show the $\chi^2$ values for the best-fitting $\Lambda$CDM, T3 and $\text{T3}_\text{ph}$ models. Our T3 and $\text{T3}_\text{ph}$ models can reconcile CMB, BAO and LSS data. As can be seen from the last two lines in Table~II, when the SZ cluster dataset is included the preference for the T3 and $\text{T3}_\text{ph}$ models is strong.
\begin{table}
\caption{\label{tab:chisq}$\chi^2$-values for $\Lambda$CDM, the T3-model and the T3ph-model for all tested datasets. The preference for T3 and $\text{T3}_\text{ph}$ is strong when cluster data is included.}
\begin{ruledtabular}
\begin{tabular}{l|rrrrr}
Dataset & $\chi^2_{\Lambda\text{CDM}}$ & $\chi^2_\text{T3}$ & $\chi^2_{\text{T3}_\text{ph}}$ &$\Delta \chi^2_\text{T3}$ & $\Delta \chi^2_{\text{T3}_\text{ph}}$\\
\hline
TT & $11261.80$ & $11265.12$ & $11265.20$ & $-3.32$ & $-3.40$\\
TT+J & $11946.80$ & $11949.54$ & $11950.34$ & $-2.74$ & $-3.54$\\
CMB & $11271.80$ & $11271.78$ & $11273.18$ & $0.02$ & $-1.38$\\
CMB+J & $11956.52$ & $11956.86$ & $11957.26$ & $-0.34$ & $-0.74$\\
CMB+B & $11274.46$ & $11275.58$ & $11274.88$ & $-1.12$ & $-0.42$\\
CMB+B+J & $11958.80$ & $11958.68$ & $11960.22$ & $0.12$ & $-1.42$\\
CMB+B+SZ & $11293.50$ & $11279.44$ & $11276.30$ & $14.06$ & $17.20$\\
CMB+B+SZ+J & $11978.38$ & $11965.84$ & $11961.90$ & $12.54$ & $16.48$\\
\end{tabular}
\end{ruledtabular}
\end{table}
\section{Conclusions}
\label{sec:conclusions}
We have identified an interacting dark energy model which can suppress structure growth and can reconcile CMB and LSS observations. It is a pure momentum transfer model and belongs to a class of theories constructed using the Lagrangian pull-back formalism for fluids --- the coupling function characterising the theory is not added at the level of the equations, but at the level of the action. In this way various pathologies and instabilities can be very easily identified; considering the ghost-free branch of the model, we investigated its observational signatures on the CMB and linear matter power spectra. For a constant coupling parameter $\beta$ and our specific choice of potential, the model exhibits structure growth for positive $\beta$ and growth suppression for negative $\beta$. Focusing on the latter case, we performed an MCMC analysis and found that using CMB and BAO data our model is as good as $\Lambda$CDM, while adding cluster data our model becomes strongly prefered. We note, however, that it still exhibits tension with local measurements of the Hubble constant. A full model selection analysis based on the Bayesian evidence is left for future work. However, we note that when the likelihood method shows preference for an extra parameter at a level of $3 \sigma$, so does the Bayesian analysis which is also quite sensitive to the priors used (see~\cite{Battye:2014qga} for a related analysis and discussion for the case of massive neutrinos).
Our work offers a promising alternative for resolving the CMB and LSS tension.
Another alternative is massive neutrinos, which have also been proposed to lift the discrepancy~\cite{Battye:2013xqa} but they increase the tension with the Hubble constant~\cite{Giusarma:2013pmn}. A recent interesting proposal was presented in~\cite{Lesgourgues:2015wza}, in which dark matter interacts with a new form of dark radiation and structure growth is damped via momentum transfer effects.
On the other hand, there is also the possibility that this tension is a result of poorly understood systematic effects. In the imminent future, a set of larger and better optical large scale structure surveys (the Dark Energy Survey, the Euclid satellite, the Large Synoptic Survey Telescope) as well as new probes with completely different methodology and systematics (e.g. 21cm intensity mapping with the Square Kilometre Array~\cite{Santos:2015gra}) will either resolve this tension or confirm the exciting prospect of new physics.
In order to take full advantage of current and future large scale structure datasets, understanding of the non-linear effects of exotic dark energy models is crucial. For example, in order to use the full range of the available data with confidence, one needs to correct the power spectrum on small (non-linear) scales. $N$-body simulations related to pure momentum transfer in the dark sector have been performed in~\cite{Baldi:2014ica, Baldi:2016zom}, based on the elastic scattering model presented in~\cite{Simpson:2010vh}. We plan to investigate non-linear effects for the negative coupling Type-3 models in future work.
To conclude, we may have discovered a whole family of models (Type-3-like models with pure momentum transfer) that can give suppression of growth and reconcile the tension between CMB and LSS. In this work we focused on a particular case, but there is a plethora of different choices that give models belonging to the same class. For example, we could use another coupling function $h(Z)$ and/or another form for the quintessence potential. However, this way the results are strongly model-dependent. Using the PPF approach developed in~\cite{Skordis:2015yra}, we can try to parametrise the free non-zero functions that define Type-3 models in a model-independent way and study their observational consequences; arguably, we should be able to constrain these free functions (or their combinations) such that they give late time growth suppression relative to $\Lambda$CDM --- which is what the available data currently prefer. This would be very important for phenomenological model building and for determining the constraining and discriminating power of future surveys.
\FloatBarrier
\section{Acknowledgments}
A.P. acknowledges support by STFC grant ST/H002774/1. T.T. acknowledges support by STFC grant ST/K00090X/1. Numerical computations were performed using the Sciama High Performance Compute (HPC) cluster which is supported by the ICG, SEPNet and the University of Portsmouth. We are indebted to Robert Crittenden for very useful comments and feedback.
We would also like to thank Ed Copeland, Kazuya Koyama, Jeremy Sakstein and Constantinos Skordis for useful discussions.
\FloatBarrier
\bibliographystyle{apsrev}
|
1,314,259,996,518 | arxiv | \section{Introduction}
A subalgebra $\mathcal{A}$ of the algebra $\mathcal{B}(\mathcal{H})$ of all bounded operators on a Hilbert space $\mathcal{H}$ is said to be \textit{inverse-closed} if every $a\in\mathcal{A}$ which is invertible in $\mathcal{B}(\mathcal{H})$ is also invertible in $\mathcal{A}$. The question whether an algebra of convolution operators on a Lie group, or simply ${\RR^n}$ is \textit{inverse-closed} is not new. In 1953, Calder{\'o}n and Zygmund \cite{cz2} showed that the class of convolution operators on $L^{2}({\RR^n})$ whose kernels are homogeneous of degree $-n$ and are locally in $L^{q}({\RR^n})$ away from the origin, has the property. Here $1<q<\infty$. Much later the result was generalized by Christ \cite{christ} who proved that similar algebras on a homogeneous group are \textit{inverse-closed}. A homogeneous group $\mathbb{G}$ is a nilpotent Lie group with dilations, a very natural generalization of the homogeneous structure on ${\RR^n}$.
Another direction has been taken by Christ and Geller \cite{christ-geller} who dealt with the algebra of operators with kernels which are homogeneous of degree $-n$ and smooth away from the identity on a homogeneous group with gradation. This algebra is \textit{inverse-closed} too. A step further has been made by G{\l}owacki \cite{87} who showed that this is so for any homogeneous group.
The kernels which are smooth away from the identity allow an interesting generalization. One can relax the homogeneity condition demanding only that the kernel satisfies the estimates
$$|\d_{x}^{\alpha}K(x)|\lesssim |x|^{-Q-|\alpha|},$$
where $Q$ is the homogeneous dimension of the group. The cancellation condition takes the form
$$|\langle K,\phi\rangle|=|\int_{\mathbb{G}}\phi(x)K(x)dx|\lesssim \Vert\phi\Vert,$$
for $\phi\in\ss (\mathbb{G})$, where $\Vert\cdot\Vert$ is a fixed seminorm in the Schwartz space $\ss (\mathbb{G})$. Such kernels $K$ are often called the \textit{Calder{\'o}n-Zygmund} kernels and the corresponding operators $\mathop{\rm Op}\nolimits (K)$ the \textit{Calder{\'o}n-Zygmund} operators. The class is closed under the composition of operators (Cor{\'e}-Geller \cite{core}) so they form an algebra. It has been proved recently (G{\l}owacki \cite{cz3}) that this algebra is \textit{inverse-closed} as well.
Let us specify the notion to the Heisenberg group which is the group under study in this paper. For the sake of simplicity, let us consider here only the one-dimensional case.
The underlying manifold of $\mathbb{H}$ is $\mathbb{R}^{3}$ which we write down as
$$\mathbb{H}=\mathbb{H}_{1}\oplus\mathbb{H}_{2}=\mathbb{R}^{2}\oplus\mathbb{R}.$$
In these coordinates the group law is
$$(w,t)\circ (v,s)=(w+v,t+s+w_{1}v_{2}),$$
where $w=(w_{1},w_{2})$, $v=(v_{1},v_{2})\in\mathbb{R}^{2}$, $t,s\in\mathbb{R}$. There are many choices of compatible dilations, but the most natural is
\[
\delta_{r}(w,t)=(rw,r^{2}t),\qquad r>0.
\]
In this setting the size condition for a \textit{Calder{\'o}n-Zygmund} kernel reads
$$|\partial_{w}^{\alpha}\d_{t}^{\beta}K(w,t)|\lesssim (|w|+|t|)^{-4-\alpha-2\beta}.$$
The cancellation condition doesn't get any simpler, so we do not repeat it here.
We are going to compare these conditions with the estimates that define \textit{flag kernels} which are the main object of study in this paper. Flag kernels were introduced by M{\"u}ller-Ricci-Stein \cite{muller} and Nagel-Ricci-Stein \cite{nagel} in their study of Marcinkiewicz multipliers (the first paper) and CR manifolds (the other one). These kernels are much more singular than the \textit{Calder{\'o}n-Zygmund} kernels. Accordingly, the definition is more complex. We consider a tempered distribution $K$ on $\mathbb{H}$ which is smooth for $w\neq 0$ and satisfies the estimates
$$|\d_{w}^{\alpha}\d_{t}^{\beta}K(w,t)|\lesssim |w|^{-2-\alpha}(|w|+|t|)^{-2-2\beta},\qquad w\neq 0,$$
as well as the following three cancellation conditions:
1) For every $\phi\in\ss (\mathbb{H}_{1})$, the distribution
$$f\mapsto\int_{\mathbb{H}}\phi(w)f(t)dwdt$$
is a \textit{Calder{\'o}n-Zygmund} kernel on $\mathbb{H}_{2}$,
2) For every $\phi\in\ss (\mathbb{H}_{2})$, the distribution
$$f\mapsto\int_{\mathbb{H}}f(w)\phi(t)dwdt$$
is a \textit{Calder{\'o}n-Zygmund} kernel on $\mathbb{H}_{1}$,
3) For every $\phi\in\ss (\mathbb{H})$,
$$|\int_{\mathbb{H}}\phi(w,t)dwdt|\lesssim 1.$$
Finally, for given $\alpha,\beta$, the estimates are uniform with respect to $\phi$ if $\phi$ stays in a bounded set in the respective Schwartz space.
The operators with flag kernels share some properties with the \textit{Calder{\'o}n-Zygmund} operators. They are bounded on $L^{p}(\mathbb{G})$-spaces and form an algebra (see M{\"u}ller-Ricci-Stein \cite{muller}, Nagel-Ricci-Stein \cite{nagel}, Nagel-Ricci-Stein-Wainger \cite{nagel-ricci}, G{\l}owacki \cite{colloquium2010}, G{\l}owacki \cite{lp}). We are, however, interested in the inversion problem for this class. Before going any further, let us pause for a moment and consider the simplest case, namely that of an Abelian group ${\RR^n}$. Then, the Fourier transform $\widehat{K}$ is a function on ${\RR^n}$ which is smooth away from the origin and satisfies the estimates
\begin{align*}
|\d^{\alpha}_{\xi}\widehat{K}(\xi)|\lesssim |\xi|^{-|\alpha|},\qquad\xi\neq 0.
\end{align*}
These estimates are, equivalent to the ones defining the \textit{Calder{\'o}n-Zygmund} kernel. If the operator $\mathop{\rm Op}\nolimits (K)$ is invertible, then
$$|\widehat{K}(\xi)|\geqslant c>0,\ \ \ 0\neq\xi\in{\RR^n},$$
and it is directly checked that $\widehat{L}=1/\widehat{K}$ satisfies analogous estimates, so that $L$ is a flag kernel such that $L\star K=K\star L=\delta_{0}$.
A similar idea works for the Heisenberg group $\mathbb{H}$. Let $\pi^{\lambda}$ denote the Schr{\"o}dinger representation of $\mathbb{H}$ with the Planck constant $\lambda\neq 0$, If $K$ is a flag kernel on $\mathbb{H}$ such that the operator $\mathop{\rm Op}\nolimits (K)$ is invertible, then, for every $\lambda\neq 0$, the operator $\pi_{K}^{\lambda}$ is invertible and can be regarded as a pseudodifferential operator in a suitable class. By the Beals theorem, the inverse belongs to the same class. Now, the estimates are uniform in $\lambda$, so one can recover the kernel of the inverse operator from the kernels of $(\pi_{K}^{\lambda})^{-1}$ and show that it is a flag kernel. Thus the algebra of the operators with flag kernels on the Heisenberg group turns out to be \textit{inverse-closed}. We believe that similar method could be used in the case of a general 2-step nilpotent Lie group.
There is a technicality in the proof we want to comment on. There exists no universal definition of the extension of the unitary representation to a space of distributions. One has to rely on specific properties of the distribution space in question. Everything works fine for distributions with compact support. A \textit{Calder{\'o}n-Zygmund} kernel can be split into a compactly supported part and a part which is square integrable, so there is no problem with the definition of $\pi_{K}^{\lambda}$. No such splitting is available for flag kernels. Instead we modify the domain of the distribution. Originally, a distribution is a functional on the Schwartz space. We introduce two other spaces on which flag kernels can be regarded as continuous functionals. The cancellation conditions are important here. We also take adventage of a functional calculus of G{\l}owacki \cite{arkiv2007}. Once $\pi_{K}^{\lambda}$ is defined for our flag kernel, we can follow the path outlined above.
\section{Preliminaries}
The main structure of this work is the Heisenberg group. As a set it is
\[\mathbb{H}^{n}={\RR^n}\times{\RR^n}\times\mathbb{R}.
\]
Elements of the group will usually be denoted by
\[
\mathbb{H}^{n}\ni h=(x,y,t)=(v,t).
\]
The group multiplication is
\[
(x,y,t)\cdot(x',y',t')=(x+x',y+y',t+t'+xy').
\]
We define a homogeneous norm on ${\HH^n}$ as
\[
|h|=\Vert v\Vert+|t|^{\frac{1}{2}}=\sum_{i=1}^{2n}|v_{i}|+|t|^{\frac{1}{2}}
\]
with the corresponding family of dilations
\[
\delta_{j}(h)=\delta_{j}(v,t)=(jv,j^{2}t),\qquad j>0,
\]
in the sense that $|\delta_{j}(h)|=j|h|$.
The set $\{\delta_{j}\}_{j>0}$ actually forms a group of automorphisms. The homogeneous dimension is the number $Q=2n+2$. We will use the designations
\[
\d^{\gamma}_{h}=\d_{v}^{\alpha}\d_{t}^{\beta}=\d_{v_{1}}^{\alpha_{1}}\d_{v_{2}}^{\alpha_{2}}...\d_{v_{2n}}^{\alpha_{2n}}\d_{t}^{\beta}
\]
and
\[
|\gamma|=|\alpha|+2\beta=\sum_{i=1}^{2n}\alpha_{i}+2\beta,
\]
where $\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{2n}),\ \alpha_{k},\beta\in\mathbb{N}$.
One of the main tools is the abelian Fourier transform defined by the formula
$$\widehat{f}(\zeta):=\int_{{\HH^n}}f(h)e^{-2\pi ih\zeta}dh,$$
where
\[
\mathbb{H}_{n}\ni \zeta=(\xi,\eta,\lambda)=(w,\lambda),
\hspace{2cm}
\mathbb{H}_{n}=\mathbb{R}_{n}\times\mathbb{R}_{n}\times\mathbb{R}.
\]
It can be first defined for Schwartz functions
\[
\mathcal{S}(\mathbb{H}^{n})=\{f\in\mathcal{C}^{\infty}(\mathbb{H}^{n}):\forall N\in\mathbb{N}\sup_{h\in{\HH^n}}\max_{|\gamma|\leqslant N}|\partial_{h}^{\gamma}f(h)|(1+|h|)^{N}<\infty\}
\]
and then lifted to the Lebesque space of the square-integrable functions
\[
L^{2}({\HH^n})=\{f:\int_{{\HH^n}}|f(h)|^{2}dh<\infty\}
\]
or to the space of tempered distributions $\ss'({\HH^n})$ wchich is the space of all continuous linear functionals on $\ss({\HH^n})$ in the sense of the usual seminorm topology.
For $S\in\ss '({\HH^n}),\ f\in\ss ({\HH^n})$ one can put
$$\langle\widehat{S},f\rangle:=\langle S,\widehat{f}\rangle.$$
Let also
\[
f^{\star}(x)=\overline{f(x^{-1})},\quad\langle S^{\star},f\rangle:=\langle S,f^{\star}\rangle,\quad S\in\ss'({\HH^n}),\ f\in\ss ({\HH^n}).
\]
By $\delta_{0}$ we will denote the Dirac distribution.
\section{Flag kernels and their convolution operators}
Automorphisms $\{\delta_{j}\}_{j>0}$ decompose our group ${\HH^n}$ into their eigenspaces
$$\mathbb{G}_{1}\oplus\mathbb{G}_{2}\ni (v,t).$$
Theorem 2.3.9 of Nagel-Ricci-Stein \cite{nagel} says that there is a one-to-one correspodence between flag kernels and their multipliers. It allows us to define flag kernels as follows.
\begin{definition}
Let ${\HH^n}$ be the Heisenberg group and
$$\mathbb{H}_{n}=\mathbb{G}_{1}^{\star}\oplus\mathbb{G}_{2}^{\star}\ni (w,\lambda)$$ the dual vector space to ${\HH^n}$. We say that a tempered distribution $K$ is a flag kernel iff its Fourier transform $\widehat{K}$ agrees with a smooth function outside of hyperspace
$\{(w,\lambda):\lambda=0\}$ and satisfies the estimates
\begin{center}
$|\partial_{w}^{\alpha}\partial_{\lambda}^{\beta}\widehat{K}(w,\lambda)|\leqslant c_{\alpha,\beta}(\Vert w\Vert+|\lambda|^{1/2})^{-|\alpha|}|\lambda|^{-\beta}$,\ \ \ all $\alpha,\beta$.
\end{center}
\end{definition}
Observe that in particular $\widehat{K}$ belongs to $L^{\infty}(\mathbb{H}_{n})$.
For $f\in\mathcal{S}({\HH^n})$, $K\in\mathcal{S}'({\HH^n})$ we define their convolution as $$K\star f(h):=\langle K,\, _{h^{-1}}\!\widetilde{f}\rangle=\int_{{\HH^n}}K(h')f(h'^{-1}h)dh',$$
where $\widetilde{f}(h)=f(h^{-1})$.
Our point of departure are the following two theorems:
\begin{theorem}
Let $K$ be a flag kernel on the Heisenberg group. Then
\[
\Vert K\star f\Vert_{2}\lesssim \Vert f\Vert_{2},\qquad f\in\ss({\HH^n}).
\]
Hence we have an $L^{2}$-bounded operator $\mathop{\rm Op}\nolimits(K)f:=K\star f$.
\end{theorem}
\begin{theorem}\label{splot}
Let $K,S$ be flag kernels on the Heisenberg group ${\HH^n}$ and $$T:=\mathop{\rm Op}\nolimits(K)\mathop{\rm Op}\nolimits(S).$$ Then, there exists a flag kernel $L$ such that $T=\mathop{\rm Op}\nolimits(L)$.
\end{theorem}
Thus the flag kernels give rise to convolution operators, bounded on $L^{2}({\HH^n})$, which form a subalgebra of $\mathcal{B}(L^{2}({\HH^n}))$. For convenience we will write $L=K\star S$.
These are theorems of Nagel-Ricci-Stein \cite{nagel} who proved them for a class of homogeneous groups which includes all two-step homogeneous groups (Theorems 2.6.B and 2.7.2). Partial results can be found in an earlier paper of M{\"u}ller-Ricci-Stein \cite{muller} (Theorem 3.1). They were subsequently generalized for all homogeneous group independently and virtually simultanously by Nagel-Ricci-Stein-Wainger \cite{nagel-ricci} and G{\l}owacki \cite{colloquium2010}.
The aim of this paper is to prove the following theorem.
\begin{theorem}\label{nasze}
Let ${\HH^n}$ be the Heisenberg group. Let $K$ be a flag kernel on ${\HH^n}$. Suppose that the operator $\mathop{\rm Op}\nolimits(K)$ is invertible on $L^{2}({\HH^n})$. Then there exists a flag kernel $L$, such that for all $f\in L^{2}({\HH^n})$
$$\mathop{\rm Op}\nolimits(K)^{-1}f=L\star f=\mathop{\rm Op}\nolimits(L)f.$$
\end{theorem}
Observe that $\mathop{\rm Op}\nolimits(K)$ is translation invariant.
Further the same holds for its inversion.
By general theory it follows that there exists a tempered distribution $L$ such that $\mathop{\rm Op}\nolimits(K)^{-1}f=L\star f$. Now it suffices to show that $L$ is a flag kernel. In the following considerations we can assume that the flag kernel $K$ is symmetric, i.e. $K=K^{\star}$. In fact if Theorem \ref{nasze} is true for such kernels, let us pick an arbitrary flag kernel $K$. Then, we can consider kernels $K^{\star}\star K$ and $K\star K^{\star}$ which are symmetric. By Theorem \ref{splot} they are both flag kernels. Therefore, by Theorem \ref{nasze}, there exist flag kernels $S,T$ such that
\[
S\star(K^{\star}\star K)=\delta_{0}
\hspace{1cm}
\&
\hspace{1cm}
(K\star K^{\star})\star T=\delta_{0}.
\]
Again, by Theorem \ref{splot} and associativity, it follows that there exist flag kernels $L_{1},L_{2}$ such that
\[
L_{1}\star K=\delta_{0}
\hspace{1cm}
\&
\hspace{1cm}
K\star L_{2}=\delta_{0}.
\]
The identity
$$L_{1}=L_{1}\star (K\star L_{2})=(L_{1}\star K)\star L_{2}=L_{2}$$
ends the proof of our theorem for an arbitrary flag kernel $K$.
\section{Schr{\"o}dinger representation}
\begin{definition}
For $\lambda\neq 0$ and $h\in{\HH^n}$ we define the family of operators $$\{\pi_{h}^{\lambda}:h\in{\HH^n},\lambda\neq 0\},$$ all acting on the same $L^{2}({\RR^n})$ by the following formula
\[
\pi_{h}^{\lambda}f(s):=\left\{
\begin{array}{ll}
e^{2\pi i\lambda t}e^{2\pi i\sqrt{\lambda}ys}f(s+\sqrt{\lambda}x);\ \lambda>0,
\cr
e^{2\pi i\lambda t}e^{2\pi i\sqrt{|\lambda|}ys}f(s-\sqrt{|\lambda|}x);\ \lambda<0.
\end{array}
\right.
\]
\end{definition}
For a Hilbert space $\mathcal{H}$, denote by $\mathcal{U}(\mathcal{H})$, $\mathcal{B}(\mathcal{H})$ the spaces of all unitary and bounded operators, respectively. It is well-known (see, e.g. Folland \cite{folland}, sec. 1.3) that, for every $\lambda\neq 0$,
$${\HH^n}\ni h\longmapsto\pi_{h}^{\lambda}\in\mathcal{U}(L^{2}({\RR^n}))$$
is a unitary representation on the Hilbert space $L^{2}({\RR^n})$.
$$L^{1}({\HH^n})\ni f\longmapsto\pi_{f}^{\lambda}\in\mathcal{B}(L^{2}({\RR^n}))$$
is a representation of $\star$-algebra $L^{1}({\HH^n})$ on the Hilbert space $L^{2}({\RR^n})$.
\section{Useful notation}
For $f,g\in\ss({\RR^n})$ we define the function
\[
c_{f,g}(x,y):=\int_{{\RR^n}}e^{2\pi iyu}f(u+x)g(u)du.
\]
In particular
\[
\widehat{c_{f,g}}(\xi,\eta)=\widehat{f}(\xi)g(\eta)e^{2\pi i\xi\eta}
.\]
Let also
\[
C_{f,g}^{\lambda}(x,y,t):=\langle\pi_{h}^{\lambda}f,g\rangle
\]
Let $\lambda>0$. One can calculate that
\begin{align*}
C_{f,g}^{\lambda}(x,y,t)=\int_{{\RR^n}}\pi_{(x,y,t)}^{\lambda}f(u)g(u)du=e^{2\pi it\lambda}c_{f,g}(\sqrt{\lambda}x,\sqrt{\lambda}y).
\end{align*}
We also have
\begin{align*}
\widehat{c_{f,g}\circ\delta_{\sqrt{\lambda}}}(\xi,\eta)&=\int\int\int e^{-2\pi ix\xi}e^{-2\pi iy\eta}e^{2\pi i\sqrt{\lambda}yu}f(u+\sqrt{\lambda}x)g(u)dudxdy
\\
&=\int\int\int\lambda^{-n/2}e^{-2\pi ix\frac{\xi}{\sqrt{\lambda}}}e^{2\pi iu\frac{\xi}{\sqrt{\lambda}}}e^{-2\pi iy\eta}e^{2\pi i\sqrt{\lambda}yu}f(x)g(u)dudxdy\\
&=\int\int\lambda^{-n/2}\widehat{f}(\frac{\xi}{\sqrt{\lambda}})e^{2\pi iu(\frac{\xi}{\sqrt{\lambda}}+\sqrt{\lambda}y)}e^{-2\pi iy\eta}g(u)dudy\\
&=\int\lambda^{-n/2}\widehat{f}(\frac{\xi}{\sqrt{\lambda}})g^{\vee}(\frac{\xi}{\sqrt{\lambda}}+\sqrt{\lambda}y)e^{-2\pi iy\eta}dy\\
&=\int|\lambda|^{-n}\widehat{f}(\frac{\xi}{\sqrt{\lambda}})g^{\vee}(y)e^{-2\pi iy\frac{\eta}{\sqrt{\lambda}}}e^{2\pi i\frac{\xi}{\sqrt{\lambda}}\frac{\eta}{\sqrt{\lambda}}}dy\\
&=|\lambda|^{-n}\widehat{f}(\frac{\xi}{\sqrt{\lambda}})g(\frac{\eta}{\sqrt{\lambda}})e^{2\pi i\frac{\xi}{\sqrt{\lambda}}\frac{\eta}{\sqrt{\lambda}}}=|\lambda|^{-n}\widehat{c_{f,g}}(\frac{\xi}{\sqrt{\lambda}},\frac{\eta}{\sqrt{\lambda}}).
\end{align*}
Moreover
\[
\widehat{C_{f,g}^{\lambda}}(\xi,\eta,r)=\widehat{c_{f,g}\circ\delta_{\sqrt{\lambda}}}\otimes\widehat{e^{2\pi i(\cdot)\lambda}}(\xi,\eta,r)=|\lambda|^{-n}\widehat{c_{f,g}}\otimes\delta_{\lambda}(\frac{\xi}{\sqrt{\lambda}},\frac{\eta}{\sqrt{\lambda}},r),
\]
where $\delta_{\lambda}$ is a Dirac distribution supported at $\lambda$. For $\lambda<0$ the above formula should be slightly modified. For such $\lambda$ one can get analogously
\[
\widehat{C_{f,g}^{\lambda}}(\xi,\eta,r)=-|\lambda|^{-n}\widehat{c_{f,g}}\otimes\delta_{\lambda}(-\frac{\xi}{|\lambda|^{1/2}},\frac{\eta}{|\lambda|^{1/2}},r).
\]
Suppose that $a$ is a function on ${\RR^n}\times{\RR^n}$ which is bounded or square-integrable. Then, the weakly defined operator
\begin{align*}
\langle Af,g\rangle &=\int\int e^{2\pi i\xi\eta}a(\xi,\eta)\widehat{f}(\xi)g(\eta)d\eta d\xi
\\
&=\int\int a(\xi,\eta)\widehat{c_{f,g}}(\xi,\eta)d\xi d\eta=\langle a,\widehat{c_{f,g}}\rangle
\end{align*}
is a continuous mapping from $\ss ({\RR^n})$ to $\ss' ({\RR^n})$. It is often denoted by $A=a(x,D)$ and called a pseudodifferential operator with the KN (Kohn-Nirenberg) symbol $a$.
\section{The class $\ss_{0}$ and the operator $\pi_{K}^{\lambda}$}
Let $g$ be a function on $\mathbb{H}_{n}$ such that $g(w,\lambda)=\phi(\lambda)$, where $\phi\in\mathcal{C}_{c}^{\infty}(\mathbb{G}_{2}^{\star})$. Then
$g^{\vee}(u,t)=\delta_{0}\otimes\phi^{\vee}(u,t)$.
Observe that if for example $f\in\ss({\HH^n})$, then
\begin{align*}
f\star g^{\vee}(h)&=\int f(hr^{-1})dg^{\vee} (r)=\int f(h-r)dg^{\vee} (r)
\\
&=\int f(r^{-1}h)dg^{\vee} (r)=g^{\vee}\star f(h),
\end{align*}
so $g^{\vee}$ is a central measure.
We will need a notion of the $\lambda$-support of a function $f$. By definition a real number $\lambda_{0}$ is not in the $\lambda$-$\mathrm{supp\,}(f)$ iff there exists $\varepsilon$ such that no point $(w,\lambda)$, where $\lambda\in (\lambda_{0}-\varepsilon,\lambda_{0}+\varepsilon)$, is in the support of $f$.
\begin{definition}\label{lambda}
We say that a Schwartz function $f$ is in $\ss_{0} ({\HH^n})$ iff
\[
(\exists\varepsilon>0)\forall\phi\in\mathcal{C}_{c}^{\infty}((-\varepsilon,\varepsilon)\cup(-1/\varepsilon,-\infty)\cup(1/\varepsilon,\infty))\,\widehat{f}\phi=0,
\]
that is, iff $\lambda$-$\mathrm{supp\,} (f)$ is bounded and does not contain $0$.
\end{definition}
\begin{lemma}
Suppose $f\in\ss_{0} ({\HH^n})$ and $K$ is a flag kernel. Then $K\star f$ is in $\ss_{0} ({\HH^n})$.
\end{lemma}
\begin{proof}
Let us define first $a\# b:=(a^{\vee}\star b^{\vee})^{\wedge}$. As $f$ is in $\ss_{0}({\HH^n})$ take $\varepsilon,\phi$ which satisfy the condition of the definition \ref{lambda}. We have
\begin{align*}
\widehat{K\star f}\phi=(K\star f\star\phi^{\vee})^{\wedge}=\widehat{K}\#\widehat{f}\phi=0,
\end{align*}
so the same $\varepsilon$ works also for $K\star f$.
It remains to explain why $K\star f$ is an element of $\ss({\HH^n})$.
Let $\psi\in\mathcal{C}_{c}^{\infty}(\mathbb{R}\setminus\{0\})$ be equal to 1 on $\lambda$-$\mathrm{supp\,}$ of $f$. Observe that $\psi^{\vee}$ can be thought of as a central measure. Thus
\begin{align*}
K\star f=K\star f\star\psi^{\vee}=K\star\psi^{\vee}\star f=K_{1}\star f,
\end{align*}
where $\widehat{K_{1}}$ is smooth and
\begin{align*}
|\d_{w}^{\alpha}\d_{\lambda}^{\beta}\widehat{K_{1}}(w,\lambda)|\leqslant c_{\alpha,\beta}(1+\Vert w\Vert+|\lambda|^{1/2})^{-|\alpha|}(1+|\lambda|)^{-\beta}.
\end{align*}
Now if we write that $a\in Sym^{N,M}({\HH^n})$ iff
\begin{align*}
|\d_{w}^{\alpha}\d_{\lambda}^{\beta}\widehat{a}(w,\lambda)|\leqslant c_{\alpha,\beta}(1+\Vert w\Vert+|\lambda|^{1/2})^{-|\alpha|-N}(1+|\lambda|)^{-\beta-M},
\end{align*}
then $K_{1}\in Sym^{0,0}({\HH^n})$, $f\in Sym^{N,M}({\HH^n})$ for all $N,M$ because it is a Schwartz function. By G{\l}owacki's symbolic calculus \cite{arkiv2007} (Theorem 6.4) governed by the metric
\[
g_{(w,\lambda)}(u,r)=\frac{\Vert u\Vert}{1+\Vert w\Vert+|\lambda|^{1/2}}+\frac{|r|}{(1+|\lambda|^{1/2})^{2}};\qquad (w,\lambda)\in\mathbb{H}_{n},\ (u,r)\in\mathbb{H}_{n}
\]
we have
\begin{align*}
K\star f&=K_{1}\star f\in Sym^{0,0}({\HH^n})\star \bigcap_{N,M}Sym^{N,M}({\HH^n})\\
&\subseteq\bigcap_{N,M}Sym^{0,0}({\HH^n})\star Sym^{N,M}({\HH^n})
\\
&\subseteq\bigcap_{N,M}Sym^{N,M}({\HH^n})\cong \ss ({\HH^n}).
\end{align*}
\end{proof}
\begin{lemma}
The class $S_{0} ({\HH^n})$ is dense in $L^{2}({\HH^n}).$
\end{lemma}
\begin{proof}
Let $g\in L^{2}({\HH^n})$ be such that $\forall f\in S_{0}({\HH^n})\ \langle g,f\rangle=0.$
Then $\langle\hat{g},\hat{f}\rangle=0$. Hence $\mathrm{supp\,}\hat{g}\subseteq\mathbb{R}^{2n}\times\{0\}.$
But it implies that $g=0$ almost everywhere.
\end{proof}
\begin{lemma}
The G{\"a}rding space $$\mathcal{G}^{\lambda}:=\{\pi_{\phi}^{\lambda}f: \phi\in S_{0}({\HH^n}), f\in L^{2}({\RR^n})\}$$
is dense in $L^{2}({\RR^n}).$
\end{lemma}
\begin{proof}
Take any $g\in L^{2}({\RR^n})$ such that for all $\phi\in S_{0},\ f\in L^{2}({\RR^n})$ we have
$\langle g,\pi_{\phi}^{\lambda}f\rangle =0.$ We will show that $g=0$ a.e.
Consider only those functions $\phi$ which can be decomposed as $\phi(x,y,t)=\phi_{1}(x,y)\phi_{2}(t).$ Then
\begin{align*}
0&=\langle g,\pi_{\phi}^{\lambda}f\rangle =\langle g,\int_{{\HH^n}}\phi(h)\pi_{h}^{\lambda}fdh\rangle=\int_{{\HH^n}}\phi(h)\langle g,\pi_{h}^{\lambda}f\rangle dh\\
&=\int_{\mathbb{R}}\left(\int_{\mathbb{R}^{2n}}\phi_{1}(x,y)\langle g,\pi_{h}^{\lambda}f\rangle dxdy\right)\phi_{2}(t)dt=\int_{\mathbb{R}}F(t)\phi_{2}(t)dt.
\end{align*}
It follows that $\langle\hat{F},\hat{\phi_{2}}\rangle=0$, which, by the structure of $\ss_{0} ({\HH^n})$, implies that
$$\mathrm{supp\,}\hat{F}\subseteq\{0\}.$$
Hence $\hat{F}=\sum_{n=0}^{N}c_{n}\delta_{0}^{(n)}.$
By the Schwarz inequality
$||F||_{\infty}\leq ||f||_{2}||g||_{2}||\phi_{1}||_{1}$. So $F\in L^{\infty}(\mathbb{R})$ and at the same time
$F(t)=\sum_{n=0}^{N}c_{n}t^{n}.$
Consequently, there is no other option than $F=const$, which means that
$$\int_{\mathbb{R}^{2n}}\phi_{1}(x,y)\langle g,\pi_{h}^{\lambda}f\rangle dxdy=c_{\phi_{2}}.$$
By the density of $S(\mathbb{R}^{2n})$ in $L^{2}(\mathbb{R}^{2n})$, we have that the expression $\langle g,\pi_{h}^{\lambda}f\rangle$ does not depend on the variable $t$.
So $(\pi_{(0,0,t)}^{\lambda}-I)g=0$ for all $t$,
which leads to a contradiction unless $g=0$ a.e.
\end{proof}
\begin{lemma}
Let $K,L$ be flag kernels such that $K\upharpoonright_{\ss_{0}({\HH^n})}=L\upharpoonright_{\ss_{0}({\HH^n})}$. Then, $K=L$.
\end{lemma}
\begin{proof}
If $\langle K,f\rangle =\langle L,f\rangle$ for $f\in\ss_{0}({\HH^n})$, then
$\langle K-L,f\rangle=0$ and so
$\langle\widehat{K-L},\widehat{f}\rangle =0$. Therefore from the definition of the class $\ss_{0}({\HH^n})$ for every $\lambda$ nonzero element of the center of $\mathbb{H}_{n}$, we have $\widehat{K-L}=0.$ Thus $\widehat{K}=\widehat{L}$ as elements of $L^{\infty}(\mathbb{H}_{n})$, so $K=L$ in $\ss'({\HH^n})$.
\end{proof}
\begin{definition}
We denote by $\mathcal{B}_{0}({\HH^n})$ the class of all smooth functions such that their Fourier transforms are bounded measures whose $\lambda$-support does not contain $0$. One can norm this space with $\Vert f\Vert_{\mathcal{B}_{0}}=\Vert\widehat{f}\Vert_{\mathcal{M}}$, where $\Vert\cdot\Vert_{\mathcal{M}}$ denotes the total variation of a measure.
\end{definition}
Observe that $\ss_{0}({\HH^n})\subset\mathcal{B}_{0}({\HH^n})$. Moreover it also contains objects of type $C_{f,g}^{\lambda}$, as $\Vert C_{f,g}^{\lambda}\Vert_{\mathcal{B}_{0}}\leqslant\Vert\widehat{c_{f,g}}\Vert_{1}$. Now as $\ss_{0}({\HH^n})$ is total for flag kernels we can extend such a kernel from $\ss_{0}({\HH^n})$ to $\mathcal{B}_{0}({\HH^n})$ by the formula
\[
\langle K,f\rangle=\int_{\mathbb{H}_{n}}\widehat{K}(w,\lambda)d\widehat{f}(w,\lambda).
\]
Continuity is gained for free as $|\langle K,f\rangle|\leqslant\Vert \widehat{K}\Vert_{\infty}\Vert\widehat{f}\Vert_{\mathcal{M}}$.
Now we can define the representation of a flag kernel. Suppose first that $K\in\ss_{0}({\HH^n})$. Then,
\begin{align*}
\langle\pi_{K}^{\lambda}f,g\rangle =\langle\int_{{\HH^n}} K(h)\pi_{h}^{\lambda}fdh,g\rangle=\int_{{\HH^n}} K(h)\langle\pi_{h}^{\lambda}f,g\rangle dh=\langle K,C^{\lambda}_{f,g}\rangle,
\end{align*}
for $f,g\in\ss ({\RR^n})$.
Hence, for every flag kernel we put $\langle\pi_{K}^{\lambda}f,g\rangle:=\langle K,C^{\lambda}_{f,g}\rangle$ as a weak definition of its representation. Observe next that if $\lambda>0$
\begin{align*}
\langle\pi_{K}^{\lambda}f,g\rangle &=\int\int K(u,t)c_{f,g}(|\lambda|^{1/2}u)e^{2\pi it\lambda}dudt\\
&=\int\int\int\widetilde{\widehat{K}}(\xi,\eta,r)|\lambda|^{-n}\widehat{c_{f,g}}(\frac{\xi}{|\lambda|^{1/2}},\frac{\eta}{|\lambda|^{1/2}})d\xi d\eta d\delta_{\lambda}(r)\\
&=\int\int\widetilde{\widehat{K}}(|\lambda|^{1/2}\xi,|\lambda|^{1/2}\eta,\lambda)\widehat{c_{f,g}}(\xi,\eta)d\xi d\eta.
\end{align*}
Similar calculation for $\lambda<0$ leads to a conclusion that $\pi_{K}^{\lambda}$ is a pseudodifferential operator with the KN symbol
$$\widetilde{\widehat{K}}(\mathrm{sgn\,}(\lambda)|\lambda|^{1/2}\xi,|\lambda|^{1/2}\eta,\lambda).$$
\begin{lemma}
Let $K$ be a flag kernel and $\phi$ in $\ss_{0} ({\HH^n})$. Then, the operators $\pi_{K\star\phi}^{\lambda}$ and $\pi_{K}^{\lambda}\pi_{\phi}^{\lambda}$ are equal.
\end{lemma}
\begin{proof}
First one can calculate that
\begin{align*}
C_{\pi_{\phi}^{\lambda}f,g}^{\lambda}(h)&=\int\pi_{h}^{\lambda}\pi_{\phi}^{\lambda}f(s)g(s)ds=\int\pi_{h}^{\lambda}\int\phi(h')\pi_{h'}^{\lambda}f(s)dh'g(s)ds\\
&=\int\phi(h')\int\pi_{hh'}^{\lambda}f(s)g(s)dsdh'=\int\widetilde{\phi}(h')C_{f,g}^{\lambda}(hh'^{-1})dh'\\
&=C_{f,g}^{\lambda}\star\widetilde{\phi}(h).
\end{align*}
Now using fact that $K\star\phi$ is in $\ss_{0} ({\HH^n})$
\begin{align*}
\langle\pi_{K\star\phi}^{\lambda}f,g\rangle& =\int K\star\phi(h)\langle\pi_{h}^{\lambda}f,g\rangle dh=\int K\star\phi(h)C_{f,g}^{\lambda}(h)dh\\
&=\int K(h)C_{f,g}^{\lambda}\star\widetilde{\phi}(h)dh=\int K(h)C_{\pi_{\phi}^{\lambda}f,g}^{\lambda}(h)dh=\langle\pi_{K}^{\lambda}\pi_{\phi}^{\lambda}f,g\rangle.
\end{align*}
\end{proof}
\begin{corollary}
Let $K_{1},K_{2}$ be flag kernels. The operators $\pi_{K_{1}\star K_{2}}^{\lambda}$ and $\pi_{K_{1}}^{\lambda}\pi_{K_{2}}^{\lambda}$ are equal.
\end{corollary}
\begin{proof}
As flag kernels form an algebra, the above lemma implies
\begin{align}\label{splatanie}
\pi_{K_{1}\star K_{2}}^{\lambda}\pi_{\phi}^{\lambda}f=\pi_{K_{1}\star K_{2}\star\phi}^{\lambda}f=\pi_{K_{1}}^{\lambda}\pi_{K_{2}\star\phi}^{\lambda}f=\pi_{K_{1}}^{\lambda}\pi_{K_{2}}^{\lambda}\pi_{\phi}^{\lambda}f,
\end{align}
where the second equality follows by the fact that $K_{2}\star\phi\in\ss_{0} ({\HH^n})$. \ref{splatanie} proves that the operators agree on vectors of type $\pi_{\phi}^{\lambda}f$ which are dense in $L^{2}({\RR^n})$ when $\phi\in\ss_{0} ({\HH^n})$, $f\in L^{2}({\RR^n})$.
\end{proof}
\section{Representations of $L^{2}$}
We start from a simple calculation of the Kohn-Nirenberg symbol of $\pi_{f}^{\lambda}$, where $f$ is a Schwartz function. Let $\lambda$ be positive. We have
\begin{align*}
\pi_{f}^{\lambda}u(s)&=\int f(x,y,t)e^{2\pi it\lambda}e^{2\pi i\sqrt{\lambda}ys}u(s+\sqrt{\lambda}x)dxdydt\\
&=\int f(x,\sqrt{\lambda}s^{\vee},\lambda^{\vee})u(s+\sqrt{\lambda}x)dx
=|\lambda|^{-n/2}\int f(\frac{x-s}{\sqrt{\lambda}},\sqrt{\lambda}s^{\vee},\lambda^{\vee})u(x)dx\\
&=|\lambda|^{-n/2}\int\int f(\frac{x-s}{\sqrt{\lambda}},\sqrt{\lambda}s^{\vee},\lambda^{\vee})e^{2\pi ix\xi}\widehat{u}(\xi)d\xi dx\\
&=\int\int f(x,\sqrt{\lambda}s^{\vee},\lambda^{\vee})e^{2\pi i\sqrt{\lambda}x\xi}e^{2\pi is\xi}\widehat{u}(\xi)d\xi dx\\
&=\int f^{\vee}(\sqrt{\lambda}\xi,\sqrt{\lambda}s,\lambda)e^{2\pi is\xi}\widehat{u}(\xi)d\xi.
\end{align*}
As we can see in this case the symbol of $\pi_{f}^{\lambda}$ is also $$a(\xi,\eta)=\widetilde{\widehat{f}}(\mathrm{sgn\,}(\lambda)|\lambda|^{1/2}\xi,|\lambda|^{1/2}\eta,\lambda).$$
Let for a moment $x,y\in{\HH^n}$ and $A$ be a Hilbert-Schmidt operator on $\ss({\HH^n})$ with a kernel $\Omega$. One can calculate that
\begin{align*}
Au(x)=\int \Omega(x,y)u(y)dy=\int\Omega(x,y)\int e^{2\pi iy\xi}\widehat{u}(\xi)d\xi dy=\int\Omega(x,\xi^{\vee})\widehat{u}(\xi)d\xi.
\end{align*}
It is easy to see that if $a$ is the KN symbol of $A$, then
$$a(\xi,\eta)=e^{-2\pi i\xi\eta}\Omega(\xi,\eta^{\vee}),$$
and by the Plancharel formula,
\begin{align*}
\Vert A\Vert_{HS}=\Vert\Omega\Vert_{2}=\int\int|\Omega(\xi,y)|^{2}d\xi dy=\int\int|e^{-2\pi i\xi\eta}\Omega(\xi,\eta^{\vee})|^{2}d\xi d\eta=\Vert a\Vert_{2}.
\end{align*}
The sign change on the first coordinate, in a situation where $\lambda$ is negative, have no impact on the obstacles with which we struggle. Thus from now on in all calculations we will disregard this difference.
\begin{lemma}\label{l2}
Let $f\in L^{2}({\HH^n})$ and $\{f_{n}\}_{n}\subset\ss_{0} ({\HH^n})$ be such that $f_{n}\rightarrow f$ in $L^{2}$. For almost every $\lambda$, there exists a subsequence $\{f_{n_{k}(\lambda)}\}_{k}$ such that $\pi^{\lambda}_{f_{n_{k}(\lambda)}}$ tend to an operator $A^{\lambda}$ in the Hilbert-Schmidt norm. $A^{\lambda}$ depends neither on the chosen sequence $f_{n}$ nor on its subsequence $f_{n_{k}(\lambda)}$. Moreover the KN symbol of $A^{\lambda}$ is $a_{\lambda}(w)=\widetilde{\widehat{f}}(|\lambda|^{1/2}w,\lambda)$.
\end{lemma}
\begin{proof}
By Plancherel's formula
\begin{align*}
\Vert f_{n}-f\Vert_{2}^{2}&=\int |\widehat{f_{n}}(w,\lambda)-\widehat{f}(w,\lambda)|^{2}dwd\lambda\\
&=\int|\lambda|^{n}\int|\widetilde{\widehat{f_{n}}}(|\lambda|^{1/2}w,\lambda)-\widetilde{\widehat{f}}(|\lambda|^{1/2}w,\lambda)|^{2}dwd\lambda\\
&=\int|\lambda|^{n}\Vert\pi_{f_{n}}^{\lambda}-A^{\lambda}\Vert_{HS}^{2}d\lambda,
\end{align*}
where $A^{\lambda}$ is the Hilbert-Schmidt operator with the symbol $(w,\lambda)\mapsto\widetilde{\widehat{f}}(|\lambda|^{1/2}w,\lambda)$.
By Fatou's lemma $\liminf\Vert\pi_{f_{n}}^{\lambda}-A^{\lambda}\Vert_{HS}=0$ for almost every $\lambda$.
Therefore, for almost every $\lambda$, there exists a subsequence $f_{n_{k}(\lambda)}$ such that $$\Vert\pi_{f_{n_{k}(\lambda)}}^{\lambda}-A^{\lambda}\Vert_{HS}\rightarrow 0.$$
\end{proof}
Let $\widehat{f}^{\lambda}(w):=\widehat{f}(|\lambda|^{1/2}w,\lambda)$.
Lemma \ref{l2} says that for every $u,v\in\ss({\HH^n})$, $\langle\widehat{f}^{\lambda}_{n},\widehat{c_{u,v}}\rangle$ tends to $\langle\widehat{f}^{\lambda},\widehat{c_{u,v}}\rangle$ a.e. which implies that $\langle\pi_{f_{n}}^{\lambda}u,v\rangle$ must have a limit. This limit is $A^{\lambda}$ an it will be denoted by $\pi_{f}^{\lambda}$. Nevertheless, for a given $L^{2}$ function, the operator exists only for a.e. $\lambda$.
\begin{lemma}
Let $K$ be a flag kernel on ${\HH^n}$. Then, for every $f\in L^{2}({\HH^n})$,
$$\pi_{K}^{\lambda}\pi_{f}^{\lambda}=\pi_{K\star f}^{\lambda}.$$
\end{lemma}
\begin{proof}
Let $\{f_{n}\}_{n}\subset\ss_{0} ({\HH^n})$ tend to $f$ in $L^{2}({\HH^n})$. By definition
$$\pi_{K\star f}^{\lambda}=\lim\pi_{K\star f_{n_{k}(\lambda)}}^{\lambda},$$ for a.e. $\lambda$. As $f_{n_{k}(\lambda)}$ converges to $f$ in $L^{2}({\HH^n})$, by Lemma \ref{l2}, $K\star f_{n_{k}(\lambda)}$ converges to $K\star f$ a.e. Hence
\begin{align*}
\pi_{K}^{\lambda}\pi_{f}^{\lambda}=\lim\pi_{K}^{\lambda}\pi_{f_{n_{k_{s}}(\lambda)}}^{\lambda}=\lim\pi_{K\star f_{n_{k_{s}}(\lambda)}}^{\lambda}=\pi_{K\star f}^{\lambda}.
\end{align*}
\end{proof}
Assume that $f\in L^{2}({\HH^n})$. Let us continue with the calculation of kernel $\Omega_{f}^{\lambda}$ of the operator $\pi_{f}^{\lambda}$. As it has been said before, we have
$$\Omega_{f}^{\lambda}(\xi,\eta^{\vee})=e^{2\pi i\xi\eta}f^{\vee}(|\lambda|^{1/2}\xi,|\lambda|^{1/2}\eta,\lambda).$$
Therefore,
\begin{align*}
\Omega_{f}^{\lambda}(\xi,y)=\int e^{-2\pi i\eta(y-\xi)}f^{\vee}(|\lambda|^{1/2}\xi,|\lambda|^{1/2}\eta,\lambda)d\eta=|\lambda|^{-n/2}f(|\lambda|^{1/2}\xi^{\vee},\frac{y-\xi}{|\lambda|^{1/2}},\lambda^{\vee}).
\end{align*}
Furthermore, by Plancherel's formula
\begin{align*}
\mathfrak{G}_{f}(\lambda):&=|\lambda|^{n}\Vert\pi_{f}^{\lambda}\Vert_{HS}^{2}=|\lambda|^{n}\Vert\Omega_{f}^{\lambda}\Vert_{2}^{2}=\int\int|f(|\lambda|^{1/2}\xi^{\vee},\frac{y-\xi}{|\lambda|^{1/2}},\lambda^{\vee})|^{2}d\xi dy\\
&=\int\int|f(x,y,\lambda^{\vee})|^{2}dxdy.
\end{align*}
The function $\mathfrak{G}_{f}$ is continuous when $f\in\ss ({\HH^n})$.
Let $A$ be linear, bounded operator on $L^{2}({\HH^n})$ (in particular a convolver), $\chi_{E}$ a characteristic function of a set $E\subset\mathbb{R}$.
Then, from the above calculation we can conclude that $$\chi_{E}(\lambda)\mathfrak{G}_{Af}(\lambda)=\mathfrak{G}_{A(\chi_{E}(\lambda)f)}(\lambda).$$
Let as recall here that $\Vert f\Vert_{2}^{2}=\int_{\mathbb{R}^{\star}}\mathfrak{G}_{f}(\lambda)d\lambda$.
\begin{lemma}\label{calki}
Let A be a linear, bounded operator on $L^{2}({\HH^n})$. Suppose that, for every $f\in L^{2}({\HH^n})$ $$\int_{\mathbb{R}^{\star}}\mathfrak{G}_{Af}(\lambda)d\lambda\geqslant c^{2}\int_{\mathbb{R}^{\star}}\mathfrak{G}_{f}(\lambda)d\lambda,$$
then, for almost every $\lambda$, $\mathfrak{G}_{Af}(\lambda)\geqslant c^{2}\mathfrak{G}_{f}(\lambda).$
\end{lemma}
\begin{proof}
Assume a contrario that for a function $g$ and $\lambda$ in a set $E$ of positive Lebesque measure we have that $\mathfrak{G}_{Ag}(\lambda)<c^{2}\mathfrak{G}_{g}(\lambda)$. Then, there exists $\varepsilon$>0 and a subset $F$ of $E$ of positive Lebesque measure such that
$\mathfrak{G}_{Ag}(\lambda)\leqslant (1-\varepsilon)c^{2}\mathfrak{G}_{g}(\lambda)$ on $F$. Therefore,
\begin{align*}
c^{2}\int_{\mathbb{R}^{\star}}\mathfrak{G}_{\chi_{F}(\lambda)g}(\lambda)d\lambda&\leqslant\int_{\mathbb{R}^{\star}}\mathfrak{G}_{A(\chi_{F}(\lambda)g)}(\lambda)d\lambda=\int_{F}\mathfrak{G}_{Ag}(\lambda)d\lambda\\
&\leqslant c^{2}(1-\varepsilon)\int_{F}\mathfrak{G}_{g}(\lambda)d\lambda
=c^{2}(1-\varepsilon)\int_{\mathbb{R}^{\star}}\mathfrak{G}_{\chi_{F}(\lambda)g}(\lambda)d\lambda,
\end{align*}
which is obviously a contradiction.
\end{proof}
The same holds true for the opposite inequality and the proof is analogous.
\begin{theorem}
Let $K$ be a symmetric flag kernel, such that $\mathop{\rm Op}\nolimits(K)$ is invertible. The family $\{\pi_{K}^{\lambda}\}_{\lambda}$ is uniformly invertible, that is all $\pi_{K}^{\lambda}$ are invertible and the family of operators $\{(\pi_{K}^{\lambda})^{-1}\}_{\lambda}$ is uniformly bounded on $L^{2}({\RR^n})$.
\end{theorem}
\begin{proof}
As $\mathop{\rm Op}\nolimits(K)$ is invertible there exists a constant $C_{K}$, such that for $f\in L^{2}({\HH^n})$ $\Vert K\star f\Vert_{2}\geqslant C_{K}\Vert f\Vert_{2}$.
Using Plancherel formula we have
\begin{align*}
\int_{\mathbb{R}^{\star}}|\lambda|^{n}\Vert\pi_{K\star f}^{\lambda}\Vert_{HS}^{2}d\lambda=\Vert K\star f\Vert_{2}^{2}\geqslant C_{K}^{2}\Vert f\Vert_{2}^{2}=C_{K}^{2}\int_{\mathbb{R}^{\star}}|\lambda|^{n}\Vert\pi_{f}^{\lambda}\Vert_{HS}^{2}d\lambda.
\end{align*}
Now by Lemma \eqref{calki}
\begin{align*}
C_{K}\Vert\pi_{f}^{\lambda}\Vert_{HS}\leqslant\Vert\pi_{K\star f}^{\lambda}\Vert_{HS}=\Vert\pi_{K}^{\lambda}\pi_{f}^{\lambda}\Vert_{HS}\leqslant\Vert\pi_{K}^{\lambda}\Vert_{2\rightarrow 2}\Vert\pi_{f}^{\lambda}\Vert_{HS}.
\end{align*}
Consider the operator $\mathcal{P}_{g,h}$; $g,h\in L^{2}({\RR^n})$, where $\Vert h\Vert_{2}\neq 0$, which acts on vectors
$u\in L^{2}({\RR^n})$ by $\mathcal{P}_{g,h}(u):=\langle u,g\rangle h$. It is easy to see that the kernel of $\mathcal{P}_{g,h}$ is $\Omega_{\mathcal{P}}(x,y)=g(x)h(y)$, so $\Vert\mathcal{P}_{g,h}\Vert_{HS}=\Vert g\Vert_{2}\Vert h\Vert_{2}$. Now
\begin{align*}
\pi_{K}^{\lambda}\mathcal{P}_{g,h}u=\langle \pi_{K}^{\lambda}u,g\rangle h=\langle u,(\pi_{K}^{\lambda})^{\star}g\rangle h=\langle u,\pi_{K^{\star}}^{\lambda}g\rangle h=\langle u,\pi_{K}^{\lambda}g\rangle h=\mathcal{P}_{\pi_{K}^{\lambda}g,h}u.
\end{align*}
Hence
\begin{align*}
\Vert\pi_{K}^{\lambda}g\Vert_{2}\Vert h\Vert_{2}=\Vert\pi_{K}^{\lambda}\mathcal{P}_{g,h}\Vert_{HS}\geqslant C_{K}\Vert\mathcal{P}_{g,h}\Vert_{HS}=C_{K}\Vert g\Vert_{2}\Vert h\Vert_{2}.
\end{align*}
Dividing both sides by $\Vert h\Vert_{2}$ we obtain that
$\Vert \pi_{K}^{\lambda}g\Vert_{2}\geqslant C_{K}\Vert g\Vert_{2}$ holds for every $g\in\ L^{2}({\RR^n})$ which, together with the fact that $\pi_{K}^{\lambda}$ is self-adjoint, implies our claim.
\end{proof}
\section{The Beals theorem and the main result}
Summing up our previous results we conclude that, for every $\lambda$, a flag kernel $K$, gives rise to an operator $\pi_{K}^{\lambda}$ which acts on $L^{2}({\RR^n})$ as a pseudodifferential operator with the Kohn-Nirenberg symbol
$a_{\lambda}(w)=\widetilde{\widehat{K}}(|\lambda|^{1/2}w,\lambda)$. Moreover,
\begin{align*}
|\d_{w}^{\alpha}a_{\lambda}(w)|&=|\partial_{w}^{\alpha}\{\widetilde{\widehat{K}}(|\lambda|^{1/2}w,\lambda)\}|\leqslant c_{\alpha}(\Vert|\lambda|^{1/2}w\Vert+|\lambda|^{1/2})^{-|\alpha|}|\lambda|^{|\alpha|/2}\\
&=c_{\alpha}(1+\Vert w\Vert)^{-|\alpha|}.
\end{align*}
Observe that these estimates do not depend on $\lambda$. In particular, by the Calder{\'o}n-Vaillancourt theorem, the family $\{\pi_{K}^{\lambda}\}_{\lambda}$ is uniformly bounded on $L^{2}({\RR^n})$.
Let us define
$$Sym^{0}(\mathbb{R}^{2n}):=\{a\in\mathcal{C}^{\infty}(\mathbb{R}^{2n}):|\partial_{w}^{\alpha}a(w)|\leqslant c_{\alpha}(1+\Vert w\Vert)^{-|\alpha|}\}.$$
In this language the family of symbols $\{a_{\lambda}\}_{\lambda}$ is bounded in $Sym^{0}(\mathbb{R}^{2n})$ with the natural seminorm topology.
The key point in our argument is the following application of a much more general theorem of Beals.
\begin{theorem}[Beals \cite{beals}, Thm. 4.7]
Let $A=a(x,D)$, where $a\in Sym^{0}(\mathbb{R}^{2n})$, be invertible on $L^{2}({\RR^n})$. Then, $A^{-1}=b(x,D)$ with $b\in Sym^{0}(\mathbb{R}^{2n})$. Each seminorm of $b$ depends only on a finite number of seminorms of $a$ and the operator norm of $A^{-1}$.
\end{theorem}
Let us denote the symbol of $(\pi_{K}^{\lambda})^{-1}$ by $b_{\lambda}$.
One can conclude that the seminorms of $b_{\lambda}$ once again do not depend on $\lambda$. So the family $\{b_{\lambda}\}_{\lambda}$ also corresponds to a bounded family in $Sym^{0}(\mathbb{R}^{2n})$. Following G{\l}owacki \cite{arkiv2007}, we say that $a$ is a weak limit of a bounded sequence $\{a_{n}\}_{n}$ in $Sym^{0}(\mathbb{R}^{2n})$ iff for every $\alpha$ the sequence $\{\d^{\alpha}a_{n}\}_{n}$ converges almost uniformly to $\d^{\alpha}a$. The twisted multiplication $\#$ is continuous in the weak sense. In the language of symbols the equation $\pi_{K}^{\lambda}(\pi_{K}^{\lambda})^{-1}=Id$ corresponds to $a_{\lambda}\#b_{\lambda}=1$.
\begin{lemma}
The family $\{b_{\lambda}\}_{\lambda}$ is weakly smooth in the parameter $\lambda$.
\end{lemma}
\begin{proof}
We proceed by induction. Let $\{\lambda_{n}\}_{n}$ converge to a nonzero $\lambda$. As $\{b_{\lambda_{n}}\}_{n}$ is bounded in $Sym^{0}(\mathbb{R}^{2n})$, we can use Arzeli-Ascoli theorem to find a weakly convergent subsequence. Let $\{b_{\lambda_{n_{k}}}\}_{k}$ tend to $b_{\lambda}(\{n_{k}\})$. We have $$1=b_{\lambda_{n_{k}}}\#a_{\lambda_{n_{k}}}\rightarrow b_{\lambda(\{n_{k}\})}\#a_{\lambda}.$$ Hence for every convergent subsequence $\{b_{\lambda_{n_{k}}}\}_{k}$, the limit must be the same and equal to $b_{\lambda}$. Therefore, it also must be the limit of $\{b_{\lambda_{n}}\}_{n}$.
Assume now that $\d_{\lambda}^{N}b_{\lambda}$ is continuous for $N<M$.
Observe that using continuity of $b_{\lambda}$ which we have just obtained, formally we have
\begin{align}\label{rozklad}
\lim_{h\rightarrow 0}\frac{b_{\lambda+h}-b_{\lambda}}{h}=\lim_{h\rightarrow 0}b_{\lambda}\#\frac{a_{\lambda}-a_{\lambda+h}}{h}\#b_{\lambda+h}=-b_{\lambda}\#\d_{\lambda}a_{\lambda}\#b_{\lambda},
\end{align}
where the right hand side is weakly continuous. Consider the set $$\Xi:=\{M=(M_{1},M_{2},M_{3}):M_{2}>0,M_{1}+M_{2}+M_{3}=M\}.$$ Iterating the decomposition (\ref{rozklad}) we obtain
\begin{align}\label{pelny}
\d_{\lambda}^{M}b_{\lambda}=\sum_{M\in\Xi}c_{M}\d_{\lambda}^{M_{1}}b_{\lambda}\#\d_{\lambda}^{M_{2}}a_{\lambda}\#\d_{\lambda}^{M_{3}}b_{\lambda}.
\end{align}
By induction hypothesis the right hand side is again weakly continuous. Hence the proof is complete.
\end{proof}
The decomposition (\ref{pelny}) actually gives more. It turns out that $b(w,\lambda):=b_{\lambda}(w)$ is also smooth if only $\lambda\neq 0$. It is a consequence of the fact that every derivative of $b(w,\lambda)$ has bounded partial derivatives outside of every set of type $\mathbb{R}^{2n}\times [-\varepsilon,\varepsilon]$.
Note that $\widetilde{\widehat{B}}(w,\lambda)=\widehat{B}(-w,-\lambda)$, so $\widetilde{\widehat{B}}$ is the Fourier transform of a flag kernel if and only if $\widehat{B}$ is.
\begin{theorem}
Let $B$ be a distribution such that $\widehat{B}(w,\lambda)=b_{\lambda}(|\lambda|^{-1/2}w)$. Then, $B$ is a flag kernel.
\end{theorem}
\begin{proof}
It is obvious from the definition that $\widehat{B}$ is smooth away from the hyperspace $\{(w,\lambda):\lambda=0\}$. We have
\begin{align*}
|(\d_{1,2,...,2n}^{\alpha}\widehat{B})(|\lambda|^{1/2}w,\lambda)||\lambda|^{|\alpha|/2}&=|\d_{1,2,...,2n}^{\alpha}\{\widehat{B}(|\lambda|^{1/2}w,\lambda)\}|=|\d_{w}^{\alpha}b_{\lambda}(w)|\\
\leqslant c_{\alpha}(1+\Vert w\Vert)^{-|\alpha|}.
\end{align*}
Therefore,
\begin{align*}
|(\d_{1,2,...,2n}^{\alpha}\widehat{B})(|\lambda|^{1/2}w,\lambda)|\leqslant c_{\alpha}(1+\Vert w\Vert)^{-|\alpha|}|\lambda|^{-|\alpha|/2}=c_{\alpha}(|\lambda|^{1/2}+\Vert|\lambda|^{1/2}w\Vert)^{-|\alpha|}.
\end{align*}
Now putting $w$ instead of $|\lambda|^{1/2}w$ we obtain
\begin{align}\label{indukcja}
|(\d_{w}^{\alpha}\widehat{B})(w,\lambda)|\leqslant c_{\alpha}(|\lambda|^{1/2}+\Vert w\Vert)^{-|\alpha|}.
\end{align}
It sufficies now to get the estimates of the derivatives with respect to $\lambda$. We can treat inequality (\ref{indukcja}) as an initial step of an induction.
First of all using the fact that $K$ is a flag kernel one can calculate that
\begin{align*}
|\d_{\lambda}^{M}a_{\lambda}(w)|&=|\d_{\lambda}^{M}\{\widetilde{\widehat{K}}(|\lambda|^{1/2}w,\lambda)\}|\\
&=\mid\sum_{1\leqslant |\beta|+j\leqslant M}\frac{c_{\beta,j}w^{\beta}}{|\lambda|^{M-j-|\beta|/2}}(\d_{1,2,...,2n}^{\beta}\d^{j}_{2n+1}\widetilde{\widehat{K})}(|\lambda|^{1/2}w,\lambda)\mid\\
&\leqslant\sum_{1\leqslant |\beta|+j\leqslant M}\frac{|c_{\beta,j}|\Vert w\Vert^{|\beta|}}{|\lambda|^{M-j-|\beta|/2}}(\Vert |\lambda|^{1/2}w\Vert+|\lambda|^{1/2})^{-|\beta|}|\lambda|^{-j}\\
&\lesssim\frac{\Vert w\Vert^{|\beta|}}{|\lambda|^{M}(1+\Vert w\Vert)^{|\beta|}}\leqslant|\lambda|^{-M}.
\end{align*}
As $\{b_{\lambda}\}_{\lambda}$ is a bounded family in $Sym^{0}(\mathbb{R}^{2n})$ let us assume that it is so for the families $\{|\lambda|^{N}\d_{\lambda}^{N}b_{\lambda}\}_{\lambda}$, where $N<M$.
Now using (\ref{pelny}) we can write
$$|\lambda|^{M}\d_{\lambda}^{M}b_{\lambda}=\sum_{M\in\Xi}c_{M}|\lambda|^{M_{1}}\d_{\lambda}^{M_{1}}b_{\lambda}\#|\lambda|^{M_{2}}\d_{\lambda}^{M_{2}}a_{\lambda}\#|\lambda|^{M_{3}}\d_{\lambda}^{M_{3}}b_{\lambda}.$$
As $M_{2}>0$ one can use an induction argument and the standard symbolic calculus to deduce that the right hand side is bounded in $Sym^{0}(\mathbb{R}^{2n})$. Therefore, the families $\{|\lambda|^{M}\d_{\lambda}^{M}b_{\lambda}\}_{\lambda}$ are bounded, for all $M\in\mathbb{N}$.
Thus, $$|\d_{w}^{\alpha}(|\lambda|^{M}\d_{\lambda}^{\beta}b_{\lambda}(w))|\leqslant c_{\alpha, M}(1+\Vert w\Vert)^{-|\alpha|},$$
so
$$|\d_{w}^{\alpha}\d_{\lambda}^{\beta}b_{\lambda}(w)|\leqslant c_{\alpha, M}(1+\Vert w\Vert)^{-|\alpha|}|\lambda|^{-M}.$$
One can calculate that
$$\d_{w}^{\alpha}\d_{\lambda}^{M}b_{\lambda}(w)=\sum_{\gamma+\delta=\alpha}\sum_{1\leqslant |\beta|+j\leqslant M}\frac{c_{\gamma,\beta,j}w^{\beta-\gamma}}{|\lambda|^{M-j-(|\beta|+|\delta|)/2}}(\d_{1,2,...,2n}^{\beta+\delta}\d^{j}_{2n+1}\widehat{B})(|\lambda|^{1/2}w,\lambda).$$
The only component of the sum on the right hand side that includes $j=M$ is the one with $\beta=\gamma=0,\ \delta=\alpha$. So, by induction hypothesis, we have
\begin{align*}
&|(\d_{1,2,...,2n}^{\alpha}\d^{M}_{2n+1}\widehat{B})(|\lambda|^{1/2}w,\lambda)||\lambda|^{|\alpha|/2}\\
&\lesssim\sum_{\gamma+\delta=\alpha}\sum_{1\leqslant |\beta|+j\leqslant M,\ \beta\neq 0}\frac{\Vert w\Vert^{|\beta|-|\gamma|}}{|\lambda|^{M-j-(|\beta|+|\delta|)/2}(\Vert |\lambda|^{1/2}w\Vert+|\lambda|^{1/2})^{|\beta|+|\delta|}|\lambda|^{j}}\\
&+|\d_{w}^{\alpha}\d_{\lambda}^{M}b_{\lambda}(w)|\\
&\lesssim\sum_{\gamma+\delta=\alpha}\sum_{1\leqslant |\beta|+j\leqslant M,\ \beta\neq 0}\frac{\Vert w\Vert^{|\beta|-|\gamma|}}{(1+\Vert w\Vert)^{|\beta|-|\gamma|}(1+\Vert w\Vert)^{|\alpha|}|\lambda|^{M}}+(1+\Vert w\Vert)^{-|\alpha|}|\lambda|^{-M}\\
&\lesssim (1+\Vert w\Vert)^{-|\alpha|}|\lambda|^{-M}.
\end{align*}
Thus,
\begin{align*}
|(\d_{1,2,...,2n}^{\alpha}\d^{M}_{2n+1}\widehat{B})(|\lambda|^{1/2}w,\lambda)|&\leqslant c_{\alpha,M}(1+\Vert w\Vert)^{-|\alpha|}|\lambda|^{-M}|\lambda|^{-|\alpha|/2}\\
&= c_{\alpha,M}(\Vert |\lambda|^{1/2}w\Vert+|\lambda|^{1/2})^{-|\alpha|}|\lambda|^{-M}.
\end{align*}
Again putting $w$ instead of $|\lambda|^{1/2}w$ we obtain
$$|(\d_{w}^{\alpha}\d^{M}_{\lambda}\widehat{B})(w,\lambda)|\leqslant c_{\alpha,M}(\Vert w\Vert+|\lambda|^{1/2})^{-|\alpha|}|\lambda|^{-M}.$$
\end{proof}
\begin{proof}[proof of Theorem \ref{nasze}]
The KN symbol of $(\pi_{K}^{\lambda})^{-1}$ is $b_{\lambda}(w)=\widehat{B}(|\lambda|^{1/2}w,\lambda)$ which is the symbol of $\pi_{\widetilde{B}}^{\lambda}$ and $\widetilde{B}$ is a flag kernel. So $(\pi_{K}^{\lambda})^{-1}=\pi_{\widetilde{B}}^{\lambda}$. Now
$$\pi_{\delta_{0}}^{\lambda}=Id=(\pi_{K}^{\lambda})^{-1}\pi_{K}^{\lambda}=\pi_{K}^{\lambda}(\pi_{K}^{\lambda})^{-1}=\pi_{\widetilde{B}}^{\lambda}\pi_{K}^{\lambda}=\pi_{K}^{\lambda}\pi_{\widetilde{B}}^{\lambda}=\pi_{\widetilde{B}\star K}^{\lambda}=\pi_{K\star \widetilde{B}}^{\lambda}.$$
So $K\star \widetilde{B}=\delta_{0}=\widetilde{B}\star K$. Now putting $L=\widetilde{B}$ for $f\in L^{2}({\HH^n})$ we achieve $K\star L\star f=L\star K\star f=f$, which is equivalent to $\mathop{\rm Op}\nolimits(K)\mathop{\rm Op}\nolimits(L)f=\mathop{\rm Op}\nolimits(L)\mathop{\rm Op}\nolimits(K)f=f$ and finally $\mathop{\rm Op}\nolimits(K)^{-1}=\mathop{\rm Op}\nolimits(L)$.
\end{proof}
\section*{Acknowledgements}
The author wishes to express his deep gratitude to P.G{\l}owacki and M.Preisner for their helpful advices in preparing the manuscript.
|
1,314,259,996,519 | arxiv |
\section{Conclusions and Future Work}
\vspace{-5pt}
This work is a preliminary study to explain the reason for similar or different application resilience between the serial and parallel executions.
For the future work, we will investigate more benchmarks and establish a model to predict application resilience for the parallel execution based on the fault injection results for the serial execution.
\vspace{-10pt}
\section{Evaluation Methodology}
\vspace{-5pt}
We employ a fault injection tool, PFSEFI to study three NAS benchmarks (CG, FT, BT) with the input problem $S$.
For serial execution fault injection, we only run one MPI process; For parallel execution fault injection, we run four MPI processes and then randomly choose one MPI process for fault injection. We inject faults into the whole application and focus on two types of instructions, i.e., floating point addition (\textit{fadd}) and floating point multiplication (\textit{fmul}), because they are the most common ones in HPC applications. To ensure statistical significance for fault injection, we gradually increase the number of fault injection tests until the fault injection result becomes stable.
The fault injection results are classified into three types: (1) Benign: the computation results of benchmarks pass the benchmarks' verification phase, it means the computation results are acceptable. But the computation results may be different from those without fault injection. (2) Silent data corruption (SDC): the computation results of benchmarks do not pass the benchmarks' verification phase; (3) Crashes: the benchmark cannot run to completion. Since the fault injection happens based on the random selection of dynamic instruction, we cannot know where the fault happens within the application code. But we can know the instruction address in the EIP register when the fault happens.
We map the instruction address into the application code via PYELFTOOLS~\cite{pyelftool:github}.
Based on the EIP information for all random fault injection points, we can know the occurrence frequency of each faulty instruction; also, we can analyze the code, and understand the difference or similarity of application resilience in serial and parallel executions.
\vspace{-10pt}
\section{EXPERIMENT RESULTS}
\vspace{-5pt}
\begin{figure}[!t]
\centering
\includegraphics[width=0.46\textwidth, height=0.15\textheight]{figures/sdc_rate_pic.pdf}
\vspace{-13pt}
\caption{SDC Rate of NPB CG,FT, BT Benchmarks.}
\vspace{-20pt}
\label{fig:sensitity_study_on_num_tests}
\vspace{-3pt}
\end{figure}
Figure~\ref{fig:sensitity_study_on_num_tests} shows the fault injection results (i.e., SDC rate of fault injection tests).
We collect 10,000 fault injection test results for each benchmark and calculate the SDC rate every 1000 fault injection tests.
The fault injection results become stable after first 6,000 tests.
\begin{figure*}[t]
\centering
\subcaptionbox{FT (Serial + Benign)}[.23\linewidth][c]{%
\includegraphics[width=.23\linewidth]{figures/ft_s_benign_pic.pdf}}\quad
\subcaptionbox{FT (Serial + SDC)}[.23\linewidth][c]{%
\includegraphics[width=.23\linewidth]{figures/ft_s_sdc_pic.pdf}}\quad
\subcaptionbox{FT (Parallel + Benign)}[.23\linewidth][c]{%
\includegraphics[width=.23\linewidth]{figures/ft_p_benign_pic.pdf}}\quad
\subcaptionbox{FT (Parallel + SDC)}[.23\linewidth][c]{%
\includegraphics[width=.23\linewidth]{figures/ft_p_sdc_pic.pdf}}
\vspace{-3pt}
\subcaptionbox{BT (Serial + Benign)}[.23\linewidth][c]{%
\includegraphics[width=.23\linewidth]{figures/bt_s_benign_pic.pdf}}\quad
\subcaptionbox{BT (Serial + SDC)}[.23\linewidth][c]{%
\includegraphics[width=.23\linewidth]{figures/bt_s_sdc_pic.pdf}}\quad
\subcaptionbox{BT (Parallel + Benign)}[.23\linewidth][c]{%
\includegraphics[width=.23\linewidth]{figures/bt_p_benign_pic.pdf}} \quad
\subcaptionbox{BT (Parallel + SDC)}[.23\linewidth][c]{%
\includegraphics[width=.23\linewidth]{figures/bt_p_sdc_pic.pdf}}
\vspace{-3pt}
\subcaptionbox{CG (Serial + Benign)}[.23\linewidth][c]{%
\includegraphics[width=.23\linewidth]{figures/cg_s_benign_pic.pdf}}\quad
\subcaptionbox{CG (Serial + SDC)}[.23\linewidth][c]{%
\includegraphics[width=.23\linewidth]{figures/cg_s_sdc_pic.pdf}}\quad
\subcaptionbox{CG (Parallel + Benign)}[.23\linewidth][c]
\includegraphics[width=.23\linewidth]{figures/cg_p_benign_pic.pdf}}\quad
\subcaptionbox{CG (Parallel + SDC)}[.23\linewidth][c]{%
\includegraphics[width=.23\linewidth]{figures/cg_p_sdc_pic.pdf}}
\vspace{-13pt}
\caption{The distribution of faulty floating point \textit{add} instructions in fault injection tests.}
\vspace{-12pt}
\label{fig:instr_dist}
\vspace{-5pt}
\end{figure*}
Figure~\ref{fig:instr_dist} shows the faulty instruction distribution for the fault injection tests on floating point \textit{add} instructions.
In particular, we find that there are no crashes happened in tests of three benchmarks; thus we only show how frequent each instruction is selected when the fault injection results are benign and SDC.
Figure~\ref{fig:instr_dist} (a)-(d)shows that for FT, the randomly selected faulty instructions in the fault injection tests for the serial and parallel executions are the same, which explains why the fault injection results for the two executions are almost the same.
The fact that the faulty instructions are the same
mainly because of the code similarity between the serial and parallel codes.
For BT(see figure~\ref{fig:instr_dist}(e)-(h)), we find that faulty instructions are widely spread across the parallel and serial executions. There is almost no instruction similarity in those faulty instructions between the serial and parallel executions.
It is because BT has complicated computation. There is no dominant computation phase where the faulty instructions can repeatedly happen.
For CG(see figure~\ref{fig:instr_dist}(i)-(l)), we find that faulty instructions are limited to a few instructions, which is very different from the cases of BT. Also, the fault injection results for the serial and parallel executions are quite different. To understand the reason for such difference, we map the faulty instructions into the source code of CG and have the following observations.
\vspace{-3pt}
\textbf{Observation 1}:
The instruction at 0x0804A03F is the most frequently selected instruction for fault injection. Such instruction appears in all cases (serial+benign), (serial+SDC),(parallel+benign) and (parallel+SDC).
This instruction is used so often in the benchmark, such that most of faults are injected into it. Also, the corruption of this instruction seems to easily cause SDC.
\vspace{-3pt}
\textbf{Observation 2}:
Some instructions only appear in (serial+benign), (parallel+benign) and (parallel+SDC), but do not appear in (serial+SDC). Those instructions include those at 0x0804A2B7, 0x0804A3BA, 0X0804A402 and 0x0804A502.
Those instructions cause fault injection result difference between the serial and parallel executions.
Figure~\ref{fig:ob2} shows the related code segment for 0x0804A2B7.
In particular, the serial and parallel executions have a different value for the variable \textit{l2npcols}, which leads to different code structure (particularly the MPI synchronization) for serial and parallel executions.
Such difference in the code structure makes the faulty injection at 0x0804A2B7 behave differently in the serial and parallel executions.
\begin{figure}[!t]
\centering
\includegraphics[width=0.45\textwidth, height=0.13\textheight]{figures/CG_code_map_observation_3_pic.pdf}
\vspace{-15pt}
\caption{Source code analysis for the observation 2}
\vspace{-15pt}
\label{fig:ob2}
\vspace{-2pt}
\end{figure}
\vspace{-3pt}
\textbf{Observation 3}: The instruction at 0x0804A163 is only shown in (parallel+bengin) and (parallel+SDC), and such instruction only exists in the parallel execution because of the following reason: the variable \textit{l2npcols} has a different value in the serial and parallel executions. Hence the two executions behave differently (Figure~\ref{fig:ob3}).
\begin{figure}[!t]
\centering
\includegraphics[width=0.45\textwidth, height=0.13\textheight]{figures/CG_code_map_observation_4_pic.pdf}
\vspace{-15pt}
\caption{Source code analysis for the observation 3}
\vspace{-20pt}
\label{fig:ob3}
\vspace{-3pt}
\end{figure}
\section{Introduction}
\vspace{-5pt}
Making system resilient to hardware and software faults is a critical design goal for future extreme scale systems.
To implement resilient HPC, we must have a good understanding of application resilience in the existence of faults.
Currently, the application level fault injection is the major method to understand application resilience.
The application level fault injection triggers random bit flip in the operand or result of a random instruction.
Typically, the statistical results of many fault injection tests, e.g., the percentage of the fault injection tests that have success application outcome, is used to evaluate the application resilience.
However, the application level fault injection can be very expensive, because HPC users need to inject a large number of faults to ensure statistical significance. Moreover, comparing with fault injection for the serial execution, fault injection for the parallel execution can be even more expensive. First, the parallel version needs more hardware resource than the serial version to deploy fault injection tests. Second, injecting faults into the parallel execution can be more difficult, because there is a larger exploration space for fault injection.
In this poster, we explore the correlation between the parallel and serial executions regarding their resilience. Our ultimate goal is that by studying the resilience of the serial execution we can derive the resilience of the parallel execution without using expensive fault injection. We aim to answer two fundamental questions. First, does the application resilience remain the same across the serial and parallel executions? Second, if the application resilience is difference between the two executions, what code structure causes such difference? We use an application-level fault injection tool named PFSEFI~\cite{PFSEFI:SIMUTOOLS16} to randomly choose dynamic instruction and then randomly flip one bit in the instruction result. After enough fault injection tests, we characterize and compare the serial and parallel execution codes based on the fault injection results.
We hope that our work can lay foundation to build a model to predict the resilience of the parallel execution only based on fault injection results in the serial execution.
\vspace{-10pt} |
1,314,259,996,520 | arxiv |
\section{Introduction}
\label{sec:intro}
Celestial mechanics, the field that deals with the motion of celestial objects, has been an active field of research since the days of Newton and Kepler.
Analytic solutions only exist for a few special cases.
Historically, the main driver for the development of perturbation theory has been the problem of planets orbiting the Sun.
Because the central body is so much more massive than the planets, it is profitable to ask how the small mutual tugs between the planets modify the Keplerian orbits they would each individually follow around the Sun in the absence of the other bodies.
This analytical approach has been, and continues to be, successful in explaining many important features of planetary orbits.
However, the Solar System is chaotic, and the rise of computing power has yielded many important insights.
There is therefore considerable interest in developing fast and accurate numerical integrators.
A large number of such integrators have been developed over the years to perform this task.
For many long term integrations, symplectic integrators have proven to be a favourable choice.
Symplectic schemes incorporate the symmetries of Hamiltonian systems, and therefore typically conserve quantities like the energy and angular momentum better than non-symplectic integrators.
For integrations of planetary systems, \cite{WisdomHolman1991}, and independently \cite{Kinoshita1991}, developed a widely used class of symplectic integrators.
The ideas of \cite{WisdomHolman1991} developed from the original ideas of the mapping method of \cite{Wisdom1981}.
We refer to these as a Wisdom-Holman mapping or a Wisdom-Holman integrator.
Since then, many authors have modified and built upon this method, and several have made their integrators publicly available to the astrophysics community \citep[e.g.][]{Chambers1997,Duncan1998}.
The Wisdom-Holman integrator exploits the intuition from perturbation theory that one can separate the problem into a system of Keplerian orbits about the Sun, modified by small perturbations among the planets.
The nuisance is that while Newton provided us the solution to the two-body problem, Poincar{\'e} showed that the remaining superimposed perturbations are not integrable.
Analytically, the traditional way forward is to average over the short-period oscillations in the problem to yield approximate solutions.
The great insight of Wisdom and Holman was that, at the same level of approximation, one can {\it add} high frequency terms.
By judicious choice of these additional frequencies, the perturbations among the planets can be transformed into trivially integrated delta functions.
The result is an exceedingly efficient integrator that has proven an indispensable tool for modern studies in celestial mechanics.
In this paper, we present results from a complete reimplementation of the Wisdom-Holman integrator.
We show how to speed up the algorithm in several ways and dramatically increase its accuracy.
Many of the improvements are related to finite double floating-point precision on modern computers \citep[IEEE754,][]{IEEE754}.
The fact that almost all real numbers cannot be represented exactly in floating-point precision leads to important consequences for the numerical stability of any algorithm and the growth of numerical round-off error.
To our knowledge, we present the first publicly available Wisdom-Holman integrator that is unbiased, i.e. the errors are random and uncorrelated.
This leads to a very slow error growth.
For sufficiently small timesteps, we achieve Brouwer's law, i.e., the energy error grows as time to the power of one half.
We have also sped up the integrator through various improvements to the integrator's Kepler-solver.
Our implementation allows for the evolution of variational equations (to determine whether orbits are chaotic) at almost no additional cost.
Additionally, we implement so-called symplectic correctors up to order eleven to increase the accurary~\citep{Wisdom1996}, allow for arbitrary unit choices, and do not tie the integration to a particular frame of reference.
We make our integrator, which we call {\sc \tt WHFast}\xspace, publicly available in its native C99 implementation and as an easy-to-use python module.
The remainder of this paper is structured as follows.
We first summarize the concepts and algorithms used in this paper, including Jacobi Coordinates, our choice of Hamiltonian splitting, the symplectic Wisdom-Holman map, symplectic correctors and the variational equations in Sect.~\ref{sec:background}.
We then go into detail discussing the improvements we have made to these algorithms in Sect.~\ref{sec:improvements}.
Numerical tests are presented in Sect.~\ref{sec:numericalresults} before we conclude in Sect.~\ref{sec:conclusions}.
\section{Background}\label{sec:background}
The Hamiltonian $\mathcal{H}$ of the gravitational $N$-body system can be written as the sum of kinetic and potential terms in Cartesian coordinates
\begin{eqnarray}
\mathcal{H} &=& \sum_{i=0}^{N-1} \frac{\mathbf{p}_i^2}{2m_i} - \sum_{i=0}^{N-1} \sum_{j=i+1}^{N-1} \frac{Gm_im_j}{|\mathbf{r}_i-\mathbf{r}_j|}.\label{eq:H}
\end{eqnarray}
One way forward toward separating out the two-body Keplerian Hamiltonians is to transform to heliocentric coordinates involving the centre-of-mass and the $\mathbf{r}_i-\mathbf{r}_0$.
However, rewriting the Cartesian momenta in terms of heliocentric momenta (which have an additional component along the centre-of-mass momentum), leads to several cross-terms.
Alternatively, Jacobi worked out a coordinate system in which the kinetic terms are particularly clean, and the kinetic energy remains a sum of squares.
For readers that may not be familiar, and because our improved accuracy is largely due to modifications of the manner in which we transform between Cartesian and Jacobi coordinates, we briefly review them \citep[see also][]{Plummer1918,SussmanWisdom2001,solarsystemdynamics}.
\subsection{Jacobi Coordinates}
Rather than reference planet positions to the central star, a planet's Jacobi coordinates are measured relative to the centre-of-mass of all bodies with lower indices.
For concreteness, consider a system of $N$ particles with masses $m_i$, $i=0,\ldots, N-1$.
Let $\mathbf{r}_i$ be the position vector of the $i$-th particle with respect to an arbitrary origin that is fixed in an inertial frame.
Here we assume that the particles are ordered such that $i=0$ corresponds to the central object, $i=1$ to the innermost object orbiting the central object and so on.
The existence of such an ordering does not restrict the architecture of the system.
For example, the coordinates of an equal-mass binary with a circumbinary particle can be expressed in Jacobi coordinates.
But note that the ordering might in general be non-unique and that it can change during an integration.
This can have important implications for a numerical scheme using Jacobi coordinates.
The Jacobi coordinate $\mathbf{r}'_i$ of the $i$-th particles is the position relative to $\mathbf{R}_{i-1}$, the centre-of-mass of all the particles interior to the $i$-th particle:
\begin{eqnarray}
\mathbf{r}'_i &=& \mathbf{r}_i - \mathbf{R}_{i-1},\quad\quad\quad\quad\text{for } \;i=1,\ldots,N-1\\
\text{where} \quad \mathbf{R}_{i} &=& \frac{1}{M_i}\sum_{j=0}^i m_j \mathbf{r}_j \quad \text{and } \quad M_{i} \;=\; \sum_{j=0}^i m_j.
\end{eqnarray}
Other quantities such as the velocity and acceleration (also the coordinates in the variational equations, see below) transform in the same way.
This is because the Jacobi coordinates are a linear function of the Cartesian coordinates, and the velocity is the time derivative of the position in both coordinates systems.
The momenta, however, transform differently\footnote{But note that we do not need to calculate the momenta explicitly in our algorithm.}.
The momentum conjugate to $\mathbf{r}'_i$ and the corresponding Jacobi mass are given by
\begin{eqnarray}
\mathbf{p}'_i = m'_i \dot{\mathbf{r}'}_i = m'_i \mathbf{v'}_i \quad\quad\text{ and } \quad\quad m'_i = m_i \frac{M_{i-1}}{M_{i}}= \frac{m_i\,M_{i-1}}{m_i + M_{i-1}}.\label{eq:p}
\end{eqnarray}
Note that the Jacobi mass $m'_i$ is the reduced mass of $m_i$ and $M_{i-1}$.
Explicit expressions for the momenta can be found by evaluating the time derivative r{Eq.~\ref{eq:p}.}
The Jacobi coordinates above are relative coordinates for $i=1,\ldots, N-1$.
For the $0$-th coordinate, a different convention is used,
\begin{eqnarray}
\mathbf{r}'_0 = \mathbf{R}_{N-1},\quad\quad\quad m'_0 = M_{N-1},\quad\quad\quad \mathbf{p}'_0 = \sum_{j=0}^{N-1} \mathbf{p}_j.
\end{eqnarray}
Thus, $\mathbf{r}'_0$ points towards the centre-of-mass of the entire system,~$\mathbf{p}'_0$~is the total momentum and $m'_0$ is the total mass.
\subsection{Hamiltonian Splitting}\label{sec:splitting}
After some algebra, we can rewrite the Hamiltonian in Eq.~\ref{eq:H} in terms of the conjugate momenta of the Jacobi coordinates \citep[e.g.][]{solarsystemdynamics,SussmanWisdom2001}.
We only rewrite the kinetic term and keep the potential term expressed as a function of the Cartesian coordinates:
\begin{eqnarray}
\mathcal{H} &=& \sum_{i=0}^{N-1} \frac{{\mathbf{p}'_i}^2}{2m'_i} - \sum_{i=0}^{N-1} \sum_{j=i+1}^{N-1} \frac{Gm_im_j}{|\mathbf{r}_i-\mathbf{r}_j|}.
\end{eqnarray}
Note that the kinetic term is still diagonal, i.e. there are no cross terms involving $\mathbf{p}_i\mathbf{p}_j$ with $i\neq j$.
Next, we add and subtract the term
\begin{eqnarray}
\Ha_\pm = \sum_{i=1}^{N-1}\frac{G m'_i M_{i} }{|\mathbf{r}'_i|}. \label{eq:hpm}
\end{eqnarray}
After grouping terms in the Hamiltonian, we arrive at
\begin{eqnarray}
\mathcal{H} &=&
\underbrace{\frac{{\mathbf{p}'_0}^2}{2m'_0}\vphantom{\sum_{i=1}^{N-1} \frac{{\mathbf{p}'_i}^2}{2m'_i} }}_{\mathcal{H}_{0}}
+\underbrace{\sum_{i=1}^{N-1} \frac{{\mathbf{p}'_i}^2}{2m'_i} -\sum_{i=1}^{N-1}\frac{G m'_i M_{i} }{|\mathbf{r}'_i|}}_{\mathcal{H}_{\rm Kepler}} \nonumber \\
&& + \underbrace{\sum_{i=1}^{N-1}\frac{G m'_i M_{i} }{|\mathbf{r}'_i|}- \sum_{i=0}^{N-1} \sum_{j=i+1}^{N-1} \frac{Gm_im_j}{|\mathbf{r}_i-\mathbf{r}_j|}}_{\mathcal{H}_{\rm Interaction}}.\label{eq:hamsplit}
\end{eqnarray}
The first term, $\mathcal{H}_0$, simply describes the motion of the centre-of-mass $\mathbf{r}'_0$ along a straight line.
For that reason this term is often ignored.
However, we keep it which will allow us to integrate particles without any restriction to a particular frame of reference.
The terms $\mathcal{H}_{\rm Kepler}$ can be split up further into a sum of
\begin{eqnarray}
\left(\mathcal{H}_{\rm Kepler}\right)_i &=& \frac{{\mathbf{p}'_i}^2}{2m'_i} - \frac{G m'_i M_{i} }{|\mathbf{r}'_i|}. \label{eq:HKepler}
\end{eqnarray}
Each of the Hamiltonians $\left(\mathcal{H}_{\rm Kepler}\right)_i$ describes the Keplerian motion of the $i$-th particle with mass $m_i$ around the centre-of-mass of all interior particles with total mass $M_{i-1}$.
After some more algebra, the interaction term can be simplified and split into two parts, one of which can be easily computed in Jacobi coordinates and the other in Cartesian coordinates of the inertial frame
\begin{eqnarray}
\Ha_{\rm Interaction} =
\sum_{i=2}^{N-1} \frac{ G{m'}_i M_i }{|\mathbf{r}'_i|}
- \sum_{i=0}^{N-1} \sum_{\substack{j=i+1\\j\neq 1}}^{N-1} \frac{Gm_im_j}{|\mathbf{r}_i-\mathbf{r}_j|}.
\end{eqnarray}
One important point to note is that our choice of $\Ha_\pm$ is slightly different from that used by \cite{solarsystemdynamics} and \cite{WisdomHolman1991}.
These authors use
\begin{eqnarray}
\left(\Ha_\pm\right)_{\rm WH1991} &=& \sum_{i=1}^{N-1}\frac{G m'_i \, \mathcal{M}_i }{|\mathbf{r}'_i|}.
\end{eqnarray}
where $\mathcal{M}_i = m_0 \frac{M_i}{M_{i-1}}$.
Their choice leads to the usual disturbing function in perturbation theory.
We conducted various tests but found no significant difference between these mass choices.
We therefore chose our prescription, Eq.\:\ref{eq:hpm}, which has a simpler physical interpretation: the mass entering Kepler's third law is simply the interior mass.
\subsection{Wisdom-Holman Mapping}
Our goal is to find a solution to the equations of motion for particles governed by the Hamiltonian in Eq.~\ref{eq:H}.
No analytic solution exists to the full Hamiltonian and we thus need to find an approximate solution.
There are many different ways to do that.
Here, we describe the idea of constructing a symplectic integrator by means of splitting the Hamiltonian into smaller parts, each of which can be easily integrated.
The introduction of Jacobi coordinates led us to the Hamiltonian splitting described in Sect.~\ref{sec:splitting}.
Analytic solutions can be found for the evolution of the system under each of the individual Hamiltonians $\Ha_0$ and $\Ha_{\rm Interaction}$.
The solution to $\Ha_0$ simply corresponds to motion along a straight line.
The solution to $\Ha_{\rm Interaction}$ is a kick step where the velocities change due the inter-particle accelerations but the position remain constant.
The solution to $\Ha_{\rm Kepler}$ is a set of two-body Kepler orbits, which can also be easily solved with an iterative algorithm.
We discuss the details related to the Kepler problem in Sect.~\ref{sec:kepler}.
Now that we have broken down the full Hamiltonian into individual Hamiltonians, to all of which we know the solution (or can easily calculate it), we can construct a symplectic integrator for the total Hamiltonian using an operator split method \citep[e.g.][]{SahaTremaine1992}.
Let us describe the evolution of particles under a Hamiltonian $\Ha$ for a time $\mathit{dt}$ using the operator notation $\hat{\Ha}(\mathit{dt})$.
The notation $\hat{\Ha}_2(\mathit{dt})\circ\hat{\Ha}_1(\mathit{dt})$ means applying operator $\hat{\Ha}_1$ first, then applying operator $\hat{\Ha}_2$.
It is easy to see that many of the operators commute with each other, i.e.
\begin{eqnarray}
\left[\hat \Ha_0, \hat \Ha_{\rm Kepler} \right] &=& 0 \\
\left[\hat \Ha_0, \hat \Ha_{\rm Interaction} \right] &=& 0\\
\left[\left(\hat \Ha_{\rm Kepler}\right)_i , \left(\hat \Ha_{\rm Kepler}\right)_j \right] &=& 0\quad \forall i,j,
\end{eqnarray}
where $[\hat{\Ha}_1,\hat{\Ha}_2] = \hat{\Ha}_1\circ \hat{\Ha}_2- \hat{\Ha}_2\circ \hat{\Ha}_1$.
This leads to the following Drift-Kick-Drift (DKD) operator splitting scheme, which we refer to as the Wisdom-Holman map:
\begin{enumerate}[labelwidth=1.5cm,labelindent=10pt,leftmargin=1.2cm]
\item [(Drift)] Evolve the system under $\hat \Ha_{\rm Kepler}(\mathit{dt}/2) \circ \hat\Ha_0(\mathit{dt}/2)$.
\item [(Kick)] Evolve the system under $\hat \Ha_{\rm Interaction}(\mathit{dt})$.
\item [(Drift)] Evolve the system under $\hat \Ha_{\rm Kepler}(\mathit{dt}/2) \circ \hat\Ha_0(\mathit{dt}/2)$.
\end{enumerate}
The ordering of $\hat \Ha_{\rm Kepler}$ and $\Ha_0$ in the first and last step doesn't matter as they commute.
The first and last steps can be combined if the system is evolved for multiple timesteps.
Note that the evolution of $\Ha_{\rm Kepler}$ and $\Ha_0$ is most easily accomplished in Jacobi coordinates.
The interaction Hamiltonian $\Ha_{\rm Interaction}$, however, contains terms that depend on both the Cartesian and Jacobi coordinates.
The simplest way to calculate these terms is to convert to Cartesian coordinates, evaluate the $\mathbf{r}_i-\mathbf{r}_j$ term, convert the accelerations back to Jacobi accelerations, and calculate the remaining terms.
\subsection{Symplectic Correctors}\label{sec:correctors}
The operator splitting method used in the symplectic integrator discussed above effectively adds high frequency terms to the Hamiltonian.
An argument often used in favour of symplectic integrators is that, although these high-frequency terms alter the Hamiltonian, they do not change the long term evolution as they average out.
However, they do lead to relatively large short term oscillations, for example in the energy error.
The idea of a symplectic corrector, first used by \cite{TittemoreWisdom1989} and fully developed by \cite{Wisdom1996}, is to remove some of these high frequency terms using perturbation theory.
The basic procedure is as follows.
Before the start of an integration, we convert from real coordinates to so-called mapping coordinates.
Then we perform the integration using our standard symplectic map.
After the simulation has finished (or whenever we need an output) we convert back from mapping to real coordinates.
The symplectic corrector operator that we use is a combination of several $\hat \Ha_{\rm Interaction}(\mathit{dt})$ and $\hat \Ha_{\rm Kepler}(\mathit{dt}) \circ \hat\Ha_0(\mathit{dt})$ operators applied for different (positive and negative) intervals $\mathit{dt}$.
If $\epsilon$ is the order of the perturbations, i.e. the mass ratio and therefore the relative magnitude of $\hat \Ha_{\rm Interaction}$ compared to $\hat \Ha_{\rm Kepler}$, then one can show that the use of symplectic correctors can lead to a scheme of order $O(\epsilon \mathit{dt}^{K} ) + O(\epsilon^2 \mathit{dt}^2)$ where $K$ is the order of the symplectic corrector \citep{MikkolaPalmer2000}.
A second order Wisdom-Holman map without symplectic correctors has an energy error of order $O(\epsilon^2 \mathit{dt}^2)$.
Because this coordinate transformation for the symplectic corrector is only performed for outputs and at the beginning and end of the simulation, its effect on the speed of the algorithm is negligible for sparse output.
A full derivation of the symplectic correctors would go beyond the scope of this paper and we refer the reader to \cite{Wisdom1996} and \cite{MikkolaPalmer2000}.
The corrector coefficients are listed in a compact form in \cite{Wisdom2006}.
We implement a third, fifth, seventh and eleventh order symplectic corrector for {\sc \tt WHFast}\xspace.
Whether the high order-symplectic correctors provide any improvement over the low-order ones depends on the mass ratios in the system.
For Jupiter-mass planets, a symplectic corrector of fifth order is no less accurate than a higher order one.
If in doubt, there is no harm done in using a higher-order corrector as the speed implications are minimal.
Thus, we implement the eleventh-order symplectic corrector by default.
\subsection{Chaos Indicators}
A powerful tool for studying the long term evolution of Hamiltonian systems is the Lyapunov characteristic number (LCN).
The inverse of the LCN is the Lyapunov timescale and gives an estimate of how fast two nearby particle trajectories diverge.
If the system is chaotic, the divergence is exponential in time and the Lyapunov timescale is finite.
Thus, measuring the LCN gives us an estimate of whether the system is chaotic and, if so, on what timescale.
A more recent approach with similar informative value is the Mean Exponential Growth factor of Nearby Orbits, or MEGNO for short \citep{Cincotta2003}.
The MEGNO, $Y(t)$, is a scalar function of time, and provides a clear picture of resonant structures and of the locations of stable and unstable periodic orbits.
There are two ways to calculate the LCN or the MEGNO.
Conceptually the simplest is to integrate an additional shadow particle for each body in the simulation, i.e. a particle with slightly perturbed initial conditions.
One can then directly measure the divergence of each particle's path from its shadow.
The second approach is to consider each body's six-dimensional displacement vector $\pmb{\delta_i}$ from its shadow (in both position and velocity) as a dynamical variable.
One can then obtain differential equations for each $\pmb{\delta_i}$ vector by applying a variational principle to the trajectories of the original bodies.
We choose to follow the latter approach, as it is both faster and numerically more robust \citep{Tancredi2001}.
In this scheme, one can imagine shadow particles with phase-space coordinates $\pmb \xi^s_i = \pmb\xi_i + \pmb\delta_i$, where $\pmb\xi_i=(\mathbf{r_i}, \mathbf{v_i})$ is the phase-space coordinate of the $i$-th original particle.
Initially, we set each component of $\pmb \delta_i$ to a small value.
We follow the work of \cite{MikkolaInnanen1999} who describe how to efficiently couple the variational equations to the original equations of motion.
This allows us to construct a symplectic integrator for the variational equations (a symplectic tangent map).
An important advantage of this method is that we only solve Kepler's equation once for each particle/shadow-particle pair (one of the most time-consuming steps in a Wisdom-Holman integrator for small particle numbers).
The MEGNO is then straightforwardly computed from the variations as \citep{Cincotta2003}
\begin{eqnarray}
Y(t) = \frac{2}{t} \int_0^t t' \frac{\sum_{i=0}^{N-1} \dot{\pmb{\delta}_i}(t') \cdot \pmb{\delta}_i(t')}{\sum_{i=0}^{N-1} \pmb{\delta}_i^2(t')} dt'.
\end{eqnarray}
If $Y(t)\rightarrow \infty$, then the system is chaotic.
For quasi-periodic orbits, the MEGNO converges to a finite value, $Y(t)\rightarrow 2$ \citep[e.g.][]{Hinse2010}.
One can obtain the Lyapunov characteristic number (LCN), the inverse of the Lyapunov timescale, from the time evolution of the MEGNO via a linear least square fit to~$Y(t)$.
\subsection{Kepler Problem with Variations}\label{sec:kepler}
In this section, we summarize how to solve the two-body Kepler problem numerically, including the variational equations.
Although the solution has been known since the days of Newton, the transcendental nature of Kepler's equation does not admit a closed-form mathematical expression.
We closely follow the work of \citep{MikkolaInnanen1999} where the reader can find additional information that we have left out.
The equivalent one-body Hamiltonian for the Kepler problem is
\begin{eqnarray}
H_{\rm Kepler} &=& \frac12 \mathbf{v}^2 - \frac{M}{|\mathbf{r}|},
\end{eqnarray}
where $M$ is the total mass of the two bodies.
For consistency with \cite{MikkolaInnanen1999}, we have dropped the primes, have scaled out $m'_i$ from $p'_i$, and rewritten Eq.~\ref{eq:HKepler} in non-dimensional form, i.e. the gravitational constant $G=1$ for the remainder of this paper.
However, we have taken care to remove any dependence on the choice of units from our implementation, so $G$ can be freely set by the user in our implementation of the algorithm.
Our task is to find the final positions and velocities $\mathbf{r}$ and $\mathbf{v}$ of a particle evolving under this Hamiltonian for some time $\mathit{dt}$, given the initial conditions $\mathbf{r}_0$ and $\mathbf{v}_0$.
Thus, we seek the effect of the operator $\hat H_{\rm Kepler}(\mathit{dt})$.
It is advantageous to solve the Kepler problem numerically using the Gauss f and g functions, which express the relevant quantities in terms of $\mathbf{r}_0$ and $\mathbf{v}_0$ \citep{WisdomHolman1991}. This avoids the computationally expensive conversion between Cartesian and classical orbital elements, and avoids coordinate singularities associated with circular orbits. We find that it is advantageous to use universal variables in this solution \citep{Stumpff1962}. This approach provides greater speed and numerical stability compared to a solution using elliptic elements. It also avoids the singularity associated with the transition from elliptic to hyperbolic motion.
To solve the analogue of Kepler's equation for the particle's position in time, we make use of several special functions.
Let us begin by defining the $c$-functions \citep{Stumpff1962} as a series expansion:
\begin{eqnarray}
c_n(z) \equiv \sum_{j=0}^\infty \frac{(-z)^j}{(n+2j)!} \label{eq:c},
\end{eqnarray}
which satisfy the recursion relation
\begin{eqnarray}
c_n(z) &=& \frac{1}{n!} - z \,c_{n+2}. \label{eq:recur}
\end{eqnarray}
The $c$-functions are related to trigonometric functions, for example
\begin{eqnarray}
c_0(z) = \cos \sqrt{z} \quad\quad \text{and}\quad\quad c_1(z) = \frac{\sin\sqrt{z}}{\sqrt{z}},
\end{eqnarray}
and thus satisfy the following relationships \citep{Mikkola1997}, which are related to the half-angle formula for trigonometric functions:
\begin{eqnarray}
c_5(z) &=& \frac1{16} \left[ c_5(z/4) + c_4(z/4) + c_3(z/4)c_2(z/4)\label{eq:crel5}\right]\\
c_4(z) &=& \frac1{8} c_3(z/4) \left[1 + c_1(z/4).\label{eq:crel4}\right]
\end{eqnarray}
Values for $c_0$ through $c_3$ are then readily computed from Eq.~\ref{eq:recur}.
Next, we introduce the so called $G$-functions \citep{StiefelScheifele1971} which in turn depend on the $c$-functions:
\begin{eqnarray}
G_n(\beta,X) \equiv X^n c_n(\beta X^2). \label{eq:G}
\end{eqnarray}
The $G$-functions also satisfy recursion relationships similar to those mentioned above for the $c$-functions.
We can easily calculate derivatives of $G_n$ by looking at the series expansion of $c_n$ \citep[see][for details]{MikkolaInnanen1999}.
With this framework, we can now write down the steps needed to find the solution to the Kepler Hamiltonian in compact form.
First, we need to calculate the following three quantities from the initial conditions $\mathbf{r}_0, \mathbf{v}_0$:
\begin{eqnarray}
\beta &=& \frac{2M}{r_0}-v_0^2\\
\eta_0 &=& \mathbf{r}_0 \cdot \mathbf{v}_0 \\
\zeta_0 &=& M-\beta r_0
\end{eqnarray}
where $r_0 = |\mathbf{r}_0|$ and $v_0 = |\mathbf{v}_0|$. Note that the semi-major axis $a$ can be written as $a=M/\beta$.
Second, we need to solve Kepler's equation which, using the above notation, takes the form
\begin{eqnarray}
r_0 X + \eta_0 G_2(\beta,X) + \zeta_0 G_3(\beta,X) - \mathit{dt} = 0. \label{eq:kepler}
\end{eqnarray}
We solve this equation for $X$.
This is a non-algebraic (i.e. transcendental) equation that we need to solve iteratively, for example using Newton's method.
In Sect.~\ref{sec:newton}, we describe our algorithm in detail.
Third, having solved Kepler's Equation, we can calculate the so called Gau{\ss} $f$ and $g$-functions as well as their time derivatives via
\begin{align}
f &= 1 - M \frac{G_2}{r_0}\quad & \dot f &= -\frac{M\,G_1}{r_0r} \label{eq:fg}&\\
g &= dt - M G_3 & \dot g &= 1-\frac{M\,G_2}{r},\label{eq:fg2}
\end{align}
where $r = r_0 + \eta_0 G_1 + \zeta_0 G_2$.
Note that all the $G$-functions depend on $\beta$ and the $X$ value found in the second step.
Fourth, we write the final positions and velocities as a linear transformation of the initial conditions using the Gau{\ss} $f$ and $g$-functions:
\begin{align}
\mathbf{r} &= f \mathbf{r}_0 + g \mathbf{v}_0
&\mathbf{v} &= \dot f \mathbf{r}_0 + \dot g \mathbf{v}_0.\label{eq:fgupdate}&&
\end{align}
This completes the solution of the Kepler problem.
To solve for the variational equations, we also make use of the $G$-functions.
Fortunately, we only need to solve Kepler's equation once (to solve for $X$).
We then get the solution for the variational equations without solving another transcendental equation and thus have only one iteration loop per timestep for both the particle and its variational counterpart.
The position and velocity components of $\pmb \delta$ at the end of the timestep,~$\delta \mathbf{r}$ and $\delta \mathbf{v}$, can be written as
\begin{eqnarray}
\delta \mathbf{r} &=& f \;\delta \mathbf{r}_0 + g \;\delta\mathbf{v}_0 + \mathbf{r}_0 \;\delta f+ \mathbf{v}_0 \;\delta g\label{eq:var1}\\
\delta \mathbf{v} &=& \dot f \;\delta \mathbf{r}_0 + \dot g \;\delta\mathbf{v}_0 + \mathbf{r}_0 \;\delta \dot f+ \mathbf{v}_0 \;\delta \dot g,\label{eq:var2}
\end{eqnarray}
where the variations $\delta f$, $\delta g$, $\delta \dot f$ and $\delta \dot g$ can be derived from Eqs.~\ref{eq:fg}-\ref{eq:fg2} (see \citealt{MikkolaInnanen1999} for the explicit expressions).
\subsection{Types of Numerical Errors}\label{sec:errors}
There are three distinct effects contributing to the energy error of a symplectic integrator \citep[see e.g.,][]{QuinnTremaine1990}.
See also \cite{ReinSpiegel2015} for a similar discussion for non-symplectic integrators.
First, there is an error term associated with the integrator itself because we are not solving the equations of motion for the Hamiltonian $\mathcal{H}$ exactly.
For symplectic integrators such as those discussed here, this error term is bound and we call it $E_{\rm bound}$.
If the mass ratio of the planets to the star is $\epsilon$, then the order of this error term is roughly $O(\epsilon\, \mathit{dt}^{2})$ for integrators without symplectic corectors and $O(\epsilon \,\mathit{dt}^{K} ) + O(\epsilon^2 \,\mathit{dt}^2)$ for those with symplectic correctors (see Sect.~\ref{sec:correctors}).
Note that $E_{\rm bound}$ is independent of time $t$.
Second, there is an error term associated with the finite precision of numbers represented on a computer.
We can only represent a small subset of all real numbers exactly in floating-point precision.
Thus after every operation such as an addition or multiplication, the computer rounds to a nearby floating-point number.
For CPUs and compilers that follow the IEEE754 standard \citep{IEEE754}, we are guaranteed to round to the nearest floating-point number.
Thus, if all operations follow the IEEE754 standard, then as long as the algorithm itself is unbiased, we expect the error to grow as the square root of the number of operations, i.e. $E_{\rm rand} \sim \sqrt{N}\sim \sqrt{t}$, where $N$ is the number of timesteps.
This is the best behaviour achievable; to do better we would have to move to extended precision or use fewer operations.
This fundamental limit is known as Brouwer's law~\citep{Newcomb1899,Brouwer1937}.
Third, if any parts of the integration algorithm are biased, the errors will be correlated.
This leads to a faster long-term energy-error growth than if errors are uncorrelated; it grows linearly with time, i.e. $E_{\rm bias}\sim N \sim t$.
For a given integrator, which of these three error terms dominates depends on the nature of the simulation, the timestep, and the total integration time (number of timesteps).
\section{Improvements}\label{sec:improvements}
The algorithms we describe in Sect.~\ref{sec:background} have been used successfully for many years.
In the following, we show how to significantly improve the speed and accuracy of the algorithms by taking special care in the implementation of several details, many of which are related to finite floating-point precision on modern computers.
For the remainder of this paper, we will assume that we work with a CPU that follows the IEEE~754 standard for floating-point arithmetic.
Most importantly, we assume that all floating-point operations follow the \textit{rounding to nearest, ties to even} rule \citep{IEEE754}.
What follows is in principle applicable to any precision.
However, we work exclusively in double floating-point precision (64 bit) which is used on almost all modern CPUs.
\subsection{Jacobi Coordinate Transformations}\label{sec:jacobi}
The evolution under the effect of the interaction Hamiltonian is most efficiently done in Cartesian coordinates.
On the other hand, the evolution of the Kepler Hamiltonian is easier in Jacobi coordinates.
We thus need an efficient way to convert to and from Jacobi coordinates.
Luckily, the conversion from Cartesian to Jacobi coordinates and back can be done efficiently in $\mathcal{O}(N)$.
We construct the algorithms from the definitions above and list them here in pseudo code.
As before, primes denote Jacobi coordinates,
Note that these algorithms work even if some of the bodies are test particles with $m_i=0$ (for $i\neq 0$).
To convert from Cartesian to Jacobi coordinates:\vspace{0.2cm}
\begin{algorithmic}
\State $\mathbf{R} \gets m_0 \cdot \mathbf{r}_0$
\For{$i \gets 1,N-1$}
\State $\mathbf{r}'_i \gets \mathbf{r}_i - \mathbf{R}/M_{i-1}$
\State $\mathbf{R} \gets \mathbf{R} \cdot (1 +m_i/M_{i-1}) + m_i\cdot\mathbf{r}'_i$
\EndFor
\State $\mathbf{r}'_0 \gets \mathbf{R} / M_{N-1}$ \Comment{This is the centre-of-mass.}
\end{algorithmic}\vspace{0.2cm}
Similarly, we construct the algorithm to convert back from Jacobi to Cartesian coordinates as follows:\vspace{0.2cm}
\begin{algorithmic}
\State $\mathbf{R} \gets \mathbf{r}'_0\cdot M_{N-1}$ \Comment{Centre of mass.}
\For{$i \gets N-1,1$} \Comment{Loop is in reverse order.}
\State $\mathbf{R} \gets (\mathbf{R}- m_i \cdot \mathbf{r}'_i) / M_i$
\State $\mathbf{r}_i \gets \mathbf{r}'_i + \mathbf{R}$
\State $\mathbf{R} \gets \mathbf{R} \cdot M_{i-1}$
\EndFor
\State $\mathbf{r}_0 \gets \mathbf{R}/m_0$\Comment{Setting the coordinate of the 0-th particle.}
\end{algorithmic}\vspace{0.2cm}
We thoroughly tested the conversions to and from Jacobi coordinates to ensure they are unbiased.
This task turns out to be much harder than we na{\"i}vely expected.
As an example, consider the following algorithm which is formally equivalent to the above but numerically much less stable.\vspace{0.2cm}
\begin{algorithmic}
\State $\mathbf{R} \gets 0$
\For{$i \gets N-1,1$} \Comment{Loop is in reverse order.}
\State $\mathbf{r}_i \gets \mathbf{r}'_0 + M_{i-1}/M_i \cdot \mathbf{r}'_i -\mathbf{R}$ \Comment{$\mathbf{r}'_0$ is the centre-of-mass.}
\State $\mathbf{R} \gets \mathbf{R} + m_i/M_i \cdot \mathbf{r}'_i$
\EndFor
\State $\mathbf{r}_0 \gets \mathbf{r}'_0 - \mathbf{R}$\Comment{Setting the coordinate of the 0-th particle.}
\end{algorithmic}\vspace{0.2cm}
In the above algorithm, we access $\mathbf{r}_0'$ multiple times and have to do a subtraction in the last step.
This significantly promotes error propagation and leads to floating-point errors that can be orders of magnitudes higher than in the other implementation.
After many timesteps, this leads to a linear secular growth in the energy error.
\subsection{Implementation of Newton's Method}\label{sec:newton}
To solve Kepler's equation (Eq.~\ref{eq:kepler}) for $X$, we need to use an iterative scheme.
We now describe our implementation of Newton's method in floating-point arithmetic.
The straightforward implementation is an iteration loop that terminates when the change to $X$ is small, e.g., \vspace{0.2cm}
\begin{algorithmic}
\State $X \gets initial\; guess$
\Repeat
\State $dX \gets -f(X)/f'(X)$
\State $X \gets X + dX$
\Until{ $|dX/X| < \eps$}.
\end{algorithmic} \vspace{0.2cm}
Here, $\eps$ is a small number just above machine precision, typically~$\eps \sim10^{-15}$.
We use a different implementation of Newton's method that is both faster and more accurate, despite the fact that it is algebraically equivalent to the above implementation. \vspace{0.2cm}
\begin{algorithmic}
\State $X \gets initial\; guess$
\State $X_{\rm prev 1} \gets \mathit{NaN}$\Comment{Any number different from $X$ works.}
\Repeat
\State $X_{\rm prev 2} \gets X_{\rm prev 1}$
\State $X_{\rm prev 1} \gets X$
\State $X \gets (X\cdot f'(X)-f(X))/f'(X)$
\Until{$X = X_{\rm{ prev 1}}$ or $X = X_{\rm{ prev 2}}$ }
\end{algorithmic} \vspace{0.2cm}
Note that the equal sign in the above breakout condition is evaluated in floating-point precision.
In comparison to the first algorithm, at each iteration step we test whether the iteration has converged by a simple comparison rather than by a slow division and absolute-value operation.
We keep track of two previous values instead of just one because for certain initial conditions, the iteration can cycle indefinitely between two nearby floating-point numbers and not converge to a single floating-point number.
Our implementation thus ensures that the value of $X$ is more accurately calculated than in the straightforward implementation using a heuristic value of $\eps$.
A further advantage of rewriting Newton's method in the above form is that the term on the right-hand-side of the last line can be simplified significantly for the Kepler problem, giving:
\begin{eqnarray}
X \gets \frac{X (\eta_0 G_1 + \zeta_0 G_2) -\eta_0 G_2 -\zeta_0 G_3+dt }{r_0+\eta_0 G_1 +\zeta_0 G_2} \label{eq:kepeq}
\end{eqnarray}
where the $G$'s on the right-hand-side all depend on $X$ and $\beta$ (see Eq.~\ref{eq:G}).
We also experimented with higher-order generalizations of Newton's method (Householder's methods). For typical cases where the orbits are not extremely elliptical ($e\lesssim 0.99$) and the timestep is much smaller than the shortest orbital period, we found Newton's method to always be fastest.
This is because when the value and derivatives of the function are easily evaluated, the precision gain from these higher-order methods does not compensate for the increased computation cost of each iteration.
In other words, while higher-order methods will converge in fewer iterations than Newton's method, the overall computation time is longer.
At large eccentricities and long timesteps, the $G$-function evaluations become expensive (one must recursively apply the quarter-angle formulas described in Sect.~\ref{sec:cs}), and higher-order methods are helpful.
For large eccentricities we use a higher order method described in detail in Sect.~\ref{sec:largeE}.
To safeguard against rare cases where Newton's method might fail, we also implemented a failsafe bisection method.
We find that the bisection method is only triggered when the timestep is comparable to the orbital period.
\subsection{The Initial Guess for Kepler's Equation: Short Timesteps}
The quantity X in Eq.~\ref{eq:kepeq} can also be expressed as
\begin{eqnarray}
X = \int_{t_0}^{t_0 + \mathit{dt}} \frac{\mathit{dt}'}{r} = \mathit{dt} \cdot{\langle r^{-1}\rangle} \label{eq:xint}
\end{eqnarray}
where $t_0$ is the time at the beginning of the timestep, and $\langle r^{-1}\rangle$ is the time-averaged value of $r^{-1}$ over the interval $[t_0, t_0+\mathit{dt}]$.
Thus, if the orbit's eccentricity $e$ is low, or more generally if the timestep is short enough that the orbital radius does not vary much, then $X \approx \mathit{dt}/r_0$.
The troublesome cases are highly eccentric orbits near pericentre where the radius changes rapidly.
For such cases, the radius varies by a factor of $1+e \approx 2$ from pericentre to a true anomaly of $90^\circ$.
We can therefore estimate the timescale over which the orbital radius varies near pericentre as
\begin{eqnarray}
T_{char} = \frac{q}{v_q} = \frac{a(1-e)}{na}\Bigg(\frac{1-e}{1+e}\Bigg)^{1/2} \sim \frac{(1-e)^{3/2}}{n}, \label{eq:tchar}
\end{eqnarray}
where $q$ is the pericentre distance, $v_q$ is the speed at pericentre and $n$ is the mean motion.
Thus, if one does not resolve pericentre passages (i.e., $n\,\mathit{dt} = \Delta M \gtrsim (1-e)^{3/2}$), $X$ will differ from $\mathit{dt}/r_0$ near pericentre (but may nevertheless conform to the simple approximation at apocentre where the body moves slowly).
More quantitatively, one can non-dimensionalize Eq.~\ref{eq:kepeq}, setting $\tilde{X} = r_0 X / \mathit{dt}$.
One can then solve the equation perturbatively, assuming the deviations from $\tilde{X} = 1$ are small.
This procedure requires that the following three non-dimensional parameters in the equation also be much smaller than unity,
\begin{align}
\chi \equiv \frac{\beta \mathit{dt}^2}{r_0^2}\quad \eta \equiv \frac{\eta_0\mathit{dt}}{r_0^2} \quad \zeta \equiv \frac{\zeta_0\mathit{dt}^2}{r_0^3}.
\end{align}
One can show that when our heuristic estimate $\Delta M \ll (1-e)^{3/2}$ is satisfied, $\chi$, $\eta$, $\zeta \ll 1$.
In this case, one can extend the solution of Eq.~\ref{eq:kepeq} to higher order.
For the initial guess in our algorithm, we go up to second order
\begin{eqnarray}
X = \frac{\mathit{dt}}{r_0} \cdot \left(1 - \frac12 \eta \right). \label{eq:xinit}
\end{eqnarray}
We experimented with higher-order initial guesses (see \citealt{Danby1987} for explicit expressions), but found these to be slower, even for small eccentricities and timesteps.
This can again be attributed to the computational efficiency of each iteration of Newton's method.
\subsection{Large Eccentricities and Timesteps} \label{sec:largeE}
The previous two sections describe an optimized algorithm for solving Kepler's equation when the timestep and eccentricities are low.
We have also developed an improved handling of high-eccentricity/long-timestep cases.
In this regime, both the solver and initial guess should be modified.
Like previous authors \citep{Conway1986, Danby1987}, we found the root-finding method of Laguerre-Conway to be most stable.
However, unlike \cite{Danby1987}, who finds the method to always converge (presumably using comparatively small timesteps), we often have to resort to bisection when the timestep is comparable to the orbital period.
Of course, such long timesteps should not be chosen anyway, since they poorly sample inter-planet interactions, and are more susceptible to timestep resonances \citep{WisdomHolman1992,ToumaWisdom1993,Rauch1999}.
We also had to modify the breakout condition used for Newton's method.
While the Laguerre-Conway algorithm sometimes also bounces between two floating-point values once it has converged, in this regime the method often executes larger-period cycles (e.g., it will periodically repeat the last eight floating-point numbers).
We therefore chose to store the values from each iteration and exit the loop whenever a result was repeated.
One way to determine which solver should be used is to check whether $\mathit{dt}$ is smaller than $T_{char}$ (Eq.~\ref{eq:tchar}).
However, because $T_{char}$ is expensive to compute from $\mathbf{r}_0$ and $\mathbf{v}_0$, we instead check how much the first iteration of Newton's method deviates from the initial guess, as a fraction of $2\pi \beta^{-1/2}$.
The latter is a natural quantity to compare against since it is the value of $X$ when the timestep is equal to the orbital period.
We found a threshold of $\sim 1\%$ to strike a good balance over a wide parameter range in timestep/eccentricity space, though the algorithm's speed is not particularly sensitive to the exact value adopted.
Finally, the method can be sped up in this regime with an improved initial guess for $X$, since $\mathit{dt}/r_0$ in Eq.~\ref{eq:xinit} blows up near pericentre as the eccentricity gets large.
\cite{Danby1983} provide a widely used initial guess using classical orbital elements but, to our knowledge, no comparably simple initial guess has been found for universal variables.
In this high-eccentricity / long timestep regime, most existing methods using universal variables choose to make the expensive conversion to orbital elements and use Danby's guess.
We instead observed that because $\langle r^{-1} \rangle = a^{-1}$ over one orbital period, $X = \mathit{dt}/a$ for a timestep of one orbit.
We find that over a relevant parameter range with timesteps logarithmically spaced between 0.03 and 1 orbital periods, and eccentricities between 0.999 and 0.9999, our improved guess is faster than converting to orbital elements and using Danby's by $\approx 30\%$.
In a manner analogous to that described in the previous section, we also solved Eq.~\ref{eq:kepeq} perturbatively around $X = \mathit{dt}/a = \beta\; \mathit{dt} / M$ in the regime $\chi \gg 1$, but we found the second-order solution to be a slower initial guess than the simple $X = \beta \; \mathit{dt} / M$.
\subsection{Implementation of $c$-functions} \label{sec:cs}
Finding a solution to Kepler's equation is done iteratively and is thus the most expensive step in solving the Kepler problem.
The iteration itself involves the calculation of multiple $G$-functions, which in turn require the calculation of $c$-functions.
Thus, it is particularly important to optimize these functions for both speed and accuracy.
When calculating chaos indicators, we need~$c_0$, $c_1$, $c_2$, $c_3$, $c_4$ and $c_5$.
If we are not integrating the variational equations, we only need $c_0$, $c_1$, $c_2$ and $c_3$.
We first ensure that $z$ is smaller than~$0.1$ to guarantee that the series expansion of $c$ in Eq.~\ref{eq:c} converges.
We do this by dividing $z$ repeatedly by 4.
Note that divisions by powers of 2 are fast and exact in floating-point arithmetic.
To calculate the series expansion, we need an inverse factorial for every term.
Calculating this inverse factorial by multiplying floating-point numbers and then implementing a floating-point division would be very slow.
We found that the fastest way to calculate the inverse factorial is to use a simple lookup table.
We checked that the series expansions of the $c$-functions converge very quickly for small $z$ and thus we only store inverse factorials up to $1/34!$ in the lookup table.
Any larger factorial would contribute less than one part in $10^{16}$ to the sum and can thus be neglected (as we work in double floating-point precision).
We always calculate the first two terms in the series expansion.
We then enter a loop and add more terms until the result no longer changes.
Because $z$ is small and the inverse factorials decrease quickly, we are assured that the series will converge to a single floating-point number.
This allows us to simply check whether the value changes from one iteration to the next, which is much faster than evaluating relative changes (cf. Sect.~\ref{sec:newton}).
Once the $c$-functions are calculated for the small $z$ value, we use the relations in Eqs.~\ref{eq:crel5}-\ref{eq:crel4} with Eq.~\ref{eq:recur} to calculate the $c$-functions for the original $z$ value.
Because this algorithm is an integral part of the integrator, we list the function to calculate $c(z)$ in pseudo code: \vspace{0.2cm}
\begin{algorithmic}
\State $n \gets 0$ \Comment{Counter for quarter-angle formula.}
\While{$z>0.1$} \Comment{Ensure that $z$ is small.}
\State $z \gets z/4$
\State $n \gets n+1$
\EndWhile
\State $c_4 \gets \frac{1}{4!} - z \cdot \frac{1}{6!}$\Comment{Hard coded first two terms for $c_4$.}
\State $c_5 \gets \frac{1}{5!} - z \cdot \frac{1}{7!}$
\State $\bar z \gets -z$
\State $p \gets \bar z$ \Comment{$p$ will the $(-z)^j$ factor in the loop.}
\State $k \gets 8$ \Comment{Third term in $c_4$ contains factor $\frac{1}{8!}$.}
\Repeat
\State {$c_{4\rm, prev} \gets c_4$} \Comment{Keep old value to check for convergence.}
\State $p \gets p \cdot \bar z$
\State $c_4 \gets c_4 + p \cdot \frac{1}{k!}$ \Comment{$1/k!$ comes from lookup table.}
\State $k \gets k+1$
\State $c_5 \gets c_5 + p \cdot \frac{1}{k!}$
\State $k \gets k+1$
\Until {$c_4 = c_{4\rm, prev}$ } \Comment{Converged?}
\State $c_3 \gets \frac16 - z \cdot c_5$\Comment{Use Eq.~\ref{eq:recur} to get $c_3$, $c_2$ and $c_1$.}
\State $c_2 \gets \frac12 - z \cdot c_4$
\State $c_1 \gets 1 - z \cdot c_3$
\While{$n>0$} \Comment{Apply quarter angle formula $n$ times.}
\State $z \gets 4 \cdot z$
\State $c_5 \gets \frac1{16}\cdot (c_5+c_4+c_3+c_2) $
\State $c_4 \gets \frac18\cdot c_3 \cdot (1-c_1)$
\State $c_3 \gets \frac16 -z \cdot c_5$
\State $c_2 \gets \frac12 -z \cdot c_4$
\State $c_1 \gets 1 -z \cdot c_3$
\State $n\gets n-1$
\EndWhile
\State $c_0 \gets 1 -z \cdot c_2$
\end{algorithmic}\vspace{0.2cm}
\subsection{Implementation of Gau{\ss} $f$ and $g$-functions}\label{sec:gauss}
The precise implementation of Gauss $f$ and $g$ functions matters for long term integrations.
The straightforward implementation following \citep{MikkolaInnanen1999} leads to the $f$ and $g$-functions in Eq.~\ref{eq:fg}.
Note that for timesteps smaller than half an orbital period, the term $MG_2/r_0$ in $f$ is small compared to the first term (which is just 1).
The same argument holds true for $\dot g$.
We can define new $\hat f$ and $\hat{\dot g}$-functions
\begin{eqnarray}
&\hat f = - M \frac{G_2}{r_0}\quad & \dot f = -\frac{M\,G_1}{r_0r} \label{eq:fghat}\\
&g = dt - M G_3 & \hat {\dot g} = -\frac{M\,G_2}{r}.
\end{eqnarray}
This allows us to rewrite the last step in solving the Kepler problem as
\begin{eqnarray}
\mathbf{r} = \left( \hat f \mathbf{r}_0 + g \mathbf{v}_0 \right) + \mathbf{r}_0 \quad\quad\quad\quad
\mathbf{v} = \left( \dot f \mathbf{r}_0 + \hat{\dot g} \mathbf{v}_0 \right) + \mathbf{v}_0.
\end{eqnarray}
Although this step is algebraically equivalent to the original Eq.~\ref{eq:fgupdate}, we achieve higher precision.
The reason is that we can now ensure that the small quantities in brackets are summed before they are added to the larger quantity (the initial value).
We implement the same trick for the variational equations, Eqs.~\ref{eq:var1} and~\ref{eq:var2}
\subsection{A full integration in Jacobi coordinates} \label{sec:intinjac}
The algorithms to convert to and from Jacobi coordinates that we describe in Sect.~\ref{sec:jacobi} are unbiased and fast.
Nevertheless, we aim to avoid as many conversion as possible.
As it turns out, we can reduce the number of conversions per timestep to two, one for the positions from Jacobi coordinates to the inertial frame, and one for the accelerations from the inertial frame to Jacobi accelerations.
But note that this is only possible under the following assumptions:
1) the particle position and velocities are not changed in-between timesteps, e.g. manually by the user or by collisions,
2) outputs are not required at every timestep,
3) variational equations are not integrated,
4) no additional velocity-dependent forces are present.
In such a case, an integration starting from an arbitrary inertial frame is achieved as follows:\vspace{0.2cm}
\begin{algorithmic}
\State calculate Jacobi coordinates
\State drift all particles under $H_{\rm Kepler}$ for half a timestep, $\mathit{dt}/2$
\While{$t<t_{\rm max}$}
\State calculate 1st part of $H_{\rm Interaction}$ in Jacobi coordinates
\State update positions in the inertial frame
\State calculate 2nd part of $H_{\rm Interaction}$ in inertial frame
\State convert accelerations from 2nd part to Jacobi accelerations
\State apply kick from Jacobi accelerations to Jacobi velocities
\If{not last timestep}
\State drift all particles under $H_{\rm Kepler}$ for a full timestep $\mathit{dt}$
\EndIf
\EndWhile
\State drift all particles under $H_{\rm Kepler}$ for half a timestep $\mathit{dt}/2$
\State update both positions and velocities in the inertial frame.
\end{algorithmic}\vspace{0.2cm}
Note that we never update the velocities in the inertial frame until the end of the simulation (or when an output is needed).
We only convert the positions and velocities to Jacobi coordinates at the very beginning and not at every timestep.
Besides the obvious speed-up, avoiding to go back and fourth between different coordinate systems reduces the build-up of round-off errors and thus makes the integrator more robust.
\subsection{LCN calculation}\label{sec:megnocalc}
To calculate the Lyapunov characteristic number and the Lyapunov timescale we need to perform a linear least square fit to the function $Y(t)$.
Thus we need the mean and the covariance of $Y(t)$.
Storing all previous values of $Y(t)$ just to calculate its mean and covariance is inefficient.
We therefore implement an efficient one-pass method described by \cite{Pebay2008}.
This method lets us calculate the LCN at every timestep in $\mathcal{O}(1)$ and has the further advantage of being numerically more robust than the standard implementation.
\begin{figure*}
\centering \resizebox{0.99\textwidth}{!}{\includegraphics{2body}}
\caption{
Tests of the Kepler-solver.
Simulation with two bodies, integrated for 100 orbits with varying eccentricity and timestep. Left column: results using the standard {\sc \tt WH}\xspace integrator. Right column: results using our new {\sc \tt WHFast}\xspace integrator. Top row: relative energy error at the end of the simulation.
Middle row: sign of the energy error at the end of the simulation.
Bottom row: average runtime for one timestep.
\label{fig:2body}}
\end{figure*}
\section{Numerical Results}\label{sec:numericalresults}
In this section, we test the speed, accuracy and numerical stability of {\sc \tt WHFast}\xspace and compare it to other publicly available and widely used integrators.
We begin by briefly defining our nomenclature for these other integrators and summarizing their properties.
{\sc \tt MERCURY}\xspace is a mixed-variable symplectic integrator implemented in fortran and provided by the {\sc \tt MERCURY}\xspace package \citep{Chambers1997}.
This Wisdom-Holman style integrator uses high-order symplectic correctors.
We directly call the fortran code without any modifications.
{\sc \tt SWIFTER-WHM}\xspace is again a classical 2nd-order Wisdom-Holman integrator without symplectic correctors \citep{WisdomHolman1991}.
We use the integrator provided by the {\sc \tt SWIFTER}\xspace package.
It is implemented in fortran and we directly call the {\sc \tt SWIFTER}\xspace executable without any modifications.
{\sc \tt SWIFTER-HELIO}\xspace is also 2nd-order symplectic integrator without symplectic correctors \citep{Duncan1998}.
It uses democratic heliocentric coordinates.
We again use the integrator provided by the {\sc \tt SWIFTER}\xspace package.
It is implemented in fortran and we directly call the {\sc \tt SWIFTER}\xspace executable without any modifications.
{\sc \tt SWIFTER-TU4}\xspace is a 4th-order symplectic integrator.
It is \emph{not} a Wisdom-Holman integrator but splits the Hamiltonian in kinetic and potential terms \citep{Gladman1991}.
We also use the integrator provided by the {\sc \tt SWIFTER}\xspace package.
It is implemented in fortran and we directly call the {\sc \tt SWIFTER}\xspace executable without any modifications.
For a more direct comparison, we also make use of an integrator that we simply refer to as {\sc \tt WH}\xspace. It is based on the {\sc \tt SWIFTER-WHM}\xspace integrator in {\sc \tt SWIFTER}\xspace but ported to C and available in the {\sc \tt REBOUND}\xspace \citep{ReinLiu2012} package.
Like the {\sc \tt SWIFTER-WHM}\xspace integrator, it is a symplectic integrator that works in the heliocentric frame, and does not implement any symplectic correctors.
Note that this is not the original integrator used by \cite{WisdomHolman1991}, which is not publicly available.
{\sc \tt WHFast}\xspace is C99 compliant. The C99 standard guarantees that floating point operations are not re-ordered by the compiler (unless one of the fast-math options is turned on).
Because of that, the final positions and velocities of particles agree down to the last bit across different platforms.
This makes {\sc \tt WHFast}\xspace platform independent and the simulation results reproducible.
We verified this on different architectures (Linux, MacOSX), different CPUs (Intel Core i5-3427U, Intel Xeon E5-2697 v2, Intel Xeon E5-2620 v3) and different compilers (Apple LLVM 6.1.0, gcc 4.4.7).
\subsection{Two-body Kepler Solver}
The kernel of every Wisdom-Holman integrator is the Kepler solver.
We describe our implementation in detail in Sections~\ref{sec:newton}-\ref{sec:gauss}.
Here, we test the Kepler solver using a two-body problem.
The two body problem is invariant with respect to rescaling of the total mass, the mass ratio, the value of the gravitational constant and the orbital period.
What does matter is the eccentricity of the orbit and the ratio of the timestep to the orbital period.
We thus scan the parameter space in those two dimensions by integrating two bodies for 100 orbital periods.
We explore an extremely wide parameter space.
The eccentricities range from zero to $0.999\,999\,99 = 1-10^{-8}$.
The range of timesteps goes from 0.1\% of the orbital period all the way up to one orbital period.
Fig.~\ref{fig:2body} shows the performance of {\sc \tt WHFast}\xspace (right column) compared to {\sc \tt WH}\xspace (left column).
The top row shows the absolute value of the relative energy error at the end of the simulation.
The middle row shows the sign of the energy error.
The bottom row shows the average runtime for a single timestep.
The vertical lines visible in the top row correspond to timestep resonances \citep{WisdomHolman1992,ToumaWisdom1993,Rauch1999}.
One can see that {\sc \tt WHFast}\xspace is significantly more accurate than the standard {\sc \tt WH}\xspace integrator for the most important parts of parameter space (eccentricities less than $\sim 0.99$).
The relative energy is conserved better by two to three orders of magnitude.
Most importantly, note that the energy error in the standard {\sc \tt WH}\xspace integrator is biased over large regions of the parameter space (there are large blue and red areas in the second row).
On the other hand, {\sc \tt WHFast}\xspace has a random energy error throughout the parameter space.
Having a biased energy error will lead to a long-term linear growth of the energy error (see below).
In the entire parameter space explored, {\sc \tt WHFast}\xspace requires less time to complete a timestep than {\sc \tt WH}\xspace.
The speed-up is typically between 20\% and 100\%.
For the integrations performed in this section, we convert to and from Jacobi coordinates at every timestep to provide a fair comparison.
Thus, the speed-up and the energy-conservation properties of {\sc \tt WHFast}\xspace are in fact even better than shown here in any actual production run (see Sect.~\ref{sec:intinjac}).
\subsection{Short Term Energy Conservation}\label{sec:shorttermenergy}
\begin{figure}
\centering \resizebox{0.99\columnwidth}{!}{\includegraphics{shorttermenergy}}
\caption{
Relative energy error in simulations of the outer Solar System after 1000~Jupiter orbits as a function of the number of steps per orbit.
\label{fig:shorttermenergy}}
\end{figure}
To compare the accuracy of the different integrators in a realistic test case, we run simulations of the outer Solar System for one thousand Jupiter orbits ($12\,000$~years).
We include the Sun and four massive bodies with approximate initial conditions corresponding to those of Jupiter, Saturn, Uranus and Neptune.
In each simulation the initial conditions and masses are randomly perturbed by 0.1\%.
In Fig.~\ref{fig:shorttermenergy}, we plot the relative energy errors at the end of the simulation as a function of the number of timesteps imposed per Jupiter orbit.
One can see that all the integrators except {\sc \tt SWIFTER-TU4}\xspace are second-order schemes.
For timesteps between 20\% and 0.1\% of the orbital period of Jupiter (50 to 1000~timesteps per orbit), their error decreases quadratically with decreasing timesteps.
This is the error term $E_{\rm bound}$ introduced in Sect.~\ref{sec:errors}.
However, decreasing the timestep also increases the number of floating point operations. There will therefore be a timestep value at which the numerical round-off error dominates over the error associated with the symplectic method itself $E_{\rm bound}$.
For that reason we find that for small timesteps, less than 0.1\% of the shortest orbital period, the errors of all integrators rise instead of decreasing further.
Thus there is an optimum timestep $\mathit{dt}_{\rm opt}$ that yields the minimum energy error.
This optimum timestep depends on the length of the integration and will be larger for longer simulations.
In Fig.~\ref{fig:shorttermenergy} one can see that the errors of {\sc \tt WH}\xspace, {\sc \tt SWIFTER-WHM}\xspace, {\sc \tt SWIFTER-HELIO}\xspace and {\sc \tt MERCURY}\xspace rise very rapidly after reaching $\mathit{dt}_{\rm opt}$, scaling as at least $\mathit{dt}^{-2}$ for the first decade.
The optimum timestep for {\sc \tt WHFast}\xspace is roughly 0.1\% of the shortest orbital period.
However, {\sc \tt WHFast}\xspace's error grows much more slowly with decreasing timestep than that of the other second-order integrators.
In fact, the error is dominated by $E_{\rm rand}$ and thus follows $\mathit{dt}^{-1/2}$ as the number of timesteps $N_{\rm steps}$ increases as $\sim\mathit{dt}^{-1}$ if we keep the total integration time constant.
Thus the behaviour of {\sc \tt WHFast}\xspace in Fig.~\ref{fig:shorttermenergy} for small timesteps can be seen as the first indication that {\sc \tt WHFast}\xspace follows Brouwer's law (see Sects.~\ref{sec:errors} and~\ref{sec:longtermtest}).
The {\sc \tt SWIFTER-TU4}\xspace integrator is the only other integrator we tested that seems to follow Brouwer's law, but it performs poorly at large timesteps.
This is expected, since unlike the other integrators, {\sc \tt SWIFTER-TU4}\xspace does not assume a Keplerian splitting and must therefore take smaller timesteps to accurately reproduce the orbital motions.
Integrators with symplectic correctors, {\sc \tt MERCURY}\xspace and {\sc \tt WHFast}\xspace, perform significantly better for long timesteps.
Their energy conservation is three orders of magnitude better ($E_{\rm bound}$ is three orders of magnitude smaller) compared to integrators without symplectic correctors.
This is due to the mass ratio of Jupiter and the Sun being roughly $10^{-3}$.
The order of the symplectic corrector is not very important for relatively high mass ratios such as these, i.e. a fifth-order symplectic corrector performs as well as an 11th-order one.
For much smaller mass ratios (when the mass ratio is less than the timestep ratio), higher-order symplectic correctors are advantageous.
Note that $\mathit{dt}_{\rm opt}$ for almost all of the integrators is $10^{-3}$~orbital periods of Jupiter, i.e. 4 days.
This is significant because Mercury's orbital period is 88~days.
Thus if we included Mercury in our simulation, we would be very restricted in our timestep choice.
We need more than 20 timesteps ($\mathit{dt}\approx4$~days) to resolve Mercury's orbit accurately.
However, if we choose choose a timestep smaller than 4 days, we start to accumulate errors in the outer Solar System.
It is worth reiterating that the simulations shown in Fig.~\ref{fig:shorttermenergy} all ran for only 1000~orbits.
If we ran a longer simulation with the same timestep, we would have more timesteps and thus accumulate more round-off errors by the end of the simulation.
One can therefore reach better energy conservation with a longer timestep.
In other words, $\mathit{dt}_{\rm opt}$ is larger for longer integration times.
\subsection{Speed Comparison}\label{sec:speedcomparison}
\begin{figure}
\centering \resizebox{0.99\columnwidth}{!}{\includegraphics{speed}}
\caption{
Relative energy error in simulations of the outer Solar System after 1000~Jupiter orbits as a function of run time.
\label{fig:speedcomparison}}
\end{figure}
We run the same simulations as in Sect.~\ref{sec:shorttermenergy} to compare the speed of the different integrators.
Fig.~\ref{fig:speedcomparison} shows the relative energy error as a function of runtime.
The results show that no matter what the desired energy error is, {\sc \tt WHFast}\xspace is the fastest integrator.
In the large timestep limit, the speed-up compared to {\sc \tt MERCURY}\xspace is roughly a factor of~5.
In the small timestep limit, $\mathit{dt} < \mathit{dt}_{\rm opt}$, we can only compare {\sc \tt WHFast}\xspace to {\sc \tt SWIFTER-TU4}\xspace, as all the other integrators' errors are significantly larger (by 4 to 5~orders of magnitude) due to numerical roundoff errors (see below).
{\sc \tt SWIFTER-TU4}\xspace is as fast for small timesteps as {\sc \tt WHFast}\xspace but, as noted above, is unsuitable for large timesteps since it is not a Wisdom-Holman integrator.
It is only shown here as a comparison.
\subsection{Long Term Energy Conservation}\label{sec:longtermtest}
\begin{figure*}
\centering \resizebox{0.99\textwidth}{!}{\includegraphics{longtermtest}}
\caption{
Relative energy error in simulations of the outer Solar System as a function of time for different symplectic integrators. The timestep for all simulations is 1.5~days.
\label{fig:longtermtest}}
\end{figure*}
Let us finally address the most important benchmark, the long term energy conservation properties of {\sc \tt WHFast}\xspace compared to other integrators in a real world test case.
In this section we only study the energy error, but other conserved properties like the angular momentum behave the same way.
In Fig.~\ref{fig:longtermtest}, we show the time evolution of the relative energy error in a simulation of the outer Solar System.
As in Sect.~\ref{sec:shorttermenergy}, we include the Sun and four massive bodies with approximate initial conditions corresponding to those of Jupiter, Saturn, Uranus and Neptune.
The timestep for all simulations is 1.5~days.
Note that this timestep is smaller than what one would typically choose for this kind of integration.
However, with a 1.5~day timestep we reach machine precision for integrators that use symplectic correctors, allowing us to better quantify the long-term behaviour of {\sc \tt WHFast}\xspace.
All the effects we discuss here are also present in simulations with longer timesteps, but they would manifest themselves in the relative energy error at a later time.
We run four simulations for each integrator and randomly perturb the initial conditions and masses by 0.1\% in each simulation.
We plot the individual simulations as thin lines, and the average error as a bold line.
The lower relative energy bound is set by machine precision for all integrators, roughly $10^{-16}$.
{\sc \tt WHFast}\xspace and {\sc \tt MERCURY}\xspace, the integrators in our sample that have symplectic correctors, almost reach this limit early on in the simulation.
The bound energy error $E_{\rm bound}$ is approximately $10^{-14}$.
The integrators without symplectic correctors, {\sc \tt SWIFTER-WHM}\xspace and {\sc \tt WH}\xspace have an energy error roughly three order of magnitudes higher $E_{\rm bound}\approx 10^{-10.5}$.
From Fig.~\ref{fig:longtermtest} it is clear that the integrators {\sc \tt MERCURY}\xspace, {\sc \tt WH}\xspace and {\sc \tt SWIFTER-WHM}\xspace show a linear behaviour in the energy error at late times.
This is due to the term $E_{\rm bias}$.
The $E_{\rm bias}$ term already dominates at early times (after 100 Jupiter orbits) for {\sc \tt MERCURY}\xspace because the symplectic correctors lower the value of $E_{\rm bound}$.
For {\sc \tt WH}\xspace and {\sc \tt SWIFTER-WHM}\xspace the $E_{\rm bias}$ term dominates after $10\,000$~Jupiter orbits.
This result shows that one or more steps in these integration algorithms are biased.
We found that the two main contributions were the inaccurate implementation of the rootfinder for Kepler's equation and the conversions to and from Jacobi coordinates.
In {\sc \tt WHFast}\xspace, $E_{\rm bias}$ is absent, showing that its implementation is completely unbiased.
Since all integrators are implemented in double floating-point precision and use the same timestep, they all have roughly the same error term $E_{\rm rand}$.
However, it is only visible in Fig.~\ref{fig:longtermtest} for the {\sc \tt WHFast}\xspace integrator.
For all other integrators the linearly growing term $E_{\rm bias}$ dominates over $E_{\rm rand}$.
If we increase the timestep, the linear error growth will show up at a later time because $E_{\rm bound}$ will be larger.
However, it is still present at all times.
Let us think of a symplectic integrator as an exact integrator for a perturbed Hamiltonian $\tilde{\mathcal{H}}$ with high frequency terms added compared to $\mathcal{H}$ in Eq.~\ref{eq:H}, see e.g. \cite{Wisdom1996}.
Then the quantity related to the energy error for $\tilde{\mathcal{H}}$, let us call this $\tilde{E}$, should be conserved exactly at all times (that is the idea of a symplectic integrator).
However, if the implementation is biased, $\tilde{E}$ will undergo a linear growth at all times.
With {\sc \tt WHFast}\xspace, we improve the conservation of $\tilde{E}$ by many orders of magnitude in any integration, regardless of timestep.
This difference could have important implication for the dynamical evolution of the system and could for example push it from a stable to an unstable region of parameter space.
We plan to study the effect of different integrators on systems near a chaotic/non-chaotic separatrix in a follow up paper.
\section{Conclusions}\label{sec:conclusions}
In this paper, we presented {\sc \tt WHFast}\xspace, a new implementation of a symplectic Wisdom-Holman integrator.
Key advantages and improvements over other publicly available implementations of symplectic integrators are:
\textit{{\sc \tt WHFast}\xspace is faster by a factor of $1.5$~to~$5$.}
Of that, a 50\% speedup comes from the improved Kepler solver, where we use a fast convergence criteria for Newton's method and an efficient implementation of $c$ and $G$-functions.
The remainder of the speedup is due to combining drift steps at the end and beginning of each timestep and to only converting to and from Jacobi coordinates when needed.
\textit{The Kepler solver is more accurate and unbiased.}
We achieve this thanks to improvements to the convergence criteria in Newton's method, a Laguerre-Conway solver for highly eccentric orbits with long timesteps, the high accuracy implementations of the $c$ and $G$-functions and a careful ordering of floating-point operations.
\textit{We remove the secular energy error that grows linearly with integration time.}
This is due to two improvements.
First, the unbiased Kepler solver.
Second, the improved and also unbiased coordinate transformations to and from Jacobi coordinates.
To our knowledge, {\sc \tt WHFast}\xspace is the first publicly available implementation of a Wisdom-Holman integrator that follows Brouwer's law over long timescales for small enough timesteps and does not show a linear growth in the energy error.
\textit{We implement variational equations that allow us to compute the Lyapunov timescale and the MEGNO.}
Our algorithm to calculate the Lyapunov timescale uses a numerically stable algorithm that is based on a one-pass covariance filter.
The variational equations do not require us to solve Kepler's equation and are thus very inexpensive to calculate.
\textit{Symplectic correctors of order 3, 5, 7, and 11 are implemented.}
These symplectic corrector allow for high-accuracy simulations of systems with small mass ratios.
Even for relatively massive planets like those in the Solar System, symplectic correctors achieve an improvement of three orders of magnitude.
For long integrations, the performance cost of symplectic correctors is negligible and so our default setting uses an 11th-order corrector.
\textit{{\sc \tt WHFast}\xspace lets the centre-of-mass move freely during an integration.}
We integrate an additional degree of freedom in order for our integrator to work in any inertial frame, i.e. one is not restricted to the heliocentric or barycentric frame.
Additionally, we do not tie our implementation to a specific choice of units.
\textit{The integrator is available as an easy to use python module.}
The module works on both python 2 and 3.
It can be installed on most Unix and MacOS systems with a single command:
\begin{lstlisting}
pip install rebound
\end{lstlisting}
\noindent The following python script imports the rebound module, adds particles to the simulation, selects an integrator and timestep and runs the integration.
\begin{lstlisting}
import rebound
rebound.add(m=1)
rebound.add(m=0.001, a=1.)
rebound.add(m=0.001, a=2., e=0.1)
rebound.integrator = 'whfast'
rebound.dt = 0.01
rebound.integrate(6.2831)
\end{lstlisting}
More complicated examples and the source code of {\sc \tt WHFast}\xspace (written in C, compliant with the C99 standard) can be found in the {\sc \tt REBOUND}\xspace package.
{\sc \tt REBOUND}\xspace includes several other integrators, collision detection algorithms, a gravity tree code and much more.
The {\sc \tt REBOUND}\xspace git repository is hosted at \url{https://github.com/hannorein/rebound}.
We also provide an experimental hybrid integrator for simulations in which close encounters occur.
The hybrid integrator switches over to a high-order non-symplectic integrator \citep[IAS15,][]{ReinSpiegel2015} during a close encounter.
A detailed discussion of this integrator and its properties will be given in a follow-up paper.
We hope that with the speed and accuracy improvements, {\sc \tt WHFast}\xspace will become the go-to integrator package for short and long-term orbit simulations of planetary systems.
\section*{Acknowledgments}
This research has been supported by the NSERC Discovery Grant RGPIN-2014-04553.
We thank Wayne Enright, Philip Sharp and Scott Tremaine for stimulating discussions and Jack Wisdom for a helpful referee report.
|
1,314,259,996,521 | arxiv | \section{Introduction and Results}
\setcounter{equation}{0}
The propagation of strings in Friedmann-Robertson-Walker (FRW)
cosmologies has been
investigated using both exact and approximative methods, see for example
Refs.[1-8]
(as well as numerical methods, which shall not be discussed here).
Except for anti de Sitter spacetime, which has negative spatial curvature, the
cosmologies that have been considered until now, have been spatially flat. In
this paper we will consider the physical effects of a non-zero (positive or
negative) curvature index on the classical and quantum strings. The
non-vanishing components of the Riemann tensor for the generic D-dimensional
FRW line element, in comoving coordinates:
\begin{equation}
ds^2=-dt^2+a^2(t)\frac{d\vec{x} d\vec{x}}{(1+\frac{K}{4}\vec{x}\vec{x})^2},
\end{equation}
are given by:
\begin{equation}
R_{itit}=\frac{-aa_{tt}}{(1+\frac{K}{4}\vec{x}\vec{x})^2},\;\;\;\;\;\;
R_{ijij}=\frac{a^2(K+a_{t}^2)}{(1+\frac{K}{4}\vec{x}\vec{x})^4};\;\;\;\;i\neq j
\end{equation}
where $a=a(t)$ is the scale factor and $K$ is the curvature index. Clearly, a
non-zero curvature index introduces a non-zero spacetime curvature; the
exceptional case provided by $K=-a_{t}^2=\mbox{const.},$ corresponds to
the Milne-Universe. From Eqs.(1.2), it is also seen that the curvature index
has to compete with the first
derivative of the scale factor. The effects of the
curvature index are therefore conveniently discussed in the family of
FRW-universes with constant scale factor, the so-called static
Robertson-Walker spacetimes. This is the point of view we take in the
present paper.
We consider both the closed ($K>0$) and the hyperbolic ($K<0$) static
Robertson-Walker spacetimes, and all our results are compared with the
already known results in the flat ($K=0$) Minkowski spacetime. We determine the
evolution of circular strings, derive the corresponding equations of state,
discuss the question of strings as self-consistent solutions to the
Einstein equations \cite{san4}, and we perform a semi-classical
quantization. We find all the stationary string configurations in these
spacetimes and we perform a canonical quantization, using the string
perturbation series approach \cite{san1}, for a static string center of mass.
\vskip 6pt
\hspace*{-6mm}The radius of a classical circular string in the spacetime
(1.1), for $a=1,$ is determined by:
\begin{equation}
\dot{r}^2+V(r)=0;\;\;\;\;\;\;\;\;V(r)=(1-Kr^2)(r^2-b\alpha'^2),
\end{equation}
where $b$ is an integration constant. This equation is solved in terms of
elliptic functions and all solutions describe oscillating strings (Fig.1
shows the potential $V(r)$ for $K>0,\;K=0,\;K<0$).
For $K>0,$ when the spatial section is a hypersphere, the string either
oscillates on one hemisphere or on the full hypersphere. The energy is
positive while the average pressure can be positive, negative or zero;
the equation of state is given by Eqs.(3.19), (3.25).
Interestingly enough, we find that the circular strings provide a
self-consistent solution to the Einstein equations with a selected
value of the curvature
index, Eq.(3.28). Self-consistent solutions to the Einstein equations with
string sources have been found previously in the form of power law
inflationary universes \cite{san4}. We semi-classically quantize the
circular strings using the stationary phase approximation method of
Ref.\cite{das}. The
strings oscillating on one hemisphere give rise to a finite number $N_-$
of states with the following mass-formula:
\begin{equation}
m_-^2\alpha'\approx \pi\;n,\;\;\;\;\;\;\;\;N_-\approx\frac{4}{\pi K\alpha'}.
\end{equation}
As in flat Minkowski spacetime, the scale of these string states is set by
$\alpha'.$ The strings oscillating on the full hypersphere give rise to
an infinity of more and more massive states with the asymptotic mass-formula:
\begin{equation}
m_+^2\approx K\;n^2.
\end{equation}
The masses of these states are independent of $\alpha',$ the scale is set
by the curvature index $K.$ Notice also that the level spacing grows
with $n.$ A similar result was found recently for strings in anti de
Sitter spacetime \cite{san5,san6}.
For $K<0,$ when the spatial section is a hyperboloid, both the energy and
the average pressure of the oscillating strings are positive. The equation
of state is given by Eq.(3.35). In this case, the strings can not provide a
self-consistent solution to the Einstein equations. After semi-classical
quantization, we find an infinity of more and more massive states. The
mass-formula is given by Eq.(4.26):
\begin{equation}
\sqrt{-Km^2\alpha'^2}\;\log\sqrt{-Km^2\alpha'^2}\approx
-\frac{\pi}{2}K\alpha'\;n
\end{equation}
Notice that the level spacing grows faster than in
Minkowski spacetime but slower than in the closed
static Robertson-Walker spacetime.
A summary of the classical and semi-classical features of the circular
strings is presented in Tables I and II. Figs.2-4 depict the mass
quantization conditions for the $K>0$ and $K<0$ cases.
\vskip 6pt
\hspace*{-6mm}On the other hand, the stationary strings are determined by:
\begin{equation}
\phi'=\frac{L}{r^2},\;\;\;\;\;\;\;\;
r'^2+U(r)=0;\;\;\;U(r)=(1-Kr^2)(\frac{L^2}{r^2}-1),
\end{equation}
where $L$ is an integration constant (Fig.5 shows the potential
$U(r)$ for $K>0,\;K=0,\;K<0$). For $K>0,$ all the stationary string
solutions describe circular strings winding around the hypersphere.
The equation of state is of the extremely unstable string type \cite{ven}.
For $K<0,$ the stationary strings are represented by infinitely long
open configurations with an angle between the two "arms" given by:
\begin{equation}
\Delta\phi=\pi-2\arctan(\sqrt{-K}L).
\end{equation}
The energy density is positive while the pressure densities are negative.
No simple equation of state is found for these solutions.
A summary of the results for the stationary strings is presented in Table III.
\vskip 6pt
\hspace*{-6mm}Finally we compute the first and second order fluctuations
around a static string center of mass, using the string perturbation series
approach \cite{san1}
and its covariant versions \cite{san3,men}. Up to second order,
the mass-formula for arbitrary values of the curvature index
(positive or negative) is identical to the well-known flat spacetime
mass-formula; all dependence on $K$ cancels out.
\section{The Static Robertson-Walker Spacetimes}
\setcounter{equation}{0}
To clarify our notation we start by reviewing a few fundamental aspects of
the Robertson-Walker spacetimes with curved spatial sections. The general
line element is:
\begin{equation}
ds^2=-dt^2+a^2(t)[d\xi^2+f^2(\xi)d\Omega^2_{D-2}],
\end{equation}
where the function $f(\xi)$ is given by:
\begin{eqnarray}
f(\xi)\hspace*{-2mm}&=&\hspace*{-2mm}\sin\xi,\;\;\;\;0\leq\xi\leq\pi,\;\;\;\;K>0
\nonumber\\
f(\xi)\hspace*{-2mm}&=&\hspace*{-2mm}\xi,\;\;\;\;0\leq\xi<\infty,\;\;\;\;K=0\\
f(\xi)\hspace*{-2mm}&=&\hspace*{-2mm}\sinh\xi,\;\;\;\;0\leq\xi<\infty,
\;\;\;\;K<0\nonumber
\end{eqnarray}
The spatial sections are closed, flat or hyperbolic depending on whether
$K$ is positive, zero or negative. Usually a non-zero curvature index $K$
is scaled to either plus or minus $1$ by a redefinition of the scale factor
$a(t),$ but for our purposes of considering the so-called static
Robertson-Walker spacetimes, it is convenient to set the constant scale-factor
equal to unity and keep $K$ arbitrary. We shall also use the coordinates
defined by setting:
\begin{equation}
r=f(\xi),\\
\end{equation}
in which case the line element takes the
form (after a rescaling):
\begin{equation}
ds^2=-dt^2+a^2(t)[\frac{dr^2}{1-Kr^2}+r^2d\Omega^2_{D-2}],
\end{equation}
Notice that in the case of closed spatial sections, the latter coordinates
cover only half of the spatial hypersurface (for $r\in[0,1/\sqrt{K}\;]$), which
is in that case a hypersphere
of radius $1/\sqrt{K}.$ Finally it is also useful to
have the comoving coordinates defined by:
\begin{equation}
r=\frac{R}{1+\frac{K}{4}R^2},
\end{equation}
with the corresponding line element:
\begin{equation}
ds^2=-dt^2+a^2(t)[\frac{dR^2+R^2d\Omega^2_{D-2}}{(1+\frac{K}{4}R^2)^2}].
\end{equation}
For a general $D$-dimensional curved spacetime with curvature index $K,$
cosmological constant $\Lambda$ and an energy-momentum
tensor of the fluid form, the Einstein equations read:
\begin{equation}
(D-1)(D-2)(K+a_t^2)=2a^2(G\rho+\Lambda),
\end{equation}
\begin{equation}
2(D-2)aa_{tt}+(D-3)(D-2)(K+a_t^2)=2a^2(\Lambda-GP),
\end{equation}
where $\rho$ is the energy density, $P$ is the pressure, $G$ is a positive
constant (essentially the gravitational constant in $D$ dimensions)
and $a_t\equiv da(t)/dt.$ In both equations the curvature index has to
compete directly with the derivative of the scale factor, thus for instance in
inflationary models (like de Sitter or power law universes), the effect of
the curvature index will soon be negligible. In this paper we are interested
precisely
in the effects of the curvature index on the classical and quantum string
propagation, and it follows that this investigation is most suitably performed
in spacetimes with vanishing $a_t,$ and not only because of simplicity. These
spacetimes of constant scale factor are denoted the static Robertson-Walker
spacetimes. By scaling we obtain $a=1,$ and the Einstein equations take the
form:
\begin{equation}
(D-1)(D-2)K=2(G\rho+\Lambda),
\end{equation}
\begin{equation}
(D-3)(D-2)K=2(\Lambda-GP).
\end{equation}
Usually energy and pressure are
supposed to be non-negative. However, considering string sources in the
Einstein equations, this is not
necessarily true \cite{san4}, not even
in flat Minkowski spacetime \cite{san5}. In fact, in the
next section we shall return to the question of self-consistent string
solutions to the Einstein equations in the case of vanishing cosmological
constant. We will consider bosonic strings with equations of motion and
constraints given by:
\begin{eqnarray}
&\ddot{x}^\mu-x''^\mu+\Gamma^\mu_{\rho\sigma}(\dot{x}^\rho\dot{x}^\sigma-
x'^\rho x'^\sigma)=0,&\nonumber\\
&g_{\mu\nu}\dot{x}^\mu x'^\nu=g_{\mu\nu}(\dot{x}^\mu\dot{x}^\nu+x'^\mu x'^\nu)
=0,&
\end{eqnarray}
where dot and prime stand for derivative with respect to $\tau$ and
$\sigma,$ respectively. For the $2+1$ dimensional
($d\Omega^2_{D-2}=d\phi^2$) metric defined by the line element (2.4), for
$a=1,$
they take the form:
\begin{eqnarray}
\ddot{t}\hspace*{-2mm}&-&\hspace*{-2mm}t''=0,\nonumber\\
\ddot{r}\hspace*{-2mm}&-&\hspace*{-2mm}r''+\frac{Kr}{1-Kr^2}(\dot{r}^2-r'^2)-
r(1-Kr^2)(\dot{\phi^2}-\phi'^2)=0,\nonumber\\
\ddot{\phi}\hspace*{-2mm}&-&\hspace*{-2mm}\phi''+\frac{2}{r}(\dot{\phi}
\dot{r}-\phi' r')=0,\nonumber\\
\hspace*{-2mm}&-&\hspace*{-2mm}\dot{t}t'+\frac{\dot{r}r'}{1-Kr^2}
+r^2\dot{\phi}\phi'=0,\nonumber\\
\hspace*{-2mm}&-&\hspace*{-2mm}(\dot{t}^2+t'^2)+\frac{1}{1-Kr^2}(\dot{r}^2+r'^2)
+r^2(\dot{\phi}^2+\phi'^2)=0.
\end{eqnarray}
\section{Circular Strings, Physical Interpretation, Self-Consistency}
\setcounter{equation}{0}
In this section we shall give a complete description of
the evolution and physical interpretation of circular string
configurations in the static Robertson-Walker spacetimes. A plane circular
string effectively lives in $2+1$ dimensions so we will drop the dimensions
perpendicular to the string plane. The line element, in the coordinates (2.4),
is then:
\begin{equation}
ds^2=-dt^2+\frac{dr^2}{1-Kr^2}+r^2d\phi^2.
\end{equation}
Circular strings have been intensively studied in static
spacetimes \cite{san2,san3,egu,san5,mik1,mik2,all1},
but until now not in the static Robertson-Walker spacetimes. The general
equations determining the evolution and dynamics of the circular strings
\cite{san3,egu} can however be used directly here too. The ansatz
$(t=t(\tau),\;r=r(\tau),\;\phi=\sigma),$ corresponding to a
circular string, leads to \cite{san3}:
\begin{equation}
\dot{t}=\sqrt{b}\alpha',\;\;\;\;\;\;\;\;\dot{r}^2+V(r)=0,
\end{equation}
where $\alpha'$ is the string tension, $b$ is a positive
integration constant with
the dimension of $(\mbox{mass})^2$ and the potential $V(r)$ is given by:
\begin{equation}
V(r)=(1-Kr^2)(r^2-b\alpha'^2).
\end{equation}
By insertion of Eqs.(3.2)-(3.3) into Eq.(3.1), we obtain the
induced line element on the world-sheet:
\begin{equation}
ds^2=r^2(\tau)(-d\tau^2+d\sigma^2),
\end{equation}
and the string length is given by:
\begin{equation}
l(\tau)=2\pi|r(\tau)|.
\end{equation}
Energy and pressure of the circular strings can be obtained from the $2+1$
dimensional spacetime energy-momentum tensor:
\begin{equation}
\sqrt{-g}T^{\mu\nu}=\frac{1}{2\pi\alpha'}\int d\tau d\sigma
(\dot{X}^\mu\dot{X}^\nu-X'^\mu X'^\nu)\delta^{(3)}(X-X(\tau,\sigma)).
\end{equation}
After integration over a spatial volume that completely encloses the string
\cite{san4},
the energy-momentum tensor for a circular string takes the form of a
fluid:
\begin{equation}
T^\mu\;_\nu=\mbox{diag.}(-\rho,P,P),
\end{equation}
where in the comoving coordinates:
\begin{equation}
\rho=\frac{1}{\alpha'}\dot{t}=\sqrt{b},
\end{equation}
\begin{equation}
P=\frac{1}{2\alpha'\dot{t}}\;\frac{\dot{R}^2-R^2}{(1+\frac{K}{4}R^2)^2}=
\frac{1}{2\sqrt{b}\alpha'^2}[\frac{\dot{r}^2}{1-Kr^2}-r^2]=
\frac{b\alpha'^2-2r^2}{2\sqrt{b}\alpha'^2}.
\end{equation}
Let us now consider separately
the three different cases of vanishing, positive and negative curvature index.
For $K=0$ we have flat Minkowski spacetime, the circular string
configurations there
and their physical interpretation were already discussed in
Ref.\cite{san5}. The circular string potential (3.3) is shown in Fig.1a and the
equations of motion (3.2) determining the string radius are solved by:
\begin{equation}
t(\tau)=\sqrt{b}\alpha'\tau,\;\;\;\;\;r(\tau)=\sqrt{b}\alpha'\cos\tau,
\;\;\;\;\;(K=0)
\end{equation}
i.e. the string motion follows a pure harmonic motion with period
$T_{\tau}=2\pi$ in
the world-sheet time. The energy and pressure, Eqs.(3.8)-(3.9), are
given by:
\begin{equation}
\rho=\sqrt{b},
\end{equation}
\begin{equation}
P=-\frac{\sqrt{b}}{2}\cos 2\tau.
\end{equation}
Notice that during an oscilation of the string, the equation of state
"oscillates" between $P=\rho/2,$ corresponding to
{\it ultra-relativistic matter}
in $2+1$ dimensions, and $P=-\rho/2,$ corresponding to
{\it extremely unstable
strings} \cite{ven}. For further discussion on this point, see
Ref.\cite{san5}.
Using the exact time-dependent pressure (3.12), it is clear from
Eqs.(2.9)-(2.10) that the circular strings do not provide a self-consistent
solution in Minkowski spacetime. If using instead the average values over one
oscillation:
\begin{equation}
<\rho>=\sqrt{b},\;\;\;\;<P>=0,
\end{equation}
corresponding to cold matter, we see that Eq.(2.10) with vanishing
cosmological constant is fulfilled, while
Eq.(2.9) leads to:
\begin{equation}
K=G\sqrt{b}.
\end{equation}
The circular strings thus generate a positive curvature index and we conclude
that even after averaging over an oscillation, they
do not provide a self-consistent solution in Minkowski spacetime.
For positive curvature index, the spatial hypersurface is a sphere and
we have to distinguish between the two cases
$\sqrt{bK}\alpha'\leq 1$ and $\sqrt{bK}\alpha'>1,$ as is clear from the
location of the zeros of the potential, see Figs.1b,1c. For
$\sqrt{bK}\alpha'\leq 1,$ the solution of Eqs.(3.2)-(3.3) is:
\begin{equation}
t_-(\tau)=\sqrt{b}\alpha'\tau,\;\;\;\;\;r_-(\tau)=
\sqrt{b}\alpha'\mbox{sn}[\tau,k_-]\;\;\;\;\;(K>0),
\end{equation}
where $\mbox{sn}[\tau,k_-]$ is the Jacobi elliptic function and the
elliptic modulus is given by:
\begin{equation}
k_-=\sqrt{bK}\alpha'\in[0,1].
\end{equation}
The
solution describes a string oscillating between zero radius and maximal
radius $r_{\mbox{max}}=\sqrt{b}\alpha'$ with period
$T_\tau=4K(k_-)$ in the world-sheet time,
where $K(k_-)$ is the complete elliptic integral of the first
kind. Since $r_{\mbox{max}}\leq 1/\sqrt{K},$ the string oscillates on one
hemisphere; it does not cross the equator. The energy and pressure are obtained
from Eqs.(3.8)-(3.9):
\begin{equation}
\rho_-=\sqrt{b}=\frac{k_-}{\sqrt{K}\alpha'},
\end{equation}
\begin{equation}
P_-=\frac{k_-}{2\sqrt{K}\alpha'}(1-2\mbox{sn}^2[\tau,k_-]).
\end{equation}
During an oscillation of the string, the equation of state "oscillates"
between $P_-=\rho_-/2$ and $P_-=-\rho_-/2.$ This is similar to the situation
in Minkowski spacetime. The average values are given by:
\begin{equation}
<\rho_->=\frac{k_-}{\sqrt{K}\alpha'},\;\;\;\;\;<P_->
=\frac{1}{2\sqrt{K}\alpha'}[\frac{k_-^2-2}{k_-}+
\frac{2}{k_-}\;\frac{E(k_-)}{K(k_-)}],
\end{equation}
so that $2\sqrt{K}\alpha'<P_->\;\in[-1,0]$ where the limit $-1$ corresponds to
$k_-\rightarrow 1$ and the limit $0$ corresponds to $k_-\rightarrow 0.$ Thus in
average the pressure is {\it negative}
(in the limit $k_-=0,$ there are no strings
at all) and the equation of state is written:
\begin{equation}
<P_->=(\gamma(k_-)-1)<\rho_->;\;\;\;\;\;\gamma(k_-)=\frac{3}{2}-
\frac{1}{k_-^2}[1-\frac{E(k_-)}{K(k_-)}].
\end{equation}
When $k_-$
increases from $k_-=0$ to $k_-=1,$ the function $\gamma(k_-)$ decreases
from $\gamma(0)=1$ to $\gamma(1)=1/2,$ that is, from {\it cold matter} type
($\gamma=1$) to {\it extremely unstable string} type ($\gamma=1/2$).
Returning now
to the Einstein equations (2.9)-(2.10) without cosmological constant, we see
that Eq.(2.9) can be fulfilled using the average values (3.19), but Eq.(2.10)
can not. We conclude that the string solutions, Eq.(3.15), for
$\sqrt{bK}\alpha'\leq 1,$ do not provide a
self-consistent solution to the Einstein equations.
We now consider the case where
$\sqrt{bK}\alpha'>1.$ The solution of Eqs.(3.2)-(3.3) is (see Fig.1c):
\begin{equation}
t_+(\tau)=\sqrt{b}\alpha'\tau,\;\;\;\;\;r_+(\tau)=\frac{1}{\sqrt{K}}
\;\mbox{sn}[\tau/k_+,k_+],\;\;\;\;\;(K>0)
\end{equation}
where the elliptic modulus is now given by:
\begin{equation}
k_+=\frac{1}{\sqrt{bK}\alpha'}\in\;]0,1[\;.
\end{equation}
This string solution is oscillating
between zero radius and maximal radius $r_{\mbox{max}}=1/\sqrt{K},$
corresponding to the radius of the hypersphere, with period
$T_\tau=4k_+K(k_+)$ in the world-sheet
time. The physical interpretation of this solution is a
string oscillating on the full hypersphere. For $\tau=0$ it starts with zero
radius on one of the hemispheres. It expands and reaches the equator
for $\tau=k_+K(k_+).$ It then crosses the equator and contracts on the other
hemisphere until it collapses to a point for $\tau=2k_+K(k_+).$ It now expands
again, crosses the equator and eventually collapses to its initial
configuration of zero radius for $\tau=4k_+K(k_+).$
The energy and pressure of this solution are given by:
\begin{equation}
\rho_+=\sqrt{b}=\frac{1}{\sqrt{K}\alpha'k_+},
\end{equation}
\begin{equation}
P_+=\frac{1-2k_+^2\mbox{sn}^2[\tau/k_+,k_+]}{2k_+\sqrt{K}\alpha'}.
\end{equation}
During an oscillation, the equation of state "oscillates" between
$P_+=\rho_+/2$ and $P_+=(1-2k_+^2)\rho_+/2.$ The average values are given by:
\begin{equation}
<\rho_+>=\frac{1}{\sqrt{K}\alpha'k_+},\;\;\;\;\;<P_+>=
\frac{1}{2k_+\sqrt{K}\alpha'}
[\frac{2E(k_+)}{K(k_+)}-1],
\end{equation}
so that $2\sqrt{K}\alpha'<P_+>\;\in\;]-1,\infty[\;$ where the limit $-1$
corresponds to
$k_+\rightarrow 1$ and the limit $\infty$ corresponds to
$k_+\rightarrow 0.$ Thus
the average pressure can be negative, zero or positive for these solutions.
The equation of state is:
\begin{equation}
<P_+>=(\gamma(k_+)-1)<\rho_+>;\;\;\;\;\;\gamma(k_+)=
\frac{1}{2}+\frac{E(k_+)}{K(k_+)}.
\end{equation}
When $k_+$ increases from $k_+=0$ to $k_+=1,$ the function
$\gamma(k_+)$ decreases
from $\gamma(0)=3/2$ to $\gamma(1)=1/2,$ that is, from
{\it ultra-relativistic matter} type
($\gamma=3/2$) to {\it extremely unstable string} type ($\gamma=1/2$).
Let us consider also the question of self-consistency in this case. Using
the average values (3.26) in the Einstein equations (2.9)-(2.10),
without cosmological constant, we find that the self-consistency conditions
are satisfied with:
\begin{equation}
K=\frac{G}{k_+\sqrt{K}\alpha'},\;\;\;\;\;2E(k_+)=K(k_+),
\end{equation}
which yield the (numerical) solution:
\begin{equation}
k_+=0.9089...,\;\;\;\;\;K=(\frac{G}{\alpha'})^{2/3}\times 1.0658...
\end{equation}
It follows that a gas of oscillating circular strings described by
Eq.(3.21) for $k_+=0.9089...,$ provides a self-consistent solution to the
Einstein equations. The solution is a spatially closed static
Robertson-Walker spacetime with scale factor normalized to $a=1$
and curvature index
$K=(\frac{G}{\alpha'})^{2/3}\times 1.0658...$
Finally we consider circular strings in the spatially hyperbolic case,
corresponding to negative curvature index. The potential is shown in
Fig.1d. and Eqs.(3.2)-(3.3) are solved by:
\begin{equation}
t(\tau)=\sqrt{b}\alpha'\tau,\;\;\;\;\;r(\tau)=\frac{k}{\sqrt{-K}}
\;\mbox{sd}[\mu\tau,k],\;\;\;\;\;(K<0)
\end{equation}
where we introduced the notation:
\begin{equation}
\mu=\sqrt{1-Kb\alpha'^2},\;\;\;\;\;k=\sqrt{\frac{-Kb\alpha'^2}{1-Kb\alpha'^2}}
\;\in\;[0,1[\;
\end{equation}
The solution describes a string oscillating between zero radius and maximal
radius $r_{\mbox{max}}$ with period $T_\tau$ in the world-sheet time:
\begin{equation}
r_{\mbox{max}}=\frac{1}{\sqrt{-K}}\frac{k}{\sqrt{1-k^2}},\;\;\;\;\;
T_\tau=4K(k)/\mu.
\end{equation}
The energy and pressure are given by:
\begin{equation}
\rho=\sqrt{b}=\frac{k}{\sqrt{-K}\alpha'\sqrt{1-k^2}},
\end{equation}
\begin{equation}
P=\frac{k-2k(1-k^2)\mbox{sd}^2[\mu\tau,k]}{2\sqrt{1-k^2}\sqrt{-K}\alpha'}.
\end{equation}
During an oscillation, the equation of state "oscillates" between
$P=\rho/2$ and $P=-\rho/2.$ This is like the situation in flat Minkowski
spacetime. The average values are given by:
\begin{equation}
<\rho>=\frac{k}{\sqrt{-K}\alpha'\sqrt{1-k^2}},\;\;\;\;\;
<P>=\frac{1}{2\sqrt{1-k^2}\sqrt{-K}\alpha'}
[-k+\frac{2}{k}(1-\frac{E(k)}{K(k)})],
\end{equation}
so that $2\sqrt{-K}\alpha'<P>\;\in[0,\infty[\;$ where the limit $0$
corresponds to
$k\rightarrow 0$ and the limit $\infty$ corresponds to $k\rightarrow 1.$ The
average pressure is always {\it positive}
(for $k=0,$ there are no strings at all).
The equation of state takes the form:
\begin{equation}
<P>=(\gamma(k)-1)<\rho>;\;\;\;\;\;\gamma(k)=\frac{1}{2}+
\frac{1}{k^2}(1-\frac{E(k)}{K(k)}).
\end{equation}
When $k$ increases from $k=0$ to $k=1,$ the function $\gamma(k)$ increases
from $\gamma(0)=1$ to $\gamma(1)=3/2,$ that is, from {\it cold matter} type
($\gamma=1$) to {\it ultra-relativistic matter}
type ($\gamma=3/2$). Clearly, these
solutions can not provide a self-consistent solution to the Einstein
equations (2.9)-(2.10).
This concludes our analysis of classical circular string configurations
in the static Robertson-Walker spacetimes. A summary of the results
is presented in Table I.
\section{Semi-Classical Quantization}
\setcounter{equation}{0}
In this section we perform a semi-classical quantization of the circular
string configurations discussed in the previous section. We use an approach
developed in field theory by Dashen et. al. \cite{das}, based
on the stationary phase approximation of
the partition function. The method can be only used for time-periodic
solutions of
the classical equations of motion. In our string problem, these
solutions include all the circular
string solutions in the static Robertson-Walker spacetimes, discussed
in the previous section.
The method has recently been used to quantize circular strings in de
Sitter and anti de Sitter spacetimes \cite{san5} also, and we shall follow
the analysis of Ref.\cite{san5} closely.
The result of the stationary phase integration is expressed in terms of
the function:
\begin{equation}
W(m)\equiv S^{\mbox{cl}}(T(m))+m\;T(m),
\end{equation}
where $S^{\mbox{cl}}$ is the action of the classical solution, $m$ is the
mass and
the period $T(m)$ is implicitly given by:
\begin{equation}
\frac{dS^{\mbox{cl}}}{dT}=-m.
\end{equation}
In string theory we must choose $T$ to be the period in a physical
time variable. In the static Robertson-Walker spacetimes, it is
convenient to take $T$ to be the period in the comoving time $t.$ From
Eq.(3.2), it follows that:
\begin{equation}
T=\sqrt{b}\alpha' T_\tau.
\end{equation}
The bound state quantization condition is \cite{das}:
\begin{equation}
W(m)=2\pi\;n,\quad n \in N_{0},
\end{equation}
$n$ being "large". We will consider the two cases of positive and negative
curvature index. The case of zero curvature index was considered
in Ref.\cite{san5}
and the results in that case will come out anyway in the limit $K\rightarrow 0$
(from above or below).
The classical action is given by:
\begin{eqnarray}
S^{\mbox{cl}}\hspace*{-2mm}&=&\hspace*{-2mm}
\frac{1}{2\pi\alpha'}\int_0^{2\pi}d\sigma\int_0^{T_\tau}d\tau\;
g_{\mu\nu}[\dot{X}^\mu\dot{X}^\nu-X'^\mu X'^\nu]\nonumber\\
\hspace*{-2mm}&=&\hspace*{-2mm}-\frac{2}{\alpha'}\int_0^{T_\tau}d\tau\;
r^2(\tau),
\end{eqnarray}
where we used Eqs.(3.2)-(3.3).
We first consider positive $K$ in the case $\sqrt{bK}\alpha'\leq 1,$ i.e.
the solution (3.15), corresponding to strings oscillating on one
hemisphere. The period in comoving time is given by:
\begin{equation}
T_-=\frac{4k_-K(k_-)}{\sqrt{K}}.
\end{equation}
The classical action over one period becomes:
\begin{equation}
S_-^{\mbox{cl}}=\frac{8}{K\alpha'}[E(k_-)-K(k_-)].
\end{equation}
A straightforward calculation yields:
\begin{equation}
\frac{dT_-}{dk_-}=\frac{4}{\sqrt{K}}\frac{E(k_-)}{1-k_-^2},\;\;\;\;\;
\frac{dS_-^{\mbox{cl}}}{dk_-}=-\frac{8}{K\alpha'}\frac{k_-E(k_-)}{1-k_-^2}.
\end{equation}
Now Eq.(4.2) leads to:
\begin{equation}
m_-=\frac{2k_-}{\sqrt{K}\alpha'},
\end{equation}
and the quantization condition (4.4) becomes:
\begin{equation}
W_-=\frac{8}{K\alpha'}[E(k_-)-(1-k_-^2)K(k_-)]=2\pi\;n
\end{equation}
This equation determines a quantization of the parameter $k_-,$
which through Eq.(4.9)
yields a quantization of the mass. A full parametric plot of $K\alpha'W_-$
as a function of $K\alpha'^2m_-^2\;$ for $k_-\in[0,1]$ is shown in Fig.2.
A fair approximation is provided by the straight line connecting the two
endpoints:
\begin{equation}
W_-\approx 2m_-^2\alpha',
\end{equation}
so that the mass quantization condition becomes:
\begin{equation}
m_-^2\alpha'\approx \pi\;n
\end{equation}
The total number of states is then estimated to be:
\begin{equation}
N_-\approx\frac{4}{\pi K\alpha'}.
\end{equation}
This is the number of quantized circular string states oscillating on
one hemisphere without crossing the equator. The circular strings oscillating
on the full hypersphere are obtained for
$\sqrt{bK}\alpha'>1,$ and are given by the solution (3.21).
Their period in comoving time is:
\begin{equation}
T_+=\frac{4K(k_+)}{\sqrt{K}}.
\end{equation}
The classical action over one period becomes:
\begin{equation}
S_+^{\mbox{cl}}=\frac{8}{K\alpha'}\;\frac{E(k_+)-K(k_+)}{k_+}.
\end{equation}
A straightforward calculation yields:
\begin{equation}
\frac{dT_+}{dk_+}=\frac{4}{\sqrt{K}}[\frac{E(k_+)}{k_+(1-k_+^2)}-
\frac{K(k_+)}{k_+}],
\;\;\;\;\;\frac{dS_+^{\mbox{cl}}}{dk_+}=-\frac{8}{K\alpha'}[
\frac{E(k_+)}{k_+^2(1-k_+^2)}-\frac{K(k_+)}{k_+^2}].
\end{equation}
Now Eq.(4.2) leads to:
\begin{equation}
m_+=\frac{2}{\sqrt{K}\alpha'k_+},
\end{equation}
and the quantization condition (4.4) becomes:
\begin{equation}
W_+=\frac{8}{K\alpha'}\frac{E(k_+)}{k_+}=2\pi\;n
\end{equation}
This equation determines a quantization of the parameter $k_+,$ which
through Eq.(4.17)
yields a quantization of the mass. A parametric plot of $K\alpha'W_+$
as a function of $K\alpha'^2m_+^2\;$ for $k_+\in\;]0,1[\;$ is shown in Fig.3.
Asymptotically, when the mass grows indefinetely, corresponding to
$k_+\rightarrow 0$ (see Eq.(4.17)), we find:
\begin{equation}
K\alpha'^2m_+^2\approx\frac{4}{k_+^2},\;\;\;\;\;K\alpha'W_+
\approx\frac{4\pi}{k_+},
\end{equation}
and the mass quantization condition becomes:
\begin{equation}
m_+^2\approx K\;n^2.
\end{equation}
Several interesting remarks are now in order: First, notice that the mass is
independent of $\alpha'.$ The {\it scale}
of these very massive states is set by
the curvature index $K.$ Secondly, since the mass is proportional to $n,$
the level spacing ($\Delta(m^2\alpha')$ as a function of $n$) {\it grows}
proportionally to $n.$ This is completely different from flat
Minkowski spacetime where the level spacing is constant. Finally, it should
be mentioned that the same behaviour for very massive strings was
found recently in anti de Sitter spacetime \cite{san5,san6}. At
this point it is
tempting to consider the partition function for a gas of strings at finite
temperature, but it must be stressed that a discussion of the thermodynamic
properties (for instance existence or non-existence of a Hagedorn
temperature) must be based on exact quantization of generic strings,
and not only on a semi-classical
quantization of special circular string configurations. It must be noticed,
however, that it has been shown recently \cite{san5,san6}, that for Minkowski,
de Sitter and anti de Sitter spacetimes, the spectrum of generic strings and
the semi-classical spectrum of circular strings are in complete agreement.
Let us now consider the semi-classical quantization of the circular strings
in the spatially hyperbolic spacetime, i.e. we return to the solutions (3.29).
The period in comoving time is:
\begin{equation}
T=\frac{4kK(k)}{\sqrt{-K}}.
\end{equation}
The classical action over one period becomes:
\begin{equation}
S^{\mbox{cl}}=\frac{8}{K\alpha'}\;\frac{E(k)-(1-k^2)K(k)}{\sqrt{1-k^2}}.
\end{equation}
A straightforward calculation yields:
\begin{equation}
\frac{dT}{dk}=\frac{4}{\sqrt{-K}}\;\frac{E(k)}{1-k^2},
\;\;\;\;\;\frac{dS^{\mbox{cl}}}{dk}=\frac{8}{K\alpha'}\;
\frac{kE(k)}{(1-k^2)^{3/2}}.
\end{equation}
Now Eq.(4.2) leads to:
\begin{equation}
m=\frac{2}{\sqrt{-K}\alpha'}\;\frac{k}{\sqrt{1-k^2}},
\end{equation}
and the quantization condition (4.4) becomes:
\begin{equation}
W=\frac{8}{K\alpha'}\frac{E(k)-K(k)}{\sqrt{1-k^2}}=2\pi\;n
\end{equation}
This equation determines a quantization of the parameter $k,$ which
through Eq.(4.24)
yields a quantization of the mass. A parametric plot of $\;-K\alpha'W$
as a function of $\;-K\alpha'^2m^2\;$ for $k\in[0,1[\;$ is shown in Fig.4.
Asymptotically, when the mass grows indefinetely, corresponding to
$k\rightarrow 1$ (see Eq.(4.24)), we find the quantization condition:
\begin{equation}
\sqrt{-Km^2\alpha'^2}\;\log\sqrt{-Km^2\alpha'^2}\approx
-\frac{\pi}{2}K\alpha'\;n
\end{equation}
Formally, it corresponds to a mass-formula in the form
$"m\log m\propto n"$, i.e. the level spacing grows faster than in
Minkowski spacetime (where it is constant) but slower than in the closed
static Robertson-Walker spacetime (where it grows proportional to $n$).
We close this section by ensuring that our results reproduce correctly the
spectrum in flat Minkowski spacetime by taking the limit $K\rightarrow 0.$
Starting for instance from Eqs.(4.9)-(4.10) in the spatially closed
universe, we find in the limit $K\rightarrow 0:$
\begin{equation}
m=2\sqrt{b},\;\;\;\;\;W=2\pi\sqrt{b}\alpha'
\end{equation}
which by Eq.(4.4) gives:
\begin{equation}
m^2\alpha'=4\;n
\end{equation}
If we subtract the intercept $-4,$ this is the well-known (exact)
mass formula for closed bosonic strings in flat Minkowski spacetime. Same
result is obtained by starting from Eqs.(4.17)-(4.18) or from
Eqs.(4.24)-(4.25).
The main conclusions of this section are presented in Table II.
\section{Stationary Strings}
\setcounter{equation}{0}
In this section we will supplement our results on exact classical string
solutions by considering the family of stationary strings. Stationary
strings certainly do not tell much about string propagation in curved
spacetimes, which is our main aim here, but since they are equilibrium
configurations, existing only when there is an exact balance between
the local gravity and the string tension, their actual shape reflects the
geometry and topology of the underlying spacetime and they provide
information about the interaction of gravity on strings. We shall follow a
recent approach \cite{all2} where the stationary strings are described by a
potential in the (stationary) radial coordinate.
The stationary string ansatz $(t=\tau,\;r=r(\sigma),\;\phi=\phi(\sigma))$
in Eq.(2.12) leads to:
\begin{equation}
\phi'=\frac{L}{r^2},\;\;\;\;\;r'^2+U(r)=0,
\end{equation}
where $L$ is an integration constant and the potential $U(r)$ is given by:
\begin{equation}
U(r)=(1-Kr^2)(\frac{L^2}{r^2}-1).
\end{equation}
Notice that we consider plane
stationary strings in the backgrounds (3.1), for $K$ positive, negative or
zero.
Possible extra transverse dimensions have been dropped for simplicity.
Beside energy and pressure, it is also interesting to consider the string
length. By insertion of the stationary string ansatz and Eqs.(5.1)-(5.2)
in the line element (3.1),
we find that the string length element is:
\begin{equation}
dl=d\sigma,
\end{equation}
thus $\sigma$ measures directly the length of stationary strings in the static
Robertson-Walker spacetimes.
For vanishing curvature index, corresponding to flat Minkowski spacetime, it is
well-known that the only stationary strings are the straight ones. Indeed, the
potential $U(r)$ is given by (Fig.5a.):
\begin{equation}
U(r)=\frac{L^2}{r^2}-1,
\end{equation}
and Eqs.(5.1) are solved by:
\begin{equation}
r(\sigma)=\sqrt{\sigma^2+L^2},
\end{equation}
\begin{equation}
\phi(\sigma)=\arctan(\sigma/L),
\end{equation}
which for $\sigma\in\;]-\infty,\;+\infty[\;$ describes an infinitely long
straight string parallel to the $y$-axis with "impact-parameter" $L.$
The string energy and pressure densities are obtained from Eq.(3.6):
\begin{equation}
\frac{d\rho}{dl}=\frac{d\rho}{d\sigma}=
\frac{d}{d\sigma}\int d^3 X \sqrt{-g}T^{00}=
\frac{1}{2\pi\alpha'},
\end{equation}
\begin{equation}
\frac{dP_y}{dl}=\frac{dP_y}{d\sigma}=
\frac{d}{d\sigma}\int d^3 X \sqrt{-g}T^{y}\;_{y}=
-\frac{1}{2\pi\alpha'},
\end{equation}
while $P_x=0.$ Obviously the integrated energy and pressure are infinite.
Eqs.(5.7)-(5.8) represent the well-known string equation of state in $1+1$
effective dimensions.
Let us now turn to the more interesting cases of non-vanishing curvature index.
For positive $K,$ the potential is given by (Fig.5b.):
\begin{equation}
U(r)=(1-|K|r^2)(\frac{L^2}{r^2}-1),
\end{equation}
For $L>1/\sqrt{K}$ there are no solutions since the "impact-parameter" $L$
is larger
than the radius of the hypersphere, thus we need only consider the case
$L\leq 1/\sqrt{K}.$ The solution of Eqs.(5.1) is:
\begin{equation}
r^2(\sigma)=\frac{1}{K}\;\frac{KL^2+\tan^2(\sqrt{K}\sigma)}
{1+\tan^2(\sqrt{K}\sigma)},
\end{equation}
\begin{equation}
\phi(\sigma)=\arctan[\frac{\tan(\sqrt{K}\sigma)}{\sqrt{K}L}].
\end{equation}
Consider first the "degenerate" cases $L=0$ and $L=1/\sqrt{K}:$
\begin{equation}
\phi=\pi/2,\;\;\;\;\;r(\sigma)=\pm\frac{1}{\sqrt{K}}\cos
(\sqrt{K}\sigma)\;;\;\;\;\;\;\;\;\;
\mbox{for}\;\;\;\;\;L=0
\end{equation}
\begin{equation}
\phi=\sqrt{K}\sigma,\;\;\;\;\;r(\sigma)=1/\sqrt{K}\;;\;\;\;\;\;\;\;\;
\mbox{for}\;\;\;\;\;L=1/\sqrt{K}
\end{equation}
Eqs.(5.12) describe a stationary circular string winding around the hypersphere
from pole to pole, while Eqs.(5.13) describe a stationary circular
string winding around the hypersphere along the equator. More generally,
the solution (5.10)-(5.11) describes a stationary circular string of radius
$1/\sqrt{K}$ winding around the hypersphere
for arbitrary values of $L,$ as can be seen as follows: The hypersphere is
parametrized by:
\begin{eqnarray}
x\hspace*{-2mm}&=&\hspace*{-2mm}\frac{1}{\sqrt{K}}\sin\xi\sin\phi\nonumber\\
y\hspace*{-2mm}&=&\hspace*{-2mm}\frac{1}{\sqrt{K}}\cos\xi\\
z\hspace*{-2mm}&=&\hspace*{-2mm}\frac{1}{\sqrt{K}}\sin\xi\cos\phi\nonumber
\end{eqnarray}
where $\sin\xi=\sqrt{K}r.$
A rotation by the angle $\theta$ in the $y-z\;$ plane leads to:
\begin{eqnarray}
\tilde{x}\hspace*{-2mm}&=&\hspace*{-2mm}x\nonumber\\
\tilde{y}\hspace*{-2mm}&=&\hspace*{-2mm}y\cos\theta-z\sin\theta\\
\tilde{z}\hspace*{-2mm}&=&\hspace*{-2mm}y\sin\theta+z\cos\theta\nonumber
\end{eqnarray}
Consider now the general solution (5.10)-(5.11)
and take $\theta=\arccos(\sqrt{K}L):$
\begin{eqnarray}
\tilde{x}\hspace*{-2mm}&=&\hspace*{-2mm}\frac{1}{\sqrt{K}}
\sin(\sqrt{K}\sigma)\nonumber\\
\tilde{y}\hspace*{-2mm}&=&\hspace*{-2mm}0\\
\tilde{z}\hspace*{-2mm}&=&\hspace*{-2mm}\frac{1}{\sqrt{K}}
\cos(\sqrt{K}\sigma)\nonumber
\end{eqnarray}
i.e. the solution (5.13). All stationary strings are identical up to rotations,
so we need only consider (say) the solution (5.13). By integrating the
components of the energy-momentum tensor, Eq.(3.6), we find:
\begin{equation}
\rho=\frac{1}{\sqrt{K}\alpha'},
\end{equation}
\begin{equation}
P_x=P_z=-\frac{1}{2\sqrt{K}\alpha'},
\end{equation}
corresponding to the {\it extremely
unstable string} type equation of state $P=-\rho/2$
in $2+1$ effective dimensions.
For negative curvature index, the potential (5.2) is:
\begin{equation}
U(r)=(1+|K|r^2)(\frac{L^2}{r^2}-1),
\end{equation}
and stationary strings can only exist for $r\geq L,$ see Fig.5c. Eqs.(5.1)
are solved by:
\begin{equation}
r^2(\sigma)=\frac{1}{K}\;\frac{KL^2-\tanh^2(\sqrt{-K}\sigma)}
{1-\tanh^2(\sqrt{-K}\sigma)},
\end{equation}
\begin{equation}
\phi(\sigma)=\pm\left\{\frac{\pi}{2}-\arctan[\sqrt{-K}L\tanh^{-1}
(\sqrt{-K}\sigma)]\right\}.
\end{equation}
For $\sigma\in\;]-\infty,\;+\infty[\;,$ the solution describes a string
stretching from spatial infinity towards $r=L$ and back towards spatial
infinity. The angle between the two "arms" is given by:
\begin{equation}
\Delta\phi=\pi-2\arctan(\sqrt{-K}L)\;\;\in\;\;]0,\pi].
\end{equation}
The total string length and energy are infinite, while
the energy density is given by:
\begin{equation}
\frac{d\rho}{d\l}=\frac{1}{2\pi\alpha'}.
\end{equation}
The pressures in the two directions are generally different due to lack of
symmetry. In the comoving coordinates (2.6) we find the following integral
expressions, using also Eq.(3.6):
\begin{equation}
P_x=-\frac{1}{2\pi\alpha'}\int_{-\infty}^{+\infty}d\sigma\;
[R'\cos\phi-R\phi'\sin\phi]^2,
\end{equation}
\begin{equation}
P_y=-\frac{1}{2\pi\alpha'}\int_{-\infty}^{+\infty}d\sigma\;
[R'\sin\phi-R\phi'\cos\phi]^2,
\end{equation}
where $R$ and $r$ are related by Eq.(2.5). Using the explicit solutions,
Eqs.(5.20)-(5.21), the pressure densities are:
\begin{equation}
\frac{dP_x}{dl}=\frac{dP_x}{d\sigma}=\frac{8KL^2}{\pi\alpha'(1-KL^2)^2}
e^{-\sqrt{-K}\mid\sigma\mid}
\end{equation}
\begin{equation}
\frac{dP_y}{dl}=\frac{dP_y}{d\sigma}=\frac{-8}{\pi\alpha'(1-KL^2)^2}
e^{-\sqrt{-K}\mid\sigma\mid}
\end{equation}
The pressure densities are negative ($K$ is negative) but asymptotically
go to zero exponentially. The integrated pressures are thus {\it finite}. In
fact, Eqs.(5.24)-(5.25) can be integrated explicitly:
\begin{equation}
P_x=\frac{2}{\pi\alpha'\sqrt{-K}(1-KL^2)}\left\{-\frac{2}{3}+\frac{1}{KL^2}-
\frac{1-KL^2}{KL^3\sqrt{-K}}[\frac{\pi}{2}-\arcsin\frac{1}{\sqrt{1-KL^2}}]
\right\},
\end{equation}
\begin{equation}
P_y-P_x=\frac{-4}{3\pi\alpha'\sqrt{-K}(1-KL^2)}\;<\;0
\end{equation}
In the two extreme limits $L\rightarrow\infty,\;L\rightarrow 0,$ we have:
\begin{equation}
P_x\rightarrow 0,\;\;\;\;\;P_y\rightarrow 0\;;\;\;\;\;\;\;\;\;
\mbox{for}\;\;\;\;\;L\rightarrow\infty
\end{equation}
\begin{equation}
P_x\rightarrow 0,\;\;\;\;\;P_y\rightarrow \frac{-4}{3\pi\alpha'\sqrt{-K}}\;;
\;\;\;\;\;\;\;\;\mbox{for}\;\;\;\;\;L\rightarrow 0
\end{equation}
where the latter case describes a straight string through $r=0.$
This concludes our discussion of the stationary strings and their physical
interpretation in the static Robertson-Walker spacetimes.
A summary of the results is presented in Table III.
\section{String Perturbation Series Approach}
\setcounter{equation}{0}
Until now, our analysis has been based on exact solutions to the string
equations of motion and constraints (2.11). However, it
is interesting to consider also approximative methods. In this section
we use the string perturbation series approach, originally developed
by de Vega and S\'{a}nchez \cite{san1}, to describe
string fluctuations around
the string center of mass. The target space coordinates are expanded as:
\begin{equation}
x^\mu(\tau,\sigma)=q^\mu(\tau)+\eta^\mu(\tau,\sigma)+\xi^\mu(\tau,\sigma)+...
\end{equation}
and after insertion of this series into Eqs.(2.12),
one solves the string equations of motion and constraints order
by order in the expansion. For a massive string, the zeroth order equations
determining the string center of mass read:
\begin{equation}
\ddot{q}^\mu+\Gamma^\mu_{\rho\sigma}\dot{q}^\rho\dot{q}^\sigma=0,\;\;\;\;\;
g_{\mu\nu}\dot{q}^\mu\dot{q}^\nu=-m^2\alpha'^2
\end{equation}
For a string with center of mass in the (say) $x-y$ plane of the
$D$-dimensional static Robertson-Walker spacetime (2.4), Eqs.(6.2) lead to:
\begin{eqnarray}
t\hspace*{-2mm}&=&\hspace*{-2mm}\sqrt{p^2+m^2}\;\alpha'\tau,\nonumber\\
\dot{\phi}\hspace*{-2mm}&=&\hspace*{-2mm}\frac{L\alpha'}{r^2},\\
\dot{r}^2\hspace*{-2mm}&=&\hspace*{-2mm}(1-Kr^2)(p^2-\frac{L^2}{r^2})
\alpha'^2,\nonumber
\end{eqnarray}
where $p$ and $L$ are integration constants with the physical interpretation
of momentum and angular momentum, respectively. Eqs.(6.3) can be easily solved
in terms of elementary functions, but we shall not need the general solutions
here. The
equations for the first and second order string fluctuations will turn
out to be quite complicated in the general case, so we shall consider here
only a static string center of mass, $p=L=0:$
\begin{equation}
t=m\alpha'\tau,\;\;\;\;\;r=\mbox{const.}\equiv r_0,\;\;\;\;\;\mbox{all angular
coordinates constant.}
\end{equation}
It is convenient to consider from the beginning only first order string
fluctuations in the directions perpendicular to the geodesic of the center
of mass. We thus introduce normal vectors $n^\mu_R,\;\;R=1,2,...,(D-1)$:
\begin{equation}
\eta^\mu=\delta x^R n^\mu_R,
\end{equation}
where $\delta x^R$ are the comoving fluctuations, i.e. the fluctuations
as seen by an observer travelling with the center of mass of the string. It
can be shown that the first order fluctuations fulfill the equations
\cite{san3}:
\begin{equation}
\ddot{C}_{nR}+(n^2\delta_{RS}-
R_{\mu\rho\sigma\nu}n^\mu_R n^\nu_S\dot{q}^\rho\dot{q}^\sigma)C^S_n=0,
\end{equation}
where $R_{\mu\rho\sigma\nu}$ is the Riemann tensor of the background and
$C^R_n$ are the modes of the fluctuations:
\begin{equation}
\delta x_R(\tau,\sigma)=\sum_n C_{nR}(\tau)e^{-in\sigma}
\end{equation}
In the present case of static Robertson-Walker spacetimes,
the Riemann tensor is non-zero but the projections appearing in Eq.(6.6)
actually vanish, as can be easily verified. It follows from the
explicit expressions of the normal vectors:
\begin{equation}
n^r=(0,\;\sqrt{1-Kr_0^2},\;0),\;\;\;\;\;n^{i}=(0,0,...,0,\frac{1}{r_0},0,...,0,0),
\end{equation}
that the first
order string fluctuations are ordinary plane waves:
\begin{equation}
\eta^{t}(\tau,\sigma)=0,
\end{equation}
\begin{equation}
\eta^r(\tau,\sigma)=\sqrt{1-Kr_0^2}\;\sum_n [A_ne^{-in(\sigma+\tau)}+
\tilde{A}_ne^{-in(\sigma-\tau)}],
\end{equation}
\begin{equation}
\eta^{i}(\tau,\sigma)=\frac{1}{r_0}\sum_n [A^{i}_n e^{-in(\sigma+\tau)}+
\tilde{A}^{i}_ne^{-in(\sigma-\tau)}].
\end{equation}
Here $\eta^{t},$ $\eta^r$ and $\eta^{i}$ denote the fluctuations
in the temporal, radial and
angular directions, respectively. The second order fluctuations are determined
by:
\begin{equation}
\ddot{\xi}^{t}-\xi''^t=0,
\end{equation}
\begin{equation}
\ddot{\xi^r}-\xi''^r=-\frac{Kr_0}{1-Kr_0^2}[(\dot{\eta}^r)^2-
(\eta'^r)^2]+r_0(1-Kr_0^2)\sum_{i}[(\dot{\eta}^{i})^2-(\eta'^{i})^2],
\end{equation}
\begin{equation}
\ddot{\xi}^{i}-\xi''^{i}=-\frac{2}{r_0}\sum_{i}[\dot{\eta}^r\dot{\eta}^{i}-
\eta'^r \eta'^{i}].
\end{equation}
These are just ordinary wave equations with source terms, and can be easily
solved. Thereafter, we have to expand also the constraint equations. In the
present case we find up to second order:
\begin{equation}
m^2\alpha'^2=\frac{1}{1-Kr_0^2}[(\dot{\eta}^r)^2+(\eta'^r)^2]+
r_0^2\sum_{i}[(\dot{\eta}^{i})^2+(\eta'^{i})^2],
\end{equation}
\begin{equation}
\frac{1}{1-Kr_0^2}\dot{\eta}^r \eta'^r+r_0^2\sum_{i}\dot{\eta}^{i}
\eta'^{i}=0.
\end{equation}
Introducing the notation $A_n=A^0_n,\;\;A^\alpha_n=(A^0_n,\;A^{i}_n),\;$
these equations read explicitly:
\begin{equation}
m^2\alpha'^2=-2\sum_\alpha\sum_{n,l}
l(n-l)[A^\alpha_{n-l}A^\alpha_l e^{-in(\sigma+\tau)}+\tilde{A}^\alpha_{n-l}
\tilde{A}^\alpha_l e^{-in(\sigma-\tau)}],
\end{equation}
\begin{equation}
\sum_\alpha\sum_{n,l}[A^\alpha_{n-l}A^\alpha_l
e^{-in(\sigma+\tau)}-\tilde{A}^\alpha_{n-l}
\tilde{A}^\alpha_l e^{-in(\sigma-\tau)}]=0,
\end{equation}
which are just the usual flat spacetime constraints. All dependence of the
curvature index $K$ (positive or negative) has canceled out in these formulae.
For $n=0,$ in particular, we get the usual flat spacetime mass-formula:
\begin{equation}
m^2\alpha'^2=2\sum_\alpha\sum_{l}
l^2[A^\alpha_{l}A^\alpha_{-l}+\tilde{A}^\alpha_{l}
\tilde{A}^\alpha_{-l}],
\end{equation}
with the constraint that there must be an equal amount of left and right
movers:
\begin{equation}
\sum_\alpha\sum_{l}
l^2[A^\alpha_{l}A^\alpha_{-l}-\tilde{A}^\alpha_{l}
\tilde{A}^\alpha_{-l}]=0.
\end{equation}
Notice that the spectrum found here (the flat spacetime spectrum) is very
different (when $K\neq 0$) from the spectrum of the circular strings, as
discussed in Section 4. This is however in no ways contradictory. The
circular string ansatz merely picks out particular states in the complete
spectrum, so there is no apriory reason to believe that the circular
string spectrum should be similar to the generic spectrum. Furthermore,
the perturbation approach used in this section is also semi-classical
in nature, in the sense that it is based on fluctuations around a special
solution, namely the static string center of mass, Eq.(6.4).
\section{Concluding Remarks}
We have solved the equations of motion and constraints for circular strings
in static Robertson-Walker spacetimes. We computed the equations of state
and found a self-consistent solution to the Einstein equations. The solutions
have been quantized semi-classically using the stationary phase approximation
method and the resulting spectra were analyzed and discussed. We also found all
stationary string configurations in these spacetimes and we computed the
corresponding physical quantities, string length, energy and pressure.
Finally we calculated the first and second order string fluctuations
around a static center of mass, using the string perturbation series approach.
\vskip 48pt
\hspace*{-6mm}{\bf Acknowledgements:}\\
A.L. Larsen is supported by the Danish Natural Science Research
Council under grant No. 11-1231-1SE
\newpage
|
1,314,259,996,522 | arxiv | \section{Introduction}
This is a personal account of a career of 50 years which strongly
overlapped with two of the many of Dick Dalitz's interests,
hypernuclei and the low energy $\bar{K}N$ system.
As a new graduate student, I first met Dick about 50 years ago
when he gave a seminar on the low energy $\bar{K}N$ system, a
preview of the subsequent Dalitz and Tuan paper~\cite{ref1}.
My first publication~\cite{ref2} concerned the observation of a
$\Sigma^+$ hyperon decaying into a proton with an associated
Dalitz pair~\cite{ref3}, a mere verification of what was already
well known; one decay mode of the $\Sigma^+$ was to $\pi^0p$.
In 1961 I joined Levi Setti's emulsion group in Chicago, the hub
of hypernuclear physics at that time, and got to know Dick very
well. While there, we made the first direct experimental estimate
of the $\Lambda$ nuclear potential well-depth, $D_{\Lambda}$, from
the decays of heavy spallation hypernuclei~\cite{ref4}. Our
value, of a little less than 30\,MeV, agreed well with Dick's
value of $26.5\pm2.5$\,MeV~\cite{ref5} obtained by extrapolating
from the then known binding energy of $_{\,\Lambda}^{13}$C.
However, it should be remarked that at this time various
theoretical estimates using $\Lambda N$ potentials derived from
the $B_{\Lambda}$ values of the $s$--shell hypernuclei all gave
values nearer to 60\,MeV. Also at this time was the determination
of the spin of $_{\Lambda}^{\,8}$Li~\cite{ref6,ref7}.
I returned to London to rejoin the European $K^-$ Emulsion
Collaboration and Dick came to England shortly afterwards, first
to Cambridge and then permanently to Oxford to join up again with
his old mentor, Rudolf Peierls.
There began a long, symbiotic, and fruitful collaboration between
Dick and us. We benefited greatly from his advice, encouragement
and theoretical input, and he knew of our results well before
publication. He was invited and came to many of our meetings, not
only those in London, but sometimes to those in Brussels and CERN
as well.
In the sixties and early seventies we worked on such topics as the
$B_{\Lambda}$ values of $p$--shell hypernuclei~\cite{ref8}, spins
of $_{\Lambda}^{\,8}$Li~\cite{ref9} and
$_{\,\Lambda}^{12}$B~\cite{ref10}, $\pi^+$ decays of
hypernuclei~\cite{ref11}, charge symmetry breaking in the
$_{\Lambda}^{\,4}$He, $_{\Lambda}^{\,4}$H mirror
doublet~\cite{ref12} and many others, always in close
collaboration with Dick.
However, I would like to concentrate on two long-running sagas in
which Dick played a pivotal role, those of double hypernuclei and
the proton--emitting $p$-wave hypernuclear states.
\section{Double Hypernuclei versus the H}
The first double hypernucleus, an example of $_{\Lambda\Lambda}^{\
10}$Be, was found by our collaborators in Warsaw in
1962~\cite{ref13} and Dick was among the first to glean properties
of the $\Lambda\Lambda$ interaction from it~\cite{ref14}. A second
event, an example of $_{\Lambda\Lambda}^{\ \ 6}$He, was reported
by Prowse in 1966~\cite{ref15} and then, for a long time, nothing.
Prowse died in 1971 and Pniewski began to express doubts about the
authenticity of the Prowse event. In 1977, Jaffe predicted the
existence of the $H$ particle, a deeply bound six quark system
$(u,u,d,d,s,s)$, the quark content of two $\Lambda$ hyperons. I
well remember questioning Jan de Swart at the Jab{\l}onna
Conference in 1979, that was it not difficult to reconcile the
existence of double hypernuclei with that of the $H$ particle? His
reply, which was echoed in many subsequent reviews by proponents
of the $H$, was that there were only two events reported in
emulsions and perhaps they had been wrongly interpreted.
Dick asked me at the end of the eighties if we could possibly
remeasure the event in order to allay these doubts. I told him
that this was not possible as the emulsion pellicle containing the
event no longer existed. However, I did have in my possession
copies of unpublished photomicrographs taken of the event by
P.\,Fowler \emph{et al.}\ in Bristol some 25 years earlier.
It should be noted that the three vertices of the event, the
$\Xi^-$ capture and the double and single hypernuclear decays were
all contained in a cube of side length 3 microns. Moreover,
nuclear emulsion is approximately one half silver bromide crystals
by volume and, as a consequence, a pellicle shrinks in height by a
factor a little more than two on processing. In order to achieve
greater resolution for photography, the emulsion was swelled by
soaking it in a saturated sugar solution made from Tate and Lyle's
golden syrup, since water has the wrong optical properties, to 3.5
times its processed thickness. Unfortunately for Dick's request,
the sugar ultimately crystallised and the emulsion was destroyed.
The photomicrographs and an independent analysis of the event made
during its photographing were published in the Royal Society paper
of 1989~\cite{ref16}.
Every attempt was made to elicit further details of the Prowse
event but without success. However, one significant fact did
emerge which persuaded me that Pniewski was right to doubt the
event. Although Prowse had moved to Wyoming and published the
event from there, his emulsion work had remained in UCLA and where
at that time Ticho, Schlein and Slater, all erstwhile hypernuclear
emulsion physicists, were based. All were contacted, but none had
any recollection of having seen the event!
There now exists the tightly constrained $_{\Lambda\Lambda}^{\ \
6}$He~\cite{ref17} with a $\Delta B_{\Lambda\Lambda}\approx
1.0\,$MeV, completely at odds with the Prowse event, and for that
matter with the original $_{\Lambda\Lambda}^{\ 10}$Be event also.
For the $_{\Lambda\Lambda}^{\ 10}$Be event, as was stated in a
footnote to a table in the paper of 1963, `It should be noted that
the value of $B_{\Lambda\Lambda}(_{\Lambda\Lambda}Z)$, and hence
$\Delta B_{\Lambda\Lambda}$, may be overestimated if the ordinary
hyperfragment is produced in an excited state', and should the
decay have occurred to a now known state of $_{\Lambda}^{\,9}$Be
at $\sim3.0\,$MeV~\cite{ref18}, the $\Delta B_{\Lambda\Lambda}$
would become $\sim 1.3\,$MeV, well compatible with the Nagara
event~\cite{ref17}. No such escape clause exists for the Prowse
$_{\Lambda\Lambda}^{\ \ 6}$He event~\cite{ref15}.
\centerline{And still there is no $H$!}
\section{Strangeness--Exchange States}
Around the late sixties we were attempting to determine the
$B_{\Lambda}$ values of $p$--shell hypernuclei. There were many
possible examples of $_{\,\Lambda}^{11}$B decays to
$\pi^-\:{}_{\,\Lambda}^{11}$C but see Table~\ref{table1}.
\begin{table}[htb]
\begin{center}
\caption{Competing two--body decay characteristics \label{table1}}
\begin{tabular}{cccc}
\hline
&pion range&recoil range& recoil\\
&($\mu$m)&($\mu$m)&\\
\hline
$_{\,\Lambda}^{11}\textrm{B}\to\pi^-\:{}_{\,\Lambda}^{11}$C&
20700&1.0&$\beta^+$\\
$_{\,\Lambda}^{10}\textrm{Be}\to\pi^-\:{}_{\,\Lambda}^{10}$B&19800&1.1&
stable\\
$_{\Lambda}^{\,7}\textrm{Li}\to\pi^-\:{}_{\Lambda}^{\,7}$Be&21600&1.8&E.C.\\
\hline
\end{tabular}
\end{center}
\end{table}
With range straggling of the pions of about 3\%, the three
hypernuclei in Table~\ref{table1} cannot be separated from decay
kinematics alone. It was noted that many of the
$_{\,\Lambda}^{11}\textrm{B}$ candidates had $K^-$ at rest
production topologies of hyperfragment + $\pi$ + one baryonic
track. The azimuthal and dip angles of all three tracks were
measured, as were the ranges of the hyperfragment and assumed
proton (the pion invariably left the emulsion before stopping).
Assuming the $K^-$ capture is on a light nucleus of the emulsion,
in this case $^{12}$C, the proposed reaction is
\begin{equation}
\label{eq1} K^- + ^{12}\textrm{C}\rightarrow \pi^- + p +
_{\,\Lambda}^{11}\textrm{B} + Q.
\end{equation}
The range of the proton determines both $T_p$ and $p_p$. With the
$Q$ value known, an iterative procedure was used to determine the
value of the pion's energy. Starting with a value of $T_{HF} =
0$, this gave $T(1)_{\pi}$, hence $\vec{p}(1)_{\pi}$,
$\vec{p}(1)_{\pi} + \vec{p}_p \rightarrow \vec{p}(1)_{HF}
\rightarrow T(1)_{HF} \rightarrow T(2)_{\pi} \rightarrow
\vec{p}(2)_{\pi}$, $\vec{p}(2)_{\pi} + \vec{p}_p\rightarrow
T(2)_{HF}$, and so on. The procedure rapidly converges; after two
iterations the pion kinetic energy is stable to 5 keV. The
compatibility with reaction (\ref{eq1}) was then tested by
comparing the measured and computed ranges and directions of the
hypernucleus. The compiled data revealed a sharp spike, of the
order 1\,MeV wide, in the pion spectrum. Such a spike suggests
that many of the events proceeded \emph{via} a two-body reaction
\begin{eqnarray}
\nonumber K^- +
^{12\!}\textrm{C}\rightarrow\pi^-+_{\,\Lambda}^{12}\textrm{C}^*&&\\
&&\hspace{-5mm}\hookrightarrow p +_{\,\Lambda}^{11}\textrm{B}
\label{eq2}
\end{eqnarray}
This was written up and sent to Nuclear Physics but the referee
demurred --- `Not every spike implies a resonance.' It was
resubmitted with the inclusion of phase space and impulse model
curves. The spike remained prominent, but still the referee was
not happy, and so it went to a second referee. The second referee
was more forthcoming, `first observation of a highly excited state
of a hypernucleus', and so it was published~\cite{ref19}. I saw
Dick shortly afterwards and thanked him for overruling the first
referee. He was taken aback and said that Nuclear Physics had no
business to divulge the referee's name. I had to assure him that
they had not. Remember, this was well before the coming of the
Word, the Microsoft Word that is. Referees wrote reports which
were typed by a secretary. However, as anyone who has received a
typewritten document from Dick will know, he was never satisfied
and always modified the text in his own neat but unmistakable
handwriting, and so it was here.
We published a further paper on the subject with some more
statistics but then left the field to go hunting for charm and
beauty particles. The subject was taken up by the counter
$(K,\pi)$ and $(\pi,K)$ spectroscopy groups at CERN, BNL and later
KEK. I was brought back to the subject by Dick, who else,
approaching me at the 1982 Heidelberg Conference. He had a
problem. Was the $\Lambda$ in $_{\,\Lambda}^{12}\textrm{C}^*$
bound or unbound? Theoreticians worry about such things. The
Brookhaven group had two values for $B_{\Lambda}$ in
$_{\,\Lambda}^{12}\textrm{C}^*$, one measured with the pion going
off in the forward direction as everyone else had done, the other
with the pion making a large, $15^{\circ}$ angle to the kaon
direction.
\begin{eqnarray*}
\theta_{K\pi} = \phantom{1}0^{\circ}\ \ &\Rightarrow&\ \
B_{\Lambda} = +0.033\pm
0.180\,\textrm{MeV},\\
\theta_{K\pi} = 15^{\circ}\ \ &\Rightarrow&\ \ B_{\Lambda} =
-0.027\pm 0.160\,\textrm{MeV}.
\end{eqnarray*}
He remarked that he had also checked our two papers and in both
the $B_{\Lambda}$ values were positive, but they did not agree
with one another! This was a problem. Dave Tovee and I went back
to our old results and discovered the cause of the discrepancy;
the input mass values necessary to our analysis, especially that
of the kaon, had changed in the Particle Data Group's listings in
the early seventies by considerably more than the stated errors.
We reanalysed all of our old data, including a large sample of
events obtained using the resonance peak to identify
$_{\,\Lambda}^{11}\textrm{B}$ events in order to study their
non-mesonic decays~\cite{ref20} and the results of this analysis
were presented at the 1985 Brookhaven Conference~\cite{ref21}. The
$Q$ value distribution in the decay
$_{\,\Lambda}^{12}\textrm{C}^*\to p +_{\,\Lambda}^{11}\textrm{B}$
is given in Fig.~\ref{fig1}~\cite{ref22}.
\begin{figure}[htb]
\begin{center}
\resizebox{0.45\textwidth}{!}{%
\includegraphics{Fig1.EPS}}
\end{center}
\caption {Distribution of the $Q$-values in the
$_{\,\Lambda}^{12}\textrm{C}^*\to p +_{\,\Lambda}^{11}\textrm{B}$
decay, treating all events as occurring on $^{12}$C.\label{fig1}}
\end{figure}
A good fit to the spectrum below 2\,MeV, the events there comprise
an essentially pure sample of $_{\,\Lambda}^{12}\textrm{C}^*$,
requires at least three Breit--Wigner distributions. One stands
alone, but two more are needed, both centred around
$Q\approx1.4\,$MeV, a very narrow one and a wider one. What do we
expect? The states are formed by the addition of a $P\fmn{3}{2}$
neutron hole and either a $P\fmn{1}{2}$ or a $P\fmn{3}{2}$
$\Lambda$ hyperon.
\begin{eqnarray*}
\bar{n}(P\fmn{3}{2}) + \Lambda(P\fmn{1}{2})&\Rightarrow& 1^+,2^+,\\
\bar{n}(P\fmn{3}{2}) + \Lambda(P\fmn{3}{2})&\Rightarrow&
0^+,1^+,2^+,3^+.
\end{eqnarray*}
The low energy $\bar{K}N$ interaction is $s$--wave, which contains
no spin flip, and so we expect only the $0^+$ and the two $2^+$
states to be produced. Since the $_{\,\Lambda}^{11}\textrm{B}$
ground state has spin $\fmn{5}{2}^+$, while the proton from the
two $2^+$ states may be emitted in the $s$--wave, that from the
$0^+$ state has to be $d$--wave. It is thus natural to assign the
observed narrow state to be the $0^+$, since the angular momentum
barrier for such a small energy release, $\sim 1\,$MeV, will
strongly inhibit its decay. With this assignment we construct the
level scheme given in Table~\ref{table2}.
\begin{table}[htb]
\begin{center}
\caption{Level scheme for the observed
$_{\,\Lambda}^{12}\textrm{C}^*$ states \label{table2}}
\begin{tabular}{cccc}
\hline
&No of&$\Gamma$&$B_{\Lambda}$\\
&events&(keV)&(MeV)\\
\hline%
$0^+$& 64 & $<100$ & $+0.14\pm0.05$\\
$2^+$& 193 & $\sim 600$ & $+0.20\pm0.05$\\
$2^+$& 48 & $\sim 150$ & $+0.95\pm0.05$.\\
\hline
\end{tabular}
\end{center}
\end{table}
To conclude, three states are expected, and three are found. The
real widths of the $2^+$ states have been determined, whereas it
is only possible to give an upper limit to the width of the $0^+$
state; there are limits even to the resolution of the emulsion
technique! The $B_{\Lambda}$ difference between the two $2^+$
states places a limit on the $\Lambda N$ spin--orbit interaction
and the relative production rates following $K^-$ capture at rest
on $^{12}$C are given. Finally, in answer to Dick's original
question, the $B_{\Lambda}$ values have been determined and in ALL
states the $\Lambda$ hyperon is bound.
The remaining problem with this analysis; why did the $0^+$ state
not decay quickly by $s$-wave proton emission to the expected
$\fmn{1}{2}^+$ excited state of $_{\,\Lambda}^{11}\textrm{B}$ was
solved recently when it was found that this state was
energetically out of reach~\cite{ref23}.
\section{Conclusions}
In conclusion, it should be emphasised that neither of these two
investigations would have been undertaken without Dick's
encyclopaedic knowledge of past results and his nagging
persistence to obtain solutions to puzzling situations. The
relevant information would otherwise have mouldered away on my
bookshelves, forgotten.
I greatly miss his friendship and his phone calls, usually at home
late on a weekend evening, asking for details of things recorded
in his notebooks of which I had presumably told him some twenty or
thirty years before. The whole hypernuclear community will sorely
miss him too.
I would like to thank the organisers for giving me the opportunity
to speak of Dick, for a very enjoyable conference and for
contributing generously towards my expenses.
|
1,314,259,996,523 | arxiv |
\subsection{Composition of Dataset}
For the selection of suitable training data we thus looked at a variety of available datasets which should provide a more balanced set of images for surface types as a whole.
In addition, composing training data from multiple datasets has the advantage of covering several different cameras which helps avoiding learning features specific to the camera-setup of a single research vehicle.
One restriction we applied was that the perspective towards the surface should be vaguely similar to the perspective of a windscreen mounted camera, to avoid applying artificial distortion of the chosen images.
For the composition of the dataset we used images from the following datasets:
\begin{itemize}
\item NREC Human Detection \& Tracking in Agriculture \cite{pezzementi2017}
\item KITTI Vision Benchmark Suite \cite{geiger2013}
\item Oxford Robocar Dataset \cite{maddern2017}
\item New College Vision and Laser Data Set \cite{smith2009}
\item Imageset published by \cite{giusti2016}
\item Image sequences captured in the Stadtpilot project by our research vehicle \emph{Leonie} \cite{nothdurft2011a}
\end{itemize}
The obtained classes available in each individual dataset are presented in Table \ref{tab:datasets}.
\bgroup
\def1.24{1.24}
\begin{table}
\caption{Available classes in the individual datasets. Numbers in parentheses denote total number of selected samples.}
\begin{tabularx}{\columnwidth}[c]{m{1.2cm}M{0.8cm}M{0.5cm}M{0.8cm}M{0.8cm}M{1cm}M{0.6cm}}\toprule
& asphalt\newline (10273) & dirt\newline (8547) & grass\newline (2887) & wet\newline asphalt \newline (3668) & cobble-\newline stone\newline(1082) & snow\newline(3075) \\ \midrule
Robocar & X & & X & X & & X \\
\rowcolor[gray]{.9} Stadtpilot & X & & & X & X & \\
NREC & & X & X & & & \\
\rowcolor[gray]{.9} New\newline College & X & X & X & & & \\
Giusti\newline et al. & X & X & X & & X & X \\
\rowcolor[gray]{.9} KIITI & X & & X & & X & \\ \bottomrule
\end{tabularx}
\label{tab:datasets}
\end{table}
\egroup
Analyzing these datasets and comparing the majority class asphalt with the minority class cobblestone yields an imbalance ratio of 10:1:
The class \emph{asphalt} consists of over 10000 images, while the class \emph{cobblestone} is represented in just over 1300 images.
To counteract the imbalance, instead of applying over or under sampling, we added further images from Google image search, following the example of \cite{krause2016} for fine grained image classification.
\subsection{Selection of Test and Training Data}
All used datasets provide frame sequences rather than a random collection of independently recorded frames.
Thus the road conditions vary only slightly between frames from a single sequence.
When dividing the selected images into test and training sets, we did not only split single sequences, but also selected images from different sequences for test where possible.
The finally used test set consisted of 300 images per class.
The remaining images were used for training, building three different training sets.
A first set only consisted of the images from the datasets mentioned above.
To create a balanced set, 700 images per class were chosen randomly.
300 images were used for validation.
For the second dataset, the classes \emph{cobblestone} and \emph{wet asphalt} were extended with images from Google image search as mentioned earlier.
The class \emph{wet asphalt} was available in the fewest sequences, while the class \emph{cobblestone} had the lowest number of samples.
As the image search resulted in too few usable samples (\emph{grass}) or a sufficient number of images was available (\emph{asphalt}), only the first two classes were extended.
Using the Google image search, the class \emph{cobblestone} was extended such that each class consisted of 2500 images.
The class \emph{wet asphalt} was extended to increase variation of images within the class.
The training set was thus more than doubled.
500 images were selected for validation.
For a third dataset all classes from the basic dataset were extended with 300 images from Google image search, which corresponds to the number of returned usable images for the class \emph{grass}.
In order to overcome the issue of lacking variation between consecutive frames in the sequences, the used sequences were subsampled, using only every $n^{th}$ frame, with $n$ depending on the length of the sequence.
\subsection{Training \& Classification Performance}
This section presents an overview about the achieved training and classification results with both implemented architectures on the three datasets (basic, image search augmentation for two classes, augmentation for all classes) described above.
Considering training performance, the InceptionV3 model terminates after seven (second training dataset) to ten (first training dataset) epochs.
The maximum validation accuracy is reached after the third epoch as shown in Figure \ref{val:acc} (left hand side).
The average validation accuracies of the models trained on the basic dataset and the second dataset are comparable.
Extending the basic dataset with images from Google image search for all classes leads to a decrease of $\approx$\SI{1.5}{\percent} in validation accuracy on training data.
The extension of the training dataset does not impact the duration of the training.
The inference run on an NVIDIA Titan X GPU takes \SI{153}{\milli\second} in average.
When evaluating the performance on the test dataset, the InceptionV3 architecture behaves differently:
Training the model on the first and second dataset resulted in a comparable test accuracy of \SI{90}{\percent}.
Extending all classes with images from image search, however, resulted in a an test accuracy of only \SI{84}{\percent}.
The behavior of the model provides a hint, that the additional variation of training data does not provide any benefit.
In contrary, the network starts to overfit
Training of the ResNet50 architecture takes longer than the training of the InceptionV3 model:
Training terminates after ten (basic dataset) to twenty epochs (second and third dataset).
In contrast to the InceptionV3 architecture, adding images from Google image search, speeds up the training process (cf.~Fig.\ref{val:acc}, right hand side).
The ResNet model trained on the first dataset achieved a lower test accuracy on the test dataset (\SI{80}{\percent}) than the corresponding InceptionV3 model.
However, the partial addition of images from google image search increased the test accuracy by \SI{4}{\percent} to an overall \SI{92}{\percent}.
The ResNet50 architecture exposes the same over-fitting behavior as InceptionV3 with performance decreased to \SI{84}{\percent} when the basic dataset is extended with images for each class.
Inference for ResNet50 takes \SI{94}{\milli\second} on the NVIDIA Titan X.
\begin{figure}[bhtp
\centering
\includegraphics[width=\columnwidth]{figures/tikz/plots_training}
\caption{Training and validation accuracy of InceptionV3 (left) and ResNet50 (right) architectures trained on the three datasets. Top to bottom: basic dataset, dataset with class \emph{cobblestone} and \emph{wet asphalt} extended from image search, dataset with all classes augmented from image search. All data is plotted until training terminated due to early stopping.}
\label{val:acc}
\end{figure}
\subsection{Analysis of Results}
Looking at the confusion matrices in Figure \ref{fig:confusion}, misclassification occurs in a pattern.
Images from the class \emph{wet asphalt} are often classified as \emph{asphalt}, but in no case with the class \emph{grass}, when augmenting the classes \emph{cobblestone} and \emph{wet asphalt}.
In contrast classification accuracy for the class \emph{cobblestone} increases by \SI{2}{\percent} and for \emph{asphalt} by \SI{12}{\percent}.
Images from the classes \emph{snow} and \emph{grass} are classified with high recall.
By examining the misclassified images, several possible causes for the misclassification could be identified.
\begin{figure
\centering
\begin{subfigure}{0.15\textwidth}
\includegraphics[width=\textwidth]{figures/data/misscolor1.png}
\caption{Classified as "grass".}
\end{subfigure}
\begin{subfigure}{0.15\textwidth}
\includegraphics[width=\textwidth]{figures/data/misscolor2.png}
\caption{Classified as "grass".}
\end{subfigure}
\begin{subfigure}{0.15\textwidth}
\includegraphics[width=\textwidth]{figures/data/misscolor3.png}
\caption{Training image for the class "grass".}
\end{subfigure}
\begin{subfigure}{0.15\textwidth}
\includegraphics[width=\textwidth]{figures/data/misscolor4.png}
\caption{Classified as "snow"}
\end{subfigure}
\begin{subfigure}{0.15\textwidth}
\includegraphics[width=\textwidth]{figures/data/misscolor5.png}
\caption{Classified as "snow"}
\end{subfigure}
\begin{subfigure}{0.15\textwidth}
\includegraphics[width=\textwidth]{figures/data/misscolor6.png}
\caption{Training image for the class \emph{snow}.}
\end{subfigure}
\caption{The first two images in each row were misclassified. The rightmost images were part of the training set.}
\label{miss:color}
\vspace{-1em}
\end{figure}
The first one is the dominating color in the images.
Within the given classes \emph{snow} and \emph{grass}, the most distinctive feature is color, as grass is commonly green and a road covered with snow is commonly white.
Color as a learned feature can thus result in a high recall score for both classes.
Evaluating the false positive classifications, the resulting images consist of images which contain these colors.
Samples are shown in Figure \ref{miss:color}.
\begin{figure}[b
\centering
\begin{subfigure}{0.2\textwidth}
\includegraphics[width=\textwidth]{figures/data/ambig1.png}
\end{subfigure}
%
\begin{subfigure}{0.2\textwidth}
\includegraphics[width=\textwidth]{figures/data/ambig2.png}
\end{subfigure}
\caption{Examples for images with ambiguous classes: remaining puddles (left), cobblestone and asphalt in the same ROI (right)}
\label{miss:ambig}
\end{figure}
\begin{figure*}
\centering
\input{figures/tikz/plots_confusion.tikz}
\caption{Evaluation on test data: Confusion matrices for trained InceptionV3 (top) and ResNet50 (bottom) architectures. Left to right: Basic dataset, dataset with classes \emph{cobblestone} and \emph{wet asphalt} extended from image search, dataset with all classes augmented from image search.}
\label{fig:confusion}
\vspace{-1em}
\end{figure*}
When evaluating images from the class \emph{dirt}, the misclassified samples partially contain patches of grass, which makes the class prone to be misclassified as \emph{grass}.
The same reason for misclassification applies to images containing overexposed regions which appear white and can thus be misclassified as \emph{snow}.
Not only the samples containing the class \emph{dirt} are prone to ambiguous texture information, as shown in Figure \ref{miss:ambig}.
If the region of interest contains transitions between two road surfaces (e.g. \emph{asphalt} and \emph{cobblestone}, as shown on the right in Figure \ref{miss:ambig}), it contains features of two classes and are therefore also prone to misclassification.
Another example for this are left-over puddles on an already dry cobblestone road (Fig. \ref{miss:ambig} on the left).
Although part of the surface is wet, the surface should be considered as \emph{cobblestone}.
Although the classifier operates on single frames, the images are part of sequences.
In order to get an impression of the stability of the classification results when applied to those sequences, we evaluated the classification on sample sequences from the \emph{Stadtpilot} project, which were not part of the training dataset.
No tracking was performed between frames.
For this classification, ResNet50 trained on the second dataset was used.
In Figure \ref{seq} three of the worst classification results in sequences are shown.
Looking at these results, it is visible that misclassification tends to appear in groups of several frames.
Fluctuations as shown in the center and bottom sequence can render the trained classifiers unsuitable for adapting control algorithms.
\begin{figure
\centering
\includegraphics[width=.99\columnwidth]{figures/tikz/plots_sequences}
%
\caption{Classification results in sequences. Each image is classified separately, no tracking is performed. Ground truth from top to bottom: cobblestone, wet asphalt, wet asphalt.}
\label{seq}
\end{figure}
\section{Introduction}
\label{sec:intro}
\input{01_introduction}
\section{Related Work}
\label{sec:related}
\input{02_related_work}
\section{Challenges Regarding Available Datasets}
\label{sec:challenges}
\input{03_challenges}
\section{Approach for Surface Classification}
\label{sec:approach}
\input{04_approach}
\section{Training Parameters}
\label{sec:architectures}
\input{05_architecture}
\section{Results}
\label{sec:results}
\input{06_results}
\section{Conclusion \& Future Work}
\label{sec:conclusion}
\input{07_conclusion}
\section*{Acknowledgement}
\label{sec:ack}
\input{08_acknowledgement}
\IEEEtriggeratref{1}
\renewcommand*{\bibfont}{\footnotesize}
\printbibliography
\end{document}
|
1,314,259,996,524 | arxiv | \section{Introduction}
Complexity theory finds its root in three different papers that, in the span of a single year, tackled the difficult question of defining a notion of \emph{feasible} computation \cite{ cobham,edmonds65,hartmanisstearns}. Interestingly, the three authors independently came up with the same answer, namely the class \FPtime of polynomial time computable functions. Indeed, while computability finds its roots in mathematical logic, i.e. very far from actual computing devices, it became quite clear at the time that the notion of \emph{computable function} does not fall into any reasonable notion of \emph{effective computability}.
Based on this, the fields of complexity theory, whose aim is the definition and classification of functions based on how much resources (e.g. time, space) are needed to compute them, quickly developed.
While progress on the classification problem was quick in the early days, new results quickly became more and more scarcer. The difficulty of the classification problem can be explained in several ways. First, from a logical point of view, the question of showing whether a complexity class cannot compute a given function corresponds to showing the negation of an existential statement. But the severe difficulty of this problem can be understood through negative results known as \emph{barriers}, i.e. results stating that currently known methods cannot solve current open problems. While three such barriers exist, we will take the stand that only two \emph{conceptual} barriers exists, the \emph{algebrization} barrier being understood as a (far-reaching) refinement of the older \emph{relativization} barrier.
\paragraph{Barriers.} For the following short discussion about barriers, let us consider the famous \Ptime vs. \NPtime problem. The relativisation barrier is based on Baker, Gill and Solovay result \cite{relativisation} that there exists (recursive) oracles $\mathcal{A,B}$ such that $\Ptime^{\mathcal{A}}=\NPtime^{\mathcal{A}}$ and $\Ptime^{\mathcal{B}}\neq\NPtime^{\mathcal{B}}$. This implies that any proof method that is oblivious to the dis/use of oracles -- in other words, that \emph{relativises} -- will not answer the \Ptime vs. \NPtime problem. The \emph{algebrization} barrier \cite{algebrisation} provides a conceptually similar but refined negative result for algebraic methods, i.e. for those proof methods that are oblivious to the dis/use of \emph{algebraic extensions} of oracles -- which are said to \emph{algebrize}. The third barrier provides a somewhat orthogonal negative result, based on a notion of \emph{natural proofs}: roughly speaking a proof is natural if it can be formulated as separating two complexity classes using a predicate on boolean functions that satisfies three properties: \emph{largeness} -- i.e. most problems in the smallest of the two classes will satisfy the predicate --, and \emph{constructibility} -- i.e. the predicate is decidable in exponential time.
Razborov and Rudich showed \cite{naturalproofs} that under the assumption of the existence of exponentially-hard random generators, a natural proof cannot be used to prove that $\Ptime\neq\NPtime$.
These three barriers altogether capture all known separation methods to date, and therefore state that proving new separation results will \enquote{require radically new techniques} \cite{algebrisation}. In the last twenty years, one research program has been thought of as the only serious proposal for developing new separation methods that would circumvent the barriers, namely Mulmuley's Geometric Complexity Theory ({\sc gct}\xspace) programme \cite{GCT1,GCT2,GCTsurvey1,GCTsurvey2}. Inspired from a geometric proof of lower bounds on a (weakened) variation of the {\sc pram}\xspace model \cite{MulmuleyPRAM}, Mulmuley {\sc gct}\xspace methods are based on involved methods from algebraic geometry. However, after twenty years of existence, the {\sc gct}\xspace program has been dented with a few negative results in the last years.
\paragraph{Graphings.} The author recently proposed a new approach to complexity theory, one which may lead to new proof methods for separation \cite{seiller-towards,seiller-tacl,seiller-goinda}. Although this claim might not be formally justified at this point, let us point out that a recent preprint uses these methods to recast and improve lower bounds in algebraic complexity \cite{seiller-pramsLB}. The principal idea behind the approach is to propose a new general mathematical theory of computation that accounts for the dynamics of programs. In essence, the guiding intuition is that a computation should be mathematically modelled as a dynamical system, in the same way physical phenomena are. Obviously, while a computation (i.e. a run of a program) is deterministic and can be represented as such, a \emph{program} is not in general: it might be e.g. probabilistic, deterministic, and represent in itself several possible runs on a given input. Seiller's proposal is therefore to work with generalisations of dynamical systems introduced under the name of \emph{graphings}. Similarly to dynamical systems, graphings come in three different flavours: discrete, topological and measurable. The distinction between those does not impact the following discussion, although the results in this paper will use \emph{measurable graphings}.
Graphings were initially introduced in the context of ergodic theory \cite{adams,gaboriaul2}, and entered the realm of theoretical computer science through work on semantics of linear logic \cite{seiller-goig,seiller-goie,seiller-goif}. The main result in this aspect is that a collection of graphings built from a monoid action onto a space $\alpha: M\curvearrowright X$ -- i.e. graphings that can be locally identified with endomorphisms of the type $\alpha(m)$ -- gives rise to a model of (fragments of) linear logic. It is in this context that Seiller discovered that the action $\alpha: M\curvearrowright X$ can be put into correspondence with complexity classes, i.e. choosing different monoid actions leads to models in which the represented programs are of limited complexity, formalising an intuition that already appeared in the more involved context of operator algebras \cite{seiller-masas}. The first result in this direction \cite{seiller-goinda} provided a correspondence between a hierarchy of group actions $\microcosm{m}_{1},\microcosm{m}_{2},\dots,\microcosm{m}_{k},\dots,\microcosm{m}_{\infty}$ and a hierarchy of complexity classes between (and including) \Regular -- the classe of regular languages -- and \coNLogspace.
\paragraph{Contributions.} The current paper provides similar characterisations of the corresponding deterministic, non-deterministic (with the usual notion of acceptance), and probabilistic hierarchies. This is an important step in the overall program, as it shows the techniques apply to several computational paradigms. From a more external point of view, the techniques provides the first implicit characterisation probabilistic complexity classes, such as \PLogspace (resp. \PPtime) of problems decidable (with unbounded error) by a probabilistic machine using logarithmic space (resp. polynomial time) in the input. Figure \ref{contribs} recapitulates\footnote{This table is not exhaustive, but shows the most common classes. In particular, this paper also characterises numerous classes not shown here, notably the classes of languages recognized by $k$-head two-way atutomata with a pushdown stack where $k$ is a fixed integer.} the known characterisations through the author's methods, showing the results of the current paper in white cells (previous results are shown in gray cells).
\begin{figure*}
\begin{center}
\begin{tabular}{|c||c|c|c|c|}
\hline
Microcosm & deterministic model & \multicolumn{2}{c|}{non-deterministic model} & probabilistic model \\\hline
$\microcosm{m}_{1}$ & \cellcolor{blue!10}$\Regular$ & \cellcolor{blue!10}\Regular & \Regular & \cellcolor{blue!10}\Stochastic \\
$\vdots$&$\cellcolor{blue!10}\vdots$&$\cellcolor{blue!10}\vdots$&$\vdots$&$\cellcolor{blue!10}\vdots$\\
$\microcosm{m}_{k}$ & \cellcolor{blue!10}$\textsc{d}_{k}$ & \cellcolor{blue!10}$\textsc{n}_{k}$& $\textsc{co-n}_{k}$ & \cellcolor{blue!10}$\textsc{p}_{k}$ \\
$\vdots$&\cellcolor{blue!10}$\vdots$&\cellcolor{blue!10}$\vdots$&$\vdots$&\cellcolor{blue!10}$\vdots$\\
$\microcosm{m}_{\infty}$ & \cellcolor{blue!10}\Logspace & \cellcolor{blue!10}\NLogspace& \coNLogspace & \cellcolor{blue!10}\PLogspace \\
\hline
$\microcosm{n}$ & \cellcolor{blue!10}\Ptime & \cellcolor{blue!10}\Ptime& \cellcolor{blue!10}\Ptime & \cellcolor{blue!10}\PPtime \\
\hline
\end{tabular}
\end{center}
\caption{Known semantic characterisations of predicate complexity classes. Our contributions are shown in blue cells.\newline \small $\textsc{d}_{k}$ (resp. $\textsc{n}_{k}$, $\textsc{p}_{k}$) is the languages decided by two-way k-heads (resp. nondeterministic, probabilistic) automata.}\label{contribs}
\end{figure*}
It is important to understand that this approach can be understood both from a logical and a computability point of view. While coming from models of linear logic, the techniques relate to \emph{Implicit Computational Complexity} ({\sc icc}\xspace) and can be seen from this perspective as a semantic variant of constraint linear logics. However the modelling of programs as graphings go beyond the usual scope of the Curry-Howard correspondence, allowing for instance the sound representation of {\sc pram}\xspace machines\footnote{Although the cited work shows how to interpret some algebraic variant of {\sc pram}s\xspace, it should be clear to the reader that the techniques used apply immediately to the usual notion of {\sc pram}s\xspace.} \cite{seiller-pramsLB}. From this perspective, one could argue for this approach as a "computing with dynamical systems", with the underlying belief that any computation might be represented as such.
\section{Interaction Graphs and Complexity}
Interaction Graphs models of linear logic were developed in order to generalise Girard's geometry of interaction constructions to account for \emph{quantitative} aspects, in particular adapting to non-deterministic and probabilistic settings. The aim of the {\sc goi}\xspace (and hence the {\sc ig}\xspace) approach is to obtain a \emph{dynamic} model of proofs and their cut-elimination procedure.
\subsection{Graphings, Execution and Measurement}
In the general setting, graphings can be defined in three different flavours: discrete, topological and measurable. In this paper, we will be working with measurable graphings, and will refer to them simply as \emph{graphings}. Graphings act on a chosen space (hence here, on a measured space); the definition of graphings makes sense for any measured space $\measured{X}$, and under some mild assumptions on $\measured{X}$ it provides a model of (at least) Multiplicative-Additive Linear Logic (\mall) \cite{seiller-goig,seiller-goie,seiller-goif}. We now fix the measure space of interest in this paper.
\begin{definition}[The Space]
We define the measure space $\measure{X}=\realN\times[0,1]^{\naturalN}\times\{\star,0,1\}^{\naturalN}$ where $\realN\times[0,1]^{\naturalN}$ is considered with its usual Borel $\sigma$-algebra and Lebesgue measure. The space $\{\star,0,1\}^{\naturalN}$ is endowed with the natural topology\footnote{I.e. the topology induced by basic \emph{cylindrical} open sets $V(w)=\{ f:\naturalN\rightarrow\{0,1\} \mid \forall i\in [\lg{w}], f(i)=w_i\}$ where $w$ is a finite word on $\{0,1\}$.}, the corresponding Borel $\sigma$-algebra and the natural measure given by $\mu(V(w))=3^{-\lg{w}}$.
\end{definition}
Borrowing the notation introduced in previous work \cite{seiller-goif,seiller-goinda}, we denote by $(x,\seq{s},\seq{\pi})$ the points in $\measure{X}$, where $\seq{s}$ and $\seq{\pi}$ are sequences for which we allow a concatenation-based notation, e.g.\ we write $(a,b)\cdot \seq{s}$ for the sequences whose first two elements are $a,b$ (and we abusively write $a\cdot\seq{s}$ instead of $(a)\cdot\seq{s}$). Given a permutation $\sigma$ over the natural numbers, we write $\sigma(\seq{s})$ the result of its natural action on the $\naturalN$-indexed list $\seq{s}$.
Graphings are then defined as objects acting on the measured space $\measured{X}$. A parameter in the construction allows one to consider subsets of graphings based on \emph{how} they act on the space. To do this, we fix a monoid of measurable maps that we call a \emph{microcosm}. Again, while graphings can be defined in full generality, some conditions on the chosen microcosms are needed to construct models of \mall. The following microcosms, which are of of interest in this paper, do satisfy these additional requirements.
\begin{definition}[Microcosms]
For all integer $i\geqslant 1$, we consider the translations
\[ \ttrm{t}_{z}:(x,\seq{s},\seq{\pi})\mapsto(x+z,\seq{s},\seq{\pi}) \]
for all integer $z$, and the permutations
\[ \ttrm{p}_{\sigma}:(x,\seq{s},\seq{\pi})\mapsto(x,\sigma(\seq{s}),\seq{\pi}) \]
for all bijection $\sigma:\naturalN\rightarrow\naturalN$ such that $\sigma(k)=k$ for all $k>i$.
We denote by $\microcosm{m}_i$ the monoid generated by those maps, and by $\microcosm{m}_{\infty}$ the union $\cup_{i>1}\microcosm{m}_{i}$.
We also consider the maps:
\[ {\tt pop}: (x,\seq{s},c\cdot\seq{pi})\mapsto (x,\seq{s},\seq{pi}) \]
\[ {\tt push}_{0}: (x,\seq{s},\seq{pi})\mapsto (x,\seq{s},0\cdot\seq{pi}) \]
\[ {\tt push}_{1}: (x,\seq{s},\seq{pi})\mapsto (x,\seq{s},1\cdot\seq{pi}) \]
\[ {\tt push}_{\star}: (x,\seq{s},\seq{pi})\mapsto (x,\seq{s},\star\cdot\seq{pi}) \]
We denote by $\microcosm{n}_i$ the monoid generated the microcosm $\microcosm{m}_i$ extended by those three maps, and by $\microcosm{n}_{\infty}$ the union $\cup_{i>1}\microcosm{n}_{i}$.
Finally, let us denote by $a\bar{+}b$ the fractional part of the sum $a+b$.
We also define the microcosms $\bar{\microcosm{m}}_{i}$ (resp. $\bar{\microcosm{n}}_{i}$) as the smallest microcosms containing $\microcosm{m}_{i}$ (resp. $\microcosm{n}_{i}$) and all translations $\ttrm{t}_{\lambda}:(x,a\cdot \seq{s})\mapsto(x, (a\bar{+}\lambda)\cdot\seq{s})$ for $\lambda$ in $[0,1]$.
\end{definition}
We are now able to define graphings. Those are formally defined as quotient of graph-like objects called \emph{graphing representatives}\footnote{In earlier works \cite{seiller-goig}, the author did not introduce separate terminologies for graphings and graphing representatives. While this can be allowed since all operations considered on graphings are compatible with the quotient, we chose here a more pedagogical approach of introducing a clear distinction between those types of objects.}. Graphing representatives are just (countable) families of weighted \emph{edges} defined by a source -- a measurable subset of $\measured{X}\times D$ where $S$ is a fixed set of \emph{states} -- and a \emph{realiser}, i.e. a map in the considered microcosm and a new state in $D$.
As in previous work \cite{seiller-goinda}, we fix the monoid of weights of graphings $\Omega$ to be equal to $[0,1]\times\{0,1\}$ with usual multiplication on the unit interval and the product on $\{0,1\}$. To simplify notations, we write elements of the form $(a,0)$ as $a$ and elements of the form $(a,1)$ as $a\cdot \mathbf{1}$. On this set of weights, we will consider the fixed parameter map $m(x,y)=xy$ in the following (used in \autoref{def:measurement}.
\begin{definition}[Graphing representative]
We fix a measure space $\measure{X}$, a microcosm $\microcosm{m}$, a monoid $\Omega$, a measurable subset $V^{G}$ of $\measure{X}$, and a finite set $S^{G}$. A ($\Omega$-weighted) \emph{$\microcosm{m}$-graphing representative} $G$ of \emph{support} $V^{G}$ and \emph{stateset} $S^{G}$ is a countable set
$$\{(S^{G}_{e},\ttrm{i}^{G}_{e},\ttrm{o}^{G}_{e},\phi^{G}_{e},\omega^{G}_{e})~|~e\in E^{G}\},$$
where $S^{G}_{e}$ is a measurable subset\footnote{As $D^{G}$ is considered as a discrete measure space, a measurable subset of the product is simply a finite collection of measurable subset indexed by elements of $D^{G}$.} of $V^{G}\times D^{G}$, $\phi^{G}_{e}$ is an element of $\microcosm{m}$ such that $\phi_{e}^{G}(S^{G}_{e})\subseteq V^{G}$, $\ttrm{i}^{G}_{e},\ttrm{o}^{G}_{e}$ are elements of $D^{G}$, and $\omega^{G}_{e}\in\Omega$ is a \emph{weight}. We will refer to the indexing set $E^{G}$ as the set of \emph{edges}. For each edge $e\in E^{G}$ the set $S^{G}_{e}\times\{\ttrm{i}^{G}_{e}\}$ is called the source of $e$, and we define the \emph{target} of $e$ as the set $T^{G}_{e}\times\{\ttrm{o}^{G}_{e}\}$ where $T^{G}_{e}=\phi^{G}_{e}(S^{G}_{e})$.
\end{definition}
The notion of graphing representatives captures more than just an action on a space. Indeed, consider a graphing representative $G$ with a single edge of source $S\disjun S'$, weight $w$ and realiser $f$. Then the graphing representative $H$ having two edges of weight $w$, realiser $f$ and respective sources $S$ and $S'$ intuitively represent the same action on $\measured{X}$ as $G$. In other words, the notion of graphing representative captures both the notion of action and a notion of representation of this action. The auuthor therefore defines a notion of \emph{refinement} which allows for the consideration of the equivalence $G\sim H$ that captures the fact that $G$ and $H$ represent the same action.
\begin{definition}[Refinements]
Let $F,G$ be graphing representatives. Then $F$ is a refinement of $G$ -- written $F\leqslant G$ -- if there exists a partition\footnote{We allow the sets $E^{F}_{e}$ to be empty.} $(E^{F}_{e})_{e\in E^{G}}$ of $E^{F}$ s.t. $\forall e\in E^{G}, \forall f,f'\in E^{F}_{e}$:
\begin{itemize}[noitemsep,nolistsep]
\item $\omega^{F}_{f}=\omega_{e}^{G}\textrm{ and }\phi^{F}_{f}=\phi_{e}^{G}$;
\item $\cup_{f\in E^{F}_{e}} S^{F}_{f} =_{a.e.} S^{G}_{e}\textrm{ and }f\neq f' \Rightarrow \mu(S^{F}_{f} \cap S^{F}_{f'})=0$.
\end{itemize}
\end{definition}
Then two graphing representatives are \emph{equivalent} if and only if they possess a common refinement. The actual notion of \emph{graphing} is then an equivalence class of the objects just defined w.r.t. this equivalence.
\begin{definition}[Graphing]
A \emph{graphing} is an equivalence class of graphing representatives w.r.t. the induced equivalence:
$$F\sim G\Leftrightarrow \exists H,~H\leqslant F\textrm{ and }H\leqslant G$$
\end{definition}
Since all operations considered on graphings were shown to be compatible with this quotienting \cite{seiller-goig}, i.e.\ well defined on the equivalence classes, we will in general make no distinction between a graphing -- as an equivalence class -- and a graphing representative belonging to this equivalence class.
\begin{definition}[Execution]\label{def:execution}
Let $F$ and $G$ be graphings such that $V^{F}=V\disjun C$ and $V^{G}=C\disjun W$ with $V\cap W$ of null measure. Their \emph{execution} $F\plug G$ is the graphing of support $V\disjun W$ and stateset $D^{F}\times D^{G}$ defined as the set of all\footnote{We refer to the author's work on graphing for the full definition. Intuitively, if $\pi$ is a path, then $[\pi]_{o}^{o}(C)$ denotes the restriction $\pi$ to the points in the source that lie outside the set $C$ and are mapped by the realiser of the path to an element outside of $C$.} $[\pi]_{o}^{o}(C)$ where $\pi$ is an element of $\altpath{F,G}$, the set of alternating path between $F$ and $G$, i.e. the set of paths $\pi=e_1e_2\dots e_n$ such that for all $i=1,\dots,n-1$, $e_i\in F$ iff $e_{i+1}\in G$.
$$
\begin{array}{rcl} F\plug G&=&\left\{\left(S^{\decoupe C}_{\pi},(\ttrm{i}_{e_{1}},\ttrm{i}_{e_{2}}),(\ttrm{o}_{e_{n-1}},\ttrm{o}_{e_{n}}),\phi_{\pi},\omega_{\pi}\right)\right.\\
&&\hspace{2em}\left.|~\pi= e_{1},e_{2},\dots,e_{n}\in\altpath{F,G}\right\}
\end{array}
$$
\end{definition}
\subsection{Measurement, Proofs and Types}
We now recall the notion of measurement. Since in the specific case that will be of interest to us, i.e. when restricting to the microcosms considered in this paper, the expression of the measurement can be simplified, we only give this simple expression and point the curious reader to earlier work for the general definition \cite{seiller-goig}.
\begin{definition}\label{def:measurement}
The measurement between two graphings (realised by measure-preserving maps) is defined as
$$\meas[]{F,G}=\sum_{\pi\in\repcirc{F,G}} \int_{\supp{\pi}} \frac{m(\omega(\pi)^{\rho_{\phi_{\pi}}(x))}}{\rho_{\phi_{\pi}}(x)}d\lambda(x)$$
where $\rho_{\phi_{\pi}(x)}=\inf\{n\in\naturalN~|~ \phi_{\pi}^{n}(x)=x\}$ (here $\inf\emptyset=\infty$), and the support
$\supp{\pi}$ of $\pi$ is the set of points $x$ belonging to a finite orbit \cite[Definition 41]{seiller-goig}.\end{definition}
The measurement is used to define linear negation. But first, let us recall the notion of \emph{project} which is the semantic equivalent of \emph{proofs}.
\begin{definition}
A project of support $V$ is a pair $(a,A)$ of a real number $a$ and a finite formal sum $A=\sum_{i\in I} \alpha_{i}A_{i}$ where for all $i\in I$, $\alpha_{i}\in\realN$ and $A_{i}$ is a graphing of support $V$.
\end{definition}
We can then define an \emph{orthogonality relation} on the set of projects. Orthogonality captures the notion of linear negation and somehow translates the correctness criterion for proof nets. Its definition is based on the measurement defined above, extended to formal weighted sums of graphings by \enquote{linearity} \cite{seiller-goiadd,seiller-goig}.
\begin{definition}
Two projects $(a,A)$ and $(b,B)$ are orthogonal -- written $(a,A)\poll{}(b,B)$ -- when they have equal support and $\meas[]{(a,A),(b,B)}\neq 0,\infty$. We also define the orthogonal of a set $E$ as $E^{\pol}=\{(b,B): \forall (a,A)\in A, (a,A)\poll{}(b,B)\}$ and write $E^{\pol\pol}$ the double-orthogonal $(E^{\pol})^{\pol}$.
\end{definition}
Orthogonality allows for a definition of types. In fact the models are defined based on two notions of types --\emph{conducts} and \emph{behaviours} \cite{seiller-goiadd}. Conducts are simple to define but while their definition is enough to define a model of multiplicative linear logic, dealing with additives requires the more refined notion of \emph{behaviour}.
\begin{definition}
A \emph{conduct} of support $V^{A}$ is a set $\cond{A}$ of projects of support $V^{A}$ such that $\cond{A}=\cond{A}^{\pol\pol}$. A \emph{behaviour} is a conduct such that for all $(a,A)$ in $\cond{A}$ (resp. $\cond{A}^{\pol}$) and for all $\lambda\in\realN$, $(a,A+\lambda\emptyset)$ belongs to $\cond{A}$ (resp. $\cond{A}^{\pol}$) as well. If both $\cond{A}$ and $\cond{A}^{\pol}$ are non-empty, we say $\cond{A}$ is \emph{proper}.
\end{definition}
Conducts provide a model of Multiplicative Linear Logic. The connectives $\otimes,\multimap$ are defined as follows: if $\cond{A}$ and $\cond{B}$ are conducts of disjoint supports $V^{A}, V^{B}$, i.e. $V^{A}\cap V^{B}$ is of null measure, then:
\[
\begin{array}{rcl}
\cond{A\otimes B}&=&\{\de{a\plug b}~|~\de{a}\in\cond{A},\de{b}\in\cond{B}\}^{\pol\pol}\\
\cond{A\multimap B}&=&\{\de{f}~|~\forall \de{a}\in\cond{A},\de{f\plug b}\in\cond{B}\}.
\end{array}
\]
However, to define additive connectives, one has to restrict the model to behaviours. In this paper, we will deal almost exclusively with proper behaviours. Based on the following proposition, we will therefore consider mostly projects of the form $(0,L)$ which we abusively identify with the underlying sliced graphing $L$. Moreover, we will use the term \enquote{behaviour} in place of \enquote{proper behaviour}.
\begin{proposition}[{\cite[Proposition 60]{seiller-goiadd}}]
If $\cond{A}$ is a proper behaviour, $(a,A)\in\cond{A}$ implies $a=0$.
\end{proposition}
Finally, let us mention the fundamental theorem for the interaction graphs construction in the restricted case we just exposed\footnote{The general construction allows for other sets of weights as well as whole families of measurements \cite{seiller-goig}.}.
\begin{theorem}[{\cite[Theorem 1]{seiller-goig}}]\label{thm:mallmodels}
For any microcosm $\microcosm{m}$, the set of behaviours provides a model of Multiplicative-Additive Linear Logic (\MALL) without multiplicative units.
\end{theorem}
\subsection{Integers, Machines and Complexity}
We now recall some definitions introduced in previous work by the author characterising complexity classes by use of graphings \cite{seiller-goinda}; interested readers will find there (and in some references therein \cite{seiller-conl,seiller-lsp}) more detailed explanations of -- and motivations for -- the definitions. In particular the representation of binary words is related to
\[ {\rm BList} := \forall X~ \oc(X\multimap X)\multimap \oc(X\multimap X)\multimap \oc(X\multimap X), \]
the type of binary lists in Elementary Linear Logic \cite{LLL,danosjoinet}.
\begin{notation}
To ease notations, we only consider words over $\Sigma=\{0,1\}$ in this paper. We write $\ext{\Sigma}$ the set $\{0,1,\star\}\times\{\In,\Out\}$. We also denote by $\vertices{\Sigma}$ the set $\ext{\Sigma}\cup\{\ttrm{a},\ttrm{r}\}$, where $\ttrm{a}$ (resp. $\ttrm{r}$) stand for $\accept$ (resp. $\reject$).
Initial segments of the natural numbers $\{0,1,\dots,n\}$ are denoted $[n]$. Up to renaming, all statesets can be considered to be of this form.
\end{notation}
\begin{notation}
As in the previous paper \cite{seiller-goinda}, we fix once and for all an injection $\Psi$ from the set $\vertices{\Sigma}$ to intervals in $\realN$ of the form $[k,k+1]$ with $k$ an integer. For all $f\in\vertices{\Sigma}$ and $Y$ a measurable subset of $[0,1]^{\naturalN}$, we denote by $\bracket{f}_{Y}^{Z}$ the measurable subset $\Psi(f)\times Y\times Z$ of $\measure{X}$, where $Y\subset[0,1]^{\naturalN}$ and $Z\subset\{\star,0,1\}^{\naturalN}$. If $Y=[0,1]^{\naturalN}$ (resp. $Z=\{\star,0,1\}^{\naturalN}$), we omit the subscript (resp. superscript). The notation is extended to subsets $S\subset\vertices{\Sigma}$ by $\support{S}=\cup_{f\in S}\support{f}$ (a disjoint union).
\end{notation}
Given a word $\word{w}=\star a_{1}a_{2}\dots a_{k}$, we denote $\graphtw{w}$ the graph with set of vertices $V^{\graphtw{w}}\times D^{\graphtw{w}}$, set of edges $E^{\graphtw{w}}$, source map $s^{\graphtw{w}}$ and target map $t^{\graphtw{w}}$ respectively defined as follows:
\begin{equation*}
\begin{array}{l}
V^{\graphtw{w}}=\ext{\Sigma} \hspace{2em} D^{\graphtw{w}}=[k] \hspace{2em} E^{\graphtw{w}}=\{r,l\}\times[k]\\
\begin{array}{rcl}
s^{\graphtw{w}}&=&(r,i)\mapsto (a_{i},\Out,i)\\
&&(l,i)\mapsto (a_{i}, \In,i)\\
t^{\graphtw{w}}&=&(r,i)\mapsto (a_{i+1},\In,i+1 \textnormal{ mod } k+1)\\
&&(l,i)\mapsto (a_{i-1}, \Out,i-1 \textnormal{ mod } k+1)
\end{array}
\end{array}
\end{equation*}
This graph is the discrete representation of $\word{w}$. Detailed explanations on how these graphs relate to the proofs of the sequent $\vdash {\rm BList}$ can be found in earlier work \cite{seiller-phd,seiller-conl}.
\begin{definition}
Let $\word{w}$ be a word $\word{w}=\star a_{1}a_{2}\dots a_{k}$ over the alphabet $\Sigma$. We define the \emph{word graphing} $\graphingtw{w}$ of support $\wordsupport$ and stateset $D^{\graphtw{w}}$ by the set of edges $E^{\graphtw{w}}$ and for all edge $e$:
\[\{(\bracket{f},i,j,\phi_{f,i}^{g,j},1)\mid e\in E^{\graphtw{w}}\}, \]
where $s^{\graphtw{w}}(e)=(f,i)$, $t^{\graphtw{w}}(e)=(g,j)$, and $\phi_{f,i}^{g,j}:(\bracket{f},x,i)\mapsto (\bracket{g},x,j)\}$.
\end{definition}
\begin{notation}
We write $\reptw{w}$ the set of word graphings for $\word{w}$. It is defined as the
set of graphings obtained by renaming the stateset $D^{\graphtw{w}}$ w.r.t. an injection $[k]\rightarrow [n]$
\end{notation}
\begin{definition}
Given a word $\word{w}$, a \emph{representation of $\word{w}$} is a graphing $\oc L$ where $L$ belongs to $\reptw{w}$. The set of representations of words in $\Sigma$ is denoted $\twwprojects$, the set of representations of a specific word $\word{w}$ is denoted $\repany{w}$.
We then define the conduct $\oc\ListType=\cond{(\twwprojects)^{\pol{}\pol{}}}$.
\end{definition}
\begin{definition}
We define the (unproper) behaviour $\NBool$ as $\cond{T}_{\resultsupport}$, where for all measurable sets $V$ the behaviour $\cond{T}_{V}$ is defined as the set of all projects of support $V$.
For all microcosms $\microcosm{m}$, we define $\pred{m}$ as the set of $\microcosm{m}$-graphings in $\oc\ListType\multimap\NBool$.
\end{definition}
\begin{definition}
An $\microcosm{m}$-graphing $G$ is \emph{finite} when it has a representative $H$ whose set of edges $E^{H}$ is finite.
\end{definition}
\begin{definition}
A \emph{nondeterministic predicate $\microcosm{m}$-machine} over the alphabet $\Sigma$ is a finite $\microcosm{m}$-graphing belonging to $\pred{m}$ with all weights equal to $1$.
\end{definition}
The computation of a given machine given an argument is represented by the \emph{execution}, i.e. the computation of paths defined in \autoref{def:execution}. The result of the execution is an element of $\NBool$, i.e.\ somehow a generalised boolean value\footnote{If one were working with \enquote{deterministic machines} \cite{seiller-towards}, it would belong to the subtype $\Bool$ of booleans.}.
\begin{definition}[Computation]
Let $M$ be a $\microcosm{m}$-machine, $\word{w}$ a word over the alphabet $\Sigma$ and $\oc L\in\oc\ListType$. The \emph{computation} of $\de{M}$ over $\oc L$ is defined as the graphing $M\plug \oc L\in\NBool$.
\end{definition}
\begin{definition}[Tests]\label{def:tests}
A \emph{test} is a family $\testfont{T}=\{\de{t}_{i}=(t_{i},T_{i})~|~i\in I\}$ of projects of support $\resultsupport$
\end{definition}
We now want to define the language characterised by a machine. For this, one could consider \emph{existential} $\mathcal{L}_{\exists}^{\testfont{T}}(M)$ and \emph{universal} $\mathcal{L}_{\forall}^{\testfont{T}}(M)$ languages for a machine $M$ w.r.t. a test $\testfont{T}$:
$$
\begin{array}{rcl}
\mathcal{L}_{\exists}^{\testfont{T}}(M)&=&\{\word{w}\in\Sigma^{\ast}~|~\forall \de{t}_{i}\in\testfont{T}, \exists \de{w}\in\repany{w}, M\plug \de{w}\poll{} \de{t}_{i}\}\\
\mathcal{L}_{\forall}^{\testfont{T}}(M)&=&\{\word{w}\in\Sigma^{\ast}~|~\forall \de{t}_{i}\in\testfont{T}, \forall \de{w}\in\repany{w}, M\plug \de{w}\poll{} \de{t}_{i}\}
\end{array}
$$
We now introduce the notion of uniformity, which describes a situation where both definitions above coincide. This collapse of definitions is of particular interest because it ensures that both of the following problems are easy to solve:
\begin{itemize}[nolistsep,noitemsep]
\item whether a word belongs to the language: from the existential definition one only needs to consider one representation of the word;
\item whether a word does not belong to the language: from the universal definition, one needs to consider only one representation of the word.
\end{itemize}
\begin{definition}[Uniformity]
Let $\microcosm{m}$ be a microcosm. The test $\testfont{T}$ is said \emph{uniform} w.r.t. $\microcosm{m}$-machines if for all such machine $M$, and any two elements $\de{w},\de{w'}$ in $\repany{w}$:
$$M\plug \de{w}\in \testfont{T}^{\pol} \text{ if and only if }M\plug \de{w}'\in \testfont{T}^{\pol}$$
Given a $\microcosm{m}$-machine $M$, we write in this case $\mathcal{L}^{\testfont{T}}(M)=\mathcal{L}_{\exists}^{\testfont{T}}(M)=\mathcal{L}_{\forall}^{\testfont{T}}(M)$.
\end{definition}
\begin{definition}
For $U\subset\mathbf{X}$, we define $\identity[U]$ as the graphing with a single edge and stateset $[0]$: $\{(\support{\ttrm{r}},0,0,x\mapsto x,1\cdot\mathbf{1})\}$.
\end{definition}
\begin{definition}
We define the test $\testdetneg$ as the set consisting of the projects $\de{t}^{-}_{\zeta}=(\zeta,\identity[\support{\ttrm{r}}])$, where $\zeta\neq0$.
\end{definition}
\begin{proposition}
The test $\testdetneg$ is uniform w.r.t. $\microcosm{m}_{\infty}$-machines.
\end{proposition}
\begin{definition}
We define the complexity class $\predcondet{m}$ as the set
\[ \{\mathcal{L}^{\testdetneg}(M)\mid M\text{ $\microcosm{m}$-machine} \}. \]
\end{definition}
We recall the main theorem of the author's previous paper \cite{seiller-goinda}, and refer to \autoref{def:cc} for the definition of the characterised complexity classes.
\begin{theorem}
For all $i\in\naturalN^{\ast}\cup\{\infty\}$, the class $\predcondet{{m}_{i}}$ is equal to \cctwconfa{i}. As particular cases, $\predcondet{{m}_{1}}=\Regular$ and $\predcondet{{m}_{\infty}}=\coNLogspace$.
\end{theorem}
\subsection{Characterising $\NLogspace$}
The starting point of this work was the realisation that one can define another test $\testdetpos$ that allows to capture the notion of acceptance in $\NLogspace$. Based on this idea, and using technical lemmas from the previous paper, we can characterise easily the hierarchy of complexity classes defined by $k$-head non-deterministic automata with the standard non-deterministic acceptance condition (i.e. there is at least one accepting run).
\begin{definition}
For $U\subset\mathbf{X}$, we define $\halfidentity[U]$ as the graphing with a single edge and stateset $[0]$: $\{(\support{\ttrm{r}},0,0,x\mapsto x,\frac{1}{2}\cdot\mathbf{1})\}$.
\end{definition}
\begin{definition}
The test $\testdetpos$ is defined as the family:
\[ \{ (0,\halfidentity[\support{\ttrm{a}}_{[0,\frac{1}{n}]^{n}\times[0,1]^{\naturalN}}])\mid n\in\naturalN \}. \]
\end{definition}
\begin{definition}
We define the complexity class $\predndet{m}$ as the set
\[ \{\mathcal{L}^{\testdetpos}(M)\mid M\text{ $\microcosm{m}$-machine in $\nmodel{\{1\}}{m}$} \}. \]
\end{definition}
This test does indeed provide the right characterisation. In fact, the sole element $(0,\halfidentity[\support{\ttrm{a}}])$ is enough to obtain soundness, since the result relies on a result from the author's previous paper \cite[Proposition 46]{seiller-goinda} (which is not stated here, as it is generalised by \autoref{prop:generalised46} below). Thus, a $k$-heads two-way automaton $\automaton{M}$ accepts a word $\word{w}$ if and only if there exists at least one alternating path between the graphing translation $\autograph{M}$ of $M$ and the word representation $\oc \graphingtw{w}$ whose source and target is $\support{\ttrm{a}}_{Y}$. Thus, $\automaton{M}$ accepts $\word{w}$ if and only if there are alternating cycles between $\autograph{M}\plug \oc \graphingtw{w}$ and $\halfidentity[\support{\ttrm{a}}]$, i.e. if and only if $\meas[]{\autograph{M}\plug \oc \graphingtw{w},\halfidentity[\support{\ttrm{a}}]}\neq 0,\infty$.
However, the whole family of tests is needed to obtain completeness. Indeed, in the general case, it might be possible that a $\microcosm{m}_i$-machine $G$ passes the test $\{(0,\halfidentity[\support{\ttrm{a}}])\}$ by taking several paths through the execution $G\plug \oc \graphingtw{w}$ (hence creating a cycle of arbitrary length). In that case, it is not clear that the existence of such a cycle can be decided with some automaton $M$. However, if $G\plug \oc \graphingtw{w}$ passes all tests in $\testdetpos$, it imposes the existence of a cycle of length 2 between $G\plug \oc \graphingtw{w}$ and $\halfidentity[\support{\ttrm{a}}]$, something that can be decided by an automaton.
A simple adaptation of the arguments then provides a proof of the following. We omit the details for the moment, as the next sections will expose a generalisation of the technique that also applies to the probabilistic case and to automata with a pushdown stack. The definitions of the complexity classes involved in the statement are given in \autoref{def:cc}.
\begin{theorem}
For all $i\in\naturalN^{\ast}\cup\{\infty\}$,
\[\predndet{m_{i}}=\text{\cctwnfa{i}}\hspace{2em} \predndet{n_{i}}=\text{\cctwnfastack{i}}.\]
In particular, $\predndet{m_{\infty}}=\Logspace$ and $\predndet{n_{\infty}}=4\Ptime$\footnote{Here we characterise $\Ptime$ and not $\NPtime$ as one may expect because non-determinism for pushdown machines do not add expressivity, as shown by Cook \cite{cookP} using memoization.}
\end{theorem}
\section{Deterministic and Probabilistic Models}
\begin{definition}
A graphing $\graphing{G}=\{S^{G}_{e},\phi^{G}_{e},\omega^{G}_{e}~|~e\in E^{G}\}$ is \emph{deterministic} if all edges have weight equal to $1$ and the following holds:
\[ \mu\left(\left\{x\in \measure{X}~|~ \exists e,f\in E^{G}, e\neq f\text{ and }x\in S_{e}^{G}\cap S_{f}^{G}\right\}\right)=0 \]
\end{definition}
\begin{remark}
The notion is quite natural, and corresponds in the case of graphs to the. requirement the out-degree of all vertices to be less or equal to 1.
\end{remark}
We now prove that the set of deterministic graphings is closed under composition, i.e. if $F,G$ are deterministic graphings, then their execution $F\plug G$ is again a deterministic graphing. This shows that the sets of deterministic and non-deterministic graphings define submodels of $\model{\Omega}{m}$.
\begin{lemma}
The set of deterministic graphings is closed under execution.
\end{lemma}
\begin{proof}
A \emph{deterministic graphing} $F$ satisfies that for every edges $e,f\in E^{F}$, $S^{F}_{e}\cap S^{F}_{f}$ is of null measure. Suppose that the graphing $F\plug G$ is not deterministic. Then there exists a Borel $B$ of non-zero measure and two edges $e,f\in E^{F\plug G}$ such that $B\subset S^{F\plug G}_{e}\cap S^{F\plug G}_{f}$. The edges $e,f$ correspond to paths $\pi_{e}$ and $\pi_{f}$ alternating between $F$ and $G$. It is clear that the first step of these paths belong to the same graphing, say $F$ without loss of generality, because the Borel set $B$ did not belong to the \emph{cut}. Thus $\pi_{e}$ and $\pi_{f}$ can be written $\pi_{e}=f_{0}\pi^{1}_{e}$ and $\pi_{f}=f_{0}\pi^{1}_{f}$. Thus the domains of the paths $\pi^{1}_{e}$ and $\pi_{f}^{1}$ coincide on the Borel set $\phi_{f_{0}}^{F}(B)$ which is of non-zero measure since all maps considered are non-singular. One can then continue the reasoning up to the end of one of the paths and show that they are equal up to this point. Now, if one of the paths ends before the other we have a contradiction because it would mean the the Borel set under consideration would be at the same time inside and outside the cut, which is not possible. So both paths have the same length and are therefore equal. Which shows that $F\plug G$ is deterministic since we have shown that if the domain of two paths alternating between $F$ and $G$ coincide on a non-zero measure Borel set, the two paths are equal (hence they correspond to the same edge in $F\plug G$).
\end{proof}
One can then check that the interpretations of proofs by graphings in earlier papers \cite{seiller-goig,seiller-goie,seiller-goif} are all deterministic. This gives us the following theorem as a corollary of the previous lemma.
\begin{theorem}[Deterministic model]
Let $\Omega$ be a monoid and $\microcosm{m}$ a microcosm. The set of $\Omega$-weighted \emph{deterministic} graphings in $\microcosm{m}$ yields a model, denoted by $\dmodel{\Omega}{m}$, of multiplicative-additive linear logic.
\end{theorem}
\begin{definition}
We define the complexity class $\preddet{m}$ as the set
\[ \{\mathcal{L}^{\testdetpos}(M)\mid M\text{ $\microcosm{m}$-machine in $\dmodel{\{1\}}{m}$} \}. \]
\end{definition}
\subsection{The Probabilistic Model}
One can also consider several other classes of graphings. We explain here the simplest non-classical model one could consider, namely that of \emph{sub-probabilistic graphings}. In order for this notion to be well-defined, one should suppose that the unit interval $[0,1]$ endowed with multiplication is a submonoid of $\Omega$.
\begin{definition}
A graphing $\graphing{G}=\{S^{G}_{e},\phi^{G}_{e},\omega^{G}_{e}~|~e\in E^{G}\}$ is \emph{sub-probabilistic} if all the edges have weight in $[0,1]$ and the following holds:
\[ \mu\left(\left\{x\in \measure{X}~|~ \sum_{e\in E^{G}, x\in S^{G}_{e}}\omega^{G}_{e}>1\right\}\right)=0 \]
\end{definition}
It turns out that this notion of graphing also behaves well under composition, i.e. there exists a \emph{sub-probabilistic} submodel of $\model{\Omega}{m}$, namely the model of \emph{sub-probabilistic graphings}.
\begin{theorem}
The set of sub-probabilistic graphings is closed under execution.
\end{theorem}
\begin{proof}
If the weights of edges in $F$ and $G$ are elements of $[0,1]$, then it is clear that the weights of edges in $F\plug G$ are also elements of $[0,1]$. We therefore only need to check that the second condition is preserved.
Let us denote by $\outset{F\plug G}$ the set of $x\in X$ which are source of paths whose added weight is greater than $1$, and by $\outset{F\cup G}$ the set of $x$ which are source of edges (either in $F$ or $G$) whose added weight is greater than $1$. First, we notice that if $x\in\outset{F\plug G}$ then either $x\in\outset{F\cup G}$, or $x$ is mapped -- through at least one edge -- to an element $y$ which is itself in $\outset{F\cup G}$. To prove this statement, let us write $\outpaths{x}$ (resp. $\outedges{x}$) the set of paths in $F\plug G$ (resp. edges in $F$ or $G$) whose source contain $x$. We know the sum of all the weights of these paths is greater than $1$, i.e. $\sum_{\pi\in\outpaths{x}}\omega(\pi)>1$. But this sum can be rearranged by ordering paths depending on theirs initial edge, i.e. $\sum_{\pi\in\outpaths{x}}\omega(\pi)=\sum_{e\in\outedges{x}}\sum_{\pi=e\rho\in\outpaths{x}^{e}}\omega(\pi)$, where $\outpaths{x}^{e}$ denotes the paths whose first edge is $e$. Now, since the weight of $e$ appears in all $\omega(e\rho)=\omega(e)\omega(\rho)$, we can factorize and obtain the following inequality.
\[ \sum_{e\in\outedges{x}}\omega(e)\left(\sum_{\pi=e\rho\in\outpaths{x}^{e}}\omega(\rho)\right)>1 \]
Since the sum $\sum_{e\in\outedges{x}}\omega(e)$ is not greater than $1$, we deduce that there exists at least one $e\in\outedges{x}$ such that $\sum_{\pi=e\rho\in\outpaths{x}^{e}}\omega(\rho)>1$. However, this means that $\phi_{e}(x)$ is an element of $\outset{F\plug G}$.
Now, we must note that $x$ is not element of a cycle. This is clear from the fact that $x$ lies in the carrier of $F\plug G$.
Then, an induction shows that $x$ is an element of $\outset{F\plug G}$ if and only if there is a (finite, possibly empty) path from $x$ to an element of $\outset{F\cup G}$, i.e. $\outset{F\plug G}$ is at most a countable union of images of the set $\outset{F\cup G}$. But since all maps considered are non-singular, these images of $\outset{F\cup G}$ are negligible subsets since $\outset{F\cup G}$ is itself negligible. This ends the proof as a countable union of copies of negligible sets are negligible (by countable additivity), hence $\outset{F\plug G}$ is negligible.
\end{proof}
\begin{theorem}[Probabilistic model]
Let $\Omega$ be a monoid and $\microcosm{m}$ a microcosm. The set of $\Omega$-weighted \emph{sub-probabilistic} graphings in $\microcosm{m}$ yields a model, which we will denote $\pmodel{\Omega}{m}$, of multiplicative-additive linear logic.
\end{theorem}
We will now show how deterministic and probabilistic complexity classes can be characterised by means of the type of predicates $\pred{m}$ in the deterministic and probabilistic models respectively. We will start by establishing soundness by showing how the computation by automata can be represented by the execution between graphings and word representations.
\section{Soundness}
The proof of the characterisation theorem \cite{seiller-goinda} relies on a representation of multihead automata as graphings. We here generalise the result to probabilistic automata with a pushdown stack. For practical purposes, we consider a variant of the classical notion of probabilistic two-way multihead finite automata with a pushdown stack obtained by:
\begin{itemize}[nolistsep,noitemsep]
\item fixing the right and left end-markers as both being equal to the fixed symbol $\star$;
\item fixing once and for all unique initial, accept and reject states
\item choosing that each transition step moves exactly one of the multiple heads of the automaton;
\item imposing that all heads are repositioned on the left end-marker and the stack is emptied before accepting/rejecting.
\item symbols from the stack are read by performing a ${\tt pop}$ instruction; if the end-of-stack symbol $\star$ is popped, it is pushed on the stack in the next transition.
\end{itemize}
It should be clear that these choices in design have no effect on the sets of languages recognised.
\begin{definition}\label{def:cc}
A \emph{$k$-heads probabilistic two-way multihead finite automata with a pushdown stack} (\cctwpfa{k}) $\automaton{M}$ is defined as a tuple $(\Sigma,Q,\rightarrow)$, where the transition function $\rightarrow$ is a map that associates to each element of $\starred{\Sigma}^{k}\times Q$ a sub-probability distribution over the set $\left(\textrm{Inst}\times Q\right)$ where $\textrm{Inst}$ is the set of instructions: $(\{1,\dots,k\}\times\{\In,\Out\})\times\{\identity,{\tt pop},{\tt push}_{1},{\tt push}_{0},{\tt push}_{\star}\}$.
The set of deterministic (resp. probabilistic) two-way multihead automata with $k$ heads is written $\twdfa{k}$ (resp. $\twpdfa{k}$) and the corresponding complexity class is noted \cctwdfa{k} (resp. \cctwpfa{k}). The set of all deterministic (resp. probabilistic) two-way multihead automata $\cup_{k\geqslant 1}\twdfa{k}$ is denoted by $\twdfas$ (resp $\twpfas$): the corresponding complexity classes \cctwdfa{$\infty$} and \cctwpfa{$\infty$} are known to be equal to $\Logspace$ and $\PLogspace$ \cite{Holzer}.
The set of $k$ heads deterministic (resp. probabilistic) two-way multihead automata with a pushdown stack is written $\twdfastack{k}$ (resp. $\twpdfastack{k}$) and the corresponding complexity class is noted \cctwdfastack{k} (resp. \cctwpfastack{k}). The set of all deterministic (resp. probabilistic) two-way multihead automata with a pushdown stack $\cup_{k\geqslant 1}\twdfastack{k}$ is denoted by $\twdfastacks$ (resp $\twpfastacks$): the corresponding complexity classes \cctwdfastack{$\infty$} and \cctwpfastack{$\infty$} are known to be equal to $\Ptime$ \cite{Macarie} and $\PPtime$.
\end{definition}
We now describe how to extend the author's translation of multihead automata as graphings to the set of all \cctwpfa{k}.
To simplify the definition, we define for all for $\automaton{t}=((\mathaccent"017E{s},q),(i,d',q'))$ the notation $\automaton{t}\in\rightarrow$ to denote that $\rightarrow(\mathaccent"017E{s},q)(i,d',q')>0$, i.e. the probability that the automaton will perform the transition $\automaton{t}$ is non-zero.
The encoding is heavy but the principle is easy to grasp. We use the stateset to keep track of the last values read by the heads, as well as the last popped symbol from the stack. The subtlety is that we also keep track of the permutation of the heads of the machine. Indeed, the graphing representation has the peculiarity that moving one head requires to use a permutation. As a consequence, to keep track of where the heads are at a given point, we store and update a permutation. Lastly, the stack is initiated with the symbol $\star$; this is done by simply restricting the source of the edges from the initial state to the subspace $V(\star)$ of sequences starting with the symbol $\star$.
\begin{definition}
Let $\automaton{M}=(\Sigma,Q,\rightarrow)$ be a \cctwpfa{k}. We define $\autograph{M}$ a graphing in $\microcosm{n}$ with dialect -- set of states -- $Q\times\mathfrak{G}_{k}\times\{\star,0,1\}^{k}\times\{\star,0,1\}$ as follows.
\begin{itemize}
\item each transition of the form $\automaton{t}=((\mathaccent"017E{s},q),(\nu,q'))$ with $q\neq\init$ and $\nu=(i,d')\times\iota$ with $\iota\neq{\tt pop}$ gives rise to a family of edges indexed by a permutation $\sigma$ and an element $u$ of $\{\star,0,1\}$:
\begin{eqnarray*}
\lefteqn{\support{(a,d)}\times\{(q,\sigma,\mathaccent"017E{s},u)\}}\\
&\longrightarrow& \support{(s_{i},d')}\times\{(q',\tau_{1,\sigma(i)}\circ\sigma,\mathaccent"017E{s}[s_{\sigma^{-1}(1)}:=s],u)\},
\end{eqnarray*}
realised by the map $\ttrm{p}_{(1,\sigma(i))}$ together with the adequate map on the stack subspace and the adequate translation on $\integerN$, and of weight $\rightarrow(\mathaccent"017E{s},q)(\nu,q')$;
\item each transition of the form $\automaton{t}=((\mathaccent"017E{s},q),(\nu,q'))$ with $q\neq\init$ and $\nu=(i,d')\times{\tt pop}$ gives rise to a family of edges indexed by a permutation $\sigma$ and an element $u$ of $\{\star,0,1\}$:
\begin{eqnarray*}
\lefteqn{\support{(a,d)}^{V(u)}\times\{(q,\sigma,\mathaccent"017E{s})\}}\\
&\longrightarrow& \support{(s_{i},d')}\times\{(q',\tau_{1,\sigma(i)}\circ\sigma,\mathaccent"017E{s}[s_{\sigma^{-1}(1)}:=s])\},u
\end{eqnarray*}
realised by the map $\ttrm{p}_{(1,\sigma(i))}$ composed with the ${\tt pop}$ map and the adequate translation on $\integerN$, and of weight $\rightarrow(\mathaccent"017E{s},q)(\nu,q')$;
\item each transition of the form $\automaton{t}=((\mathaccent"017E{s},q),(\nu,q'))$ with $q=\init$ and $\nu=(i,d')\times\iota$ with $\iota\neq{\tt pop}$ gives rise to a family of edges indexed by an element $v\in\{\ttrm{a},\ttrm{r}\}$ and an element $u$ of $\{\star,0,1\}$:
\begin{eqnarray*}
\lefteqn{\support{\ttrm{v}}^{V(\star)}\times\{(\textrm{init},\textrm{Id},\mathaccent"017E{\star},u)\}}\\
&\longrightarrow& \support{(s_{i},d')}\times\{(q',\tau_{1,\sigma(i)},\mathaccent"017E{s}[s_{\sigma^{-1}(1)}:=s],u)\},
\end{eqnarray*}
realised by the map $\ttrm{p}_{(1,\sigma(i))}$ together with the adequate map on the stack subspace and the adequate translation on $\integerN$, and of weight $\rightarrow(\mathaccent"017E{s},q)(\nu,q')$;
\item each transition of the form $\automaton{t}=((\mathaccent"017E{s},q),(\nu,q'))$ with $q=\init$ and $\nu=(i,d')\times{\tt pop}$ gives rise to a family of edges indexed by an element $v\in\{\ttrm{a},\ttrm{r}\}$ and an element $u$ of $\{\star,0,1\}$:
\begin{eqnarray*}
\lefteqn{\support{\ttrm{v}}^{V(\star)}\times\{(\textrm{init},\textrm{Id},\mathaccent"017E{\star},u)\}}\\
&\longrightarrow& \support{(s_{i},d')}\times\{(q',\tau_{1,\sigma(i)},\mathaccent"017E{s}[s_{\sigma^{-1}(1)}:=s],u)\},
\end{eqnarray*}
realised by the map $\ttrm{p}_{(1,\sigma(i))}$ together with the ${\tt pop}$ map on the stack subspace and the adequate translation on $\integerN$, and of weight $\rightarrow(\mathaccent"017E{s},q)(\nu,q')$.
\end{itemize}
\end{definition}
We now generalise the key lemma from the previious paper \cite{seiller-goinda}. This result will be essential for all later results stated in this paper. The proof is a simple but lengthy induction.
\begin{proposition}\label{tracespaths}\label{prop:generalised46}
Let $\automaton{M}$ be a $\twpfa{k}$. Alternating paths of odd length between $\autograph{M}$ and $\oc \graphingtw{w}$ of source $\support{\ttrm{a}}_{Y}$ with\footnote{To understand where the subset $Y$ comes from, we refer the reader to the proof of Lemma \ref{technicallemma}.} $Y=[0,\frac{1}{\lg(\word{w})}]^{k}\times[0,1]^{\naturalN}$ are in a weight-preserving bijective correspondence with the non-empty computation traces of $\automaton{M}$ given $\word{w}$ as input.
\end{proposition}
\begin{corollary}\label{pathaccept}
The automaton $\automaton{M}$ accepts $\word{w}$ with probability $p$ if and only if $p$ is equal to the sum of the weights of alternating paths between $\autograph{M}$ and $\oc \graphingtw{w}$ of source and target $\support{\ttrm{a}}$.
\end{corollary}
We now define the probabilistic tests. These will be used to characterise probabilistic classes thanks to the lemma that follows and which relates the probability that a computation accepts with the orthogonality.
\begin{definition}
For $\eta>0$, we define the test $\testprob[\epsilon]$ as the set
\[ (\log(1-\frac{1}{2}.u),\halfidentity\support{\ttrm{a}}^{V(\star^n)}_{[0,\frac{1}{n}]^{n}\times[0,1]^{\naturalN}}])\mid u\in [0,\epsilon], n\in\naturalN \}. \]
\end{definition}
\begin{lemma}
The sum of the weights of alternating paths between $\autograph{M}$ and $\oc \graphingtw{w}$ of source and target $\support{\ttrm{a}}$ is greater than $\epsilon$ if and only if $\autograph{M}\plug\oc \graphingtw{w}\poll \testprob[\epsilon]$.
\end{lemma}
\begin{proof}
Let us write the weights of alternating paths between $\autograph{M}$ and $\oc \graphingtw{w}$ of source and target $\support{\ttrm{a}}$ as $p_0,p_1,\dots, p_k$. We use here a result from the first work on Interaction Graphs \cite{seiller-goim} showing that in the probabilistic case the measurement of two graphs $\meas{G,H}$ is equal to the measurement of the graphs $\meas{\hat{G},\hat{H}}$ where $\hat{.}$ fusion the edges with same source and target into a single edge by summing the weights \cite[Proposition 16]{seiller-goim}. Therefore, $\meas[]{\autograph{M}\plug\oc \graphingtw{w}, \testprob[\eta]}$ is equal to $\eta-\log(1-m(\frac{1}{2}\cdot\mathbf{1}.(\sum p_i)))=\eta-\log(1-\frac{1}{2}(\sum p_i))$. Now, $\sum p_i>\epsilon$ if and only if $1-\frac{1}{2} (\sum p_i)<1-\frac{1}{2}\epsilon$, if and only if $-\log(1-\frac{1}{2} (\sum p_i))>-\log(1-\frac{1}{2}\epsilon)$. I.e. $\sum p_i>\epsilon$ if and only if $\log(1-\frac{1}{2}\epsilon)-\log(1-\frac{1}{2} (\sum p_i))>0$. This gives the result, i.e. $\sum p_i>\epsilon$ if and only if $\autograph{M}\plug\oc \graphingtw{w}\poll (\log(1-\frac{1}{2}.u),\halfidentity\support{\ttrm{a}}_{[0,\frac{1}{n}]^{n}\times[0,1]^{\naturalN}}])$ for all $u\in[0,\epsilon]$.
\end{proof}
\begin{definition}
We define the complexity class $\predprob{m}$ as the set
\[ \{\mathcal{L}^{\testprob[\frac{1}{2}]}(M)\mid M\text{ $\microcosm{m}$-machine in $\pmodel{[0,1]}{m}$} \}. \]
\end{definition}
Using the preceding lemma and the definitions of the complexity classes, we obtain the following theorem.
\begin{theorem}
For all $i\in\naturalN^{\ast}\cup\{\infty\}$,
\[\text{\cctwdfa{i}}\subseteq\preddet{m_{i}}\hspace{2em} \text{\cctwdfastack{i}}\subseteq\preddet{n_{i}}\]
\[\text{\cctwpfa{i}}\subseteq\predprob{m_{i}}\hspace{2em} \text{\cctwpfastack{i}}\subseteq\predprob{n_{i}}\]
\end{theorem}
\section{Completeness}
We here generalise a technical lemma from the previous paper \cite[Lemma4.14]{seiller-goinda} to include probabilities and pushdown stacks. The principle is the following. By the author's proof, the computation of a $\microcosm{m}_i$-machine given an input $w$ can be simulated by a computation of paths between finite graphs. This can be extended with probabilistic weights in a straightforward manner. Now, the operations on stacks could be thought of as breaking this result, since stacks are arbitrarily long. However, this can be dealt with by considering weight within the monoid $\Theta$ generated by $\{0,1,\star,c\}$ and the relations $c0=c1=c\star=\epsilon$ where epsilon is the empty sequence, thus the neutral element of $\Theta$. We will thus obtain that the computation of a $\microcosm{n}_i$-machine given an input $w$ can be simulated by a computation of paths between finite graphs with weights in $\Omega\times\Theta$.
\begin{lemma}[Technical Lemma]\label{technicallemma}
Let $M$ be a $\microcosm{n}_{\infty}$-machine. The computation of $M$ with a representation $\oc W$ of a word $\word{w}$ is equivalent to the execution of a finite\footnote{Whose size depends on both the length of $\word{w}$ and the smallest $k$ such that $M$ is a $\microcosm{m}_{k}$ machine and (TODO).} $\Omega\times\Theta$-weighted graph $\bar{M}$ and the graph representation $\graphtw{w}$ of $\word{w}$.
\end{lemma}
\begin{proof}
The proof of this lemma follows the proof of the restricted case provided in earlier work \cite{seiller-goinda}. Based on the finiteness of $\microcosm{n}_{\infty}$-machines, there exists an integer $N$ such that $M$ is a $\microcosm{n}_{N}$-machine. We now pick a word $\word{w}\in\Sigma^{k}$ and $(0,W_{\word{w}})$ the project $(0,\oc \graphingtw{w})$. All maps realising edges in $M$ or in $\oc \graphingtw{w}$ are of the form $\phi\times\identity[\bigtimes_{i=N+1}^{\infty}{[0,1]}]\times \psi$ -- i.e. they are the identity on copies of $[0,1]$ indexed by natural numbers $>N$. So we can consider the underlying space to be of the form $\integerN\times[0,1]^{N}\times\{\star,0,1\}^{\naturalN}$ instead of $\measure{X}$ by just replacing realisers $\phi\times\psi$. Moreover, the maps $\phi$ here act either as permutations over copies of $[0,1]$ (realisers of edges of $M$) or as permutations over a decomposition of $[0,1]$ into $k$ intervals (realisers of $\oc \graphingtw{w}$). Consequently, all $\phi$ act as permutations over the set of $N$-cubes $\{\bigtimes_{i=1}^{N}[k_i/k,(k_{i}+1)/k]~|~0\leqslant k_i\leqslant k-1\}$, i.e. their restrictions to $N$-cubes are translations.
Based on this, one can build two (thick\footnote{Thick graphs are graphs with dialects, where dialects act as they do in graphings, i.e.\ as control states.}) graphs $\bar{M}$ and $\bar{W}_{\word{w}}$ over the set of vertices $\ext{\Sigma}\times\{\bigtimes_{i=1}^{N}[k_i/k,(k_{i}+1)/k]~|~0\leqslant k_i\leqslant k-1\}$ as in the proof of the restricted lemma \cite{seiller-goinda}. The only difference is that we keep track of weights and encode the stack operations as elements of $\Theta$ (we use the identification: $[[{\tt push}_{1}]]=1$, $[[{\tt push}_{0}]]=0$, $[[{\tt push}_{\star}]]=\star$, $[[{\tt pop}]]=c$). There is an edge in $\bar{M}$ of source $(s,(k_{i})_{i=1}^{N},d)$ to $(s',(k'_{i})_{i=1}^{N},d')$ and weight $(p,[[\psi]])$ if and only if there is an edge in $M$ of source $\bracket{s}\times\{d\}$ and target $\bracket{s'}\times\{d'\}$, of weight $p$ and whose realisation is $\phi\times\psi$ where $\phi$ sends the $N$-cube $\bigtimes_{i=1}^{N}[k_i/k,(k_{i}+1)/k]$ onto the $N$-cube $\bigtimes_{i=1}^{N}[k'_i/k,(k'_{i}+1)/k]$. There is an edge (of weight $(1,\epsilon)$) in $\bar{W}_{\word{w}}$ of source $(s,(k_{i})_{i=1}^{N},d)$ to $(s',(k'_{i})_{i=1}^{N},d')$ if and only if $d=d'$, $k_i=k'_i$ for $i\geqslant 2$ and there is an edge in $W_{\word{w}}$ of source $\bracket{s}\times[k_1/k,(k_1+1)/k]\times[0,1]^{\naturalN}$ and target $\bracket{s'}\times[k'_1/k,(k'_1+1)/k]\times[0,1]^{\naturalN}$.
Then one checks that there exists an alternating path between $M$ and $\oc \graphingtw{w}$ of weight $p$ and whose stack operation is equal to $\psi$ if and only if there exists an alternating path between $\bar{M}$ and $\graphtw{w}$ of weight $(p,[[\psi]])$.
\end{proof}
This lemma will be useful because of the following proposition \cite[Proposition 4.15]{seiller-goinda}.
\begin{proposition}\label{orthogonalityandcycles}
For any $\microcosm{n}_\infty$-machine $M$ and word representation $\oc W$, $M\plug\oc W$ is orthogonal to $\testprob[\epsilon]$ if and only if the sum of the weights in $\Omega$ of alternating paths of $\Theta$-weight $\epsilon$ between $M$ and $\oc W\otimes \identity[\support{\ttrm{a}}]$ from $\support{\ttrm{a}}$ to itself is greater than $\epsilon$.
\end{proposition}
\begin{proof}
Using the \emph{trefoil property} for graphings \cite{seiller-goig}, which in this case becomes
\[\meas{(0,M)\plug(0,\oc W),(t,T)}=\meas{(0,M),(0,\oc W)\plug(t,T)}.\]
Since the support of $\oc W$ and the test are disjoint, we have the equality $(0,\oc W)\plug(t,T)=(0,\oc W)\otimes(t,T)$. Hence $M\plug\oc W$ is orthogonal to $\testprob[\epsilon]$ if and only if $M$ is orthogonal to $(0,\oc W)\otimes (\log(1-\frac{1}{2}.u),\halfidentity\support{\ttrm{a}}^{V(\star^n)}_{[0,\frac{1}{n}]^{n}\times[0,1]^{\naturalN}}])$ for all $u\in [0,\epsilon]$.
Thus $M\plug\oc W$ is orthogonal to $\testdetneg$ if and only if $\log(1-\frac{1}{2}.u)+\meas{M,\oc W\otimes \halfidentity\support{\ttrm{a}}^{V(\star^n)}_{[0,\frac{1}{n}]^{n}\times[0,1]^{\naturalN}}]}\neq 0,\infty$ for all $u\in[0,\epsilon]$, i.e. if and only if there are no alternating cycles between $M$ and $\oc W\otimes \halfidentity\support{\ttrm{a}}^{V(\star^n)}_{[0,\frac{1}{n}]^{n}\times[0,1]^{\naturalN}}]$ of weight $a\cdot \mathbf{1}$ with $a\leqslant \epsilon$. Moreover, since no weights in $M$ and $\oc W$ are equal to $\lambda\cdot\mathbf{1}$, such cycles need to go through $\support{\ttrm{r}}$.
\end{proof}
We will now define an automaton that will compute the same language as a given $\microcosm{n}_{\infty}$-machine $M$. Notice one subtlety here: a $\microcosm{n}_i$ machine can use the ${\tt push}_{\star}$ instruction at any given moment. Thus the automata to be defined works with a ternary stack -- over the alphabet $\{\star,0,1\}$ -- and not a binary one. This is fine because from any automaton with a ternary stack one can define an automaton on a binary stack recognising the same language (very naively, using a representation of the ternary alphabet as words of length 2, this simply multiplies the number of state by a factor of 2).
Now, a key element in the result which has not yet appeared is that the shrinking of the support of the tests as $n$ grows implies that the cycles considered in the previous proof need to go through the first $N$-cube in the finite graphs $\bar{M}$ and $\graphtw{w}$. The same trick implies that the stack is emptied during the path, i.e. the weight of the path alternating between the finite graphs $\bar{M}$ and $\graphtw{w}$ is required to be equal to $(p,c^i)$. All in all, for any $\microcosm{n}_\infty$-machine $M$ and word representation $\oc W$, $M\plug\oc W$ is orthogonal to $\testprob[\epsilon]$ if and only if there exists a path of weight $(p,c^i)$ with $p>\epsilon$ from $\support{a}_{\bigtimes_{i=1}^{N}[0,1/k]}$ -- the first $N$-cube on $\ttrm{a}$ -- to itself. It is then easy to define an automata $\autograph{M}$ that computes the same language as $M$ by simply following the transitions of $\bar{M}$, and check that this automata accepts a word $w$ with probability $p$ if and only if there is a path of weight $(p,c^i)$ with $p>\epsilon$ from $\support{a}_{\bigtimes_{i=1}^{N}[0,1/k]}$ -- the first $N$-cube on $\ttrm{a}$ -- to itself. This leads to the following proposition.
\begin{proposition}
Let $G$ be a $\microcosm{n}_i$-machine. The automaton $\autograph{G}$ is such that for all word $w$ and all word representation $!W$ of $w$, the sum of the weights in $\Omega$ of alternating paths of $\Theta$-weight $\epsilon$ between $G$ and $\oc W\otimes \identity[\support{\ttrm{a}}]$ from $\support{\ttrm{a}}$ to itself is greater than $\epsilon$ if and only if $\autograph{G}$ accepts $w$ with probability greater than $\epsilon$.
\end{proposition}
Putting together this result and the main theorem of the last section, we obtain.
\begin{theorem}
For all $i\in\naturalN^{\ast}\cup\{\infty\}$,
\[\preddet{m_{i}}=\text{\cctwdfa{i}}\hspace{2em} \preddet{n_{i}}=\text{\cctwdfastack{i}}\]
\[\predprob{m_{i}}=\text{\cctwpfa{i}}\hspace{2em} \predprob{n_{i}}=\text{\cctwpfastack{i}}\]
\end{theorem}
\begin{corollary}
In particular,
\[ \predprob{m_{\infty}}=\PLogspace,\hspace{2em} \predprob{n_{\infty}}=\PPtime. \]
\end{corollary}
\section{Conclusion and Perspectives}
We have shown how to extend the author's method to capture numerous complexity classes between regular languages and polynomial time, showing how the method applies as well to probabilistic computation. This provides the first examples of implicit characterisations of probabilistic complexity classes. This is however related to bounded error classes, and it will be natural to try and characterise bounded-error classes. In particular, it should be possible to capture both $\BPL$ and $\BPP$ from the present work. We expect to be able to do so using the rich notion of type provided by Interaction graphs models. As an example, let us explain how the characterisations above can be expressed through types in the models.
We can define the language associated to a $\microcosm{m}$-machine $M$ and a test $\testfont{T}$ as a type. Indeed, we say a word $w$ is in the langage defined by $M$ if and only if $M\plug \de{w}\poll \de{t}$ for all $\de{t}\in\testfont{T}$. Using standard properties of the execution and orthogonality \cite{seiller-goig}, this can be rephrased as $M\plug \de{t}\poll \de{w}$. Thus, $M$ defines a set of projects $\{ M\plug \de{t} \mid \de{t}\in\testfont{T}\}$ which \emph{tests} natural numbers, i.e. elements of $\ListType$.
\begin{definition}
Let $M$ be a $\microcosm{m}$-machine and $\testfont{T}$ be a test. We define the type:
\[ \LangType[M]{\testfont{T}}=(\oc \ListType^{\pol}\cup\{M\plug \de{t} \mid \de{t}\in\testfont{T}\})^{\pol} \]
\end{definition}
In fact, $\LangType[M]{\testfont{T}}$ can also be defined as an intersection type, by noting $M(\testfont{T})=\{M\plug \de{t} \mid \de{t}\in\testfont{T}\}$.
\begin{lemma}
\[ \LangType[M]{\testfont{T}} = M(\testfont{T})^{\pol}\cap\oc \ListType \]
\end{lemma}
The type represents a language in the following fashion.
\begin{proposition}
Let $M$ be a $\microcosm{m}$-machine and $\testfont{T}$ be a test.
\[ \de{w}\in \LangType[M]{\testfont{T}} \Leftrightarrow \exists w\in \mathcal{L}^{\testfont{T}}(M), \de{w}\in\repany{w} \]
\end{proposition}
Now, this is particularly interesting when one considers that the model allows for the definition of (linear) \emph{dependent types}. Indeed, if $\cond{A}(\de{u})$ is a family of types (we suppose here that $\de{u}$ ranges over the type $\cond{U}$), the types $\sum_{\de{u}: \cond{U}}\cond{A}(u)$ and $\prod_{\de{u}: \cond{U}}\cond{A}(u)$ are well defined:
\[
\begin{array}{rcl}
\sum_{\de{u}: \cond{U}}\cond{A}(u) & = & \{ \de{u}\otimes \de{a}\mid \de{u}\in\cond{U}, \de{a}\in\cond{A}(\de{u}) \}^{\pol\pol}\\
\prod_{\de{u}: \cond{U}}\cond{A}(u) & = & \{ \de{f} \mid \forall \de{u}\in\cond{U}, \de{f\plug u}\in\cond{A}(\de{u}) \}
\end{array}
\]
In the probabilistic model, we can use the following type to characterise\footnote{In fact, this type is more than $\PPtime$ and the latter should be defined as a quotient to identify those $M$ such that $\LangType[M]{\testfont{T}}$..} $\PPtime$:
\[\sum_{M: \oc\ListType\multimap\NBool} \LangType[M]{\testprob[\frac{1}{2}]}.\]
Indeed, we have that:
\[\cond{A}\in\PPtime \Leftrightarrow \exists M: \oc\ListType\multimap\NBool, \cond{A}=\LangType[M]{\testprob[\frac{1}{2}]}. \]
Noting that $\testprob[\frac{1}{2}]$ can be defined as a countable intersection (thus a universal quantification), it is equal to $\forall n\in\naturalN, \testprob[\frac{1}{2}+\frac{1}{n}]$. The above type then becomes:
\[\sum_{M: \oc\ListType\multimap\NBool} \forall n\in\naturalN,~ M(\testprob[\frac{1}{2}+\frac{1}{n}])^{\pol}\cap\oc \ListType.\]
This type could then be used, through a quotient, to represent $\PPtime$ in the model $\pmodel{[0,1]}{n_\infty}$, and $\PLogspace$ in the model $\pmodel{[0,1]}{m_\infty}$.
We expect to provide types characterising bounded error predicates in the same way, providing characterisations of $\BPP$ in the model $\pmodel{[0,1]}{n_\infty}$, and $\BPL$ in the model $\pmodel{[0,1]}{m_\infty}$.
\bibliographystyle{abbrv}
|
1,314,259,996,525 | arxiv | \section{Introduction}
Calculations of x-ray absorption spectra (XAS) at finite temperature
(FT) have been carried out routinely in recent years. These studies range from relatively low temperatures up to a few hundred K (LT),
to the
warm dense matter (WDM) regime at high-temperatures (HT), where $T$
is of order the Fermi temperature $T_F$ \cite{PEYRUSSE2008, RECOULES2009, CHO2011, DORCHIES2015,
ENGELHORN2015, OGITSU2018, BOLIS2019, JOURDAIN2020, ZI2021}. A
standard FT approach is to apply Fermi's golden rule with initial and
final states calculated using conventional density functional theory
(DFT) and Fermi occupation factors.
Since DFT is a ground state theory,
FT quasiparticle corrections to DFT
are essential for HT excited state calculations \cite{FALEEV2006, HOLLEBON2019}.
However, even at extreme temperatures,
e.g., many thousands of K, self-consistent field calculations of the
core-hole state have sometimes been done with ground state
exchange-correlation functionals $\varepsilon_{xc}[\rho]$ \cite{RECOULES2009, CHO2016, DENOEUD2014, JOURDAIN2020}. This
ground-state approximation can be unreliable in that its validity
depends strongly on the system state
and its properties. Some properties
are only weakly sensitive to the temperature dependence of
exchange and correlation, at least for low temperatures
well below the WDM
regime. Others, such as the electrical conductivity and x-ray absorption spectra (XAS), are strongly temperature
dependent \cite{Karasiev2016, KDT16}.
Nevertheless, the use of temperature-dependent
free-energy exchange-correlation functionals $f_{xc}[\rho,T]$ \cite{IIT1987, IIT2017, TANAKA2016,PDW2000, KSDT2014, GDTTFB2017,KDT16} alone ignores effects like inelastic losses. For example,
the electron inelastic scattering effect, which results in an energy-dependent broadening, is often included via a post-processing step by convolution
of the absorption cross-section using an empirical model (e.g. Seah-Dench formalism \cite{XSPECTRA2013,SEAHDENCH1979}), or the imaginary part of the self-energy \cite{CUSHING2018}.
However, the finite temperature dependence of the energy-dependent broadening is typically neglected.
Another common approach for XAS calculations has been the use of
the real-space multiple scattering (RSMS) method, which is also referred to as
the real-space Green's function (RSGF) method \cite{REHR2000}. This approach
is the real-space analog of the Korringa-Kohn-Rostoker (KKR) approach
\cite{KORRINGA1947,KOHN1954,DUPREE1961,BEEBY1967,MORGAN1966}. The
method treats excited quasiparticle states via an energy-dependent self-energy, and
also takes into account the dynamic response of the system to the suddenly
created core-hole. The self-energy can be viewed as an
energy-dependent, non-local analog of the exchange-correlation
potential in DFT \cite{Casida1995}. The FT generalization of the
self-energy can be done formally via the
Matsubara formalism. For example, \citeauthor{BENEDICT2002} used the
approach to investigate the effect of $T$ on the spectral function in
jellium and aluminum, e.g., on optical properties of solid-density
Al. Alternatively, as discussed by Kas et al. \cite{KAS2017}, the FT
self-energy can be calculated using a generalization of the Migdal
approximation \cite{ALLEN1983}, analogous to the FT $GW$ approximation
of Hedin \cite{HEDIN1965}.
Our main goal in this work is to discuss the effects of the
FT GW self-energy on XAS, an approach that heretofore has not been
explored in detail \cite{TAN2021}.
In particular, to facilitate the calculations we introduce a
parametrization of the quasiparticle FT GW self-energy within the
$G_0 W_0$ scheme \cite{MartinReiningCeperley2016}. As illustrations
we apply the approach to the XAS for two systems with $T$ up to 10
eV (i.e. $T$ of order $10^5$ K). Our calculations demonstrate that thermal broadening due to the imaginary part of the self-energy
are significant above $T
\approx 1$ eV. Although lattice vibrations also are strongly
temperature dependent, that behavior is dependent on the lattice
temperature $T_l$ which can differ from the electronic temperature $T$
in non-equilibrium states, as discussed in a previous
work \cite{TAN2021}.
The remainder of the paper is organized as follows. Section
\ref{section:review}.\ provides an brief overview of the real-space Green's
function approach to XAS and its dependence
on the self-energy $\Sigma$. In Section \ref{section:result},
we highlight the FT corrections to XAS with a few examples and
in Section \ref{section:conclusion}, we present a brief summary
and conclusions. Throughout we use Hartree atomic units
$q_e = \hbar = m = 1$, with $q_e=e$ the electron charge. Thus
energies are in Hartree and distances in Bohr,
unless otherwise noted. For temperature we use either K or eV, with 1
eV $\approx$ 11,604 K. Electron densities are expressed in Wigner-Seitz
radii $r_s = (3/4\pi \rho)^{1/3}$.
\section{Theory Summary}
\label{section:review}
\subsection{Finite-temperature X-ray Absorption}
Formally the zero temperature X-ray absorption cross section
is defined via Fermi's golden rule as
\begin{eqnarray}
\sigma(\omega) = 4\pi^2\frac{\omega}{c} \sum_{i,f} |\langle \Psi_i | \hat{\xi}\cdot \mathbf{R} | \Psi_f \rangle|^2 \delta_{\Gamma}(\omega + E_i -E_f),
\end{eqnarray}
where $|\Psi_i\rangle$ and $|\Psi_f\rangle$ are the many-body
initial and final states, $\hat{\xi}$ is the polarization of the
incident photon, and $\mathbf{R}$ is the many-body position
operator. Then within the single-particle (quasiparticle) approximation with dipole interactions and the
sudden approximation, the zero temperature XAS becomes
\begin{eqnarray}
\label{eqn:xas}
\sigma_s(\omega) = 4\pi^2 \frac{\omega}{c} \sum_{i,f} |\langle i | {d} | f \rangle|^2 \delta_{\Gamma}(\omega + \varepsilon_i -\varepsilon_f),
\end{eqnarray}
where $\varepsilon_i$ and $\varepsilon_f$ are the energies of the
quasiparticle initial $|i\rangle$ and final $|f\rangle$ levels and many-body shake-up factors $S_0^2 \approx 1$ are ignored. The
$\delta_\Gamma$ factor denotes a Lorentzian of width $\Gamma$ which
includes both quasiparticle and core-hole lifetime broadening.
Here, the transition operator $ {d=\hat\xi\cdot {\bf r}}=$ is the single-particle
electric dipole operator.
The one-particle states $|i\rangle$ and $|f\rangle$ can be obtained from
Hartree-Fock theory or Kohn-Sham DFT. For the treatment via DFT,
see e.g., Refs.\ \citenum{OANA2013,TAILLEFUMIER2002,GOUGOUSSIS2009}.
For x-ray
absorption, the number of final states $|f\rangle$ required to compute
the dipole matrix element has an impact on the computational efficiency of
evaluating Eq.\ (\ref{eqn:xas}). The present work uses the RSMS approach to alleviate this
bottleneck. In RSMS, we replace the summation over the
final states $|f\rangle$ with the retarded single-electron Green's
function $G(\omega)$ in a basis of local site-angular momentum states
$|Lj\rangle$ \cite{REHR2000},
\begin{eqnarray}
G^{jj'}_{LL'}(\omega) = \sum_f \frac{\langle Lj|f\rangle\langle f| L'j'\rangle}{\omega-\varepsilon_f + i\eta} \; .
\end{eqnarray}
In this expression, $j$ is the index of a given site $\mathbf{R}_j$
and $L=(l,m)$ are the angular momentum quantum numbers. The initial
states $|i\rangle$ are calculated with the ground state Hamiltonian $H = p^2/2 + v(r)$ while
the final states $|f\rangle$ are described by the quasiparticle Hamiltonian $H' =
p^2/2 + v_f(r) + \Sigma(r,E)$, where $v(r)$ is the self-consistent one-electron Hartree potential,
$v_f$ is the final state one electron Hartree potential in the presence of a
screened core hole, and $\Sigma$ is the dynamically screened
quasiparticle self-energy discussed in detail below.
The imaginary part of the quasiparticle self-energy $\Sigma$ accounts for
the mean free path of the photoelectron.
Within the quasiparticle local density approximation (QPLDA) \cite{SHAM1966}, the self-energy is given by \cite{MUSTRE1991}
\begin{eqnarray}
\Sigma(\textbf{r}, E,T=0) &=& v^{LDA}_{xc}(\rho(\textbf{r})) + \Sigma_{GW}(\rho(\textbf{r}), E, T=0)\nonumber\\
& & - \Sigma_{GW}(\rho(\textbf{r}), E_F, T=0) \; .
\end{eqnarray}
Here $\Sigma_{GW}$ is the $GW$ self-energy calculated at the $G_0W_0$ level of refinement, that is, without self-consistent iteration of $G$ or $W$ \cite{HEDIN1965}.
For simplicity from here onward, we drop the spatial dependence $\textbf{r}$.
For the FT generalization, we replace the $T=0$ $GW$ self-energy with the finite-temperature $GW$ self-energy, $\Sigma_{GW}(T)$, and
introduce $T$-dependent Fermi occupation numbers, $f(\varepsilon) = 1/[\exp{\{\beta(\varepsilon-\mu)\}} + 1]$ for the initial and
final states in Eq.\ (\ref{eqn:xas}).
In addition, the ground state exchange-correlation
potential $v^{LDA}_{xc}$ is replaced by its FT generalization
$v^{LDA}_{xc}(T)$. Thus, the finite-temperature QPLDA self-energy is
\begin{eqnarray}
\Sigma(E, T) &=& v^{LDA}_{xc}(\rho,T) + \Sigma_{GW}(\rho,E, T) \nonumber\\
& & - \textnormal{Re}\ [\Sigma_{GW}(\rho,\mu_T, T)]
\end{eqnarray}
Lastly, by using $G^{jj'}_{LL'}(\omega)$ in Eq.\ (\ref{eqn:xas}) in place of the sum over final states $|f\rangle$, the FT quasiparticle cross section can be re-expressed as:\cite{ANKUDINOV1998}
\begin{eqnarray}
\sigma_{qp}(\omega) &=& - 4\pi^2 \frac{\omega}{c} {\textnormal{Im}} \sum_{iLL'} \langle i | \hat{d} G^{00}_{LL'}(\omega+\varepsilon_i) \hat{d}^\dagger | i \rangle \nonumber\\
& & \times f(\varepsilon_i) \big[ 1 - f(\omega + \varepsilon_i) \big],
\end{eqnarray}
Here, we denote the absorbing atom by the index 0.
\subsection{Finite-temperature Self-energy $\Sigma$}
The finite-$T$ quasiparticle electron self-energy within the $GW$ approximation is defined formally \cite{ALLEN1983,MAHAN2000} by the expression
\begin{eqnarray}
\Sigma^M_{GW}({\mathbf{k}},i\omega_m) &=& -\frac{1}{\beta} \int \frac{d^3{\mathbf{q}}}{(2\pi)^3} \sum_{n=-\infty}^{\infty} G_0^M ({\mathbf{k}}-{\mathbf{q}},i\omega_m-i\nu_n) \nonumber\\
& & \times W^M({\mathbf{q}},i\nu_n) \;.
\end{eqnarray}
%
Here $G_0^M$ is the one-electron Matsubara Green's function, $W^M=
\epsilon^{-1} v$ is the screened Coulomb interaction, and
$\omega_m=2(m+1)\pi k_B T$, $\nu_n=2n\pi k_B T$ are the Matsubara
frequencies, where $\epsilon$ is the dielectric function and $v$ is
the bare Coulomb potential. The screened interaction $W^M$ can be
expressed in terms of its spectral representation as
\begin{eqnarray}
W^M({\mathbf{q}}, i\nu_n) &=& v({\mathbf{q}}) + \int_{-\infty}^{\infty} d\omega' \frac{D({\mathbf{q}}, \omega')}{i\nu_n - \omega' + i\eta\, {\textnormal{sgn}}(\omega')}
\end{eqnarray}
where $v({\mathbf{q}})=4\pi/q^2$ is the bare Coulomb potential in Fourier
representation, and $D({\mathbf{q}},\omega) = (1/\pi) |{\textnormal{Im}}\, W^M_c({\mathbf{q}}, \omega)|\,{\textnormal{sgn}}(\omega)$ is the anti-symmetric
(in frequency) bosonic excitation spectrum. $W^M_c = W^M - v$ is the correlation part of the screened interaction.
Our choice of the electron gas dielectric function reflects a balance between the
level of physics included and computational feasibility. Thus for
simplicity, we use the random phase approximation (RPA), which is
analogous to the FT generalization of the Lindhard function \cite{ARISTA1984},
\begin{eqnarray}
\epsilon({\mathbf{q}}, \omega, T) &=& 1 + 2v({\mathbf{q}}) \int \frac{d^3{\mathbf{k}}}{(2\pi)^3} \frac{f(\varepsilon_{{\mathbf{k}}-{\mathbf{q}}})-f(\varepsilon)}{\omega - \varepsilon_{{\mathbf{k}}-{\mathbf{q}}} + \varepsilon_{\mathbf{k}} + i\eta},
\end{eqnarray}
where $f(\varepsilon) = 1/[\exp{\{\beta(\varepsilon-\mu)\}} + 1]$ is
the Fermi-Dirac occupation factor, and $\mu = \mu(T)$ is the chemical potential.
The real part of $\epsilon({\mathbf{q}}, \omega, T)$ is obtained from the
imaginary part via a Kramers-Kronig transform.
From an analytic continuation to the real-$\omega$ axis,
the FT GW retarded self-energy $\Sigma_{GW}$ is given by the Migdal
approximation \cite{ALLEN1983}
\begin{eqnarray}
\label{eqn:ft_sigma}
&&\Sigma_{GW}({\mathbf{k}}, \omega, T) = \Sigma_{X}({\mathbf{k}}, \omega, T) + \int_0^{\infty} d\omega' \int \frac{d^3{\mathbf{q}}}{(2\pi)^3} D({\mathbf{q}},\omega') \nonumber\\
&\times& \bigg[ \frac{f(\varepsilon_{{\mathbf{k}}-{\mathbf{q}}}) + N(\omega')}{\omega+\omega'-\varepsilon_{{\mathbf{k}}-{\mathbf{q}}} + i\eta}
+ \frac{ 1 - f(\varepsilon_{{\mathbf{k}}-{\mathbf{q}}}) + N(\omega')}{\omega-\omega'-\varepsilon_{{\mathbf{k}}-{\mathbf{q}}} + i\eta} \bigg],
\end{eqnarray}
where $\Sigma_{X}({\mathbf{k}},\omega,T) = \int [d^3{\mathbf{q}}/(2\pi)^3] f(\varepsilon_{{\mathbf{k}}-{\mathbf{q}}})v_{\mathbf{q}}$ is the exchange part of self-energy and $N(\omega) = 1/[\exp\{\beta\omega\}- 1]$ is the Bose factor. The poles of the Green's function $G^M$ contribute to the Fermi occupations whereas the poles of the screened interaction $W^M$ contribute to the Bose factor.
Calculations of the imaginary part of
$\Sigma_{GW}({\mathbf{k}}, \omega, T)$ involves a singe integral over the
magnitude of ${\mathbf{q}}$ but to obtained the real part
we need to perform a Kramers-Kronig transform resulting in a double integral.
In typical RSGF XAS
calculations \cite{FEFF10}, tens of thousands of self-energy evaluations
are required.
Thus that calculation of quasiparticle self-energy $\Sigma_{GW}( k,k^2/2,T)$ immediately
becomes a computational
bottleneck.
To circumvent that difficulty, we model $\Sigma_{GW}(k,k^2/2,T)$ via low-order polynomial fits
to numerical calculation of $\Sigma_{GW}$ based on Eq.\ (\ref{eqn:ft_sigma}), on a grid up to $T = 2 T_F$, where $T_F=E_F/k_B$. The
form of the fitting functions is described in detail in the Appendices. A
comparison between the QPLDA $\Sigma_{GW}$ and the fits is shown in
Fig.\ \ref{fig:gw_vs_ksdt} for the homogeneous electron gas.
For simplicity, we approximate the self-energy for levels $k < k_F$ with $v_{xc}(T)$ independent of $k$.
\begin{figure}
\includegraphics[width=0.45\textwidth]{re_sigma_comparison_ksdt.pdf}
\caption{The real part of the $\Sigma_{GW}$ (dashed) and our parametrization (solid) for various densities: $r_s$ = 1.0, 2.0, 3.0, 4.0 at $T/T_F$ = 0.01 (blue) and 1.0 (red). For reference the LDA-$v_{xc}$(T) \cite{KSDT2014} values are denoted as dots at $k_F$.}
\label{fig:gw_vs_ksdt}
\end{figure}
\section{Results and Discussion}
\label{section:result}
In this section, we present our results showing effects of the
use of the FT $\Sigma_{GW}(T>0)$ instead of the zero $T$
values $\Sigma_{GW}(T=0)$.
Note that $\Sigma_{GW}(T=0)$ depends implicitly on the
electronic temperature $T$ through the self-consistent electron
density at that temperature. The FT SCF calculations are carried out using an extension of the original RSMS code, which is now implemented in FEFF10 \cite{FEFF10, FEFF9}. For the FT exchange
correlation-potential, we use the KSDT tabulation $v^{LDA}_{xc}(T)$ \cite{KSDT2014,KTD2019PRB} for both calculations.
In the FEFF10 calculations, we compute the atomic part with the ground state exchange potential, whereas we use $\Sigma_{GW}(T)$ for the fine structure.
As noted in Ref. \cite{RECOULES2009, JOURDAIN2020}, corrections to the core energy levels are needed for $T \gtrsim 1$ eV. We compute the core level shifts using the
all-electron full-potential linearized plane wave code FLEUR \cite{FLEUR, FLPAW1982, FLEUR2018}.
Within these calculations, we ignore the explicit
temperature dependence in the LDA exchange correlation functional \cite{PZ1981}.
As a side note, FLEUR uses non-overlapping muffin-tin potentials whereas FEFF uses overlapping muffin-tin potentials.
In the RSMS formalism, the decomposition of the Green's function $G$
into a central atom $G^c$ contribution and a multiple-scattering
$G^{sc}$ contribution allows us to describe the XAS
in terms of the atomic background $\sigma_0$
and the oscillatory fine structure $\chi$, i.e., $\sigma = \sigma_0 (1 + \chi)$. For many XANES calculations, it is found
that the atomic background matches the experimental results better
when calculated without the self-energy corrections due to the
overestimation of the exchange within the muffin-tin (MT) potential
approximation \cite{REHR1994}.
Fig.\ \ref{fig:atomic_background} shows the effect of using the ground-state potential versus the use of $\Sigma_{GW}(T)$ for the atomic background.
The pre-edge is dominated by the atomic background and thus is sensitive to the choice of exchange potential.
The pre-edge amplitude is reduced by
$\approx 28\%$ due to the temperature correction of $\Sigma_{GW}(T)$.
More pump-probe experimental XAS measurements for
$T \approx 10$ eV
are required to validate the FT self-energy effect for the atomic background.
Nonetheless, the FT self-energy corrections are important for the description of HT fine-structure.
\begin{figure}
\includegraphics[width=0.45\textwidth]{cu_atomic_background_effect.pdf}
\caption{$\textnormal{L}_3$-edge XAS for Cu (lattice constant $a = 3.61$ \AA \cite{KITTEL2005}) at electronic temperature $T=0$ eV and 2 eV.
The solid curves denote the absorption with ground state atomic background while
the dots represent the absorption with $\Sigma_{GW}(T>0)$ self-energy atomic background. The experimental measurement at ambient condition is shown as black cross.\cite{KIYONO1978}}
\label{fig:atomic_background}
\end{figure}
As a first example, we consider the FT K-edge x-ray absorption
near-edge spectrum (XANES) for aluminum (fcc Al, lattice constant $a = 4.05$ \AA \cite{KITTEL2005}). Al is a
prototypical nearly-free electron system in the sense that
the electronic density
of states (DOS) in the conduction band has a nearly square root like
dispersion at the bottom of the band.
Fig.\ \ref{fig:1} shows the comparison of the Al K-edge
spectrum at different temperatures including or excluding explicit electronic T-dependent effects in the self-energy, namely, using $\Sigma_{GW}(T=0)$ or $\Sigma_{GW}(T>0)$.
When restricting the temperature T solely to that which is introduced through the density ($\Sigma_{GW}(T=0)$ case), we observe the broadening of the edge and no shift in the edge position. The shift in the core levels compensates for the shift in the valence density. On the other hand, for $T$ up to about 1 eV, the finite
temperature self-energy correction is negligible, and the temperature-independent electron self-energy model is a good
approximation. However, as temperature grows to order $\approx$ 10 eV,
the fine structures are smoothed by the large broadening ($\approx$ 3
eV) associated with shortened electronic excitation lifetime. The
shift in quasiparticle shift due to the finite temperature self-energy
correction is small for the near-edge region, and only becomes
significant between 10 eV and 20 eV above the chemical potential.
As a further illustration of the explicit $T$-dependence of the FT
quasiparticle self-energy, we compute the quasiparticle energy
correction $\Delta_k = \varepsilon_{qp} - \varepsilon_{k}$. Here, the quasiparticle energy $\varepsilon_{qp}$ is the solution to $\varepsilon_{qp}(k') = \varepsilon_{k} + \Sigma_{GW}(k', \varepsilon_{qp}(k'))$ and $k = \sqrt{2(E-\mu)}$ is the photoelectron wavenumber. We compare
the real part of $\Delta$ in Fig.\ \ref{fig:1a} and imaginary part in
Fig.\ \ref{fig:1b} for different self-energies at the interstitial density:
$\Sigma_{GW}(T=0)$ and $\Sigma_{GW}(T>0)$.
Note that the real part of $\Delta$
shows a strong temperature dependence between 5 eV and 20 eV near the
plasmon onset. For the imaginary part of $\Delta$, the broadening effect becomes important above $T$ = 1 eV.
\begin{figure}
\includegraphics[width=0.45\textwidth]{al_k_edge.pdf}
\caption{
$\textnormal{K}$-edge XAS for fcc aluminum ($a = 4.05$ \AA) using different self-energies:
$T$-independent GW self-energy $\Sigma_{GW}(T=0)$ (solid curves) and $T$-dependent GW self-energy $\Sigma_{GW}(T>0)$ (dots).}
\label{fig:1}
\end{figure}
\begin{figure}
\includegraphics[width=0.45\textwidth]{al_k_edge_correction_real.pdf}
\caption{The quasiparticle corrections Re $\Delta_k$ for aluminum at temperatures $T=0, 1, 2$ and $5$ eV.
The calculations used KSDT $v_{xc}(T)$ and different self-energies: $\Sigma_{GW}(T=0)$ (solid), and $\Sigma_{GW}(T>0)$ (dashed).}
\label{fig:1a}
\end{figure}
\begin{figure}
\includegraphics[width=0.45\textwidth]{al_k_edge_correction_imag.pdf}
\caption{The quasiparticle corrections Im $\Delta_k$ for aluminum at temperatures $T=0, 1, 2$ and $5$ eV.
The calculations used KSDT $v_{xc}(T)$ and different self-energies: $\Sigma_{GW}(T=0)$ (solid), and $\Sigma_{GW}(T>0)$ (dashed).}
\label{fig:1b}
\end{figure}
As a second example, we present results for a noble transition metal
(fcc copper, lattice constant $a = 3.61$ \AA \cite{KITTEL2005}) for which the $d$-bands are essentially full. Unlike the
K-edge, the L-edge probes the highly-localized $d$-bands of Cu near
the chemical potential. At high temperatures, the pre-edge peak
increases in amplitude due to the decreasing $d$-state
occupation \cite{JOURDAIN2020, TAN2021}. Fig.\ \ref{fig:copper_l_edge}
shows the $\textnormal{L}_{3,2}$-edge XAS up to $T=5$ eV. The
temperature dependence of $\textnormal{Im}\ \Sigma_{GW}(T)$ results in
changes to pre-edge peaks at
$T \gtrapprox 2$ eV. Consequently,
the estimation of temperature based on the pre-edge area method or
direct spectrum fitting will deviate more from the $\Sigma_{GW}(T=0)$
model as temperature increases.
\begin{figure}
\includegraphics[width=0.45\textwidth]{cu_l_edge.pdf}
\caption{$\textnormal{L}_{3,2}$-edge XAS for Cu ($a = 3.61$ \AA) finite electronic temperature $T=0, 1, 2$ and $5$ eV, where the structure reflects that of the unfilled $d$-bands.
The solid curves denote the $\Sigma_{GW}(T=0)$ self-energy results while
the dashes represent the $\Sigma_{GW}(T>0)$ self-energy results.}
\label{fig:copper_l_edge}
\end{figure}
\section{Summary, Conclusions, and Outlook}
\label{section:conclusion}
Our parametrization of the FT GW electron self-energy enables
efficient calculations of XAS at finite $T$, from LT of a few hundred K, up to the WDM regime
with $T$ at least 10 eV. Our strategy uses the QPLDA $G_0W_0$ level of
refinement for the self-energy, with the RPA dielectric function in conjunction with the
KSDT finite-$T$ LDA exchange-correlation functional. Specifically,
the FT self-energy for a system is approximated using the uniform
electron gas with density equal to that of the local density. This is a significant
simplification, as direct calculations using the exact loss function
of the system for the entire energy range of typical XAS experiments would be
computationally formidable.
A finite-temperature
SCF procedure for the XAS calculations is carried out in the complex energy plane in terms of
the FT one-electron Green’s function. The procedure includes the FT
exchange-correlation potential, approximated here by the KSDT
parametrization. Important FT XAS effects include
the smearing of the absorption edge and the presence of
peaks in the spectrum below the $T = 0$ K Fermi energy. The FT
exchange-correlation potential has only a small effect on XAS at low temperatures $T \ll
T_{F}$ compared to the effect of Fermi smearing. The FT self-energy is also
important for XAS, accounting for both temperature dependent
shifts and final-state
broadening.
To illustrate its efficacy,
the approach was applied to calculations of XANES for
crystalline Al and Cu at normal density. Above $T>1$ eV, the
fine structures experience substantial broadening in the K-edges, corresponding to a
reduction of the quasiparticle lifetime with increasing $T$.
Going forward, a computationally efficient approximation beyond
the uniform electron gas dielectric function would be to use a
many-pole model \cite{KAS2007, KAS2009}, which is an extension of
the Hedin-Lundqvist single plasmon-pole model \cite{LUNDQVIST1967,HEDIN1971}. {\it Ab-initio} dielectric
functions also can be obtained from modern electronic structure
codes. Such a finite-temperature generalization of the many-pole model is currently
under development.
\acknowledgments
The contributions from JJK and JJR are supported
by the Theory Institute for Materials and Energy Spectroscopies (TIMES) at SLAC funded by the U.S. DOE, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under contract EAC02-76SF0051.
TST and SBT are supported by DOE grant DE-SC0002139.
|
1,314,259,996,526 | arxiv | \section{Introduction} \label{sec:intro}
The last hundred years have provided a great impulse towards the knowledge of the history and fate of the Universe. The field equations of general relativity formulated by \citet{einstein1915feldgleichungen} on one hand, the \citet{friedmann1922krummung}, \citet{lemaitre1927lemaitre,lemaitre1931expansion}, \citet{robertson1933relativistic} and \citet{walker1937milne} (FLRW) model on the other, along with several cosmological distances defined from luminosity and angles subtended by objects, constitute the basis of the \textit{standard model}. The matching between theoretical predictions and observational data provides information about the different components of the Universe (i.e., curvature, radiation, matter and dark energy).\\
Since the discovery of the universe expansion by \citet{hubble1929relation}, many astronomical surveys have been performed to determine the evolution of the Universe. Different test have been proposed to confront expansion against other theories as the \textit{tired light} (\citet{zwicky1929redshift}). The most conclusive test is the time dilation of Type Ia supernovae light curves that was suggested by \citet{wilson1939} and verified by \citet{leibundgut1996time} and \citet{goldhaber2001timescale}. Another relevant test was proposed by Tolman in 1934 which predicts a surface brightness dropping as $\sim(1+z)^{-4}$ with redshift $z$ for an expanding universe. The values obtained for the exponent $n$ in different data analysis (\citet{hoyle1956second}, \citet{sandage1961ability}, \citet{petrosian1976surface}, \citet{meier1976optical}, \citet{sandage1991surface}, \citet{pahre1996tolman}, \citet{lubin2001tolman},\citet{sandage2010tolman}) differ from the predicted value of $n=-4$ and thus it is assumed the existence of a non negligible galaxy evolution effect.\\
Recently, \textit{Expansion Lensing} (EL) was presented (\citet{2020arXiv2003.05307}). EL predicts a new luminosity-angular distances relation $D_L=D_A(1+z)$ unlike $D_L=D_A(1+z)^2$ assumed by the \textit{standard model}. In the same sense, the Tolman's surface brightness-redshift relation changes from $\mu\sim(1+z)^{-4}$ to $\mu\sim(1+z)^{-2}$. \\
In this paper, a method denominated \textit{Cosmic Vector Inference} (CVI) is developed to determine experimentally the luminosity-angular distance relation. The method has been applied to 1.2 million galaxies from SDSS DR15 sample (\citet{aguado2019fifteenth}). On the other hand, the mean surface brightness of galaxy samples from SDSS and VIPERS (\citet{guzzo2014vimos}) surveys have been computed from spectra. This amount allows one to evaluate the galaxy evolution on \textit{Tolman's model} against the \textit{expansion lensing} paradigm. Relevant conclusions can be extracted from this comparison.\\
The rest of the paper is organized as follow. Section~\ref{sec:st} introduces some basic elements of the \textit{standard model} of cosmology and define the \textit{cosmic index} that relates luminosity and angular distances. In Section~\ref{sec:cl} the \textit{Expansion Lensing} paradigm is described. Section~\ref{sec:lax} develops the method \textit{Cosmic Vector Inference} and applies it to determine de cosmic index. Section~\ref{sec:SurfaceBrightness} describes the surface brightness measurement from the spectra, and analise possible galaxy evolution according to the values predicted by the \textit{Tolman's model} and \textit{Expansion Lensing} paradigm. The conclusions are presented in Section~\ref{sec:conclusions}.\\
\section{Standard Model of cosmology: cosmic index $n_c=2$}
\label{sec:st}
The Stardard Model of cosmology compiles the current knowledge related to the begining, evolution and fate of the Universe. The model is based on the \textit{Friedmann-Lemaitre-Robertson-Walker} (FLRW) metric Eq.~\ref{eq:flrw}, which is a solution of Einstein's field equation of \textit{General Relativity} describing a homogeneous and isotropic expanding universe,
\begin{eqnarray}
\label{eq:flrw}
-c^2d\tau^2=-c^2dt^2+a(t)^2\left[\frac{dr^2}{1-kr^2}+r^2d\Omega^2\right]
\end{eqnarray}
\noindent
where k describes the curvature and a(t) is the scale factor responsible of the universe expansion.\\
Along with the main equations of cosmology there are several distances defined to link the theory with the observational data. Let us to reproduce here a brief summary of the distances and its relation with cosmological models described by relative densities $\Omega_M$, $\Omega_r$, $\Omega_\Lambda$, $\Omega_k$ for matter, radiation, cosmological constant and curvature respectively (\citet{hogg1999distance}).\\
\noindent
Let E(z) be the function defined as:
\begin{eqnarray}
\label{eq:ez}
E(z)=\sqrt{\Omega_K(1+z)^2+\Omega_\Lambda+\Omega_M(1+z)^3+\Omega_r(1+z)^4}
\end{eqnarray}
\noindent
The \textit{Line of sight Comoving Distance} $D_C$ is defined by
\begin{eqnarray}
\label{eq:comovingDistance}
D_C(\Omega_i)=D_H\int_{0}^{z}\frac{dz'}{E(z')}
\end{eqnarray}
\noindent
where $\Omega_i$ remarks the dependence from relative densities and where
\begin{eqnarray}
\label{eq:D_H}
D_H=c/H_0=3000 h^{-1} Mpc
\end{eqnarray}
\noindent
is the Hubble distance. \\
The \textit{Transverse Comoving Distance} $D_M$ is defined by
\begin{eqnarray}
\label{eq:transverseComovingDistance}
D_M=
\left \{
\begin{array}{ccc}
D_H\frac{1}{\sqrt\Omega_k}\sinh[\sqrt\Omega_kD_C/D_H]&for&\Omega_k>0\\
D_c&for&\Omega_k=0\\
D_H\frac{1}{\sqrt{|\Omega_k|}}\sin[\sqrt{|\Omega_k|}D_C/D_H]&for&\Omega_k<0
\end{array}
\right \}
\end{eqnarray}
\noindent
On the other hand, the \text{Luminosity Distance} defines the relation between the bolometric flux energy $f$ received at earth from an object to its bolometric luminosity L by means of
\begin{eqnarray}
\label{eq:fluxEnergy}
f= \frac{L}{4\pi D_L^2}
\end{eqnarray}
\noindent
being
\begin{eqnarray}
\label{eq:dl_dm_st}
D_L= D_M(1+z) \qquad \qquad(standard\ model)
\end{eqnarray}
Eq.~\ref{eq:dl_dm_st} provides the link between a measurable amount $D_L$ and the densities of the components of the Universe through $D_M(\Omega_M, \Omega_r, \Omega_\Lambda,\Omega_k)$.\\
The \textit{Angular Diameter Distance} $D_A$ is defined as the ratio between the size of the object $S$ and its angular size $\theta$
\begin{eqnarray}
\label{eq:angularDiameterDistanceTh}
D_A=\frac{S}{\theta}
\end{eqnarray}
\noindent
The \textit{Angular Diameter Distance} is related to the transverse comoving distance by
\begin{eqnarray}
\label{eq:da_dm}
D_M=D_A(1+z)
\end{eqnarray}
\noindent
and taking into account Eq.~\ref{eq:dl_dm_st} we have
\begin{eqnarray}
\label{eq:dl_da_st}
D_L= D_A(1+z)^2 \qquad (standard\ model)
\end{eqnarray}\\
For reasons stated below, let us to denominate the (1+z) exponent of this luminosity-angular relation as \textit{cosmic index} $n_c$, being $n_c=2$ for the current standard model.
\noindent
Finally the surface brightness ($\mu$) is given by
\begin{eqnarray}
\label{eq:sb_st}
\mu=l_S(1+z)^{-4} \qquad (standard\ model)
\end{eqnarray}
\noindent
where $l_S$ is the luminosity of the source per area unit and time unit.
Note that $\mu$ only depends on the redshift when $l_S$ is constant.\\
\noindent
Finding $l_S$ in Eq.~\ref{eq:sb_st} one has
\begin{eqnarray}
\label{eq:ls_st}
l_S=\mu(1+z)^{4} \qquad (standard\ model)
\end{eqnarray}
\\
\section{Expansion Lensing: cosmic index $n_c=1$}
\label{sec:cl}
Expansion Lensing is a new paradigm for the luminosity-angular distances relation. It is common in many fields of physics the \textit{euclidean} inverse-square law, e.g., the flux decreases as the inverse of square of the distance between the source and the observer. Although the static euclidean geometry is not applicable to describe an expanding universe, \citet{2020arXiv2003.05307} demonstrates that the inverse-square law is still applicable within the FLRW geometry. Then, the relation between the different cosmological distances can be rewritten within the \textit{expansion lensing} paradigm as
\begin{eqnarray}
\label{eq:dl_dm_cl}
D_L= D_M \qquad (expansion\ lensing)
\end{eqnarray}
\begin{eqnarray}
\label{eq:dl_da_cl}
D_L= D_A(1+z) \qquad (expansion\ lensing)
\end{eqnarray}
\noindent being therefore the cosmic index $n_c=1$.\\
\noindent The flux can be expressed as
\begin{eqnarray}
\label{eq:inversSquareLaw}
f= \frac{L}{4\pi D_L^2}=\frac{L}{4\pi D_M^2}
\end{eqnarray}
\noindent
In the same way, \textit{expansion lensing} also affects to surface brightness ($\mu$) that is transformed from Eq.~\ref{eq:sb_st} to
\begin{eqnarray}
\label{eq:sb_cl}
\mu=l_S(1+z)^{-2} \qquad (expansion\ lensing)
\end{eqnarray}
\noindent
where $l_S$ is the luminosity of the source per area and time units.
\noindent Finding $l_S$ in Eq.~\ref{eq:sb_cl} one obtains
\begin{eqnarray}
\label{eq:ls_cl}
l_S=\mu(1+z)^{2} \qquad (expansion\ lensing)
\end{eqnarray}
\section{Empirical determination of cosmic index: $n_c=1.0\pm0.05$}
\label{sec:lax}
In this section we develop \textit{Cosmic Vector Inference}, a direct method to measure the \textit{cosmic index} in the luminosity-angular distances relation.
The luminosity distance can be expressed as
\begin{eqnarray}
\label{eq:dl}
d_L=\sqrt{\frac{L}{4\pi f_L}}
\end{eqnarray}
\noindent
and since the angular distance corresponds to the luminosity distance at emission, it can be expressed also as
\begin{eqnarray}
\label{eq:da}
d_A=\sqrt{\frac{L}{4\pi f_A}}
\end{eqnarray}
\noindent
where $f_A$ corresponds to the flux that would be measured in a static universe.
Dividing both expressions we have
\begin{eqnarray}
\label{eq:dlda}
\frac{d_L}{d_A}=\sqrt{\frac{f_A}{f_L}}
\end{eqnarray}
On the other hand, the luminosity-angular distances relation is given by
\begin{eqnarray}
\label{eq:ci}
(1+z)^{n_c}=\frac{d_L}{d_A}
\end{eqnarray}
\noindent
where $n_c=2$ for the \textit{standard model} and $n_c=1$ for \textit{expansion lensing}. Substituting Eq.~\ref{eq:dlda} in Eq.~\ref{eq:ci} we have
\begin{eqnarray}
\label{eq:lax}
(1+z)^{n_c}=\left[\frac{f_A}{f_L}\right]^{1/2}
\end{eqnarray}
\noindent Taking base10 logarithm in both sides of the equation we have
\begin{eqnarray}
\label{eq:lax1}
n_c \log(1+z)=\frac{1}{2}log\left(\frac{f_A}{f_L}\right)
\end{eqnarray}
Let us to consider a new class of magnitudes, \textit{natural magnitudes}, defined as minus base10 logarithm of flux, dropping the usual meaningless 2.5 factor.
In this scope, we can define: \\
\noindent- Luminosity magnitude as $m_L=-\log{f_L}$ \\
- Angular magnitude as $m_A=-\log{f_A}$ \\
- Cosmic magnitude as $m_c=m_L-m_A$ \\
- Natural magnitudes \\
\indent e.g., $\mathbf{m}=(u/2.5, g/2.5, r/2.5,i/2.5,z/2.5,1)$ \\
\noindent where \textit{ugriz} correspond to common SDSS magnitudes and the last component 1 as been added for convenience.\\
\noindent In this way we have
\begin{eqnarray}
\label{eq:lax2}
2n_c \log{(1+z)}=m_L-m_A
\end{eqnarray}
\noindent or
\begin{eqnarray}
\label{eq:lax3}
2n_c \log{(1+z)}=m_c
\end{eqnarray}
\noindent defining the \textit{cosmic magnitude} as $m_c=m_L-m_A$.\\
On the other hand, a tentative expression can be considered to relate cosmic magnitude $m_c$ and measured magnitudes $\mathbf{m}$.
Thus, we assume there exists a vector $\mathbf{V_c}$ on magnitude space, denominated \textit{cosmic vector}, where the projection of $\mathbf{m}$ produces the \textit{cosmic magnitude} $m_c$ for each galaxy.
\begin{eqnarray}
\label{eq:lax4}
m_c=\mathbf {m \cdot V_c}
\end{eqnarray}
\noindent or substituting ~\ref{eq:lax3} in ~\ref{eq:lax4} we have
\begin{eqnarray}
\label{eq:lax5}
2n_c \log{(1+z)}=\mathbf {m \cdot V_{n_c}}
\end{eqnarray}
\noindent where $\mathbf {V_{n_c}}$ denote the value of $\mathbf {V_c}$ for $n_c$. \\
To assess the validity of such assumption, we need a galaxy sample with spectroscopic redshift and properly measured photometric magnitudes in several bands. A regression can be applied on this sample to determine $\mathbf{V_{n_c}}$ for different values of the \textit{cosmic index} $n_c$. The success of $n_c$ determination requires two conditions: \\
\indent1. High correlation between both sizes of Eq.~\ref{eq:lax5} to trust on the results. Note that $n_c$ is solely a multiplicative factor on left size of Eq.~\ref{eq:lax5}, thus it is expected the same correlation for all values of $n_c$ \\
\indent2. Then, assuming this high correlation is achieved, what is the true value of $n_c$? The true value of $n_c$ is the one that produces a \textit{cosmic vector}$ \mathbf {V_c}$ that meets
\begin{eqnarray}
\label{eq:lax6}
||{\mathbf{V_{n_c}}}|| =1
\end{eqnarray}
\noindent since it gives the real value of the cosmic magnitude, i.e., the true projection of measured magnitudes on a normalized cosmic vector.\\
\begin{figure}
\centering
\leavevmode
\includegraphics[width=0.48\textwidth]{cosmicRedshift_el_st.png}
\caption{Cosmic redshift reconstruction}
\label{fig:cosmicRedshift1}
\end{figure}%
\begin{figure}
\centering
\leavevmode
\includegraphics[width=0.35\textwidth]{cosmicIndexDetermination_05.png}
\caption{Cosmic redshift reconstruction along the cosmic index $n_c$.}
\label{fig:cosmicRedshift2}
\end{figure
In order to apply Cosmic Vector Inference to compute the \textit{cosmic index} by Eq.~\ref{eq:lax5}, we resort to SDSS DR15 that provides simultaneously spectrum and photometric measurements for about 1.2 million of luminous galaxies. To ensure a uniform treatment for all galaxies, we select \textit{De Vaucouleurs} magnitude (deVMag) which achieves accurate measurement of the flux of the bulge, the most luminous part of the galaxies.\\
We have applied Eq.~\ref{eq:lax5} to this sample obtaining high correlation ($c=0.91$) between both sides of the equation for all values of $n_c$. Thus, the first condition to properly measure $n_c$ is met. Regarding the second condition, alternatively to verifying Eq.~\ref{eq:lax6} to determine the true value of $n_c$, we can inversely reconstruct the value of the redshift $z_c$. To properly evaluate $m_{n_c}$ projection we first normalize $\mathbf{V_{n_c}}$
\begin{eqnarray}
\label{eq:lax7}
\mathbf{v_{n_c}}=\frac{\mathbf{V_{n_c}}}{||\mathbf{V_{n_c}}||}
\end{eqnarray}
\noindent and then projects $\mathbf{m}$ over $\mathbf{v_{n_c}}$ to obtain $m_{n_c}$
\begin{eqnarray}
\label{eq:lax8}
m_{n_c}=\mathbf {m \cdot v_{n_c}}
\end{eqnarray}
\noindent Finding $z$ in Equation ~\ref{eq:lax3}
\begin{eqnarray}
\label{eq:lax9}
z_c=10^\frac{m_{n_c}}{2n_c}-1
\end{eqnarray}
Fig.~\ref{fig:cosmicRedshift1} shows the results in the determination of the cosmic index $n_c$. We can see the cosmic redshift reconstruction for $n_c=2$ \textit{(standard model)} and $n_c=1$ \textit{(expansion lensing)}. In Fig.~\ref{fig:cosmicRedshift2} the redshift reconstruction has been extended along several $n_c$ values obtaining the true value for $n_c=1$, which corresponds to \textit{the expansion lensing} paradigm.
\section{Surface Brightness: Tolman's model vs Expansion Lensing}
\label{sec:SurfaceBrightness}
Few years after the discovery of the universe expansion, Tolman proposed the \textit{surface brightness} ($\mu$) test to differentiate between static and expanding universes. According to this prediction, the \textit{surface brightness} should decrease as Eq.~\ref{eq:sb_st} for an expanding universe. On the contrary, the \textit{expansion lensing} paradigm predicts a new \textit{surface brightness} relation in an expanding universe following Eq.~\ref{eq:sb_cl}. In this section, the \textit{surface brightness} of different galaxy samples is computed. The comparison of the measured \textit{surface brightness} $\mu$ with the \textit{Tolman's model} and \textit{expansion lensing} predictions provides constraints for galaxy evolution within each model. The feasibility of such galaxy evolution can be then analysed.
\subsection{Measuring the \textit{surface brightness} from spectra}
\label{subsec:MeasuringSurfaceBrightness}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{spec.png}
\caption{galaxy spectra}
\label{fig:y equals x}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{bulge_restframe_spectrum_vs_z.png}
\caption{restframe galaxy spectra}
\label{fig:three sin x}
\end{subfigure}
\hfill
\captionsetup{labelfont=bf} \captionof{figure}{(a) Rest frame flux density (left) (b) Rest frame flux density resampled in wavelength and averaged by redshifts bins (right)}
\label{fig:restframespec}
\end{figure*}
Let $f_o(\lambda_o)$ be the observed spectrum (i.e. the flux density measured in $erg/cm^2/s/$\text{\AA}) of a galaxy. Since the spectrum is taken with a constant fiber aperture, it can be converted to surface brightness ($\mu(\lambda_o)$) by dividing by the aperture. In what follows, let us to equate $f_o(\lambda_o)$ to $\mu(\lambda_0)$ by introducing (in the spectrum units) a constant $\beta$ representing the aperture normalization to $arcsec^{-2}$. On the other hand, let z be the measured galaxy redshift. The surface brightness within the observed band $b_0$ would be given by
\begin{eqnarray}
\label{eq:sb0}
\mu_o=\int_{b_o}{f_o(\lambda_o)d\lambda_o}
\end{eqnarray}
To compare the surface brightness of galaxies with different redshifts, it is convenient to blue-shift by an amount (1+z) the observed spectrum $f_o(\lambda_o)$ to a common rest-frame emission band $b_r$. To avoid distorting the data analyses with this transformation, the surface brightness $\mu$ represented in the common rest-frame $\mu_r$ should be equal to the measured one in the observation band $\mu_o$. Therefore, performing a change of variable on Eq. ~\ref{eq:sb0} we have
\begin{eqnarray}
\label{eq:redshiftrelation}
\lambda_o=(1+z)\lambda_r
\end{eqnarray}
\begin{eqnarray}
\label{eq:dzredshiftrelation}
d\lambda_o=(1+z)d\lambda_r
\end{eqnarray}
\begin{eqnarray}
\label{eq:sb}
\mu_o=\int_{b_o}{f_o(\lambda_o)d\lambda_o}=\int_{b_r}(1+z){f_o(\lambda_r)d\lambda_r}=\int_{b_r}{f_r(\lambda_r)d\lambda_r}=\mu
\end{eqnarray}
\noindent
and then
\begin{eqnarray}
\label{eq:fluxdensity_blueshift}
f_r(\lambda_r)=(1+z)f_o(\lambda_r)
\end{eqnarray}
\noindent
Therefore, Eq.~\ref{eq:redshiftrelation} and Eq.~\ref{eq:fluxdensity_blueshift} allow one to obtain the spectra in the common rest-frame band.
To prevent the \textit{surface brightness-redshift} relationship $\mu$(z) from the effects of galaxy evolution, one needs to find a source with constant luminosity along a large redshift period (i.e. a standard candle). The best known standard candle are supernovae Type Ia since they provide a very uniform luminosity. Though they have been used extensively in the last decades, the technique is complex and there are some complications in their measurements associated to the eventuality and standardization issues as is related in \citet{riess1998observational} and \citet{perlmutter1999measurements}.
In the same sense, Luminous Red Galaxies (LRGs) constitutes a very uniform and homogeneous set of galaxies that provides high luminosity up to redshift of cosmological interest. Though LRGs are not recognized standard candles, the study of the \textit{surface brightness} on this sample allows one to constrain and analyze the galaxy evolution within the \textit{Tolman's model} and \textit{expansion lensing} paradigm.
\subsection{Surface brightness on SDSS}
For luminosity studies on SDSS sample, we are interested on galaxies composed uniquely by bulge ---mostly LRGs--- since they represent a very luminous and homogeneous sample composed of very stable stars and hence foreseeable low luminosity mean evolution. Thus, it was selected a subcatalog by setting the selection parameter $fracDeV=1$, which account for exclusive \textit{de Vaucouleurs} profile and hence bulge-shape galaxies ($\sim127.000$ galaxies).
\subsubsection{Surface brightness-redshift relation}
Let us to apply Eq.~\ref{eq:redshiftrelation} and Eq.~\ref{eq:fluxdensity_blueshift} to obtain the rest-frame spectrum on bulge-SDSS sample. After shifting the sample to rest frame, all galaxy spectra become aligned (Fig.~\ref{fig:restframespec}(a)). Such alignment can be better visualized by spectra normalization in wider wavelength bins and averaging in redshift bins (Fig.~\ref{fig:restframespec}(b)). The \textit{surface brightness} $\mu$ can be obtained by integrating the spectra in a common rest-frame emission band. The SDSS spectrum of galaxies was taken between $(3650-10400)$\text{\AA}. The secure integration interval should not be larger than $\lambda_{max}/(1+z_{max}))$ to ensure that all rest-frame spectra have valid measured data. Thus, it has been selected a conservative wavelength integration interval $b_r=(3940-5200)$\text{\AA} that meets the above restriction up to $z_{max}=1$ and includes some characteristic LRG features as the $4000$\text{\AA} break and the absortion lines between $(5160-5200)$\text{\AA} corresponding to low evolution stars. Then, the surface brightness $\mu$ in this band can be obtained by
\begin{eqnarray}
\label{eq:sbrestframe}
\mu=\int_{b_o(z)}{f_o(\lambda_o)d\lambda_o}=\int_{b_r}f_r(\lambda_r)d\lambda_r
\end{eqnarray}
then
\begin{eqnarray}
\label{eq:sbrestframe_data}
\mu=\int_{3940}^{5200}f_r(\lambda_r)d\lambda_r
\end{eqnarray}
\begin{figure}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{sb_bulge_sdss.png}
\caption{spectrum}
\label{fig:sb_sulge}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{nz_sb_bulge_sdss.png}
\caption{restframe spectrum}
\label{fig:nz_sb}
\end{subfigure}
\hfill
\caption{bulge-SDSS sample: (a) Surface brightness ($\mu$) in $(3600-5700)$\text{\AA} emission band. Comparison with the behaviour predicted by \textit{Tolman's model} and \textit{expansion lensing}. Note how $\mu$ follows the \textit{expansion lensing} prediction for $z>0.47$. (b) Relation between the number of galaxies N(z) and the luminosity per surface unit ($l_S$) for \textit{expansion lensing}. Note the low passive evolution of $l_S$ for $z>=0.47$ and its growth for $z<0.47$ coinciding with the drop in number density of bulge galaxies. It may be explained by dry mergers increasing the luminosity of bulge-galaxies and consequently reducing its number.}
\label{fig:SB_luminosityvsNz}
\end{figure}
Averaging the surface brightness $\mu$ of the different bulge-galaxies in redshift bins one obtains $\mu(z)$ (Fig.~\ref{fig:SB_luminosityvsNz}(a)). In this plot, it was also represented the surface brightness prediction by \textit{Tolman's model} (Eq.~\ref{eq:sb_st}) and by \textit{expansion lensing} paradigm (Eq.~\ref{eq:sb_cl}), assuming in both cases a constant value of $l_S$.
The difference between a prediction and $\mu(z)$ would correspond to $l_S$ variation and hence galaxy evolution. The shadow along the line corresponds to galaxy dispersion. There is a notable divergent behaviour of $\mu$ for $z<0.47$ and $z>0.47$. Let one to focus by now on $z>0.47$. In this case, $\mu(z)$ is very close to \textit{expansion lensing} prediction and far from \textit{Tolman's model} one. As we show below in Section ~\ref{sec:Luminosity per surface unit}, the closeness to \textit{expansion lensing} prediction indicates low passive evolution of bulge-SDSS sample as expected due to the low start formation rate.
\noindent
\subsubsection{bulge-SDSS dry mergers}
Nevertheless, low passive evolution of bulge-SDSS sample in the emission wavelength band studied $(3960-5200)$\text{\AA} does not explain the large break observed in Fig.~\ref{fig:SB_luminosityvsNz}(a) for $z<0.47$. Thus, we need an explanation different from spectral evolution for this break. Let us to hypothesize an explanation. Fig.~\ref{fig:SB_luminosityvsNz}(b) shows the luminosity per surface unit $l_S$ for \textit{expansion lensing} (Eq. ~\ref{eq:ls_cl}) vs the number of galaxies per redshift bin N(z). It can be appreciated that increments in the luminosity slope corresponds to drops in the number of galaxies N(z). Although N(z) depends on many factors including the spectroscopic selection function, it seems probable that the increase in luminosity slope of the bulge-SDSS sample be due to dry mergers (i.e., gas-poor galaxies merging with low star formation but significant stellar mass growth (\citet{bell2006dry}). More clues about dry merging are given below.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{bulge_ls_cl_vs_z.png}
\caption{bulge-SDSS sample luminosity spectra}
\label{fig:ls_bulge_cl}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{bulge_ls_st_vs_z.png}
\caption{bulge-SDSS sample luminosity spectra}
\label{fig:ls_bulge_st}
\end{subfigure}
\hfill
\caption{Luminosity per surface unit $l_S$ for bulge-SDSS sample. (a) Expansion Lensing: the behaviour of $l_S$ can be explained by minor galaxy spectral evolution from high to low redshift and by dry mergers at the lowest redshift bin, increasing substantially the luminosity but maintaining the spectrum shape as should be expected by mergers of the same galaxy type. (b) Tolman's model: the luminosity drop simultaneously in all wavelengths from z=0.85 to z=0.45. This behaviour seems unfeasible by common galaxy evolution mechanisms, pointing to some unknown factor of cosmological origin, i.e. the fault of the Tolman's model to explain the observations.}
\label{fig:ls}
\end{figure*}
\subsubsection{Luminosity per area unit}
\label{sec:Luminosity per surface unit}
Another clue about the reality of \textit{expansion lensing} and the unfeasibility of \textit{Tolman's model} surface brightness prediction becomes patent by the luminosity per surface area $l_S$. This amount is not an observable but can be derived from the spectrum of galaxies for both paradigms. Let one consider $\mu(\lambda)\simeq f_r(\lambda_r)$ (see subsection ~\ref{subsec:MeasuringSurfaceBrightness}). Substituting in Eq.~\ref{eq:sb_st} and Eq.~\ref{eq:sb_cl} one obtains
\begin{eqnarray}
\label{eq:ls_st_lambda}
l_S(\lambda) \simeq f_r(\lambda_r)(1+z)^{4} \qquad (Tolman's\ model)
\end{eqnarray}
\begin{eqnarray}
\label{eq:ls_cl_lambda}
l_S(\lambda) \simeq f_r(\lambda_r)(1+z)^{2}\ (expansion\ lensing)
\end{eqnarray}
Fig.~\ref{fig:ls} shows the luminosity spectra $l_S(\lambda)$ of bulge-SDSS sample averaged in redshifts bins for (a) \textit{expansion lensing} (b) \textit{Tolman's model}. \textit{Expansion lensing} plot shows moderate luminosity spectra evolution along the transit between redshift bins, except for the lowest redshift bin where the luminosity grows appreciably. Note how this large growth maintains the spectrum shape, which is a clue of the merging of similar galaxies (i.e. dry merging of bulge-SDSS galaxies). On the contrary, the luminosity spectrum decays parallel ---i.e. simultaneously in all wavelengths--- for Tolman's model from $z=0.86$ to $z=0.45$ and then grows for $z<0.45$. While the final growth could be explained by dry merging, the parallel decay of the luminosity in all wavelengths discard the galaxy spectral evolution, pointing to some unknown factor of cosmological origin, i.e., the fault of the Tolman's model to explain the observations.
\subsection{Surface brightness on VIPERS}
The VIMOS Public Extragalactic Redshift Survey (VIPERS) was conceived to study the large-scale distribution and evolution of galaxies at $0.5<z<1.2$. In this paper we focus on W1 field of VIPERS PDR-2 (\citet{scodeggio2018vimos}) that provides spectrum and redshift measurements for about $\sim60.000$ galaxies to $iAB<22.5$.
\subsubsection{$\mu(z)$-redshift relation}
The spectra were measured at the band $b=(5500-9500)$\text{\AA}. Since our selected rest-frame band is $b_0=(3960-5200)$\text{\AA}, the minimum and maximum redshifts with valid data at this band are
\begin{eqnarray}
\label{eq:zmin}
z_{min}=\frac{5500}{3960}-1=0.39
\end{eqnarray}
\noindent
and
\begin{eqnarray}
\label{eq:zmax}
z_{max}=\frac{9500}{5200}-1=0.83
\end{eqnarray}
Fig.~\ref{fig:surfaceBrightnessLg2} shows the surface brightness $\mu$ of the VIPERS sample as a function of redshift. Note that within small fluctuations due to possible mergers and residual spectral evolution, $\mu(z)$ follows the prediction of \textit{expansion lensing} for $z>0.6$. On the contrary, as occurs with SDSS samples, VIPERS $\mu(z)$ transits far from surface brightness Tolman's model prediction.
\begin{figure}
\centering
\leavevmode
\includegraphics[width=0.45\textwidth]{sb_vipers.png}
\caption{Tolman's model vs \textit{expansion lensing}: Surface Brightness ($\mu$) in common rest-frame emission wavelength band $(3960-5200)$\text{\AA} for VIPERS sample. Note how $\mu(z)$ evolves close to \textit{expansion lensing} prediction for $z>0.6$.}
\label{fig:surfaceBrightnessLg2}
\end{figure}%
\section{Conclusions}
\label{sec:conclusions}
Early after the discovery of the universe expansion, Tolman proposed a surface brightness test as a mean to differentiate an expanding from a non-expanding universe. The test predicts the relation $\mu\sim(1+z)^{-4}$ for an expanding universe. Recently, \textit{expansion lensing} --a novel cosmological paradigm-- was presented providing a new assessment of the flux received from cosmological sources. Thus, \textit{expansion lensing} predicts surface brightness given by $\mu\sim(1+z)^{-2}$ and a luminosity-angular distances relation given by $d_L=d_A(1+z)^{n_c}$, with cosmic index $n_c=1$ rather than the value $n_c=2$ established by the standard model.
In this paper, empirical evidences of the reality of the \textit{expansion lensing} paradigm are presented. On one hand, the method Cosmic Vector Inference is developed to determine the cosmic index. The method has been applied to the public SDSS DR15 catalog obtaining a value of $n_c=1.0\pm0.05$. On the other hand, the surface brightness-redshift relation has been derived and analysed from DR15 SDSS and PDR-2 VIPERS spectroscopic data releases. The results also provide arguments favoring \textit{expansion lensing} over \textit{Tolman's model}.
Based on these results, a deep revision of methods involving luminosity as cosmological probes have to be performed under the new Expansion Lensing look. The Hubble constant and the density components of the Universe (i.e. dark matter, dark energy) have to be reassessed on such luminosity probes.
\begin{acknowledgements}
Funding support for this work was provided by the Autonomous Community
of Madrid through the project TEC2SPACE-CM (S2018/NMT-4291).
This paper uses data from public SDSS DR-15.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges
support and resources from the Center for High-Performance Computing at
the University of Utah. The SDSS web site is www.sdss.org.
SDSS-IV is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the
Brazilian Participation Group, the Carnegie Institution for Science,
Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics,
Instituto de Astrof\'isica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) /
University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory,
Leibniz Institut f\"ur Astrophysik Potsdam (AIP),
Max-Planck-Institut f\"ur Astronomie (MPIA Heidelberg),
Max-Planck-Institut f\"ur Astrophysik (MPA Garching),
Max-Planck-Institut f\"ur Extraterrestrische Physik (MPE),
National Astronomical Observatories of China, New Mexico State University,
New York University, University of Notre Dame,
Observat\'ario Nacional / MCTI, The Ohio State University,
Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group,
Universidad Nacional Aut\'onoma de M\'exico, University of Arizona,
University of Colorado Boulder, University of Oxford, University of Portsmouth,
University of Utah, University of Virginia, University of Washington, University of Wisconsin,
Vanderbilt University, and Yale University.
This paper uses data from the VIMOS Public Extragalactic Redshift Survey (VIPERS). VIPERS has been performed using the ESO Very Large Telescope, under the "Large Programme" 182.A-0886. The participating institutions and funding agencies are listed at http://vipers.inaf.it
\end{acknowledgements}
\bibliographystyle{aa}
|
1,314,259,996,527 | arxiv | \section{Introduction}
The production of heavy flavors, like bottom and charm, is a cornerstone high-energy collider process. It offers a wealth of information about the Standard Model and represents an excellent tool for probing the QCD dynamics. Heavy flavor production has been extensively studied at past and present high-energy lepton and/or hadron colliders as well as in nuclear collisions where heavy flavors are a prominent probe of the underlying nuclear dynamics.
Heavy flavors are copiously produced at the LHC. Indeed, the $b\bar b$ and $c\bar c$ cross sections are among the largest at this collider. Such large production rates enable detailed and very precise measurements in wide kinematic ranges. The theoretical description of these processes, currently at next-to leading order (NLO) in QCD, is lagging in precision behind the experimental needs. For improving the precision of theory predictions the inclusion of the NNLO QCD corrections is mandatory.
When discussing the production of a heavy flavor of mass $m$ at a hadron collider, it is instructive to distinguish two kinematic regimes: the low $p_T$ regime where $p_T\sim m$ and the high $p_T$ one where $p_T\gg m$. The low $p_T$ production of a heavy flavor can be described in fixed order perturbation theory as an expansion in powers of the strong coupling constant evaluated at the scale $m$, i.e. ${\alpha_s}(m)$, and including the full dependence of the heavy quark mass $m$. For bottom, and especially charm, this expansion converges slowly since ${\alpha_s}(m)$ is not much smaller than unity. This expectation was confirmed by the recent fully-differential NNLO QCD calculation of $b\bar b$ production at the quark level \cite{Catani:2020kkl}. Such behavior is to be contrasted with $t\bar t$ production which is very similar technically but the smallness of ${\alpha_s}(m_t)$ leads to a well-converging perturbative expansion \cite{Czakon:2015owf,Czakon:2016dgf} through NNLO in QCD.
The description of heavy flavor production at high $p_T$ involves a different set of challenges. Fixed order perturbation theory is no longer adequate there since large quasi-collinear logarithms $\log(p_T/m)$ appear to all orders in perturbation theory and need to be resummed. The resummation of these logs can be consistently carried out in the so-called perturbative fragmentation function (PFF) formalism \cite{Mele:1990cw}. Unlike the low $p_T$ case, a calculation of heavy flavor production at high $p_T$ is performed with a massless heavy quark since in the high-energy limit all terms that are power suppressed with $m$ are negligible while the mass-independent terms as well as the logarithmically enhanced ones are automatically accounted for by the PFF formalism. The current state of the art is NLO with next to leading logarithmic (NLL) accuracy. The goal of the present paper is to extend, for the first time, this description at hadron colliders to NNLO in QCD.
A generic application to heavy flavor production that is valid in all kinematic regimes would require the merging of the low $p_T$ and high $p_T$ descriptions mentioned above. This has been achieved at NLO in QCD within the so called FONLL approach \cite{Cacciari:1998it}. Some of the recent hadron collider applications include refs.~\cite{Cacciari:2012ny,Cacciari:2015fta}. Its generalization to NNLO goes beyond the scope of this paper.
As a first application of this formalism at NNLO in QCD we compute several $B$-hadron differential distributions in top quark pair production and decay at the LHC $pp \to t\bar t+X\to B+X$. The reason for choosing this process is twofold: first, $B$-production is central to top quark physics and $B$-hadron related observables are a great tool for precise top quark mass determination at hadron colliders. Second, in $t\bar t$ events the top quark mass provides a natural large hard scale such that for almost all distributions of interest the power suppressed effects $\sim(m_b)^n$ are negligible. This makes this process an ideal application for the massless $b$ quark PFF formalism used in this work. $B$-hadron production in other processes, like open $B$ production at high $p_T$, would be a straightforward extension of the current work and we hope to report on it in future publications.
This work is organized as follows: in sec.~\ref{sec:general} we discuss the general features of the formalism for calculations with an identified hadron. In sec.~\ref{sec:computational-framework} we explain our calculational framework. In sec.~\ref{sec:PFF-NPFF} we introduce the $B$ fragmentation functions used in this work. Sec.~\ref{sec:applications} is devoted to phenomenological LHC applications. We study in detail $B$-hadron distributions in top quark decay and in $t\bar t$ production and decay. We also propose an observable which we find suitable for extracting $B$-hadron fragmentation functions from LHC data. Several appendices contain additional results. In appendix \ref{sec:tt-xsec-fragmentation} we give the structure of the NNLO cross section for the process $pp \to t\bar t+X\to B+X$. In appendix \ref{sec:collinear} we give in explicit form the general expressions for the collinear counterterms needed for any NNLO hadron collider process with fragmentation. Appendices \ref{sec:e+e-} and \ref{sec:sum-rules} present two highly non-trivial checks of our calculational setup: the calculation of $B$ production in $e^+e^-$ collisions which is compared to the exact analytic result and the fulfillment of sum rules in top quark decay.
\section{Fragmentation: the general framework}\label{sec:general}
A typical calculation in perturbative QCD involves final states with QCD partons, which are clustered into jets, and colourless particles such as leptons. By clustering particles into jets, information is lost about the properties of the individual particles. On the experimental side, it also introduces jet energy scale uncertainties, which can dominate the total uncertainty on jet-based observables (see e.g. ref.~\cite{Aad:2015nba}), but are largely absent when instead measuring a single hadron's momentum (e.g. ref.~\cite{Khachatryan:2016pek}). As an alternative to this usual approach of jet-based observables, it therefore seems appealing to instead consider observables involving the momentum of a single hadron, $h$.
Perturbation theory alone cannot describe non-perturbative phenomena like the transition from partons to hadrons, called fragmentation. The solution is to factorise the non-perturbative aspects into fragmentation functions \cite{Berman:1971xz} in analogy to how parton distribution functions are introduced to describe transitions from hadrons to partons in the initial state. The fragmentation functions depend on the hadron $h$ but are otherwise universal and can thus be extracted from experimental data.
The theoretical description of the production of an identified hadron proceeds as follows. Standard tools and techniques are used to describe the production of on-shell partons. The partonic calculation is then extended by fragmenting the final-state partons, one at a time, into the observed hadron $h$ which has a well-defined momentum $p_h$. In practice, fragmentation corresponds to multiplying the fragmenting parton's momentum with a momentum fraction between $0$ and $1$, and then integrating the partonic cross section over it with a weight given by the corresponding fragmentation function. This procedure is equivalent to convolving the differential partonic cross sections with fragmentation functions:
\begin{equation}
\frac{d\sigma_h}{dE_h}(E_h) = \sum_i \bigg(D_{i\to h}\otimes \frac{d\sigma_i}{dE_i}\bigg)(E_h) \equiv \sum_i \int_{0}^1\frac{dx}{x} D_{i\to h}(x)\frac{d\sigma_i}{dE_i}\bigg(\frac{E_h}{x}\bigg)\;,
\label{eq:factorized-x-section}
\end{equation}
where the summation over $i$ is over all partons in the final state. $D_{i\to h}$ is the fragmentation function for the transition $i\to h$. Although the hadron's energy $E_h$ is used as an example here, any observable linear in the hadron's momentum can be utilized.
The kinematics of the collinear fragmentation process can be represented as follows
\begin{equation}
i(p_i) \to h(p_h) + X(p_i-p_h)\;,\;\;\;\; p_h^\mu= xp_i^\mu\;,\;\;\;\; x\in[0,1]\;,
\label{eq:splitting-kinematics}
\end{equation}
where the momenta of particles have been indicated in brackets and $X$ represents the particles produced in the fragmentation process which are not explicitly described by the fragmentation function, i.e. all particles in the jet initiated by $i$ other than the observed hadron $h$. Essentially, this means that one relates the hadron's momentum to that of a single parton, the latter being an infrared-unsafe quantity.
As the above discussion indicates, the partonic cross section for producing a parton $i$ is infrared unsafe and therefore contains uncancelled divergences. These are collinear divergences which factorise into lower-order contributions to the cross section and process-independent splitting functions. Because of this general and process-independent structure, it is possible to absorb the uncancelled divergences into the fragmentation functions via collinear renormalisation \cite{Ellis:1991qj}:
\begin{equation}
D_{i\to h}^{\text{bare}}(x) = \sum_j \big(\hat\Gamma_{ij}\otimes D_{i\to h}\big)(x)\;,
\label{eq:collinear-ren}
\end{equation}
where the sum is over all partons. The collinear counterterms $\hat\Gamma_{ij}$ are functions of $x$ and can be specified, not uniquely, within perturbation theory. In practice a choice is made about the finite terms contained in these counterterms. Such a choice implies that the IR renormalized coefficient and fragmentation functions, $d\sigma_i$ and $D_{i\to h}$, are individually scheme dependent however their convolution $d\sigma_h$ is not, as one may expect from an observable. As for parton distribution functions, it is standard practice to define the counterterms $\hat\Gamma_{ij}$ in the $\overline{\rm MS}$ scheme.
The collinear renormalisation eq.~(\ref{eq:collinear-ren}) introduces scale dependence into the renormalised fragmentation functions, which is described by the (time-like) Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations \cite{Altarelli:1977zs,Dokshitzer:1977sg,Gribov:1972ri}:
\begin{equation}
\mu_{Fr}^2\frac{dD_{i\to h}}{d\mu_{Fr}^2}(x,\mu_{Fr}) = \sum_j \big(P^{\text{T}}_{ij}\otimes D_{j\to h}\big)(x,\mu_{Fr})\;,
\label{eq:DGLAP}
\end{equation}
where $P^{\text{T}}_{ij}$ are the time-like splitting functions, known through NNLO \cite{Mitov:2006ic,Moch:2007tx,Almasy:2011eq}, and $\mu_{Fr}$ is the fragmentation factorisation scale, or simply the fragmentation scale. Because fragmentation functions are extracted from experiment at a certain scale, it is necessary to relate fragmentation functions evaluated at two different scales. This is achieved by solving the DGLAP equations eq.~(\ref{eq:DGLAP}). The initial conditions necessary for fully specifying the solution are discussed in sec.~\ref{sec:PFF-NPFF}. The solution of the DGLAP equation has the additional benefit that any large logarithms of the ratio of two scales are resummed with a logarithmic accuracy given by the order of the splitting functions used.
\section{Computational approach}\label{sec:computational-framework}
Fixed order calculations are typically performed using a subtraction scheme. The purpose of a subtraction scheme is to ensure that in any singular limit of the kinematics, the singularities of physical cross sections are matched by those of the relevant subtraction terms and that in those limits, the corresponding final states are indistinguishable. These are the requirements for the numerical integrability of the cross section. If the singular behaviour of the cross section is not matched by its subtraction terms, then a numerically non-integrable singularity remains. If the singular behaviours of the contributions match, but the kinematics are distinct, then the fully inclusive cross section is numerically integrable, but differential and fiducial cross sections may not be. Schematically, a cross section differential in some observable $O$ can be written as
\begin{align}
\frac{d\sigma}{dO} &= \sum_n\int_n d\sigma_n(\{y_i\}_n)\delta(O(\{y_i\}_n)-O) \notag\\
&= \sum_n\int_n \bigg(d\sigma_n\delta(O(\{y_i\}_n)-O)-\sum_m d\sigma_n^m\delta(O^m(\{y_i\}_n)-O)\notag\\
&\phantom{{}= \sum_n\int_n \bigg(d\sigma_n\delta(O(\{y_i\}_n)-O)}+\sum_m d\sigma_n^m\delta(O^m(\{y_i\}_n)-O)\bigg)\;,
\label{eqn:SubtScheme}
\end{align}
where $n$ denotes the number of final-state particles, $d\sigma_n$ is the fully differential $n$-particle cross section, $\{y_i\}_n$ is the set of $n$-particle phase space parameters to be integrated over, e.g.\ the set of momentum components of the particles, $d\sigma_n^m$ is a subtraction term, the integral $\int_n$ is over the full $n$-particle phase space and the dependence of $d\sigma_n$ and $d\sigma_n^m$ on $\{y_i\}_n$ has been omitted on the second and third lines for brevity. As usual, the intent is to integrate the combination of terms on the second line fully numerically, while the integration over the singular behaviour of the terms on the third line is performed analytically.
Due to conceptual differences, a subtraction scheme has to be modified with respect to the case without fragmentation in order to perform calculations involving fragmentation. Such modifications have been made to the sector-improved residue subtraction scheme \cite{Czakon:2010td,Czakon:2011ve,Czakon:2014oma,Czakon:2019tmo} and its implementation in the {\tt Stripper} library, enabling the calculations presented here. The additional complications due to fragmentation have been discussed in the past in the context of NLO subtraction schemes, see e.g. ref.~\cite{Catani:1996vz}, and no further complications are introduced beyond NLO. Nonetheless, the required modifications will also be discussed here for consistency and completeness. All of the necessary changes can be identified by considering which additional requirements fragmentation effectively puts on the calculation. Writing the fragmentation equivalent of eq.\ \eqref{eqn:SubtScheme}, these requirements become apparent:
\begin{align}
\frac{d\sigma}{dO} &= \sum_n\int_0^1dx\int_n d\sigma_n(\{y_i\}_n)D(x)\delta(O(\{y_i\}_n,x)-O) \notag\\
&= \sum_n\int_0^1dx\int_n \bigg(d\sigma_n(\{y_i\}_n)D(x)\delta(O(\{y_i\}_n,x)-O)\notag\\
&\phantom{{}= \sum_n\int_0^1dx\int_n \bigg(}-\sum_m d\sigma_n^m(\{y_i\}_n)D(\tilde{x}^m(\{y_i\}_n,x))\delta(O^m(\{y_i\}_n,x)-O)\notag\\
&\phantom{{}= \sum_n\int_0^1dx\int_n \bigg(}+\sum_m d\sigma_n^m(\{y_i\}_n)D(\tilde{x}^m(\{y_i\}_n,x))\delta(O^m(\{y_i\}_n,x)-O)\bigg)\;,
\label{eqn:SubtSchemeFrag}
\end{align}
where for simplicity a single fragmentation function contributes. A realistic cross section would simply be a sum over such contributions. The functions $\tilde{x}^m$ will be discussed later. The two differences with respect to eq.\ \eqref{eqn:SubtScheme} are multiplication by the fragmentation function and the dependence of the observable on the momentum fraction $x$. For subtraction terms, the dependence of the observable on the phase space parameters changes as well and this point will be discussed first.
In typical calculations without fragmentation, all partons are clustered into jets and all observables depend only on the kinematics of the partons indirectly through the kinematics of jets. For a collinear limit, this means the relative magnitude of the momenta of the collinear partons is irrelevant, as only their sum enters the observable. Because of this, it is sufficient for the kinematics of the subtraction term to correspond to the exact collinear configuration, replacing the collinear partons by a single parton carrying the appropriate combination of their conserved quantities, such as flavour and momentum, as illustrated in fig.\ \ref{fig:CollLimit}. If one of the collinear partons fragments, then the magnitude of its momentum does enter observables, as it is directly related to the momentum of the final hadron via the rescaling by the momentum fraction. For the example shown in fig.\ \ref{fig:CollLimit}, this implies the requirement
\begin{equation}
x_i p_i = p_h = x_{ij} (p_i + p_j)\;,
\end{equation}
where $x_i$ and $x_{ij}$ are the momentum fractions for the parton $i$ and the combination of $i$ and $j$, respectively. Similarly, the flavour of the fragmenting parton determines the size of the contribution through the fragmentation function. When moving to the subtraction kinematics, it is thus necessary to retain the information on the contribution from the fragmenting parton to the total momentum of the collinear partons, which for the example above is e.g.\ the ratio between $p_i$ and $p_i+p_j$, and the flavour of the fragmenting particle.
\begin{figure}[t]
\centering
\includegraphics[width=1\textwidth]{fig/FragCollDiagram.pdf}
\caption{Observable kinematics in a collinear limit. When the partons $i$ and $j$ become collinear, the jets they initiate become indistinguishable from a jet initiated by a single particle carrying the sum of their momenta (blue shaded regions). If a single hadron $h$ is identified in the final state (red), then the fraction of the fragmenting particle's momentum carried by $h$ is smaller for the combination of $i$ and $j$ than for $i$, since the momentum $p_h$ does not change.}
\label{fig:CollLimit}
\end{figure}
There is an important point to stress here concerning soft limits, as the situation is slightly different. A singular soft limit occurs when the total energy of a flavourless set of partons -- containing gluons and equal numbers of quarks and anti-quarks of each flavour -- becomes small. The standard observation is that a configuration containing a zero-energy flavourless set of partons cannot be distinguished from one where this set is removed from the final state. The kinematics of the subtraction term thus corresponds to the exact soft configuration, removing the zero-energy flavourless set of partons from the final state. The statement that zero-energy, flavourless sets of partons can be removed from a final state without changing any observable is no longer true if one of those partons fragments, as the hadron is assumed to always be observable on its own. One could in principle proceed as for the collinear case and construct the subtraction kinematics as usual, keeping the information about the flavour and momentum, the latter being zero by definition, of the fragmenting parton in the soft limit. However, this yields contributions where the hadron always carries zero momentum. Not only is this an unphysical configuration, as the hadron has a non-zero mass, the factorisation of the cross section into the hard process and a fragmentation function only applies if the hard scale of the hadron, e.g.\ its transverse momentum, is much larger than its mass \cite{Collins:1989gx}. Additionally, fragmentation functions are divergent as the momentum fraction goes to zero, so even if these soft limits of the partonic cross section were regulated, the hadronic cross section would still be divergent. Because of this, there are no (integrated) subtraction terms regulating the soft limit of a fragmenting parton.
By considering exact singular limits, it has been explained that the kinematics of subtraction terms must be modified. The exact dependence of the kinematics on the full phase space parametrisation is arbitrary, only in singular limits must the kinematics of the cross section and its corresponding subtraction terms match, i.e.:
\begin{equation}
O^m(\{y_i\}_n,x) \to O(\{y_i\}_n,x)\;,
\label{eqn:ObservableLimit}
\end{equation}
where the limit is any limit which is supposed to be regulated by $d\sigma_n^m$ and eq.\ \eqref{eqn:ObservableLimit} should hold for all infrared-safe observables, where the momentum of the hadron is considered an infrared-safe quantity within this framework. Without fragmentation, the analytic integration performed to obtain the integrated subtraction term relies on the fact that $O^m$ does not depend on the parameters integrated over. As explained above, this may not be the case when one of the partons fragments. An example would be a subtraction term which regulates both a collinear and a soft singularity, which is a part of e.g.\ the Catani-Seymour dipole subtraction scheme \cite{Catani:1996vz}, with one of the partons fragmenting. In this case, the energy of the hadron would depend on the energy of the soft parton, so it would depend on the parameter parameterising the soft limit, spoiling the ability to perform this integration fully analytically. An implementation of a subtraction scheme containing such subtraction terms would therefore require laborious modifications before general fragmentation computations can be performed.
Here a critical simplification exists for the sector-improved residue subtraction scheme with respect to many other subtraction schemes. If there are no subtraction terms which regulate more than one type of singularity, i.e.\ every subtraction term is designed to counter a singularity occurring as all elements of a single set of phase space parameters simultaneously approach a singular point, then the kinematics of each subtraction term can be chosen to always match those of the cross section in the subtraction term's characteristic singular limit. Because this is a constant with respect to the variables which parameterise the subtraction term and are integrated over analytically in the integrated subtraction term, the integrated subtraction is unchanged by the introduction of fragmentation (aside from an overall factor given by the fragmentation function), avoiding the need to redo any analytic integration previously performed for the original subtraction scheme. For this reason, the sector-improved residue subtraction scheme is particularly suited for the extension to fragmentation, since it does not contain any subtraction terms which regulate multiple singularities \cite{Czakon:2014oma}.
Aside from changes to the kinematics of the final state, the inclusion of fragmentation also modifies the size of the contributions of different phase space points via multiplication by the fragmentation function, as shown in eq.\ \eqref{eqn:SubtSchemeFrag}. There is a certain amount of freedom when it comes to the point at which the fragmentation function is evaluated for a subtraction term, written in eq.\ \eqref{eqn:SubtSchemeFrag} as the functions $\tilde{x}^m(\{y_i\}_n,x)$. The only strictly necessary condition is that a subtraction term matches the singular behaviour of the cross section in certain singular limits. This requires that in any singular limit, the fragmentation function is evaluated at the same point for both the cross section and its corresponding subtraction terms, i.e.:
\begin{equation}
\tilde{x}^m(\{y_i\}_n,x) \to x\;,
\end{equation}
where the limit is again any limit which is supposed to be regulated by $d\sigma_n^m$. The most simple choice
\begin{equation}
\tilde{x}^m(\{y_i\}_n,x) \equiv x
\label{eqn:SubtMomentumFraction}
\end{equation}
is made here. Note that in order to reuse the integrated subtraction terms from the case without fragmentation as explained above, $\tilde{x}^m(\{y_i\}_n,x)$ must fulfill an additional condition: it should not depend on the parameters parameterizing the singular limit regulated by $d\sigma_n^m$. This is trivially satisfied by the choice shown in eq.\ \eqref{eqn:SubtMomentumFraction}.
The modifications discussed up until now are sufficient to perform calculations with fragmentation, but often lead to suboptimal numerical convergence. The reason for this is that while the kinematics of the cross section and one of its subtraction terms match in the singular limit, they do not in the remainder of the phase space. It is thus possible for both contributions to be large with opposite signs, but instead of mostly cancelling each other, the contributions are added to different bins of a calculated histogram. This missed-binning increases the fluctuations within individual bins, increasing their Monte Carlo uncertainty for a given number of events and thus reducing the rate of numerical convergence. To mitigate this, one can rescale the momentum fraction $x$ for each contribution on an event-by-event basis, such that the value of an observable of choice is always identical for all contributions for any given event. If this reference observable is now binned in a histogram, then missed-binning cannot occur by definition, potentially vastly improving the numerical convergence.
The final difference with respect to calculations without fragmentation is the introduction of collinear renormalisation counterterms for fragmentation functions. These are conceptually identical to those for PDFs and it is well-known how to obtain them in terms of splitting functions. The only difference with respect to the renormalisation of PDFs is the need to use time-like splitting functions, which differ from the space-like ones starting at NLO \cite{Curci:1980uw,Furmanski:1980cm}. For completeness, in appendix \ref{sec:collinear} we present the explicit expressions for the collinear counterterms while in appendix \ref{sec:tt-xsec-fragmentation} we give in some detail the structure of the cross section for the process $pp\to t\bar t+X\to B+X$.
\section{Perturbative and Non-Perturbative Fragmentation Functions for Heavy Flavor Fragmentation}\label{sec:PFF-NPFF}
The fragmentation functions used in this paper are based on the perturbative fragmentation function approach \cite{Mele:1990cw}, in which all fragmentation functions for the production of heavy-flavoured hadrons can be related to a single non-perturbative fragmentation function (NPFF) via convolutions with perturbatively calculable coefficients, called perturbative fragmentation functions (PFFs):
\begin{equation}
D_{i\to h}(x) = \big(D_{i\to q}\otimes D_h^{\text{NP}}\big)(x)\;,
\end{equation}
where $i$ can be any parton, $h$ is the heavy-flavoured hadron and $q$ is the heavy quark. The heavy-quark PFFs were originally derived at NLO \cite{Mele:1990cw} and have since been computed at NNLO as well \cite{Melnikov:2004bm,Mitov:2004du}. The only ingredient required to compute FFs for the production of heavy-flavoured hadrons is thus the NPFF. Typically, NPFFs are extracted from $e^+e^-$ data, however, theoretically motivated ones also exist \cite{Braaten:1994bz,Aglietti:2006yf}.
In the remainder of this work we will be interested in the case where the heavy quark is the bottom, i.e. $q=b$, and the heavy-flavored hadron is a $b$-flavored one, i.e. $h=B$.
At present, no such extraction at NNLO employing the PFF approach is available in the literature. For this reason, three different sets of FFs were obtained from two different extractions, each set corresponding to a different compromise. A third extraction, which follows an approximation of the PFF approach, was presented in ref.~\cite{Salajegheh:2019ach}, but has not been used here.
The first two sets of FFs are based on the extraction of ref.~\cite{Fickinger:2016rfd}. The FF of that paper is not based on the PFF approach, instead relying on effective field theory calculations. Nonetheless, a NPFF was extracted at NLO and NNLO, including NNLL and N$^3$LL large-$x$ resummation, respectively. Unfortunately, due to the different approach to the computation of FFs, there is no simple relation between the FF of that paper and one computed within the PFF approach. A reasonable conversion from one type of FF to the other has to be chosen. Another important point is that the extracted FF corresponds to the non-singlet (NS) combination, i.e.\ the difference between the bottom and anti-bottom FFs:
\begin{equation}
D_B^{\text{NS}}(\mu_{Fr},x) = D_{b\to B}(\mu_{Fr},x)-D_{\bar{b}\to B}(\mu_{Fr},x)\;.
\end{equation}
The set of FFs used most centrally in this paper is labelled ``FFKM". Its initial conditions are obtained by taking the extracted non-singlet function of ref.~\cite{Fickinger:2016rfd} evaluated at the initial scale $\mu_{Fr}=\mu_0$ with $\mu_{0}=m_b=4.66$ GeV, then calculating the FFs other than the bottom-quark FF from the PFFs and the extracted NPFF and, finally, adding the anti-bottom FF to the non-singlet one to obtain the full bottom FF:
\begin{align}
D_{i\to B}(\mu_{0},x) &= \big(D_{i\to b}\otimes D_B^{\text{NP}}\big)(\mu_{0},x)\;,\;\;i\neq b\;,\\
D_{b\to B}(\mu_{0},x) &= D_B^{\text{NS}}(\mu_{0},x)+D_{\bar{b}\to B}(\mu_{0},x)\;.
\end{align}
The FFs at any other scale $\mu_{Fr} > \mu_0$ are then obtained by evolving these initial conditions using the DGLAP evolution library {\tt APFEL} \cite{Bertone:2013vaa}.
An alternative construction labelled ``FFKM(2)" is to proceed as for the FFKM set, but as a final step the non-singlet contribution at each scale is replaced by the non-singlet contribution at that scale as provided by the authors of ref.~\cite{Fickinger:2016rfd}. This is not equivalent to the FFKM set, since the FF of ref.~\cite{Fickinger:2016rfd} does not satisfy the non-singlet DGLAP evolution equation.
The third and final set of FFs, labelled ``CNO", is obtained by taking the extraction of ref.~\cite{Cacciari:2006vy}. This extraction was performed using the PFF approach, but only at NLO including NLL large-$x$ resummation. This time $\mu_{0}=m_b=4.75$ GeV.
\begin{table}
\centering
\begin{tabular}{| c || c | c | c | c |}
\hline
FF set & NPFF & PFF & Large-$x$ & DGLAP\\\hline\hline
FFKM/FFKM(2) @ NLO & NLO & NLO & NNLL & NLL\\\hline
FFKM/FFKM(2) @ NNLO & NNLO & NNLO & N$^3$LL & NNLL\\\hline
CNO @ NLO & NLO & NLO & NLL & NLL\\\hline
CNO @ NNLO & NLO & NNLO & NLL & NNLL\\\hline
\end{tabular}
\caption{The differences between the FFKM/FFKM(2) sets and the CNO set at NLO and NNLO in terms of perturbative and logarithmic orders. NPFF refers to the perturbative order at which the extraction of non-perturbative parameters was performed. PFF refers to the perturbative order of the PFFs used. The column ``large-$x$" shows the logarithmic order of the resummation of logarithms of $1-x$, while the column labelled ``DGLAP" indicates the logarithmic order of the DGLAP resummation.}
\label{tab:FragFuncOrders}
\end{table}
NLO and NNLO versions of all three sets were constructed. The perturbative and logarithmic orders of different components of the fragmentation functions are shown in \linebreak table\ \ref{tab:FragFuncOrders}. All FFs are symmetrised with respect to particles and anti-particles. The scale evolution is always performed using {\tt APFEL}, where the value and running of $\alpha_s$ are always chosen to match those of the PDF set used at the same order. As an alternative to performing the evolution with {\tt APFEL}, the {\tt MELA} \cite{Bertone:2015cwa} library could have been used instead, as was e.g.\ done in ref.~\cite{Ridolfi:2019bch} to perform a detailed study of the evolution of heavy-quark fragmentation functions. For simplicity, neither {\tt MELA} nor the results of ref.~\cite{Ridolfi:2019bch} have been used here.
In order to be able to estimate uncertainties due to the errors on the extracted FFs, multiple versions of all sets were constructed, corresponding to taking the extracted non-perturbative parameters and independently varying them by one standard deviation. Since there is only one parameter for the FFKM and FFKM(2) sets, this leads to three variations each, while the CNO set involves two parameters, leading to 9 variations. For the CNO set, correlations between the parameters are ignored.
All three FFs were found to be within reasonable agreement with each other, suggesting none of the individual compromises are particularly significant.
\section{Applications}\label{sec:applications}
\subsection{$b$-fragmentation in top-quark decay}\label{sec:top-decay}
As a first application we consider the process $t\to B+W+X$ with the subsequent decay $W\to \ell+\nu$ in NNLO in QCD. We work with top quark pole mass $m_t = 172.5\;\text{GeV}$. We use fixed scale choices for the renormalization and fragmentation scales: $\mu_R=\mu_{Fr}=m_t/2$. The rationale for this scale choice is discussed in the next section. Scale variation is done following the standard 7-point scale variation approach: $1/2 \le \mu_{R}/\mu_{Fr} \le 2$. Perturbative calculations for top decay at any accuracy (LO, NLO or NNLO) are always convolved with FF at NNLO. In all cases the value of the strong coupling ${\alpha_s}$ is taken from the {\tt LHAPDF} interface \cite{Buckley:2014ana} as produced by the {\tt NNPDF3.1} NNLO pdf set \cite{Ball:2017nwa}. Further details about this process and its setup can be found in appendix \ref{sec:tt-xsec-fragmentation} as well as in ref.~\cite{Czakon:2020qbd}.
In all observables discussed in this section we implement an energy cutoff of $E(B) > 5$ GeV. This cutoff helps us avoid the low $x$ region of the FFs. Excluding this region is not consequential for this work since in our implementation all power corrections $\sim (m_b)^n, n\ge 2$, are neglected and our predictions are not valid in the very low $x$ region anyway.
As a check on our implementation we have verified that our calculation satisfies the momentum conservation sum rule, see appendix \ref{sec:sum-rules} for details.
We study the following observables: the invariant mass of the lepton and the hadron $m(B\ell)$ and the energy fraction of the $B$-hadron to its maximum energy $E(B)/E(B)_{\text{max}}$, where
\begin{equation}
E(B)_{\text{max}} = \frac{m_t^2-m_W^2}{2 m_t}\,.
\label{eq:EB-max}
\end{equation}
The observables are shown in fig.~\ref{fig:top-decay-order}. In both cases we show the absolute distributions at different perturbative orders for the FFKM NNLO fragmentation function. The lower panel shows the ratio to the NLO result. The colored bands correspond to 7-point scale variation. In fig.~\ref{fig:top-decay-scale} we show a breakdown of the NNLO scale variation due to $\mu_R$ and $\mu_{Fr}$. Each one of these scales is varied (3-point variation) while the other scale is kept fixed at its central value. Similarly, fig.~\ref{fig:top-decay-fragmentation} shows the fragmentation function variation for the default FFKM fragmentation function at NNLO. Also shown are the central predictions at NNLO based on the other two FF sets: FFKM(2) and CNO.
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{fig/decay_mlB_main-FFKM_NNLO.pdf}
\includegraphics[width=0.49\textwidth]{fig/decay_Efrac_main-FFKM_NNLO.pdf}
\caption{Absolute differential top decay width as a function of the invariant mass $m(B\ell)$ (left) and the energy fraction $E(B)/E(B)_{\text{max}}$ (right). All curves are convoluted with the same FF: FFKM at NNLO. Shown is comparison for different perturbative orders: LO, NLO and NNLO.}
\label{fig:top-decay-order}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{fig/decay_mlB_scaleVar-FFKM_NNLO.pdf}
\includegraphics[width=0.49\textwidth]{fig/decay_Efrac_scaleVar-FFKM_NNLO.pdf}
\caption{As in fig.~\ref{fig:top-decay-order} but showing the scale variation of the NNLO prediction: $\mu_R$-only vs. total scale variation (upper plot) and $\mu_{Fr}$-only vs. total scale variation (lower plot).}
\label{fig:top-decay-scale}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{fig/decay_mlB_FF_Var-NNLO.pdf}
\includegraphics[width=0.49\textwidth]{fig/decay_Efrac_FF_Var-NNLO.pdf}
\caption{As in fig.~\ref{fig:top-decay-order} but showing the fragmentation function variation of the default FF FFKM at NNLO. Shown also are the central predictions for the other two FF at NNLO: FFKM(2) and CNO.}
\label{fig:top-decay-fragmentation}
\end{figure}
The invariant mass differential width $m(B\ell)$ is of particular interest since it is suitable for extracting the top quark mass with high-precision \cite{Kharchilava:1999yj}. It has previously been studied with NLO precision in ref.~\cite{Biswas:2010sa}.
The normalized energy spectrum is also interesting in top mass determinations since it directly exposes the fragmentation function. Therefore it allows one to directly assess the sensitivity of this observable to $b$-fragmentation and its potential for measuring NPFF's. This observable has been studied in NLO+NLL QCD in \cite{Corcella:2001hz,Cacciari:2002re,Corcella:2005dk,Kniehl:2012mn,Nejad:2013fba,Nejad:2016epx}. The analytic expressions of the coefficient functions for both $m_b=0$ and $m_b\neq 0$ are known through NLO in QCD.
\subsection{$b$-fragmentation in top-quark pair-production and decay at the LHC}\label{sec:top-prod-and-decay}
In this section we present our predictions for the following $B$-hadron distributions in dilepton $t\bar t$ events at the LHC: the invariant mass of the $B$-hadron and charged lepton $m(B\ell)$ as well as $B$-hadron's energy $E(B)$. These two distributions are the $t\bar t$ equivalents of the distributions discussed in sec.~\ref{sec:top-decay} in the context of top quark decay. The advantage of working with $m(B\ell)$ and $E(B)$ is that they are defined in the detector frame and are, therefore, directly measurable without the need for reconstructing frames associated with the top quark. Both $m(B\ell)$ and $E(B)$ are of prime interest in the context of top quark mass determination at the LHC and have been extensively studied in the past in NLO QCD \cite{Kharchilava:1999yj,Biswas:2010sa,Agashe:2012bn,Agashe:2016bok}.
The setup of the present calculation, which is closely related to the one in ref.~\cite{Czakon:2020qbd}, see also appendix \ref{sec:tt-xsec-fragmentation}, is as follows. We utilize the pdf set {\tt NNPDF3.1}. Its order is chosen in such a way that it matches the order of the perturbative calculation. The value of the strong coupling constant is obtained from the {\tt LHAPDF} library as provided by the {\tt NNPDF3.1} pdf set. The order of the strong coupling constant evolution in the perturbative calculation is matched to the order of the pdf while the order of the coupling in the FF evolution is matched to the order of the FF.
The pdf variation utilizes the so-called reduced pdf set, see ref.~\cite{Czakon:2020qbd} for details. Our predictions are based on fixed central scales
\begin{equation}
\mu_R=\mu_F=\mu_{Fr}={m_t\over 2}\,.
\label{eq:scales}
\end{equation}
The reasons behind this scale choice are as follows. A fixed scale choice is well-justified in the kinematic ranges considered in this work. Furthermore, the use of fixed scales (instead of dynamic scales) can simplify the interpretation of the results especially when there are many scales and perturbative orders. The specific value of the central scale, $m_t/2$, is motivated by the study \cite{Czakon:2016dgf} on stable $t\bar t$ production. One may wonder if a central scale $m_t$ and not $m_t/2$ is more appropriate for the description of top decay. While both choices are equally suitable in principle and can be implemented in practice, we decided to use the scale choice (\ref{eq:scales}) in this first work on $b$-fragmentation in $t\bar t$ production and decay in order to make the interpretation of the scale variation of the prediction as transparent as possible since this way all three scales appearing in this calculation have the same central values.
Scale variation is defined through a 15-scale variation, i.e. scaling up and down the {\it common} central scale by a factor of 2, subject to the constraints
\begin{eqnarray}
&& 1/2 \le \mu_{R}/\mu_{F} \le 2\,, \nonumber\\
&& 1/2 \le \mu_{R}/\mu_{Fr} \le 2\,, \label{eq:scales-var}\\
&& 1/2 \le \mu_{F}/\mu_{Fr} \le 2\,.\nonumber
\end{eqnarray}
We use the $G_F$ scheme with the following parameters
\begin{eqnarray}
m_W &=& 80.385 \; \text{GeV} \,,\nonumber\\
\Gamma_W &=& 2.0928\;\text{GeV}\,,\nonumber\\
m_Z &=& 91.1876 \; \text{GeV} \,,\nonumber\\
\Gamma_Z &=& 2.4952\;\text{GeV}\,,\nonumber\\
G_F &=& 1.166379\cdot10^{-5} \;\text{GeV}^{-2} \,,\nonumber\\
\alpha &=&\frac{\sqrt{2} G_F}{\pi} m_W^2\left(1-(m_W/m_Z)^2\right)\,.
\label{eq:GF-scheme}
\end{eqnarray}
Defining $\xi=(m_W/m_t)^2$, the leading order top-quark width is computed from
\begin{equation}
\Gamma_t^{(0)} = G_F \frac{m_t^3}{8\pi\sqrt{2}}(1-\xi)^2(1+2\xi) = 1.48063 \;\text{GeV}~ ({\rm for}~ m_t = 172.5\;\text{GeV})\,.
\end{equation}
Our calculations are subject to typical phase space cuts:
\begin{equation}
p_T(B) \geq 10\, {\rm GeV} ~~~,~~~ |\eta(B)| \leq 2.4 \,.
\label{eq:cuts}
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{fig/mlB_main-FFKM_NNLO.pdf}
\includegraphics[width=0.49\textwidth]{fig/EB_main-FFKM_NNLO.pdf}
\caption{Absolute differential top-quark pair production and decay cross section as a function of the invariant mass $m(B\ell)$ (left) and the $B$-hadron energy $E(B)$ (right). All curves are convoluted with the same FF: FFKM at NNLO. Comparisons for LO, NLO and NNLO are shown.}
\label{fig:ttbar-order}
\vskip 8mm
\includegraphics[width=0.49\textwidth]{fig/mlB_scaleVar-FFKM_NNLO.pdf}
\includegraphics[width=0.49\textwidth]{fig/EB_scaleVar-FFKM_NNLO.pdf}
\caption{As in fig.~\ref{fig:ttbar-order} but showing the scale variation of the NNLO prediction: $\mu_R$-only vs. total (upper plot), $\mu_F$-only vs. total (middle plot) and $\mu_{Fr}$-only vs. total scale variation (lower plot).}
\label{fig:ttbar-scales}
\vskip 8mm
\includegraphics[width=0.49\textwidth]{fig/mlB_PDF_FF_Var-NNLO.pdf}
\includegraphics[width=0.49\textwidth]{fig/EB_PDF_FF_Var-NNLO.pdf}
\caption{As in fig.~\ref{fig:ttbar-order} but showing the fragmentation and pdf variations of the default FFKM FF. Also shown are the central predictions for the other two FF at NNLO: FFKM(2) and CNO.}
\label{fig:ttbar-fragmentation}
\end{figure}
The predictions for the absolute distributions $m(B\ell)$ and $E(B)$ through NNLO in QCD are shown in fig.~\ref{fig:ttbar-order}. The bands around the three central predictions indicate their 15-point scale variation. For both distributions we observe that the reduction of the scale uncertainty when going both from LO to NLO and from NLO to NNLO is substantial. The NNLO scale variation is about couple of percent in most bins. Notably, for the scale choice (\ref{eq:scales}) the NNLO scale variation is asymmetric, unlike the LO and NLO ones. Because of this asymmetry it is more useful to quantify the total width of the NNLO scale variation band which never exceeds 10\% and, in fact, in most bins is about half that value. This implies that the corrections due to missing higher order effects are probably at the one-percent level and thus rather small.
We also observe that the size of the higher order corrections in both observables is moderate and in all cases the higher-order corrections are contained within the corresponding lower order scale variation band. The only exception is the lowest bin of the $E(B)$ distribution however it is worth keeping in mind that this bin is strongly impacted by the cuts (\ref{eq:cuts}). The NNLO/NLO K-factor is rather small and tends to be within $5$\% for most bins in both distributions. It has a non-trivial shape relative to the NLO predictions once one accounts for the small size of the NNLO scale uncertainty band.
The region of the $m(B\ell)$ distribution above about 150 GeV is impacted by corrections beyond the narrow width approximation which is utilized in this work (see ref.~\cite{Czakon:2020qbd} for details). The monotonic increase in the shape of the NNLO/NLO $K$-factor of the $E(B)$ distribution suggests that at NNLO the maximum of that distribution is shifted towards higher values of $E(B)$ relative to NLO. Although in this paper we are not able to quantify this shift with sufficient precision, we note that it may significantly affect any extraction of the top quark mass based on the proposal in refs.~\cite{Agashe:2012bn,Agashe:2016bok}. A more precise estimate of this effect is possible but it will require a dedicated and more refined calculation which we leave for a future work.
With the help of fig.~\ref{fig:ttbar-scales} one can assess the origin of the scale variation in these two observables at NNLO. To that end we have shown a breakdown of the scale variation due to one scale at a time (the other two being fixed at their central values) and compared to the total scale variation eq.~(\ref{eq:scales-var}). It immediately becomes apparent that the bulk of the scale variation is due to the renormalization scale $\mu_R$. The second largest contribution is due to the fragmentation scale $\mu_{Fr}$ while the contribution due to the factorization scale alone is tiny.
In fig.~\ref{fig:ttbar-fragmentation} we compare at NNLO the three main sources of uncertainty for these two distributions: scale, pdf and fragmentation uncertainties. The variations shown are for the default FFKM fragmentation function. As an alternative measure for the fragmentation uncertainty we show the central predictions based on the two alternative FFs: FFKM(2) and CNO. It is evident from this figure that scale variation is the dominant source of uncertainty. This is true for all bins of both distributions. The second largest uncertainty is the one due to fragmentation followed by the pdf uncertainty. The differences between the three fragmentation functions tends to be consistent with the estimate of the fragmentation uncertainty although in some bins that difference is as large as twice the value of the fragmentation uncertainty estimate.
In summary, the total uncertainty of the NNLO predictions for the $m(B\ell)$ and $E(B)$ distributions is within 5\% for almost all bins and is dominated by the scale uncertainty. While in this first NNLO work on this subject we have considered the 15-point scale variation eq.~(\ref{eq:scales-var}) around the central scale eq.~(\ref{eq:scales}) as the most straightforward generalization of the usual restricted scale variation in processes involving a single factorization scale, it may be beneficial to revisit this in the future and try to assess the impact and merits of a more restrictive scale variation and/or different dynamic or fixed scale choices.
\subsection{Extraction of $B$-hadron FFs from $t\bar t$ events}
The focus of the previous discussions was on predictions for LHC observables given a set of fragmentation functions. Due to the limitations of the existing extractions from $e^+e^-$ data one may naturally ask the question if LHC data can be used to improve the extraction of non-perturbative FFs. In this section we address this question in the context of $b$-fragmentation in $t\bar t$ events. As it will become clear shortly, this study can easily be extended to other processes like direct $b$ production.
In principle, one can use any well-measured LHC $B$-hadron distribution to fit the NPFF. In order to increase the sensitivity to the NPFF it would be ideal if one uses distributions that are as closely related to the FF's as possible. An example for such a distribution is the $B$ energy spectrum in top quark decay discussed in sec.~\ref{sec:top-decay}. The only drawback of this distribution is that it requires the reconstruction of the decaying top quark and, thus, cannot be measured directly. It is therefore preferable to have distributions with similar sensitivity to NPFF that are directly defined in the lab frame.
In this work we propose one such distribution: the ratio $p_T(B)/p_T(j_B)$ of the transverse momentum of the identified $B$-hadron with respect to the transverse momentum of the jet that contains it. We cluster jets with the anti-$k_T$ algorithm \cite{Cacciari:2008gp} with radius $R = 0.8$. We require that this jet fulfills $p_T(B) \leq p_T(j_B)$ and $|\eta(j_B)| < 2.4$, consistent with the cuts in eq.~(\ref{eq:cuts}). Note that both the $B$-hadron and its fragmentation remnants are included in this jet-clustering, see the discussion around eq.~(\ref{eq:splitting-kinematics}).
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{fig/jetratio_main-FFKM_NNLO.pdf}%
\includegraphics[width=0.5\textwidth]{fig/jetratio_PDF_FF_Var-NNLO.pdf}
\caption{Absolute differential cross section as a function of the transverse momentum ratio $p_T(B)/p_T(j_B)$ in top-quark pair production and decay. Comparison of the FFKM NNLO FF for different perturbative orders showing scale variation (left) and comparison of FF, pdf and scale uncertainties (right). PDFs are matched to the corresponding perturbative order. The scale variation bands are based on 15-point scale variation.}
\label{fig:prod_jetratio}
\end{figure}
The differential $p_T(B)/p_T(j_B)$ distribution is shown in fig.~\ref{fig:prod_jetratio}. The shape and behavior of this observable at different perturbative orders is fairly similar to the $E(B)/E(B)_{\rm max}$ distribution in top decay shown in fig.~\ref{fig:top-decay-order}. Higher order corrections are largely consistent with the scale uncertainty bands of the lower perturbative order. The size of scale variation at NNLO is below 5\% except for large values of $p_T(B)/p_T(j_B)$ where it starts to increase. We have checked that, just like in the case of $m(B\ell)$ and the $B$-hadron energy $E(B)$ distributions shown in fig.~\ref{fig:ttbar-scales}, the scale variation in this observable is driven by the renormalization scale and in much smaller degree, by the fragmentation scale $\mu_{Fr}$. The variation due to $\mu_F$ alone is negligible.
From fig.~\ref{fig:prod_jetratio} one can also conclude that for intermediate and large values of $p_T(B)/p_T(j_B)$ the uncertainty of this observable is driven by the uncertainty in the non-perturbative fragmentation function. For values $p_T(B)/p_T(j_B) \lesssim 0.5$ the total uncertainty is dominated by the scale variation. The pdf uncertainty is negligible throughout the kinematic range. These observations imply that this observable has strong potential for constraining FF at NNLO in QCD at intermediate and large values of $x$.
We next probe the sensitivity of the $p_T(B)/p_T(j_B)$ distribution to the following parameters: the jet algorithm, the jet size and the $B$-hadron $p_T$ cut. Our aim is to determine optimal values for these parameters which will facilitate the extraction of the fragmentation function.
In fig.~\ref{fig:prod_jetratio_jetalg} we show the $p_T(B)/p_T(j_B)$ distribution for three different jet algorithms: anti-$k_T$, $k_T$ \cite{Catani:1993hr,Ellis:1993tq} and flavour-$k_T$ \cite{Banfi:2006hf}. For ease of the comparison all jet algorithms have the same jet size $R=0.4$. For each jet algorithm we show the LO, NLO and NNLO corrections, including their scale variation. The pattern of higher-order corrections is almost identical for the three jet algorithms. The three algorithms produce very similar distributions. This can be seen in the top left plot which shows a comparison of the three jet algorithms at NNLO. There we see that the anti-$k_T$ and $k_T$ algorithms lead to almost identical behavior. The flavour-$k_T$ algorithm also produces almost identical distribution for values of $p_T(B)/p_T(j_B)$ above about 0.6, but starts to deviate from the other two jet algorithms for lower values. Still the difference between the flavour-$k_T$ and the other two algorithms is much smaller than the NNLO scale uncertainty. These comparisons indicate that from the viewpoint of this observable all three jet algorithms, anti-$k_T$, $k_T$ and flavour-$k_T$, are suitable for the extraction of NPFF in $t\bar t$ events.
Another comment about the use of the anti-$k_T$ and $k_T$ algorithms in this calculation is in order. It is well known \cite{Banfi:2006hf} that starting from NNLO, flavorless jet algorithms are not automatically infrared (IR) safe when applied to flavored problems. To achieve IR safety of jets in the flavored context, dedicated jet algorithms are needed. One such proposal is the flavour-$k_T$ algorithm of ref.~\cite{Banfi:2006hf}. Related ideas have been discussed in refs.~\cite{Buckley:2015gua,Dai:2018ywt}.
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{fig/jetratio_JetVar_sep1.pdf}
\includegraphics[width=0.49\textwidth]{fig/jetratio_JetVar_sep2.pdf}\\
\includegraphics[width=0.49\textwidth]{fig/jetratio_JetVar_sep3.pdf}
\includegraphics[width=0.49\textwidth]{fig/jetratio_JetVar_sep4.pdf}
\caption{As in fig.~\ref{fig:prod_jetratio} but comparing different jet-algorithms: anti-$k_T$, $k_T$ and flavour-$k_T$.}
\label{fig:prod_jetratio_jetalg}
\end{figure}
The use of the anti-$k_T$ and $k_T$ algorithms is justified in the present work because of the special nature of the observables computed here. Unlike a typical fixed order calculation, in this work we cluster not just partons but the $B$-hadron and its accompanying remnants. Since by construction all collinear singularities have been regulated at the level of the partonic cross-section, a jet algorithm is no longer needed to ensure IR finiteness of the calculation. In this sense our calculation is closer to an experimental setup than to a typical fixed order partonic jet calculation. Since the fixed-order part of the $B$-hadron production cross-section contains terms of the type $\log^n(m)$ we expect that they will also be present in the corresponding jet calculation. However due to the NNLL DGLAP resummation they are likely to not play any role.
We next consider the effect of the jet size $R$. In fig.~\ref{fig:prod_jetratio_R} we compare predictions based on the anti-$k_T$ algorithm with jet sized $R=0.2, 0.4, 0.6, 0.8$. We observe an expected pattern of higher order corrections: as the jet size decreases, the observable becomes less inclusive which results in decreased perturbative convergence. This is manifested through the increase of scale uncertainty at all orders considered in this calculation as well as larger $K$-factors. From this we concluded that from the viewpoint of theory, larger jet sizes are better for extracting fragmentation functions from the $p_T(B)/p_T(j_B)$ distribution.
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{fig/jetratio_RVar_sep1.pdf}
\includegraphics[width=0.49\textwidth]{fig/jetratio_RVar_sep2.pdf}\\
\includegraphics[width=0.49\textwidth]{fig/jetratio_RVar_sep3.pdf}
\includegraphics[width=0.49\textwidth]{fig/jetratio_RVar_sep4.pdf}
\caption{As in fig.~\ref{fig:prod_jetratio} but comparing different jet sizes $R=0.2, 0.4, 0.6$ and $0.8$.}
\label{fig:prod_jetratio_R}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{fig/jetratio_pTVar_sep1.pdf}
\includegraphics[width=0.49\textwidth]{fig/jetratio_pTVar_sep2.pdf}\\
\includegraphics[width=0.49\textwidth]{fig/jetratio_pTVar_sep3.pdf}
\includegraphics[width=0.49\textwidth]{fig/jetratio_pTVar_sep4.pdf}
\caption{As in fig.~\ref{fig:prod_jetratio} but comparing different values of the $p_T(B)$ cut: $p_T(B) > 10, 20, 30$ GeV.}
\label{fig:prod_jetratio_PTcut}
\end{figure}
Finally, we consider the impact of the low $p_T(B)$ cut. To that end in fig.~\ref{fig:prod_jetratio_PTcut} we show the $p_T(B)/p_T(j_B)$ distribution computed for three different values of this cut: $10, 20$ and $30$ GeV. We show the LO, NLO and NNLO distribution for each $p_T(B)$-cut as well as a comparison of the three cuts at NNLO. In all cases we use same jet algorithm: anti-$k_T$ with $R=0.4$. We observe that the intermediate-to-large $p_T(B)/p_T(j_B)$ region is not very much affected by the value of the low $p_T(B)$ cut which, in turn, means that the extracted fragmentation function at intermediate or large values of $x$ is not very sensitive to this cut. From the top-left plot in fig.~\ref{fig:prod_jetratio_PTcut} we observe that in this region the NNLO scale variation for all cut values is approximately the same.
On the other hand, the value of the cut has a strong impact on the distribution at low $p_T(B)/p_T(j_B)$. As the $p_T(B)$ cut is lowered, the distribution becomes divergent in fixed order perturbation theory. This is consistent with the observed behavior of the distribution, which for smaller values of the $p_T(B)$ cut starts to show the typical signs of bad perturbative convergence: larger scale variation bands and increased $K$-factors. Finally, one should keep in mind that our calculation is performed with a massless $b$ quark and therefore misses corrections $\sim (m_b)^n$ for $n\geq 2$. For this reason it would be incomplete at low values of $p_T(B)$. For these reasons we conclude that if experimentally viable, a larger $p_T(B)$ cut would be preferable since it leads to more stable predictions and since any missing $b$ mass corrections are automatically rendered negligible or at least significantly reduced in importance.
\section{Conclusions}\label{sec:conclusions}
Heavy flavor production at hadron colliders has traditionally demanded improved theoretical precision which matches the large statistics accumulated at colliders like the Tevatron and the LHC. In processes like $b$ and $c$ production, identified $b$- or $c$-flavored hadrons are copiously produced with transverse momenta much larger than their masses. For such kinematics the heavy quark mass plays the role of an infrared regulator. In an appropriately defined formalism, like the perturbative fragmentation function one we utilize in the present work, such mass effects could be consistently neglected.
In this work we extend for the first time the PFF formalism at hadron colliders to NNLO QCD. The novelty of the present work is that it develops a general, numeric, fully-flexible computational framework for perturbative cross sections for hadron collider processes with identified hadrons in NNLO QCD. Our work also benefits from the fact that all process-independent contributions needed for the description of heavy flavor fragmentation in NNLO -- like perturbative fragmentation functions, splitting functions and extracted from data non-perturbative fragmentation functions -- are available in the literature. Our framework is able to compute fully differential distributions with a single identified heavy hadron plus additional jets and non-strongly interacting particles. As a first application we compute the NNLO QCD corrections to $B$-hadron production in $t\bar t$ production with dilepton decays. The predicted realistic differential distributions significantly benefit from the inclusion of the NNLO QCD corrections.
There are a number of ways the current work can be extended and we plan to pursue those in the near future. For example, one can compute open $B$ production at high $p_T$. The framework developed here can be extended in a straightforward way to charm production as well.
One of the bottlenecks in this approach is the availability of high-quality non-perturbative fragmentation functions. These have previously been extracted from $e^+e^-$ data but the precision is not on par with current demand. In addition, the existing fragmentation functions are not fully compatible with our approach. To correct for this we intend to extract in the future non-perturbative fragmentation functions from $e^+e^-$ data within our framework.
In this work we have also studied the prospect of using LHC data for extracting $B$-hadron fragmentation functions. To that end we have proposed, and studied in detail, a distribution which we find to be particularly well suited for this task: the ratio of the $p_T$ of the $B$ hadron to the $p_T$ of the jet containing it. In the course of this study we have paid particular attention to the thorny problem of flavored jets in NNLO QCD.
Finally, an all-encompassing description of heavy flavor production in NNLO QCD will require the merging of fixed order calculations at low $p_T$ with the high $p_T$ description considered here. It is perhaps not too hard to envisage such a solution which, for example, builds on the FONLL approach at NLO. NNLO calculations with full mass dependence are possible as was recently demonstrated in ref.~\cite{Catani:2020kkl}. While such a merging is beyond the scope of the present work it represents a natural future extension of the present work.
\begin{acknowledgments}
The work of M.C. was supported by the Deutsche Forschungsgemeinschaft under grant 396021762 - TRR 257. The work of T.G. was supported by the Deutsche Forschungsgemeinschaft (DFG) under grant 400140256 - GRK 2497: The physics of the heaviest particles at the Large Hadron Collider. The research of A.M. and R.P. has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme (grant agreement no. 683211). A.M. was also supported by the UK STFC grants ST/L002760/1 and ST/K004883/1.
\end{acknowledgments}
|
1,314,259,996,528 | arxiv | \section{Introduction}
Thanks to the development of sensors, GPS devices, and satellite systems,
a wide variety of spatial data are being accumulated,
including climate~\cite{stralberg2015projecting,wang2016locally},
traffic~\cite{zheng2014urban,yuan2011t}, economic, and social data~\cite{haining1993spatial,shadbolt2012linked}.
Analyzing such spatial data is critical
in various fields, such as
environmental sciences~\cite{jerrett2005spatial,hession2011spatial},
urban planning~\cite{yuan2012discovering}, socio-economics~\cite{smith2014poverty,rupasingha2007social},
and public security~\cite{bogomolov2014once,wang2016crime}.
Collecting data on some attributes is difficult if the attribute-specific sensing devices are very expensive,
or experts that have extensive domain knowledge are required to observe data.
Also, collecting data in some regions is difficult if they are not readily accessible.
To counter these problems,
many spatial regression methods have been proposed~\cite{gao2006empirical,gao2006estimation,ward2018spatial};
they predict missing attribute values given data observed at some locations in the region.
Although Gaussian processes (GPs)~\cite{williams2006gaussian,banerjee2008gaussian} have been successfully used for spatial regression,
they fail when the data observed in the target region are insufficient.
In this paper,
we propose a few-shot learning method for spatial regression.
Our model learns
from spatial datasets on various attributes in various regions,
and predicts values when observed data in the target task is scant,
where both the attribute and region of the target task are different from those in the training datasets.
Figure~\ref{fig:framework} illustrates the framework of the proposed method.
Some attributes in some regions are expected to exhibit similar spatial patterns to the target task.
Our model uses the knowledge learned from such attributes and regions
in the training datasets to realize prediction in the target task.
\begin{figure}[t!]
\centering
\includegraphics[width=22em]{task.png}
\caption{Our framework. In a training phase, our model learns from training datasets containing various attributes from various regions. In a test phase, our model predicts spatial values of a target attribute in a target region given a few observations; the target attribute and region are not present in the training datasets.}
\label{fig:framework}
\end{figure}
Our model uses a neural network to embed a few labeled data
into a task representation.
Then, target spatial data are predicted based on a GP
with neural network-based mean and kernel functions
that depend on the inferred task representation.
Using the task representation yields
a task-specific prediction function.
By basing the modeling on GPs,
the prediction function can be rapidly adapted to small labeled data in a closed form
without iterative optimization,
which enables efficient back-propagation through the adaptation.
As the mean and kernel functions employ neural networks,
we can flexibly model spatial patterns in various attributes and regions.
By sharing the neural networks across different tasks in our model,
we can learn from multiple attributes and regions,
and use the learned knowledge to handle new attributes and regions.
The neural network parameters are estimated
by minimizing the expected prediction performance when a few observed data are given,
which is calculated using training datasets
by an episodic training framework~\cite{ravi2016optimization,santoro2016meta,snell2017prototypical,finn2017model,li2019episodic}.
The main contributions of this paper are as follows:
1) We present a framework of few-shot learning for spatial regression.
2) We propose a GP-based model that uses neural networks to
learn spatial patterns from various attributes and regions.
3) We empirically demonstrate that the proposed method performs well
in few-shot spatial regression tasks.
\begin{comment}
\begin{enumerate}
\item We present a problem of few-shot learning for spatial regression.
\item We propose a GP-based model that uses neural networks to
learn spatial patterns from various attributes and regions.
\item We empirically demonstrate that the proposed method performs well
in few-shot spatial regression tasks.
\end{enumerate}
\end{comment}
\section{Related work}
\label{sec:related}
GPs, or kriging~\cite{cressie1990origins}, have been widely used for spatial
regression~\cite{banerjee2008gaussian,luttinen2009variational,park2011domain,stein2012interpolation,gu2012spatial}.
They achieve high prediction performance at locations that are close to the observed locations.
However, if the target region is large and only a few observed data are given,
performance falls at locations far from the observed locations.
For improving generalization performance,
neural networks have been used
for mean and/or kernel functions of GPs~\cite{wilson2011gaussian,huang2015scalable,calandra2016manifold,wilson2016deep,wilson2016stochastic,iwata2017improving,iwata2019efficient}.
However, these methods require a lot of training data.
Many few-shot learning, or meta-learning, methods have been
proposed~\cite{schmidhuber:1987:srl,bengio1991learning,ravi2016optimization,andrychowicz2016learning,vinyals2016matching,snell2017prototypical,bartunov2018few,finn2017model,li2017meta,kimbayesian,finn2018probabilistic,rusu2018meta,yao2019hierarchically,edwards2016towards,garnelo2018conditional,kim2019attentive,hewitt2018variational,bornschein2017variational,reed2017few,rezende2016one}.
However, they are not intended for spatial regression.
On the other hand, our model is based on GPs, which have been
successfully used for spatial regression.
Some few-shot learning methods based on GPs have been proposed~\cite{harrison2018meta,tossou2019adaptive,fortuin2019deep}.
Adaptive learning for probabilistic connectionist architectures (ALPaCA)~\cite{harrison2018meta}
and adaptive deep kernel learning~\cite{tossou2019adaptive}
incorporate the information in small labeled data in kernel functions using neural networks,
but they assume zero mean functions.
Although meta-learning mean functions~\cite{fortuin2019deep}
use a neural network for the mean function,
the mean function does not change outputs depending on the given small labeled data.
On the other hand, the proposed method uses a neural network-based mean function
that outputs task-specific values by extracting a task representation from the small labeled data.
The effectiveness of our mean function is shown in the ablation study in our experiments.
A summary of related methods is shown in the appendix.
Our model is related to
conditional neural processes (NPs)~\cite{garnelo2018conditional,garnelo2018neural}
as both use neural networks for task representation inference
and for prediction with inferred task representations.
However, since NP prediction is based on fully parametric models,
they are less flexible in adapting to the given target observations
than GPs, which are nonparametric models.
In contrast, our GP-based model enjoys the benefits of the nonparametric approach,
swift adaptation to the target observations,
even though the mean and kernel functions are modeled parametrically.
Our model is also related to similarity-based meta-learning methods,
such as
matching networks~\cite{vinyals2016matching} and prototypical networks~\cite{snell2017prototypical},
since the kernel function represents similarities between data points.
Although existing similarity-based meta-learning methods were designed for classification tasks,
our model is designed for regression tasks.
The proposed method is also related to model-agnostic meta-learning
(MAML)~\cite{finn2017model}
in the sense that
both methods trains models
so that the expected error on unseen data is minimized
when adapted to a few observed data.
For the adaptation, MAML requires costly back-propagation through iterative gradient descent steps.
On the other hand, the proposed method achieves an efficient adaptation
in a closed form using a GP framework.
Ridge regression differentiable discriminator (R2-D2)~\cite{bertinetto2018meta}
is a neural network-based meta-learning method, where
the last layer is adapted by solving a ridge regression problem in a closed form.
Although R2-D2 and \cite{lee2019meta} adapt with a linear model,
the proposed method adapts with a nonlinear GP model,
which enables us to adapt to complicated patterns more flexibly.
Transfer learning methods, such as
multi-task GPs~\cite{yu2005learning,bonilla2008multi,wei2017source}
and co-kriging~\cite{myers1982matrix,stein1991universal},
have been proposed;
they transfer knowledge derived from source tasks to target tasks.
However, they do not assume a few observations in target tasks.
In addition, since these methods use target data to
learn the relationship between source and target tasks,
they require computationally costly re-training given new tasks that
are not present in the training phase.
On the other hand, the proposed method
can be applied to unseen tasks
by inferring task representations
from a few observations without re-training.
\section{Proposed method}
\label{sec:proposed}
\subsection{Task}
\begin{figure}[t!]
\centering
\includegraphics[width=25em]{model.png}
\caption{Our model. Each pair of location vector $\vec{x}_{n}$ and attribute value $y_{n}$ in a few labeled data set (support set) is fed to neural network $f_{\mathrm{z}}$. By averaging the outputs of the neural network, we obtain task representation $\vec{z}$. The task representation and query location vector $\vec{x}$ are fed to neural networks $f_{\mathrm{b}}$, $f_{\mathrm{k}}$, and $f_{\mathrm{m}}$ to calculate kernel $k$ and mean $m$. Attribute value $\hat{y}$ of the query location vector is predicted by using the kernel and mean based on a GP. Shaded nodes represent observed data.}
\label{fig:model}
\end{figure}
In a training phase, we are given spatial datasets for $|\mathcal{R}|$ regions,
$\mathcal{D}=\{\mathcal{D}_{r}\}_{r\in\mathcal{R}}$,
where $\mathcal{R}$ is the set of regions, and
$\mathcal{D}_{r}$ is the dataset for region $r$.
For each region, there are $|\mathcal{C}_{r}|$ attributes,
$\mathcal{D}_{r}=\{\mathcal{D}_{rc}\}_{c\in\mathcal{C}_{r}}$,
where $\mathcal{C}_{r}$ is the set of attributes in region $r$,
and $\mathcal{D}_{rc}$ is the dataset of attribute $c$ in region $r$.
The attribute sets can be different across the regions.
Each dataset consists of
a set of location vectors and attribute values,
$\mathcal{D}_{rc}=\{(\vec{x}_{rcn},y_{rcn})\}_{n=1}^{N_{rc}}$,
where $\vec{x}_{rcn}\in\mathbb{R}^{2}$
is a two-dimensional vector specifying the location of the $n$th point, e.g., longitude and latitude,
and
$y_{rcn}\in\mathbb{R}$
is the scalar value on attribute $c$ at that location.
In a test phase, we are given
a few labeled observations in a target region,
$\mathcal{D}_{r^{*}c^{*}}=\{(\vec{x}_{r^{*}c^{*}n},y_{r^{*}c^{*}n})\}_{n=1}^{N_{r^{*}c^{*}}}$,
where target region $r^{*}$ is not one of the regions in the training datasets,
$r^{*}\notin\mathcal{R}$,
and target attribute $c^{*}$ is not contained in the training datasets,
$c^{*}\notin\mathcal{C}_{r}$ for all $r\in\mathcal{R}$.
Our task is to predict target attribute value $\hat{y}_{r^{*}c^{*}}$ at location $\vec{x}_{r^{*}c^{*}}$
in the target region.
Location vector $\vec{x}_{rcn}$ represents
the relative position of the point in region $r$.
We used longitudes and latitudes normalized with zero mean for the location vectors in our experiments.
Spatial data sometimes include auxiliary information such as elevation.
In that case, we can include the auxiliary information
in $\vec{x}_{rcn}\in\mathbb{R}^{M+2}$, where $M$ is the number of additional types of auxiliary information.
\subsection{Model}
Let $\mathcal{S}=\{(\vec{x}_{n},y_{n})\}_{n=1}^{N}$
be a few labeled observations, which are called the {\it support set}.
We here present our model for predicting attribute scalar value
$\hat{y}$ at location vector $\vec{x}$,
which is called the {\it query},
given support set $\mathcal{S}$.
This model is used for training as described in Section~\ref{sec:learning}
as well as target spatial regression in a test phase.
Figure~\ref{fig:model} illustrates our model.
Our model infers task representation $\vec{z}$
from support set $\mathcal{S}$ as described in Section~\ref{sec:inference}.
Then, using the inferred task representation $\vec{z}$,
we predict attribute scalar value $\hat{y}$ of location vector $\vec{x}$
by a neural network-based GP as described in Section~\ref{sec:prediction}.
We omit indices for regions and attributes for simplicity in this subsection.
\subsubsection{Inferring task representation}
\label{sec:inference}
First, each pair of the location vector and attribute value, ($\vec{x}_{n},y_{n}$), in the support set
is converted into $K$-dimensional latent vector $\vec{z}_{n}\in\mathbb{R}^{K}$
by a neural network:
$\vec{z}_{n}=f_{\mathrm{z}}([\vec{x}_{n},y_{n}])$,
where $f_{\mathrm{z}}$ is a feed-forward neural network with
the $(M+3)$-dimensional input layer and $K$-dimensional output layer,
and $[\cdot,\cdot]$ represents the concatenation.
Second, the set of latent vectors $\{\vec{z}_{n}\}_{n=1}^{N}$ in the support set
are aggregated to $K$-dimensional latent vector $\vec{z}\in\mathbb{R}^{K}$ by averaging:
$\vec{z}=\frac{1}{N}\sum_{n=1}^{N}\vec{z}_{n}$,
which is a representation of the task extracted from support set $\mathcal{S}$.
We can use other aggregation functions,
such as summation~\cite{zaheer2017deep}, attention-based~\cite{kim2019attentive},
and recurrent neural networks~\cite{vinyals2016matching}.
\subsubsection{Predicting attribute values}
\label{sec:prediction}
Our prediction function assumes
a GP with neural network-based mean and kernel functions
that depend on the inferred task representation $\vec{z}$.
In particular, the mean function is modeled by
\begin{align}
m(\vec{x};\vec{z}) = f_{\mathrm{m}}([\vec{x},\vec{z}]),
\label{eq:mean}
\end{align}
where
$f_{\mathrm{m}}$ is a feed-forward neural network that outputs a scalar value.
The kernel function is modeled by
\begin{align}
k(\vec{x},\vec{x}';\vec{z}) &= \exp\left(-\parallel f_{\mathrm{k}}([\vec{x},\vec{z}])-f_{\mathrm{k}}([\vec{x}',\vec{z}])\parallel^{2}
\right)
+f_{\mathrm{b}}(\vec{z})\delta(\vec{x},\vec{x}'),
\label{eq:kernel}
\end{align}
where
$f_{\mathrm{k}}$ is a feed-forward neural network,
$f_{\mathrm{b}}$ is a feed-forward neural network with that outputs
a positive scalar value,
$\delta(\vec{x},\vec{x}')=1$ if $\vec{x}$ and $\vec{x}'$ are identical, and zero otherwise.
The kernel function is positive definite since
it is a Gaussian kernel
and $f_{\mathrm{b}}(\vec{z})$ is positive.
By incorporating task representation $\vec{z}$
in the mean and kernel functions using neural networks,
we can model nonlinear functions that depend on the support set.
In GPs, zero mean functions are often used
since the GPs with zero mean functions can approximate an arbitrary continuous function,
if given enough data~\cite{micchelli2006universal}.
However, GPs with zero mean functions predict zero at areas
far from observed data points~\cite{iwata2017improving},
which is problematic in few-shot learning.
Modeling the mean function by a neural network~(\ref{eq:mean})
allows us to predict values effectively even in areas
far from observed data points in a target region
due to the high generalization performance of neural networks.
Location vector $\vec{x}$ is transformed
by neural network $f_{\mathrm{k}}$ before computing the kernel function by the Gaussian kernel in (\ref{eq:kernel}).
The use of the neural network yields flexible modeling of the
correlation across locations depending on the task representation.
The noise parameter is also modeled by neural network $f_{\mathrm{b}}$,
which enables us to infer the noise level from the support set without re-training.
The predicted value for query $\vec{x}$ is given by
\begin{align}
\hat{y}(\vec{x},\mathcal{S};\bm{\Phi})=f_{\mathrm{m}}([\vec{x},\vec{z}])+\vec{k}^{\top}\vec{K}^{-1}(\vec{y}-\vec{m}),
\label{eq:prediction}
\end{align}
where
$\vec{K}$ is the $N\times N$ matrix of the kernel function evaluated between location vectors
in the support set,
$\vec{K}_{nn'}=k(\vec{x}_{n},\vec{x}_{n'})$,
$\vec{k}$ is the $N$-dimensional vector of the kernel function
between the query and support set, $\vec{k}=(k(\vec{x},\vec{x}_{n}))_{n=1}^{N}$,
$\vec{y}$ is the $N$-dimensional vector of attribute values in the support set,
$\vec{y}=(y_{n})_{n=1}^{N}$,
$\vec{m}$ is the $N$-dimensional vector of the mean function evaluated on locations in the support set,
$\vec{m} = (f_{\mathrm{m}}([\vec{x}_{n},\vec{z}]))_{n=1}^{N}$,
and $\bm{\Phi}$ are the parameters of neural networks $f_{\mathrm{z}}$, $f_{\mathrm{m}}$, $f_{\mathrm{k}}$, and $f_{\mathrm{b}}$.
An advantage of our model is that
the predicted value given the support set is analytically calculated
without iterative optimization,
by which we can minimize the expected prediction error efficiently based on gradient-descent methods.
When noise $f_{\mathrm{b}}(\vec{z})$ is small,
the predicted value approaches the observed values
at locations close to the observed locations.
This property of GPs is beneficial for few-shot regression without re-training.
If a neural network without GPs is used for the prediction function,
the predicted values might differ from the observations
even at the observed locations
when re-training based on the observations is not conducted.
The first term in (\ref{eq:prediction}) is similar to conditional neural processes,
where a neural network is used for the prediction function.
The second term in (\ref{eq:prediction}) is related to similarity-based
meta-learning methods since the second term uses the similarities between the query and support set
that are calculated by the kernel function.
Therefore, our model can be seen as an extension of
the conditional neural process and similarity-based meta-learning approach,
where both of them are naturally integrated within a GP framework.
When $\vec{x}$ is far from (close to) the observed locations,
the first (second) term becomes dominant due to kernel $\vec{k}$~\cite{iwata2017improving}.
This is reasonable since similarity-based approaches are more reliable
when there are observations nearby.
The variance of the predicted attribute value of the query is given by
$\mathbb{V}[y|\vec{x},\mathcal{S};\bm{\Phi}] = k(\vec{x},\vec{x};\vec{z})-\vec{k}^{\top}\vec{K}^{-1}\vec{k}$.
\subsection{Learning}
\label{sec:learning}
We estimate neural network parameters $\bm{\Phi}$
by minimizing the expected prediction error
on a query set given a support set
using an episodic training framework~\cite{ravi2016optimization,santoro2016meta,snell2017prototypical,finn2017model,li2019episodic}.
Although training datasets $\mathcal{D}$ contain many observations,
they should be used in a way that closely simulates the test phase.
Therefore, with the episodic training framework,
support and query sets are generated
by a random subset of training datasets $\mathcal{D}$
for each training iteration.
In particular, we use the following objective function:
\begin{align}
\hat{\bm{\Phi}}=
\arg \min_{\bm{\Phi}} \mathbb{E}_{r\sim\mathcal{R}}[\mathbb{E}_{c\sim\mathcal{C}_{r}}[
\mathbb{E}_{(\mathcal{S},\mathcal{Q})\sim\mathcal{D}_{rc}}[
L(\mathcal{S},\mathcal{Q};\bm{\Phi})]],
\end{align}
where $\mathbb{E}$ represents an expectation,
\begin{align}
L(\mathcal{S},\mathcal{Q};\bm{\Phi})
= \frac{1}{N_{\mathrm{Q}}}
\sum_{(\vec{x},y)\in\mathcal{Q}}
\parallel \hat{y}(\vec{x},\mathcal{S};\bm{\Phi})-y \parallel^{2},
\label{eq:E}
\end{align}
is the mean squared error on query set $\mathcal{Q}$ given support set $\mathcal{S}$,
and $N_{\mathrm{Q}}$ is the number of instances in the query set.
Usually, GPs are trained by maximizing the marginal likelihood
of training data (support set),
where test data (query set) are not used.
On the other hand,
the proposed method minimizes the prediction error
on a query set when a support set is observed,
by which we can simulate a test phase and
learn a model that improves the prediction performance on target tasks.
When we want to predict the density, we can use the following negative predictive
log-likelihood
\begin{align}
L(\mathcal{S},\mathcal{Q};\bm{\Phi})
= -\frac{1}{N_{\mathrm{Q}}}
\sum_{(\vec{x},y)\in\mathcal{Q}}
\mathcal{N}(y|\hat{y}(\vec{x},\mathcal{S};\bm{\Phi}),\mathbb{V}[y|\vec{x},\mathcal{S};\bm{\Phi}]),
\label{eq:E2}
\end{align}
instead of the mean squared error (\ref{eq:E}),
where $\mathcal{N}(y|\mu,\sigma^{2})$ is the Gaussian distribution with mean $\mu$ and variance $\sigma^{2}$.
This is related to training GPs with the log pseudo-likelihood~\cite{williams2006gaussian},
where the leave-one-out predictive log-likelihood is used as the objective function.
The training procedure of our model is shown in
Algorithm~\ref{alg}.
The computational complexity for evaluation loss (\ref{eq:E})
is $O(N_{\mathrm{Q}}N_{\mathrm{S}}^{3})$, where $N_{\mathrm{S}}$ is
the number of instances in the support set
since we need the inverse of the kernel matrix whose size is $N_{\mathrm{S}}\times N_{\mathrm{S}}$.
In few-shot learning, the number of target observed data is very small,
and so a very small support size $N_{\mathrm{S}}$ is used in training.
Therefore, our model can be optimized efficiently
with the episodic training framework.
This is in contrast to the high computational complexity of training for standard GP regression,
which is cubic in the number of training instances.
\begin{comment}
In each iteration,
we randomly generate support and query sets (Lines 2 -- 5) from dataset $\mathcal{D}_{rc}$ by
randomly selecting region $r$ and attribute $c$
for simulating a test phase.
Given the support and query sets so generated, we calculate the loss (Line 6) by (\ref{eq:E}) or (\ref{eq:E2}).
We update model parameters by using any of the stochastic gradient-descent methods, such as ADAM~\cite{kingma2014adam} (Line 7).
By training the model using randomly generated support and query sets,
the trained model can predict values with a wide variety
of observed location distributions,
attributes and regions in a test phase.
\end{comment}
\begin{algorithm}[t]
\caption{Training procedure of our model.}
\label{alg}
\begin{algorithmic}[1]
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\REQUIRE{Spatial datasets $\mathcal{D}$,
support set size $N_{\mathrm{S}}$, query set size $N_{\mathrm{Q}}$}
\ENSURE{Trained neural network parameters $\bm{\Phi}$}
\WHILE{not done}
\STATE Randomly sample region $r$ from $\mathcal{R}$
\STATE Randomly sample attribute $c$ from $\mathcal{C}_{r}$
\STATE Randomly sample support set $\mathcal{S}$ from $\mathcal{D}_{rc}$
\STATE Randomly sample query set $\mathcal{Q}$ from $\mathcal{D}_{rc}\setminus\mathcal{S}$
\STATE Calculate loss by Eq.~(\ref{eq:E}) or (\ref{eq:E2}), and its gradients
\STATE Update model parameters $\bm{\Phi}$ using the loss and its gradients
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\section{Experiments}
\label{sec:experiments}
\subsection{Data}
We evaluated the proposed method using the following three spatial datasets:
NAE, NA, and JA.
NAE and NA were
the climate data in North American, which were obtained from \url{https://sites.ualberta.ca/~ahamann/data/climatena.html}.
As the location vector,
NA used longitude and latitude.
With NAE, elevation in meters above sea level was additionally used in the location vector.
For attributes, we used 26 bio-climate values,
such as the annual heat-moisture index and climatic moisture deficit.
We generated 1829 non-overlapping regions covering North America,
where the size of each region was 100km $\times$ 100km,
and attribute values were observed at 1km $\times$ 1km grid squares
in each region.
300 training, 18 validation, and 549 target regions
were randomly selected without replacement from the 1829 regions.
Also, 26 attributes were randomly splitted into
20 training, five validation, and one target attributes.
JA was the climate data in Japan, which was obtained from
\url{http://nlftp.mlit.go.jp/ksj/gml/datalist/KsjTmplt-G02.html}.
We used seven climate attributes, such as
precipitation and maximum temperature.
The data contained 273 regrions,
where attribute values were observed at 1km $\times$ 1km grid square,
and there were at most 6,400 locations in a region.
The 273 regions were randomly splitted into
200 training, 28 validation, 45 target regions.
Also, seven attributes were randomly splitted into
four training, two validation, and one target attributes.
The details of the datasets were described in the appendix.
For all data, in each target region,
values on a target attribute at five locations were observed,
and values at the other locations were used for evaluation.
\subsection{Results}
We compared the proposed method
with the conditional neural process (NP),
Gaussian process regression (GPR),
neural network (NN),
fine-tuning with NN (FT), and
model-agnostic meta-learning with NN (MAML).
The detailed settings of the proposed method and comparing methods
are described in the appendix.
\begin{table}[t!]
\centering
\caption{(a) Test mean squared errors and (b) test log likelihoods averaged over ten experiments. Values in bold typeface are not statistically significantly different at the 5\% level from the best performing method in each row according to a paired t-test.}
\label{tab:err}
\begin{tabular}{cc}
\begin{minipage}{0.5\textwidth}
\centering
(a) Test mean squared errors\\
{\tabcolsep=0.25em
\begin{tabular}{lrrrrrr}
\hline
Data & Ours & NP & GPR & NN & FT & {\small MAML}\\
\hline
NAE & {\bf 0.316} & 0.348 & 0.476 & 0.963 & 0.497 & 0.710 \\
NA & {\bf 0.552} & 0.593 & 0.701 & 0.986 & 0.746 & 0.824 \\
JA & {\bf 0.653} & 0.756 & 0.703 & 1.016 & 0.871 & 0.873 \\
\hline
\end{tabular}}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\centering
(b) Test log likelihoods\\
{\tabcolsep=0.25em
\begin{tabular}{rrrrrr}
\hline
Ours & NP & GPR & NN & FT & {\small MAML}\\
\hline
{\bf -0.987} & -1.025 & -1.067 & -1.281 & -1.271 & -1.256 \\
{\bf -1.166} & -1.190 & -1.230 & -1.322 & -1.311 & -1.306 \\
{\bf -1.227} & -1.307 & {\bf -1.220} & -1.333 & -1.598 & -1.314 \\
\hline
\end{tabular}}
\end{minipage}
\end{tabular}
\end{table}
The test mean squared errors (a) and
test log likelihoods (b) in the target tasks
averaged over ten experiments are shown in Table~\ref{tab:err}.
Here, for the test mean squared error evaluations,
all methods were trained with the mean squared error objective function in Eq.~(\ref{eq:E}),
and for the test log likelihood evaluations,
all methods were trained with the negative log likelihood objective function in Eq.~(\ref{eq:E2}).
The proposed method achieved the best performance in all cases except for
the test likelihood with JA data.
NP was worse than the proposed method
because its predictions were poor
when task representations were not properly inferred.
On the other hand,
the proposed method performed well with any tasks in at least areas close to the observations,
as its GP framework offers a smooth nonlinear function that passes over the observations.
GPR was worse than the proposed method
since GPR only shares kernel parameters across different tasks.
In contrast,
the proposed method shares neural networks across different tasks,
which enables us to
learn flexible spatial patterns in various attributes and regions
and use them for target tasks.
NN suffered the worst performance since it cannot use the target data.
Fine-tuning (FT) decreased the error,
but it remained worse than that of the proposed method.
This is because
FT consisted of two separate steps: pretrain and fine-tuning,
and did not learn how to transfer knowledge.
In contrast, the proposed method trained the neural network
in a single step so that test performance
is maximized when the support set is given in the episodic training framework.
MAML performance was low
since it was difficulty in learning the parameters
that fine-tuned well with just a small number of epochs
with various regions and attributes,
where target function shapes vary drastically.
Note that due to the high computational complexity of MAML,
where it demands that the gradients of many gradient-descent steps be calculated,
MAML makes it infeasible to use a large number of fine-tuning epochs.
On the other hand,
with the proposed method,
since predicted values given the support set
are calculated analytically based on a GP,
the neural networks are optimized efficiently in terms of fitting the support set,
and therefore the trained model
attained high prediction performance for various attributes and regions.
Since the number of training attributes was small with JA data,
and training data were insufficient to train neural networks,
the test likelihood of the proposed method was not significantly different from that of GPR.
\begin{figure*}[t!]
\centering
{\tabcolsep=0em
\begin{tabular}{ccc}
\includegraphics[width=15em]{err_r9.pdf}&
\includegraphics[width=15em]{err_r8.pdf}&
\includegraphics[width=15em]{err_r6.pdf}
\end{tabular}}
\caption{Average test mean squared errors with (a) different target support sizes, (b) different numbers of training attributes, and (c) different numbers of training regions on NAE data. The bar shows the standard error.}
\label{fig:mse}
\end{figure*}
Figure~\ref{fig:mse}(a)
shows the average test mean squared errors with different numbers of target support sizes
with the proposed method, NP, and GPR.
We omitted the results with NN, FT, and MAML since
their performance was low as shown in Table~\ref{tab:err}.
All methods yielded decreased error
as the target support size increased.
The proposed method achieved low errors with different target support sizes
since it uses neural networks to learn the relationship
between support and query sets using the training datasets.
NP achieved low error rates when the target support size was small.
However, NP had higher error rates than GPR when the size was ten.
Since NP used a fixed trained neural network to incorporate the support set information,
it was difficult to adapt prediction functions to a large support set.
In contrast, since the proposed method and GPR
can adapt them easily to support sets by calculating the posterior in a closed form,
their errors were effectively decreased as the target support size increased.
Figure~\ref{fig:mse}(b) shows
the average test mean squared errors with different numbers of training attributes.
The errors with the proposed method and NP decreased
as the training attribute numbers increased.
This is reasonable since
the possibility that tasks similar to target tasks
are included in the training datasets
increases as the number of training attributes increases.
Since GPR shared only kernel parameters
across different tasks,
its performance was not improved even when many attributes were used.
Figure~\ref{fig:mse}(c) shows
the average test mean squared errors with different numbers of training regions.
The errors with the proposed method and NP decreased
as the training regions increased.
The computational time of the proposed and comparing methods was described in the appendix.
\begin{figure*}[t!]
\centering
{\tabcolsep=0em\begin{tabular}{ccccccc}
\multicolumn{5}{c}{True attribute values}\\
\includegraphics[width=8em]{i0j4m0p0_plt10_y.png}&
\includegraphics[width=8em]{i0j3m0p0_plt12_y.png}&
\includegraphics[width=8em]{i0j6m0p0_plt27_y.png}&
\includegraphics[width=8em]{i0j7m0p0_plt25_y.png}&
\includegraphics[width=8em]{i0j19m0p0_plt11_y.png}\\
\multicolumn{5}{c}{Ours}\\
\includegraphics[width=8em]{i0j4m0p0_plt10_esty.png}&
\includegraphics[width=8em]{i0j3m0p0_plt12_esty.png}&
\includegraphics[width=8em]{i0j6m0p0_plt27_esty.png}&
\includegraphics[width=8em]{i0j7m0p0_plt25_esty.png}&
\includegraphics[width=8em]{i0j19m0p0_plt11_esty.png}\\
\multicolumn{5}{c}{NP}\\
\includegraphics[width=8em]{i0j4m2p0_plt10_esty.png}&
\includegraphics[width=8em]{i0j3m2p0_plt12_esty.png}&
\includegraphics[width=8em]{i0j6m2p0_plt27_esty.png}&
\includegraphics[width=8em]{i0j7m2p0_plt25_esty.png}&
\includegraphics[width=8em]{i0j19m2p0_plt11_esty.png}\\
\multicolumn{5}{c}{GPR}\\
\includegraphics[width=8em]{i0j4m4p0_plt10_esty.png}&
\includegraphics[width=8em]{i0j3m4p0_plt12_esty.png}&
\includegraphics[width=8em]{i0j6m4p0_plt27_esty.png}&
\includegraphics[width=8em]{i0j7m4p0_plt25_esty.png}&
\includegraphics[width=8em]{i0j19m4p0_plt11_esty.png}\\
(a) PAS &
(b) PPTsm & (c) EMT & (d) DD18 & (e) bFFP \\
\\
\end{tabular}}
\caption{Predictions for five attributes and regions of
target tasks yielded by the proposed method, NP, and GPR.
The top row shows the true attribute values.
Red circles indicate observed locations. Values below each plot shows the mean squared error.}
\label{fig:viz}
\end{figure*}
Figure~\ref{fig:viz} visualizes the predictions
for five attributes and regions
of target tasks
with the proposed method, NP, and GPR.
The proposed method attained appropriate predictions in various attributes and regions.
NP did not necessarily
output predicted values that were similar to the observations.
For example, in Figure~\ref{fig:viz}(a,NP),
the predicted values of NP at two left observed locations
differed from the true value.
On the other hand, the proposed method and GPR
predicted values similar to the observation at the locations.
Since GPR could not extract the rich knowledge present in the training datasets,
it sometimes failed to predict values.
For example, in Figure~\ref{fig:viz}(a,GPR),
the predicted values differed from the true values in the lower area.
In contrast,
the proposed method and NP predicted values at the area well
using neural networks.
The proposed method improved prediction performance by adopting
both advantages of GPs and neural networks.
\begin{table}[t!]
\centering
\caption{Ablation study. Test mean squared errors and test log likelihoods on target tasks with NAE data. ErrObj is the proposed method with the mean squared error objective function in Eq.~(\ref{eq:E}). LikeObj is that with the log likelihood objective function in Eq.~(\ref{eq:E2}), MarLikeObj is that with the marginal likelihood on the support set, NoSptMean is that with neural netwok-based mean function without the support information, $m(\vec{x})=f_{\mathrm{m}}(\vec{x})$, and
ZeroMean is that with zero mean function.}
\label{tab:ablation}
\begin{tabular}{lrrrrr}
\hline
Evaluation measurement & ErrObj & LikeObj & MarLikeObj & NoSptMean & ZeroMean \\
\hline
Test mean squared error & {\bf 0.316} & 0.329 & 0.476 & 0.386 & 0.394\\
Test log likelihood & -1.025 & {\bf -0.987} & -1.479 & -1.086 & -1.088\\
\hline
\end{tabular}
\end{table}
Table~\ref{tab:ablation} shows the results of the ablation study of the proposed method.
In terms of the test mean squared error,
the proposed method with the mean squared error objective function (ErrObj) was better than
that with the likelihood objective function (LikeObj).
In terms of the test log likelihood, LikeObj was better than ErrObj.
These results imply that the objective function should be selected properly depending on the applications.
The proposed method with the marginal likelihood objective function on the support set (MarLikeObj)
was worse than ErrObj and LikeObj.
This result demonstrates the effectiveness to use the test performance for the objective function
for few-shot learning although standard GPs are usually trained with the marginal likelihood of training data.
The proposed with mean function without the support information (NoSptMean)
and that with zero mean function (ZeroMean)
performed worse than the proposed method.
This result indicates the importance to use non-zero mean functions that incorporate the support information,
and the advantage of the proposed method over existing GP-based meta-leaning methods
those that use zero mean functions~\cite{harrison2018meta,tossou2019adaptive}
and those that do not use the support information~\cite{harrison2018meta}.
\section{Conclusion}
\label{sec:conclusion}
We proposed a few-shot learning method for spatial regression.
The proposed method can predict
attribute values given a few observations,
even if the target attribute and region are not included in the training
datasets.
The proposed method
uses a neural network to infer a task representation from a few observed data.
Then, it uses the inferred task representation to calculate the predicted values
by a neural network-based Gaussian process framework.
Experiments on climate spatial data
showed that the proposed method
achieved better prediction performance than existing methods.
For future work, we want to apply our framework
to other types of tasks, such as spatio-temporal regression,
regression for non-spatial data, and classification.
\bibliographystyle{abbrv}
|
1,314,259,996,529 | arxiv | \chapter*{Preface}
This book represents an account and contextualization of a research program
that was executed by the authors and their collaborators over the period
of several years. Both authors began their careers in classical geometric group theory and began collaborating on
projects about right-angled Artin groups and their relationship with mapping class groups of surfaces, at a time
when the first author was a visiting assistant professor at Tufts University and while the second
author was a graduate student at Harvard University.
Around 2014, the authors became interested in a question attributed by M.~Kapovich to V.~Kharlamov,
one that asks which right-angled Artin groups can act faithfully by smooth
diffeomorphisms on the circle. Because of the profusion of right-angled Artin subgroups of mapping class groups
and because of
the robust persistence of right-angled Artin groups under passing to finite index subgroups,
an answer to Kharlamov's question had the potential
to shed light on a question of F.~Labourie that he posed in his 1998 ICM
talk and that was reiterated by B.~Farb and J.~Franks in several places:
are there finite index subgroups of (sufficiently complicated) mapping class groups acting smoothly
on the circle?
At first, it seemed to the authors that an easy solution to Kharlamov's question was available, and that all right-angled Artin groups admitted
such faithful actions. It was soon pointed out by J.~Bowden that the actions constructed by the authors and H.~Baik in fact failed
to be differentiable at certain accumulations of fixed points, and so only furnished faithful smooth actions on the real line.
After approximately two
years of work, inspired by Bowden's observation, Baik and the authors managed to prove that that most right-angled Artin groups
do not admit faithful $C^2$ actions on the compact interval nor on the circle. As a consequence, Labourie's question could be completely
answered in the case of $C^2$ actions of finite index subgroups of mapping class groups.
About a year and a half later, the authors were able to develop a tool that they dubbed the \emph{$abt$--Lemma}, which is a certain
combinatorial--algebraic obstruction for smoothness of a group action on a compact one--manifold. As a result, they were able
to answer Kharlamov's question by giving a concise characterization of right-angled Artin groups acting
faithfully by smooth diffeomorphisms on the circle.
Because of the technical details
involved in the analysis of smooth actions of right-angled Artin groups on one--manifolds, the authors became interested
in Thompson's group $F$, which occurs naturally in these contexts. Inspired by the dynamical theory of Thompson's group,
the authors investigated the class of
chain groups in subsequent work with Y.~Lodha.
These objects form a highly diverse class of groups of homeomorphisms of the interval, while nevertheless exhibiting remarkable uniformity
properties.
Over the years that the preceding story unfolded, the authors had long wondered about the existence of finitely generated groups of
homeomorphisms which can act by $C^k$ diffeomorphisms, but which admit no faithful actions by $C^{k+1}$ diffeomorphisms, especially
in the case $k\geq 2$.
The question of the existence of such groups was posed explicitly by A.~Navas in his 2018 ICM talk. Having built a
foundation consisting of the theory of right-angled Artin group actions on one--manifolds and the theory of chain groups, the authors
were able to construct explicit examples of finitely generated groups that act on the circle and on the interval with prescribed
\emph{critical regularities}\index{critical regularity}, in the following strong sense:
every smoother action of a finite index subgroup factors through an abelian quotient.
More recent work of the authors with C.~Rivas has improved this state of knowledge, particularly concerning right-angled Artin group
and mapping class group actions on the circle. It is now known that
for all $\epsilon>0$, a finite index subgroup of a mapping class group (that is not virtually free--by--cyclic) does not
admit a faithful action by $C^{1,\epsilon}$ diffeomorphisms on the interval nor on the circle.
The historical development of mathematical ideas is rarely systematic, which is why we recount the story of the authors' contribution to
the theory in this book in a chronological way, so as to complement the systematic exposition in the body of the monograph. Moreover, the
drive within academia to publish research articles that continually break new ground is in direct tension with the cultural and
intellectual need of the mathematics community
to have access to complete, detailed, and polished accounts of mathematical subjects. It is thus that we approached the project of writing
this book: we construct a coherent narrative that gives a systematic overview of the theory of critical regularity of groups, and at the same
time provide sufficient background and context so that a reasonably experienced graduate student could traverse the gap from introductory
courses to the cutting edge in the span of these pages. The result is a book that is half lecture notes, half research monograph.
Herein, the reader will find all the results mentioned above, with complete proofs. In many cases, the proofs we have provided are
more efficient and more general than the ones given in the original publications. Notably, the precise statement of the result that furnishes
finitely generated groups of prescribed critical regularity is substantially stronger than the one in the authors' original paper.
In addition to the account of the authors' contributions,
there is an exposition on background and contextual results that are scattered through both
the published literature as well as various unpublished sources.
One chapter is dedicated to what is known as Denjoy's theory of diffeomorphisms.
Many excellent books and expository articles give proofs of Denjoy's Theorem, though our presentation is tailored to our agenda.
In the commentary surrounding Denjoy's Theorem, we have included a self--contained discussion of invariant and stationary measures.
We avoid long digressions into amenability and measurable dynamics, proceeding merely with our narrative needs in mind.
The section on
differentiable Denjoy counterexamples includes results of the authors which have hitherto appeared only in research journals.
A large chapter of the book
is devoted to results of F.~Takens, R.~Filipkiewicz, and M.~Rubin, which for us mostly serve to contextualize
the theory of critical regularity. To the authors'
knowledge, this will be the first instance of a treatment of these results as parts of a coherent whole. Moreover, the authors hope that
this chapter will serve to clear up certain inaccuracies, misattributions, and inefficiencies present in the literature.
One chapter of the book, together with the appendices, introduces standard tools for analyzing group actions on one--manifolds. The exposition
therein concentrates on the $C^0$, the $C^1$, and the $C^2$ theories. We have recounted proofs of nearly all the results, sometimes
reproducing proofs that already exist in the literature, and other times streamlining known proofs.
The remaining chapters of the book were written mostly in service to proving the existence of groups with prescribed critical regularity,
and to a commentary on the applications of the construction.
The authors sincerely hope that the reader will find this book intellectually stimulating and satisfying, and humbly wish that they
may attract the attention
of fresh minds to this beautiful subject.
\vspace{\baselineskip}
\begin{flushright}\noindent
Seoul and Charlottesville,\hfill {\it Sang-hyun Kim}\\
June 2021\hfill {\it Thomas Koberda}\\
\end{flushright}
\chapter*{Acknowledgements}
The authors thank Adam Clay, Benson Farb, Andreas Holmsen, Nam-Gyu Kang, Yash Lodha, Curtis T.~McMullen, Crist\'obal Rivas and
William Winston for helpful discussions and corrections.
The authors are especially grateful to Andrés Navas for his inspiring book and pioneering works in the direction of this work.
The authors are also grateful to Elena Griniari for guiding the authors throughout the creation of this book.
The first author is supported by Samsung Science and Technology Foundation (SSTF-BA1301-51).
The second author is partially supported by an Alfred P. Sloan Foundation Research Fellowship, by NSF Grant DMS-1711488, and by
NSF Grant DMS-2002596.
\tableofcontents
\mainmatter
\chapter{Introduction}\label{ch:intro}
\begin{abstract}
In this monograph, we give an account of the relationship between the algebraic structure of finitely generated and countable
groups and the regularity with which they act on manifolds. We concentrate on the case of one--dimensional manifolds, culminating
with a uniform construction of finitely generated groups acting with prescribed regularity on the compact interval and on the circle. We develop
the theory of dynamical obstructions to smoothness, beginning with classical results of Denjoy, to more recent results of Kopell, to
modern results such as the $abt$--Lemma. We give a classification of the right-angled Artin groups that have finite critical
regularity, and discuss their exact critical regularities in many cases, and we compute the
virtual critical regularity of most mapping class groups of orientable surfaces.\end{abstract}
This book is a discussion of the relationship between algebraic structure of groups, dynamics of group actions, and regularity of diffeomorphisms. Particularly,
we are interested in finitely generated, or more generally countable groups, acting on a fixed compact manifold, and how the level
of differentiability of the action controls the possible algebraic structure of the groups.
The following diagram illustrates the basic slogans of this book.
\[\begin{tikzcd}
& \textrm{Group theory} \arrow[dr,dash,"\textrm{Regularity}"] \\
\textrm{Dynamics} \arrow[ur,dash,"\textrm{Structure of orbits}"] \arrow[rr,dash,"\textrm{Propagation of derivatives in orbits}"]
&& \textrm{Analysis}
\end{tikzcd}
\]
At one of the vertices, we have group theory, which for us means the description of the algebraic structure of groups. We are
primarily concerned with infinite groups that are classically of interest in geometric group theory and related areas, such as mapping
class groups of hyperbolic surfaces, right-angled Artin groups, and the Higman--Thompson type groups such as Thompson's group $F$.
These groups are so interesting to us because they embody a rich combination of commutativity and non--commutativity,
that is, \emph{partial commutativity}\index{partial commutativity}.
Algebraically,
partial commutativity is interesting and informative, because it is tractable (in the sense that many defining relations are easy to write
down and understand intuitively) and because it is complicated enough to exhibit a wide range of phenomena; indeed, the theory
of right-angled Artin groups shows that in a precise sense, ``group with partial commutation" are at least as complicated as the class
of all finite graphs, which in turn are a model for all of finitary mathematics. Partial commutativity is also useful for studying groups
up to commensurability. Indeed, commutation of group elements is robust under replacing a group element with a nonzero power.
Thus, partial commutativities in an ambient group generally propagate to finite index subgroups. This is very helpful in understanding
groups whose finite index subgroups are too complicated for contemporary technology, such as mapping class groups of hyperbolic surfaces.
Much of this book is concerned with groups of diffeomorphisms. Commutation
of diffeomorphisms, as we will explore in great detail, is a rare phenomenon, and imposes a high
degree of structure on pairs of diffeomorphisms
which exhibit it, especially in higher regularity. In fact,
the following could be considered as a principle that informs a large portion of the writing of this
book:
\begin{prin}\label{prin:main}
The Mean Value Theorem makes it hard for diffeomorphisms of a compact manifold to commute with each other.
\end{prin}
Principle~\ref{prin:main} illustrates an interaction between the vertices labeled ``group theory" and ``analysis", though it
conceals the important role of the third vertex. Principle~\ref{prin:main}
highly constrains actions of groups that exhibit a high degree of partial commutativity, by forcing group
elements to have a lot of fixed points, and then higher derivatives at accumulation points of the fixed points
(which is where compactness comes in!) often fail to be continuous, due
to the orbit structure of the group action.
Dynamics' domain is that of the global structure of orbits, and interacts in an essential way with both the structure of a group that is acting,
and with the topology of the underlying manifold. All of our discussion would be greatly hobbled, if not completely invalidated, were the
manifolds under consideration not restricted to one dimension. The interval and circle are ordered structures, and their topologies are
completely determined by the orders. The consequence of this interaction between the order relation and the topology is roughly that,
away from fixed points, homeomorphisms of the interval and of the circle have a direction. This last statement means that the combinatorics
of orbits are much more tractable than they would be on an arbitrary manifold, or even worse, on an arbitrary topological space. Without
a systematic way to analyze and classify possible orbit structures, the edge connecting the ``group theory" and the ``analysis" vertices
would be inscrutable.
By investigating the three vertices and edges of the diagram, we are able to carry out a number of constructions and to illustrate several
completed mathematical programs. Among the highlights of the book are the following:
\begin{itemize}
\item
A complete classification of right-angled Artin groups which admit faithful actions on the interval and the circle with regularity at least two,
in terms of their defining graphs.
\item
A complete classification of mapping class groups that (virtually) admit faithful actions on the interval and the circle with regularity at least two,
in terms of the topology of the underlying surface.
\item
A uniform construction of finitely generated groups with prescribed \emph{critical regularity}\index{critical regularity}
when acting on the interval or on the circle. Here,
the critical regularity of a group $G$ is the supremum of the real numbers $r=k+\tau$ with $k\in\bZ_{>0}$ and $\tau\in [0,1)$, where $G$
admits a faithful action by $C^k$ diffeomorphisms whose $k^{th}$ derivatives are $\tau$--H\"older continuous.
\item
New, robust constructions of codimension one foliations on closed $3$--manifolds subject to mild topological hypotheses, which have
prescribed regularity properties.
\item
Several technical obstructions to $C^1$ smoothability of group actions that do not rely on Thurston's Stability Theorem, which start
with a relatively weak hypothesis (the orbit structure is subjected to a naturally occurring dynamical constraint) and end with a relatively strong
conclusion (the action was not differentiable).
\end{itemize}
We move on now beyond philosophical musings to discuss the content and context of this book in more precise terms.
\subsection{Some conventions and notation}
Throughout this book, we let $\bZ_{>0}$ denote the set of positive integers,
and let $\bN$ denote the set of \emph{natural numbers}\index{natural numbers}. That is, we have
\[
\bN:=\{0,1,2,\ldots\}=\{0\}\cup \bZ_{>0}.\]
We denote by $I$ a compact interval; typically we mean $I=[0,1]$ unless specified otherwise.
A map or manifold is said to be \emph{smooth}\index{smooth manifold} if it is $C^\infty$.
\section{Groups of manifold diffeomorphisms}
In this section, we give an overview of the main actors in the sequel. A reader familiar with the basics of manifold theory and homeomorphism
groups can safely skim or skip this section.
One of the basic objects in this book is a \emph{manifold}\index{manifold} $M$.
For us, a manifold is always connected and second countable. Since $M$
is connected and hence path--connected, the \emph{dimension}\index{dimension of a manifold}
of $M$ will be well-defined, as this quantity is locally constant.
The entire discussion of this book occurs in relation to, and usually entirely within, the group $\Homeo(M)$ of homeomorphisms
\[\phi\colon M\longrightarrow M.\] When $M$ is orientable, there is a natural subgroup $\Homeo_+(M)$ consisting of orientation
preserving homeomorphisms of $M$. When $\Homeo_+(M)$ and $\Homeo(M)$ do not coincide, we will usually restrict our attention
to $\Homeo_+(M)$, since this latter group is just an index two subgroup in the full group of homeomorphisms.
Manifolds, \emph{a priori}, are only topologically locally homeomorphic to Euclidean spaces, and there are many subtle questions about
differentiable manifolds, piecewise geometric manifolds, and the relationships between these categories.
Since these are not the concern
of this book, we will always assume that manifolds are smooth, which is to say that they are implicitly equipped with an atlas of charts
whose transition functions are smooth (that is, $C^\infty$) local diffeomorphisms.
With the sole exception of Chapter~\ref{sec:filip-tak}, we will be working in low dimensions. Within the context of low dimensions, we will
be working almost exclusively in dimension one. Dimension two will only occur in relation to mapping classes of surfaces, and dimension
three will occur only in the context of the existence of non-smoothable codimension one foliations. Thus, the distinction between the
topological category, the $C^1$ category, the smooth category, and the piecewise linear category of manifolds is nil, for our purposes
(cf.~\cite{Moise77book,Thurston1997book}).
Thus, for a manifold $M$, we will be justified in filtering $\Homeo(M)$ by subgroups that are analytically defined. The most coarse filtration
we will use is by integral regularities. We will write $\Diff^k(M)$ for the group of $C^k$ diffeomorphisms of $M$. Thus, $f\in\Diff^k(M)$
if and only if $f$ and $f^{-1}$ are continuous and have continuous derivatives up to order $k$. That $\Diff^k(M)$ is indeed a group
is a consequence of the chain rule. We thus obtain a descending chain of subgroups \[\Homeo(M)>\Diff^1(M)>\cdots>\Diff^k(M)>\cdots.\]
It is convenient to allow $\Diff^0(M)=\Homeo(M)$, and we will implicitly adopt this convention throughout.
When $M$ is orientable, then for all $k$ we can also decorate the group $\Diff^k(M)$
with the plus sign by restricting to the orientation preserving diffeomorphisms, identifying $\Diff_+^k(M)$ with a subgroup of $\Homeo_+(M)$.
The group $\Diff^{\infty}(M)$ is defined
by \[\Diff^{\infty}(M)=\bigcap_{k\geq 1} \Diff^k(M),\] and $\Diff_+^{\infty}(M)$ is defined in the obvious way.
We remark that there are interesting groups of diffeomorphisms of manifolds that lie even deeper than $\Diff^{\infty}(M)$. For instance,
if $M$ has an analytic or algebraic or symplectic structure, then one can consider the group of diffeomorphisms that respect this structure. If
$M$ is equipped with a smooth Riemannian metric then one can consider the isometries of $M$, which is to say the diffeomorphisms
which respect this metric. One can consider other geometric structures that are not necessarily metric, strictly speaking, such as projective
structures, and thus get many more interesting groups of diffeomorphisms. These kinds of groups, while rich in examples and questions,
are beyond the scope of this book.
One group which will occur a number of times and which does not fit easily into any of the groups of diffeomorphisms we have defined so far
is the group $\PL_+(M)\le \Homeo_+(M)$
of \emph{piecewise linear}\index{piecewise linear} homeomorphisms of $M$, where $M\in\{I,\bR,S^1\}$. This name, though standard,
is a bit of a misnomer, since elements of $\PL(M)$ are technically \emph{piecewise affine}\index{piecewise affine}.
An element $f\in\PL_+(M)$ described as being locally defined
by affine maps of $M$ (where here the affine structure on $S^1$ is by the identification of $S^1=\bR/\bZ$) outside of a finite set of
points (called the \emph{breakpoints}\index{breakpoint} of $f$).
The derivative of $f$ is locally constant away from the breakpoints of $f$ and is usually
not continuous at the breakpoints, but we insist that $f$ be continuous. The fact that $\PL_+(M)$ is a group is an easy exercise.
The relationship between $\PL_+(M)$ and $\Diff_+^{\infty}(M)$ will be investigated in Chapter~\ref{ch:chain-groups}.
We can refine the filtration of $\Homeo(M)$ by diffeomorphism groups of integral regularity to non--integral regularities. Let
$f\colon X\longrightarrow Y$ be continuous, where $X$ and $Y$ are metric spaces.
Recall that $f$ is \emph{H\"older continuous}\index{H\"older continuity}
with exponent $\tau>0$ if there exists a constant $C>0$ such that for all $a,b\in X$, we have
\[d_Y(f(a),f(b))\leq C\cdot \left(d_X(a,b)\right)^{\tau}.\] The function $x\mapsto x^{\tau}$ is called a
\emph{modulus of continuity}\index{modulus of continuity}. The most interesting values of $\tau$ to consider lie in $(0,1]$, where the case
of $\tau=1$ is called \emph{Lipschitz continuity}\index{Lipschitz continuity}.
We say $f$ is \emph{locally $\tau$--H\"older continuous}\index{local H\"older continuity}
if it is H\"older continuous with the exponent $\tau$ in some neighborhood at each point.
A local modulus of continuity coincides with a global one if the space is compact
or if the function is compactly supported, i.e.~when the function is uniformly continuous.
For $\tau\in(0,1)$, we write $\Diff^{k,\tau}(M)$ for the set of $C^k$ diffeomorphisms of $M$ whose $k^{th}$ derivatives
are locally $\tau$--H\"older
continuous.
It is true that for $k\geq 1$, the set $\Diff^{k,\tau}(M)$ does indeed form a group, though some argument is needed to establish
this fact; we relegate further details to Appendix~\ref{ch:append1}.
The elements of this group are called $C^{k,\tau}$ diffeomorphisms.
For $\tau=1$, we write $\Diff^{k,\mathrm{Lip}}(M)$ for the $C^k$ diffeomorphisms
of $M$ whose $k^{th}$ derivatives are locally Lipschitz continuous.
We emphasize again that a regularity ($C^k$ or $C^{k,\tau}$) is a local property.
This convention makes the theory more consistent; for instance, $C^2$ diffeomorphisms are
necessarily $C^{1,\mathrm{Lip}}$ although they may not have globally Lipschitz first derivatives.
The set of homeomorphisms $\Diff^{k,\mathrm{Lip}}(M)$ of $M$ forms a group, even when $k=0$.
While it is true that Lipschitz maps are differentiable almost everywhere,
the distinction between almost everywhere differentiability and everywhere differentiability is significant. Whereas it is typical practice to write
$\Diff^{k+\tau}(M)$ for $\Diff^{k,\tau}(M)$, one must be careful when $\tau=1$.
Indeed, note that
\[\Diff^{k+1}(M)\neq\Diff^{k,1}(M)=
\Diff^{k,\mathrm{Lip}}(M).\]
These various groups of homeomorphisms and diffeomorphisms of manifolds have their own intrinsic topology, coming
inherited from the Whitney $C^k$
topology on all differentiable functions $M\longrightarrow M$.
Oftentimes, these groups can be thought of as infinite dimensional
Lie groups that are locally Banach or locally Fr\'echet, though we will usually not need this aspect of their structures.
Sometimes
the point--set topology of these groups does play a role, however, as we have already suggested. Indeed, these groups are often
not connected, this being typified by the existence of an orientation--reversing homeomorphism of $M$. Even restricting to
orientation preserving homeomorphisms, the group $\Homeo_+(M)$ and its differentiable subgroups may not be connected. This arises
from the fact manifolds often have nontrivial \emph{mapping class groups}\index{mapping class group},
which is to say groups of isotopy classes of homeomorphisms.
The mapping class group $\Mod(M)$ is generally defined to be $\pi_0(\Homeo(M))$, which has the structure of a group since $\Homeo(M)$
itself has the structure of a group. The topology on $\Mod(M)$ is the quotient topology, which in many cases of interest is simply the
discrete topology. In some parts of this book, primarily in Chapter~\ref{sec:filip-tak}, we will need connectedness of groups of homeomorphisms
and diffeomorphisms; for this reason, when it is relevant, we will restrict to the connected component of the identity in these groups.
A further complication arises for noncompact manifolds, which we will address when necessary: in dealing with noncompact
manifolds, we will usually consider only groups of compactly supported homeomorphisms or diffeomorphisms when investigating the
noncompact manifolds for their own sakes. Occasionally, we will consider noncompactly supported homeomorphisms of noncompact
manifolds that are lifts of a homeomorphism of a compact quotient. Of course, for compact manifolds, these preceding remarks are
generally irrelevant.
The connected component of the identity oftentimes has nontrivial topology, such as a nontrivial fundamental group. For the interval,
the group of orientation preserving homeomorphisms and $C^k$ diffeomorphisms is contractible, where as for the circle the corresponding
group is homotopy equivalent to the circle itself. This nontriviality of the fundamental group of $\Homeo_+(S^1)$ will be used to
pass freely between homeomorphisms of the circle and periodic homeomorphisms of the real line.
\section[The Mather--Thurston Theorem, the Epstein--Ling Theorem]{The Mather--Thurston Theorem, the Epstein--Ling Theorem, and lattice-like rigidity}
The Mather--Thurston Theorem, while itself not a subject of detailed discussion in this book,
has deep implications regarding Haefliger's classifying spaces for foliations and the homology of diffeomorphism groups.
In its essence, the theorem says that diffeomorphism groups of manifolds are algebraically rigid,
in the sense that the homomorphisms between them are rather limited.
We state a version of this fact that is particularly helpful for contextualizing the content
of this monograph.
\begin{thm}[See~\cite{Mather1,Mather2,Mather3,Thurston1974BAMS}]\label{thm:mather-thurston}
Let $M^n$ be a smooth connected boundaryless manifold of dimension $n\ge1$.
For $k\in\bZ_{>0}\cup\{\infty\}$, let $\Diff_c^k(M)_0 $ denote the group of compactly
supported diffeomorphisms of $M$ that are isotopic to the identity by a compactly supported isotopy. Then the group $\Diff_c^k(M)_0 $ is
simple, provided that $k\neq n+1$.
\end{thm}
We remark that in the literature, the Mather--Thurston Theorem often refers to the perfectness of the groups $\Diff_c^k(M)_0 $, i.e.~the fact
that
\[[\Diff_c^k(M)_0,\Diff_c^k(M)_0 ]=\Diff_c^k(M)_0 .\] The simplicity of the full group $\Diff_c^k(M)_0$ then follows from much more general
principles due to Higman, Epstein--Ling, and others; see Chapter~\ref{sec:filip-tak} for a detailed discussion of the simplicity of commutator
subgroups of diffeomorphism groups, and the discussion of Theorem~\ref{thm:epstein-ling-intro} below.
For the reader who is not familiar with Theorem~\ref{thm:mather-thurston}, the hypothesis on the dimension may appear bizarre,
and has to do with finding a fixed point of a certain operator. For finite $k$, when $k\neq n+1$ then this operator can be shown to admit
a fixed point, but the fixed point finding procedure fails otherwise. It is not clear whether this failure of the method of proof when $k=n+1$
is an artifact or a feature; indeed, Mather proves in~\cite{Mather3} that the group $\Diff_c^{1+\mathrm{bv}}(\bR)$ of compactly supported
diffeomorphisms of $\bR$ whose first derivatives have bounded variation is not perfect. By decomposing the distributional derivative
$D(\log f')$ for $f\in \Diff_c^{1+\mathrm{bv}}(\bR)$ into its regular and singular parts and by integrating against the regular part of the
measure, Mather shows that one obtains a surjective homomorphism from $\Diff_c^{1+\mathrm{bv}}(\bR)$ to $\bR$.
We direct the reader to~\cite{CKK2019} for a detailed exposition of
Theorem~\ref{thm:mather-thurston} in the case $n=1$.
More classically, it was known from the work of Epstein and Ling that the commutator subgroup
of $\Diff_c^k(M)_0$ is simple.
\begin{thm}\label{thm:epstein-ling-intro}
Let $M^n$ be a smooth connected boundaryless manifold of dimension $n$.
For $k\in\bZ_{>0}\cup\{\infty\}$, let $\Diff_c^k(M)_0 $ be as in Theorem~\ref{thm:mather-thurston}. Then the group
\[\Diff_c^k(M)_0 '=[\Diff_c^k(M)_0 ,\Diff_c^k(M)_0 ]\]
is simple.
\end{thm}
We shall prove Theorem~\ref{thm:epstein-ling-intro}
in Chapter~\ref{sec:filip-tak}. The argument that $\Diff_c^k(M)_0 '$ is simple is not sensitive to the dimension
of $M$, and is general enough to be applicable to a wide range of other groups. The step from Theorem~\ref{thm:epstein-ling-intro}
to Theorem~\ref{thm:mather-thurston} is in computing the homology group $H_1(\Diff_c^k(M)_0 ,\bZ)$, where $\Diff_c^k(M)_0 $ is viewed
as a discrete group.
As a consequence of Theorem~\ref{thm:epstein-ling-intro}
and Theorem~\ref{thm:mather-thurston}, we have the following observation:
\begin{cor}\label{cor:diffeo-homo}
Let $M^m$ and $N^n$ be smooth connected boundaryless manifolds of dimensions $m$ and $n$, and let $k,\ell\in\bZ_{>0}\cup\{\infty\}$.
If \[\phi\colon\Diff_c^k(M)_0 \longrightarrow\Diff_c^{\ell}(N)_0\]
is a non-injective homomorphism, then its image is abelian; if in addition $k\neq \dim M+1$, the image is trivial.
\end{cor}
In practical applications, it is not always so easy to show that the map $\phi$ in Corollary~\ref{cor:diffeo-homo} fails to be injective. Sometimes
it is injective: indeed, if $M=N$ and if $\ell\leq k$, then there is a natural inclusion between
the diffeomorphism groups. Were one to try to prove that
there is no injective homomorphism between these diffeomorphism groups when $\ell>k$ from first principles, it is not immediately apparent
how to proceed. Why, after all, should there be no way to find a simultaneous smoothing of all $C^k$ diffeomorphisms of $M$? The answer
is that one cannot, and we will return to this question in Section~\ref{sec:filip-tak-intro} below;
the reader may also consult~\cite{HurtadoGT15,Kramer11,Mann2015,Mann2016GT,Mann-Notices,RoseSol07,Rybicki95}
for some background results in this direction.
Before proceeding, we will take this opportunity to clarify some terminology that is relevant to the discussions in this book. There are
two distinct notions of smoothability of groups of homeomorphisms or diffeomorphisms, namely
\emph{topological smoothability}\index{topological smoothability} and
\emph{algebraic smoothability}\index{algebraic smoothability}.
If $G$ is a group of homeomorphisms or diffeomorphisms of a manifold $M$, then a topological smoothing
of $G$ into $\Diff^{k,\tau}(M)$ is given by a topological conjugacy of $G$ into $\Diff^{k,\tau}(M)$.
That is, there is a homeomorphism
$h\co M\longrightarrow M$ such that \[hGh^{-1}\le \Diff^{k,\tau}(M).\]
If the regularity of the elements of $G$ is lower than $(k,\tau)$ then clearly
$h$ cannot be an element of $\Diff^{k,\tau}(M)$, and it is best to simply record $h$ as a homeomorphism. We remark that
\emph{in the literature
on group actions, a smoothing of a group is usually a topological smoothing}.
An algebraic smoothing of $G$ is simply an injective homomorphism into $\Diff^{k,\tau}(M)$. Clearly a topological smoothing is an
algebraic smoothing, though the converse many not hold. A topological smoothing of $G$ respects all the dynamical properties of the action
of $G$, whereas an algebraic smoothing is blind to everything but the algebraic structure of $G$. There will be several places in this book
where we will explicitly be interested in topological smoothings, and these will be clearly noted as such; see Corollary~\ref{c:slp-0}, for instance. Otherwise, the primary
interest herein will be algebraic smoothings of groups.
One of the primary goals of this book is to give examples of finitely generated groups of diffeomorphisms that have a prescribed level
of regularity. We will defer stating the full statement of the relevant results until Section~\ref{sec:critreg-intro} below, stating only a relevant
consequence.
\begin{prop}\label{prop:unsmooth}
For $k\in\bZ_{>0}$ and $\tau\in [0,1)$
and for $M\in\{I,S^1\}$, there exists a finitely generated nonabelian group $G_{k,\tau}$ with a simple commutator subgroup
such that there is an injective homomorphism
\[\phi\colon G_{k,\tau}\longrightarrow \Diff_+^{k,\tau}(M),\] but such that
for every finite index subgroup $H_{k,\tau}\le G_{k,\tau}$, every homomorphism \[\psi\colon H_{k,\tau}\longrightarrow
\Diff_+^{\ell,\eps}(M)\] has abelian image, where $\ell\in\bZ_{>0},\epsilon\in[0,1)$ and $k+\tau<\ell+\eps$.
\end{prop}
The conclusion of Proposition~\ref{prop:unsmooth} cannot be improved to triviality of the image. This is a consequence of Thurston's
Stability Theorem, which will be discussed in Appendix~\ref{ch:append3}. Thurston proves that a finitely generated group of diffeomorphisms
of $\Diff^1[0,1)$ always surjects to $\bZ$ and hence cannot be simple.
Proposition~\ref{prop:unsmooth} gives a finitary (in the sense that it is determined by finitely many group elements) obstruction to the
algebraic smoothability of a group. We thus obtain the following consequence from Theorem~\ref{thm:mather-thurston} and
Theorem~\ref{thm:epstein-ling-intro}:
\begin{cor}\label{cor:diffeo-homo-fg}
Let $M\in\{\bR,S^1\}$, and suppose that $k+\tau<\ell+\eps$. Then every homomorphism \[\Diff_c^{k,\tau}(M)_0\longrightarrow\Diff_+^{\ell,\eps}(M)\]
has trivial image, with the possible exception when $k=2$ and $\tau=0$, in which case every such homomorphism has abelian image.
\end{cor}
The absence of an analogue of Corollary~\ref{cor:diffeo-homo-fg} in higher dimensions is one of the motivations for developing a theory
of critical regularity for finitely generated groups in dimensions two and higher. In dimension one, the question of whether $\Diff_c^2(\bR)$
is simple remains open, and our finitary methods do not shed any light on this question.
The groups furnished by Proposition~\ref{prop:unsmooth} enjoy rigidity properties that are reminiscent of various rigidity results for
lattices in higher rank Lie groups. The most general of these is Margulis' Superrigidity Theorem. We state a simplified version of this
result, since the full generality would not illuminate the principle any further.
\begin{thm}[Margulis Superrigidity, simplified statement~\cite{MargulisBook1991,Witte2015}]\label{thm:margulis-super}
Let $\Gamma$ be a non-cocompact lattice in $\mathrm{SL}_{k\ge3}(\bR)$, and let \[\rho\colon \gam\longrightarrow\mathrm{GL}_n(\bR)\]
be a homomorphism. Then there is a continuous homomorphism \[\widehat\rho\colon \mathrm{SL}_k(\bR)\longrightarrow\mathrm{GL}_n(\bR)\]
such that $\rho$ and $\widehat\rho$ coincide on a finite index subgroup of $\Gamma$.
\end{thm}
Here, a \emph{lattice}\index{lattice} is a discrete subgroup in $\mathrm{SL}_k(\bR)$ whose covolume (with respect to the Haar measure) is
finite.
Theorem~\ref{thm:margulis-super} relies on a great deal of structure to which we simply do not have access. The analogy between the
groups furnished by Proposition~\ref{prop:unsmooth} and nonuniform (i.e.~non-cocompact) lattices in $\mathrm{SL}_k(\bR)$ is
simply that homomorphisms from them to a group of more regular diffeomorphisms are extremely limited.
\begin{cor}\label{cor:super-analogue}
Let $G_{k,\tau}$ be as in Proposition~\ref{prop:unsmooth}.
Suppose $k+\tau<\ell+\eps$. If $H\le \Diff_+^{\ell,\eps}(M)$ is a subgroup, then every homomorphism \[G_{k,\tau}\longrightarrow H\] has
abelian image.
\end{cor}
\section{The Takens--Filipkiewicz Theorem and Rubin's Theorem}\label{sec:filip-tak-intro}
Another facet to the line of inquiry pursued in this book emerges from the following basic question: to what degree is a mathematical object
determined by its group of symmetries? If $M$ is a manifold, it is reasonable then to wonder the degree to which $M$ is determined by
its natural group of symmetries, which in the context of the foregoing discussion would be its group of homeomorphisms $\Homeo(M)$.
If $M$ has further structure, such as a smooth structure, a complex structure, a symplectic structure, etc.~then one can pose an analogous
question as to the degree to which $M$ with the further structure is preserved by the group of symmetries of $M$ that preserve that structure.
In 1963, Whittaker~\cite{Whittaker1963} proved that an abstract isomorphism \[\phi\colon\Homeo(M)\longrightarrow\Homeo(N)\] between
two arbitrary manifolds arises from a homeomorphism between $M$ and $N$. That is,
there is a homeomorphism $\alpha\colon M\longrightarrow N$ such that for all $f\in\Homeo(M)$, we have that
\[\phi(f)=\alpha\circ f\circ\alpha^{-1}.\]
This reconstruction theorem completely determines
isomorphisms between homeomorphism groups of manifolds.
While this book is about smooth structures on manifolds, we remark that
Whittaker's theorem is also a trivial consequence of Rubin's Theorem, as stated below.
It turns out that the smooth structure
of a manifold is completely determined by the group $\Diff^r_c(M)_0$ up to $C^r$ diffeomorphism, at least when $M$ has no boundary.
The first result we will discuss in this direction is as below, which is the weaker of the two; a special case of this theorem is originally due to F.~Takens.
\begin{thm}[Takens' Theorem]\label{thm:takens-intro}
If a set-theoretic bijection
$w\colon M\longrightarrow N$
between smooth connected boundaryless manifolds $M$ and $N$
conjugates $\Diff_c^{p}(M)_0$ to $\Diff_c^{p}(N)_0$
for some $p\in\bN\cup\{\infty\}$,
then $w$ is a $C^p$ diffeomorphism.\end{thm}
Theorem~\ref{thm:takens-intro} is an example of a reconstruction theorem that takes as an input a relatively weak hypothesis (i.e.~a
set theoretic bijection inducing a certain group isomorphism) and promotes that bijection to a map with a high level of structure
(i.e.~a $C^r$ diffeomorphism).
We will (nearly) replicate the complete
original proof of Theorem~\ref{thm:takens-intro} in the case of one--dimensional
manifolds in Section~\ref{sec:takens} and $p=\infty$. The only part of the proof we will not spell out is a result of Sternberg on local linearization of
diffeomorphisms with hyperbolic fixed points. We will not reproduce Takens' argument in higher dimensions.
One reason is to avoid introducing a large amount of additional background.
Another is that Taken's theorem is subsumed by Filipkiewicz's Theorem, which itself can be further generalized as follows:
\begin{thm}[Takens--Filipkiewicz Theorem]\label{thm:filip-intro}
Let $M$ and $N$ be smooth connected boundaryless manifolds.
If there exists a group isomorphism
\[ \Phi\co \Diff_c^{p}(M)_0\longrightarrow \Diff_c^{q}(N)_0\]
for some $p,q\in\bN\cup\{\infty\}$,
then
we have that
$p=q$
and that
there exists a $C^p$ diffeomorphism
$w\colon M\longrightarrow N$ inducing $\Phi$ by conjugation.
\end{thm}
Whereas Takens' Theorem presumes the existence of a bijection between $M$ and $N$ that induces an isomorphism between the
corresponding diffeomorphism groups, Filipkiewicz starts with a very weak assumption (i.e.~the groups
$\Diff_c^p(M)_0 $ and $\Diff_c^q(N)_0$ are
isomorphic) and ends with a strong conclusion (i.e.~$p=q$ and $M$ and $N$ are diffeomorphic via a $C^p$ diffeomorphism that
induces the isomorphism). As a consequence, Filipkiewicz's Theorem implies that there are no exotic automorphisms of $\Diff_c^p(M)_0 $
for a smooth manifold $M$; they are all induced by $C^p$ diffeomorphisms of $M$.
In our exposition of Filipkiewicz's result, we will retain a version of his argument for the sake of historical record, though in the main body
of the exposition, we will obtain Filipkiewicz's Theorem as a consequence of a simplified version of a much more general result due to Rubin.
\begin{thm}[Rubin's Theorem]\label{thm:rubin-intro}
If we have an isomorphism between two locally dense groups of homeomorphisms on perfect, locally compact, Hausdorff topological spaces,
then there exists a homeomorphism of those spaces that conjugates the isomorphism.
\end{thm}
In Theorem~\ref{thm:rubin-intro}, let $G$ be a group of homeomorphisms of a locally compact Hausdorff topological
space $X$ and let $U\sse X$. We write $G[U]\le G$ for the subgroup consisting of homeomorphisms that are the identity outside of $U$.
We say $G$ is \emph{locally dense}\index{locally dense action}
if for each point $x\in X$ and for each open neighborhood $U$ of $x$,
the closure of the orbit $G[U].x$ has nonempty interior. We will give a self-contained proof of Rubin's Theorem, which the authors hope
the reader will find intellectually satisfying.
Our presentation of Filipkiewicz's Theorem is based on Rubin's Theorem, which will imply that an isomorphism between
diffeomorphism groups of manifolds automatically induces a homeomorphism between them. To upgrade this homeomorphism to
diffeomorphism of the desired regularity, one uses a simultaneous continuity result for actions of $\bR^n$ on manifolds that is due to
Bochner--Montgomery, and which itself is often
incorrectly attributed to Montgomery--Zippin. The reader will find a full account of the details in Section~\ref{sec:boch-mont}.
Returning to the main thread of discussion, let $M$ be a given manifold.
Even though for $p\neq q$ we have that $\Diff_c^p(M)_0 $ and
$\Diff_c^q(M)_0$ are not isomorphic to each other as
algebraic groups, it is very difficult to imagine distinguishing between two such enormous infinite--dimensional continuous topological groups,
at least from an algebraic point of view. There are many different ways that one could hope to distinguish between two groups
$G$ and $H$, which is to say in an attempt
to find a checkable certificate that they are not isomorphic to each other. One such certificate might come from looking at the class of
subgroups of $G$ and $H$, and to find a subgroup of $G$ that does not occur as a subgroup of $H$, or vice versa. The simplest
subgroups to consider which have any hope of distinguishing $G$ and $H$ would be the finitely generated subgroups, and so we can
formulate a motivating question.
\begin{que}\label{que:subgroup-main}
Let $p< q$. Do the finitely generated subgroups of $\Diff^p_c(M)_0$ distinguish it from $\Diff^q_c(M)_0$? That is, is there a finitely generated group
of $C^p$ diffeomorphisms of $M$ that is not algebraically $C^q$ smoothable, i.e.~realized as a subgroup of $\Diff^q_c(M)_0$? More generally,
can we distinguish between $\Diff^p_c(M)_0$ and $\Diff^q_c(N)_0$ by finitely generated subgroups in order to conclude that either $p\neq q$ or
$p=q$ and $M$ is not $C^p$--diffeomorphic to $N$?
\end{que}
Returning to the results in this book that we have already announced, we restate Proposition~\ref{prop:unsmooth} in a way that
gives a satisfactory answer to Question~\ref{que:subgroup-main}, at least in the case of one--dimensional compact manifolds.
In the case of $M=I$ or $M=S^1$, every element of $\Homeo_+(M)$ is already isotopic to the identity, and compactness of isotopies
is automatic by the compactness of the ambient manifold, and so we can safely suppress the
$0$--subscripts in the notation for diffeomorphism
groups. We will write $r=k+\eps$ and $s=\ell+\tau$, where $k,\ell\in\bZ_{>0}$ and where $\eps,\tau\in [0,1)$, and $M$ and $N$ will
both denote $I$ or $S^1$.
\begin{prop}\label{prop:unsmooth-refined}
For compact connected one--manifolds $M$ and $N$,
the following conclusions hold.
\begin{enumerate}[(1)]
\item
If $r<s$ then there is a finitely generated subgroup $G_r \le \Diff^r_+(M)$ such that no finite index subgroup $G_r$ is isomorphic to a subgroup
of $\Diff^s_+(M)$.
\item
For all $r$ and $s$ and $M\neq N$, there is a finitely generated subgroup $G_{r,M}\le \Diff_+^r(M)$ such that no finite index subgroup of
$G_{r,M}$ is isomorphic to a subgroup of $\Diff_+^s(N)$.
\end{enumerate}
\end{prop}
Indeed, the answer to Question~\ref{que:subgroup-main} is yes, in the case of compact one--manifolds. The reader may object to a claim
that the subgroups $G_r$ or $G_{r,M}$ are truly ``certificates", in the sense of some sort of easy checkability. Even though each group $G_r$
or $G_{r,M}$ is finitely generated, Proposition~\ref{prop:unsmooth-refined} implicitly furnishes an uncountable collection of pairwise
non--isomorphic subgroups, for example $\{G_r\}_{r\in\bR_{\geq 1}}$. There are only countably many isomorphism classes of finitely
presented groups, and in fact only countably many isomorphism classes of recursively presented groups (i.e.~finitely generated groups
whose relations are enumerable by a Turing machine) since there are only countably many different Turing machines. It follows that
the vast majority of the groups $\{G_r\}_{r\in\bR_{\geq 1}}$ are not even recursively presentable,
which leads to a completely valid philosophical
question as to what it means to record such a group. The reader will find in the proof of Proposition~\ref{prop:unsmooth-refined}
(or, really, the proof of the most general results from which Proposition~\ref{prop:unsmooth-refined} follows), we will write down
generators for the groups $\{G_r\}_{r\in\bR_{\geq 1}}$, though again the claim that we are really ``writing down" diffeomorphisms is debatable,
since we will express them as certain limits which are not truly explicit. The proof that $G_r$ admits no homomorphism with
nonabelian image to $\Diff_+^s(M)$ for $s>r$ will inevitably have to take a detour through some infinitary methods.
Even the assertion that most of the groups $\{G_r\}_{r\in\bR_{\geq 1}}$ are not recursively
presentable is non--constructive, since it says nothing
about $G_r$ for any specific value of $r$ and rather relies on the set theoretic properties of the indexing set. If one restricts to a countable
subset of the indexing set, we do not know if the corresponding groups are recursively presentable. We are left with the following question:
\begin{que}\label{que:fp}
For $M\in \{I,S^1\}$ and $k\in\bN$, is there a finitely presentable subgroup $G_k\le \Diff_+^k(M)$ that is not isomorphic to
a subgroup of $\Diff_+^{k+1}(M)$? What about a recursively presentable subgroup?
\end{que}
We direct the reader to the last paragraph of Appendix C for an answer when $k=0$.
Unfortunately, the methods given in this book do not give any insight into Question~\ref{que:fp}, at least as far as the authors can see.
The question of whether or not there exists an easily checkable certificate to distinguish two diffeomorphism groups remains, in this sense,
open.
\section{Critical regularity: history and overview}\label{sec:critreg-intro}
Much of the discussion in this book will be framed in terms of \emph{critical regularity}\index{critical regularity},
which we introduce and survey in this section.
The critical regularity of a group should be thought of as the sharp bound on the level of smoothness with which the group
can act on a smooth manifold, or more precisely, the upper limit on the algebraic smoothability of a group.
\subsection{Definitions and remarks}\label{ss:defn-rem}
To formulate a precise definition, let $G$ be a group, and let $M$ be a smooth manifold. The \emph{critical regularity}
of $G$ is defined to be \[\CR_M(G)=\sup\{r\in\bR_{\geq 0}\mid G\le \Diff_c^r(M)_0\}=\sup\{k+\eps\mid G\le \Diff_c^{k,\eps}(M)_0\}.\]
Here, $G$ is an abstract group, and so
$G\le \Diff^r(M)$ is shorthand for there existing an injective homomorphism $G\longrightarrow\Diff^r(M)$.
One could in principle extend the definition of critical regularity to the full group of $C^r$ diffeomorphisms of $M$, though in the cases
$M=I$ and $M=S^1$ that are the primary subject of this book, restricting the definition is the same as assuming that the diffeomorphisms
under consideration are orientation preserving.
The definition of critical regularity we have given here can be taken to be the more refined
\emph{algebraic critical regularity}\index{algebraic critical regularity}, which
is then contrasted with the \emph{topological critical regularity}\index{topological critical regularity},
which we define now for later use. The topological critical regularity
refers not to a group but to a group with an action on $M$, and is then the supremum of regularities $r$ for which the action is topologically
conjugate into a group of $C^r$ diffeomorphisms.
From the definition of (algebraic) critical regularity, we have that:
\[\textrm{if}\quad G\le \Homeo_0(M)\quad \textrm{then}\quad \CR_M(G)\geq 0.\] If there is no injective homomorphism $G\longrightarrow
\Homeo_0(M)$ then we have \[\CR_M(G):=\inf\{\}=-\infty.\] When $G\le \Diff^{\infty}_c(M)_0$ we have $\CR_M(G)=\infty$.
Thus, \emph{a priori}, the possible
range of $\CR_M$ is $\{-\infty\}\cup [0,\infty]$.
For an arbitrary given manifold $M$, it is not clear that this range is fully populated; that is, it is often
very difficult to decide if for finite values of $r$ if there is a group $G$ such that $\CR_M(G)=r$, especially if
one insists that $G$ be a countable
or finitely generated group.
Note from Filipkiewicz's Theorem that
\[
\CR_M(\Diff^p_c(M)_0)=\CR_M(\Diff^p(M))=p\]
for all smooth connected boundaryless $n$--manifold $M$ and for all $p\in\bN$.
The set of actual values achieved by $\CR_M(G)$ for finitely generated groups $G$ is called the \emph{critical regularity spectrum}\index{critical
regularity spectrum} of $M$.
As the critical regularity of a group is a supremum, the statement that \[\CR_M(G)=r\] does not
necessarily mean that there is an injective homomorphism
$G\longrightarrow\Diff^r_c(M)_0$, but rather only that there is an injective homomorphism into $\Diff^s_c(M)_0$ for all $s<r$. Thus, the
finite part of the critical regularity
spectrum of $M$ bifurcates into the \emph{achieved spectrum}\index{achieved spectrum} and the
\emph{unachieved spectrum}\index{unachieved spectrum}, where $r$ lies in the achieved
spectrum if there is in fact an injective homomorphism
$G\longrightarrow\Diff^r_c(M)_0$.
For the rest of this section, we will survey the state of knowledge about critical regularity for $M\in\{I,S^1\}$. The first examples of
groups of homeomorphisms of $M$ that one produces generally have infinite critical regularity. Indeed, a single homeomorphism
of $M$ generates a copy of $\bZ$ (or possibly $\bZ/n\bZ$ in the case $M=S^1$), and even if this homeomorphism is not continuously
differentiable, the group generated by it is algebraically smoothable. In the case of $M=I$, the group generated by the homeomorphism
is even topologically smoothable, though in the case of $M=S^1$, it may not be topologically smoothable.
From a flow on $M$, one can build free abelian groups of diffeomorphisms of arbitrarily large rank. Since one can build a smooth flow
from a smooth vector field on $M$, the critical regularity of a free abelian group is infinite. For free abelian groups of rank two or more,
one sees the difference between algebraic smoothability and topological smoothability of groups acting on $I$.
Whereas given a faithful action
by $\bZ^n$ for $n\geq 2$, one can always find another faithful action of this group that is as smooth as one likes, though one might not
be able to conjugate a given faithful action of $\bZ^n$ to a smoother one; see Kopell's Lemma (Theorem~\ref{thm:kopell}), for instance.
Nonabelian free groups also have infinite critical regularity, at least if they have at most countable rank. This fact is somewhat harder to
see than the infinite critical regularity of free abelian groups, and results essentially from a Baire Category type argument. We will
spell out some of the details in Corollary~\ref{cor:free-abundant}.
For groups that are not abelian or free, the computation of critical regularity quickly becomes difficult, as it can be hard to produce
faithful actions of a given group in the first place, and understanding the regularity properties of the group action usually involves very
subtle convergence questions and correspondingly subtle techniques. The definition of critical regularity implicitly quantifies over all
possible actions of a group on $M$, and so an additional layer of complication is added by the fact that it can be quite difficult to tell
if one has addressed all possible action of a group. To deal with this complication, one has to have good control over the orbit structure
for group actions on $M$, and this is where dynamical methods play an essential role.
\subsection{Cyclic groups, topological versus algebraic critical regularity}
We have already alluded to one of the earliest manifestations of the interplay between regularity, dynamics, and group actions, and
has to do with one of the simplest groups (i.e.~$\bZ$) acting freely on the circle $S^1$. Here, a \emph{free action}\index{free action}
is one where
no nontrivial group element fixes a point in $S^1$. The easiest examples of such actions come from the action of $S^1$ on itself, when
the manifold is viewed as an abelian group. Writing $S^1=\bR/\bZ$, an arbitrary element of $S^1$ acts additively on $S^1$, and the
subgroup generated by that element clearly acts freely. It is not difficult to see that an element $\theta\in S^1$ will have infinite order
as a homeomorphism of $S^1$ if and only if $\theta$ is not rational under the identification $S^1=\bR/\bZ$.
We write the corresponding element of $\Homeo_+(S^1)$ as $R_{\theta}$ and call it an \emph{irrational rotation}\index{irrational rotation}.
Clearly not every infinite order homeomorphism of $S^1$ acting freely is equal to an irrational rotation. Indeed, if $f\in\Homeo_+(S^1)$ is
arbitrary then $\form{ f^{-1}R_{\theta}f}\cong\bZ$ will be a free action of $\bZ$ on $S^1$ which will not be a rotation itself unless
$f$ commutes with $R_{\theta}$, which is quite rare among homeomorphisms that are not themselves rotations. Nevertheless, this is
a somewhat ``trivial" perturbation of a rotation, since $f^{-1}R_{\theta}f$ differs from $R_{\theta}$ only via an identification of $S^1$ with
itself, which is to say $f^{-1}R_{\theta}f$ is \emph{topologically conjugate}\index{topological conjugacy} to $R_{\theta}$.
For an arbitrary element $g\in\Homeo_+(S^1)$, one can define an invariant called its \emph{rotation number}\index{rotation number},
and written $\rho(g)$.
This invariant is quite old, having been introduced by Poincar\'e, and has applications in geometric group theory and bounded cohomology
of groups as well as in dynamics~\cite{FrigerioBook,Ghys1987,Ghys2001,KKM2019}.
The rotation number takes values in $S^1=\bR/\bZ$, and it measures the tendency of $g$ to make points in $S^1$ wind about the circle.
We will avoid giving a precise definition of the rotation number here, as the definition is not so straightforward, and since
it will be discussed in great detail in Chapter~\ref{sec:denjoy}.
Several of the most important features of the rotation number are that it is invariant under topological conjugacy,
that it is nonzero precisely in the absence of a fixed point, that it is a homomorphism when restricted to the cyclic group generated
by a single homeomorphism (i.e.~it is \emph{homogeneous}\index{homogeneity}), and that it satisfies
$\rho(R_{\theta})=\theta$.
It is therefore natural to wonder, at least for irrational values of $\theta$, whether or not
the rotation number is a complete topological conjugacy invariant; that is, if $\rho(g)=\theta$, is it true that $g$ is topologically conjugate
to $R_{\theta}$?
The answer, perhaps surprisingly at first, is that it depends on the regularity of $g$. In broad strokes, we have the following:
\begin{thm}[Denjoy's theory of rotations, coarse summary]\label{thm:denjoy-intro}
If $\theta\in\bR/\bZ$ is irrational, then the following conclusions hold.
\begin{enumerate}[(1)]
\item
There exist continuous, and even $C^1$ diffeomorphisms $g$ of $S^1$ such that $\rho(g)=\theta$ but such that $g$ is not topologically
conjugate to $R_{\theta}$.
\item
If $g\in\Diff_+^2(S^1)$ and $\rho(g)=\theta$, then $g$ is topologically conjugate to $R_{\theta}$.
\end{enumerate}
\end{thm}
For a commonly satisified regularity assumption on $g$ that guarantees topological conjugacy to a rotation, one can relax
twice--differentiability to $\Diff_+^{1+\mathrm{bv}}(S^1)$, the group of $C^1$ diffeomorphisms of $S^1$ whose first derivatives have
bounded variation. Chapter~\ref{sec:denjoy} will give a self--contained and complete account of Theorem~\ref{thm:denjoy-intro}, and
will in the process will provide a full proof.
As we have mentioned before, there is a rather large gap between $\Diff_+^1(S^1)$ and
$\Diff_+^2(S^1)$, or even $\Diff_+^{1+\mathrm{bv}}(S^1)$, one that is manifested
not least by considering all possible diffeomorphisms whose derivatives satisfy
various H\"older continuity conditions. It is still an area of active research to determine the exact conditions on the derivatives of a
diffeomorphism of $S^1$ which guarantee topological conjugacy to a rotation, or conversely that allow for the construction of diffeomorphisms
with prescribed rotation numbers that are not conjugate to rotations (so--called \emph{Denjoy counterexamples}\index{Denjoy
counterexample}).
Before proceeding, we recall the notion of $\alpha$--continuity, which is a generalization of Lipschitz and H\"older continuity. Let
\[\alpha\colon\bR_{\geq0}\longrightarrow\bR_{\geq0}\] be a homeomorphism that is concave as a function. We say that a real valued function
$f$ is $\alpha$--continuous if there is a universal constant $C$ such that \[|f(x)-f(y)|\leq C\cdot \alpha(|x-y|)\] for all $x$ and $y$ in the domain
of $f$. Note that $\tau$--H\"older continuity is just $\alpha$--continuity for $\alpha(x)=x^{\tau}$. We call $\alpha$ a \emph{concave modulus of
continuity}\index{concave modulus of continuity}.
To give the reader an idea of some of what is known, we will give a full proof of the following result due to the authors in
Chapter~\ref{sec:denjoy}. In broad strokes, this theorem
shows that there are many Denjoy counterexamples that are very close to being $C^2$:
\begin{thm}\label{thm:int-denjoy}
Let $\alpha$ be a concave modulus of continuity and let $\theta\in\bR/\bZ$ be irrational.
If \[\int_{(0,1]}\frac{dx}{\alpha(x)}<\infty,\] then there exists a Denjoy counterexample $f\in\Diff_+^1(S^1)$ such that $f'$ is $\alpha$--continuous.
\end{thm}
\subsection{Foliations}
We now move beyond cyclic groups to describe the provenance and state of knowledge concerning critical regularity for finitely generated
groups, and more generally, countable groups.
One of the original motivations for studying regularity of group actions arises from foliation theory. A \emph{foliation}\index{foliation}
on a manifold $M$
of dimension $n$ is an equivalence relation on $M$, where the equivalence classes are given by $p$--dimensional immersed submanifolds,
for $p\leq n$. Thus, a foliated manifold locally looks like a direct product decomposition $\bR^n\cong\bR^p\times \bR^q$, where $q=n-p$,
and the immersed $p$--dimensional manifolds are called \emph{leaves}\index{leaf of a foliation} of the foliation.
The numbers $p$ and $q$ are the \emph{dimension}\index{dimension of a foliation} and
\emph{codimension}\index{codimension of a foliation} of the foliation, respectively.
Imagine the French \emph{mille
feuille}\index{mille feuille} (thousand leaves, or thousand sheets) pastry.
The data describing a foliation is a coherent atlas, much like one defining a manifold, which keeps track of the local product decomposition,
and which satisfies suitable compatibility conditions. We will not recall the precise definition of a foliation here, postponing it
instead to Section~\ref{sec:foliation}.
Foliation theory originates in differential topology, with its foundational result perhaps being
Ehresmann's Submersion Theorem~\cite{Ehresmann51,DundasBook},
which was seminal not only in the theory of foliations but also in the study of fibrations and fiber bundles. Foliation theory developed into
a vast and deep subject, with connections to relativity, differential geometry, $C^*$--algebras, noncommutative geometry, and dynamical
systems. We will not attempt to summarize this story here, and we will only note the facts we need
when they are required in Chapter~\ref{ch:app}. We refer the reader to~\cite{CandelConlonI,CandelConlonII} for an encyclopedic reference on
foliation theory.
Even when $M$ is a smooth manifold, the data of the foliation may have a degree of regularity which is much lower than that of $M$.
Indeed, even when $M$ is a smooth manifold, one can find foliations on $M$ whose transition functions (i.e.~the ones encoding the local
product decomposition) are not differentiable, and such that there is no homeomorphism (of foliated manifolds) to another manifold
wherein the target foliation is differentiable.
The smoothness of a foliation is closely related to a certain pseudo-group associated to the foliation, called its
\emph{holonomy}\index{holonomy}. In
Section~\ref{sec:foliation}, we will describe a particular type of construction of foliations, called
\emph{suspension of a group action}\index{suspension of a group action},
which realizes many groups as holonomy groups of foliations. The suspension of a group action is essentially the same as the construction
of a flat bundle $E$ with fiber $F$ and base space $M$,
whose monodromy group lies in $\Diff^r(M)$ and some suitable regularity $r$, and results in a
foliated bundle with exactly that regularity. More precisely, the data of the foliation is specified by a representation \[\phi\colon\pi_1(F)
\longrightarrow \Diff^r(M),\] and the homeomorphism class of the foliated bundle is completely determined by the topological conjugacy class
of $\phi$. In particular, the construction of suspended group actions that are unsmoothable beyond some regularity $r$ is equivalent to
constructing representations $\phi$ into diffeomorphism groups that are not topologically smoothable. This is one specific reason that
regularity of group actions is of interest in foliation theory.
Since topological smoothability is a stronger condition than algebraic smoothability, it is perhaps not surprising that it is somewhat easier
to find examples of groups with a given topological critical regularity (i.e.~a particular action of the group cannot be topologically smoothed
to a higher degree of regularity) than it is to find examples of groups with a given algebraic critical regularity (i.e.~no action in higher degree
of regularity exists). In Section~\ref{sec:foliation}, we will discuss in detail a construction due to Tsuboi of such a group with given topological
critical regularity; in all degrees, the group is fixed, and so the algebraic critical regularity of the abstract group in question is infinite,
but the particular actions Tsuboi constructs are not topologically conjugate to smoother ones. Another construction in a very similar
vein was also found by Cantwell and Conlon. We will delay a discussion of the details until Chapter~\ref{ch:app}.
Obtaining control on the topological critical regularity of a group action already has consequences for foliation theory, as we see, though
one can ask if a certain abstract group occurs as the holonomy of a codimension one foliation, for example. When it comes to
analyzing the smoothness of a foliation with such a holonomy group, the critical regularity of the group plays an essential role.
\subsection{Nilpotent groups}
We have seen that free abelian groups may have finite topological critical regularity, even in rank one, but their (algebraic) critical
regularity as we are discussing it is infinite. The next level of algebraic complication for groups is nilpotence. It is not
difficult to show that finitely generated torsion--free nilpotent groups are \emph{orderable}\index{orderable group},
which is to say that they admit total orderings
that are compatible with the group structure. We include a quick proof of this fact for the convenience of the reader, and it will be clear
that the method of proof generalizes to a wide class of groups.
\begin{prop}
Let $N$ be a finitely generated torsion--free nilpotent group. Then $N$ admits a left invariant ordering, which is to say a total ordering $\leq$
such that for all $a,b,c\in N$, we have $a\leq b$ if and only if $ca\leq cb$.
\end{prop}
\begin{proof}
We proceed by induction on the \emph{Hirsch length}\index{Hirsch length}
of $N$, or the \emph{polycyclic length}\index{polycyclic length} of $N$. That is, we look at the longest subnormal
chain of subgroups \[N=N_0>N_1>\cdots>N_k=\{1\},\] such that $N_i$ is normal in $N_{i-1}$ for all suitable indices, and so that
$N_{i-1}/N_i\cong\bZ$. The length of this series is defined to be $k$.
It is a straightforward exercise to show that such a sequence exists and that it terminates.
Clearly, we have that the group $\bZ$ admits a left invariant ordering, as we may consider the inherited ordering from $\bR$. Next, if
\[1\longrightarrow K\longrightarrow G\longrightarrow Q\longrightarrow 1\] is a short exact sequence of groups such that $K$ and $Q$
are both left orderable, then so is $G$. Indeed, we let $g_1,g_2\in G$. We we say $g_1\leq g_2$ if the images of these elements in $Q$
satisfy $\overline{g_1}\leq \overline{g_2}$ in the ordering on $Q$, or if $g_1^{-1}g_2\in K$ and $1\leq g_1^{-1}g_2$ in the ordering on $K$.
The reader may check that this is a left invariant ordering on $G$. It is clear now that $N$ admits a left invariant ordering.
\end{proof}
Among other things, we see that if $N$ is finitely generated, torsion--free, and nilpotent,
then $N\le \Homeo_+(M)$ for $M\in\{I,S^1\}$. The above result extends to solvable groups as well.
We direct the reader to
Appendix~\ref{ch:append2} for more detail on orderability, and in particular
a proof of the fact that a countable group admitting a left invariant ordering admits an injective homomorphism into $\Homeo_+[0,1]$.
While actions by homeomorphisms arising from orderability are usually not differentiable, it is not clear \emph{a priori} that nilpotent groups
do not have infinite critical regularity. It turns out that their critical regularity is finite, by a
1976 result of Plante--Thurston~\cite{PT1976}.
\begin{thm}\label{thm:pt-intro}
If $M\in\{I,S^1\}$ then every nilpotent subgroup of $\Diff_+^{1+\mathrm{bv}}(M)$ is abelian.
\end{thm}
As an immediate consequence, using the fact that $\Diff^2_+(M)\le \Diff_+^{1+\mathrm{bv}}(M)$, we have:
\begin{cor}
If $N$ is a nonabelian, torsion--free nilpotent group, then $\CR(N)\leq 2$. If $\CR(N)=2$ then the critical regularity is not achieved.
\end{cor}
We will give a full proof of Theorem~\ref{thm:pt-intro} in Chapter~\ref{ch:c2-thry}. If $N$ is a finitely generated torsion--free nilpotent group
that is not abelian, we have now that its critical regularity is finite. The problem of determining the critical regularity precisely is very
challenging, and only partial results are known.
The first question to ask is whether or not a finitely generated, nonabelian, torsion--free nilpotent group $N$ can be realized as a group
of diffeomorphisms of $M$. The answer is yes, though the proof is far from obvious.
\begin{thm}\label{thm:jor-farb-franks}
Every finitely generated, torsion--free nilpotent group embeds into $\Diff_+^1(M)$.
\end{thm}
It is remarkable that Theorem~\ref{thm:jor-farb-franks} appeared much later than Theorem~\ref{thm:pt-intro}, appearing first in~\cite{FF2003}.
Related constructions were given by Jorquera~\cite{Jorquera}. It is not very difficult to produce explicit, faithful actions of a
finitely generated, torsion--free
nilpotent group via appeals to homeomorphisms that take certain intervals to certain other intervals, and it is in gluing these homeomorphisms
together in a way that is smooth which necessitates most of the work in proofs of Theorem~\ref{thm:jor-farb-franks}. To illustrate these ideas
to the reader, we will give the constructions
carried out in Farb--Franks' and Jorquera's papers in Section~\ref{sec:nilpotent}, and partially write down the diffeomorphisms
they use. We will explain the steps needed to complete the proof, but we will not reproduce the calculations.
Whereas Farb--Franks and Jorquera produce an action of a given nilpotent group (or in Jorquera's case, an action of a certain direct limit
of nilpotent groups), a remarkable result of Parkhe gives a general orbit structure theory for
\emph{all} nilpotent group actions on $I$ and $S^1$.
We will not reproduce the statement of Parkhe's structure theorem here, which the reader may find as Theorem~\ref{thm:parkhe-main}.
As a consequence, Parkhe is able to deduce a lower bound on the critical regularity of nilpotent groups that is uniform in the growth
rate. Here, the \emph{growth rate}\index{growth of groups}
of a finitely generated group measures the number of elements in the group that can be written as products
of at most a fixed number $n$ of generators. To define the growth rate precisely, let $S$ be a finite generating set for a group $G$ such
that $S=S^{-1}$ (i.e.~$S$ is closed under taking inverses). We write $b_S(n)$ for the number of distinct elements of $G$ that can be
written as a product of at most $n$ elements of $S$. Whereas $b_S$ depends on $S$, its asymptotic behavior does not. We say
that $G$ has \emph{polynomial growth}\index{polynomial growth} of degree at most $d$ if $b_S(n)=O(n^d)$.
Here, the notation $O(n^d)$ is the usual big-oh notation, meaning that the growth rate of the group
as a function of $n$ is bounded above by $C\cdot n^d$, for a suitable constant $C$.
Clearly, an infinite group has at least polynomial growth, and no group
can have growth rate faster than an exponential function of $n$. A group $G$ has \emph{exponential growth}\index{exponential growth}
if the growth rate is bounded below by an exponential function, and has \emph{intermediate growth}\index{intermediate growth}
if it has superpolynomial and
subexponential growth. We refer the reader to~\cite{dlHarpe2000} for more background.
A foundational result of Gromov characterizes finitely
generated groups that have a nilpotent subgroup of finite index as precisely those that have polynomial growth;
see Subsection~\ref{ss:parkhe} for more detail and references.
Parkhe proves the following,
which will occur also as Theorem~\ref{thm:parkhe-conj} below:
\begin{thm}\label{thm:parkhe-intro}
Let $N\le \Homeo_+(M)$ be a finitely generated group of polynomial growth $O(n^d)$. Then for all $\tau<1/d$, the group $N$ is topologically
conjugate into $\Diff_+^{1,\tau}(M)$.
\end{thm}
Theorem~\ref{thm:parkhe-intro} has several
interesting consequences. For one,
in conjunction with Theorem~\ref{thm:pt-intro}, it says that the critical regularity of a finitely generated, torsion--free, nonabelian nilpotent groups
always lies in the interval $(1,2]$. Moreover, it says that if $N$ acts on $M$ by homeomorphisms, then this action is secretly differentiable;
a failure to be $C^1$ is simply concealed in a poor choice of identification of $M$ with itself. Finally, it says that a foliated bundle given by
suspending a nilpotent group action on a compact one--manifold is always smoothable to a $C^{1,\tau}$ action for a suitable $\tau$ that
is bounded below by a coarse geometric invariant of the group. In the interest of space, we will not give a complete proof of Parkhe's results.
Whereas Parkhe's work gives uniform lower bounds on critical regularity for nilpotent groups that complement the Plante--Thurston Theorem,
the bounds produced by Theorem~\ref{thm:parkhe-intro} are generally not optimal; one need only consider the case of abelian groups.
Even for nilpotent groups, the precise determination of the critical regularity of nilpotent groups is very difficult and known only in a few
cases. To state them, let $N_m\le \mathrm{SL}_{m+1}(\bZ)$ denote the group of unipotent upper triangular matrices. That is, $N_m$ consists of
all integer $(m+1)\times (m+1)$ matrices which are upper triangular, and whose only diagonal entry value is $1$. It is a standard fact that $N_m$ is
nilpotent, and that every finitely generated torsion--free nilpotent group can be realized as a subgroup of $N_m$ for some sufficiently large $m$.
The group $N_1$ is isomorphic to $\bZ$, and the group $N_2$ is isomorphic to the integral Heisenberg group \[\mathrm{Heis}=\form{ x,y,z\mid [x,y]=z,\,
[z,x]=[z,y]=1}.\] The following result is due to Jorquera--Navas--Rivas (cf.~Theorem~\ref{thm:jnr} below):
\begin{thm}\label{jnr-intro}
We have $\CR_I(N_3)=1.5$.
\end{thm}
It is unknown whether the critical regularity of $N_3$ is achieved. The following result is due to Castro--Jorquera--Navas
(cf.~Theorem~\ref{thm:cjn} below):
\begin{thm}\label{cjn-intro}
For all $d\geq 2$, there exists a metabelian group of nilpotence degree $d$ whose critical regularity is equal to $2$. The critical regularity
for these groups is not achieved. In particular, the integral Heisenberg group satisfies $\CR_M(H)=2$ for $M\in\{I,S^1\}$.
\end{thm}
Here, a group is metabelian if its commutator subgroup is abelian. The nilpotence degree of a group is the length of its lower central series.
We will not prove Theorems~\ref{jnr-intro} nor~\ref{cjn-intro}, mostly for reasons of space.
The results mentioned here document the current state of knowledge concerning critical regularity for nilpotent groups.
\subsection{Right-angled Artin groups and mapping class groups}
There are very few classes of groups for which critical regularity is understood to the degree that it is for nilpotent groups. One class
for which a significant amount of progress has been made is \emph{right-angled Artin groups}\index{right-angled Artin group}.
A right-angled Artin group is determined by
a finite simplicial graph $\Gamma$ with vertex set $V$ and edge set $E$. The right-angled Artin group \[A(\gam)=\form{ V(\gam)\mid
[v,w]=1\textrm{
if and only if } \{v,w\}\in E(\gam)}.\] A reader unfamiliar with right-angled Artin groups may check that the class accommodates free
abelian groups, nonabelian free groups, and many groups in between. It turns out that right-angled Artin groups exhibit a combination
of diversity and uniformity of behavior. The reader may consult~\cite{Charney2007,KK2013,KK2013b,Koberda-Yale,Wise2012} for instance,
for background on the subject for which we will not have space to discuss
in detail.
Since the class of right-angled Artin groups contains free abelian groups and free groups, clearly some right-angled Artin groups have infinite
critical regularity. It is not very difficult to show that free products of free abelian groups, and direct products of free products of free abelian
groups are all right-angled Artin groups with infinite critical regularity (see Section~\ref{sec:raag} below). The underlying graphs of these
right-angled Artin groups are disjoint unions of complete graphs (free products of free abelian groups) and joins of these kinds of graphs
(direct products of free products of free abelian groups). In general right-angled Artin groups are residually
torsion--free nilpotent~\cite{DK1992a},
whence it will follow from Theorem~\ref{thm:ff2003} below that they admit faithful $C^1$ actions on $M$:
\begin{prop}\label{prop:raag-c1-intro}
Every finitely generated right-angled Artin group embeds into $\Diff_+^{1}(M)$.
\end{prop}
For complicated defining graphs, it is somewhat harder, or more directly, impossible, to construct smooth actions of right-angled Artin
groups. Historically, the first graph shown to be an obstruction to infinite critical regularity for right-angled Artin groups was the path $P_4$
of length three~\cite{BKK2019JEMS}. See Figure~\ref{f:p4-intro}.
\begin{figure}[h!]
\tikzstyle {bv}=[black,draw,shape=circle,fill=black,inner sep=1pt]
\begin{center}
\begin{tikzpicture}[main/.style = {draw, circle}]
\node[main] (1) {$a$};
\node[main] (2) [right of=1] {$b$};
\node[main] (3) [right of=2] {$c$};
\node[main] (4) [right of=3] {$d$};
\draw (1)--(2)--(3)--(4);
\end{tikzpicture}%
\caption{The graph $P_4$, one of the simplest defining graphs of a right-angled Artin group with finite critical regularity.}
\label{f:p4-intro}
\end{center}
\end{figure}
Note that \[A(P_4)=\form{ a,b,c,d\mid [a,b]=[b,c]=[c,d]=1}.\] A reader familiar with three--manifold topology will observe that
$A(P_4)$ is the fundamental group of a link complement in $S^3$, precisely the link consisting of a chain of four unknots, successive
ones linked with linking number one. We have the following result, which will appear as Theorem~\ref{thm:a4-c2} below:
\begin{thm}\label{thm:bkk-intro}
There is no injective homomorphism \[A(P_4)\longrightarrow\Diff_+^{1+\mathrm{bv}}(M).\] In particular, $\CR_M(A(P_4))\leq 2$.
\end{thm}
Graphs which contain no copy of $P_4$ as a full subgraph are called \emph{cographs}\index{cograph}, and right-angled Artin groups on
cographs are characterized by not containing copies of $A(P_4)$ as subgroups, as we will show in Section~\ref{sec:raag}.
Cographs are organized in a hierarchy called the \emph{cograph hierarchy}\index{cograph hierarchy}.
Roughly, all cographs are generated recursively from a singleton
vertex via the graph operations of disjoint union and join; dually, right-angled Artin groups on cographs are generated recursively from $\bZ$
via the group operations of direct product and free product.
It is not true that a right-angled Artin group on a cograph has infinite critical regularity; in fact, it turns out that the group $(F_2\times\bZ)*\bZ$
is an obstruction to infinite critical regularity, and its presence characterizes right-angled Artin groups with finite critical regularity.
\begin{thm}\label{thm:raag-class-intro}
Let $\Gamma$ be a finite simplicial graph. Then the following are equivalent.
\begin{enumerate}[(1)]
\item
$\CR_M(A(\gam))<\infty$.
\item
$\CR_M(A(\gam))\leq 2$.
\item
We have $(F_2\times\bZ)*\bZ\le A(\gam)$.
\item
We have that $A(\gam)$ is not a direct product of free products of free abelian groups.
\end{enumerate}
\end{thm}
Theorem~\ref{thm:raag-class-intro} is strictly stronger than Theorem~\ref{thm:bkk-intro} and will be proved fully in this book.
It follows essentially from Theorem~\ref{thm:f2-int} and Theorem~\ref{thm:f2-circ}, together with the
remainder of the discussion in Section~\ref{sec:raag}. The reader may also consult Theorem~\ref{thm:kharlamov} for a more refined
statement.
The precise value of the critical regularity of right-angled Artin groups is unknown in general. It seems like as good a guess as any other
that the critical regularity should depend on the combinatorics of the defining graph, though there is strong evidence that when the
critical regularity of a right-angled Artin group is finite, then it should be exactly equal to $1$. The following will appear as
Theorem~\ref{thm:kkr-2020} below:
\begin{thm}\label{kkr-intro}
For a compact connected one--manifold $M$, we have $\CR_M((F_2\times F_2)*\bZ)=1$.
\end{thm}
If $\Gamma$ is a graph that contains a square as a full subgraph and if its complement graph is connected,
then Theorem~\ref{kkr-intro} implies that
$A(\gam)$ also has critical regularity one. Graphs $\Gamma$ for which $(F_2\times F_2)*\bZ$ is not a subgroup of $A(\gam)$ resist
an easy characterization, unlike cographs and graphs for which $A(\gam)$ contains $(F_2\times\bZ)*\bZ$. The critical regularity
of $(F_2\times\bZ)*\bZ$ remains tantalizingly open, though it would be not be inappropriate to hope for the intellectually satisfying
answer that it should also be exactly one, in which case right-angled Artin group critical regularities would bifurcate neatly.
\begin{que}
Let $\Gamma$ be a finite simplicial graph. Is it true that \[\CR_M(A(\gam))\in\{1,\infty\}?\]
\end{que}
Critical regularity questions for mapping class groups of surfaces are also understood to some degree, mostly because of their
entanglement with right-angled Artin groups. Let $S$ be an orientable surface with genus $g$ and $n$ punctures,
marked points, or boundary components.
We let $\Mod(S)$ denote the \emph{mapping class group}\index{mapping class group}
of $S$, consisting of isotopy classes of orientation preserving homeomorphisms
of $S$. A comprehensive overview of mapping class groups and their properties can be found in~\cite{FM2012}.
Classical results of Nielsen, Thurston, and Handel show that if $S$ has a marked point, then $\Mod(S)$ acts faithfully on the circle,
and if $S$ has a boundary component then $\Mod(S)$ acts faithfully on the interval. These facts are discussed as Theorem~\ref{thm:nielsen}
and Theorem~\ref{thm:thurston-handel} below.
In~\cite{FF2001}, Farb and Franks proved that if the genus of $S$ is sufficiently large, then $\CR_M(\Mod(S))\leq 2$. For reasons that
we will explain shortly, it is natural to replace the full mapping class group with groups that are
\emph{commensurable}\index{commensurability}
with mapping class groups
(i.e.~groups with finite index subgroups that are isomorphic to a finite index subgroup of the mapping class group).
Since their introduction by Dehn~\cite{Dehn-collected,Birman-MCG},
many authors have observed that mapping class groups of surfaces share many properties with
lattices in semisimple Lie groups. Here as before, a \emph{lattice}\index{lattice}
$\Gamma$ in a Lie group $G$ is a discrete subgroup such that $G/\gam$ has
finite volume with respect to the Haar measure on $G$.
It is known that the mapping class group itself is not a lattice in a semisimple Lie group, and it shares
some properties with rank one lattices, and some properties with higher rank lattices (see~\cite{FLM01}, for example).
Actions of lattices on compact manifolds are the subject of the \emph{Zimmer Program}\index{Zimmer Program},
which asserts that ``large" groups cannot act on
``small" manifolds in interesting ways. Here, a ``small" manifold is usually a compact manifold of dimension $n$, and a ``large" group
is an irreducible lattice in a semisimple Lie group whose rank is sufficiently large in comparison with $n$. Recent years have seen huge
progress on this program, and we refer the reader to~\cite{BM1999,Witte1994,Ghys1999,BFH20,BFH16} for further information.
For actions on the circle and interval, ``large" has usually meant an irreducible lattice of rank at least two. One of the first results regarding
lattices and actions on compact one--manifolds was obtain by Witte Morris:
\begin{thm}[See~\cite{Witte1994}]\label{thm:wm-intro}
Let $n\geq 3$ and let $\gam\le \mathrm{SL}_n(\bZ)$ be a subgroup of finite index.
Then every homomorphism
\[\gam\longrightarrow\Homeo_+(M)\] has finite image, for $M\in\{I,S^1\}$,
\end{thm}
For more general lattices, we have the following result that was obtained by Ghys~\cite{Ghys1999} and Burger--Monod~\cite{BM1999}:
\begin{thm}\label{thm:bm-intro}
For an irreducible lattice $\Gamma$ in a semisimple Lie group of rank at least two, we have the following.
\begin{enumerate}[(1)]
\item
Every $C^0$ action of $\Gamma$ on $S^1$ has a finite orbit.
\item
Every $C^1$ action of $\Gamma$ on $S^1$ factors through a finite group.
\end{enumerate}
\end{thm}
Theorem~\ref{thm:bm-intro} implies that there are no interesting differentiable actions of higher rank lattices on the circle, and that
after possibly passing to a finite index sublattice, every continuous action of a higher rank lattice on the circle is actually just an action
on the interval. As we have suggested already, and as the reader will find from consulting Appendix~\ref{ch:append2}, the existence
of nontrivial actions of higher rank lattices on the circle thus reduces to finding left invariant orderings on lattices. This is generally
a very difficult problem; recent progress has been made by Deroin--Hurtado~\cite{DeroinHurt}.
Lattices in rank one Lie groups, however, often do admit highly regular faithful actions on compact one manifolds. A lattice in
$\mathrm{PSL}_2(\bR)$, for instance, admits a faithful analytic action on the circle, since the group $\mathrm{PSL}_2(\bR)$
is itself a group of analytic diffeomorphisms of the circle. All hyperbolic $3$--orbifold groups, which are precisely the lattices
in $\mathrm{PSL}_2(\bC)$, admit faithful $C^1$ actions on the circle and on the interval. This is a consequence of the work
of Agol, Wise, and Kahn--Markovic~\cite{Agol2008,Agol2013,Wise2011,Wise2012,KM2012}
that shows that these groups embed in right-angled Artin groups (possibly after
passing to a finite index subgroup), combined with Theorem~\ref{thm:ff2003}.
Since a finite index subgroup of a lattice is again a lattice, any analogy between mapping class groups and lattices in semisimple Lie groups
should only consider the \emph{commensurability class}\index{commensurability class} of the mapping class group, which is to say of the
mapping class group up to passing to a finite index
subgroup. The difficulty here is that finite index subgroups of the mapping class group are very opaque.
See Section~\ref{sec:mcg} for a more detailed discussion.
Right-angled Artin groups furnish a tool to study finite index subgroups of mapping class groups because of the following straightforward
observation: if $\Gamma$ is a finite simplicial graph, if $G$ is a group, and if $A(\gam)\le G$, then every finite index subgroup of $G$ contains
an isomorphic copy of $A(\gam)$. Moreover, most mapping class groups contain copies of $A(P_4)$ or $(F_2\times \bZ)*\bZ$. This
can be seen easily from a result of the second author that exhibits a profusion of right-angled Artin subgroups of mapping class groups.
See Theorem~\ref{thm:mcg-raag} below. As a consequence, even though mapping class groups often do admit faithful continuous
actions on compact one--manifolds, these actions are never $C^{1+\mathrm{bv}}$ or better. The following result will be discussed and
contextualized further in Section~\ref{sec:mcg}.
\begin{thm}\label{thm:bkk-mcg-intro}
Let $S$ be an orientable surface and let $M\in\{I,S^1\}$. Then one of the two mutually exclusive conclusions holds.
\begin{enumerate}[(1)]
\item
There is a finite index subgroup $G\le \Mod(S)$ such that $G$ is a product of a free group and an abelian group. In this case,
$\CR_M(G)=\infty$.
\item
For all finite index subgroups $G\le \Mod(S)$, we have $\CR_M(G)\leq 2$.
\end{enumerate}
\end{thm}
Thus, mapping class groups again exhibit behavior of a lattice of intermediate rank, that is something between rank one and higher
rank. They often admit faithful continuous actions on compact one--manifolds,
but not very smooth ones.
For most mapping class groups, one can improve the bound in Theorem~\ref{thm:bkk-mcg-intro} to $\CR_M(G)\leq 1$. We will postpone
further discussion until Section~\ref{sec:mcg}.
\subsection{Property (T) and intermediate growth}\label{sec:ggt}
Most of the obstructions to smooth actions on one--manifolds we have considered up to this point arise from
commutation, and how partially commutative phenomena interact with differentiability.
There are some deep properties of groups which do not have much to do with partial commutativity but which nevertheless seem
to be incompatible with smooth actions on one--manifolds. We mention relevant results here for completeness of the overview of the
theory, though we will not discuss them in further detail in the body of the book.
\subsubsection{Kazhdan's property (T)}
Property (T), introduced by Kazhdan~\cite{Kazhdan64},
is a somewhat abstract property on the unitary dual of a group. Qualitatively, groups with property (T)
are ``large" groups, and are hence expected not to admit interesting actions on ``small" manifolds, according to the principles guiding
the Zimmer program. Indeed, property (T) is a property enjoyed by irreducible lattices in higher rank semisimple Lie groups, and was
used to prove many facts about them.
To give a rough definition, the set of unitary representations of a group has a natural topology on it, called the Fell topology. A group has
property (T) if the trivial representation is isolated in this topology, which gives an etymology of the name of the property. There are many
other equivalent characterizations of property (T). A more intuitive one, in the authors' opinion, is that $G$ has property (T) if and only if
any isometric affine action of $G$ on a Hilbert space has a fixed point. There is yet another characterization of property (T) in terms of
cohomology; we avoid discussing this subject at length here, directing the reader instead to the books~\cite{bekka-valette,Zimmer84}.
Given the conjectural picture that lattices in higher rank should admit no interesting actions on compact one--manifolds, it is natural
to ask whether groups with the more abstract Kazhdan's property (T) can admit such actions. Progress in this direction is made
by Navas~\cite{Navas02}:
\begin{thm}\label{thm:navas-kazhdan}
Let $G$ be a countable group with property (T), and let $\tau>1/2$. If \[\phi\colon G\longrightarrow\Diff_+^{1,\tau}(S^1)\] is a homomorphism
then $\phi$ has finite image. In particular, if $G$ is an infinite countable group with property (T) then $\CR_{S^1}(G)\leq 1.5$.
\end{thm}
Bader--Furman--Gelander--Monod~\cite{BFGM2007AM} show that if $G$ is as in Theorem~\ref{thm:navas-kazhdan}, then
$G$ is not a subgroup of $\Diff_+^{1,1/2}(S^1)$. It remains an open question as to whether a group with Kazhdan's property (T) can
admit an infinite image action by homeomorphisms on the circle or on the interval.
\subsubsection{Intermediate growth}
Recall that if $G$ is finitely generated by a symmetric generating set $S$ then we can define the growth function $b_S(n)$, measuring
the number of elements of $G$ that can be written as a product of at most $n$ elements of $S$. All the explicit groups we have considered
up to this point are either of polynomial growth (i.e.~virtually nilpotent groups) or of exponential growth (everything else). Since both
these classes of groups contain groups with infinite critical regularity and with finite critical regularity, it is hard to suppose that the
growth rate of a group should be related to critical regularity in any way.
It turns out that groups of intermediate growth, however, always have finite critical regularity when they act on the interval or on the circle.
Groups of intermediate growth are not trivial to construct in the first place, the first of which being constructed by Grigorchuk
(see~\cite{dlHarpe2000}). Groups of intermediate growth admitting faithful actions on the interval were produced by
Grigorchuk--Machi~\cite{GM1993}. Navas~\cite{Navas2008GAFA} proved the following result:
\begin{thm}\label{thm:navas-intermediate}
Let $\tau>0$, let $M\in\{I,S^1\}$, and let $G$ be a finitely generated subgroup of $\Diff_+^{1,\tau}(M)$ with subexponential growth. Then $G$ is
virtually nilpotent. In particular, a finitely generated group $G$ of intermediate growth satisfies $\CR_M(G)\leq 1$.
\end{thm}
Navas also proved that $\Diff_+^1(M)$ contains subgroups of intermediate growth, and so the critical regularity bound
given by Theorem~\ref{thm:navas-intermediate} is achieved by some
groups of intermediate growth. The essential connection between growth of groups and one--dimensional dynamics is through
free semigroups. If $G$ is a group of subexponential growth then $G$ cannot contain a free semigroup on two generators.
Navas is then able to exploit the absence of the free semigroup in $G$ to exclude
\emph{crossed homeomorphisms}\index{crossed homeomorphisms} from
any action of $G$ on $M$. Manifestations of crossed homeomorphisms will arise in Chapter~\ref{ch:c2-thry}, and their absence
is one characterization of Conradian group actions. We will not discuss this part of the story any further, since Navas already
gives an account in~\cite{Navas2011}.
\subsection{Lipschitz lower bounds}
In the preceding sections, we have glossed over groups of critical regularity between zero and one. All the explicit examples we have
discussed have had critical regularity at least one. There is a good reason for this, which arises from a result of
Deroin--Kleptsyn--Navas~\cite{DKN2007}:
\begin{thm}\label{thm:dkn-intro}
If $M\in\{I,S^1\}$, then
every countable subgroup of $\Homeo_+(M)$ is topologically conjugate into a group of bi--Lipschitz
homeomorphisms of $M$.
\end{thm}
Since the Lipschitz modulus of continuity is stronger than every H\"older modulus of continuity, it is reasonable to say in light of
Theorem~\ref{thm:dkn-intro} that:
\begin{cor}\label{cor:lip-bound}
Let $G$ be a countable group. If \[\CR_M(G)\geq 0\quad \textrm{then}\quad \CR_M(G)\geq 1.\]
\end{cor}
Corollary~\ref{cor:lip-bound} is not an unreasonable statement, especially
since bi--Lipschitz homeomorphisms are $C^1$ almost everywhere. However, it leads to the antinomy that groups of critical regularity
one might still admit no differentiable actions on $M$. Indeed, such groups are discussed in Appendix~\ref{ch:append3}.
From the dynamical point of view that we develop in this book, the quantitative behavior of a bi--Lipschitz homeomorphism is no
different from that of a diffeomorphism. More generally, the quantitative behavior of a $C^k$ diffeomorphism whose $k^{th}$ derivative
is Lipschitz is no different from that of a $C^{k+1}$ diffeomorphism. In the authors' paper~\cite{KK2020crit}, it was unclear whether
the classes of finitely generated groups of integer critical regularity $k\geq 2$ were truly bifurcated into $(k,\mathrm{Lip})$ and $(k+1)$
diffeomorphisms; that is, the methods used could not produce a group of $C^k$ diffeomorphisms with Lipschitz $k^{th}$ derivatives that
were not $C^{k+1}$ diffeomorphisms already. We will comment on this further in Subsection~\ref{ss:prescribed} below.
Theorem~~\ref{thm:dkn-intro} implies that for one--manifolds, the spectrum of critical regularities of countable groups is contained in
$\{-\infty\}\cup [1,\infty]$. Another technical complication that Theorem~\ref{thm:dkn-intro} avoids is that the set of homeomorphisms
of $M$ that are $\tau$--H\"older continuous do not form a group;
for $\tau,\tau'\in[0,1)$, the composition of a $C^{\tau}$ and a $C^{\tau'}$ homeomorphism
need only be $C^{\tau\tau'}$. Of course, groups of such homeomorphisms exist, but a foundational
issue arises in analyzing compositions of such homeomorphisms. It is a nontrivial fact then that for $\tau\in [0,1)$
and for integers $k\geq 1$, the set $\Diff_+^{k,\tau}(M)$
does indeed form a group, as we will show in Appendix~\ref{ch:append1}.
We will provide a complete proof of Theorem~\ref{thm:dkn-intro} in this book, and it appears as Theorem~\ref{thm:lip-conj} in the body
of the monograph.
\subsection{Groups of prescribed critical regularity}\label{ss:prescribed}
The reader will note that in most of the discussion in the preceding sections that dealt with explicit, the natural classes of groups
that occur all have critical regularity in the set $\{1,2,\infty\}$. Before the appearance of obstructions to topological smoothing,
many authors even asserted that there seemed to be little qualitative difference between foliations (and hence, via suspensions,
group actions) in regularity $C^2$ and $C^{\infty}$ (see Cantwell--Conlon's preprint~\cite{CC-unpublished}, for instance).
Even after the appearance of topological obstructions, such as Tsuboi's and Cantwell--Conlon's as we have mentioned already,
known groups of non-integral algebraic critical regularity were few, and there were no known examples of groups with integral critical
regularity three or more. The question of the existence of such groups was explicitly posed by Navas in his 2018 ICM
address~\cite{Navas-icm}, which
was written just before the paper~\cite{KK2020crit} appeared as a preprint.
We are now ready to state the existence theorem for groups of specified critical regularity in its
(nearly) fullest generality, as will occur again
as Theorem~\ref{t:optimal-all} below and the various variations on it.
The condition on the integral below may seem \emph{ad hoc}, though in fact there is some significant
leeway in making these choices, as we will see in Chapter~\ref{ch:optimal}
\begin{thm}\label{thm:crit-intro}
Let $k\in\bZ_{>0}$,
and let $\alpha,\beta$ be concave moduli of continuity satisfying
\[
\int_{(0,1]} \frac1x \left( \frac{\beta(x)}{\alpha(x)}\right)^{1/k}dx<\infty.\]
If $k=1$, we further assume that $\beta$ is \emph{sub-tame}\index{sub-tame modulus} in the sense that
\[\lim_{t\to 0^+}\sup_{x>0}\frac{\beta(tx)}{\beta(x)}=0.\]
Then there is a finitely generated subgroup \[G=G_{\alpha,\beta,M}\le \Diff_+^{k,\alpha}(M)\] such that the following conclusions hold.
\begin{enumerate}[(1)]
\item
The commutator subgroup $[G,G]$ is nonabelian and simple.
\item
For all finite index subgroups $H\le G$, we have $[G,G]'\le H$.
\item
For all finite index subgroups $H\le G$ and for all \[\psi\colon H\longrightarrow\Diff_+^{k,\beta}(M),\] we have that the image of
$\psi$ is abelian.
\end{enumerate}
\end{thm}
A large portion of this book is devoted to giving a complete proof of Theorem~\ref{thm:crit-intro}. For H\"older moduli of continuity, we will
write $r=k+\tau$ and $\GG^r(M)$ for the set of isomorphism types of countable subgroups of $\Diff_+^{k,\tau}(M)$.
Applying the above theorem to the H\"older modulus $x^r$ and another modulus that is a slight perturbation or $x^r$ (as in Lemma~\ref{l:omega-st}),
we obtain the result below.
\begin{cor}\label{cor:crit-intro}
Let $r\geq 1$. Then the following conclusions hold.
\begin{enumerate}[(1)]
\item
The set \[\GG^r\setminus\bigcup_{s>r}\GG^s\] contains continuum many isomorphism types of finitely generated groups and
countable simple groups.
\item
The set \[\left(\bigcap_{s<r}\GG^s\right)\setminus \GG^r\] contains continuum many isomorphism types of finitely generated groups and
countable simple groups.
\end{enumerate}
\end{cor}
For $r=1$, the second part of Corollary~\ref{cor:crit-intro} should be interpreted as groups admitting bi--Lipschitz but non--differentiable
actions. Note that the two parts of Corollary~\ref{cor:crit-intro} furnish groups for which the critical regularity is achieved and for which
it is not achieved, respectively.
We remark that the simple groups in Corollary~\ref{cor:crit-intro} are necessarily infinitely generated, as follows from Thurston's
Stability Theorem (see Appendix~\ref{ch:append3}).
In short, as to whether there exist groups acting with prescribed algebraic critical regularity on the interval and on the circle, the answer
is yes, and there are tons of examples. As might be expected, there are many more questions raised than there are answered by
Theorem~\ref{thm:crit-intro}. For instance:
\begin{itemize}
\item
What happens in higher dimension? Can finitely generated subgroups of diffeomorphism groups give insight into exotic smooth
structures on manifolds?
\item
For which moduli of continuity (even just restricting to H\"older moduli) are there finitely presented examples of groups with a prescribed
critical regularity? What about recursively presented examples?
\item
What are the coarse geometric properties of the groups furnished in Theorem~\ref{thm:crit-intro}? Do they form pairwise
distinct quasi--isometry classes?
\item
Is there a finitely generated group acting faithfully by $C^k$ diffeomorphisms on $M$ for all $k$, but that admits no faithful
$C^{\infty}$ action on $M$?
\end{itemize}
The last of these questions was explicitly posed in~\cite{KK2020crit}, and it seems rather difficult.
We close this section with a brief overview of another recent perspective on Theorem~\ref{thm:crit-intro} due to
Mann--Wolff~\cite{Mann:aa}, which uses different methods
to build groups with prescribed critical regularity. The basic idea is that one can combine groups that are differentially rigid with
diffeomorphisms with prescribed regularity properties in order to get groups with given critical regularity.
\emph{Differential rigidity}\index{differential rigidity}
of a group $G$ means that $G\le \Diff^{\infty}(M)$ and that above some cutoff $r$ and $s\geq r$, all faithful $C^{s}$--actions of $G$ on $M$
are conjugate to the $C^{\infty}$ action by a $C^{s}$ diffeomorphism. Examples of differentially rigid group actions
on the interval (with cutoff $r=1$) appear in~\cite{BMNR2017MZ}, and on the circle (with cutoff $r=3$) appear in~\cite{Ghys93IHES}.
Suppose now we wish to construct a group with prescribed critical regularity $r=k+\tau$.
Roughly speaking, taking a differentially rigid group and adjoin a properly supported $C^r$ diffeomorphism
$f$, calling the resulting group $G_f$.
Then arbitrary realization of $G_f$ as a group of $C^s$ diffeomorphisms forces the differentially rigid group to be $C^s$--conjugate to
a $C^{\infty}$ action. If the orbit structure of the differentially rigid group is sufficiently rigid, this conjugacy has to send $f$ to a
conjugate of itself by a $C^s$ diffeomorphism, which will again be only $C^r$. It follows then that $G_f$ has critical regularity exactly
$r$.
While Mann--Wolff's construction does not achieve the algebraic control that Theorem~\ref{thm:crit-intro} does, it is remarkable for
other reasons. For one, it shows that the set of
isomorphism classes of finitely generated subgroups of $\Diff_+^{k,\mathrm{Lip}}(M)$ and $\Diff_+^{k+1}(M)$
do not coincide, something which comparisons of moduli of continuity in Theorem~\ref{thm:crit-intro} cannot achieve. Moreover,
their result shows that the set of isomorphism classes of subgroups of $\Diff_+^{k,\tau}(M)$ is as rich and diverse as the set
$\Diff_+^{k,\tau}(M)$ itself.
\section{What this book is about}
The purpose of this book is to give a self--contained account of the construction of groups with a prescribed critical regularity.
In the interest of a coherent and motivated discussion, we have included a large number of peripheral and related results, all of which
serve to contextualize and complete the theory we are describing. In the interest of space and readability, we have been compelled to make
choices to omit many subjects. Some discussion of what is omitted can be found in the next section.
Some of the material we have decided to include is already described in other books and surveys, and so there is some overlap
between this book and existing literature. Our primary aim in recounting such results is to illustrate how the theory of critical regularity fits in the
theory of dynamical systems and geometric group theory. Our secondary aim is to provide a reference that is useable as a standalone volume,
and sometimes repetition is necessary. We summarize the content of the book as follows.
\begin{enumerate}
\item
Chapter~\ref{sec:denjoy} gives a full account of dynamics and regularity for single homeomorphisms of the circle. One of the highlights
of the chapter is a complete proof of Denjoy's Theorem. There are many other secondary results and background that are developed along
the way. These include:
\begin{itemize}
\item
The existence of invariant measures for solvable group actions on the circle.
\item
The existence of stationary measures
for diffusion semigroups.
\item
Bi--Lipschitz conjugacy of countable groups of homeomorphisms.
\item
Poincar\'e's theory of rotation numbers.
\item
Unique ergodicity of homeomorphisms with irrational rotation number.
\item
Additivity of the rotation number on subgroups with an invariant measure.
\item
The construction of $C^1$ Denjoy counterexamples for a large class of moduli of continuity.
\end{itemize}
\item
Chapter~\ref{sec:filip-tak} gives an account of the results of Takens, Filipkiewicz, and Rubin,
which in particular show that the abstract isomorphism type of the
$C^p$ diffeomorphism group of a manifold determines the manifold up to $C^p$ diffeomorphism. Thus, while classical, this chapter
illustrates the continuous version of the existence of groups with prescribed critical regularity.
Takens' Theorem is proved in its original form in dimension one,
with the exception of one linearization result of Sternberg. Filipkiewicz's Theorem is proved in a generalized form; one proof of Filipkiewicz's Theorem
is obtained from Rubin's Theorem, which itself is also proved in its entirety. Among the additional results
established along the way are the Bochner--Montgomery result on simultaneous continuity, and the simplicity of the
commutator subgroup of $\Diff_c^p(M)_0 $ for arbitrary $M$ and $p$; the latter of these figures prominently
in the discussion surrounding the Mather--Thurston Theorem, and also in the discussion of chain groups.
\item
Chapter~\ref{ch:c2-thry} develops many of the fundamental tools and methods used to investigate groups of
$C^1$ and $C^2$ diffeomorphisms
of $I$ and $S^1$. Some of the ideas and methods contained therein are classical or semi--classical, and some were developed in recent
years by the authors. Among the highlights of the discussion are:
\begin{itemize}
\item
Kopell's Lemma on commuting diffeomorphisms (classical).
\item
The Plante--Thurston Theorem (classical).
\item
Farb--Franks, Jorquera, and Parkhe's theory of (residually) nilpotent groups acting on compact one manifolds (modern).
\item
The Two-jumps Lemma, which is developed by the authors and Baik (semi--classical)
and which generalizes an earlier result of Bonatti--Crovisier--Wilkinson.
\item
The $abt$--Lemma (modern).
\end{itemize}
\item
Chapter~\ref{ch:chain-groups} develops the theory of chain groups. These are a family of finitely generated groups of homeomorphisms
of the interval that exhibit a range of uniformity and diversity, and which are also an important technical tool in constructing groups with
specified critical regularity. Among the topics discussed here are the following:
\begin{itemize}
\item
Covering distances in groups of homeomorphisms.
\item
Algebraic and dynamical properties of chain groups.
\item
Ghys--Sergiescu's smooth realization of Thompson's group $F$.
\end{itemize}
\item
Chapter~\ref{ch:slp} is one of the two mostly technical chapters in the book. The purpose of this chapter is to establish the
so--called \emph{Slow Progress Lemma}, which is a highly technical result that makes precise the intuition that ``more regular"
diffeomorphisms are ``less expansive". The Slow Progress Lemma originally appeared in~\cite{KK2020crit}, in a weaker form than
that given in this book.
\item
Chapter~\ref{ch:optimal} is the other technical chapter in the book. Its function is to construct diffeomorphisms with optimal expansivity
properties, and to use them to construct finitely generated groups with prescribed critical regularity. In this chapter, the ideas from
Chapters~\ref{ch:c2-thry},~\ref{ch:chain-groups}, and~\ref{ch:slp} all come together to furnish the announced groups.
\item
Chapter~\ref{ch:app} reaps the benefits of the theory developed in the previous chapters. Among the results established there are:
\begin{itemize}
\item
The construction of foliations on closed $3$--manifolds (satisfying mild topological hypotheses) that have a prescribed critical regularity.
\item
Characterization of right-angled Artin groups and mapping class groups that have finite critical regularities.
\end{itemize}
\item
The appendices to the book cover relevant results which, in the opinion of the authors, do not fit as neatly into the narrative of the book
but which they believe to be central enough to merit inclusion. Some of these, such as H\"older's Theorem, are so central that one could
reasonably argue that they belong in the main body of the book, though because the proof known to the authors relies on orderability, we
have relegated it to the appendices. The topics in the appendices include:
\begin{itemize}
\item
Medvedev's Theorem on smooth moduli of continuity,
and the Muller--Tsuboi trick on flattening interval diffeomorphisms at the boundary.
\item
Background on orderable groups and H\"older's Theorem.
\item
The Thurston Stability Theorem.
\end{itemize}
\end{enumerate}
The topics included in the book undoubtedly reflect the personal biases and preferences of the authors. We have striven to provide
as complete and nuanced a picture of the theory of critical regularity as possible, and we have included many references in places
where we omit details.
\section{What this book is not about}
As we have already mentioned, we have excluded a large number potential topics of discussion.
This book is not a general treatise on dynamical systems, nor is it even a general reference on groups acting on the circle.
Here we note some major omissions from our monograph, all of which are discussed at length in other sources.
\begin{enumerate}
\item
Orderability of groups. Many excellent books on the subject already exist~\cite{BMR77,CR2016,DDRW08,DNR2014,Navas2011},
and we only need orderability in a few places. All the relevant
background for us is contained in Appendix~\ref{ch:append2}.
\item
The structure of full diffeomorphism groups, viewed as discrete groups.
This is the discussion surrounding the Mather--Thurston Theorem, and besides
the simplicity of commutator subgroups and the Takens--Filipkiewicz--Rubin theory,
we will not be including an exposition on full diffeomorphism groups at all.
\item
$C^0$ theory. There is much to be said about actions of many types of groups, from closed surface groups to mapping class groups,
hyperbolic manifold groups, simple groups, and beyond on the circle and on the interval. There is also a vast theory of piecewise linear
groups acting on the interval and the circle. We will mostly exclude these topics from our account.
\item
Groups of analytic diffeomorphisms. The theory of subgroups of $\mathrm{PSL}_2(\bR)$ and its covers is well--developed (e.g.~\cite{KKM2019,KatokBook}), and we will
not discuss it. In general, groups of analytic diffeomorphisms do not furnish interesting examples in the circle of ideas on which
we are focussing.
For us, partial commutativity yields the most interesting examples and subsequent theory; among analytic diffeomorphisms, commutativity
is transitive, and so partially commutative groups of analytic diffeomorphisms are too simple to be interesting from our perspective.
\item
Solvable groups of diffeomorphisms. There is a structure theory for solvable subgroups of diffeomorphism groups of the
interval~\cite{Navas04solv},
which since it is peripheral to the discussion of critical regularity and since it is worked out in detail in~\cite{Navas2011}, we have decided
to omit it. Among the highlights of this theory are that polycyclic subgroups of $\Diff_+^{1+\mathrm{bv}}[0,1]$ are metabelian, but there
are solvable groups of diffeomorphisms of arbitrary long derived series length. We also direct the reader to~\cite{BursWilk04} for a discussion
of the rigidity of solvable group actions on the circle and a classification of solvable groups of analytic diffeomorphisms of the circle.
\item
Lattices in Lie groups. There is much to say about lattices in semisimple Lie groups in higher rank, and how these can act on manifolds;
indeed, this is a major concern of the Zimmer program. Other than what has already been said in the introduction of this book,
we have opted to avoid a discussion of this vast and rich subject.
\item
Geometric group theory. Many excellent books on geometric group theory and various classes of groups of interest in geometric group
theory already exist~\cite{dlHarpe2000,ClayMarg17,DM2018,Loeh2017,GGTIAS,Bowditch06,GH1990,GGT1990,BH1999}.
Right-angled Artin groups, mapping class groups, and nilpotent groups will be
sources of examples for us, though we will
only reference facts about them as needed.
\end{enumerate}
\section{What we will assume of the reader}
The authors have taken great pains to make this book as self-contained as possible. The writing contained herein should be accessible
to a beginning--to--intermediate level graduate student in mathematics. We will assume familiarity with standard notions from point--set
topology, and insofar as algebraic topology is concerned, we will only assume familiarity with the fundamental group, basic
covering space theory, and the classification of surfaces.
We will assume only the most basic notions about differentiable manifolds. Familiarity with hyperbolic geometry
is helpful but not necessary.
We will require some more sophisticated background in analysis,
to the level which is typically covered in a first graduate course in measure theory
and functional analysis. We will assume that the reader is comfortable with measure theory, the Riesz Representation Theorem for
functionals on the Banach space of continuous functions on a compact Hausdorff topological space, and the Banach--Alaoglu Theorem.
Sufficient background is found in the first few chapters of Rudin's standard textbook~\cite{Rudin87book}
and the first few chapters of~\cite{Zimmer-funct}.
We will assume that the reader is familiar with infinite groups and standard constructions with them. We will make free use of group
presentations and manipulations of them, and the reader should be familiar with the basics of groups acting on trees, to the level
of the early chapters of~\cite{Serre1977}. We will make free reference to nilpotent and solvable groups, as well as to the lower central
and derived series of a group. Familiarity with notions from coarse geometry and geometric group theory such as quasi--isometry
is useful but not necessary.
\chapter{Denjoy's Theorem and exceptional diffeomorphisms of the circle}\label{sec:denjoy}
\begin{abstract}
This chapter is a mostly self-contained account of the Denjoy--Herman--Yoccoz theory of circle diffeomorphisms, together with some
new contributions of the authors. Many beautiful expositions of Denjoy's theory are available in the literature -- the reader may consult
~\cite{athanassopoulos,Navas2011}, for instance. We do not pretend to improve on the work of those authors, and we reproduce many of the
ideas therein for completeness' sake and for the convenience of the reader.\end{abstract}
\section{The minimal set and the rotation number}
Let $f\in\Homeo_+(S^1)$. One of the most basic questions one can ask about $f$ concerns a characterization of the \emph{dynamics}
of $f$. Dynamics, broadly speaking, investigates the long term behavior of systems endowed with a prescribed transformation rule.
In the case of a homeomorphism $f\in\Homeo_+(S^1)$, or more generally a subgroup $\gam\le \Homeo_+(S^1)$, the system consists of a phase
space, namely the circle $S^1$, and the transformation rule is encoded by the function (or functions) $f$ (or $\Gamma$).
For a dynamical system like this, dynamical questions might be of the following form:
\begin{enumerate}
\item
What are the global fixed points of the group $\Gamma$? That is, characterize the points $x\in S^1$ such that $\gamma(x)=x$ for all $\gamma\in
\gam$.
\item
What are the finite orbits (i.e.~periodic points) of $\Gamma$?
\item
What are the long term itineraries of points (under the action of $\Gamma$) like?
\item
What are the $\Gamma$--invariant, closed subsets of $S^1$?
\item
What sorts of probability measures (or measure classes) on $S^1$ are $\Gamma$--invariant?
\end{enumerate}
This chapter will give a complete or at least extensive answer to all of these questions, at least in the case where $\gam\cong\bZ$ is
generated by a single homeomorphism of $S^1$. We will begin by introducing the minimal set of a group of homeomorphisms of $S^1$,
and then introduce the rotation number as a fundamental and powerful dynamical invariant of a homeomorphism. The interaction between
the minimal set, rotation number, and analytic concerns (i.e.~regularity) will be the subject of the remainder of the chapter.
\subsection{Minimal sets and exceptional diffeomorphisms}
We begin by giving an answer to the following question: let $\gam\le \Homeo_+(S^1)$. What are the possible closed, $\Gamma$--invariant subsets
of $S^1$? Are there canonical such $\Gamma$--invariant subsets?
\subsubsection{Periodic orbits}
To get a feeling for the foregoing question and possible answers to it, we note some examples.
First, consider the homeomorphism of the real line given by
$x\mapsto x+1$. This is clearly orientation preserving, and by adding a point at infinity, we obtain an element $f\in\Homeo_+(S^1)$ that has
exactly one fixed point. This point has ``parabolic" dynamical behavior, in the sense that points move away from infinity in the far negative
part of $\bR$, and move towards infinity in the far positive part of $\bR$. To get a sense of what the
proper and nonempty closed invariant subsets are, we note that infinity is clearly invariant. Moreover, infinity together with the orbit of a point
in $\bR$ is closed and invariant. More generally, one can consider a closed subset of $[0,1]$,
take the union of its translates by $x\mapsto x+1$
and infinity, and thus produce a closed invariant subset of $S^1$.
For a slightly more complicated example that retains some of
the same flavor, we may consider a rotation of $S^1$ by a rational number. Clearly, every point of such a homeomorphism is periodic. One can combine such a rotation with an arbitrary homeomorphism
of the real line by noting that the interval between two consecutive points in an orbit of the rotation
(in the direction of rotation) is homeomorphic
to $\bR$.
Thus, one can take an arbitrary homeomorphism of $\bR$ and propagate it around the circle via the rational rotation,
arranging for example (if the homeomorphism of $\bR$ has no fixed points) a homeomorphism
of $S^1$ that has exactly one periodic orbit. The resulting homeomorphism's closed invariant subsets can be analyzed in a manner similar
to the previous example.
\subsubsection{Generalities on rotations}
The preceding discussion introduces the notion of a rotation of $S^1$. Certain concrete realizations of the circle allow for straightforward
descriptions of rotations. For instance, if $S^1$ is realized as the unit complex numbers in $\bC$, then rotation by an angle $\theta$
is realized by mutiplication by the complex number $e^{i\theta}$. If the circle is realized as $\bR/\bZ$, then rotation by an angle
$\theta$ is simply realized by addition of $\theta \pmod{\bZ}$. The centrality of rotations as
one of the most basic examples of a homeomorphism
of the circle arises from a more abstract characterization of the circle as a one--dimensional compact Lie group. The circle,
viewed as an abstract group, is circularly orderable (see the Appendix~\ref{ss:circular}), and there is a unique topology that is compatible
with this circular order (in that open intervals in the circular order are a basis of open sets in the topology). It turns out in fact that this topology
determines a smooth manifold structure on the circle that is unique up to diffeomorphism (see~\cite{Tao-Hilbert}, for instance).
With this setup, rotations of the circle are merely the left (or right) regular action of the circle on itself, viewed as a group. A
fundamental question that is investigated in this chapter is, how does one distinguish rotations of the circle (i.e.~homeomorphisms of
the circle arising from the action of the circle on itself) via their dynamics?
\subsubsection{Irrational rotations and a digression on ergodic theory}
To introduce a central actor in the subsequent discussion,
consider a rotation of $S^1$ through an irrational number.
Not only are there
no periodic points, but in fact every orbit is dense. A homeomorphism for which every orbit is dense is called
\emph{minimal}\index{minimal homeomorphism}. Equivalently, the
closure of every orbit is the whole phase space, and so the only invariant closed subspaces are the empty set and the whole space.
\begin{prop}\label{prop:irr-rot-min}
Let $f\in\Homeo_+(S^1)$ be a rotation through an irrational number. Then $f$ is minimal.
\end{prop}
\begin{proof}
Note that the \emph{radial}\index{radial metric} (i.e.~arclength) metric on $S^1$ is
invariant by $f$, so that $f$ acts by an \emph{isometry}\index{isometry}
of the radial metric.
Now, since the angle $\theta$ through which
$f$ rotates is irrational, no point has a finite orbit under $f$. This is simply because if $f^n(x)=x$ then, writing
$S^1$ additively as $\bR/\bZ$, we have \[f^n(x)=x+n\theta\equiv x\pmod {\bZ},\] which means that $n\theta$ is an integer, a
contradiction. A straightforward compactness argument shows that, if $\OO$ is the orbit of $f$ on
$x$ and $\epsilon>0$, then there are two points
in $\OO$ at distance at most $\eps$ from each other.
Now, suppose that $\OO$ is not dense. Let $U\sse S^1$ be a component
of $S^1\setminus\overline{\OO}$, and assume that $U$
has length $\ell>0$. Choose points \[x_n=f^n(x),\quad x_m=f^m(x)\] such that the radial distance between $x_n$ and $x_m$ is less than
$\ell$. Then $f^{n-m}(x)$ is at a distance smaller than $\ell$ from $x$. Now, since both endpoints of $U$ are accumulation points of the closure
of $\OO$, there is an $N$ such that $f^N(x)$ is as close to either endpoint of $U$ as we like. It follows then that $f^{N+n-m}(x)$ will lie in
$U$, for a suitable $N$. Therefore, $\OO$ is dense.
\end{proof}
Irrational rotations have yet another important property, closely related to their minimality: they are \emph{ergodic}\index{ergodic}.
Recall that the radial
length function extends to a unique Borel measure on $S^1$, which is a multiple of Lebesgue measure on $S^1$.
Since rotations act by isometries on $S^1$ with respect to the radial metric and because Lebesgue measure is determined by its value
on open intervals,
we have that rotations also preserve Lebesgue measure. A measurable map $f\colon S^1\longrightarrow S^1$ is called \emph{ergodic}
if the only measurable
$f$--invariant subsets of $S^1$ have zero measure or full measure. To see the relationship between ergodicity and minimality, the reader
will be able to show (after absorbing Theorem~\ref{thm:minimal-set}) that a homeomorphism of $S^1$ that preserves Lebesgure measure
and that is ergodic with respect to Lebesgue measure is necessarily minimal.
\begin{prop}\label{prop:irr-rot-erg}
Let $f\in\Homeo_+(S^1)$ be an irrational rotation. Then $f$ is ergodic with respect to Lebesgue measure.
\end{prop}
\begin{proof}
Let $g\in L^2(S^1)$ be a square--integrable function that is invariant under rotation, i.e.~$g\circ f=g$. It suffices to show that $g$ is constant
almost everywhere with respect to Lebesgue measure. This will suffice to prove the ergodicity of $f$,
since one can set $g$ to be the indicator function
of a measurable set.
Now, we expand $g$ in $L^2(S^1)$ by Fourier theory, viewing $S^1=\bR/\bZ$.
For $n\in \bZ$, we have that the $n^{th}$ Fourier coefficient of $g$ is
\[c_n=\int_{S^1}g(t)e^{-2\pi i n t}\,dt,\] so that \[\sum_{n\in \bZ} c_n e^{2\pi i n t}\longrightarrow g\] in $L^2(S^1)$. Now, if $g$ is invariant under
$f$, we must have that the Fourier coefficients of $g$ and $g\circ f$ are the same. Viewing the circle additively and computing, we have
\[c_n=\int_{S^1}g(t+\theta)e^{-2\pi i n t}\,dt=\int_{S^1} g(t)e^{- 2\pi i n (t-\theta)}\,dt=e^{2\pi i n\theta}c_n.\] Since $\theta$ is irrational, we have
that $e^{2\pi i n\theta}\neq 1$ for all $n\neq 0$, so that all Fourier coefficients vanish, other than the zeroth one. It follows that $g$ is constant
almost everywhere.
\end{proof}
In fact, even more is true about irrational rotations: they are \emph{uniquely ergodic}\index{uniquely ergodic}.
That is, they admit a unique invariant probability
measure on $S^1$, which therefore must be Lebesgue measure. If a probability measure $\lambda$ is the unique invariant measure for
a measurable
transformation
$f$, then $f$ is automatically ergodic. To see this, if $A$ and $B$ measurably partition the phase space and both have positive measure
with respect to $\lambda$, then
one can build a measure $\lambda_A$, which is just $\lambda$ restricted to $A$ and rescaled to have measure one, and such that
$\lambda_A$ assigns measure zero to $B$. A measure $\lambda_B$ can be built similarly.
Clearly both $\lambda_A$ and $\lambda_B$
are invariant, as is every convex combination of them. So, a uniquely ergodic transformation is automatically ergodic.
The proof of Proposition
\ref{prop:irr-rot-erg} actually shows that if $\mu$ and $\lambda$ are
arbitrary Borel probability measures that are invariant under a rotation through an
irrational number, and if $f$ is a continuous function on $S^1$, then \[\int_{S^1}f\,d\mu=\int_{S^1}f\,d\lambda.\] Indeed,
these integrals are merely the $0^{th}$ Fourier coefficients of $f$. Since Borel measures are separated by integrating against continuous
functions (by Lusin's Theorem for example~\cite{Rudin87book}), it follows that $\mu=\lambda$.
\begin{cor}\label{cor:leb-ue}
Let $f\in\Homeo_+(S^1)$ be a rotation through an irrational number. Then $f$ is uniquely ergodic with respect to Lebesgue measure.
\end{cor}
In Subsection~\ref{ss:rot} below, we will introduce the rotation number, and we will show that in fact all
homeomorphisms with irrational rotation number
(of which irrational rotations themselves are examples) are uniquely ergodic (see Theorem~\ref{thm:irr-ue}). This fact is a crucial ingredient
in Denjoy's theory of circle diffeomorphisms as we will develop it here.
One of the general and useful properties of a uniquely ergodic transformation $T$ of a phase space $X$ is that every sequence of
probability measures that converges to a $T$--invariant measure must converge to the unique $T$--invariant probability measure on $X$.
One such sequence of probability measures can be described as follows, and is particularly intuitive when $T$ is
a homeomorphism of a compact Hausdorff topological space (as we shall
assume).
For $n\in\bN$ and $x\in X$, let $\mu_{n,x}$ be a probability measure on $X$ defined by
\[\mu_{n,x}(A)=\frac{1}{2n+1}\left|A\cap \left\{\bigcup_{i=-n}^n \{f^i(x)\}\right\}\right|,\] where $A$ is a Borel subset of $X$. Note that the
set $\{\mu_{n,x}\}_{n\ge1}$ is precompact by the Banach--Alaoglu Theorem (see~\cite{Zimmer-funct}, for instance),
and so we may extract a weak--$\ast$ limit $\mu$.
A straightforward manipulation shows that $\mu$ is $T$--invariant and is hence the unique $T$--invariant probability measure on $X$.
We will spell out these ideas in more detail in Lemma~\ref{lem:kaku-abel}.
\subsubsection{Continuous Denjoy counterexamples}
There are examples of homeomorphisms of $S^1$ which have neither periodic points, nor are they minimal. In fact, these are the generic
types of homeomorphisms, though imagining them at first is not always so easy. One of the most important examples comes from
a \emph{blow-up}\index{blow-up} of an orbit of a minimal homeomorphism.
We now describe this construction for an irrational rotation of the circle, though
the method is broadly applicable. This type of example is called
a~\emph{(continuous) Denjoy counterexample}\index{Denjoy counterexample}, for reasons which will become
more apparent later in this chapter.
Let $f$ be an irrational rotation, and let $\OO$ be the orbit of a point $x\in S^1$.
Let $\{\ell_n\}_{n\in\bZ}$ be a collection of positive real numbers
such that \[\sum_{n\in\bZ}\ell_n<\infty.\] We construct a sequence of intervals $J_n=[0,\ell_n]$ for $n\in\bZ$, and homeomorphisms
\[\phi_{n,m}\colon J_n\longrightarrow J_m\] given by \[\phi_{n,m}(x)=\ell_n^{-1}\cdot \ell_m\cdot x.\] Now, let $X$ be the space obtained
by inserting a copy of $J_n$ at $f^n(x)\in\OO$. More precisely, we cut $S^1$ open at $x_n$, and glue the two endpoints of $J_n$ to
the two preimages of $x_n$. The resulting topological space is easily seen to be homeomorphic to $S^1$. One now defines a map
$f_X\colon X\longrightarrow X$ by setting $f_X(y)=f(y)$ if $y\notin\OO$, and $f_X(y)=\phi_{n,n+1}(y)$ if $y\in J_n$. It is again
straightforward to check that $f_X$ defines a homeomorphism of $X\cong S^1$, and evidently $f_X$ has no periodic points. We claim that
\[C=X\setminus\bigcup_{n\in\bZ} J_n^0\] is homeomorphic to a Cantor set, where here $J_n^0$ denotes the interior of $J_n$. Because
this argument is on the long side and because the result is important, we isolate it formally.
\begin{prop}\label{prop:blowup-cantor}
The set $C$ is homeomorphic to a Cantor set.
\end{prop}
The reader may compare the statement and proof of Proposition~\ref{prop:blowup-cantor} to that of Theorem~\ref{thm:minimal-set} below.
\begin{proof}[Proof of Proposition~\ref{prop:blowup-cantor}]
Note first that $C$ is compact, since $X$ is compact Hausdorff and $C$ is closed. It is also clear that $C$ is nonempty.
The subspace $C$ is clearly metrizable since it inherits the radial metric from $X\cong S^1$. To complete
the verification that $C$ is homeomorphic to a Cantor set, it suffices to show that it is perfect and totally disconnected. To see that $C$ is
totally disconnected, note that there is a map $\kappa \colon X\longrightarrow S^1$ given by collapsing the intervals $\{J_n\}_{n\in\bZ}$ each to
a point. This map is one--to--one except on the endpoints of the collapsed intervals, where it is two--to--one. If $y\in\OO$ then it is clear
that the two points in the preimage $\kappa^{-1}(y)$ are separated by open sets. Now, if $y,z\in S^1$ are arbitrary and distinct
then $\OO$ meets
both components of $S^1\setminus\{y,z\}$, and so that arbitrary points
\[y_X\in\kappa^{-1}(y),\quad z_X\in\kappa^{-1}(z)\] are separated by open sets that meet the complement of $C$.
It follows that $C$ is totally disconnected.
To see that $C$ is perfect, let $y\in S^1$ be arbitrary, and choose $y_X\in\kappa^{-1}(y)$. There is a sequence of points $\{x_n\}\sse\OO$
such that $x_n\longrightarrow y$, by Proposition~\ref{prop:irr-rot-min}.
Note that for each $M\in\bZ$, there is an $N$ such that
for all $n\geq N$, the only points in $\OO$ that meet the (shorter) interval $(x_n,y)$ are contained in \[\OO_M=\{f^m(x)\mid |m|\geq M\}.\]
Since the total length of the intervals $\{J_n\}_{n\in\bZ}$ is finite, we have that for each $\eps>0$ there is an $M\gg 0$ such that
\[\sum_{|m|\geq M}\ell_m<\eps.\] It follows that the radial distance between $\kappa^{-1}(x_n)$ and $y_X$ tends to zero as $n$ tends to
infinity. Therefore, $y_X$ is not an isolated point of $C$, and the proof is complete.
\end{proof}
It is clear that the set $C$ and the union $J=\cup J_n^0$
of the interiors of the intervals $\{J_n\}_{n\in\bZ}$ are both invariant under $f_X$.
The set $J$ is called the \emph{wandering set}\index{wandering set} and is
characterized by the fact that if $I\sse J_n^0$ for some $n\in\bZ$ then $f^m(I)\cap I\neq\varnothing$ implies $m=0$. The maximal
connected components of $J$ are called \emph{wandering intervals}\index{wandering interval}.
It turns out
that $C$ is the unique nonempty closed invariant subset of $X$ on which $f_X$ acts minimally, but we will not justify this claim just yet.
\subsubsection{Minimal sets for general group actions}
The preceding examples give examples of the typical minimality phenomena observed among groups of homeomorphisms of $S^1$. This
is made precise by the following basic result.
\begin{thm}[cf.~\cite{Navas2011}, Theorem 2.1.1 or ~\cite{athanassopoulos}, Proposition 2.5]\label{thm:minimal-set}
Let \[\gam\le \Homeo_+(S^1).\] Exactly one of the following conclusions holds.
\begin{enumerate}[(1)]
\item
There is a finite $\Gamma$--orbit.
\item
The group $\Gamma$ acts minimally on $S^1$.
\item
There is a unique, nonempty, closed, invariant $C\sse S^1$ such that $\Gamma$ acts minimally on $C$ and such that $C$ is homeomorphic
to a Cantor set. Moreover, if $\OO$ is an arbitrary orbit of $\Gamma$ then the closure $\overline{\OO}$ contains $C$.
\end{enumerate}
\end{thm}
\begin{proof}
We use Zorn's Lemma, applied to the partially ordered set $\PP$ consisting of closed, nonempty, $\Gamma$--invariant subsets of $S^1$, ordered
by reverse inclusion. We have
that $\PP$ is obviously nonempty, since $S^1\in\PP$. Chains have upper bounds in $\PP$, since closed subsets of $S^1$ are compact.
In particular, if $\{K_i\}_{i\in I}$ is a collection of nested compact subsets of $S^1$ then the finite intersection property characterization of
compactness implies that \[\bigcap_{i\in I} K_i\neq\varnothing.\] Zorn's Lemma allows us to extract a minimal element $C\in\PP$.
Observe first that the $\Gamma$--action on $C$ is (topologically) minimal.
This is more or less immediate, since otherwise there would be a point $x\in C$
whose orbit $\OO(x)$ is not dense in $C$. Then, the closure $\overline{\OO(x)}$ is a closed $\Gamma$--invariant subset of $S^1$ that is
properly contained in $C$, violating the minimality (with respect to the partial order) of the choice of $C$.
We write $C'$ for the derived set of $C$. Recall that this is the set of accumulation points of $C$, or equivalently the subset of $C$ obtained
by discarding all isolated points. Observe that $C'$ is a closed and $\Gamma$--invariant subset of $S^1$, and so $C'$ is either empty or equal to
$C$, by the minimality of the choice of $C$. If $C'$ is empty then $C$ is discrete and hence finite, which means that $\Gamma$ has a finite orbit.
Otherwise, we have that $C'=C$, and so we consider the boundary $\partial C$. Again, $\partial C$ is closed and $\Gamma$--invariant, and so
either \[\partial C=C'\quad \textrm{or} \quad \partial C=\varnothing.\]
The only way the latter equality can hold is if $C'=C=S^1$, in which case the original
$\Gamma$--action on $S^1$ is minimal.
Thus, we are left with the case \[C'=C=\partial C.\] In this case, we have that $C$ is compact (since it is closed),
perfect (since it has no isolated points), and totally disconnected (since it is equal to its boundary and hence contains no
intervals). It follows that
$C$ is homeomorphic to a Cantor set.
It remains to show that $C$ is contained in the closure of the orbit of an arbitrary point $x\in S^1$. This will also prove that $C$ is unique,
since $C$ is then just the intersection of all orbit closures.
Let $y\in C$ and $x\in S^1$ be arbitrary. If $x\in C$ as well, then the fact that the $\Gamma$--action on $C$ is minimal implies that there
is a sequence of elements $\{\gamma_i\}_{i\ge1}\sse\gam$
such that \[\gamma_i(x)\longrightarrow y\] as $i$ tends to infinity. Otherwise, we may
assume that $x$ is contained in an open interval $J\sse S^1\setminus C$. Writing \[\{x_1,x_2\}=\partial J\sse C,\] we have that
the $\Gamma$ orbits of both $x_1$ and $x_2$ are dense in $C$. So, there are group elements $\{\gamma_i\}_{i\ge1}\sse\gam$ such that
$\gamma_i(x_1)\neq\gamma_j(x_1)$ for $i\neq j$ and such that
\[\gamma_i(x_1)\longrightarrow y\] as $i$ tends to infinity. Since $\gamma_i(J)$ is an interval in $S^1\setminus C$ for all $i$, and since
$\Gamma$ acts by orientation preserving homeomorphisms, we have that \[\gamma_i(J)\cap\gamma_j(J)=\varnothing\] for $i\neq j$. It follows
then that the length $|\gamma_i(J)|$ tends to zero as $i$ tends to infinity, so that the radial distance between $\gamma_i(x)$ and
$\gamma_i(x_1)$ also tends to zero. The triangle inequality immediately implies that the radial distance from $\gamma_i(x)$ and $y$
also tends to zero, so that the $\Gamma$--orbit of $x$ accumulates on all of $C$. This completes the proof of the theorem.
\end{proof}
The set $C$ furnished by Theorem~\ref{thm:minimal-set} in the absence of a finite orbit is called the \emph{minimal set}\index{minimal set}.
If $C\neq S^1$,
we call the set $C$ the \emph{exceptional minimal set}\index{exceptional minimal set}
of the $\Gamma$--action on $S^1$. If \[\gam\cong\bZ=\form{ f}\] admits an
exceptional minimal set, then we call $f$ \emph{exceptional}\index{exceptional homeomorphism}.
The continuous Denjoy counterexample we have introduced in this subsection
is an example of an exceptional homeomorphism of the circle.
\subsection{The rotation number}\label{ss:rot}
The rotation number is a basic dynamical invariant of a homeomorphism of $S^1$ that measures how the orbit of a point travels around
the circle. The definition is simple enough. Let $f\in\Homeo_+(S^1)$. Choose an arbitrary lift $\tilde f\in\Homeo_+^{\bZ}(\bR)$.
Choose an arbitrary $x\in\bR$ and let \[\rot(f)=\lim_{n\to\infty}\frac{F^n(x)}{n}\in\bR/\bZ.\]
The point $\rot(f)\in S^1$ is defined to be the \emph{rotation number}\index{rotation number} of $f$.
Of course, there are a number of things to check to make sense of this definition.
We need to verify the existence of the limit,
and the independence of the definition from the choice of $\tilde f$ and $x$.
For this, we first record a basic fact as below.
Let $G$ be a group.
A real valued function $\varphi$ on $G$ is called a \emph{quasimorphism}\index{quasimorphism}
if
there exists a $C\in\bR$ such that for all $g,h\in G$, we have
\[ |\varphi(gh)-\varphi(g)-\varphi(h)|\le C.\]
The infimum of such a constant $C$ is the \emph{defect}\index{defect of a quasimorphism} of the quasimorphism.
This definition is a mild variation on a
\emph{subadditive sequence}\index{subadditive}, which is easiest to define for
\[
\varphi\co \bN\longrightarrow\bR_+,\]
in which case we would simply require
$\varphi(m+n)\leq \varphi(m)+\varphi(n)$. For subadditive functions, the following result is known as Fekete's Lemma.
\begin{lem}[cf.~\cite{Navas2011}, Lemma 2.2.1, for instance]\label{lem:subadditive}
If $f\co \bN\longrightarrow\bR$ is a quasimorphism,
then
the limit \[\rho=\lim_{n\to\infty} f(n)/n\] exists and is the unique
real number for which the sequence $\{f(n)-n\rho\}_{n\ge1}$ is bounded.
\end{lem}
\begin{proof}
The idea is to realize $\rho$ as the intersection of a nested sequence of closed intervals, whereby the finite intersection property
characterization of compactness says that the intersection of these intervals is nonempty. With this in mind, let $C$ denote the defect
of the quasimorphism $f$ and set \[J_n=\left[\frac{f(n)-C}{n},\frac{f(n)+C}{n}\right].\]
If $k\geq 1$ is an integer then we apply the triangle inequality to conclude that
\[k f(n)-k C
\le f(k n)-C \le
f(k n)+C\leq k f(n)+k C.\]
Dividing through by $k n$, we have $J_{kn}\sse J_n$.
It follows that every finite collection \[\{J_{n_1},\ldots,J_{n_i}\}\] has nonempty intersection,
and so the intersection \[J=\bigcap_{n\ge1} J_n\neq\varnothing,\] by the finite intersection property characterization of compactness.
If $\rho\in J$ then clearly $|f(n)-n\rho|\leq C$, so that the sequence $\{f(n)-n\rho\}_{n\ge1}$ is bounded. We claim that $J$ consists of
exactly one point. Indeed, suppose that $\delta\neq \rho$ are two distinct elements of $J$. Then $|\rho-\delta|>0$, which we can compute
\[|f(n)-n\delta|=|f(n)+n(\rho-\delta)-n\rho|\geq n|\rho-\delta|-C,\] since $|f(n)-n\rho|$ is bounded by $C$. It follows then that $|f(n)-n\delta|$
tends to infinity, a contradiction. Thus, $\rho$ is unique. That \[\rho=\lim_{n\to\infty}\frac{f(n)}{n}\] is now immediate.
\end{proof}
The following is immediate from Lemma~\ref{lem:subadditive},
and we leave the detail to the reader.
The map $\bar\varphi$ will be called the \emph{homogenization}\index{homogenization} of $\varphi$.
\begin{lem}[cf.~\cite{Calegari2009,Navas2011,KKM2019}]\label{lem:qm}
If $\varphi$ is a real--valued quasimorphism defined on a group $G$,
then for each $g\in G$ the limit \[\bar \varphi(g)=\lim_{n\to\infty} \varphi(g^n)/n\] exists.
Furthermore, the following hold.
\be[(1)]
\item The value $\bar \varphi(g)$ is the unique
real number for which the sequence $\{\varphi(g^n)-n\bar \varphi(g)\}_{n\ge1}$ is bounded;
\item $\bar\varphi(g^m)=m\bar\varphi(g)$ for all $g\in G$ and $m\in\bZ$;
\item $\bar\varphi(hgh^{-1})=\bar\varphi(g)$ for all $g,h\in G$.
\ee
\end{lem}
Now let us now consider an arbitrary $F\in\Homeo_+^\bZ(\bR)$, where here $\Homeo_+^\bZ(\bR)$ denotes the group of $\bZ$--periodic
homeomorphisms of $\bR$.
For a given point $x\in \bR$,
we define
\[\varphi_x(F):=F( x)- x\in\bR.\]
We note the following.
\be[(i)]
\item If $x-y\in\bZ$ then $\varphi_x(F)=\varphi_y(F)$.
\item For $x,y\in\bR$, we have $\varphi_x(F)-\varphi_y(F)< 1$.
\item The map $F\mapsto \varphi_x(F)$ is a quasi-morphism on $\Homeo_+^\bZ(\bR)$.
\ee
Part (i) is obvious by letting $y=x+m$ and noting $F(y)=F(x)+m$.
For part (ii), we may first assume by part (i) that $y\le x<y+1$. Then we have
\[
\varphi_x(F)-\varphi_y(F)=F(x)-F(y)-(x-y)< F(y+1)-F(y)-(y-y)=1.\]
It is now easy to deduce part (iii), since for $F,G\in\Homeo^\bZ_+(\bR)$ we have
\[
\varphi_x(F\circ G)
=F\circ G(x)-x
= \varphi_{G(x)}(F)+\varphi_x(G)\in(\varphi_x(F)+\varphi_x(G)-1,\varphi_x(F)+\varphi_x(G)+1).\]
By Lemma~\ref{lem:qm} we can homogenize $\varphi_x$ and obtain
a homogeneous class function $\bar\varphi\co \Homeo_+^\bZ(\bR)\longrightarrow\bR$
defined by
\[
\bar\varphi(F):=\lim_{n\to\infty}\varphi_x(F^n)/n=\lim_{n\to\infty} (F^n(x)-x)/n
=\lim_{n\to\infty} F^n(x)/n.\]
The condition (ii) above implies that $\bar\varphi$ is independent of the choice of $x\in\bR$, justifying the reference-point--free notation.
Let us now suppose $f\in \Homeo_+(S^1)$.
If $F$ and $G$ are two lifts of $f$ in $\Homeo^\bZ_+(\bR)$ then
the value $F(x)-G(x)$ is a fixed integer, and hence
$\bar\varphi(F)$ and $\bar\varphi(G)$ differ by that integer.
Quotienting by $\bZ$, we obtain a well-defined map
\[
\rot(f):=\bar\varphi(F)=\lim_{n\to\infty} \frac{F^n(x)}n \in\bR/\bZ,\]
where $F$ is an arbitrary lift of $f$, and $x\in\bR$ is also arbitrary.
As a matter of terminology, we will say that the rotation number of a homeomorphism is \emph{rational}\index{rational rotation}
if it belongs to $\bQ/\bZ$, and \emph{irrational}\index{irrational rotation} otherwise.
Let us summarize important properties of the rotation number.
\begin{prop}\label{prop:rot-easy}
The rotation number \[\rot\colon\Homeo_+(S^1)\longrightarrow S^1\] enjoys the following properties.
\begin{enumerate}[(1)]
\item
Homogeneity: that is, for all $f\in\Homeo_+(S^1)$ and $n\in\bZ$, we have \[\rot(f^n)=n\rot(f).\]
\item
Class function: $f,g\in\Homeo_+(S^1)$ then $\rot(g^{-1}fg)=\rot(f)$.
\item
If $R_{\theta}$ is a rotation of $S^1$ by an angle $\theta\in S^1$ then $\rot(R_{\theta})=\theta$.
\item
The map $f$ has a fixed point if and only if $\rot(f)=0$.
\item The map $f$ has a periodic point if and only if $\rot(f)$ is rational;
moreover, all periodic points have the same period.
\end{enumerate}
\end{prop}
\begin{proof}
The first three parts are immediate consequences of Lemma~\ref{lem:qm}.
For the fourth part of the proposition, the well-definedness of the rotation number implies that we need only show that if $\rot(f)=0$ then
$f$ has a fixed point. To show the contrapositive, let $f\in\Homeo_+(S^1)$ be fixed-point-free. Let $F$ be the unique left $f$ such that
$F(0)\in (0,1)$. Because $f$ has no fixed points, the function $F(x)-x$ does not achieve values in the set $\bZ$.
Since $F$ is continuous and periodic, it
satisfies the conclusions of the extreme value theorem, whence we may assume that there is an $\epsilon$ such that
\[0<\eps\leq F(x)-x\leq 1-\epsilon\] for all $x\in\bR$.
Observe that for all $n$ we have the estimate \[0<n\cdot\eps\leq \sum_{i=0}^{n-1}F(F^i(0))-F^i(0)=
F^n(0)-0\leq n(1-\eps).\] Dividing through by $n$, we see that $F^n(0)/n$ is bounded away from $0$ and $1$, whence the rotation
number of $f$ is nonzero.
The homogenity of $\rot$ now implies that
$\rot(f)$ is rational if and only if $f^n$ fixes a point for some nonzero $n\in\bZ$.
To see that all periodic points have the same period, let $x,y\in S^1$ have periods $n$ and $m$ respectively,
with $n<m$. Then $f^n$ admits $x$ as a fixed point, and so $\rot(f^n)=0$. However, $f^n(y)\neq y$ and $f^{nm}(y)=y$, and since
$f$ is orientation preserving, we have that $f^n$ has no global fixed points, which is a contradiction.\end{proof}
The continuous Denjoy counterexample we have introduced above gives an example of a homeomorphism of $S^1$ with irrational rotation
number, as the reader may check, and which is obviously not minimal. It turns out that examples of this ilk, which is to say ones constructed
by blowing up orbits, are essentially the only examples of homeomorphisms of the circle which have irrational rotation number and which
are exceptional.
\begin{thm}\label{thm:irr-rot-semi}
Let $f\in\Homeo_+(S^1)$ be such that $\rot(f)=\theta$ is irrational. Then there exists a $\bZ$--periodic map \[H\colon\bR\longrightarrow\bR\]
such that:
\begin{enumerate}[(1)]
\item
We have $H(0)=0$.
\item
The map $H$ is continuous and nondecreasing.
\item
The map $H$ descends to a self--map \[h\colon S^1\longrightarrow S^1\] such that:
\begin{enumerate}[(i)]
\item
The map $h$ is a degree one, orientation preserving, continuous surjection.
\item
If $R_{\theta}$ denotes rotation by $\theta$, we have $h\circ f=R_{\theta}\circ h$.
\item
The map $h$ is a homeomorphism if and only if $f$ is minimal.
\end{enumerate}
\end{enumerate}
\end{thm}
The maps $H$ and $h$ are oftentimes called \emph{semi-conjugacies}\index{semi-conjugacy}.
We will give the proof of Theorem~\ref{thm:irr-rot-semi} below,
after Theorem~\ref{thm:kaku-mar}.
The notion of a semi-conjugacies is intuitively straightforward, because topology in one dimension is so limited.
If $H$ is a semi-conjugacy, then it is not difficult to see that the preimage $H^{-1}(x)$ of a point is either a singleton point or a compact
interval. Thus, a semi-conjugacy behaves like a homeomorphism near some points (and therefore restricts to a bijection
on those points), and the remaining intervals are collapsed to
points.
Before closing this subsection, we give the following fact, which shows that the rotation number of a homeomorphism
is a semi-conjugacy invariant.
\begin{prop}\label{prop:semi-rot}
Let $f,g\in\Homeo_+(S^1)$ be semi-conjugate. Then $\rot(f)=\rot(g)$.
\end{prop}
\begin{proof}
Let $h$ be a semi-conjugacy from $f$ to $g$, and let $\{F,G\}$ be arbitrary lifts of $\{f,g\}$ to $\Homeo_+^{\bZ}(\bR)$ and similarly
let $H$ be a lift of $h$ (which of course may not be a homeomorphism). Note that
\[H\circ F^n\equiv G^n\circ H \pmod{\bZ}\] for all $n\in\bZ$. It is convenient to write $\Delta(x)=H(x)-x$. Then, we can compute:
\[F^n(x)+\Delta(F^n(x))\equiv G^n(H(x))-H(x)+x+\Delta(x)\pmod{\bZ}.\] It follows that \[\frac{F^n(x)-x)}{n}+\frac{\Delta(F^n(x))}{n}\equiv
\frac{G^n(H(x))-H(x)}{n}+\frac{\Delta(x)}{n}\pmod{\bZ}.\] Since both $\Delta\circ F^n$ and $\Delta$ are bounded, the limits of the two sides
as $n\to\infty$ are equal, as claimed.
\end{proof}
\subsection{Rotation numbers, invariant measures, and amenability}
Another important perspective on rotation numbers for circle homeomorphisms is provided from the angle of invariant probability measures.
Suppose $\mu$ is a probability measure on $S^1$, and suppose that $\mu$ is invariant under the action of a group $\gam\le \Homeo_+(S^1)$.
Let $f\in\gam$.
For $x\in S^1$, let \[\rho_{\mu}(f)=\mu[x,f(x))\pmod{\bZ}.\]
The very notation $\rho_{\mu}$ suggests that it should be independent of the choice of
$x$, which indeed it is thanks to the invariance of $\mu$ under the action of $\Gamma$. Indeed, choose a point $y$ such that $y<f(x)<f(y)$ if
it exists, where these points are viewed as lying in the interval $[0,1)$.
Then, we compute: \[\mu[y,f(y))=\mu[y,f(x))+\mu[f(x),f(y)).\] Since \[\mu[f(x),f(y))=\mu[x,y),\] we obtain \[\mu[y,f(y))=\mu[x,f(x)).\] We leave
the verification of the general case to the reader. The interest in introducing $\rho_{\mu}$ comes from the following basic result which will
be used several times in the sequel.
\begin{thm}[cf.~\cite{Navas2011}, Theorem 2.2.10]\label{thm:invt-homo}
Let $\mu$ be an invariant probability measure for a group $\gam\le \Homeo_+(S^1)$. The the following hold.
\begin{enumerate}[(1)]
\item
For all $f\in \gam$, we have that $\rho_{\mu}(f)=\rot(f)$.
\item
The rotation number furnishes a homomorphism \[\rot\colon\gam\longrightarrow S^1.\]
\end{enumerate}
\end{thm}
\begin{proof}
We write \[\exp\colon [0,1)\longrightarrow S^1\] for the exponentiation map $x\mapsto e^{2\pi ix}$. The map $\exp$ has a measurable inverse,
and we write such an inverse $\exp^{-1}$.
Let $\mu$ be as in the statement of the theorem and let $f\in\gam$. First, renormalize $\mu$ to have total measure $1$. We pull back $\mu$
along $\exp$ (that is, push forward along $\exp^{-1}$) to obtain a measure $\exp^{-1}_*\mu$ on $[0,1)$, and we extend this measure
periodically to obtain a measure $\nu$ on $\bR$ that is invariant under an arbitrary lift $F$ of $f$. The reader may check that the measure
$\nu$ has the property that for all $x\in \bR$ and $k\in\bN$, we have \[\nu[x,x+ k)= k.\]
We may thus make some straightforward estimates as
follows. If $F$ is a lift of $f$ and if \[F^n(x)\in [x+ k,x+ k+1)\] then we have \[F^n(x)-x-1\leq k\leq \nu[x,F^n(x))\leq k+
1\leq F^n(x)-x+1.\]
Dividing by $n$ and passing to the limit, we see that \[\lim_{n\to\infty}\frac{F^n(x)-x}{n}=\lim_{n\to\infty}\frac{\nu[x,F^n(x))}{n}.\]
Breaking up $[x,F^n(x))$ along points in the $F$--orbit of $x$, we see that the right hand side of this last equality coincides with
\[\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\nu[F^k(x),F^{k+1}(x))=\lim_{n\to\infty}\frac{1}{n}\cdot n\cdot\nu[x,F(x))=\nu[x,F(x)),\]
where here we
are using the $F$--invariance of $\nu$. It follows immediately now that $\rho_{\mu}(f)=\rot(f)$, which establishes the first claim of the theorem.
To see that the rotation number is a homomorphism when restricted to $\Gamma$, it suffices to see that
\[\rho_{\mu}(f\circ g)=\rho_{\mu}(f)+
\rho_{\mu}(g).\] Note that
\begin{align*}\rho_{\mu}(f\circ g)=\mu[x,fg(x))\pmod{\bZ}\\ \equiv\mu[x,g(x))+\mu[g(x),fg(x))\pmod{\bZ}\\ \equiv\rho_{\mu}(f)+
\rho_{\mu}(g)\pmod{\bZ},\end{align*} as required.
\end{proof}
The rotation number is not a homomorphism when restricted to an arbitrary subgroup of $\Homeo_+(S^1)$, as the reader may have
suspected. For an explicit example of a such a subgroup, we consider \[\PSL_2(\bR)\le \Homeo_+(S^1)\] where $\PSL_2(\bR)$ acts
on $S^1\cong \bR\bP^1$ by fractional linear transformations \[\begin{pmatrix}a&b\\c&d \end{pmatrix}\colon x\mapsto\frac{ax+b}{cx+d}.\] The
subgroup $\Gamma\le \PSL_2(\bR)$ generated by the matrices
\[\left\{\begin{pmatrix}2&0\\0&\frac{1}{2} \end{pmatrix},\begin{pmatrix}t&0\\0&t^{-1} \end{pmatrix},\begin{pmatrix}1&1\\0&1 \end{pmatrix}\right\},\]
where here $t\in\bR$ is arbitrarily chosen so that $\form{ \log 2,\log t}$ is a dense additive subgroup of $\bR$,
has the property that the three generators of $\Gamma$ each have fixed points in $S^1$ and hence have rotation number zero. If the rotation
number were a homomorphism when restricted to $\Gamma$ then the rotation number would be trivial for all elements of $\Gamma$.
However,
general facts about Lie groups imply that $\Gamma$ is in fact dense in $\PSL_2(\bR)$. Indeed, since $\Gamma$ is not virtually solvable (as it
contains a nonabelian free group), it is Zariski dense in $\PSL_2(\bR)$. Since $\Gamma$ is not discrete in $\PSL_2(\bR)$ by the choice
of $t$, its topological closure is a Lie subgroup of positive dimension which must therefore be all of $\PSL_2(\bR)$. It follows that
$\Gamma$ meets every nonempty open subset of $\PSL_2(\bR)$, and in particular there is an element $\gamma\in\gam$
with trace contained in the
interval $(-2,2)$. It is not difficult to show that such an element $\gamma$ is conjugate to the Euclidean rotation
\[\begin{pmatrix}\cos\theta &-\sin\theta\\ \sin\theta&\cos\theta \end{pmatrix}\] for some $\theta\notin \bZ$, as
\[\tr(\gamma)=2\cos\theta\in (-2,2).\] It follows that the rotation number of $\gamma$ is nonzero, a contradiction.
So far, the results concerning invariant measures and rotation numbers have been conditional, in the sense that they assume the existence
of an invariant measure before they can be applied. Most subgroups of $\Homeo_+(S^1)$ do not admit invariant measures, though if one is
willing to impose some algebraic conditions, they will. In general if $G$ is a countable discrete group,
one says that $G$ is \emph{amenable group}\index{amenable group} if every
action of $G$ on a compact Hausdorff topological space admits an invariant probability measure. Examples of amenable groups include
abelian groups and solvable groups, and the class of amenable groups is closed under extensions, subgroups, quotients, and direct limits.
Amenability is not a primary concern of this book, and we will content ourselves to direct the interested reader
to~\cite{Lub94,Paterson88,Runde02,Zimmer84} for more on amenability.
We will prove amenability for solvable groups since the amenability of these groups, especially with respect to applying
Theorem~\ref{thm:invt-homo}, will arise in later chapters. Specifically, we will prove a variation on the classical Kakutani--Markov
Fixed Point Theorem.
\begin{thm}\label{thm:kaku-mar}
Let $G$ be a countable discrete solvable group acting by homeomorphisms on a compact, Hausdorff space $X$. Then $G$
preserves a probability measure on $X$.
\end{thm}
The proof of Theorem~\ref{thm:kaku-mar} requires some functional analysis which we will simply assume here, directing the reader
to~\cite{Rudin87book,Zimmer-funct} for the relevant background.
Consider the space $C(X)$ of complex--valued continuous functions on $X$.
The Riesz Representation
Theorem implies that the dual space $V=C(X)^*$ of bounded linear functionals on $C(X)$ is the space of complex regular
Borel measures on $X$ of finite total measure,
and the space of positive linear
functionals on $C(X)$ consists precisely of
positive regular Borel measures on $X$, with the pairing being given by \[(\mu,f)\mapsto\int_X f\,d\mu.\] For our purposes, the following
statement is sufficient, which the reader may find as Theorem 2.14 in~\cite{Rudin87book}.
\begin{thm}[Riesz Representation Theorem]\label{thm:riesz}
Let $X$ be a compact Hausdorff topological space, and let $\Lambda$ be positive linear functional on $C(X)$ (i.e.~$\Lambda(f)\ge 0$
whenever $f(x)\geq 0$ for all $x$). Then there is a regular Borel measure $\mu$ such that $\mu(X)<\infty$ and such that
\[\Lambda(f)=\int_X f\,d\mu\] for all $f\in C(X)$.
\end{thm}
General (bounded) functionals on $C(X)$ can be recovered as complex linear combinations of measures on $X$.
The natural topology to consider on $V$ is the weak--$\ast$ topology, so that $\{\mu_n\}_{n\ge1}$ converges to $\mu$ if for every
$f\in C(X)$, we have \[\int_X f\,d\mu_n\longrightarrow\int_X f\,d\mu.\] The fundamental property of the weak--$\ast$ topology that
we will use is the following.
\begin{thm}[Banach--Alaoglu Theorem, Theorem 1.1.28 of~\cite{Zimmer-funct}]\label{thm:banach-al}
Let $E$ be a normed linear space. Then the unit ball in $E^*$ is compact in the weak--$\ast$ topology.
\end{thm}
We set $\PP(X)\sse V=C(X)^*$ to be the subset consisting of regular Borel probability measures on $X$. The subset $\PP(X)\sse V$ is compact
in the weak--$\ast$ topology, by the Banach--Alaoglu Theorem, since it is a closed subset of the unit ball.
Moreover, it is clear from its definition that $\PP(X)$ is convex. We remark briefly
that $\PP(X)$ is obviously nonempty if $X$ has at least one point, since a Dirac mass at a point of $X$ lies in $\PP(X)$.
Now let $G$ be a group acting on a compact Hausdorff space $X$ by homeomorphisms. We will always assume that $G$ is equipped with the discrete topology,
so that there are no issues with continuity when it comes to the associated actions on $V$ and $\PP(X)$. The group $G$ acts on the
space of all measures on $X$ by pushforward. Since $G$ acts continuously on $X$, the group $G$ preserves the Borel measures and the
regular measures. It is clear that $G$ preserves the set of positive regular Borel measures on $X$, and among those it preserves the
probability measures. Thus, $\PP(X)$ is $G$--invariant.
The following fact is sometimes called the Kakutani--Markov Fixed Point Theorem, when one allows $G$ to be a general abelian topological
group leaving invariant a compact, convex subset of a topological vector space whose topology is defined by a sufficient family of seminorms
on which $G$ acts continuously (see~\cite{Zimmer-funct}, for example).
\begin{lem}[cf.~\cite{Zimmer-funct}, Theorem 2.1.5]\label{lem:kaku-abel}
Let $G$ be an abelian group and $X$ be a compact Hausdorff space.
If $Y\sse\PP(X)$ is a nonempty, compact, convex, $G$--invariant set,
then there is a measure $\mu\in Y$ that is $G$--invariant. Moreover, the set $Y_G$ of $G$--invariant measures in
$Y$ is convex and compact.
\end{lem}
\begin{proof}
For notational convenience, we write \[\phi\colon G\longrightarrow \Aut(V)\] for the action of $G$ on $V$. For $g\in G$ and $n\in\bZ_{>0}$,
we obtain operators
\[A_{n,g}=\frac{1}{n}\sum_{i=0}^{n-1}\phi(g^i),\] which are clearly continuous
endomorphisms of $V$. Since $Y$ is convex and invariant under the
action of $G$, it is immediate to check that $A_{n,g}$ preserves $Y$.
Consider the semigroup $S$ generated by \[\{A_{n,g}\mid n\in\bZ_{>0},\, g\in G\}.\] An arbitrary element $T\in S$ is simply a finite composition
of operators of the form $A_{n,g}$. Since $Y$ is invariant under each of endomorphisms, we have that $T(Y)\sse Y$ for
arbitrary elements $T\in S$. Note that since $G$ is an abelian group, the semigroup $S$ is commutative.
{\bf Claim 1:} We have that \[Y_G=\bigcap_{T\in S} T(Y)\neq\varnothing.\]
It is clear that $Y_G$ coincides with the intersection above.
Since each $T$ is continuous and since $Y$ is compact, it suffices to
verify the finite intersection property characterization of compactness.
Since $Y$ is $S$--invariant, for $T_1,T_2\in S$ we have that
\[
T_1\circ T_2(Y)\sse T_1(Y)\]
and that
\[T_1\circ T_2(Y)=T_2\circ T_1(Y)\sse T_2(Y).\]
In general, if \[\{T_1,\ldots,T_n\}\sse S\] are arbitrary, then we have that
\[T_1\circ\cdots\circ T_n(Y)\sse\bigcap_i T_i(Y).\] It follows that $Y_G$ is nonempty as claimed.
{\bf Claim 2:} If $\mu\in Y_G$ then $\mu$ is $G$--invariant.
By the very definition of $Y_G$, we have $\mu=s(\lambda)$ for some $s\in S$ and
$\lambda\in Y$.
It suffices to consider the case that $s= A_{n,g}$.
Observe then that \[\phi(g)(\mu)-\mu=\frac{1}{n}\left(\phi(g^n)(\lambda)-\lambda\right).\] Let $f\in C(X)$
with $\sup_{x\in X} |f(x)|\leq 1$.
Evidently, by the triangle inequality, we have that \[\left| \int_X f\,d(\phi(g)(\lambda))-\int_X f\,d\lambda\right| \leq \frac{2}{n},\] since $\lambda$
is a probability measure. Since $n$ is arbitrary, we have that \[\left| \int_X f\,d(\phi(g)(\mu))-\int_X f\,d\mu\right| =0\] for all continuous
functions $f$, so $\mu$ is $G$--invariant.
{\bf Claim 3:} The set $Y_G$ is convex and compact. These properties of $Y_G$ follow easily from the definitions.
\end{proof}
We immediately obtain:
\begin{cor}\label{cor:homeo-invt-meas}
Let $f\in\Homeo_+(S^1)$. Then $f$ preserves a Borel probability measure on $S^1$.
\end{cor}
This gives an alternative definition of a rotation number, by the formula given in Theorem~\ref{thm:invt-homo}.
We can prove Theorem~\ref{thm:kaku-mar} for solvable groups by bootstrapping Lemma~\ref{lem:kaku-abel}, and inducting on the length
of the derived series of $G$.
\begin{proof}[Proof of Theorem~\ref{thm:kaku-mar}]
Let $D_0=G$ and $D_k=[D_{k-1},D_{k-1}]$ denote terms of the derived series of $G$. By assumption, $G$ is solvable and so for some
$n$ we have $D_{n+1}=\{1\}$. We proceed by induction on $n$. The set $Y_n$ consisting of $D_n$--invariant Borel probability measures on
$X$ is nonempty, compact, and convex by Lemma~\ref{lem:kaku-abel}, and this is the base case of the induction.
We assume that the set $Y_1$ consisting of $D_1$--invariant Borel probability measures on $X$ is nonempty, compact, and convex.
{\bf Claim 1:} The set $Y_1$ is $D_0$--invariant. Indeed, let $\mu\in Y_1$, let $g\in G$, and consider $\phi(g)(\mu)$.
This is clearly a Borel probability
measure on $X$. If $h\in D_1$ then
the normality of $D_1$ in $D_0$ implies that
\[\phi(h)(\phi(g)(\mu))=\phi(hg)(\mu)=\phi(g)(\phi(g^{-1}hg)(\mu))=\phi(g)(\mu),\] so that $\phi(g)(\mu)$
is $D_1$--invariant. This proves the claim.
{\bf Claim 2:} The set $Y_0$ of $D_0$--invariant Borel probability measures is nonempty. Indeed, the action of $D_0$ on $Y_1$
factors through the quotient $D_0/D_1$, since $D_1$ acts trivially on $Y_1$. We have that $Y_1$ is nonempty, compact, and convex,
and $D_0/D_1$ is abelian. Lemma~\ref{lem:kaku-abel} shows that the set $Y_0$ is nonempty.
\end{proof}
We are now ready to give a proof of the fact that every orientation preserving homeomorphism of $S^1$ with irrational rotation number is
semi-conjugate to an irrational rotation.
\begin{proof}[Proof of Theorem~\ref{thm:irr-rot-semi}]
We begin with a setup identical to that in Theorem~\ref{thm:invt-homo}.
Let $f$ admit an invariant probability measure $\mu$ as furnished by Corollary~\ref{cor:homeo-invt-meas} which as before we renormalize
to have total mass $1$,
and let \[\exp\colon [0,1)\longrightarrow S^1\] be the usual exponential map. The map
$\exp^{-1}$ is measurable. So, we let $\nu$ be the pushforward of $\mu$ by $\exp^{-1}$, and we extend $\nu$ periodically to all of $\bR$.
Since $g$ has no periodic points by assumption, $\mu$ has no atoms and hence every point
has measure zero. Clearly
$\nu$ is also atomless. Then by construction, we have $\nu(F(X))=\nu(X)$ for every lift $F$ of $f$ and every Borel set $X\sse \bR$. We
set \[H(x)=\int_0^x\, d\nu,\] so that $H(0)=0$ and $H$ is periodic with respect to translation by $1$. Since $\nu$ has no atoms, the function
$H$ is continuous, and since $\nu$ is a positive measure, we have that $H$ is non-decreasing.
We now claim that $H$ semi-conjugates $f$ to a rotation, which in light of Proposition~\ref{prop:semi-rot} must be rotation by $\rot(f)=\theta$.
Computing, we note that \[H(F(x))-H(x)=\int_0^{F(0)}\,d\nu+\int_{F(0)}^{F(x)}\,d\nu-\int_0^x\, d\nu.\] Since $\nu$ is $F$--invariant, the last
two terms cancel out, so that \[H(F(x))-H(x)=H(F(0))-H(0)\] for all $x$. It follows that $H\circ F$ is a translation of $\bR$, so that pushing
$H$ down to a self-map $h$ of $S^1$, we have that $h\circ f$ is a rotation.
Note that if $h$ is a homeomorphism then $f$ must be minimal since minimality is preserved under conjugacy. Conversely, if $f$ is minimal, let
$x\in S^1$ be a point such that every neighborhood of $x$ has positive measure with respect to $\mu$. Such a point exists since otherwise
$S^1$ would admit a finite cover by open sets with zero measure. We have that the orbit of $x$ is dense, so that every
nonempty open set has positive
measure. It follows then that the map $H$ is strictly increasing, so that $h$ must be a homeomorphism.
\end{proof}
\subsection{Invariant measures and free subgroups}
To close this section,
we mention one more result about invariant measures due to Margulis, though we will not give a proof here. Recall that a group satisfies the
\emph{Tits alternative}\index{Tits alternative} if every subgroup is either virtually solvable or contains a
nonabelian free group. The terminology arises from Tits'
famous result that linear groups enjoy the Tits alternative~\cite{tits-jalg}.
Homeomorphism groups of one--manifolds do not enjoy the Tits alternative, though we are not quite ready to show this yet. Thompson's
group $F$ is a group of homeomorphisms of the interval which is neither virtually solvable nor contains a nonabelian free subgroup; see
Theorem~\ref{thm:f-subgp} below. A version of the Tits alternative for homeomorphisms of the circle was established by Margulis:
\begin{thm}[See~\cite{marg-CR00,Ghys2001} and also Theorem 2.3.2 in~\cite{Navas2011}]\label{thm:marg-ghys}
Let \[G\le \Homeo_+(S^1)\] be a subgroup. Then either $G$ preserves a probability measure on $S^1$, or $G$ contains a nonabelian free
subgroup.
\end{thm}
Theorem~\ref{thm:marg-ghys} is only interesting for groups acting on the circle without global fixed points, since a Dirac mass concentrated
at a global fixed point is an invariant probability measure. Also, whereas the Tits alternative is a true dichotomy, Theorem~\ref{thm:marg-ghys}
is not. There are many interesting actions by groups containing nonabelian free subgroups that preserve nonatomic probability measures,
and this is closely related to the theory of \emph{Conradian orderings}\index{Conradian ordering},
which we will not discuss in detail in this book. The interested
reader may begin by consulting~\cite{navasrivas09}, for instance.
\section{Denjoy's theorem}
In this section, we prove the following result of Denjoy:
\begin{thm}[Denjoy's Theorem]\label{thm:denjoy}
Let $f\in\Diffb(S^1)$, and suppose that $\rot(f)=\theta$ is irrational. Then $f$ is conjugate to the rotation $R_{\theta}$.
\end{thm}
Thus, Denjoy's Theorem relates exceptional minimal sets, rotation numbers, and analysis in a decisive way. The moment $f$ is a
diffeomorphism and the derivative of $f$ has bounded variation (a conclusion which is satisfied, for example, when $f$ is twice differentiable),
then $f$ cannot be an exceptional diffeomorphism. We emphasize that Denjoy's Theorem is false if $f$ is merely assumed to be a
diffeomorphism, so that smooth Denjoy counterexamples exist. The exact regularities in which there exist Denjoy counterexamples is a subtle
and complicated question, and is the subject of Section~\ref{sec:denjoy-crit} below. The account we follow here uses an equidistribution
phenomenon called the Denjoy--Koksma inequality, which combines with certain derivative estimates to yield Denjoy's Theorem. Another
approach based on Sacksteder's Theorem and the existence of hyperbolic fixed points can be found in~\cite{Navas2011}.
\subsection{Rational approximations of real numbers}
If $f\in\Homeo_+(S^1)$ has rational rotation number, then clearly $f$ need not be conjugate to a rational rotation, regardless of the level of
regularity one imposes on $f$. Thus, one must use irrationality in some essential way. Indeed, rational approximation of arbitrary
real numbers by rational numbers plays a key role in the proof of Denjoy's Theorem. Specifically, we have the following classical result:
\begin{thm}[Dirichlet's Diophantine Approximation Theorem]\label{thm:dirichlet-1}
Let \[\theta\in\bR\setminus\bQ\] and $N\in\bZ_{>0}$. There exist relatively prime integers $p$ and $q$ with $1\leq q<N$ such that
\[|q\cdot\theta-p|<\frac{1}{N}.\]
\end{thm}
The rational number $p/q$ is called a \emph{rational approximation}\index{rational approximation} of $\theta$.
We restrict to irrational $\theta$, since the result for rational values of $\theta$ is not relevant to our purposes, and because the proof is
easier to the point of being almost trivial.
\begin{proof}[Proof of Theorem~\ref{thm:dirichlet-1}]
This is just a consequence of the pigeonhole principle. We consider the set of real numbers \[S=\{0,1,\{n\theta\}\mid 1\leq n\leq N-1\},\]
where here $\{x\}$ denotes the fractional part of $x$, the unique real number $\eps\in [0,1)$ such that $x-\eps$ is an integer.
Clearly $|S|=N+1$ and $S\sse [0,1]$. Cutting $[0,1]$ into disjoint intervals of length $1/N$, the pigeonhole principle implies that at least
one such interval meets $S$ in two points. Observe that since $\theta$ is not rational, none of the points \[\{k/N\mid 1\leq k\leq N-1\}\] meet
the set $S$, so that the distance between two such points is strictly less than $1/N$.
If $s_1,s_2\in S$ are two such points then \[s_1-s_2=q\cdot\theta-p\] for a suitable choice of integers
$p$ and $q$, where $0<q<N$. It follows that \[|q\cdot\theta-p|<\frac{1}{N}.\] If $p$ and $q$ fail to be relatively prime then we may simply divide
through by the greatest common divisor, thus obtaining the claimed result.
\end{proof}
We will use the following version of Dirichlet's theorem:
\begin{cor}\label{cor:dirichlet}
Let $\theta\in\bR\setminus\bQ$. There exists a sequence $\{(p_n,q_n)\}_{n\ge1}$ with $p_n\in\bZ$ and $q_n\in\bZ_{>0}$ for all $n$, such that:
\begin{enumerate}[(1)]
\item
For all $n$, we have $p_n$ and $q_n$ are relatively prime.
\item
We have \[\lim_{n\to\infty} q_n=\infty.\]
\item
We have \[\left| \theta-\frac{p_n}{q_n}\right|<\frac{1}{q_n^2}.\]
\end{enumerate}
\end{cor}
\begin{proof}
Let $N_1\in\bZ_{>0}$ be arbitrary, and let $(p_1,q_1)$ be a pair as furnished by Theorem~\ref{thm:dirichlet-1}, so that \[\left|\theta-\frac{p_1}{q_1}
\right|<\frac{1}{N_1\cdot q_1}<\frac{1}{q_1^2}.\] We then choose a natural number \[N_2>\frac{1}{|\theta-p_1/q_1|}.\]
Theorem~\ref{thm:dirichlet-1} furnishes a new pair $(p_2,q_2)$,
with $N_2$ as the input parameter. Note that we may estimate \[\left|\theta-\frac{p_2}{q_2}\right|<\frac{1}{N_2\cdot
q_2}<\left|\theta-\frac{p_1}{q_1}\right|\cdot\frac{1}{q_2}\leq \left|\theta-\frac{p_1}{q_1}\right|.\] It follows that $p_2/q_2$ is a strictly better
approximation of $\theta$ that $p_1/q_1$. Thus, we may recursively produce a sequence $\{(p_n,q_n)\}_{n\ge1}$ as claimed, and since
for all $i$ we have $p_{i+1}/q_{i+1}$ is a better approximation to $\theta$ than $p_i/q_i$, it follows that $q_n\to\infty$ as $n\to\infty$.
\end{proof}
The usefulness of rational approximations is that they give some control over the distribution of orbits of an irrational rotation for powers
that do not exceed the denominator of the approximating rational number.
Again, let \[\exp\colon [0,1]\longrightarrow S^1\] be the exponentiation map
$\exp(t)=e^{2\pi i t}$.
\begin{cor}\label{cor:dirichlet-circle}
Let $\theta\in (0,1)$ be irrational, and let $p/q\in (0,1)$ be a rational approximation of $\theta$. Consider the sequence of points
\[S_q=\{\exp( k\theta)\mid 1\leq k\leq q\}.\] Then for all $0\leq j<q$ we have \[\left|S_q\cap \exp\left(\left[\frac{ j}{q},\frac{(j+1)}{q}\right]
\right)\right|=1.\]
\end{cor}
\begin{proof}
First, observe that \[e^{ k\theta}\in \exp\left(\left[\frac{ kp}{q},\frac{(k+1)p}{q}\right]\right)\quad \textrm{or}\quad e^{ k\theta}\in
\exp\left(\left[\frac{ (k-1)p}{q},\frac{ kp}{q}\right]\right),\] according to whether $\theta$ is greater than or smaller than $p/q$,
since we have \[\left|k\theta-\frac{kp}{q}\right|<\frac{k}{q^2}<\frac{1}{q}.\] The intervals
\[\exp\left(\left[\frac{ k}{q},\frac{(k+1)}{q}\right]
\right)\] have disjoint interiors, as $k$ varies between $0$ and $q-1$, since $p$ and $q$ are relatively prime, and each contains exactly
one point in $S_q$. This establishes the corollary.
\end{proof}
\subsection{The Denjoy--Koksma inequality and unique ergodicity}\label{sss:denjoy-koksma}
A key observation about about
homeomorphisms with irrational rotation number is the Denjoy--Koksma inequality, which says that if $\phi$ is a reasonably nice function
on $S^1$, then averaging the values of $\phi$ on orbits of the homeomorphism converges to the average value of $\phi$. The reader
familiar with ergodic theory will notice a resemblance with Birkhoff's Ergodic Theorem~\cite{EW11book,petersen83}.
Here, ``reasonably
nice" means \emph{bounded variation}\index{bounded variation}.
Recall that a function $\phi\colon S^1\longrightarrow \bR$ has bounded variation if
\[
\Var(\phi;S^1):=\sup_{n\ge1}\sum_{\{x_0,\cdots,x_n\}\sse S^1} |\phi(x_i)-\phi(x_{i-1})|\] is finite, where here
\[x_0<x_1<\cdots <x_n<x_0,\] where the indices are considered
modulo $n$, and where the ordering is the cyclic ordering on $S^1$ determined by a choice of orientation.
\begin{thm}[Denjoy--Koksma inequality]\label{thm:denjoy-koksma}
Suppose $f\in\Homeo_+(S^1)$ with irrational rotation number $\theta$. Let $p/q$ be a rational approximation of \[{\theta}\in (0,1),\]
and let $\phi\colon S^1\longrightarrow\bR$ be a function of bounded variation. If $x\in S^1$ is arbitrary and if $\mu$
is an $f$--invariant Borel probability measure, then, we have
\[\left|\frac{1}{q}\sum_{k=0}^{q-1}\phi(f^k(x))-\int_{S^1}\phi\,d\mu\right|\leq \frac{\Var(\phi;S^1)}{q}.\]
\end{thm}
Even though the Denjoy--Koksma inequality superficially resembles the Birkhoff Ergodic Theorem, we note that the factor of $q$ is of
crucial importance in the application of the inequality. Thus, the Denjoy--Koksma inequality should be viewed as
having both number--theoretic
and dynamical content.
\begin{proof}[Proof of Theorem~\ref{thm:denjoy-koksma}]
We follow the argument in~\cite{athanassopoulos}.
Theorem~\ref{thm:irr-rot-semi} furnishes a semi-conjugacy $h$ between $f$ and the rotation $R_{\theta}$ by $\theta$. Observe that
the pushforward $\nu=h_*\mu$ of $\mu$ by $h$ is an invariant measure for $R_{\theta}$, and thus
by Corollary~\ref{cor:leb-ue} must be Lebesgue measure, which we normalize for the purpose of this proof to have total mass one.
Choose an arbitrary point $z_0\in S^1$, let \[\{z_0,z_1,\ldots,z_{q-1}\}\sse S^1\] be the orbit of $z_0$ under the rotation by
$1/q$, and let \[\{x_0,\ldots,x_{q-1}\}\sse S^1=h^{-1}(S^1)\]
be points such that $x_i\in h^{-1}(z_i)$ for all $i$. In the circular order on $S^1$, we may
clearly assume that \[x_0<x_1<\cdots<x_{q-1}<x_0,\] and so we may consider the oriented interval $[x_i,x_{i+1}]$, where the indices are
again regarded modulo $q$. By definition, we have \[\int_{[x_i,x_{i+1}]}\,d\mu=\int_{[z_i,z_{i+1}]}\,d\nu=\frac{1}{q}.\] By Corollary
\ref{cor:dirichlet-circle}, for all $1\leq k\leq q$, there is a unique $i=i(k)$ such that \[R_{\theta}^k(h(x_0))\in [z_i,z_{i+1}].\] We may thus
write \[\left|\frac{1}{q}\sum_{k=1}^{q}\phi(f^k(x_0))-\int_{S^1}\phi\,d\mu\right|=
\left|\sum_{k=1}^{q}\left(\frac{1}{q}\phi(f^k(x_0))-\int_{[x_{i(k)},x_{i(k)+1}]}\phi\,d\mu\right)\right|.\] Writing $c_k=\phi(f^k(x_0))$
and $J_k=[x_{i(k)},x_{i(k)+1}]$, this last
expression is bounded above by \[\sum_{k=1}^q\left|\int_{J_k}(c_k-\phi)\,d\mu\right|\leq
\frac{1}{q}\sum_{k=1}^q\sup_{x,y\in J_k}|\phi(x)-\phi(y)|.\] The right hand side of this
last expression is clearly bounded above by $\Var(\phi;S^1)/q$, whence the
inequality follows after setting $x=f^{-1}(x_0)$.
\end{proof}
The Denjoy--Koksma inequality has the following corollary, which was promised after Proposition~\ref{prop:irr-rot-erg}.
\begin{thm}\label{thm:irr-ue}
Let $f\in\Homeo_+(S^1)$ have irrational rotation number. Then $f$ is uniquely ergodic.
\end{thm}
\begin{proof}
Let $\theta$ be the rotation number of $f$ and let \[\frac{p_n}{q_n}\longrightarrow{\theta}\] be a sequence of rational approximations
such that $q_n\to\infty$ as $n\to\infty$. Theorem~\ref{thm:denjoy-koksma} implies that if $\phi$ has bounded variation and $\mu$
is an $f$--invariant Borel probability measure, then
\[\lim_{n\to\infty}\frac{1}{q_n}\sum_{k=0}^{q_n-1}\phi\circ f^k=\int_{S^1}\phi\,d\mu,\] where this convergence is uniform on $S^1$. If
$\lambda\neq\mu$ were another $f$--invariant Borel probability measure, then there would be a Borel subset $A\sse S^1$ such that
$\lambda(A)\neq\mu(A)$. The classical result Lusin's Theorem from measure theory~\cite{Rudin87book}
implies that there is a continuous function $\phi$
(which can easily be arranged to have bounded variation) approximating the characteristic function of $A$ and
such that \[\int_{S^1} \phi\,d\mu\neq\int_{S^1}\phi\,d\lambda.\] This is clearly a contradiction.
\end{proof}
\subsection{Completing the proof of Denjoy's Theorem}
We now complete the last steps in proving Denjoy's Theorem. In this subsection, we always assume that $f\in\Diff_+^1(S^1)$ is an
orientation preserving diffeomorphism. The next lemma says that the average value of the logarithm of the derivative of a diffeomorphism
is zero.
\begin{lem}[cf.~\cite{athanassopoulos}, Proposition 3.3]\label{lem:log-int}
Let $f$ have an irrational rotation number and let $\mu$ be the unique $f$--invariant Borel probability measure. Then
\[\int_{S^1}\log (f')\,d\mu=0.\]
\end{lem}
\begin{proof}
Note that \[\log((f^n)')=\sum_{k=0}^{n-1}(\log (f'))\circ f^k,\] as follows from the chain rule. It follows that \[\lim_{n\to\infty}\frac{1}{n}\log((f^n)')=
\int_{S^1}\log (f')\,d\mu,\] where the equation is valid when the left hand side is evaluated at an arbitrary point of $S^1$.
Indeed, since $f$ is uniquely ergodic with respect to $\mu$ by Theorem~\ref{thm:irr-ue}, the sequence of probability measures given by
measuring the (renormalized) number of points in the $f$--orbit of an arbitrary
point that meet a given subset of $S^1$ has a unique weak--$\ast$
limit that must be $\mu$ (see the remarks after Corollary~\ref{cor:leb-ue} above). The left hand side of the equation is
the limit of a sequence of
integrals of $\log(f')$ with respect to these measures.
Now, suppose that \[\int_{S^1}\log(f')\,d\mu>0.\] We must have that $(f^n)'$ tends to $+\infty$, uniformly on $S^1$.
If $F$ represents an arbitrary lift of $f$ to $\bR$ then \[\int_{S^1}(f^n)'(\theta)\,d\lambda=\int_0^{1}(F^n)'(x)\,dx\longrightarrow +\infty,\] where
$\lambda$ denotes Lebesgue measure on $S^1$ and where $dx$ denotes Lebesgue measure on $\bR$.
If, on the other hand \[\int_{S^1}\log(f')\,d\mu<0,\] we have that $(f^n)'$ tends to $0$, uniformly on $S^1$, so that
\[\int_{S^1}(f^n)'(\theta)\,d\lambda=\int_0^{1}(F^n)'(x)\,dx\longrightarrow 0.\] However, $f^n$ is a diffeomorphism, and so the integral
over $S^1$ of its derivative is simply $1$. In either of the previous two cases, we obtain a contradiction.
\end{proof}
The following is the last crucial observation before the proof of Denjoy's Theorem.
\begin{lem}[cf.~\cite{athanassopoulos}, Proposition 3.5]\label{lem:denjoy-var}
Suppose $\rot(f)=\theta$ is irrational, that $p/q$ is a rational approximation of $\theta$, and that $\log(f')$ has bounded variation $V$.
Then \[|\log((f^q)')|\leq V.\]
\end{lem}
\begin{proof}
The Denjoy--Koksma inequality shows that for $x\in S^1$ arbitrary, we have \[\left|\sum_{k=0}^{q-1}\log(f'(f^k(x)))-q\cdot\int_{S^1}\log(f')\,d\mu
\right|\leq V.\]
By Lemma~\ref{lem:log-int}, the second term on the left hand side is zero. The chain rule says that \[\sum_{k=0}^{q-1}\log(f'(f^k(x)))=
\log((f^q)').\] The lemma is now immediate.
\end{proof}
The conclusion of Lemma~\ref{lem:denjoy-var} is equivalent to the statement that \[e^{-V}\leq (f^q)'\leq e^V.\]
Finally, we can give a proof of Denjoy's Theorem.
\begin{proof}[Proof of Theorem~\ref{thm:denjoy}]
If $f$ is minimal then $f$ is conjugate to the rotation $R_{\theta}$, by Theorem~\ref{thm:irr-rot-semi}. So, we may assume that $f$ is not
minimal. We have that $f$ has no periodic points by Proposition~\ref{prop:rot-easy}, and so Theorem~\ref{thm:minimal-set} implies
that there is a minimal invariant Cantor set $C\sse S^1$. If $J\sse S^1\setminus C$ is a component the $J$ is an open interval, and
we have $f^n(J)\cap J=\varnothing$ for $n\neq 0$.
Let \[\frac{p_n}{q_n}\longrightarrow{\theta}{}\]
be a sequence of rational approximations, where $q_n\to\infty$ as $n\to\infty$, as furnished by
Corollary~\ref{cor:dirichlet}. Clearly, we may assume that $q_n<q_m$ for $n<m$.
Lemma~\ref{lem:denjoy-var} bounds the derivative of $f^{q_n}$ away from zero,
independently of anything other than the variation $V$ of $\log (f')$.
Therefore, the Mean Value Theorem implies that \[\frac{|f^{q_n}(J)|}{|J|}\geq e^{-V},\] where $|J|$ denotes the length of the interval $J$.
We finally obtain \[+\infty=\sum_{n\ge1}e^{-V}|J|\leq \sum_{n\ge1}|f^{q_n}(J)|\leq 1,\] which is a contradiction.
\end{proof}
\section{Exceptional diffeomorphisms and integrable moduli}\label{sec:denjoy-crit}
This section is devoted to investigating the failure of Denjoy's Theorem in the case of elements $f\in\Homeo_+(S^1)$ that do not lie in
$\Diffb(S^1)$. We have already explicitly constructed elements of $\Homeo_+(S^1)$ that have no periodic points and exceptional minimal
sets, which we called continuous Denjoy counterexamples. These homeomorphisms are defined by equivariant families of affine
homeomorphisms on the complement of the exceptional minimal set,
and so it might be rather difficult to force a continuous Denjoy counterexample to have desirable regularity properties.
Indeed, Denjoy's Theorem implies that a continuous Denjoy counterexample cannot have a first derivative of bounded variation.
If $f$ is a $C^2$ diffeomorphism then $f'$ and $(f^{-1})'$ are both differentiable, and so that the total variation of $\log (f')$
for example is just
\[\int_{S^1}\left|\frac{f''}{f'}\right|\,d\lambda,\] where $\lambda$ is Lebesgue measure normalized to have total mass $1$.
This integral is finite since $f'$ is continuous and therefore
is bounded away from zero on $S^1$.
We will show that many (but not all) regularities that are weaker than $C^2$ do admit Denjoy counterexamples, which is to say exceptional
diffeomorphisms with prescribed irrational rotation number.
\subsection{Stationary measures and Lipschitz homeomorphisms}
This subsection will largely follow Section 2.3.2 in~\cite{Navas2011}. We reproduce just the details we need in order to give a complete
account of critical regularity in the sequel.
Recall that $f\in\Homeo_+(S^1)$ (or more generally, a real valued function on a metric space)
is called \emph{Lipschitz}\index{Lipschitz map} if there is a constant $C$ such that for all $x,y\in S^1$, we have
\[|f(x)-f(y)|\leq C|x-y|,\] where here as usual the circle is considered as the additive reals modulo $\bZ$.
If an identical inequality also holds for $f^{-1}$, we say that $f$ is \emph{bi-Lipschitz}\index{bi-Lipschitz map}.
The infimum of such constants $C$ for a function $f$
is called the \emph{Lipschitz constant}\index{Lipschitz constant} of $f$.
It is not difficult to show that the set of bi-Lipschitz homeomorphisms of $S^1$ forms a subgroup of $\Homeo_+(S^1)$, as the
Lipschitz constants just multiply under composition.
It is not difficult to produce a continuous Denjoy counterexample that is bi-Lipschitz, simply by choosing the lengths of the wandering
intervals $\{J_n\}_{n\in\bZ}$ with sufficient care. We will prove a fact originally observed by
Deroin--Kleptsyn--Navas~\cite{DKN2007} that establishes a much more
general fact: an arbitrary countable subgroup $\gam\le \Homeo_+(S^1)$ is topologically conjugate to group of bi-Lipschitz homeomorphisms
of $S^1$. This level of generality is not necessary for us to produce bi-Lipschitz exceptional diffeomorphisms,
but the existence of a such a conjugacy is an important part of the story of critical regularity of groups
that we will recount later in this book.
\begin{thm}[See~\cite{Navas2011}, Proposition 2.3.15; cf.~\cite{Calegari2007}, Theorem 2.118]\label{thm:lip-conj}
Suppose $\gam\le \Homeo_+(S^1)$ is countable.
Then there is an element $\phi\in\Homeo_+(S^1)$ such that $\phi^{-1}\gam \phi$ is a group of bi-Lipschitz
homeomorphisms of $S^1$.
\end{thm}
Before proving Theorem~\ref{thm:lip-conj}, we develop some basic facts about stationary measures. Let $\bP$ be a probability measure
on $\Gamma$, where $\Gamma$ is a countable group. We write $\supp\bP$ for the \emph{support}\index{support of a probability
measure on a group}
of $\bP$, which is the set
\[\supp\bP=\{\gamma\in\gam\mid\bP(\gam)>0\}.\] Recall that the support of a Borel measure on a topological space $X$
is typically defined to be the largest closed subset $C$ of $X$ such that every open subset of $C$ has positive measure. To avoid inconsistent
definitions, we will implicitly endow $\Gamma$ with the discrete topology.
To avoid the degeneracy of a probability measure that is essentially supported on
a proper subgroup of $\Gamma$, one usually requires $\bP$ to be \emph{nondegenerate}\index{nondegenerate probability measure}.
That is, $\form{\supp\bP}=\gam$. The group
action of $\Gamma$ on $S^1$ and the measure $\bP$ give rise to a bounded operator (which we restrict to real valued continuous functions)
\[\Delta\colon C(S^1)\longrightarrow C(S^1)\] defined by
\[\Delta(f)(x)=\int_{\gam}f(\gamma(x))\,d\bP(\gamma).\] The intuitive meaning of $\Delta$ is clear: it averages the values of a function $f$
among its orbit under $\Gamma$, weighted by the measure $\bP$. Since the measure $\bP$ is a probability measure, one can think
of $\Delta$ as modeling a random evolution of functions under a $\Gamma$ action, and this
is why $\Delta$ is often called a \emph{diffusion operator}\index{diffusion operator}.
To proceed, we consider the dual operator to $\Delta$, which is defined as the formal adjoint
\[\Delta^*\colon C(S^1)^*\longrightarrow C(S^1)^*,\] acting on bounded linear functionals on $C(S^1)$.
By the Riesz Representation Theorem (Theorem~\ref{thm:riesz}), we have that the space of positive functionals on $C(S^1)$
coincides with the space
of positive Borel measures on $S^1$ with finite total measure. So, if $\mu$ is such a measure on $S^1$ then $\Delta^*\mu$ is also
a measure on $S^1$, defined by \[\int_{S^1}f\,d(\Delta^*\mu)=\int_{S^1}\Delta(f)\,d\mu=\int_{S^1}\int_{\gam}f(\gamma(x))\,d\bP(\gamma)d\mu.\]
The measure $\Delta^*\mu$ is called the \emph{convolution}\index{convolution of measures}
of $\bP$ and $\mu$, and is also written $\bP*\mu$. A measure $\mu$
on $S^1$ such that $\bP*\mu=\mu$ is called \emph{stationary}\index{stationary measure} for the data of $\Gamma$ and $\bP$.
One can make several immediate observations about convolution. First, suppose $f\in C(S^1)$ is a positive function, i.e.~$f(x)\geq 0$ for
all points $x\in S^1$. It is immediate that
\[\int_{S^1}\Delta(f)\,d\mu\geq 0,\] since at every point $x\in S^1$ and for every $\gamma\in\gam$, the value of $f(\gamma(x))$ is non-negative.
It follows that convolution with $\bP$ preserves the set of positive Borel measures of finite total mass on $S^1$. Integrating the
constant function $f(x)\equiv
1$ against $\Delta^*\mu$ shows that $\Delta^*$ preserves the set of Borel probability measures on $S^1$.
\begin{lem}\label{lem:stat-meas}
Let $\{\gam,\bP\}$ be as above. Then there exists a Borel probability measure $\mu$ on $S^1$ that is stationary for $\bP$.
\end{lem}
\begin{proof}
The convolution operator preserves the nonempty, convex, and compact set of Borel probability measures on $S^1$,
by the remarks preceding the lemma.
To prove that there exists a fixed point of the convolution operator, one can simply follow
the proof of Lemma~\ref{lem:kaku-abel} above. Instead of using a group $G$, we simply use a semigroup $S$ generated by a single element
(namely $\Delta^*$).
The proof goes through without any difficulty, and we leave the details to
the reader.
\end{proof}
The support of a stationary measure\index{support of a measure}
$\mu$ for $\{\gam,\bP\}$ turns out to be closely related to the dynamical trichotomy given in
Theorem~\ref{thm:minimal-set}.
\begin{lem}[See~\cite{Navas2011}, Lemma~2.3.14]\label{lem:stat-support}
Let $\{\gam,\bP\}$ be as above, with $\bP$ nondegenerate,
and let $\mu$ be a stationary probability measure on $S^1$ for this data. Then the following conclusions hold.
\begin{enumerate}[(1)]
\item
If $\mu$ has an atom then $\Gamma$ has a finite orbit.
\item
If $\Gamma$ acts minimally then $\mu$ has full support; that is, $\mu$ gives positive measure to every open subset of $S^1$.
\item
If $\Gamma$ admits an exceptional minimal set $C$ then $\mu(C)=1$.
\end{enumerate}
\end{lem}
\begin{proof}
Suppose first that $x\in S^1$ is a point with $\mu(x)>0$, and where $x$ has maximal mass.
Since $\mu$ is stationary, we have that
\[\mu(x)=\int_{\gam}\mu(\gamma^{-1}(x))\,d\bP(\gamma).\]
If $\mu(\gamma^{-1}(x))<\mu(x)$ for some
$\gamma$
then we obtain \[\sum_{\gamma\in\gam}\mu(\gamma^{-1}(x))\bP(\gamma)<\mu(x),\] a contradiction.
Thus, $\mu(\gamma^{-1}(x))=\mu(x)$ for $\gamma\in\supp\bP$. Since $\supp\bP$ generates
all of $\Gamma$, we see that $\mu$ is constant on the $\Gamma$--orbit of $x$. Since $\mu$ is a probability measure, it follows that the $\Gamma$--orbit
of $x$ is finite.
The support of $\mu$ is a closed subset of $S^1$ by definition, and it is also $\Gamma$--invariant, as is a straightforward consequence of the
stationarity of $\mu$ and the non-degeneracy of $\bP$. It follows that if $\Gamma$ acts minimally on $S^1$ then $\supp\mu=S^1$.
Now, suppose that $\Gamma$ admits an exceptional minimal set $C$. Then $C\sse\supp\mu$, by
the minimality of $C$. It suffices to show that
if $J\sse S^1\setminus C$ is an interval then $\mu(J)=0$.
Assume for a contradiction that $\mu(J)>0$. Without loss of generality, $J$ is a maximal connected component of the
complement of $C$, so that for
all $\gamma\in\gam$ we either have \[\gamma(J)\cap J=\varnothing\quad \textrm{ or }\quad \gamma(J)=J.\] As in the case of an atomic
measure, we have \[\mu(J)=\int_{\gam}\mu(\gamma^{-1}(J))\,d\bP(\gamma),\] by stationarity. It follows that the $\Gamma$--orbit of $J$ is
finite. However, if $x\in\partial J$ then the $\Gamma$--orbit of $x$ is dense in $C$, which is the desired contradiction.
\end{proof}
In the case of a minimal action, it follows that the stationary measure $\mu$ can be pushed forward to Lebesgue measure by a circle
homeomorphism:
\begin{lem}\label{lem:leb-push}
Let $\mu$ be a probability measure on $S^1$ with no atoms and whose support coincides with $S^1$.
Then there is a homeomorphism $\phi\in\Homeo_+(S^1)$
such that $\phi_*\mu$ is Lebesgue measure.
\end{lem}
\begin{proof}
As we have seen several times already, the measure $\mu$ induces a periodically defined measure $\nu$ on $\bR$. One can define
a homeomorphism of $\bR$ by
\[\Phi(x)=\int_0^x \,d\nu.\]
Since $\mu$ has no atoms, neither does $\nu$, and $\Phi$ is continuous. Since $\mu$ is fully supported and
since an open interval separates every pair of distinct points in $\bR$, the map $\Phi$ is injective. Since $\Phi$ is periodic, it is in fact a
homeomorphism and descends to a homeomorphism $\phi$ of $S^1$.
The homeomorphism $\phi$ pushes $\mu$ forward to Lebesgue
measure.
\end{proof}
We can now prove that an arbitrary countable group action on $S^1$ is conjugate to a bi-Lipschitz action.
\begin{proof}[Proof of Theorem~\ref{thm:lip-conj}]
First, note that one may assume $\Gamma$ acts on $S^1$ minimally. If not, then one can add an irrational rotation to $\Gamma$ to obtain a larger
(but still countable) group of homeomorphisms.
We set $\bP$ to be a non-degenerate probability measure on $\Gamma$ that is symmetric, which is to say $\bP(\gamma)=\bP(\gamma^{-1})$
for all $\gamma\in\gam$. Lemma~\ref{lem:stat-meas} says that there exists a stationary measure $\mu$ on $S^1$ associated to
$\Gamma$ and $\bP$. Writing $\supp\bP\sse\gam$ for the support of the measure $\bP$, the stationarity of $\mu$ implies that
\[\sum_{\gamma\in\supp\bP}\mu(\gamma^{-1}(J))\bP(\gamma)=\mu(J)\] for all intervals $J\sse S^1$. For an arbitrary $\gamma\in \gam$,
we obtain that \[\mu(\gamma(J))\bP(\gamma^{-1})\leq \mu(J),\] and so
since $\bP$ is assumed to be symmetric, we get \[\mu(\gamma(J))\leq\frac{\mu(J)}{\bP(\gamma)},\] provided $\gamma$
lies in the support of $\bP$.
By Lemma~\ref{lem:stat-support} and Lemma~\ref{lem:leb-push},
we have that there is an element $\phi\in\Homeo_+(S^1)$ such that $\phi_*\mu=\lambda$,
where $\lambda$ denotes Lebesgue measure. We claim that $\phi$ conjugates $\Gamma$ to a group of bi-Lipschitz homeomorphisms.
To this end, let $\gamma\in\supp\bP\sse \gam$, let $C=\bP(\gamma)^{-1}$, and let $J\sse S^1$ be an interval. We estimate:
\[|\phi\circ\gamma\circ\phi^{-1}(J)|=\mu(\gamma\circ\phi^{-1}(J))\leq C\mu(\phi^{-1}(J))=C|J|,\] where here as before the absolute value
denotes Euclidean length. The first and last equalities are justified since $\phi$ pushes $\mu$ forward to Lebesgue measure, and the
inequality is justified in the previous paragraph. It follows that $\phi$ conjugates a generating set for $\Gamma$ to a set of bi-Lipschitz
homeomorphisms of $S^1$, which establishes the result.
\end{proof}
We remark that a verbatim analogue of Theorem~\ref{thm:lip-conj} holds for the interval. Since $\Homeo_+[0,1]$ can be embedded in
$\Homeo_+(S^1)$ by identifying the endpoints of the interval, it is immediate that one can conjugate countable subgroups of $\Homeo_+[0,1]$
to bi-Lipschitz homeomorphisms. Some care needs to be taken to show that the conjugacy can be realized within $\Homeo_+[0,1]$. We leave
the details to the reader.
\subsection{Smooth Denjoy counterexamples and the spectrum of their moduli of continuity}\label{ss:smooth-denjoy}
We now concentrate on exceptional diffeomorphisms $f\in\Diff_+^1(S^1)$. By definition, the derivatives of $f$ and $f^{-1}$ are
continuous functions. Bohl~\cite{Bohl1916} and Denjoy~\cite{Denjoy1932} have constructed exceptional diffeomorphisms of the circle,
and their results were extended significantly by Herman~\cite{Herman1979}.
In light of Denjoy's Theorem, there is a dividing line (or dividing lines) in the regularity properties of the derivatives $f'$ and $(f^{-1})'$
that distinguish between the possibility of exceptional diffeomorphisms and the proscription thereof.
To introduce a precise setup, let \[\alpha\colon [0,\infty)\longrightarrow [0,\infty)\] be a homeomorphism that is concave as a function.
Sometimes we will refer to $\alpha$ as a \emph{concave modulus of continuity}\index{concave modulus of continuity}.
Let $X$ and $Y$ be metric spaces.
We say a continuous map
\[
f\co X\longrightarrow Y\]
is $\alpha$--continuous, or $C^\alpha$, if there exists a $C>0$
such that for all $x,y\in X$ we have
\[d_Y(f(x),f(y))|\leq C\cdot\alpha(d_X(x,y)).\]
The infimum of possible values of $C$ is called the $\alpha$--norm of $f$, and is written $[f]_{\alpha}$.
We let $C^\alpha(X,Y)$ denote the space of $\alpha$--continuous maps from $X$ to $Y$.
The reader will note that if $\alpha(x)=x$ then $\alpha$--continuity is the same as Lipschitz continuity.
More generally, if $\alpha(x)=x^{\tau}$ for $\tau\in (0,1]$ then $\alpha$--continuity is the same as $\tau$--H\"older continuity.
The reason for considering concave moduli of continuity is because nonconstant functions generally do not satisfy non-concave modulus
of continuity bounds. The reader may check, for instance, that if $\tau>1$ then only constant functions are $\tau$--H\"older continuous.
We say that $f\in\Diff_+^{1,\alpha}(S^1)$ if $f$ is a diffeomorphism and if \[[f']_{\alpha},[(f^{-1})']_{\alpha}<\infty.\]
If $f\in\Diff_+^1(S^1)$
is arbitrary then $f\in\Diff_+^{1,\alpha}(S^1)$ for some suitable \emph{smooth} concave modulus $\alpha$.
This follows from the following somewhat more general fact.
A \emph{geodesic}\index{geodesic} in a metric space $X$ is a continuous path
\[ \gamma\co [0,D]\longrightarrow X\]
such that \[d_X(\gamma(s),\gamma(t))=|s-t|\] for all $s,t\in[0,D]$.
A \emph{geodesic space}\index{geodesic metric space} is a metric space every pair of points in which can be joined by a geodesic.
\begin{lem}[cf.~\cite{CKK2019}, Proposition 2.7]\label{lem:modulus-existence}
Let $X$ and $Y$ be metric spaces. If $X$ is a geodesic space and if $f\co X\longrightarrow Y$ is a uniformly continuous function,
then there exists a concave modulus\[\beta\co[0,\infty)\longrightarrow [0,\infty)\] which is smooth on $(0,\infty)$ such that the following hold:
\be[(i)]
\item $f$ is $\beta$--continuous;
\item if $\alpha$ is a concave modulus such that $f$ is $\alpha$--continuous,
then there exist constants $\delta>0$ and $C\geq 1$ such that
\[ \beta(x)\le C \alpha(x)\] for all $x\in[0,\delta]$.
\ee
\end{lem}
The concave modulus $\beta$ above is called an \emph{smooth optimal concave modulus of continuity for $f$}\index{optimal modulus of
continuity}.
We will postpone a detailed proof of the above lemma to Appendix~\ref{ch:append1}.
\begin{rem}
The regularity of a function is naturally treated as a local property.
More precisely, we say a map $f\co X\longrightarrow Y$ is \emph{locally $\alpha$--continous}\index{locally $\alpha$--continuous}
if for each $x_0\in X$ there exists a ball $B$ of finite diameter centered at $x_0$
such that the restriction $f\restriction_B$ is $\alpha$--continuous. We denote by
\[
C^\alpha(X,Y)\]
the space of locally $\alpha$--continuous maps from $X$ to $Y$.
On the other hand,
it is an easy exercise~\cite[Lemma 1.1]{CKK2019} to see that if $X$ is compact then a locally $\alpha$--continuous map from $X$ to $Y$ is $\alpha$--continuous.
As in this book we focus on the compact spaces $S^1$ and $[0,1]$ or on compactly supported diffeomorphisms,
the notion of local $\alpha$--continuity coincides with the notion of $\alpha$--continuity most of the time.
We note also that in the above lemma, with an extra hypothesis that $X$ is compact the optimal modulus $\beta$ satisfies
\[
C^\beta(X,Y)=\bigcap\left\{ C^\alpha(X,Y)\mid \alpha\text{ is a concave modulus such that }f\in C^\alpha(X,Y)\right\}.\]
\end{rem}
Lemma~\ref{lem:modulus-existence} shows that \[\Diff_+^1(S^1)=\bigcup_{\alpha}\Diff_+^{1,\alpha}(S^1),\] as $\alpha$ ranges over
all concave moduli of continuity. Therefore we may formulate the following question:
\begin{que}\label{que:denjoy-counter}
For $\theta$ irrational and $\alpha$ a concave modulus of continuity,
does there exist an exceptional diffeomorphism $f\in\Diff^{1,\alpha}_+(S^1)$ with rotation number $\theta$?
\end{que}
Some partial answers to Question~\ref{que:denjoy-counter} are immediate. For instance, if $\alpha(x)=x$ then there are no such
diffeomorphisms. Indeed, if $f'$ is Lipschitz then it is also of bounded variation, so that $\log(f')$ has bounded variation; therefore
Denjoy's Theorem says that if $\rot(f)=\theta$ then $f$ is conjugate to $R_{\theta}$.
Weakening the modulus of continuity to
$\tau$--H\"older continuity for $\tau\in (0,1)$ then the answer is completely different: Herman~\cite{Herman1979}
proved that for $\alpha=x^{\tau}$
there are exceptional diffeomorphisms in $\Diff^{1,\alpha}_+(S^1)$ for all irrational rotation numbers. Herman's methods applied to a wider
range of moduli. For instance, he produced exceptional diffeomorphisms with arbitrary irrational rotation number for the modulus
\[\alpha(x)=x(\log 1/x)^{1+\eps},\] where $\epsilon>0$ is arbitrary.
This last modulus is stronger than every $\tau$--H\"older modulus of
continuity for $\tau<1$ since for small values of $x$ we hae \[x(\log 1/x)^{1+\epsilon}\leq C\cdot x^{\tau}\] for a suitably chosen constant
$C$, but is weaker than the Lipschitz modulus, as the reader may check.
A definitive answer to Question~\ref{que:denjoy-counter} is by no means the end of the discussion about the existence and nonexistence
of exceptional diffeomorphisms of the circle. Sullivan~\cite{Sullivan1992} and Hu--Sullivan~\cite{HuSu1997},
for example, studied other regularity conditions one can place
on the derivative of a diffeomorphism with irrational rotation number in order to guarantee conjugacy to an irrational rotation, including the
\emph{Zygmund condition}\index{Zygmund condition} and
\emph{bounded quadratic variation}\index{quadratic variation}. We will not be discussing this aspect of the theory here, as it would
cause us to stray too far from the main narrative, and instead we direct the reader to the literature.
The primary result that we shall discuss for the remainder of this section is the following, which recovers Herman's results and furnishes
new moduli of continuity that admit exceptional diffeomorphisms.
\begin{thm}[See~\cite{KK2020-DCDS}, Theorem 1.2]\label{thm:kk-denjoy}
Let $\theta$ be an irrational rotation number, and suppose $\alpha$ is a concave modulus of continuity such that
\[\int_0^1\frac{dx}{\alpha(x)}<\infty.\] Then there exists an exceptional $f\in\Diff^{1,\alpha}(S^1)$ with $\rot(f)=\theta$.
For all $\eps>0$, we may arrange \[\sup_{x\in S^1} |f(x)-R_{\theta}(x)|+\sup_{x\in S^1} |f'(x)-1|<\eps,\]
so that $f$ is $\eps$--close to $R_{\theta}$ in
the $C^1$--topology.
\end{thm}
Notice that for \[\alpha(x)\in\{x(\log 1/x)^{1+\epsilon},\, x^{\tau}\mid \eps>0,\,\tau<1\},\] we have that $1/\alpha(x)$ is integrable near zero.
We can even get integrability of $1/\alpha(x)$ near zero for
\begin{align*}
\alpha(x)=x(\log 1/x)(\log\log 1/x)^{1+\epsilon}\\ \alpha(x)=x(\log 1/x)(\log\log 1/x)(\log\log\log 1/x)^{1+\epsilon}\\
\alpha(x)=x(\log 1/x)(\log\log 1/x)(\log\log\log 1/x)(\log\log\log\log 1/x)^{1+\epsilon}\ldots,
\end{align*}
and this furnishes many moduli of continuity, each stronger than the previous, that admit exceptional diffeomorphisms. Of course,
the function $1/x$ is not integrable near zero, consistent with the nonexistence of exceptional diffeomorphisms with Lipschitz moduli.
The function $\alpha(x)=x(\log 1/x)$ is not integrable near zero, and it is a well-known open question within the field whether or not there
exist exceptional diffeomorphisms with this modulus of continuity. It is even suspected that the answer to this last question depends on the
diophantine properties of the rotation number $\theta$ in question. The reader is directed to
Section 4.1.4 of~\cite{Navas2011} for a more detailed discussion.
\subsubsection{Herman's construction of exceptional diffeomorphisms}
The main idea behind the construction of exceptional diffeomorphisms is to reduce the problem to function theory. That is, in order to
construct the diffeomorphism $f$, one should construct the function $g=f'-1$ by hand, and then $f$ is obtained just by integration. The
function $g$ will consist of a sequence of bumps that are sufficiently smooth, and that are supported on a carefully chosen shrinking
family of intervals, which are then glued into the circle as in our original description of a continuous Denjoy counterexample. To see
how this might work, we have the following fact.
\begin{prop}[See~\cite{KK2020-DCDS}, Proposition 4]\label{prop:herman-1}
Let $\alpha$ be a concave modulus of continuity, and let \[\{J_i=(x_i,y_i)\}_{i\in\bZ}\] be a collection of disjoint subintervals of $S^1$. Suppose
that $g$ is a positive, $\alpha$--continuous function on $S^1$ that satisfies the following conditions:
\begin{enumerate}[(1)]
\item
We have
\[\int_{S^1} g\,d\theta=1;\]
\item
For all $i\in\bZ$, we have \[\int_{J_i}g\,d\theta=|J_{i+1}|;\]
\item
For all $i$, we have \[\int_{x_{i-1}}^{x_i}g\,d\theta=|x_{i+1}-x_i|.\]
\end{enumerate}
Then the function defined by \[f(x)=x_1+\int_{x_0}^x g\,d\theta\] is an element of $\Diff_+^{1,\alpha}(S^1)$ that satisfies $f(J_i)=J_{i+1}$ for
all $i\in\bZ$.
\end{prop}
\begin{proof}
This is a straightforward verification. First, check that $f'$ is $\alpha$--continuous and that $f$ is a bijection, and that $(f^{-1})'$ is
$\alpha$--continuous by an easy application of the inverse function theorem. One also has, by an easy
calculation, that \[x_i+\int_{x_{i-1}}^x g\,d\theta=x_{i+1}+\int_{x_i}^x g\,d\theta,\] and so $f$ has the desired properties.
\end{proof}
Let $\{J_i\}_{i\in\bZ}$ be a sequence of compact intervals in $S^1$. We say that $\{J_i\}_{i\in\bZ}$ is
\emph{circular order preserving}\index{circular order} if
\[J_i<J_k<J_{\ell}\quad \textrm{if and only if}\quad J_{i+1}<J_{k+1}<J_{\ell+1}.\] Recall that we use the notation $\{x\}\in(0,1)$ for the fractional
part of $x\in\bR$.
The continuous Denjoy examples as we have
originally constructed them can be obtained via integration as in Proposition~\ref{prop:herman-1}, though not by integrating a function
but rather an atomic measure. Let $\{\ell_i\}_{i\in\bZ}$ be a collection of positive lengths, whose total sum adds up to at most $1$,
and let $\theta$ be an irrational number. Let $\mu$ be the measure on $S^1$ defined by
\[\mu=\left(1-\sum_{i\in\bZ}\ell_i\right)\lambda+\sum_{i\in\bZ}\ell_i\delta_{i\cdot\theta},\] where $\lambda$ denotes Lebesgue measure
(or total length $1$) and $\delta$ denotes a Dirac mass. Writing
$x_i=\mu([0,\{i\cdot\theta\})$ for $i\in\bZ$, we may write \[J_i=[x_i,x_i+\ell_i].\]
It is easy to see that the collection of intervals $\{J_i\}_{i\in\bZ}$ is a
disjoint, circular order preserving collection of compact intervals in $S^1$.
Let $f\in\Diff_+^{1,\alpha}(S^1)$ be an exceptional diffeomorphism, where as before the function $\alpha$ is a concave modulus of continuity.
Let $C$ denote the exceptional minimal set, and let $J\sse S^1\setminus C$ be a maximal open subinterval. We have that for
all $n\in\bZ\setminus \{0\}$, the equality $f^n(J)\cap J=\varnothing$. We let $J=J_0$ and write $J_i=f^i(J)$ for $i\in\bZ$. For compactness
of notation, we write $\ell_i=|J_i|$ for the length of $J_i$.
The following estimate is a crucial in analyzing smooth exceptional diffeomorphisms.
\begin{lem}[Fundamental Estimate;~\cite{KK2020-DCDS}, Lemma 3.1]\label{lem:fund-est}
Let $f$ be as above. Suppose that \[\lim_{n\to\infty}\frac{\ell_{n+1}}{\ell_{n}}=1.\] Then
\[\sup_{n\in\bZ}\frac{1}{\alpha(\ell_n)}\left|1-\frac{\ell_{n+1}}{\ell_n}\right|<\infty.\]
\end{lem}
To unpack the meaning of Lemma~\ref{lem:fund-est}, recall that $\ell_n\longrightarrow 0$ as $n\to\infty$, and similarly
\[\left|1-\frac{\ell_{n+1}}{\ell_n}\right|\longrightarrow 0\] as $n\to\infty$ as the successive ratios of lengths of the $\{J_i\}_{i\in\bZ}$ tends to one.
Thus, in order for $f$ to lie in $\Diff_+^{1,\alpha}(S^1)$ with this simple limiting behavior for lengths of wandering intervals, then
$\alpha(\ell_n)$ must control the size of $(1-\ell_{n+1}/\ell_n)$.
\begin{proof}[Proof of Lemma~\ref{lem:fund-est}]
Let $C$ denote the minimal set of $f$. We claim that the derivative satisfies $f'\equiv 1$ on $C$. Indeed, this follows from the Mean Value
Theorem. For all $n\in\bZ$ there exists $p_n\in J_n$ such that $f'(p_n)=\ell_{n+1}/\ell_n$. By Theorem~\ref{thm:minimal-set} and since
\[\lim_{n\to\pm\infty}\ell_n=0,\]
for an arbitrary $x\in C$, we may extract a subsequence $S_x=S\sse \{p_n\}_{n\in\bZ}$ where $f'\longrightarrow 1$ along $S$ and
$S$ converges to $x$. It follows from the continuity of the derivative that $f'$ is identically $1$ on $C$.
Now, note that if $x\in\partial J_n$ then \[\left|1-\frac{\ell_{n+1}}{\ell_n}\right|=|f'(x)-f'(p_n)|\leq [f']_{\alpha}\cdot\alpha(\ell_n).\] The lemma
now follows.
\end{proof}
Lemma~\ref{lem:fund-est} shows where much of the the difficulty lies in constructing exceptional diffeomorphisms
with a particular modulus of continuity.
In order to build such diffeomorphisms, one needs to find a sequence of finite lengths of finite total length, so that the inequality in
the lemma holds. This may not always be possible. For instance, if $\ell_{i+1}/\ell_i\to 1$ as $i\to\infty$, then one can show that if $f'$ is
$\alpha$--continuous then there is a positive constant $A$ such that $\alpha(\ell_i)\geq A/i$. This shows that one cannot find such an
exceptional diffeomorphism such that $f'$ is Lipshitz, or even $(x(\log 1/x))$--continuous.
See~\cite{KK2020-DCDS} for a more thorough discussion of this point.
The fundamental estimate in Lemma~\ref{lem:fund-est} above can be used in conjunction with Proposition~\ref{prop:herman-1} to produce
many smooth Denjoy counterexamples, once lengths of wandering intervals are carefully chosen. The reader will note that the following
proposition provides a kind of converse to the fundamental estimate.
\begin{prop}[\cite{KK2020-DCDS}, Proposition 5, cf.~\cite{Herman1979} and Chapter 12 of~\cite{KH1995}]\label{prop:herman-2}
Let $\alpha$ be a concave modulus of continuity, and let \[J_k=[x_k,y_k],\, k\in\bZ\] be a sequence of disjoint, circular order preserving intervals
in $S^1$. Suppose furthermore that for all $i\in\bZ$ we have \[\lambda\left( [x_{i-1},x_i]\setminus \bigcup_{k\in\bZ} J_k\right)=
\lambda\left( [x_{i},x_{i+1}]\setminus \bigcup_{k\in\bZ} J_k\right),\] where as before $\lambda$ denotes Lebesgue measure.
Writing $\ell_k=|J_k|$ as before, assume that \[\sup_k\frac{1}{\alpha(\ell_k)}
\left(1-\frac{\ell_{k+1}}{\ell_k}\right)<\infty,\] and that \[\inf_{k}\frac{\ell_{k+1}}{\ell_k}>0.\]
Then there exists an $f\in\Diff^{1,\alpha}_+(S^1)$ such that $f(J_k)=J_{k+1}$ for all $k$.
\end{prop}
We note that the condition that the infimum of successive lengths is bounded away from zero is automatically satisfied if the successive
ratios converge to $1$, for example.
\begin{proof}[Proof of Proposition~\ref{prop:herman-2}]
For $k\in\bZ$, let $\rho_k$ be an arbitrary smooth function on $[0,1]$ satisfying \[\int_{[0,1]}\rho_k\,dx=1,\] and which satisfies
\[1-\left(1-\frac{\ell_{k+1}}{\ell_k}\right)\rho_k(x)>0\] for all $x\in [0,1]$. We will not write down a formula for $\rho_k$, as a construction
of such functions is standard from differential topology. Since \[\inf_k\frac{\ell_{k+1}}{\ell_k}>0\] by assumption, we may assume that
$\rho_k$ is uniformly bounded, independently of $k$, and in fact we may choose a uniform Lipschitz constant.
We write \[g(x)=1-\sum_{k\in\bZ}\left(1-\frac{\ell_{k+1}}{\ell_k}
\rho_k\left(\frac{x-x_k}{\ell_k}\right)\right).\] Note that $g$ is positive by construction.
Integrating, $g$ over $S^1$ results in a telescoping sum that gives a total integral of $1$. The remaining hypotheses of
Proposition~\ref{prop:herman-1} are checked similarly. To get the result via Proposition~\ref{prop:herman-1}, it suffices to
check that $g$ is $\alpha$--continuous.
Note that \[\sup_{x,y\in J_k}\frac{|g(x)-g(y)|}{\alpha(|x-y|)}=1-\frac{\ell_{k+1}}{\ell_k}
\sup_{s,t\in [0,1]}\frac{|\rho_k(s)-\rho_k(t)|}{\alpha(\ell_k|s-t|)}.\] Note that
\[\frac{|\rho_k(s)-\rho_k(t)|}{\alpha(\ell_k|s-t|)}\leq C_k\frac{|s-t|}{\alpha(\ell_k|s-t|)},\] where here $C_k$ denotes the Lipschitz constant of
$\rho_k$. The concavity of $\alpha$ implies that the function $x/\alpha(x)$ is monotone increasing, so we get that
\[C_k\frac{|s-t|}{\alpha(\ell_k|s-t|)}\left(1-\frac{\ell_{k+1}}{\ell_k}\right)\leq \frac{C}{\alpha(\ell_k)}\left(1-\frac{\ell_{k+1}}{\ell_k}\right),\] where $C$
is the supremum of the Lipschitz constants of $\{\rho_k\}_{k\in\bZ}$. This bounds the $\alpha$--norm of $g$ on the intervals $\{J_k\}_{k\in\bZ}$,
and since $g$ is identically $1$ outside of the union of these intervals, we see that $[g]_{\alpha}$ is bounded.
\end{proof}
As a consequence, we have the following:
\begin{cor}\label{cor:herman-3}
Let $\alpha$ be a concave modulus of continuity. Suppose there exists a sequence $\{\ell_k\}_{k\in\bZ}$
of positive real numbers such that \[\sum_{k\in\bZ}\ell_k\leq
1\] and such that \[\sup_k\frac{1}{\alpha(\ell_k)}\left(1-\frac{\ell_{k+1}}{\ell_k}\right)<\infty,\] then there exists an exceptional diffeomorphism
$f\in\Diff_+^{1,\alpha}(S^1)$ with a wandering interval $J\sse S^1$ such that $|f^k(J)|=\ell_k$ for all $k$.
\end{cor}
\subsubsection{Integrability of moduli and exceptional diffeomorphisms}\label{ss:integrability}
We are now ready to prove Theorem~\ref{thm:kk-denjoy}.
\begin{proof}[Proof of Theorem~\ref{thm:kk-denjoy}]
By Propostition~\ref{prop:herman-2} and Corollary~\ref{cor:herman-3},
in order to construct the exceptional diffeomorphism $f$, it suffices to use the integrability of $\alpha$ near zero to construct
suitable lengths. We will use the modulus $\alpha$ to construct the lengths of the wandering intervals as functions of their index.
By Lemma~\ref{lem:modulus-existence}, we may assume that $\alpha$ is a smooth function. Write
\[K=\max\{2,1/\alpha(1)\}\quad \textrm{and}\quad v(x)=x^2\alpha(1/x).\]
Observe that for $t\geq 1$ we have $v(x/t)\geq v(x)/t^2$. Indeed, we compute:
\[v(x/t)=(x^2/t^2)\alpha(t/x)\geq (x^2/t^2)\alpha(1/x)=v(x)/t^2.\] Since $\alpha$ is concave, we have $x/\alpha(x)$ is monotone increasing, and
therefore \[x\alpha(1/x)=\frac{\alpha(1/x)}{1/x}\] is also monotone increasing and hence has a nonnegative derivative.
Therefore, we can compute \[(v(x)/x)'=\left(\frac{\alpha(1/x)}{1/x}\right)'\geq 0.\] It follows that when $x\geq K$ then \[v(x)\geq x\cdot v(1)\geq
x/K.\]
We also have that $\alpha(x)/x$ has nonpositive derivative, so that \[\left(\frac{\alpha(x)}{x}\right)'=\frac{x\alpha(x)'-\alpha(x)}{x^2}\leq 0.\]
It follows that \[\alpha(x)\geq x\alpha'(x)>0\] for all $x$, and so that in particular \[v(x)=x^2\alpha(1/x)\geq x\alpha'(1/x)>0.\]
We combine these observations to see that
\[xv'(x)=x(2x\alpha(1/x)-\alpha'(1/x))=2v(x)-x\alpha'(1/x)\in [v(x),2v(x)].\]
Armed with these estimates, we can define the lengths of the wandering intervals. We write \[\ell_k=\frac{1}{v(|k|+K)}\] for $k\in\bZ$. The
integrability assumption on $\alpha$ is precisely what allows us to choose such lengths to be finite in total. Namely, we have
\[\int_K^{\infty}v(x)\,dx=\int_0^{1/K}\frac{dx}{\alpha(x)}<\infty.\] So, we may set \[\sum_{k\in\bZ}\ell_k\leq 1,\] at the cost of possibly
increasing the value of $K$.
To complete the construction of $f$, we need only to estimate the size of \[\frac{1}{\alpha(\ell_k)}\left(1-\frac{\ell_{k+1}}{\ell_k}\right),\]
and show that it is bounded independently of $k$.
For $k\in \bZ$, we will write $j=|k|$ for ease of notation. We have shown that \[v(j+K)\geq \frac{j+K}{K},\] and observe that
\[v(j+K\pm 1)\geq v\left(\frac{j+K}{2}\right)\geq\frac{v(j+K)}{4}.\] The second inequality follows from the inequality $v(x/t)\geq v(x)/t^2$ that
holds for $t\geq 1$, as we have already established.
We may now apply the Mean Value Theorem to a suitable point \[y\in [j+K,j+K+1]\quad \textrm{or} \quad y\in [j+K-1,j+K].\] We have
\begin{align*}
&\frac{1}{\alpha(\ell_j)}\left|1-\frac{\ell_{j+1}}{\ell_j}\right|=\frac{1}{\ell_j^2v(1/\ell_j)}\left|1-\frac{v(j+K)}{v(j+K\pm 1)}\right|\\
&=\frac{v(j+K)^2}{v\circ v(j+K)}\cdot \frac{v'(y)}{v(j+K\pm1)}=\frac{v(j+K)}{v\circ v(j+K)}\cdot\frac{v(j+K)}{v(j+K\pm 1)}\cdot v'(y)\\
&\leq \frac{j/K+1}{v(j/K+1)}\cdot 4\cdot\frac{2v(y)}{y}\leq \frac{j/K+1}{v(2j+2K)/(2K)^2}\cdot\frac{8v(y)}{y}\\
&=32K\cdot \frac{j+K}{y}\cdot\frac{v(y)}{v(2j+2K)}\leq 64K.
\end{align*}
This furnishes a uniform bound for \[\frac{1}{\alpha(\ell_k)}\left(1-\frac{\ell_{k+1}}{\ell_k}\right),\] independently of $k$ and implies the
existence of the claimed exceptional diffeomorphism.
To complete the proof of the result, we will show that $f$ can be chosen as close to the rotation $R_{\theta}$ as we like, in the $C^1$
topology. To see that $f$ can be chosen to be $C^0$--close to $R_{\theta}$, we may simply increase the cutoff $K$ so that
\[\sum_{k\in\bZ}\ell_k\approx 0.\]
Achieving this, we have that for each $\eps>0$, we may choose a cutoff so that
\[\sup_{x\in S^1}|f(x)-R_{\theta}(x)|<\epsilon.\]
To get $f$ to be $C^1$--close to $R_{\theta}$, we need to show that for each $\eps>0$ we may arrange \[\sup_{x\in S^1}|f'(x)-1|<\eps.\]
We have that $f'(x)-1$ is identically zero outside of the wandering intervals, and using the construction of $f$ from
Proposition~\ref{prop:herman-2}, we see that on the component $J_k$ wandering set, the function $f'(x)-1$ is simply given by
\[\left(1-\frac{\ell_{k+1}}{\ell_k}\right)\rho_k\left(\frac{x-x_k}{\ell_k}\right).\]
The function $\rho_k$ is bounded independently of $k$. We have that $\ell_{k+1}/\ell_k\to 1$ as $k\to\infty$, and so by setting the cutoff
$K$ to be sufficiently large, we may arrange for $|f'(x)-1|<\eps$ on $J_k$, which completes the proof.
\end{proof}
\subsubsection{On the lengths of wandering intervals}
We end this chapter with a remark on an open problem in the theory of exceptional diffeomorphisms. If $f\in\Diff^1_+(S^1)$ is an exceptional
diffeomorphism with minimal set $C$, it might not generally be the case that $f'\equiv 1$ on $C$. That is, the estimate in
Lemma~\ref{lem:fund-est} might fail. In that case, it is not true that $\ell_{k+1}/\ell_k\to 1$ for successive wandering intervals, as $k\to\infty$.
Without
this assumption, the construction of exceptional diffeomorphisms we have carried out here does not work at all.
There is another angle from which to probe lengths of wandering intervals for exceptional diffeomorphisms. Consider the set
$\{\ell_k\}_{k\in\bZ}$, and reorder it according to decreasing lengths. Thus, we get a sequence $\{\lambda_i\}_{i\ge1}$ of real numbers,
where $\lambda_i$ is the $i^{th}$ largest of the real numbers $\{\ell_k\}_{k\in\bZ}$. This sequence has very little to do with the original
order of the lengths $\{\ell_k\}_{k\in\bZ}$. It is a well-known open question in the field as to whether or not $\lambda_{i+1}/\lambda_i\to 1$
as $i\to\infty$. This question was originally posed by D.~McDuff, and she was able to prove that $1$ is an accumulation point of successive
quotients of lengths, and that the quotients are bounded independently of $i$. The reader is directed
to~\cite{McDuff1981,athanassopoulos,KK2020-DCDS} for a more detailed discussion.
\chapter{Full diffeomorphism groups determine the diffeomorphism class of a manifold}\label{sec:filip-tak}
\begin{abstract}In this chapter, we establish that the algebraic structure of the full $C^p$ diffeomorphism group of a smooth connected boundaryless manifold can recover the regularity $p$, and the manifold up to $C^p$ diffeomorphisms.
This was proved by Filipkiewicz in 1980s, following a preceding, weaker result of Takens. Filipkiewicz's proof uses a strategy pursued by Whittaker,
and conclude
by the Bochner--Montgomery Theorem regarding Hilbert's Fifth Problem.
Instead of using Whittaker's method, we give a proof using a powerful reconstruction theorem due to Rubin. We also give a generalization of Filipkiewicz original approach as an alternative proof. This chapter is mostly self-contained, containing complete proofs of Bochner--Montgomery Theorem and Rubin's Theorem.\end{abstract}
In this chapter we leave the world of one-manifolds and discuss general diffeomorphism groups of manifolds, before returning
to one--manifolds in the next chapter. We include this discussion now because it gives a natural context to introduce several tools
which will be useful in later chapters.
One of the main goals of this monograph will be to give a self--contained proof that the isomorphism types of
finitely generated subgroups
of $\Diff_+^p(M)$ determine both $M$ and $r$, at least in the case where $M$ is a $1$--manifold. It is therefore natural to wonder
the degree to which this result generalizes to higher dimensional manifolds. We will begin with some musings about critical regularity of
finitely generated groups acting on manifolds in dimension two or more. The bulk of this chapter will be devoted to results of F.~Takens and
R.~Filipkiewicz, which show that for general manifolds, the isomorphism type of the group $\Diff_c^p(M)_0$, where here $1\leq p\leq\infty$,
determines the manifold $M$ up to $C^p$--diffeomorphism.
We make a standing assumption throughout this book that, unless explicitly
noted to the contrary,\emph{ all manifolds are Hausdorff and second countable, hence admit partitions of unity.}
We will also let $M$ denote a smooth connected boundaryless manifold of dimension $d$.
We will recall the original proofs that were given by Takens and Filipkiewicz in their original papers. Filipkiewicz's paper is not self-contained,
requiring background results in the algebraic structure of diffeomorphism groups, together with a subtle regularity result due to
Bochner--Montgomery. We will include a complete account of the simplicity of commutator subgroups of diffeomorphism groups of manifolds,
and a complete proof of the Bochner--Montgomery Theorem. From the Bochner--Montgomery Theorem, we will be able to prove
Takens' Theorem, and we will give Takens' proof in one dimension. To prove Filipkiewicz's Theorem, we will give a self--contained account
of Rubin's Theorem, which is an extremely powerful reconstruction result for objects with sufficiently complicated groups of automorphisms.
We will retain a polished version of Filipkiewicz's result as a matter
of historical record and to provide his perspective on this flavor of reconstruction result.
\section{Diffeomorphism groups of general manifolds and critical regularity}
For a $C^p$ manifold $M$ with $p\in\bZ_{>0}\cup\{0,\infty\}$, we let
$\Diff^p(M)$ denote the group of $C^p$--diffeomorphisms of $M$.
The group of compactly supported diffeomorphisms in $\Diff^p(M)$ is denoted as
$\Diff^p_c(M)$.
If $f\in\Diff^p_c(M)$ is $C^p$--diffeotopic to the identity via a diffeotopy supported in a compact set, then we write $f\in\Diff^p_c(M)_0$.
All these groups are equipped with the Whitney $C^p$--topology~\cite{Hirsch1994}, although we will essentially care only about its $C^1$--topology.
\begin{rem}\label{rem:diffeotopy}
The group $\Diff_c^p(M)_0$ coincides with the identity component of $\Diff_c^p(M)$, although this fact does not play a role for our purpose of this chapter. See~\cite[Corollary 1.2.2]{Banyaga1997} for a proof.\end{rem}
The ultimate goal of this chapter is to prove the following result.
\begin{thm}[Generalized Takens--Filipkiewicz Theorem]\label{thm:filip-gen}
Let $M$ and $N$ be smooth connected boundaryless manifolds, and let $p,q\in\bZ_{>0}\cup\{0,\infty\}$.
If there exists a group isomorphism
\[ \Phi\co G\longrightarrow H\]
for some groups $G,H$ satisfying
\[
\Diff_c^{p}(M)_0\le G\le \Diff^{p}(M)\]
and
\[
\Diff_c^{q}(N)_0\le H\le \Diff^{q}(N)\]
then we have that
$p=q$
and that
there exists a $C^p$ diffeomorphism
$w\colon M\longrightarrow N$ satisfying
\[\Phi(g)=w\circ g\circ w^{-1}\]
for all $g\in G$.\end{thm}
The case $p=q=0$ of the above theorem is due to Whittaker~\cite{Whittaker1963}.
Takens~\cite{Takens79} originally proved the theorem when $p=q=\infty$
and when $G$ and $H$ are full $C^\infty$ diffeomorphism groups of respective manifolds, with an additional assumption that such a
map $w$ is already given as a bijection, and not \emph{a priori} by a
diffeomorphismi. Filipkiewicz extended this to the theorem below, which is also a special case of Theorem~\ref{thm:filip-gen}~\cite{Filip82}.
\begin{thm}[Filipkiewicz's Theorem]\label{thm:filip}
If $p,q\in\bZ_{>0}\cup\{\infty\}$ and if
\[\Phi\co\Diff^p(M)\longrightarrow\Diff^q(N)\] is a group isomorphism,
then we have that $p=q$ and that there exists a $C^p$ diffeomorphism
$w\colon M\longrightarrow N$ satisfying
\[\Phi(g)=w\circ g\circ w^{-1}\]
for all $g\in \Diff^p(M)$.
\end{thm}
We end this introduction by formulating a conjecture that is natural in light of the previous remarks (cf.~Question~\ref{que:subgroup-main}
from the introduction).
\begin{con}\label{con:critreg-hd}
Let $M$ and $N$ be compact connected $C^p$ manifolds.
Then $M$ and $N$ are $C^p$--diffeomorphic if and only if the
set of isomorphism classes of finitely generated subgroups
of $\Diff^p_c(M)_0$ and $\Diff^p_c(N)_0$ coincide.
\end{con}
We remark that, as discussed in the introduction,
homeomorphisms in dimension one are controlled by the fact that, away from fixed points, they always have a well--defined
direction. This structure imposed by the ambient topology makes many of the arguments made in this book possible. In higher dimension,
such structure is lost, and behavior of diffeomorphism groups of even two--dimensional manifolds is much more complicated than the
one--dimensional case.
Conjecture~\ref{con:critreg-hd}, if correct, has the potential to yield a fresh perspective on classical questions in differential topology,
especially relating to exotic differentiable structures, and can give reformulations of these problems in terms of combinatorial group theory.
Conjecture~\ref{con:critreg-hd} in its stated form is wide open. As such, we will not comment on it any further here.
\section{Generalities from differential topology}
In this section, let us collect standard facts on manifolds and their diffeomorphism groups. Two main results to be
established are the existence of a compactly supported smooth group action of $\bR^d$ on $M$,
and the fragmentation of $C^p$ diffeomorphisms on $M$. Canonical references include~\cite{Hirsch1994},
~\cite{deRham1984} and~\cite{PS1970}.
\subsection{Open balls in manifolds}\label{s:balls}
Let $g$ be a homeomorphism of a topological space $X$.
To avoid confusion with the existing literature, we use a notation
\begin{align*}
\suppo g &:=\supp g=X\setminus \Fix g,\\
\suppc g&:=\overline{\supp g}.
\end{align*}
\begin{rem}
The latter of the above is called the \emph{closed support}\index{closed support} of $g$,
which is often denoted simply as $\supp g$ in the literature of differential topology. Though, some authors reserve the
notation $\supp g$ for the \emph{open support}\index{open support} $\suppo g$, as we have done in other chapters of this book.
\end{rem}
Let $X$ be a topological space, and let $G\le\Homeo(X)$ be a subgroup.
We write $G_c$ the subgroup of $G$ consisting of compactly supported homeomorphism groups.
When $G$ is given with a topology a priori,
we let $G_0$ denote the identity component of $G$.
Consequently, $G_{c0}$ is interpreted as $(G_c)_0$.
Let $M$ be a smooth connected boundaryless manifold of dimension $d$.
In the case when $X=M$
and when $G=\Diff^p(M)$, the group of $C^p$ diffeomorphisms of $M$,
we use the notation $\Diff_c^p(M)$, $\Diff^p(M)_0 $ and $\Diff_c^p(M)_0$ slightly changing the positions of the subscripts.
We let $B^d(a;r)\sse\bR^d$ denote the open ball with radius $r$ centered at $a\in\bR^d$.
It will be useful for us to have an open basis of a manifold $M$ that is invariant under $\Diff^p(M)$, defined by the following open sets.
\bd\label{defn:ball}
For a manifold $M$ of dimension $d$, we say an open set $U\sse M$ is a \emph{$C^p$--open ball}\index{$C^p$--open ball}
if there exists a $C^p$--embedding
\[ h\co \bR^d\longrightarrow M\] satisfying that $U=h(B^d(0;1))$.
\ed
We denote by $\BB^p(M)$ the collection of all the $C^p$ open balls in $M$.
We simply say $U\in\BB^p(M)$ is a \emph{$C^p$--open ball}, or simply an \emph{open ball} if the meaning is clear.
Let $G$ be a group acting on a set $X$. Then $G$ induces the \emph{pseudo-group topology}\index{pseudo-group topology}
on $X$, for which the basic open sets are
subsets $U\sse X$ for which $U=\supp g$ for some $g\in G$.
The lemma below shows that the affine action of $\bR^d$ on the Euclidean space can be
conjugated into the compactly supported smooth diffeomorphism group of $M$.
This lemma implies that the original Euclidean topology of $M$, the pseudo-group topology coming from
$\Diff^p_c(M)_0$, and the topology generated by $\BB^p(M)$ all coincide.
In particular, the collection $\BB^p(M)$ is a $\Diff^p(M)$--invariant open basis of $M$.
\begin{lem}[\cite{deRham1984}]\label{lem:affine}
Let $d\ge1$.
Then there exists a smooth diffeomorphism
\[
h\co B^d(0;1)\longrightarrow\bR^d\]
such that for each fixed $v\in\bR^d$ the map
\[
a_v(x):=\begin{cases}
h^{-1}\left( v+h(x)\right), &\text{ if }x\in B^d(0;1),\\
x,& \text{ otherwise.}\end{cases}\]
is a smooth diffeomorphism of $\bR^d$.
In particular, the map
\[v.x:=a_v(x)\]
defines a smooth group action of $v\in \bR^d$ on $x\in \bR^d$ supported in $B^d(0;1)$, which is conjugate to the translation action $x\mapsto x+v$.
\end{lem}
\bp
We will provide only the recipe of the map $h$ as the computational verification is elementary and worked out in~\cite[Section 15]{deRham1984};
one can also give an alternative proof that uses Theorem~\ref{thm:bs-cpt}.
Pick an arbitrary smooth diffeomorphism
\[
\phi\co [0,1)\longrightarrow[0,\infty)\]
such that $\phi(x)=x$ for $x\in(0,1/3]$
and such that
\[
\phi(x)=e^{1/(1-x)^2}\]
for $x\in[2/3,1)$.
Define $h\co B^d(0;1)\longrightarrow\bR^d$ by
\[
h(x):= \phi(|x|)\cdot \frac{x}{|x|}\]
and $h(0)=0$. If $X\co\bR^d\longrightarrow\bR^d$ is the constant vector field having the value $v$, then one can check that the pull back
\[
h^*(X)(p):=(D_p h)^{-1} X\circ h(p)\]
extends to a smooth vector field on $\bR^d$ supported in $B^d(0;1)$.
\ep
Let us now consider a general context of topological spaces.
Let $X$ be a topological space.
If $G\le\Homeo(X)$ and if $U\sse X$, then we let $G_U$ be the elements $g\in G$ such that \[\suppc g\sse U.\]
\bd\label{defn:lt-ld}
We say a group $G$ of homeomorphisms on a space $X$ is \emph{locally transitive (LT)}\index{locally transitive} if every point in
$X$ has a local basis consisting of open sets $U$ such that the group $G_U$ acts transitively on $U$.\ed
In other words, we require that for each pair $x,y\in U$ there exists $g\in G$ such that $\suppc g\sse U$ and such that $g(x)=y$.
It is trivial that if $G$ acts locally transitively on $X$ then the pseudo-group topology recovers the original topology on $X$.
Lemma~\ref{lem:affine} gives a rigorous proof of the following, which is somewhat obvious anyway from the intuition.
\begin{lem}\label{lem:ltld-diff}
An arbitrary subgroup of $\Homeo(M)$ containing $\Diff_c^\infty(M)_0$ is locally transitive.\end{lem}
\subsection{Fragmentation of diffeomorphisms}\label{ss:frag}
We say a group $G\le\Homeo(X)$ is \emph{fragmented}\index{fragmentation} if for every open cover $\UU$ of $X$, we have that
\[G=\form{ G_U\mid U\in\UU}.\]
As the reader might have guessed, we wish to show that $\Diff^p_c(M)_0$ is fragmented. This fact can be traced at least back to M.~Hirsch.
\begin{lem}\label{lem:frag}
If $p\in\bZ_{>0}\cup\{\infty\}$, then the group $\Diff^p_c(M)_0$ is fragmented.
\end{lem}
\begin{proof}[cf.~\cite{PS1970,Banyaga1997}]
Let $\UU$ be an arbitrary cover of $M$.
Pick an element $g\in\Diff_c^p(M)_0$, which is isotopic to the identity via diffeomorphisms $\{g_t\}_{t\in[0,1]}$
supported in a compact set $K\sse M$.
Since every topological group is generated by an arbitrary neighborhood of the identity, it suffices for us to find a
$C^1$--neighborhood $V\sse\Diff_K^p(M)_0$ of the identity that consists of fragmented diffeomorphisms;
it will then follow that $g$ is fragmented.
We have some finite cover \[\{U_1,\ldots,U_m\}\sse \UU\] of $K$.
Since $M$ is paracompact, we have a partition of unity $\{\phi_i\}_{1\leq i\leq m}$ subordinate to the cover $\{U_1,\ldots,U_m\}$ of $K$.
We set $\phi_0=0$ and
\[\omega_j(x)=\sum_{i=0}^j \phi_i(x).\]
Note that $\omega_m(x)=1$ on $K$ and $\omega_i(x)=\omega_{i-1}(x)$ if $i\ge1$ and $x\in M\setminus U_i$.
Assuming we have some identity neighborhood $V\sse \Diff^p_K(M)_0$ at hand, pick $h\in V$ and a $C^p$--diffeotopy \[\{h_t\}_{t\in[0,1]}
\sse\Diff^p_K(M)_0;\] see Remark~\ref{rem:diffeotopy} or the paragraph below.
For $i=1,\ldots,m$ we have $C^p$ maps
\[h_i(x):=h_{\omega_i(x)}(x)\]
that are homotopic to the identity.
The set $\Diff^p_K(M)$ can be included in the set of all $C^p$
functions from $M$ to itself
\[C^p_K(M,M)\]
that are the identity outside $K$; see~\cite{Hirsch1994}.
Moreover, $\Diff^p_K(M)$ is defined by $C^1$--open conditions
in $C^p_K(M,M)$.
If $V$ is small enough a priori then one can require that each $h_i$ is $C^1$--close enough to the identity after defining a priori
\[
h_t(x)=\exp_x (t \exp^{-1}_x(h(x)))\]
by the Riemannian exponential map. Hence, each $h_i$ is a $C^p$--diffeomorphism.
Observe that $h_{i-1}(x)=h_i(x)\in\Diff_K^p(M)_0$
whenever $x\notin U_i$. It follows that $h_{i-1}^{-1}h_i$ is the identity outside of $U_i$. Finally, \[h=(h_0^{-1}h_1)(h_1^{-1}h_2)\cdots
(h_{m-1}^{-1}h_m),\] which furnishes the desired fragmentation of $h\in V$.
\end{proof}
\section{Simplicity of commutator subgroups}
The following is a well-known result due essentially to D.~B.~A. Epstein, which will be required in the proof of Filipkiewicz's Theorem and
which we will prove here for the sake of completeness.
We follow the general outline of the authors' expository work with J.~Chang in~\cite{CKK2019},
based on the simplification of Epstein's method by W.~Ling~\cite{Ling1984}.
\begin{thm}[Epstein~\cite{Epstein1970}]\label{thm:epstein}
If $M$ is a smooth connected boundaryless manifold
and if $p\in\bZ_{>0}\cup\{\infty\}$,
then the group $[\Diff^p_c(M)_0,\Diff^p_c(M)_0]$ is simple.
\end{thm}
In fact, one can strengthen Theorem~\ref{thm:epstein} in various contexts to diffeomorphism groups with non-integral regularities.
See~\cite{CKK2019} for a self-contained account, for instance.
It is a crucial point in Theorem~\ref{thm:epstein} to restrict to the connected component of the identity of $\Diff^p(M)$. For instance,
if $\Sigma$ is a closed orientable surface of genus at least two then $[\Diff^p(\Sigma),\Diff^p(\Sigma)]$ admits an infinite discrete group
as a quotient; indeed the group $\Diff^p(\Sigma)/\Diff^p_0(\Sigma)$ is the group of isotopy classes of $C^p$ diffeomorphisms of
$\Sigma$, which is identified with the mapping class group $\Mod(\Sigma)$. This latter group has finite abelianization, which implies that
$[\Diff^p(\Sigma),\Diff^p(\Sigma)]$ surjects to a finite index subgroup of $\Mod(\Sigma)$.
There are two main ideas for the proof of Theorem~\ref{thm:epstein}.
The first is Higman's Theorem, which is a very useful result for studying the simplicity of
groups that act on a set in a highly transitive manner.
In order to apply techniques in the vein of Higman's Theorem
to groups of diffeomorphisms, one needs to be able to generate the full group by elements which are supported on sufficiently
small subsets of the manifold, which is where the technology of fragmentation comes into play; this will be the second
main idea of this section.
\subsection{Higman's Theorem}
Higman's Theorem has applications
beyond diffeomorphism groups, and is a standard tool in the study of groups of piecewise linear homeomorphisms. In the authors' opinion,
it is not nearly as well known as it should be, and so we will include a complete discussion of it.
Recall our convention that for an element $g$ in a permutation group $\Sym(X)$ of a set $X$, we denote its
\emph{support}\index{support of a permutation} by
\[\supp g=X\setminus\Fix g.\]
In a later subsection when $X$ is equipped with a topology, we will make a distinction between $\supp g$ and its closure.
\begin{thm}[Higman's Theorem]\label{thm:higman}
Let $G$ be a nontrivial group acting faithfully on a set $X$. Suppose that for all triples $\{r,s,t\}$ of nonidentity elements of $G$, there is an element
$g\in G$ such that \[g(\supp r\cup \supp s)\cap tg(\supp r\cup\supp s)=\varnothing.\] Then:
\begin{enumerate}[(1)]
\item
The commutator subgroup $G'=[G,G]$ is nonabelian and simple.
\item
Every proper quotient of $G$ is abelian.
\item
The center of $G$ is trivial.
\end{enumerate}
\end{thm}
Generally speaking, if $G$ satisfies the hypotheses of Higman's Theorem then $G'$ could be an infinitely generated simple group,
even if $G$ happens to be finitely generated.
Higman's Theorem is tailor-made to investigate certain groups of homeomorphisms of topological objects. The moment that for all
nontrivial $t\in G$ there
always exists a group element $g\in G$ that shrinks and moves the total supports of two nontrivial elements of $G$ so that $t$ moves the
shrunken and translated supports of themselves, then $G$ satisfies the conclusions of Higman's Theorem.
Higman's Theorem is sufficient to prove Theorem~\ref{thm:epstein} for
Euclidean spaces (see Remark~\ref{rem:euclidean})
although one could face
some issues which have to do with the global topology in the case of a general manifold $M$.
For two group elements $t$ and $g$ we write as usual
\[
t^g:=g^{-1}tg.\]
\begin{proof}[Proof of Theorem~\ref{thm:higman}]
We prove a sequence of claims which will allow us to then easily obtain the conclusion of the theorem.
\begin{claim}
The commutator subgroup $G'$ is nonabelian.\end{claim}
First note that $G$ itself must be nonabelian.
If $G$ were abelian, we set $r=s=t$ to be
nontrivial elements of $G$. For each $g\in G$
we have
\[
g(\supp r\cup \supp s)\cap tg(\supp r\cup\supp s)
= g\supp r\cap gt\supp r=g\supp r\ne\varnothing.\]
This violates the hypothesis on $G$.
To see $G'$ is nonabelian, we now choose $r=s=t$ to be nontrivial elements of $G'$ and let $g\in G$ be a corresponding element from the hypothesis.
For each $x\in g\supp r=\supp grg^{-1}$, we have that
\[x\not\in rg\supp r=\supp \left(rgr(rg)^{-1}\right).\]
It follows that
\[
x= rgr(rg)^{-1}(x)\ne rgrg^{-1}r^{-1} \left(gr^{-1}g^{-1}(x)\right).\]
This implies that $r$ and $grg^{-1}$ are non-commuting elements in $G'$,
proving the claim.
\begin{claim}
For all nontrivial $\{r,s,t\}\sse G$, there is an element $g\in G$ such that \[\left[r,t^gs(t^g)^{-1}\right]=1.\]\end{claim}
By applying $g^{-1}$ to the hypothesis of Higman's Theorem,
we see that there is an element $g\in G$ satisfying \[\supp r\cap t^g(\supp s)=\varnothing,\]
so that $r$ and $t^gs(t^g)^{-1}$ have disjoint supports. This proves the claim.
\begin{claim}If $T$ is a nontrivial normal subgroup of $G$ then $T$ contains $G'$.\end{claim}
Let $t$ be a nontrivial element of $T$, and let $r,s\in G$ be arbitrary and nontrivial. By the second claim, there is an element $g\in G$ such that
$(t^g)^{-1}r t^g$ and $s$ commute. It follows that
\[[r,s]=[r\cdot (t^g)^{-1}r^{-1}t^g,s].\]
Since the right hand side is a product of conjugates of $t^{\pm1}$ and $t^{-1}$,
we deduce that $[r,s]\in T$. It follows that
$G'\le T$, as claimed.
From the first and the third claims, we readily see that $G''=G'$ and that every proper quotient of $G$ is abelian.
One can also promote the given hypothesis to further require that $g$ belongs to the commutator subgroup.
\begin{claim}For all nontrivial $\{r,s,t\}\sse G$, there exists a $u\in G'$ such that
\[u(\supp r\cup \supp s)\cap t(u(\supp r\cup\supp s))=\varnothing.\]
\end{claim}
Let $g\in G$ be as in the hypothesis of the theorem for the triple $\{r,s,t\}$. If $g\in G'$ we are done, so we may assume $g\notin G'$,
and so $g$ is nontrivial. Applying the second claim for the triple
$(t,g,t)$, we have an element $h\in G$ such
that \[[t,{t^h}g(t^h)^{-1}]=1.\] Set $u:=[t^h,g^{-1}]\in G'$.
Since \[[t,ug^{-1}]=[t,t^hg^{-1}(t^h)^{-1}]=1,\] we have that
$t^u=t^g$.
It follows that $u$ gives us a suitable element of $G'$ since
\[(\supp r\cup \supp s)\cap t^u(\supp r\cup\supp s)
=(\supp r\cup \supp s)\cap t^g(\supp r\cup\supp s)=
\varnothing.\]
This proves the claim.
We can now complete the proof of the theorem. Let $Z$ be the center of $G$. We have that $Z$ is an abelian and normal subgroup of $G$.
By the third claim, if $Z$ is nontrivial then $Z$ contains $G'$. This latter group is nonabelian by the first claim, so $Z$ is indeed trivial.
It only remains to show that $G'$ is simple.
Suppose that $T$ is a nontrivial normal subgroup of $G'$, and let $t\in T$ be a nontrivial element. If $r,s\in G'$ are
arbitrary nontrivial elements, then the last claim shows that for some $u\in G'$ we have \[[r,{t^u}s(t^u)^{-1}]=1.\]
Applying the same trick as in the third claim,
we see that \[[r,s]=[r\cdot (t^u)^{-1}rt^u,s]\in T.\] It follows then that
\[G''=[G',G']\le T.\] As we have seen $G'=G''$, we conclude $T=G'$. This proves the simplicity of $G'$.
\end{proof}
Higman's Theorem is formulated in an essentially combinatorial manner. For us, it is useful to recast it in a topological manner,
and in the process making more precise the remarks following the statement of Theorem~\ref{thm:higman}. The
following definition appears in the authors' work with Lodha~\cite{KKL2019ASENS} (cf.~\cite{CKK2019}).
\begin{defn}
Let $X$ be a topological space and let $G\le \Homeo(X)$ be a subgroup. We say that $G$ acts
\emph{compact-open--transitively}\index{compact-open--transitive}, or
\emph{CO--transitively}\index{CO--transitive},
if for each proper compact subset $A\sse X$ and for each nonempty open $B\sse X$, there is an element
$g\in G$ such that $g(A)\sse B$.
\end{defn}
In the case where $X=S^1$, being CO--transitive is equivalent to being
\emph{minimal and strongly proximal}\index{minimal and strongly proximal} as defined in~\cite{BF2014ICM}.
\begin{lem}[See~\cite{KKL2019ASENS}]\label{lem:co-trans}
Let $G$ be a group of compactly supported homeomorphisms acting on a noncompact Hausdorff topological space $X$. If $G$ acts
CO--transitively on $X$ then $G'$ is nonabelian and simple, every proper quotient of $G$ is abelian, and the center of $G$ is trivial.
\end{lem}
\begin{proof}
Since $X$ is noncompact and Hausdorff, it follows easily that $G$ is nontrivial. To apply Theorem~\ref{thm:higman}, let $\{r,s,t\}$ be
nontrivial elements of $G$. We may find a nonempty open set $U\sse X$ such that $t(U)\cap U=\varnothing$, since $X$ is
Hausdorff. Let $A$ be a compact set such that \[\supp r\cup\supp s\sse A.\] Since the action of $G$ is CO--transitive, we know that
there is an element $g\in G$ such that $g(A)\sse U$. As $g(A)\cap t(g(A))=\varnothing$, we obtain
\[g(\supp r\cup\supp s)\cap tg(\supp r\cup\supp s)=\varnothing.\] Thus, the hypotheses of Theorem~\ref{thm:higman} are satisfied and
the conclusion of the lemma follows.
\end{proof}
\begin{rem}\label{rem:euclidean}
The property of being CO--transitive transfers from a subgroup to a supergroup.
Since the action of $\Diff^\infty_c(\bR^d)_0$ on $\bR^d$ is CO--transitive,
and so are the actions of the groups in the family
\[
\left\{
\Diff^k_c(\bR^d)_0,\Diff^k_c(\bR^d)\right\}
\]
for all $k\in\bN\cup\{\infty\}$. It follows that
$[G,G]$ is simple for such a group $G$ in the family.
Actually, the group
\[\Diff^k_c(\bR^d)_0\]
is known to be simple for $k\ne d+1$, while the case $k=d+1$ is still open at the time of this writing (cf.~\cite{Mather1,Mather2}).
\end{rem}
\subsection{The Epstein--Ling Theorem}\label{sss:methods}
As remarked above, in order to apply a result like Higman's Theorem to general diffeomorphism groups of manifolds,
we need to be able to write
a diffeomorphism as a product of ones which have small support. When the manifold in question is a Euclidean space then the topology
of the manifold does not get in the way of CO--transitivity. For a general manifold, one needs to be more careful.
Recall that a Hausdorff topological space $X$ is \emph{paracompact}\index{paracompact}
if for every open cover $\VV$ of $X$, there is a refinement
$\UU$ such that for all $A,B\in\UU$, either $A\cap B=\emptyset$ or there is a third $C\in\VV$ such that $A\cup B\sse C$. We will
always assume that manifolds are paracompact. This kind of refinement is sometimes called an \emph{open star refinement}\index{open
star refinement} of $\VV$.
Regarding the simplicity of the commutator subgroups of certain homeomorphism groups,
the following topological property played a key role in Epstein's work~\cite{Epstein1970} and in Ling's simplification~\cite{Ling1984}.
\bd
Let $X$ be a topological space.
We say a group $G\le \Homeo(X)$ is \emph{transitive--inclusive}\index{transitive--inclusive}
if there exists a basis of open sets $\BB$ for the topology of $X$
such that for all basic open sets $U,V\in\BB$
some element $g\in G$ satisfies $g(U)\sse V$.\ed
\begin{thm}[Epstein--Ling~\cite{Epstein1970,Ling1984}]\label{thm:trans-i}
If $X$ is a paracompact space and if a subgroup $G$ of $\Homeo(X)$ is fragmented and transitive--inclusive, then
$[G,G]$ is simple.
\end{thm}
The reader will note the strong resemblance between Theorem~\ref{thm:trans-i} and Lemma \ref{lem:co-trans}, which is not accidental.
Let us give a proof of the above theorem in the remainder of this section.
For this, we let $X$ and $G$ be as in Theorem~\ref{thm:trans-i},
and let $\BB$ be an open basis of $X$ with respect to which $G$ is transitive--inclusive.
We will make an additional assumption that $G$ is nonabelian, without loss of generality.
In the sequel, for a group $G$, we will write $G^{(n)}$ for the $n^{th}$ term of the \emph{derived series}\index{derived series}
of $G$. Thus, $G=G^{0}$,
and $G^{(k+1)}=[G^{(k)},G^{(k)}]$.
\begin{lem}[\cite{CKK2019}, Lemma 3.13]\label{lem:derived}
If $x\in X$, if $n\geq 0$ is an integer, and if $U\sse X$ is an
open neighborhood of $x$ then there is an element $g\in G_U^{(n)}$ such that $g(x)\neq x$.
\end{lem}
Lemma~\ref{lem:derived} in particular implies that $G_U$ cannot be solvable.
\begin{proof}[Proof of Lemma~\ref{lem:derived}]
Since $G$ is nonabelian, $X$ has at least three points. Since $X$ is Hausdorff and transitive--inclusive, the action of $G$ on $X$ does not
have a global fixed point. It follows that there is an element $g\in G$ such that $g(x)\neq x$. The open sets $\{U,X\setminus \{x\}\}$ form an
open cover of $X$, and so we may fragment $g$ with respect to this cover. It follows immediately that there is an element $g_U\in G_U$
such that $g_U(x)\neq x$.
By induction, suppose that we have shown that there exists an element $g\in G_U^{(n-1)}$ that does not fix $x$. There is a neighborhood
$V\sse U$ of $x$ such that $g(V)\cap V=\varnothing$, since $X$ is Hausdorff. Observe that by induction there is an element
$h\in G_V^{(n-1)}$ such that $h(x)\neq x$. Note that \[g(h(x))\neq g(x)=h(g(x)).\] The product $(ghg^{-1})h^{-1}$ is therefore nontrivial
and lies in $G_U^{(n)}$, which establishes the lemma.
\end{proof}
\begin{lem}[\cite{CKK2019}, Lemma 3.14]\label{lem:higman-epstein}
We have the following.
\begin{enumerate}[(1)]
\item
Every proper quotient of $G$ is abelian.
\item
We have $G'=G''$.
\item
The center of $G$ is trivial.
\end{enumerate}
\end{lem}
\begin{proof}
Let $K\le G$ be a normal subgroup and let $t\in K$ be a nontrivial element. We may find an element $V(t)\in\BB$ such that $t(V(t))\cap V(t)=
\varnothing$. If $V\in \VV$ is arbitrary, we let $f,g\in G_V$ be arbitrary. By transitive--inclusivity, we have that there exists an element
$h\in G$ such that $h(V)\sse V(t)$. In particular, we have \[h(\supp f)\cap t(h(\supp g))\sse h(V)\cap t(h(V))=\varnothing.\] The reader
will note the similarity between this expression and the hypotheses of Theorem~\ref{thm:higman}.
Write $k=(t^h)(g^{-1})(t^h)^{-1}$. Computing, we find that $[f,k]=1$. Moreover, $[f,g]$ lies in the normal closure of $t$, since
\[[f,g]=[f,gk]=[f,[g,t^h]],\] and this latter commutator is a product of conjugates of $t$ and $t^{-1}$. Since $K$ is normal, we have
that $G_V'=G_V^{(1)}\le K$.
If $f,g\in G$ are arbitrary, then we apply fragmentation with respect to the chosen open star refinement $\UU$ of $\VV$. We have that
$f$ is a (finite) product of elements $\{f_i\}_{i\in I}$ with $f_i\in G_{U_i^f}$ for suitable elements $U_i^f\in\UU$, and similarly $g$ is a product
of element $\{g_j\}_{j\in J}$ with $g_j\in G_{U_j^g}$ for suitable $U_j^g\in \UU$.
Standard manipulation of commutators shows that in a group $G$ and for arbitrary elements $f,g,h\in G$,
there exist elements $u,v\in\form{ f,g,h}$ such that \[[fg,h]=[f,h]\cdot [g,h]^u,\quad [f,gh]=[f,g]\cdot [f,h]^v.\] Applying this observation
to the commutator of $[f,g]$, we have that there exist suitable elements
\[\{u_{i,j}\}_{i\in I,j\in J}\sse \form{ \{f_i\}_{i\in I},\{g_j\}_{j\in J}}\] such that $[f,g]$ is a product of commutators of the form
$[f_i,g_j]^{u_{i,j}}$.
Note that the commutators $[f_i,g_j]$ is trivial if $U_i^f\cap U_j^g=\varnothing$. Thus, the commutators is nontrivial only if
$U_i^f\cap U_j^g\neq \varnothing$. Now, we use the paracompactness. The fact that $\UU$ is an open star refinement of $\VV$ implies
that if $U_i^f\cap U_j^g\neq\varnothing$ then there is an open set $V\in\VV$ such that $f_i,g_j\in G_{V_{i,j}}$. We have already shown that
$G_{V_{i,j}}'\le K$, whence it follows that $[f,g]\in K$. It follows that $G'\le K$.
The second claim of the lemma follows from the first. Indeed, $G''$ is nontrivial, since $G$ is nonsolvable by Lemma~\ref{lem:derived}. We
have that $G''$ is a normal subgroup of $G$ and hence $G''$ contains $G'$. The other inclusion is immediate.
The third claim follows similarly. The center $Z$ of $G$ is an abelian normal subgroup and hence must contain $G'$ if it is
nontrivial. Since $G$ is not
solvable, $G'$ is nonabelian. Thus, $Z$ is trivial.
\end{proof}
For each $x\in X$, there is an open $V_x\in\BB$ containing $x$ and an element
$g_x\in G$ such that $g_x(V_x)\cap V_x=\varnothing$. We let \[\VV=\{V_x\mid x\in X\}\sse \BB\] and set $\UU$ to be an open star refinement of $\VV$.
Before proving Theorem~\ref{thm:trans-i}, we will require one more technical lemma which shows that for the cover $\UU$ (i.e.~ the
chosen open star refinement of the cover $\VV$ above), the action of the group $G$ can be simulated by the action of the group $G'$.
\begin{lem}[\cite{CKK2019}, Lemma 3.15]\label{lem:simulate}
Let $G$ be as before, let $g\in G$, and let $U\in \UU$. There exists an element $h\in G'$ such that $g(U)=h(U)$.
\end{lem}
\begin{proof}
We may fragment $G$ for the covering $\UU$, and so there exist open sets \[\{U_1,\ldots,U_m\}\sse \UU\] such that $g$ is a finite
product of elements $g_i\in G_{U_i}$, say $g=g_m\cdots g_1$.
We now produce elements $h_i\in G$ as follows. If $U_i\cap U=\varnothing$ then let $h_i$ be the identity. If $U_i\cap U\neq\varnothing$
then there is an element $V\in\VV$ such that $U_i\cap U\sse V$, and we may choose $h_i$ such that $h_i(V)\cap V=\varnothing$,
so that in particular $U\cap h_i(U_i)=\varnothing$.
Set \[k_i=(g_{i-1}\cdots g_1)h_i\in G.\] Since $h_i(U_i)\cap U=\varnothing$, we obtain that \[k_i(U_i)\cap (g_{i-1}\cdots g_1(U))=\varnothing.\]
It follows from these calculations that \[\supp(k_ig_ik_i^{-1})\cap (g_{i-1}\cdots g_1(U))=\varnothing.\] So,
\[\prod_{i=m}^1[g_i,k_i](U)=\prod_{i=m}^2 [g_i,k_i]\cdot g_1\cdot (k_1g_1^{-1}k_1^{-1})(U)=\prod_{i=m}^2 [g_i,k_i]\cdot g_1(U).\]
By an easy induction on $m$, we have that \[\prod_{i=m}^1[g_i,k_i](U)=g(U),\] which establishes the lemma.
\end{proof}
We now show that for a fragmented, transitive--inclusive group $G\le \Homeo(X)$, the commutator subgroup $G'$ is simple.
\begin{proof}[Proof of Theorem~\ref{thm:trans-i}]
It suffices to prove that the normal closure of a nontrivial element of $G'$ coincides with all of $G$.
Retaining the notation from the previous lemmas, let $H$ be the subgroup generated
by $G_{g(U)}^{(2)}$, where $g$ varies over $G$ and $U$ varies over $\UU$. By Lemma~\ref{lem:simulate}, we have that $g(U)=h(U)$ for
some suitable $h\in G'$, so that $H$ coincides with the subgroup generated by $G_{h(U)}^{(2)}$, where $U\in\UU$ and $h\in G'$. Observe that
for $k\in G$, we have that \[kG_{g(U)}^{(2)}k^{-1}=G_{k(g(U))}^{(2)}\le H,\] so that $H$ is in fact normal in $G$. Lemma~\ref{lem:higman-epstein}
then implies that $H=G'$.
Let $s\in G'$ be an arbitrary nontrivial element, which we wish to show normally generates $G'$, where here we mean normally in $G'$
and not normally in $G$. Let $V(s)\in\BB$ be a basic open set such
that $V(s)\cap s(V(s))=\varnothing$. If $U\in\UU$ is arbitrary, it suffices to show that $G_U^{(2)}$ is contained in the normal closure of $s$.
Indeed, then $G_{g(U)}^{(2)}$ will be contained in the normal closure of $s$ for all $g\in G$ and all $U\in \UU$, whence $s$ normally generates
$G'$.
So, let $f,g\in G_U'$ be arbitrary. Transitive--inclusivity implies that there is an element $h\in G$ such that $h(U)\sse V(s)$, and by
Lemma~\ref{lem:simulate}, we may assume that $h\in G'$. We have that \[h(\supp(f))\cap s(h(\supp g))\sse h(U)\cap s(h(U))=\varnothing,\]
so that \[[f,g]=[f,[g,s^h]],\] and as have argued before, this last commutator is a product of conjugates of $s$ and $s^{-1}$. It follows that
$[f,g]$ lies in the normal closure of $s$, as required.
\end{proof}
Now we can finally prove the simiplicity of the commutator subgroup of $\Diff^k_c(M)_0$.
\begin{proof}[Proof of Theorem~\ref{thm:epstein}]
We have that $\Diff^k_c(M)_0$ is fragmented by Lemma \ref{lem:frag}. The manifold $M$ is paracompact by assumption, and
the faithfulness of the action of $\Diff^k_c(M)_0$ is by definition.
We may choose a basis for the topology of $M$ consisting of the $C^k$--open balls (Definition~\ref{defn:ball}).
If $U$ and $V$ are such open balls, we wish to find a compactly
supported diffeomorphism of $M$ that sends $U$ into $V$. Since $M$ is connected, there is an embedded path
$\gamma$ from $U$ to $V$, whereby there is a
tubular neighborhood $N$ of $\overline{U\cup V\cup \gamma}$ such that $\overline{N}$ is compact and a diffeomorphism supported in $N$
that sends $U$ to $V$. This diffeomorphism then extends to $M$ by the identity. Thus, the action of $\Diff^k_c(M)_0$ is transitive--inclusive.
The theorem now follows from Theorem~\ref{thm:trans-i}.
\end{proof}
\begin{cor}[{cf.~\cite[Theorem 2.5]{Filip82}}]\label{cor:minimal}
The subgroup \[[\Diff^k_c(M)_0,\Diff^k_c(M)_0]\] is the unique nontrivial minimal normal subgroup
of the four groups below:
\[
\Diff_c^k(M)_0,
\Diff_c^k(M),
\Diff^k(M)_0,
\Diff^k(M).
\]
\end{cor}
\begin{proof}
Let $G_{c0}, G_{c},G_0$ and $G$ denote the given four groups respectively.
Note that $G_{c0}$ is nonabelian, fragmented, and transitive--inclusive.
By
part (1) of Lemma~\ref{lem:higman-epstein},
the group $[G_{c0},G_{c0}]$ contains
every nontrivial normal subgroup of $G_{c0}$.
This implies the conclusion for the case of $G_{c0}$.
Assume $G$ is one of the remaining three groups, and let $1\ne N\unlhd G$.
It suffices for us to show that $N$ contains $ [G_{c0},G_{c0}]$.
Pick an $C^k$ open ball $U\sse M$ and $h\in N$ such that $U$ and $h(U)$ are disjoint.
We can find
\[f,g\in \Diff_c^k(U)_0\le G_{c0}\unlhd G\] such that $[f,g]\ne1$. For instance,
we can pick distinct points \[\{a,b,c,d\}\sse U\] and use the path--transitivity (Lemma~\ref{lem:t3}) of $G_{c0}$ to find
$f,g\in\Diff_c^k(U)_0$ such that
\[
f(a)=b, f(b)=c, g(a)=a, g(b)=c.\]
Then we have
\[ gf(a)=c=f(b)\ne fg(a).\]
Setting $\bar f:=[f,h]$ and $\bar g:=[g,h]$ we see that $\bar f, \bar g\in N\cap G_{c0}$.
Moreover, we have
\[ [F,G](x)=[f,g](x)\]
for all $x\in U$. It follows that
\[
1\ne [F,G]\in N\cap [G_{c0},G_{c0}]\unlhd [G_{c0},G_{c0}].\]
Since $ [G_{c0},G_{c0}]$ is simple
by
Theorem~\ref{thm:trans-i}, it must coincide with $N\cap [G_{c0},G_{c0}] $.
\end{proof}
We will later see that all of the four groups in the above corollary are pairwise non-isomorphic (Corollary~\ref{cor:normal}).
\section{The Bochner--Montgomery Theorem on continuous group actions}\label{sec:boch-mont}
The Hilbert's fifth problem, in one interpretation, asks whether or not every locally Euclidean topological group is a Lie group.
This was confirmed affirmatively by Montgomery and Zippin~\cite{MZ1952} and Gleason~\cite{Gleason1952}.
An analogous result for a transformation group (that is, an action of a Lie group $G$ on a manifold $X$ which defines a simultaneously
continuous map $(g,x)\mapsto g.x$) was proved before by Bochner and Montgomery~\cite{BM1945}; we remark this result is often
attributed to Montgomery and Zippin in the literature, although this is not quite right.
\begin{thm}[Bochner--Montgomery, Theorem 4 of~\cite{BM1945}]\label{thm:BM}
Let $k\in\bN\cup\{\infty\}$, and let $X$ be a $C^k$--manifold.
If a Lie group $G$ acts on $X$ by $C^k$--diffeomorphisms
such that the map $H\co G\times X\longrightarrow X$ given by
\[ H\co (g,x)\mapsto g.x\]
is continuous, then $H(g,x)$ is a $C^k$ map on $(g,x)$ with respect to an analytic parameterization of $g\in G$.\end{thm}
We will give a proof of the above theorem only for the case of Euclidean group actions, based on the original proof of Bochner and Montgomery~\cite{BM1945,MZ1955}. This special case is sufficient for our purpose of proving Filipkiewicz's theorem (Theorem~\ref{thm:filip}).
Moreover, it still captures all the essential ideas of the theorem in the full generality while maintaining relatively simple notation.
As usual, a \emph{manifold}\index{manifold} for us means a Hausdorff, paracompact, locally Euclidean space.
\begin{thm}[Bochner--Montgomery theorem for Euclidean actions]\label{thm:BM-simple}
Let $d,k\in\bZ_{>0}$, and let $X$ be a smooth connected boundaryless $n$--manifold.
If
\[H\co \bR^d\times X\longrightarrow X\]
is a continuous map inducing an action of $\bR^d$ on $X$ by $C^k$--diffeomorphisms, then $H$ is a $C^k$ map.
\end{thm}
Note that the case $k=\infty$ is an immediate consequence.
\subsection{Analytic prerequisites}
Let us first collect some classical facts on real analysis.
The following result (this version being due to G.~Peano)
describes a condition that allows for switching the order of partial derivatives.
\begin{thm}[{Clairaut's Theorem, \cite[Chapter 9]{Rudin-PMA}}]\label{thm:clairaut}
If $f$ is a real--valued continuous function on $\bR^2$ such that
the maps
\[
D_1f,\, D_2f,\, D_{12}f
\]
exist and are continuous everywhere,
then $D_{21}f$ exists everywhere and coincides with $D_{12}f$.\end{thm}
Here, the subscripts denote the indices of the variables that are being differentiated.
The interchangeability of integration and differentiation
is an easy consequence of the Clairaut's theorem.
\begin{thm}[{Leibniz Integral Rule, \cite[Chapter 9]{Rudin-PMA}}]\label{thm:leibniz}
If $f(x,y)$ is a real--valued continuous function on $\bR^2$ such that
$D_1f$
exists and is continuous everywhere,
then for all $a<b$ we have that
\[
\int_a^b D_1f(x,y)dy = \frac{d}{dx} \int_a^b f(x,y)dy.\]
\end{thm}
Recall that a \emph{$G_\delta$ set}\index{$G_{\delta}$ set} is a countable intersection of open sets.
The following classical result is a consequence of the Baire Characterization Theorem of
Baire class one functions and the Baire Category Theorem.
We include a succinct proof, following the lines of~\cite{Kechris1995}, for the convenience of the reader.
\begin{thm}[{Baire's Simple Limit Theorem, cf.~\cite[Theorem 24.14]{Kechris1995}}]\label{thm:baire-simple}
If $\{f_i\}_{i\ge1}$ is a sequence of real--valued continuous functions on a locally compact Hausdorff topological space $Z$ such that the pointwise limit
\[
f(z):=\lim_{i\to\infty} f_i(z)\]
exists for all $z\in Z$, then the continuity set
\[
\operatorname{Cont}(f):=\{z\in Z\mid f\text{ is continuous at }z\}\]
is a dense $G_\delta$ subset of $Z$.\end{thm}
\bp
Pick a countable open basis $\{U_i\}_{i\ge1}$ of $\bR$. Observe that
\[f^{-1}(U_i)=\bigcup\left\{
\bigcap_{j\ge N} f_j^{-1}(\bar U_m)
\middle\vert
{m,N\ge1\text{ and }\bar U_m\sse U_i}
\right\}
.\]
It follows that the set
\[
Z\setminus\operatorname{Cont}(f)
=\bigcup_i \left( f^{-1}(U_i)\setminus \Int f^{-1}(U_i)\right)\]
is a countable union of closed sets with empty interior.
The proof is completed by an application of the Baire Category Theorem.
\ep
\subsection{Standing assumptions}
To prove Theorem~\ref{thm:BM-simple}, we will make the following assumptions.
We fix positive integers $d$ and $k$, and set
\[
T:=\bR^d\]
We let $X$ be a smooth connected boundaryless $n$--manifold.
The coordinates of points in $T$ will be called \emph{time variables}\index{time variable},
while those in $X$ are called \emph{spatial variables}\index{spatial variable}, specified locally.
We fix a continuous map
\[
H\co T\times X\longrightarrow X\]
which defines a $C^k$--action of $T$ on $X$;
we also write $g(x):=H(g,x)$ for $g\in T$ and $x\in X$ when there is little danger of confusion. We write
\[H=(H_1,\ldots,H_n)\]
in a suitable local coordinate of $X$.
For an arbitrary function
\[
F\co T\times X\longrightarrow\bR\]
and for $t=(t_1,\ldots,t_d)$ and $x=(x_1,\ldots,x_n)$,
we write
\begin{align*}
D_jF(t,x)&:=\frac{\partial F}{\partial x_j}(t,x),\\
\bar D_qF(t,x)&:=\frac{\partial F}{\partial t_q}(t,x),
\end{align*}
whenever they are defined in suitable local coordinates.
We denote by $e_j$ the $j^{th}$ standard basis vector in $\bR^d$.
\subsection{On spatial derivatives}
The following consequence of Baire's Simple Limit Theorem
plays a crucial role in transferring the regularity of spatial variables to that of time variables.
It roughly says that
\emph{first spatial derivatives are simultaneously continuous at almost every time moment}.
Let us denote $B^n(a;r)\sse\bR^n$ (or $B(a;r)$ more simply) the open ball with radius $r$ centered at $a\in\bR^n$.
\begin{lem}\label{lem:MZ-baire}
If $F$ is a real--valued continuous map on $T\times \bR^n$ such that the map
\[
x\mapsto D_1F(g,x)\]
exists and is continuous for each $g\in T$,
then for each fixed $x_0\in \bR^n$, the set
\[
Z^*(x_0):=\{g_0\in T\mid D_1F(g,x)\text{ is continuous at the point }(g_0,x_0)\}\]
is a dense $G_\delta$ subset in $T$.
\end{lem}
\bp
Let us set $f:=D_1F$ and
\[
H(g,x,t):=\frac1t\left( F(g,x+te_1)-F(g,x)\right).\]
The map $H$ is continuous on $(g,x,t)$ for $t\ne0$; moreover,
\[
H(g,x,t)=f(g,x+t\theta)\]
for some $\theta=\theta(g,x,t)\in[0,1]$.
Since
\[f(g,x)=\lim_{t\to0} H(g,x,t),\]
Baire's Simple Limit Theorem implies that
the set of continuity points
\[
\operatorname{Cont}(f)\sse T\times\bR^n\]
is a dense $G_\delta$ set.
If we denote by $p_1$ and $p_2$ the projections from $T\times\bR^n$ to the two factors, we see that
\[
Z^*(x_0)=p_1\left(p_2^{-1}(x_0)\cap\operatorname{Cont}(f)\right)\]
is a $G_\delta$ subset of $T$ as well.
Let us fix an arbitrary relatively compact open set $Z_0^*$ of $T$.
We claim that there exists some $g_0\in Z_0^*$,
open neighborhoods $\{\OO_i\}$ of $x_0$,
open neighborhoods $\{Z_i^*\}$ of $g_0$
such that
\[
\diam f(Z_i^*\times\OO_i)\le 1/i.\]
Once this claim is proved, we see that $(g_0,x_0)\in\operatorname{Cont}(f)$
and that $g_0\in Z^*(x_0)$.
Since the choice of $Z_0^*$ is arbitrary, we conclude that $Z^*(x_0)$ is dense.
It only remains for us to show the claim. We fix $r=1/2$.
Since $x\mapsto f(g,x)$ is continuous at each $g\in T$, we have
\[
Z_0^*=\bigcup_{\delta>0} \left\{ g\in Z_0^* \middle\vert
f\left(g,B(x_0,\delta)\right)\sse B(x_0,r/5)\right\}.\]
By the Baire Category Theorem, there exists some $\delta_1>0$
such that for
\[
\OO_1:=B(x_0,\delta_1),\]
the following set is not nowhere dense:
\[
Y_1:=\{g\in Z_0^*\mid f(g,\OO_1)\sse B(x_0,r/5)\}.\]
In other words, we can find a compact neighborhood $Z_1\sse Z_0^*$ such that
$Y_1\cap Z_1$ is dense in $Z_1$.
Applying Baire's Simple Limit Theorem to the sequence $\{H(g,x_0,1/i)\}_i$, we see that $\Int\, Z_1$ contains a continuity point $g_1$
of the map $g\mapsto f(g,x_0)$.
We have an open neighborhood $Z_1^*\sse Z_1$ of $g_1$ such that
\[
f(Z_1^*,x_0)\sse B\left( f(g_1,x_0),r/5\right).\]
Let $(h,x)\in Z_1^*\times\OO_1$.
We establish the following inequalities for some $t>0$ and $g\in Y_1\cap Z_1^*$.
\begin{align*}
|f(h,x)-H(h,x,t)|&\le r/5,\\
|H(h,x,t)-H(g,x,t)|&\le r/5,\\
|H(g,x,t)-f(g,x)|&\le r/5,\\
|f(g,x)-f(g,x_0)|&\le r/5,\\
|f(g,x_0)-f(g_1,x_0)|&\le r/5.\end{align*}
The first inequality holds for all sufficiently small $t>0$, depending on $h$ and $x$. We use the density of $Y_1\cap Z_1^*$ in $Z_1^*$
to find $g$ satisfying the second. The third is guaranteed by reducing $t>0$ further, depending on $g$.
The fourth follows from the choice of $\OO_1$. The choices of $g_1$ and $Z_1^*$, for which we used Baire's Simple Limit Theorem,
makes the last inequality hold.
Combining the five inequalities, we have that
\[
f(Z_1^*\times\OO_1)\sse B(f(g_1,x_0),r).\]
Inductively, we can find open sets $Z_i^*$ and elements
$g_i\in T$ such that
\[ g_i\in Z_i^*\sse \overline{Z_i^*}\sse Z_{i-1}^*,\]
and such that
\[
f(Z_i^*\times\OO_i)\sse B(f(g_i,x_0),r/i)\]
for some open neighborhood $\OO_i$ of $x_0$.
Then we have that
\[\diam f(Z_i^*\times\OO_i)\le 2r/i=1/i.\]
Choosing an element
\[
g_0\in \bigcap_i Z_i^*= \bigcap_i \overline{Z_i^*},\]
we obtain the claim.
\ep
We show that \emph{first spatial derivatives are simultaneously continuous}:
\begin{lem}\label{lem:spatial-c1}
The map $D_jH_i$ is continuous on $ T\times X$ for all $i,j$.
\end{lem}
\bp
Let us fix $x_0$. By Lemma~\ref{lem:MZ-baire} we can find densely many $g_0\in T$ such that
$D_jH_i(g,x)$ is continuous at $(g_0,x_0)$ for all $i,j$.
For a small vector $h\in T$ and for all $x\in X$ we have
\[
H_i(g_0+h,x)=H_i(g_0,h(x)).\]
In a suitable coordinate neighborhood, we have
\[
(D_jH_i)(g_0+h,x)=\sum_k D_k H_i(g_0,h(x))\cdot (D_jH_k)(h,x).\]
Some parentheses above are not strictly necessary for the correct interpretation of the above equation,
but we have added them nevertheless for readers' convenience.
As $(h,x)$ approaches $(0,x_0)$ the left hand side approaches $D_jH_i(g_0,x_0)$
since $(g_0,x_0)$ is a point of continuity for $D_j H_i$.
The matrix \[\left( D_jH_i(g_0,h(x))\right)_{i,j=1,\ldots,n}\] is continuous at $(h,x)=(0,x_0)$
since we have made a standing assumption that $H(h,x)$ is continuous,
and that $D_kH_i$ is continuous in the spatial variables;
in particular, this matrix is nonsingular for $(h,x)$ near $(0,x_0)$
as $x\mapsto H(g_0,x)$ is a $C^k$--diffeomorphism.
It follows that $(0,x_0)$ is a continuity point of $D_jH_k$ for all $x_0\in X$.
Replacing $g_0$ by an arbitrary $g\in T$,
a very similar argument shows that $(g,x_0)$ is a continuity point of $D_jH_k$.\ep
Bootstrapping the above argument, we have that \emph{$k$--th spatial derivatives are simultaneously continuous}:
\begin{lem}\label{lem:spatial-ck}
For $j_1,\ldots,j_k\in\{1,\ldots,n\}$,
the $k$--th partial derivative
\[D_{j_1}\cdots D_{j_k}H_i\]
is continuous on $T\times X$.
\end{lem}
\bp
The existence of such a spatial partial derivative is a part of the standing assumption.
We use the Baire's Simple Limit Theorem again to find
a point $(g_0,x_0)\in T\times X$ at which all such spatial partial derivatives are simultaneously continuous.
One can then proceeds inductively, in a similar manner as in the proof of Lemma~\ref{lem:spatial-c1};
namely, one establishes the continuity at $(0,x_0)$, and then at $(g,x_0)$ for an arbitrary $g\in T$. This completes the proof.
\ep
\subsection{On time derivatives}
We now turn our attention to the time derivatives, and prove that
\emph{the first time derivatives exist and spatially $C^{k-1}$ at time zero}:
\begin{lem}\label{lem:time-c1-0}
For all $i$ and $q$,
the time derivative at time zero
\[\bar D_qH_i(0,x)\]
exists and is $C^{k-1}$ with respect to the spatial variable $x\in X$.
Moreover, if $h$ is sufficiently close to $0$ then
for all $x\in X$ we have
\[
H_i(he_q,x)-H_i(0,x)=\sum_j \left(\int_0^h D_jH_i(\tau e_q,x)d\tau\right) \cdot \bar D_q H_j(0,x).\]
\end{lem}
As usual, the above expression and our computation below make sense in suitable local coordinates.
\bp
We may assume $q=1$.
Consider the auxiliary function
\[
T_i(h,x):=\int_0^h H_i(\tau e_1,x)d\tau.\]
By Lemma~\ref{lem:spatial-c1} the map $D_jH_i$ exists and is simultaneously continuous.
It follows from the Leibniz Integral Rule (Theorem~\ref{thm:leibniz}) that
\[
D_jT_i(h,x)=\int_0^h D_jH_i(\tau e_1,x)d\tau.\]
For each $t$ sufficiently close to $0$, we can find $z=z_i(t)$
on the line segment between $x$ and $H(te_1,x)$ such that
\[T_i(h,H(te_1,x))-T_i(h,x)
=\sum_j D_j T_i(h,z_i(t))\cdot (H_j(te_1,x)-x_j).\]
The left hand side can be rewritten as
\[\int_t^{h+t} H_i(\tau e_1,x)d\tau - \int_0^h H_i(\tau e_1,x)d\tau
=\int_0^t\left( H_i((h+\tau)e_1,x)-H_i(\tau e_1,x)\right)d\tau.\]
After defining the matrix
\[
A(h,y)=\left(A(h,y)_{ij}\right):=\left(D_jT_i(h,y)\right),
\]
we conclude that
\begin{equation}\label{eq:aij}
\frac1t \int_0^t\left( H_i((h+\tau)e_1,x)-H_i(\tau e_1,x)\right)d\tau
=
\sum_j A(h,z_i(t))_{ij}\cdot \frac{H_j(te_1,x)-H_j(0,x)}{t}.\end{equation}
The matrix $A(h,z_i(t))$ is arbitrarily close to $h\cdot\Id$ if $h$ and $t$ are sufficiently close to $0$; in particular,
it is non-singular for such $h$ and $t$.
It follows that
\[
\frac{H_i(te_1,x)-H_i(0,x)}{t}
=\sum_j \left(A(h,z_i(t))^{-1}\right)_{ij}\cdot\frac1t \int_0^t\left( H_j((h+\tau)e_1,x)-H_j(\tau e_1,x)\right)d\tau.\]
Sending $t$ to $0$, we obtain that
\[
\bar D_1 H_i(0,x)
=\sum_j \left(A(h,x)^{-1}\right)_{ij} (H_j(he_1,x)-x_j).\]
Since the map
\[
A_{ij}(h,x)=D_jT_i(h,x)=\int_0^h D_jH_i(\tau e_1,x)d\tau\]
is $C^{k-1}$ on $x$, so is $\bar D_1H_i(0,x)$.
By sending $t\to 0$ in the equation~\eqref{eq:aij}, we obtain
the second conclusion of the lemma as well.
\ep
Recall our abbreviation
\[
g(x):=H(g,x)\]
for $g\in T$ and $x\in X$.
We now establish that \emph{the first time derivatives of $H$ exist, are simultaneously continuous,
and are spatially $C^{k-1}$ everywhere}:
\begin{lem}\label{lem:time-c1}
For all $q$ and $i$, the map $\bar D_qH_i$
exists and is continuous on $T\times X$; moreover, it
satisfies
\[
\bar D_q H_i(g,x)=\bar D_q H_i(0,g(x))\]
for $(g,x)\in T\times X$;
furthermore, the map $x\mapsto \bar D_qH_i(g,x)$ is $C^{k-1}$ with respect to the spatial variable $x\in X$ for each $g\in T$.
\end{lem}
\bp
We may only consider the case $q=1$.
Using Lemma~\ref{lem:time-c1-0} we have that
\begin{align*}
\frac{H_i(g+he_1,x)-H_i(g,x)}h
&=
\frac{H_i(he_1,g(x))-H_i(0,g(x))}h\\
&=\sum_j \frac1h
\left(\int_0^h D_jH_i(\tau e_1,g(x))d\tau\right) \cdot \bar D_1 H_j(0,g(x)).
\end{align*}
Sending $h\to 0$, we see from
Lemma~\ref{lem:spatial-c1}
that
\[
\bar D_1 H_i(g,x)
=\sum_j D_jH_i(0,g(x))\cdot \bar D_1 H_j(0,g(x))
=\bar D_1H_i(0,g(x)).\]
Here, we used that $H(0,y)=y$ for all $y\in X$.
The second conclusion also follows
since $y\mapsto \bar D_1H_i(0,y)$ is $C^{k-1}$
by Lemma~\ref{lem:time-c1-0},
and since $x\mapsto g(x)$ is $C^k$ by the standing assumption.
\ep
We are now ready to complete the proof of the Bochner--Montgomery Theorem for Euclidean actions.
\bp[Proof of Theorem~\ref{thm:BM-simple}]
By Lemmas~\ref{lem:spatial-c1} and~\ref{lem:time-c1}, we have that $H$ is $C^1$ on $T\times X$.
Assume inductively that $H$ is $C^{K-1}$ for some $1\le K\le k$.
We saw that
\[
\bar D_q H_i(g,x)=\bar D_q H_i(0,g(x)).\]
Since $(g,x)\mapsto g(x)$ is $C^{K-1}$
and since the map $y\mapsto \bar D_j H_i(0,y)$ is $C^{k-1}$,
the map $(g,x)\mapsto \bar D_j H_i(g,x)$ is $C^{K-1}$.
It remains to consider a $(K-1)^{th}$ partial derivative mixed with time and spatial variables, say written as $D_0$, of the map $D_jH_i$.
The map $D_jH_i(g,x)$ is $C^{k-1}$ with respect to the spatial variables. If $D_0$ involves a time variable, then the Clairaut's
Theorem (Theorem~\ref{thm:clairaut})
gives the desired conclusion when combined the preceding paragraph; that is, $D_0 D_j H_i$ exists and is continuous.
This proves that $H_i$ is $C^K$, and by induction, that $H$ is $C^k$.
\ep
\section{Takens' Theorem}\label{sec:takens}
For the purposes of this section, let $M$ and $N$ be
smooth, connected, boundaryless manifolds.
The (generalized) Takens--Filipkiewicz theorem below itself falls slightly short of proving that the group $\Diff^{r}(M)$ determines
$M$ up to $C^{r}$ diffeomorphism,
but which illustrates an important step in proving that fact, namely that a bijection between two manifolds inducing a bijection between
diffeomorphism groups is itself a diffeomorphism; cf.~Section~\ref{sec:filip}.
The result for full diffeomorphism groups was originally proved by Takens when $p=q=\infty$ and when $M$ and $N$ have the
same dimensions.
R.~P.~Filipkiewicz extended this to all $p,q\in\bZ_{>0}\cup\{\infty\}$ for
$\Diff^p(M)$ and $\Diff^q(N)$.
\begin{thm}[Generalized Takens' Theorem]\label{thm:takens}
Let $M$ and $N$ be smooth connected boundaryless manifolds, and let $p,q\in\bZ_{>0}\cup\{0,\infty\}$.
Suppose we have groups $G$ and $H$ satisfying
\[
\Diff_c^{p}(M)_0\le G\le \Diff^{p}(M)\]
and
\[
\Diff_c^{q}(N)_0\le H\le \Diff^{q}(N).\]
If a set--theoretic bijection
$w\colon M\longrightarrow N$ satisfies
\[ w G w^{-1} =H,\]
then we have that
$p=q$
and that
$w$ is a $C^p$ diffeomorphism.
\end{thm}
As observed by Takens, Theorem~\ref{thm:takens} implies the following statement: in the group of set--theoretic bijections
$M\longrightarrow M$, the group $\Diff^p(M)$ is equal to its own normalizer.
We will present two different proofs of the theorem, after establishing that $w$ is a homeomorphism by elementary considerations.
The first proof in this book uses the Bochner--Montgomery Theorem, based on~\cite{Filip82}. This readily gives a simple and full
account of Theorem~\ref{thm:takens2}.
For the second proof, which comes from Takens' original article~\cite{Takens79}, we reduce the
case to maps between Euclidean spaces and then argue locally that the bijection $\phi$ is a diffeomorphism.
This gives a more computationally explicit argument.
We present only a part of the proof for the sake of brevity, avoiding the introduction of many new concepts of smooth dynamics; namely,
after reducing to the Euclidean case we will consider only the dimension one case with full diffeomorphism groups.
\begin{rem}
It is very interesting to note that both of the proofs presented here are based on non-constructive existence theorems from real analysis.
The first proof is deduced from the Bochner--Montgomery theorem, which in turn is based on Baire's Simple Limit Theorem.
The second is based on Lebesgue's theorem on the almost everywhere differentiability of monotone functions, together with
Sternberg's Linearization Theorem, which in turn is a consequence of the Schauder--Tychonoff Fixed Point Theorem
(see Chapter V of~\cite{DSbook1}).
In both proofs, one finds a reasonably regular point by applying the suitable nonconstructive result,
and then ``transfers'' this regularity to other points using the transitivity of group actions.
\end{rem}
\subsection{Promoting a bijection to a homeomorphism}\label{ss:bij-homeo}
The following lemma in particular implies the ``easy'' part of Takens--Filipkiewicz Theorem. Namely, it asserts that the map
$w$ given in the hypothesis of the Takens' Theorem must be a homeomorphism.
\begin{lem}\label{lem:bij-homeo}
Let $X_1$ and $X_2$ be topological spaces,
and let $G_i\le\Homeo(X_i)$ be locally transitive for $i=1,2$.
If there exists an isomorphism
\[
\Phi\co G_1\longrightarrow G_2\]
and a bijection
\[
w\co X_1\longrightarrow X_2\]
such that
\[wgw^{-1}=\Phi(g)\]
for all $g\in G_1$,
then $w$ is a homeomorphism.
\end{lem}
\bp
Let $\UU_i$ be a local basis as in the definition of local transitivity for $i=1,2$.
Consider an arbitrary $x\in X_1$ and its open neighborhood $U\in\UU_1$.
By the local transitivity one has some $g\in G_1$ such that
\[
x\in \suppo g\sse \suppc g\sse U.\]
This implies that the collection
\[
\{\suppo g\mid g\in G_1\}\]
is an open basis of $X_1$.
Furthermore, we have that
\[
w(\suppo g)
=\suppo wgw^{-1}
=\suppo \Phi(g)\]
for all $g\in G_1$. This implies that $w$ is open. By symmetry,
we conclude that $w$ is a homeomorphism.\ep
\subsection{Takens' theorem via Bochner--Montgomery}
It is relatively a simple task to deduce Theorem~\ref{thm:takens}
from the Bochner--Montgomery Theorem (Theorem~\ref{thm:BM-simple}),
which we do in this subsection. We will prove the following stronger result,
and show how this implies Theorem~\ref{thm:takens}.
\begin{thm}\label{thm:takens2}
Let $p\in\bZ_{>0}\cup\{\infty\}$, and let $M$ and $N$ be smooth, connected, boundaryless manifolds.
If $w\colon M\longrightarrow N$ is a homeomorphism such that
\[ w^{-1} \Diff^\infty_c(N)_0 w \sse \Diff^p(M),\]
then $w^{-1}$ is a $C^p$ map.
\end{thm}
\bp
We let $d=\dim M=\dim N$.
As the statement is purely local, it suffices to prove the desired regularity $C^p$ at a given fixed point, say $x_0\in M$ and $y_0:=w(x_0)$.
Pick a sufficiently small smooth open ball $U\sse N$ containing $y_0$ so that $U$ and $w^{-1}(U)$ are
contained in charts of $N$ and $M$. By Lemma~\ref{lem:affine} we can find
a smooth embedding $h\co \bR^d\longrightarrow N$
such that $h(\bR^d)=U$ and such that $h(0)=y_0$.
This embedding induces a smooth action $\bR^d$ on $N$.
Namely, for $v\in \bR^d$ and $y\in N$ we define
\[
B_v(y):=
\begin{cases}
h\left(v+h^{-1}(y)\right),&\text{ if }y\in U,\\
y,& \text{otherwise}.\end{cases}\]
Now we can investigate the relationship between $w$ and $B$.
For $v\in\bR^d$ and $x\in M$, we define
\[
A_v:=w^{-1}\circ B_v\circ w \in\Diff^p(M).\]
The continuity of the map
\[
(v,x)\mapsto A_v(x)= w^{-1}\left(
h\left(v+h^{-1}(w(x))\right)
\right)
\]
is obvious.
Bochner--Montgomery Theorem implies that the map
\[ (v,x)\mapsto A_v(x)\]
must be a $C^p$ map.
Note that $w^{-1}(B_v(y_0))=A_v(x_0)$ for all $v\in\bR^d$.
Since
the map
\[
B_v(y_0)\mapsto v\]
is a smooth chart about $y_0$
and
the map
\[
v\mapsto A_v(x_0)\]
is $C^p$, we conclude that $w^{-1}$ is a $C^p$ map.
\ep
We can now complete the first proof of the Takens' Theorem.
\bp[Proof of Theorem~\ref{thm:takens}]
Since $\Diff^\infty_c(M)_0$ and $\Diff^\infty_c(N)_0$ are locally transitive, we have that $w$ is a homeomorphism by
Lemma~\ref{lem:bij-homeo}.
Applying Theorem~\ref{thm:takens2}, we also have that $w$ and $w^{-1}$ are $C^q$ and $C^p$ maps, respectively.
By symmetry, we may assume $p\le q$, so that $w$ is a $C^p$ diffeomorphism.
It only remains to show $p=q$.
If $p<q$, then the conjugation by the $C^p$--diffeomorphism $w$ induces a group isomorphism between
$\Diff^p_c(M)_0$ and $\Diff^p_c(N)_0$.
This is a restriction of $\Phi$, whose image $H$ is contained in $\Diff^q(N)$. This is contradiction,
since $\Diff^p_c(N)_0\not\sse \Diff^q(N)$ for $q>p$.
\[\begin{tikzcd}
G\arrow{r}{\Phi} \arrow[dash]{d}{} &
H=H\cap \Diff^q(N)\\
\Diff^p_c(M)_0\arrow[swap]{r}{f\mapsto wfw^{-1}}& \Diff^p_c(N)_0
\end{tikzcd}
\]
We thus conclude that $p=q$.
\ep
\subsection{Addendum: Takens' original proof}
In this section, we give an alternative, partial proof of Theorem~\ref{thm:takens} based on Takens' original argument~\cite{Takens79}.
Namely, we will give a proof of the following result for $p\ge2$ and $\dim M=\dim N=1$. The hypothesis on $p$ arises from the
hypotheses in Sternberg's Linearization Theorem.
\begin{thm}[Takens Theorem~\cite{Takens79}]\label{thm:takens3}
Let $M$ and $N$ be smooth connected boundaryless manifolds.
If $w\colon M\longrightarrow N$
is a set--theoretic bijection satisfying that
\[
w \Diff^\infty(M)w^{-1}= \Diff^\infty(N),\]
then $w$ is a smooth diffeomorphism.\end{thm}
Takens' proof of Theorem~\ref{thm:takens3} can be broken up into several smaller pieces.
The first is to consider the case $M=\bR^d$, as is a common first step in such results.
This step requires some care to avoid issues with exotic smooth structures on Euclidean spaces when $d=4$.
\begin{thm}[\cite{Takens79}, Theorem 2]\label{thm:takens-reduction}
If a bijection $\phi\colon\bR^d\longrightarrow\bR^d$ satisfies that
\[
\phi^{-1}\Diff_+^\infty(\bR^d)\phi=\Diff_+^\infty(\bR^d),\]
then $\phi$ is a smooth--diffeomorphism.
\end{thm}
For the purposes of Theorem~\ref{thm:takens3}, the global topology of $M$ and $N$ are irrelevant. This is made precise by the following
fact:
\begin{lem}\label{lem:takens-reduction}
Theorem~\ref{thm:takens3} follows from Theorem~\ref{thm:takens-reduction}.
\end{lem}
The second step is to prove that if $d=1$ then for $M=N=\bR$, the map $\phi$ in Theorem~\ref{thm:takens3} is
automatically a diffeomorphism.
This is an essentially one--dimensional argument, as it relies fundamentally on the almost everywhere differentiability of a monotone
real--valued function on $\bR$.
The third step is the general case, which uses the second step as a bootstrap. The proof is less self-contained, and relies on invariant
manifold theory. It is not possible for us to give a self--contained proof of the general case
here, as it would take us too far afield into smooth dynamics.
We will therefore content ourselves to giving a proof of Theorem~\ref{thm:takens3} in dimension one only.
\subsubsection{Reduction to Euclidean spaces}
The proof of Lemma~\ref{lem:takens-reduction} is mostly formal, though as we alluded to above, there is one issue to be avoided, namely
the existence of exotic differential structures on $\bR^d$.
To begin, we note from Lemma~\ref{lem:bij-homeo} that $\phi$ is necessarily a homeomorphism, for $M$ and $N$ arbitrary.
If $W\sse M$ is diffeomorphic to $\bR^d$, then clearly
$\phi(W)$ is homeomorphic to $\bR^d$. The possibility of exotic smooth structures precludes concluding that $\phi(W)$ is diffeomorphic to $\bR^d$ and thus that $\phi$ is a diffeomorphism.
\begin{proof}[Proof of Lemma~\ref{lem:takens-reduction}]
It suffices to show that for every point $x\in M$, there is an open neighborhood $W$ containing $x$ such that $\phi$ is a diffeomorphism
when restricted to $W$. Without loss of generality, we may assume that $W$ is diffeomorphic to $\bR^d$. We write $V=\phi(W)\sse N$.
Let $D\sse V$ be a closed ball, and let $E=\phi^{-1}(D)$. We let $\psi\in\Diff^\infty(M)$ be such that $E\sse\psi(E)^{\circ}$, the
interior of $\psi(E)$, and such that \[W=\bigcup_{i\geq 0}\psi^i(E).\] The conditions on $\phi$ imply that $\phi\psi\phi^{-1}=\eta$ is
a diffeomorphism of $N$ such that $D\sse\eta(D)^{\circ}$, and \[V=\bigcup_{i\geq 0}\eta^i(D).\] Because $\eta^i(D)$ is diffeomorphic to
a closed ball for all $i$ and is compactly contained in the interior of $\eta^{i+1}(D)$, we see from the
Isotopy Extension Theorem (\cite{Palais1960,Lima64}) that
\[\eta^{i+1}(D)\setminus\eta^i(D)^{\circ}\cong S^{d-1}\times [0,1].\] From this it follows easily
that $V$ is in fact diffeomorphic to $\bR^d$. This argument is symmetric with respect to replacing $\phi$ by its inverse,
and so we may conclude that $\phi$ is a diffeomorphism.
\end{proof}
\subsubsection{Theorem~\ref{thm:takens-reduction} for $n=1$}
Establishing Theorem~\ref{thm:takens} for $\bR$ (and hence for $1$--manifolds) again breaks into two steps. First, we show that
if $\phi$ is as in the hypotheses, then $\phi$ and $\phi^{-1}$ both have a first derivative everywhere in $\bR$.
Then, we bootstrap this fact to conclude that
$\phi$ and $\phi^{-1}$ are smooth. For the remainder of this subsection, we will assume that $\phi$ is as in
Theorem~\ref{thm:takens}, with $M=N=\bR$.
\begin{lem}[\cite{Takens79}, Lemma 3.1]\label{lem:phi-der}
For all $x\in \bR$, the functions $\phi$ and $\phi^{-1}$ both have nonzero first derivative at $x$.
\end{lem}
\begin{proof}
Since $\phi$ is a homeomorphism, we have that $\phi$ and $\phi^{-1}$ are monotone
real-valued functions on $\bR$.
A standard fact about such monotone functions is that they are differentiable almost everywhere.
If $f\colon\bR\longrightarrow\bR$ is a diffeomorphism and $x\in \bR$, then $\phi^{-1}f\phi$ is differentiable at $x$ and has a nonzero
derivative. Recording this fact, we have that
\[(\phi^{-1}f\phi)'(x)=\lim_{h\to 0}\frac{(\phi^{-1}f\phi)(x+h)-(\phi^{-1}f\phi)(x)}{h}.\]
The ratio in the limit can be formally written as a product of three ratios, all of which are defined for $h\neq 0$:
\[
q_1=\frac{(\phi^{-1}f\phi)(x+h)-(\phi^{-1}f\phi)(x)}{(f\phi)(x+h)-(f\phi)(x)},\] \[q_2=\frac{(f\phi)(x+h)-(f\phi)(x)}{\phi(x+h)-\phi(x)},\]
\[q_3=\frac{\phi(x+h)-\phi(x)}{h}.\]
We will argue that for all three ratios, the limits
as $h\to 0$ exist and are nonzero. The continuity of $\phi$ and the fact that $f$ is a diffeomorphism of $\bR$ implies that the limit
as $h\to 0$ of $q_2$ exists and is nonzero, since we are merely expressing a change of coordinates for $f$.
Since $\phi$ is differentiable almost everywhere, we may select a point $x$ so that the limit for $q_3$ exists. Suppose
first that $\phi'(x)=0$. We then compute the limit of $q_1$, and conclude that if $\phi'(x)=0$ then $\phi^{-1}$ does not have a derivative
at $(f\phi)(x)$. Since $f$ is arbitrary and diffeomorphisms of $\bR$ act transitively on $\bR$, we obtain that $\phi^{-1}$ is nowhere
differentiable. This contradicts the fact that $\phi^{-1}$ is almost everywhere differentiable. So, $\phi'(x)\neq 0$, whence $\phi^{-1}$
has a finite and nonzero first derivative at $(f\phi)(x)$. Again using the transitivity of the diffeomorphism group, we conclude that $\phi^{-1}$
has nonzero first derivative everywhere. Symmetrically, we find the same conclusion for $\phi$.
\end{proof}
We now upgrade Lemma~\ref{lem:phi-der} to prove that $\phi$ is actually $C^{\infty}$ with a $C^{\infty}$ inverse. We have the following
technical construction:
\begin{lem}[\cite{Takens79}, Lemma 3.2]\label{lem:phi-conj}
Let $2\le p\le\infty$, and let $\phi\colon \bR\longrightarrow\bR$ be a homeomorphism such that $\phi$ and $\phi^{-1}$ are differentiable for all points in $\bR$.
Suppose furthermore that $f_1$ and $f_2$ are $C^{p}$ orientation preserving diffeomorphisms of $\bR$ such that:
\begin{enumerate}[(1)]
\item
We have $f_1(0)=0$;
\item
We have $f_1'(0)<1$;
\item
For each $x\in\bR$, we have that $f_1^k(x)\longrightarrow 0$ as $k\longrightarrow\infty$;
\item
We have $\phi^{-1}f_1\phi=f_2$.
\end{enumerate}
Then $\phi\in\Diff^{p}(\bR)$.
\end{lem}
In the statement of Lemma~\ref{lem:phi-conj}, $\phi$ is not assumed to be a $C^1$ diffeomorphism, since we do not assume that the
derivatives of $\phi$ and $\phi^{-1}$ are continuous.
\begin{proof}[Proof of Theorem~\ref{thm:takens-reduction} for $n=1$, assuming Lemma~\ref{lem:phi-conj}]
Let $f_1(x)=x/2$ and let $\phi$ be as in the statement of the theorem. Then Lemma~\ref{lem:phi-der} implies that $\phi$ and $\phi^{-1}$
are differentiable at all points of $\bR$. By assumption, we have that \[f_2=\phi^{-1}f_1\phi\in\Diff^{\infty}(\bR).\]
Lemma~\ref{lem:phi-conj} immediately implies the desired conclusion.
\end{proof}
Before we can give the proof of Lemma~\ref{lem:phi-conj}, we will require a result due to Sternberg, which implies that the diffeomorphisms
$f_1$ and $f_2$ in the statement of the lemma are $C^{p}$--conjugate to linear diffeomorphisms. We state Sternberg's result here
in the very special case of one--manifolds, and we omit the proof since the statement itself is believable,
since accessible proofs are available in the literature, since a full proof would
take us far afield, and since we have already given a complete proof of Takens' result, and since the result will
also be subsumed by Filipkiewicz's result in the next section.
\begin{thm}[See~\cite{Sternberg57,Sternberg58}]\label{thm:sternberg}
Let $2\le p\le\infty$.
If $f\colon\bR\longrightarrow\bR$ is a $C^p$ diffeomorphism with $f(0)=0$ and $|f'(0)|\neq 1$, then = the
diffeomorphism $f$ is locally $C^{p}$--conjugate to
a linear diffeomorphism. That is, there exists a $C^p$ diffeomorphism $\psi$ such that $\psi f\psi^{-1}$ is given by a linear map on some
neighborhood $U$ that contains $0$.
\end{thm}
\begin{proof}[Proof of Lemma~\ref{lem:phi-conj}]
Recall we have assumed that $p\ge2$.
We first prove that $\phi$ is a $C^p$ diffeomorphism in a neighborhood of $0$.
By replacing $\phi$ by a translate if necessary, we may assume that $\phi(0)=0$. So, we have that $f_2(0)=0$.
Since dynamics are
preserved under conjugacy, we have that $f_2^k(x)\longrightarrow 0$ for all $x\in \bR$.
Using the Chain Rule, we see that
\[
D_0:=f_1'(0)=f_2'(0)<1.\]
Theorem~\ref{thm:sternberg} implies the existence
of diffeomorphisms $\psi_i\in\Diff^{p}(\bR)$ for $i\in\{1,2\}$ that fix $0$ and that conjugate $f_i$ to a linear map in a neighborhood
$U$ of the origin, i.e.~
$\psi_i f_i\psi_i^{-1}$ is linear on $U$ for $i\in\{1,2\}$.
We define a map $\eta\colon\bR\longrightarrow\bR$ by $\psi_1\phi\psi_2^{-1}$. It is clear from the definitions that $\eta$ and
$\eta^{-1}$ are both differentiable at zero. We claim that $\eta$ is linear near the origin, whence it will follow that $\phi$ is a
$C^p$ diffeomorphism near the origin.
Note that $D_0$ is the slope of the linearization of $f_i$ for
$i\in\{1,2\}$.
Without loss of generality we may assume that $1\in U$, and we write $\eta(1)=\alpha$.
Choose $\beta\in U$, and write $\gamma=\eta(\beta)$.
Considering the $k$--th power $D_0^k$ of $D_0$, we have
\begin{align*}
\eta(D_0^k)=\eta(\psi_2f_2^k\psi_2^{-1}(1))=\psi_1\phi\psi_2^{-1}\psi_2f_2^k\psi_2^{-1}(1)=\\
=\psi_1f_1^k\psi_1^{-1}\psi_1\phi\psi_2^{-1}(1)=D_0^k\eta(1)=D_0^k\alpha.
\end{align*}
Similarly, we obtain that $\eta(D_0^k\cdot\beta)=D_0^k\cdot\gamma$.
Since $D_0<1$, we may compute as follows:
\begin{align*}
\eta'(0)&=\lim_{k\to\infty}\frac{\eta(D_0^k)-\eta(0)}{D_0^k}=\alpha,\\
\eta'(0)&=\lim_{k\to\infty}\frac{\eta(D_0^k\cdot \beta)-\eta(0)}{D_0^k\cdot\beta}=\frac{\gamma}{\beta}.
\end{align*}
It follows that $\alpha=\gamma/\beta$. Since $\beta$ is arbitrary
and $\eta$ is continuous, we have that $\eta(\beta)=\beta\eta(1)$ for all $\beta\in U$. This implies that $\eta$ is linear in a neighborhood of
$0$, as claimed.
To prove that $\phi$ is a $C^p$ diffeomorphism everywhere, we simply change coordinates. If $f_1$ is a diffeomorphism which satisfies
the hypotheses of the lemma with $y$ in place of $0$, we can translate $\phi$ in order to have $\phi$ fix $y$ as well.
Then, we can conjugate the entire picture by a translation to move the fixed point to $0$
and without changing the absolute value of the derivative of $f_1$.
\end{proof}
This completes the proof of Theorem~\ref{thm:takens} for $1$--manifolds and concludes our discussion of Takens' result.
\section{Rubin's Theorem}\label{sec:rubin}
In this section, we give a (simplified, for our purposes)
self--contained proof of Rubin's powerful reconstruction theorem~\cite{Rubin1989,Rubin1996}. This will
then give an efficient proof of Filipkiewicz's Theorem.
Let $G$ be a group acting on a space $X$.
Recall our notation below that is used throughout this book:
\begin{align*}
\supp g&:=X\setminus\Fix g,& \text{ if }g\in G,\\
G[U]&:=\{g\in G\mid \supp g\sse U\},& \text{ if }U\sse X.
\end{align*}
The group $G[U]$ is sometimes called the \emph{rigid stabilizer}\index{rigid stabilizer} of $U\sse X$.
\bd\label{d:locally-moving}
Let $X$ be a topological space, and let $G\le\Homeo(X)$.
\be[(1)]
\item
We say $G$ is \emph{locally moving}\index{locally moving action}
if for each nonempty open set $U\sse X$ the group $G[U]$ is nontrivial.
\item
We say $G$ is \emph{locally dense}\index{locally dense action}
if for each point $x\in X$ and for each open neighborhood $U$ of $x$,
the closure of the orbit $G[U].x$ has nonempty interior.
\ee
\ed
Note that a locally dense action is locally moving if the space is
\emph{perfect}\index{perfect space} (i.e.~has no isolated points).
\begin{thm}\label{thm:rubin}
Let $X_1$ and $X_2$ be perfect, locally compact, Hausdorff topological spaces,
and let $G_i\le\Homeo(X_i)$ be locally dense groups, for $i=1,2$. Suppose that there exists an isomorphism of groups
\[
\Phi\co G_1\longrightarrow G_2.\]
Then there exists a homeomorphism \[\phi\co X_1\longrightarrow X_2\]
such that for all $g\in G_1$ and for all $x\in X_1$ we have
\[
\phi(g.x)=\Phi(g).\phi(x).\]\end{thm}
An account of this result, following the original arguments given by Rubin~\cite{Rubin1989,Rubin1996}, will occupy the remainder
of this section.
\subsection{First order expressibility of rigid stabilizers}
Suppose $A$ is a subset of the space $X$.
We denote the interior and the closure of a set $A\sse X$ by
$\Int_X A=\Int A$ and $\cl_X A=\cl A$, respectively.
The notion of the \emph{extended support}\index{extended support} of a homeomorphism will be essential for our discussion,
and we define it as
\[\suppe g:=\Int\cl \supp g,\]
for $g\in G$.
Note the inclusion
\[
\supp g\sse \suppe g\sse \cl{\supp g}.\]
For a subset $A$ of a group $G$, we let
\[
Z_G(A):=\{h\in G\mid [a,h]=1\text{ for all }a\in A\}, \text{ if }A\sse G.\]
We also write $Z_G(g):=Z_G(\{g\})$.
We remark that the reader may think of $Z_G(A)$ as a \emph{definable set}\index{definable set} in the sense of model theory
(see~\cite{hinman-book,marker-book,tz-book} for background, for instance). This is not truly a definable set in the sense of classical
model theory in general since $A$ is allowed
to be infinite, and so there may not be a single formula with parameters in $A$ that defines $Z_G(A)$.
But this subtle point will
not cause any trouble for us since we will only consider the case where $A$ is defined by a first order formula.
A remarkable fact that is crucially used in the proof of Rubin's Theorem is that,
for a locally moving group $G$,
the rigid stabilizer of the extended support of a $g\in G$ can be expressed in
terms of a first order formula in the language of group theory (i.e.~the language that admits a single binary operation and
a distinguished constant corresponding to the identity; sometimes it is useful to include the [definable] inversion function).
To state this sentence concretely, for each $f\in G$ we define
\begin{align*}
\eta_G(f):=\{g\in G \mid & \text{ for all }h\in G\setminus Z_G(f)\text{ there exist }f_1,f_2\in Z_G(g)\\
&\text{ such that }1\ne [[h,f_1],f_2]\in Z(g)\}.\end{align*}
We remark that for each $f\in G$, the set $\eta_G(f)$ is indeed definable from the parameter $f$, and the reader is encouraged
to write the sentence down in formal syntax.
\begin{thm}\label{thm:rubin-supp}
If $G$ is a locally moving group of homeomorphisms on a Hausdorff topological space,
then for all $f\in G$ we have that
\[
G[\suppe f]=Z_G\left(\left\{ g^{12}\mid g\in \eta_G(f)\right\}\right).\]
\end{thm}
We will establish Theorem~\ref{thm:rubin-supp} in this subsection.
Let us first make elementary observations from point set topology.
For a set $A$ in a space $X$, we let $\comp_X A=\comp A=X\setminus A$,
and
\[
\ext_X A=\ext A=A^\perp=\comp \cl A.\]
The latter of the above is called the
\emph{exterior}\index{exterior of a set} of $A$.
We say $A$ is a \emph{regular open set}\index{regular open set} of $X$ if
\[
A = A^{\perp\perp}.\]
We let $\Ro(X)$ denote the set of all regular open sets.
\begin{lem}\label{lem:ro-basic}
Let $X$ be a topological space.
\be[(1)]
\item For all subset $A$ of $X$, we have
\begin{align*}
A^{\perp}&= \Int \comp A,\\
A&\sse A^{\perp\perp}=\Int \cl A.
\end{align*}
\item\label{p:ppp} If $A\sse X$ is open, then $A^{\perp}=A^{\perp\perp\perp}$; in particular, we have $A^\perp\in\Ro(X)$.
\item If $A\sse X$ is open, then $\cl A = \cl \left(A^{\perp\perp}\right)$.
\item\label{p:ab-disj} If $A$ and $B$ are disjoint open subsets of $X$,
then $A^{\perp\perp}$ and $B$ are also disjoint.
\item\label{p:uv-perp}
Let $U\in\Ro(X)$.
If an open set $V\sse X$ satisfies
that $V\cap U^\perp=\varnothing$,
then $V\sse U$.
\item\label{p:uv-haus}
If $X$ is Hausdorff and if $x_0$ and $x_1$ are distinct points in $X$
then there exist a disjoint pair of regular open neighborhoods $U_i$ of $x_i$ for $i=0,1$.
\ee
\end{lem}
\bp
All are simple to check. For instance, to see (\ref{p:ab-disj}), it suffices to note
\[A^{\perp\perp}=\Int \cl A\sse \Int \cl \comp B=\Int \comp B\sse \comp B.\]
For (\ref{p:uv-perp}), we note
\[
V=\Int V \sse\Int \comp U^{\perp}= U^{\perp\perp}=U.\]
To see (\ref{p:uv-haus}), pick a disjoint pair of open neighborhoods $V_i$ of $x_i$ for $i=0,1$. By part (\ref{p:ab-disj}) we have that
\[
V_i\cap V_{1-i}^{\perp\perp}=\varnothing.\]
Parts (\ref{p:ppp}) and (\ref{p:uv-perp}) then imply that
\[ V_i \sse V_{1-i}^\perp.\]
Setting $U_i:=V_i^{\perp\perp}\cap V_{1-i}^\perp\in\Ro(X)$,
we see that
\[
U_0\cap U_1\sse V_0^{\perp\perp}\cap V_0^{\perp}=\varnothing,\]
and that
\[ x_i\in V_i\cap V_{1-i}^{\perp}\sse U_i.\]
The rest are simple mental exercises, the fun of which we will not ruin for the reader.
The book \cite{GH2009} is also a good reference.
\ep
By the lemma above, we see that $\suppe g$ is a regular open set whenever $g\in\Homeo(X)$.
Let us now note some basic facts about locally moving group actions.
\begin{lem}\label{lem:LM-basic}
Let $G$ be a locally moving group faithfully acting on a Hausdorff topological space $X$.
\be[(1)]
\item\label{p:move-supp}
If $U$ is a nonempty open subset of $\supp f$ for some $f\in G$ then there exists some nontrivial element
$g\in G[U]\setminus Z_G(f)$ such that $\supp g\cap f\supp g=\varnothing$.
\item\label{p:move-cap}
If $A$ is a nonempty open subset of $\bigcap_{i=1}^m \supp g_i$ for suitable elements \[\{g_1,\ldots,g_m\}\sse G,\] then there exists a
nonempty open subset $B\sse A$ such that
\[
B\cap \left(\bigcup_{i=1}^m g_i (B)\right)=\varnothing.\]
\item\label{p:move-fn}
For each open set $U\sse X$ and for each nonzero integer $n$, there exists an element $g\in G[U]$ and a
nonempty open set $V\sse U$ such that
\[
\{V, g(V), g^2(V),\ldots, g^n(V)\}\]
is a disjoint collection of open sets.
\item\label{p:move-fg}
If $f,g\in G\setminus\{1\}$ satisfy $\supp f\sse\supp g$,
then there exists some $h\in G[\supp f]$ such that $[hfh^{-1},g]\ne1$.
\item\label{p:move-sse}
For two regular open sets $U,V$ of $X$, we have that
\[
U\sse V\Longleftrightarrow G[U]\le G[V].\]
\ee
\end{lem}
\bp
(\ref{p:move-supp}) Pick $x\in U$ such that $f(x)\ne x$. We can find some open neighborhood $V$ of $x$ such that $V\cap f(V)=\varnothing$.
Since $G$ is locally moving we have some $g\in G[V]\le G[U]$. Then
\[
\supp g\cap f\supp g\sse V\cap f(V)=\varnothing.\]
Since $f\supp g\ne\supp g$, we have $g\not\in Z_G(f)$.
(\ref{p:move-cap})
Pick $a\in A$. Since $a\ne g_i(a)$ for each $i$, we can find some open neighborhood $B_i$ of $a$ such that $B_i\cap g_i(B_i)=\varnothing$.
Then $B:=\bigcap_i B_i$ satisfies the conclusion.
(\ref{p:move-fn})
This will be strengthened in Theorem~\ref{thm:lawless}.
Let us give a short proof here.
Since $U$ is Hausdorff and $G[U]$ is locally moving, it suffices to consider the case that $U=X$.
The case $n=1$ is trivial since $X$ is Hausdorff.
Let us assume the conclusion for $n$, and inductively prove the case $n+1$.
We have some nonempty open set $V\sse X$ and some element $g\in G$ such that
$\{g^i(V)\}_{0\le i\le n}$ is a disjoint collection.
If we have some $v\in V$ such that $g^{n+1}(v)\ne v$, then some open neighborhood of $v$ satisfies the conclusion for $n+1$.
Hence, we may assume that $g^{n+1}\restriction_V=\Id_V$. Pick an arbitrary element $h\in G[V]$, and a suitable nonempty open
set $W\sse V$ such that $W\cap h(W)=\varnothing$.
We note that
\[ (gh)^i(W)=g^i\circ h(W)\sse g^i(V)\]
forms a disjoint collection
for $i\in\{0,\ldots,n\}$. Since
\[
(gh)^{n+1}(W)\cap W = g^{n+1}h(W)=h(W)\]
is disjoint from $W$, we have that
\[
\{W,gh(W),\ldots,(gh)^{n+1}(W)\}\]
is a disjoint collection.
We just found an open set $W$ satisfying the conclusion with the element $gh\in G$.
(\ref{p:move-fg}) We may assume $[f,g]=1$, for otherwise we can set $h:=1$.
By the above, we can choose some nonempty open set $A\sse\supp f$ such that
\[
A\cap(f(A)\cup g(A))=\varnothing.\]
Depending on whether or not $\supp (fg)$ intersects $A$,
we can find a nonempty open set $B\sse A$ such that
one of the following holds:
\begin{itemize}
\item
$B\cap fg(B)=\varnothing$;
\item
$\supp (fg)\cap B=\varnothing$.\end{itemize}
Pick a nontrivial element $h\in G[B]$ and nonempty open set $C\sse B$ such that
\[
\{C,h(C),h^2(C)\}\]
is a disjoint collection; see Lemma~\ref{lem:LM-basic} (\ref{p:move-fn}).
By the choice of $A$, we see that
\begin{align*}
x_1&:=g\cdot hfh^{-1}(hC)=ghf(C)=gf(C)=fg(C),\\
x_2&:=hfh^{-1}\cdot g(hC)=hfg(hC).\end{align*}
In the case when $B\cap fg(B)=\varnothing$, we have that
\[x_2=fg(hC)\ne fg(C)=x_1.\]
If $\supp (fg)\cap B=\varnothing$, then we have
\[x_1=fg(C)=C\ne h^2(C)= hfgh(C)=x_2.\]
This completes the proof.
(\ref{p:move-sse}) The forward direction is trivial. For the reverse direction,
assume that $U\not\sse V$. By Lemma~\ref{lem:ro-basic} (\ref{p:uv-perp}), we have that
$W:=U\cap V^{\perp}\ne\varnothing$. By the locally moving hypothesis, we have some $1\ne f\in G[W]\le G[U]$.
Since \[W\sse V^{\perp}=\comp\cl V\sse \comp V,\]
we see that $\supp f\cap V=\varnothing$. In particular, we have $G[U]\not\sse G[V]$.
\ep
Note the following simple combinatorial fact.
\begin{lem}\label{lem:sets}
Let $m\ge1$ be an integer and let $Z$ be a set.
Suppose a collection of subsets \[\{A_{ij}\sse Z\}_{0\le i\le m,1\le j\le m}\]
has the property that for each $j=1,\ldots,m$
the sets
\[
A_{0j}, A_{1j},\ldots,A_{mj}\]
are pairwise disjoint.
Then we have that
\[\bigcap_{i=0}^m \left(\bigcup_{j=1}^m A_{ij}\right)=\varnothing.\]
\end{lem}
\bp
Let $x\in Z$. We imagine to have $m$ pigeons and $m+1$ pigeonholes.
We let the $j$--th pigeon stay in the $i$--th hole if $x\in A_{ij}$; it is clear from the hypothesis that each pigeon can stay in at most one hole.
For some $0\le i\le m$, the $i$--th hole will be vacant.
This implies that $x$ does not belong to $\bigcup_j A_{ij}$.\ep
The following lemma asserts that the group theoretic condition $g\in\eta_G(f)$ ``almost'' detects the topological condition that $f$ and $g$ have disjoint supports.
\begin{lem}\label{lem:eta}
Let $G$ be a locally moving group faithfully acting on a Hausdorff topological space $X$.
For two elements $f,g\in G$ we have the following conclusions.
\be[(1)]
\item
If $\supp f\cap\supp g=\varnothing$, then $g\in \eta_G(f)$.
\item
If $g\in\eta_G(f)$ then
\[\supp f\cap \supp g^3\cap \supp g^4=\varnothing.\]
\ee
\end{lem}
We will often omit the subscript $G$ in $Z_G$ or $\eta_G$ when the meaning is clear.
\bp[Proof of Lemma~\ref{lem:eta}]
(1) In order to show $g\in\eta(f)$, pick an arbitrary $h\in G\setminus Z(f)$.
In particular, we have $\supp f\cap \supp h\ne\varnothing$.
By Lemma~\ref{lem:LM-basic} (\ref{p:move-supp}) we can pick elements $f_1,f_2\in G$
satisfying the following conditions:
\begin{itemize}
\item $f_1\in G[\supp f\cap \supp h]\setminus Z(h)$ and $\supp f_1 \cap h\supp f_1=\varnothing$;
\item $f_2\in G[\supp f_1]\setminus Z(f_1)$ and $\supp f_2 \cap f_1\supp f_2=\varnothing$.
\end{itemize}
Since $\supp f_2\sse \supp f_1\sse \supp f$, we have $f_1,f_2\in Z(g)$.
Note that
\[\supp hf_1h^{-1}=h\supp f_1\] is disjoint from $\supp f_1$.
It follows that
\[
[[h,f_1],f_2]=[hf_1h^{-1}\cdot f_1^{-1},f_2]=[f_1^{-1},f_2]
=\left(f_1^{-1}f_2f_1\right)\cdot f_2^{-1}\in Z(g).\]
The last term of the equality is nontrivial, since it is written as the product of two nontrivial homeomorphisms with
disjoint supports. This shows that $g\in \eta_G(f)$.
(2) Let us assume $g\in \eta(f)$. Assume for contradiction that
\[ A:=\supp f\cap \supp g^3\cap \supp g^4\ne\varnothing.\]
In particular, we have $A\sse \bigcap_{i=1}^4 \supp g^i$.
By Lemma~\ref{lem:LM-basic} (\ref{p:move-cap}) there exists a
nonempty open set $B\sse A$ satisfying
\[
B\cap \left( \bigcup_{i=1}^4 g^i(B)\right)=\varnothing.\]
It follows that $\{g^i (B)\}_{0\le i\le 4}$ is a disjoint family of nonempty open sets.
By the same lemma, we can also find an element $h\in G[B]\setminus Z(f)$.
Using the hypothesis that $g\in\eta(f)$, we know of the existence of elements $f_1, f_2\in Z(g)$ such that
\[ u:=[[h,f_1],f_2]\in Z(g)\setminus\{1\}.\]
Setting $S:=\{1,f_1,f_2,f_2f_1\}\sse Z(g)$, we easily see that
\[
\bigcup_{i=0}^4 g^i(\supp u) = \supp u
=
\supp \left(h\cdot (f_1 h^{-1}f_1^{-1})\cdot (f_2f_1 h f_1^{-1}f_2^{-1})(f_2 h^{-1}f_2^{-1})\right)\sse S(B),\]
and that
\[
\supp u\sse \bigcap_{i=0}^4 \bigcup_{s\in S} g^{-i}s(B).\]
This contradicts Lemma~\ref{lem:sets}, since for each $s\in S$ the collection
\[\{ g^{-i}s(B)=sg^{-i}(B)\}_{0\le i\le 4}\]
consists of disjoint sets.
\ep
We are now ready to express the set $G[\suppe f]$ in the language of group theory. It will be convenient for us to introduce the notation
\[\xi_G^m(f):=Z_G\left(\left\{ g^m\mid g\in \eta_G(f)\right\}\right),\]
which is defined for every group $G$ and every element $f\in G$.
\bp[Proof of Theorem~\ref{thm:rubin-supp}]
Let $m\in\bZ_{>0}$.
Let us first pick arbitrary $q\in G[\suppe f]$ and $g\in \eta(f)$.
Since \[\supp g^{12}\sse \supp g^3\cap\supp g^4,\] we see from Lemma~\ref{lem:eta} that
\[\supp f\cap\supp\left(g^{12}\right)=\varnothing.\]
Lemma~\ref{lem:ro-basic} implies that
\[\suppe f\cap\supp\left(g^{12}\right)=\varnothing.\]
Since $\supp q\sse \suppe f$, we see that $q$ commutes with $g^{12}$.
We have thus shown that \[G[\suppe f]\le \xi_G^{12}(f)\le\bigcup_{m\ge1}\xi_G^m(f).\]
To see the opposite inclusion, pick arbitrary $m\in\bZ_{>0}$ and $q\in G\setminus G[\suppe f]$.
Since $\supp q\not\sse\suppe f$, Lemma~\ref{lem:ro-basic} (\ref{p:uv-perp}) implies that the open set
\[
A:=\supp q\cap \left(\suppe f\right)^\perp\]
is nonempty. We can therefore pick some $g\in G[A]$ such that $g^m\ne1$,
using Lemma~\ref{lem:LM-basic} (\ref{p:move-fn}).
Summarizing, we have
\[
\varnothing\ne \supp g^m\sse \supp g\sse A\sse \supp q.\]
Using part (\ref{p:move-fg}) of the same lemma, we can pick
some $h\in G\left[\supp g^m\right]$ such that
$q$ does not commute with $hg^mh^{-1}$.
We claim that the elements $f$ and $hgh^{-1}$
have disjoint supports.
Indeed,
applying Lemma~\ref{lem:ro-basic}, we see that
\begin{align*}
\supp hgh^{-1}&=h\supp g=\supp g
\sse A\sse (\suppe f)^\perp\\
&=(\supp f)^{\perp\perp\perp}=(\supp f)^{\perp}
\sse \comp\supp f.\end{align*}
The claim is thus proved.
Lemma~\ref{lem:eta} implies that $hgh^{-1}\in\eta(f)$,
and hence that $q\not\in \xi_G^m(f)$.
This implies that $G\setminus G[\suppe f]\sse G\setminus \xi_G^m(f)$, for an arbitrary $m\in\bZ_{>0}$.
So far, we have proved that
\[
\bigcup_{m\ge1} \xi_G^m(f)\le G[\suppe f]\le \xi_G^{12}(f)\le\bigcup_{m\ge1} \xi_G^m(f).\]
This completes the proof.
\ep
\begin{rem}\label{rem:rubin-supp}
\be[(1)]\item In the above we have established a purely group theoretic statement \[
Z_G\left(\left\{ g^{m}\mid g\in \eta_G(f)\right\}\right)\sse Z_G\left(\left\{ g^{12}\mid g\in \eta_G(f)\right\}\right)\]
for all $m\ge1$. The mysterious number 12 seems to be intrinsically related to the locally moving hypothesis of $G$ (and of course
comes out of the proof of Theorem~\ref{thm:rubin-supp} as the least common multiple of $3$ and $4$).
\item
We used in the proof the fact that if $\supp q\setminus \suppe f\ne\varnothing$
then $\supp q\setminus \supp f$ contains a nonempty open set.
The same is not true under the weaker hypothesis that $\supp q\setminus\supp f\ne\varnothing$.
This is one of the places where an extended support turns out to be more useful for us than an open support.
\ee
\end{rem}
From now on, we write
\[
\xi_G(f):=\xi_G^{12}(f)=Z_G\left(\left\{ g^{12}\mid g\in \eta_G(f)\right\}\right)\le G.\]
We can now express inclusions between the sets
\[\{\suppe f\mid f\in G\}\] purely in terms of group theory.
\begin{cor}\label{cor:rubin-supp}
Let $G$ be a locally moving group of homeomorphisms on a Hausdorff topological space $X$. Then for all $f,g\in G$,
we have $\suppe f\sse \suppe g$ if and only if $\xi_G(f)\le\xi_G(g)$.
\end{cor}
\bp
This is a trivial consequence of Theorem~\ref{thm:rubin-supp} and Lemma~\ref{lem:LM-basic} (\ref{p:move-sse}).
\ep
\subsection{From groups to Boolean algebras}\label{sec:gp-bool}
The next step in the proof of Rubin's Theorem is to extract a bijection between regular open sets of spaces that respects their
\emph{Boolean structures}\index{Boolean structure}.
Let $B$ be a set containing two distinguished elements $0$ and $1$.
If there exist binary operations $\wedge,\vee$
and a unary operation $\perp$ satisfying the following natural set-theoretic axioms, then we say the structure
$(B,\wedge,\vee,\perp,0,1)$ is a \emph{Boolean algebra}\index{Boolean algebra}.
\begin{itemize}
\item both of $\wedge$ and $\vee$ are commutative and associative;
\item $\wedge$ and $\vee$ are distributive, in the sense that
\begin{align*}
u\wedge(v\vee w) &=(u\wedge v)\vee(u\wedge w),\\
u\vee(v\wedge w) &=(u\vee v)\wedge(u\vee w).\end{align*}
\item we have $u\vee(u\wedge v)=u=u\wedge(u\vee v)$.
\item $u\vee u^\perp=1$ and $u\wedge u^{\perp}=0$.
\end{itemize}
Note that
$0,1,\wedge,\vee,\perp$ are axiomatized to reflect the behavior of the set theoretic objects $\varnothing,X,\cap,\cup,\comp$, respectively.
See~\cite{GH2009} or~\cite{Jech2003} for a concise introduction to Boolean algebras.
A Boolean algebra $B$ naturally comes with
a partial order $\le$ defined
by
\[v\le u\Longleftrightarrow v\wedge u^{\perp}=0.\]
It is helpful to regard $v\wedge u^{\perp}$ as the ``subtraction'' of $u$ from $v$.
If a Boolean algebra $B$ admits a least upper bound and a greatest lower bound for every nonempty
subset $A\sse B$ then we say $B$ is \emph{complete}\index{complete Boolean algebra}.
We say $B$ is \emph{atomless}\index{atomless Boolean algebra} if it does not contain a nonzero minimal element.
For the purpose of this book, one may only consider the Boolean algebra of regular open sets defined as follows.
Let $X$ be a Hausdorff topological space.
Recall that we denote the collection of regular open sets in $X$ by $\Ro(X)$, namely the sets $U\sse X$ satisfying
\[ U = \Int\cl U=U^{\perp\perp}.\]
The set $\Ro(X)$ is equipped with the unary operation $\perp$, and a binary operation $\vee$ defined by
\[
U\vee V:=(U\cup V)^{\perp\perp}.\]
Setting \[\wedge:=\cap,\quad 0:=\varnothing,\quad 1:=X,\]
we have that $\Ro(X)$ acquires the structure of a Boolean algebra.
A particularly useful fact for us is that the ``subtraction'' of two regular open sets can be defined by
\[
V\cap U^{\perp}\in\Ro(X).\]
We have seen in Lemma~\ref{lem:ro-basic} (\ref{p:uv-perp}) that if $V\not\sse U$ for regular open sets $U,V$ then $V\cap U^{\perp}\ne\varnothing$.
In other words, the Boolean partial order for $\Ro(X)$ coincides with the set theoretic inclusion $\sse$.
Let us note that a self-homeomorphism induces a Boolean automorphism, in the following strong sense.
\begin{lem}\label{lem:top-bool}
Let $X$ is a Hausdorff topological space, and let $G\le\Homeo(X)$.
If $U\in\Ro(X)$,
then we have that
\[G[U] = \{g\in G\mid g(V)=V\text{ for all regular open set }V\sse U^\perp\}.\]
\end{lem}
\bp
If $g\in G[U]$, then $g$ restricts to the identity on $U^\perp$ and so, $g$ belongs to the right handside.
For the reverse direction, suppose $g\not\in G[U]$. By Lemma~\ref{lem:ro-basic} we can find some $z\in \supp g\setminus U^\perp$.
By the same Lemma~\ref{lem:ro-basic} we can pick a pair of disjoint regular open sets $V_0$ and $V_1$ of $z$ and $g(z)$, respectively.
Since $g(z)\in g(V_0)$ we see that $g(V_0)\ne V_0$. This implies that $g$ does not belong to the right hand side, completing the proof.\ep
By setting $U=\varnothing$ we see that the group $\Homeo(X)$ acts naturally and faithfully on the Boolean algebra $\Ro(X)$
by Boolean automorphisms. For $G\le\Homeo(X)$ we continue to denote this new action on $\Ro(X)$ still as $G$.
In particular, the group $G[U]\le G$ can be described purely in terms of the Boolean algebra $\Ro(X)$,
the Boolean automorphic action of $G$ and its element $U$.
For a family $F\sse \Ro(X)$ we define the supremum and the infimum of this family by
\begin{align*}
\sup F&:=\left(\bigcup F\right)^{\perp\perp}\in \Ro(X),\\
\inf F&:=\left(\bigcap F\right)^{\perp\perp}\in \Ro(X).
\end{align*}
The Boolean algebra $\Ro(X)$ is {complete},
since these are indeed the least upper bounds and the greatest lower bounds
with respect to the inclusion.
If we further assume that $X$ is a perfect, then $\Ro(X)$ is atomless.
Observe that if $G$ is a locally moving group of homeomorphisms on $X$ then $X$ must be perfect.
We now prove that an abstract group isomorphism between two locally moving groups of homeomorphisms of Hausdorff topological spaces
induces a Boolean isomorphism of regular open sets of those spaces, that is equivariant with respect to this group isomorphism.
\begin{thm}\label{thm:gp-bool2}
Let $X_1$ and $X_2$ be Hausdorff topological spaces.
If a group isomorphism \[\Phi\co G_1\longrightarrow G_2\] is given,
where $G_i\le\Homeo(X_i)$ are locally moving subgroups for $i=1,2$,
then there uniquely exists a Boolean isomorphism \[\Psi\co \Ro(X_1)\longrightarrow \Ro(X_2)\]
such that the following diagram of set-theoretic maps are commutative for all $g\in G_1$:
\[\begin{tikzcd}
G_1 \arrow{r}{\suppe}\arrow{d}[swap]{\Phi} & \Ro(X_1)\arrow{r}{g}\arrow{d}{\Psi}&
\Ro(X_1)\arrow{d}{\Psi}\arrow{r}{G_1[\cdot]} & \operatorname{Subgroups}(G_1)\arrow{d}{\Phi}\\
G_2 \arrow{r}{\suppe} & \Ro(X_2)\arrow{r}{\Phi(g)}& \Ro(X_2)\arrow{r}{G_2[\cdot]} & \operatorname{Subgroups}(G_2).\end{tikzcd}
\]
\end{thm}
The first horizontal maps are sending $g$ to $\suppe g$, for $g$ in $G_1$ or $G_2$.
The third horizontal maps send a regular open set $U\in\Ro(X_i)$ to the corresponding rigid stabilizer $G_i[U]$.
The bijection $\Psi$ above respects the Boolean objects and operations
\[ \cap, \vee,\perp,\varnothing,X_i.\]
Note that suprema and infima are respected as well, since the partial order $\sse$ is respected.
The horizontal maps in the middle square are actions of $G_i$ on the Boolean algebras $\Ro(X_i)$.
The deduction of Theorem~\ref{thm:gp-bool2} from Theorem~\ref{thm:rubin-supp}
is relatively simple and may be regarded as a Boolean algebra ``language game''. We remark one logical subtlety:
Theorem~\ref{thm:rubin-supp} characterizes rigid stabilizers in terms of first order formulae with parameters, and
Theorem~\ref{thm:gp-bool2} follows in some sense because of ``preservation of logical implication". Whereas the language of
group theory is retained, the logic that is being used to establish the implication is different.
Indeed, since suprema and quantification over subgroups will be
required, passage to second order logic is warranted in the proof of
Theorem~\ref{thm:gp-bool2}.
Suppose $(Q,\le)$ is a partially ordered set. For each $q\in Q$ we write
\[Q\restriction_q:=\{x\in Q\mid x\le q\}.\]
The main example that are considered here is the case when $Q$ is the set of nonzero elements in a Boolean algebra.
We say that a subset $P$ of $(Q,\le)$ is \emph{dense}\index{dense subset of a partial order}
if every $q$ in $Q$ satisfies $Q\restriction_q\cap P\ne\varnothing$.
If $B$ is a complete Boolean algebra and if $P\sse B\setminus\{0\}$ is a dense subset,
then every element $b\in B\setminus\{0\}$ can be expressed as
\[ b:=\sup (B\restriction_b\cap P);\] cf.~Chapter 25 of~\cite{GH2009}.
We note the following result on Boolean algebras.
\begin{lem}\label{lem:ba-complete}
Let $B$ be a complete Boolean algebra,
and let $P\sse B\setminus\{0\}$ be a dense subset.
If there exists an order--preserving dense embedding
\[f\co P\longrightarrow B'\setminus\{0\}\]
for some complete Boolean algebra $B'$,
then $f$ uniquely extends to a Boolean isomorphism from $B$ to $B'$.
\end{lem}
Lemma~\ref{lem:ba-complete} can be succinctly expressed by the slogan that
every \emph{separative}\index{separative subset of a partially ordered set}
dense partially ordered set admits a unique completion to a Boolean algebra. Here, a \emph{separative} partially ordered
set is one that is a dense subset in the set of nonzero elements of a Boolean algebra. We omit the proof, which is elementary to verify.
The interested reader is directed to~{\cite[Theorem 14.10]{Jech2003}} for a further discussion.
\bp[Proof of Theorem~\ref{thm:gp-bool2}]
For each $i=1,2$ we let
\[
P_i:=\{\suppe g\mid g\in G_i\setminus\{1\}\}\sse \Ro(X_i)\setminus\{\varnothing\}.\]
Since $G_i$ is locally moving, we see that $P_i$ is dense in $\Ro(X_i)\setminus\{\varnothing\}$. Define a map
\[\Psi\co P_1\longrightarrow P_2\]
by the formula
\[
\Psi(\suppe g):=\suppe \Phi(g).\]
Recall the notation $\xi_G(g)$ from the previous subsection.
Using the fact that
\[\Phi(\xi_{G_1}(g))=\xi_{G_2}(\Phi(g)),\]
we can apply Corollary~\ref{cor:rubin-supp}
and deduce that $\Psi$ is a well-defined, order--preserving bijection
between $P_1$ and $P_2$.
Since $\Ro(X_i)$ is complete, we see from Lemma~\ref{lem:ba-complete}
that $\Psi$ uniquely extends to a Boolean isomorphism
\[\Psi\co \Ro(X_1)\longrightarrow\Ro(X_2).\]
The first square in the diagram commutes by definition.
The third one commutes since for $g\in G_1$ and $U\in\Ro(X_1)$ we have
\[g\in G_1[U]
\Longleftrightarrow \suppe g\sse U
\Longleftrightarrow \suppe \Phi(g)\sse \Psi(U)
\Longleftrightarrow \Phi(g)\in G_2[\Psi(U)].\]
It suffices now to prove that $\Psi$ intertwines the two actions.
For $\suppe f\in P_1$ and $g\in G_1$ we note
\[
\Psi(g.\suppe f)=\Psi(\suppe gfg^{-1})=\suppe \Phi(gfg^{-1})=\Phi(g)\suppe\Phi(f).\]
Note also that $P_i$ is dense $\Ro(X_i)$, and that $\Psi$ respects the suprema.
Since $\Ro(X_i)$ is complete, so is the proof.
\ep
\subsection{From Boolean algebras to topologies}
So far, we have extracted a Boolean isomorphism between regular open sets of Hausdorff topological spaces,
under the hypothesis that they admit isomorphic locally moving groups. We now consider the stronger hypotheses of local compactness
of the spaces, together with local density of groups, and deduce that the spaces are in fact homeomorphic. For this,
we will need to identify a point in terms of regular open sets, thus crucially using the notion of an ultrafilter.
Let $B=B(\wedge,\vee, \perp,0,1)$ be a Boolean algebra. Recall that a \emph{filter}\index{filter} is a nonempty subset $F\sse B$ such that
the following two conditions hold:
\be[(i)]
\item If $a,b\in F$ then $a\wedge b\in F$;
\item If $a\in F$ and $a\le b$ for $b\in B$, then $b\in F$.
\ee
A proper filter $F\subsetneq B$ is an \emph{ultrafilter}\index{ultrafilter} if it satisfies the following additional condition:
\be[(i)]
\addtocounter{enumi}{2}
\item For all $b\in B$ either $b\in F$ or $b^\perp\in F$.
\ee
It is routine to check that a proper filter of $B$ is an {ultrafilter} if and only if it is maximal.
A subset $F_0$ of $B$ is said to have the \emph{finite intersection property}\index{finite intersection property for a filter}, if we have
\[
b_1\wedge \cdots \wedge b_k\ne0\]
for all $k\ge1$ and $b_i\in F_0$. It is a standard fact that each subset of $B$ satisfying the
finite intersection property is contained in an ultrafilter of $B$~\cite[Chapter 20, Exercise 12]{GH2009}. Proofs of this fact
use the Axiom of Choice, but the existence of ultrafilters is weaker than the Axiom of Choice.
The \emph{Stone space}\index{Stone space} $\CS(B)$ is the set of all ultrafilters in $B$, equipped with the subspace topology of $2^B$.
Although the topology of $\CS(B)$ is not of our concern in this book,
we note that this space is compact and totally disconnected; it is even perfect if $B$ is atomless.
The following lemma hints at the idea of identifying points in a space with ultrafilters of regular open sets.
The set of all open neighborhoods of a point $x$ in a space $X$ will be denoted by $\Nbr_X(x)=\Nbr(x)$.
\begin{lem}\label{lem:haus-filter}
Let $X$ be a Hausdorff topological space. For each ultrafilter $F$ on the Boolean algebra $\Ro(X)$, we define
\[
A_F:=\bigcap_{U\in F}\cl U.\]
The following conclusions hold.
\be[(1)]
\item\label{p:single} The set $A_F$ contains at most one point.
\item If $F$ contains a relatively compact regular open set, then $A_F\ne\varnothing$.
\item If $x\in A_F$ then $\operatorname{Nbr}(x)\cap\Ro(X)\sse F$.
\item\label{p:fx} For each $x\in X$ there exists some $F\in \Ro(X)$ such that $x\in A_{F}$.
\ee
\end{lem}
\bp
(1) Suppose $x,y$ are distinct points in $A_F$. We can find disjoint regular open neighborhoods $U,V$ of them by part
(\ref{p:uv-haus}) of Lemma~\ref{lem:ro-basic}. We may choose $V:=U^\perp$.
By maximality of the ultrafilter, we have either $U\in F$ or $U^\perp\in F$. If $U\in F$ then we have that $\cl U\cap U^\perp=\varnothing$
and that $y\not\in A_F$, a contradiction. The case $U^\perp\in F$ is similar.
(2) Let $V\in F$ be relatively compact.
Since $F$ satisfies the finite intersection property as a filter,
the family of compact sets
\[
\{ \cl (U\cap V) \mid U\in F\}\]
satisfies the finite intersection property as a collection of subspaces in $X$.
It follows that
\[
\varnothing
\ne
\bigcap
\{ \cl (U\cap V) \mid U\in F\}\sse A_F.\]
(3) Let $V\in\Nbr(x)$ be regular open. For each finite subcollection \[\{U_1,\ldots,U_k\}\sse F,\]
we have that $U_0:=\bigcap_i U_i\in F$. Moreover, we have that
\[x\in A_F\cap V \sse(\cl U_0)\cap V.\]
It follows that $U_0\cap V\ne\varnothing$. We have that
$F\cup\{V\}\sse \Ro(X)$ satisfies the finite intersection property.
Since $F$ is an ultrafilter, we see that $V\in F$.
(4) Using the finite intersection property
one can find an ultrafilter $F_x$ containing the set $\Nbr(x)\cap\Ro(X)$.
Assume for contradiction that the closure of some $U\cap F_x$
does not contain $x$. In other words, we have $x\in U^\perp$.
Then we have that $U^\perp\in \Nbr(x)\cap\Ro(X)\sse F_x$, which contradicts that $F_x$ is an ultrafilter.
\ep
The preceding lemma justifies the following definition.
\bd
Let $X$ be a Hausdorff topological space.
We say $F\in\CS(\Ro(X))$ is a \emph{topological filter}\index{filter!topological} of $X$ if
\[\bigcap_{U\in F}\cl U\neq\varnothing.\] We denote $\TF(X)$ the set of all topological filters of $X$.
We write
\[
\rho_X\co \TF(\Ro(X))\longrightarrow X\]
for the surjection defined by the formula
\[\{\rho_X(F)\}=\bigcap_{U\in F}\cl U.\]
\ed
A topological filter is defined by the topology of the space being considered.
Rubin defined the following collection of ultrafilters, which is given purely in terms of a group action on a Boolean algebra.
It will turn out that those two concepts coincide in the setting of Rubin's Theorem.
\bd
Let $B$ be a Boolean algebra, and let $G$ be a group of Boolean automorphisms of $B$. We say an ultrafilter $F$ of $B$ is a
\emph{$G$--local filter}\index{local filter} if there exists some $b\in F$ such that \[B\restriction_b\sse \{0\}\cup G.F.\]
We will write $\CS(B;G)$ for the set of all $G$--local filters in $B$.
\ed
It will be convenient for us to introduce the following construction.
\begin{notation}
Let $H$ be an automorphism group of a Boolean algebra $B$.
For an ultrafilter $F\in\CS(B)$, we define
\[
H\left\{F^\perp\right\}:=
\bigcup_{w\in F}
\{
g\in H\mid
g(a)=a\text{ for all }a\in B\restriction_w\}.\]\end{notation}
This is indeed a group, as can be seen by applying the finite intersection property of $F$.
In the case where $G\le\Homeo(X)$, we can also regard $G$ as a group of Boolean automorphisms of $\Ro(X)$ as in
Lemma~\ref{lem:top-bool}.
Then, for an ultrafilter $F\in\CS(\Ro(X))$, we obtain
\[
G\left\{F^\perp\right\}=
\bigcup_{W\in F} G\left[W^\perp\right].\]
The following theorem on a single space $X$ will provide us a connection from a Boolean isomorphism to a topological homeomorphism.
\begin{thm}\label{thm:bool-homeo}
Suppose $X$ is a locally compact, Hausdorff, perfect topological space,
and suppose that $G$ is a locally dense subgroup of $\Homeo(X)$.
Then the following conclusions hold.
\be[(1)]
\item\label{p:tf} We have that \[ \TF(X) = \CS(\Ro(X),G).\]
\item\label{p:rho-x}
For each $x\in X$, we have that
\[
\Nbr(x)\cap\Ro(X)
=\bigcap \rho_X^{-1}(x)
.\]
\item\label{p:rhoi-rho}
For each $F\in \TF(X)$, we have that
\[
\TF(X)\setminus\rho_X^{-1}\circ \rho_X(F)
= \CS\left(\Ro(X), G\left\{F^\perp\right\}\right).\]
\ee
\end{thm}
\begin{rem}When we use the notation $\CS(\Ro(X),H)$ for some group $H\le\Homeo(X)$,
we are regarding $H$ as a group of Boolean automorphisms of $\Ro(X)$; this notation (as well as its ambiguity) is
justified by Lemma~\ref{lem:top-bool}.\end{rem}
\bp[Proof of Theorem~\ref{thm:bool-homeo}, parts (\ref{p:tf}) and (\ref{p:rho-x})]
(\ref{p:tf})
Suppose $F\sse\Ro(X)$ is $G$--local filter of $\Ro(X)$.
By definition, we can find some $U\in F$ such that
\[\Ro(X)\restriction_U \sse \{\varnothing\}\cup G.F.\]
By local compactness, we can find a relatively compact nonempty regular open set $V\sse U$.
Since $V\in G.F$, we see from Lemma~\ref{lem:haus-filter} that $F\in\TF(X)$.
Conversely, suppose $F\in \TF(X)$ and let $x:=\rho_X(F)$.
By local density, we have a nonempty regular open set $U$ contained in the closure of $G.x\sse X$.
Consider an arbitrary $V\in \Ro(X)\restriction_U\setminus\{\varnothing\}$.
Since $V$ is inside the closure of $G.x$, we can find a $g\in G$ such that
$g.x\in V$.
By the same lemma as above, we have that
\[
x\in g^{-1}(V) \in \Nbr(x)\cap \Ro(X)\sse F.\]
This implies that $F$ is $G$--local.
(\ref{p:rho-x})
We see from Lemma~\ref{lem:haus-filter} that
\[\Nbr(x)\cap\Ro(X)\sse F\]
for all $F\in \TF(X)$ satisfying $\rho_X(F)=x$.
Conversely, suppose $U$ is an element of $\bigcap \rho_X^{-1}(x)$.
Assume for a contradiction that $x\not\in U$.
For an arbitrary regular open neighborhood $V$ of $x$, we have that
$V\not\sse U$, which implies $V\cap U^\perp=\varnothing$.
The family
\[
F_0:=\left(\Nbr(x)\cap\Ro(X)\right)\bigcup \left\{ U^\perp\right\}\]
enjoys the finite intersection property, and hence extends to some $F\in \CS(\Ro(X))$.
Note that the intersection of $\cl V$ for all $V\in F_0$ is already a singleton, namely $\{x\}$.
This implies that $\rho_X(F)=x$, and by hypothesis, $U\in F$.
This implies $\{U,U^\perp\}\sse F$, which is a contradiction.
\ep
We postpone the proof of the part (\ref{p:rhoi-rho}) above, first observing a general fact.
\begin{lem}\label{lem:pointed}
Let $X$ be a perfect Hausdorff topological space.
Fix a point $\infty\in X$, and let $Y:=X\setminus\{\infty\}$.
Then the following conclusions hold.
\be[(1)]
\item
There exists a Boolean isomorphism
\[\psi\co \Ro(X)\longrightarrow\Ro(Y)\]
satisfying
\begin{align*}
\psi(U)&:=U\cap Y,&\text{ if }U\in\Ro(X),\\
\psi^{-1}(V)&:=\Int_X\cl_X V,&\text{ if }V\in\Ro(Y).\end{align*}
and inducing a bijection
\[
\TF(X)\setminus\rho_X^{-1}(\infty) \longrightarrow \TF(Y).\]
\item If $G\le\Homeo(X)$ is locally dense, then for each $F\in\rho_X^{-1}(\infty)$
the group
\[
G\left\{F^\perp\right\}\] fixes the point $\infty$ and acts locally densely on $Y$.
\ee
\end{lem}
\bp
Note first that every $U\in\Ro(X)$ satisfies
\begin{align*}
(\Int_X U)\cap Y&=\Int_Y (U\cap Y),\\
(\cl_X U)\cap Y&=\cl_Y (U\cap Y),\\
(\ext_X U)\cap Y&=\ext_Y (U\cap Y),\\
\Int_X \cl_X (U\cap Y)&=U.
\end{align*}
It is then easy to see then that $\psi$ as defined in the lemma is an isomorphism.
Suppose $F\in\TF(X)$ satisfies \[y:=\rho_X(F)\ne \infty.\]
Then we have the equalities
\[
y\in\bigcap_{U\in F} \cl_X(U)\cap Y=\bigcap_{U\in F} \cl_Y(U\cap Y)
=\bigcap_{V\in \psi(F)} \cl_Y V,\]
which imply that $ \psi(F)\in \TF(Y)$.
Conversely if $F\in \TF(Y)$ then
\[\infty\ne \rho_Y(F)\in \bigcap_{U\in F}\cl_Y(U)
\sse
\bigcap_{U\in F}\cl_X(U)
=\bigcap_{V\in\psi^{-1}(F)} \cl_X(V)
.\]
This implies that $ \psi^{-1}(F)\in\TF(X)\setminus\rho_X^{-1}(\infty)$.
To see part (2), let $g\in G\left\{F^\perp\right\}$. We have some $W\in F$
such that $\supp g\sse W^\perp$.
Since $\infty\in \cl_X W$, we obtain $g\in \Fix\infty$.
Lastly, to verify the local density let us pick an arbitrary point $\infty\ne y\in Y$ and its open neighborhood $V\sse Y$.
By shrinking $V$ if necessary (using the fact that $Y$ is locally compact and Hausdorff), we may assume that $V$ is regular
and that $\infty\in \ext_X V$.
In particular, we see that $G[V].y\sse V$
and that $G[V]$ fixes $\infty$.
By the local density of $G$, some element $V_0\in\Ro(X)$ satisfies
\[
\varnothing\ne V_0\sse \cl_X \left(G[V].y\right)\sse \cl_X V=\comp_X \ext_X V.\]
This implies that $\infty\not\in V_0$ and that
\[
V_0\sse Y\cap \cl_X \left(G[V].y\right)=\cl_Y \left(G[V].y\right).\]
We see that the action of $G\left\{F^\perp\right\}$ on $Y$ is locally dense.
\ep
\bp[Proof of Theorem~\ref{thm:bool-homeo}, part (\ref{p:rhoi-rho})]
Set $\infty:=\rho_X(F)\in X$ and $Y:=X\setminus\{\infty\}$.
By part (\ref{p:tf}) of Theorem~\ref{thm:bool-homeo} and by Lemma~\ref{lem:pointed}, we have a sequence of bijections
\[
\begin{tikzcd}
\TF(X)\setminus\rho_X^{-1}(\infty)
\arrow{r}{\psi} &
\TF(Y)\arrow{d}{=} \\
\CS(\Ro(X),G[F^\perp]) &
\CS(\Ro(Y),G[F^\perp])\arrow{l}[swap]{\psi^{-1}}
\end{tikzcd}\]
This completes the proof of the theorem.
\ep
The deduction of Rubin's Theorem from Theorem~\ref{thm:bool-homeo} will be mostly formal, just like that of Theorem~\ref{thm:gp-bool2} from Theorem~\ref{thm:rubin-supp}.
\bp[Proof of Theorem~\ref{thm:rubin}]
We have constructed a Boolean isomorphism
\[
\Psi\co \Ro(X_1)\longrightarrow\Ro(X_2)\]
in Theorem~\ref{thm:gp-bool2} that is equivariant with respect to the group actions $G_i$.
We have the following equalities
for $F\in \TF(X_1)$:
\begin{align*}
&\Psi(\TF(X_1))
=\TF(X_2),\\
&\Psi(\rho_{X_1}^{-1}\circ\rho_{X_1}(F))
=\rho_{X_2}^{-1}\circ\rho_{X_2}(\Psi(F)),\\
&\Psi(\Nbr_{X_1}(\rho_{X_1}(F))\cap\Ro(X_1))
=\Nbr_{X_2}(\rho_{X_2}(\Psi(F))\cap\Ro(X_2).
\end{align*}
Indeed, the sides of each equality above can be expressed purely in terms of the Boolean actions $G_i$ on the Boolean algebras
$\Ro(X_i)$ by Theorem~\ref{thm:bool-homeo}. Thus, they are preserved under the equivariant Boolean isomorphism $\Psi$.
We now define a map $\phi\co X_1\longrightarrow X_2$ by
\[
\phi(\rho_{X_1}(F))=\rho_{X_2}(\Psi(F)),\]
for $F\in\TF(X_1)$. This is a well-defined bijection by the observation above.
For a regular open set $U\sse X_1$, we have
\begin{align*}
\phi(U)&=\{ \phi(y)\mid y\in U\}
=\{\phi\circ\rho_{X_1}(F)\mid F\in\TF(X_1)\text{ and }\rho_{X_1}(F)\in U\}\\
&=\{\rho_{X_2}\circ\Psi(F)\mid F\in\TF(X_1)\text{ and }U\in\bigcap\rho_{X_1}^{-1}\circ\rho_{X_1}(F)\}\\
&=\{\rho_{X_2}(F')\mid
F'\in\TF(X_2)\text{ and }\rho_{X_2}(F')\in\Psi(U)\}=\Psi(U).
\end{align*}
This implies that $\phi$ is an open map, and by symmetry, a homeomorphism.
\ep
\begin{rem}
\be[(1)]
\item
In Rubin's Theorem, one cannot drop the perfectness hypothesis; for instance, one may let $G_1=G_2=\bZ/2\bZ$ nontrivially act on two finite discrete spaces of different cardinalities.
\item
Similarly, one cannot weaken the local density hypothesis to mere local movement.
To see an example, consider the action of $\Homeo_+[0,1]$ on the compact interval $X_1=[0,1]$ and also its one-point compactification $X_2=S^1$.
Both actions are locally moving, but the spaces are not homeomorphic.
\item One can weaken the hypothesis of local compactness
to \emph{regional compactness}\index{regional compactness},
which means that every nonempty open set contains some nonempty compact neighborhood. We direct the reader to
\cite{Rubin1996} for details.
\ee
\end{rem}
\subsection{Applications to manifold homeomorphism groups}
We can now
combine Rubin's Theorem (\ref{thm:rubin})
with Taken's Theorem (\ref{thm:takens})
and obtain a simple proof of (generalized) Filipkiewicz's Theorem (Theorem~\ref{thm:filip-gen}).
\bp[Proof of Filipkiewicz's Theorem]
Note that for a smooth connected boundaryless manifold $X$
and for $p\in\bZ_{>0}\cup\{0,\infty\}$, the group
$\Diff_c^p(X)_0$ acts on $X$ locally densely.
This is an immediate consequence of Lemma~\ref{lem:affine}.
Therefore, the groups $G$ and $H$ given in the hypothesis act locally densely on $M$ and $N$ respectively.
Rubin's Theorem implies that there exists a homeomorphism $w\co M\longrightarrow N$ intertwining these actions.
We conclude from Takens' Theorem (Theorem~\ref{thm:takens}) that
$p=q$ and that $w$ is a $C^p$ diffeomorphism.
\ep
Ben Ami and Rubin gave a different reconstruction theorem, as we now describe.
We will omit the proof, which relies on Theorem~\ref{thm:gp-bool2}
and which resembles that of Rubin's Theorem.
We remark that in the paper of Ben Ami and Rubin,
the spaces are only assumed to be regular;
however, the statement below is equivalent to the original, since the perfectness and local
compactness are consequences of the group theoretic hypotheses,
even when only regularities are assumed.
\begin{thm}[\cite{BAR2010}]\label{thm:bar}
Let $X_1$ and $X_2$ be perfect, locally compact, Hausdorff topological space.
Suppose we have groups
\[H_i\le G_i\le \Homeo(X_i)\]
such that $H_i$ is fragmented and has no global fixed points for $i=1,2$.
Assume further that $\cl(G_i.x)$ has nonempty interior for $i=1,2$ and for all $x\in X_i$.
If there exists a group isomorphism
\[
\Phi\co G_1\to G_2,\]
then there exists a homeomorphism
\[
\phi\co X_1\to X_2\]
such that
\[
\phi\circ g=\Phi(g)\circ\phi\]
for all $g\in G_1$.\end{thm}
\bp[Deduction of Filipkiewicz's Theorem from Theorem~\ref{thm:bar}, for $p,q\ge1$]
Setting \[ H_1:=\Diff_c^p(M)_0, \quad H_2:=\Diff_c^q(N)_0,\]
we note that $H_i$ is fragmented (Lemma~\ref{lem:frag}). Furthermore, arbitrary orbits of these groups have nonempty interior.
It follows from Theorem~\ref{thm:bar} that there exists a homeomorphism $X_1\to X_2$ that is equivariant with the given group isomorphism
$G_1\to G_2$. The rest of the proof is identical as above.
\ep
Another easy application of Rubin's Theorem is that the PL homeomorphism group determines the ambient PL manifold
up to homeomorphisms~\cite{Rubin1989}. One also sees that the minimal action of the Thompson's group $F$
(more generally, a minimal chain group) on an interval is unique up to homeomorphism~\cite{KKL2019ASENS}; cf.~Section~\ref{sec:chain}.
Let us also note another consequence.
\begin{cor}\label{cor:normal}
Let $p\in\bZ_{>0}\cup\{\infty\}$.
\be[(1)]
\item
If two distinct normal subgroups of $\Diff^p(M)$ contain $\Diff^p_c(M)_0$,
then they are not isomorphic as groups.
\item
If $p\ne\dim M+1$,
then two distinct normal subgroups of $\Diff^p(M)$ are never isomorphic as groups.
\ee
\end{cor}
\bp
For part (1), assume for contradiction that for two such normal groups $N_1$ and $N_2$
there exists an isomorphism $\Phi\co N_1\longrightarrow N_2$.
By Theorem~\ref{thm:filip-gen}, this isomorphism is the restriction of an inner automorphism of $\Diff^p(M)$.
By normality, we obtain that $N_1=N_2$.
In part (2) note first that $\Diff^p_c(M)_0$ is simple (Theorem~\ref{thm:mather-thurston}); see~\cite{Mather1,Mather2,Thurston1974BAMS}. By Corollary~\ref{cor:minimal}, every nontrivial normal subgroup of $\Diff^p(M)$ contains
the group
\[
\left[\Diff^p_c(M)_0,\Diff^p_c(M)_0\right]
=\Diff^p_c(M)_0.\]
Part 1 now implies that two distinct nontrivial normal subgroups can never be isomorphic.
\ep
In particular, the four groups in Corollary~\ref{cor:minimal} are pairwise non-isomorphic.
\subsection{Locally moving groups obey no law}\label{ss:lawless}
The local movement hypothesis puts strong restrictions on algebraic structure of the underlying group.
Let $G$ be a group.
Each word $w=w(a_1,\ldots,a_k)$ in the rank--$k$ free group $F(x_1,\ldots,x_k)$
can be regarded as a map
\[
w\co G^k\to G\]
sending $(g_1,\ldots,g_k)$ to $w(g_1,\ldots,g_k)$.
We say $G$ obeys a nontrivial \emph{law}\index{law} if $w(G^k)=\{1\}$ for some integer $k\ge1$ and
nontrivial reduced word $w\in F(a_1,\ldots,a_k)$.
For example, abelian
groups obey the law $a_1^{-1}a_2^{-1}a_1a_2$.
Brin and Squier proved that Thompson's group $F$ does not obey a law~\cite{BS1985}; see also Section~\ref{ss:sub-chain}.
Ab\'ert gave a short proof of this fact by establishing the result below.
\begin{thm}[{Ab\'ert, \cite{Abert2005}}]\label{thm:abert}
Let $G$ be a permutation group of a set $X$.
Assume that for all finite subsets $Y\sse X$ and for all $z\in X\setminus Y$,
there exists $g\in G$ such that $g\restriction_Y=\Id$ and such that $g(z)\ne z$.
Then $G$ does not satisfy a law.
\end{thm}
Generalizing the proof of Lemma~\ref{lem:LM-basic} (\ref{p:move-fn}) given by Rubin~\cite{Rubin1996}, one can see that the local movement condition is closely related; this was observed in Nekrashevych's note~\cite{Nekrashevych-note}.
\begin{thm}\label{thm:lawless-top}
A locally moving group of homeomorphisms of a Hausdorff topological space does not obey a law.
\end{thm}
As an example, the group of $C^p$ diffeomorphisms of a smooth connected boundaryless manifold obeys no laws.
One can also see that the commutator subgroup of a locally moving group acting on a Hausdorff topological space is locally moving~\cite{Nekrashevych-note}. To see this, suppose $G\le\Homeo(X)$ is locally moving and let $\varnothing \ne U\in \Ro(X)$. Since $H:=G[U]$ is locally moving,
this latter group is not abelian by Theorem~\ref{thm:lawless-top}.
In particular, the group $[H,H]$ is nontrivial. It follows that \[[G,G][U]=G'[U]\] is nontrivial.
The same argument shows that an arbitrary term in the lower central or derived series of $G$ is locally moving, and in particular, nontrivial.
One can also formulate a similar result on lawlessness in the context of a Boolean algebra.
We will deduce both of the preceding theorems from the following unifying fact.
\begin{thm}\label{thm:lawless}
Let $G$ be a permutation group of a set $X$.
Assume that for each finite subset $S\sse G$ containing $1$ and for each $g\in G$,
there exist elements $h\in G$ and $y\in X$ such that the following conditions hold.
\be[(i)]
\item $\#S(y)=\max_{x\in X}\#S(x)$;
\item $S(y)\setminus\{y\}\sse \Fix h$;
\item $h(y)\not\in gS(y)$.
\ee
Then $G$ does not obey a law.
\end{thm}
\bp[Proof of Ab\'ert's Theorem from Theorem~\ref{thm:lawless}]
We assume that $(G,X)$ satisfies the hypothesis of Ab\'ert's Theorem.
In order to verify conditions (i) through (iii) of Theorem~\ref{thm:lawless},
we let $S\sse G$ be a finite subset containing $1$, and let $g\in G$.
Pick $y\in X$ realizing the condition (i).
We set
\[
Z:= S(y)\setminus\{y\}.\]
We claim that the orbit $C:=G[X\setminus Z].y$ is infinite.
If $C$ is finite, then consider the group
\[H:= \{r\in G[X]\mid r(s)=s\text{ for all }s\in Z\cup C\setminus\{y\}\}.\]
Applying the hypothesis of Ab\'ert's Theorem to the finite set
\[Z\cup C\setminus\{y\}\sse X\setminus\{y\},\] we see that $H$ cannot fix $y$.
On the other hand, the group $H\le G[X\setminus Z]$ stabilizes $C$ setwise.
As $H$ fixes $C\setminus\{y\}$ pointwise,
we have a contradiction.
By the above claim, we can now pick an element $h\in G[X\setminus Z]$ such that
\[ h(y) \not\in gS(y).\]
This completes the verification of conditions (i) through (iii).\ep
\bp[Proof of Theorem~\ref{thm:lawless-top} from Theorem~\ref{thm:lawless}]
Let us again verify the conditions of Theorem~\ref{thm:lawless}.
We first pick $x$ such that $\#S(x)=\max_{z\in X}\#S(z)$.
By considering a subset $S_0\sse S$ if necessary, we may assume that
$s(x)\ne s'(x)$ for all distinct $s,s'\in S$;
we can further assume for some open neighborhood $U$ of $x$ that
\[sU\cap s'U=\varnothing,\]
as we have seen in the proof of Lemma~\ref{lem:LM-basic}.
If $x\not\in gS(x)$, then the conditions are trivially satisfied by setting $y:=x$ and $h:=\Id$.
We assume $x=gq(x)$ for some $q\in S$; note that such a $q$ is unique.
We may shrink $U$ to another open set $V$ containing $x$
such that $V$ and $gs(V)$ are disjoint for all $s\in S\setminus\{q\}$,
and such that $gq(V)\sse U$.
We set $h\in G[V]$ and pick $y\in \supp h$.
We first note that both $(h,y)$ and $(\Id,y)$ satisfy the conditions (i) and (ii).
Indeed, from the fact that $s(V)$ and $s'(V)$ are disjoint for all distinct $s,s'\in S$,
the condition (i) is satisfied.
The condition (ii) is trivial for $(\Id,y)$.
For each $s\in S\setminus\{1\}$ we have that $U\cap s(U)=\varnothing$, and that
$s(y)\not\in \supp h\sse V$. This implies that $hs(y)=s(y)$ and verifies (ii) for $(h,y)$.
We claim that either $y\not\in gS(y)$ or $h(y)\not\in gS(y)$.
Indeed, assume first that $y\in gS(y)$. Since $g(S\setminus\{q\})V\cap V=\varnothing$,
we have that
\[
h(y)\ne y=gq(y).\]
We also have that $h(y)\not\in g(S\setminus\{q\})y$, implying $h(y)\not\in gS(y)$.
For the case $h(y)\in gS(y)$, we similarly obtain
\[
y\ne h(y)=gq(y),\]
and that $y\not\in gS(y)$. The claim is proved.
Summarizing, either $(\Id,y)$ or $(h,y)$ satisfies the condition (iii).\ep
Let us now verify the main result of this subsection.
\bp[Proof of Theorem~\ref{thm:lawless}]
To show that $G$ obeys no law, it suffices to show that it obeys no law in a free group $F_2$ on two generators
$\{a,b\}$, since every finitely generated free
group embeds in a free group on two generators.
Let $\ell$ be a positive integer, and let $w(a,b)=b_\ell\cdots b_1 \in F_2$ be a nontrivial reduced word.
In particular, we have $b_i\in\{a,b\}^{\pm1}$.
We set
\[
w_i(a,b):=b_i\cdots b_1\]
and $w_0:=1$.
We use the induction on the length $\ell$
to find elements $u,v\in G$ and a point $y\in X$ such that
the points in the collection
\[
\{w_i(u,v)(y)\}_{0\le i\le \ell}\]
are all distinct.
If $\ell=1$, we have $w(a,b)$ is a single letter.
Note that $G$ is nontrivial by condition (iii).
We can define $w(u,v)$ to be an arbitrary nontrivial element of $G$
and $y$ to be a point nontrivially moved by that element.
Assume now that the conclusion holds for $\ell-1$.
We have some $u,v\in G$
and a point $x_0\in X$ such that
the points
\[
x_i:= w_i(u,v)(x_0)\]
are all distinct for $i\in\{0,1,\ldots,\ell-1\}$.
We assume $x_\ell:=w(u,v)(x_0)=x_j$ for some $j\le \ell-1$, for otherwise the proof is done.
Without loss of generality, we may assume $b_\ell=a$.
Setting $x:=x_{\ell-1}$ we have $x_\ell=u(x)$.
Let us set
\[
S:=\{w_i(u,v)w_{\ell-1}(u,v)^{-1}\mid 0\le i\le \ell-1\}.\]
Applying the hypothesis to $(S,g=u^{-1})$ we obtain $h\in G$ and $y\in X$. In particular,
\[
\#S(y)=\ell=\#S.\]
We set
\[
y_i:=w_i(u,v)w_{\ell-1}(u,v)^{-1}(y)\]
for $0\le i\le \ell$. The points $y_0,\ldots,y_{\ell-1}$ are all distinct.
By the condition (ii), we have $h(y)_i=y_i$ for $0\le i\le \ell-2$.
Using the condition that $b_{\ell-1}\ne a^{-1}$, we easily see for each $0\le i\le \ell-1$ that
\[w_i(uh,v)(y_0)=w_i(u,v)(y_0)=y_i.\]
Moreover, we have that
\[
w_\ell(uh,v)(y_0)=uh(y)\not\in\{y_0,\ldots,y_{\ell-1}\}.\]
We thus obtain that \[\#\{w_i(uh,v)y_0\mid 0\le i\le \ell\}=\ell+1,\] as required.
\ep
Small amount of additional work will imply the following result on the abundance of free subgroups.
We note that a similar idea can be found in Ghys' article~\cite{Ghys1999}, where he proves that a generic two-generator subgroup of $\Homeo_+(S^1)$ is nonabelian free.
Recall that if a space is complete metrizable or locally compact Hausdorff
then it is \emph{Baire}\index{Baire space}, i.e. satisfies the conclusion of the Baire Category Theorem.
\begin{cor}\label{cor:free-abundant}
Let $G$ be a topological group, let $X$ be a space,
and let
\[ G\longrightarrow \Homeo(X)\]
be a continuous injective homomorphism.
Assume that for each nonempty open set $U\sse X$
and for each identity neighborhood $\VV\sse G$
we have that
\[
G[U]\cap\VV\ne\{1\}.\]
If $G$ and $X$ are Baire spaces,
then a generic pair of elements in $G$ generate a nonabelian free group.
\end{cor}
Here, a \emph{generic}\index{generic pair} pair means it belongs to a comeager subset of $G\times G$.
For instance, generic pair of diffeomorphisms in $\Diff^\infty(M)$ will generate $F_2$ for a smooth connected manifold $M$,
since the diffeomorphism group
is Baire (because it is a Fr\'echet space~\cite{conway-funct}).
Note that the group $G$ above is locally moving.
\bp[Proof of Corollary~\ref{cor:free-abundant}]
Since $G$ is Baire, so is the space \[Y=G\times G\times X.\]
For each nontrivial reduced word $w(a,b)\in F_2:=F(a,b)$ we
consider the open subset $U_w\sse Y$ given by triples
\[U_w=\{(u,v,x)\mid w(u,v)(x)\ne x\}.\]
If we can show that $U_w$ is dense for all $1\neq w\in F_2$,
then the Baire Category Theorem implies that
the $G_\delta$ set
\[\Delta=\bigcap_{1\neq w\in F_2} U_w\] is comeagre.
If $(u,v,x)\in \Delta$ then for all nontrivial $w\in F_2$, we have that $w(u,v)(x)\neq x$,
and so the homeomorphisms $u$ and $v$ generate a free subgroup of $G$. This implies the conclusion.
It now remains for us to show that $U_w$ is dense, or equivalently, $Z_w:=Y\setminus U_w$ has empty interior.
The idea is already contained in the proofs of Theorems~\ref{thm:lawless-top} and~\ref{thm:lawless}, so we will reuse notations and observations from those proofs.
We pick $(u,v,x_0)\in Z_w$, and consider arbitrary open neighborhoods $\Id\in \VV\sse G$ and $x_0\in U\sse X$.
We wish to prove that $(u\VV,v\VV,U)$ contains an element of $U_w$, implying that $(u,v,x_0)$ is not an interior point of $Z_w$.
The case that $w$ is a single letter is obvious, as we can find $y\in U$ and $h\in G[U]\cap \VV$ such that $h(y)\ne y$,
and hence, either $w(y)\ne y$ or $wh(y)\ne w(y)=y$.
Inductively, let $w$ have length $\ell>1$.
We may assume that \[(u,v,x_0)\in Z_w\cap\left(\bigcap_g U_g\right),\] where $g$ ranges over all the reduced words shorter than $w$.
We define \[w_0=1,w_1,\ldots,w_\ell=w\] to be exactly the same as in Theorem~\ref{thm:lawless}.
We also assume $w=a w_{\ell-1}$, as before.
The points $x_i:=w_i(u,v)(x_0)$ are all distinct for $0\le i<\ell$. We have that $x_\ell=x_j$ for some $j<\ell$ since $(u,v,x_0)\in Z_w$.
As noted in the proof of Theorem~\ref{thm:lawless-top}, we can find an open neighborhood $V_0$ of $x_0$ such that $\{V_i:=w_i(u,v)(V_0)\}_{0\le i\le \ell-1}$ are all disjoint.
We can pick $h\in G[V_{\ell-1}]\cap \VV$ such that $h(y)\ne y$ for some $y\in V_{\ell-1}$.
As was seen in the proof of Theorem~\ref{thm:lawless}, we deduce that
either $(u,v,w_{\ell-1}(u,v)^{-1}(y))$ or
$(uh,v,w_{\ell-1}(u,v)^{-1}(y))$ does not belong to $Z_w$, completing the proof.\ep
\section{The original proof of Whittaker--Filipkiewicz}\label{sec:filip}
In this section we give another proof of Theorem~\ref{thm:filip-gen}, based on the original argument of
Whittaker~\cite{Whittaker1963} and its generalization by Filipkiewicz~\cite{Filip82}. This method is much more
involved than Rubin's Theorem and also significantly more limited in scope, but we present the original proof for the showcase of the
idea of detecting stabilizer groups by group theoretic properties. Moreover, it has a slight advantage of dealing with possibly
non-locally compact spaces, where Rubin's Theorem does not apply.
We will restrict ourselves to the differentiable world, namely the cases $p,q\ge1$, so that we may
employ fragmentation techniques.
Throughout, we will let $M$ and $N$ be smooth connected boundaryless manifolds.
As we saw in the introduction of this chapter, this theorem implies the result of R.~P.~Filipkiewicz (Theorem~\ref{thm:filip}),
which shows that an isomorphism between the diffeomorphism groups of $M$ and of $N$ is induced by a diffeomorphism
between these manifolds themselves.
The idea of the proof of Theorem~\ref{thm:filip-gen} is to first find a bijection between $M$ and $N$, conjugation by which realizes the given
isomorphism $\Phi$ between the diffeomorphism groups. The bijection itself arises from a bijection between stabilizers of
points in $M$ and $N$. Specifically, one considers $\Stab_G(x)$ for a point $x\in M$, and concludes that there
is a point $y\in N$ such that $\Phi(\Stab_G(x))=\Stab_H(y)$. The assignment $x\mapsto y$ turns out to be a bijection
realizing the isomorphism $\Phi$.
As in Lemma~\ref{lem:bij-homeo}, the resulting bijection will necessarily be
continuous. The remaining work is in showing that the bijection is in fact a $C^p$ diffeomorphism,
and this last bit utilizes Theorem~\ref{thm:BM-simple} (the Bochner--Montgomery Theorem).
This process can be more conceptually conveyed using abstract properties of group actions.
A \emph{circular order}\index{circular order} on a set $X$ roughly means that
arbitrary three distinct points $x,y,z$ in $X$
can be listed as
\[
x<y<z<x\] after a suitable permutation, and moreover this ordering is transitive;
see Section~\ref{ss:circular} for a precise definition.
The reader may only consider the case $X=S^1$ for the purpose of this section.
\bd
A \emph{dense order (without bounds)}\index{dense order} on a set $X$ is
a total or circular order $\le$ such that
every interval in $X$ contains a point in $X$
and such that $X$ does not have a maximum or a minimum.
\ed
We will often drop the phrase ``without bounds'' when discussing orders.
By a classical result of Cantor, a countable set with a dense total order without bounds is order isomorphic to $\bQ$.
The same condition for a circular order yields $\bQ/\bZ$.
For a set $X$, we let
$\Sym(X)$ denote the group of all permutations of $X$,
This group naturally acts on the configuration space of $n$ distinct points
\[
\operatorname{Conf}_n(X):=\{(x_1,\ldots,x_n)\mid x_i\in X\text{ and }x_i\ne x_j\}.\]
If $X$ is given with a linear order $\le$, then we define
\[
\operatorname{Conf}^+_n(X):=\{(x_1,\ldots,x_n)\mid x_i\in X\text{ and }
x_1<x_2<\ldots<x_n\}.\]
If $X$ is equipped with a circular order $\le$, then we let
\[
\operatorname{Conf}^+_n(X):=\{(x_1,\ldots,x_n)\mid x_i\in X\text{ and }
x_1<x_2<\ldots<x_n<x_1\}.\]
When $X$ is equipped with either of these orders, then a linear or circular order preserving permutation of $X$
naturally acts on $\operatorname{Conf}^+_n(X)$.
\bd\label{defn:n-trans}
Let $X$ be a set.
\be[(1)]
\item A group $G\le\Sym(X)$ is \emph{$n$--transitive}\index{$n$--transitive}, or \emph{$T(n)$}, if
it acts transitively on $\operatorname{Conf}_n(X)$.
\item Suppose $X$ is equipped with a total or circular order.
A group $G\le\Sym(X)$ is \emph{positively $n$--transitive}\index{positively $n$--transitive},
or \emph{$T^+(n)$}, if it acts transitively on $\operatorname{Conf}^+_n(X)$.
\ee\ed
For brevity, we adopt the convention that when we discuss homeomorphisms of one--manifolds,
``$n$--transitivity'' actually means positive $n$--transitivity unless stated to the contrary.
\begin{rem}
\be[(1)]
\item A $2$--transitive action on a circularly ordered set is positively 2--transitive by definition.
\item A positively 3--transitive action on a set with a dense, total or circular order is automatically order preserving.
\ee
\end{rem}
Roughly based on Banyaga's exposition~\cite{Banyaga1997}, we introduce the following axiomatic definitions for topological actions.
\bd\label{defn:axioms}
Let $X$ be a topological space, and let $G\le\Homeo(X)$.
\be[(1)]
\item The group $G$ is \emph{locally dilative (LD)}\index{locally dilative}
if every point in $X$ has an open local basis consisting of open sets $U$ such that some element in $G_U$ has a connected closed support.
\item
The group $G$ is \emph{weakly fragmented (WF)}\index{weak fragmentation}
if
for every open cover $\VV$
of $X$ the group generated by the collection
\[
\big\{
\left[
G_V,G_V
\right]\;
\big\vert\;
V\in\VV
\big\}
\]
contains a nontrivial normal subgroup of $G$.
\ee
\ed
\begin{rem}\label{rem:banyaga}
\be[(1)]
\item
The property (LD) is much weaker than the \emph{property (B)}\index{property (B)} defined by Banyaga~\cite{Banyaga1997},
which states that some element $g\in G$ satisfies $\suppo g = U\setminus\{x_0\}$ for each element $U$ of the local basis at $x_0$.
\item
The property (WF) is trivially implied by the \emph{property (L)}\index{property (L)} of Banyaga, which states that
\[
\form*{ \left[ (G_V)_0,(G_V)_0\right]\middle \vert V\in\VV}=[G_{c0},G_{c0}],
\]
assuming $G$ is a $C^k$ diffeomorphism group of a manifold
and $\VV$ is a subcover of $C^k$ open balls.
\item Banyaga defined $G\le\Homeo(M)$ to be \emph{path--transitive}\index{path transitive} if every neighborhood of a given path
$c\co[0,1]\longrightarrow M$ admits some $g\in G$ with $\suppc g\sse U$ and such that $g(c(0))=c(1)$.
The path--transitivity of $G$ easily implies $(LT)$ and $T(n)$ or $T^+(n)$ (depending on the dimension of $M$).
\item\label{p:ptrans} Conversely, if $X$ is path--connected then every locally transitive action is path--transitive. This is obvious from a typical Lebesgue number argument.
\ee
\end{rem}
We can now state a generalization of Filipkiewicz Theorem.
Banyaga~\cite{Banyaga1997} proved the same conclusion with the stronger assumption that the properties (B), (L), and path--transitivity hold for the groups $G_1$ and $G_2$.
\begin{thm}\label{thm:isom-homeo}
Let $X_1$ and $X_2$ be connected Hausdorff topological spaces.
For each $i\in\{1,2\}$, we assume one of the following:
\be[(A)]
\item
A group $G_i$ acts faithfully and 3--transitively on $X$.
\item
A group $G_i$ acts faithfully and positively 3--transitively on $X_i$
with respect to some dense total or circular order on $X_i$.
\ee
Assume further that each $G_i$ is locally transitive, locally dilative, and weakly fragmented.
If there exists a group isomorphism
\[
\Phi\co G_1\longrightarrow G_2,\]
then there exists a homeomorphism
\[w\co X_1\longrightarrow X_2\]
such that each $g\in G_1$ satisfies
\[
\Phi(g)=wgw^{-1}.\]\end{thm}
We emphasize that it is not required a priori that the spaces $X_1$ and $X_2$ are simultaneously ordered,
even when one of them is assumed to be ordered.
We also do not need to assume that $X_i$ is locally compact, in comparison with Rubin's Theorem (Theorem~\ref{thm:rubin}).
\begin{rem}The 3--transitivity assumption is redundant when $X_1$ and $X_2$ are connected manifolds of dimension at least two. This is because for all distinct $p,q\in \Conf_3(X_i)$
one can find disjoint paths $P_1,P_2,P_3\sse X_i$ such that the path $(P_1,P_2,P_3)\sse \Conf_3(X_i)$ joins $p$ and $q$. One can then apply part~(\ref{p:ptrans}) of Remark~\ref{rem:banyaga} to see that the local transitivity hypothesis readily implies the 3--transitivity. The positive 3--transitivity assumption can also be dropped for a similar reason when $X_i$ is a connected one--manifold.\end{rem}
\begin{prop}\label{prop:wf}
Let $H$ be a group, and let $p\in\bZ_{>0}\cup\{\infty\}$.
\be[(1)]
\item If we have
\[\Diff^\infty_c(M)_0\le H\le\Homeo(M),\]
then $H$ is locally transitive, locally dilative and 3--transitive (positively, if $M$ is a one--manifold).
\item If we have
\[\Diff^p_c(M)_0\le H\le\Diff^p(M),\]
then $H$ is weakly fragmented.
\ee
\end{prop}
The previous proposition implies that all of the four groups
\[
\Diff^p(M),\Diff^p_c(M),\Diff^p_{c}(M)_0,\Diff^p(M)_0\]
are locally transitive, locally dilative, 3--transitive and weakly fragmented.
The Generalized Filipkiewicz's theorem as stated in Theorem~\ref{thm:filip-gen}
is an immediate consequence of Theorem~\ref{thm:isom-homeo}
and of Takens' Theorem (Theorem~\ref{thm:takens}).
\begin{rem}
Similar conclusions to Theorem~\ref{thm:isom-homeo} hold for the groups of contactomorphisms, symplectomorhisms
(when the manifolds are compact), volume-form preserving smooth diffeomoprhisms, and ``good-measure'' preserving homeomorphisms,
by establishing the properties described in the theorem above. The most difficult parts in the processes are usually the property (WF).
We will not delve into these groups as they require many definitions beyond the scope of this book. See~\cite{Banyaga1997} for details.
\end{rem}
\subsection{Transitivity, dilativity and weak fragmentation}\label{ss:ltld}
In this subsection, we establish the properties (LT), (LD), and $T(n)$ in Proposition~\ref{prop:wf}.
As these properties transfer from a subgroup $H\le G$ to a bigger group $G$,
it suffices to verify them for the smallest group considered in the proposition, that is
\[
\Diff_c^\infty(M)_0.\]
For a subset $U\sse M$, we abbreviate that
\[ \Diff^k_0(U):=\Diff^k_c(U)_0=(\Diff^k_c(M)_U)_0.\]
Returning to the proof of Proposition~\ref{prop:wf},
we observe from the above lemma that for each given $C^\infty$ open ball $U\sse M$
there exists $g\in\Diff_c^\infty(M)_0$ whose open support is $U$ itself
and which moves between two points given a priori.
It follows that the group
\[
\Diff_c^\infty(M)_0\]
has the properties (LT) and (LD). The property $T(n)$ of this group is given below.
\begin{lem}[\cite{Banyaga1997}]\label{lem:t3}
For each path $c\co I\longrightarrow M$
and for each open neighborhood $U$ of $c(I)$ there exists
$g\in \Diff_0^\infty(U)$
such that $g(c(0))=c(1)$.
In particular, $\Diff_c^\infty(M)_0$ is $n$--transitive for all $n$.
\end{lem}
\bp
If $c(I)$ is contained in a $C^\infty$ open ball $U_0$ in $M$,
and if $U_0\sse U$, then Lemma~\ref{lem:affine} obviously implies
the existence of $g\in\Diff_0^\infty(U_0)$ such that $g(c(0))=c(1)$.
In general, we may divide $c(I)$ into finitely small pieces $c_1,\ldots,c_m$, each piece of which is contained in a $C^\infty$ open ball $U_i$ inside $U$.
Composing maps $g_i$ sending the initial point of $\gamma_i$ to the terminal point as above, we have a map $g$ satisfying the first conclusion.
The second conclusion $T(n)$ is an easy inductive consequence of the first, along with the hypothesis that $M$ is path--connected.
\ep
The weak fragmentation property in Proposition~\ref{prop:wf}
is a consequence of a stronger result, that is Proposition~\ref{prop:wf2}.
This gives yet another perspective on fragmentation of homeomorphisms, which will later be crucial in the proof of Filipkiewicz's Theorem.
\begin{prop}[{cf. \cite[Theorem 2.2]{Filip82}}]\label{prop:wf2}
If $p\in\bZ_{>0}\cup\{\infty\}$ and if $\UU$ is an open cover of $M$,
then the collection of groups
\[
\left\{
\left[
\Diff^p(U)_0,\Diff^p(U)_0
\right]\;
\middle\vert\;
U\in\UU
\right\}\]
contains the group
\[\left[\Diff^p_c(M)_0,\Diff^p_c(M)_0\right].\]
\end{prop}
The weak fragmentation property in Proposition~\ref{prop:wf} would be an easy consequence.
\bp[Proof of Part 2 of Proposition~\ref{prop:wf}, assuming Proposition~\ref{prop:wf2}]
Let $H$ be as in the proposition,
and let $\VV$ be an open cover of $M$.
Denote by $G_1$ the group generated by the collection
\[
\left[
H_V,H_V
\right]\]
for $V\in\VV$.
It suffices for us to prove that $G_1$ contains
\[K_0:=\left[\Diff^p_c(M)_0,\Diff^p_c(M)_0\right]\]
since $K_0$ is nontrivial and normal in $H$.
Since the collection $\BB^p(M)$
of $C^p$ open balls generates the topology of $M$,
the collection
\[
\UU:=\{U\in \BB^p(M)\mid U\sse V\text{ for some }V\in \VV\}\]
still covers $M$.
The group $G_2$ generated by the collection
\[
\left[
H_U,H_U
\right]\]
for $U\in\UU$ is contained in $G_1$.
Since each $U\in\UU$ is relatively compact
and since $\Diff^p_c(M)_0\le H$,
the group $G_2$ contains the group $K_L$
generated by the collection of the groups
\[
\left[
\Diff^p(U)_0,\Diff^p(U)_0
\right]\]
for $U\in\UU$.
This completes the proof that $H$ is weakly fragmented, as $K_0\le K_L$ by Proposition~\ref{prop:wf2}.\ep
The proof of Proposition~\ref{prop:wf2} requires two ingredient. The first is the Fragmentation Lemma (Lemma~\ref{lem:frag}), and the second is the ``shrinking lemma'' of Filipkiewicz.
This latter lemma asserts that the closed unit ball in a Euclidean space
can be contracted onto a smaller ball
by the multiplication of commutators of smooth diffeomorphisms
supported in smaller open balls.
Recall our notation that $B^d(a;r)$ denote the radius--$r$ ball centered at $a\in\bR^d$.
\begin{lem}[Filipkiewicz's Shrinking Lemma]\label{lem:filip-comm}
If $\UU$ is an open cover of the closed unit ball $B$ in $\bR^d$,
then for each $a\in (0,1]$
there exist elements \[\{f_i,g_i\}_{1\leq i\leq m}\sse\Diff_c^{\infty}(\bR^d)_0\] satisfying the following:
\be[(1)]
\item
For each $i$, there is a $U_i\in\UU$ such that $f_i,g_i\in\Diff^k_0(U_i)$.
\item
We have that \[[f_m,g_m]\cdots [f_1,g_1](B)\sse {B^d(0;a)}.\]
\end{enumerate}
\end{lem}
\bp
We let $A\sse[0,1]$ be the set of $a\in[0,1]$ satisfying the conclusion. This set is nonempty as it contains $1$.
Taking the infimum $a$ of this set, it suffices for us to show $a=0$.
For a contradiction, assume $a>0$.
Let us denote the sphere of radius $r>0$ as $S_r:=\partial B^d(0;r)$.
We claim that there exist elements $f_i, g_i$ satisfying the first condition of the lemma such that
\[\prod_i[f_i,g_i] \left(S_{a+\delta/2}\right)\sse S_{a-\delta/2}\] for some $\delta>0$.
The desired contradiction would then follow from that $a+\delta/2\in A$.
We let $N_\delta$ denote the open $\delta$--neighborhood of $S_a$.
Pick a Lebesgue number $\epsilon\in(0,a/2)$ of the cover $\UU$ for $B$.
There exist points $x_1,\ldots,x_m\in S_a$
such that the balls
\[ V_i:=B^d(x_i;\epsilon/2)\] cover this sphere.
For some sufficiently small $\delta\in(0,\epsilon/4)$,
we also have that
\[
S_a\sse \overline{N_\delta}\sse \bigcup_i V_i\sse N_{\epsilon/2}.\]
We can pick an element
$f\in\Diff^\infty(\Int N_\delta)_0$ satisfying
\[ f\left(S_{a+\delta/2}\right)\sse S_{a-\delta/2}.\]
Each $U_i:=B(x_i;\epsilon)$ is contained in some element of $\UU$.
Moreover,
we can find an element $g_i\in\Diff_0^\infty(U_i)$ such that
\[g_i(V_i)\sse U_i\cap B(0;a-\epsilon/2)\sse\bR^d\setminus N_{\epsilon/2}.\]
Applying Lemma~\ref{lem:frag} to the manifold $\Int N_\delta$, we can find
a fragmentation
\[
f=f_r\cdots f_1\]
such that each $f_i$ is supported in
some set $W_i$ that coincides with one of $\{V_1,\ldots,V_m\}$.
Rearranging, we may assume $r=m$ and $W_i=V_i$ for each $i=1,\ldots,m$.
The map
\[
[f_i,g_i]=f_i\cdot (g_i f_i g_i^{-1})\]
is supported in $V_i\cup g_i(V_i)$.
If $x\in N_{\epsilon/2}$, then we have that $x\not\in g_i(V_i)$ and that
\[
[f_i,g_i](x)=f_i(x)\in N_{\epsilon/2}.\]
Inductively, we see that
\[
[f_i,g_i]\cdots[f_1,g_1](x)=\prod_i f_i(x)\in N_{\epsilon/2}.\]
In particular, we have that
\[[f_m,g_m]\cdots[f_1,g_1](x)=f(x).\]
This proves the aforementioned claim.
\ep
Using Lemma~\ref{lem:filip-comm}, we can now establish Proposition~\ref{prop:wf2}.
This also gives yet another perspective on fragmentation of homeomorphisms (cf.~Lemma~\ref{lem:frag}).
\begin{proof}[Proposition~\ref{prop:wf2}]
As in our proof for the weak fragmentation part of Proposition~\ref{prop:wf},
we may assume that $\UU\sse\BB^p(M)$.
We also define $K_L$ and $K_0$ as in that proof.
Let \[K:=\form{ [\Diff_0^p(U),\Diff_0^p(U)]\mid U\in\BB^p(M)}.\]
Since $\UU\sse\BB^p(M)$ we have that
\[K_L\le K\le K_0.\]
Moreover, $K$ is nontrivial and normal in $\Diff^p(M)$
since
$\BB^p(M)$ is $\Diff^p(M)$--invariant.
By Theorem~\ref{thm:epstein} and Corollary~\ref{cor:minimal}, we have that $K=K_0$.
It only remains to show that $K\le K_L$.
Let $d$ be the dimension of $M$
and let $U\in\BB^p(M)$.
There exists a $C^p$ embedding
\[ h\co \bR^d\longrightarrow M\]
such that $U=h(B^d(0;1))$.
For each $x\in \overline U$, there is some neighborhood $U_x\in\UU$ of $x$ such that
\[[\Diff^p_0(U_x),\Diff^p_0(U_x)]\le K_L\] by the definition of $K_L$.
We may assume $U_x\sse h(B^d(0;2))$, without loss of generality.
Since $\overline{U}$ is compact, we can select finitely many of these neighborhoods, say $\{U_1,\ldots,U_m\}$, that cover $\overline{U}$.
We let $V_i=h^{-1}(U_i)$ for $i\in\{1,\ldots,m\}$. We have \[B^d(0;1)\sse\bigcup_{i=1}^m V_i.\] Relabeling if necessary, we assume that
$B^d(0;a)\sse V_1\sse\bR^d$,
for some small $a>0$.
Lemma~\ref{lem:filip-comm} implies the existence of commutators
$\{[f_j,g_j]\}_{1\leq j\leq r}$, where $f_j,g_j\in \Diff^{\infty}_0(V_{i(j)})$ for some suitable labeling function $i(j)$, and such that
\[[f_r,g_r]\cdots[f_1,g_1](B^d(0;1))\sse B^d(0;a).\]
Since all the diffeomorphisms $f_j$ and $g_j$ are supported in the interior of
$B^d(0;2)$, we can conjugate $f_j$ and $g_j$ by $h$ to define $C^p$ diffeomorphisms on the image of $h$, and extending to
all of $M$ by the identity. Writing $\tilde{f}_j$ and $\tilde{g}_j$ for the respective resulting diffeomorphisms of $M$, we obtain that
$\tilde{f}_j$ and $\tilde{g}_j $ are supported on $U_{i(j)}$, and \[[\tilde{f}_r,\tilde{g}_r]\cdots [\tilde{f}_1,\tilde{g}_1]\in K_L\]
sends $U$ into $U_1$. Thus,
$[\Diff_0^p(U),\Diff_0^p(U)]$ is conjugate by an element of $K_L$ into $[\Diff_0^p(U_1),\Diff_0^p(U_1)]\le K_L$.
This implies the conclusion of the proposition.
\end{proof}
\begin{rem}
Strictly speaking, it suffices to show $K_L=K$ in the preceding proof for the purpose of establishing the weak fragmentation
properties in Proposition~\ref{prop:wf};
in other words, the fact that $K=K_0$ is not necessary for this purpose.
This is because the group $K$ in the proof is readily seen to be normal and nontrivial in $\Diff^p(M)$.
In that sense, Higman's Theorem and the Epstein--Ling Theorem are not necessary ingredients for the proof of Theorem~\ref{thm:filip-gen},
although those two theorems make the exposition more concrete by pinpointing exactly what the elements of $K$ are.
\end{rem}
\subsection{The pre-stabilizer subgroup}
The key to proving Theorem~\ref{thm:isom-homeo} is to show that $\Phi$ sends point stabilizers to point stabilizers,
from which one can build a
well--defined bijection from $X_1$ to $X_2$. The way one shows that $\Phi$ sends point stabilizers to point stabilizers is to characterize point
stabilizers as maximal proper subgroups in the corresponding homeomorphism groups.
Let us first consider group actions on abstract sets, without any reference to any topology.
The next definition records some useful double coset decompositions and their consequences for sufficiently transitive groups actions on sets.
\bd\label{defn:prestab}
Let $G$ be a group. A nontrivial proper subgroup $K$ of $G$ is called a \emph{pre-stabilizer subgroup}\index{pre-stabilizer subgroup}
if all of the following hold.
\be[(i)]
\item
For all $f\in G\setminus K$, we have that
\[
G\setminus K =KfK\cup Kf^{-1}K.\]
\item For all $f\in G\setminus K$ and $g\in KfK$ satisfying $fg,gf\not\in G\setminus K$, we have that
\[ fg, gf\in KfK.\]
\item
For all $f\in G\setminus K$ and $g_0,g_1\in KfK$
there exist some \[s_i,t_i\in f^{-1}Kg_i\cap K\]
for $i=0,1$ such that
$
s_0s_1=t_1t_0$.
\ee
\ed
Being a pre-stablizer subgroup is a purely group theoretic property, which is obviously preserved under group isomorphisms.
We first note a simple consequence.
\begin{lem}\label{lem:prestab-maximal}
A pre-stabilizer subgroup of a group $G$ is a maximal proper subgroup of $G$.
\end{lem}
\bp
Suppose we have
\[
K\lneq H\le G\]
for some pre-stabilizer subgroup $K$ of $G$.
Fix $h\in H\setminus K$. Then for all $g\in G\setminus K$ we have
\[
h\in KgK\cup Kg^{-1}K\sse HgH\cup Hg^{-1}H.\]
This implies that $g\in H$. We have shown that
\[
G\setminus K\sse H,\]
which implies that $H$ is not a proper subgroup.
\ep
The properties of Definition~\ref{defn:prestab} are possessed by point-stablizer groups of sufficiently transitive actions.
\begin{lem}[{cf.~\cite[Lemma 6]{Whittaker1963}}]\label{lem:prestab}
Assume one of the following.
\be[(A)]
\item
$X$ is an infinite set and
a group $G\le \Sym(X)$ acts 3--transitively on $X$.
\item
$X$ is a set equipped with a dense order that is total or circular,
and a group $G\le \Sym(X)$ acts positively 3--transitively on $X$.
\ee
Then for each $x\in X$ the stabilizer group
\[
\Stab_G(x)\]
is a pre-stabilizer subgroup of $G$.
\end{lem}
\bp
Set $K=\Stab_G(x)$. It is an easy consequence of (positive) 3--transitivity that $1\ne K\ne G$. For instance,
if $X$ is circularly ordered,
then we pick some points $y,z,w$ such that
\[ x<y<z<w<x.\]
We have some $g\in G$ such that \[g(x)=x,\quad g(y)=z,\quad g(w)=w.\]
This implies that $g\in K\ne 1$. The other cases are similar.
Let us now consider the three cases separately.
{\bf Case 1: $G\le\Sym(X)$.}
We verify the three properties of Definition~\ref{defn:prestab}.
For condition (i) of the definition, we make a stronger claim that
\[
G\setminus K = KfK.\]
Indeed, if $g\in G\setminus K$ then there exists an element $\sigma\in K$ such that $\sigma(g(x))=f(x)$, by 2--transitivity.
It follows that $\tau=f^{-1}\sigma g\in K$ and that $g=\sigma^{-1}f\tau\in KfK$.
This claim implies condition (ii) as well.
For condition (iii), let us set
\[u=f^{-1}(x),\quad v_0=g_0^{-1}(x),\quad v_1=g_1^{-1}(x).\]
By the 2--transitivity of the action of $G$, the set below is nonempty:
\[f^{-1}Kg_i\cap K=
\{\sigma\in G\mid \sigma(x)=x\text{ and }\sigma(v_i)=u\}.\]
Choosing a point $p\in X$, we will define maps based on the diagram below.
\[\begin{tikzcd}
p \arrow{r}{\sigma_0}\arrow[two heads]{d}[swap]{\sigma_1} & v_1\arrow[two heads]{d}{\sigma_1}
& x\arrow[loop right]{}{\sigma_0,\sigma_1} \\
v_0 \arrow{r}{\sigma_0} & u
\end{tikzcd}
\]
Namely, we use the 3--transitivity to find
$\sigma_0,\sigma_1\in K$ such that
\[\sigma_i(p)=v_{1-i},\quad \sigma_i(v_i)=u.\]
For such a $\sigma_i$ to exist,
it is sufficient and necessary to require the following:
\begin{itemize}
\item $p=v_0$ if and only if $v_1=u$;
\item $p=v_1$ if and only if $v_0=u$.
\end{itemize}
After setting $\tau_0:=\sigma_0$ and
\[
\tau_1:=\sigma_0\sigma_1\sigma_0^{-1}\]
we obtain the condition (iii).
{\bf Case 2-1: $X$ is equipped with a total order $\le$.}
The proof is very similar to Case 1.
Given three distinct points $x,y,z$ of $X$, we say $y,z\in X$ are on the \emph{same side} of $x\in X$
if
either \[ x<y\text{ and }x<z\] or
either \[ y<x\text{ and }z<x.\]
To see condition (i) of Definition~\ref{defn:prestab}, let $g\in G\setminus K$.
If $f(x)$ and $g(x)$ are on the same side of $x$, then we can find $\sigma$ as in Case 1 and conclude that $g\in KfK$;
otherwise the points $f^{-1}(x)$ and $g(x)$ are on the same side and hence, $g\in Kf^{-1}K$.
For condition (ii), note that $f(x)$ and $g(x)$ are on the same side of $x$ by the hypothesis. For instance, assume
$f(x)<x$ and $g(x)<x$. Then we have
\[fg(x)<f(x)<x\quad \textrm{and}\quad gf(x)<g(x)<x.\] Condition (ii) follows from the preceding paragraph.
When applying the same proof of condition (iii) in Case 1 to the present case,
note first that $u,v_0,v_1$ are on the same side of $x$.
The only thing that remains to check is the existence of $\sigma_i$. For this, it suffices to impose the following additional conditions below:
\begin{itemize}
\item $p>v_1 \text{ if and only if }v_0>u$;
\item $p>v_0 \text{ if and only if }v_1>u$;
\item $p,u,v_0,v_1$ are on the same side of $x$.
\end{itemize}
One may succinctly rewrite the first two conditions as
\[
(p-v_0)(v_1-u)>0\text{ and }(p-v_1)(v_0-u)>0\]
in the special case when $X=\bR$.
Such a point $p$ exists by the density of $X$.
The rest of the proof proceeds in the exactly same manner as Case 1.
{\bf Case 2-2: $X$ is equipped with a circular order $\le$.}
Since $G$ is $2$--transitive in this case,
the verification of the two conditions (i) and (ii) is exactly the same as in Case 1.
For the condition (iii), one may follow the proof for Case 2-1 and simply ignore the requirement that
\begin{center}
$p,u,v_0,v_1$ are on the same side of $x$.
\end{center}
This verifies that $\Stab_G(x)$ is a pre-stabilizer subgroup in all cases.
\ep
We now consider locally transitivity and dilativity in relation to pre-stabilizer subgroups.
\begin{lem}\label{lem:four-ball}
Let $X$ be an infinite connected Hausdorff topological space,
on which a group $G$ acts faithfully by homeomorphisms.
Assume that $K\le G$ is a pre-stabilizer subgroup.
\be[(1)]
\item\label{p:three-orbit} If a normal subgroup $N$ of $G$ admits an orbit $N.x\sse X$ of cardinality at least two,
then there exists an open neighborhood $U$ of $x$ such that
\[
\left[G_U,G_U\right]\le N.\]
\item\label{p:four-ball} If $G$ is locally dilative, then
some nontrivial element in $K$ fixes a nonempty open set.
\item\label{p:singleton}
If $G$ is locally transitive, and if $A\sse X$ is a proper, nonempty, closed, $K$--invariant subset, then $\partial A$ is a singleton
and $K=\Stab_G(\partial A)$.
\ee
\end{lem}
\bp
To prove part~\ref{p:three-orbit}, pick an element $g$ of $N$
and a nonempty open neighborhood $U\sse X$ of $x$ such that $U\cap g(U)=\varnothing$.
Let $h_1,h_2\in G_U$ be arbitrary.
Since
\[\suppo gh_1^{-1}g^{-1}
= g\suppo h_1 \]
is disjoint from $U$, we see that
\[[[h_1,g],h_2]=[h_1gh_1^{-1}g^{-1},h_2]=[h_1,h_2].\]
By normality, the above commutator belongs to $N$, which proves the first part.
For part~\ref{p:four-ball}, let us fix four disjoint nonempty open sets $U_0,\ldots,U_3$ in $X$.
By local dilativity, we have some $g_i\in G$ such that
\[\suppc g_i=\suppc g_i^{-1}\] is a connected subset of $U_i$.
We see that the element $f:=g_2g_3$ is conjugate to none of $g_i^{\pm1}$, by comparing the number of
connected components of respective closed supports.
Set $V:=U_2\cup U_3$.
Assume for contradiction that the fixed point set of each nontrivial element in $K$ has empty interior.
Since $\Fix g_i$ contains an open set $U_j$ for $j\ne i$, we have that $f,g_i\not \in K$.
Replacing $g_i$ by $g_i^{-1}$ if necessary, we may
apply the coset decompositions from Definition~\ref{defn:prestab}
and further require that
\[
g_0,g_1\in KfK.\]
We can also find some
\[ s_i,t_i\in f^{-1}Kg_i\cap K\]
for $i=0,1$ such that
\[s_0s_1=t_1t_0.\]
We claim that if $\tau\in f^{-1}Kg_i\cap K$ for some $i=0,1$,
then \[\tau^{-1}(X\setminus \overline V) \sse \suppc g_i\sse U_i.\]
Indeed, pick $\sigma\in K$ such that $\tau=f^{-1}\sigma g_i\in K$.
If there exists a point
\[
x\in \tau^{-1}(X\setminus \overline V)\cap (X\setminus \suppc g_i)\]
then we would have
\[
\tau(x) = f\tau(x)= \sigma g_i(x)
= \sigma(x).\]
This implies that $x$ is an interior point of $\Fix \sigma^{-1}\tau$.
Since $f$ and $g_i$ are not conjugate we have that
$\sigma^{-1}\tau$ a nontrivial element of $K$, a contradiction.
Applying the above claim to $s_i$ and $t_i$, we compute
\[
s_1^{-1}s_0^{-1}\left(X\setminus \overline V\right)
\sse s_1^{-1}\left({U_0}\right)
\sse s_1^{-1}\left(X\setminus \overline{V}\right)\sse {U_1}.\]
Similarly, we have $t_0^{-1}t_1^{-1}$ maps
$X\setminus \overline V$ into ${U_0}$.
This contradicts $s_1^{-1}s_0^{-1}=t_0^{-1}t_1^{-1}$.
Part~\ref{p:four-ball} of the lemma follows.
Let us prove part~\ref{p:singleton}.
Since $X$ is connected, the set $A$ is not open. In particular, $\partial A$ is nonempty. If we show that $\partial A$ is a singleton,
then the maximality of a pre-stabilizer group (Lemma~\ref{lem:prestab-maximal}) would imply that $K=\Stab_{G}(\partial A)$.
Here, note the trivial fact that $G(\partial A)\ne\partial A$.
Assume for contradiction that $\partial A$ contains distinct points $x_0$ and $x_1$. Pick disjoint open neighborhoods
$U_0$ and $U_1$ of $x_0$ and $x_1$. By local transitivity, we have some $h_i\in G_{U_i}$ such that \[h_i(x_i)\in U_i\setminus A.\]
Since $A$ is $K$--invariant, we have $h_i\not\in K$.
By the definition of a pre-stabilizer group, we have
\[
h_0\in K h_1^{\pm1}K.\]
Setting $g_0:=h_0$ and $g_1:=h_1^{\pm1}$, we have $g_0\in K g_1 K$.
Since $g_0g_1(x_0)=g_0(x_0)\not\in A$, we have that
\[f:= g_1g_0=g_0g_1\not\in K.\]
It follows that $g_i\in K f K$ for $i=0,1$, again applying the definition of a pre-stabilizer group.
For $i=0,1$ we have
\[
x_i\in A_i:= g_i^{-1}(X\setminus A)\cap A.\]
It follows that $A_i\sse \suppo g_i\sse U_i$. Set
\[B:=A_0\cup A_1=f^{-1}(X\setminus A)\cap A.\]
We claim that every $\sigma$ in the set
\[
f^{-1} K g_i\cap K,\]
satisfies $\sigma(A_i)=B$. To see this, let us write
\[
\sigma=f^{-1}\tau g_i\]
for some $\tau\in K$. Then
\[
f\sigma(A_i)=\tau g_i(A_i)\sse \tau(X\setminus A)=X\setminus A.\]
We have that
\[\sigma(A_i)\sse f^{-1}(X\setminus A).\]
Since $\sigma(A_i)\sse A$, we see from the definition of $B$ that
$\sigma(A_i)\sse B$.
We also see that
\[
g_i\sigma^{-1}(B)=\tau^{-1}f(B)\sse \tau(X\setminus A)=X\setminus A.\]
This implies that $\sigma^{-1}(B)\sse A_i$. This proves the claim.
Since $A_0$ and $A_1$ are nonempty, so is $B$.
Using the condition for a pre-stabilizer group, we can find
$s_i,t_i\in f^{-1}Kg_i$ such that $s_0s_1=t_1t_0$.
As in the previous part, we have
\[
s_1^{-1}s_0^{-1}(B)
\sse s_1^{-1}(A_0)
\sse s_1^{-1}(B)\sse A_1.\]
Similarly, we have $t_0^{-1}t_1^{-1}$ maps
$B$ into $A_0$.
This is a contradiction, completing the proof of the lemma.\ep
\subsection{Reconstructing a homeomorphism from pre-stabilizer groups}
We can now establish the aforementioned generalization of Filipkiewicz's theorem, namely Theorem~\ref{thm:isom-homeo}.
\bp[Proof of Theorem~\ref{thm:isom-homeo}]
Fix a $G_i$--invariant open basis $\UU_i$ of $X_i$.
For each $y\in X_2$, we set
\[
F^1_y:=\Phi^{-1}(\Stab_{G_2}(y))\le G_1.\]
Similarly for $x\in X_1$ put
\[
F^2_x:=\Phi(\Stab_{G_1}(x))\le G_2.\]
These are pre-stabilizer subgroups
in respective groups, by Lemma~\ref{lem:prestab}.
Define families of open sets
\[
\AAA_x^i:=\left\{ U\in \UU_i \middle\vert \left[ (G_i)_U,(G_i)_U\right]\le F_x^i\right\}\]
for each $i=1,2$ and $x\in X_{3-i}$.
For each $x\in X_1$, $f\in G_1$ and $U\in \AAA_x^2$, we have
\begin{align*}
& \left[ (G_2)_{\Phi(f)U}, (G_2)_{\Phi(f)U}\right]=
\Phi(f) \left[ (G_2)_U,(G_2)_U\right]\Phi(f)^{-1}\\
&\le
\Phi(f)F^2_{x}\Phi(f)^{-1}
=\Phi\left(\Stab_{G_1}(f(x))\right)
=F^2_{f(x)}.\end{align*}
For each $x\in X_{3-i}$, we set
\[
E_x^i=X_i\setminus \bigcup \AAA_x^i.\]
Then $E_x^i$ is a closed $F_x$--invariant subset of $X_i$.
\begin{claim}
For each $i=1,2$ and $x\in X_i$ the set $E_x^i$ is nonempty.
\end{claim}
We may consider the case $i=2$ only, by symmetry.
Assume for contradiction that $E_x^2$ is nonempty for some $x\in X_1$.
The collection $\AAA_x^2$ is an open cover of $X_2$.
Since $G_2$ is weakly fragmented, we have that
$F_x^2$ contains a nontrivial normal subgroup $K_2$ of $G_2$.
Let $y\in X_1$ be arbitrary. By the simple transitivity (as is implied by the 3--transitivity) of $G_1$,
we have that some $f\in G_1$ sends $x$ to $y$.
Then we see that
\[
K_2
=
\Phi(f)K_2\Phi(f^{-1})
= \Phi(f) F^2_x \Phi(f^{-1})=F^2_y.\]
It follows that
\[
K_2\le \bigcap_{y\in X_1}F^2_y
=\Phi\left(\bigcap_{y\in X_1}\Stab_{G_1}(y)\right).\]
This is a contradiction, since the right hand side is trivial.
The claim is thus proved.
\begin{claim}
There exists some $i\in\{1,2\}$ and $x\in X_{3-i}$ such that
$\AAA_x^i$ is nonempty.
\end{claim}
Let us pick $y\in X_2$. We have noted above that $F_y^1$ is a pre-stabilizer subgroup of $G_1$.
By part~\ref{p:four-ball} of Lemma~\ref{lem:four-ball},
there exists some nontrivial $g_y\in F_y^1$ fixing some open set, say $A_y\in \UU_1$.
Since $g_y$ is nontrivial, we have a proper nonempty closed set
\[
B:=\Fix \Phi(g_y)\ni y.\]
From this,
we define two natural subgroups of $G_1$. Namely, we let
\[H=\Phi^{-1}(\Stab_{G_2}(B)),\quad K=\Phi^{-1}(G_2[X_2\setminus B]).\]
Here, $\Stab_{G_2}(B)$ is the
set-wise stabilizer of $B$.
Observe that $y\in B$ and that
\[ g_y\in K\unlhd H\le G_1.\]
Since $g_y$ is centralized by $G_1[A_y]$,
we have that $\Phi(G_1[A_y])$ fixes
$B$ set-wise, so that $G_1[A_y]\le H$.
We now finish the proof of the claim by analyzing $K$.
Assume first that for some $a\in A_y$, the orbit $K.a$ has cardinality at least two. By part~\ref{p:three-orbit} of Lemma~\ref{lem:four-ball},
we can find an open set $A_y'\in \UU$ such that
\[ a\in A_y'\sse A_y\] and such that
\[
\left[
(G_1)_{A_y'},(G_1)_{A_y'}
\right]
\le
\left[
H_{A_y'},H_{A_y'}
\right]\le K\le F_y^1.\]
This fits the definition of $\AAA_y^1$, which is now shown to be nonempty.
Lastly, assume that $K$ fixes all points in $A_y$.
Pick an arbitrary $a\in A_y$ so that $K\le\Stab_{G_1}(a)$.
Since $B$ is a proper closed subset of $X_2$, we have some $V\in \UU_2$ contained in $X_2\setminus B$.
We see that
\[
[(G_2)_V,(G_2)_V]\le (G_2)_V\le (G_2)_{X\setminus B}= \Phi(K)\le F_a^2.\]
This implies that $V\in \AAA_a^2$, completing the proof of the claim.
By symmetry, we may assume that $\AAA_a^2$ is nonempty for some $a\in X_1$. It follows that
$E_a^2$ is a proper nonempty closed $F_a$--invariant subset of $X_2$.
Applying Lemma~\ref{lem:four-ball} to the group $F_a$ and to the set $E_a^2$,
we obtain a point $b\in E_a^2$ such that
\[\Phi(\Stab_{G_1}(a))=\Stab_{G_2}(b).\]
We see that there exists a unique bijection
\[
w\co X_1\longrightarrow X_2\]
satisfying
\[ w(g.a)=\Phi(g).b\]
for all $g\in G_1$. Indeed, $w$ is well-defined since whenever $g.a=h.a$, the preceding paragraph implies that
$\Phi(g).b=\Phi(h).b$. Similarly, $w$ is invertible and is therefore a bijection.
It is immediate to see the relation \[\Phi(g)=wgw^{-1}\] for $g\in G_1$.
Note also that
\[
\Phi(\Stab_{G_1}(x))=\Stab_{G_2}(w(x))\]
for all $x\in X_1$.
By Lemma~\ref{lem:bij-homeo}, we conclude that $w$ is a homeomorphism as claimed.\ep
This concludes the second proof of the Takens--Filpkiewicz Theorem, and thus the chapter.
\chapter{The $C^1$ and $C^2$ theory of diffeomorphism groups}\label{ch:c2-thry}
\begin{abstract}This chapter concentrates on the interplay of analytic and combinatorial methods for investigating finitely generated groups
of homeomorphisms of one--manifolds, with an emphasis on obstructions to $C^1$ and $C^2$ actions. Much (though not all) of this
chapter is self-contained. Whereas all the results presented herein are useful in various contexts, the $abt$--Lemma is the most important
theorem insofar as future applications are concerned, and its proof occupies the majority of the content. The reader will note that an
overarching theme in this chapter is that regularity greatly constrains the orbit structure of group actions in the presence of partial
commutation.\end{abstract}
\section{Kopell's Lemma}
Kopell's Lemma, formulated by N.~Kopell in~\cite{Kopell1970}, is one of the earliest and most powerful tools for controlling the relationship
between the algebraic structure of groups and regularity of group actions on one--manifolds. It says, roughly, that two commuting
diffeomorphisms have to have equal or disjoint supports.
Originally formulated for $C^2$ actions, a generalization to $C^{1+\mathrm{bv}}$ actions was given by A.~Navas in his thesis.
We give a proof here which is a mild generalization using an estimate due to Polterovich and Sodin~\cite{PS2004}.
Recall our convention from the introduction that the regularities we consider are
always thought of as local properties of functions. This convention also applies to
diffeomorphisms whose derivatives have bounded variation. The distinction between global and local estimates did not play a role in
Chapter~\ref{sec:denjoy} where we only considered compact one--manifolds. In general, a
$C^k$ diffeomorphism $f$ of a possibly non-compact one--manifold $M$ is said to be $C^{k,\mathrm{bv}}$
if each point in $M$ has a neighborhood $U$ where $f^{(k)}$ has a bounded total variation, denoted by
$\operatorname{Var}\left(f^{(k)};U\right)$. This way, we have that a $C^{k+1}$
diffeomorphism of a one manifold is necessarily $C^{k,\mathrm{bv}}$.
In the theorem below as well, each element $f$ in $\Diffb[0,1)$ satisfies \[ \Var(f';[0,\delta])<\infty\] for all $\delta\in(0,1)$.
We emphasize again that the supremum
\[
\sup_{\delta>0}\Var(f';[0,\delta])\] is allowed to be infinite.
\begin{thm}[Kopell's Lemma]\label{thm:kopell}
Let $f\in\Diffb[0,1)$ and $g\in\Diff_+^1[0,1)$ be commuting diffeomorphisms such that \[\Fix(f)\cap (0,1)=\varnothing\quad \textrm{and}
\quad\Fix(g)\cap (0,1)\neq\varnothing.\] Then $g$ is the identity.
\end{thm}
To establish the above, let us note two elementary facts about $C^1$ diffeomorphisms.
\begin{lem}\label{lem:diff-fix}
If $g$ is a $C^1$ diffeomorphism of a one--manifold $M$,
and if $x$ is an accumulation point of $\Fix g$ in $M$,
then we have that $g'(x)=1$.
\end{lem}
\bp
Let $\{x_n\}_{n\ge1}$ be a sequence of fixed points of $g$ converging to $x$. The Mean Value Theorem implies that for some $y_n$
lying between $x$ and $x_n$, we have \[g'(y_n)=\frac{g(x_n)-g(x)}{x_n-x}=1.\]
By the continuity of $g'$ we have that
\[
g'(x) = \lim_{n\to\infty} g'(y_n)=1,\]
which establishes the lemma.
\ep
The following is essentially observed by Polterovich and Sodin~\cite{PS2004}.
\begin{lem}\label{lem:inf-finite}
If $g$ is a nontrivial orientation preserving $C^1$ diffeomorphism of a bounded interval $J$
then we have that
\[
\sum_{n\in\bZ}\inf\left\{\left(g^n\right)'(x)\mid x\in J\right\} <\infty.\]
\end{lem}
\bp
Switching $g$ and $g^{-1}$ if necessary
we may assume to have some $c\in J$ such that $g(c)>c$.
Put $K:=[c,g(c)]$. By the Mean Value Theorem, we have some $x_n\in K$
such that
\[
|J|\ge \sum_{n\in\bZ} \abs*{g^n(K)}= \sum_{n\in\bZ}|K|\cdot \left(g^n\right)'(x_n)
\ge
|K|\cdot \sum_{n\in\bZ}\inf\left\{\left(g^n\right)'(x)\mid x\in J\right\}.\]
This implies that the given infinite sum is at most $|J|/|K|$.
\ep
As was implicitly used in the proof of Lemma~\ref{lem:inf-finite},
whenever $f$ is a fixed point free diffeomorphism of $(0,1)$, then $f$ admits a ~\emph{fundamental domain}\index{fundamental domain},
which is a half-open interval
$J\sse (0,1)$ such that $f^n(J)\cap J=\varnothing$ if $n\in\Z\setminus\{0\}$, and such that \[(0,1)=\bigcup_{n\in\Z} f^n(J).\] In fact,
one may set $J=[c,f(c))$ or $J=[f(c),c)$ for an arbitrary $c\in (0,1)$.
The following lemma considers a more general situation than Kopell's Lemma, as it
only assumes that $g$ is $C^1$ in the open interval $(0,1)$.
Roughly, it shows that
\[ \liminf_{y\to+0} \left(g^n\right)'(y)\]
is less than $1/|n|$ for almost all $n\in\bZ$, even when $g$ is not differentiable at $0$; the interested may wish to compare
with Lemma~\ref{l:sum-density}.
\begin{lem}\label{lem:kopell0}
If $f\in\Diffb[0,1)$ and $g\in\Diff_+^1(0,1)$ are nontrivial, commuting diffeomorphisms
such that \[\Fix(f)\cap (0,1)=\varnothing\quad \textrm{and}
\quad\Fix(g)\cap (0,1)\neq\varnothing.\]
Then we have the estimate
\[
\sum_{n\in\bZ} \liminf_{y\to+0} \left(g^n\right)'(y)<\infty.\]
\end{lem}
\bp
Replacing $f$ by its inverse if necessary, we assume that $f(x)<x$ for all $x\in (0,1)$.
Pick a fundamental domain $J=[f(c),c)$ of $f$, for some $c\in (0,1)$ to be chosen later.
It is easy to see that $\log f'$ has bounded variation on $[0,1)$ whenever $f'$ does as well.
So, for each $n\ge0 $ and for each $a,b\in J$ we have that
\[
\log\frac{(f^n)'(a)}{(f^n)'(b)}
=\sum_{i=0}^{n-1} \abs*{\log|f'(f^i(a))|-\log |f'(f^i(b))|}
\le v:=\Var\left(f;[0,c]\right)<\infty.\]
Let us now require \emph{a priori} that $c\in\Fix g$.
We have that $g(J)=J$ since \[gf(c)=fg(c)=f(c).\]
Moreover, for arbitrary $m,n\in\bZ$ and $x\in J$,
we obtain the estimate
\[
\left(g^n\right)'(x) = \frac{\left(f^m\right)'(x)}{ \left(f^m\right)'\left( g^n(x)\right)}\cdot \left(g^n\right)'\left( f^m(x)\right)\ge e^{-v} \cdot \left(g^n\right)'\left( f^m(x)\right).\]
Here, the last inequality comes from the fact that $g^n(x)\in J$.
By fixing $n\in\bZ$ and sending $m\to\infty$, we see that
\[
\left(g^n\right)'(x) \ge
e^{-v} \liminf_{y\to+0} \left(g^n\right)'(y).\]
Assuming $g$ is nontrivial, we now have that
\[ \sum_n\liminf_{y\to+0} \left(g^n\right)'(y)
\le \sum_n e^v \inf\left\{\left(g^n\right)'(x) \mid x\in J\right\} <\infty,\]
by Lemma~\ref{lem:inf-finite}.
\ep
\begin{proof}[Theorem~\ref{thm:kopell}]
Let $c\in(0,1)$ be a fixed point of $g$.
Since $f$ and $g$ commute, the points $\{f^n(c)\}_{n\in\bZ}$ are also fixed by $g$. In particular, $0$ is an accumulation of $\Fix g =\Fix g^n$.
Lemma~\ref{lem:diff-fix} implies that $\left(g^n\right)'(0)=1$ for all $n$.
By Lemma~\ref{lem:kopell0}, this implies that $g$ must be trivial.\ep
\iffalse
We suppose for a contradiction that $g$ is not the identity.
We have that $g$ has at least one fixed point $p\in (0,1)$.
Choosing a fixed point in $\partial\Fix(g)\cap (0,1)$
and replacing $g$ by its inverse if necessary, we may suppose
that $g^k(x)\to p$ as $k\to\infty$ for some $x$ near $p$. We may choose $x$ and $p$ to lie within a single fundamental domain for $f$,
so that the expression \[\left|\frac{(f^n)'(x)}{(f^n)'(g^m(x))}\right|\] is bounded away from both infinity and zero, independently of
$n,m\geq 0$. As $g(p)=p$, for all $\epsilon>0$, we may find an $m>0$ such that \[\frac{|g^m(x)-p|}{|x-p|}<\epsilon.\] Thus,
by choosing $m$ sufficiently large, we may arrange $|(g^m)'(x)|<\delta$ for a prescribed bound $\delta>0$.
We have that the fixed points of $g$ accumulate at $0$, whence it follows from the continuity of $g'$ and the Mean Value Theorem
that $(g^m)'(0)=1$ for all $m$. For a fixed $\delta>0$, we may choose $n>0$ large enough so that $|(g^m)'(f^n(x))|\geq 1-\delta$.
We therefore have the estimate \[\left|\frac{(g^m)'(x)}{(g^m)'(f^n(x))}\right|\leq\frac{\delta}{1-\delta}.\] The constant $\delta$ was
arbitrary, whence from the preceding inequalities we obtain a contradiction.
\end{proof}
\fi
One of the most immediate applications of Kopell's Lemma is the Plante--Thurston theorem~\cite{PT1976},
which asserts that $C^{1+\mathrm{bv}}$ actions of nilpotent groups on the interval or on the circle factor
through abelian quotients. We closely follow an argument given by Navas~\cite{Navas2011} (cf.~\cite{KoberdaSurv20}).
\begin{thm}[Plante--Thurston Theorem]\label{thm:plante-thurston}
If $M\in\{I,S^1\}$ then every nilpotent subgroup of $\Diff_+^{1+\mathrm{bv}}(M)$ is abelian.
\end{thm}
\begin{proof}
Let $N\le \Diff_+^{1+\mathrm{bv}}(M)$ be nilpotent.
{\bf Case 1: $M=[0,1]$.} It suffices to prove that $N$ acts freely on $I$, by H\"older's Theorem (Theorem~\ref{thm:holder}).
Clearly we may assume that $N$ has
no global fixed points in $(0,1)$, and we will use this to derive a contradiction. Since $N$ is nilpotent there is a nontrivial element
$g$ contained in the center $Z(N)\le N$.
Suppose now that $f\in N$ is nontrivial and has at least one fixed point in $(0,1)$. Since $f$ is assumed to
be nontrivial, we may choose a fixed point $x\in \partial\Fix(f)\cap (0,1)$.
We have that $g(x)=x$. Indeed otherwise we may iterate $g$ on $x$ and extract limit points $a$ and $b$ of $\{g^n(x)\}_{n\in\Z}$,
and $[a,b]$ is a nondegenerate $g$--invariant interval. Note that by construction, $g$ acts on $(a,b)$ without fixed points.
Since $f$ commutes with $g$, we have that $[a,b]$ is also $f$--invariant.
Kopell's Lemma now implies that $f$ is the identity, whence we obtain $g(x)=x$.
The previous argument shows that if the action of $N$ is not free then every nontrivial element of $Z(N)$ has a fixed point in $(0,1)$.
So, let $y\in\partial\Fix(g)\cap (0,1)$. If $h\in N$ is nontrivial, then the argument of the previous paragraph (switching the roles of
$f$ and $g$ with $g$ and $h$ respectively) proves that $h(y)=y$. Since $h$ is arbitrary, $y$ is a global fixed point of the action of $N$,
a contradiction.
{\bf Case 2: $M=S^1$.} First, we have that $N$ preserves a probability measure $\mu$ on $S^1$, as follows from the amenability of $N$.
Indeed, $N$ is nilpotent and hence solvable, and so Theorem~\ref{thm:kaku-mar} implies that $N$ preserves a probability measure on $S^1$.
Theorem~\ref{thm:invt-homo} implies that the
rotation number thus gives a homomorphism from $N$ to the circle group. If there is an element $\nu\in N$ with irrational
rotation number, then $\nu$ is topologically conjugate to an irrational rotation, by Denjoy's Theorem (Theorem~\ref{thm:denjoy}.
Since $\nu$ is uniquely ergodic by Theorem~\ref{thm:irr-ue},
it follows that the invariant measure $\mu$
is the pushforward of Lebesgue measure by a homeomorphism (see Lemma~\ref{lem:leb-push}). In this case, $N$ is conjugate to a group
of rotations of the circle and is therefore abelian. We leave the proof that $N$ is conjugate to a group of rotations as an exercise for the
reader, or refer the reader to Proposition 1.1.1 of~\cite{Navas2011}.
We may therefore assume that all elements of $N$ have rational rotation number.
If $\nu\in N$ has a fixed point in $S^1$ and if $J\sse S^1$ is an interval such that $J\cap \nu J=\varnothing$,
then $J$ cannot be given positive
measure under the $N$--invariant measure on $S^1$. It follows easily that the support of $\mu$ is contained in the intersection of all
periodic orbits of $N$.
If $N$ is nonabelian and nilpotent, then by taking a commutator of elements in the penultimate term of the lower central series of $N$,
we may find a nontrivial central element $h\in Z(N)$ which is a commutator in $N$.
We write $h=[f,g]$. Since the rotation number is a homomorphism on $N$, we have that the rotation number of $h$ is zero, whence
$h$ has a fixed point (see Proposition~\ref{prop:rot-easy}).
An easy computation shows that \[h^{mn}=f^{-n}g^{-m}f^ng^m.\] We suppose that $x_0$ is a point in the support
of $\mu$, so that $x_0$ is a periodic point of both $f$ and $g$. For suitably chosen
values of $n$ and $m$, we see that $x_0$ is a fixed point of the subgroup $N_0=\form{ f^n, g^m}$. We have that $N_0$ is
nilpotent and nonabelian. Indeed, if $N_0$ were abelian then $h^{mn}$ would be the identity. Since $h$ is nontrivial and fixes a point
in $S^1$, it must have infinite order by Proposition~\ref{prop:l-order-homeo}, a contradiction.
Cutting the circle open at $x_0$, we see that the subgroup of $N_0$
acts on the resulting compact interval. The case where $M=[0,1]$ now implies that $N_0$ is abelian, and this is the desired contradiction.
It follows that $N$ must be abelian.
\end{proof}
\section{(Residually) nilpotent groups acting by $C^1$ diffeomorphisms}\label{sec:nilpotent}
In this section, we will showcase some constructions of faithful $C^1$ actions of groups on $I$ and $S^1$. This section will therefore be
more expository than the others in this book, and the reader will find less detailed (or entirely absent) proofs the results discussed here.
The Plante--Thurston Theorem gives the first nontrivial algebraic restriction of groups that admit faithful representations into $\Diffb(M)$.
This means that the critical regularity of a nonabelian nilpotent group is at most two. It is therefore an interesting question to determine where
exactly the critical regularity of nonabelian nilpotent groups lies.
We have that $\Homeo_+[0,1]$ is torsion--free since a finite cyclic group cannot admit a left invariant ordering
(see Proposition~\ref{prop:l-order-homeo}), and combining this with H\"older's Theorem for the circle (Theorem~\ref{thm:holder-circle})
and the characterization of fixed points via rotation number (Propositions~\ref{prop:rot-easy} and~\ref{prop:rot-easy}), we have
that an arbitrary finite subgroup of $\Homeo_+(S^1)$ is cyclic.
If $N$ is a finitely generated nilpotent group, then certainly $N$ might have torsion. However, it is not difficult to prove that in $N$, the
product of two torsion elements is again torsion, and so the torsion elements of $N$ form a normal subgroup which is itself finite, as is
easily checked. It
follows that $N$ admits a torsion--free quotient with finite kernel. Since finitely generated nilpotent groups can be shown to be
residually finite (since they admit faithful homomorphisms into matrix groups, finitely generated subgroups of which are always
residually finite),
$N$ admits a finite index subgroup that is torsion--free. So, in studying actions of finitely generated nilpotent groups on
manifolds, one loses relatively little of the dynamical and algebraic richness by assuming that $N$ is torsion--free.
Now, if $N$ is an arbitrary torsion--free countable nilpotent group, then $N$ admits a left invariant ordering and hence embeds
into both $\Homeo_+[0,1]$ and $\Homeo_+(S^1)$ (Proposition~\ref{prop:l-order-homeo} and Proposition~\ref{prop:c-order-homeo}).
To see that $N$ is orderable,
it suffices to prove the claim for finitely generated subgroups of $N$, since
a group admits a left invariant ordering if and only if every finitely generated subgroup admits such an ordering. Then, we have that
$N$ admits quotient by a finitely generated
central subgroup $Z$ such that $N/Z$ is again torsion--free, and such that the lower central series of $N/Z$ is
strictly shorter than that of $N$. Since finitely generated torsion--free abelian groups admit left invariant orderings, we may assume that
$Z$ and $N/Z$ admit such orderings by induction. We then order $N$ lexicographically; that is $n_1<n_2$ if the image of $n_1$ is less
than the image of $n_2$ in $N/Z$. If the images are equal then $n_1<n_2$ if $n_1^{-1}n_2$ is positive in $Z$. It is easy to check that
the resulting ordering is left invariant on $N$.
A natural question then follows: if $N$ is finitely generated and torsion--free nilpotent, can $N$ be realized as a subgroup of $\Diff_+^1(M)$
for $M\in\{I,S^1\}$? The answer turns out to be yes, by a result of Farb--Franks~\cite{FF2003}, though this
fact is far from obvious. One can find a further generalization of this result in~\cite{Jorquera,CJN2014,JNR2018}.
\subsection{The Farb--Franks Theorem and the universal nilpotent group}
We will give an outline of the argument due to Farb and Franks that ansers the previous question.
Let $N_m\le \SL_{m+1}(\bZ)$ denote the group of lower triangular matrices
with integer entries, and where all diagonal entries are $1$. It is a standard computation that $N_m$ is nilpotent, and that the lower central
series of $N_m$ has length $m$.
We have a finite generating
\[\{u_{1,m},\ldots,u_{m,m}\}\]
of the group $N_m$,
where the matrix
$u_{i,m}\in\SL_{m+1}(\bZ)$ consists of ones down the diagonal, and whose only nonzero entry is a one in position $(i,i-1)$.
If $N$ is a finitely generated, torsion--free, nilpotent group, then there is an $m$ for which $N$ embeds into $N_m$, as follows from
a classical result of Mal'cev (see~\cite{Raghunathan1972}, for instance). One way to show this is to build a certain nilpotent real Lie group
$N\otimes\bR$, called the \emph{Mal'cev completion}\index{Mal'cev completion}
of $N$. Then one shows that $N\otimes\bR$ embeds in the group $N_m(\bR)$
of lower triangular real $(m+1)\times (m+1)$ matrices with $1$ on the diagonal, and that $N$ lies in the integer points of this image.
Another perspective on this fact is as follows.
Recall, the \emph{Hirsch length}\index{Hirsch length} (or, the \emph{polycyclic length}\index{polycyclic length}) of a group $G_0$ is the
the length of a subnormal
series \[\{G_k\leq G_0\}_{k\ge1}\] such that $G_i/G_{i+1}$ is infinite cyclic.
For a group $N$ as in the preceding paragraph, one can find an exact sequence of the form
\[1\longrightarrow K\longrightarrow N\longrightarrow \bZ\longrightarrow 1\]
such that $K$ is a nilpotent group whose Hirsch length is strictly shorter than that of $N$.
By induction we have that $K$ embeds in $N_m$ for some $m$.
Since $\bZ$ is cyclic and torsion--free, the surjection $N\longrightarrow \bZ$ splits, and so $N$ has the structure of a semidirect product of
$K$ with $\bZ$. Choosing a generator $t$ for $\bZ$, we have that conjugation by $t$ is an automorphism of $K$. This automorphism
must act on the group $H_1(K,\bQ)$ by a unipotent matrix. This will guarantee that $N$ is nilpotent, as
is easily verified by computing commutators
in the semidirect product $\yt{K}$ of $H_1(K,\bZ)$ with $\bZ$, the group $\yt{K}$ also necessarily being
nilpotent since it is quotient of $N$. It is then possible
to embed $K$ in a (possibly larger) $N_m$, and to realize $t$ as global conjugation by a lower triangular matrix in $N_m$.
Let $P$ be a group theoretic property. We say a group $G$ is \emph{residually $P$}\emph{residual property}
if for each nontrivial element $g\in G$
there exists some quotient $Q$ of $G$ having the property $P$ such that the image of $g$ in $Q$ is nontrivial.
Note that $G$ is residually $P$ if and only if $G$ embeds into the direct product of groups that have the property $P$.
For instance, a residually torsion--free nilpotent group is a group $G$ such that every nontrivial element of $G$ survives in a
torsion--free nilpotent quotient of $G$. The class of residually torsion--free nilpotent groups is quite extensive, and includes
free groups, fundamental groups of closed surfaces, right-angled Artin groups, pure braid groups, and Torelli groups of
surfaces; see~\cite{BP2009JKTR} for instance.
In~\cite{FF2003}, Farb and Franks proved the following remarkable result and initiated the study of
smooth one--manifold actions of nilpotent groups, which resulted in many interesting
discoveries~\cite{Navas2008GAFA,Jorquera,ParkheETDS,CJN2014,JNR2018}.
\begin{thm}[\cite{FF2003}]\label{thm:ff2003}
Every finitely generated residually torsion--free nilpotent group embeds into $\Diff_+^1[0,1]$.
\end{thm}
Recall $\Diff^1_K(\bR)$ denotes the group of $C^1$--diffeomorphisms of $\bR$ supported in $K\sse\bR$.
The key step of the proof is the following.
\begin{prop}[\cite{FF2003}]\label{prop:ff2003}
For each $m\in\bN$ and $\epsilon>0$, there exists an embedding
\[
\rho=\rho_{m,\epsilon}\co N_m\longrightarrow \Diff^1_{[0,1]}(\bR)\]
such that for all $i=1,\ldots,m$ we have
\[
|\rho(u_{i,m})'-1|\le \epsilon.\]
\end{prop}
\bp[Proof of Theorem~\ref{thm:ff2003}, assuming Proposition~\ref{prop:ff2003}]
Let $G$ be a residually torsion-free nilpotent group with a finite generating set $\{a_1,\ldots,a_k\}$. We have a sequence of homomorphisms
\[
\phi_i\co G\longrightarrow N_{n(i)}\]
for each $i=1,2,\ldots$
and for some positive integers $n(1),n(2),\ldots$
such that \[\bigcap_i\ker\phi_i=\{1\}.\]
By Proposition~\ref{prop:ff2003}, for each interval $K_i:=[1/(i+1),1/i]$
we can pick an embedding
\[
\psi_i\co N_{n(i)}\longrightarrow \Diff^1_{K_i}(\bR)\]
such that
\[
|\left(\psi_i\circ\phi_i(a_j)\right)'-1|\le 1/i\]
for all $i\ge1$ and $1\le j\le k$.
Let $g\in G$.
We define
\[\phi\co G\longrightarrow \Homeo_{[0,1]}(\bR)\]
by the infinite product
\[
\phi(g):=\prod_i \psi_i\circ\phi_i(g).\]
This map $\phi$ is injective since each $g\in G$ survives under some $\phi_i$.
Since $\supp\phi(g)$ is contained in the disjoint union
\[K_0:=\bigsqcup_i \Int(K_i),\] it is obvious that $\phi(g)$ is $C^1$ on $K_0$.
Since
\[\psi_i(h)'(\partial K_i)=\{1\}\] for all $h\in N_{n(i)}$, we also have that
$\phi(g)$ is $C^1$ at $x\in 1/i$, for each $i\ge1$.
The only place remaining to check the $C^1$ regularity of $\phi(g)$ is $x=0$.
Pick a point $t\in K_i$. We first have
\[
1\le\frac{\phi(g)(t)}{t} \le \frac{1/i}{1/(i+1)},\]
which implies that $\phi(g)'(0)=1$.
Moreover, if we let $\ell$ be the word length of $g$ in the given generating set
then
\[
(1-1/i)^{\ell}\le |\phi(g)'(t)|\le (1+1/i)^\ell.\]
We see that $\lim_{t\to+0} \phi(g)'(t)=1$, which establishes that $\phi(g)$ is $C^1$ at $x=0$.
\ep
Let us now sketch some of the ideas behind the proof of the Proposition~\ref{prop:ff2003}.
The starting observation in~\cite{FF2003} is that the group $N_m$ acts faithfully on the real line. An even stronger conclusion
can be formulated as follows:
there exists a natural embedding $\iota\co N_m\longrightarrow N_{m+1}$ defined by
\[
\iota(A)_{ij}:=\begin{cases}
A_{ij},&\text{ if }1\le i,j\le m+1\\
\delta_{ij},&\text{ if }i=m+2\text{ or }j=m+2,\end{cases}
\]
where $\delta_{ij}$ denotes the Kronecker delta.
There also exists a left inverse $\pi$ of $\iota$ which forgets the $(m+2)$--th row and column.
From these maps $\iota$ and $\pi$, we can define direct and inverse limits
\begin{align*}
\Lambda&:=\varinjlim N_m,\\
\Gamma&:=\varprojlim N_m.\end{align*}
Each element of the group $\Gamma$ can be regarded as an $\bZ_{>0}\times\bZ_{>0}$ lower triangular matrix with ones on the diagonal.
We have natural inclusions
\[
N_m\hookrightarrow \Lambda\hookrightarrow \Gamma\hookrightarrow \prod_m N_m.\]
It is simple to check that the group $\Gamma$ acts faithfully on
\[
X_\infty:=\{(1,x_0,x_1,\ldots)\mid x_i\in\bZ\}\sse \bZ^{\bN}.\]
It is important for us that the lexicographical order on $X_\infty$ is preserved by the action of $\Gamma$.
Let us introduce a notation that for two intervals $I,J\sse\bR$ we write $I\le J$ if $\sup I\le\inf J$. We partition $I^*=[0,1]$ into compact intervals $I_i$ as an interior--disjoint union
\[
I^*=\bigcup_{i\in\bZ} I_i\bigcup\{0,1\}\]
such that $I_i\le I_j$ for $i\le j$.
For each vector $v\in\bZ^m$ with $m\ge1$ we partition again
\[
I_v=
\partial I_v\cup
\bigcup_{i\in\bZ} I_{v,i}\]
as an interior--disjoint union.
We will also choose the lengths of the intervals such that
\[
\lim_{m\to\infty} \sup_{v\in\bZ^m} |I_v|=0.\]
If $v=(x_0,x_1,\ldots)\in\bZ^{\bN}$ then we have nested compact intervals
\[
I_{x_0}\supseteq I_{x_0,x_1}\supseteq I_{x_0,x_1,x_2}\supseteq\cdots.\]
So, there uniquely exists a point $z_v$ in the union of the above compact intervals.
We have a faithful action of $\Gamma$ on the set $\{z_v\mid v\in\bZ^{\bN}\}$.
Namely, for each $A\in\Gamma$ and $v\in\bZ^{\bN}$, we define
\[
A(z_v)=z_{A.v},\]
where $A.v$ is the image of $v\in X_\infty$ under $A$;
in other words, we have an infinite matrix multiplication
\[
A\begin{pmatrix} 1\\ v\end{pmatrix}=\begin{pmatrix} 1\\ A.v\end{pmatrix}.\]
This action preserves the lexicographical order on $\bZ^{\bN}$,
and hence, naturally extends to the whole interval $I$, so that
\[\Gamma\hookrightarrow\Homeo_+[0,1].\]
From this we see that the uncountable group $\Gamma$ admits a left invariant ordering; see Proposition~\ref{prop:l-order-homeo}.
Because $\Gamma$ accommodates all finitely generated torsion--free nilpotent groups, we have that $\Gamma$ itself is not a nilpotent group.
We remark that every finitely generated residually torsion--free nilpotent group embeds into $\Gamma$ using similar ideas.
We now return to the proof of Proposition~\ref{prop:ff2003}.
For this we fix an integer $m\ge1$.
We will consider the collection of intervals
\[
\II_m:=\bigcup_{v\in\bZ^m}\{I_v\}.\]
The goal is to define an action of $N_m$ on this collection of intervals
in a consistent way with the action of $\Gamma$,
and extend this action to an action on $I$. For this, it will be useful to have a uniform way of
choosing a diffeomorphism for an arbitrary pair of intervals.
We use a certain equivariant family defined by Yoccoz (cf.~\cite{Navas2011}, Definition 4.1.23).
An \emph{equivariant family}\index{equivariant family} of homeomorphisms
\[\{\phi_{a,b}\colon [0,a]\longrightarrow [0,b]\mid a,b>0\},\] is one which satisfies
\[\phi_{b,c}\circ\phi_{a,b}=\phi_{a,c}\] for all $a,b,c>0$. We have already used one equivariant family when we constructed the first
examples of continuous Denjoy counterexamples in Chapter~\ref{sec:denjoy}; the family we used there was particularly simple, as they
were simply linear scalings. As we remarked then, such equivariant families are perfectly good for constructing continuous actions,
though they do not have good smoothness properties since the slope of $\phi_{a,b}$ near zero is $b/a$, and so if two different such
families are operating on intervals that share a boundary point, the derivatives on both sides will not agree unless the two families
scale by the same factor. This lattermost condition, as one might imagine, is usually too much to require.
Yoccoz's more sophisticated family is defined as follows. We define \[\phi_a\colon (0,a)\longrightarrow\bR\] by setting
\[\phi_a(x)=-\frac{1}{a}\cot\left(\frac{\pi\cdot x}{a}\right).\] The reader can check that this is a homeomorphism that preserves the usual
order on both $(0,a)$ and $\bR$.
The map $\phi_{a,b}$ is defined by $\phi_b^{-1}\circ \phi_a$, and thus furnishes a $C^{\infty}$
diffeomorphism $(0,a)\longrightarrow (0,b)$, and is easily seen to be equivariant. In fact, the equivariance of this family has very little
to do with the specific nature of the map $\phi_a$; the definition of $\phi_{a,b}$ is used to analyze the asymptotic regularity properties
of the family. For $0<x<a$ we can compute
\[
\phi_{a,b}(x) =\frac{b}2-\frac{b\arctan(b/a\cdot \cot(\pi x/a))}{\pi}
=
\int_0^x \left( 1 -
\frac{1-b^2/a^2}{1+b^2/a^2\cdot \cot^2(\pi x/a)}\right)dx.\]
An easy calculation using the chain rule shows that \[\phi'_{a,b}(x)=\frac{(\phi_a(x))^2+1/a^2}{(\phi_a(x))^2+1/b^2}.\] In particular,
\[\lim_{x\to 0}\phi'_{a,b}(x)=\lim_{x\to a}\phi'_{a,b}(x)=1,\] and so one can extend $\phi_{a,b}$ to a $C^1$ diffeomorphism $[0,a]
\longrightarrow [0,b]$. One can in fact prove that $\phi_{a,b}$ is a $C^2$ diffeomorphism, with second derivatives vanishing at $0$ and $a$.
The relevant properties of the family $\{\phi_{a,b}\}_{a,b>0}$ can be summarized as follows:
\begin{prop}\label{prop:yoccoz}
The equivariant family $\{\phi_{a,b}\}_{a,b>0}$ has the following properties.
\begin{enumerate}[(1)]
\item
For $a,b>0$, we have $\phi'_{a,b}(0)=\phi'_{a,b}(a)=1$.
\item
We have that \[\sup_{x\in[0,a]}|\phi'_{a,b}(x)-1|=\left|\frac{b^2}{a^2}-1\right|.\]
\item
For all $x\in [0,a]$, we have that \[|\phi''_{a,b}(x)|\leq \pi\frac{(\phi_a(x))^2+1/a^2}{(\phi_a(x))^2+1/b^2}\left|\frac{1}{b^2}-\frac{1}{a^2}\right|b.\]
\end{enumerate}
\end{prop}
We omit the details of Proposition~\ref{prop:yoccoz}, instead directing the reader to the discussion in~\cite{Navas2011,FF2003,KK2020-DCDS}.
Now, to construct group actions on $I$, we use the Yoccoz family. If $I=[y,y+a]$ and $J=[z,z+b]$ are compact intervals with $a,b>0$,
then we obtain an identification of these intervals via the family
in the obvious way, namely by \[\phi^I_J\colon I\longrightarrow J,\quad \phi^{I}_{J}(x)=
\phi_{a,b}(x-y)+z.\]
Let us now assume that $A\in N_m\le\SL_{m+1}(\bZ)$,
and $x\in I$.
Suppose first that $x\in I_v$ for some $v\in\bZ^m$.
Write $w:=A.v\in\bZ^m$, for the action of $A$ on $\bZ^m$ described above.
Then we define
\[
A(x):=\phi^{I_v}_{I_w}(x).\]
We can write $I^*=\bigcup_{v\in\bZ^m}I_v\cup J$ for some countable set $J$,
and hence, this action $A$ extends to the whole interval $I^*$.
By Proposition~\ref{prop:yoccoz}, the moment
we piece elements in the Yoccoz equivariant family in a way that is continuous (which will be immediate from the construction),
the result will automatically be a diffeomorphism in the interior of the interval, at least away from the accumulation points of intervals
of the form $I_v$.
There is an issue with continuity of the derivative
at the accumulation points,
which is why the choices of lengths of the intervals (which we will not specify here)
and the particulars of the Yoccoz equivariant family are especially relevant.
By the faithfulness of the action of $\Gamma$ on $I$, we see that
the action $x\mapsto A(x)$ defines an embedding
\[
N_m\longrightarrow\Homeo_+[0,1].\]
The rest of the proof of Proposition~\ref{prop:ff2003} consists of
technical assignment of lengths of $I_v$
and verifying (one by one!) the regularity of each generator of $N_m$.
We will not reproduce proofs of the more time--consuming calculations, though the ambitious reader may wish to try
them. See~\cite{FF2003,Jorquera,CJN2014,JNR2018} for related computations.
We will conclude the discussion by simply pointing out a key step for the proof.
\begin{lem}\label{lem:ff2003}
Let $m\ge2$ and $\epsilon>0$.
Then for some assignment of lengths for $\{I_v\}_{v\in\bZ^m}$
and for the action
of
\[
N_m=\{ u_{1,m},\ldots,u_{m,m}\}\]
defined as above,
we have that
\[
\sup \left\{
|u_{i,m}'(x)-1|
\middle\vert
{i=1,\ldots,m}\text{ and }{x\in I^*}
\right\}<\epsilon.\]
\end{lem}
\subsection{Topological conjugacy of virtually nilpotent group actions}\label{ss:parkhe}
Here we will briefly summarize some stronger results about nilpotent groups acting on one--manifolds, due to Parkhe, Castro, Jorquera,
Navas, and Rivas,
which appear in~\cite{ParkheETDS,CJN2014,JNR2018}.
In~\cite{ParkheETDS}, Parkhe proves the following remarkable
structure theorem for nilpotent groups actions on $I$ and $S^1$, which says that the construction of the action in Theorem~\ref{thm:ff2003}
is a common feature of all nilpotent group actions on one--manifolds:
\begin{thm}[See Theorem 1.1 in~\cite{ParkheETDS}]\label{thm:parkhe-main}
Let $M$ be a one--manifold and let $N\le \Homeo_+(M)$ be a finitely generated subgroup that contains a nilpotent group with finite index.
Then there exists countable collections of open sets $\{I_k\}_{k\ge1}$ and $\{J_k\}_{k\ge1}$ which satisfy the following conditions:
\begin{enumerate}[(1)]
\item
We have \[I=\bigcup_{k\ge1} I_k\quad \textrm{and}\quad J=\bigcup_{k\ge1} J_k\] are disjoint from each other.
\item
For $k\neq\ell$, we have \[I_k\cap I_{\ell}=J_k\cap J_{\ell}=\varnothing.\]
\item
For each $k$, the sets $I_k$ and $J_k$ are $N$--invariant.
\item
We can write \[I_k=\bigcup_{\ell} I_{k,\ell},\] where $I_{k,\ell}$ is a nonempty open interval and where the union defining $I_k$ is disjoint, such that
for all indices $\ell$ and $\ell'$, there
exists an element $g\in N$ satisfying that \[g(I_{k,\ell})=I_{k,\ell'},\] and that the stabilizer of $I_{k,\ell}$ in $N$ is trivial.
\item
For all $k$, the action of $N$ on $J_k$ is minimal, and the stabilizer in $N$ of each component of $J_k$ is abelian.
\item
We have $I\cup J$ is dense in $M$.
\end{enumerate}
\end{thm}
The components of $I$ are the ones occurring in the construction in the proof of Theorem~\ref{thm:ff2003}, and the components of $J$
account for the possibility that the group $N$ can surject to an abelian group of rank at least two, which can act minimally on the interval
and on the circle.
The usefulness of Theorem~\ref{thm:parkhe-main} to the problem of smoothability may not be immediately apparent, though it is evident
that it highly constrains the structure of orbits of actions by \emph{virtually}\index{virtual property}
nilpotent groups (i.e.~ones that contain nilpotent groups with finite
index).
To state the salient corollary to Theorem~\ref{thm:parkhe-main}, we recall a basic notion from geometric group theory. Let $G$ be a group
generated by a finite set $S$ satisfying $S=S^{-1}$. There is a natural metric on $G$, given by defining \[|g|_S=\min\{n\mid g=s_1\cdots s_n,\,
s_i\in S\},\] and then by defining $d_S(g,h)=|g^{-1}h|_S$. This metric depends on $S$, but its bi--Lipschitz equivalence class does not.
For $n\ge1$, we write \[B_{S}(n)=\{g\mid\,\, |g|_S\leq n\},\] and $b_{S}(n)=|B_{S}(n)|$.
The function $b_{S}(n)$ counts how many distinct group
elements in $G$ can be written as a product of at most $n$ generators. Of course $b_S(n)$ depends on $S$, but its coarse behavior does not.
We say that a finitely generated group $G$ has \emph{polynomial growth}\index{polynomial growth}
if $b_S(n)$ is dominated by a polynomial function $p(n)$ for
some (equivalently all) generating sets $S$. Since the coefficients of $p$ are sensitive to the choice of $S$ and since only the highest
degree term of $p$ is salient, we say that $G$ has polynomial growth of degree at most $d$ if $b_S(n)=O(n^d)$.
For a reader familiar with nilpotent groups, it is not difficult to prove that a finitely generated nilpotent group has polynomial growth, and
more generally that a finitely generated virtually nilpotent group has polynomial growth. It is a foundational result of Gromov that
the converse is also true.
\begin{thm}[See~\cite{gromov-poly,Kleiner-poly,TaoShalom-poly}]\label{thm:gromov-poly}
Let $N$ be a finitely generated group that has polynomial growth. Then $N$ is virtually nilpotent.
\end{thm}
Parkhe deduces the following consequence of Theorem~\ref{thm:parkhe-main}:
\begin{thm}[Theorem 1.4 in~\cite{ParkheETDS}]\label{thm:parkhe-conj}
Let $N\le \Homeo_+(M)$ be a finitely generated group of polynomial growth $O(n^d)$. Then for all $\tau<1/d$, the group $N$ is topologically
conjugate into $\Diff_+^{1,\tau}(M)$.
\end{thm}
Theorem~\ref{thm:parkhe-conj} shows that there are essentially no non--$C^1$ actions of a finitely generated nilpotent group $N$ on $M$;
any failure of differentiability results from a poor identification of the (abstract) support of $N$ with a subset of $M$, and that this can
be fixed by changing coordinates suitably.
Parkhe proves Theorem~\ref{thm:parkhe-conj} from Theorem~\ref{thm:parkhe-main} by choosing group elements whose $C^{1,\tau}$--norms
are bounded uniformly on every component of their supports, not unlike Jorquera. The diffeomorphisms are built from an equivariant
family which is more or less the same as Yoccoz's equivariant family.
Parkhe's result also has the remarkable consequence that the critical regularity of a finitely generated torsion--free nilpotent group
is always strictly greater than one. While this is a uniform result that holds for all nilpotent groups, the lower bound $1+1/d$ on the critical
regularity of a torsion--free nilpotent group of polynomial growth of degree $d$ is not sharp; this can be seen immediately from the
fact that torsion--free abelian groups act faithfully by $C^{\infty}$ diffeomorphisms, but $\bZ^d$ has growth $\sim n^d$.
Theorem~\ref{thm:parkhe-conj} does not furnish an upper bound on the critical regularity of a torsion--free nilpotent group, though the
Plante--Thurston Theorem (Theorem~\ref{thm:plante-thurston}) gives $2$ as an \emph{a priori} upper bound.
Recall that we write $N_n\le \SL_n(\bZ)$ for the group of unipotent upper triangular integer matrices.
Theorem~\ref{thm:parkhe-conj} is complemented by Theorem A of~\cite{CJN2014}, which shows
that the Farb--Franks construction of a faithful $C^1$ action of $N_n$ on $M$ is not topologically conjugate into $\Diff_+^{1,\tau}(M)$
for $\tau>2/((n-1)(n-2))$, provided $n\geq 4$.
Some better upper bounds
can be given for specific nilpotent groups.
\begin{thm}[~\cite{CJN2014}, Theorem C]\label{thm:cjn}
For all $\tau<1$ and $d\geq 1$, the group $\Diff_+^{1,\tau}(I)$ contains a metabelian group of nilpotence degree $d$.
\end{thm}
Here, a group is \emph{metabelian}\index{metabelian group}
if its commutator subgroup is abelian. Theorem~\ref{thm:cjn} implies that the integral Heisenberg
group $N_3$, which has polynomial growth of degree $4$, has critical regularity exactly two. Theorem~\ref{thm:cjn} does not say
anything about the groups $N_n$ for $n\geq 4$, since these groups are no longer metabelian.
\begin{thm}[Theorems A and B of~\cite{JNR2018}]\label{thm:jnr}
The group $N_4$ has critical regularity exactly $1+1/2$. That is, there is an injective homomorphism \[N_4\longrightarrow \Diff_+^{1,\tau}(I)\]
for all $\tau<1/2$ and no such homomorphism for $\tau>1/2$.
\end{thm}
It is unknown if there is an injective homomorphism of $N_4$ into $\Diff_+^{1,1/2}(I)$, and this is probably a very difficult question to resolve.
The critical regularity of $N_n$ for $n\geq 5$ is also unknown at the time of this book's writing.
\section{The Two-jumps lemma and the $abt$--lemma}\label{sec:abt}
In this section we turn away from constructions of $C^1$ actions of groups, and we
formulate and prove two fundamental dynamical obstructions for certain homeomorphisms (and certain groups
of homeomorphisms) to
be smoothable.
\subsection{The Two-jumps lemma}
The two-jumps lemma was originally formulated by H.~Baik and the authors. It can be viewed as a generalization of an unpublished result
of Bonatti--Crovisier--Wilkinson (\cite[Proposition 4.2.25]{Navas2011}), which concerns crossed $C^1$ diffeomorphisms. The statement is somewhat complicated, but
the result is very useful and the proof is
a fairly straightforward application of the Mean Value Theorem.
\begin{thm}[Two-jumps lemma; see~\cite{BKK2019JEMS}, and~\cite{KK2018JT}, Lemma 2.11]\label{thm:two-jumps}
Let $M$ denote either the compact interval or the circle, and let $f,g\co M\longrightarrow M$ be continuous maps.
Suppose $(s_i), (t_i)$ and $(y_i)$ are infinite sequences of points in $M$
such that for each $i\ge1$, one of the following two conditions hold:
\be[(i)]
\item
$f(y_i)\le s_i = g(s_i) < y_i < t_i = f(t_i) \le g(y_i)$;
\item
$g(y_i)\le t_i = f(t_i) < y_i < s_i = g(s_i) \le f(y_i)$.
\ee
If $|g(y_i)-f(y_i)|$ converges to $0$ as $i$ goes to infinity,
then $f$ or $g$ fails to be $C^1$.
\end{thm}
Figure~\ref{f:fg} illustrates the case (i) of the Two-jumps Lemma.
\begin{figure}[h!]
\tikzstyle {bv}=[black,draw,shape=circle,fill=black,inner sep=1pt]
\centering
\begin{tikzpicture}[>=stealth',auto,node distance=3cm, thick]
\draw (-4,0) node (1) [bv] {} node [below] {\small $f(y_i)$}
-- (-2,0) node (2) [bv] {} node [below] {\small $s_i$}
-- (0,0) node (3) [bv] {} node [below] {\small $y_i$}
-- (2,0) node (4) [bv] {} node [below] {\small $t_i$}
-- (4,0) node (5) [bv] {} node [below] {\small $g(y_i)$};
\draw (-1.1,-.4) node {\small $g$};
\draw (1.1,.4) node {\small $f$};
\path (3) edge [->,bend right,red] node {} (1);
\path (3) edge [->>,bend right,blue] node {} (5);
\draw [->>, blue] (2) edge [out = 200,in=-20,looseness=50] (2);
\draw [->, red] (4) edge [out = 20,in=160,looseness=50] (4);
\end{tikzpicture}%
\caption{Two--jumps Lemma.}
\label{f:fg}
\end{figure}
\begin{proof}[Proof of Theorem~\ref{thm:two-jumps}]
Suppose the contrary, so that $f'$ and $g'$ are both continuous on $M$.
For each index $i$, we let $I_i$ be the closed interval whose endpoints are the points $f(y_i)$ and $g(y_i)$. We set $A_i$ and $B_i$
to be closed intervals characterized by the conditions \[I_i=A_i\cup B_i,\quad A_i\cap B_i= y_i,\quad f(y_i)\in A_i,\, g(y_i)\in B_i.\]
In Figure~\ref{f:fg}, the interval $A_i$ makes up the left half of the full interval and $B_i$ makes up the right half.
The assumptions of the theorem imply that $s_i\in A_i$ and $t_i\in B_i$, though not necessarily in the interiors of these intervals.
Since $M$ is compact, by passing to a subsequence if necessary, we may assume that $\{y_i\}_{i\ge1}$ converges to a point $y\in M$.
Since by assumption the lengths of the intervals $I_i$ tend to zero as $i$ tends to infinity, we have that $y$ is fixed by both $f$ and $g$.
By Lemma~\ref{lem:diff-fix}, we have that $f'(y)=g'(y)=1$.
The Mean Value Theorem implies the existence of points $u_i\in (s_i,y_i)$ and $v_i\in (y_i, t_i)$ such that
\[g'(u_i)=\frac{g(y_i)-s_i}{y_i-s_i}= 1+\frac{|B_i|}{y_i-s_i}\geq 1+\frac{|B_i|}{|A_i|},\] and such that
\[f'(v_i)=\frac{t_i-f(y_i)}{t_i-y_i}= 1+\frac{|A_i|}{t_i-y_i}\geq 1+\frac{|A_i|}{|B_i|}.\] Note that \[\lim_{i\to\infty} u_i=\lim_{i\to\infty} v_i=y.\]
Multiplying these two expressions together, we see that
\[f'(v_i)g'(u_i)\geq 2+\frac{|B_i|}{|A_i|}+\frac{|A_i|}{|B_i|}\geq 4,\] since an easy calculation shows that
\[\frac{|B_i|}{|A_i|}+\frac{|A_i|}{|B_i|}=\frac{|A_i|^2+|B_i|^2}{|A_i|\cdot |B_i|}\geq 2.\] Lemma~\ref{lem:diff-fix} again shows that
\[\lim_{i\to\infty} f'(v_i)g'(u_i)\longrightarrow 1,\] a contradiction.
\end{proof}
\subsection{The $abt$--lemma}\label{ss:abt}
Suppose we are given three diffeomorphisms of the interval or of the circle, say $\{a,b,t\}$, subject to the proviso that $a$ and $b$ commute.
If these three diffeomorphisms are chosen otherwise in a sufficiently ``generic" manner, it would seem likely that the abstract
group $\form{ A,B,T}$ they generate
should be one which $A$ and $B$ commute and in which there are no other relations, which is to say $\bZ^2*\bZ$. The main feature of the
$abt$--lemma is that this intuition is completely mistaken, at least in the case where the supports of $a$ and $b$ are disjoint. More
precisely:
\begin{thm}[The $abt$--lemma]\label{thm:abt}
Let $M$ denote the interval $I$ or the circle $S^1$, and suppose $a,b,t\in\Diff^1_+(M)$. Suppose that \[(\supp a)\cap(\supp b)=\varnothing.\]
Then the abstract group $\form{ a,b,t}$ is not isomorphic to $\bZ^2*\bZ$.
\end{thm}
In fact, one can often say significantly more beyond the mere fact that the group $\form{ a,b,t}$ is not isomorphic to $\bZ^2*\bZ$.
Typically, the group $\form{ a,b,t}$ will contain a copy of the lamplighter group $\bZ\wr\bZ$, whence it is clear that no isomorphism
with $\bZ^2*\bZ$ can exist.
Before embarking on a proof of Theorem~\ref{thm:abt}, we first show how it can fail for homeomorphisms.
\begin{prop}\label{prop:z2z-real}
There exist homeomorphisms $a,b,t\in\Homeo_+(M)$ such that \[(\supp a)\cap (\supp b)=\varnothing,\] and such that
$\form{ a,b,t}\cong \bZ^2*\bZ$.
\end{prop}
\begin{proof}[Sketch]
Since the interval is a two--point compactification of $\bR$ and since $I$ is a submanifold of $S^1$, it suffices to give a proof of the proposition
for $M=\bR$.
Moreover, for each sequence $\{\ell_n\}_{n\ge1}$ of positive real numbers, we may choose
a sequence $\{I_n\}_{n\ge1}$ of disjoint
intervals with $|I_j|=\ell_j$. Thus, it suffices to find an action of $\bZ^2*\bZ$ on $\bR$ satisfying the hypotheses of the proposition such
that each nontrivial $g\in\bZ^2*\bZ$ acts nontrivially on some interval $I_g$ whose endpoints are global fixed points of $\bZ^2*\bZ$.
Let $1\neq g=T^{n_k}W_k\cdots T^{n_1} W_1$ be a reduced expression for $g\in\bZ^2*\bZ=\form{ A,B,T}$, where each
\[W_i=A^{p_i}B^{q_i}\in \form{ A,B}=\bZ^2.\] For each $i$, choose a generator $X_i\in\{ A,B\}$ such that the exponent $r_i$ of
$X_i$ in $W_i$ is nonzero.
Now, let $J=[0,2k+1]$, let $J_i=[2i-2,2i]$, and let $K_i=[2i-1,2i+1]$
for $1\leq i\leq k$. We now define the action of $\bZ^2*\bZ$. We set $T$ to fix $\partial K_i$
for each $i$, and we set $T$ to act on each $K_i$ by an arbitrary homeomorphism subject to the requirement that \[T^{n_i}(2i-3/4)\geq 2i+3/4.\]
Similarly, we set $X_i$ to act on each $J_i$ by an arbitrary homeomorphism subject to the requirement that \[X_i^{r_i}(2i-7/4)\geq 2i-1/4.\]
We set $Y_i=\{A,B\}\setminus X_i$ to act by the identity on $J_i$. It is straightforward to check now that $g(1/4)\geq 2k+3/4$, so that
$g$ does not act by the identity on $J$.
\end{proof}
It is an informative exercise to generalize Proposition~\ref{prop:z2z-real} to a general right-angled Artin group
$A(\gam)$ (see Section~\ref{sec:raag} below). We sketch here
a few hints to guide the reader. First, consider a word $w$ written in the vertices of the defining graph $\Gamma$
and their inverses. The word $w$ represents the
identity in $A(\gam)$ if and only if it can be reduced to the identity by applying the following moves (see~\cite{cartier-foata,cgw2009,hm1995}):
\begin{itemize}
\item
Free reductions;
\item
Swapping of commuting generators, i.e.~ $v_1v_2\mapsto v_2v_1$ whenever $\{v_1,v_2\}$ is an edge of $\Gamma$.
\end{itemize}
Next, one can express $w$ as a product $w=w_kw_{k-1}\cdots w_1$ of subwords which satisfy the following two conditions:
\begin{itemize}
\item
For $1\leq i\leq k$, any two generators occurring in $w_i$ commute with each other;
\item
For $1\leq i\leq k-1$, given a generator $v_i$ occurring in $w_i$, there is a generator $v_{i+1}$ occurring in $w_{i+1}$ which does not
commute with $v_i$.
\end{itemize}
This last claim can be proved by implementing a sorting algorithm on the generators occurring in $w$. Given this setup, it is fairly
straightforward to prove that $A(\gam)$ acts faithfully by orientation preserving homeomorphisms of $\bR$.
Another variation of Proposition~\ref{prop:z2z-real} is the observation that for all subgroups $G$ and $H$
of $\Homeo_+(\bR)$ the free product $G\ast H$ embeds into $\Homeo_+(\bR)$.
This property is not true for $\Diff_+^1(I)$ or $\Diff_+^1(S^1)$, as we will see in Corollary~\ref{c:bs12}.
However, a similar ping-pong argument can show that
for all subgroups $G$ and $H$ of $\Diff_+^\infty(I)$, the free product $G\ast H$ admits an
\emph{eventually injective}\index{eventually injective} sequence of homomorphisms into $\Diff_+^\infty(I)$ in the following sense.
One can also describe this property as that $G\ast H$ is \emph{residually $\Diff_+^\infty(I)$.}\index{residual property}
\begin{lem}[cf. \cite{BKK2014,KKFreeProd2017}]\label{l:cinfty-free-product}
If $G_0$ and $G_1$ are subgroups of $\Diff_+^\infty(I)$,
then for each nontrivial $g\in G_0\ast G_1$
there exists a representation
\[
\phi=\phi_g\co G_0\ast G_1\longrightarrow \Diff_+^\infty(I)\]
having a connected support such that $\phi(g)\ne1$. Furthermore, one can require that
$\phi$ is injective on $G_0$ and on $G_1$.
\end{lem}
The proof will follow the classical ping-pong argument of Klein~\cite{Klein1883MA}, Maskit~\cite{Maskit1988Springer} and
Tits~\cite{tits-jalg}. Indeed, the nontriviality of $g$ for some new action will be certified by exhibiting a point that is moved
consecutively by factors of $g$. We will pay extra care so that the support of this new action is connected.
The following notation will be used for the remainder book.
\begin{notation}\label{notation:diff-J}
For an interval $J$ in $\bR$, we define
\[
\Homeo_J(\bR):=\{f\in \Homeo_+(\bR)\mid \supp f\sse J\}.\]
We similarly define
$\Diff_J^{k}(\bR)$ and $\Diff_J^{k,\alpha}(\bR)$
for an integer $k\ge1$ and for a concave modulus $\alpha$.
\end{notation}
If $J$ is bounded, one may identify each element of $\Diff_J^{k}(\bR)$ with a $C^{k}$
diffeomorphism of $J$ that is $C^k$--tangent to the identity at $\partial J$.
\bp[Proof of Lemma~\ref{l:cinfty-free-product}]
We let $I=[0,1]$. By Muller--Tsuboi trick (Theorem~\ref{t:muller-tsuboi}), we may assume that
\[ G_0, G_1\le\Diff_I^\infty(\bR).\]
It will be convenient for us to have a notation for the modulo--two remainder:
\[p(i)=i-2\floor{i/2}\]
The conclusion for the case $g\in {G_0}\cup {G_1}$ is easy to prove, so we will assume to have
\[
g =v_{2k-1}\cdots v_1v_0\]
for some integer $k\ge1$ and for some nontrivial elements \[v_i\in G_{p(i)}\le\Diff_I^\infty(\bR).\]
Let $J_i$ be the closure of a connected component of $\supp G_{p(i)}$ such that $v_i\restriction_{J_i}$ is nontrivial.
Using Muller--Tsuboi trick again, we can find a $C^0$--conjugate $H_{p(i)}\le\Diff_{ J_i}^\infty(\bR)$ of $G_{p(i)}\restriction_{J_i}$.
Note that the inversion $\sigma(x)=1-x$ of $I$ conjugates $\Diff_I^\infty(\bR)$ to itself.
We can construct an action
\[
\phi\co G_0\ast G_1\longrightarrow \Diff_{[0,2k+1]}^\infty(\bR)\]
satisfying the following.
\begin{itemize}
\item $\supp\phi(G_0)=\bigcup_{i=0}^{k-1} (2i,2i+2) \cup (2k+3/5,2k+4/5)$;
\item $\supp\phi(G_1)=\bigcup_{i=1}^{k} (2i-1,2i+1) \cup (1/5,2/5)$;
\item $\phi(G_0)\!\!\!\restriction_{[2k+3/5,2k+4/5]}$ is $C^\infty$--conjugate to $G_0\!\!\!\restriction_I$;
\item $\phi(G_1)\!\!\!\restriction_{[1/5,2/5]}$ is $C^\infty$--conjugate to $G_1\!\!\!\restriction_I$;
\item $\phi(G_i)\!\!\!\restriction_{[i,i+2]}$ is $C^\infty$--conjugate to $H_{p(i)}\!\!\!\restriction_{J_i}$
or to $\left(\sigma H_{p(i)} \sigma\right)\!\!\!\restriction_{\sigma ( J_{i})}$
for $i=0,\ldots,2k-1$
\item $\phi(v_i)(i+1/2)=i+3/2$ for $i=0,\ldots,2k-1$.
\end{itemize}
In particular, we see that $\phi(g)(1/2)=2k+1/2$, and the conclusion follows.
\ep
\subsubsection{Compact supports}\label{ss:compact supports}
Let $H\le \Homeo_+[0,1]$. Recall that $\supp H$ consists of $x\in M$ for which there exists a $h\in H$ with $h.x\neq x$, which is an open
subset of $(0,1)$.
If $1\neq g\in H$, we say that $g$ is \emph{compactly supported (in $H$)}\index{compactly supported homeomorphism},
or has \emph{compact support}, if
$\overline{\supp g}$ is contained in $\supp H$. That is, $\supp g$ is a precompact subset of $\supp H$.
Elements with compact support are useful for producing ``unexpected" commutations among diffeomorphisms. This idea goes back
to the Zassenhaus Lemma (see~\cite{Raghunathan1972}),
and was crucial in Brin--Squier's proof that the group of piecewise linear homeomorphisms of the interval
does not contain a nonabelian free group (see Theorem~\ref{thm:f-subgp} below, where we will give a proof of the Brin--Squier Theorem).
This idea will also play an important role in the construction of finitely generated groups of diffeomorphisms in
Chapter~\ref{ch:optimal}, especially in Theorem~\ref{t:optimal-group}.
As a warmup, we consider the following.
\begin{prop}[See~\cite{KK2018JT}, Lemma 3.3]\label{prop:compact-conn}
Suppose $H\le \Homeo_+[0,1]$ with $\supp H$ connected, and suppose that $g\in H$ has compact support. Then there exists a nontrivial element
$t\in H$ such that $[g,tgt^{-1}]=1$.
\end{prop}
\begin{proof}
Since $\supp H$ is connected, there is a collection $\{J_1,\ldots,J_k\}$ of intervals such that:
\begin{enumerate}[(i)]
\item
We have \[\overline{\supp g}\sse\bigcup_{i=1}^k J_i.\]
\item
We have $J_i\cap J_{i+2}=\varnothing$ for $i<k-1$.
\item
We have $\inf J_i<\inf J_{i+1}<\sup J_i<\sup J_{i+1}$ for $i\le k-1$.
\item
For all $i$, the interval $J_i$ is a component of the support of an element $h_i\in H$.
\end{enumerate}
A collection satisfying the conditions (ii) and (iii) above will be called as a $k$--chain; see Definition~\ref{defn:chain}.
Write $K=\overline{\supp g}$. We may clearly assume that $z_1=\inf K\in J_1$ and $\sup K\in J_k$. There are
integral exponents $\{n_i\}_{1\leq i\leq k}$
such that:
\begin{itemize}
\item
We have $z_2=h_1^{n_1}(z_1)\in J_2$.
\item
Having inductively defined $z_i$, we have $z_{i+1}=h_i^{n_i}(z_i)\in J_{i+1}$ for all $i<k$.
\item
We have $h_k^{n_k}(z_k)\geq\sup K$.
\end{itemize}
We then set $t=h_k^{n_k}\cdots h_1^{n_1}$. It is easy to see then that $\supp g$ and $\supp tgt^{-1}$ are disjoint, whence the conclusion
of the proposition.
\end{proof}
In fact, Proposition~\ref{prop:compact-conn} implies that $H$ contains the \emph{lamplighter group}\index{lamplighter group}
\[
\bZ\wr\bZ= \left(\bigoplus_{\bZ}\bZ\right)\rtimes\bZ.\]
Recall that this is the
\emph{wreath product}\index{wreath product} of $\bZ$ with $\bZ$.
This is built by taking \[N:=\bigoplus_{\bZ}\bZ,\] and letting $\bZ$ act on this direct sum
by shifting the index (which is to say via the natural action of $\bZ$ on itself), thus forming a semidirect product of $\bZ$ and $N$.
Letting $\lambda$ be a generator of the copy of $\bZ$ acting on $N$ and letting $\tau$ be a generator of the summand of $N$ corresponding to
$0\in\bZ$, it is easy to see the following group presentation:
\[\bZ\wr\bZ=\form{\tau,\lambda\mid \left[\tau,\lambda^i\tau \lambda^{-i}\right]=1\text{ for all }i\in\bZ}.\]
This wreath product is a finitely generated, nonabelian, solvable group
(in fact, it is a metabelian group), that is not finitely presentable. All but the last of these claims is an easy consequence of the definitions
save the last, which we will not discuss further here.
Lamplighter groups occur
quite commonly in the group of homeomorphisms of the interval.
Indeed, let $g\in\Homeo_+(\bR)$ be a homeomorphism such that $\supp g\sse (0,1)$, and let $f$ be translation by one. Then $f$ and $g$
are easily seen to generate a subgroup of $\Homeo_+(\bR)\cong \Homeo_+[0,1]$ that is isomorphic to $\bZ\wr\bZ$. Under the hypotheses
of Proposition~\ref{prop:compact-conn}, it is clear that the elements $g,t\in H$ furnished by the proposition generate a lamplighter group.
Proposition~\ref{prop:compact-conn} has the following corollary for homeomorphisms of the circle.
\begin{prop}\label{prop:comm-circle}
Let \[\{a,b,c,d\}\sse\Homeo_+(S^1)\] be nontrivial elements such that \[(\supp a)\cap(\supp b)=\varnothing\quad \textrm{and}\quad
(\supp c)\cap(\supp d)=\varnothing.\] Writing $G=\form{ a,b,c,d}$, if $\supp G= S^1$ then $G$ contains a copy of $\bZ\wr\bZ$.
\end{prop}
We leave the proof of Proposition~\ref{prop:comm-circle} as an exercise for the reader. To proceed with the proof of Theorem~\ref{thm:abt},
we need to state and prove some rather opaque facts about supports of commutators of homeomorphisms. It is difficult to give a
satisfactory intuitive explanation of the estimates we give; the reader should perhaps just keep in mind that we are merely aiming to force
a configuration of intervals under the hypotheses of Theorem~\ref{thm:abt} wherein the Two-jumps Lemma (Theorem~\ref{thm:two-jumps})
becomes applicable.
The reader may keep the following picture in their mind: suppose $\{a,b,t\}$ are as in the statement of Theorem~\ref{thm:abt}, with the
supports of $a$ and $b$ disjoint. It is not difficult to see that for $\form{ a,b,t}$ to be isomorphic to $\bZ*\bZ^2$, then both
$\supp a$ and $\supp b$ must have infinitely many components. Now, let $c=t^{-1}at$ and $d=t^{-1}bt$. If
$\form{ a,b,t}\cong\bZ*\bZ^2$ then $\form{ a,b,c,d}\cong\bZ^2*\bZ^2$. For the action of $\bZ^2*\bZ^2$ on $I$ via $\{a,b,c,d\}$
to have any hope of being faithful, then on infinitely many components of support, the diffeomorphisms $\{a,b,c,d\}$ have to translate points
a definite fraction of the length of their supports. This is a violation of the Mean Value Theorem (which is implicit in the Two-jumps Lemma).
The first technical result is the following.
\begin{lem}[\cite{KK2018JT}, Lemma 3.5]\label{lem:supp-comm}
Let $X$ be a Hausdorff topological space, and let $f,g\in\Homeo(X)$. Then, we have \[\overline{\supp [f,g]}\sse\supp f\cup\supp g\cup
\overline{\supp f\cap\supp g}.\]
\end{lem}
\begin{proof}
Suppose first that \[x\notin \supp f\cup\supp g\cup
\overline{\supp f\cap\supp g}.\] Then \[f(x)=g(x)=x,\] and so $x\notin \supp [f,g]$. It suffices to show that there is a neighborhood of $x$
that is in the fixed point set of $[f,g]$.
There is an open neighborhood $U$ containing $x$ such that \[U\cap (\supp f\cap\supp g)=\varnothing,\] since $x$ is not in the
closure of $\supp f\cap\supp g$. There is a further open sub-neighborhood $V$ of $x$ for which we have \[f^{\pm1}(V)\cup g^{\pm 1}(V)
\sse U.\] We claim that $V$ is fixed by $[f,g]$. If $y\in V$, then it is easy to check that $[f,g](y)=y$ by checking the cases
\[y\in V\cap\supp f,\quad y\in V\cap\supp g,\quad y\in V\cap \Fix f\cap\Fix g.\] This implies that \[x\notin\overline{\supp[f,g]},\] which
proves the lemma.
\end{proof}
The next lemma is a crucial estimate, which will allow us to finally apply the Two-jumps Lemma. It is established by a brute force
calculation.
\begin{lem}[\cite{KK2018JT}, Lemma 3.6]\label{lem:supp-phi}
Let $b,c,d$ be bijections of a set $X$ such that
\[\supp c\cap\supp d=\varnothing,\]
and write
\[\phi= [c,bdb^{-1}].\]
Then, we have
\[\supp\phi\sse\supp b\cup cb\left(\supp b\cap\supp d\right)
\cup db^{-1}\left(\supp b\cap\supp c\right).\]
\end{lem}
\begin{proof}
We will follow the notation in~\cite{KK2018JT} and write $\yt g=\supp g$ for $g\in\{b,c,d\}$. Note that
\[[c,bdb^{-1}]=cbd(cb)^{-1}\cdot bd^{-1}b^{-1}=c\cdot b\cdot db^{-1}c^{-1}(db^{-1})^{-1}\cdot b^{-1}.\]
We have that \[\supp\phi\sse \left(\yt{c}\cup b(\yt d)\right)\cap
\left(cb(\yt d)\cup b(\yt d)\right)\cap\left(\yt c\cup\yt b\cup db^{-1}(\yt c)\right).\]
Some tedious calculations, using the facts that \[b(\yt d)\sse \yt b\cup\yt d\quad
\textrm{and}\quad \yt c\cap\yt d=\varnothing,\] all taken together imply that
\[ \left(\yt{c}\cup b(\yt d)\right)\cap \left(cb(\yt d)\cup b(\yt d)\right)\cap\left(\yt c\cup\yt b\cup db^{-1}(\yt c)\right)\sse
\left(\yt c\cap cb(\yt d)\right)\cup\yt b\cup\left(\yt d\cap db^{-1}(\yt c)\right).\] We leave the details of these calculations
as an exercise for the reader.
Observe that it now suffices to prove the inclusions \[\yt c\cap \left(cb(\yt d)\right)\sse cb\left(\yt b\cap\yt d\right),\] and
\[\yt d\cap \left(db^{-1}(\yt c)\right)\sse db^{-1}\left(\yt b\cap\yt c\right).\] We will prove the first inclusion, as the second
will follow by the symmetry of $c$ and $d$.
So, let $x\in X$, and suppose that \[cb(x)\in\yt c\cap cb(\yt d).\] It follows then the $x\in\yt d$ and $cb(x)\in\yt c$. We have
that $x\notin\Fix b$, since $\yt c\cap\yt d=\varnothing$ and since $b(x)\in\yt c$. It follows that $x\in\yt b$, and so \[cb(c)\in cb(\yt b\cap\yt d).\]
This proves the desired inclusion, and with it, the lemma.
\end{proof}
The foregoing lemmata were purely combinatorial, and made no reference to diffeomorphisms. We will now use the
differentiability hypothesis to obtain a diffeomorphism whose algebraic provenance forces it to be compactly supported. We first see
that $\phi$ as in Lemma~\ref{lem:supp-phi}, if built on the interval $I$ using $C^1$
diffeomorphisms, is compactly supported ``modulo $\supp b$".
\begin{lem}[\cite{KK2018JT}, Lemma 3.7]\label{lem:diff-phi-b}
If we have orientation--preserving $C^1$ diffeomorphisms $b,c,d$ of a compact interval $I$ such that
\[\supp c\cap\supp d=\varnothing,\]
and if we write
\[\phi= [c,bdb^{-1}],\]
then we have that \[\overline{\supp\phi\setminus\supp b}\sse \supp c\cup\supp d.\]
\end{lem}
\begin{proof}
For the purposes of this proof, we will write $\BB$ for the set of components of $\supp b$ and $\CC$ for the set of components of $\supp c$.
For $B\in\BB$, we will write \[J_B=B\cup cb(B\cap\supp d)\cup db^{-1}(B\cap\supp c).\] Note that
\[\supp\phi\sse \bigcup \{J_B\mid B\in\BB\}=\bigcup\{J_B\setminus B\mid B\in\BB\}\cup\supp b,\] as follows from
Lemma~\ref{lem:supp-phi}. Observe also that \[\overline{J_B\setminus B}\sse \overline{c(B)\setminus B}\cup\overline{d(B)\setminus B}
\sse \supp c\cup\supp d.\]
{\bf Claim: the collection \[\BB_0=\{B\in\BB\mid J_B\neq B\}\] consists of finitely many intervals.} This claim is where we will use the
differentiability hypothesis.
The conclusion of the Lemma follows from this claim. Indeed, we have \[\overline{\supp\phi\setminus\supp b}\sse \overline{\bigcup\{
J_B\setminus B\mid B\in\BB_0\}},\] as follows formally. If $\BB_0$ is a finite union of intervals then closure commutes with unions, and
so we get \[\overline{\supp\phi\setminus\supp b}\sse \bigcup\{\overline{J_B\setminus B}\mid B\in\BB_0\}\sse \supp c\cup \supp d.\]
To prove the claim,
we set \[\BB_1=\{B\in\BB\mid cb(B\cap\supp d)\setminus B\neq\varnothing\},\] and we write \[\BB_2=\{B\in\BB\mid db^{-1}(B\cap\supp c)
\setminus B\neq\varnothing\}.\] Observe that $\BB_0=\BB_1\cup\BB_2$, and suppose that this collection is infinite. We will suppose
that $\BB_1$ is infinite and derive a contradiction, the argument for $\BB_2$ being essentially the same.
Thus, we assume that \[\{B_i\}_{i\ge1}\sse \BB_1\] are distinct, and for all $i\ge1$ there are points $x_i\in B_i\cap \supp d$ such
that $cb(x_i)\notin B_i$.
In particular, we have that $b(x_i)\in\supp c$ for $i\ge1$. We can find elements \[\{C_i\}_{i\ge1}\sse \CC\] such that for all $i$, we have \[b(x_i), cb(x_i)\in C_i.\]
Let $J_i=[x_i, cb(x_i)]$, possibly with the endpoints switched. Note that $b(x_i)$ lies in the interior of $J_i$. It follows that the interval
$(b(x_i),cb(x_i)]$ meets $\partial B_i$, and that the interval $[x_i,b(x_i))$ meets $\partial C_i$.
The claim now follows from the Two-jumps Lemma (Theorem~\ref{thm:two-jumps}), setting \[f=b^{-1},\quad g=c,\quad s_i=\partial C_i\cap B_i,
\quad t_i=\partial B_i\cap C_i,\quad y_i=b(x_i).\] This completes the proof of the lemma.
\end{proof}
As we have suggested before, the following lemma will conclude the proof of Theorem~\ref{thm:abt}.
\begin{lem}[\cite{KK2018JT}, Lemma 3.8]\label{lem:z2-z2}
Let $M\in\{I,S^1\}$, and let \[\{a,b,c,d\}\sse \Diff_+^1(M).\] Suppose that \[\supp a\cap\supp b=\supp c\cap\supp d=\varnothing.\]
Then the group $\form{ a,b,c,d}$ is not isomorphic to the group $\bZ^2*\bZ^2$.
\end{lem}
Indeed, let $\{a,b,t\}\sse \Diff_+^1(M)$ satisfy the hypotheses of Theorem~\ref{thm:abt}, and suppose that \[\form{ a,b,t}\cong\bZ*
\bZ^2.\] Then we have \[\form{ a,b,t^{-1}at,t^{-1}bt}\cong\bZ^2*\bZ^2.\] Setting $c=t^{-1}at$ and $d=t^{-1}bt$, we have that
\[\supp c\cap\supp d=\varnothing.\] It follows that the hypotheses of Lemma~\ref{lem:z2-z2} are satisfied, but the conclusion
is contradicted. It remains therefore to establish Lemma~\ref{lem:z2-z2}.
Before establishing Lemma~\ref{lem:z2-z2}, we record two basic facts about $\bZ^2*\bZ$, which we will prove fully here for the
purposes of self-containment.
\begin{prop}\label{prop:2g-hopf}
The following are properties of $\bZ^2*\bZ^2$.
\begin{enumerate}[(1)]
\item
If $g,h\in\bZ^2*\bZ^2$ then $\form{ g,h}$ is either free or abelian. In particular, $\bZ^2*\bZ$ contains no subgroup isomorphic to
$\bZ\wr\bZ$.
\item
Every surjective endomorphism of $\bZ^2*\bZ^2$ is an isomorphism; that is, $\bZ^2*\bZ^2$ is \emph{Hopfian}\index{Hopfian group}.
\end{enumerate}
\end{prop}
Both of the conclusions of Proposition~\ref{prop:2g-hopf} hold for general right-angled Artin groups, though we will not need such
generality here. See~\cite{Baudisch1981,KK2015GT,LM2010GD} for a discussion of two--generated subgroups,
and~\cite{dlHarpe2000,KS2020rfrp} for a
discussion of Hopficity and residual finiteness. The proof of Proposition~\ref{prop:2g-hopf} draws on ideas in~\cite{dlHarpe2000,KS2020rfrp}
especially.
\begin{proof}[Proof of Proposition~\ref{prop:2g-hopf}]
For the first item, we use a special case of
the Kurosh Subgroup Theorem, which says that a finitely generated subgroup of a free product $G*H$ of two groups
$G$ and $H$ is of the form \[K_1*K_2\cdots*K_n* F,\] where each $K_i$ is isomorphic to a finitely generated subgroup of $G$ or of $H$, and
where $F$ is a finitely generated free group. The Kurosh Subgroup Theorem itself follows easily from the covering space theory of
a wedge (i.e.~one--point union) of an arbitrary collection of connected cell complexes.
Thus, if \[K=\form{ g,f}\le \bZ^2*\bZ^2\] then if $K$ is itself not cyclic or trivial then either
$K\cong\bZ^2$, or $K$ is a free group of rank two.
For the second item, it suffices to show that $\bZ^2*\bZ^2$ is \emph{residually finite}\index{residually finite},
which is to say that every nontrivial element
$1\neq g\in\bZ^2*\bZ^2$ does not lie in the kernel of a homomorphism \[\phi_g\colon \bZ^2*\bZ^2\longrightarrow F,\] where
$F$ is a finite group. Indeed, suppose that \[\phi\colon \bZ^2*\bZ^2\longrightarrow \bZ^2*\bZ^2\] is a surjection with kernel $K$.
Since $\bZ^2*\bZ^2$ is finitely generated, there are only finitely many subgroups of $\bZ^2*\bZ^2$ of finite index $n$, say
$\{G_1,\ldots,G_m\}$. Then the subgroups \[\{\phi^{-1}(G_i)\mid 1\leq i\leq m\}\] coincide with the groups $\{G_1,\ldots,G_m\}$.
It follows that $K$ lies in the intersection of all finite index subgroups of $\bZ^2*\bZ^2$. So, if $\bZ^2*\bZ^2$ is residually finite then $K$
is trivial.
To see that $\bZ^2*\bZ^2$ is residually finite, we endow the usual $2$--dimensional torus $T$ with the standard flat metric coming from the
square. We set $X=X_0$ to be the one--point union of two copies of $T$, so that $\pi_1(X)\cong \bZ^2*\bZ^2$. We write
$p$ for the wedge point in $X$. We extend the metric
on $T$ to $X$ in the obvious (i.e.~the $\ell^1$) way. We define a tower
of covers $\{X_i\}_{i\ge1}$, where $X_{i+1}$ is the cover of $X_i$ classified by the surjection \[\pi_1(X_i)\longrightarrow
H_1(X_i,\bZ/2\bZ).\]
Consider a shortest--length homotopically nontrivial loop $\gamma_i$ on $X_i$. If $\gamma_i$ lies entirely in a component of the
preimage of one of the two tori $T$ making up $X$, then clearly the homotopy class $[\gamma_i]$ is nontrivial in $H_1(T,\bZ)$
and is hence nontrivial in $H_1(X_i,\bZ)$. Since $\gamma_i$ is shortest, this homotopy class will obviously be primitive, and hence
will be nontrivial after reducing modulo two. It follows then that $\gamma_i$ does not lift to $X_{i+1}$.
If $\gamma_i$ does not lie in a component of the preimage of one of the two tori, then $\gamma_i$ traverses at least one component
of the preimage of $p$ in $X_i$. Since $\gamma_i$ is a shortest curve on $X_i$, there must be one such component of the preimage of $p$,
say $q$,
which is traversed exactly once. Traversing $q$
(with orientation) represents a nontrivial integral cohomology class, and counting the number of times a loop traverses $q$ modulo two
represents a nontrivial $\bZ/2\bZ$--valued cohomology class. It thus follows that $\gamma_i$
represents a nontrivial homology class in $H_1(X_i,\bZ/2\bZ)$. Again it follows that $\gamma_i$ does not lift to $X_{i+1}$.
Thus, an easy induction shows that the shortest loop on $X_i$ has length at least $i+1$. This proves that the intersection of all
finite index subgroups of $\pi_1(X)$ is trivial, whence $\pi_1(X)$ is residually finite.
\end{proof}
We now complete the proof of Lemma~\ref{lem:z2-z2} and hence of Theorem~\ref{thm:abt}.
\begin{proof}[Proof of Lemma~\ref{lem:z2-z2}]
We write \[\bZ^2*\bZ^2\cong \form{ w,x,y,z\mid [w,x]=[y,z]=1}.\] We define a surjection
\[\tau\colon\bZ^2*\bZ^2\longrightarrow\Diff_+^1(M)\]
by the rules \[w\mapsto a,\quad x\mapsto b,\quad y\mapsto c,\quad z\mapsto d.\] By Proposition~\ref{prop:2g-hopf}, it
suffices to show that $\form{ a,b,c,d}$ is not abstractly isomorphic to $\bZ^2*\bZ^2$. We assume the contrary,
namely that $\tau$ is an isomorphism, and so derive
a contradiction.
Since $\bZ^2*\bZ^2$ contains no copy of $\bZ\wr\bZ$, we may assume that $M\neq S^1$ by Proposition~\ref{prop:comm-circle}.
So, we will assume that $M=I$. Write \[\phi=[c,bdb^{-1}],\quad \psi=[\phi,a].\]
Applying Lemma~\ref{lem:supp-comm}, we see that \[\overline{\supp\psi}\sse \supp\phi\cup \supp a\cup\overline{\supp \phi\cap
\supp a}.\]
Lemma~\ref{lem:diff-phi-b} now implies that \[\overline{\supp\phi\cap\supp a}\sse \overline{\supp\phi\setminus\supp b}\sse
\supp c\cup\supp d.\] We conclude that \[\overline{\supp\psi}\sse \supp G,\] so that $\psi$ is compact supported in
$\supp G$. Finally, we have that $\psi$ is the image
under $\tau$ of the element \[[[y,xzx^{-1}],w]\in\bZ^2*\bZ^2.\] This element is clearly seen to
be nontrivial in $\bZ^2*\bZ^2$, and so we have that if $\tau$ is an isomorphism then $\psi$ is nontrivial in $\Diff_+^1(I)$.
Proposition~\ref{prop:compact-conn} implies that $\bZ^2*\bZ^2$ contains a copy of $\bZ\wr\bZ$, which is a contradiction.
\end{proof}
The following is an immediate consequence from the proof of Lemma~\ref{lem:z2-z2}, after setting
$c=a^t$ and $d=b^t$.
We will employ this result crucially in the construction of groups of \emph{optimally expanding diffeomorphisms}\index{optimally expading
diffeomorphism} in Chapter~\ref{ch:optimal}.
\begin{lem}\label{l:ucsd}
If $a,b,t$ are orientation--preserving $C^1$ diffeomorphisms of a compact interval $I$
and if $a$ and $b$ have disjoint supports,
then the closure of the support of the element
\[
\left[ [a^t,b b^t b^{-1}],a\right]\]
is contained in the support of the group $\form{a,b,t}$.
\end{lem}
\section{Crossed homeomorphisms and commutation}\label{sec:crossed}
One of the main difficulties in using Theorem~\ref{thm:abt} is in ensuring that the hypotheses are met. Typically, one does not
expect two homeomorphisms of a on--manifold to have disjoint supports, even if they commute with each other.
We briefly recall
a standard construction: let $\tau\in\Homeo_+(\bR)$ be given by $\tau(x)=x+1$, and let $\sigma_0$ be an arbitrary homeomorphism
supported on $(0,1)$. The homeomorphism \[\sigma=\prod_{i\in\bZ} \tau^{-i}\sigma_0\tau^i\] commutes with $\tau$ by construction,
whereas the (closures of the) supports of both $\tau$ and $\sigma$ are the whole real line. To give applications of Theorem~\ref{thm:abt}.
we will give specific conditions under which the hypotheses will be satisfied; see Section~\ref{sec:misc} below, and especially
Section~\ref{sec:raag} below, where we will give a complete description of right-angled Artin groups that admit
faithful actions on $I$ and $S^1$ of regularity $C^{1+\mathrm{bv}}$ and better.
We first note an easy and almost immediate consequence of Theorem~\ref{thm:abt}. If $G\le \Homeo_+[0,1]$, we say that
the action of $G$ is \emph{overlapping}\index{overlapping action}
if for arbitrary nontrivial elements $g_1,g_2\in G$, we have \[\supp g_1\cap\supp g_2
\neq\varnothing.\]
\begin{prop}\label{prop:overlapping}
Let $G\le \Diff_+^1(I)$ be a subgroup such that $G*\bZ\le \Diff_+^1(I)$. Then the action of $G$ is overlapping.
\end{prop}
Overlapping actions of groups are closely related to crossed elements, Conradian actions, and Conradian orderability of groups
(cf.~Appendix~\ref{ch:append2} below).
Following Navas and Rivas~\cite{navasrivas09}, let $(\Omega,<)$ be a totally ordered space, let $G$ be a group of order preserving
permutations of $\Omega$, and let $f,g\in G$. We say that $f$ and $g$ are \emph{crossed}\index{crossed homeomorphisms}
if there are points
$u<w<v$ in $\Omega$ such that:
\begin{enumerate}[(1)]
\item
We have $g^n(u)<w<f^n(v)$ for all $n\in\bZ$.
\item
There is an $N\in\bZ$ such that $g^N(v)<w<f^N(u)$.
\end{enumerate}
If a group $G$ acts on $\Omega$ with no crossed pairs of elements, then the action of $G$ is said to be
\emph{Conradian}\index{Conradian ordering}.
A pair of homeomorphisms of $I$ are crossed if they are crossed as order preserving permutations of the interval.
The reader will find many examples of groups generated by crossed homeomorphisms in Chapter~\ref{ch:chain-groups} below. It is
a useful exercise for the reader to draw several examples of crossed homeomorphisms of the interval, in order to develop an intuition
for the concept.
Conradian subgroups $G\le \Homeo_+[0,1]$ are generally easier to investigate than arbitrary subgroups. For one, it is easy to show that
if $G$ is Conradian and if $f,g\in G$ then if $J_x$ is a component of $\supp x$ for $x\in \{f,g\}$, then either $J_f\cap J_g=\varnothing$,
or there is an inclusion relation between $J_f$ and $J_g$.
The existence of crossed diffeomorphisms in a subgroup $G\le \Diff_+^1(I)$ also has some remarkable consequences. If $h\in\Diff_+^1(I)$
and if $p\in\Fix h$, we say that $p$ is a \emph{hyperbolic}\index{hyperbolic fixed point}
fixed point for $h$ if $|h'(p)|\neq 1$. The following fact can be found
in~\cite{DKN2007,BF2015}.
\begin{prop}\label{prop:hyp-fp}
Let $f,g\in\Diff_+^1(I)$ be crossed diffeomorphisms. Then there exists an element $h$ in the positive semigroup generated by $f$ and $g$
that has a hyperbolic fixed point.
\end{prop}
It is not difficult to imagine how a proof of Proposition~\ref{prop:hyp-fp} might go; one would use the definition of crossed homeomorphisms
and the contraction mapping principle to find a fixed point for an element in the positive semigroup generated by $f$ and $g$. One then
argues from some sort of uniform contraction and the $C^1$ hypothesis that the derivative at the fixed point must be either
strictly greater than one or strictly less than one. We omit further details.
To illustrate more precisely how crossed diffeomorphisms figure into the algebraic theory of diffeomorphisms, we have the following
fact:
\begin{prop}[See~\cite{Navas2011}, Proposition 4.2.2.25, and~\cite{KKR2020}, Lemma 3.10]\label{prop:crossed-comm}
Let $c\in\Diff_+^1(I)$ be a diffeomorphism such that $\supp c=(0,1)$. Then the centralizer of $c$ in $\Diff_+^1(I)$ has no pairs of
crossed diffeomorphisms.
\end{prop}
\begin{proof}
Suppose the contrary, so that $f$ and $g$ centralize $c$ and are crossed. Then the supports and dynamics of $f$ and $g$ are $c$--invariant,
and it is easy to check that then the hypotheses of Theorem~\ref{thm:two-jumps} are satisfied. It follows that either $f$ or $g$ fails to be
$C^1$.
\end{proof}
Often, quite a bit of manipulation is needed to show that a particular faithful group action on $I$ is forced to be Conradian, and in the interest
of space and focus we will not spell out many more details, directing the interested reader to~\cite{KKR2020} and the further
references therein, for instance. As we have already mentioned, once an action is known to be Conradian, one can often analyze the
combinatorics of orbits in order to prove that a purported faithful action cannot be faithful after all.
We close this section by stating the main results of~\cite{KKR2020} and~\cite{KKR2021},
a special case of which we will revisit as Theorem~\ref{thm:kkr-2020}
below. We recall quickly for the convenience of the reader that the \emph{derived series}\index{derived series}
$\left\{G^{(n)}\right\}_{n\in\bN}$ of $G$ is defined by $G^{(0)}=G$
and \[G^{(n+1)}=[G^{(n)},G^{(n)}].\] The group $G$ is \emph{not solvable of degree at most $n$}\index{non-solvable group}
if $G^{(n)}\neq \{1\}$.
\begin{thm}[See~\cite{KKR2020}, Theorem 1.1 and~\cite{KKR2021}, Theorem 3.1]\label{thm:kkr2020-gen}
Let $G$ and $H$ be groups.
\begin{enumerate}[(1)]
\item
Suppose $G$ and $H$ are not solvable of degree at most $n\geq 3$, let $M\in\{I,S^1\}$,
and suppose that $\tau\in [0,1)$ satisfied \[\tau(1+\tau)^{n-2}\geq 1.\]
Then there is no faithful homomorphism \[(G\times H)*\bZ\longrightarrow\Diff_+^{1,\tau}(M).\]
\item
If $G$ and $H$ are not solvable, then for all $\tau>0$ there is no faithful homomorphism
\[(G\times H)*\bZ\longrightarrow\Diff_+^{1,\tau}(M).\]
\end{enumerate}
\end{thm}
In Theorem~\ref{thm:kkr2020-gen}, the symmetry in the non--solvability of $G$ and $H$ is necessary. We do not know how to make
such nontrivial statements about $C^{1,\tau}$ actions of groups of the form $G\times\bZ$, for example, when $G$ has a relatively
long derived series. It is this technical difficulty which makes a full computation of the critical regularity of all right-angled Artin groups
out of reach with the present state of the available technology.
\section{Groups of $C^{1+\mathrm{bv}}$ diffeomorphisms}\label{sec:misc}
Here we record some further useful facts about elements of $\Diffb(S^1)$ which synthesize the foregoing discussion, and which
will be useful for us in the sequel. The following is an easy consequence of Kopell's Lemma (Theorem~\ref{thm:kopell}).
\begin{prop}[See Lemma 2.6 in~\cite{KK2018JT}, for instance]\label{prop:disj-ab}
The following hold.
\begin{enumerate}[(1)]
\item
Let $M\in\{I,S^1\}$ and let $a,b\in\Diffb(M)$ be commuting elements such that \[\Fix a,\Fix b\neq\varnothing.\]
Write $J_a$ and $J_b$ for components of $\supp a$ and $\supp b$
respectively. Then either $J_a=J_b$ or $J_a\cap J_b=\varnothing$.
\item
Let $a,b,c\in\Diffb(I)$ with $\Fix(a)\cap (0,1)=\varnothing$. Then commutation is transitive in the following sense:
if $a$ commutes with $b$ and $a$ commutes with $c$ then $b$ commutes with $c$.
\end{enumerate}
\end{prop}
\begin{proof}
(1) Suppose first that $M=I$. Suppose that $J_a\cap J_b=\varnothing$. It is easy to see that there must be an inclusion
relation among $J_a$ and $J_b$, say $J_a\sse J_b$. Since
that $\partial J_b$ does not meet $\supp a$, we have that both $a$ and $b$ act on $J_b$, and $a$ acts with a fixed point in the interior
of $J_b$. Kopell's Lemma (Theorem~\ref{thm:kopell}) implies that $a$ must restrict to the identity on $J_b$, a contradiction.
If $M=S^1$, let $J_b$ be a component of $\supp b$, meeting $J_a$, a component of $\supp a$. Without loss of generality, $\partial J_a$
meets $J_b$. Since $a$ and $b$ commute, we have that if $x\in (\partial J_a)\cap J_b$, then $b^nx\to\partial J_b$ as $n\to\infty$, and each
of the points in this orbit is fixed by $a$ since $a$ and $b$ commute. We are then immediately reduced to the case $M=I$.
(2) Suppose that $b$ and $c$ do not commute. Then the group generated by $\form{ b,c}$ is not abelian, and therefore
cannot act freely on $(0,1)$, by H\"older's Theorem (Theorem~\ref{thm:holder}). It follows that there is a nontrivial $s\in \form{ b,c}$
with a fixed point in $(0,1)$. Since $[s,a]=1$, Theorem~\ref{thm:kopell} implies that $s$ is the identity, a contradiction.
\end{proof}
The two parts of Proposition~\ref{prop:disj-ab} are sometimes called the \emph{disjointness criterion}\index{disjointness criterion}
and the \emph{abelian criterion}\index{abelian criterion},
respectively.
The most substantial result in this section is the following, which can be found as Lemma 2.8 in~\cite{KK2018JT}.
\begin{prop}\label{prop:tame}
Let $f\in\Diffb(S^1)$ be an infinite order element, and let $Z\le\Diffb(S^1)$ denote its centralizer. We have the following statements.
\begin{enumerate}[(1)]
\item
If $\Fix f\neq\varnothing$ and if $H\le Z$ is generated by elements with nonempty fixed point sets, then every element $h\in H$ satisfies
$\Fix h\neq\varnothing$. Furthermore, we have \[\supp f\cap\supp H'=\varnothing.\]
\item
If the rotation number of $f$ is irrational then $Z$ is abelian.
\item
If the rotation number of $f$ is rational then the rotation number of each element of $Z$ is rational.
\item
If $f$ has rational rotation number then the rotation number restricted to $Z$ is a homomorphism.
In particular, if $h\in Z'$ then $\Fix h\neq\varnothing$.
\end{enumerate}
\end{prop}
\begin{proof}
(1)
Let $J$ be a component of $\supp f$. We have that $H$ acts on $J$, by Proposition~\ref{prop:disj-ab}. Since every element of $H$ preserves
$\partial J$, we have that every element of $H$ has a nonempty set of fixed points. Proposition~\ref{prop:disj-ab} shows that the restriction of
the commutator subgroup of $H$ to $J$ is the identity.
(2)
Denjoy's Theorem (Theorem~\ref{thm:denjoy}) implies that $f$ is topologically conjugate to an irrational rotation. If $1\neq g\in\Homeo_+(S^1)$
commutes with an irrational rotation then $\Fix g=\varnothing$. Indeed, we have that $\Fix g$ is closed and $f$--invariant,
and $f$ is minimal, whence
it follows that $\Fix g= S^1$. It follows that the centralizer of $f$ acts freely on the circle and is
therefore abelian by H\"older's Theorem for the circle (Theorem~\ref{thm:holder-circle}).
(3)
Suppose for a contradiction that $g\in Z$ has irrational rotation number. Then by the previous item, we have that no nontrivial element in
the centralizer of $g$ can have a fixed point. Since $f$ has infinite order,
it follows that $f$ has no periodic points and hence has irrational rotation number, a contradiction.
(4)
Passing to a positive power if necessary, we may assume that $\Fix f\neq\varnothing$. Indeed, passing to a power of $f$ cannot
decrease the size of the centralizer, so assuming $\Fix f\neq\varnothing$ is not a loss of generality. Let $G$ denote the centralizer of $f$, and
let $K\sse G$ denote the set of elements of $G$ with zero rotation number. Item 1 of the proposition implies that $K$ is actually
a group. Since the rotation number depends only on the conjugacy class of a homeomorphism in $\Homeo_+(S^1)$
(Proposition~\ref{prop:rot-easy}), we have that
$K$ is normal subgroup of $G$.
We have that $\Fix f$ is $G$--invariant, and \[X=\partial \Fix f\] is a closed, $G$--invariant subset of $\Fix f$. We claim that $X$ is pointwise
fixed by $K$. Indeed, this follows from the fact that if $J$ is a component of $\supp f$ then an arbitrary element $k\in K$ fixes $\partial J$,
by Item 1. The set $X$ consists of the closure of the union of the sets $\partial J$, as $J$ ranges over $\supp f$. We thus obtain a map
\[\phi\colon G/K\longrightarrow \Homeo_+(X),\] just defined by restricting the action of $G$ on $S^1$. Since $g\in G\setminus K$ if and only if
$g$ has nonzero rotation number, Corollary~\ref{cor:holder-extend} implies that the action of $G/K$ on $X$ extends to
a map \[\Phi\colon G/K\longrightarrow\Homeo_+(S^1),\]
that this action is free, and that the rotation number is an injective homomorphism \[G/K\longrightarrow S^1.\] If $gK\in G/K$ then it is
straightforward to check that the rotation number of $\Phi(gK)$ coincides with the rotation number of $g$, completing the proof.
\end{proof}
In part (2) of Proposition~\ref{prop:tame}, one can in fact assert that $Z$ is conjugate into the group of rotations of the circle, though we will
not spell this out further. Similarly, in part (4) we may eschew the hypothesis that $f$ have rational rotation number.
\section{Classification of $C^1$--actions of solvable Baumslag--Solitar groups}\label{sec:bs}
For nonzero integers $p$ and $q$, we define the corresponding Baumslag--Solitar group\index{Baumslag--Solitar group} by
\[
\operatorname{BS}(p,q):=\form{a,e\mid ae^pa^{-1}=e^q}.\]
Having a Baumslag--Solitar subgroup is often an obstruction for a group from admitting certain geometric structures.
For instance, word-hyperbolic groups, $\operatorname{CAT}(0)$ groups, and 3--manifold groups do not contain
$\operatorname{BS}(p,q)$ for $0<|p| < |q|$ (see~\cite{Shalen-TAIA, Gromov1987, BH1999}). It is also known that
$\operatorname{BS}(p,q)$ does not admit a faithful $C^2$ action on a compact interval for $1<p<q$ (see~\cite{FF2001}).
In this section, we will classify $C^1$--actions of \emph{non-abelian solvable}\index{Baumslag--Solitar groups}
Baumslag--Solitar groups; that is, we will consider the group
\[B:= \operatorname{BS}(1,m)\]
for a fixed $m>1$. As a result, we will deduce that every $C^1$--action of $\bZ\times\operatorname{BS}(1,2)$ on a compact
connected one--manifold admits two nontrivial elements of the group with disjoint supports (Corollary~\ref{c:bs12}).
This result will be crucially used in Chapter~\ref{ch:optimal}, namely for the construction of optimally expanding diffeomorphism groups.
We remark that the group $\operatorname{BS}(1,-m)$ is also solvable, but it does not yield an interesting $C^1$
action for us (see Proposition~\ref{p:bs-negative}). We also note that for $p$ and $q$ different from $\pm1$, the group
$\operatorname{BS}(p,q)$ are non-solvable and contains $F_2$~\cite{KarrassSolitar1971CJM}.
\subsection{Topological smoothing of the standard affine action}\label{ss:bs-smooth}
The group of orientation preserving affine maps on the real line can be described as
\[\Aff_+(\bR)=e^\bR\ltimes \bR=
\left\{\begin{pmatrix} e^s&t\\ 0 & 1\end{pmatrix}\middle\vert s,t\in\bR\right\}.\]
This is naturally a subgroup of $\Diff_+^\infty(\bR)$, and hence of $\Diff_+^\infty(0,1)$.
After a suitable topological smoothing, we can make this group act smoothly on a compact interval:
\begin{thm}\label{thm:bs-cpt}
There exists an injective group homomorphism
\[
\Aff_+(\bR)\longrightarrow \Diff_{[0,1]}^\infty(\bR)\]
defined as a topological conjugation by a homeomorphism $(0,1)\longrightarrow\bR $.
\end{thm}
More precisely, we will build a homeomorphism
$\psi\co (0,1)\longrightarrow\bR $ such that for each $g\in \Aff_+(\bR)$
the map
\[\psi^{-1} g\psi\co (0,1)\longrightarrow(0,1)\]
extends to a map in $\Diff_{[0,1]}^\infty(\bR)$, the group of smooth real line diffeomorphisms fixing all the points outside $[0,1]$.
\bp[Proof of Theorem~\ref{thm:bs-cpt}]
Let us first consider a smooth diffeomorphism $\varphi_0\co (0,1)\longrightarrow\bR$ satisfying $\varphi_0(x) =-1/x$ near $x=0$
and that $\varphi_0(1-x)=\varphi_0(x)$ for all $x\in(0,1)$.
Then the affine transformation $f(x) = sx+t$ is conjugated to the map
\[
F(x):=\varphi_0^{-1}\circ f\circ\varphi_0(x) = \frac{x}{s-tx}\]
Since $s>0$ the map $F$ is smooth near $x=0$.
It follows by symmetry at $x=1$ that $F\in \Diff_+^\infty[0,1]$.
By the Muller--Tsuboi trick (Theorem~\ref{t:muller-tsuboi}),
there exists a smooth diffeomorphism $\varphi$ of $(0,1)$ (which is not differentiable at $\{0,1\}$) such that
\[\varphi^{-2}\Diff_+^\infty[0,1]\varphi^{2}\le\Diff_{[0,1]}^\infty(\bR).\]
Then the map
$\psi:=\varphi_0\varphi^2$
satisfies the condition above.
\ep
\begin{rem}Alternatively, one can view $\Aff_+(\bR)$ as a group of matrices acting on $\bR^2$
and preserving the $x$--axis. Projectivizing, this group acts on a compact interval contained in the unit circle. This action can be seen to be smooth~\cite{Tsuboi1987}.\end{rem}
The following definition will be handy for us.
\bd\label{defn:tangent-id}
Let $r\in\bZ_{>0}\cup\{\infty\}$.
A $C^r$--diffeomorphism $f$ of a manifold $M$
is said to be \emph{$C^r$--tangent to the identity}\index{tangent to the identity} at $p\in M$
if the following hold.
\begin{itemize}
\item $f(p)=p$;
\item $D^if(p)=D^i(\Id)(p)$ for $i=1,\ldots,r$.\end{itemize}\ed
Here, $D^i$ denotes the $i$--th derivative.
In the context of one--manifold actions, this simply means that
$f(p)=p$, $f'(p)=1$ and $\der{f}{i}(p)=0$ for $i=2,\ldots,r$.
The group $\operatorname{BS}(1,m)=\form{a,e\mid aea^{-1}=e^m}$ admits the \emph{standard affine action}\index{affine group action} generated by
\[\bar a(x) =mx,\quad \bar e(x)=x+1.\]
By the theorem above, we see that this action is topologically conjugate to a smooth action on $[0,1]$ that is $C^\infty$--tangent to the identity at the endpoints.
We will continue to use the symbols $\bar a$ and $\bar e$ to represent the standard affine generators of $\operatorname{BS}(1,m)$.
\subsection{$C^1$--actions on intervals}
Let us now describe a topological classification of nonabelian $C^1$--actions of solvable Baumslag groups, which
is a special case of results in~\cite{BMNR2017MZ}; see~\cite{TriestinoNote} for a nice exposition particularly for
Baumslag--Solitar groups. A similar result can be found in~\cite{CC-unpublished}.
\begin{thm}\label{t:bmnr}
Let $m\ge2$ be an integer. Suppose we have a representation
\[
\rho\co \operatorname{BS}(1,m)\longrightarrow \Diff_+^1[0,1].\]
Then each $J\in\pi_0\supp \rho(e)$ is preserved by the whole image of $\rho$,
and the restriction of the action $\rho$ on $J$ is topologically conjugate to the standard affine action of $\operatorname{BS}(1,m)$.
Moreover, the derivative of $\rho(a)$ at its unique fixed point in $J$ is $m$.
\end{thm}
We prove the preceding theorem in this subsection.
By the following simple observation, we see that the conclusion is vacuous when $\rho$ is not faithful, i.e. $\rho(e)=1$:
\begin{lem}\label{l:faithful}
Let $m\ge2$ be an integer.
If \[\phi\co \operatorname{BS}(1,m)\longrightarrow Q\]
is a quotient such that the image of $e$ has an infinite order, then $\phi$ is an isomorphism.
\end{lem}
\bp
Pick an arbitrary element $g\in \ker\phi$. Up to conjugation, we can write $g=e^pa^q$ for some $p,q\in\bZ$. We have
\[
1=\phi(aga^{-1})=\phi(e^{mp}a^q)=\phi(e)^{(m-1)p}.\]
This implies that $p=0$ and $\rho(a)^q=1$.
We have that
\[
\rho(e) = \rho(a^qea^{-q})=\rho(e^{m^q}).\]
It follows that $m^q=1$ and $q=0$. We have shown $\ker\phi=\{1\}$.
\ep
Set
\[
H:=\fform{e}=\form{a^{-k}ea^k\mid k\ge0}\unlhd \BS(1,m).\]
We have an isomorphism from the additive group $\bZ[1/m]$ to $H$ that maps $1/m^k$ to $a^{-k}ea^k$.
For each $r\in \bZ[1/m]$, we let $e[r]$ to denote its image under this isomorphism.
Note that for $n\in\bZ$ we have $e[n]=e^n$.
Recall that an action $\rho\co K\longrightarrow \Homeo_+(\bR)$ of a group $K$ is \emph{semi-conjugate}\index{semi-conjugacy}
to another action
$\rho'\co K\longrightarrow\Homeo_+(\bR)$ if for some monotone increasing surjective (hence, continuous) map $q\co \bR\longrightarrow \bR$
we have a commutative diagram
\[\begin{tikzcd}
\bR\arrow{r}{\rho(g)} \arrow{d}{q} &
\bR\arrow{d}{q}\\
\bR\arrow{r}{\rho'(g)}& \bR
\end{tikzcd}
\]
for all $g\in K$. The map $q$ is called a {semi-conjugacy}.
One may call $\rho$ above as a \emph{blow-up}\index{blow-up} of $\rho'$ and define two representations are
\emph{semi-conjugately equivalent} (or, \emph{semi-conjugate} for short) if they have a common blow-up (see~\cite{KKM2019}, cf.~Theorem~\ref{thm:irr-rot-semi}).
Let us first consider the setting of topological actions by $\operatorname{BS}(1,m)=\form{a,e}$.
\begin{lem}\label{l:bmnr-c0}
If $\BS(1,m)$ acts topologically on $\bR$ such that $\Fix e=\varnothing$, then this action is semi-conjugate to the standard affine action.
\end{lem}
\bp
For all $r\in \bZ[1/m]$, the element $e[r]$ has no fixed point since $e[r]$ and $e$ has a nontrivial common power.
After a topological conjugacy, we may assume $e(x)=x+1$ (up to inverting $e$ to $e^{-1}$ if necessary).
Define a map $q\co H.0\longrightarrow \bZ[1/m]$ by
\[
q\left(e[r]\right)=r.\]
It is obvious that the map $q$ is well-defined, strictly increasing and surjective.
We consider its continuous extension
\[
\bar q\co \bR\longrightarrow \bR\]
satisfying
\[
\bar q(x) :=\sup q\left(H.0\cap(-\infty,x]\right)=\inf q\left(H.0\cap[x,\infty)\right).\]
It is not hard to see that $\bar q$ is well-defined, monotone increasing and surjective.
One can also see from the density of $\bZ[1/m]$ in $\bR$ that
\[
\bar q(e[r].x)=\bar q(x)+r\]
for all $x\in\bR$ and $r\in \bZ[1/m]$.
Let us set $\sigma:=\bar q(a.0)$. Then we have that
\[
\bar q(a e[r].0)=\bar q (a e[r] a^{-1}\cdot a.0)=\bar q(e[{mr}]\cdot a.0)=mr+\sigma=m\bar q(e[r].0)+\sigma.\]
By the density of $\bZ[1/m]$ again, we have that
\[\bar q(a.x)=m\bar q(x)+\sigma\]
for all $x$. Hence, the map $\bar q$ semi-conjugates the given action to the affine action
\[
\bar a(x)=mx+\sigma,\quad \bar e(x)=x+1.\]
We can then further conjugate $\bar a(x)$ to $x\mapsto m x$, as desired.
\ep
The last ingredient of the proof of Theorem~\ref{t:bmnr} is the following.
\begin{prop}\label{p:bmnr}
If $\rho\co \BS(1,m)\longrightarrow\Diff_+^1[0,1]$ is a faithful representation
and if $\rho$ does not have a global fixed point in $(0,1)$,
then the following hold.
\be[(1)]
\item\label{p:p:bmnr-a} $\Fix\rho(a)\cap(0,1)\ne \varnothing$ and $\Fix\rho(e)\cap(0,1)=\varnothing$;
\item $\rho(\BS(1,m))\restriction_{(0,1)}$ is conjugate to the standard affine action;
\item\label{p:bmnr-m} At the unique fixed point $x_0$ of $\rho(a)$, we have that $\rho(a)'(x_0)=m$.
\ee
\end{prop}
The essence of the proof for the above proposition is the following counting argument.
\begin{lem}\label{l:counting}
Let $C>0$ be a constant, let $\mathcal{N}\sse\bZ$ be an infinite set,
let $K\le\Diff_{[0,1]}^1(\bR)$ be a group with a finite generating set $S$,
and let $v\in K$ be a $C^1$ diffeomorphism satisfying $v(x)>x$ for all $x\in (0,1)$.
Assume we have subsets
\[ S_1, S_2, \ldots\sse K\]
and compact intervals
\[ J_1, J_2, \ldots\sse(0,1)\]
satisfying the following three conditions for all $N\in\mathcal{N}$ and for all $n\ge1$:
\be[(i)]
\item\label{l:p:disj} The intervals in the set $\{g v^N(J_n)\mid g\in S_n\}$ have pairwise disjoint interiors;
\item\label{l:p:suffix} Each $g\in S_n$ can be written as
\[g=s_\ell\cdots s_2s_1\]
for some $\ell\le Cn$ and $s_i\in S\cup S^{-1}$ such that
\[ \left(\bigcup_{j=1,\ldots,\ell} s_j\cdots s_2s_1 v^N J_n \right)\sse [0,1]\setminus\left( v^{-|N|}\sup J_n, v^{|N|} \inf J_n\right)\]
\item\label{l:p:definite} The set $\bigcap_n \bigcup_{g\in S_n}gv^NJ_n$ has a nonempty interior.
\ee
Then we have that
\[ \lim_{n\to\infty} \left(\#S_n \cdot |J_n|\right)^{1/n}=1.\]
\end{lem}
The element $s_j\cdots s_2s_1$ above is called a \emph{suffix}\index{suffix} of $g$.
\begin{rem}\label{r:counting}
Note that the hypothesis of the lemma is invariant under a topological conjugacy.
The conclusion is also topologically invariant in the special case when all $J_n$'s coincide with a fixed interval $J_0$, since we have
\[\lim_{n\to\infty} |J_n|^{1/n}=\lim_{n\to\infty} |J_0|^{1/n}=1.\]
Therefore in this special case, we can weaken the hypothesis so that $K\le\Homeo_+[0,1]$ is only topologically conjugate into
$\Diff_+^1[0,1]$, and conclude that
\[\lim_{n\to\infty} \left(\# S_n\right)^{1/n}=1.\]
\end{rem}
\iffalse
The hypothesis of the above lemma still holds after a topological conjugation of the action of $K$.
Also, in the special case when $J_0:=\bigcap_n J_n$ has a nonempty interior (or more strongly, when all $J_n$'s coincide)
we can replace all $J_n$ by $J_0$ and see from the conclusion that the growth rate of $\# S_n$ is one.
Using Muller--Tsuboi trick again, in this special case we deduce from the lemma an obstruction
for $K\le\Homeo_+(\bR)$ to be topologically conjugate into $\Diff_+^1[0,1]$ as follows.
\begin{cor}\label{c:counting}
Let $K\le\Homeo_+(\bR)$ be a group with a finite generating set $S$.
Assume that for each $N>0$ and for each $n\ge1$ we have a compact interval
\[J_n\sse\bR\setminus(-N,N)\]
and a set $S_n\sse K$ with the following properties:
\be[(i)]
\item the interiors of the intervals $\{g J_n\mid g\in S_n\}$ are pairwise disjoint;
\item each $g\in S_n$ can be written as
\[g=s_\ell\cdots s_2s_1\]
for some $\ell\le n$ and $s_i\in S\cup S^{-1}$ such that \[s_j\cdots s_2s_1 J_n\sse\bR\setminus(-N,N)\] for all $j$;
\item the set $\bigcap_n (S_nJ_n)$ has a nonempty interior.
\ee
If $K$ is topologically conjugate into $\Diff_+^1[0,1]$,
then we have
\[ \lim_{n\to\infty} \left(\#S_n\right)^{1/n}=1,\]
\end{cor}
\fi
Intuitively speaking, the lemma addresses the situation where some set $S_n$ in the radius $O(n)$ ball of the Cayley graph
disjointly translates the interval $v^N(J_n)$ in such a way that
the set $S_n\cdot v^N\cdot J_n$ has a definite length depending only on $N$,
and such that every suffix $h$ of $g\in S_n$ places $v^N(J_n)$ no farther from the boundary $\{0,1\}$ of $I$ than $v^N(J_n)$ itself.
If this happens arbitrarily near from the boundary (that is, for $|N|\gg0$) then
the growth rate of $S_n$ is equal to that of $1/|J_n|$.
It will be enlightening for the reader to give a proof of Proposition~\ref{p:bmnr} first, in order to illustrate concrete instances of the lemma.
\bp[Proof of Proposition~\ref{p:bmnr}, assuming Lemma~\ref{l:counting}]
As before, we simply write $g$ to denote the image of $g\in B$ under $\rho$ when the meaning is clear.
We will apply Lemma~\ref{l:counting} by setting \[K=\BS(1,m),\quad S=\{a,e\},\quad C=m+1,\] and varying the other parameters.
The conclusions of parts (1) and (2) are invariant under topological conjugacy. By Muller--Tsuboi trick,
we may assume that the image of $\rho$ is in $\Diff_{[0,1]}^1(\bR)$ for those parts and apply Lemma~\ref{l:counting}.
To prove part (1),
let us first note that the former half of the statement $\Fix a\cap(0,1)\ne\varnothing$ implies the latter half $\supp e =(0,1)$.
Indeed, if $\supp e\ne(0,1)$ then one can find some $x_0\in \Fix e\cap(0,1)$,
together with either the forward or the backward limit of $\{a^k(x_0)\}_k$ that is a global fixed point in the set $\Fix a\cap(0,1)$.
This contradicts the absence of global fixed points.
It therefore suffices for us to deduce a contradiction after assuming $\Fix a\cap(0,1)=\varnothing$.
We may assume $a(x)>x$; otherwise we can conjugate $\rho$ by the inversion $\sigma(x)=1-x$
and apply the same argument. Let $J\in\pi_0\supp e$; we may further assume $e(x)>x$ on $J$ by
replacing $e$ by $e^{-1}$ if necessary.
Pick $J_0$ be a compact interval in $J$ such that $J_0$ and $e(J_0)$ are disjoint.
For each $n\ge1$ we set
\[ S_n:=\left\{ e[r] \mid r\in(-m^n,0]\cap\bZ\right\}\sse H\]
and $J_n:=J_0$.
We claim that the hypothesis of the lemma is satisfied with $v:=a$, for all $N<0$.
Indeed, the condition (i) follows from that
\[ e^{-1} a^N(J_0)\cap a^N(J_0)
=a^N e^{-m^{-N}}(J_0)\cap a^N(J_0)=\varnothing.\]
For each $g\in S_n$ there exists some $k_0,\ldots,k_{n-1}\in\{0,\ldots,m-1\}$ such that
\[g= e\left[-\sum_{0\le i\le n-1} k_i m^i\right]= \prod_{i=0}^{n-1} a^i e^{-k_i} a^{-i}
=
a^{n-1} e^{-k_{n-1}}a^{-1} e^{-k_{n-2}}\cdots a^{-1}e^{-k_0}.
\]
The word length of $g$ is at most
\[2(n-1)+\sum k_i \le (m+1)n.\]
Then for every suffix $h$ of the word $g$ and for every $x\in a^N(J)$, we have that \[h(x)\in a^{N-i}(J)\] for some $i\in\{0,1,\ldots,n-1\}$.
This verifies the condition (ii). The condition (iii) is trivial since $J_n=J_0$ is independent of $n$
and since each $S_n$ contains the identity.
As mentioned in Remark~\ref{r:counting}, we can compute
\[\lim_{n\to\infty} \left(\#S_n\cdot |J_n|\right)^{1/n}=\lim_{n\to\infty} \left(\#S_n\right)^{1/n}=m.\]
Since this contradicts Lemma~\ref{l:counting}, we conclude that $\Fix a\cap(0,1)\ne\varnothing$.
From now on, we will assume $e(x)>x$ without loss of generality.
To prove part (2), it suffices for us to show that the action $\rho(B)\restriction_{(0,1)}$ is minimal since we already know that
$\rho(B)$ is semi-conjugate to the standard affine action.
Assume the contrary, and pick a wandering interval $J_0$ of this action. Now, apply Lemma~\ref{l:counting} for $v=e$, $J_n:=J_0$
and
\[ S_n:=\left\{ e[r] \middle\vert r\in[0,1)\cap \frac1{m^{n}}\bZ\right\}\sse H.\]
For all sufficiently large $N>0$ we have that $a(x)>x$ whenever $x\in a^N J_0$, by semi-conjugacy.
The conditions (i) and (iii) are obvious since $\rho(H)$ is a blow-up of a free affine action
and since $J_n=J_0$ is independent of $n$.
For part (ii), note that
each $g\in S_n$ can be written as
\[
g = e\left[\sum_{1\le i\le n} k_i m^{-i}\right]= \prod_{i=1}^{n} a^{-i} e^{k_i} a^{i}
=
a^{-n} e^{k_{n}}a e^{k_{n-1}}\cdots ae^{k_1}a.
\]
The word length of $g$ is again at most $(m+1)n$.
We also see that for each suffix $h$ of $g\in S_n$, we have that $ha^N(\inf J_0)\ge \inf a^N J_0$.
This contradicts the lemma again, since we have
\[\lim_{n\to\infty} \left(\#S_n\cdot |J_n|\right)^{1/n}=\lim_{n\to\infty} \left(\#S_n\right)^{1/n}=m.\]
Finally, we establish part (3). Let us first assume that $\rho(B)\le\Diff_{[0,1]}^1(\bR)$.
We saw in part (2) that there exists a conjugacy $q\co (0,1)\longrightarrow\bR$ from $\rho$ to the standard affine action.
We set \[x_0:=q^{-1}(0),\quad J_0=q^{-1}[0,1]\quad \textrm{and}\quad
J_n:=q^{-1}[0,1/m^n]=a^{-n}(J_0).\]
We let $v=e$ and $S_n$ be the same as part (2) above,
and claim that the conditions (i) through (iii) hold for these choices and for all positive $N$.
Note that all these conditions are invariant under topological conjugacy and easy to check for the standard affine actions.
For instance, the condition (iii) is verified by the computation
\[
q\left( \bigcap_{n\ge1} \bigcup_{g\in S_n} ge^N J_n\right)
= \bigcap_{n\ge1}[N,N+1]=[N,N+1].\]
Using the Mean Value Theorem, it is elementary to see that
\[a'(x_0)=\lim_{n\to\infty} \left(\frac{|J_0|}{|a^{-n}(J_0)|}\right)^{1/n}=\lim_{n\to\infty} \frac1{|J_n|^{1/n}}.\]
We obtain from the lemma that
\[1=\lim_{n\to\infty} \left(\#S_n\cdot |J_n|\right)^{1/n}=\lim_{n\to\infty}m/ a'(x_0),\]
and that $a'(x_0)=m$.
In order to complete the proof, assume that $\rho(\BS(1,m))\le\Diff_+^1[0,1]$.
By the Muller--Tsuboi trick, we obtain a $C^\infty$ homeomorphism $h$ such that
\[ h\rho(\BS(1,m))h^{-1}\le\Diff_{[0,1]}^1(\bR).\]
We have seen so far that at the unique interior fixed point $x_0$ of $h\rho(a)h^{-1}$,
one can compute
\[h'\circ\rho(a)\circ h^{-1}(x_0)\cdot \rho(a)'\circ h^{-1}(x_0)/h'\circ h^{-1}(x_0)=m.\]
The point $h^{-1}(x_0)=y_0$ is the unique interior fixed point of $\rho(a)$
and satisfies
\[h'(y_0)\cdot \rho(a)'(y_0)/h'(y_0)=m.\]
This proves that $\rho(a)'(y_0)=m$.
\ep
To complete the proof of Theorem~\ref{t:bmnr}, it remains to show Lemma~\ref{l:counting}.
\bp[Proof of Lemma~\ref{l:counting}]
Assume the hypotheses of the lemma.
We will fix an arbitrarily small $\epsilon>0$, and let $n$ vary over $\bZ_{>0}$.
Since $K$ is $C^1$--tangent to the identity at the boundary,
there exists some $N\in\mathcal{N}$ such that for each element $s\in S\cup S^{-1}$
and for each point
\[
x\in [0,1]\setminus\left( v^{-|N|}\sup J_n, v^{|N|} \inf J_n\right)\]
we have that
\[
\abs*{s'(x)-1}\le\epsilon.\]
By conditions~(\ref{l:p:disj}) and (\ref{l:p:definite}), we have some $c_N>0$ such that
\[
c_N \le \sum_{g\in S_n} \abs*{gv^NJ_n}\le 1\]
for all $n$.
For each $g=s_\ell\cdots s_1\in S_n$, there exists some $z=z_{N,n}\in v^NJ_n$ such that
\[
|gv^NJ_n|/|v^NJ_n| = g'(z) = \prod_{j=\ell}^1 s_j'\left(s_{j-1}\cdots s_1(z)\right).\]
Using condition~(\ref{l:p:suffix}), we have
\[
\# S_n\cdot |v^NJ_n|\cdot (1-\epsilon)^{Cn}\le
\sum_{g\in S_n} |gv^NJ_n|
\le \# S_n\cdot |v^NJ_n|\cdot (1+\epsilon)^{Cn}.\]
We also have
\[
\left(\inf v'\right)^N|J_n|\le |v^NJ_n| \le\left (\sup v'\right)^N|J_n|.\]
Combining the above inequalities, we obtain
\[
1=\liminf_{n\to\infty} c_N^{1/n}
\le
\liminf_{n\to\infty} \left(\# S_n\cdot \left(\sup v'\right)^N\cdot |J_n|\right)^{1/n}\cdot (1+\epsilon)^C,\]
together with
\[
\limsup_{n\to\infty} \left(\# S_n\cdot \left(\inf v'\right)^N\cdot|J_n|\right)^{1/n}(1-\epsilon)^C
\le
1.\]
Since the choice of $\epsilon>0$ is arbitrary, we conclude that
\[\liminf_{n\to\infty} \left(\# S_n\cdot |J_n|\right)^{1/n}=\limsup_{n\to\infty} \left(\# S_n\cdot |J_n|\right)^{1/n}=1.\]
This completes the proof of the lemma.
\ep
In~\cite{BMNR2017MZ}, a different proof (in a more general setting) was given to part (1) of Proposition~\ref{p:bmnr}.
The proof employs an idea of Thurston (see~\cite{Thurston1974Top}, cf. Appendix~\ref{ch:append3}) that
diffeomorphisms of a compact interval are ``asymptotically translations'' at the endpoints.
Similar arguments are repeatedly used over
the literature including~\cite{McCarthy2010,KKT2020}. To give the reader a feeling for this new idea, we
describe a self-contained proof along these lines.
\bp[Alternative proof of Proposition~\ref{p:bmnr} part (1)]
As we have seen in the previous proof of this proposition,
it suffices for us to show that $a$ must fix some point in $(0,1)$.
Assume $a(x)>x$ for all $x\in (0,1)$ for contradiction.
For all $g\in B$ and $x\in(0,1)$, we introduce the notation
\[
\Delta_xg=g(x)-x.\]
Pick an arbitrary $\epsilon>0$.
There exists a $\sigma\in(0,1)$ such that
\[ s'(y)\in(1-\epsilon,1+\epsilon)\]
for all $s\in\{a^{\pm1},e^{\pm1}\}$ and
for all $y\in(0,1)\setminus(\sigma,1-\sigma)$.
We choose $x$ to be sufficiently smaller than $\sigma$ so that
$e^m(x)$ and $a^{\pm1}(x)$ are also less than $\sigma$.
Then we have
\[
| \Delta_xe^m -m\Delta_xe |\le \sum_{i=1}^{m-1} \abs*{(e^i(e(x))-e^i(x))-(e(x)-x)}
\le
\epsilon'\Delta_xe \]
for $\epsilon'=(m-1) ((1+\epsilon)^{m-1}-1)$.
This inequality may be regarded as an asymptotic linearization \[\Delta_xe^m \approx m \Delta_xe .\]
We also have some $z$ between $x$ and $e^m(x)$ so that
\begin{align*}
\abs*{\Delta_{a^{-1}(x)}e -\Delta_xe^m }
&=\abs*{ (a^{-1}e^m(x) - a^{-1}(x)) - (e^m(x)-x)}
=|\Delta_{e^m(x)}a^{-1}-\Delta_x a^{-1}|\\
&=|(a^{-1})'(z)-1|\cdot\Delta_xe^m \le \epsilon\cdot (m+\epsilon')\Delta_xe .\end{align*}
By choosing $\epsilon,\epsilon'>0$ to be sufficiently small from the beginning,
we obtain that
\[
\abs*{\Delta_{a^{-1}(x)}e }
\ge m|\Delta_xe |-
\abs*{\Delta_{a^{-1}(x)}e -m\Delta_xe }\ge
\left(m-
1/2\right)\Delta_xe .\]
Since $a^{-1}(x)$ is even closer to $0$ than $x$, we can iterate the estimate and see that
\[
\abs*{\Delta_{a^{-k}(x)}e }\ge (m-1/2)^k\Delta_xe \]
for all $k\ge1$.
Since $\supp e$ is $a$--invariant, there exists an $x$ arbitrarily close to $0$ such that $\Delta_xe \ne0$.
We then have a contradiction, since the right-hand side diverges to the infinity as $k\to\infty$,
while $a^{-k}(x)$ converges to $0$.
\ep
We conclude this subsection by noting that the solvable Baumslag--Solitar group with a negative sign
\[
\operatorname{BS}(1,-m)=\form{a,e\mid aea^{-1}=e^{-m}}\]
does not admit any interesting $C^1$--faithful actions on a compact interval for $m>1$.
We thank Crist\'obal Rivas for teaching us this proof.
\begin{prop}\label{p:bs-negative}
For $m>1$, every orientation--preserving $C^1$--action of $\operatorname{BS}(1,-m)$ on $[0,1]$ is abelian.
\end{prop}
\bp
We may assume that the action does not have a global fixed point in the interior $(0,1)$.
We claim that the image of $e$ is trivial.
If $e(x)>x$ for all $x\in (0,1)$, we would obtain a contradiction since
\[
a a^{-1}(x) < aea^{-1}(x)=e^{-m}(x)<x.\]
By symmetry, it follows that $\Fix e\cap(0,1)\ne\varnothing$.
The rest of the proof goes almost identically to that of Proposition~\ref{p:bmnr} (\ref{p:p:bmnr-a}).
We consider the same set of words $S_n$, and
note a slightly different expansion
\[g= e\left[-\sum_{0\le i\le n-1} k_i m^i\right]= \prod_{i=0}^{n-1} a^i e^{-(-1)^ik_i} a^{-i}
=
a^{n-1} e^{-(-1)^{n-1}k_{n-1}}a^{-1} e^{-k_{n-2}}\cdots a^{-1}e^{-k_0}.
\]
Then the same estimates goes through and Lemma~\ref{l:counting} again applies to yield a contradiction.
We conclude that the image of $e$ is trivial.
\ep
\subsection{$C^1$--actions on circles}
We now consider $C^1$--actions of
\[
\operatorname{BS}(1,2)=\form{a,e\mid aea^{-1}=e^2}\]
on the circle.
First, we note the existence of a finite orbit.
\begin{thm}[\cite{GL2011}]\label{t:gl2011}
Every faithful $C^1$--action of $\operatorname{BS}(1,2)$ on a circle
admits a finite orbit.
\end{thm}
\begin{rem}\label{r:gl2011}
In the preceding theorem, note first that $e$ must fix a point since
\[\rot e = \rot \left(aea^{-1}\right)=\rot e^2=2\rot e.\]
In particular, if the action of $e$ has finite order then it must be trivial.
It follows that the faithfulness of the action is equivalent to the nontriviality of the action of $e$.
Moreover, if $x_0$ belongs to a finite orbit of $\BS(1,2)$ as in the conclusion of the theorem, then $x_0\in \Fix e$.
It follows that for some $m>0$, the point $x_0$ is fixed by the finite index subgroup \[\form{a^m,e}\cong\operatorname{BS}(1,2^m).\]
\end{rem}
\bp[Proof of Theorem~\ref{t:gl2011}]
We will use a counting argument as in Lemma~\ref{l:counting}.
We have noted that $\Fix e\ne\varnothing$.
Let us pick $I_0\in\pi_0\supp e$.
Since
\[ a \Fix e = \Fix \left(aea^{-1}\right)=\Fix e^2=\Fix e,\]
we have that $a^k(I_0)\in\pi_0\supp e$ for all $k$.
\begin{claim}
We have that $a^{-m}(I_0)=I_0$ for some $m>0$.\end{claim}
Once the claim is proved,
we can deduce the conclusion of the theorem by choosing a point $x_0\in\partial I_0$ and noting that
\[ \#\left(\form{a,e}x_0\right)=\#\left(\form{a}x_0\right)\le m.\]
In order to prove the claim,
assume for contradiction that
$\{a^{-k}(I_0)\}_{k\ge 0}$ is a disjoint collection of open intervals in $S^1$.
Let $\epsilon>0$ be arbitrary.
Note that
\[ \lim_{k\to\infty} \abs*{a^{-k}(I_0)}=0.\]
By the uniform continuity of $e'$ and $\log a'$,
whenever $k$ is sufficiently large and $x,y\in a^{-k}(I_0)$,
we have that
\[
{|e'(x)-1|}<\epsilon\]
and
\[
\abs*{a'(x)/a'(y)-1}<\epsilon.\]
After replacing $I_0$ by $a^{-k}(I_0)$ for some $k\gg0$ if necessary,
we may assume that the above two inequalities hold for all $k\ge 0$
and for all $x,y\in a^{-k}(I_0)$.
We then proceed in a manner similar to that of Proposition~\ref{p:bmnr} (1).
Pick an arbitrary $n\ge1$.
We fix an open interval $J_0\sse I_0$ such that $e(J_0)\cap J_0=\varnothing$,
and set
\[ S_n:=\{e^i\mid 0\le i<2^n\}.\]
Then $S_n(J_0)$ is a disjoint collection of open intervals in $I_0$. As in the proof of Proposition~\ref{p:bmnr}, we can write
\[
e^i=a^{n-1} e^{k_{n-1}}a^{-1}\cdots e^{k_1}a^{-1} e^{k_0}\]
for some $k_i\in\{0,1\}$.
For each suffix $h$ of the above word, the interval $h(I_0)$ is contained in $a^{-t}(I_0)$ for some $t\ge0$.
Hence,
we can find some $x_j,y_j,z_j\in a^{-j}(I_0)$ for $j=0,1,\ldots,n-1$ such that
\[
\abs*{e^i(J_0)}=\prod_{j=0}^{n-1} \left(e^{k_j}\right)'(x_j) \cdot \prod_{j=0}^{n-1}a'(y_j)/a'(z_j) \cdot|J_0| \]
It follows that
\[
1\ge \sum_{i=0}^{2^n-1} \abs*{e^i(J_0)}\ge 2^n (1-\epsilon)^{2n} |J_0| .\]
By choosing $2(1-\epsilon)^2>1$ and letting $n\to\infty$, we have the desired contradiction.
The claim is now proved.
\ep
The following corollary asserts that $\bZ\times\operatorname{BS}(1,2)$ admits pair of elements with
``universally disjoint supports'', meaning they have disjoint supports for arbitrary $C^1$--actions on a compact connected one--manifold.
This will be a key ingredient for our construction of an optimally expanding diffeomorphism group.
\begin{cor}\label{c:bs12}
Let $M^1$ be a compact connected one--manifold. Then for every representation
\[
\bZ\times \operatorname{BS}(1,2)=\form{c}\times \form{a,e\mid aea^{-1}=e^2}\longrightarrow\Diff_+^1(M^1),\]
there exists some $k\ge 1$ depending only on the image of $\form{a,e}$ such that
\[
\supp c^k\cap \supp e=\varnothing.\]
In particular, for every representation
\[ \rho\co (\bZ\times \operatorname{BS}(1,2))\ast \bZ=(\form{c}\times\form{a,e})\ast\form{d}\longrightarrow\Diff_+^1(M^1),\]
the support of the image of the group element
\[
u_0:=\left[ [c^d,e e^d e^{-1}],c\right]\]
is contained in some compact subset of $\supp\rho$.
\end{cor}
Note that if $M^1=I$ then we can choose $k=1$ since \[\supp c=\supp c^2=\cdots\] in this case.
Note also that the second part of the conclusion obviously follows from the first part
and from a consequence of the $abt$--Lemma, namely Lemma~\ref{l:ucsd}.
We will spend the remainder of this section in establishing the above corollary, beginning with the case $M^1=I$.
\begin{lem}\label{l:bs1m}
If $m\ge2$ is an integer, then for an arbitrary action
\[
\bZ\times\operatorname{BS}(1,m)=\form{c}\times\form{a,e}\longrightarrow\Diff_+^1[0,1],\]
the supports of $c$ and $e$ are disjoint.\end{lem}
\bp
We may assume the action does not have a global fixed point in $(0,1)$.
Assume for contradiction that $J_0\in\pi_0\supp c$ and $J_1\in\pi_0\supp e$ nontrivially intersect.
We then have either $J_0\sse J_1$ or $J_1\sse J_0$.
We note from Theorem~\ref{t:bmnr} that the restriction of the action of $\operatorname{BS}(1,m)$ to
$J_1$ is topologically conjugate to the standard affine action.
Moreover, there exists $p\in J_0\cap \Fix a$ with $a'(p)=m$.
Since $c^k(p)$ is fixed by $a$ for all $k\in\bZ$
and since $\Fix a\cap J_1$ is a singleton, we have that $J_0\ne J_1$.
If $J_1\subsetneq J_0$, then $\{c^k(p)\}_{k\in\bZ}$ is an infinite set of hyperbolic fixed points of $a$ with derivative $m$.
This is a contradiction, since at accumulation points of $\Fix a$ the derivative of $a$ should be one. Hence, we have $J_0\subsetneq J_1$.
Since the action of $\operatorname{BS}(1,m)$ on $J_1$ is topologically conjugate to the standard affine action, we should have $p\in \Fix c$.
Moreover, the action of $H=\fform{e}$ on $J_1$ is minimal and preserves $\Fix c\cap J_1$. This implies that
$\Fix c\cap J_1$ is dense in $J_1$, which contradicts the assumption $J_0\sse J_1$.
\ep
By plugging $m=2$, we obtain the conclusion of the corollary for the case $M^1=I$.
For the case of a circle, we prove a stronger result below.
\begin{lem}[\cite{KK2020crit}]\label{l:bs-z1}
Suppose we have a representation
\[ \operatorname{BS}(1,2)=\form{a,e}\longrightarrow\Diff_+^1(S^1).\]
We denote the centralizer by
\[ Z^1:=\{c\in \Diff_+^1(S^1)\mid [c,g]=1\text{ for all }g\in \operatorname{BS}(1,2)\}.\]
Then there exists a normal subgroup $Z_0\unlhd Z^1$ such that $Z^1/Z_0$ is a finite cyclic group and such that
\[
\supp Z_0\cap \supp e=\varnothing.\]
\end{lem}
\bp
We may assume that $\BS(1,2)$ acts faithfully, for otherwise $e$ acts trivially and there is nothing to show (cf.~Remark~\ref{r:gl2011}).
So, we may regard $\operatorname{BS}(1,2)$ as a subgroup of $\Diff_+^1(S^1)$.
Moreover, there exists an $m\ge1$ such that
\[
B_0:=\form{a^m,e}\cong\operatorname{BS}(1,2^m)\]
fixes a point. Applying Theorem~\ref{t:bmnr} to $B_0$, we see that each component $J\sse\supp B_0$ contains a point $p_J$ satisfying
\[
\left(a^m\right)'(p_J)=2^m.\]
We conclude that $\supp B_0$ has only finitely many components.
Let $X$ be the union of the boundary points of all the components of $\supp B_0$.
Since $Z^1$ preserves $\Fix B_0$, it permutes the finite set $X$ preserving the circular order on $X$. Therefore, there exists a representation
\[
\phi\co Z^1\longrightarrow \bZ/k\bZ\]
for $k:=\#X$; this is also an easy instance of Corollary~\ref{cor:holder-extend}.
Write $Z_0:=\ker\phi$. Then for every $z\in Z_0$ and for every $x_0\in X$, we obtain an action
\[
\form{z}\times\operatorname{BS}(1,2^m)\longrightarrow \Diff_+^1(S^1\setminus\{x_0\}).\]
By Lemma~\ref{l:bs1m}, the support of $z$ is disjoint from that of $e$.
\ep
We thus obtain a proof of Corollary~\ref{c:bs12} by noting that
\[
c^{K}\in Z_0,\]
where here $K=\# (Z^1/Z_0)$ in the above lemma.
\chapter{Chain groups}\label{ch:chain-groups}
\begin{abstract}In this chapter, we develop some ideas about a particularly natural class of diffeomorphism groups, which the authors and Lodha
named \emph{chain groups}\index{chain groups} in~\cite{KKL2019ASENS}.
Roughly, a chain of intervals is a sequence of open intervals $\{J_1,\ldots,J_n\}$,
with $J_i\sse I$ for all $i$, such that $J_i\cap J_k\neq\varnothing$ if and only if $|i-k|\leq 1$.
One then considers the subgroup $G$
of $\Homeo_+[0,1]$ generated by homeomorphisms $\{f_1,\ldots,f_k\}$ such that $\supp f_i\subseteq J_i$. The group $G$ is called a chain
group if a certain mild dynamical condition is met.
It turns out that chain groups exhibit a combination of uniform properties, together with a remarkable diversity of behaviors. The flexibility
afforded by the diverse phenomena one can observe among chain groups, together with a unified theory that makes them tractable, makes
chain groups a powerful tool that plays a critical role in the construction of groups of homeomorphisms of a given critical regularity.\end{abstract}
\section{Chains and covering distances}
For a group $G$ acting on a topological space $X$, we recall the definition of the \emph{support}\index{support of a group element}
(which in this chapter will
always mean \emph{open support}\index{open support} unless otherwise noted)
of $g\in G$, that is to say the set $\supp g:=X\setminus\Fix g$.
We let $\suppc g$ denote its closure. We denote
\[\supp G:=\bigcup_{g\in G}\supp g.\]
If $X$ is an interval then each component of $\supp g$ is called a \emph{supporting interval}\index{supporting interval} of $g$.
In the case where $X=I$ is a (usually compact) interval and where $G$ is finitely generated,
we will construct a combinatorial function that behaves much like a metric,
called the \emph{covering distance}\index{covering distance}.
Roughly speaking, this metric measures the ``topological complexity'' of coverings of a closed interval by supporting intervals of the generators.
In general, a collection of subspaces in a topological space is said to have \emph{finite intersection multiplicity}\index{finite intersection
multiplicity}
if each point in the space belongs to at most finitely many elements of the collection (cf.~\cite{dMvS1993}).
\bd\label{defn:chain}
We say that a finite sequence of intervals
\[I_1,I_2,\ldots,I_m\]
in the real line is an \emph{$m$--chain}\index{$m$--chain} if
\[ \inf I_j <\inf I_{j+1} <\sup I_j<\sup I_{j+1}\]
for each $j=1,2,\ldots,m-1$.
\ed
An infinite chain is also naturally defined by an infinite sequence of intervals such that every consecutive subsequence of
$m$ intervals forms an $m$--chain. The parameter $m$ will sometimes be called the \emph{length}\index{chain length} of the chain.
\begin{lem}\label{l:m-chain-finite}
Fix $m\in\bZ_{>0}$.
If a collection $\VV$ of open intervals in $\bR$ has finite intersection multiplicity, then so does the collection
\[
\VV_m:=\{ J \mid J=J_1\cup\cdots \cup J_m\text{ for some }m\text{--chain }\{J_1,\ldots,J_m\}\text{ in }\VV\}.\]
\end{lem}
\bp
The proof is a simple induction, the base case $m=1$ being trivial. Suppose the conclusion holds for chains of length $m-1$ or less,
and assume for a contradiction that a point $x_0$ belongs to infinitely many distinct open intervals each of which is the union of an $m$--chain.
By the hypothesis, there exists an interval $J\in \VV$ containing $x_0$ and a positive integer $j\le m$ such that $J$ appears as the
$j^{th}$ term of infinitely many $m$--chains whose unions are all distinct.
Since there exist only finitely many intervals in $\VV_{j-1}$ and in $\VV_{m-j}$ containing $\inf J$ and $\sup J$ respectively,
we obtain the desired contradiction.
\ep
For a topological space $X$ and its covering $\UU$, we define
the \emph{covering length}\index{covering length} of a subset $A\sse X$ as the integer
\[
\CL_\UU(A):=\min\{ m \mid A\sse U_1\cup\cdots \cup U_m\text{ for some }U_i\in \UU\}.\]
Below we prove that under certain conditions, the covering length function for an interval is
topologically conjugate to the floor or the ceiling function.
\begin{lem}\label{l:cover}
Let $\mathcal{V}$ be a collection of bounded open intervals in $\bR$ with finite intersection multiplicity.
If $\VV$ covers $[0,\infty)$, then there exists a homeomorphism $h:[0,\infty)\longrightarrow[0,\infty)$ such that the following hold for all $x>0$:
\begin{itemize}
\item $\CL_{\mathcal{V}}[0,x)=\lceil h(x)\rceil$;
\item $\CL_\VV[0,x]=\lfloor h(x)\rfloor+1$.
\end{itemize}
\end{lem}
\bp
Consider the monotone function $f(z):=\CL_\VV[0,z)$, defined for $z>0$.
By Lemma~\ref{l:m-chain-finite}, for each $m>0$ we can define
\[ z_m := \max \{ \sup J \mid J\text{ is the union of an }m\text{--chain from }\VV\text{ such that }0\in J\}<\infty.\]
Note that $f(z_1)=1$ and $f(z_m)\le m$.
Assume inductively that $f(z_{m-1})=m-1$. By considering an interval from $\VV$ that contains $z_{m-1}$, we see that
$f(z_{m-1}+\delta)=m$ for all sufficiently small $\delta>0$. It follows that \[f(z_m)\ge f(z_{m-1}+\delta)=m.\] That is, for all $m>0$ we have
\[f(z_{m-1},z_m]=\{m\}.\]
Choosing a homeomorphism $h\co [0,\infty)\longrightarrow[0,\infty)$ satisfying $h(z_m)=m$, we obtain the first conclusion.
After noting that for all $z\in[z_{m-1},z_m)$ we have
\[
\CL_\VV[0,z]=m,\] the proof is complete.
\ep
\begin{rem}
One cannot drop the finite intersection multiplicity hypothesis. Indeed, if $\mathcal{V}$ consists of all intervals of the form $(-1,n)$ for $n\ge1$
then $\CL_{\mathcal{V}}[0,x)=1$ for all $x>0$.
\end{rem}
Let us now assume that $G\le\Homeo_+[0,1]$ is a group generated by a finite set $V$. The collection of open intervals
\[
\VV:=\bigcup_{v\in V} \pi_0\supp v\]
is a covering of $\supp G$ with finite intersection multiplicity.
For $x,y\in \Int I$, we define the \emph{covering distance}\index{covering distance}
\[
d_V(x,y) := \CL[x,y]\in\bN\cup\{\infty\}.\]
As mentioned above, the covering length is a measure of topological complexity. If
$G$ acts without global fixed points in $\Int I$, and if $f\in G$ is a compactly supported homeomorphism
(that is, the closure of $\supp f$ is a compact set in $\Int I$),
then $\CL({\suppc f})<\infty$.
For two compactly supported homeomorphisms, the covering length of their supports allows one to
quantitatively assert that one of them has a ``larger" support
than the other.
For $g\in G$, we define its \emph{syllable length (with respect to $V$)}\index{syllable length} as
\[
\|g\|_{\mathrm{syl}}:=\inf\{ \ell \mid g = v_\ell^{n_\ell} \cdots v_2^{n_2}v_1^{n_1} \text{ for some }v_i\in V\text{ and }n_i\in\bZ\}.\]
The lemma below asserts that the covering length of the closure of an interval is the minimum syllable length of a
group element that can move the interval off itself. Note that the former quantity depends on the action (dynamics),
while the latter one is purely group theoretically defined.
This observation may be considered as a manifestation of a recurring theme of the book, which is relating dynamical
features of finitely generated group actions to group theoretic properties.
\begin{lem}\label{l:covering-translation-distance}
Let $G\le\Homeo_+[0,1]$ be a group with a finite generating set $V$,
and let $J\sse \supp G$ be a (compact or non-compact) nondegenerate interval.
Then we have that
\[\CL(\bar J)=\min\{ \|g\|_{\mathrm{syl}}\mid g\in G\text{ and }g(J)\cap J=\varnothing\}.\]
\end{lem}
\bp
In this proof, the covering distance and the syllable length are both defined by $V$.
We let $x<y$ be the endpoints of $J$.
Put $m:=\CL(\bar J)$.
There exists a minimal length $m$--chain
\[ U_1, U_2, \ldots,U_m\]
(with the indices in the given order)
such that $U_i\in\pi_0\supp v_i$ for some $v_i\in V$
and such that \[\bigcup_i U_i\supset [x,y]=\bar J.\]
Depending on whether $v_i$ moves points in $U_i$ to the right or to the left, one can find integers $n_i\gg0$ or $n_i\ll0$ such that
\[
v_i^{n_i}\cdots v_1^{n_1}(x)\in U_i\cap U_{i+1}\]
for each $i<m$. Since $y\in U_m$, we can pick $n_m$ such that
$g_0:=v_m^{n_m}\cdots v_1^{n_1}$ moves $x$ further to the right of $y$.
Since $\|g_0\|_{\mathrm{syl}}\le m$, we have that
\[m\ge\min\{ \|g\|_{\mathrm{syl}}\mid g\in G\text{ and }g(J)\cap J=\varnothing\}.\]
For the opposite inequality,
pick an arbitrary $g\in G$ satisfying $ g(J)\cap J=\varnothing$.
Without loss of generality, we may assume that
\[ x<y \le g(x).\]
Writing $g=v^{n_\ell}_\ell \cdots v_1^{n_1}$ for some $v_i\in V$ and $n_i\in \bZ$,
we see that
\[
[x,y]\sse [x,g(x)]\sse\bigcup_i \supp v_i.\]
This implies that $m=\CL(\bar J)\le \ell\le\|g\|_{\mathrm{syl}}$. This completes the proof.
\ep
We will make use of covering lengths crucially in Chapters~\ref{ch:slp} and~\ref{ch:optimal}.
\section{Generalities on chain groups}\label{sec:chain}
Let $\{J_1,\ldots,J_m\}$ be an $m$--chain, let \[J=\bigcup_{i=1}^m J_i\sse I,\] and let $\{f_1,\ldots,f_m\}\sse \Homeo_+(J)$ such that
$\supp f_i\sse J_i$. Following~\cite{KKL2019ASENS}, the group \[\form{ f_1,\ldots,f_m}=G\le \Homeo_+(J)\] is called a
\emph{pre-chain group}\index{pre-chain group}.
The group $G$ is called an \emph{$m$--chain group}\index{chain group} (or simply a \emph{chain group}) if for $i<m$ we have that
\[f_{i+1}^{\pm1}(f_i^{\pm 1}(\inf J_{i+1}))\geq\sup J_i,\] where by the exponent $\pm 1$ we mean that this inequality holds for some choice
of exponents.
\begin{figure}[h!]
\centering
\begin{tikzpicture}[ultra thick,scale=.5]
\draw [red] (4,0) -- (12,0);
\draw [dashed,purple] (12,.8) -- (12,-.3);
\draw (9,.8) node [above] {\small $z$};
\draw [dashed,purple] (9,.8) -- (9,-.3);
\draw (12,.8) node [above] {\small $y$};
\draw (7,0) node [above] {\small $J_1$};
\draw [blue] (9,.5) -- (17,.5);
\draw (14,.5) node [above] {\small $J_2$};
\end{tikzpicture}
\caption{A chain of two intervals. The condition defining a chain group says that if $z=\inf J_2$ is the left endpoint of $J_2$
and $y=\sup J_1$ is the right endpoint of $J_1$, then
$f_2f_1(z)$ lies at least as far to the right as $y$.}
\label{f:coint}
\end{figure}
At first glance, this condition may look rather bizarre, though if $\supp f_i=J_i$ for all $i$ then the condition is dynamically
robust, in the sense that it holds after replacing each $f_i$ by $f_i^N$ for some $N\gg 0$. Indeed, this is obvious since (up to replacing
$f_i$ by its inverse) we have that $f_i^n(x)\to\sup J_i$ as $n\to\infty$, for all $x\in J_i$.
It turns out that in a chain group, the subgroup $\form{ f_i,f_{i+1}}$ is isomorphic to Thompson's group $F$. We will demystify
some of these definitions and apparent coincidences in this section.
We remark that if $f_1$ and $f_2$ are homeomorphisms generating a $2$--chain group, then $f_1$ and $f_2$ are
\emph{crossed}\index{crossed interval}
in the sense of Section~\ref{sec:crossed} above.
\subsection{Two--chain groups and Thompson's group $F$}\label{ss:2-chain}
\emph{Thompson's group $F$}\index{Thompson's group $F$}
is a central object in the study of the group $\mathrm{PL}_+(I)$, the group of orientation preserving piecewise
linear homeomorphisms of the interval. The name \emph{piecewise linear}\index{piecewise linear}
is a bit of a misnomer, since elements of $\mathrm{PL}_+(I)$ are
locally described by affine functions. An element $f\in\PL_+(I)$ is required to be continuous,
and the derivative $f'$ is locally constant on the complement
of finitely many points in $I$, which we call the \emph{breakpoints}\index{breakpoint} of $f$.
The group $F$ is defined to be the full group of
orientation preserving piecewise linear homeomorphisms of $I=[0,1]$ such that all breakpoints of each element of $F$
are dyadic rationals, so that
all derivatives are powers of $2$. In this subsection we will discuss the following basic fact, which illustrates the ubiquity
with which Thompson's group is found within the study of groups of homeomorphisms of one-manifolds:
\begin{prop}\label{prop:two-chain}
Let $G$ be a two--chain group. Then $G$ is isomorphic to Thompson's group $F$.
\end{prop}
It is worth noting that there are no assumptions on $G$ other than it being a two--chain group. In particular, there are no restrictions
on the regularity of the generators of $G$; once a certain dynamical hypothesis is verified, the algebraic conclusion follows.
Since $(0,1)$ is homeomorphic to $\bR$, it is not surprising that the group $F$ is isomorphic to an easily describable subgroup of
$\Homeo_+(\bR)$. Indeed, it turns out that if we set
\[
a(x)=x+1\quad\text{and}\quad
b(x)=
\begin{cases}
x&\text{ if }x\leq 0,\\
2x&\text{ if }0<x< 1,\\
x+1&\text{ if }1\leq x\\
\end{cases}
\]
then the subgroup of $\Homeo_+(\bR)$ generated by $a$ and $b$ is isomorphic to $F$. The subgroup $\form{ a,b}\le \Homeo_+(\bR)$
can be conjugated into $\mathrm{PL}_+(I)$ by an explicit homeomorphism $h$. See Figure~\ref{fig:gens}.
\begin{figure}
\centering
\tikzstyle {a}=[postaction=decorate,decoration={%
markings,%
mark=at position .65 with {\arrow{stealth};}}]
{\begin{tikzpicture}[scale=.75]
\draw (0,0) -- (0,.5) -- (4,.5) -- (4,0) --cycle;
\draw [a] (2,.5) -- (1,0);
\draw [a] (3,.5) -- (2,0);
\draw [right] (0,.5) node [above] {\tiny $0$};
\draw [right] (4,.5) node [above] {\tiny $1$};
\draw (2,0.5) node [above] {\tiny $\frac12$} (3,.5) node [above] {\tiny $\frac34$};
\draw [below] (1,0) node {\tiny $\frac14$} (2,0) node {\tiny $\frac12$};
\draw [below] (4,0) node {\tiny $1$};
\draw [below] (0,0) node {\tiny $0$};
\draw [below] (2,-1) node {\small $A$};
\end{tikzpicture}}
\quad\quad
{\begin{tikzpicture}[scale=.75]
\draw (0,0) -- (0,.5) -- (4,.5) -- (4,0) --cycle;
\draw [a] (2,.5) -- (2,0);
\draw [a] (3,.5)--(2.5,0) ;
\draw [a] (3.5,.5)--(3,0) ;
\draw [right] (0,.5) node [above] {\tiny $0$};
\draw [right] (4,.5) node [above] {\tiny $1$};
\draw [below] (0,0) node {\tiny $0$};
\draw [below] (4,0) node {\tiny $1$};
\draw [below] (2,0) node {\tiny $\frac12$} (3,0) node {\tiny $\frac34$} (2.5,0) node {\tiny $\frac58$};
\draw [above] (2,.5) node {\tiny $\frac12$} (3,.5) node {\tiny $\frac34$} (3.5,.5) node {\tiny $\frac78$};
\draw [below] (2,-1) node {\small $B$};
\end{tikzpicture}}
\quad\quad
{\begin{tikzpicture}[scale=.75]
\draw (0,0) -- (0,.5) -- (4,.5) -- (4,0) --cycle;
\foreach \i in {1,3,3.5} \draw [a] (\i,.5)--(\i,0);
\draw [right] (0,.5) node [above] {\tiny $0$};
\draw [right] (4,.5) node [above] {\tiny $1$};
\draw [below] (0,0) node {\tiny $-\infty$};
\draw [below] (4,0) node {\tiny $\infty$};
\draw [below] (1,0) node {\tiny $-1$};
\draw [below] (3,0) node {\tiny $1$} (3.5,0) node {\tiny $2$} ;
\draw [above]
(1,.5) node {\tiny $\frac14$} (3,.5) node {\tiny $\frac34$} (3.5,.5) node {\tiny $\frac78$};
\draw [below] (2,-1) node {\small $h$};
\end{tikzpicture}}
\caption{Elements $A,B\in\mathrm{PL}_+(I)$ generating Thompson's group $F$ and a conjugating homeomorphism $h$.}
\label{fig:gens}
\end{figure}
We have that \[a=hA^{-1}h^{-1}\quad \textrm{and}\quad b=hB^{-1}h^{-1},\] and $F\cong\form{ A,B}$. In fact, one can say much more;
it turns out that \[F\cong\form{ A,B\mid [AB^{-1}, A^{-1}BA],\, [AB^{-1},A^{-2}BA^2]}\] is a presentation for $F$. Since these claims about
presenting Thompson's group are not the primary concern of this book, we will omit proofs, directing the reader instead to some of the
standard references about Thompson's group, such as~\cite{CFP1996,BurilloBook}. The presentation for $F$
we have given here is essentially the only fact about
$F$ that we will not justify in this book.
Observe that there is a natural map \[D\colon \mathrm{PL}_+(I)\longrightarrow \bZ^2\] that takes a homeomorphism
and computes the right derivative at $0$ and the left derivative at $1$. Since $0$ and $1$ are global fixed points of the action of
$\mathrm{PL}_+(I)$, the chain rule implies that $D$ is in fact a homomorphism to the multiplicative group $\bR_+^2$.
Since elements of $\mathrm{PL}_+(I)$ only have finitely many
breakpoints, the derivative is continuous at all but finitely many points; it follows that breakpoints cannot accumulate,
and hence the homomorphism $D$ is well--defined since the derivative is constant in an open neighborhood of $0$ and $1$. For this reason,
the homomorphism $D$ is said to be computing derivatives of \emph{germs}\index{germ} of $\mathrm{PL}$ homeomorphisms; we will discuss
this concept in more generality shortly.
By considering
homeomorphisms that are the identity near one endpoint of $I$ but not the other, it is easy to see that $D$ is surjective. Moreover,
it is easy to see that the restriction of $D$ to $F$ is surjective. Since $F$ is two--generated, it follows that the commutator subgroup
$[F,F]$ consists of exactly the elements of $F$ that are the identity near $\{0,1\}$, which is to say homeomorphisms whose support
is compactly contained in $(0,1)$.
Note that since dyadic rational numbers are dense, there is an infinite sequence of elements
$\{f_n\}_{n\ge1}\sse F$ such that \[{\suppc f_n}\sse \supp f_{n+1}\] for all $n$, and such that if $J\sse (0,1)$ is a compact
interval then $J\sse \supp f_n$ for $n\gg 0$. It follows that $[F,F]$ is not finitely generated.
The following fact has been encountered in another guise, and is an important feature of $F$.
\begin{lem}\label{lem:f-simple}
The group $F$ has trivial center, the commutator subgroup $F'=[F,F]$ is simple, and every proper quotient of $F$ is abelian.
\end{lem}
\begin{proof}
It is a straightforward exercise to show that $F'$ acts CO--transitively on $(0,1)$. It now follows from
Lemma~\ref{lem:co-trans} that $F''=[F',F']$ is nonabelian and simple, that the center of $F'$ is trivial, and that every proper quotient of $F'$ is
abelian.
Now, if $f\in F'$ is arbitrary, the ${\suppc f}$ is compactly contained in $(0,1)$. Thus, there is a compact interval $J\sse (0,1)$ with
dyadic endpoints such that ${\suppc f}$ is contained in the interior of $J$. The subgroup $F_J\le \PL_+(J)$ consisting of homeomorphisms
with dyadic breakpoints and derivatives being powers of $2$ is isomorphic to $F$, since $J$ is homeomorphic to $[0,1]$ via an affine
homeomorphism that preserves dyadic rationals. Since $f\in F_J$ and since $f$ is compactly supported in $J$, we have that
$f\in F_J'=[F_J,F_J]$. Since $F_J$ consists of homeomorphisms compactly supported in $(0,1)$, we have that $F_J\le F'$. It follows that
if $f\in F'$ then $f\in F''$, so that $F'\le F''$ and hence $F'=F''$.
It is straightforward now to conclude that the center of $F$ is trivial. To see that every proper quotient of $F$ is abelian, let $K\le F$ be a
nontrivial normal subgroup. It follows that $[K,F]$ is a normal subgroup of $F'$, so that either $K$ commutes with every element of $F$
or $F'\le [K,F]$. The first possibility is ruled out since the center of $F$ is trivial, and the second possibility implies that $F/K$ is abelian.
\end{proof}
In the proof of Lemma~\ref{lem:f-simple}, we implicitly argued that $F'$ is perfect. This is part of a completely general phenomenon, which
we record now for future reference.
\begin{lem}\label{lem:chain-comm}
Let $G$ be a chain group. Then $G'$ is perfect and $G$ has trivial center.
\end{lem}
We recall that if $G$ acts by homeomorphisms on a Hausdorff topological space $X$ and if $x\in X$ is a global fixed point of $G$, we say
that the \emph{germ} of the $G$--action at $x$ is the equivalence relation on $G$ that says $g_1\sim g_2$ if $g_1$ and $g_2$ agree in
an open neighborhood of $x$.
\begin{proof}[Proof of Lemma~\ref{lem:chain-comm}]
Let $G$ be generated $\{f_1,\ldots,f_m\}$, where $\supp f_i=J_i$ and $\{J_1,\ldots,J_m\}$ forms an $m$--chain. Clearly we may assume
$m\geq 3$. Without loss of generality,
$\supp G=(0,1)$.
Let $u\in G$ be a nontrivial central element. We have that $\supp u$ is invariant under $G$ and accumulates at $0$ and $1$. Observe that
the germ of the action of $G$ at $0$ coincides with the cyclic group generated by $f_1$, and similarly the germ at $1$ coincides with the
cyclic group generated by $f_m$. Since the set $\supp u$ is invariant under $G$, we have that $\supp u=(0,1)$. Conjugating $f_1$ by a power
of $u$ results in a homeomorphism that does not commute with $f_1$,
since $0=\inf J_1$ is fixed by $u$ and since $\sup J_1$ is not. This is a
contradiction, so the center of $G$ is trivial.
Now, we have that for $1\leq i\leq m-1$, the group $\form{ f_i,f_{i+1}}$ is isomorphic to $F$. Lemma~\ref{lem:f-simple} implies
that $F''=F'$, so \[[f_i,f_{i+1}]\in \form{ f_i,f_{i+1}}'=\form{ f_i,f_{i+1}}''\le G''.\] Since $G'$ is normally generated by elements of the
form $[f_i,f_{i+1}]$, we have that $G'=G''$ and so $G'$ is perfect.
\end{proof}
We can now prove the main result of this subsection.
\begin{proof}[Proof of Proposition~\ref{prop:two-chain}]
Let $\{J_1,J_2\}$ be a two--chain, as in Figure~\ref{f:coint}. Write $y=\sup J_1$ and $z=\inf J_2$. We write $G=\form{ f,g}$ for the
two--chain group in question, so that \[\supp f=J_1,\quad \supp g= J_2,\] and so that $g(f(z))\geq y$.
Observe that $k=g\circ f$ moves $J_2$ off of $J_1$, so that if $\supp h\sse J_2$ then \[(\supp khk^{-1})\cap J_1=\varnothing.\] In
particular, we have that \[[f^{-1},kg^{-1}k^{-1}]=[f^{-1},k^2g^{-1}k^{-2}]=1\] in the group $\form{ f,g}$.
Setting $f^{-1}g^{-1}=A$ and $g^{-1}=B$, we see that $AB^{-1}=f^{-1}$ and $A^{-1}=k$. It follows that $A$ and $B$
generate $G$, and we have
\[[f^{-1},kg^{-1}k^{-1}]=[AB^{-1},A^{-1}BA],\quad [f^{-1},k^2g^{-1}k^{-2}]=[AB^{-1},A^{-2}BA^2].\]
It follows from the finite presentation of $F$ above that the map
\[\phi\colon F\longrightarrow G\] given by \[A\mapsto f^{-1}g^{-1}\quad \textrm{and}\quad B\mapsto g^{-1}\]
is a well-defined homomorphism of groups
that is clearly surjective.
It is immediate that $G$ is not abelian, since
on the one hand we have \[f(g(\inf J_2))=f(\inf J_2),\] but on the other hand since $f(\inf J_2)>\inf J_2$, we have that
\[g(f(\inf J_2))\neq f(\inf J_2).\] It follows immediately from Lemma~\ref{lem:f-simple} that $\phi$ is an isomorphism from $F$ to $G$.
\end{proof}
We remark that for $m$--chains with $m\geq 3$,
there is a similar result, except one replaces the group $F$ with the \emph{Higman--Thompson group}\index{Higman--Thompson group}
$F_m$.
These groups form a natural generalization of Thompson's group $F$, and have the following uniform infinite presentations (though
they are all finitely presented):
\[ F_m\cong \form{ \{g_i\}_{i\ge1}\mid g_j^{g_i}=g_{j+m-1}\,\textrm{ for all } 0\leq i\leq j}.\] It turns out that the group
$F_2$ is just the group $F$, and the ambitious reader may try and prove this for themself.
Let $\{J_1,\ldots,J_m\}$ be an $m$--chain, and let $\{f_1,\ldots,f_m\}$ generate an $m$--chain group with $\supp f_i=J_i$.
If
\[f_m\cdots f_1(\inf J_2)\geq\sup J_{m-1},\] then we have that $\form{ f_1,\ldots,f_m}$ generates the $m^{th}$ Higman--Thompson
group $F_m$. Note that if $\{f_1,\ldots,f_m\}$ generate an $m$--chain group then for all sufficiently large $N\gg 0$, we have that
$\{f_1^N,\ldots,f_m^N\}$ generate a copy of $F_m$. Thus, the group $F_m$ is the \emph{dynamical stabilization}\index{dynamical
stabilization} of all $m$--chain groups.
It is easy to see that the standard action of the Thompson's group $F\le\PL_+(I)$ on $(0,1)$ minimal, i.e. every orbit is dense
under the action.
We note a stronger fact regarding the \emph{positive $n$--transitivity}\index{positively $n$--transitive},
as defined in Definition~\ref{defn:n-trans}.
This fact is well-known;
see~\cite[Theorem 3.1.2]{BurilloBook} for instance.
\begin{prop}\label{p:n-transitive}
For all $n\ge1$, the standard action of $F$ on $(0,1)$ is positively $n$--transitive on dyadic rationals.
In particular, every interval $U\sse (0,1)$ can be expanded arbitrarily close to $(0,1)$ by an element of $F$.
\end{prop}
The first statement means that given a pair of $n$--tuples
\[ x_1<x_2<\cdots<x_n\]
and \[y_1<y_2<\cdots<y_n\]
of dyadic rationals in $(0,1)$
there exists $g\in F$ such that $g(x_i)=y_i$ for all $i$.
The second statement is an obvious consequence of the first by the density of dyadic rationals.
\bp[Proof of Proposition~\ref{p:n-transitive}]
We follow an argument in~\cite{BurilloBook}. Set $x_0=y_0=0$ and $x_{n+1}=y_{n+1}=1$.
We claim that there exists a piecewise linear homeomorphism $g_i\co [x_i,x_{i+1}]\to[y_i,y_{i+1}]$
with breakpoints in $\bZ[1/2]$ and with slopes in $2^\bZ$.
Then the map $g$ in the conclusion can be obtained by concatenating $g_0,\ldots,g_n$.
To see the claim, we fix $i$ and write \[x_{i+1}-x_i = p/2^m\quad\textrm{and}\quad y_{i+1}-y_i=q/2^n\quad \textrm{for some}\quad p,q,m,n\in\bZ_{>0}.\]
We may assume $p\le q$, for otherwise one can consider the inverse of $g_i$.
Write
\[
p/2^m =(p-1)/2^m + \sum_{i=1}^{q-p} 1/2^{m+i}+ 1/2^{m+q-p}.\]
So, $x_{i+1}-x_i$ can be partitioned into $q$ intervals whose lengths lie in $2^\bZ$. By mapping each of these intervals
to an interval of the form $[y_i+j/2^n,y_i+(j+1)/2^n]$, we obtain $g_i$ as claimed.
\ep
\subsection{Smooth realizations of Thompson's group}\label{ss:smooth}
This subsection follows Section 1.5.2 in~\cite{Navas2011}. The account here differs very little from Navas' exposition, and we include
it here because the smooth, locally dense realization of Thompson's group $F$ is critical in subsequent chapters.
Proposition~\ref{prop:two-chain} shows that an arbitrary two--chain group is automatically isomorphic to Thompson's group $F$. In particular,
it realizes many $C^{\infty}$ copies of $F$ inside of $\Homeo_+[0,1]$. While the smoothness of a copy of $F$ will be a critical tool in our
discussion of critical regularity later in this book, we will need to exercise some further control over the dynamical behavior of the
action of $F$ in a way that is not immediate as a consequence of smoothness.
Let $G\le \Diff_+^{\infty}(I)$ be a two--chain group such that $\supp G=(0,1)$.
If $G$ acts minimally on $(0,1)$ then Rubin's Theorem (Theorem~\ref{thm:rubin}) implies that this action is
topologically conjugate to the standard piecewise linear action of $F$. However, there is little
reason to believe that the action is minimal. If a given $C^{\infty}$ action of $F$ as a two--chain group
fails to be minimal, then one can apply a semi-conjugacy to obtain a minimal action (see Theorem~\ref{thm:min-set-r} below), though
in the process one may lose smoothness.
It turns out, however, that there is a copy of $F$
inside of $\Diff_+^{\infty}(I)$ that does act minimally on $(0,1)$, and therefore this action is topologically conjugate to the standard realization of
$F$ inside of $\PL_+(I)$. This fact was originally established by Ghys--Sergiescu~\cite{GS1987}.
We remark that W.~P.~Thurston proved that $F$ is conjugate into the group $\Diff^{1+\mathrm{Lip}}_+(I)$,
using Farey sequences and maps that are piecewise defined to be in $\PSL(2,\bZ)$ (see Section 1.5.1 of~\cite{Navas2011}).
Since we will use a $C^{\infty}$
action of $F$ on $I$ that is minimal on $(0,1)$ in an essential way, we will explain the construction of such an action, following
Navas' exposition of Ghys--Sergiescu's work in Section 1.5.2 of~\cite{Navas2011} (cf.~Subsection~\ref{sss:tsuboi}).
The idea is to associate representations of $F$ to homeomorphisms of $\bR$ that satisfy some
relatively mild hypotheses.
For the rest of this subsection, we will write $h\in\Homeo_+(\bR)$ for a homeomorphism that satisfies the following conditions.
\begin{enumerate}[(A)]
\item
We have $h(0)=0$.
\item
$h(x+1)=h(x)+2$ for all $x\in \bR$.
\item $|h(x)-h(y)|>|x-y|$ for all $x,y\in\bR$.
\end{enumerate}
An example of a homeomorphism of $\bR$ that satisfies the preceding three conditions is the map $x\mapsto 2x$.
For each such homeomorphism $h$, we will build an action \[\varphi_h\colon F\longrightarrow\Homeo_+(\bR)\] of $F$ on the real line.
We will write $\Aff_+^2(\bR)$ for the group or orientation preserving affine homeomorphisms that preserve the set of dyadic rational numbers.
It will be helpful to write $\bQ_2\le \bR$ for the additive subgroup of $\bR$ consisting of all dyadic rational numbers.
For $y\in\bR$,
we define $\lambda,\tau_y\in \Homeo_+(\bR)$ by
\begin{align*}
\lambda(x)&:=2x,\\
\tau_y(x)&:=x+y.\end{align*}
As in Section~\ref{ss:bs-smooth},
we have several equivalent expressions for $\Aff_+^2(\bR)$
\[
\Aff_+^2(\bR)=2^\bZ\times \bQ_2=
\left\{\begin{pmatrix} 2^n&p/2^q\\ 0 & 1\end{pmatrix}\middle\vert n,p,q\in\bZ\right\}=\form{a,e\mid aea^{-1}=e^2}.\]
The verification of the equivalence is trivial using the correspondence
\[
\lambda=\begin{pmatrix} 2&0\\ 0 & 1\end{pmatrix}=a,\qquad
\tau_1=\begin{pmatrix} 1&1\\ 0 & 1\end{pmatrix}=e.\]
The presentation for the group $\form{a,e}$ above is called the \emph{Baumslag--Solitar group} of type $(1,2)$, and denoted as $\BS(1,2)$; see Section~\ref{sec:bs} for an extensive discussion on this group.
The following is transparent almost only by navigating the definitions.
\begin{lem}\label{lem:aff-homo}
There uniquely exists a homeomorphism $\eta_h\in\Homeo_+(\bR)$
satisfying the following.
\be[(i)]
\item $\eta_h(0)=0$;
\item $\eta_h\tau_1=\tau_1\eta_h$;
\item $\eta_h\lambda=h\eta_h$;
\item $\eta_h(\bQ_2)$ is dense.
\ee
\end{lem}
\bp
Let us define a map
\[\Psi_h\colon \Aff_+^2(\bR)\longrightarrow\Homeo_+(\bR)\]
by the formula
$\Psi_h(\lambda)=h$ and that $\Psi_h(\tau_1)=\tau_1$.
To see that $\Psi_h$ is well-defined and a homomorphism, it suffices to see
that the relation of $\BS(1,2)\cong \Aff_+^2(\bR)$ are preserved. Indeed, we note
for each $x\in \bR$ that
\[h\tau_1h^{-1}(x)=h(h^{-1}(x)+1)=x+2=\tau_1^2(x).\]
Let us now define
$\eta_h (g(0))=\Psi_h(g)(0)$ for all $g\in \Aff_+^2(\bR)$, which is easily seen to be well-defined and makes the diagram below commutes for all $g\in \Aff_+^2(\bR)$:
\[\begin{tikzcd}
\bQ_2\arrow{r}{g} \arrow{d}{\eta_h } &
\bQ_2\arrow{d}{\eta_h }\\
\bR\arrow{r}{\Psi_h(g)}& \bR
\end{tikzcd}
\]
The map $\eta_h$ preserves the order. Indeed, let us pick dyadic rationals $p/2^q$ and $p'/2^q$ for some $p<p'$.
We have that
\[
\eta_h (p/2^q)=\eta_h (\lambda^{-q}\tau_1^p\lambda^q(0))
=h^{-q}\tau_1^p h^q(0)=h^{-q}(p)
< h^{-q}(p')=\eta_h (p'/2^q), \]
implying that the order is preserved. We also see from the above computation that $\eta_h \restriction_\bZ$ is the identity.
We claim that the set $\eta_h(\bQ_2)$ is dense in $\bR$. Once this claim is established, we can simply extend $\eta_h$ to the closure of $\eta_h(\bQ_2)$ continuously. The uniqueness of $\eta_h$ will also follow since the condition
$\eta_h (g(0))=\Psi_h(g)(0)$ is necessary from that $\eta_h$ conjugates $\lambda$ to $h$.
To prove the claim, we assume that the set
\[
C:=\overline{\eta_h(\bQ_2)}=\overline{\{ h^{-q}(p)\mid p,q\in\bZ\}}\]
is proper in $\bR$.
The set $C$ is also $1$--periodic since $\eta_h$ commutes with $\tau_1$.
In particular, we can choose a component $J$ of $\bR\setminus A$ with a maximal length.
On the other hand, we have that $h(C)=C$ since the set $\{h^{-q}(p)\mid p,q\in\bZ\}$ is $h$--invariant. It follows that $h(J)$ is also a component of $\bR\setminus C$.
By the condition (C) above we have
\[|h(J)|>|J|,\]
which contradicts the maximality.
\ep
Extending the above definition, we continue to denote by
\[\Psi_h\co \Homeo_+(\bR)\to\Homeo_+(\bR)\]
the conjugation by $\eta_h$.
Namely,
\[
\Psi_h(g)=\eta_h g\eta_h^{-1}\]
for $g\in \Homeo_+(\bR)$.
We consider the restriction
\[
\varphi_h:=\Psi_h\restriction_F.\]
In other words, we let $\varphi_h$ be the topological conjugacy of the standard action of $F$ on $[0,1]$ by $\eta_h$, extending the map by the identity outside $[0,1]$.
Observe that if $h(x)=2x$ then $\varphi_h(F)$ is just the usual action of $F$ by piecewise linear homeomorphisms on $F$.
Indeed, let
$F=\form{ a,b}$ be the usual generators of $F$ (see Subsection~\ref{ss:2-chain}, especially Proposition~\ref{prop:two-chain}).
We have that $\varphi_h(a)=\Psi_h(\tau_1)=\tau_1=a$. Similarly we can check that
$\varphi_h(b)$ agrees with the identity on $(-\infty,0)$ and with $\tau_1$ on $(1,\infty)$, and agrees with $h$ on $(0,1)$. Thus, $\varphi_h(b)=b$.
We recall the meaning of $C^r$--tangency to the identity from Definition~\ref{defn:tangent-id}.
\begin{thm}[\cite{GS1987}]\label{thm:phi-smooth}
If $h\in\Diff^r_+(\bR)$ satisfies the conditions (A), (B) and (C) above,
and if
\addtocounter{enumi}{3}
\be[(D)]
\item
$h$ is $C^r$--tangent to the identity at $0$
for some $r\in{\bZ_{>0}}\cup\{\infty\}$,
\ee
then we have
\[
\varphi_h(F)\le\Diff_+^r(\bR).\]
\end{thm}
\begin{proof}
Let $\{a_n\}_{n\in\bZ}$ be an increasing sequence containing breakpoints of $g$, and let $\{b_n:=\eta_h(a_n)\}_{n\in\bZ}$. We write $f_n\in\Aff_+^2(\bR)$ describing $g$ on the interval $[a_n,a_{n+1}]$.
Since $h\in\Diff^r_+(\bR)$, it is immediate from the definitions that
\[\Psi_h(f_n)\in\form{\tau_1,h}\le \Diff^r_+(\bR)\]
for all $n$.
To prove that $\varphi_h(g)\in\Diff^r_+(\bR)$, we need
to check smoothness at each of the points $\{b_n\}_{n\in\bZ}$. Evidently, this is equivalent to the statement of equality of derivatives
\[\Psi_h(f_{n-1})^{(i)}(b_n)=\Psi_h(f_{n})^{(i)}(b_n),\] for all $i\leq r$.
Since $f_{n-1}(a_n)=f_n(a_n)$ by the continuity of $g$, we have that the
affine map $\tau_{a_n}^{-1}f_{n-1}^{-1}f_n\tau_{a_n}$ fixes the origin, and is hence given by multiplication by a power of two, say $2^p$.
Computing, we have that \[\Psi_h(\tau_{a_n}^{-1}f_{n-1}^{-1}f_n\tau_{a_n})=h^p.\] Taylor's Theorem says that $h$ is given
near the origin by the function $x+\omega(x)$, where all derivatives of $\omega$ at $0$ vanish up
to order $r$. It follows that \[\Psi_h(\tau_{a_n}^{-1}f_{n-1}^{-1}f_n\tau_{a_n})^{(i)}(0)=
\begin{cases}
1&\text{ if }i= 1,\\
0&\text{ if }2\leq i\leq r.\\
\end{cases}\]
It follows that the Taylor expansions of $\Psi_h(f_{n-1})$ and $\Psi_h(f_{n})$ agree up to order $r$ at $b_n$.
\end{proof}
\begin{cor}\label{cor:ghys-serg}
There is an embedding $F\longrightarrow\Diff_+^{\infty}(I)$ such that the action of $F$
on $(0,1)$ is minimal.
\end{cor}
\begin{proof}
We embed $F$ into $\PL_+(I)$ in the usual way, and extend by the identity to all of $\bR$. This realizes $F$ as a group of homeomorphisms
of $\bR$ and are locally described by elements of $\Aff_+^2(\bR)$, which are supported on $[0,1]$. Applying $\varphi_h$
for $h\in\Diff_+^{\infty}(\bR)$
satisfying the conditions (A) through (D) above,
we obtain a group of diffeomorphisms
of $\bR$ that are the identity outside of a compact interval. Restricting to this interval, we obtain an embedding \[F\longrightarrow\Diff_+^{\infty}
(I)\] such that each element of $F$ has derivatives of all orders that coincide with the identity at $0$ and $1$. The fact that this action of
$F$ is minimal on $(0,1)$ follows from the fact that the usual action of $F$ by piecewise linear homeomorphisms if minimal, and the
fact that $\varphi_h(F)$ is topologically conjugate to the usual action.
\end{proof}
\begin{rem}
In Lemma~\ref{lem:aff-homo}, if the condition
\be[(A)]
\setcounter{enumi}{2}
\item $|h(x)-h(y)|>|x-y|$ for all $x,y\in\bR$,
\ee
is dropped then the set $\eta_h(\bQ_2)$ may not be dense
and so, $\eta_h\co\bQ_2\to\bR$ may not extend to a continuous map of the real line.
We can still extend $\eta_h^{-1}$ to a periodic monotone increasing continuous map $\eta_h^{-1}\co\bR\to\bR$ in this case.
A smooth action
\[
\Psi_h\co F\to\Diff_{[0,1]}^\infty(\bR)\]
can also still be defined in a piecewise manner, and moreover,
$\eta_h^{-1}$ semi-conjugates this action to the standard piecewise linear action.
This way, one can construct a faithful smooth exceptional action of $F$ on $[0,1]$.
The construction of this subsection works equally well for the Thompson's group $G$, which piecewise linearly acts on the circle
with dyadic breakpoints and with slopes powers of two~\cite{GS1987}.
\end{rem}
\subsection{Subgroups of chain groups}\label{ss:sub-chain}
In light of Proposition~\ref{prop:two-chain}, one might be tempted to believe that the structure of chain groups is similar to that of
$F$, with perhaps a na\"ive guess being that a general $m$--chain group is isomorphic to the Higman--Thompson group $F_m$. Though this
first guess is evidenced by the dynamical stabilization for chain groups we have already observed,
it is not right. In fact, the diversity of subgroups
of chain groups is so wild as to furnish continuum many isomorphism types of $m$--chain groups for all $m\geq 3$.
\begin{thm}[cf.~\cite{KKL2019ASENS}, Theorem 1.4]\label{thm:chain-subgp}
Let \[\gam=\form{ f_1,\ldots,f_n}\le \Homeo_+(\bR)\] be an arbitrary finitely generated subgroup.
\begin{enumerate}[(1)]
\item
The group $\Gamma$ is a subgroup of an $(n+2)$--chain group.
\item
If $\supp f_1$ has finitely many components then $\Gamma$ can be realized as a subgroup of an $(n+1)$--chain group.
\end{enumerate}
\end{thm}
Theorem~\ref{thm:chain-subgp} can be contrasted with the following fact about $F$, one that was originally due to Brin and Squier~\cite{BS1985},
and which severely limits the possible subgroups of $F$.
\begin{thm}\label{thm:f-subgp}
The group $F$ contains no nonabelian free subgroups and obeys no nontrivial law.
\end{thm}
The proof of Theorem~\ref{thm:f-subgp} is not difficult once one has made the correct observations, and the idea behind this theorem can be traced all the way back to at least the Zassenhaus Lemma (see~\cite{Raghunathan1972}). The reader may also compare with Proposition~\ref{prop:compact-conn}.
The same ideas will recur later in this book in a stronger form.
We will give a proof here based on~\cite{BS1985}, more in this line of ideas.
In the local context of Theorem~\ref{thm:f-subgp} only, we will use the notation $\bF_2$ for the free group
of rank two, in order to distinguish it from Thompson's group.
\begin{proof}[Proof of Theorem~\ref{thm:f-subgp} (Sketch)]
Observing that $F\le\Homeo_+[0,1]$ is locally moving (Definition~\ref{d:locally-moving}), we immediately deduce from
Theorem~\ref{thm:lawless} that $F$ obeys no law. We even note from the paragraph following the corollary that $F'$ obeys no law either.
Alternatively, we can repeat the ideas in the proof of
Proposition~\ref{prop:z2z-real} to show that $F$ obeys no law.
Briefly, using that argument for every nontrivial element
of the free group $w\in\bF_2$, we can construct a finite sequence of compactly supported homeomorphisms
(i.e.~bumps) which together witness the fact that $w$ is not a law obeyed in $\Homeo_+(\bR)$.
It is not difficult to realize these bumps as elements of the commutator subgroup $F'$, which shows that $F'$ (and therefore $F$) obeys no law.
To show that $F$ contains no copy of $\bF_2$, let $f,g\in F$ be arbitrary elements that generate a subgroup $G\le F$.
Write $U=\supp f\cup\supp g$. Because $f$ and $g$ are piecewise linear, we have that $U$ consists of finitely many components,
and every element of the derived subgroup $G'\le G$ is the identity near $\partial U$.
Now, let $h\in G'$, and suppose that $V=\supp h$ meets $k>0$ components of $U$, say $\{U_1,\ldots,U_k\}$. Writing $V_1=V\cap U_1$,
there is a nontrivial elements $g_1\in G$ such that $g_1(V_1)\cap V_1=\varnothing$. It follows that $\supp [h,h^{g_1}]$ meets
$U$ in at most $k-1$ components. It follows that we may recursively construct a sequence of nested commutators which eventually
meet $U$ in no components, which is to say that the commutator becomes the identity. Since this commutator can easily be arranged to
be a reduced word in the generators $f$ and $g$ of $G$, this proves that $\form{ f,g}\ncong\bF_2$.
\end{proof}
We will postpone the proof of Theorem~\ref{thm:chain-subgp} until later in this chapter, as it will be useful to develop some further
dynamical and algebraic tools first.
\section{Simplicity}
Observe that since chain groups are natural generalizations of Thompson's group $F$, it is reasonable to suspect that they might share
some dynamical properties, even if their algebraic properties are not exactly analogous, as we have seen from
Theorems~\ref{thm:chain-subgp}
and Theorem~\ref{thm:f-subgp}. Since the usual realization of $F\le \PL_+(I)$ is CO--transitive when restricted to $F'$, we were able to prove
that $F'$ is simple (see Lemma~\ref{lem:f-simple}).
It is not immediate, however, that arbitrary chain groups have simple commutator subgroups. Indeed, the simplicity of $F'$ follows from
the fact that $F$ has a particularly nice realization. There are many other realizations of $F$ that are not CO--transitive. Indeed,
consider the usual action of $F$ on $[0,1]$, and blow up the orbit of an arbitrary point $x\in (0,1)$, as we did to construct continuous
Denjoy counterexamples. That is, we glue in intervals of finite total length along points in the orbit of $x$ in order to get a new action
of $F$ on $I$ that has a wandering set $J$.
Let $y\in (0,1)\setminus J$, and let $K$ be a small compact neighborhood of $y$. It is not the
case that for all open sets $\varnothing\neq U\sse (0,1)$ there is an element of $F$ that sends $K$ into $U$. Indeed, if $U$ is a connected
component of $J$ and $f\in F$, then $f(K)\cap U\neq\varnothing$ implies that $f(K)\cap\partial U\neq\varnothing$. Thus, such actions of
$F$ are clearly not CO--transitive. If the original orbit that was blown up avoided the endpoints of the intervals defining a two--chain group,
then this blown up copy of $F$ can be realized as a two--chain group. Through these observations
we see that the simplicity of $F$ really depended on
a minimal action of $F$ on $(0,1)$.
It is not surprising then that whether or not one can predict the simplicity of the commutator subgroup $G'$ of
a chain group $G$ depends on the
particular realization of the chain group that is used.
In order to proceed, we need to analyze the general behavior of minimal sets for group actions
on the interval $I$, which unfortunately does not have as nice an answer as for
group actions on the circle (see Theorem~\ref{thm:minimal-set}).
The reason that the argument for the circle does not generalize verbatim to the interval is that $(0,1)$ is not compact. Since the circle
case relies in an essential way on compactness (via the finite intersection property), another argument is required.
In the end, one does find that for every finitely generated subgroup $G\le \Homeo_+[0,1]$,
there is a closed subset $C\sse (0,1)\cong\bR$ on which
$G$ acts minimally, though unfortunately this set may not be unique. Precisely, we have the following.
\begin{thm}[\cite{Navas2011}, Proposition 2.1.12]\label{thm:min-set-r}
Let $G\le \Homeo_+[0,1]$ be a finitely generated group. Then there is a nonempty, closed, invariant subset $C\sse (0,1)$
on which $G$ acts minimally.
Moreover, let $C'$ denote the derived subset of $C$ and let $\partial C$ be the boundary of $C$. Then we have
exactly one of the following conclusions:
\begin{enumerate}[(1)]
\item
We have $C'=\varnothing$. Then, $C$ is a discrete subset of $(0,1)$. If $C$ is finite then it consists of global fixed points of $G$. Otherwise,
$C$ is a bi-infinite sequence of points that accumulates exactly at $\{0,1\}$.
\item
We have $\partial C=\varnothing$. Then $C=(0,1)$ and $G$ acts minimally on $(0,1)$.
\item
We have $C'=\partial C=C$. Then, if $J\sse (0,1)$ is an open interval that is compactly contained in $(0,1)$,
we have that $\overline{J\cap C}$ is a Cantor set. Moreover, there is a continuous, monotone function $h\colon (0,1)\to (0,1)$
and a quotient \[q\colon G\longrightarrow Q\] such that for all $g\in G$, we have \[q(g)\circ h=h\circ g.\] The map $h$ is surjective when
restricted to $C$ and constant on components of $(0,1)\setminus C$, and so $Q$ acts minimally on $(0,1)$.
\end{enumerate}
\end{thm}
As in the discussion surrounding Theorem~\ref{thm:irr-rot-semi}, the map $h$ is called a \emph{semi-conjugacy}\index{semi-conjugacy}.
As in the case of the circle, we say that group actions as in the last conclusion are \emph{exceptional}\index{exceptional minimal set},
and that $C$ is an \emph{exceptional
minimal set}.
\begin{proof}[Proof of Theorem~\ref{thm:min-set-r}]
Let $G$ be generated by a finite set $S$, which is assumed to be symmetric (i.e.~$S$ is closed under taking inverses). If there exists a
global fixed point for $G$ then clearly the conclusion of the theorem is true. Note that the global fixed set of $G$ can have nonempty
interior, and every point in the fixed point set is a closed, minimal, invariant subset. This illustrates the fact that the set $C$ may not be unique.
We assume for the rest of the proof that $G$ has no global fixed points. Let $x\in (0,1)$ is arbitrary, and let $\OO$ be the $G$--orbit of $x$.
We have that $\sup\OO$ and $\inf\OO$ are both global fixed points of $G$ and must therefore
coincide with the set $\{0,1\}$. Indeed, let $y=\sup\OO$,
and let $g\in G$. If $g(y)\neq y$ the $g(y)<y$, which implies that $g^{-1}(y)>y$, a contradiction. This also shows that if $C$ is discrete then
it consists of a bi-infinite sequence of points accumulating at $0$ and $1$.
To apply Zorn's Lemma as in Theorem~\ref{thm:minimal-set}, we build a compact subset $J$ of $(0,1)$ such that the closure of every orbit
meets $J$.
To this end,
if $x\in (0,1)$, let \[y=\max \{s(x)\mid s\in S\}.\] We claim that an arbitrary $G$--orbit $\OO$ must meet the interval $[x,y]$. Indeed, we have
that \[\{\sup\OO,\inf\OO\}=\{0,1\},\] and so there are points $w<x<y<z$ such that $w,z\in\OO$. Let $g(w)=z$. Writing $g$ as a product of
elements of $G$, we have \[g=s_k\cdots s_1.\] We let $\ell$ be the last index so that \[v=s_{\ell}\cdots s_1(w)<x.\] Then we have that
\[x<s_{\ell+1}(v)<y,\] since $s_{\ell+1}(x)\leq y$ and $s_{\ell}$ is order preserving. Note that this is where we use the fact that $S$ is a
symmetric generating set.
Finally, let $J=[x,y]$. We write $\PP$ for the set of nonempty, $G$--invariant subsets of $(0,1)$, and for $C_1,C_2\in\PP$, write $C_1\geq C_2$
if \[C_1\cap J\sse C_2\cap J.\] If $C\in\PP$ is arbitrary, we have that $C\cap J\neq\varnothing$. The finite intersection property again
implies that chains have upper bounds in $\PP$, and hence Zorn's Lemma furnishes a maximal element, on which $G$ must act minimally.
Indeed, otherwise the closure of a $G$--orbit would be a proper closed invariant subset and would violate maximality.
We can now analyze the structure of $C$. Suppose first that $C$ contains a nonempty open interval $K$. The minimality of the $G$--action on
$C$ implies that if $x\in C$ is arbitrary, then there is an element $g\in G$ such that $g(x)\in K$, and so that $G$--invariance of $C$ implies that
a neighborhood of $x$ is contained in $C$. In particular, $\partial C=\varnothing$ and $C= (0,1)$, and so $G$ acts minimally on $(0,1)$.
Thus if $C\neq (0,1)$ then $C$ is totally disconnected. The minimality of the $G$--action on $C$ again implies that either every point of $C$
is isolated, or every point of $C$ is an accumulation point. If every point of $C$ is isolated then $C$ is discrete and accumulates only
at $0$ and $1$. If one point of $C$ is an accumulation point then $C$ has no isolated points and hence is perfect. It follows then that
if $J$ is a nonempty open interval that is compactly contained in $(0,1)$ then $\overline{J\cap C}$ is a Cantor set.
In this last case, let $\mu$ be a $\sigma$--finite, regular, nonatomic Borel measure that is supported on $C$. Then the map
\[h\colon (0,1)\longrightarrow \bR\] given by \[h(x)=\frac{1}{2}+\int_{1/2}^x\,d\mu\] is a continuous map that is a surjection when
restricted to $C$, and which is constant on components of $(0,1)\setminus C$. For $y=h(x)$, we set $\overline{g}(y)=h(g(x))$.
This is well-defined, since $h$ is at most two--to--one, and is one--to--one outside of the set of boundary points of components of
$(0,1)\setminus C$. If $J$ is such a component then so if $g(J)$, and so $G$ preserves fibers of $h$. This establishes the well-definedness
of the action of $\overline{g}$.
Finally, we set $q(g)=\overline{g}$, which defines a quotient $Q$ of $G$.
\end{proof}
It is not difficult to show that $Q$ might be a proper quotient of $G$. Indeed, choose a
finitely generated group $G$ and a $G$--action on $(0,1)$ with an exceptional
minimal set $C$, and let $J$ be a component of $(0,1)\setminus C$. Choose an arbitrary finitely generated minimal $H$--action on $J$,
and propagate it to the $G$--orbit of $J$ be conjugation by $G$. Then $\form{ G,H}$ act on $(0,1)$, and $C$ is still an exceptional
minimal set. Building $h$ as in the proof of Theorem~\ref{thm:min-set-r} furnishes a quotient $Q\cong G$ of $\form{ G,H}$ that
is a proper quotient.
We now return to the subject of chain groups. First, a preliminary definition. Let $X$ be a Hausdorff topological space, and let
$G\le \Homeo(X)$ be a subgroup.
Following~\cite{KKL2019ASENS}, we say that $G$ is \emph{locally CO--transitive}\index{locally CO--transitive}
if for each proper compact $A\sse X$, there exists
a point $x\in X$ such that for an arbitrary open neighborhood $U$ of $x$, we have $g(A)\sse U$ for some $g\in G$.
\begin{lem}[\cite{KKL2019ASENS}, Lemma 3.6]\label{lem:chain-arb-dyn}
Let $G$ be an $m$--chain group, with $m\geq 2$.
\begin{enumerate}[(1)]
\item
There is a point $x\in\supp G$ such that every $G$--orbit accumulates at $x$.
\item
For $g\in G$ and $A\sse \supp G$ nonempty and compact, there is an element $h\in G'$ such that $g$ and $h$ agree as functions on $A$.
\item
Every $G$--orbit is also a $G'$--orbit.
\item
The action of $G'$ is locally CO--transitive.
\end{enumerate}
\end{lem}
\begin{proof}
Write $\{f_1,\ldots,f_m\}$ for generators of $G$, with $\supp f_i=J_i$. Without loss of generality, we may assume that
\[\bigcup_{i=1}^m J_i=(0,1).\]
Clearly, every $G$--orbit accumulates at $x=\sup J_1\in (0,1)$. This establishes the first part of the lemma.
For the second part of the lemma, write \[g=s_k\cdots s_1,\] where for each $i$ we have \[s_i\in\{f_1^{\pm1},\ldots,f_m^{\pm1}\},\]
and let $A\sse (0,1)$ be nonempty and compact. Observe that there exist nonempty open intervals \[U_1=(0,a)\sse J_1\quad
\textrm{and}\quad
U_2=(b,1)\sse J_m\] such that for $1\leq i\leq k$, we have \[(s_i\cdots s_1(A))\cap (U_1\cup U_2)=\varnothing.\] It is a straightforward
exercise, using the connectedness of $\supp G$, to find elements $g_i\in G$ such that $g_i(J_i)\sse U_1\cup U_2$.
Let \[h=\left(\prod_{i=1}^k g_is_i^{-1}g_i^{-1}\right)g.\] It is immediate that $h\in G'$, and an easy calculation shows that $h$ agrees
with $g$ on $A$.
The third part of the lemma follows from the second, letting $A$ be a single point.
For the fourth part, since local transitivity of $G'$ concerns the behavior of $G'$ on compact subsets of $(0,1)$, the second part of the
lemma implies that we may as well prove that $G$ is locally CO--transitive. We let $x=\sup J_1$ as in the first part, and we fix an open
neighborhood $U$ of $x$. Again, if $A\sse (0,1)$ is nonempty and compact then we may find an element of
$g\in G$ such that $\sup A\in J_1$, so that we have $g(A)\sse J_1$. Applying a sufficiently large power of $f_1$ sends $g(A)$ into
$U$, thus completing the proof.
\end{proof}
As a consequence of the preceding discussion, we have the following basic structural result about chain groups.
\begin{thm}[\cite{KKL2019ASENS}, Theorem 3.7]\label{thm:chain-dichotomy}
Let $G$ be an $m$--chain group, for $m\geq 2$. Then exactly one of the following conclusions holds.
\begin{enumerate}[(1)]
\item
The action of $G$ is minimal.
\item
The action of $G$ admits a unique exceptional minimal invariant set.
\end{enumerate}
In the first of these conclusions, we have that the commutator subgroup $G'$ is simple. In the second of these conclusions,
we have that $G$ surjects onto an $m$--chain group that acts minimally.
\end{thm}
\begin{proof}
As before, we assume that $\supp G=(0,1)$, and we write
$\{f_1,\ldots,f_m\}$ for generators of $G$, with $\supp f_i=J_i$. Suppose first that the action of $G$ is not minimal.
Then since $G$ obviously has no global fixed points,
Theorem~\ref{thm:min-set-r} implies that $G$ admits an (\emph{a priori} non--unique) exceptional minimal set $C$.
In the case of an exceptional minimal set, we have from Lemma~\ref{lem:chain-arb-dyn}, there is a point $x\in (0,1)$ that meets the
closure of an arbitrary orbit $\OO$ of $G$. It follows that the closure of the $G$--orbit of $x$ is contained in every
nonempty, closed, $G$--invariant
subset of $(0,1)$. It follows that the exceptional minimal set $C$ is unique. Theorem~\ref{thm:min-set-r} furnishes a quotient $Q$ of
$G$ acting on $(0,1)$ minimally. To complete the analysis of this case, it suffices to show that $Q$ is again an $m$--chain group.
Let $J_i$ and $J_{i+1}$ be two consecutive intervals in the $m$--chain defining $G$. We have that
\[\inf J_i<\inf J_{i+1}<\sup J_i<\sup J_{i+1},\]
by definition. Write \[K_1=(\inf J_i,\inf J_{i+1}),\quad K_2=(\inf J_{i+1},\sup J_i),\quad K_3=(\sup J_i,\sup J_{i+1}).\]
It is a straightforward exercise to show that $C\cap K_{\ell}$ is infinite for $\ell\in \{1,2,3\}$. Since the map $h$ semi-conjugating the action
of $G$ to the action of $Q$ is at most two--to--one, we have that $h(J_i)$ and $h(J_{i+1})$ form a two--chain of nondegenerate intervals.
It follows that $Q$ is indeed an $m$--chain group.
If $G$ is a minimal chain group, then Lemma~\ref{lem:chain-arb-dyn} implies that the action of $G'$ on $(0,1)$ is also minimal and
CO--transitive. Indeed, the CO--transitivity can be shown as follows. If $A\sse (0,1)$ is compact and $x=\sup J_1$, then for each $a<x$,
there is a $g_a\in G$ such that $g_a(A)\sse (a,x)$. Since $G$ act minimally, for a
nonempty open $U$, there is an $h$ such that $h(x)\in U$.
Continuity implies that there is an $a_0$ such that $h((a_0,x])\sse U$. Then, $h\cdot g_{a_0}$ sends $A$ into $U$.
Now, elements of $G'$ have the property that their supports are compactly contained in $(0,1)$, since the germs of the $G$--action at
$0$ and $1$ are abelian groups. By Lemma~\ref{lem:chain-comm}, we have that $G'$ is perfect. Lemma~\ref{lem:co-trans} now implies
that $G'$ is simple.
\end{proof}
Theorem~\ref{thm:chain-dichotomy} is indeed a true dichotomy, at least for $m$--chain groups with $m\geq 3$. Precisely, we have the
following result, which we will not prove here but which the reader may try and prove for themself.
\begin{prop}[See~\cite{KKL2019ASENS}, Proposition 4.8]\label{prop:non-simple}
For $m\geq 3$, there exists an $m$--chain group with a non--simple commutator group.
\end{prop}
Proposition~\ref{prop:non-simple} is obviously false for $m=2$. The construction in the proposition is carried out
by blowing up an orbit, much like in the
construction of a continuous Denjoy counterexample, and ``inserting" a group action in the wandering set. The construction fails when $m=2$
because one needs to build a nontrivial element in the chain group that fixes the whole wandering set, and this is not possible to achieve
with $F$.
\section{The chain group trick and the rank trick}
We now illustrate some algebraic and combinatorial tricks that can be performed in order to embed groups of homeomorphisms into
chain groups. These methods are quite flexible and broadly applicable, and will be useful in our proofs of the existence of finitely generated
groups with exotic regularity properties and which only admit abelian proper quotients.
For this section, it will be convenient to pass from $\Homeo_+[0,1]$ to $\Homeo_+(\bR)$, where of course the latter is isomorphic to the latter
by compactifying the real line with the points $\pm\infty$.
\subsection{Maximal rank and the chain group trick}
If $G$ is a group and $H$ is a subgroup with some interesting properties, we might hope that in homomorphic images of $G$, some relevant
features of $H$ are preserved. Chain groups provide a setting in which to arrange for pairs $(G,H)$ like this.
For instance, if $G$ is a
chain group that acts minimally (either on $(0,1)$ or on $\bR$) and if $H$ lies in the commutator subgroup $G'$, then every nonabelian
quotient of $G$ will contain an isomorphic copy of $H$ by Theorem~\ref{thm:chain-dichotomy}.
We will retain notation from Subsection~\ref{ss:2-chain} and remind the reader that
\[
a(x)=x+1\quad\text{and}\quad
b(x)=
\begin{cases}
x&\text{ if }x\leq 0,\\
2x&\text{ if }0<x< 1,\\
x+1&\text{ if }1\leq x\\
\end{cases}
\]
\begin{lem}[\cite{KKL2019ASENS}, Lemma 4.1]\label{lem:comm-embed}
Let $\gam\le \Homeo_+(0,1)$ be an $n$--generated group for $n\geq 1$, and suppose that \[\gam/\gam'\cong\bZ^n.\] Then the group $\Gamma$
embeds as a subgroup of \[\form{ \gam,a}'\le \Homeo_+(\bR).\]
\end{lem}
\begin{proof}
Choose generators $\{g_1,\ldots,g_n\}$ for $\Gamma$ such that $\gam/\gam'$ is freely generated as an abelian group
by the images of these generators.
Write \[h_i=a^ig_ia^{-1}\in\Homeo_+(i,i+1).\] The homeomorphism $h_i$ is just $g_i$, translated over by $i$. We now set $k_i=g_ih_i^{-1}$,
and we let \[K=\form{ k_1,\ldots,k_n}\le \form{ \gam,a}'.\] We have that $K\cong \gam$.
Indeed, we view $\bZ^n$ as the free abelian group
on $\{g_1,\ldots,g_n\}$, and we have a map
\[\phi\colon \gam\longrightarrow \gam\times\bZ^n,\] given by $g\mapsto (g,g^{-1})$.
Writing $p_i$ for the projections onto the two factors for $i\in\{1,2\}$,
we have $p_i\circ\phi$ is surjective and $p_1\circ\phi$ is an isomorphism. The homeomorphism $h_i$
acts on the interval $(i,i+1)$ by a copy of $\bZ$, and this copy of $\bZ$ is a well-defined quotient of $\Gamma$, given by sending
$g_i$ to $h_i^{-1}$ and sending $g_j$ to the identity for $j\neq i$. It follows that the map \[\psi\colon \gam\longrightarrow K\] is simply identifying
$\phi(g_i)$ with $k_i$, and this map is now clearly an isomorphism.
\end{proof}
Adjoining $a$ to a rather flexible class of groups of homeomorphisms of $\bR$ yields a profusion of chain groups. To this end, write $\DD$
for the set of homeomorphisms $g$ of $\bR$ that have the following property.
\[
g(x)=
\begin{cases}
x&\text{ if }x\leq 0,\\
g(x)\in (x,x+1)&\text{ if }0<x< 1,\\
x+1&\text{ if }1\leq x\\
\end{cases}
\]
The homeomorphism $b$ from Subsection~\ref{ss:2-chain} is an element of $\DD$.
\begin{lem}[\cite{KKL2019ASENS}, Lemma 4.2]\label{lem:dd-chain}
Let $\{g_1,\ldots,g_n\}\sse \DD$. Then the group \[\form{ g_1,\ldots,g_n,a}\le \Homeo_+(\bR)\]
is isomorphic to an $(n+1)$--chain group.
\end{lem}
Before giving a proof of Lemma~\ref{lem:dd-chain}, note that we did not assume that $\{g_1,\ldots,g_n\}$ are distinct homeomorphisms.
In particular, there is no harm in assuming, say, \[g_1=\cdots=g_{n-1}=b.\] Lemma~\ref{lem:dd-chain} immediately implies the following
curious fact, which illustrates the protean nature of chain groups.
\begin{cor}
For all $n\geq 2$, Thompson's group $F$ is isomorphic to an $n$--chain group.
\end{cor}
In fact, much more is true:
\begin{thm}[\cite{KKL2019ASENS}, Theorem 4.7]\label{thm:chain-protean}
Let $n\geq m\geq 2$, and let $G$ be an $m$--chain group. Then $G$ is isomorphic to an $n$--chain group.
\end{thm}
We will not prove Theorem~\ref{thm:chain-protean} as it will not be relevant in the sequel, but an ambitious reader should be able to prove it
for themself without significant difficulty. Another bizarre phenomenon in the theory of chain groups which follows from
Theorem~\ref{thm:chain-protean} concerns the abelianizations of chain groups. Recall that for Thompson's group $F$, we have
$F/F'\cong\bZ^2$, where this isomorphism comes from computing the germs at the endpoints of the interval (or at $\pm\infty$ for an
action on $\bR$). For a general chain group $G$, it is much more difficult to compute the rank of the abelianization of $G$, and even harder
to
compute the full abelianization of $G$. Of course, we have that $G/G'$ surjects to $\bZ^2$, since we can compute the germs of the action
at $\{\inf\supp G,\sup\supp G\}$. However, the abelianization might be somewhat larger. Recall the
Higman--Thompson groups $\{F_n\}_{n\geq 2}$
from Subsection~\ref{ss:2-chain}, which are realizable as $n$--chain groups. From the presentation we gave, one can check easily that
$F_n/F_n'\cong\bZ^n$. Moreover, Theorem~\ref{thm:chain-protean} shows that $F_n$ is isomorphic to an $m$--chain group for all $m\geq n$.
This implies that the abelianization of an $m$--chain group can have rank anywhere from $2$ to $m$. We do not know if the
abelianization of a chain group can have torsion, but it seems not unlikely that they can.
\begin{proof}[Proof of Lemma~\ref{lem:dd-chain}]
Let \[S=\{g_1,\ldots,g_n\},\] and write \[f_0=g_1^{-1}a,\quad f_n=a^{n-1}g_na^{-n+1}.\] By construction, $f_0$ agrees with $a$ near $-\infty$
and with the identity to the right of $1$, and $f_n$ agrees with $a$ at $\infty$ and with the identity to the left of $n-1$.
Write \[f_i=(a^ig_{i+1}^{-1}a^{-i})(a^{i-1}g_ia^{-i+1}),\quad\textrm{for}\,\, 1\leq i\leq n-1.\] Notice that \[\form{ \{f_0,\ldots,f_n\}}=\form{ S,a
}.\]
Observe that for $1\leq i\leq n-1$, we have \[\supp f_i=(i-1,i+1).\] It follows that setting $J_i=\supp f_i$ for $0\leq i\leq n$, we have
that $\{J_0,\ldots,J_n\}$ forms an $(n+1)$--chain. To verify that $\{f_1,\ldots,f_n\}$ generate a chain group, we note that
\[f_{i+1}f_i(i)=i+1.\] Since \[i=\inf J_{i+1}\quad \textrm{and}\quad i+1=\sup J_i,\] this completes the proof.
\end{proof}
The proof of the following lemma is an easy computation, and we leave it as an exercise for the reader.
Recall from Notation~\ref{notation:diff-J} that for an interval $J\sse \bR$ we write
\[\Homeo_J(\bR):=\{ f\in \Homeo_+(\bR)\mid \supp f\sse J\}.\]
\begin{lem}[\cite{KKL2019ASENS}, Lemma 4.4]\label{lem:expansion}
Let $J:=(1/4,1/2)$.
If $f\in\DD$ satisfies $f(J)=(1/2,1)$, then $fg\in\DD$ for all $g\in\Homeo_J(\bR)$.
\end{lem}
We are finally able to give the proof of Theorem~\ref{thm:chain-subgp} we claimed above. We will actually prove a stronger statement:
\begin{thm}\label{thm:chain-subgp-full}
Let $\Gamma$ be a subgroup of $\Homeo_+(\bR)$ generated by $\{f_1,\ldots,f_n\}$, for $n\geq 1$.
\begin{enumerate}[(1)]
\item
The group $\Gamma$ embeds into an $(n+2)$--chain group $G$ such that $G'$ is simple.
\item
If $\supp f_1$ has finitely many components, then we may arrange in addition for $G$ to be an $(n+1)$--chain group.
\end{enumerate}
In either case, if we have that $\gam/\gam'\cong\bZ^n$ then we may arrange for $\Gamma$ to embed in the derived subgroup $G'$.
\end{thm}
\begin{proof}
We still let $J:=(1/4,1/2)$.
Write $S=\{f_1,\ldots,f_n\}$. Without loss of generality, we may assume that $G\le \Homeo_J(\bR)$, as this latter group is isomorphic
to $\Homeo_+(\bR)$.
Note that Lemma~\ref{lem:expansion} shows that $b\cdot S\sse \DD$, whence that group
\[G=\form{ S,b,a}=\form{ b\cdot S,b,a}\] is abstractly isomorphic to a chain group by Lemma~\ref{lem:dd-chain}.
Since the group $\form{ a,b}\cong F$ is topologically conjugate to the usual realization of $F$ as a subgroup of $\PL_+(I)$ and since
the latter group acts minimally on $(0,1)$, we have that $\form{ a,b}$ acts minimally on $\bR$. Therefore, $G$ acts minimally on
$\bR$, and Theorem~\ref{thm:chain-dichotomy} implies that $G'$ is simple. This proves the first assertion of the theorem.
If $f_1$ has finitely many components, then we may conjugate $S$ within the group $\Homeo_J(\bR)$ in order to arrange for
$f_1$ to be piecewise linear with dyadic breakpoints. In particular, in this way we may arrange for $f_1$ to be an element of
$F=\form{ a,b}$. Setting $S_0=S\setminus\{f_1\}$, we set \[G=\form{ S_0,b,a},\] which again is
isomorphic to a $(n+1)$--chain group by Lemma~\ref{lem:dd-chain}, and the action on $\bR$ remains minimal. This proves the
second assertion of the theorem.
For the third assertion, if $\gam/\gam'\cong\bZ^n$ then we may realize \[\gam\le \form{ \gam,a}'\le \form{ \gam,a,b}',\]
by Lemma~\ref{lem:comm-embed}. The conclusion follows immediately.
\end{proof}
We state the following fact, which in~\cite{KK2020crit} is called the chain group trick, and which will be useful in the sequel. We leave
the details of the proof to the reader.
\begin{cor}[The chain group trick]\label{cor:chain-gp-trick}
Let $\gam\le \Homeo_+(\bR)$ be an $m$--generated group such that $\supp\gam$ is compactly contained in $(0,1)$, and let $F_{GS}$
be a smooth realization of Thompson's group $F$ on $[0,1]$ that acts minimally on $(0,1)$, as furnished by
Corollary~\ref{cor:ghys-serg}. Write
\[G=\form{ \gam,F_{GS}}.\] Then the following conclusions hold.
\begin{enumerate}[(1)]
\item
The group $G$ is an $(m+2)$--chain group acting minimally on $(0,1)$, so that $G'$ is simple and every proper quotient of $G$ is abelian.
\item
If $\gam/\gam'$ is free abelian, then there is an embedding of $\Gamma$ into $G'$.
\end{enumerate}
\end{cor}
We remark that in the second part of Corollary~\ref{cor:chain-gp-trick}, we do not assume that $\gam/\gam'$ has rank $m$. The reader will
observe that since $F_{GS}\le \Diff_+^{\infty}([0,1])$, we have that the regularity of $G$ coincides with the regularity of $\Gamma$.
Since $\Gamma$
is compactly supported and since
each element in $F_{GS}$ is tangent to the identity at $\partial I$,
we have that the action of $G$ extends by the identity to $\bR$, without any loss of regularity. By identifying $0$ and $1$, we also obtain an action
of $G$ on $S^1$ without any loss of regularity.
\subsection{The rank trick}\label{ss:rank-trick}
In results such as Theorem~\ref{thm:chain-subgp-full}, and more fundamentally in Lemma~\ref{lem:comm-embed}, the maximal rank
hypothesis on the abelianization can be somewhat annoying. In this subsection, we illustrate a technical tool which will be of use later,
and which helps satisfy maximal rank hypotheses.
\begin{lem}[The rank trick; cf.~\cite{KK2020crit}, Lemma 6.1]\label{lem:rank-trick}
Let $G$ be a group such that $G/G'$ is a finitely generated torsion-free abelian group.
If we have a representation
\[\rho\colon G\longrightarrow \Homeo_c(\bR),\] then there exists another representation
\[\tau\colon G\longrightarrow\form{ \rho(G),\Diff_c^{\infty}(\bR)}\] that has the following properties:
\begin{enumerate}[(1)]
\item
We have $\tau$ agrees with $\rho$ on $G'$;
\item
We have $\tau(G)/(\tau(G))'\cong G/G'$.
\end{enumerate}
\end{lem}
\begin{proof}
Let \[G/G'\cong\bZ^m,\] where $m\geq 1$; note that the case where $G=G'$ is trivial by setting $\rho=\tau$. We choose arbitrary
compactly supported
$C^{\infty}$ diffeomorphisms $\{f_1,\ldots,f_m\}$ such that \[\supp f_i\cap \supp f_j=\varnothing\] for $i\neq j$, and such that
\[\supp f_i\cap\supp\rho(G)=\varnothing\] for all $i$. Observe that there is a natural surjective map \[\phi\colon G\longrightarrow
\form{ f_1,\ldots,f_m}\cong \bZ^m.\] We simply set \[\tau\colon G\longrightarrow \form{ \rho(G),\Diff_+^{\infty}(\bR)},\quad
\tau(g)=\rho(g)\phi(g).\]
It is immediate that $\supp\tau(G)$ has compact closure in $\bR$, and that $\tau$ agrees with $\rho$ on $G'$.
The abelianization of $\tau(G)$ agrees with the abelianization of $G$, since we have a surjective composition of maps
\[G\longrightarrow \tau(G)=\rho(G)\times\form{ f_1,\ldots,f_m}\longrightarrow \form{ f_1,\ldots,f_m}\cong\bZ^m,\]
where the last arrow is just the projection onto the second coordinate. The reader may compare with the proof of
Lemma~\ref{lem:comm-embed} above.
\end{proof}
The reader will note that the hypothesis that ${\suppc \rho(G)}$ is compact can be relaxed; one need only assume that
$\supp \rho(G)$ be bounded from one side, and in the conclusion of the lemma, $\supp\tau(G)$ will also be bounded from the same side.
\chapter{The Slow Progress Lemma}\label{ch:slp}
\begin{abstract}In this chapter, we give a proof of the Slow Progress Lemma. This result, stated as Theorem~\ref{t:slp},
asserts that a certain iteration of smoother group elements makes slower progress of the orbit in the sense of covering distances than
corresponding iterations of less smooth group elements.
This lemma is a key dynamical ingredient of our main result (Theorem~\ref{t:optimal-all}), and also serves as a bridge between
analytic information (i.e. regularity) and dynamical data (i.e. the asymptotics of covering distances) associated to a group action.
The version of the Slow Progress Lemma we present here is significantly stronger and quantitatively more precise than the one
in~\cite{KK2020crit}.\end{abstract}
\section{Statement of the result}
Recall that a \emph{concave modulus of continuity}\index{concave modulus of continuity} is simply a concave homeomorphism of $[0,\infty)$.
Often, a concave modulus is specified only near zero, though
such a local definition can always be replaced by a smooth concave modulus defined on the whole interval $[0,\infty)$;
see Lemma~\ref{lem:medvedev} for more detail.
We say a concave modulus $\beta$ is \emph{sup-tame}\index{sup-tame modulus} if
\[\lim_{t\to 0+} \sup_{x>0} t\beta(x)/\beta(tx)=0.\]
On the other hand, $\beta$ is said to be \emph{sub-tame}\index{sub-tame modulus} if
\[\lim_{t\to 0+} \sup_{x>0} \beta(tx)/\beta(x)=0.\]
Intuitively, a sup-tame modulus is the one that is ``not too small'' near zero;
on the other hand, one may say that a sub-tame modulus is ``sufficiently small''.
For instance, if $\tau\in(0,1)$ then $\beta(x)=x^\tau$ is both sup-and sub-tame.
The Lipschitz modulus $\beta(x)=x$ is the smallest possible modulus;
a Lipschitz continuous function is $\alpha$--continuous for all concave moduli $\alpha$.
One may easily check that the Lipschitz modulus is sub-tame but not sup-tame.
A ``big'' concave modulus such as the one named $\beta$ below is sup-tame but not sub-tame:
\[
\beta(x) =1/\log(1/x).\]
Note that the above definition of $\beta$ makes sense only for small $x>0$.
We say that $(k,\beta)$ is a \emph{tame pair}\index{tame pair},
and write $\beta\succ_k0$,
if $k\in{\bZ_{>0}}\cup\{0\}$
and if $\beta$ is a concave modulus such that one of the following holds:
\begin{itemize}
\item $k\ge2$;
\item $k=1$ and $\beta$ is sub-tame (``sufficiently small'');
\item $k=0$ and $\beta$ is sup-tame (``not too small'').
\end{itemize}
For $k\ne1$, Mather~\cite{Mather1,Mather2} proved the simplicity of the group $\Diff_c^{n+k}(M^n)_0$ of compactly supported $C^{n+k}$
diffeomorphisms of an $n$--manifold $M$ that are isotopic to the identity by compactly supported isotopies. One can actually
extend Mather's arguments to prove that whenever $\beta\succ_k0$, the following group is simple (cf.~\cite{CKK2019}):
\[\Diff_c^{n+k,\beta}(M^n)_0:=\left\{ f\in \Diff_c^{n+k}(M^n)_0\mid f^{(n+k)}\text{ is locally }\beta\text{--continuous}\right\}.\]
Though Mather did not actually used this term, it is interesting to note that the precisely same notion of tame pairs
appear again in quite a different setting as in Theorem~\ref{t:slp} below.
As usual, we let $I$ denote a compact, nondegenerate interval.
For a set $V\sse\Homeo_+(I)$, we will write $\CD_V(x,y)$ for the corresponding covering distance on $I$. That is,
\[
\CD_V(x,y):=\min\left\{m \mid [x,y]\sse J_1\cup \cdots\cup J_m\text{ for some }J_i\in\bigcup_{ v\in V} \pi_0\supp v\right\}.\]
Let $\{a_p\}_p$ and $\{b_p\}_p$ be real sequences indexed by a subset $P$ of $\bN$.
If there exists some $C>0$ such that $a_p\le Cb_p$ for all $p\in P$,
then we write
\[a_p\preccurlyeq b_p.\]
We also add a phrase \emph{for almost all $i\in P$} if
\[
\lim_{N\to\infty}\frac{\#\{ p\in P \mid a_p\le C b_p\}\cap[1,N]}{\#(P\cap[1,N])}=1.\]
We will often consider the case $P=\bN$.
\begin{thm}[Slow Progress Lemma]\label{t:slp}
Let $k\in{\bZ_{>0}}$ and let $\beta$ be a concave modulus such that $\beta\succ_k0$.
Suppose we have a sequence $\{N_i\}_{i\ge1}\sse {\bZ_{>0}}$ satisfying \[N_i\preccurlyeq i^{k-1}/\beta(1/i)\] for almost all $i\ge1$.
Suppose furthermore that $V$ is a finite set such that either \[V\sse \Diff_+^{k,\beta}(I)\quad\textrm{or}\quad
V\sse \Diff_+^{k,\mathrm{bv}}(I),\]
and let $\{v_i\}_{i\ge1}$ be a sequence of elements of $V$.
Then there exists a positive constant $\epsilon$ depending only on $k$ such that
for every point $x_0$ in $I$, we have
\[
\limsup_{i\to\infty} \frac1i \CD_V\left(x_0,v_i^{N_i} \cdots v_2^{N_2} v_1^{N_1}(x_0)\right)\le1-\epsilon.\]
\end{thm}
We will prove that one can set $\epsilon:=1/(4k+2)$.
As a special case of the Slow Progess Lemma, one obtains the same conclusion if $N_i\preccurlyeq i^k$ and if
$V\sse\Diff_+^{k+1}(I)$ or $V\sse\Diff_+^{k+\mathrm{bv}}(I)$, since the hypothesis of the theorem holds for $\beta(x)=x$.
\begin{rem}
In~\cite{KK2020crit}, a more restrictive, qualitative version of the Slow Progress Lemma was established.
Namely, it was previously proved that
\[
\lim_{i\to\infty} \left(i - \CD_V\left(x_0,v_i^{N_i} \cdots v_2^{N_2} v_1^{N_1}(x_0)\right)\right)=\infty,\]
under an additional hypothesis that the natural density of
\[
\{i\in\bZ_{>0}\mid v_i=v\}\]
is well-defined for each $v\in V$. In Theorem~\ref{t:slp}, this additional hypothesis is dropped and the conclusion is strengthened.
\end{rem}
Observe that the hypotheses (and hence also the conclusion) of the Slow Progress Lemma are invariant under topological conjugacy.
Thus, one obtains the following topological non-smoothability criterion for group actions.
\begin{cor}\label{c:slp-0}
Let $k\in{\bZ_{>0}}$ and $\beta\succ_k0$.
Suppose $\{N_i\}_{i\ge1}\sse \bN$ is a sequence such that
\[N_i\preccurlyeq i^{k-1}/\beta(1/i)\]
for almost all $i\ge1$. Assume we have an action
\[
\psi\co G\to\Homeo_+(I)\]
of a group $G$ generated by a finite set $V$,
and a point $x_0\in I$
such that for some sequence
$\{v_i\}_{i\ge1}$ of elements of $V$,
we have
\[\limsup_{i\to\infty} \frac1i {\CD_V\left(x_0,\psi\left(v_i^{N_i} \cdots v_2^{N_2} v_1^{N_1}\right)(x_0)\right)}=1.\]
Then $\psi$ is not topologically conjugate into $\Diff_+^{k,\beta}(I)$ nor into $\Diff_+^{k,\mathrm{bv}}(I)$.
\end{cor}
\section{Natural densities}
For a set $A$, we with write $\#A$ for its cardinality.
\bd\label{d:density}
Let $P\sse \bN$ be a set.
The \emph{upper density}\index{upper density} of $P$ is defined as
\[\overline{d}_\bN(P):=\limsup_{n\to\infty} \#(P\cap [1,n])/n.\]
Similarly, the \emph{lower density}\index{lower density} of $P$ is
\[\underline{d}_\bN(P):=\limsup_{n\to\infty} \#(P\cap [1,n])/n.\]
If $\overline{d}_\bN(P)$ coincides with $\underline{d}_\bN(P)$, this number is called the \emph{natural density}\index{natural density} of $P$.
\ed
The following is well-known~\cite{Salat1964MZ}; a particularly simple proof was given by Moser~\cite{Moser1958}, which we will reproduce
here for the convenience of the reader.
\begin{lem}\label{l:sum-density}
Let $Q\sse\bN$ be a set. If \[\sum_{q\in Q} 1/q<\infty,\] then $d_\bN(Q)=0$.\end{lem}
\begin{proof}
For $n\in\bN$, we write $Q_n=Q\cap [1,n]$. Write \[\chi(n)=\sum_{q\in Q_n} 1,\quad H(n)=\sum_{q\in Q_n}\frac{1}{q},\] with $H(0)=0$ by
convention.
Observe that \[\chi(n)=\sum_{i=1}^n i\cdot (H(i)-H(i-1)).\]
Thus, we have that \[\frac{\chi(n)}{n}=H(n)-\frac{1}{n}\sum_{i=1}^n H(i-1).\] Note that the second term on the right hand sum is the $n^{th}$
Ces\`aro sum of the sequence $\{1/q\}_{q\in Q}$, and so converges to $\lim_{n\to\infty}H(n)$ if this latter
limit exists. If \[\sum_{q\in Q} 1/q<\infty,\]
we obtain $\lim_{n\to\infty}\chi(n)/n=0$, so that $Q$ has density zero.
\end{proof}
A set with a high upper density contains a long sequence of consecutive numbers. More precisely, we have the following.
\begin{lem}\label{l:upper-density}
For each subset $P\sse\bN$ and for each $C\in\bZ_{>0}$, we have that
\[
\overline{d}_\bN\left( P\cap (P-1)\cap\cdots\cap (P-C+1)\right)\ge 1-C\left(1-\overline{d}_\bN(P)\right).\]\
\end{lem}
\bp
Set $\epsilon:=1-\overline{d}_\bN(P)$.
For all $\epsilon'>\epsilon$, there exist infinitely many $N$ such that
\[\#(P\cap[1,N])>(1-\epsilon')N.\]
For such an $N$ we have
\[\#((P-1)\cap [1,N])=\#(P\cap [2,N+1])> (1-\epsilon')N-1.\]
Using the equation $\#(A\cap B)=\#A+ \#B-\#(A\cup B)$, we obtain
\[\#(P\cap (P-1)\cap [1,N])> (1-2\epsilon')N-1.\]
We inductively deduce that
\[\#(P\cap (P-1)\cap \cdots\cap (P-C+1)\cap [1,N]) > (1-C\epsilon')N-(1+2+\cdots+(C-1)).\]
As there are infinitely many such $N$, we have that
\[\overline{d}_\bN\left( P\cap (P-1)\cap\cdots\cap (P-C+1)\right)\ge 1-C\epsilon'.\]
Since $\epsilon'>\epsilon$ was arbitrary, the proof is complete.
\ep
If $\{d_i\}_i$ is a discrete walk in $\bR$ that can either stay put, move forward, or move backward at each step,
then we show below that the limit superior of the average speed $d_n/n$ can be realized by the upper density of the times at which
new ``progress'' is made.
\begin{lem}\label{l:p-almost}
If $\{d_i\}_i$ is a sequence of nonnegative integers such that $|d_{i+1}-d_i|\le 1$ for each $i\ge0$,
then we have that
\[
\overline{d}_\bN\left\{n\in\bN \middle\vert d_n > \max_{0\le m<n} d_m\right\}
= \limsup_{i\to\infty} {d_i}/{i}.\]
\end{lem}
\bp
We may normalize the sequence and assume $d_0=0$. We are only interested in the case when
\[P:=\left\{n\in\bN \mid d_n > \max_{0\le m<n} d_m\right\}\]
is an infinite set, for otherwise the claim is trivial.
Let us enumerate $P$ in increasing order, so that
\[P=\{p_1<p_2<\cdots\}.\] For all sufficiently large $n\gg0$, we can find $j\in\bN$ such that $p_j\le n<p_{j+1}$. Then
\[
\#(P\cap[1,n]) = j = d_{p_j}\ge d_n.\]
Dividing by $n$ and taking limit superiors over $n$, we obtain $\overline{d}_\bN(P)\ge \limsup_n d_n/n$.
To prove the opposite inequality, we may assume $\overline{d}_\bN(P)>0$
and pick an arbitrary $\epsilon'\in(0,\overline{d}_\bN(P))$. For each $p_j\le m<p_{j+1}$,
note that
\[ \#(P\cap[1,m])/m = j/m\le j/p_j.\]
By the definition of an upper density, we can find infinitely many $j$ such that
\[
\epsilon'<j/p_j = d_{p_j}/p_j.\]
It follows that \[ \epsilon'\le \limsup_i \frac{d_i}{i},\] and the proof is complete.
\ep
\begin{rem}
If the sequence $\{d_i\}_i$ above is chosen as a random walk on $\bR$, then by the Law of Iterated Logarithm~\cite{Kolmogorov29},
one has that
$\limsup_i d_i/i=0$ \emph{almost surely}.
\end{rem}
\section{Probabilistic dynamical behavior}
The main idea of the Slow Progress Lemma can be roughly summarized by the phrase that ``smoother diffeomorphisms are slower''. Here,
smoother diffeomorphisms are those of higher regularity. The ``slowness'' of such diffeomorphisms refers to the fact that the
displacement $|f(x)-x|$ is small in a certain
probabilistic sense described in Definition~\ref{d:expansive} and Lemma~\ref{l:prob-dyn} below.
\bd\label{d:k-fixed}
Let $k\in{\bZ_{>0}}$, and let $f\in\Homeo_+(I)$.
We say $f$ is \emph{$k$--fixed}\index{$k$--fixed homeomorphism} on an interval $J\sse I$
if at least one of the following conditions holds:
\begin{itemize}
\item $f$ fixes more than $k$ points in $ J$.
\item $ J$ contains an accumulation point of $\Fix f$.
\end{itemize}
\ed
We will denote the identity function as $\Id(x)=x$. Note
that \[\Id^{(0)}(x)=x,\quad \Id^{(1)}(x)=1,\quad \Id^{(i)}(x)=0\quad \textrm{for $i>1$}.\]
\begin{lem}\label{l:k-fixed}
Let $k\in{\bZ_{>0}}$.
If $f\in\Diff_+^k(I)$ is $k$--fixed on a compact interval $J\sse I$,
then for each $i\in\{0,1,\ldots,k\}$ there exists a point $s_i\in J$ such that
\[
f^{(i)}(s_i)=\Id^{(i)}(s_i).\]
\end{lem}
\bp
Let us first assume that $J$ contains more than $k$ fixed points of $f$. By the Mean Value Theorem (or Rolle's Theorem), one inductively
sees for each $i\in\{0,1,2,\ldots,k\}$ that
\[
\#\left\{ s\in J\mid f^{(i)}(s)=\Id^{(i)}(s)\right\}>k-i.\]
Thus, the proof is complete in this case.
Assume instead that $J$ contains an accumulation point $x_0$ of $\Fix f$.
We may further assume that $x_0\in \partial J$, for otherwise the proof is trivial by the previous paragraph.
As above, for each $i\in\{0,1,\ldots,k\}$, we see that
$x_0$ is an accumulation point of
\[\{ s\in J\mid f^{(i)}(s)=\Id^{(i)}(s)\}.\]
So, we may set $s_i=x_0$ for all $i\le k$.
\ep
For two subsets $A$ and $B$ of $\bR$, we let \[d(A,B)=\inf_{a\in A,b\in B} d(a,b)\] denote the distance between $A$ and $B$.
\bd\label{d:expansive}
Let $f\co J\longrightarrow J$ be a homeomorphism of an interval $J$.
We say that $f$ is \emph{$\delta$--expansive on $J$}\index{$\delta$--expansive homeomorphism} for some $\delta>0$ if
\[\sup_{x\in \Int J} \frac{|f(x)-x|}{d\left(\{x,f(x)\},\partial J\right)}\ge \delta.\]
We say that $f\co J\longrightarrow J$ is \emph{at most $\delta$--expansive on $J$} if
\[\sup_{x\in \Int J} \frac{|f(x)-x|}{d\left(\{x,f(x)\},\partial J\right)}\le \delta.\]
\ed
For instance, if $f$ acts on $J=[a,b]$ such that some $x\in (a,b)$ satisfies
\[\frac{f(x)-x}{x-a}\ge\delta,\]
then $f$ is $\delta$--expansive on $J$.
Note that the inverse of a $\delta$--expansive homeomorphism is also $\delta$--expansive.
For a concave modulus $\beta$ and a $C^\beta$--continuous map $f$,
we define its $\beta$--norm as
\[
[f]_\beta:=\sup_{x\ne y}\frac{|f(x)-f(y)|}{\beta\left(|x-y|\right)}.\]
\begin{lem}\label{l:expansive}
Let $k\in{\bZ_{>0}}$, and let $f\in \Diff_+^k(I)$.
If $f$ is $k$--fixed
on some compact interval $J\sse I$,
then $f$ is at most $\delta_1$--expansive on $J$ for
\[\delta_1:= |J|^{k-1} \left(\sup_{x\in J} \left|f^{(k)}(x) - \Id^{(k)}(x)\right|+\sup_{x\in J} \left|\left(f^{-1}\right)^{(k)}(x) - \Id^{(k)}(x)\right|\right).\]
If we further assume that $f\in \Diff_+^{k,\beta}(I)$ for some concave modulus $\beta$,
then $f$ is at most $\delta_2$--expansive on $J$ for
\[\delta_2:= |J|^{k-1}\beta(|J|)\left(\left[f^{(k)}\right]_\beta+\left[\left(f^{-1}\right)^{(k)}\right]_\beta\right).\]
\end{lem}
\bp
By Lemma~\ref{l:k-fixed}, we have some $s_i\in J$ for each $i\in\{0,1,\ldots,k\}$ such that $f^{(i)}(s_i)=\Id^{(i)}(s_i)$.
Let $x\in \Int J=(a,b)$. We compute that
\begin{align*}
f(x) - x &=\int_{s_0}^x (f'(t_1)-1)dt_1= \int_{s_0}^x \int_{s_1}^{t_1} f''(t_2)dt_2\;dt_1=\cdots\\
&=\int_{s_0}^{x}\int_{s_1}^{t_1}\cdots\int_{s_{k-1}}^{t_{k-1}} \left(f^{(k)}(t_k)-\Id^{(k)}(t_k)\right)\; dt_k\cdots dt_1.
\end{align*}
It follows that
\[ |f(x)-x| \le \sup_{t\in J}\left|f^{(k)}(t)-\Id^{(k)}(t)\right| \cdot |x-s_0|\cdot |J|^{k-1} \le \delta_1 |x-s_0|.\]
After choosing $s_0$ to be either $a$ or $b$, and applying the same argument to $f^{-1}$ instead of
$f$, we conclude that $f$ is at most $\delta_1$--expansive.
A similar argument shows that
\begin{align*}
|f(x) - x| &\le \int_{s_0}^{x}\int_{s_1}^{t_1}\cdots\int_{s_{k-1}}^{t_{k-1}} \left|f^{(k)}(t_k)-f^{(k)}(s_k)\right|\; dt_k\cdots dt_1\\
&\le \left[f^{(k)}\right]_\beta\cdot\beta(|J|) \cdot |x-s_0|\cdot |J|^{k-1}\le \delta_2 |x-s_0|,
\end{align*}
which implies the second required inequality.
\ep
Let us also note a simple consequence of the above proof for a later use (Section~\ref{sss:tsuboi}).
\begin{lem}\label{l:expansive2}
Let $k\in{\bZ_{>0}}$, and let $f\in \Diff_+^k(I)$.
If $f$ is $k$--fixed on some compact interval $J\sse I$,
then we have
\[
\sup_{x\in J} |f'(x)-1|
\le
\sup_{t\in J}\left|f^{(k)}(t)-\Id^{(k)}(t)\right|
\cdot |J|^{k-1}.\]
\end{lem}
\bp
Continuing to use the notation as above, we have
\[ |f'(x)-1| \le \sup_{t\in J}\left|f^{(k)}(t)-\Id^{(k)}(t)\right| \cdot |x-s_1|\cdot |J|^{k-2},\]
which implies the conclusion.
\ep
\begin{lem}\label{l:N-delta-exp}
Suppose that $f\in\Homeo_+(I)$ preserves a compact interval $J\sse I$.
If $f^N$ is $\delta$--expansive on $J$ for some $N\in\bN$ and $\delta>0$, then $f$ is $\frac1N\log(1+\delta)$--expansive on $J$.
\end{lem}
\bp
Let us write $J=[a,b]$. We can find some $x_0\in(a,b)$ such that either
\[
\delta\le \frac{f^N(x_0)-x_0}{x_0-a}
\]
or
\[
\delta\le \frac{f^N(x_0)-x_0}{b-f^N(x_0)},
\]
possibly after switching $f$ with $f^{-1}$. We may further assume to have the former of the above two cases,
as the latter case can be treated similarly.
Then we have
\[
1+\delta \le \frac{f^N(x_0)-a}{x_0-a}=\prod_{i=0}^{N-1} \frac{f^{i+1}(x_0)-a}{f^{i}(x_0)-a}.\]
Hence, for some $x_1:=f^{i}(x_0)$ we have that
\[
(1+\delta)^{1/N}\le \frac{f(x_1)-a}{x_1-a}.\]
We have an estiamte
\[
\frac{f(x_1)-x_1}{x_1-a}= \frac{f(x_1)-a}{x_1-a}-1 \ge(1+\delta)^{1/N}-1 \ge \frac{\log(1+\delta)}{N},\]
which implies the conclusion.
\ep
\begin{lem}\label{l:decomposition}
Every collection of intervals in the real line with intersection multiplicity $K>0$ admits a partition into $K$ subcollections,
each of which consists of disjoint intervals.
\end{lem}
Here, we recall that the \emph{intersection multiplicity}\index{intersection multiplicity} of a collection $X$ of intervals
refers to the maximum number of intervals in $X$ that can contain a given point.
We thank Andreas Holmsen for teaching us the following argument.
\bp[Proof of Lemma~\ref{l:decomposition}]
Let $X$ be the given collection of intervals, and let $\Gamma$ be its \emph{intersection graph}\index{intersection graph};
that is, the vertices of $\Gamma$
correspond to intervals in $X$,
and two vertices are adjacent if the two corresponding intervals have nonempty intersection.
Helly's Theorem (for $\bR$) says that
if $Y$ is a finite collection of intervals in $\bR$ such that every pair of elements of $Y$ intersect, then \[\bigcap_{J\in Y} J\neq\varnothing.\] We leave
the justification of Helly's Theorem as an exercise for the reader.
By Helly's theorem for $\bR$, we see that $K$ is the maximal size of a clique in $\Gamma$.
It now suffices for us to show that the chromatic number $K'$ of $\Gamma$ coincides with $K$.
It is obvious that $K'\ge K$.
It is also well-known that the intersection graph of finitely many intervals is \emph{chordal}\index{chordal graph} (i.e.~every cycle of
length $4$ or more admits a chord), and hence
\emph{perfect}~\cite{golumbic2004}\index{perfect graph}; this means that $K=K'$ when $X$ is finite.
So, we may assume $X$ is infinite. By the De Brujin--Erd\"os Theorem (\cite{dBE1951}, cf.~Proposition~\ref{prop:db-e} below),
we can find a finite subcollection
$X'$ of $X$ such that the intersection graph $\Gamma'$ of $X'$ has the same chromatic number $K'$.
Also, the maximal size of a clique in $\Gamma'$ is at most $K$. By the result for the case when $X$ is finite,
we conclude that $K'\le K$.
\ep
For the convenience of the reader, we give a quick proof of the De Brujin--Erd\"os Theorem,
originally given by Gottschalk~\cite{Gottschalk51}.
\begin{prop}\label{prop:db-e}
Let $\Gamma$ be a graph, all of whose finite subgraphs are $k$--colorable. Then $\Gamma$ is itself $k$--colorable.
\end{prop}
\begin{proof}
Let $X=k^{V(\gam)}$ be the space of all assignments of $k$ colors to the vertices of $\Gamma$, where $k=\{0,1,\ldots,k-1\}$
is given the discrete topology.
Then $X$ is compact by Tychonoff's Theorem. Valid colorings of finite subgraphs of $\Gamma$ are easily seen to be closed
subsets $X$.
If $F_1$ and $F_2$ are finite subgraphs of $\Gamma$ and if $X_{F_1}$ and $X_{F_2}$ are closed subsets of $X$ corresponding
to valid $k$--colorings of $F_1$ and $F_2$ respectively, then $X_{F_1}\cap X_{F_2}$ corresponds to valid $k$--colorings of $F_1\cup F_2$.
Moreover, this intersection is nonempty
by hypothesis. The finite intersection property characterization of compactness shows that if every finite subgraph of $\Gamma$ admits a valid
coloring, then the intersection of the corresponding closed subsets of $X$ is nonempty and consists precisely of valid $k$--colorings of $\Gamma$.
\end{proof}
\begin{lem}\label{l:mod-compare}
Let $k\in{\bZ_{>0}}$ and let $\beta$ be a concave modulus that satisfies $\beta\succ_k0$.
Suppose $P$ is a subset of $\bN$.
If two positive sequences $\{a_p\}_{p\in P}$ and $\{b_p\}_{p\in P}$ satisfy
\[a_p^{k-1}\beta(a_p)\preccurlyeq b_p^{k-1}\beta(b_p),\]
then we have that $a_p\preccurlyeq b_p $.
\end{lem}
\bp
Suppose not. We can find an increasing sequence $\{p_i\}$ in $P$ such that the sequence
$t_{p_i}:=b_{p_i}/a_{p_i}$ converges to zero.
Since $(k,\beta)$ is a tame pair, we obtain that
\[
\frac{b_{p_i}^{k-1}\beta\left(b_{p_i}\right)}
{a_{p_i}^{k-1}\beta\left(a_{p_i}\right)}
=
t_{p_i}^{k-1}\cdot\frac{\beta\left(t_{p_i}a_{p_i}\right)}
{\beta\left(a_{p_i}\right)}\to 0.
\]
This contradicts the given hypothesis.
\ep
We can now prove the main result of this section.
\begin{lem}\label{l:prob-dyn}
Let $k\in{\bZ_{>0}}$ and let $\beta$ be a concave modulus that satisfies $\beta\succ_k0$.
Let $P\sse \bN$.
Suppose $V\sse\Homeo_+(I)$ is a finite set.
Assume that for each $p\in P$ we are given a natural number $N_p\in \bN$, an element $v_p\in V$, and a compact interval $J_p$
such that the following hold:
\be[(i)]
\item
$N_p\preccurlyeq p^{k-1}/\beta(1/p)$ for $p\in P$;
\item $v_p$ is $k$--fixed on $J_p$;
\item\label{p:fim} The collection $\{J_p\mid p\in P\}$ has finite intersection multiplicity.
\ee
If either $V\sse \Diff_+^{k,\beta}(I)$ or $V\sse\Diff_+^{k,\mathrm{bv}}(I)$,
then for all $\delta>0$ we have that
\[
d_\bN\left\{ p\in P\mid v_p^{N_p}\text{ is }\delta\text{--expansive on }J_p\right\}=0.\]
\end{lem}
\bp
By the condition (\ref{p:fim}) above and by Lemma~\ref{l:decomposition}, the collection $\{J_p\mid p\in P\}$ can be partitioned into $K$
subcollections of disjoint intervals for some $K>0$. In particular, \[\sum_p |J_p|\le K|I|.\]
Let $Q\sse P$ be the set the density of which is claimed to be zero.
For each $q\in Q$, Lemma~\ref{l:N-delta-exp} implies that $v_q$ is $\frac{\log(1+\delta)}{N_q}$--expansive on $J_q$.
{\bf Case 1: $V\sse\Diff_+^{k,\beta}(I)$.}
We obviously have that
\[
\max_{v\in V} \left(\left[ v^{(k)}\right]_\beta +\left[ \left(v^{-1}\right)^{(k)}\right]_\beta
\right)<\infty.\]
By Lemma~\ref{l:expansive}, we have for $q\in Q$ that
\[
\left(\frac1q\right)^{k-1}\beta\left(\frac1q\right)
\preccurlyeq \frac{\log(1+\delta)}{N_q}
\preccurlyeq|J_q|^{k-1}\beta(|J_q|).\]
From Lemma~\ref{l:mod-compare} we see that
\[\frac{1}{q} \preccurlyeq |J_q|,\]
and that
\[
\sum_{q\in Q}1/q<C\sum_{q\in Q} |J_q|
\le CK|I|<\infty\]
for some $C>0$.
Lemma~\ref{l:sum-density} implies that $d_\bN(Q)=0$.
{\bf Case 2: $V\sse\Diff_+^{k,\mathrm{bv}}(I)$.}
Note from the concavity that \[\frac{x}{\beta(x)}\le \frac{1}{\beta(1)}\] for $x\le 1$. It follows for $q\in Q$ that
\[
\frac1{q^k}
\preccurlyeq \frac{\beta(1/q)}{q^{k-1}} \preccurlyeq \frac1{N_q}.\]
Now, let $q\in Q$.
By Lemmas~\ref{l:k-fixed} and~\ref{l:expansive}, we have that
\begin{align*}
\frac{\log(1+\delta)}{N_q}&
\le |J_q|^{k-1}
\left(\sup_{x\in J_q} \left|v_q^{(k)}(x) - \Id^{(k)}(x)\right|+\sup_{x\in J_q} \left|\left(v_q^{-1}\right)^{(k)}(x) - \Id^{(k)}(x)\right|\right)\\
&\le
|J_q|^{k-1} \left(
\operatorname{Var}\left(v_q^{(k)};J_q\right)+\operatorname{Var}\left(\left(v_q^{-1}\right)^{(k)};J_q\right)\right).
\end{align*}
By H\"older's inequality, we have some $C,C'>0$ such that
\begin{align*}
\sum_{q\in Q}\frac1q
&\le
\sum_{q\in Q}\frac{C}{N_q^{1/k}}
\le C'
\sum_{q\in Q} |J_q|^{1-1/k} \left(\operatorname{Var}\left(v_q^{(k)};J_q\right)+
\operatorname{Var}\left(\left(v_q^{-1}\right)^{(k)};J_q\right)\right)^{1/k}\\
&\le C'
\left(\sum_{q\in Q} |J_q|\right)^{1-1/k}
\left(\sum_{q\in Q} \operatorname{Var}\left(v_q^{(k)};J_q\right)+
\sum_{q\in Q} \operatorname{Var}\left(\left(v_q^{-1}\right)^{(k)};J_q\right)\right)^{1/k}\\
&\le C'
\left(K\cdot |I|\right)^{1-1/k}
\left(
K\cdot
\sum_{v\in V\cup V^{-1}}\operatorname{Var}\left(v^{(k)};I\right)
\right)^{1/k}<\infty.
\end{align*}
It follows again from Lemma~\ref{l:sum-density} that $d_\bN(Q)=0$.
\ep
\section{Proof of the Slow Progress Lemma}
We let $k,\beta,V,\{v_i\}$ and $\{N_i\}$ be as in the hypotheses of the Slow Progress Lemma,
such that one of the following holds.
\be[(i)]
\item $V\sse\Diff_+^{k,\beta}(I)$;
\item $V\sse \Diff_+^{k,\mathrm{bv}}(I)$.
\ee
Fix $x_0\in I$, and define
\[x_n:=v_n^{N_n}\cdots v_1^{N_1}(x_0).\]
We also define
\[P:=\left\{ n\in\bN\mid \CD_V(x_0,x_{n})>\max_{0\le m< n}\CD_V(x_0,x_m)\right\}.\]
By Lemma~\ref{l:p-almost}, we have that
\[\limsup_{n\to\infty} \frac{\CD_V(x_0,x_n)}n=\overline{d}_\bN(P)=:1-\epsilon\]
for some $\epsilon\in[0,1]$. It suffices for us to show that $\epsilon\ge 1/(4k+2)$.
We may assume $\epsilon\in[0,1)$ and $P$ is infinite. By symmetry that we can also assume that the map
$p\mapsto x_p$ is order--preserving (instead of order--reversing) for $p\in P$.
Pick $p_0\gg0$ so that $\CD_V(x_0,x_p)>2k+1$ for all $p\in P\cap [p_0,\infty)$.
For such a $p$, we let $J_p$ be the unique connected component of
$\supp v_p$
containing $[x_{p-1},x_p]$.
We have at least $k$ fixed points of $v_p$ in $(x_0,x_p)$. Indeed, if
\[ r:=\#(\Fix v_p\cap (x_0,x_p))\]
for some $r<k$, then $(x_0,x_p)$ minus those $r$ fixed points can be covered by $r+1$ intervals in $\pi_0\supp v_p$.
Combining these $r+1$ intervals with $r+2$ intervals from \[\bigcup_{v\in V}\pi_0\supp v\] that cover
$\{x_0, x_p\}$ and the $r$ fixed points, we would have a contradiction that
\[\CD_V(x_0,x_p)\le 2r+3\le 2k+1.\]
From the previous paragraph, we can define for each $p\ge p_0$ in $P$ the smallest compact intervals
$L_p$ and $R_p$ of the form $[x, \sup J_p]$ and $[\inf J_p,y]$ respectively, such that $v_p$ is $k$--fixed on $L_p$ and on $R_p$.
For each $\delta>0$ and $C\in\bN$, we set
\begin{align*}
P_\delta&:=\left\{p\in P\cap(p_0,\infty)\middle\vert v_p^{N_p}\text{ is at most }\delta\text{--expansive on }L_p \text{ and }R_p\right\},\\
P_{\delta,C}&:=\bigcap_{0\le i\le C-1} (P_\delta-i).
\end{align*}
By Lemmas~\ref{l:upper-density} and~\ref{l:prob-dyn}, we have that
\[
\overline{d}_\bN(P_{\delta,C}) \ge 1- C(1-\overline{d}_\bN(P_\delta))=1-C\epsilon.\]
Set $C:=4k+2$ and $\delta:=1/C$.
Assume for contradiction that \[\epsilon<\frac{1}{4k+2}=\frac{1}{C}.\] Then $P_{\delta,C}$ is nonempty and there exists some $p\in P_{\delta,C}$.
For each $i=-1,0,\ldots,C-1$, we have that
\[ \CD_V(x_0,x_{p+i})=\CD_V(x_0,x_p)+i.\]
Let $0\le i\le C/2-1=2k$. Since
\[\CD_V(x_{p+i},\sup R_{p+i})\le 2k<\CD_V(x_{p+i},x_{p+i+2k+1}),\] we have that
\[
\sup R_{p+i}<x_{p+i+2k+1}\le x_{p+C-1}.\]
Since $v_{p+i}^{N_{p+i}}$ is at most $\delta$--expansive on $R_{p+i}$ we see
\[ x_{p+i}-x_{p+i-1}\le \delta (\sup R_{p+i}-x_{p+i})< \delta (x_{p+C-1} - x_p).\]
Similarly for $C/2\le i\le C-1$, we have that
\[
\inf L_{p+i}>x_{p+i-1-2k-1}\ge x_{p-1},\]
and that
\[ x_{p+i}-x_{p+i-1} \le \delta (x_{p+i-1}-\inf L_{p+i})
< \delta (x_{p+C-2} - x_{p-1}).\]
Summing up, we obtain
\[x_{p+C-1}-x_{p-1} =\sum_{i=0}^{C-1} (x_{p+i}-x_{p+i-1})< C\delta (x_{p+C-1}-x_{p-1}).\]
We thus obtain a contradiction, since $C\delta=1$.
This completes the proof of the Slow Progress Lemma.
Let us now record a consequence of the slow progress lemma that will be useful for us later.
\begin{cor}\label{c:slp}
Let $k,\beta$ and $\{N_i\}$ be as in the hypothesis of the Slow Progress Lemma.
We let $G$ be a group with a finite generating set $V$,
and let
\[
\psi\co G\to \Homeo_+(I)\]
be a representation which is topologically conjugate either into
$ \Diff_+^{k,\beta}(I)$ or $\Diff_+^{k,\mathrm{bv}}(I)$.
Pick a sequence
$\{v_i\}_{i\ge1}$ of elements
from $V$ and set
\[w_i := v_i^{N_i}\cdots v_1^{N_1}.\]
If $K\sse\supp\psi$ is a compact interval,
then for all sufficiently large $i\ge1$
we have that
\[
\CL(\psi(w_i)K)<2i.\]
\end{cor}
\bp
The conclusion is only concerned with the action $\psi$ restricted
to the component of $\supp\psi$ containing $K$.
So, for brevity we may assume that $I=[0,1]$ and $\supp\psi=(0,1)$.
We set $K=[x,y]$.
We simply write $\CD$ and $\CL$ for the covering distance and the covering length defined by the set $\psi(V)$.
Put $T:=\CD(x,y)>0$.
By the Slow Progress Lemma, for all sufficiently large $i$
the covering distances
$\CD(x,\psi(w_i)x)$
and $\CD(y,\psi(w_i)y)$ will be smaller than $i-T$.
So, we have that
\[
\CD(\psi(w_i)x,\psi(w_i)y)< 2(i-T)+\CD(x,y)=2i-T<2i,\] as required.
\ep
\chapter{Algebraic obstructions for general regularities}\label{ch:optimal}
\begin{abstract}
The main goal of this chapter is to construct finitely generated groups of $C^{k,\alpha}$ diffeomorphisms of a compact
one--manifold that cannot be embedded into the group of $C^{k,\beta}$ diffeomorphisms for $\beta$ ``sufficiently smaller'' than $\alpha$.
There are two crucial ingredients for such a construction. One is the Slow Progress Lemma from the previous chapter,
which roughly asserts that for finitely generated group actions by diffeomorphisms, the smoother the action the slower it
expands covering lengths of an interval.
The other is an explicit construction of a finitely generated $C^{k,\alpha}$ diffeomorphism group that expands covering
lengths faster than the rate allowed by the Slow Progress Lemma for $C^{k,\beta}$ diffeomorphism groups.
We will establish a result for this second ingredient in the case of a compact interval, and generalize the result to circles using
the material from preceding chapters.
We conclude the chapter with consequences regarding H\"older regularities $C^r$ for $r\in[1,\infty)$. \end{abstract}
\section{Statement of the results}\label{s:statement}
For convenience of notation, we will write
$\CM$ for the set of all concave moduli.
Recall that we write $\beta\succ_k0$ for $\beta\in\CM$ if either $k\ge2$, or $k=1$ and $\beta$ is sub-tame.
For a manifold $M$, we will often use the notation
\[
\Diff_+^{k,\beta'}(M)\]
with $\beta'\in\{\beta,\mathrm{bv}\}$. This group
is either $\Diff_+^{k,\beta}(M)$ or $\Diff_+^{k,\mathrm{bv}}(M)$,
depending on the choice of $\beta'$.
The precise goal of this chapter is to establish the following theorem.
\begin{thm}\label{t:optimal-all}
If $k\in{\bZ_{>0}}$ and $\alpha,\beta\in\CM$ satisfy
that $\beta\succ_k0$, and that
\[
\int_0^1 \frac1x\left(\frac{\beta(x)}{\alpha(x)}\right)^{1/k}dx<\infty,\]
then there exists a finitely generated nonabelian group
\[ R=R(k,\alpha,\beta)\le\Diff_+^{k,\alpha}(I)\]
such that
every homomorphism
\[
[R,R]\longrightarrow\Diff_+^{k,\beta'}(M^1)\]
is trivial for $\beta'\in\{\beta,\mathrm{bv}\}$ and for $M^1\in\{I,S^1\}$.
Moreover, $R$ can be chosen so that $[R,R]$ is simple and every proper quotient of $R$ is abelian.
\end{thm}
By the last condition, we see that every homomorphism from $R$
to $\Diff_+^{k,\beta'}(M^1)$ has abelian image.
\begin{rem}
We note that the conclusion regarding
$\Diff_+^{k,\mathrm{bv}}(M^1)$
does not involve the modulus $\beta(x)$.
Indeed, to obtain the desired conclusion, it suffices to pick $\beta(x)$ as small as possible; that is,
we may assume $\beta(x)=x$ and verify the hypothesis of the theorem.
\end{rem}
\begin{exmp}\label{ex:hoelder-critical}
If we consider the H\"older moduli
\[
\alpha(x)=x^r,\quad\beta(x)=x^s\]
for some $0<r<s\le1$, then the hypothesis of Theorem~\ref{t:optimal-all} is satisfied for all $k\ge1$.
We thus obtain a finitely generated subgroup of $\Diff_+^{k,r}(I)$ that never embeds into $\Diff_+^{k,s}(I)$.\end{exmp}
We will actually deduce a much stronger consequence than Example~\ref{ex:hoelder-critical} regarding $\Diff_+^{k,r}(M)$. See Section~\ref{s:hoelder} for further discussion.
\begin{cor}\label{c:optimal-all}
For each real number $r\ge1$ we have the following.
\be[(1)]
\item
There exists a finitely generated nonabelian group $G_r\le\Diff_+^r(I)$ such that every homomorphism
\[ G_r\longrightarrow \bigcup_{s>r} \Diff_+^s(M^1)\]
has an abelian image for $M^1\in\{I,S^1\}$.
\item
There exists a finitely generated nonabelian group \[H_r\le\bigcup_{s<r}\Diff_+^s(I)\] such that every homomorphism
\[
H_r\longrightarrow \Diff_+^r(M^1)\] has abelian image for $M^1\in\{I,S^1\}$.
\ee
Furthermore, we can require that $[G_r,G_r]$ and $[H_r,H_r]$ are simple.
\end{cor}
We will often write
\[\log^s(x):=\left(\log x\right)^s.\]
The groups $R(k,\alpha,\beta)$ in Theorem~\ref{t:optimal-all} will be chosen from a family of representations $\phi(k,\alpha,\{\ell_i\})$
of a fixed group
\[
G^\dagger:=(\bZ\times\BS(1,2))\ast F_2,\] for a suitably chosen countable set of parameters $\{\ell_i\}$.
More precisely, we will consider a set of intervals whose lengths $\{\ell_i\}$ satisfy the following mild restrictions.
\bd\label{d:tame}
A positive real sequence $\{\ell_i\}_{i\ge1}$ is said to be (a sequence of) \emph{admissible lengths}\index{admissible lengths}
if \[\sum_i \ell_i <\infty\]
and if
\[
0< \inf \ell_{i+1}/\ell_i \le \sup \ell_{i+1}/\ell_i <\infty.\]
\ed
The reader may find it useful to keep in mind the following concrete examples:
\begin{itemize}
\item $\ell_i = 1/\left({i\log^2i}\right)$;
\item $\ell_i = 1/\left(i\log i (\log\log i)^{1+\epsilon}\right)$ for $\epsilon>0$.
\end{itemize}
We have $\lim_i \ell_{i+1}/\ell_i=1$ in both of the cases.
As we have noted in the introduction of this chapter, the main content of the proof for Theorem~\ref{t:optimal-all}
is the following construction of an ``optimally expanding'' diffeomorphism group.
\begin{thm}\label{t:optimal-group}
Let $k\in{\bZ_{>0}}$, let $\alpha\in\CM$, and let $\{\ell_i\}$ be admissible lengths.
Then there exists a representation
\[\phi=\phi(k,\alpha,\{\ell_i\})\co G^\dagger\longrightarrow\Diff_+^{k,\alpha}(I)\]
satisfying the following:
\begin{itemize}
\item[]
If a concave modulus $\beta\succ_k0$ satisfies
\[\beta(1/i) \cdot(1/i)^k\preccurlyeq \alpha(1/i)\cdot \ell_i^k\]
for almost all $i\ge1$,
and if $\beta'\in\{\beta,\mathrm{bv}\}$,
then for every representation of the form
\[
\psi\co G^\dagger \longrightarrow \Diff_+^{k,\beta'}(I),\]
we have that
\[
[G^\dagger,G^\dagger]\cap(\ker\psi\setminus\ker\phi)\ne\varnothing.\]
\end{itemize}
\end{thm}
\begin{rem}
The main result of~\cite{KK2020crit} asserts the conclusion of Theorem~\ref{t:optimal-group} under the hypothesis that
\[ \lim_{x\to+0} \frac{\beta(x)\log^K (1/x)}{\alpha(x)}=0\]
for all $K>0$.
This hypothesis obviously implies\[\beta(1/i) \cdot (1/i)^k \preccurlyeq \alpha(1/i) \cdot \ell_i^k\]after choosing admissible lengths
of the form $\ell_i := \frac1{i\log^2 i}$. That is, Theorem~\ref{t:optimal-group} recovers the main result of~\cite{KK2020crit}.
\end{rem}
\begin{rem}\label{r:non-Lipschitz}
If $\alpha$ is the Lipschitz modulus, i.e. if $\alpha(x)=x$
then the theorem is vacuous.
Indeed, from $\sum_i \ell_i<\infty$, we have that $\frac1{i\ell_i}$ is unbounded.
If $\beta$ satisfies the hypothesis, then
\[
\liminf_{x\to+0} \frac{\beta(x)}{x}
\le \liminf_{i\to\infty} \frac{\beta(1/i)}{1/i}
\le C\liminf_{i\to\infty} (i\ell_i)^k=0.\]
As $\beta(x)/x$ is monotone decreasing we have that $\beta(x)\equiv0$, which is a contradiction.
\end{rem}
The majority of this chapter is devoted to the proof of Theorem~\ref{t:optimal-group}.
We will then establish Theorem~\ref{t:optimal-all} in Section~\ref{s:circle}
by applying the Rank Trick and the Chain Group Trick from Chapter~\ref{ch:chain-groups}.
\section{Strategy for the proof of Theorem~\ref{t:optimal-group}}\label{s:strategy}
The proof of Theorem~\ref{t:optimal-group} involves results from several different parts of this book
and leverages somewhat complicated interactions between group theoretic, dynamical and analytic features.
Lemma~\ref{l:covering-translation-distance} relates syllable lengths (group theory) with covering distances (dynamics).
The Slow Progress Lemma is a bridge between the growth of covering distances (dynamics) and regularity (analysis).
In this section, we will collect essential consequences of the three main tools, stated for topological actions for ease of
understanding, and explain
how the proof of Theorem~\ref{t:optimal-group} will proceed.
We hope that this will allow the reader to grasp the main idea behind the proof before we embark on explicating the details.
To begin, suppose that $G$ is a group with a finite generating set $V$.
We will assume that $G$ contains a ``universally compactly supported element'' $u_0$ in the following sense.
The reader will recall that we have seen an instance of such a group in Corollary~\ref{c:bs12}.
\be[(A)]
\setcounter{enumi}{0}
\item\label{p:ucsd} \emph{There exists a nontrivial element $u_0\in [G,G]$
that is compactly supported under every action $\rho\co G\longrightarrow\Homeo_+(I)$. }
\ee
Assume now that we are given two actions
\[\phi,\psi\co G\longrightarrow \Homeo_+(I).\]
Let us denote the covering lengths corresponding to $\phi(V)$ and $\psi(V)$ as
$\CL_\phi$ and $\CL_\psi$, respectively.
We will require from the construction of $\phi$ that
\be[(A)]
\setcounter{enumi}{1}
\item\label{p:nontrivial} \emph{$\supp\phi$ is connected, and $\phi(u_0)$ is nontrivial.}
\ee
We will also make the following assumption,
which is true for instance when $\phi(G)\le\Diff_+^\infty(\Int I)$ as illustrated in Proposition~\ref{prop:disj-ab}.
\be[(A)]
\setcounter{enumi}{2}
\item\label{p:noncommute}\emph{
If the images of $g_1,g_2\in G$ under $\phi$
are compactly supported and commute with each other,
and if $J_i$ is a component of $\supp\phi(g_i)$ for $i=1,2$,
then either $J_1=J_2$ or $J_1\cap J_2=\varnothing$.}
\ee
We assume furthermore that the following conditions hold for some choice of elements $\{w_i\}_{i\ge1}\sse G$.
\be[(A)]
\setcounter{enumi}{3}
\item\label{p:small} \emph{For each compact nondegenerate interval $K\sse\supp\psi$
and for all sufficiently large $i\ge1$, we have
\[\CL_\psi(\psi(w_i)K)<2i.\]}
\item\label{p:large} \emph{For each compact nondegenerate interval $K\sse\supp\phi$ there exists some $g\in G$ such that
\[\CL_\phi(\phi(w_ig)K)>2i\]}
whenever $i\ge1$.
\ee
In this abstract setting (without a concrete $G$ or $\phi$) we can deduce the conclusion of Theorem~\ref{t:optimal-group}.
\begin{lem}\label{lem:optimal-abstract}
Under the hypotheses (\ref{p:ucsd}) through (\ref{p:large}), the following set is nonempty:
\[ [G,G]\cap\ker\psi\setminus\ker\phi.\]
\end{lem}
A key idea of the proof is that if $G$ acts on an open interval
the syllable length $\|g\|_{\mathrm{syl}}$ of $g\in G$
can be understood as the ``displacement energy'' of $g$,
which gives an upper bound for the covering length of an interval that can be moved off itself by $\rho(g)$.
A precise statement of this observation is given in Lemma~\ref{l:covering-translation-distance}.
\bp[Proof of Lemma~\ref{lem:optimal-abstract}]
We may assume $\psi(u_0)\ne1$ for, otherwise there is nothing to show
by condition~(\ref{p:nontrivial}).
Set $u_1:=u_0$.
By condition~(\ref{p:ucsd}), we can find a minimal, finite collection
\[ U_1, U_2, \ldots, U_m\]
of components of $\supp\psi$ so that
\[{\suppc \psi(u_1)}\sse U_1\cup\cdots\cup U_m.\]
Let $J_1$ be the closure of a component of $\supp\phi(u_1)$.
Using condition~(\ref{p:large}), we can find
some $g_1\in G$ such that
\[\CL_\phi(\phi(w_ig)J_1)>2i\]
for all $i\ge1$. Setting $u_1':=g_1u_1g_1^{-1}$ we see that
$J_1':=\phi(g)J_1$ is the closure of a component of $\phi(u_1')$.
Since each $U_i$ is $\psi(G)$--invariant, the homeomorphism $\psi(u_1')$ is still
compactly supported in $\bigcup_i U_i$.
Pick a compact interval $K_1$ such that
\[\supp\psi(u_1')\cap U_1\sse K_1\sse U_1.\]
Applying condition~(\ref{p:small}) we can fix some $i\gg0$ such that
\[\CL_\psi(\psi(w_i)K_1)<2i.\]
By Lemma~\ref{l:covering-translation-distance}, there exists some $h_1\in G$ with syllable length less than $2i$
such that $\psi(h_1)$ moves $\psi(w_i)K_1$ off itself.
If $\phi(h_1w_i)(J_1')$ is not equal to $\phi(w_i)(J_1')$,
then we set $v=1$; otherwise,
pick some $v\in V$ such that $\partial \phi(h_1w_i)(J_1')\cap \supp\phi(v)\ne\varnothing$ .
For this choice of $v\in\{1\}\cup V$, we have that
\[\phi(v^{\pm1}h_1)\phi(w_i)(J_1')\ne \phi(w_i)(J_1').\]
We claim that some $s\in\{-1,1\}$ satisfies
\[\psi(v^sh_1w_i)K_1\cap \psi(w_i)K_1=\varnothing.\]
Indeed, consider the case when
\[p:=\sup \psi(h_1w_i)K_1<\inf \psi(w_i)K_1.\]
For some $s\in\{-1,1\}$, the map $\psi(v^{s})$ will move $p$ to the left or stabilize it.
In this case, the interval
$\psi(v^{s} h_1w_i)K_1$ will still be disjoint from $ \psi(w_i)K_1$.
The remaining cases can be treated in the same manner, and thus the claim is proved.
It follows from the claim that
\[U_1\cap \supp\psi\left[w_iu_1'w_i^{-1},h_1'w_iu_1'(h_1'w_i)^{-1}\right]=\varnothing\]
for $h_1':=v^s h_1$.
Since $\|h_1'\|_{\mathrm{syl}}\le 2i$,
Lemma~\ref{l:covering-translation-distance} implies that $\phi(h_1')$ cannot move $\phi(w_i)(\Int J_1')$ off itself.
Note that $\phi(w_i)(\Int J_1')$ is a component of $\supp\phi(w_iu_1'w_i^{-1})$.
So, we see from condition~(\ref{p:noncommute}) that the image of
\[u_2:=
\left[w_iu_1'w_i^{-1},
\left(h_1'w_i \right) u_1' \left(h_1'w_i \right)^{-1}\right]\in [G,G]\]
under the action $\phi$ is nontrivial.
Note also that $\phi(u_2)$ and $\psi(u_2)$ are still compactly supported
since $u_2\in\fform{u_1}$.
Since $\psi(u_2)$ acts trivially on $U_1$,
its support is now contained in $U_2\cup\cdots \cup U_m$.
After rearranging if necessary, we assume that $\psi(u_2)$ acts nontrivally on $U_2$.
Proceeding as above, we continue producing
\[J_2, g_2,u_2', J_2',K_2,h_2,h_2',\]
in this order, using the given hypotheses.
We then find an element $u_3\in [G,G]\setminus\ker\phi$
such that the support of $\psi(u_3)$ is precompact in $U_3\cup\cdots\cup U_m$.
This process terminates in $i\le m$ steps, and we eventually find
some element \[u_{i+1}\in ( [G,G]\cap\ker\psi)\setminus\ker\phi.\]
Note that the final word $u_{i+1}$ in the above proof belongs $\fform{u_0}$.
\ep
The preceding proof is yet another instance of finding a kernel element of a representation by taking successive
commutators; see the beginning of Subsection~\ref{ss:compact supports} for relevant remarks.
In this chapter, we will prove Theorem~\ref{t:optimal-group} by
verifying conditions (A) through (E) of the above lemma
for the group
\[
G:=G^\dagger=(\bZ\times \BS(1,2))\ast F_2=
(\form{c}\times\form{a,e\mid aea^{-1}=e^2})\ast\form{b,d}.\]
Condition~(\ref{p:ucsd}) holds for
\[u_0:=\left[
[c^d,ee^de^{-1}],c\right]\in G^\dagger\]
by Corollary~\ref{c:bs12}.
Under the hypotheses of the theorem, we let
\[ N_i :=\ceil*{\frac1{\ell_i^{k-1}\alpha(\ell_i)}},\]
and set
\[ w_i:= v_i^{N_i}\cdots v_2^{N_2}v_1^{N_1}\in G^\dagger\]
for \[a=v_1=v_3=\cdots\quad \textrm{and}\quad b=v_2=v_4=\cdots.\]
Let $\beta$ and $\psi$ be as given in the theorem.
Note from Lemma~\ref{l:sum-density} that the set \[P:=\{i\in\bN\mid i\ell_i\le 1\}\]
has natural density one.
For each $i\in P$, we have
\[\frac{\ell_i}{\alpha(\ell_i)}\le\frac{1/i}{\alpha(1/i)}.\]
It follows for $i\in P$ that
\[
\frac{N_i\beta(1/i)}{i^{k-1}}\preccurlyeq \left( \frac{\beta(1/i)}{(i\ell_i)^k\alpha(1/i)}\cdot
\frac{i\ell_i\alpha(1/i)}{\alpha(\ell_i)}\right)\le \left( \frac{\beta(1/i)}{(i\ell_i)^k\alpha(1/i)}\right).
\]
Therefore, we have
\[
N_i\preccurlyeq i^{k-1}/\beta(1/i)\quad\text{ for almost all }i.\]
So, we can apply Corollary~\ref{c:slp} and see that condition~(\ref{p:small}) holds.
From the next section on, we will construct a representation
\[\phi\co G^\dagger\longrightarrow\Diff_+^{k,\alpha}(I)\cap\Diff_+^\infty(\Int I).\]
In particular, condition (\ref{p:noncommute}) will be automatic by Proposition~\ref{prop:disj-ab}.
It will then remain to engineer $\phi$ so that
the two conditions (\ref{p:nontrivial}) and (\ref{p:large}) also hold, which will finish the proof of the theorem.
We informally call $\phi(G^\dagger)$ as an \emph{optimally expanding group}\index{optimally expanding group}
because of condition (\ref{p:large}), in contrast with (\ref{p:small}).
\section{A single diffeomorphism of optimal expansion}\label{s:optimal-single}
In order to find a representation $\phi$ satisfying condition~(\ref{p:large}),
we will describe a method of constructing a single diffeomorphism that moves ``sufficiently fast'' on each supporting intervals.
Let us fix a constant
\begin{equation}\delta_0\in[9/10,1),\end{equation}
and set $D_0:=(1-\delta_0)/2\in(0,1/20]$.
The following condition is a key dynamical feature that will provide us a necessary condition for the regularity of a diffeomoprhism.
\bd\label{d:fast}
Let $f\co J\longrightarrow J$ be a homeomorphism of an interval $J$.
We say $f$ is \emph{$\delta$--fast}\index{$\delta$--fast} for some $\delta$ if there exists some $x_0\in I$ such that $|f(x_0)-x_0|\ge \delta |J|$.
\ed
This notion is stronger than the $\delta'$--expansiveness (Definition~\ref{d:expansive}) for some $\delta'>0$.
\begin{lem}\label{l:fast-expansive}
If $\delta\in(0,1)$,
then a $\delta$--fast homeomorphism of an interval is $\delta'$--expansive
for
\[
\delta':=2\delta/(1-\delta).\]
\end{lem}
\bp
Let $f$ denote a $\delta$--fast homeomorphism of an interval $J$.
We may assume that $f(x_0)-x_0 \ge \delta |J|$ for some $x_0\in J$.
For brevity, let us write
\[
A := x_0-\inf J, \quad
B :=f(x_0)-x_0,\quad
C := \sup J - f(x_0).\]
Then we have that $B/(A+B+C)\ge \delta$. By elementary computation,
we deduce that
\[ \max \left(B/A, B/C\right)\ge 2/\left(1/\delta-1\right).\]
This implies that $f$ is $\delta'$--expansive.
\ep
The starting point of our construction is to explicitly find a single $C^{k,\alpha}$--diffeomorphism that is $\delta_0$--fast, with
suitably bounded $C^k$ and $C^{k,\alpha}$ norms. Then we will suitably concatenate such diffeomorphisms in order to obtain
sequence of diffeomorphisms that can expand an arbitrary open interval intersecting the union of supports.
Recall our notation that for an interval $J\sse \bR$, we write
\[
\Diff_J^{k,\alpha}(\bR):=\{f\in \Diff_+^{k,\alpha}(\bR)\mid \supp f\sse J\}.\]
If $J$ is bounded, one may identify each element of $\Diff_J^{k,\alpha}(\bR)$ with a $C^{k,\alpha}$
diffeomorphism of $J$ that is $C^k$--tangent to the identity at $\partial J$.
Pick a smooth function $\Phi\co \bR\longrightarrow\bR$ satisfying the following conditions:
\begin{itemize}
\item $\Phi(x)=-1$ for $x\le 0$.
\item $\Phi(x)+\Phi(1-x)=0$; in particular, $\Phi(x)=1$ for $x\ge1$ and $\Phi(1/2)=0$.
\item $\Phi(x)$ is strictly increasing on $(0,1)$.
\end{itemize}
The precise choice of $\Phi$ will not matter for us. For instance, one may construct $\Phi$ from a
smooth bump function supported in $(0,1)$ as follows:
\begin{equation}
\Phi(x) := 2\frac{\int_{-\infty}^x e^{-1/(t(1-t))} \chi_{(0,1)}(t)dt}{\int_{\bR} e^{-1/(t(1-t))} \chi_{(0,1)}(t)dt} -1.
\end{equation}
For a function $f$, we let $\|f\|$ denote its uniform norm.
Let $r\ge0$ be an integer.
If $f$ is a real-valued $C^r$--map defined on some set $U$, then its $C^r$--norm (or $C^r$--metric) is
\[
\| f\|_{C^r}:=\sup_{0\le i\le r} \abss*{f^{(i)}}.\]
We define the $C^{r,\alpha}$--norm of $f$ by
\[[f]_{r,\alpha}:=\left[f^{(r)}\right]_\alpha=\sup_{x\ne y\text{ in }U}\frac{ |f(x)-f(y)|}{\alpha(|x-y|)}.\]
We define a constant
\begin{equation}
K_0=K_0(k):=\frac1{D_0^{k+1}}\abss*{\Phi}_{C^{k+1}}\end{equation}
and pick $\ell_0=\ell_0(k,\alpha,\delta_0)>0$ such that
\begin{equation}\ell_0+K_0\alpha(\ell_0)\le 1.\end{equation}
Let us consider an arbitrary $\ell\in(0,\ell_0]$.
We put $\Delta:=\delta_0\ell^k\alpha(\ell)$, and
define
\[
g_\ell(x) := \frac\Delta2 \left( \Phi\left(\frac{t}{D_0\ell}\right)+\Phi\left(\frac{\ell-t}{D_0\ell}\right)\right).\]
Then we have the following.
\be[(i)]
\item $g_\ell(x)=0$ outside $(0,\ell)$;
\item $g_\ell(x)=\Delta$ on $[D_0\ell,(1-D_0)\ell]$;
\item $g_\ell$ is strictly increasing on $(0,D_0\ell)$ and strictly decreasing on $((1-D_0)\ell,\ell)$;
\item\label{p:gkal} $\left[{g_\ell^{(k)}}\right]_\alpha\le K_0$.
\ee
The observations (i) through (iii) are immediate.
To establish (\ref{p:gkal}), we first note that
\begin{align*}
\left[g_\ell^{(k)}\right]_\alpha
&\le \Delta \left[ \left(\Phi\left(\frac{t}{D_0\ell}\right)\right)^{(k)}\right]_\alpha
= \frac{\Delta}{(D_0\ell)^k}
\sup_{0\le x<y\le D_0\ell} \frac{\abs*{\Phi^{(k)}\left(\frac{y}{D_0\ell}\right)-\Phi^{(k)}\left(\frac{x}{D_0\ell}\right)}}{\alpha(y-x)}\\
&\le
\frac{\delta_0 \alpha(\ell)}{D_0^k} \cdot
\abss*{\Phi^{(k+1)}}\cdot
\sup_{0\le x<y\le D_0\ell} \ \frac{y-x}{D_0\ell \alpha(y-x)}.\end{align*}
Using the fact that $x/\alpha(x)$ is monotone increasing, we have that
\[
\left[g_\ell^{(k)}\right]_\alpha
\le
\frac{\alpha(\ell)}{D_0^k\alpha(D_0\ell)}
\abss*{\Phi^{(k+1)}}
\le
\frac{1}{D_0^{k+1}}
\abss*{\Phi^{(k+1)}}=K_0.\]
Using $g_\ell$, let us construct a $C^{k,\alpha}$ diffeomorphism supported on a single interval that is $\delta_0$--fast.
\begin{lem}\label{l:optimal-diffeo}
For each $\ell\in(0,\ell_0]$ there exists some $f=f_\ell\in\Diff_+^\infty(\bR)$ such that the following hold.
\be[(i)]
\item $f(x)=\Id$ outside $(0,\ell)$;
\item $f(x)>x$ on $(0,\ell)$;
\item\label{p:fkal} $\left[f\right]_{k,\alpha}\le K_0$;
\item\label{p:ck} $\|f-\Id\|_{C^k}\le K_0\alpha(\ell)$;
\item\label{p:fnfast} $f^N$ is $\delta_0$--fast for all $N\ge 1 / (\ell^{k-1}\alpha(\ell))$.
\ee
\end{lem}
\bp
Setting $f:=g_\ell+\Id$,
we immediately see the conditions (i) through (iii).
The condition~(\ref{p:ck}) is actually a consequence of (iii). Indeed, for each $x\in[0,\ell]$ we have
\[\left|f^{(k)}(x)-\Id^{(k)}(x)\right|=\left|f^{(k)}(x)-f^{(k)}(0)\right|\le K_0\alpha(\ell).\]
For $1\le i<k$, we have
\begin{align*}
|f^{(i)}(x)-\Id^{(i)}(x)|&\le
\int_{s_1=0}^x\int_{s_2=0}^{s_1} \cdots\int_{s_{k-i}=0}^{s_{k-i-1}} |f^{(k)}(s_{k-i})-\Id^{(k)}(s_{k-i})|
ds_{k-i} \cdots ds_1\\
&\le
\ell^{k-i} K_0\alpha(\ell)\le K_0\alpha(\ell).
\end{align*}
We see that $f$ is a diffeomorphism since
\[
f'(x)\ge 1- \|f'-1\|\ge 1 - K_0\alpha(\ell)>0.\]
It only remains to show part (\ref{p:fnfast}).
We first set
\[
N_0:=\left\lceil \frac1{\ell^{k-1}\alpha(\ell)}\right\rceil=\left\lceil \frac{(1-2D_0)\ell}{\Delta}\right\rceil.\]
We then have that
\[
D_0\ell+(N_0-1)\Delta<(1-D_0)\ell\le D_0\ell+N_0\Delta.\]
Using an induction on $i=1,\ldots, N_0$, we see that
\[
f^i(D_0\ell) = g\circ f^{i-1}(D_0\ell) + f^{i-1}(D_0\ell)=\Delta+ f^{i-1}(D_0\ell)=D_0\ell+ i\Delta.\]
For all $N\ge N_0$, it follows that
\[
f^N(D_0\ell)-D_0\ell
\ge
f^{N_0}(D_0\ell)-D_0\ell= N_0\Delta\ge (1-2D_0)\ell =\delta_0\ell.\]
This proves part (\ref{p:fnfast}).
\ep
The following general observation is an easy consequence of the concavity of $\alpha$.
Here, we emphasize that $\prod_{i=1}^m f_i$ denotes a composition of diffeomorphisms.
\begin{lem}\label{l:disj-kal}
Let $\{J_i\}_{1\le i\le m}$ be a disjoint collection of compact intervals in $\bR$.
If $f_1,\ldots,f_m$ are orientation-preserving $C^{k,\alpha}$ diffeomorphisms of the real line
satisfying $\supp f_i\sse J_i$ for each $i$, then
\[
\left[\prod_{i=1}^m f_i\right]_{k,\alpha}\le 2\sup_i [f_i]_{k,\alpha}.\]
\end{lem}
\bp
Let us set $F:=\prod_{i=1}^m f_i$ and $K:=\sup_i[f_i]_{k,\alpha}$.
Pick real numbers $x<y$. It suffices for us to show that
\[ \abs*{ F^{(k)}(y) - F^{(k)}(x)} \le 2K\alpha(y-x).\]
This is immediate when $x,y\in J_i$ for some $i$.
Suppose $x\in J_i$ and $y\in J_j$ for some $i\ne j$.
We can find $x_0\in\partial J_i$ and $y_0\in\partial J_j$ such that
\[x\le x_0<y_0\le y.\]
Since $f_t=\Id$ at $\partial J_t$ for each $t$, we have
\begin{align*}
\abs*{ F^{(k)}(y) - F^{(k)}(x)}
&\le\abs*{ f_j^{(k)}(y) - f_j^{(k)}(y_0)} +\abs*{ \Id^{(k)}(y_0) - \Id^{(k)}(x_0)} +\abs*{ f_i^{(k)}(x_0) - f_i^{(k)}(x)}\\
&\le K\alpha(y-y_0)+K\alpha(x_0-x)\le 2K\alpha(y-x).
\end{align*}
If $x\in J_i$ and $y\not\in \bigcup_j J_j$ then we can find $x_0\in \partial J_i$ such that
$x\le x_0<y$. Then we have
\[ \abs*{ F^{(k)}(y) - F^{(k)}(x)}
=\abs*{ \Id^{(k)}(y) - F^{(k)}(x)}
=\abs*{F^{(k)}(x_0) - F^{(k)}(x)}
\le K\alpha(y-x).\]
\ep
The following is a standard consequence of Arzel\`a--Ascoli theorem. We briefly sketch the proof
for readers' convenience; a detailed proof can be found in~\cite{CKK2019}.
\begin{lem}[cf. {\cite[Lemma 4.19]{CKK2019}}]\label{l:ck-compact}
Let $I\sse\bR$ be a compact interval, let $r\in(0,\infty)$ and let $\epsilon\in(0,1)$.
Then the set
\[
\{f\in \Diff_I^{k,\alpha}(\bR)\mid [f]_{k,\alpha}\le r\text{ and }\|f'-1\|\le \epsilon\}\]
is isometric to a compact convex subset of the Banach space
\[C_I^{k,\alpha}(\bR):=\{g\in C^{k,\alpha}(\bR) \mid g(x)=0\text{ for }x\not\in I\},\]
both equipped with the $C^k$--metrics.
\end{lem}
\bp
Let us set
\[
B:=\{g \in C_I^{k,\alpha}(\bR)\mid [g]_{k,\alpha}\le r\text{ and }\|g'\|\le\epsilon\}.\]
Then Arzel\`a--Ascoli theorem implies that $B$ is compact with respect to the $C^k$--topology in the Banach space of
$C^k$--maps supported on $I$.
In particular, the map $f\mapsto f-\Id$ isometrically maps
the first set given in the hypothesis
onto the $C^k$--compact convex set $B$. We note that the condition $\|f'-1\|\le\epsilon$ was used to guarantee that this map is surjective.
\ep
We can now proceed to the main construction of this section.
\begin{thm}\label{t:optimal-diffeo}
Let $k\in{\bZ_{>0}}$, let $\alpha\in\CM$,
and let $\delta_0\in[9/10,1)$.
If $\{J_i\}_{i\ge1}$ is a collection of disjoint compact intervals contained in some compact subset of $\bR$,
then there exists $f\in \Diff_+^{k,\alpha}(\bR)$ supported in $\bigcup_i J_i$
such that the following hold for all $i$:
\be[(i)]
\item $f(x)>x$ for $x\in J_i\setminus\partial J_i$;
\item $f^N$ is $\delta_0$--fast on $J_i$ if $N\ge 1/\left(|J_i|^{k-1}\alpha(|J_i|)\right)$;
\item If an open interval $U\sse \bR$ intersects finitely many $J_i$'s, then $f$ is $C^\infty$ on $U$.
\ee
\end{thm}
\bp
We first reduce the proof to the case when $|J_i|\le \ell_0$ for all $i\ge1$.
Pick a sufficiently large $i_0$ such that $|J_i|\le \ell_0$ for $i\ge i_0$.
It is trivial that we can find $F_0\in \Diff_+^{\infty}(\bR)$ supported in $\bigcup_{i< i_0} J_i$
such that the conditions (i) through (iii) hold for $i< i_0$.
If we can find $F$ supported in $\bigcup_{i\ge i_0} J_i$ such that
the conditions (i) through (iii) hold for $i\ge i_0$, then $f:=F\circ F_0$ will satisfy the desired conclusion.
So, we may now assume that $\{J_i\}$ is an infinite collection and that $i_0=1$.
Let us consider a compact interval $I$ the interior of which contains the closure of $\bigcup_i J_i$.
Using Lemma~\ref{l:optimal-diffeo}, we can find $f_i\in\Diff_+^\infty(\bR)$ supported in $J_i$ such that
\begin{itemize}
\item $f_i(x)>x$ on $J_i\setminus\partial J_i$;
\item $[f_i]_{k,\alpha}\le K_0$;
\item $f_i^N$ is $\delta_0$--fast for $N\ge 1/ \left( |J_i|^{k-1}\alpha(|J_i|)\right)$.
\end{itemize}
Since $\{J_i\}$ is a disjoint collection, we have a well-defined homeomorphism
\[
f:=\prod_{i=1}^\infty f_i. \]
The conditions (i) and (ii) are obvious for $f$.
Let us now define a sequence $\{F_m\}$ of $C^\infty$ diffeomorphisms supported in $I$ as follows.
\[
F_m:= \prod_{1\le i\le m} f_i.\]
The condition (iii) follows from that $f=F_m$ on $U$ for some $m\gg0$.
By Lemma~\ref{l:disj-kal} we have
\[[F_m]_{k,\alpha}\le 2K_0.\]
We also have that
\[
\|F_m'-1\|=\sup_{i\le m} \|f_i'-1\|\le K_0\alpha(\ell_0)<1.\]
By Lemma~\ref{l:ck-compact}, it follows that $F_m$ converges to $f$ in the $C^k$--metric, and that $f\in \Diff_+^{k,\alpha}(I)$.
Since the closure of $\bigcup_i J_i$ is contained in the interior of $I$, we can extend $f$ outside $I$ by the identity.
In particular, we have $f\in \Diff_+^{k,\alpha}(\bR)$.
\ep
\section{Construction of optimally expanding diffeomorphism groups}\label{s:optimal}
As in the hypotheses of Theorem~\ref{t:optimal-group}, let us fix
$k\in{\bZ_{>0}}$, $\alpha\in\CM$,
and a sequence of admissible lengths $\{\ell_i\}_{i\ge1}$.
We will now construct a representation $\phi$ as promised in Section~\ref{s:strategy}.
There are positive constants $c_1, c_2$ satisfying
\[
c_1\le \ell_{i+1}/\ell_i \le c_2\]
for all $i\ge1$.
We let $\kappa>0$ be small enough that $\kappa\ell_1<1$
and that
\[\frac{\ell_i}{\ell_i+\ell_{i+1}}\ge \frac1{1+c_2}>\kappa\]
for all $i\ge1$.
We then pick a constant $\delta_0\in[9/10,1)$ sufficiently near from $1$ so that
\[ \frac{\ell_{i+1}}{\ell_i+\ell_{i+1}}\kappa \ge \frac{c_1\kappa}{1+c_1}>1-\delta_0,\]
and we set \[ N_i:=\ceil*{\frac1{\ell_i^{k-1}\alpha(\ell_i)}}.\]
\subsection{Specifying the images of generators}\label{ss:specifying}
The symbols $a,b,c,d,e$ will mean the corresponding generators of
\[G^\dagger=(\bZ\times \BS(1,2))\ast F_2=(\form{c}\times\form{a,e\mid aea^{-1}=e^2})\ast\form{b,d}.\]
The images of the generators under $\phi$ will be supported in some intervals from a collection $\FF$ defined below. See Figure~\ref{fig:config} also.
\begin{figure}[h!]
{
\tikzstyle {Av}=[blue,draw,shape=circle,fill=red,inner sep=1pt]
\tikzstyle {Bv}=[red,draw,shape=circle,fill=blue,inner sep=1pt]
\tikzstyle {Cv}=[brown,draw,shape=circle,fill=Maroon,inner sep=1pt]
\tikzstyle {Dv}=[teal,draw,shape=circle,fill=PineGreen,inner sep=1pt]
\tikzstyle {Ev}=[violet,draw,shape=circle,fill=Plum,inner sep=1pt]
\tikzstyle {A}=[blue,postaction=decorate,decoration={%
markings,%
mark=at position .7 with {\arrow[blue]{stealth};}}]
\tikzstyle {B}=[red,postaction=decorate,decoration={%
markings,%
mark=at position .7 with {\arrow[red]{stealth};}}]
\tikzstyle {C}=[brown,postaction=decorate,decoration={%
markings,%
mark=at position .7 with {\arrow[brown]{stealth};}}]
\tikzstyle {D}=[teal,postaction=decorate,decoration={%
markings,%
mark=at position .7 with {\arrow[teal]{stealth};}}]
\tikzstyle {E}=[violet,postaction=decorate,decoration={%
markings,%
mark=at position .7 with {\arrow[violet]{stealth};}}]
\begin{tikzpicture}[ultra thick,scale=.38]
\path (1,0) edge [A] node {} (4,0);
\draw (2.5,0) node [below] {\small $b$};
\path (3,1) edge [C] node {} (6,1);
\draw (4.5,1) node [above] {\small $c$};
\path (5,0) edge [D] node {} (8,0);
\draw (6.5,0) node [below] {\small $d$};
\path (7,1) edge [B] node {} (10,1);
\draw (8.5,1) node [above] {\small $a$};
\path (9,0) edge [A] node {} (12,0);
\draw (10.5,0) node [below] {\small $b$};
\path (11,1) edge [B] node {} (14,1);
\draw (12.5,1) node [above] {\small $a$};
\path (13,0) edge [A] node {} (16,0);
\draw (14.5,0) node [below] {\small $b$};
\path (-1.7,1) edge node {} (1.7,1);
\draw (0,1) node [above] {\small $a,c,d,e$};
\path (-1,0) edge [A] node {} (-4,0);
\draw (-2.5,0) node [below] {\small $b$};
\path (-3,1) edge [C] node {} (-6,1);
\draw (-4.5,1) node [above] {\small $c$};
\draw (16,.2) node [above] {$\cdots$};
\draw (-16,.2) node [above] {$\cdots$};
\path (-5,0) edge [D] node {} (-8,0);
\draw (-6.5,0) node [below] {\small $d$};
\path (-7,1) edge [B] node {} (-10,1);
\draw (-8.5,1) node [above] {\small $a$};
\path (-9,0) edge [A] node {} (-12,0);
\draw (-10.5,0) node [below] {\small $b$};
\path (-11,1) edge [B] node {} (-14,1);
\draw (-12.5,1) node [above] {\small $a$};
\path (-13,0) edge [A] node {} (-16,0);
\draw (-14.5,0) node [below] {\small $b$};
\draw [thin,black] (1.7,1) -- (1.7,2) node [above,black] {\tiny $3/4$};
\draw [thin,black] (-1.7,1) -- (-1.7,2) node [above,black] {\tiny $-3/4$};
\end{tikzpicture}}
\caption{The infinite chain
$\mathcal{F}$
and the diffeomorphism supported in each interval of $\mathcal{F}$. }
\label{fig:config}
\end{figure}
\begin{claim}
There exists an infinite chain of open intervals
\[ \FF= \{ \ldots, L_{-2},L_{-1},J_{-3},J_{-2},J_{-1},J_0,J_1,J_2,J_3,L_1,L_2,\ldots\},\]
in the given order, specified by the following conditions:
\begin{itemize}
\item $J_i=(i-3/4,i+3/4)$ for $-3\le i\le 3$;
\item $|J_3\cap L_1|=\kappa\ell_1$;
\item $|L_i\cap L_{i+1}|=\kappa\ell_{i+1}$ for $i\ge1$;
\item $L_{-i}=L_i$ for $i\ge1$.
\end{itemize}
\end{claim}
\bp
It is clear that $\{J_i\}_{-3\le i\le 3}$ forms a 7-chain.
To see that $\FF$ is a chain, it remains to verify the following conditions.
\be[(i)]
\item $|J_2\cap J_3|+|J_3\cap L_1|<|J_3|$;
\item $|J_3\cap L_1|+|L_1\cap L_2|<|L_1|$;
\item $|L_i\cap L_{i+1}|+|L_{i+1}\cap L_{i+2}|<|L_{i+1}|$ for $i\ge1$.
\ee
The inequality (i) follows from \[ 1/2 + \kappa\ell_1 < 1/2+1=|J_3|.\]
The inequalities (ii) and (iii) are consequences of the bound
\[
\kappa\ell_i + \kappa \ell_{i+1}<\ell_i.\]\ep
We now define $I$ to be the closure $\bigcup\FF$, which is a compact interval in $\bR$.
Recall we have chosen in Section~\ref{s:strategy} that
\[u_0:=\left[
[c^d,ee^de^{-1}],c\right]\in \form{a,c,d,e}.\]
Using Theorem~\ref{thm:bs-cpt} and Lemma~\ref{l:cinfty-free-product},
we can find a representation
\[
\phi_0\co G^\dagger\longrightarrow \Diff_{J_0}^\infty(\bR)\]
such that $\phi_0(b)=1$
and such that
$\phi_0(u_0)\ne1$. We can further require that $\supp\phi_0=J_0$, verifying condition~(\ref{p:nontrivial}) in Section~\ref{s:strategy}.
Let $b^+\in\Diff_{J_1}^\infty(\bR)$ satisfy that $b^+(x)>x$ for $x\in J_1$.
Using Corollary~\ref{cor:ghys-serg}, we can find $c^+\in \Diff_{J_2}^\infty(\bR)$ and
$d^+\in\Diff_{J_3}^\infty(\bR)$ such that the following hold:
\begin{itemize}
\item $c^+(x)>x$ for $x\in J_2$;
\item $d^+(x)>x$ for $x\in J_3$;
\item the action of $\form{c^+,d^+}$ on $J_2\cup J_3$ is topologically conjugate to the standard action of Thompson's group $F$ on $(0,1)$.
\end{itemize}
We define $b^-,c^-,d^-$ by symmetry
\[
v^-(x)=-v^+(-x)\]
for $v\in\{b,c,d\}$ and $x\in \bR$. We have another representation
\[
\phi_1\co G^\dagger\longrightarrow \Diff_I^\infty(\bR)\]
defined by $\phi_1(a)=\phi_1(e)=1$
and by
\[
\phi_1(v)=v^+v^-\]
for $v\in\{b,c,d\}$.
We apply Theorem~\ref{t:optimal-diffeo}
to the intervals
\[ L_1,L_3,L_5,\ldots\]
to obtain \[a^+\in \Diff_I^{k,\alpha}(\bR)\cap \Diff_+^\infty( I\setminus\partial I)\]
such that
$a^+(x)>x$ for each $x\in \bigcup_{i\ge1} L_{2i-1}$
and $a^+(x)=x$ otherwise; moreover, we require that
$\left(a^+\right)^{N_{2i-1}}$ is $\delta_0$--fast on $L_{2i-1}$ for all $i\ge1$.
We define $a^-(x)=-a^+(-x)$.
Similarly, we define \[b^+\in \Diff_I^{k,\alpha}(\bR)\cap \Diff_+^\infty( I\setminus\partial I)\] by applying the same theorem to the intervals
$\{L_{2i}\}_{i\ge1}$ and set $b^-(x)=-b^+(-x)$. Then we define
\[
\phi_2\co G^\dagger\longrightarrow \Diff_I^\infty(\bR)\]
by $\phi_2\form{c,d,e}=1$
and by
\[
\phi_2(v)=v^+v^-\]
for $v\in\{a,b\}$.
Summing up, we have a representation
\[
\phi\co G^\dagger\longrightarrow \Diff_I^\infty(\bR)\]
defined by
\[\phi(v)=\phi_0(v)\phi_1(v)\phi_2(v)\]
for all $v\in\{a,b,c,d,e\}$. For all such $v$, the supports of $\phi_0(v), \phi_1(v), \phi_2(v)$ are all disjoint and so, the representation $\phi$ is well-defined.
After rescaling if necessary, we may assume $I=[0,1]$.
\subsection{Verifying the optimal expansion of $\phi$}\label{ss:top-smooth}
As above, we let $v_{2i-1}:=a$ and $v_{2i}:=b$ and for $i\ge1$.
Since the action of $\phi(v_i)^{N_i}$ is $\delta_0$--fast on $L_i$, it is
$2\delta_0/(1-\delta_0)$--expansive by Lemma~\ref{l:fast-expansive}.
We have already seen that
\[
N_i\preccurlyeq i^{k-1}/\beta(1/i)\quad\text{ for almost all }i.\]
By applying Lemma~\ref{l:prob-dyn} with $V=\{\phi(a)\}$ or $V=\{\phi(b)\}$,
we deduce that
the elements \[\phi(a),\phi(b)\not\in \Diff_+^{k,\beta}(I)\cup\Diff_+^{k,\mathrm{bv}}(I).\]
We will now use covering distance estimates as dynamical obstructions that prohibits $\phi(G^\dagger)$ from being topologically conjugate into $\Diff_+^{k,\beta}(I)$
or into $\Diff_+^{k,\mathrm{bv}}(I)$.
Let us first make an easy general observation.
\begin{lem}\label{l:1-delta}
Let $\ell>0$ and $\delta\in(0,1)$.
Suppose $f\in\Homeo_+(0,\ell)$ is $\delta$--fast
and satisfies $f(x)\ge x$ for all $x\in(0,\ell)$.
Then for each \[ s\in((1-\delta)\ell,\ell),\] we have that
$f(s)>\delta\ell$.
\end{lem}
\bp
Pick $z\in(0,\ell)$ so that $f(z)-z \ge\delta\ell$. We have
\[
z\le f(z)-\delta\ell < (1-\delta)\ell < s.\]
It follows that $f(s)>f(z)\ge z+\delta\ell\ge \delta\ell$.
\ep
Returning to the proof of Theorem~\ref{t:optimal-group},
we will from now on define \[
w_i := v_i^{N_i}\cdots v_2^{N_2} v_1^{N_1}\in G^\dagger.\]
\begin{lem}\label{l:jumps}
If $s_1\in I$ satisfies
\[\inf L_1+(1-\delta_0)\ell_1 < s_1 < \inf L_1+\kappa\ell_1,\]
then for all $i\ge1$ we have that $\phi(w_i)(s_1)\in L_i\cap L_{i+1}$.
\end{lem}
\bp
Set $s_{i+1}:=\phi(w_i)s_1$ for $i\ge1$. We will prove a stronger claim that
\[ s_i - \inf L_i\in ( (1-\delta_0)\ell_i, \kappa\ell_i)\]
for all $i\ge1$. The case $i=1$ is obvious from the hypothesis.
Suppose the claim is proved for $i\ge1$.
Since $\phi(v_i)^{N_i}$ is $\delta_0$--fast on $L_i$,
we can apply Lemma~\ref{l:1-delta} to see that
\begin{align*}
s_{i+1}&=\phi(v_i)^{N_i} s_i > \inf L_i +\delta\ell_i =\sup L_i-(1-\delta_0)\ell_i\\
&=\inf L_{i+1}+\kappa\ell_{i+1}-(1-\delta_0)\ell_i>\inf L_{i+1}+(1-\delta_0)\ell_{i+1}.
\end{align*}
Since $s_{i+1}\in L_i\cap L_{i+1}$, the claim is proved.
\ep
Let $\CD_\FF$ denote the covering distance on $I$ defined by the intervals in $\FF$.
The above lemma implies that
\[
\CD_\FF(s_1,\phi(w_i)(s_1))=i\text{ or }i+1.\]
By the Slow Progress Lemma (Theorem~\ref{t:slp}), we see that the action $\phi$ is
topologically conjugate neither into $\Diff_+^{k,\beta}(I)$ nor $\Diff_+^{k,\mathrm{bv}}(I)$.
What will be important for us is that every nondegenerate interval $U$ in $I$ can be arbitrarily expanded under the action of $\phi$:
\begin{lem}\label{l:expand}
For every nondegenerate interval $U\sse I$ there exists some $g\in G^\dagger$ such that
\[\phi(g)(U)\cap L_1\cap J_3\ne\varnothing\ne \phi(g)(U)\cap L_{-1} \cap J_{-3}.\]
\end{lem}
From now on,
we will write $g(x)$ to mean $\phi(g)(x)$ whenever the meaning is clear from the context.
\bp[Proof of Lemma~\ref{l:expand}]
First note that the action $\phi$ on the interior of $I$ is sufficiently transitive; namely,
for $x\in \Int I$ and for all $J$ in the collection $\FF$ there exists some $g\in G^\dagger$ that moves $x$ into $J$.
This immediately follows from that $\FF$ is an infinite chain and from that the restriction of the action
$\form{a,c,d,e}$ onto $J_0$ has no global fixed points. So, we may assume that some
$x_1\in U$ belongs to $L_1\cap J_3$ possibly after moving $U$ by an element of $G^\dagger$
if necessary. Let us consider four (overlaping) cases.
\emph{Case 1. Some $x_{-1}\in U$ belongs to $\bigcup_{i\le-1} L_i$.}\\
The conclusion is trivial in this case since $\inf U\le \sup L_{-1}$ and $\sup U\ge \inf L_1$.
\emph{Case 2. Some $x_{-1}\in U$ belongs to $\bigcup_{-3\le i\le -1} J_i$.}\\
For some $P,Q,R\ge0$, the element \[g=d^Rc^Qb^P\] moves $x_{-1}$ into $L_{-1}\cap J_{-3}$.
Since $g(x_1)=d^R(x_1)\ge x_1$, we have that
\[
g(U)\cap (L_{\pm1}\cap J_{\pm3})\ne\varnothing.\]
\emph{Case 3. Some $x_{-1}\in U$ belongs to $J_0\cup J_1$.}\\
For some $P,Q\ge0$ and $w\in \form{a,c,d,e}$, the element $g_1=b^Q w b^{-P}$ moves $x_{-1}$ into $J_{-1}\cap J_{-2}$.
Then we have \[g_1(x_1)=b^Qw(x_1)\in J_2\cup J_3\cup L_1\cup L_2.\]
Using that $\form{c,d}\cong F$ acts on $J_2\cup J_3$ minimally, we can find $g_2\in \form{c,d}$ such that $g_2g_1(x_1)\in L_1\cup L_2$.
Since \[g_2g_1(x_{-1})\in \bigcup_{-3\le i\le -1} J_i,\] the proof reduces to Case 2.
\emph{Case 4. Some $x_{-1}\in U$ belongs to $J_2\cup J_3$.}\\
Using the 2--transitivity of $F$ as in Proposition~\ref{p:n-transitive},
we can find $g\in \form{c,d}$ such that $g(x_{-1})\in J_1\cap J_2$ and $g(x_1)\in J_3\cap L_1$.
The proof reduces to Case 3.
Since $x_1\in J_3\cap L_1$, the above argument exhaust all possible cases.
\ep
The above lemma verifies condition~(\ref{p:large}) in Section~\ref{s:strategy}, hence completing the proof of Theorem~\ref{t:optimal-group}.
\begin{lem}\label{l:non-commute}
For every nondegenerate interval $K\sse\supp\phi$,
there exists some $g\in G^\dagger$
such that
\[
\CL(\phi(w_i g) K)>2i\]
whenever $i\ge1$.
\end{lem}
\bp
By Lemma~\ref{l:expand}, we can find a $g\in G^\dagger$
such that some $s_1\in L_1\cap J_3$ and $s_{-1}:=-s_1$ both belong to $\phi(g)(K)$.
Lemma~\ref{l:jumps} then implies that
\[\CL(\phi(w_i g) K)\ge
\CL\left(\phi(w_i)[s_{-1},s_1]\right)>2i\] for all $i$.\ep
\section{Promotion to circles}\label{s:circle}
In this section, we will promote optimally expanding groups for compact intervals to those for circles.
The key idea of such a process is the following lemma.
\begin{lem}\label{l:circle-bs12}
Let $k\ge1$ be an integer and let $\beta$ be a concave modulus.
If a torsion-free simple group $H$ does not admit an embedding into $\Diff_+^{k,\beta}(I)$,
then $H\times\BS(1,2)$ does not admit an embedding into $\Diff_+^{k,\beta}(S^1)$.
\end{lem}
\bp
As usual, let us write \[\BS(1,2)=\form{a,e\mid aea^{-1}=e^2}.\]
Suppose we have a faithful representation
\[
\psi\co H\times\BS(1,2)\longrightarrow \Diff_+^{k,\beta}(S^1).\]
By Corollary~\ref{c:bs12}, there exists an integer $k\ge1$ depending on $\psi$ such that
\[
H_0:=\form{h^k\mid h\in H}\unlhd H\]
fixes all the points in $\supp e$.
In particular, we have a restriction map
\[\psi\restriction_{H_0}\co H_0\longrightarrow \Diff_+^{k,\beta}(I).\]
On the other hand, the group $H_0$ is nontrivial since $H$ is torsion-free.
By simplicity, we have that $H_0=H$, contradicting that $H$ does not faithfully act on $I$ by $C^{k,\beta}$ diffeomorphisms.
\ep
The following theorem generalizes the discussion so far for Theorem~\ref{t:optimal-group} to all compact connected one--manifolds.
Theorem~\ref{t:optimal-all} will then be an immediate consequence.
\begin{thm}\label{t:optimal-all2}
Let $k\in{\bZ_{>0}}$, let $\alpha\in\CM$, and let $\{\ell_i\}$ be admissible lengths.
There exists a finitely generated nonabelian group
\[ Q=Q(k,\alpha,\{\ell_i\})\le\Diff_+^{k,\alpha}(I)\]
such that
if a concave modulus $\beta\succ_k0$ satisfies
\[\beta(1/i) \cdot(1/i)^k\preccurlyeq \alpha(1/i)\cdot \ell_i^k\]
for almost all $i\ge1$,
and then every homomorphism
\[
[Q,Q]\longrightarrow\Diff_+^{k,\beta'}(M^1)\]
is trivial for $\beta'\in\{\beta,\mathrm{bv}\}$ and for $M^1\in\{I,S^1\}$.
Furthermore, $Q$ can be chosen so that $[Q,Q]$ is simple and every proper quotient of $Q$ is abelian.
\end{thm}
\bp
Given the parameter $(k,\alpha,\{\ell_i\})$, we apply Theorem~\ref{t:optimal-group}
to define
\[\phi_1:=\phi(k,\alpha,\{\ell_i\})\co G^\dagger\longrightarrow\Diff_c^{k,\alpha}(\bR).\]
We will fix a concave modulus $\beta\succ_k0$ satisfies
\[\beta(1/i) \cdot(1/i)^k\preccurlyeq \alpha(1/i)\cdot \ell_i^k\]
for almost all $i\ge1$. We let $\beta'\in\{\beta,\mathrm{bv}\}$.
Let $T_1$ be the image of $\phi_1$, which is a five--generated group.
We will successively upgrade $T_1$ through a sequence of finitely generated groups
such that each group in the sequence enjoys stronger algebraic properties than the previous.
The Chain Group Trick~\ref{cor:chain-gp-trick} will be used twice for this purpose,
but it requires that the group being promoted to have torsion-free abelianization.
So, we will apply the Rank Trick (Lemma~\ref{lem:rank-trick}) twice
to guarantee that the group's abelianization is torsion-free.
We now spell out this sketched argument a bit more.
Using the Rank Trick to $\phi_1$,
we can promote $\phi_1$ to another representation
\[
\phi_2\co G^\dagger \longrightarrow \form{T_1,\Diff_c^\infty(\bR)}\le\Diff_c^{k,\alpha}(\bR)\]
such that $\phi_1(g)=\phi_2(g)$ for $g\in [G^\dagger,G^\dagger]$,
and such that the abelianization of $T_2:=\phi_2(G^\dagger)$ is torsion-free.
We may assume ${\suppc T_2}\sse(0,1)$.
By the Chain Group Trick,
we see that
\[
T_3:=\form{T_2,\rho_{\mathrm{GS}}(F)}\]
is a seven--chain group acting minimally on $(0,1)$.
Furthermore, there exist an embedding
\[u\co T_2\into [T_3,T_3].\]
\begin{claim}
Every homomorphism $[T_3,T_3]\longrightarrow\Diff_+^{k,\beta'}(I)$ is trivial.
\end{claim}
Suppose we have a homomorphism
\[\rho\co[T_3,T_3]\longrightarrow \Diff_+^{k,\beta'}(I).\]
Since $[T_3,T_3]$ is simple, it suffices to show that $\rho$ is not injective.
Comparing the map $\phi_1$ with the composition
\[\begin{tikzcd}
G^\dagger\arrow{r}{\phi_2} &
T_2\arrow[hookrightarrow]{r}{u} &
{[T_3,T_3]}\arrow{r}{\rho}& \Diff_+^{k,\beta'}(I),
\end{tikzcd}
\]
we see from Theorem~\ref{t:optimal-group} that some $g\in[G^\dagger,G^\dagger]$ satisfies $\rho\circ u\circ \phi_2(g)=1$ but $\phi_1(g)\ne1$.
Since $u\circ\phi_2(g)=u\circ\phi_1(g)\ne1$, the claim is proved.
The group $T_3$ may not have a torsion-free abelianization.
So, we will apply the Rank Trick to the quotient $F_7\onto T_3$
in order to obtain a representation
\[
\phi_4\co F_7\longrightarrow \Diff_c^{k,\alpha}(\bR)\]
such that the abelianization of $\phi_4(F_7)$ is torsion-free.
Setting $T_4:=\phi_4(F_7)$, we can further require that
\[
[T_3,T_3]=[T_4,T_4].\]
We define the nine--generated group
\[
T_5:=T_4\times \BS(1,2).\]
\begin{claim}
There does not exist a faithful representation
\[
T_5\longrightarrow \Diff_+^{k,\beta'}(S^1).\]
\end{claim}
Suppose we have a homomorphism
\[
\rho\co T_5\longrightarrow \Diff_+^{k,\beta'}(S^1).\]
Applying the preceding claim
and Lemma~\ref{l:circle-bs12} to
\[
H:=[T_3,T_3]=[T_4,T_4],\]
we see that $H\times \BS(1,2)$ cannot embed into $\Diff_+^{k,\beta'}(I)$.
This is a contradiction, and the claim is proved.
Since $T_5$ has a torsion-free abelianization, we can apply the Chain Group Trick
and embed $T_5$ into the commutator subgroup of some eleven--chain group
\[
R=R(k,\alpha,\beta)\le\Diff_c^{k,\alpha}(\bR)\]
acting minimally on its support.
By Theorem~\ref{thm:chain-dichotomy} and by the last claim above, we complete the proof of Theorem~\ref{t:optimal-all2}.
\ep
\bp[Proof of Theorem~\ref{t:optimal-all}]
Assuming the hypotheses of Theorem~\ref{t:optimal-all}, we set
\[
\ell_i:= \frac1i \cdot \left(\frac{\beta(1/i) }{\alpha(1/i)}\right)^{1/k}.\]
\begin{claim}
The lengths $\{\ell_i\}$ are admissible.\end{claim}
By the concavity of $\alpha$, we see that
\[
1\le \alpha(y)\le \frac{y}{x}\cdot \alpha(x)\]
for all $0<x\le y$.
In particular, we have for $i\ge1$ that
\[
1\le
\frac{\alpha(1/i)}{\alpha(1/(i+1))}\le\frac{i+1}{i}\le 2.\]
Since $\beta(x)$ satisfies the same bounds, we see that
$\{ \ell_i/\ell_{i+1}\}$ is bounded, and bounded away from zero.
Moreover, whenever $i\le y\le i+1$ we have that
\[
\frac1y\left(\frac{\beta(1/y)}{\alpha(1/y)}\right)^{1/k}
\ge
\frac1{i+1}\left(\frac{\beta(1/(i+1))}{\alpha(1/i)}\right)^{1/k}
\ge
\frac1{2^{1+1/k}}\cdot \frac1i \left(\frac{\beta(1/i)}{\alpha(1/i)}\right)^{1/k}.
\]
Since the leftmost term is integrable on $[1,\infty)$, we have that \[\sum_i\ell_i<\infty.\]
The claim is thus proved.
By the above claim, we can apply Theorem~\ref{t:optimal-all2}
and see that the group
\[R(k,\alpha,\beta):=Q(k,\alpha,\{\ell_i\})\]
satisfies the conclusion of the theorem.\ep
\begin{rem}\label{rem:g1}
Theorem~\ref{t:optimal-all} appears weaker than Theorem~\ref{t:optimal-all2}.
There is a subtle point that highlights this difference when $k=1$.
Consider the concave modulus
\[
\alpha(x)=\frac1{\log(1/x)}.\]
Applying Theorem~\ref{t:optimal-all2}, we can find some group
\[
G_1:=Q(1,\alpha,\{1/(i\log^2 i)\})\le\Diff_+^{1,\alpha}(I)\le\Diff_+^1(I),\]
such that
\[
[G_1,G_1]\not\into \Diff_+^{1,s}(M^1)\]
for all real number $0<s<1$. This is because $\beta_s(x)=x^s$ is sub-tame,
and because
\[
x^s \log^2(1/x) \preccurlyeq 1/\log(1/x)\]
for all $x$ sufficiently near from 0.
The existence of such a group $G_1$ does not immediately follow from Theorem~\ref{t:optimal-all}. Indeed,
an obvious choice of a concave modulus $\beta(x)$
satisfying
\[\int_0^1 \frac1{x}\left(\frac{\beta(x)}{\alpha(x)}\right)<\infty\]
and $\beta(x)\succcurlyeq x^s$ for all $s\in(0,1)$
would be
\[\beta(x):= \alpha(x)/\log^2(1/x).\]
However, this modulus $\beta$ is not sub-tame and one cannot apply this theorem for the pair $(\alpha,\beta)$.
\end{rem}
\section{Continua of groups of prescribed critical H\"older regularity}\label{s:hoelder}
Recall from Section~\ref{ss:defn-rem} that the critical regularity of a group $G$ with respect to a manifold $M$ is defined as
\[\CR_M(G):=\sup\{r\ge1\mid G\into\Diff_c^r(M)_0\}\in\{-\infty\}\cup[1,\infty].\]
In this section,
we prove that for each real number $r\ge1$ and for every compact connected one--manifold $M$
there exist continuum--many distinct isomorphism classes of finitely generated groups $G$ such that
\[
\CR_M(G)=r.\] Here, \emph{continuum}\index{continuum} refers to the cardinality of $\bR$.
We will actually prove a much stronger result below, which obviously implies Corollary~\ref{c:optimal-all} as well.
\begin{thm}\label{t:continuum}
For each real number $r\ge1$, there exist continua $X_r, Y_r$ of finitely generated groups such that the following hold.
\be[(i)]
\item
Each group in $X_r\cup Y_r$ is a subgroup of $\Homeo_+[0,1]$,
which has a simple commutator group
and every proper quotient of which is abelian.
\item No two groups in $X_r\cup Y_r$ have isomorphic commutator subgroups.
\item
For each group $A\in X_r$ and for each $M^1\in\{I,S^1\}$, we have that $A\le\Diff_+^r(I)$ and that
\[
[A,A]\not\into \bigcup_{s>r} \Diff_+^s(M^1).\]
\item
For each group $B\in Y_r$ and for $M^1\in\{I,S^1\}$, we have that \[B\le\bigcap_{r<s} \Diff_+^r(I)\] and that
\[
[B,B]\not\into \Diff_+^r(M^1).\]
\ee
\end{thm}
In order to construct such continua, we will apply Theorem~\ref{t:optimal-all} for suitable pairs of concave moduli
in the following form:
\[\omega_{s,t,u}(x):=\frac{x^s}{\log^u(1/x)} \exp\left(- t \sqrt{\log(1/x)}\right)
.\]
Roughly speaking,
$\omega_{s,t,u}$ is a small perturbation of the H\"older modulus $x^s$, as we can write
\[\omega_{s,t,u}(x)=
{\exp\left(-s\log(1/x) - t \sqrt{\log(1/x)}-u\log\log(1/x)\right)}.\]
\begin{lem}\label{l:omega-st}
The following hold for $u\ge 0$.
\be[(1)]
\item
The map $\omega_{s,t,u}$ is a concave modulus (defined near $x=0$) in the following cases:
\be[(i)]
\item $s=0$ and $t>0$.
\item $0<s<1$ and $t\in\bR$; in this case, $\omega_{s,t,u}$ is sub-tame.
\item $s=1$ and $t\le0$; in this case, $\omega_{s,t,u}$ is sub-tame.
\ee
\item
Let $(s,t)$ and $(S,T)$ be in the range described part (1).
If $(s,t)<(S,T)$ in the lexicographical order, then we have that
\[\lim_{x\to+0}\omega_{S,T,0}(x)/\omega_{s,t,u}(x)=0.\]
\ee
\end{lem}
\bp
(1)
The precise verification of the concavity is a bit tedious, but one can first have an intuition by noting that the behavior of $\omega_{s,t,u}$ for $s>0$ is governed by $x^s$ near $x=0$.
Rigorously, we compute (by Wolfram Mathematica 12) that
\[
\omega'_{s,t,u}(x)=
\frac{1}{2} x^{s-1} e^{-t \sqrt{\log \left(\frac{1}{x}\right)}} \log ^{-u-1}\left(\frac{1}{x}\right) \left(2 s \log \left(\frac{1}{x}\right)+t \sqrt{\log \left(\frac{1}{x}\right)}+2 u\right).\]
If $s>0$ or $s=0$ and $t>0$, then we see that $\omega_{s,t,u}'>0$ near $x=0$ from that
\[ \sqrt{\log(1/x)}=o(\log (1/x)).\]
Similarly, to determine the sign of $\omega_{s,t,u}''$, we first write
\[
\omega''_{s,t,u}(x)=
\frac{1}{4} x^{s-2} e^{-t \sqrt{\log \left(\frac{1}{x}\right)}} \log ^{-u-2}\left(\frac{1}{x}\right)F(x),\]
where
\begin{align*}
F(x)=&4 (s-1) s \log ^2\left(\frac{1}{x}\right)
+2 (2 s-1) t \log ^{\frac{3}{2}}\left(\frac{1}{x}\right)+
\left(8 s u+t^2-4 u\right)\log \left(\frac{1}{x}\right)
\\
&+(4 t u+t) \sqrt{\log \left(\frac{1}{x}\right)}+4 u (u+1).\end{align*}
Hence, if $0<s<1$ then $F(x)<0$ and $\omega''$ is concave.
The same is true when $s=0$ and $t>0$, or $s=1$ and $t<0$.
To verify that $\omega_{s,t,u}$ is sub-tame in the cases (ii) and (iii),
let us write
\[
\frac{\omega_{s,t,u}(vx)}{\omega_{s,t,u}(x)}=v^s \exp\left(-t\left(\sqrt{\log(1/(vx))}-
\sqrt{\log(1/x)}\right)\right)\cdot \left(\frac{\log(1/x)}{\log(1/v)+\log(1/x)}\right)^u.\]
Here, we assume $v>0$ is small enough.
If $t\ge 0$ it is obvious that
\[
\frac{\omega_{s,t,u}(vx)}{\omega_{s,t,u}(x)}\le v^s.\]
If $t<0$ we see
\[\frac{\omega_{s,t,u}(vx)}{\omega_{s,t,u}(x)}\le v^s\exp(-t \sqrt{\log(1/v)}).\]
In either case, we see that $\omega_{s,t,u}(x)$ is sub-tame.
Part (2) is straightforward after noticing that
\[
\log(1/x)\gg \sqrt{\log(1/x)}\gg \log\log(1/x).\]\ep
\bp[Proof of Theorem~\ref{t:continuum}]
Let us write $r = k+s$ for $k=\floor{r}\ge1$.
\emph{Case $0<s<1$.}
By Lemma~\ref{l:omega-st} and Theorem~\ref{t:optimal-all}, we can define collections of finitely generated groups as below.
\begin{align*}
X_r&:=\{ R(k,\omega_{s,t,0},\omega_{s,t,k+1}) \mid t>0)\},\\
Y_r&:=\{ R(k,\omega_{s,t,0},\omega_{s,t,k+1}) \mid t<0\}.
\end{align*}
For $t>0$, we have
\[
\bigcup_{ S\in(s,1]}\Diff_+^{k,S}(M^1)\le
\Diff_+^{k,\omega_{s,t,k+1}}(M^1)\le
\Diff_+^{k,\omega_{s,t,0}}(M^1)\le \Diff_+^{k,s}(M^1).\]
If $t<0$, then
\[
\Diff_+^{k,s}(M^1)
\le
\Diff_+^{k,\omega_{s,t,k+1}}(M^1)\le
\Diff_+^{k,\omega_{s,t,0}}(M^1)\le
\bigcap_{ S\in(0,s)}\Diff_+^{k,S}(M^1).\]
Thus, parts (i), (iii), and (iv) of the theorem are readily implied by Theorem~\ref{t:optimal-all}.
For real numbers $t<T$, we saw in Lemma~\ref{l:omega-st} that
\[\omega_{s,T,0}(x)\le\omega_{s,t,k+1}(x)\]
for all $x>0$ near $0$. In particular, we have
\[
\Diff_+^{k,\omega_{s,T,0}}(M^1)\le \Diff_+^{k,\omega_{s,t,k+1}}(M^1).\]
This implies that
\[
[R(k,\omega_{s,t,0},\omega_{s,t,k+1}),R(k,\omega_{s,t,0},\omega_{s,t,k+1})]\not\into
R(k,\omega_{s,T,0},\omega_{s,T,k+1}).\]
This proves part (ii).
\emph{Case $s=0$ and $k\ge2$.}
We have $r=k\ge2$. Similarly to the previous case, we let
\begin{align*}
X_k&:=\{ R(k,\omega_{0,t,0},\omega_{0,t,k+1}) \mid t>0)\},\\
Y_k&:=\{ R(k-1,\omega_{1,t,0},\omega_{1,t,k+1}) \mid t<0\}.
\end{align*}
These are well-defined since $\omega_{0,t,k+1}\succ_k 0$
and
$\omega_{1,t,k+1}\succ_{k-1}0$.
The conclusion is again obvious.
Since $\omega_{1,t,k+1}(x)\ge x$ for $t<0$ and for small $x>0$, we have an even stronger fact:
\[
[R(k-1,\omega_{1,t,0},\omega_{1,t,k+1}),R(k-1,\omega_{1,t,0},\omega_{1,t,k+1})]\not\into \Diff_+^{k-1,\mathrm{Lip}}(M^1).\]
\emph{Case $s=0$ and $k=1$.}
Recall in Remark~\ref{rem:g1} we have found
\[
G_1\le\Diff_+^1(I)\]
such that $[G_1,G_1]\not\into\Diff_+^{1,V}(M^1)$ for all $V>0$.
For each $V>1$, let us fix $G_V\in X_V$.
Let us consider $T\ge1$. We apply the Rank Trick for the natural surjection $F_m\onto G_T$ with some $m\ge2$,
and find a finitely generated group $\tilde G_T\le\Diff_+^1(I)$ such that
\[
[G_T,G_T]=[\tilde G_T,\tilde G_T],\]
and such that the abelianization of $\tilde G_T$ is free abelian.
Since $\tilde G_T$ is finitely generated, we have that
\[\tilde G_T\not\into \bigcup_{S>T}\Diff_+^{S}(M^1).\]
Let $V>1$.
We apply the Chain Group Trick to the product $\tilde G_1\times\tilde G_V$
and obtain a minimal chain group $\Gamma(V)\le\Diff_+^1(I)$ satisfying that
\[
\tilde G_1\times\tilde G_V\into [\Gamma(V),\Gamma(V)].\]
The group $[\Gamma(V),\Gamma(V)]$ does not embed into \[\bigcup_{S>1}\Diff_+^S(M^1),\]
since neither does $\tilde G_1$.
If there were only countably many isomorphism classes in the collection
\[\{[\Gamma(S),\Gamma(S)]\mid S\ge1\},\]
then the same is true for
the collection of finitely generated subgroups in that collection.
This would contradict that
$\CR_I(\tilde G_S)=S$ for all $S\ge1$.
It follows that there exists a continuum $X^*\sse(1,\infty)$ such that
whenever $S\ne T$ belong to $X^*$ we have
\[
[\Gamma(S),\Gamma(S)]\not\cong[\Gamma(T),\Gamma(T)].\]
This completes the construction of $X_1$.
We now construct the collection $Y_1$.
Note that every countable subgroups of $\Homeo_+(M^1)$ is topologically conjugate into
$\Diff_+^{\mathrm{Lip}}(M^1)$ by~\cite{DKN2007}; see Theorem~\ref{thm:lip-conj} above.
So, the condition (iv) of the current theorem is now equivalent to that $B\le\Homeo_+(M^1)$ and
\[
[B,B]\not\into\Diff^1_+(M^1)\]
for each $B$ in the collection $Y_1$ that we are about to build.
We start with an observation of Kropholler and Thurston
that the following is a nontrivial finitely generated perfect orderable group:
\[\tilde\Delta=\form{a,b,c,t\mid a^2=b^3=c^7=abc=t}\le\widetilde{\PSL}(2,\bR)\le\Homeo_+(\bR).\]
We direct the reader to Appendix~\ref{ch:append3} and to~\cite{Thurston1974Top,Bergman1991PJM} for a further discussion of this
example and its perfectness.
Note that $\tilde\Delta$ is the fundamental group of a Seifert fibered 3--manifold, which fibers over the Fuchsian $(2,3,7)$--triangle group
\[
\Delta=\form{a,b,c\mid a^2=b^3=c^7=abc=1}\le\PSL(2,\bR)\le\Homeo_+(S^1)\]
By Thurston Stability (Theorem~\ref{thm:thurston-stab}), the group $\tilde\Delta$ does not embed into $\Diff_+^1(I)$.
We feed $T_2:=\tilde\Delta$ to our proof of Theorem~\ref{t:optimal-all}.
We proceed to obtain $T_3,T_4,T_5$ and $R$ as in Section~\ref{s:circle},
and see that $R$ is a minimal chain group on $(0,1)$
and $[R,R]\not\into\Diff_+^1(S^1)$.
Seting $H_1:=R$, follow the construction of $X_1$.
Namely, we pick some $H_S\in Y_S$ for each $S>1$, and build finitely generated groups $\tilde H_1,\tilde H_S$ and $\Lambda(S)$ such that
\[
\tilde H_1\times\tilde H_S\into[\Lambda(S),\Lambda(S)]\not\into\Diff_+^1(M^1).\]
In particular, $\Lambda(S)$ will be chosen to be a minimal chain group.
By the same reasoning, we can find a continuum $Y^*\sse(0,1)$ such that
for every distinct pair $S,T$ in $Y^*$ we have
\[
[\Lambda(S),\Lambda(S)]\not\cong [\Lambda(T),\Lambda(T)].\]
This completes the construction of $Y_1$.\ep
\begin{rem}
Calegari~\cite{Calegari2006TAMS} provided an example a group $Q\le\Homeo_+(S^1)$
that admits no embedding into $\Diff_+^1(S^1)$.
He extended the embedding
\[
\tilde\Delta\into \Homeo_+(\bR)\into\Homeo_+(S^1)\]
to a homomorphism
\[
\rho\co \form{\tilde\Delta,x,y,z\mid xax^{-1}=a^2,yby^{-1}=b^2,zcz^{-1}=c^2}\longrightarrow\Homeo_+(S^1),\]
and proved that the image of $\rho$ is such an example.
Theorem~\ref{t:continuum} generalizes his result to provide continuum--many such examples.
\end{rem}
\chapter{Applications}\label{ch:app}
\begin{abstract}In this chapter, we will outline some of the main consequences of the machinery developed in this book, following mostly
applications that are discussed in~\cite{KK2020crit,BKK2019JEMS,KK2018JT}. We will begin with foliation theory, which
was the driving force behind the study of regularity of group actions on manifolds for much of the latter's history, and then move on
to the problem of computing the critical regularity of various classes of groups.
This chapter will be of a different flavor than most of the rest of the book, serving more as a survey in a more informal tone.
As such, less detailed proofs and a more
informal exposition will be given.\end{abstract}
One of the major consequences of
the construction of groups with specified critical regularity is that they yield strong unsmoothability
results for certain foliations on $3$--manifolds, coming from foliations on trivial $I$--bundles over surfaces whose monodromy groups
are groups with specified critical regularity (see Corollary~\ref{cor:tsuboi-ref} and Corollary~\ref{cor:3-mfld}).
We will also describe some results that illustrate the interplay between algebraic structure of groups and possible regularities of group
actions one one--manifolds, concentrating on right-angled Artin groups and mapping class groups of surfaces. The main results in that
discussion will by Theorem~\ref{thm:kharlamov} and Theorem~\ref{thm:mcg-c2}.
Throughout this chapter, we will adopt the following standing convention. If $ r\geq 1$ is a real number of the form $ r=k+\tau$, where
$k$ is an integer and $\tau\in[0,1)$, then we will write $C^{ r}$ and $\Diff^{ r}$ for $C^{k,\tau}$
and $\Diff^{k,\tau}$, respectively. This will be notationally convenient for the
discussion of constructions that are uniform in $ r$, so that we do not need to decompose each real number as a sum of its floor and
fractional part.
\section{Foliation theory}\label{sec:foliation}
One of the original sources of interest in regularity of group actions comes from the theory of foliations on manifolds.
In fact, the Plante--Thurston Theorem (Theorem~\ref{thm:plante-thurston}) and the Thurston Stability Theorem
(Theorem~\ref{thm:thurston-stab}) were both originally formulated as results about foliations, and their content as facts about regularity
is mostly abstracted after the fact. In this section, we
shall go in the other direction, by discussing the applicability of critical regularity to the non--smoothability of
certain codimension one foliations on manifolds.
\subsection{Foliations and suspensions of group actions}
A foliation $\FF$ on an $n$--manifold $M$ is a local decomposition of $M$ modeled on the direct product decomposition
$\bR^n\cong\bR^p\times\bR^q$, where $p,q\geq 1$ and $p+q=n$. Precisely, let $\{x_1,\ldots,x_n\}$ be coordinates for $\bR^n$;
the manifold $M$ is equipped with a
$q$--dimensional \emph{foliation
atlas}\index{foliation atlas},
which is a collection of charts \[\varphi_i\colon U_i\longrightarrow \bR^n\] for some suitable cover $\UU=\{U_i\}_{i\in I}$ of $M$,
so that the transition functions $\varphi_{ij}=\varphi_i\circ\varphi_j^{-1}$ decompose in coordinates as
\[\varphi_{ij}(x_1,\ldots,x_n)=(\varphi_{ij}^1(x_1,\ldots,x_n), \varphi_{ij}^2(x_{q+1},\ldots,x_n)),\] where
\[\varphi^1_{ij}\colon\bR^n\longrightarrow
\bR^q,\quad \varphi^2_{ij}\colon\bR^p\longrightarrow\bR^p.\]
The intuitive meaning behind this decomposition is that a foliation is locally modeled by a submersion $\bR^n\longrightarrow\bR^q$. More
generally, a submersion from an $n$--dimensional manifold $M$ to a $q$--dimensional manifold induces a foliated structure on $M$ via
the Implicit Function Theorem. Transition maps between foliation charts preserve leaves of the foliation, which is why $\varphi^1_{ij}$
is allowed to depend on all variables but $\varphi^2_{ij}$ may not.
Thus locally, $M$ is an $\bR^p$--worth of $q$--dimensional
immersed manifolds. The natural number $p$ is the \emph{codimension}\index{codimension of a foliation} of the foliation,
and $q$ is the \emph{dimension}\index{dimension of a foliation} of the foliation.
The regularity of the functions \[\{\varphi_{ij}^k\}_{i,j\in I, k\in\{1,2\}},\] which is always assumed to be at least $C^0$,
is the \emph{regularity}\index{regularity of a foliation} of the foliation. The regularities of $\varphi_{ij}^1$ and $\varphi_{ij}^2$
need not coincide, and so the regularity of the foliation is generally an ordered pair. When one regularity is finite and the other is infinite,
then the regularity is simply the finite member of the pair, and this will usually be the case for us; specifically, unless otherwise noted, we
will always assume that $\varphi_{ij}^1$ is $C^{\infty}$, and so the regularity of the foliation is simply the regularity of the functions
$\varphi_{ij}^2$ occurring in the atlas.
The regularity of the foliation
may be much lower than the regularity of $M$. Indeed, $(M,\FF)$ may be such that $M$ itself may admit a $C^{\infty}$ atlas, but the
foliation atlas describing $\FF$ is merely required to be $C^0$. If $ r\geq 1$, a \emph{$C^{ r}$--smoothing}\index{smoothing of a foliation}
of $(M,\FF)$ is a homeomorphism
\[f\colon (M,\FF)\longrightarrow (M',\FF'),\] where $\FF'$ is a $C^{ r}$ foliation.
Beyond the basic definitions, we will not survey much of the vast theory of foliations, and instead we
will content ourselves with directing the reader
to some standard references such as~\cite{CandelConlonI,CandelConlonII}.
We will use one fundamental construction in the theory of foliations, namely that of the \emph{suspension of a group action}\index{suspension
of a group action}. Let $B$ and $M$
be fixed connected, smooth manifolds, and let $\yt B\longrightarrow B$ be the universal covering map. Fix a representation
\[\psi\colon\pi_1(B)\longrightarrow\Diff_+^{ r}(M).\] Notice that $\yt B\times M$ is naturally a manifold of dimension \[n=\dim B+\dim M,\]
and admits a foliation of dimension $\dim B$ and codimension $\dim M$.
We also have a natural action of $\pi_1(B)$ on
$\yt B\times M$ via the diagonal action, where $g\in\pi_1(B)$ acts on $\yt B$ by a deck transformation and on $M$ via $\psi(g)$.
Observe that since the action of $\pi_1(B)$ on $\yt B$ is free and properly discontinuous, the induced action on $\yt B\times M$ is as
well, and so \[\yt B\times M\longrightarrow E(\psi)=(\yt B\times M)/\pi_1(B)\] is a covering map. The space $E(\psi)$ is called the
\emph{suspension} of $\psi$. It has the structure of a $C^{ r}$ fiber bundle, which is in turn a foliation
$\FF$; this is called a \emph{$C^r$ foliated bundle}\index{foliated bundle}.
If $g\in\pi_1(B)$ then $g$ can be represented by a loop in
$B$.
If $b_0\in \yt B$ is a basepoint, then the fiber over $b_0$ is identified with the fiber over $g.b_0$ by the map $\psi(g)$. The map $\psi$ is
called the \emph{monodromy representation}\index{monodromy representation} of the bundle.
Suspensions of group actions give rise to rich families of examples.
Consider, for instance, an action of the fundamental group $\pi_1(S)$ of a surface $S$ on the interval $I$, and suppose that there
is a point $x\in I$ whose orbit is free; that is, the map $\pi_1(S)\longrightarrow I$ given by $g\mapsto g(x)$ is injective. Then
the suspension of the action of $\pi_1(S)$ will be homeomorphic to $S\times I$, though the induced foliation will not be the product foliation:
instead, it will admit a contractible leaf that is identified with the universal cover $\yt{S}$ of $S$. The reader is directed to~\cite{SoutoMarq18}
for some interesting issues surrounding smooth foliations of $S\times I$.
Suspensions of group actions can be understood systematically, as they
are classified up to homeomorphism by the conjugacy class of the monodromy representation:
\begin{prop}[See Theorem 3.1.5 of~\cite{CandelConlonI}]\label{prop:suspension-unique}
Let $B$ be a smooth manifold, let $M$ be a manifold of regularity at least $C^r$,
and let \[\psi,\psi'\colon\pi_1(B)\longrightarrow\Diff_+^{ r}(M).\] Then $E(\psi)$ is homeomorphic
as a bundle to $E(\psi')$ if and only if $\psi$ and
$\psi'$ are conjugate by a homeomorphism of $M$. The bundles $E(\psi)$ and $E(\psi')$ are $C^{r}$--diffeomorphic for
$r$ if and only if $\psi$ and $\psi'$ are conjugate by a $C^{r}$ diffeomorphism.
\end{prop}
In Proposition~\ref{prop:suspension-unique}, a bundle homeomorphism $E(\psi)\longrightarrow E(\psi')$ is a homeomorphism
of the total spaces that takes fibers to fibers. We leave the proof of the proposition as an exercise for the reader.
\subsection{Non-smoothability of codimension one foliations}\label{sss:tsuboi}
Proposition~\ref{prop:suspension-unique} gives a tool for building foliations that are $C^0$ but not $C^1$, or in general
$C^r$ for some $ r\geq 1$ but not $C^{s }$ whenever $s > r$. The first examples of group actions
on $M=I$ that are $C^k$ for some $k\in{\bZ_{>0}}$ but not $C^{k+1}$ were constructed by Tsuboi~\cite{Tsuboi1987},
and
independently
by Cantwell and Conlon~\cite{CC1988}.
We sketch Tsuboi's construction here, since it is simple and easy to understand, and since it makes good use of the Muller--Tsuboi
trick (Lemma~\ref{l:muller-tsuboi}).
Consider the maps
\[
\tau_t(x)=x+t,\quad \lambda_t(x)=e^tx\]
for $t\in\bR$.
These generate the orientation preserving affine transformation group $\Aff_+(\bR)$.
Recall from Theorem~\ref{thm:bs-cpt} that we have an injection
\[
\Aff_+(\bR)\longrightarrow\Diff_{[0,1]}^\infty(\bR),\]
defined using a certain topological conjugacy $\psi\co (0,1)\longrightarrow\bR$.
More precisely, the affine map $x\mapsto ax+b$ gets sent first to \[x\mapsto \frac{x}{a-bx},\] and then
this map is made infinitely tangent to the identity at the endpoints by the Muller--Tsuboi trick.
The map $\psi$ is chosen to be smooth in the open interval $(0,1)$.
For the rest of this subsection,
we will denote by $A_0\le \Diff_{[0,1]}^\infty(\bR)$ the image of such an embedding.
The vector fields $\partial/\partial x$ and $x\partial/\partial x$ generate the flows $\tau_t(x)$ and
$\lambda_t(x)$. We write $\tau^*$ and $\lambda^*$ for the pull-backs of these two vector fields by $\psi$.
Theorem~\ref{thm:bs-cpt} then amounts to showing that
\begin{align*}
T(t,x)&:=\Phi_{\tau^*}(t,x)=\psi^{-1}\circ\tau_t\circ\psi(x),\\
L(t,x)&:=\Phi_{\lambda^*}(t,x)=\psi^{-1}\circ\lambda_t\circ\psi(x)\end{align*}
belong to $A_0\le \Diff_{[0,1]}^\infty(\bR)$ for each $t$.
Here, $\Phi_\rho(t,x)$ denotes the flow of an arbitrary vector field $\rho\co\bR\longrightarrow\bR$.
More rigorously, the maps $T$ and $L$ are first defined on $\bR\times (0,1)$.
Theorem~\ref{thm:bs-cpt} then implies that for each $t\in \bR $ the maps $x\mapsto T(t,x)$ and
$x\mapsto L(t,x)$ extend to smooth maps on $x\in \bR$, by the identity outside of $(0,1)$. It is also
clear from the definition that the extended maps are simultaneously continuous on $(t,x)$.
Hence, we can apply the Bochner--Montgomery Theorem (Theorem~\ref{thm:BM-simple})
to deduce that $T$ and $L$ are smooth on $\bR\times\bR$.
In order to describe Tsuboi's example, let us now fix an integer number $r\geq 1$, and $\eps>0$; we further require that \[\eps(r-1)<1\] in the case when $r\ge2$.
We select a strictly increasing sequence of points $\{y_n\}_{n\in\bN}\sse [0,1]$ such that $y_0=0$, which converges to $1$, and which satisfies \[y_n-y_{n-1}=n^{-1-\eps}\] for $n\gg 0$. Fix an affine homeomorphism
\[\omega_n\co [0,1]\longrightarrow J_n:=[y_{n-1},y_n].\]
We also fix a positive sequence $\{u_n\}$ such that $u_n\in(0,1/n^r)$.
We define $\bar T, \bar L\in\Homeo_+[0,1]$ by requiring that $\bar T(y_n)=\bar L(y_n)=y_n$
and that
\begin{align*}
\bar T\restriction_{J_n}&:=
\omega_n\circ T(u_n, \underbar{\phantom{x}})\circ \omega_n^{-1},\\
\bar L\restriction_{J_n}&:=
\omega_n\circ L\left(\frac{\log 2}{n^r}, \underbar{\phantom{x}}\right)\circ \omega_n^{-1},
\end{align*}
for each $n$.
It is clear that by setting $\bar T(1)= \bar L(1)=1$, we obtain a homeomorphism of $[0,1]$.
Since $A_0\le \Diff_{[0,1]}^{\infty}(I)$, the homeomorphisms $\bar T$ and $\bar L$ are $C^{\infty}$ at $y_n$ for all $n$, and
so lie in $\Diff_+^{\infty}(-\infty,1)$. The particular choices involving the number $r$ will guarantee that the resulting homeomorphisms $\bar T$ and $\bar L$ are in fact $C^r$ at the point $1$; this follows from a computation that we now spell out.
For each smooth vector field $\rho\co \bR\longrightarrow\bR$ supported in $[0,1]$
and for each $i\ge1$, we set
\[
P_{i,\rho}:=\sup\left\{ \abs*{
\frac\partial{\partial t}
\left(
\frac{\partial^i\Phi_\rho(t,x)}{\partial x^i}
\right)
}\co
{-1\le t\le 1, x\in\bR}\right\}.
\]
We let $P_i:=\max(P_{i,\tau^*},P_{i,\lambda^*})$.
For each $x\in [0,1]$, we have
\[
\abs*{\frac{\partial T}{\partial x}(t,x)-1}
=\abs*{\frac{\partial T}{\partial x}(t,x)-\frac{\partial T}{\partial x}(0,x)}
\le P_1\cdot t.\]
We similarly obtain the following estimates for all $i\ge1$:
\begin{align*}
\abs*{\frac{\partial^i T}{\partial x^i}(t,x)-\delta_{i1}}
&\le P_i\cdot t,\\
\abs*{\frac{\partial^i L}{\partial x^i}(t,x)-\delta_{i1}}
&\le P_i\cdot t,\end{align*} where $\delta_{i1}$ denotes the Kronecker delta.
We now have the following estimates for each $n,i\ge1$:
\begin{align*}
\sup_{J_n} \abs*{
\bar T^{(i)}-\delta_{i1}
}&\le P_i\cdot \frac{u_n}{|J_n|^{i-1}}\le P_i\cdot n^{(i-1)(1+\epsilon)-r},\\
\sup_{J_n} \abs*{
\bar L^{(i)}-\delta_{i1}
}&\le P_i\cdot \frac{\log 2}{n^r} \cdot \frac{1}{|J_n|^{i-1}}\le \log 2 \cdot P_i\cdot n^{(i-1)(1+\epsilon)-r}.
\end{align*}
It follows that for each $1\le i\le r$, the $C^i$--distances from the identity to $\bar T$ and to $\bar L$ on $J_n$ tend to zero as $n\to\infty$.
Thus, we conclude that \[\bar T, \bar L\in \Diff_{[0,1]}^r(\bR).\]
The restriction of the group $\form{\bar T,\bar L}$ to each interval $J_n$ is simply an action of an affine group.
As \[\lambda_s^i \tau_t\lambda_s^{-i}=\tau_{e^{si} t}\] for all $s,t\in\bR$ and $i\in\bZ$,
we have that
\[
[\bar T,\bar L^i\bar T\bar L^{-i}]=1.\]
Recall from Section~\ref{sec:abt} that the \emph{lamplighter group}\index{lamplighter group} is a metabelian group defined by the
presentation
\[\bZ\wr\bZ:=\form{ \tau,\lambda\mid \left[\tau,\lambda^i\tau\lambda^{-i}\right]=1\text{ for all }i\in\bZ}.\]
We have just constructed a representation
\[\psi_r\colon\bZ\wr\bZ\longrightarrow\Diff_+^r(I),\] defined by $\tau\mapsto \bar T$ and
$\lambda\mapsto \bar L$.
We note the following non-$C^k$ criterion, which is due to Tsuboi.
\begin{lem}\label{lem:tsuboi-2}
Let $k\ge2$ be an integer.
Let $\{x_n\}_{n\in\bN}\sse I$ be a strictly increasing sequence such that $x_0=0$ and such that $\lim_{n\to\infty} x_n=1$. If
$f\in\Diff_+^k[0,1]$ and if $f(x_n)=x_n$ for all $n$, then we have \[\sum_{n=0}^{\infty}
\sup_{x\in [x_n,x_{n+1}]}
|f'(x)-1|^{1/(k-1)}<k\cdot \sup_{x\in[0,1]} |f^{(k)}(x)|^{1/(k-1)}<\infty.\]
\end{lem}
\bp
Set $P:=\sup_{x\in[0,1]} |f^{(k)}(x)|$. We note that on the interval
\[
J_{n,k}:=[x_n,x_{n+k}]\]
the map $f$ is $k$--fixed (Definition~\ref{d:k-fixed}).
It follows from Lemma~\ref{l:expansive2} that
\begin{align*}
&
\sum_{n=0}^{\infty}
\sup_{[x_n,x_{n+1}]}
|\bar f'(x)-1|^{1/(k-1)}
\le
\sum_{n=0}^{\infty}
\sup_{J_{n,k}}
|\bar f'(x)-1|^{1/(k-1)}
\\
&\le
\sum_n \sup_{J_{n,k}} |\bar f^{(k)}(x)|^{1/(k-1)} \cdot |J_{n,k}| \le k\cdot P^{1/(k-1)}<\infty.
\end{align*}
This completes the proof.\ep
We now argue that $\psi_r$ is not conjugate to a representation into $\Diff_+^{r+1}[0,1]$.
Indeed, suppose the contrary.
We call the corresponding $C^{r+1}$ diffeomorphisms $\yt T$ and $\yt L$, and their common fixed points
$\{z_n\}_{n\in\bN}$.
For convenience of notation, write $T_n$ and $L_n$ for the restrictions of $\yt T$ and $\yt L$ to $K_n=[z_{n-1},z_n]$.
We claim that there exists some $v_n\in K_n$ such that
\[
\left(L_n^{n^r}\right)'(v_n)=2.\]
Indeed,
from the equation
\[
\lambda_s \tau_t\lambda_s^{-1}=\tau_{te^s}\]
we have that
\[
L_n^{n^r} T_n L_n^{-n^r} = T_n^2.\]
In other words, we have a representation
\[
\BS(1,2)=\form{a,e\mid aea^{-1}=e^2}\longrightarrow\Diff_{K_n}^\infty(\bR)\]
defined as $a\mapsto L_n^{n_r}$ and $e\mapsto T_n$.
We see from Proposition~\ref{p:bmnr}
\ref{p:bmnr-m} that the existence of the required point $v_n$, as claimed.
From the claim and from the chain rule, we can find some point
$w_n\in K_n$ such that \[L_n'(w_n)\geq 2^{1/n^r}.\]
On the other hand, since we are assuming $\form{\yt T,\yt L}\le\Diff^{r+1}_+[0,1]$ we can apply Lemma~\ref{lem:tsuboi-2} and have that
\[
\sum_{n\ge1}
|2^{1/n^r}-1|^{1/r}\le
\sum_{n\ge1} |L_n'(w_n)-1|^{1/r}<\infty.\]
Note the estimate \[(1+t)^u-1\ge ut/2\] for $t\in[0,1]$ and $u\in[0,1]$.
Therefore, we have
\[
\sum_{n\ge1}
|2^{1/n^r}-1|^{1/r}
= \sum_{n\ge1} |(1+1)^{1/n^r}-1|^{1/r}\ge 2^{-1/r}\cdot \sum_{n\ge1} 1/n.\]
This is a contradiction, proving that $\form{\bar T,\bar L}\le\Diff_+^r[0,1]$ is not topologically conjugate into $\Diff_+^{r+1}[0,1]$.
We now have the following.
\begin{thm}
For each integer $r\ge1$ there exists a faithful representation
\[
\psi_r\co \bZ\wr\bZ\longrightarrow \Diff_+^r[0,1]\]
that is not topologically conjugate into $\Diff_+^{r+1}[0,1]$.\end{thm}
\bp
It only remains for us to show the faithfulness of the representation.
Consider an arbitrary nontrivial element $g\in \bZ\wr\bZ$. We can write $g$ in a normal form
\[
g = \prod_{i=-N}^N \lambda^i \tau^{k_i}\lambda^{-i} \cdot \lambda^m\]
for $N\ge0$ and suitable integers $k_i$ and $m$.
Let us define for each $n\ge1$ that
\[
c_n:=
\left( \sum_{i=-N}^N \left(2^{1/n^r}\right)^ik_i
\right)u_n.\]
On each interval $J_n$, the map $g$ is topologically conjugate to the affine map
\[
x\mapsto 2^{m/n^r}x + c_n.\]
If $m\ne0$ then $\psi_r(g)$ is topologically conjugate to an
affine map with a nontrivial scaling part on each interval $J_n$.
This implies that $\psi_r(g)\ne1$.
Let us now assume that $m=0$ and that $k_i\ne0$ for some $i$. Clearly, we may assume $k_N\ne0$.
The map $\psi_r(g)$ is topologically conjugate to the affine translation by $c_n$
on each interval $J_n$. Since the polynomial equation
\[
\sum_{i=-N}^N k_i x^i=0\]
has at most finitely many solutions, the value $c_n$ cannot be zero for all $n$.
Together, these observations imply that $\psi_r(g)\ne1$, and that $\psi_r$ is faithful.
\ep
We remark that in Tsuboi's original paper~\cite{Tsuboi1987}, the exact isomorphism type (namely, $\bZ\wr\bZ$) of the group $\form{\bar T,\bar L}$ is not specified; rather, he only noted that it is a quotient of the group
\[
\form{a,b\mid [a,[a,b]]=1}.\]
As an application, let us consider a closed orientable surface $S_g$ of genus $g$. The preceding theorem can be combined with
Proposition~\ref{prop:suspension-unique} and allows us to conclude:
\begin{cor}\label{cor:tsuboi}
Let $g\geq 2$. For all integer $r\geq 1$, there exists a codimension one
$C^r$ foliation on $S_g\times I$ that is not smoothable to a $C^{r+1}$ foliation.
\end{cor}
Our results on the existence of groups with prescribed critical regularity implies that for $g\geq 5$ and $ r\in\bR_{\geq 1}$, there
is a representation \[\psi_{ r}\in\Hom(\pi_1(S_g),\Diff_{[0,1]}^{ r}(\bR))\] that is not topologically conjugate to a representation into
\[\bigcup_{s > r}\Diff_{[0,1]}^{ r}(\bR).\]
We thus obtain a different perspective Corollary~\ref{cor:tsuboi}. In fact,
more is true: we get monodromy representation for $C^{ r}$ foliated structures on $S_g\times I$ that occur in no $C^{s }$ foliated
structure, for $s > r$.
\begin{cor}\label{cor:tsuboi-ref}
Let $g\geq 2$. For all $ r\geq 1$, there exists a
codimension one $C^{ r}$ foliation on $S_g\times I$ that is not smoothable to a $C^{s }$ foliation
whenever $s > r$.
\end{cor}
More precisely, the foliations furnished in Corollary~\ref{cor:tsuboi-ref} come from $I$--bundles over $S$ whose monodromy groups
do not arise from a $C^{s }$ group action on the interval, for $s > r$.
We can push the non--smoothability of foliated structures on $I$--bundles of the form $S\times I$ (where here $S$ is a surface)
even farther, producing non--smoothable foliations of closed $3$--manifolds subject to mild topological hypotheses.
Let $M$ be a closed, orientable $3$--manifold, and suppose that $H_2(M,\bZ)\neq 0$. By Poincar\'e duality, we have that
$H^1(M,\bZ)\neq 0$, and a nontrivial element $\phi\in H^1(M,\bZ)$ is classified by a smooth map \[\Phi\colon M\longrightarrow S^1.\]
Standard transversality arguments imply that $M$ admits an embedded, two--sided surface $S=\Phi^{-1}(p)$
for some suitable $p\in S^1$, possibly after modifying $\Phi$ by a homotopy, and that $S$ is Poincar\'e dual
to $\phi$. Fairly standard methods, which we shall not spell out in further detail here, imply that we may increase the genus of $S$
arbitrarily, so that for all $g\gg 0$, the manifold $M$ admits an embedded, two--sided copy of $S_g=S$. Since $S$ is two--sided, this
means that a tubular neighborhood of $S$ is homeomorphic to a trivial $I$--bundle $S\times I$.
Removing a copy of $S\times (0,1)$ from $M$ results in a manifold with boundary $N$. The following is a result of
Goodman~\cite{Goodman1975} (cf.~\cite{Thurston1976AM,CC1982}), whose proof we will not provide here.
\begin{prop}\label{prop:goodman}
Let $N$ be as above. The manifold $N$ admits a smooth foliation $\FF$ with both components of $\partial N$ as leaves.
\end{prop}
By gluing in a copy of $S\times I$ with an unsmoothable $C^{ r}$ foliation as furnished by Corollary~\ref{cor:tsuboi-ref}, we obtain
the following:
\begin{cor}\label{cor:3-mfld}
Let $M$ be a closed, orientable $3$--manifold with $H_2(M,\bZ)\neq 0$. Then there exists a codimension one $C^{ r}$ foliation
on $M$ that is not homeomorphic to a $\bigcup_{s > r} C^{s }$ foliation.
\end{cor}
\section{Right-angled Artin groups}\label{sec:raag}
Let $\Gamma$ be a finite, simplicial graph. That is, $\Gamma$ has a finite set of vertices $V(\gam)$, and a finite set of
undirected edges $E(\gam)$, so that
the $1$--complex given by the data of $V(\gam)$ and $E(\gam)$ forms a simplicial complex. One defines the
\emph{right-angled Artin group}\index{right-angled Artin group}
\[A(\gam)=\form{V(\gam)\mid [v_i,v_j]=1\,\textrm{ whenever }\, \{v_i,v_j\}\in E(\gam)}.\]
The reader will easily check that the class of right-angled Artin groups contains free abelian groups and free groups, and can be
seen as interpolating between these two classes of groups.
\subsection{Abelian groups, free groups, and smooth actions}
Part of our interest in right-angled Artin groups comes from their actions on one--manifolds.
For some right-angled Artin groups, it is very easy to find actions on one--manifolds of high regularity.
For instance,
consider the free abelian group $\bZ^n$. Suppose that $0\neq X$ is a smooth vector field on $M\in\{I,S^1\}$. In the case where $M=I$,
assume that $X$ vanishes at $\partial I$.
By integrating this vector field, we get a flow on $M$, which is to say a continuous injective homomorphism
\[\phi\colon \bR\longrightarrow\Diff_+^{\infty}(M).\] It follows that for all times $t,s\in\bR$, the corresponding diffeomorphisms $\phi(t)$ and
$\phi(s)$ commute with each other. By choosing $n$ times that are linearly independent over $\bQ$, say $\{t_1,\ldots,t_n\}$, we have that
the diffeomorphisms $\{\phi(t_1),\ldots,\phi(t_n)\}$ form a subgroup of $\Diff_+^{\infty}(M)$ that is isomorphic to $\bZ^n$.
At the other extreme, consider the free group $F_n$ of rank $n$. It is a bit less obvious how to find copies of
$F_n$ inside of $\Diff_+^{\infty}(M)$,
though as it happens these subgroups are very common. We already observed this fact in Corollary~\ref{cor:free-abundant}.
For right-angled Artin groups that are not as structurally uncomplicated as free groups or abelian groups, it is significantly less
obvious why (or indeed if) there should be smooth actions on $I$ or on $S^1$. The reader will recall from Subsection~\ref{ss:abt}
that one can build faithful actions of $A(\gam)$ by homeomorphisms on $\bR$, by building a homomorphism $\phi_w$ for every element
$1\neq w\in A(\gam)$ that witnesses that $\phi_w(w)$ is nontrivial. Once such a sequence of homomorphisms is constructed, it is not
difficult to improve them to homomorphisms to $\Diff_+^{\infty}(\bR)$. Thus, we obtain:
\begin{thm}[See~\cite{BKK2014}]\label{thm:bkk-israel}
Let $\Gamma$ be a finite simplicial graph. Then there is an injective homomorphism \[A(\gam)\longrightarrow\Diff_+^{\infty}(\bR).\]
\end{thm}
If one tries to compactify the construction in Theorem~\ref{thm:bkk-israel} in order to get a faithful homomorphisms into $\Diff_+^{\infty}(I)$,
one encounters an analytic difficulty. The reader can check without great difficulty that, even though this construction furnishes a homomorphism
from $A(\gam)$ into $\Homeo_+(I)$, the Mean Value Theorem precludes this action from being $C^1$. By pushing the methods of constructing
$C^{\infty}$ actions outlined here to their extreme, we can prove the following.
\begin{prop}[cf.~\cite{KK2018JT}]\label{prop:c-infty-raag}
Let $A(\gam)$ be a right-angled Artin group that decomposes as a direct product of free products of free abelian groups. Then
there is an injective homomorphism \[A(\gam)\longrightarrow\Diff_+^{\infty}(M),\] where $M\in\{I,S^1\}$.
\end{prop}
The graphs under the purview of Proposition~\ref{prop:c-infty-raag} are easy to describe combinatorially; these ideas will be developed
in Subsection~\ref{ss:p4} below. We will not justify some of the claims we make about passing between graphs and groups, though
the reader will not find them to be controversial. More details can be found in~\cite{Charney2007,KK2018JT,koberda21survey}.
A right-angled Artin group $A(\gam)$ is abelian if and only if the defining graph $\Gamma$ is complete. The group $A(\gam)$ is a free product
of free abelian groups if and only if every connected component of $\Gamma$ is complete. We have that $A(\gam)$ decomposes as a
(nontrivial) direct
product if and only if $\Gamma$ decomposes as a
(nontrivial) \emph{join}\index{join of graphs}, which is to say $V(\gam)=V(J_1)\cup V(J_2)$, where the graphs $J_1$ and $J_2$
spanned by $V(J_1)$ and $V(J_2)$ share no vertices (and are both nonempty),
and where every vertex $v_1\in V(J_1)$ spans an edge with every vertex
$v_2\in V(J_2)$. Therefore, we have that $A(\gam)$ decomposes as a direct product of free products of free abelian groups if and only
if $\Gamma$ decomposes as a (possibly trivial) join, where each component of the join is a disjoint union of complete graphs.
\begin{proof}[Proof of Proposition~\ref{prop:c-infty-raag}]
In the preceding paragraphs, we have shown how to obtain nonabelian free groups and free abelian groups in $\Diff_+^{\infty}(I)$, and
it is easy to see that one can arrange such an action to be supported on an arbitrary prescribed nondegenerate subinterval $J\sse I$.
Now, let \[A(\gam)\cong A(\gam_1)\times\cdots\times A(\gam_k),\] where $A(\gam_i)$ is a free product of free abelian groups for
$1\leq i\leq k$. We will choose $k$ disjoint intervals $\{J_1,\ldots,J_k\}$ of $I$ such that $(\partial J_i)\cap(\partial J_{\ell})=\varnothing$
for $i\neq\ell$.
Let $N_i$ be the maximal rank of a free abelian subgroup of $A(\gam_i)$. An easy applications of the
Kurosh Subgroup Theorem~\cite{Serre1977} shows that the group $\bZ*\bZ^{N_i}$ contains all groups of the form
\[\bZ^{n_1}*\cdots*\bZ^{n_{\ell}},\] provided that \[N_i\geq \max \{n_1,\ldots,n_{\ell}\}.\] Clearly, to prove the result for $M=I$, it suffices
to establish that $\bZ*\bZ^N\le \Diff_+^{\infty}(I)$ for $N\in\bN$,
in such a way that this group extends by the identity past the endpoints of $I$.
Indeed,
then we can build an action of $\bZ*\bZ^{N_i}$ on $J_i$ for $1\leq i\leq k$, furnishing a faithful action of $A(\gam)$ by $C^{\infty}$
diffeomorphisms on $I$. Clearly we may then identify the endpoints of $I$ to get an action on $S^1$.
Choosing a smooth flow on $I$ whose derivatives of all orders agree with those of the identity at $0$ and $1$, we may find a
group of diffeomorphisms isomorphic to $\bZ^N$. Clearly we may assume that the vector field giving rise to this flow has no zeros
in the interior of the interval, so that this copy of $\bZ^N$ acts freely on $I$. We then consider all diffeomorphisms of $I$
whose derivatives of all orders agree with those of the identity at $0$ and $1$.
By a Baire Category argument very similar to that used
to find a copy of $F_2\le \Diff_+^{\infty}(I)$ above, a generic choice of diffeomorphism of $I$ will
combine with the abelian group we have produced in order to generate a copy of $\bZ*\bZ^N$. We leave the remaining details to
the reader.\end{proof}
In the proof of Proposition~\ref{prop:c-infty-raag}, the fact that $\bZ^N$ is acting freely is essential; it can be weakened to an assumption
of \emph{full support}\index{fully supported homeomorphism},
i.e.~that a nontrivial element of $\bZ^N$ cannot have an interval's worth of fixed points. If one does not have
such an assumption, then the induction fails. Indeed, let $t$ generate the free $\bZ$--factor of $\bZ*\bZ^N$. If $x\in I$ and $t(x)$ is fixed
by $g\in\bZ^N$ then $(t^{-1}gt)(x)=x$. If $t(x)$ lies in the interior of $\Fix(g)$ then a small perturbation of $t$ will not prevent $x$ from being
a fixed point of $t^{-1}gt$. Thus, the set of ``good" choices of $t$ and $x$ is no longer dense in the set $\Diff_+^{\infty}(I)\times I$, and
the Baire Category Theorem is not applicable. This issue is not an artifact of the proof; indeed, we will see in the next section that
the right-angled Artin groups covered by Proposition~\ref{prop:c-infty-raag} are exactly the ones which embed in $\Diff_+^{\infty}(I)$.
\subsection{$P_4$, the cograph hierarchy, and $C^{1+\mathrm{bv}}$ actions}\label{ss:p4}
Given that this natural construction cannot have good regularity properties, we are motivated to consider the following question, which
M.~Kapovich attributes to Kharlamov~\cite{Kapovich2012}.
\begin{que}\label{que:kharlamov}
For which graphs $\Gamma$ is there an injective homomorphism \[A(\gam)\longrightarrow \Diff^{\infty}(M),\] where $M\in\{I,S^1\}$?
\end{que}
Proposition~\ref{prop:c-infty-raag} gives us a partial answer to Question~\ref{que:kharlamov}.
The machinery developed in this book allows us to give a complete answer.
Historically, Question~\ref{que:kharlamov} was resolved in two steps. Strictly speaking, the second step subsumed the first, and so it
will suffice to only carry out the second. It will be useful for us to retain the historical progression of the resolution of
Question~\ref{que:kharlamov}, because of connections to mapping class group we will explore in Section~\ref{sec:mcg} below.
For $n\geq 1$, we will write $P_n$ for the path of length $n-1$, which is a graph with exactly $n$ vertices which can be thought of
as ordered linearly, with the linear ordering inducing the edge relation. We have illustrated the graph $P_4$ in Figure~\ref{f:p4}.
\begin{figure}[h!]
\tikzstyle {bv}=[black,draw,shape=circle,fill=black,inner sep=1pt]
\begin{center}
\begin{tikzpicture}[main/.style = {draw, circle}]
\node[main] (1) {$a$};
\node[main] (2) [right of=1] {$b$};
\node[main] (3) [right of=2] {$c$};
\node[main] (4) [right of=3] {$d$};
\draw (1)--(2)--(3)--(4);
\end{tikzpicture}%
\caption{The graph $P_4$.}
\label{f:p4}
\end{center}
\end{figure}
The graphs $\{P_n\}_{n\geq 1}$ play an important role in the theory of right-angled Artin groups. Trivially, we have $A(P_1)\cong\bZ$ and
$A(P_2)\cong\bZ^2$. Slightly less trivially, we have $A(P_3)\cong F_2\times \bZ$. The right-angled Artin group $A(P_4)$ is the first
interesting right-angled Artin group on a path. It was observed by Droms~\cite{Droms1987} that if $A(\gam)$ is a right-angled Artin group
then every finitely generated subgroup of $A(\gam)$ is a right-angled Artin group if and only if $\Gamma$ has no full subgraph isomorphic
to the path $P_4$ and no full subgraph isomorphic to the cycle of length four $C_4$. It is true though nonobvious that $\Gamma$ contains
one of these graphs as a full subgraph if and only if $A(\gam)$ contains one of the corresponding right-angled Artin groups as a
subgroup (see~\cite{Kambites2009,KK2013}, cf.~Proposition~\ref{prop:gam-p4}).
The right-angled Artin group $A(P_4)$ turns out to be universal for right-angled Artin groups on finite forests, in the following sense:
\begin{prop}[See~\cite{KK2013}]\label{prop:raag-forest}
Let $F$ be a finite forest. Then there is an injective homomorphism $A(F)\longrightarrow A(P_4)$.
\end{prop}
The first progress on resolving Question~\ref{que:kharlamov} was obtained by the authors with Baik.
\begin{thm}[See~\cite{BKK2019JEMS}]\label{thm:a4-c2}
There is no injective homomorphism \[A(P_4)\longrightarrow \Diffb(M),\] where $M\in\{I,S^1\}$.
\end{thm}
It follows that if $A(\gam)$ admits a faithful action on a compact one--manifold $M$ with regularity $C^{1+\mathrm{bv}}$ or higher
then not only is $P_4$ not a full subgraph of $\Gamma$, but in fact $A(\gam)$ cannot contain a copy of $A(P_4)$ as a subgroup. This latter
condition may seem weaker than the former, but it turns out that they are equivalent to each other.
\begin{prop}[See~\cite{KK2013}]\label{prop:gam-p4}
There is an injective homomorphism $A(P_4)\longrightarrow A(\gam)$ if and only if $P_4$ occurs as a full subgraph of $\Gamma$.
\end{prop}
Proposition~\ref{prop:gam-p4} says that if $A(\gam)$ contains a copy of $A(P_4)$ then it contains an ``obvious" copy of $A(P_4)$ (though
of course not every copy of $A(P_4)$ need be ``visible" in the graph). Thus, to fully answer Question~\ref{que:kharlamov}, we need only
consider graphs containing no full paths of length three. It turns out that these are well--understood by graph theorists.
A finite, simplicial graph $\Gamma$ is called a \emph{cograph}\index{cograph}
if it contains no full copy of $P_4$. Such graphs fall within an inductive hierarchy,
which makes them much easier to understand. The key observations are as follows: if $\gam_1$ and $\gam_2$ are cographs, then so is
their disjoint union. Moreover, it is easy to check by hand that the join $\gam_1*\gam_2$ is also a cograph. We recall briefly that
$\gam_1*\gam_2$ can be thought of as the disjoint union of $\gam_1$ and $\gam_2$, augmented by adding an edge between each vertex
of $\gam_1$ and each vertex of $\gam_2$. It turns out that these two operations completely characterize cographs. We build a
hierarchy of graphs as follows.
\begin{enumerate}
\item
Let $\KK_0$ be a singleton vertex.
\item
For $n=2i+1$ for $i\geq 0$, we set $\KK_n$ to consist of finite joins of elements in $\KK_{j}$ for $j\leq n-1$.
\item
For $n=2i+2$ for $i\geq 0$, we set $\KK_n$ to consist of finite disjoint unions of elements in $\KK_{j}$ for $j\leq n-1$.
\item
We set \[\KK=\bigcup_{n\geq 0} \KK_n.\]
\end{enumerate}
We have the following fact, which is well--known from graph theory~\cite{cograph1,cograph2,cograph3}.
The reader may consult~\cite{KK2013} for an algebraic proof that
uses right-angled Artin groups and relies fundamentally on Proposition~\ref{prop:gam-p4}.
\begin{prop}\label{prop:cograph}
If $\Gamma$ is a cograph then $\gam\in\KK_n$ for some $n\geq 0$.
\end{prop}
Proposition~\ref{prop:cograph} gives an algebraic description of right-angled Artin groups on cographs.
\begin{prop}\label{prop:cograph-alg}
The class of groups \[\{A(\gam)\mid \gam \textrm{ is a cograph}\}\] coincides with the
smallest class of finitely generated groups that contains
$\bZ$, is closed under taking direct products, and is closed under taking free products.
\end{prop}
Clearly if $\gam\in\KK_0$ then $A(\gam)\cong \bZ$. If $\gam\in\KK_1$ then $A(\gam)$ is a free abelian group. If $\gam\in\KK_2$ then
$A(\gam)$ is a free product of free abelian groups. If $\gam\in\KK_3$ then $A(\gam)$ is a direct product of free products of free abelian
groups. The groups under the purview of Proposition~\ref{prop:c-infty-raag} therefore coincide exactly with $A(\gam)$ for $\gam\in\KK_3$.
It turns out that the group $(F_2\times\bZ)*\Z$, which has a defining graph $\Lambda$ given by a path of length two together with
an isolated vertex (see Figure~\ref{f:k4}), characterizes right-angled Artin groups whose defining graphs lie in $\KK_4$.
\begin{figure}[h!]
\tikzstyle {bv}=[black,draw,shape=circle,fill=black,inner sep=1pt]
\begin{center}
\begin{tikzpicture}[main/.style = {draw, circle}]
\node[main] (1) {$a$};
\node[main] (2) [right of=1] {$b$};
\node[main] (3) [right of=2] {$c$};
\node[main] (4) [right of=3] {$d$};
\draw (1)--(2)--(3);
\end{tikzpicture}%
\caption{The defining graph $\Lambda$ for $(F_2\times\bZ)*\bZ$.}
\label{f:k4}
\end{center}
\end{figure}
\begin{prop}[See~\cite{KK2018JT}]\label{prop:k4}
We have that $\gam\in\KK_n$ for $n\geq 4$ if and only if $A(\gam)$ contains a copy of $(F_2\times\bZ)*\bZ$.
\end{prop}
The reader might note that $P_4$ does not contain a full subgraph isomorphic to $\Lambda$. However, the reader may check the following
facts.
Consider the labeling of the vertices of $P_4$ as in Figure~\ref{f:p4}. Let \[\phi_d\colon A(P_4)\longrightarrow\bZ/2\bZ\] be the
homomorphism sending $d$ to the generator of $\bZ/2\bZ$, and where $\{a,b,c,d^2\}\sse\ker\phi_d$. Then $\ker\phi_d$ is itself
isomorphic to a right-angled Artin group whose defining graph is a path of length four, with an extra degree vertex sprouting from the
middle vertex. Thus, we get an explicit realization \[A(P_5)\cong \form{ a,b,c,b^d,a^d}.\] Since $\Lambda$ is a full subgraph
of $P_5$, we get that $(F_2\times\bZ)*\bZ$ is a subgroup of $A(P_4)$.
\begin{proof}[Proof of Proposition~\ref{prop:k4}]
Clearly if $\gam\in\KK_n$ for $n\leq 1$ then $(F_2\times\bZ)*\bZ$ does not occur as a subgroup of $A(\gam)$.
It is an easy consequence of the Kurosh Subgroup Theorem that if $\gam\in\KK_2$ then $(F_2\times\bZ)*\bZ$
also does not occur as a subgroup
of $A(\gam)$. To see that if $\gam\in\KK_3$ then $A(\gam)$ does not contain a copy of $(F_2\times\bZ)*\bZ$, let $G$ and $H$ be
arbitrary groups and let \[\psi\colon (F_2\times\bZ)*\bZ\longrightarrow G\times H\] be a homomorphism. Let $p_G$ and $p_H$
be the projections of $G\times H$ onto the two factors. It is an easy exercise in combinatorial group theory that if
$\psi$ is injective then either $p_g\circ\psi$ or $p_H\circ\psi$ is injective. Thus, if $\gam\in\KK_n$ for $n\leq 3$ then $(F_2\times\bZ)*\bZ$
is not a subgroup of $A(\gam)$.
To establish the other direction, suppose that $\Gamma$ in $\KK_n\setminus\KK_{n-1}$ for $n\geq 4$.
Decomposing $\Gamma$ maximally into its join factors,
we have that some join factor of $\Gamma$ also lies in $\KK_{n-1}\setminus\KK_{n-2}$ for $n-1\geq 4$.
Indeed, if each join factor of $\Gamma$ lies in $\KK_3$
then $\Gamma$ already lies in $\KK_3$. Let $\gam_0$ be a join factor of $\Gamma$ that does not lie in $\KK_3$.
Then $\gam_0$ is a nontrivial disjoint union of graphs,
and at least one of the components lies in \[\bigcup_{n\geq 3}\KK_n\setminus\KK_2.\]
Let $\gam_0^1$ be such a component, and let $\gam_0^2$
be another component of $\gam_0$. An arbitrary vertex of $\gam_0^2$ generates a copy of $A(\gam_0^1)*\bZ$ with $\gam_0^1$.
Since $\gam_0^1$ is connected and lies in \[\bigcup_{n\geq 3}\KK_n\setminus\KK_2,\]
we have that $A(\gam_0^1)$ is nonabelian and hence
$\gam_0^1$ admits two nonadjacent vertices.
If $v$ and $w$
are two vertices of $\gam_0^1$ that are not adjacent, then an arbitrary path between them (which exists since $\gam_0^1$ is connected)
gives a copy of $A(\Lambda)\cong (F_2\times\bZ)*\bZ$ in $A(\gam)$.
\end{proof}
The group $(F_2\times\bZ)*\bZ$ is a poison subgroup for group actions on compact one--manifolds in regularity at least $C^{1+\mathrm{bv}}$.
We begin with the case of the interval.
\begin{thm}\label{thm:f2-int}
There is no injective homomorphism $(F_2\times\bZ)*\bZ\longrightarrow\Diffb(I)$.
\end{thm}
\begin{proof}
Suppose the contrary, and let such an injective homomorphism be given. We will write \[(F_2\times\bZ)*\bZ=\form{ a,b,c,t},\]
where $\form{ a,b} \cong F_2$, where $c$ commutes with $a$ and $b$, and where the cyclic subgroup generated by $t$
splits off as a free factor. We will abuse notation and use the names for the generators of $(F_2\times\bZ)*\bZ$ to denote diffeomorphisms
of $I$.
Let $J_c$ be a component of the support of $c$, and let $J_s$ be a component of the support of $s\in\form{ a,b}$.
If $J_c\cap J_s\neq\varnothing$
then we must have $J_c=J_s$. This follows from an easy application of Kopell's Lemma
(Theorem~\ref{thm:kopell}), and explicitly from Proposition~\ref{prop:disj-ab}.
If $J_c=J_s$ for some nontrivial $s\in\form{ a,b}$ then $J_c=J_a$ or $J_c=J_b$ for
suitable components of the support of $a$ or of $b$.
If $s$ is a nontrivial product of commutators, then we must in fact have $J_c=J_a=J_b$. If $s$ has a fixed point in the interior of $J_c$
then Theorem~\ref{thm:kopell} again implies that $s$ is the identity, and so $\form{ a,b}$ acts freely on $J_c$. It follows from
H\"older's Theorem (see~\ref{thm:holder}) that the action of $\form{ a,b}$ on $J_c$ factors through the abelianization.
It follows that if $s$ lies in the commutator subgroup of $\form{ a,b}$ then each component $J_s$ of $\supp s$ is disjoint from $\supp c$.
Setting $s=[a,b]$, we have that $\form{ s,c}\cong\bZ^2$ and \[(\supp s)\cap(\supp c)=\varnothing.\] It follows from the $abt$--Lemma
(Theorem~\ref{thm:abt}) that \[\form{ s,c,t} \ncong\bZ^2*\bZ.\] Since the elements $\{s,c,t\}$ generate a copy of
$\bZ^2*\bZ\le (F_2\times\bZ)*\bZ$, it follows that the initial action of $(F_2\times\bZ)*\bZ$ was not
faithful.
\end{proof}
The case $M=S^1$ is slightly more complicated, since elements of $\Diffb(S^1)$ may have no fixed points, and so the support of a
diffeomorphism may be the whole circle and not just a disjoint union of intervals.
\begin{thm}\label{thm:f2-circ}
There is no injective homomorphism $(F_2\times\bZ)*\bZ\longrightarrow\Diffb(S^1)$.
\end{thm}
Before proving Theorem~\ref{thm:f2-circ}, we reduce to the case where the $abt$--Lemma can be applied.
\begin{lem}\label{lem:tame}
Let \[\phi\colon F_2\times\bZ\longrightarrow\Diffb(S^1)\] be an injective homomorphism. Then there exists a copy $\bZ^2\le F_2\times\bZ$
whose image under $\phi$ is generated by diffeomorphisms $a$ and $b$ such that \[(\supp a)\cap (\supp b)=\varnothing.\]
\end{lem}
\begin{proof}
Write \[F_2\times\bZ\cong\form{ a,b}\times\form{c},\] and as before we identify the names of the generators with elements in
$\Diffb(S^1)$. Proposition~\ref{prop:tame} shows that the rotation number of $c$ must be rational. In particular,
after replacing $c$ by a positive power if necessary, we may assume that $\Fix c\neq\varnothing$ (Proposition~\ref{prop:rot-easy}).
By Item 4 of Proposition~\ref{prop:tame}, we have that the rotation number restricts to a homomorphism
\[F_2=\form{ a,b}\longrightarrow S^1,\] and so if $h\in F_2'$ then $h$ has a nonempty fixed point set.
Applying Proposition~\ref{prop:tame} again, we have that \[(\supp c)\cap(\supp F_2'')=\varnothing.\] Since $F_2$ is a nonabelian free group,
we have that $F_2''$ is nontrivial. The conclusion of the lemma follows.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:f2-circ}]
Suppose the contrary. Lemma~\ref{lem:tame} furnishes a copy of $\bZ^2*\bZ$ in the image of the homomorphism such that $\bZ^2$ is
generated by diffeomorphisms with disjoint support. This violates Theorem~\ref{thm:abt}.
\end{proof}
Combining all the preceding discussion, we have the following complete answer to Question~\ref{que:kharlamov}:
\begin{thm}\label{thm:kharlamov}
Let $\Gamma$ be a finite simplicial graph. Then the following are equivalent.
\begin{enumerate}[(1)]
\item
We have $A(\gam)\le \Diff_+^{\infty}(I)$.
\item
We have $A(\gam)\le \Diff_+^{\infty}(S^1)$.
\item
We have $A(\gam)\le \Diffb(I)$.
\item
We have $A(\gam)\le \Diffb(S^1)$.
\item
We have $\gam\in\KK_3$.
\end{enumerate}
\end{thm}
It is a trivial consequence of the fact that right-angled Artin groups are residually torsion--free nilpotent~\cite{DK1992a}
and Theorem~\ref{thm:ff2003}
that for a finite simplicial graph $\Gamma$, we have $A(\gam)\le \Diff_+^1(M)$ for $M\in\{I,S^1\}$. Determining the exact
critical regularity of right-angled Artin groups is an open question in general. The following is the state of the art at the time of
this book's writing.
\begin{thm}[\cite{KKR2020,KKR2021}]\label{thm:kkr-2020}
Suppose $(F_2\times F_2)*\bZ\le A(\gam)$. Then for $\tau>0$, we have that $A(\gam)$ admits no faithful $C^{1+\tau}$ action on $I$
or $S^1$.
\end{thm}
In particular, Theorem~\ref{thm:kkr-2020} implies that if $\Gamma$ has a square and is not a nontrivial join then the critical regularity of
$A(\gam)$ on a compact one--manifold is exactly one. The proof of Theorem~\ref{thm:kkr-2020} is beyond the scope of this book.
\section{Mapping class groups}\label{sec:mcg}
Part of the authors' interest in groups acting on one manifolds arose naturally from their investigations of the relationship between mapping
class groups of surfaces and right-angled Artin groups. Let $S$ be an orientable surface of finite type. That is, $S$ is an orientable,
two--dimensional real manifold such that the Euler characteristic of $S$ is finite. It is a standard fact from combinatorial topology that
$S$ is entirely determined by its genus $g$, its number of punctures $n$, and its number of boundary components $b$.
The mapping class group $\Mod(S)$ is defined to be the group of isotopy classes of orientation preserving homeomorphisms of $S$,
i.e. \[\Mod(S)=\pi_0(\Homeo_+(S)).\] Observe that, by definition, the group $\Mod(S)$ is identified with a group of outer automorphisms
of $\pi_1(S)$. We only obtain a group of outer automorphisms, since usually there is no canonical basepoint that is preserved by
homeomorphisms. If one chooses a preferred
marked point $p\in S$ and requires all homeomorphisms and isotopies to preserve $p$, then
one obtains a subgroup of $\Aut(\pi_1(S,p))$. It is a deep result of Dehn--Nielsen--Baer that if $S$ is closed, then the mapping class group
of $S$ is canonically identified with a subgroup of $\Out(\pi_1(S))$ of index two. If one chooses a marked point, then the mapping class group
of $S$ preserving $p$ is canonically identified with a subgroup of $\Aut(\pi_1(S))$ of index two. These results
admit suitable (but not necessarily obvious) generalizations to surfaces
that are not closed~\cite{FM2012}.
\subsection{Continuous actions}
The relationship between mapping class groups and homeomorphisms arises from the following
result originally due to Nielsen~\cite{Nielsen1927,HT1985}.
\begin{thm}\label{thm:nielsen}
Let $S$ be an orientable surface of finite type such that $g\geq 2$, with $n=1$ and $b=0$. Then there is an injective homomorphism
\[\phi\colon\Mod(S)\longrightarrow\Homeo_+(S^1).\]
\end{thm}
\begin{proof}[Sketch of proof of Theorem~\ref{thm:nielsen}]
We view $S$ as a closed surface of genus $g$ with a single marked point $p$. Let $\psi\in \Mod(S)$. We choose $\Psi\in\Homeo_+(S)$
lifting $\psi$, such that $\Psi(p)=p$. We lift $\Psi$ to the universal cover $\bH^2$ of $S$, so that we get a homeomorphism
$\yt\Psi\in\Homeo_+(\bH^2)$ commuting the action of $\pi_1(S)$ on $\bH^2$. The standard Morse Lemma from hyperbolic geometry
implies that $\yt\Psi$ induces a homeomorphism $\partial\yt\Psi$ on $\partial\bH^2\cong S^1$, which is unique. Since $S$ is compact,
modifying $\Psi$ by a homotopy does not affect $\partial\yt\Psi$, and so this latter
map depends only on the mapping class $\psi$, and a choice
of lift to $\bH^2$. By choosing a preferred preimage $q\in\bH^2$ lying over $p$, we obtain a preferred lift of $\Psi$ fixing $q$.
This furnishes a homomorphism from $\Mod(S)$ to $\Homeo_+(S^1)$, where $\phi(\psi)$ is $\partial\yt\Psi$ for the preferred lift.
If $\psi\in\Mod(S)$ is nontrivial then there is a nontrivial free homotopy class on $S$ that is not fixed by $\psi$. Taking a geodesic representative
$\gamma$ of such a free homotopy class, we have that the endpoints in $\partial\bH^2$ of $\gamma$ and $\psi(\gamma)$ do not coincide.
It follows that $\psi$ acts nontrivially on $\partial\bH^2$ and hence induces a nontrivial homeomorphism of $S^1$.
\end{proof}
One can say significantly more about $\phi$. For one, the action of $\Mod(S)$ on $S^1$ under $\phi$ is minimal. The regularity properties
of $\phi$ are somewhat more difficult to understand. One fact which is known is that the image of $\phi$ consists of
\emph{quasi--symmetric}\index{quasi--symmetric homeomorphism}
homeomorphisms.
Here, a homeomorphism $f\in\Homeo_+(S^1)$ is called quasi--symmetric if there exists an increasing
function \[\omega\colon [0,\infty)\longrightarrow [0,\infty)\] such
that for all triples
of distinct points $\{x,y,z\}\sse S^1$, we have \[\frac{|f(x)-f(y)|}{|f(x)-f(z)|}\leq\omega\left(\frac{|x-y|}{|x-z|}\right).\] The fact that
the map $\phi$ in Theorem~\ref{thm:nielsen} is a consequence of the fact if $f$ is a quasi--isometry of hyperbolic space $\bH^2$,
then the map $\partial f$ induced on $S^1$ is quasi--symmetric (see~\cite{kap-lect}, for instance).
Quasi--symmetry is a generalized notion of bi--Lipschitz
homeomorphism, and we have from Theorem~\ref{thm:lip-conj} that one can find an action of $\Mod(S)$ on $S^1$ by bi--Lipschitz
homeomorphisms.
Theorem~\ref{thm:nielsen} has the consequence that mapping class groups of surfaces preserving a marked point are \emph{circularly
orderable}\index{circular ordering},
which for countable groups is equivalent to being a subgroup of $\Homeo_+(S^1)$. See Appendix~\ref{ch:append2}. Circular
orderability of groups of homeomorphisms of the circle implies that mapping class groups of closed surfaces (without marked points)
cannot act faithfully on the circle, since they contain non-cyclic torsion. Since this torsion disappears in a finite index subgroup, it
is unknown whether or not there exists a finite index subgroups of the closed mapping class group acting faithfully on the circle.
If $S$ has a boundary component $B$ then we may consider the mapping class group of $S$ preserving $B$ pointwise, which we will
write $\Mod(S,B)$. It turns out that the mapping class group $\Mod(S,B)$ can be made to act faithfully on the real line, by a result
that is analogous to Theorem~\ref{thm:nielsen}. When $S$ has a boundary component, then one can pick a preferred lift of the boundary
component to $\bH^2$, which then becomes an invariant copy of the real line connecting two points in $\partial \bH^2$, and the mapping
class group acts on it faithfully.
\begin{thm}[See~\cite{HT1985}]\label{thm:thurston-handel}
If $S$ has a boundary component $B$ then there is an injective homomorphism \[\Mod(S,B)\longrightarrow\Homeo_+(\bR).\]
\end{thm}
Thus, mapping class groups of surfaces with boundary are \emph{orderable}\index{linear ordering},
which like in the case of the circle, characterizes countable
subgroups of $\Homeo_+[0,1]$ (see Appendix~\ref{ch:append2}). The mapping class group of a surface with a marked point has torsion
(though only of a cyclic kind), and hence is not orderable. This torsion disappears in a finite index subgroup of the mapping class group,
and it is an open problem to determine whether this mapping class group admits a finite index subgroup acting faithfully on the interval.
\subsection{Smoothing to $C^2$ and beyond and connections to right-angled Artin groups}
It was a conjecture for over a decade whether or not the action of
a finite index subgroup of $\Mod(S)$ could be smoothed, which is to say topologically conjugated
to a differentiable action. A more general question allows one to abandon the constraints of Nielsen's action,
and so, one can ask: is there a finite index subgroup $G\le \Mod(S)$ and an injective homomorphism $G\longrightarrow\Diff_+^k(M)$,
for $M\in\{I,S^1\}$?
This question can be fit into the general setup of the Zimmer Conjecture, which roughly asserts that ``large" groups cannot
act in interesting ways on ``small" compact manifolds. The meaning of ``large" and ``small" are usually context dependent; for instance,
one can consider a group to be large if it is a lattice in a real simple Lie group of rank $n\geq 2$, and a compact manifold to be small if
is has dimension smaller than $n$.
A lattice in a Lie group, as a mathematical object, is only well--defined up to finite index; so, one usually considers a group
up to commensurability, for the purposes of Zimmer's Conjecture.
Interesting actions are generally continuous actions (though one often considers actions of higher
regularity, or actions that in addition preserve some extra data like a measure or a form) that do not factor through finite quotients.
For mapping class groups, deciding whether they are large or small is a matter of philosophical debate. Nielsen's action does furnish
a faithful action of a mapping class groups on the circle. On the other hand, results of Ghys, Farb--Franks, and
Parwani~\cite{FF2001,Ghys1999,Parwani2008} have shown
that the full mapping class group does not admit faithful $C^2$ actions on $I$ or $S^1$ when the genus of $S$ is at least three, and
does not admit faithful $C^1$ actions when the genus of $S$ is at least $6$. These latter results rely on the full
power of the mapping class group, and the methods are not robust under passing to finite index subgroups.
Usually, finite index subgroups of mapping class groups are quite difficult to understand. For our purposes, right-angled Artin groups
come to the rescue. Let $G$ be an arbitrary group, and let $A(\gam)\le G$ be a right-angled Artin subgroup. It turns out that every finite index
subgroup of $G$ also contains a copy of $A(\gam)$. Indeed, if $H\le G$ has index $n$, then there is an $N$ such
that for all $g\in G$, we have $g^N\in H$. If $\{v_1,\ldots,v_k\}$ are the vertices of a graph $\Gamma$ viewed as generators of $A(\gam)$, the
the elements $\{v_1^N,\ldots,v_k^N\}$ also generate a group isomorphic to $A(\gam)$. In short,
the property of containing a right-angled Artin group
of a particular isomorphism type is stable under passing to finite index subgroups.
Since by Theorem~\ref{thm:a4-c2} we know that $A(P_4)$ is not a subgroup of $\Diffb(M)$ in order to rule out a faithful action of
a finite index subgroup of a mapping class group by $C^{1+\mathrm{bv}}$ diffeomorphisms on $M$, it would suffice to find copies of
$A(P_4)$ inside of these mapping class groups. These are furnished in abundance by a result of the second author.
Let $f\in\Mod(S)$ be a mapping class. We say that $f$ is \emph{reducible}\index{reducible mapping class} if there is a
nonempty collection $R$ of (isotopy classes of)
essential, nonperipheral (i.e.~not parallel to a boundary component of puncture), simple
closed curves on $S$ such that $f$ fixes $R$. The collection $R$ is called a \emph{reduction system}\index{reduction system}
for $f$. The minimal such $R$
(with respect to inclusion) always exists~\cite{BLM1983} and is called the \emph{canonical reduction system}\index{canonical reduction
system} for $f$.
We say that $f$ is \emph{pure}\index{pure mapping class} if whenever $S_0\sse S\setminus
R$ is a component, then
$f$ preserves $S_0$, preserves $R$ component-wise,
and the restriction of $f$ to $S_0$ is either the identity on $S_0$ (away from the boundary) or acts on $S_0$
irreducibly. It is a standard fact that every mapping class in $\Mod(S)$ admits a power that is pure.
If $f$ is pure and $\gamma$ is a component of $R$, then $f$ may act nontrivially in an annular neighborhood of $\gamma$. Namely,
$f$ may cut open $S$ along $\gamma$ and reglue with a full twist. Such a mapping class is called a \emph{Dehn twist}\index{Dehn twist}.
If $f$ is pure, we say that $f$ has \emph{connected support}\index{connected support}
if $f$ satisfies one of the following conditions:
\begin{enumerate}[(1)]
\item
The mapping class $f$ is a Dehn twist about a simple closed curve.
\item
The mapping class $f$ is not a Dehn twist about a simple closed curve, and the restriction of $f$ to $S\setminus R$ is irreducible
for exactly one component $S_0\sse S\setminus R$.
Moreover, we require that, essentially, $f$ does not perform Dehn twists about curves in its
canonical reduction system that do not border on $S_0$. In other words, $R=\partial S_0$.
\end{enumerate}
We direct the reader to~\cite{FM2012} for a more detailed discussion of the foregoing facts about mapping class groups.
We say that a collection of pure mapping classes with connected support is \emph{irredundant}\index{irredundant mapping classes}
if no two mapping classes in it
generate a virtually cyclic subgroup of $\Mod(S)$.
Let $X=\{f_1,\ldots,f_k\}$ be
irredundant pure mapping classes with connected support. We build a graph $\gam_X$ by taking one vertex for each element of $X$,
and drawing an edge between two vertices if the two mapping classes commute in $\Mod(S)$. The graph $\gam_X$
is called the \emph{co--intersection}\index{co--intersection graph}
graph; it is so named since generally two vertices will be adjacent in $\gam_X$ if and only if their supports are disjoint, up to isotopy
in $S$.
The following result, whose proof is beyond the scope of this book,
gives a systematic way of producing right-angled Artin groups in mapping class groups.
\begin{thm}[See~\cite{Koberda2012}]\label{thm:mcg-raag}
Let $X=\{f_1,\ldots,f_k\}$ be
irredundant pure mapping classes with connected support with co--intersection graph $\gam_X$. Then there exists an $N>0$ such
that for all $n\gg N$, we have \[\form{ f_1^n,\ldots,f_k^n}\cong A(\gam_X)\le \Mod(S).\]
\end{thm}
Theorem~\ref{thm:mcg-raag} was the first comprehensive algebraic results about subgroups of mapping class groups generated by
sufficiently high powers of mapping classes. There are many results that are closely related to Theorem~\ref{thm:mcg-raag}, some which
preceded it~\cite{CP2001,CW2007}, some which were contemporary and focussed more on the geometric aspects of the embedding
$A(\gam_X)\longrightarrow\Mod(S)$~\cite{CLM2012},
and some which focussed on effectivization~\cite{Seo2021,Runnels2021}. For the sake of space,
we will not survey the literature on this topic in greater depth.
In order to find copies of $A(P_4)$ in the largest variety of mapping class groups, we consider copies of $A(P_4)$ generated by the
mapping classes that take up the ``least amount of space", which is to say they each are supported on a single annulus. If
we consider four distinct (i.e.~pairwise non--isotopic), essential, nonperipheral, simple closed
curves $\{\gamma_1,\ldots,\gamma_4\}$ which have the property that $\gamma_i\cap\gamma_j\neq\varnothing$
if and only if $|i-j|=1$ (where this intersection is minimized over the isotopy classes of these curves), then the curves
$\{\gamma_1,\ldots,\gamma_4\}$ form a \emph{chain}\index{chain of curves}
of four curves. Considering the Dehn twists $\{T_1,\ldots,T_4\}$ about these
curves, we may apply Theorem~\ref{thm:mcg-raag} to find powers of these twists which generate a right-angled Artin group.
The resulting group is in fact $A(P_4)$; this is a consequence of the fact that the co--intersection graph of $\{T_1,\ldots,T_4\}$ is
isomorphic to the graph whose vertices are $\{\gamma_1,\ldots,\gamma_4\}$ and whose adjacency relation is given by
nontrivial intersection. In other words, the graph $P_4$ is self--complementary.
If $S_{g,n}$ denotes the surface of genus $g$ with $n$ punctures and boundary components (as for the purpose of this discussion,
the difference between punctures and boundary components is immaterial), then there is a natural measure of complexity of
$S_{g,n}$ given by \[c(S_{g,n})=3g-3+n.\] The reader may note that $c(S_{g,n})$ resembles the Euler characteristic, though it is not
exactly the same thing. An elementary exercise in surface topology shows that $c(S_{g,n})$ is exactly the size of a maximal
collection of (isotopy classes of) distinct, pairwise disjoint, essential, nonperipheral, simple closed curves on $S$.
The following is also an easy exercise in combinatorial topology:
\begin{prop}\label{prop:surface-complex}
Let $S=S_{g,n}$. Then $S$ admits a chain of four curves if and only if $c(S)\geq 2$.
\end{prop}
Thus, the moment that $S$ admits two disjoint simple closed curves, it admits a chain of four of them. We can characterize such surfaces
as ones where $g\geq 2$, or $g= 1$ and $n\geq 2$ or $g=0$ and $n\geq 5$.
It turns out that, if $c(S)\leq 1$ then every right-angled Artin
subgroup of $\Mod(S)$ is (virtually) a product of a free group and an abelian group,
and is therefore relatively uninteresting from the point of view
of regularity of group actions on one-manifolds.
We have illustrated two different
chains of four curves in Figure~\ref{f:chain-surface}.
\begin{figure}[htb!]
\centering
{
\begin{tikzpicture}[scale=.6,rotate=90]
\draw (-.5,0) node {};
\draw (0,-2) -- (9,-2);
\draw (0,2) -- (9,2);
\draw (4.5,.5) edge [out=0,in=90] (5,0) edge [out=180,in=90] (4,0) ;
\draw (4.5,-.2) edge [out=0,in=-135] (5.1,.1) edge [out=180,in=-45] (3.9,.1);
\draw (4.5,2) edge [brown,thick,out=-75,in=75] (4.5,.5) edge [brown,thick,dashed,out=-105,in=105](4.5,.5);
\draw (4.5,-.2) edge [ucl2dkgreen,thick,out=-75,in=75] (4.5,-2) edge [ucl2dkgreen,thick,dashed,out=-105,in=105] (4.5,-2);
\draw (1,2) edge [red,thick,out=-80, in=180] (4.5,-.7);
\draw [dashed] (1,2) edge [red,thick,out=-100, in=180] (4.5,-1.35);
\draw (8,2) edge [red,thick,out=-100, in=0] (4.5,-.7);
\draw [dashed] (8,2) edge [red,thick,out=-80, in=0] (4.5,-1.35);
\draw [dashed] (1,-2) edge [blue,thick,out=80, in=-180] (4.5,.9);
\draw (1,-2) edge [blue,thick,out=100, in=-180] (4.5,1.35);
\draw [dashed] (8,-2) edge [blue,thick,out=100, in=0] (4.5,.9);
\draw (8,-2) edge [blue,thick,out=80, in=0] (4.5,1.35);
\node [ucl2dkgreen] at (5,-1.7) {\footnotesize $c$};
\node [brown] at (5,1.7) {\footnotesize $b$};
\node [blue] at (8.35,-1) {\footnotesize $d$};
\node [red] at (8.35,1) {\footnotesize $a$};
\end{tikzpicture}
}
$\qquad\qquad$
{\begin{tikzpicture}[scale=.55,rotate=90]
\draw (2,-2) -- (11,-2);
\draw (2,2) edge [out=180,in=180] (2,-2) edge (11,2);
\draw (2,.5) edge [out=0,in=90] (2.5,0) edge [out=180,in=90] (1.5,0) ;
\draw (2,-.2) edge [out=0,in=-135] (2.6,.1) edge [out=180,in=-45] (1.4,.1);
\draw (6,.5) edge [out=0,in=90] (6.5,0) edge [out=180,in=90] (5.5,0);
\draw (6,-.2) edge [out=0,in=-135] (6.6,.1) edge [out=180,in=-45] (5.4,.1);
\draw (10,.5) edge [out=0,in=90] (10.5,0) edge [out=180,in=90] (9.5,0);
\draw (10,-.2) edge [out=0,in=-135] (10.6,.1) edge [out=180,in=-45] (9.4,.1);
\draw (6,1.5) edge [blue,thick,out=180, in=30] (2.27,.41) edge [thick,blue,out=0, in=90] (7.5,0);
\draw (6,-1.5) edge [thick,blue,out=180, in=-30] (2.27,-.16) edge [thick,blue,out=0, in=-90] (7.5,0);
\draw (6,.95) edge [thick,blue,dashed, out=180, in=10] (2.27,.41);
\draw [thick,blue,dashed, out=-10, in=180] (2.27,-.16) edge (6,-.95);
\draw (7,0) edge [thick,blue,dashed,out=90, in=0] (6,.95) edge [thick,blue,dashed,out=-90, in=0] (6,-.95);
\draw (9.73,.41) edge [thick,red,out=150, in=0] (6,1.2) edge [thick,red,dashed, out=170, in=0] (6,.75);
\draw (9.73,-.16) edge [thick,red,out=-150, in=0] (6,-1.2) edge [thick,red,dashed, out=-170, in=0] (6,-.5);
\draw (4.5,0) edge [thick,red,out=-90, in=180] (6,-1.2) edge [thick,red,out=90, in=180] (6,1.2);
\draw (5,0) edge [thick,red,dashed,out=90, in=180] (6,.75) edge [thick,red,dashed,out=-90, in=180] (6,-.5);
\draw (3.5,2) edge [brown,thick,out=-75,in=75] (3.5,-2) edge [brown,thick,dashed,out=-105,in=105](3.5,-2);
\draw (8.5,2) edge [ucl2dkgreen,thick,out=-75,in=75] (8.5,-2) edge [ucl2dkgreen,thick,dashed,out=-105,in=105] (8.5,-2);
\node [brown] at (3,-1.5) {\footnotesize $b$};
\node [blue] at (4.8, 1.7) {\footnotesize $d$};
\node [red] at (7.7,1.4) {\footnotesize $a$};
\node [ucl2dkgreen] at (9,-1.5) {\footnotesize $c$};
\end{tikzpicture}
}
$\qquad\qquad$
{
\begin{tikzpicture}[scale=.48,rotate=90]
\draw (1,-2) -- (11.5,-2);
\draw (1,2) edge [out=180,in=180] (1,-2) edge (11.5,2);
\draw (3,1) edge [out=0,in=90] (3.5,.5) edge [out=180,in=90] (2.5,.5) ;
\draw (3,.3) edge [out=0,in=-135] (3.6,.6) edge [out=180,in=-45] (2.4,.6);
\draw (6,1) edge [out=0,in=90] (6.5,.5) edge [out=180,in=90] (5.5,.5);
\draw (6,.3) edge [out=0,in=-135] (6.6,.6) edge [out=180,in=-45] (5.4,.6);
\draw (9,1) edge [out=0,in=90] (9.5,.5) edge [out=180,in=90] (8.5,.5);
\draw (9,.3) edge [out=0,in=-135] (9.6,.6) edge [out=180,in=-45] (8.4,.6);
\draw (1,2) edge [brown,thick,out=-80, in=180] (4.5,-.7);
\draw [dashed] (1,2) edge [brown,thick,out=-100, in=180] (4.5,-1.35);
\draw (8,2) edge [brown,thick,out=-100, in=0] (4.5,-.7);
\draw [dashed] (8,2) edge [brown,thick,out=-80, in=0] (4.5,-1.35);
\node [brown] at (2.3, 1.4) {\footnotesize $S_1$};
\draw (4,2) edge [blue,thick,out=-80, in=180] (7.5,-.7);
\draw [dashed] (4,2) edge [blue,thick,out=-100, in=180] (7.5,-1.35);
\draw (11,2) edge [blue,thick,out=-100, in=0] (7.5,-.7);
\draw [dashed] (11,2) edge [blue,thick,out=-80, in=0] (7.5,-1.35);
\node [blue] at (9.7, 1.4) {\footnotesize $S_2$};
\end{tikzpicture}
}
\caption{Various ways to realize $A(P_4)$ as a subgroup of $\Mod(S)$. The first two pictures illustrate chains of four curves, where in
the first picture they can be nonseparating in $S$, and in the second picture they are all separating in $S$. The third picture illustrates
the first two surfaces in a chain of subsurfaces of $S$.}
\label{f:chain-surface}
\end{figure}
We thus obtain a corollary to Theorem~\ref{thm:mcg-raag}, Proposition~\ref{prop:surface-complex}, and Theorem~\ref{thm:a4-c2},
which completely answers the $C^2$--smoothability question for finite index subgroups of mapping class groups of surfaces:
\begin{thm}\label{thm:mcg-c2}
Let $M\in\{I,S^1\}$. There exists a finite index subgroup $G\le \Mod(S)$ and an embedding $G\longrightarrow \Diffb(M)$ if and only if
$c(S)\leq 1$.
\end{thm}
\subsection{Smooth actions of related groups}
Theorem~\ref{thm:mcg-raag} gives us analogues of Theorem~\ref{thm:mcg-c2} for many other classes of groups other than $\Mod(S)$.
Mapping class groups of surfaces with punctures or boundary components can be identified with groups of automorphisms (or outer
automorphisms) of free groups. So, Theorem~\ref{thm:mcg-c2} carries over verbatim with mapping class groups replaced by $\Aut(F_n)$
and $\Out(F_n)$ for $n\geq 3$.
The mapping class group of a surface $S$ admits a natural linear representation, arising from the action of $\Mod(S)$ on the
first homology
of $S$. Since the homology of $S$ is an abelian invariant, the action of the automorphism group of $\pi_1(S)$ on $H_1(S,\bZ)$
factors through $\Out(\pi_1(S))$. We therefore have a map \[\rho\colon \Mod(S)\longrightarrow \GL_n(\bZ),\] where $n$ is the
rank of $H_1(S,\bZ)$. It is not difficult to see that the image of $\rho$ is infinite; indeed, if $\gamma$ is a simple closed curve whose
homology class $[\gamma]$ is nonzero, then the Dehn twist $T$ about $\gamma$ is easily seen to have an infinite order action on
$H_1(S,\bZ)$. If $S$ is a closed surface of genus $g\geq 1$, then in fact the image of $\rho$ is the full group $\mathrm{Sp}_{2g}(\bZ)$,
the $2g\times 2g$ symplectic group over $\bZ$ (see Chapter 6 of~\cite{FM2012}).
The symplectic form preserved by the action of $\Mod(S)$ is the algebraic intersection
pairing on $H_1(S,\bZ)$.
The kernel of the map $\rho$ is called the \emph{Torelli group}\index{Torelli group} of $S$, and is written $\II(S)$.
Unless one has had some experience dealing with mapping class groups
of surfaces, it may not be obvious that $\rho$ is not injective, and that
consequently $\II(S)$ is nontrivial. Indeed, it is true and not completely trivial that
if $S$ has genus one and at most one puncture, then the map $\rho$ is injective.
The moment that $S$ admits an essential simple closed curve
$\gamma$ that is nonperipheral and separating (i.e.~$S\setminus\gamma$ is disconnected), then $\II(S)$ is nontrivial. Indeed, it is
easily checked that if $\gamma$ is such a curve then the Dehn twist about $\gamma$ acts trivially on $H_1(S,\bZ)$. Some argument
is needed to show that the Dehn twist about $\gamma$ represents a nontrivial mapping class, though this can be established
without too much difficulty by computing
a lift of the Dehn twist to a finite cover of $S$ and showing that the lift acts nontrivially on the homology of the
cover (see~\cite{KoberdaGD,HadariGT20,KobMan15,LiuJAMS20} for instance).
If $S$ admits a chain of separating curves, then $\II(S)$ contains a copy of $A(P_4)$. If $S$ is closed and of genus at least three, or
has genus two and two punctures or boundary components, then such a chain exists. See the middle picture in Figure~\ref{f:chain-surface}.
We thus have the following consequence:
\begin{cor}\label{cor:torelli}
Let $S=S_{g,n}$, and let $G\le \II(S)$ be a finite index subgroup. Then $G$ admits no faithful action by $C^{1+\mathrm{bv}}$ diffeomorphisms
on $M$, for $M\in\{I,S^1\}$, provided that one of the following conditions holds:
\begin{enumerate}[(1)]
\item
We have $g\geq 3$.
\item
We have $g=2$ and $n\geq 2$.
\end{enumerate}
\end{cor}
One can dig even deeper in the mapping class group than the Torelli group and obtain conclusions analogous to Corollary~\ref{cor:torelli}.
For simplicity, assume that $S$ has a fixed marked point $p$ that is preserved by $\Mod(S)$, so that $\Mod(S)$ is naturally a subgroup
of $\Aut(\pi_1(S))$. Write $\gamma_i(\pi_1(S))$ for the $i^{th}$ term of the lower central series, where $i\geq 1$. That is,
$\gamma_1(\pi_1(S))=\pi_1(S)$ and \[\gamma_{i+1}(\pi_1(S))=[\pi_1(S),\gamma_i(\pi_1(S))].\] We write
\[N_i=\pi_1(S)/\gamma_{i+1}(\pi_1(S)),\]
so that $N_i$ is the largest $i$--step nilpotent quotient of $\pi_1(S)$. It is a fact that we have alluded to before that
\[\bigcap_i \gamma_i(\pi_1(S))=\{1\}.\] This last equality is implied by the assertion that $\pi_1(S)$ is residually torsion--free nilpotent.
We will not prove this fact here, and instead direct the reader to~\cite{MKS04} or a combination of~\cite{DK1992a}
and~\cite{CW2004}, or leave it as a good exercise.
Since the subgroup $\gamma_i(\pi_1(S))$ is invariant under all automorphisms of $\pi_1(S)$, we obtain a map
\[\rho_i\colon \Aut(\pi_1(S))\longrightarrow\Aut(N_i).\] Restricting the map $\rho_i$ to $\Mod(S)$, we write $\JJ_i(S)=\ker(\rho_i)\cap\Mod(S)$.
The sequence of nested subgroups $\{\JJ_i(S)\}_{i\ge1}$ is called the \emph{Johnson filtration}\index{Johnson filtration}
of $S$. Note that $\JJ_1(S)=\II(S)$.
Since the intersection of the terms
of the lower central series of $\pi_1(S)$ intersect in the identity, we obtain \[\bigcap_i\JJ_i(S)=\{1\}\] as a formal consequence.
Nontriviality of the terns in the Johnson filtration for $i\geq 2$,
however, is not a formal consequence of the definitions. It is true that, if $S$ has genus
at least two, then each term of $\JJ_i(S)$ is nontrivial, and $\JJ_{i+1}(S)$ has infinite index in $\JJ_i(S)$. If $S_0\longrightarrow S$
is an inclusion of connected surfaces, then (ignoring some basepoint issues) we obtain a map \[\JJ_i(S_0)\longrightarrow\JJ_i(S)\] for
all $i$.
These considerations allow us to find many copies of $A(P_4)$ inside of each term of the Johnson filtration of $S$, provided that
$S$ is large enough. A connected subsurface $S_0\sse S$ is \emph{essential}\index{essential subsurface}
if $S_0$ is not contractible and if the inclusion map is $\pi_1$--injective.
Like in the case of a collection of simple closed curves, a collection
$\{S_1,\ldots,S_k\}$ of pairwise non--isotopic essential subsurfaces of $S$ is a \emph{chain}\index{chain of subsurfaces}
if $S_i\cap S_j\neq\varnothing$ if and only
if $|i-j|\leq 1$, where this intersection is minimized within the respective isotopy classes. See the third picture in Figure~\ref{f:chain-surface}.
The preceding discussion implies easily that if $S$ admits a chain of four essential subsurfaces,
each of which has genus at least two, then $\JJ_i(S)$ contains a copy of $A(P_4)$ for all $i\geq 1$. Such a chain exists, provided that the
genus of $S$ is at least five.
\begin{cor}\label{cor:johnson}
Let $i\geq 1$, let $S=S_{g,n}$, and let $G\le \JJ_i(S)$ be a finite index subgroup.
Then $G$ admits no faithful action by $C^{1+\mathrm{bv}}$ diffeomorphisms
on $M$, for $M\in\{I,S^1\}$, provided that $g\geq 5$.
\end{cor}
\subsection{$C^{1+\tau}$ actions}
To analyze regularities that are weaker than $C^{1+\mathrm{bv}}$, we need to consider more complicated subgroups of $\Mod(S)$.
We have that $\Mod(S)$ contains a copy of $(F_2\times F_2)*\bZ$ provided that:
\begin{enumerate}
\item
The genus of $S$ is at least two.
\item
The genus of $S$ is one and $S$ has at least three punctures or boundary components.
\item
The genus of $S$ is zero and $S$ has at least six punctures or boundary components.
\end{enumerate}
This is easily seen from the fact that these surfaces' mapping class groups contain copies of $F_2\times F_2$ generated by
the squares of two pairs of noncommuting
Dehn twists supported on disjoint surfaces,
say $\{T_1,\ldots,T_4\}$, and by adding another Dehn twist that commutes with none of $\{T_1,\ldots,T_4\}$. The fact that
sufficiently high powers of these Dehn twists then generate a copy of $(F_2\times F_2)*\bZ$ follows from Theorem~\ref{thm:mcg-raag}.
Theorem~\ref{thm:kkr-2020} implies that for all $\tau>0$, the group $(F_2\times F_2)*\bZ$ cannot act on the interval $I$
or $S^1$ by $C^{1+\tau}$
diffeomorphisms. Thus, we obtain:
\begin{cor}\label{cor:mcg-tau}
Let $S=S_{g,n}$, with $g\geq 2$, or $g= 1$ and $n\geq 3$, or $g=0$ and $n\geq 6$, and let $G\le \Mod(S)$ be a finite index subgroup.
Then for $\tau>0$, there is no faithful homomorphism $G\longrightarrow\Diff^{1+\tau}(M)$ for $M\in\{I,S^1\}$.
\end{cor}
In particular, if $S$ is sufficiently complicated and
if $\Mod(S)$ admits a finite index subgroup with a faithful $C^0$ action on $M$, then the critical regularity of that finite index
subgroup of $\Mod(S)$ is exactly one.
When $S$ is closed, or when it has a puncture or marked point and $M=I$, then the content of
Corollary~\ref{cor:mcg-tau} is unclear since we do not know if there
are such subgroups $G$ acting faithfully by homeomorphisms. However, if $S$ has a boundary component then
Theorem~\ref{thm:thurston-handel} does furnish a faithful action of $\Mod(S)$ on $I$ by homeomorphisms, which lends content to
Corollary~\ref{cor:mcg-tau}.
\subsection{$C^1$ actions and Ivanov's Conjecture}
For $C^1$--diffeomorphisms, there is little one can say for finite index subgroups of $\Mod(S)$, given the current state of knowledge.
For the whole mapping class group,
Mann--Wolff proved that for $S=S_{g,1}$, every action of $\Mod(S)$ on $S^1$
is semi-conjugate to Nielsen's action~\cite{MWGT20}. Previously, Parwani~\cite{Parwani2008} showed that every $C^1$ action of $\Mod(S)$ on $S^1$ is trivial,
provided that $g\geq 6$. The basic
idea behind Parwani's proof is the following fact:
\begin{thm}[See~\cite{Parwani2008}, Theorem 1.4]\label{thm:parwani-prod}
Let $G$ and $H$ be finitely generated groups with trivial abelianizations. If \[\phi\colon G\times H\longrightarrow\Diff^1_+(S^1)\]
is a homomorphism, then the restriction of $\phi$ to $\{1\}\times H$ or $G\times\{1\}$ is trivial.
\end{thm}
The basic idea behind the proof of Theorem~\ref{thm:parwani-prod} is to find an invariant measure for $G\times H$, and in the absence
of one, to find an invariant measure for one the factors $\{1\}\times H$ or $G\times\{1\}$. In either case, one finds a global fixed point
for the action of $\{1\}\times H$ or $G\times\{1\}$, which by Thurston's Stability Theorem~\ref{thm:thurston-stab} implies that the
action of one of these two factors is trivial.
If $g\geq 6$ then $\Mod(S)$ contains two disjoint essential subsurfaces $S_1$ and $S_2$, each of which has genus at least $3$.
We immediately have that the inclusion of subsurfaces $S_1$ and $S_2$ into $S$ induces an injection
\[\Mod(S_1)\times \Mod(S_2)\longrightarrow \Mod(S).\] A basic result about mapping class groups shows that if $S=S_{g,n}$ has
genus at least three then $\Mod(S)$ has trivial abelianization (see for example Chapter 5 of~\cite{FM2012}, also~\cite{Harer83}).
Combining with Theorem~\ref{thm:parwani-prod}, we see that
if $S$ has genus at least six then $\Mod(S)$ does not admit a faithful action on $S^1$ by $C^1$ diffeomorphisms.
To get triviality
of an arbitrary $C^1$ action of $\Mod(S)$, one simply argues that a generating set for $\Mod(S)$ must lie in the kernel of the action.
For closed surfaces, this follows from the fact that a Dehn twist about a nonseparating simple closed curve on $S$ normally generates
the whole mapping class group of $S$. For
surfaces with punctures, some extra argument is needed.
As we have suggested already,
Parwani's result relies on the fact that most mapping class groups have trivial abelianizations. This fact is a consequence of several
facts which fail for finite index subgroups of mapping class groups. The first is that the whole mapping class group is generated by
the conjugacy class of a Dehn twist about a nonseparating simple closed curve, provided that the surface is closed and has positive
genus. The other is that there is a particular relation among seven Dehn twists, known as the \emph{lantern relation}\index{lantern
relation}. In the abelianization
of the whole mapping class group in genus three or more, this relation can be made to read that the image of the cube of a Dehn twist
is equal to its fourth power.
Both of these fact fail for finite index subgroups of $\Mod(S)$,
and it is not known whether if $G\le \Mod(S)$ has finite index then $G$ has finite abelianization. The positive version of this last statement
is known as \emph{Ivanov's Conjecture}\index{Ivanov's Conjecture},
and it follows from Parwani's work that if Ivanov's Conjecture is true then there is no faithful
$C^1$ action by a finite index subgroup of $\Mod(S)$ for $g\geq 6$.
|
1,314,259,996,530 | arxiv | \section{Introduction}
\IEEEPARstart{T}{he} gaze estimation is a process of identifying the line-of-sight of the pupils at a particular instant. Eye gaze provides an important information about human visual attention and cognitive process~\cite{mason2004look,sun2014toward,wang2017deep}. It has a wide range of interactive applications including human-robot interaction~\cite{ghosh2018speech,zhou2022rfnet,sharma2021gaze}, student engagement detection~\cite{kaur2018prediction}, video games~\cite{barr2007video,cheng2013gaze}, driver attention modelling~\cite{fridman2016driver}, psychology research~\cite{birmingham2009human}, etc.
Eye gaze estimation techniques can be broadly classified into two types: \textit{intrusive} and \textit{non-intrusive}. The intrusive technique requires physical contact with user skin or eyes. It includes usage of head-mounted devices, electrodes, or sceleral coils~\cite{xia2007ir,robinson1963method,tsukada2011illumination}. These devices provide accurate gaze estimation but can cause an unpleasant user experience. On the other hand, the non-intrusive technique does not require physical contact~\cite{leo2014unsupervised}. The image processing based gaze estimation methods come under the non-intrusive category. These methods face several challenges, such as occlusion, illumination condition, head pose, specular reflection etc. To overcome these limitations, most of the gaze estimation methods were conducted in constrained environments like fixation of head pose, illumination conditions, camera angle, etc. Moreover, if the method is supervised it require a lot of high-resolution labeled images along with fast and accurate pupil-center localization.
Eye gaze is generally estimated in terms of 2D/3D location or angle in subject's visual space. With the success of supervised deep learning techniques, much progress has been witnessed in most computer vision problems. This is primarily due to the availability of large-sized labeled databases (e.g.: Gaze360~\cite{gaze360_2019}, Eth-X-Gaze~\cite{zhang2020eth}, EVE~\cite{Park2020ECCV} etc.). Furthermore, it has been observed that the labeling of complex vision tasks especially 3D gaze is a noisy and erroneous process. Labelling of 3D gaze dataset requires participant's cooperation and complicated setup.
Over the past few years, an active research effort is dedicated towards unsupervised, self-supervised and weakly-supervised methods for many real-world applications as it lessen the requirement to acquire the labeled data. Moreover, these methods has recently demonstrated application specific promising results as well~\cite{kolesnikov2019revisiting}. Self-supervised learning techniques are based on a defined \emph{pretext} task which mostly formulated using unlabeled data. In this paper, we define relative pupil location as a pretext task to learn rich representation. The pretext task is mainly inspired by the commonalities between humans' facial features as they shift their gaze from one direction to another. Based on this heuristic, we identify the possible gaze zones. Here, the gaze zones are divided into three regions, i.e, left, right, center. Our pretext task detects the coarse region of interest (aka possible visual attention of the subject) which in turns serves as pseudo labels for self supervised learning. Further, we propose an `Ize-Net' architecture that consists of capsule layer based CNN for learning a discriminating eye-gaze representation. Further, this higher-level semantic understanding is utilized to solve the downstream task. In our case, the downstream tasks include 2D/3D location/angle of eye gaze, visual attention estimation and driver gaze estimation. In brief, we first train our proposed `Ize-Net' model for solving the pretext tasks to learn rich representations which can further be used for solving the downstream tasks of interest. The experimental results show the effectiveness of our technique in predicting the eye gaze as compared to supervised techniques.
This manuscript \textit{subsumes our earlier work~\cite{dubey2019unsupervised}}. The major changes are as follows: 1) We analyze the effect of learning representation from the eye region only; 2) We add two relevant datasets (MPII and RT-GENE) in the experiment section; 3) We re-evaluate the label through voting and analyze its effect; 4) We adapt our model for driver gaze estimation task (i.e. downstream task); 4) We validate the performance of the `Pretext task' over CAVE dataset.
The \textbf{main contributions} of this paper are as follows:
\begin{itemize}
\item To the best of our knowledge, we propose \textit{RAZE, a Region guided self supervised gAZE representation learning framework}, one of the first self-supervised technique for eye gaze estimation. The representation learning is guided by a heuristic based auxiliary function i.e. pseudo gaze zone labels.
\item We automatically collect and annotate a dataset (Figure~\ref{fig:sample_images}) of 1,54,251 facial images of 100 different subjects from YouTube videos. The experimental results suggest that this heuristic based annotation method can extract substantial training data for learning robust gaze representation.
\item We propose a capsule layer based deep neural network, `Ize-Net', which is trained on the proposed dataset. The experimental results show that self-supervised techniques can be used for learning rich representation for eye gaze.
\item We demonstrate the effectiveness of learned features for solving downstream tasks as follows: 2D/3D location/angle in subject's visual space, visual attention estimation and driver gaze estimation.
\end{itemize}
The remainder of this paper is organized as follows: Section~\ref{sec:Related Work} describes the relevant prior works. Section~\ref{sec:Proposed Method} presents the details of the proposed pupil-center localization and gaze estimation methods. In Section~\ref{sec:Experiments}, we empirically study the performance of the proposed approach. Section~\ref{sec:Conclusion and Future Work} contains the conclusion, limitation and future work.
\begin{figure}[t]
\centering
\subfloat{\includegraphics[width = 1in,height=1in]{id1_672_p-center_h-center.png}}\hspace{0.01in}
\subfloat{\includegraphics[width = 1in,height=1in]{id58_9049_p-left_h-right.png}}\hspace{0.01in}
\subfloat{\includegraphics[width = 1in,height=1in]{id31_82_p-left_h-right.png}} \\
\subfloat{\includegraphics[width = 1in,height=1in]{id91_5410_p-right_h-left.png}}\hspace{0.01in}
\subfloat{\includegraphics[width = 1in,height=1in]{id58_215_p-right_h-left.png}}\hspace{0.01in}
\subfloat{\includegraphics[width = 1in,height=1in]{id39_2137_p-right_h-left.png}}\hspace{0.01in}
\\
\subfloat{\includegraphics[width = 1in,height=1in]{id68_1855_p-right_h-left.png}}\hspace{0.01in}
\subfloat{\includegraphics[width = 1in,height=1in]{id46_5415_p-right_h-left.png}}\hspace{0.01in}
\subfloat{\includegraphics[width = 1in,height=1in]{id55_1156_p-right_h-center.png}}\hspace{0.01in}
\caption{Sample images from proposed dataset. Here, we can see that there is huge variation in illumination, facial attributes of subjects, specular reflection, occlusion, etc. First and second rows from top show images for which the gaze region is correctly estimated and third row shows images where gaze region is not correctly estimated. First row; left image subject is looking towards left region. First row; middle image subject is looking towards right region. First row; right image subject is looking towards central region. Second row contains images of challenging scenarios like, occlusion and specular reflection; for which we get correct gaze region estimation. Last row contains images of scenarios where our method fails due to insufficient information for determining correct gaze region. (Image Source: YouTube creative commons)}
\label{fig:sample_images}
\end{figure}
\section{Related Work}
\label{sec:Related Work}
\subsection{Eye Gaze Estimation}
A thorough analysis of gaze estimation literature is mentioned in a recent survey~\cite{ghosh2021Automatic}. Prior works on eye gaze estimation can be broadly classified into hand-crafted and appearance-based methods. We also discuss prior works on pupil center localization as it is relevant to our pretext task.
\subsubsection{Hand-crafted methods} utilize the prior knowledge based on eye anatomy to determine feature values which further help in gaze estimation. Christoph Rasche~\cite{rasche2013curve} propose a labeling functions to identify curved, inflexion and straight segments. With respect to eye gaze, the detection of subject's pupil-centers from simple pertinent features based on shape, geometry, color, and symmetry. These features are then used to extract eye movement information. Morimoto et al.~\cite{morimoto2000pupil} assume a flat cornea surface and proposed a polynomial regression method for gaze estimation. In another interesting work, Zhu et al.~\cite{zhu2002subpixel} extract intensity feature from an image and used a Sobel edge detector to find pupil-center. The gaze direction is further determined via linear mapping function. The main drawback of this method is that the detected gaze direction is sensitive to the head pose; therefore, the users must stabilize their heads. Similarly, Torricelli et al.~\cite{torricelli2008neural} perform the iris and corner detection to extract the geometric features mapped to the screen coordinates by the general regression neural network. Valenti et al.~\cite{valenti2012accurate,valenti2011combining} estimate the eye gaze by combining the information of eye location and head pose.
\subsubsection{Appearance-based gaze estimation} methods do not explicitly extract the features; instead, these utilize the whole facial/eye image for gaze estimation. Additionally, these methods normally do not require cameras' geometry information and calibration~\cite{lu2017appearance} since the gaze mapping is directly performed on the image content. Fully supervised gaze estimation methods usually require a large number of images to train the estimator. To reduce the training cost, Lu et al.~\cite{lu2014learning} propose a decomposition scheme. It includes the initial gaze estimation and the subsequent compensations for the gaze estimation to perform effectively using training samples. Huang et al.~\cite{huang2015tabletgaze} propose an appearance-based gaze estimation method in which the video captured from the tablet was processed using HoG features and Linear Discriminant Analysis (LDA). Lu et al.~\cite{lu2010novel} propose an eye gaze tracking system which extracted the texture features from the eye regions using the local pattern model. Then a the Support Vector Regressor is utilized to obtain the gaze mapping function. Zhang et al.~\cite{zhang2017mpiigaze} propose GazeNet, which was a deep gaze estimation method. Williams et al.~\cite{williams2006sparse} propose a sparse and semi-supervised Gaussian process model to infer the gaze, which simplifies the process of collecting training data. In brief, the statistical inference based mapping is performed based on K nearest neighbor~\cite{huang2015tabletgaze}, support vector regression~\cite{smith2013gaze}, random forest~\cite{huang2015tabletgaze} and deep learning methods~\cite{krafka2016eye,zhang2017s,zhang2017mpiigaze,jyoti2018automatic,FischerECCV2018,cheng2020gaze,D_2021_CVPR,lrd2022parks}.
Several studies~\cite{sugano2014learning,benfold2011unsupervised,zhang2017everyday,santini2017calibme,park2019few,karessli2017gaze,he2019device,lu2015gaze} explore gaze estimation in unsupervised and semi-supervised settings to reduce the burden of data annotation. These approaches are mainly based on `learning-by-synthesis'~\cite{sugano2014learning}, hierarchical generative models~\cite{wang2018hierarchical}, conditional random field~\cite{benfold2011unsupervised}, unsupervised gaze target discovery~\cite{zhang2017everyday}, gaze redirection~\cite{yu2019improving}, multi-task learning/MTGLS~\cite{ghosh2021mtgls}, weakly supervised using via `Looking At Each Other (LAEO)'~\cite{kothari2021weakly}, cross-modal supervision~\cite{ghosh2022av} and few-shot learning~\cite{park2019few}. MTGLS~\cite{ghosh2021mtgls} framework leverages complementary signals via the line of sight of the pupil, the head-pose and the eye dexterity.
In literature, the domain specific knowledge is also leveraged to get strong complimentary information. These information includes facial landmark~\cite{yu2018deep}, screen saliency~\cite{Park2020ECCV,wang2019inferring}, depth~\cite{lian2019rgbd}, headpose~\cite{zhu2017monocular}, segmentation mask~\cite{wu2019eyenet} and uncertainty~\cite{gaze360_2019}. Unlike this, our study focuses on automatic gaze region labeling as pretext task to reduce the annotation burden as well as infer coarse to fine gaze adaptation.
\subsubsection{Pupil Center Localization}
Prior works on pupil-center localization can be broadly classified into two categories based on active and passive techniques~\cite{leo2014unsupervised}. The active pupil-center localization methods utilize dedicated devices to locate the pupil-center by infrared camera~\cite{xia2007ir}, contact lenses~\cite{robinson1963method} and head-mounted devices~\cite{tsukada2011illumination}. These devices require a pre-calibration phase to perform accurately. These are generally very expensive and cause an uncomfortable user experience. The passive eye localization methods try to gather information from the supplied image/video-frame regarding the pupil-center. Valenti et al.~\cite{valenti2012accurate} have used identical images to infer circular patterns and used machine learning for the prediction task. An open eye can be peculiarly defined by its shape and its components like iris and pupil contours. The structure of an open eye can be used to localize it in an image. Such methods can be broadly divided into voting-based methods~\cite{kim1999vision,perez2003precise} and model fitting methods~\cite{daugman2003importance,hansen2005eye}. Although these methods seem very intuitive, but it fails to provide good accuracy in real world secnarios. Several machine learning based pupil-center localization methods have also been proposed. One such method was proposed by Campadelli et al.~\cite{campadelli2009precise}, in which they used two Support Vector Machines (SVM) and trained them on properly selected Haar wavelet coefficients. Markuvs et al.~\cite{markuvs2014eye} use randomized regression trees for pupil localization. Prior works on pupil-center localization is mainly based on geometric feature which gives accurate results for images captured under an controlled environment. The geometric models are mainly based on physical measurements; it generalizes quite easily to new subjects with very few prior annotated data.
\subsection{Self-supervised Learning Paradigm}
Self-supervised learning attracts many researchers for its superior performance gain on different vision based emerging topics in the past few years~\cite{zhao2022unsupervised}. Self-supervised representation learning mainly leverages input data itself for supervision and infers for any relevant downstream tasks. One recent study~\cite{goyal2019scaling} shows that by leveraging various attributes of the data (for example: input data size), self-supervised technique can largely match or even exceed the performance of supervised pre-training on a variety of tasks such as object detection, surface normal estimation (3D) etc. Kocabas et al.~\cite{kocabas2019self} show that even without any 3D ground truth data and the knowledge of camera extrinsics, multi view images can be leveraged to obtain self supervision. Definition of appropriate pretext task is very crucial for self-supervised learning. Misra et al.~\cite{misra2020self} develop pretext-invariant representation learning that learns invariant representations based on pretext tasks. A recent survey~\cite{jing2020self} on self-supervised approach depicts the potential to explore this domain.
In gaze representation learning domain, Yu et al.~\cite{yu2019unsupervised} uses subject specific gaze redirection as a pretext task to learn strong representation. Swapping Affine Transformations (SwAT)~\cite{farkhondeh2022towards} is the extended version of Swapping Assignments Between Views (SwAV), a popular self supervised learning framework. It is used for gaze representation learning using different augmentation techniques. Following this trend, our approach also defines a pretext task of gaze region classification based on relative pupil location to learn efficient representation for eye gaze estimation.
\begin{table*}[t]
\centering
\caption{A statistical overview of gaze datasets in literature.}
\label{subject_comparison}
\scalebox{1.0}{
\begin{tabular}{c||c|c|c|c|c|c|c|c|c|c|c}
\toprule[0.4mm]
\rowcolor{mygray}
\textbf{Datasets} & \textbf{\begin{tabular}[c]{@{}c@{}}Gi4E\\~\cite{villanueva2013hybrid}\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}RT-GENE\\~\cite{FischerECCV2018}\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}CAVE\\~\cite{smith2013gaze}\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}OMEG\\~\cite{he2015omeg}\end{tabular}} &\textbf{\begin{tabular}[c]{@{}c@{}}MPIIGaze\\~\cite{zhang15_cvpr}\end{tabular}}&\textbf{\begin{tabular}[c]{@{}c@{}}TabletGaze\\~\cite{huang2015tabletgaze}\end{tabular}}&\textbf{\begin{tabular}[c]{@{}c@{}}GazeCapture\\~\cite{cvpr2016_gazecapture}\end{tabular}}&\begin{tabular}[c]{@{}c@{}}\textbf{Gaze 360}\\~\cite{gaze360_2019} \end{tabular}& \textbf{\begin{tabular}[c]{@{}c@{}}ETHX-Gaze \\ \cite{zhang2020eth} \end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}EVE\\ \cite{Park2020ECCV} \end{tabular}} &\textbf{\begin{tabular}[c]{@{}c@{}}RAZE \end{tabular}} \\ \hline \hline
Subjects & 103 & 15 & 56 & 50 & 15 & 41 & 1450 & 238 & 110 & 54 & 100 \\
\begin{tabular}[c]{@{}c@{}} Total Images\end{tabular} & 1K & 122K & 5K & 44K & 213K & 100K & 2445K & 172K & 1083K & 12308K & 154K \\
\bottomrule[0.4mm]
\end{tabular}}
\end{table*}
\section{Method}
\label{sec:Proposed Method}
In this section, we describe the overview of the proposed self-supervised gaze region estimation method. Accurate gaze direction estimation usually depends on several factors such as exact locations of the pupil centers, head-pose, eye blink and subject specific appearance. However, the existing benchmark datasets are curated in constrained environments. Thus, instead of limiting ourselves to these data, we web-crawled YouTube videos having creative common licence. Our proposed frame-work, RAZE is guided by pseudo-gaze zone classification objective which can further be adapted to other downstream tasks. Figure~\ref{fig:pipeline} refects the overview of the proposed framework.
\subsection{Representation Learning Framework}
\noindent \textbf{Preliminaries.}
Given a detected face $\mathbf{x}$ from dataset $\mathcal{D}$, we localize the pupil-centers (i.e. $(p_{x}^{l},p_{y}^{l})$ and $(p_{x}^{r},p_{y}^{r})$) of the concerned subject at first. Further, the relative position of the pupils are utilized as a pretext task to estimate the eye gaze region $\mathbf{e} \in \mathbb{R}^3$ (i.e. \emph{left}, \emph{right} and \emph{center}) of the subject. The RAZE framework learn the meaningful representation of the eye region via `Ize-Net' network parameterised by $\mathcal{F}_{\phi}$ . $\mathcal{F}_{\phi}$ maps the input $\mathbf{x}$ to feature space $\mathbf{z}$ by $\mathcal{F}_{\phi}:\mathbf{x} \to \mathbf{z}$, where $ \mathbf{z}\in \mathbb{R}^d $. Later, the latent representation is mapped to the label space by $\mathcal{F}_{\theta}:\mathbf{z} \to \mathbf{e}$. The workflow of the whole self-supervised paradigm is summarized in Algorithm~\ref{alg:RAZE}. The rest of the section contains details of each stages mentioned in Algorithm~\ref{alg:RAZE}.
\noindent \textbf{Pupil-Center Localization.}
The first stage of our proposed method is pupil center localization. Accurate pupil-center localization plays an important role in eye gaze estimation. We take face image as input and extract eye-regions from this image, using the facial landmarks obtained by the Dlib-ml library~\cite{king2009dlib}. Further processing is performed on the extracted eye images. We localize the pupil-center using a three stage method, i.e., blob center detection~\cite{lin2011pupil} and CHT~\cite{daway2018pupil}, and take the average of the pupil-centers obtained by both of the methods to calculate the final pupil-center. The steps of the proposed pupil-center localization method are as follows (See Algorithm~\ref{alg:pupil_loc}):
\begin{enumerate}
\item Extract eyes using facial landmark information.
\item Apply OTSU thresholding on the extracted eyes to take the advantage of unique contrast property of eye region while pupil circle detection.
\item Apply the method of blob center detection on extracted iris contours to calculate 'primary' pupil-centers.
\item Crop regions near these centers, to perform the center rectification task. The crop length is decided by applying equation (1).
\begin{equation}
\texttt{Crop}\ \texttt{len.} = \dfrac{\texttt{Height}\ \texttt{of}\ \texttt{eye}\ \texttt{contour}}{\texttt{2}} + \texttt{offset}
\end{equation}
\item Compute Adaptive thresholding and apply Canny edge detector~\cite{canny1986computational} to make the iris region more prominent.
\item Apply CHT over the edged image to find secondary pupil-centers.
\item Compute average of primary and secondary pupil-centers to finalize the value for pupil-centers.
\end{enumerate}
The detected pupil centers are utilized for the pretext task which is described next.
\begin{algorithm}[tb]
\caption{Training Procedure for RAZE}
\label{alg:RAZE}
\begin{algorithmic}[1]
\Require{$\mathcal{F}_\phi$, $\mathcal{F}_{\theta}$, and $\mathcal{D}$}
\For{\texttt{$\mathbf{n}$ epochs}} \Comment{RAZE Training}
\State $\mathbf{e} \gets \texttt{Heuristic }(\mathbf{x})$ \Comment{Pretext Task}
\State $\mathbf{z}\gets \mathcal{F}_{\phi}(\mathbf{x})$
\State $\mathbf{e'}\gets \mathcal{F}_{\theta}(\mathbf{z})$
\State $\mathcal{L}_0 = \mathcal{L}_{\texttt{gaze-region}} (\mathbf{e}, \mathbf{e'})$
\State $\{\phi, \theta\} \gets \triangledown_{\{\phi, \theta\}} \mathcal{L}_0$
\EndFor
\For{\texttt{$\mathbf{n}$ epochs}} \Comment{Downstream Adaptation}
\State $\mathbf{z} \gets \mathcal{F}_{\phi}(\mathbf{x'})$\Comment{$x' \in \mathcal{D}$}
\State $\mathbf{y'} \gets \mathcal{F}_{\theta} (\mathbf{z})$
\State $\mathcal{L}_1 = \mathcal{L}_{\texttt{FT/LP}} (\mathbf{y}, \mathbf{y'})$ \Comment{Dataset specific Fine-Tuning or Linear Probing}
\State $\{\phi$ ,/or $\theta\}\gets \triangledown_{\{\phi,/or\theta\}} \mathcal{L}_1$
\EndFor
\end{algorithmic}
\end{algorithm}
\noindent \textbf{Pretext Task: Heuristic for Eye Gaze Region Estimation.}
Pretext task is the second step of our proposed self supervised paradigm. The relative position of the pupil-centers is the most decisive feature of the face to determine gaze direction. Eye, head movement and their relative motion determines the direction of the `coarse-level' eye gaze. Thus, by using the relative position of both the pupil-centers, we can determine the possible regions where the subject is looking. When a subject looks towards his/her left, both the eyes' iris shift towards left. To utilize this unique characteristic, we compare the angles formed when we join the left pupil-center with the nose and nose with vertical; with the angle formed when we join the right pupil-center with the nose and nose with vertical. These angles are demonstrated in Figure~\ref{fig:pipeline} as angles $\theta_1$ and $\theta_2$. For a subject to look towards his/her left region, the left eye angle $\theta_1$ has to be bigger than the right eye angle $\theta_2$. This intuitive heuristic is used to detect the coarse-level gaze region (left, right, or center) in which the subject is looking. Empirically, the proposed method is immune to head movements within the range of \ang{-10} to \ang{10}.
The eye corners remain fixed with the eye movement. We utilize the eye corner points given by the Dlib-ml library to determine the head pose direction, in the same way as we determine the eye gaze region. The angles used to determine the head pose direction are demonstrated in Figure~\ref{fig:pipeline} as angles $\theta_3$ and $\theta_4$. For example, when the subject's head pose is left the $\theta_4$ is greater than $\theta_3$. By using this pretext task, we collect and annotate a large scale YouTube data described later.
\begin{figure*}[t]
\centering
\includegraphics[height=5cm,width=\linewidth]{pipeline.png}
\caption{Overview of the proposed pipeline. From left to right, we show \textit{(a) Region Guided Self Supervision via Pseudo Labels:} The proposed RAZE module first perform pseudo labelling of the detected faces based on facial landmarks. Angles $\theta_1$ and $\theta_2$ are used to estimate eye gaze region and angles $\theta_3$ and $\theta_4$ are used for head pose estimation. (Refer Sec.~\ref{sec:Proposed Method}-A Pretext Task for more details); \textit{(b) Self Supervised Representation Learning:} RAZE framework consists of the backbone network aka `Ize-Net' which maps input image to the label space.Few label space examples are also shown in yellow bounding box (Refer Sec.~\ref{sec:Proposed Method}-A for more details); \textit{(c) Inference:} We use Linear Probing (LP), Fine-Tuning (FT) for adapting to different datasets and tasks. }
\label{fig:pipeline}
\end{figure*}
\noindent \textbf{Overall RAZE Loss.}
Algorithm~\ref{alg:RAZE} describes the training procedure of the proposed RAZE. The overall learning is guided by the following objective functions: $\mathcal{L}_{\texttt{gaze-region}}= \mathcal{L}_{\texttt{{ce}}}$.
\noindent Here, $\mathcal{L}_{\texttt{{ce}}}$ is the standard cross entropy loss for three gaze zone/regions.
\subsection{Evaluation Protocol for Self-Supervision}
Following the standard evaluation protocols for self supervised learning paradigms, we also adopt Linear Probing (LP)~\cite{zhang2016colorful,he2020momentum,caron2021emerging} and Fine-Tuning (FT) for downstream adaptation~\cite{caron2021emerging}. For LP, we incorporate data augmentation strategy in terms of random resize, random crops and flipping horizontally during training phase. In gaze estimation, the ground truth gaze labels change its sign while performing horizontal flipping operation. While adapting via LP the weights $\phi$ is frozen and only the label space parameters i.e. $\mathcal{F}_{\theta}$ are updated. The downstream adaptation process is enforced by the appropriate loss function for different tasks. To be more specific, for 3D gaze estimation the following loss is incorporated $\mathcal{L}_{\texttt{3D\ gaze}} = \frac{\mathbf{g}}{||\mathbf{g}||_2}. \frac{\mathbf{g'}}{||\mathbf{g'}||_2}$ where, $\mathbf{g}$ and $\mathbf{g'}$ are ground truth and predicted labels.
For FT, instead of $\mathcal{F}_{\theta}$, all of the parameters of RAZE are updated. However, the training is started with the pretext tasked based pre-trained weights.
\begin{algorithm}[tb]
\caption{Pupil Center Localization}
\label{alg:pupil_loc}
\begin{algorithmic}[1]
\For{\texttt{$\mathbf{n}$ images}} \Comment{Pupil Center Localization}
\State $\mathbf{Eyes} \gets \texttt{Dlib-ml}\ (\mathbf{x})$ \Comment{ Eye localization via Facial landmarks}
\State $\mathbf{Iris}\gets \texttt{OTSU}\ (\mathbf{Eyes})$ \Comment{OTSU Method}
\State $\mathbf{P_{p}}\gets \texttt{Blob\ Center\ Detection}\ (\mathbf{Iris})$ \Comment{`Primary' Pupil-Center}
\State $\texttt{Crop-Length} = \dfrac{\texttt{Height}\ \texttt{of}\ \texttt{eye}\ \texttt{contour}}{\texttt{2}} + \texttt{offset}$
\State ROI $\gets$ Crop regions near Pupil-Center
\State Adaptive Thresholding (ROI) \Comment{Iris Center
Rectification}
\State $\mathbf{P_{s}}\gets$ CHT (Canny Edge (ROI)) \Comment{`Secondary' Pupil-Centers}
\State $\mathbf{\mathbf{P_{c}}}\ =\ \dfrac{\mathbf{P_{p}} + \mathbf{P_{s}}}{2}$
\EndFor
\end{algorithmic}
\end{algorithm}
\section{Experimental Protocols}
\label{sec:Experiments}
For all of our experiments, we use the Keras deep learning library with the Tensorflow backend. The proposed deep model for eye gaze estimation was trained and tested on Titan Xp GPU.
\begin{table}[t]
\centering
\caption{The categorical distribution of the proposed dataset.}
\label{data_stat}
\scalebox{1.0}{
\begin{tabular}{l||c|c|c|c}
\toprule[0.4mm]
\rowcolor{mygray}
\textbf{RAZE Dataset} & \textbf{Center} & \textbf{Left} & \textbf{Right} & \textbf{Total} \\ \hline \hline
Train set & 32,450 & 38,230 & 37,338 & 108,018 \\
Validation set & 14,008 & 16,584 & 15,641 & 46,233 \\
Total & 46,458 & 54,814 & 52,979 & 1,54,251 \\
\bottomrule[0.4mm]
\end{tabular}}
\end{table}
\noindent \textbf{Benchmark Datasets.} We evaluate the proposed method RAZE on five benchmark datasets: \textbf{CAVE}~\cite{smith2013gaze}, \textbf{MPII}~\cite{zhang2017mpiigaze}, \textbf{TabletGaze}~\cite{huang2015tabletgaze}, \textbf{RT-GENE}~\cite{fischer2018rt} and \textbf{DGW}~\cite{ghosh2021speak2label}. CAVE~\cite{smith2013gaze} dataset has 5,880 high resolution images of 56 subjects. The dataset is collected in a constrained lab environment. The data is labelled for 21 different gaze directions and head-poses for each subject. MPII~\cite{zhang2017mpiigaze} dataset is collected from 15 subjects performing everyday activity before a laptop. The dataset contains 213,659 images collected over a three-month window. TabletGaze~\cite{huang2015tabletgaze} is relatively unconstrained dataset of 51 subjects. The gaze direction is mapped with 4 different postures and 35 gaze locations. This dataset is also collected in an indoor environment. Similarly, RT-GENE dataset~\cite{fischer2018rt} is also recorded in a naturalistic environment. The ground truth annotation is assigned using a motion capture system connected with eye-tracking glasses. DGW~\cite{ghosh2021speak2label} is a large scale driver gaze zone estimation dataset. DGW contains data from 338 subjects fixating their gaze `inside a car' scenario with variation in illumination, occlusion etc. We validate the proposed pupil-center localization method (See Algorithm~\ref{alg:pupil_loc}) on BioID dataset~\cite{jesorsky2001robust}. BioID is a publicly available dataset which contains 1,521 frontal face images of 23 subjects.
\noindent \textbf{Automatic Dataset Collection Paradigm.}
In recent years, several gaze estimation datasets have been proposed~\cite{gaze360_2019,FischerECCV2018}. Most of the datasets are collected in more or less restricted environment. Moreover, few of these datasets may contain very little of images in terms of head poses, illumination, number of images, collection duration per subject and camera quality. To demonstrate the adaptability of our proposed self supervised method, we collect a dataset containing 154,251 facial images belonging to 100 different subjects from YouTube (having creative common license). The overall statistic of our dataset is shown in Table~\ref{data_stat}. We download different types of videos from YouTube. These videos belong to different categories, where a single (or multiple) subject(s) is seen on the screen at a time, like news reporting, makeup tutorials, speech videos, doing meditation etc. We have considered every third frame of the collected videos for dataset creation. The dataset has been split into training and validation sets with 70\% and 30\% uniform partitions over the subjects for the training purpose. The overview of our proposed dataset is shown in Figure~\ref{fig:sample_images}. In this figure, we can observe that our dataset contains a huge variety of images with varying illumination, occlusion, blurriness, color intensity, etc. Table~\ref{subject_comparison} provides the comparison of the state-of-the-art gaze datasets with our proposed dataset. \textit{Please note that the dataset is available upon request.}
\noindent \textbf{Implementation Details.}
\textit{1. Network Architecture:} The architecture of the proposed `Ize-Net' network is shown in Figure~\ref{fig:pipeline}. The network uses a primary capsule component combined with a series of convolution layers. The motivation of using capsule block stems from the superior performance of capsule networks~\cite{sabour2017dynamic} in handling relative location of an object's parts. Our network is trained using images of size $128\times128\times3$. We take the entire face as input instead of only the eye region. According to~\cite{zhang2017s}, gaze can be more accurately predicted when the entire face is considered. Our proposed network contains five convolution layers. Each convolution is followed by batch normalization and max-pooling. For batch normalization, we use 'ReLU' as the activation function. For max-pooling kernel of size ($2\times2$) was used. The stride of ($1\times1$) is considered for each layer. After the convolution layers, we append primary capsule, whose job is to take the features learned by convolution layers and produce combinations of the features to consider face symmetry into account. The primary capsule output is flattened and fed to fully-connected layers of dimension 1024 and 512. In the end, we apply softmax activation to produce the final output which is gaze regions (i.e. left, right and center).
\noindent\textit{2. Linear Probing(LP) and Fine Tuning(FT) details:} To linear probe the base model for prospective datasets, we add two Fully-Connected (FC) layers (dimension 256) at the end of the proposed Ize-Net network. For LP, we demonstrate the impact of weight freezing (at different level) on gaze estimation performance. The last 8 layers, last 12 layers, and complete network are fine-tuned in succession for the empirical analysis of results. For fine-tuning the network on the Tablet Gaze dataset, we used a learning rate of 0.0001 with 10 epochs, and for the other datasets, we used a learning rate of 0.0001 with 15 epochs. During fine-tuning, the mean square error loss function as well as cosine similarity is implemented following the respective evaluation protocols mentioned in prior literature. We fine-tune the Ize-Net on the DGW dataset using the SGD optimizer for 20 epochs with a learning rate of 0.0001, the decay of $1 \times e ^{-6}$ per epoch and momentum of 0.9.
We additionally evaluate a weighted nearest neighbour classifier (k-NN)~\cite{caron2021emerging} on the DGW data. The weights of the Ize-Net is frozen and the penultimate layer's feature is extracted for training. The k-NN classifier uses similarity matching operation along with voting strategy in the latent space to get the predicted label. Empirically, this analysis works for $\sim$ 13-15 NN over several iterations.
\noindent \textbf{Evaluation Metrics.}
For quantitative evaluation of the gaze region estimation, we use class-wise accuracy (in \%). Following each database's evaluation protocol, we follow `leave-one-person-out' for MPII, cross-validation for CAVE and TabletGaze; and 3-fold evaluation for RT-GENE dataset. Additionally, we compute angular error (in \textdegree) except for the TabletGaze dataset, for which we compute the error in cm (similar to ~\cite{huang2015tabletgaze}). To compare with the state-of-the-art methods, we use similar evaluation protocols mentioned in those studies.
\begin{table*}[t]
\centering
\caption{\textbf{Results on Tablet Gaze (in cm)} with comparison to baselines~\cite{smith2013gaze}. Effectiveness of learnt features in Ize-Net (Pre-trained on the collected data) is demonstrated by the fine tuning the network and by training a SVR over various FC layer features. * methods are supervised.}
\label{tg_compare}
\scalebox{0.9}{
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c}
\toprule[0.4mm]
\rowcolor{mygray}
\begin{tabular}[c]{@{}c@{}} \textbf{Methods} \end{tabular}& \begin{tabular}[c]{@{}c@{}}\textbf{Raw pixels*}\\~\cite{huang2015tabletgaze}\end{tabular} & \begin{tabular}[c]{@{}c@{}}\textbf{LoG*}\\~\cite{huang2015tabletgaze} \end{tabular} & \begin{tabular}[c]{@{}c@{}}\textbf{LBP*}\\~\cite{huang2015tabletgaze} \end{tabular} & \begin{tabular}[c]{@{}c@{}}\textbf{HoG*}\\~\cite{huang2015tabletgaze} \end{tabular} & \begin{tabular}[c]{@{}c@{}}\textbf{mHoG*}\\~\cite{huang2015tabletgaze} \end{tabular} & \begin{tabular}[c]{@{}c@{}} \textbf{\cite{jyoti2018automatic}}* \end{tabular}& \begin{tabular}[c]{@{}c@{}}\textbf{RAZE}\\ \textbf{(Full Network}\\ \textbf{ Fine Tuning})\end{tabular} & \begin{tabular}[c]{@{}c@{}}\textbf{RAZE} \\ \textbf{(last} \\ \textbf{12 layers} \\\textbf{fine-tuning)} \end{tabular} & \begin{tabular}[c]{@{}c@{}}\textbf{RAZE} \\ \textbf{(last} \\ \textbf{8 layers} \\\textbf{fine-tuning)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}\textbf{RAZE} \\ \textbf{(last} \\ \textbf{8 layers} \\\textbf{fine-tuning)}\\ \textbf{with eye patch}\end{tabular} & \begin{tabular}[c]{@{}c@{}}\textbf{RAZE} \\ \textbf{Layer (34)} \\ \textbf{+ SVR} \end{tabular} & \begin{tabular}[c]{@{}c@{}}\textbf{RAZE} \\ \textbf{Layer (31)} \\ \textbf{+ SVR} \end{tabular} \\ \hline \hline
k-NN & 9.26 & 6.45 & 6.29 & 3.73 & 3.69 & \multirow{4}{*}{2.61}& \multirow{4}{*}{2.36} & \multirow{4}{*}{3.31} & \multirow{4}{*}{3.26} &
\multirow{4}{*}{2.80}
& \multirow{4}{*}{2.42}
& \multirow{4}{*}{2.48} \\ \cline{1-6}
RF & 7.2 & 4.76 & 4.99 & 3.29 & 3.17 & & & & & & & \\ \cline{1-6}
GPR & 7.38 & 6.04 & 5.83 & 4.07 & 4.11 & & & & & & & \\ \cline{1-6}
SVR & - & - & - & - & 4.07 & & & & & & & \\
\bottomrule[0.4mm]
\end{tabular}}
\end{table*}
\begin{table*}[!htbp]
\centering
\caption{\textbf{Results on the CAVE dataset} (Pre-trained on the collected data) using the angular deviation, calculated as $ \texttt{mean}\ \texttt{error}\ $(in \textdegree)$\pm\ \texttt{standard\ deviation} $ (in \textdegree). It is interesting to note that the eye patch region based learnt representation performs best. * methods are supervised.}
\label{cave_compare}
\scalebox{1}{
\begin{tabular}{c|c|c|c|c|c}
\toprule[0.4mm]
\rowcolor{mygray}
\textbf{Calibration} & \multirow{1}{*}{\textbf{Method}} & \multicolumn{2}{c|}{\textbf{$\ang{0}$ yaw angle}} & \multicolumn{2}{c}{\textbf{Full Dataset}} \\ \cline{1-1} \cline{3-6}
\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}5 point system\\ (cross arrangement)\end{tabular}} & \cellcolor{mygray} & \cellcolor{mygray}\textbf{X} &\cellcolor{mygray} \textbf{Y} & \cellcolor{mygray}\textbf{X} & \cellcolor{mygray} \textbf{Y} \\ \cline{2-6}
& Skodras et al.~\cite{skodras2015visual}* & $ 2.65 \pm 3.96 $ & $ 4.02 \pm 5.82 $ & N/A & N/A \\ \cline{2-6}
& Jyoti et al.~\cite{jyoti2018automatic}* & $ 2.03 \pm 3.01 $ & $ 3.47 \pm 3.99 $ & N/A & N/A \\ \cline{2-6}
& \textbf{RAZE} (full face) & $ 2.94\pm 2.16 $ & $ 2.74\pm1.92 $ & $ 1.67\pm1.19 $ & $ 1.74\pm1.57 $ \\ \cline{2-6}
& \textbf{RAZE} (eye patch) & $ 2.65 \pm 1.70 $ & $ 2.16 \pm 1.44 $ & $ 0.98 \pm 0.74 $ & $ 1.05 \pm 0.73 $ \\
\bottomrule[0.4mm]
\end{tabular}}
\end{table*}
\begin{table}[t]
\centering
\caption{\textbf{Results on RT-GENE dataset~\cite{fischer2018rt} (in \textdegree)} which is pre-trained on the RAZE data. * methods are supervised.}
\label{tab:rt-gene}
\begin{tabular}{c|c|c|c|c}
\toprule[0.4mm]
\rowcolor{mygray}
\textbf{\begin{tabular}[c]{@{}c@{}}Single Eye\\~\cite{zhang15_cvpr}* \end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Spatial \\ weights \\ CNN\\~\cite{zhang2017s}*\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Spatial \\ weights\\ CNN \\ (ensemble)~\cite{fischer2018rt}*\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}4 model \\ ensemble~\cite{fischer2018rt}*\end{tabular}} & \textbf{RAZE} \\ \hline \hline
13.4 & 8.7 & 8.7 & 7.7 & 6.1 \\
\bottomrule[0.4mm]
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{\textbf{Results on MPII dataset~\cite{fischer2018rt} (in \textdegree)} which is pre-trained on the RAZE data. * methods are supervised.}
\label{tab:mpii}
\scalebox{1.0}{
\begin{tabular}{c|c|c|c|c|c|c}
\toprule[0.4mm]
\rowcolor{mygray}
\textbf{\begin{tabular}[c]{@{}c@{}}Single\\ Eye\\~\cite{zhang15_cvpr}*\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}iTracker \\~\cite{krafka2016eye}*\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Two\\ Eyes \\~\cite{fischer2018rt}*\end{tabular} } & \textbf{\begin{tabular}[c]{@{}c@{}}iTracker \\ (AlexNet)\\~\cite{krafka2016eye}*\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Single\\ Face \\~\cite{fischer2018rt}*\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Spatial \\ weights\\ CNN \\~\cite{zhang2017s}*\end{tabular}} & \textbf{RAZE} \\ \hline \hline
6.7 & 6.2 & 6.2 & 5.6 & 5.5 & 4.8 & 5.0 \\
\bottomrule[0.4mm]
\end{tabular}}
\end{table}
\begin{table}[t]
\centering
\caption{Fine-tuning result on DGW dataset~\cite{ghosh2021speak2label} for driver gaze estimation. * methods are supervised.}
\label{tab:speak2label}
\begin{tabular}{l||c|c}
\toprule[0.4mm]
\rowcolor{mygray}
\textbf{Method} & \textbf{Val. Accuracy} & \textbf{Test Accuracy} \\ \hline \hline
Vasli et al.~\cite{vasli2016driver}* & {52.60} & {50.41} \\
Tawari et al.~\cite{tawari2014driver}* & {51.30} & {50.90} \\
Fridman et al.~\cite{fridman2015driver}* & {53.10} & {52.87} \\
\begin{tabular}[c]{@{}c@{}}Vora et al.~\cite{vora2017generalizing} (Alexnet face)* \end{tabular} & {56.25 } & {57.98 } \\
\begin{tabular}[c]{@{}c@{}}Vora et al.~\cite{vora2017generalizing} (VGG face)*\end{tabular} & {58.67} & {58.90 } \\
SqueezeNet~\cite{iandola2016squeezenet}* & {59.53} & {59.18 } \\
Ghosh et al.~\cite{ghosh2021speak2label}* & {60.10} & {60.98} \\
Inception V3~\cite{DBLP:journals/corr/SzegedyVISW15}* & {67.93} & {68.04} \\
Vora et al.~\cite{vora2018driver}* & { 67.31} & {68.12} \\
ResNet-152~\cite{DBLP:journals/corr/HeZRS15}* & {68.94} & {69.01 } \\
\begin{tabular}[c]{@{}c@{}}Yoon et al.~\cite{yoon2019driver} (Face + Eyes)* \end{tabular} & {70.94} & {71.20} \\
\begin{tabular}[c]{@{}c@{}}Stappen et al.~\cite{stappen2020x}*\\\end{tabular} & {71.03} & {71.28} \\
\begin{tabular}[c]{@{}c@{}}Lyu et al.~\cite{lyu2020extract}*\\\end{tabular} & {85.40} & {81.51} \\
\begin{tabular}[c]{@{}c@{}}Yu et al.~\cite{yu2020multi}* \\\end{tabular} & {80.29} & {82.52} \\
RAZE (k-NN) & 62.50 & 63.82 \\
RAZE (LP) & 72.10 & 73.02 \\
RAZE (FT) & 80.50 & 81.82 \\
\bottomrule[0.4mm]
\end{tabular}
\end{table}
\section{Results}
We conduct comprehensive quantitative and qualitative analysis to validate our method on five publicly available benchmark datasets. We have also performed extensive ablation studies to show the impact of different components of the proposed pipeline.
\subsection{Downstream Task Specific Adaptation}
The `Ize-Net' network is trained on the proposed dataset for the task of gaze region estimation. We adapt the proposed method on 3D gaze estimation and driver gaze zone estimation tasks described below.
\noindent \textbf{`Coarse-to-fine' gaze estimation:} The learned data representation is linear-probed (LP) and fine-tuned (FT) on four benchmark gaze estimation datasets (i.e. TabletGaze~$\rightarrow$ Table~\ref{tg_compare}, CAVE~$\rightarrow$ Table~\ref{cave_compare}, MPII~$\rightarrow$ Table~\ref{tab:mpii} and RT-GENE~$\rightarrow$ Table~\ref{tab:rt-gene}) for determining the exact gaze location. Here, gaze location indicates the 3D/2D location/gaze-angle of the concerned subject.
In TABLE~\ref{tg_compare}, we incorporate the weight freezing strategy at different levels to determine the optimal layer for rich feature extraction. The last 8 layers, last 12 layers, and complete network are fine-tuned in succession for the empirical analysis of results. The empirical analysis suggest that the full network fine-tuning performs best for downstream adaptation. Even it outperforms supervised state-of-the-art~\cite{jyoti2018automatic} significantly ($2.61$cm $\rightarrow$ $2.36$ cm, $\sim$9.57\%) in person independent setting. To demonstrate that the network learned efficient features, we further trained a Support Vector Regressor (SVR) over the features learned in 31\textsuperscript{st} layer and 34\textsuperscript{th} layer for TabletGaze dataset. As depicted in TABLE~\ref{tg_compare}, the low gaze prediction errors of SVR confirms that the learned features are highly efficient.
Similarly, RAZE outperforms supervised methods~\cite{skodras2015visual,jyoti2018automatic} on CAVE dataset with 0\textdegree\ yaw angle and it is interesting to note that pre-training on `in-the-wild' data stabilizes the standard deviation significantly. Also it is quite intuitive that the eye patch based region performs the better as compared to the whole face as input. The reason being the noise introduction due to other facial parts. For experiments, we try our best to follow the protocols discussed in~\cite{skodras2015visual} and~\cite{huang2015tabletgaze}. However, there can be a few differences in frame extraction and selection.
Similarly, we perform downstream adaptation experiments on RT-GENE and MPII datasets~\cite{zhang15_cvpr,fischer2018rt}. We use the similar evaluation protocol mentioned in~\cite{zhang15_cvpr,fischer2018rt}. The result comparison with the state-of-the-art methods are depicted in TABLE~\ref{tab:rt-gene} and~\ref{tab:mpii} respectively. We use eye patch as input for both RT-GENE and MPII dataset. For RT-GENE dataset, our self-supervised method performs better than the baseline and the state-of-the-art methods ($7.7$\textdegree\ $\rightarrow$ $6.1$\textdegree, $\sim$20.77\%). For MPII dataset, our method (angular error: 5.0\textdegree) also compatible with supervised spatial weight CNN method (angular error: 4.8\textdegree). The results on the four benchmark datasets indicate that our method learns discriminative and rich representation.
\noindent \textbf{Driver Gaze Estimation:} Another application specific downstream task is driver gaze estimation. The network is adapted for driver gaze zone estimation on DGW dataset. The hyper-parameters and other relevant details of the network is described in experiment section. We evaluate the performance of Ize-Net network by cross-validating it's performance some with other gaze estimation task. We choose Driver Gaze in the Wild (DGW)~\cite{ghosh2021speak2label} data for this purpose. It performs automatic labeling by adding domain knowledge during the data recording process and generate a large scale gaze zone estimation dataset. TABLE~\ref{tab:speak2label} shows the comparison between performance of the baseline model proposed in~\cite{ghosh2021speak2label} with Ize-Net. It is observed that our approach outperforms several supervised models with a large margin which indicates that our model learns relevant representative features.
\begin{figure}[t]
\centering
\subfloat{\includegraphics[width = 0.9in,height=0.5in]{left_eye.png}}
\subfloat{\includegraphics[width = 0.9in,height=0.5in]{left_eye_1.png}}
\subfloat{\includegraphics[width = 0.9in,height=0.5in]{right_eye.png}}
\subfloat{\includegraphics[width = 0.9in,height=0.5in]{right_eye_1.png}}
\caption{Results of pupil-center localization method. Green, blue and pink colors represent the pupil-centers as mentioned in Algorithm~\ref{alg:pupil_loc} (Image Source:~\cite{smith2013gaze} best viewed in color).}
\label{fig:pupil_eye}
\end{figure}
\subsection{Ablation Studies}
\subsubsection{Choice of Pupil Localization}
The pupil-center detection is performed using OTSU thresholding with blob center detection and CHT. To perform CHT, we crop the image around the pupil-center which we detect using OTSU thresholding and blob-center. We use offset of 5 pixels to crop the image. The evaluation protocol is mentioned in equation~\ref{eqn:error}, is same as the one used in~\cite{jesorsky2001robust}.
\begin{equation}
\label{eqn:error}
e=\dfrac{max(d_l-d_r)}{\lVert C_l-C_r \rVert}
\end{equation}
where, $e$ is the error term, $d\textsubscript{l}$ and $d\textsubscript{r}$ are the Euclidean distances between the localized pupil-centers and the ground truth ones; $C\textsubscript{l}$ and $C\textsubscript{r}$ are left and right pupil-centers respectively in the ground truth.
\noindent \textbf{Quantitative Analysis:} Table~\ref{pupil_val} shows the comparison of the proposed method with some of the state-of-the-art methods. This table shows that our method is absolutely accurate in \begin{math} e \leq 0.10 \end{math} and \begin{math} e \leq 0.25 \end{math} cases, but it does not perform well enough when \begin{math} e \leq 0.05 \end{math}. The reason behind this is the inaccurate circle detection by CHT, which propagates the error while averaging primary and secondary pupil-centers (See Algorithm~\ref{alg:pupil_loc}).
\noindent \textbf{Qualitative Analysis:} Empirically, we observe that the pupil-center localization accuracy is increased by taking an average of pupil-centers calculated by the above two methods. Few sample results of pupil-center localization have been shown in Figure~\ref{fig:pupil_eye}. The blue, green, and pink dots represent the pupil-center obtained by our primary method, secondary method and their average, respectively.
\begin{table}[t]
\centering
\caption{Comparison of proposed pupil-center localization method on BioID dataset~\cite{jesorsky2001robust} with other state-of-the-art methods.}
\label{pupil_val}
\scalebox{1}{
\begin{tabular}{l|c|c|c}
\toprule[0.4mm]
\rowcolor{mygray}
\multirow{1}{*}{\textbf{Methods}} & \multicolumn{3}{c}{\textbf{Accuracy (\%)}} \\ \cline{2-4}
\cellcolor{mygray} &\cellcolor{mygray} \textbf{\begin{math} e \leq 0.05 \end{math}} & \cellcolor{mygray} \textbf{\begin{math} e \leq 0.10 \end{math}} & \cellcolor{mygray} \textbf{\begin{math} e \leq 0.25 \end{math}} \\ \hline \hline
\textbf{Ours} & \textbf{56.97} & \textbf{100.00} & \textbf{100.00} \\
Poulopoulos et al.~\cite{poulopoulos2017new} & 87.10 & 98.00 & 100.00 \\
Leo et al.~\cite{leo2014unsupervised} & 80.70 & 87.30 & 94.00 \\
Campadelli et al.~\cite{campadelli2006precise} & 62.00 & 85.20 & 96.10 \\
Cristinacce et al.~\cite{cristinacce2004multi} & 57.00 & 96.00 & 97.10 \\
Asadifard et al.~\cite{asadifard2010automatic} & 47.00 & 86.00 & 96.00 \\
\bottomrule[0.4mm]
\end{tabular}}
\end{table}
\subsubsection{Choice of Gaze Heuristic}
In order to evaluate the performance of the proposed heuristic, we compare the ground truth gaze direction derived from the CAVE dataset with the heuristic based gaze direction. The overall accuracy is approximately 87\%. The heuristic mostly fails to infer the direction when the head movement is beyond $\pm 10$\textdegree.
\subsubsection{Choice of Network Architecture}
The efficiency of the proposed eye gaze region estimation is validated on the CAVE dataset~\cite{smith2013gaze}. For this purpose, we map the angular value labels of CAVE dataset images into left, right, and central gaze regions based on the sign (positive and negative) of the gaze point. The validation results are shown in Table~\ref{result}. We also evaluate the performance of Alexnet~\cite{krizhevsky2012imagenet} and VGG-Face~\cite{parkhi2015deep} networks on the collected new dataset. AlexNet and VGG-face give 88.22\% and 84.30\% validation accuracy, respectively. We use Stochastic Gradient Descent (SGD) optimizer with categorical cross-entropy as the loss function for training both the networks. The learning rate and momentum are assigned 0.01 and 0.9 values, respectively. For quantitative anaysis, we use full face images as well as eye patch as input. From empirical analysis, it is observed that eye-patch usually performs better than full face as input. The reason behind this is that the eye patch region provide more relevant information for the gaze inference.
\subsubsection{Performance of Ize-Net Network on Pretext Task}
For training the proposed Ize-Net network, we initialize the network weights with `glorot normal' distribution. We use the SGD optimizer with a learning rate of 0.001 with the decay of $1 \times e ^{-6}$ per epoch. We use categorical cross-entropy as the loss function to train the proposed network. As mentioned in TABLE~\ref{result}, it gives 91.50\% accuracy on the validation data of the proposed dataset. The proposed network outperforms the efficiency of AlexNet and VGG-face networks. The primary reason behind the better performance of Ize-Net is the presence of the primary capsule. This enables the network to consider the geometry of the face into account during gaze region prediction. The consideration of face geometry is in accordance with the proposed heuristic used to label the collected dataset's images. We validate the performance of the proposed network on the CAVE dataset. The angular labels of CAVE dataset images have been mapped into three gaze regions. Post categorizing the images into their corresponding gaze regions, we fine-tune the Ize-Net for the entire CAVE dataset to cross-check this network's performance. We fine-tune our network for 10 epochs with 0.0001 learning rate~\cite{smith2013gaze}. As mentioned in TABLE~\ref{result}, our network gives 82.80\% five-fold cross-validation accuracy on CAVE dataset.
\begin{table}[b]
\centering
\caption{Validation of our proposed heuristic and Ize-Net network for CAVE dataset and proposed dataset.}
\label{result}
\scalebox{1}{
\begin{tabular}{l||c|c}
\toprule[0.4mm]
\rowcolor{mygray}
\textbf{Method/ Network} & \textbf{CAVE} & \textbf{\begin{tabular}[c]{@{}c@{}}RAZE\\ Dataset\end{tabular}} \\ \hline \hline
Eye Gaze heuristic & 60.37\% & N/A \\
Alexnet (full face) & N/A & 88.22\% \\
VGG-Face (full face) & N/A & 84.30\% \\
Ize-Net (full face) & 82.80\% & 91.50\% \\
Ize-Net (eye patch) & 88.80\% & 95.98\% \\
\bottomrule[0.4mm]
\end{tabular}}
\end{table}
\begin{table}[!ht]
\centering
\caption{Validation results of the proposed method with voting based label smoothing.}
\label{tab:voting}
\begin{tabular}{l||c|c}
\toprule[0.4mm]
\rowcolor{mygray}
\textbf{Method/Network} & \textbf{CAVE} & \textbf{RAZE Dataset} \\ \hline \hline
\textbf{Eye Gaze Heuristic} & 62.79\% & NA \\
\textbf{Alexnet (Full Face)} & NA & 89.45\% \\
\textbf{VGG-Face (Full Face)} & NA & 85.66\% \\
\textbf{Ize-Net (Full Face)} & 81.34\% & 90.82\% \\
\textbf{Ize-Net (Eye Patch)} & 86.25\% & 89.73\% \\
\bottomrule[0.4mm]
\end{tabular}
\end{table}
\subsection{Voting based Label Smoothing Strategy}
We introduce label based voting in time domain (here, time domain means along the time axis of the input video) to smooth the gaze trajectory. We organize image frames in the order of appearance in the corresponding video. We select the gaze labels of five neighboring frames (in successive order) and calculate the voting over 3-zones (left, right, and central). The labels are assigned according to the max-voting strategy. The results of these experiments are shown in Table~\ref{tab:voting}. We compare the gaze estimation results with label smoothing (Table~\ref{tab:voting}) and without label smoothing (Table~\ref{result}). As compared to the gaze estimation on image frames without label smoothing, there is around 1-2\% increment in the accuracy for CAVE dataset as well as our dataset. The increment in accuracy percentage suggests that label smoothing introduced more robustness in the data labeling.
\subsection{Generalization Capability of Self-Supervised Method}
We evaluate the generalization capability of our proposed method. For this purpose, we conduct experiments by pre-training on the train part and further validate it for the downstream task of gaze estimation. We train RAZE framework on CAVE and MPII datasets to validate the performance of our self-supervised method. The results are shown in TABLE~\ref{tab:general}. The results depict the generalization capability of our proposed method.
\begin{table}[htbp]
\caption{Performance of the state-of-the-art method on CAVE and MPII datasets. * methods are supervised.}
\label{tab:general}
\centering
\begin{tabular}{l||c|c|c}
\toprule[0.4mm]
\rowcolor{mygray}
Methods & Pre-train & CAVE & MPII\\ \hline \hline
Park et al.~\cite{park2018deep} & CAVE/MPII & 3.80\textdegree & 4.50\textdegree \\
Jyoti et al.~\cite{jyoti2018automatic}* & CAVE &\textbf{2.22\textdegree} & -- \\
Yu et al.~\cite{yu2019unsupervised} &CAVE & 3.42\textdegree & -- \\ Cheng et al.~\cite{cheng2020coarse}* &MPII & -- & \textbf{4.10\textdegree} \\
RAZE & CAVE/MPII &2.40\textdegree & 4.20\textdegree \\ \bottomrule[0.4mm]
\end{tabular}
\end{table}
\section{Conclusion, Limitations and Future Work}
\label{sec:Conclusion and Future Work}
In this paper we propose a method for learning a rich eye gaze representation by using self-supervised learning. At first, we define the pretext task by utilizing the relative position of pupil-centers and annotate the images on three gaze region i.e. left, right, or center. To learn a rich representation, we collect a large dataset of the facial image. We also propose a capsule layer based CNN network, `Ize-Net', which is trained on the collected dataset. The learned representation is transferred into two downstream tasks. The quantitative and qualitative results indicates that the proposed method learns rich representation.
Currently, the proposed method performs eye gaze estimation for near frontal images. We have selected the images in the dataset based on only the roll head pose angles. It is important to note here that images with varying yaw angle (within a certain range) of head pose also looks frontal. The current work does not take the variation in the yaw angle into consideration while calculating the eye gaze. Since the current approach utilizes humans' symmetrical facial features to detect the gaze-direction; the amount of error will be very less due to yaw angle variation. In the future, we plan to utilize the head pose and other relevant information while estimating the gaze region.
\section*{Acknowledgment}
We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research.
\bibliographystyle{ieee}
|
1,314,259,996,531 | arxiv | \section{Introduction}
Investigation of protein and colloidal systems
has focused the attention of the scientific community on the phase-diagram behavior of short-range attractive potentials and of the role of the range of interaction in controlling the thermodynamic and dynamic properties of the system\cite{Lekkerkerker,Pellicane2004,Rosenbaum1996,Lomakin1996, Poon1997}.
Colloidal particles, due to their nano or microscopic size are often characterized by effective interactions\cite{likos-review} whose range is significantly smaller than the particle diameter. Under these conditions, it has been argued that the actual shape of the potential is irrelevant and that the thermodynamics\cite{Noro2000}, as well as the dynamics\cite{Foffi2005}, of different systems approximately satisfy an extended law of corresponding states\cite{VliegenthartLekkerkerker2000}. This law allows for a comparison between different systems, once the effective diameter of the particle is known (i.e., when the repulsive part of the interaction can be mapped into an equivalent hard-sphere diameter~\cite{Barker1967}). It has been proposed that the second virial coefficient, normalized by
the corresponding hard-sphere second virial coefficient, $B_2^*$
may act as a proper scaling variable. Therefore systems with equal second virial coefficient and effective diameter should have similar thermodynamical properties.
The adhesive hard-sphere (AHS) potential\cite{Baxter1968}, the limiting behavior of an infinitesimal interaction range coupled to infinite interaction strength such that $B_2$ is finite, has also received significant attention.
For this potential $B_2^*=1-1/4\tau$, where $\tau$, which acts as an effective scaled temperature, is the so-called stickiness parameter. Despite the known thermodynamic anomalies\cite{Stell1991}, an analytic evaluation of the (metastable) critical point within the
Percus-Yevick closure with both the energy and the compressibility routes for this potential is available\cite{Watts1971,Baxter1968}.
In the energy route the critical
point is located at $B_2^{*c}=-1.1097$ ($\tau_c=0.1185$) and number density $\rho=0.609$.
The availability of analytic predictions for this model has favored its application in the interpretation of experimental data for several disparate colloid (and protein) systems\cite{TartagliaJPCM,verduin,CaccamoPhysRep,Pellicane2004,Rosenbaum1996,Lomakin1996, Poon1997},
an application whose validity has been reinforced by the
extended law of corresponding states. For this reason, it
is important to try to accurately estimate the properties of the AHS model as a reference, to support existing predictions or to suggest improvements to available theoretical approaches. Numerical simulations of the AHS model
have been attempted in the past\cite{seatonglandt1987, kranendonkfrenkel,Lee2001}.
A recent effort in the direction of evaluating the phase diagram of the model has been provided by Miller and Frenkel\cite{Miller2003}, based on an ingenious identification of the appropriate Monte Carlo (MC) moves for this
potential\cite{seatonglandt1987, kranendonkfrenkel}. Their study
provides an estimate of the location of the critical point
at $B_2^{*c}=-1.21(1)$ ($\tau_c=0.1133(5)$) and $\rho_c=0.508(10)$.
In this article we propose a different approach to the
numerical evaluation of the critical properties of the
AHS model, based on extrapolation of standard grand
canonical MC simulation results for a sequence of square well (SW)
potentials with progressively smaller attraction ranges $\delta$
(down to $\delta=0.005$, in units of the hard-sphere diameter $\sigma$).
The SW potential is defined as:
\begin{equation}
\label{eq:SWpotential}
U(r) = \left\{
\begin{array}{ll}
\infty & \textrm{if $r\leq\sigma$}\\
-\epsilon & \textrm{if $\sigma < r\leq\sigma+\delta \sigma$}\\
0 & \textrm{if $r>\sigma+\delta \sigma$}
\end{array} \right.
\end{equation}
\noindent
The SW fluid has been profusely studied \cite{Vega1992,DelRio2002,LopezRendon2006,pagangunton,liukumar, Elliot1999,Chang1994} for $\delta>0.1$. It has been shown\cite{pagangunton,liukumar} that for $\delta\lesssim 0.25$
gas-liquid separation becomes metastable with respect to
the fluid-solid equilibrium. Despite its metastable character,
investigation of smaller $\delta$ values retains its importance, since the crystallization time is often much longer than the
experimental one and
gas-liquid phase separation is readily accessed (an effect facilitated by the
large difference between the fluid density and the crystal density and, in experiments, by the intrinsic sample polydispersity) .
Despite the importance of the SW model in relation to
the AHS potential,
no studies of the $\delta$-dependence of the critical point location has been previously reported for very small
$\delta$. This is in large part due to the fact that for smaller and smaller $\delta$, the critical temperature significantly decreases. Indeed, according to the constant $B_2^{*c}$ prediction of Noro and Frenkel\cite{Noro2000}, it should vary as
\begin{equation}
\label{eq:Tc}
\frac{k_B T_c}{\epsilon}=\left[\ln \left (1+\frac{1-B_2^{*c}}{(1+\delta)^3-1}\right )\right]^{-1}
\end{equation}
\noindent
making bond-breaking (changes of the particle energy of order $\epsilon$) events rarer and rarer in the simulation.
Moreover, the size of the translational step in the MC code is of the order of $\delta$. On the other hand,
the location of the critical point becomes more and more metastable which, in principle, poses a limit to the smallest
$\delta$ which can be studied.
Despite these numerical difficulties, we have been able
to estimate the location of the critical point down to
$\delta=0.005 \sigma$. We present here the $\delta$ dependence of
the critical temperature and density and the values of the second virial coefficient and energy at the critical point.
In all cases, a short linear extrapolation to $\delta=0$ provides novel accurate estimates of the corresponding quantities for the AHS model.
\section{Monte Carlo simulations}
We have simulated the SW system in the grand canonical (GC) ensemble in order to locate the gas-liquid critical point for different $\delta$ values. The critical point is identified by mapping the grand canonical density distribution onto the universal Ising model distribution, following the method described by Bruce and Wilding\cite{Bruce1992}. Histogram rewighting \cite{Wilding1997} was used to achieve an accurate estimate of the critical point, and the field mixing parameter was always found to be negligible.
We define a Monte Carlo step as one hundred trial moves, with
an average of 95\% translation and 5\% trial insertions or removals of one particle in the system. A translational move is defined as
a displacement in a random direction by a random amount between $\pm \delta/2$. We simulate different realizations of the same system (at the same chemical potential
and temperature) to improve statistics. Simulations
lasted more than $10^7$ MC steps. We have studied
cubic boxes of side 5$\sigma$ and/or 8$\sigma$, to estimate the importance of finite-size effects.
Additionally, for $\delta=0.05$ we have studied several other box sizes
to estimate the deviation of the value of the critical parameter
for the bulk limit case. Proper finite size studies for $\delta<0.05$ are at this moment numerically prohibitive. Histogram reweighting was used to map the density distribution of the fluid onto the universal critical order parameter distribution of the Ising model\cite{Wilding1997}, thereby reaching an accurate estimate of the critical point. The field mixing parameter was always been found to be negligible.
We have occasionally observed a transition to a more dense stable phase, signaling that indeed the
values of the chemical potential studied admit metastable fluid solutions. In all cases where a transition to a
more dense phase was observed, the simulation was interrupted and the 10\% of configurations saved just before the transition were disregarded.
This procedure is shown graphically in Fig.-\ref{fig:densevol}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=15 cm, clip=true]{graph0}
\caption{Density evolution for a single run of a SW fluid with $\delta$=0.01, $T^{*}=0.233$, and $\mu/kT=-2.41$ (close to the critical point). At long time, a transition to a
dense phase is observed and, consequently the simulation on the right side of the vertical line
is disregarded.}
\label{fig:densevol}
\end{center}
\end{figure}
\section{Results}
Fig.-\ref{fig:TcSWAHS}-(a) and (b)
show the $\delta$ dependence of
the critical temperature and the corresponding critical
second virial coefficient
$B_2^{*c}$. For SW, the virial can be calculated as
\begin{equation}
B_2^*(T)=1-((\delta+1)^3-1)(e^{1/T}-1)
\end{equation}
For small $\delta$, a clear linear dependence
of $B_2^{*c}$ is observed.
The data are well represented by a functional form
$B_2^{*c}(\delta)= -1.174 - 1.774~\delta $. The extrapolated value of $B_2^{*c}$ for $\delta=0$ ($B_2^{*c}=-1.174$) is slightly higher than the value $-1.21(1)$ estimated by Miller and Frenkel for the AHS potential. The $\delta$ dependence of $B_2^{*c}$ formally violates the idea that $B_2^{*c}$ is the correct scaling variable for collapsing the phase-diagram of different short-range attractive potentials onto a single master curve. Nevertheless, the constant $B_2^{*c}$ approximation is sufficiently good to explain the $T_c$ dependence, due to the non-linearity of the transformation (Eq.~\ref{eq:Tc}). To prove this point we show in Fig.-\ref{fig:TcSWAHS}-(a) $T_c$ predicted according to $B_2^{*c}=-1.174$. In this representation, the assumption of constant $B_2^{*c}$ is sufficient to describe the $\delta$ dependence of $T_c$ up to $\delta=0.05$, with an error less than 1$\%$ (growing with $\delta$).
Next we compare the energy per particle of the system at the critical point in
Fig.-\ref{fig:TcSWAHS}-(c), to provide a measure of the number of contacts per particle.
This quantity also shows a linear dependence on $\delta$.
\begin{figure}[htbp]
\centering
\includegraphics[width=14 cm, clip=true]{graph1}
\caption{
Dependence of the critical temperature (a), of the second virial coefficient (b) and of the potential energy (c) on the range of the potential $\delta$.
Circles and crosses label simulation data of this work with box sides 5$\sigma$ and 8$\sigma$ respectively. The line in (a) is the theoretical prediction for the critical temperature provided by Eq.\ref{eq:Tc}. The line in (b) corresponds to the best linear fit of the simulation data for $\delta\leq0.10$ ($-1.174-1.774\delta$).
Open squares correspond to the simulation data from Ref.~\cite{Lomakin1996}. Filled squares are the AHS $B_2^{*c}$ result from Ref.~\cite{Miller2003}.
The inset presents the whole range of $\delta$ values studied.
}
\label{fig:TcSWAHS}
\end{figure}
The study of the $\delta$ dependence of the critical density $\rho_c$ has received considerably less attention than the $\delta$ dependence of $T_c$. Recently Ref.\cite{Foffi2007} suggested a plausible
relation for the $\delta$ dependence of $\rho_c$ in the limit of
small well width:
\begin{equation}
\rho_c(\delta)=\frac{\rho_c(0)}{(1+\delta/2)^3},
\label{eq:rhoc}
\end{equation}
The relation is
based on the hypothesis that $\rho_c$ should be constant if measured using the average distance between two bonded particles $(1+\delta/2)\sigma$ as the unit of length.
The relation was also supported by a potential energy landscape
interpretation of the generalized law of corresponding states\cite{foffipre} which shows that configurations with the same Boltzmann weight are generated under an isotropic scaling (to change the inter-particle distances preserving the same bonding pattern) and a simultaneous change of both $\delta$ and $T$ such that the bond free-energy remains constant.
Figure~\ref{fig:RHOcSW}-(a) shows the calculated evolution of the critical density with the range of the interaction. It also reports
previous estimates for the same system\cite{Lomakin1996} as well as the
critical density for the AHS model from Ref.\cite{Miller2003}. As the range of the SW potential is reduced, the critical density becomes higher, since a higher local density is required to generate bonded configurations.
Data for $\delta<0.1 \sigma$ are properly represented by
Eq.~\ref{eq:rhoc}, with a resulting fitting parameter $\rho(0)=0.552$. This value is significantly higher than the AHS critical density
$\rho_c=0.508$ reported in Ref.-\cite{Miller2003}.
\begin{figure}[htbp]
\centering
\includegraphics[width=15 cm, clip=true]{graph2}
\caption{ Evolution of the critical density (a), and chemical potential (b) with the range of the potential $\delta$.
Circles and crosses correspond to box sides $L=5\sigma$ and $L=8\sigma$ respectively. Empty squares in (a) are data from Ref.~\cite{Lomakin1996}.
The lines correspond to the best fit of (a) $\rho_c$ for $\delta\leq0.10$ ($0.5519/(1+\delta/2)^3$) and (b) $\mu_c/k_BT_c$ for $\delta\le0.10$ ($-2.394-1.431\delta$).
Filled squares represent AHS results from Ref.~\cite{Miller2003}. The inset presents the whole range of $\delta$ values studied in this work. }
\label{fig:RHOcSW}
\end{figure}
For completeness we show in Figure~\ref{fig:RHOcSW}-(b)
the $\delta$ dependence of $\mu_c/k_BT_c$, where
$\mu_c$ is the value of the chemical potential at the critical point.
$\mu_c/k_BT_c$ also shows a linear dependence, with an intercept
at $-2.394$, corresponding to a critical activity $\exp(\mu_c/k_BT_c)=0.091$,
to be compared with the corresponding value of
$0.087$ of Miller and Frenkel.
It is known that the finite size of the system modifies the position of the critical point~\cite{Wilding1995}. A precise estimate of the critical point requires a complete finite-size scaling study to extrapolate the results to an infinite system. We have not attempted to perform such a
careful study since it would be computationally prohibitive for the small ranges studied here and we have limited
ourselves to four different box sizes ($L=5$, 6, 8 and 10) for $\delta=0.05$ only.
The results are reported in table~\ref{tab:fsize}. As already suggested by the minor differences in the $L=5$ and $L=8$ data shown in the previous figures, no significant changes in the critical parameters are observed.
From the scatter in the data (similar to that
found by Miller and Frenkel\cite{Miller2003}) it is possible to estimate errors in the critical parameters.
The resulting values of and errors in the critical parameters are $T_c=0.3658\pm 0.0005$, $\rho_c=0.513\pm 0.008$ and $\mu_c=0.9064\pm 0.0008$. These values of the critical temperature and density suggest that the Percus-Yevick energy route \cite{Watts1971} is even better than previously thought, despite the fact that the compressibility route \cite{Baxter1968} results seem to be more often used and cited.
\begin{table}[htdp]
\caption{Critical parameters for the SW system of $\delta=0.05$ obtained with four different boxes of side $L=5$, 6, 8, 10.}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$L$ & $T_c$ & $\rho_c$ & $\mu_c$ \\
\hline
5 & 0.3660 & 0.516 & -0.9042 \\
6 & 0.3660 & 0.516 & -0.9051 \\
8 & 0.3658 & 0.513 & -0.9062 \\
10 & 0.3657 & 0.511 & -0.9069 \\
\hline
\end{tabular}
\end{center}
\label{tab:fsize}
\end{table}%
\section{Discussion}
Our study provides a set of values for the limiting AHS case, based on an accurate extrapolation of the critical parameters of the SW potential to
$\delta \rightarrow 0$. These values are
outside the error bars of Miller and Frenkel's investigation. In particular, both the critical density
and the critical virial appear to be higher than the previous estimates.
The special techniques employed in the simulation of the AHS system~\cite{Miller2003,Miller2004,kranendonkfrenkel} only consider moves that
make or break up to three contacts. A particle can readily gain more than three contacts, since higher coordination states are established by a succession of such moves. Apparently, however, the constraint on the possible moves
disfavors the formation of small nuclei of solid phases, since crystallization was extremely rarely observed with this algorithm, and then only in fluids with a very high mean reduced density (greater than 0.9).
In the SW simulation, the transient solid-like nuclei are more readily formed (and indeed we do
occasionally observe crystallization during the simulation) suggesting that the SW simulations sample
a larger region of configuration space than that accessible with the
AHS algorithm. This could indeed explain why the critical density extrapolated from the
SW simulations is significantly higher than that calculated previously.
To support this interpretation we have compared the distribution of the number of contacts per particle
(proportional to the energy of the particle) for the AHS and a SW with $\delta=0.01$
at the same virial coefficient (slightly above the critical one) and same density for three different state points. The results of MC simulations in the NVT canonical ensemble, for different densities are reported in Fig.~\ref{fig:cn+SW}. The distributions of the number of contacts per particle are coincident for low densities, but discrepancies appear as the density is increased, confirming that the algorithm used in Ref.~\cite{Miller2003} explores configurations with a somewhat smaller coordination number than does the standard MC SW simulation. Figure~\ref{fig:cn+SW} nevertheless confirms that the AHS algorithm permits the formation of high coordination states.
To avoid any potential artifact due to the {\em a priori} unknown mapping in the density
between the AHS and the SW potential, we have also repeated the calculation at a lower density,
scaled according to Eq.~\ref{eq:rhoc}. However, as shown in Fig.~\ref{fig:cn+SW}, even when the density is scaled to account for the different
bond distance in the AHS and SW models, at high density the disagreement between the two set of
simulations remains.
\begin{figure}[htbp]
\centering
\includegraphics[width=15 cm, clip=true]{cn+swb}
\caption{Probability distribution of the number of contacts per particle for the AHS system and the SW with $\delta=0.01$, for different densities. The comparison has been made at a slightly supercritical stickiness parameter $\tau$=0.120, in the case of the AHS, and the corresponding temperature provided by Eq.~\ref{eq:Tc} for the SW of $\delta=0.01$. The simulations were performed in the NVT canonical ensemble with volume $V=512\sigma^3$. Symbols correspond to AHS simulation data, for $\rho$=0.195 (circles), 0.488 (squares), 0.781 (diamonds). Dashed lines correspond to SW simulation data and dotted lines to SW simulations at the rescaled densities (see Eq.~\ref{eq:rhoc}) 0.192, 0.481, 0.770, respectively. }
\label{fig:cn+SW}
\end{figure}
\begin{table}[htdp]
\caption{Critical point parameters for the SW fluid for width $\delta$, resulting from simulations of boxes of side $L=5$ (top part of the table) and 8~$\sigma$ (bottom part of the table) respectively.}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$\delta$ &$T^*_c$&$\rho_c$&$\mu_c$\\
\hline
0.005 & 0.2007 & 0.542 & -0.4817 \\
0.01 & 0.2328 & 0.540 & -0.5614 \\
0.02 & 0.2769 & 0.538 & -0.6693 \\
0.03 & 0.3106 & 0.530 & -0.7575 \\
0.04 & 0.3398 & 0.522 & -0.8333 \\
0.05 & 0.3660 & 0.516 & -0.9042 \\
0.10 & 0.4780 & 0.478 & -1.2120 \\
\hline
0.05 & 0.3658 & 0.513 & -0.9062 \\
0.10 & 0.4780 & 0.478 & -1.2138 \\
0.20 & 0.667 & 0.421 & -1.7812 \\
0.30 & 0.847 & 0.376 & -2.3505 \\
0.40 & 1.029 & 0.339 & -2.9525 \\
0.50 & 1.220 & 0.310 & -3.6002 \\
0.60 & 1.430 & 0.287 & -4.3051 \\
0.70 & 1.665 & 0.272 & -5.0890 \\
0.80 & 1.940 & 0.263 & -5.9636 \\
\hline
\end{tabular}
\end{center}
\label{default}
\end{table}%
\section{Conclusions}
We have reported a simulation study aimed at evaluating the dependence of the critical parameters of the
short-range square well potential for interaction ranges approaching zero, down to $\delta=0.005 \sigma$. The resulting values for $B_2^{*c}$ and $\beta_c \mu_c$ in the range $0.005< \delta < 0.1$ are very well described by a linear dependence on $\delta$, providing an estimate for the $\delta \rightarrow 0$ limit. From the resulting value of $B_2^{*c}$ at $\delta=0$, it is possible to evaluate also
the corresponding critical value of the AHS model via the relation $B_2^{*}=1-1/4\tau$.
In the same range, the critical density is well represented by the previously proposed functional form of Eq.~\ref{eq:rhoc}.
More precisely, our extrapolation for the AHS limit is:
\begin{eqnarray}
B_2^{*c}(\delta=0)=-1.174\pm 0.002 \\
\tau_c = 0.1150\pm 0.0001 \\
\rho_c(\delta=0) = 0.552 \pm 0.001 \\
\beta_c \mu_c(\delta=0)=-2.394 \pm 0.001
\end{eqnarray}
where the error bars are based on the spread of results from the different box sizes in this study.
Three important observations are in order:
i) Even for short-range interactions, $B_2^{*c}$ shows a slight linear dependence on $\delta$,
apparently contradicting the law of corresponding states. While this is technically correct,
one must also remember that the small changes of $B_2^{*c}$ with $\delta$, in the range
$0< \delta < 0.05$ are not sufficient to produce any significant observable effect in the
critical temperature estimated using a constant $B_2^{*c}$ assumption via Eq.~\ref{eq:Tc}.
This is due to the highly non-linear relationship between the two quantities, explaining the
success of the law of corresponding states in the interpretation of simulation and experimental data of short-ranged attractive potentials.
From a more academic point of view, the law of corresponding states
may still hold but with a scaling variable more complicated than the reduced virial coefficient itself and with a proper scaling of the density.
ii) The ``best'' critical parameters of the $\delta=0$ case are found to be different from those
reported by Miller and Frenkel~\cite{Miller2003}. We believe this discrepancy is
related to an incomplete mapping of the configuration space which manifests
itself in a less complete sampling of the dense region. The extreme rarity of any hint of crystallization
in the simulations of Miller and Frenkel is consistent with this
proposed explanation.
iii) The higher critical temperature and density established by the SW extrapolation suggest that the Percus-Yevick energy route offers a more accurate estimate of the AHS critical point than the Percus-Yevick compressibility route.
We acknowledge support from MCRTN-CT-2003-504712 and J. L. MERG-CT-2006-046453. We thank D. Frenkel and G. Foffi for helpful discussions.
|
1,314,259,996,532 | arxiv | \section{Introduction}
\label{ug1}
A {\it Kuranishi space\/} is a topological space with a {\it
Kuranishi structure}, defined by Fukaya and Ono \cite{FuOn,FOOO}.
Let $Y$ be an orbifold and $R$ a ${\mathbin{\mathbb Q}}$-algebra. In the book
\cite{Joyc1} the author develops {\it Kuranishi homology\/}
$KH_*(Y;R)$ and {\it Kuranishi cohomology\/} $KH^*(Y;R)$. The
(co)chains in these theories are of the form $[X,\boldsymbol f,\boldsymbol G]$ where
$X$ is a compact Kuranishi space, $\boldsymbol f:X\rightarrow Y$ is a strongly
smooth map, and $\boldsymbol G$ is some extra {\it gauge-fixing data}. We
prove $KH_*(Y;R)$ is isomorphic to singular homology
$H_*^{\rm si}(Y;R)$, and $KH^*(Y;R)$ is isomorphic to
compactly-supported cohomology $H^*_{\rm cs}(Y;R)$. We also define {\it
Kuranishi bordism\/} $KB_*(Y;R)$ and {\it Kuranishi cobordism}
$KB^*(Y;R)$, for $R$ a commutative ring.
This paper is a brief introduction to selected parts of
\cite{Joyc1}. The length of \cite{Joyc1} (presently 290 pages) is
likely to deter people from reading it, but the main ideas can be
summarized much more briefly, and that is what this paper tries to
do. A User's Guide, say for a car or a computer, should give you a
broad overview of how the machine actually works, and instructions
on how to use it in practice, but it should probably not tell you
exactly where the carburettor is, or how the motherboard is wired.
This paper is written in the same spirit. It should provide you with
sufficient background to understand the sequels
\cite{AkJo,Joyc2,Joyc3} (once I get round to writing them), and also
to decide whether Kuranishi (co)homology is a good tool to use in
problems you are interested in.
Here are the main areas of \cite{Joyc1} that we will {\it not\/}
cover. In \cite{Joyc1} we also define {\it effective Kuranishi
(co)homology\/} $KH_*^{\rm ef},KH^*_{\rm ec}(Y;R)$, a variant on
Kuranishi homology that has the advantage that it works for all
commutative rings $R$, including $R={\mathbin{\mathbb Z}}$, and is isomorphic to
$H_*^{\rm si},H^*_{\rm cs}(Y;R)$, but has the disadvantage of restrictions on
the Kuranishi spaces $X$ allowed in chains $[X,\boldsymbol f,\boldsymbol G]$, which
limits its applications in Symplectic Geometry. Similarly,
\cite[Ch.~5]{Joyc1} actually defines five different kinds of
Kuranishi (co)bordism, but below we consider only the simplest. The
applications to Symplectic Geometry in \cite[Ch.~6]{Joyc1} are
omitted.
Kuranishi (co)homology is intended primarily as a tool for use in
areas of Symplectic Geometry involving $J$-holomorphic curves, and
will be applied in the sequels \cite{AkJo,Joyc2,Joyc3,Joyc4}.
Kuranishi structures occur naturally on many moduli spaces in
Differential Geometry. For example, if $(M,\omega)$ is a compact
symplectic manifold with almost complex structure $J$ then the
moduli space ${\mathbin{\smash{\,\,\overline{\!\!\mathcal M\!}\,}}}_{g,m}(M,J,\beta)$ of stable $J$-holomorphic curves
of genus $g$ with $m$ marked points in class $\beta$ in $H_2(M;{\mathbin{\mathbb Z}})$ is
a compact Kuranishi space with a strongly smooth map $\prod_i{\bf
ev}_i:{\mathbin{\smash{\,\,\overline{\!\!\mathcal M\!}\,}}}_{g,m}(M,J,\beta)\rightarrow M^m$. By choosing some gauge-fixing data
$\boldsymbol G$ we define a cycle $[{\mathbin{\smash{\,\,\overline{\!\!\mathcal M\!}\,}}}_{g,m}(M,J,\beta),\prod_i{\bf ev}_i,\boldsymbol
G]$ in $KC_*(M^m;{\mathbin{\mathbb Q}})$ whose homology class $\bigl[[{\mathbin{\smash{\,\,\overline{\!\!\mathcal M\!}\,}}}_{g,m}(M,\allowbreak
J,\allowbreak\beta),\prod_i{\bf ev}_i,\boldsymbol G]\bigr]$ in $KH_*(M^m;{\mathbin{\mathbb Q}})\cong
H^{\rm si}_*(M^m;{\mathbin{\mathbb Q}})$ is a {\it Gromov--Witten invariant\/} of
$(M,\omega)$, and is independent of the choice of almost complex
structure~$J$.
In the conventional definitions of symplectic Gromov--Witten
invariants \cite{FuOn,LiTi,Ruan,Sieb}, one must define a {\it
virtual cycle\/} for ${\mathbin{\smash{\,\,\overline{\!\!\mathcal M\!}\,}}}_{g,m}(M,J,\beta)$. This is a complicated
process, involving many arbitrary choices: first one must perturb
the moduli space, morally over ${\mathbin{\mathbb Q}}$ rather than ${\mathbin{\mathbb Z}}$, so that it
becomes something like a manifold. Then one must triangulate the
perturbed moduli space by simplices to define a cycle in the
singular chains $C_*^{\rm si}(M^m;{\mathbin{\mathbb Q}})$. The Gromov--Witten invariant is
the homology class of this virtual cycle. By using Kuranishi
(co)homology as a substitute for singular homology, this process of
defining virtual cycles becomes much simpler and less arbitrary.
{\it The moduli space is its own virtual cycle}, and we eliminate
the need to perturb moduli spaces and triangulate by simplices.
The real benefits of the Kuranishi (co)homology approach come not in
closed Gromov--Witten theory, where the moduli spaces are Kuranishi
spaces without boundary, but in areas such as open Gromov--Witten
theory, Lagrangian Floer cohomology \cite{FOOO}, Contact Homology
\cite{EES}, and Symplectic Field Theory \cite{EGH}, where the moduli
spaces are Kuranishi spaces {\it with boundary and corners}, and
their boundaries are identified with fibre products of other moduli
spaces.
In the conventional approach, one must choose virtual chains for
each moduli space, which must be compatible at the boundary with
intersection products of choices of virtual chains for other moduli
spaces. This business of boundary compatibility of virtual chains is
horribly complicated and messy, and a large part of the 1385 pages
of Fukaya, Oh, Ohta and Ono's work on Lagrangian Floer cohomology
\cite{FOOO} is devoted to dealing with it. Using Kuranishi
cohomology, because we do not perturb moduli spaces, choosing
virtual chains with boundary compatibility is very easy, and
Lagrangian Floer cohomology can be reformulated in a much more
economical way, as we will show in \cite[\S 6.6]{Joyc1}
and~\cite{AkJo}.
An important feature of these theories is that {\it Kuranishi
homology and cohomology are very well behaved at the (co)chain
level}, much better than singular homology, say. For example,
Kuranishi cochains $KC^*(Y;R)$ have a supercommutative, associative
cup product $\cup$, cap products also work well at the (co)chain
level, and there is a well-behaved functor from singular chains
$C_*^{\rm si}(Y;R)$ to Kuranishi chains $KC_*(Y;R)$. Because of this,
the theories may also have applications in other areas which may not
be directly related to Kuranishi spaces, but which need a
(co)homology theory of manifolds or orbifolds with good
(co)chain-level behaviour. In \cite{Joyc3} we will apply Kuranishi
(co)chains to reformulate the String Topology of Chas and Sullivan
\cite{ChSu}, which involves chains on infinite-dimensional loop
spaces. Another possible area is Costello's approach to Topological
Conformal Field Theories \cite{Cost}, which involves a choice of
chain complex for homology, applied to moduli spaces of Riemann
surfaces.
As well as Kuranishi homology and cohomology, in \cite[Ch.~5]{Joyc1}
we also define {\it Kuranishi bordism\/} $KB_*(Y;R)$ and {\it
Kuranishi cobordism\/} $KB^*(Y;R)$. These are simpler than Kuranishi
(co)homology, being spanned by $[X,\boldsymbol f]$ for $X$ a compact
Kuranishi space {\it without boundary} and $\boldsymbol f:X\rightarrow Y$ strongly
smooth, and do not involve gauge-fixing data. In contrast to
Kuranishi (co)homology which is isomorphic to conventional homology
and compactly-supported cohomology, these are new topological
invariants, and we show that they are very large --- for instance,
if $Y\ne\emptyset$ and $R\otimes_{\mathbin{\mathbb Z}}{\mathbin{\mathbb Q}}\ne 0$ then $KB_{2k}(Y;R)$ is infinitely
generated over $R$ for all~$k\in{\mathbin{\mathbb Z}}$.
In \cite[\S 6.2]{Joyc1} we define new Gromov--Witten type invariants
$[{\mathbin{\smash{\,\,\overline{\!\!\mathcal M\!}\,}}}_{g,m}(M,\allowbreak J,\allowbreak\beta),\prod_i{\bf ev}_i]$ in Kuranishi bordism
$KB_*(M^m;{\mathbin{\mathbb Z}})$. Since these are defined in groups over ${\mathbin{\mathbb Z}}$, not
${\mathbin{\mathbb Q}}$, the author expects that Kuranishi (co)bordism will be useful
in studying {\it integrality properties} of Gromov--Witten
invariants. In \cite[\S 6.3]{Joyc1} we outline an approach to
proving the Gopakumar--Vafa Integrality Conjecture for
Gromov--Witten invariants of Calabi--Yau 3-folds, which the author
hopes to take further in~\cite{Joyc4}.
\medskip
\noindent{\it Acknowledgements.} I am grateful to Mohammed Abouzaid,
Manabu Akaho, Kenji Fukaya, Ezra Getzler, Shinichiroh Matsuo,
Yong-Geun Oh, Hiroshi Ohta, Kauru Ono, Paul Seidel, Ivan Smith and
Dennis Sullivan for useful conversations. This research was
partially supported by EPSRC grant EP/D07763X/1.
\section{Kuranishi spaces}
\label{ug2}
{\it Kuranishi spaces\/} were introduced by Fukaya and Ono
\cite{FOOO,FuOn}, and are important in Symplectic Geometry because
moduli spaces of stable $J$-holomorphic curves in symplectic
manifolds are Kuranishi spaces. We use the definitions and notation
of \cite[\S 2]{Joyc1}, which have some modifications from those of
Fukaya--Ono.
\subsection{Manifolds and orbifolds with corners and g-corners}
\label{ug21}
In \cite{Joyc1} we work with four classes of manifolds, in
increasing order of generality: {\it manifolds without boundary},
{\it manifolds with boundary}, {\it manifolds with corners}, and
{\it manifolds with generalized corners\/} or {\it g-corners}. The
first three classes are fairly standard, although the author has not
found a reference for foundational material on manifolds with
corners. Manifolds with g-corners are new, as far as the author
knows. The precise definitions of these classes of manifolds are
given in \cite[\S 2.1]{Joyc1}. Here are the basic ideas:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item An $n$-{\it dimensional manifold without boundary\/} is
locally modelled on open sets in ${\mathbin{\mathbb R}}^n$.
\item An $n$-{\it dimensional manifold with boundary\/} is
locally modelled on open sets in ${\mathbin{\mathbb R}}^n$ or $[0,\infty)\times{\mathbin{\mathbb R}}^{n-1}$.
\item An $n$-{\it dimensional manifold with corners\/} is
locally modelled on open sets in $[0,\infty)^k\times{\mathbin{\mathbb R}}^{n-k}$ for
$k=0,\ldots,n$.
\item A {\it polyhedral cone\/} $C$ in ${\mathbin{\mathbb R}}^n$ is a subset of the
form
\begin{equation*}
C=\bigl\{(x_1,\ldots,x_n)\in{\mathbin{\mathbb R}}^n: a_1^ix_1+\cdots+a_n^ix_n\geqslant 0,\;\>
i=1,\ldots,k\bigr\},
\end{equation*}
where $a^i_j\in{\mathbin{\mathbb R}}$ for $i=1,\ldots,k$ and~$j=1,\ldots,n$.
An $n$-{\it dimensional manifold with g-corners\/} is roughly
speaking locally modelled on open sets in polyhedral cones $C$ in
${\mathbin{\mathbb R}}^n$ with nonempty interior $C^\circ$. Since $[0,\infty)^k\times{\mathbin{\mathbb R}}^{n-k}$
is a polyhedral cone $C$ with $C^\circ\ne\emptyset$, manifolds with corners
are examples of manifolds with g-corners. In fact the subsets in
${\mathbin{\mathbb R}}^n$ used as local models for manifolds with g-corners are more
general than polyhedral cones, but this extra generality is only
needed for technical reasons in the proof of Theorem \ref{ug3thm2}.
\end{itemize}
Here are some examples. The line ${\mathbin{\mathbb R}}$ is a 1-manifold without
boundary; the interval $[0,1]$ is a 1-manifold with boundary; the
square $[0,1]^2$ is a 2-manifold with corners; and the octahedron
\begin{equation*}
O=\bigl\{(x_1,x_2,x_3)\in{\mathbin{\mathbb R}}^3:\md{x_1}+\md{x_2}+\md{x_3}\leqslant\nobreak 1\bigr\}
\end{equation*}
in ${\mathbin{\mathbb R}}^3$ is a 3-manifold with g-corners. It is not a manifold with
corners, since four 2-dimensional faces of $O$ meet at the vertex
$(1,0,0)$, but three 2-dimensional faces of $[0,\infty)^3$ meet at the
vertex $(0,0,0)$, so $O$ near $(1,0,0)$ is not locally modelled on
$[0,\infty)^3$ near~$(0,0,0)$.
Manifolds $X$ with boundary, corners, or g-corners have a
well-behaved notion of {\it boundary\/} $\partial X$. To motivate the
definition, consider $[0,\infty)^2$ in ${\mathbin{\mathbb R}}^2$. If we took
$\partial\bigl([0,\infty)^2\bigr)$ to be the subset
$\bigl([0,\infty)\times\{0\}\bigr)\cup\bigl(\{0\}\times[0,\infty)\bigr)$ of
$[0,\infty)^2$, then $\partial\bigl([0,\infty)^2\bigr)$ would not be a manifold
with corners near $(0,0)$. Instead, we take
$\partial\bigl([0,\infty)^2\bigr)$ to be the {\it disjoint union\/} of the
two boundary strata $[0,\infty)\times\{0\}$ and $\{0\}\times[0,\infty)$. This is a
manifold with boundary, but now $\partial\bigl([0,\infty)^2\bigr)$ is {\it
not a subset of\/} $[0,\infty)^2$, since two points in
$\partial\bigl([0,\infty)^2\bigr)$ correspond to $(0,0)$ in~$[0,\infty)^2$.
We define the {\it boundary\/} $\partial X$ of an $n$-manifold $X$ with
(g-)corners to be the set of pairs $(p,B)$, where $p\in X$ and $B$
is a local choice of connected $(n\!-\!1)$-dimensional boundary
stratum of $X$ containing $p$. Thus, if $p$ lies in a codimension
$k$ corner of $X$ locally modelled on $[0,\infty)^k\times{\mathbin{\mathbb R}}^{n-k}$ then $p$
is represented by $k$ distinct points $(p,B_i)$ in $\partial X$ for
$i=1,\ldots,k$. Then $\partial X$ is an $(n\!-\!1)$-manifold with
(g-)corners. Note that $\partial X$ is not a subset of $X$, but has a
natural immersion $\iota:\partial X\rightarrow X$ mapping $(p,B)\mapsto p$. Often
we suppress $\iota$, and talk of restricting data on $X$ to $\partial X$,
when really we mean the pullback by~$\iota$.
If $X$ is a $n$-manifold with (g-)corners then $\partial^2X$ is an
$(n\!-\!2)$-manifold with (g-)corners. Points of $\partial^2X$ may be
written $(p,B_1,B_2)$, where $p\in X$ and $B_1,B_2$ are distinct
local boundary components of $X$ containing $p$. There is a natural,
free involution $\sigma:\partial^2X\rightarrow\partial^2X$ mapping
$\sigma:(p,B_1,B_2)\mapsto(p,B_2,B_1)$, which is orientation-reversing
if $X$ is oriented. This involution is important in questions to do
with extending data defined on $\partial X$ to $X$. For example, if
$f:\partial X\rightarrow{\mathbin{\mathbb R}}$ is a smooth function, then a necessary condition for
there to exist smooth $g:X\rightarrow{\mathbin{\mathbb R}}$ with $g\vert_{\partial X}\equiv f$ is
that $f\vert_{\partial^2X}$ is $\sigma$-invariant, and if $X$ has corners
(not g-corners) then this condition is also sufficient.
{\it Orbifolds\/} are a generalization of manifold, which allow
quotients by finite groups. Again, we define orbifolds {\it without
boundary}, or {\it with boundary}, or {\it with corners}, or {\it
with g-corners}, where orbifolds without boundary are locally
modelled on quotients ${\mathbin{\mathbb R}}^n/\Gamma$ for $\Gamma$ a finite group acting
linearly on ${\mathbin{\mathbb R}}^n$, and similarly for the other classes. (We do not
require $\Gamma$ to act {\it effectively}, so we cannot regard $\Gamma$ as
a subgroup of GL$(n,{\mathbin{\mathbb R}})$.) Orbifolds (then called $V$-manifolds)
were introduced by Satake \cite{Sata}, and a book on orbifolds is
Adem et al.\ \cite{ALR}. Note however that the right definition of
smooth maps of orbifolds is not that given by Satake, but the more
complex notion of {\it morphisms of orbifolds\/} in \cite[\S
2.4]{ALR}. When we do not specify otherwise, by a manifold or
orbifold, we always mean a manifold or orbifold with g-corners, the
most general class.
Let $X,Y$ be manifolds of dimensions $m,n$. {\it Smooth maps}
$f:X\rightarrow Y$ are continuous maps which are locally modelled on smooth
maps from ${\mathbin{\mathbb R}}^m\rightarrow{\mathbin{\mathbb R}}^n$. A smooth map $f$ induces a morphism of
vector bundles ${\rm d} f:TX\rightarrow f^*(TY)$ on $X$, where $TX,TY$ are the
tangent bundles of $X$ and $Y$. For manifolds $X,Y$ without
boundary, we call a smooth map $f:X\rightarrow Y$ a {\it submersion} if ${\rm d}
f:TX\rightarrow f^*(TY)$ is a surjective morphism of vector bundles. If
$X,Y$ have boundary or (g-)corners, the definition of submersions
$f:X\rightarrow Y$ in \cite[\S 2.1]{Joyc1} is more complicated, involving
conditions over $\partial^kX$ and $\partial^lY$ for all $k,l\geqslant 0$. When $\partial
Y=\emptyset$, $f$ is a submersion if ${\rm d}(f\vert_{\partial^kX}):T(\partial^kX)\rightarrow
f\vert_{\partial^kX}^*(TY)$ is surjective for all~$k\geqslant 0$.
Let $X,X',Y$ be manifolds (in any of the four classes above) and
$f:X\rightarrow Y$, $f':X'\rightarrow Y$ be smooth maps, at least one of which is a
submersion. Then the {\it fibre product\/} $X\times_{f,Y,f'}X'$ or
$X\times_YX'$ is
\e
X\times_{f,Y,f'}X'=\bigl\{(p,p')\in X\times X':f(p)=f'(p')\bigr\}.
\label{ug2eq1}
\e
It turns out \cite[Prop.~2.6]{Joyc1} that $X\times_YX'$ is a submanifold
of $X\times X'$, and so is a manifold (in the same class as $X,X',Y$).
When $X,X',Y$ have boundary or (g-)corners, the complicated
definition of submersion is necessary to make $X\times_YX'$ a
submanifold over~$\partial^kX,\partial^{k'}X',\partial^lY$.
Fibre products can be defined for orbifolds \cite[\S 2.2]{Joyc1},
but there are some subtleties to do with stabilizer groups. To
explain this, first consider the case in which $U,U',V$ are
manifolds, and $\Gamma,\Gamma',\Delta$ are finite groups acting on $U,U',V$
by diffeomorphisms so that $U/\Gamma$, $U'/\Gamma'$, $V/\Delta$ are
orbifolds, and $\rho:\Gamma\rightarrow\Delta$, $\rho':\Gamma'\rightarrow\Delta$ are group
homomorphisms, and $f:U\rightarrow V$, $f':U'\rightarrow V$ are smooth $\rho$- and
$\rho'$-equivariant maps, at least one of which is a submersion.
Then $f,f'$ induce smooth maps of orbifolds $f_*:U/\Gamma\rightarrow V/\Delta$,
$f'_*:U'/\Gamma'\rightarrow V/\Delta$, at least one of which is a submersion.
It turns out that the right answer for the orbifold fibre product is
\e
(U/\Gamma)\times_{f_*,V/\Delta,f'_*}(U'/\Gamma')=\bigl((U\times U')\times_{f\times f',V\times
V,\pi}(V\times\Delta)\bigr)/(\Gamma\times\Gamma').
\label{ug2eq2}
\e
Here $\pi:V\times\Delta\rightarrow V\times V$ is given by $\pi:(v,\delta)\mapsto
(v,\delta\cdot v)$, and $(U\times U')\times_{f\times f',V\times V,\pi}(V\times\Delta)$ is the
fibre product of smooth manifolds, and $\Gamma\times\Gamma'$ acts on the
manifold $(U\times U')\times_{V\times V}(V\times\Delta)$ by diffeomorphism
$(\gamma,\gamma'):\bigl((u,u'),(v,\delta)\bigr) \mapsto\bigl((\gamma\cdot
u,\gamma'\cdot u'),(\rho(\gamma)\cdot
v,\rho'(\gamma')\delta\rho'(\gamma')^{-1})\bigr)$, so that the quotient is an
orbifold. Now \eq{ug2eq2} coincides with \eq{ug2eq1} for $X=U/\Gamma$,
$X'=U'/\Gamma'$, $Y=V/\Delta$ only if one of $\rho:\Gamma\rightarrow\Delta$,
$\rho',\Gamma'\rightarrow\Delta$ are surjective; otherwise the projection from
\eq{ug2eq2} to \eq{ug2eq1} is a finite surjective map, but not
necessarily injective.
This motivates the definition of fibre products of orbifolds. Let
$X,X',Y$ be orbifolds, and $f:X\rightarrow Y$, $f':X'\rightarrow Y$ be smooth maps,
at least one of which is a submersion. Then for $p\in X$ and $p'\in
X'$ with $f(p)=q=f(p')$ in $Y$ we have morphisms of stabilizer
groups $f_*:\mathop{\rm Stab}\nolimits(p)\rightarrow\mathop{\rm Stab}\nolimits(q)$, $f'_*:\mathop{\rm Stab}\nolimits(p')\rightarrow\mathop{\rm Stab}\nolimits(q)$. Thus
we can form the double coset space
\begin{align*}
&f_*(\mathop{\rm Stab}\nolimits(p))\backslash\mathop{\rm Stab}\nolimits(q)/f'_*(\mathop{\rm Stab}\nolimits(p'))\\
&=\bigl\{ \{f_*(\gamma)\delta f_*(\gamma'):\gamma\in\mathop{\rm Stab}\nolimits(p),\;\>
\gamma'\in\mathop{\rm Stab}\nolimits(p')\}:\delta\in\mathop{\rm Stab}\nolimits(q)\bigr\}.
\end{align*}
As a set, we define
\e
\begin{split}
X\times_{f,Y,f'}X'=\bigl\{(p,p',\Delta):\,&\text{$p\in X$, $p'\in X'$,
$f(p)=f'(p')$,}\\
&\Delta\in f_*(\mathop{\rm Stab}\nolimits(p))\backslash\mathop{\rm Stab}\nolimits(f(p))/f'_*(\mathop{\rm Stab}\nolimits(p'))\bigr\}.
\end{split}
\label{ug2eq3}
\e
We give this an orbifold structure in a natural way, such that if
$(U,\Gamma,\phi)$, $(U',\Gamma',\phi'),(V',\Delta',\psi')$ are orbifold
charts on $X,X',Y$ with $f(\mathop{\rm Im}\phi),f'(\mathop{\rm Im}\phi')\subseteq\mathop{\rm Im}\psi$
then we use \eq{ug2eq2} to define an orbifold chart on~$X\times_YX'$.
\subsection{Kuranishi structures on topological spaces}
\label{ug22}
Let $X$ be a paracompact Hausdorff topological space throughout.
\begin{dfn} A {\it Kuranishi neighbourhood\/} $(V_p,E_p,s_p,
\psi_p)$ of $p\in X$ satisfies:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(i)] $V_p$ is an orbifold, which may
or may not have boundary or (g-)corners;
\item[(ii)] $E_p\rightarrow V_p$ is an orbifold vector bundle over $V_p$,
the {\it obstruction bundle};
\item[(iii)] $s_p:V_p\rightarrow E_p$ is a smooth section, the {\it
Kuranishi map}; and
\item[(iv)] $\psi_p$ is a homeomorphism from $s_p^{-1}(0)$ to
an open neighbourhood of $p$ in $X$, where $s_p^{-1}(0)$ is the
subset of $V_p$ where the section $s_p$ is zero.
\end{itemize}
\label{ug2def1}
\end{dfn}
\begin{dfn} Let $(V_p,E_p,s_p,\psi_p),(\tilde V_p,\tilde E_p,\tilde s_p,
\tilde\psi_p)$ be two Kuranishi neighbourhoods of $p\in X$. We call
$(\alpha,\hat\alpha):(V_p,\ldots,\psi_p)\rightarrow(\tilde V_p,\ldots,\tilde\psi_p)$ an
{\it isomorphism\/} if $\alpha:V_p\rightarrow\tilde V_p$ is a diffeomorphism and
$\hat\alpha:E_p\rightarrow\alpha^*(\tilde E_p)$ an isomorphism of orbibundles, such
that $\tilde s_p\circ\alpha\equiv\hat\alpha\circ s_p$
and~$\tilde\psi_p\circ\alpha\equiv\psi_p$.
We call $(V_p,\ldots,\psi_p),(\tilde V_p,\ldots,\tilde\psi_p)$ {\it
equivalent\/} if there exist open neighbourhoods $U_p\!\subseteq\!
V_p$, $\tilde U_p\!\subseteq\!\tilde V_p$ of $\psi_p^{-1}(p),
\tilde\psi_p^{-1}(p)$ such that
$(U_p,E_p\vert_{U_p},s_p\vert_{U_p},\psi_p\vert_{U_p})$ and $(\tilde
U_p,\tilde E_p\vert_{\tilde U_p},\tilde s_p\vert_{\tilde U_p},\tilde\psi_p
\vert_{\tilde U_p})$ are isomorphic.
\label{ug2def2}
\end{dfn}
\begin{dfn} Let $(V_p,E_p,s_p,\psi_p)$ and $(V_q,E_q,s_q,
\psi_q)$ be Kuranishi neighbourhoods of $p\in X$ and
$q\in\psi_p(s_p^{-1}(0))$ respectively. We call a pair
$(\phi_{pq},\hat\phi_{pq})$ a {\it coordinate change\/} from
$(V_q,\ldots,\psi_q)$ to $(V_p,\ldots,\psi_p)$ if:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(a)] $\phi_{pq}:V_q\rightarrow V_p$ is a smooth embedding of orbifolds;
\item[(b)] $\hat\phi_{pq}:E_q\rightarrow\phi_{pq}^*(E_p)$ is an embedding of
orbibundles over~$V_q$;
\item[(c)] $\hat\phi_{pq}\circ s_q\equiv s_p\circ\phi_{pq}$;
\item[(d)] $\psi_q\equiv \psi_p\circ\phi_{pq}$; and
\item[(e)] Choose an open neighbourhood $W_{pq}$ of
$\phi_{pq}(V_q)$ in $V_p$, and an orbifold vector subbundle $F_{pq}$
of $E_p\vert_{W_{pq}}$ with $\phi_{pq}^*(F_{pq})=\hat\phi_{pq}
(E_q)$, as orbifold vector subbundles of $\phi_{pq}^*(E_p)$ over
$V_q$. Write $\hat s_p:W_{pq}\rightarrow E_p/F_{pq}$ for the projection of
$s_p\vert_{W_{pq}}$ to the quotient bundle $E_p/F_{pq}$. Now
$s_p\vert_{\phi_{pq}(V_q)}$ lies in $F_{pq}$ by (c), so $\hat
s_p\vert_{\phi_{pq}(V_q)}\equiv 0$. Thus there is a well-defined
derivative
\begin{equation*}
{\rm d}\hat s_p:N_{\phi_{pq}(V_q)}V_p\rightarrow
(E_p/F_{pq})\vert_{\phi_{pq}(V_q)},
\end{equation*}
where $N_{\phi_{pq}(V_q)}V_p$ is the normal orbifold vector bundle
of $\phi_{pq}(V_q)$ in $V_p$. Pulling back to $V_q$ using
$\phi_{pq}$, and noting that $\phi_{pq}^*(F_{pq})=\hat\phi_{pq}
(E_q)$, gives a morphism of orbifold vector bundles over~$V_q$:
\e
{\rm d}\hat s_p:\frac{\phi_{pq}^*(TV_p)}{({\rm d}\phi_{pq})(TV_q)}\longrightarrow
\frac{\phi_{pq}^*(E_p)}{\hat\phi_{pq}(E_q)}\,.
\label{ug2eq4}
\e
We require that \eq{ug2eq4} should be an {\it isomorphism\/} over
$s_q^{-1}(0)$.
\end{itemize}
\label{ug2def3}
\end{dfn}
Here Definition \ref{ug2def3}(e) replaces the notion in
\cite[Def.~5.6]{FuOn}, \cite[Def.~A1.14]{FOOO} that a Kuranishi
structure {\it has a tangent bundle}.
\begin{dfn} A {\it germ of Kuranishi neighbourhoods at\/} $p\in X$
is an equivalence class of Kuranishi neighbourhoods
$(V_p,E_p,s_p,\psi_p)$ of $p$, using the notion of equivalence in
Definition \ref{ug2def2}. Suppose $(V_p,E_p,s_p,\psi_p)$ lies in
such a germ. Then for any open neighbourhood $U_p$ of
$\psi_p^{-1}(p)$ in $V_p$,
$(U_p,E_p\vert_{U_p},s_p\vert_{U_p},\psi_p\vert_{U_p})$ also lies in
the germ. As a shorthand, we say that some condition on the germ
{\it holds for sufficiently small\/} $(V_p,\ldots,\psi_p)$ if
whenever $(V_p,\ldots,\psi_p)$ lies in the germ, the condition holds
for $(U_p,\ldots,\psi_p\vert_{U_p})$ for all sufficiently small
$U_p$ as above.
A {\it Kuranishi structure\/} $\kappa$ on $X$ assigns a germ of
Kuranishi neighbourhoods for each $p\in X$ and a {\it germ of
coordinate changes\/} between them in the following sense: for each
$p\in X$, for all sufficiently small $(V_p,\ldots,\psi_p)$ in the
germ at $p$, for all $q\in\mathop{\rm Im}\psi_p$, and for all sufficiently small
$(V_q,\ldots,\psi_q)$ in the germ at $q$, we are given a coordinate
change $(\phi_{pq},\hat\phi_{pq})$ from $(V_q,\ldots,\psi_q)$ to
$(V_p,\ldots,\psi_p)$. These coordinate changes should be compatible
with equivalence in the germs at $p,q$ in the obvious way, and
satisfy:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(i)] $\mathop{\rm dim}\nolimits V_p-\mathop{\rm rank} E_p$ is independent of $p$\/; and
\item[(ii)] if $q\in\mathop{\rm Im}\psi_p$ and $r\in\mathop{\rm Im}\psi_q$ then
$\phi_{pq}\circ\phi_{qr}=\phi_{pr}$ and~$\hat\phi_{pq}\circ
\hat\phi_{qr}=\hat\phi_{pr}$.
\end{itemize}
We call $\mathop{\rm vdim}\nolimits X=\mathop{\rm dim}\nolimits V_p-\mathop{\rm rank} E_p$ the {\it virtual dimension\/}
of the Kuranishi structure. A {\it Kuranishi space\/} $(X,\kappa)$ is a
topological space $X$ with a Kuranishi structure $\kappa$. Usually we
refer to $X$ as the Kuranishi space, suppressing~$\kappa$.
\label{ug2def4}
\end{dfn}
Loosely speaking, the above definitions mean that a Kuranishi space
is locally modelled on the zeroes of a smooth section of an orbifold
vector bundle over an orbifold. Moduli spaces of $J$-holomorphic
curves in Symplectic Geometry can be given Kuranishi structures in a
natural way, as in~\cite{FuOn,FOOO}.
\subsection{Strongly smooth maps and strong submersions}
\label{ug23}
In \cite[Def.~2.24]{Joyc1} we define {\it strongly smooth maps\/}
$\boldsymbol f:X\rightarrow Y$, for $Y$ an orbifold.
\begin{dfn} Let $X$ be a Kuranishi space, and $Y$ a smooth
orbifold. A {\it strongly smooth map\/} $\boldsymbol f:X\rightarrow Y$ consists of,
for all $p\in X$ and all sufficiently small $(V_p,E_p,s_p,\psi_p)$
in the germ of Kuranishi neighbourhoods at $p$, a choice of smooth
map $f_p:V_p\rightarrow Y$, such that for all $q\in\mathop{\rm Im}\psi_p$ and
sufficiently small $(V_q,\ldots,\psi_q)$ in the germ at $q$ with
coordinate change $(\phi_{pq},\hat\phi_{pq})$ from
$(V_q,\ldots,\psi_q)$ to $(V_p,\ldots,\psi_p)$ in the germ of
coordinate changes, we have $f_p\circ\phi_{pq}=f_q$. Then $\boldsymbol f$
induces a continuous map $f:X\rightarrow Y$ in the obvious way.
We call $\boldsymbol f$ a {\it strong submersion\/} if all the $f_p$ are
submersions, that is, the maps ${\rm d} f_p:TV_p\rightarrow f_p^*(TY)$ are
surjective, and also when $V_p$ has boundary or corners,
$f_p\vert_{\partial V_p}:\partial V_p\rightarrow Y$ is a submersion, and the
restriction of $f_p$ to each codimension $k$ corner is a submersion
for all~$k$.
\label{ug2def5}
\end{dfn}
There is also \cite[Def.~2.25]{Joyc1} a definition of strongly
smooth maps $\boldsymbol f:X\rightarrow Y$ for $X,Y$ Kuranishi spaces, which we will
not give. A {\it strong diffeomorphism\/} $\boldsymbol f:X\rightarrow Y$ is a
strongly smooth map with a strongly smooth inverse. It is the
natural notion of isomorphism of Kuranishi spaces.
\subsection{Boundaries of Kuranishi spaces}
\label{ug24}
We define the {\it boundary\/} $\partial X$ of a Kuranishi space $X$,
which is itself a Kuranishi space of dimension~$\mathop{\rm vdim}\nolimits X-1$.
\begin{dfn} Let $X$ be a Kuranishi space. We shall define a
Kuranishi space $\partial X$ called the {\it boundary\/} of $X$. The
points of $\partial X$ are equivalence classes
$[p,(V_p,\ldots,\psi_p),B]$ of triples $(p,(V_p,\ldots,\psi_p),B)$,
where $p\in X$, $(V_p,\ldots,\psi_p)$ lies in the germ of Kuranishi
neighbourhoods at $p$, and $B$ is a local boundary component of
$V_p$ at $\psi_p^{-1}(p)$. Two triples
$(p,(V_p,\ldots,\psi_p),B),(q,(\tilde V_q,\ldots,\tilde\psi_q),C)$ are
{\it equivalent\/} if $p=q$, and the Kuranishi neighbourhoods
$(V_p,\ldots,\psi_p),\smash{(\tilde V_q,\ldots,\tilde\psi_q)}$ are
equivalent so that we are given an isomorphism
$(\alpha,\hat\alpha):(U_p,\ldots, \psi_p\vert_{U_p})\rightarrow(\tilde U_q,\ldots,
\tilde\psi_q\vert_{\tilde U_q})$ for open $\psi_p^{-1}(p)\in U_p\subseteq
V_p$ and $\tilde\psi_q^{-1}(q)\in\tilde U_q\subseteq\tilde V_q$, and
$\alpha_*(B)=C$ near $\tilde\psi_q^{-1}(q)$.
We can define a unique natural topology and Kuranishi structure on
$\partial X$, such that $(\partial V_p,E_p\vert_{\partial V_p},s_p\vert_{\partial
V_p},\psi'_p)$ is a Kuranishi neighbourhood on $\partial X$ for each
Kuranishi neighbourhood $(V_p,\ldots,\psi_p)$ on $X$, where
$\psi'_p:(s_p\vert_{\partial V_p})^{-1}(0)\rightarrow\partial X$ is given by
$\psi'_p:(q,B)\mapsto[\psi_p(q),(V_p,\ldots\psi_p),B]$ for
$(q,B)\in\partial V_p$ with $s_p(q)=0$. Then $\mathop{\rm vdim}\nolimits(\partial X)=\mathop{\rm vdim}\nolimits X-1$,
and $\partial X$ is compact if $X$ is compact.
\label{ug2def6}
\end{dfn}
In \S\ref{ug21} we explained that if $X$ is a manifold with
(g-)corners then there is a natural involution
$\sigma:\partial^2X\rightarrow\partial^2X$. The same construction works for orbifolds,
and for Kuranishi spaces. That is, if $X$ is a Kuranishi space then
as in \cite[\S 2.6]{Joyc1} there is a natural strong diffeomorphism
$\boldsymbol\sigma:\partial^2X\rightarrow\partial^2X$ with $\boldsymbol\sigma^2=\boldsymbol\mathop{\rm id}\nolimits_X$. If $X$ is
oriented as in \S\ref{ug26} below then $\boldsymbol\sigma$ is
orientation-reversing.
\subsection{Fibre products of Kuranishi spaces}
\label{ug25}
We can define {\it fibre products\/} of Kuranishi spaces
\cite[Def.~2.28]{Joyc1}, as for fibre products of manifolds and
orbifolds in~\S\ref{ug21}.
\begin{dfn} Let $X,X'$ be Kuranishi spaces, $Y$ an orbifold, and
$\boldsymbol f:X\rightarrow Y$, $\boldsymbol f':X'\rightarrow Y$ be strongly smooth maps inducing
continuous maps $f:X\rightarrow Y$ and $f':X'\rightarrow Y$. Suppose at least one of
$\boldsymbol f,\boldsymbol f'$ is a strong submersion. We shall define the {\it
fibre product\/} $X\times_YX'$ or $X\times_{\boldsymbol f,Y,\boldsymbol f'}X'$, a Kuranishi
space. As a set, the underlying topological space $X\times_YX'$ is given
by~\eq{ug2eq3}.
Let $p\in X$, $p'\in X'$ and $q\in Y$ with $f(p)=q=f'(p')$. Let
$(V_p,E_p,s_p,\psi_p)$, $(V'_{\smash{p'}},E'_{\smash{p'}},
s'_{\smash{p'}},\psi'_{\smash{p'}})$ be sufficiently small Kuranishi
neighbourhoods in the germs at $p,p'$ in $X,X'$, and $f_p:V_p\rightarrow Y$,
$f'_{\smash{p'}}:V'_{\smash{p'}}\rightarrow Y$ be smooth maps in the germs
of $\boldsymbol f,\boldsymbol f'$ at $p,p'$ respectively. Define a Kuranishi
neighbourhood on $X\times_YX'$ by
\e
\begin{split}
\bigl(V_p\times_{f_p,Y,f'_{\smash{p'}}}&V'_{\smash{p'}},\pi_{V_p}^*(E_p)\oplus
\pi_{V'_{\smash{p'}}}^*(E'_{\smash{p'}}),\\
&(s_p\circ\pi_{V_p})\oplus(s'_{\smash{p'}}\circ\pi_{V'_{\smash{p'}}}),
(\psi_p\circ\pi_{V_p})\times(\psi'_{\smash{p'}}\circ\pi_{V'_{\smash{p'}}})\times
\chi_{pp'}\bigr).
\end{split}
\label{ug2eq5}
\e
Here $V_p\times_{f_p,Y,f'_{\smash{p'}}}V'_{\smash{p'}}$ is the fibre
product of orbifolds, and $\pi_{V_p},\pi_{V'_{\smash{p'}}}$ are the
projections from $V_p\times_YV'_{\smash{p'}}$ to $V_p,V'_{\smash{p'}}$.
The final term $\chi_{pp'}$ in \eq{ug2eq5} maps the biquotient terms
in \eq{ug2eq3} for $V_p\times_YV'_{\smash{p'}}$ to the same terms in
\eq{ug2eq3} for the set $X\times_YX'$. Coordinate changes between
Kuranishi neighbourhoods in $X,X'$ induce coordinate changes between
neighbourhoods \eq{ug2eq5}. So the systems of germs of Kuranishi
neighbourhoods and coordinate changes on $X,X'$ induce such systems
on $X\times_YX'$. This gives a {\it Kuranishi structure\/} on $X\times_YX'$,
making it into a {\it Kuranishi space}. Clearly
$\mathop{\rm vdim}\nolimits(X\times_YX')=\mathop{\rm vdim}\nolimits X+\mathop{\rm vdim}\nolimits X'-\mathop{\rm dim}\nolimits Y$, and $X\times_YX'$ is compact
if $X,X'$ are compact.
\label{ug2def7}
\end{dfn}
\subsection{Orientations and orientation conventions}
\label{ug26}
In \cite[\S 2.7]{Joyc1} we define {\it orientations\/} on Kuranishi
spaces. Our definition is basically equivalent to Fukaya et al.\
\cite[Def.~A1.17]{FOOO}, noting that our Kuranishi spaces correspond
to their Kuranishi spaces with a tangent bundle.
\begin{dfn} Let $X$ be a Kuranishi space. An {\it orientation\/} on $X$
assigns, for all $p\in X$ and all sufficiently small Kuranishi
neighbourhoods $(V_p,E_p,s_p,\psi_p)$ in the germ at $p$,
orientations on the fibres of the orbibundle $TV_p\oplus E_p$ varying
continuously over $V_p$. These must be compatible with coordinate
changes, in the following sense. Let $q\in\mathop{\rm Im}\psi_p$,
$(V_q,\ldots,\psi_q)$ be sufficiently small in the germ at $q$, and
$(\phi_{pq},\hat\phi_{pq})$ be the coordinate change from
$(V_q,\ldots,\psi_q)$ to $(V_p,\ldots,\psi_p)$ in the germ. Define
${\rm d}\hat s_p$ near $s_q^{-1}(0)\subseteq V_q$ as in~\eq{ug2eq4}.
Locally on $V_q$, choose any orientation for the fibres of
$\phi_{pq}^*(TV_p)/({\rm d}\phi_{pq})(TV_q)$, and let
$\phi_{pq}^*(E_p)/\hat\phi_{pq}(E_q)$ have the orientation induced
from this by the isomorphism ${\rm d}\hat s_p$ in \eq{ug2eq4}. These
induce an orientation on $\frac{\phi_{pq}^*(TV_p)}{
({\rm d}\phi_{pq})(TV_q)}\oplus\frac{\phi_{pq}^*(E_p)}{
\hat\phi_{pq}(E_q)}$, which is independent of the choice for
$\phi_{pq}^*(TV_p)/({\rm d}\phi_{pq})(TV_q)$. Thus, these local choices
induce a natural orientation on the orbibundle
$\frac{\phi_{pq}^*(TV_p)}{({\rm d}\phi_{pq})(TV_q)}\oplus
\frac{\phi_{pq}^*(E_p)}{\hat\phi_{pq}(E_q)}$ near $s_q^{-1}(0)$. We
require that in oriented orbibundles over $V_q$ near $s_q^{-1}(0)$,
we have
\e
\begin{split}
\phi_{pq}^*\bigl[TV_p\oplus E_p\bigr]\cong (-1)^{\mathop{\rm dim}\nolimits V_q(\mathop{\rm dim}\nolimits
V_p-\mathop{\rm dim}\nolimits V_q)}\bigl[TV_q\oplus E_q\bigr]\oplus&\\
\bigl[\textstyle\frac{\phi_{pq}^*(TV_p)}{({\rm d}\phi_{pq})(TV_q)}\oplus
\frac{\phi_{pq}^*(E_p)}{\hat\phi_{pq}(E_q)}\bigr]&,
\end{split}
\label{ug2eq6}
\e
where $TV_p\oplus E_p$ and $TV_q\oplus E_q$ have the orientations assigned
by the orientation on $X$. An {\it oriented Kuranishi space\/} is a
Kuranishi space with an orientation.
\label{ug2def8}
\end{dfn}
Suppose $X,X'$ are oriented Kuranishi spaces, $Y$ is an oriented
orbifold, and $\boldsymbol f:X\rightarrow Y$, $\boldsymbol f':X'\rightarrow Y$ are strong
submersions. Then by \S\ref{ug24}--\S\ref{ug25} we have Kuranishi
spaces $\partial X$ and $X\times_YX'$. These can also be given orientations
in a natural way. We shall follow the orientation conventions of
Fukaya et al.~\cite[\S 45]{FOOO}.
\begin{conv} First, our conventions for manifolds:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(a)] Let $X$ be an oriented manifold with boundary $\partial
X$. Then we define the orientation on $\partial X$ such that
$TX\vert_{\partial X}={\mathbin{\mathbb R}}_{\rm out}\oplus T(\partial X)$ is an isomorphism of
oriented vector spaces, where ${\mathbin{\mathbb R}}_{\rm out}$ is oriented by an
outward-pointing normal vector to~$\partial X$.
\item[(b)] Let $X,X',Y$ be oriented manifolds, and $f:X\rightarrow
Y$, $f':X'\rightarrow Y$ be submersions. Then ${\rm d} f:TX\rightarrow f^*(TY)$ and ${\rm d}
f':TX'\rightarrow (f')^*(TY)$ are surjective maps of vector bundles over
$X,X'$. Choosing Riemannian metrics on $X,X'$ and identifying the
orthogonal complement of $\mathop{\rm Ker}{\rm d} f$ in $TX$ with the image $f^*(TY)$
of ${\rm d} f$, and similarly for $f'$, we have isomorphisms of vector
bundles over $X,X'$:
\e
TX\cong \mathop{\rm Ker}{\rm d} f\oplus f^*(TY) \quad\text{and}\quad TX'\cong
(f')^*(TY)\oplus\mathop{\rm Ker}{\rm d} f'.
\label{ug2eq7}
\e
Define orientations on the fibres of $\mathop{\rm Ker}{\rm d} f$, $\mathop{\rm Ker}{\rm d} f'$ over
$X,X'$ such that \eq{ug2eq7} are isomorphisms of oriented vector
bundles, where $TX,TX'$ are oriented by the orientations on $X,X'$,
and $f^*(TY),(f')^*(TY)$ by the orientation on $Y$. Then we define
the orientation on $X\times_YX'$ so that
\begin{equation*}
T(X\times_YX')\cong \pi_X^*(\mathop{\rm Ker}{\rm d} f)\oplus
(f\circ\pi_X)^*(TY)\oplus\pi_{X'}^*(\mathop{\rm Ker}{\rm d} f')
\end{equation*}
is an isomorphism of oriented vector bundles. Here $\pi_X:X\times_YX'\rightarrow
X$ and $\pi_{X'}:X\times_YX'\rightarrow X'$ are the natural projections, and
$f\circ\pi_X\equiv f'\circ\pi_{X'}$.
\end{itemize}
These extend immediately to orbifolds. They also extend to the
Kuranishi space versions in Definitions \ref{ug2def6} and
\ref{ug2def7}; for Definition \ref{ug2def7} they are described in
\cite[Conv.~45.1(4)]{FOOO}. An algorithm to deduce Kuranishi space
orientation conventions from manifold ones is described in~\cite[\S
2.7]{Joyc1}.
\label{ug2conv}
\end{conv}
If $X$ is an oriented Kuranishi space, we often write $-X$ for the
same Kuranishi space with the opposite orientation. Here is
\cite[Prop.~2.31]{Joyc1}, largely taken from Fukaya et
al.~\cite[Lem.~45.3]{FOOO}.
\begin{prop} Let\/ $X_1,X_2,\ldots$ be oriented Kuranishi spaces,
$Y,Y_1,\ldots$ be oriented orbifolds, and\/ $\boldsymbol f_1:X_1\rightarrow
Y,\ldots$ be strongly smooth maps, with at least one strong
submersion in each fibre product below. Then the following hold, in
oriented Kuranishi spaces:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[{\rm(a)}] If\/ $\partial Y=\emptyset,$ for\/ $\boldsymbol f_1:X_1\rightarrow Y$ and\/
$\boldsymbol f_2:X_2\rightarrow Y$ we have
\begin{equation}
\begin{gathered}
\partial(X_1\times_YX_2)=(\partial X_1)\times_YX_2\amalg (-1)^{\mathop{\rm vdim}\nolimits X_1+\mathop{\rm dim}\nolimits
Y}X_1\times_Y(\partial X_2)\\ \text{and}\quad X_1\times_YX_2=(-1)^{(\mathop{\rm vdim}\nolimits
X_1-\mathop{\rm dim}\nolimits Y)(\mathop{\rm vdim}\nolimits X_2-\mathop{\rm dim}\nolimits Y)}X_2\times_YX_1.
\end{gathered}
\label{ug2eq8}
\end{equation}
\item[{\rm(b)}] For\/ $\boldsymbol f_1:X_1\rightarrow Y_1,$ $\boldsymbol f_2:X_2\rightarrow Y_1\times
Y_2$ and\/ $\boldsymbol f_3:X_3\rightarrow Y_2,$ we have
\begin{equation}
(X_1\times_{Y_1}X_2)\times_{Y_2}X_3=X_1\times_{Y_1}(X_2\times_{Y_2}X_3).
\label{ug2eq9}
\end{equation}
\item[{\rm(c)}] For\/ $\boldsymbol f_1:X_1\rightarrow Y_1\times Y_2,$ $\boldsymbol f_2:X_2\rightarrow
Y_1$ and\/ $\boldsymbol f_3:X_3\rightarrow Y_2,$ we have
\end{itemize}
\begin{equation}
X_1\times_{Y_1\times Y_2}(X_2\times X_3)=(-1)^{\mathop{\rm dim}\nolimits Y_2(\mathop{\rm dim}\nolimits Y_1+\mathop{\rm vdim}\nolimits X_2)}
(X_1\times_{Y_1}X_2)\times_{Y_2}X_3.
\label{ug2eq10}
\end{equation}
\label{ug2prop}
\end{prop}
\subsection{Coorientations}
\label{ug27}
To define Kuranishi cohomology in \S\ref{ug4} we we will also need a
notion of {\it relative orientation} for (strong) submersions. We
call this a {\it coorientation},~\cite[\S 2.8]{Joyc1}.
\begin{dfn} Let $X,Y$ be orbifolds, and $f:X\rightarrow Y$ a submersion. A
{\it coorientation\/} for $(X,f)$ is a choice of orientations on the
fibres of the vector bundle $\mathop{\rm Ker}{\rm d} f$ over $X$ which vary
continuously over $X$. Here ${\rm d} f:TX\rightarrow f^*(TY)$ is the derivative
of $f$, a morphism of vector bundles, which is surjective as $f$ is
a submersion. Thus $\mathop{\rm Ker}{\rm d} f$ is a vector bundle over $X$, of
rank~$\mathop{\rm dim}\nolimits X-\mathop{\rm dim}\nolimits Y$.
Now let $X$ be a Kuranishi space, $Y$ an orbifold, and $\boldsymbol f:X\rightarrow
Y$ a strong submersion. A {\it coorientation\/} for $(X,\boldsymbol f)$
assigns, for all $p\in X$ and all sufficiently small Kuranishi
neighbourhoods $(V_p,E_p,s_p,\psi_p)$ in the germ at $p$ with
submersion $f_p:V_p\rightarrow Y$ representing $\boldsymbol f$, orientations on the
fibres of the orbibundle $\mathop{\rm Ker}{\rm d} f_p\oplus E_p$ varying continuously
over $V_p$, where ${\rm d} f_p:TV_p\rightarrow f_p^*(TY)$ is the (surjective)
derivative of $f_p$.
These must be compatible with coordinate changes, in the following
sense. Let $q\in\mathop{\rm Im}\psi_p$, $(V_q,\ldots,\psi_q)$ be sufficiently
small in the germ at $q$, let $f_q:V_q\rightarrow Y$ represent $\boldsymbol f$, and
$(\phi_{pq},\hat\phi_{pq})$ be the coordinate change from
$(V_q,\ldots,\psi_q)$ to $(V_p,\ldots,\psi_p)$ in the germ. Then we
require that in oriented orbibundles over $V_q$ near $s_q^{-1}(0)$,
we have
\e
\begin{split}
\phi_{pq}^*\bigl[\mathop{\rm Ker}{\rm d} f_p\oplus E_p\bigr]\cong (-1)^{\mathop{\rm dim}\nolimits V_q(\mathop{\rm dim}\nolimits
V_p-\mathop{\rm dim}\nolimits V_q)}\bigl[\mathop{\rm Ker}{\rm d} f_q\oplus E_q\bigr]\oplus&\\
\bigl[\textstyle\frac{\phi_{pq}^*(TV_p)}{({\rm d}\phi_{pq})(TV_q)}\oplus
\frac{\phi_{pq}^*(E_p)}{\hat\phi_{pq}(E_q)}\bigr]&,
\end{split}
\label{ug2eq11}
\e
by analogy with \eq{ug2eq6}, where $\frac{\phi_{pq}^*(TV_p)}{
({\rm d}\phi_{pq})(TV_q)}\oplus\frac{\phi_{pq}^*(E_p)}{\hat\phi_{pq}(E_q)}$
is oriented as in Definition~\ref{ug2def8}.
\label{ug2def9}
\end{dfn}
Suppose now that $Y$ is oriented. Then an orientation on $X$ is
equivalent to a coorientation for $(X,\boldsymbol f)$, since for all $p,
(V_p,\ldots,\psi_p),f_p$ as above, the isomorphism $TV_p\cong
f_p^*(TY)\oplus \mathop{\rm Ker}{\rm d} f_p$ induces isomorphisms of orbibundles
over~$V_p$:
\e
\bigl(TV_p\oplus E_p\bigr)\cong f_p^*(TY)\oplus\bigl(\mathop{\rm Ker}{\rm d} f_p\oplus
E_p\bigr).
\label{ug2eq12}
\e
There is a 1-1 correspondence between orientations on $X$ and
coorientations for $(X,\boldsymbol f)$ such that \eq{ug2eq12} holds in
oriented orbibundles, where $TV_p\oplus E_p$ is oriented by the
orientation on $X$, and $\mathop{\rm Ker}{\rm d} f_p\oplus E_p$ by the coorientation for
$(X,\boldsymbol f)$, and $f_p^*(TY)$ by the orientation on $Y$. Taking the
direct sum of $f_p^*(TY)$ with \eq{ug2eq11} and using \eq{ug2eq12}
yields \eq{ug2eq6}, so this is compatible with coordinate changes.
In \cite[Conv.~2.33]{Joyc1} we give our conventions for
coorientations of boundaries and fibre products. These correspond to
Convention \ref{ug2conv} under the 1-1 correspondence between
orientations on $X$ and coorientations for $(X,\boldsymbol f)$ above when
$Y$ is oriented. So the analogue of Proposition \ref{ug2prop} holds
for coorientations. In particular, for strong submersions $\boldsymbol
f_a:X_a\rightarrow Y$ with $(X_a,\boldsymbol f_a)$ cooriented for $a=1,2,3$, taking
$\partial Y=\emptyset$ in \eq{ug2eq13}, we have
\begin{gather}
\begin{split}
\bigl(\partial(X_1\times_YX_2),\boldsymbol\pi_Y\bigr)&\cong\bigl((\partial
X_1)\times_YX_2,\boldsymbol\pi_Y\bigr)\,\amalg \\
&\qquad\quad (-1)^{\mathop{\rm vdim}\nolimits X_1+\mathop{\rm dim}\nolimits Y} \bigl(X_1\times_Y(\partial
X_2),\boldsymbol\pi_Y\bigr),
\end{split}
\label{ug2eq13}\\
\bigl(X_1\times_YX_2,\boldsymbol\pi_Y\bigr)\cong(-1)^{(\mathop{\rm vdim}\nolimits X_1-\mathop{\rm dim}\nolimits Y)(\mathop{\rm vdim}\nolimits
X_2-\mathop{\rm dim}\nolimits Y)}\bigl(X_2\times_YX_1,\boldsymbol\pi_Y\bigr),
\label{ug2eq14}\\
\bigl((X_1\times_YX_2)\times_YX_3,\boldsymbol\pi_Y\bigr)\cong
\bigl(X_1\times_Y(X_2\times_YX_3),\boldsymbol\pi_Y\bigr).
\label{ug2eq15}
\end{gather}
Similarly, if $X_1$ is oriented, $\boldsymbol f_1:X_1\rightarrow Y$ is strongly
smooth, $\boldsymbol f_2:X_2\rightarrow Y$ is a cooriented strong submersion, and
$\partial Y=\emptyset$ then
\e
\partial(X_1\times_YX_2)\cong \bigl((\partial X_1)\times_YX_2\bigr)\amalg (-1)^{\mathop{\rm vdim}\nolimits
X_1+\mathop{\rm dim}\nolimits Y}\bigl(X_1\times_Y(\partial X_2)\bigr)
\label{ug2eq16}
\e
in oriented Kuranishi spaces.
\section{Kuranishi homology}
\label{ug3}
{\it Kuranishi homology\/} \cite[\S 4]{Joyc1} is a homology theory
of orbifolds $Y$ in which the chains are isomorphism classes $[X,\boldsymbol
f,\boldsymbol G]$, where $X$ is a compact, oriented Kuranishi space, $\boldsymbol
f:X\rightarrow Y$ is strongly smooth, and $\boldsymbol G$ is some extra data called
{\it gauge-fixing data}. It is isomorphic to singular homology
$H_*^{\rm si}(Y;R)$.
\subsection{Gauge-fixing data}
\label{ug31}
Let $X$ be a compact Kuranishi space, $Y$ an orbifold, and $\boldsymbol
f:X\rightarrow Y$ a strongly smooth map. A key ingredient in the definition
of Kuranishi homology in \cite{Joyc1} is the idea of {\it gauge
fixing data\/} $\boldsymbol G$ for $(X,\boldsymbol f)$ studied in \cite[\S
6]{Joyc1}. Define $P=\coprod_{k=0}^\infty {\mathbin{\mathbb R}}^k/S_k$, where the
symmetric group $S_k$ acts on ${\mathbin{\mathbb R}}^k$ by permuting the coordinates
$x_1,\ldots,x_k$. For $n=0,1,2,\ldots$, define $P_n\subset P$ by
$P_n=\coprod_{k=0}^n{\mathbin{\mathbb R}}^k/S_k$. Gauge-fixing data $\boldsymbol G$ for $(X,\boldsymbol
f)$ consists of a cover of $X$ by Kuranishi neighbourhoods
$(V^i,E^i,s^i,\psi^i)$ for $i$ in a finite indexing set $I$,
together with smooth maps $f^i:V^i\rightarrow Y$ representing $\boldsymbol f$ and
maps $G^i:E^i\rightarrow P_n\subset P$ for some $n\gg 0$, and continuous
partitions of unity $\eta_i:X\rightarrow[0,1]$ and $\eta_i^j:V^j\rightarrow[0,1]$,
satisfying many conditions. One important condition, responsible for
Theorem \ref{ug3thm1}(b) below, is that each $G^i:E^i\rightarrow P$ should
be a {\it finite\/} map, that is, $(G^i)^{-1}(p)$ is finitely many
points for all~$p\in P$.
Users of Kuranishi homology do not need to know exactly what
gauge-fixing data is, so we will not define it. Here are the
important properties of gauge-fixing data, which are proved
in~\cite[\S 3]{Joyc1}.
\begin{thm} Consider pairs\/ $(X,\boldsymbol f)$, where\/ $X$ is a compact
Kuranishi space, $Y$ an orbifold, and\/ $\boldsymbol f:X\rightarrow Y$ a strongly
smooth map. In\/ {\rm\cite[\S 3.1]{Joyc1}} we define
\begin{bfseries}gauge-fixing data\end{bfseries}\/ $\boldsymbol G$ for such
pairs $(X,\boldsymbol f),$ with the following properties:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[{\rm(a)}] Every pair $(X,\boldsymbol f)$ admits a (nonunique) choice
of gauge-fixing data $\boldsymbol G$. If\/ $\Gamma\subseteq\mathop{\rm Aut}(X,\boldsymbol f)$ is a
finite subgroup then we can choose $\boldsymbol G$ to be $\Gamma$-invariant.
\item[{\rm(b)}] For all pairs\/ $(X,\boldsymbol f)$ and choices of
gauge-fixing data\/ $\boldsymbol G$ for\/ $(X,\boldsymbol f),$ the automorphism
group\/ $\mathop{\rm Aut}(X,\boldsymbol f,\boldsymbol G)$ of isomorphisms\/ $(\boldsymbol a,\boldsymbol
b):(X,\boldsymbol f,\boldsymbol G)\rightarrow(X,\boldsymbol f,\boldsymbol G)$ is finite.
\item[{\rm(c)}] Suppose $\boldsymbol G$ is gauge-fixing data for $(X,\boldsymbol
f)$ and\/ $\Gamma$ is a finite group acting on $(X,\boldsymbol f,\boldsymbol G)$ by
isomorphisms. Then we can form the quotient\/ $\tilde X=X/\Gamma,$ a
compact Kuranishi space, with projection $\boldsymbol\pi:X\rightarrow\tilde X,$ and\/
$\boldsymbol f$ pushes down to $\boldsymbol{\tilde f}:\tilde X\rightarrow Y$ with\/ $\boldsymbol f=\boldsymbol{\tilde
f}\circ\boldsymbol\pi$. As in\/ {\rm\cite[\S 3.4]{Joyc1},} we can define
gauge-fixing data\/ $\boldsymbol{\tilde G}$ for\/ $(\tilde X,\boldsymbol{\tilde f}),$ which
is the natural push down\/ $\boldsymbol\pi_*(\boldsymbol G)$ of\/ $\boldsymbol G$ to~$\tilde
X$.
\item[{\rm(d)}] If\/ $\boldsymbol G$ is gauge-fixing data for\/ $(X,\boldsymbol
f),$ it has a restriction\/ $\boldsymbol G\vert_{\partial X}$ defined in\/
{\rm\cite[\S 3.5]{Joyc1},} which is gauge-fixing data for\/~$(\partial
X,\boldsymbol f\vert_{\partial X})$.
\item[{\rm(e)}] Let\/ $X$ be a compact, oriented Kuranishi space
with corners (not g-corners), $\boldsymbol f:X\rightarrow Y$ be strongly smooth,
and\/ $\boldsymbol\sigma:\partial^2X\rightarrow\partial^2X$ be the natural involution described
in\/ {\rm\S\ref{ug24}}. Suppose $\boldsymbol H$ is gauge-fixing data for
$(\partial X,\boldsymbol f\vert_{\partial X})$. Then there exists gauge-fixing data\/
$\boldsymbol G$ for\/ $(X,\boldsymbol f)$ with\/ $\boldsymbol G\vert_{\partial X}=\boldsymbol H$ if and
only if\/ $\boldsymbol H\vert_{\partial^2X}$ is invariant under\/ $\boldsymbol\sigma$. If
also\/ $\Gamma$ is a finite subgroup of\/ $\mathop{\rm Aut}(X,\boldsymbol f),$ and\/ $\boldsymbol
H$ is invariant under\/ $\Gamma\vert_{\partial X},$ then we can choose\/
$\boldsymbol G$ to be\/ $\Gamma$-invariant.
\item[{\rm(f)}] Let\/ $Y,Z$ be orbifolds, and\/ $h:Y\rightarrow Z$ a smooth
map. Suppose $X$ is a compact Kuranishi space, $\boldsymbol f:X\rightarrow Y$ is
strongly smooth, and\/ $\boldsymbol G$ is gauge-fixing data for $(X,\boldsymbol f)$.
Then as in {\rm\cite[\S 3.7]{Joyc1},} we can define gauge-fixing
data\/ $h_*(\boldsymbol G)$ for $(X,h\circ\boldsymbol f)$. It satisfies $(g\circ
h)_*(\boldsymbol G)=g_*(h_*(\boldsymbol G))$.
\end{itemize}
\label{ug3thm1}
\end{thm}
In order to define a working homology theory, perhaps the most
important property is Theorem \ref{ug3thm1}(b). In \cite[\S
4.9]{Joyc1} we show that if we define {\it na\"\i ve Kuranishi
(co)homology} $KH_*^{\rm na},KH^*_{\rm na}(Y;R)$ as in \S\ref{ug32} and
\S\ref{ug42} but omitting all (co-)gauge-fixing data, then
$KH_*^{\rm na}(Y;R)=0=KH^*_{\rm na}(Y;R)$ for all orbifolds $Y$ and
${\mathbin{\mathbb Q}}$-algebras $R$. For compact $Y$, we do this by constructing an
explicit cochain whose boundary is the identity cocycle $[Y,\mathop{\rm id}\nolimits_Y]$
in $KC^*_{\rm na}(Y;R)$. This explicit cochain involves a cocycle $[X,\boldsymbol
f]$ whose automorphism group $\mathop{\rm Aut}(X,\boldsymbol f)$ is infinite. Including
(co-)gauge-fixing data prevents this from happening, as it ensures
that all automorphism groups $\mathop{\rm Aut}(X,\boldsymbol f,\boldsymbol G)$ are finite.
\subsection{Kuranishi homology}
\label{ug32}
We can now define the {\it Kuranishi homology\/} of an orbifold,
\cite[\S 4.2]{Joyc1}.
\begin{dfn} Let $Y$ be an orbifold. Consider triples $(X,\boldsymbol f,\boldsymbol
G)$, where $X$ is a compact, oriented Kuranishi space, $\boldsymbol f:X\rightarrow
Y$ is strongly smooth, and $\boldsymbol G$ is gauge-fixing data for $(X,\boldsymbol
f)$. Write $[X,\boldsymbol f,\boldsymbol G]$ for the isomorphism class of $(X,\boldsymbol
f,\boldsymbol G)$ under isomorphisms $(\boldsymbol a,\boldsymbol b):(X,\boldsymbol f,\boldsymbol G)\rightarrow(\tilde
X,\boldsymbol{\tilde f},\boldsymbol{\tilde G})$, where $\boldsymbol a$ must identify the
orientations of $X,\tilde X$, and $\boldsymbol b$ lifts $\boldsymbol a$ to the
Kuranishi neighbourhoods $(V^i,\ldots,\psi^i),(\tilde
V^i,\ldots,\tilde\psi^i)$ in~$\boldsymbol G,\boldsymbol{\tilde G}$.
Let $R$ be a ${\mathbin{\mathbb Q}}$-algebra, for instance ${\mathbin{\mathbb Q}},{\mathbin{\mathbb R}}$ or ${\mathbin{\mathbb C}}$. For each
$k\in{\mathbin{\mathbb Z}}$, define $KC_k(Y;R)$ to be the $R$-module of finite
$R$-linear combinations of isomorphism classes $[X,\boldsymbol f,\boldsymbol G]$ for
which $\mathop{\rm vdim}\nolimits X=k$, with the relations:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(i)] Let $[X,\boldsymbol f,\boldsymbol G]$ be an isomorphism class,
and write $-X$ for $X$ with the opposite orientation. Then in
$KC_k(Y;R)$ we have
\begin{equation*}
[X,\boldsymbol f,\boldsymbol G]+[-X,\boldsymbol f,\boldsymbol G]=0.
\end{equation*}
\item[(ii)] Let $[X,\boldsymbol f,\boldsymbol G]$ be an isomorphism class. Suppose
that $X$ may be written as a disjoint union $X=X_+\amalg X_-$ of
compact, oriented Kuranishi spaces, and that for each Kuranishi
neighbourhood $(V^i,\ldots,\psi^i)$ for $i\in I$ in $\boldsymbol G$ we may
write $V^i=V^i_+\amalg V^i_-$ for open and closed subsets $V^i_\pm$
of $V^i$, such that $\mathop{\rm Im}\psi^i\vert_{V^i_+}\subseteq X_+$ and
$\mathop{\rm Im}\psi^i\vert_{V^i_-}\subseteq X_-$. Then we may define
gauge-fixing data $\boldsymbol G\vert_{X_\pm}$ for $(X_\pm,\boldsymbol
f\vert_{X_\pm})$, with Kuranishi neighbourhoods $(V^i_\pm,
E^i\vert_{V^i_\pm},s^i\vert_{V^i_\pm},\psi^i\vert_{V^i_\pm})$ for
$i\in I$ with $V^i_\pm\ne\emptyset$. In $KC_k(Y;R)$ we have
\begin{equation*}
[X,\boldsymbol f,\boldsymbol G]=[X_+,\boldsymbol f\vert_{X_+},\boldsymbol G\vert_{X_+}]+[X_-,\boldsymbol
f\vert_{X_-},\boldsymbol G\vert_{X_-}].
\end{equation*}
\item[(iii)] Let $[X,\boldsymbol f,\boldsymbol G]$ be an isomorphism class, and
suppose $\Gamma$ is a finite group acting on $(X,\boldsymbol f,\boldsymbol G)$ by
orientation-preserving automorphisms. Then $\tilde X=X/\Gamma$ is a
compact, oriented Kuranishi space, with a projection $\boldsymbol\pi:X\rightarrow\tilde
X$. As in Theorem \ref{ug3thm1}(c), $\boldsymbol f,\boldsymbol G$ push down to a
strong submersion $\boldsymbol\pi_*(\boldsymbol f)=\boldsymbol{\tilde f}:\tilde X\rightarrow Y$ and
gauge-fixing data $\boldsymbol\pi_*(\boldsymbol G)=\boldsymbol{\tilde G}$ for $(\tilde X,\boldsymbol{\tilde
f})$. Then
\begin{equation*}
\bigl[X/\Gamma,\boldsymbol\pi_*(\boldsymbol f),\boldsymbol\pi_*(\boldsymbol G)\bigr]=\frac{1}{\md{\Gamma}}
\,\bigl[X,\boldsymbol f,\boldsymbol G\bigr]
\end{equation*}
\end{itemize}
in $KC_k(Y;R)$. Elements of $KC_k(Y;R)$ will be called {\it
Kuranishi chains}.
Define the {\it boundary operator\/} $\partial:KC_k(Y;R)\rightarrow
KC_{k-1}(Y;R)$ by
\begin{equation*}
\partial:\textstyle\sum_{a\in A}\rho_a[X_a,\boldsymbol f_a,\boldsymbol G_a]\longmapsto
\textstyle\sum_{a\in A}\rho_a[\partial X_a,\boldsymbol f_a\vert_{\partial X_a},\boldsymbol G_a
\vert_{\partial X_a}],
\end{equation*}
where $A$ is a finite indexing set and $\rho_a\in R$ for $a\in A$.
This is a morphism of $R$-modules. Clearly, $\partial$ takes each
relation (i)--(iii) in $KC_k(Y;R)$ to the corresponding relation in
$KC_{k-1}(Y;R)$, and so $\partial$ is well-defined.
Recall from \S\ref{ug21} and \S\ref{ug24} that if $X$ is an oriented
Kuranishi space then there is a natural orientation-reversing strong
diffeomorphism $\boldsymbol\sigma:\partial^2X\rightarrow\partial^2X$, with $\boldsymbol\sigma^2=
\boldsymbol\mathop{\rm id}\nolimits_{\partial^2X}$. If $[X,\boldsymbol f,\boldsymbol G]$ is an isomorphism class then
this $\boldsymbol\sigma$ extends to an isomorphism $(\boldsymbol\sigma,\boldsymbol\tau)$ of
$(\partial^2X,\boldsymbol f\vert_{\partial^2X},\boldsymbol G\vert_{\partial^2X})$. So part (i) in
$KC_{k-2}(Y;R)$ yields
\e
[\partial^2X,\boldsymbol f\vert_{\partial^2X},\boldsymbol G\vert_{\partial^2X}]+[\partial^2X,\boldsymbol
f\vert_{\partial^2X},\boldsymbol G\vert_{\partial^2X}]=0 \quad\text{in
$KC_{k-2}(Y;R)$.}
\label{ug3eq1}
\e
As $R$ is a ${\mathbin{\mathbb Q}}$-algebra we may multiply \eq{ug3eq1} by ${\ts\frac{1}{2}}$ to get
$[\partial^2X,\boldsymbol f\vert_{\partial^2X},\boldsymbol G\vert_{\partial^2X}]=0$. Therefore
$\partial\circ\partial=0$ as a map~$KC_k(Y;R)\rightarrow KC_{k-2}(Y;R)$.
Define the {\it Kuranishi homology group\/} $KH_k(Y;R)$ of $Y$ for
$k\in{\mathbin{\mathbb Z}}$ to be
\begin{equation*}
KH_k(Y;R)=\frac{\mathop{\rm Ker}\bigl(\partial:KC_k(Y;R)\rightarrow KC_{k-1}(Y;R)\bigr)}{
\mathop{\rm Im}\bigl(\partial:KC_{k+1}(Y;R)\rightarrow KC_k(Y;R)\bigr)}\,.
\end{equation*}
Let $Y,Z$ be orbifolds, and $h:Y\rightarrow Z$ a smooth map. Define the {\it
pushforward} $h_*:KC_k(Y;R)\rightarrow KC_k(Z;R)$ for $k\in{\mathbin{\mathbb Z}}$ by
\begin{equation*}
h_*:\textstyle\sum_{a\in A}\rho_a\bigl[X_a,\boldsymbol f_a,\boldsymbol G_a\bigr]\longmapsto
\textstyle\sum_{a\in A}\rho_a\bigl[X_a,h\circ\boldsymbol f_a,h_*(\boldsymbol G_a)\bigr],
\end{equation*}
with $h_*(\boldsymbol G_a)$ as in Theorem \ref{ug3thm1}(f). These take
relations (i)--(iii) in $KC_k(Y;R)$ to (i)--(iii) in $KC_k(Z;R)$,
and so are well-defined. They satisfy $h_*\circ\partial=\partial\circ h_*$, so
they induce morphisms of homology groups $h_*:KH_k(Y;R)\rightarrow
KH_k(Z;R)$. Pushforward is functorial, that is, $(g\circ h)_*=g_*\circ
h_*$, on chains and homology.
\label{ug3def}
\end{dfn}
\subsection{Singular homology and Kuranishi homology}
\label{ug33}
Let $Y$ be an orbifold, and $R$ a ${\mathbin{\mathbb Q}}$-algebra. Then we can define
the {\it singular homology groups\/} $H_k^{\rm si}(Y;R)$, as in Bredon
\cite[\S IV]{Bred}. Write $C_k^{\rm si}(Y;R)$ for the $R$-module spanned
by {\it smooth singular $k$-simplices\/} in $Y$, which are smooth
maps $\sigma:\Delta_k\rightarrow Y$, where $\Delta_k$ is the $k$-simplex
\begin{equation*}
\Delta_k=\bigl\{(x_0,\ldots,x_k)\in{\mathbin{\mathbb R}}^{k+1}:x_i\geqslant 0,\;\>
x_0+\ldots+x_k=1\bigr\}.
\end{equation*}
As in \cite[\S IV.1]{Bred}, the boundary operator
$\partial:C_k^{\rm si}(Y;R)\rightarrow C_{k-1}^{\rm si}(Y;R)$ is given by
\begin{equation*}
\partial:\textstyle\sum_{a\in A}\rho_a\,\sigma_a\longmapsto \textstyle\sum_{a\in
A}\sum_{j=0}^k(-1)^j\rho_a(\sigma_a\circ F_j^k),
\end{equation*}
where $F_j^k:\Delta_{k-1}\rightarrow\Delta_k$, $F_j^k:(x_0,\ldots,x_{k-1})
\mapsto(x_0,\ldots,x_{j-1},0,x_j,\ldots,x_{k-1})$ for
$j=0,\ldots,k$. Then $\partial^2=0$, and $H_*^{\rm si}(Y;R)$ is the homology
of~$\bigl(C_k^{\rm si}(Y;R),\partial\bigr)$.
Define $R$-module morphisms $C_k^{\rm si}(Y;R)\rightarrow KC_k(Y;R)$ for $k\geqslant
0$ by
\begin{equation*}
\textstyle\Pi_{\rm si}^{\rm Kh}:\sum_{a\in A}\rho_a\sigma_a\longmapsto \textstyle\sum_{a\in
A}\rho_a\bigl[\Delta_k,\sigma_a,\boldsymbol G_{\Delta_k}\bigr],
\end{equation*}
where $\boldsymbol G_{\Delta_k}$ is an explicit choice of gauge-fixing data for
$(\Delta_k,\sigma_a)$ given in \cite[\S 4.3]{Joyc1}. These satisfy
$\partial\circ\Pi_{\rm si}^{\rm Kh}=\Pi_{\rm si}^{\rm Kh}\circ\partial$, and so induce $R$-module
morphisms
\e
\Pi_{\rm si}^{\rm Kh}:H_k^{\rm si}(Y;R)\longrightarrow KH_k(Y;R).
\label{ug3eq2}
\e
Here is \cite[Cor.~4.10]{Joyc1}, one of the main results
of~\cite{Joyc1}.
\begin{thm} Let\/ $Y$ be an orbifold and\/ $R$ a ${\mathbin{\mathbb Q}}$-algebra. Then
$\Pi_{\rm si}^{\rm Kh}$ in\/ \eq{ug3eq2} is an isomorphism, so that\/
$KH_k(Y;R)\cong H_k^{\rm si}(Y;R),$ with\/ $KH_k(Y;R)=\{0\}$
when\/~$k<0$.
\label{ug3thm2}
\end{thm}
The proof of Theorem \ref{ug3thm2} in \cite[App.~A--C]{Joyc1} is
very long and complex, taking up a third of \cite{Joyc1}. The
problem is to construct an inverse for $\Pi_{\rm si}^{\rm Kh}$ in
\eq{ug3eq2}. This is related to Fukaya and Ono's construction of
{\it virtual cycles\/} for compact, oriented Kuranishi spaces
without boundary in \cite[\S 6]{FuOn}, and uses some of the same
ideas. But dealing with boundaries and corners of the Kuranishi
spaces in Kuranishi chains, and the relations in the Kuranishi chain
groups $KC_*(Y;R)$, increases the complexity by an order of
magnitude.
The basic idea of the proof is to take a class $\alpha\in KH_k(Y;R)$
and represent it by cycles $\sum_{a\in A}\rho_a[X_a,\boldsymbol f_a,\boldsymbol
G_a]$ with better and better properties, until eventually we
represent it by a cycle in the image of $\Pi_{\rm si}^{\rm Kh}:
C_k^{\rm si}(Y;R)\rightarrow KC_k(Y;R)$, so showing that \eq{ug3eq2} is
surjective. Here (somewhat oversimplified) are the main steps:
firstly, by `cutting' the $X_a$ into small pieces $X_{ac}$ for $c\in
C_a$, we show we can represent $\alpha$ by a cycle $\sum_{a\in
A}\sum_{c\in C_a}\rho_a[X_{ac},\boldsymbol f_{ac},\boldsymbol G_{ac}]$ such that
$(X_{ac},\boldsymbol f_{ac},\boldsymbol G_{ac})$ is the quotient of a triple
$(\acute X_{ac},\boldsymbol{\acute f}_{ac},\boldsymbol{ \acute G}_{ac})$ by a finite
group $\Gamma_{ac}$, where $\acute X_{ac}$ has {\it trivial
stabilizers}.
Thus by Definition \ref{ug3def}(iii) we can represent $\alpha$ by a
cycle $\sum_{a,c}\rho_a\md{\Gamma_{ac}}^{-1}[\acute X_{ac},\allowbreak
\smash{\boldsymbol{\acute f}_{ac},\boldsymbol{\acute G}_{ac}}]$ involving only
Kuranishi spaces $\smash{\acute X_{ac}}$ with trivial stabilizers.
Such spaces can be deformed to manifolds with g-corners (by
single-valued perturbations, not multisections). So secondly, we
show we can represent $\alpha$ by a cycle $\sum_{a,c}\rho_a
\md{\Gamma_{ac}}^{-1}[\tilde X_{ac},\tilde f_{ac},\boldsymbol{\tilde G}_{ac}]$ in which
the $\tilde X_{ac}$ are manifolds, and $\smash{\tilde f_{ac}:\tilde X_{ac}\rightarrow
Y}$ are smooth maps. Then thirdly, we triangulate the $\smash{\tilde
X_{ac}}$ by simplices $\Delta_k$, and so prove that we can represent
$\alpha$ by a cycle in the image of $\Pi_{\rm si}^{\rm Kh}$. In this third step
it is vital to work with manifolds {\it with g-corners}, as in
\S\ref{ug21}, not just manifolds with corners, since otherwise we
would not be able to construct the homology between
$\sum_{a,c}\rho_a\md{\Gamma_{ac}}^{-1} [\tilde X_{ac},\tilde f_{ac},\boldsymbol{\tilde
G}_{ac}]$ and the singular cycle.
In the proof we use the fact that $R$ is a ${\mathbin{\mathbb Q}}$-{\it algebra\/} in
two different ways. When we replace $[X_{ac},\boldsymbol f_{ac},\boldsymbol G_{ac}]$
by $\md{\Gamma_{ac}}^{-1}[\acute X_{ac},\boldsymbol{\acute f}_{ac},\boldsymbol{\acute
G}_{ac}]$ we must have $\md{\Gamma_{ac}}^{-1}\in R$, so we need
${\mathbin{\mathbb Q}}\subseteq R$. And when we deform $\acute X_{ac}$ to manifolds
$\tilde X_{ac}$, to make $\sum_{a,c}\rho_a\md{\Gamma_{ac}}^{-1}[\tilde
X_{ac},\tilde f_{ac},\boldsymbol{\tilde G}_{ac}]$ a cycle, we should ensure our
perturbations are preserved by the automorphism groups
$\smash{\mathop{\rm Aut}(\acute X_{ac},\boldsymbol{\acute f}_{ac},\boldsymbol{\acute G}_{ac})}$.
In fact this may not be possible, if $\mathop{\rm Aut}(\acute X_{ac},\boldsymbol{\acute
f}_{ac},\boldsymbol{\acute G}_{ac})$ has fixed points. So instead, we choose
one perturbation $\tilde X_{ac}$, and then take the average of the
images of this perturbation under $\mathop{\rm Aut}(\acute X_{ac},\boldsymbol{\acute
f}_{ac},\boldsymbol{\acute G}_{ac})$. This requires us to divide by
$\bmd{\mathop{\rm Aut}(\acute X_{ac},\boldsymbol{\acute f}_{ac},\boldsymbol{\acute G}_{ac})}$,
so again we need ${\mathbin{\mathbb Q}}\subseteq R$. Also, for this step it is
necessary that automorphism groups $\mathop{\rm Aut}(X,\boldsymbol f,\boldsymbol G)$ should be
{\it finite}, as in Theorem \ref{ug3thm1}(b), and this was the
reason for introducing gauge-fixing data.
The theorem means that in many problems, particularly areas of in
Symplectic Geometry involving moduli spaces of $J$-holomorphic
curves, we can use Kuranishi chains and homology instead of singular
chains and homology, which can simplify proofs considerably, and
also improve results.
\section{Kuranishi cohomology}
\label{ug4}
We now discuss the Poincar\'e dual theory of {\it Kuranishi
cohomology\/} $KH^*(Y;R)$, which is isomorphic to
compactly-supported cohomology $H^*_{\rm cs}(Y;R)$. It is defined using a
complex of Kuranishi cochains $KC^*(Y;R)$ spanned by isomorphism
classes $[X,\boldsymbol f,\boldsymbol C]$ of triples $(X,\boldsymbol f,\boldsymbol C)$, where $X$ is
a compact Kuranishi space, $\boldsymbol f:X\rightarrow Y$ is a {\it cooriented
strong submersion}, and $\boldsymbol C$ is {\it co-gauge-fixing data}.
As is usual for cohomology, Kuranishi cohomology has an associative,
supercommutative {\it cup product\/} $\cup:KH^k(Y;R)\times KH^l(Y;R)\rightarrow
KH^{k+l}(Y;R)$, and there is also a {\it cap product\/}
$\cap:KH_k(Y;R)\times KH^l(Y;R)\rightarrow KH_{k-l}(Y;R)$ relating Kuranishi
homology and Kuranishi (co)homology, which makes $KH_*(Y;R)$ into a
module over $KH^*(Y;R)$. More unusually, we can define $\cup,\cap$
naturally on Kuranishi (co)chains, and $\cup$ is associative and
supercommutative on $KC^*(Y;R)$, and $\cap$ makes $KC_*(Y;R)$ into a
module over~$KC^*(Y;R)$.
\subsection{Co-gauge-fixing data}
\label{ug41}
Let $X$ be a compact Kuranishi space, $Y$ an orbifold, and $\boldsymbol
f:X\rightarrow Y$ a strong submersion. Kuranishi cohomology is based on the
idea of {\it co-gauge-fixing data} $\boldsymbol C$ for $(X,\boldsymbol f)$. This is
very similar to gauge-fixing data $\boldsymbol G$ in \S\ref{ug31}, and
consists of a finite cover of $X$ by Kuranishi neighbourhoods
$(V^i,E^i,s^i,\psi^i)$ for $i\in I$, submersions $f^i:V^i\rightarrow Y$
representing $\boldsymbol f$ and maps $C^i:E^i\rightarrow P_n\subset P$ for some
$n\gg 0$, and partitions of unity $\eta_i:X\rightarrow[0,1]$
and~$\eta_i^j:V^j\rightarrow[0,1]$.
Here are the important properties of co-gauge-fixing data, which are
proved in \cite[\S 3]{Joyc1}. Part (g) makes cup products work on
Kuranishi cochains, and part (h) makes cap products work. It was
difficult to find a definition of (co-)gauge-fixing data for which
properties (a)--(h) all hold at once; a large part of the complexity
of \cite[\S 3]{Joyc1} is due to the author's determination to ensure
that cup products should be associative and supercommutative {\it at
the cochain level}. This is not essential for a well-behaved
(co)homology theory, but is extremely useful in the
applications~\cite{AkJo,Joyc2,Joyc3}.
\begin{thm} Consider pairs\/ $(X,\boldsymbol f),$ where\/ $X$ is a compact
Kuranishi space, $Y$ an orbifold, and\/ $\boldsymbol f:X\rightarrow Y$ a strong
submersion. In\/ {\rm\cite[\S 3.1]{Joyc1}} we define
\begin{bfseries}co-gauge-fixing data\end{bfseries}\/ $\boldsymbol C$ for such
pairs $(X,\boldsymbol f)$. It satisfies the analogues of Theorem
{\rm\ref{ug3thm1}(a)--(e),} and also:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[{\rm(f)}] Let\/ $Y,Z$ be orbifolds, and\/ $h:Y\rightarrow Z$ a smooth,
proper map. Suppose $X$ is a compact Kuranishi space, $\boldsymbol f:X\rightarrow Z$
is a strong submersion, and\/ $\boldsymbol C$ is co-gauge-fixing data for
$(X,\boldsymbol f)$. Then the fibre product\/ $Y\times_{h,Z,\boldsymbol f}X$ is a
compact Kuranishi space, and\/ $\boldsymbol\pi_Y:Y\times_ZX\rightarrow Y$ is a strong
submersion. As in {\rm\cite[\S 3.7]{Joyc1},} we can define
co-gauge-fixing data\/ $h^*(\boldsymbol C)$ for $(Y\times_ZX,\boldsymbol\pi_Y)$. It
satisfies $(g\circ h)^*(\boldsymbol C)=h^*(g^*(\boldsymbol C))$.
\item[{\rm(g)}] Let\/ $X_1,X_2,X_3$ be compact Kuranishi spaces, $Y$
an orbifold, $\boldsymbol f_i:X_i\rightarrow Y$ be strong submersions for\/
$i=1,2,3,$ and\/ $\boldsymbol C_i$ be co-gauge-fixing data for\/ $(X_i,\boldsymbol
f_i)$ for\/ $i=1,2,3$. Then\/ {\rm\cite[\S 3.8]{Joyc1}} defines
co-gauge-fixing data $\boldsymbol C_1\times_Y\boldsymbol C_2$ for\/ $(X_1\times_{\boldsymbol
f_1,Y,\boldsymbol f_2}X_2,\boldsymbol\pi_Y)$ from\/~$\boldsymbol C_1,\boldsymbol C_2$.
This construction is \begin{bfseries}symmetric\end{bfseries}, in
that it yields isomorphic co-gauge-fixing data for\/
$(X_1\times_YX_2,\boldsymbol\pi_Y)$ and\/ $(X_2\times_YX_1,\boldsymbol\pi_Y)$ under the
natural isomorphism\/ $X_1\times_YX_2\cong X_2\times_YX_1$. It is also
\begin{bfseries}associative\end{bfseries}, in that it yields
isomorphic co-gauge-fixing data for\/
$\bigl((X_1\!\times_Y\!X_2)\!\times_Y\!X_3,\boldsymbol\pi_Y\bigr)$ and\/
$\bigl(X_1\!\times_Y\!(X_2\!\times_Y\!X_3),\boldsymbol\pi_Y \bigr)$
under\/~$(X_1\!\times_Y\!X_2)\!\times_Y\!X_3\!\cong\!X_1\!\times_Y\!
(X_2\!\times_Y\!X_3)$.
These properties also have straightforward generalizations to
multiple fibre products involving more than one orbifold\/ $Y,$ such
as\/ \eq{ug2eq9} and\/~\eq{ug2eq10}.
\item[{\rm(h)}] Let\/ $X_1,X_2$ be compact Kuranishi spaces, $Y$
an orbifold, $\boldsymbol f_1:X_1\rightarrow Y$ be strongly smooth, $\boldsymbol f_2:X_2\rightarrow
Y$ be a strong submersion, $\boldsymbol G_1$ be gauge-fixing data for
$(X_1,\boldsymbol f_1),$ and\/ $\boldsymbol C_2$ be co-gauge-fixing data for
$(X_2,\boldsymbol f_2)$. Then\/ {\rm\cite[\S 3.8]{Joyc1}} defines
gauge-fixing data $\boldsymbol G_1\times_Y\boldsymbol C_2$ for\/ $(X_1\times_{\boldsymbol f_1,Y,\boldsymbol
f_2}X_2,\boldsymbol\pi_Y)$ from\/ $\boldsymbol G_1,\boldsymbol C_2$. If also $\boldsymbol f_3:X_3\rightarrow
Y$ is a strong submersion and\/ $\boldsymbol C_3$ is co-gauge-fixing data
for $(X_3,\boldsymbol f_3)$ then the natural isomorphism
$(X_1\!\times_Y\!X_2)\!\times_Y\!X_3\cong X_1\!\times_Y\!(X_2\!\times_Y\!X_3)$
identifies $(\boldsymbol G_1\!\times_Y\!\boldsymbol C_2)\!\times_Y\!\boldsymbol C_3$ and\/~$\boldsymbol
G_1\!\times_Y\!(\boldsymbol C_2\!\times_Y\!\boldsymbol C_3)$.
\end{itemize}
\label{ug4thm1}
\end{thm}
\subsection{Kuranishi cohomology}
\label{ug42}
Here is our definition of Kuranishi cohomology~\cite[\S 4.4]{Joyc1}.
\begin{dfn} Let $Y$ be an orbifold without boundary. Consider
triples $(X,\boldsymbol f,\boldsymbol C)$, where $X$ is a compact Kuranishi space,
$\boldsymbol f:X\rightarrow Y$ is a strong submersion with $(X,\boldsymbol f)$ {\it
cooriented}, as in \S\ref{ug27}, and $\boldsymbol C$ is {\it co-gauge-fixing
data\/} for $(X,\boldsymbol f)$, as in \S\ref{ug41}. Write $[X,\boldsymbol f,\boldsymbol C]$
for the isomorphism class of $(X,\boldsymbol f,\boldsymbol C)$ under isomorphisms
$(\boldsymbol a,\boldsymbol b):(X,\boldsymbol f,\boldsymbol C)\rightarrow(\tilde X,\boldsymbol{\tilde f},\boldsymbol{\tilde C})$,
where $\boldsymbol a$ must identify the coorientations of~$(X,\boldsymbol f),(\tilde
X,\boldsymbol{\tilde f})$, and $\boldsymbol b$ lifts $\boldsymbol a$ to the Kuranishi
neighbourhoods $(V^i,\ldots,\psi^i),(\tilde V^i,\ldots,\tilde\psi^i)$
in~$\boldsymbol C,\boldsymbol{\tilde C}$.
Let $R$ be a ${\mathbin{\mathbb Q}}$-algebra. For $k\in{\mathbin{\mathbb Z}}$, define $KC^k(Y;R)$ to be
the $R$-module of finite $R$-linear combinations of isomorphism
classes $[X,\boldsymbol f,\boldsymbol C]$ for which $\mathop{\rm vdim}\nolimits X=\mathop{\rm dim}\nolimits Y-k$, with the
analogues of relations Definition \ref{ug3def}(i)--(iii), replacing
gauge-fixing data $\boldsymbol G$ by co-gauge-fixing data $\boldsymbol C$. Elements
of $KC^k(Y;R)$ are called {\it Kuranishi cochains}. Define
${\rm d}:KC^k(Y;R)\rightarrow KC^{k+1}(Y;R)$ by
\e
{\rm d}:\textstyle\sum_{a\in A}\rho_a[X_a,\boldsymbol f_a,\boldsymbol C_a]\longmapsto
\textstyle\sum_{a\in A}\rho_a[\partial X_a,\boldsymbol f_a\vert_{\partial X_a},\boldsymbol C_a
\vert_{\partial X_a}].
\label{ug4eq1}
\e
As in Definition \ref{ug3def} we have ${\rm d}\circ{\rm d}=0$. Define the {\it
Kuranishi cohomology groups} $KH^k(Y;R)$ of $Y$ for $k\in{\mathbin{\mathbb Z}}$ to be
\begin{equation*}
KH^k(Y;R)=\frac{\mathop{\rm Ker}\bigl({\rm d}:KC^k(Y;R)\rightarrow KC^{k+1}(Y;R)\bigr)}{
\mathop{\rm Im}\bigl({\rm d}:KC^{k-1}(Y;R)\rightarrow KC^k(Y;R)\bigr)}\,.
\end{equation*}
Let $Y,Z$ be orbifolds without boundary, and $h:Y\rightarrow Z$ be a smooth,
proper map. Define the {\it pullback\/} $h^*:KC^k(Z;R)\rightarrow KC^k(Y;R)$
by
\e
h^*:\textstyle\sum_{a\in A}\rho_a\bigl[X_a,\boldsymbol f_a,\boldsymbol C_a\bigr]\longmapsto
\textstyle\sum_{a\in A}\rho_a [Y\times_{h,Z,\boldsymbol f_a}X_a,\boldsymbol\pi_Y,h^*(\boldsymbol C_a)],
\label{ug4eq2}
\e
where $h^*(\boldsymbol C_a)$ is as in Theorem \ref{ug4thm1}(f), and the
coorientation for $(X_a,\boldsymbol f_a)$ pulls back to a natural
coorientation for $(Y\times_ZX_a,\boldsymbol\pi_Y)$. These $h^*:KC^k(Z;R)\rightarrow
KC^k(Y;R)$ satisfy $h^*\circ{\rm d}={\rm d}\circ h^*$, so they induce morphisms of
cohomology groups $h^*:KH^k(Z;R)\rightarrow KH^k(Y;R)$. Pullbacks are
functorial, that is, $(g\circ h)^*=h^*\circ g^*$, on both cochains and
cohomology.
\label{ug4def1}
\end{dfn}
Here for simplicity we restrict to orbifolds $Y$ {\it without
boundary}. When $\partial Y\ne\emptyset$, the definition of $KH^*(Y;R)$ is more
complicated \cite[\S 4.5]{Joyc1}: if $\boldsymbol f:X\rightarrow Y$ is a strong
submersion then we must split $\partial X=\partial_+^{\boldsymbol f}X\amalg\partial_-^{\boldsymbol
f}X$, where roughly speaking $\partial_+^{\boldsymbol f}X$ is the component of
$\partial X$ lying over $Y^\circ$, and $\partial_-^{\boldsymbol f}X$ the component of
$\partial X$ lying over $\partial Y$. Then $\boldsymbol f\vert_{\partial_+^{\boldsymbol f}X}$ is a
strong submersion $\boldsymbol f_+:\partial_+^{\boldsymbol f}X\rightarrow Y$, and $\boldsymbol
f\vert_{\partial_-^{\boldsymbol f}X}=\iota\circ\boldsymbol f_-$, where $\boldsymbol f_-:\partial_-^{\boldsymbol
f}X\rightarrow\partial Y$ is a strong submersion, and $\iota:\partial Y\rightarrow Y$ is the
natural immersion. In \eq{ug4eq1} we must replace $[\partial X_a,\boldsymbol
f_a\vert_{\partial X_a},\boldsymbol C_a \vert_{\partial X_a}]$ by~$[\partial_+^{\boldsymbol
f_a}X_a,\boldsymbol f_{a,+},\boldsymbol C_a \vert_{\partial_+^{\boldsymbol f_a}X_a}]$.
In \cite[\S 4.7]{Joyc1} we define {\it cup\/} and {\it cap
products}.
\begin{dfn} Let $Y$ be an orbifold without boundary, and $R$ a
${\mathbin{\mathbb Q}}$-algebra. Define the {\it cup product\/} $\cup:KC^k(Y;R)\times
KC^l(Y;R)\rightarrow KC^{k+l}(Y;R)$ by
\begin{equation*}
[X,\boldsymbol f,\boldsymbol C]\cup[\tilde X,\boldsymbol{\tilde f},\boldsymbol{\tilde C}]=
\smash{\bigl[X\times_{\boldsymbol f,Y,\boldsymbol{\tilde f}}\tilde X,\boldsymbol\pi_Y,\boldsymbol C\times_Y
\boldsymbol{\tilde C}\bigr]},
\end{equation*}
extended $R$-bilinearly. Here $\boldsymbol\pi_Y:X\times_{\smash{\boldsymbol f,Y,\boldsymbol{\tilde
f}}}\tilde X\rightarrow Y$ is the projection from the fibre product, which is a
strong submersion as both $\boldsymbol f,\boldsymbol{\tilde f}$ are, and $\boldsymbol
C\times_Y\boldsymbol{\tilde C}$ is as in Theorem \ref{ug4thm1}(g). Then $\cup$
takes relations (i)--(iii) in both $KC^k(Y;R)$ and $KC^l(Y;R)$ to
the same relations in $KC^{k+l}(Y;R)$. Thus $\cup$ is well-defined.
Theorem \ref{ug4thm1}(g) and \eq{ug2eq13}--\eq{ug2eq15} imply that
for $\gamma\in KC^k(Y;R)$, $\delta\in KC^l(Y;R)$ and $\epsilon\in KC^m(Y;R)$ we
have
\begin{gather}
\gamma\cup\delta=(-1)^{kl}\delta\cup\gamma,
\label{ug4eq3}\\
{\rm d}(\gamma\cup\delta)=({\rm d}\gamma)\cup\delta+(-1)^k\gamma\cup({\rm d}\delta)\;\>\text{and}\;\>
(\gamma\cup\delta)\cup\epsilon=\gamma\cup(\delta\cup\epsilon).
\label{ug4eq4}
\end{gather}
Therefore in the usual way $\cup$ induces an associative,
supercommutative product $\cup:KH^k(Y;R)\times KH^l(Y;R)\rightarrow
KH^{k+l}(Y;R)$ given for $\gamma\in KC^k(Y;R)$ and $\delta\in KC^l(Y;R)$
with ${\rm d}\gamma={\rm d}\delta=0$ by
\begin{equation*}
(\gamma+\mathop{\rm Im}{\rm d}_{k-1})\cup(\delta+\mathop{\rm Im}{\rm d}_{l-1})=(\gamma\cup\delta)+\mathop{\rm Im}{\rm d}_{k+l-1}.
\end{equation*}
Now suppose $Y$ is compact. Then $\mathop{\rm id}\nolimits_Y:Y\rightarrow Y$ is a (strong)
submersion, with a trivial coorientation giving the positive sign to
the zero vector bundle over $Y$. One can define natural
co-gauge-fixing data $\boldsymbol C_Y$ for $(Y,\mathop{\rm id}\nolimits_Y)$ such that
$[Y,\mathop{\rm id}\nolimits_Y,\boldsymbol C_Y]\in KC^0(Y;R)$, with ${\rm d}[Y,\mathop{\rm id}\nolimits_Y,\boldsymbol C_Y]=0$, and
for all $[X,\boldsymbol f,\boldsymbol C]\in KC^k(Y;R)$ we have
\begin{equation*}
[Y,\mathop{\rm id}\nolimits_Y,\boldsymbol C_Y]\cup [X,\boldsymbol f,\boldsymbol C]=[X,\boldsymbol f,\boldsymbol
C]\cup[Y,\mathop{\rm id}\nolimits_Y,\boldsymbol C_Y]=[X,\boldsymbol f,\boldsymbol C].
\end{equation*}
Thus $[Y,\mathop{\rm id}\nolimits_Y,\boldsymbol C_Y]$ is the identity for $\cup$, at the cochain
level. Passing to cohomology, $\bigl[[Y,\mathop{\rm id}\nolimits_Y,\boldsymbol C_Y]\bigr]\in
KH^0(Y;R)$ is the identity for $\cup$ in $KH^*(Y;R)$. We call
$\bigl[[Y,\mathop{\rm id}\nolimits_Y,\boldsymbol C_Y]\bigr]$ the {\it fundamental class\/}
of~$Y$.
Define the {\it cap product\/} $\cap:KC_k(Y;R)\times KC^l(Y;R)\rightarrow
KC_{k-l}(Y;R)$ by
\begin{equation*}
\smash{[X,\boldsymbol f,\boldsymbol G]\cap[\tilde X,\boldsymbol{\tilde f},\boldsymbol{\tilde C}]=\bigl[X
\times_{\boldsymbol f,Y,\boldsymbol{\tilde f}}\tilde X,\boldsymbol\pi_Y,\boldsymbol G\times_Y\boldsymbol{\tilde C}\bigr]},
\end{equation*}
extended $R$-bilinearly, with $\boldsymbol G\times_Y\boldsymbol{\tilde C}$ as in Theorem
\ref{ug4thm1}(h). For $\gamma\in KC_k(Y;R)$ and $\delta,\epsilon\in KC^*(Y;R)$,
the analogue of \eq{ug4eq4}, using Theorem \ref{ug4thm1}(h) and
\eq{ug2eq16},~is
\begin{equation*}
\partial(\gamma\cap\delta)=(\partial\gamma)\cap\delta+(-1)^{\mathop{\rm dim}\nolimits Y-k}\gamma\cup({\rm d}\delta),
\quad(\gamma\cap\delta)\cap\epsilon=\gamma\cap(\delta\cup\epsilon).
\end{equation*}
If also $Y$ is compact then $\gamma\cap[Y,\mathop{\rm id}\nolimits_Y,\boldsymbol C_Y]=\gamma$. Thus
$\cap$ induces a {\it cap product\/} $\cap:KH_k(Y;R)\times KH^l(Y;R)\rightarrow
KH_{k-l}(Y;R)$. These products $\cap$ make Kuranishi chains and
homology into {\it modules} over Kuranishi cochains and cohomology.
Let $Y,Z$ be orbifolds without boundary, and $h:Y\rightarrow Z$ a smooth,
proper map. Then \cite[Prop.~3.33]{Joyc1} implies that pullbacks
$h^*$ and pushforwards $h_*$ are compatible with $\cup,\cap$ on
(co)chains, in the sense that if $\alpha\in KC_*(Y;R)$ and $\beta,\gamma\in
KC^*(Z;R)$ then
\e
h^*(\beta\cup\gamma)=h^*(\beta)\cup h^*(\gamma) \quad\text{and}\quad
h_*(\alpha\cap h^*(\beta))=h_*(\alpha)\cap\beta.
\label{ug4eq5}
\e
Since $\cup,\cap,h^*,h_*$ are compatible with ${\rm d},\partial$, passing to
(co)homology shows that \eq{ug4eq5} also holds for $\alpha\in
KH_*(Y;R)$ and $\beta,\gamma\in KH^*(Z;R)$. If $Z$ is compact then $Y$
is, with $h^*\bigl([Z,\mathop{\rm id}\nolimits_Z,\boldsymbol C_Z]\bigr)=[Y,\mathop{\rm id}\nolimits_Y,\boldsymbol C_Y]$ in
$KC^0(Y;R)$ and $h^*\bigl(\bigl[[Z,\mathop{\rm id}\nolimits_Z,\boldsymbol C_Z]\bigr]\bigr)\allowbreak=
\bigl[[Y,\mathop{\rm id}\nolimits_Y,\boldsymbol C_Y]\bigr]$ in~$KH^0(Y;R)$.
To summarize: Kuranishi cochains $KC^*(Y;R)$ form a {\it
supercommutative, associative, differential graded\/ $R$-algebra},
and Kuranishi cohomology $KH^*(Y;R)$ is a {\it supercommutative,
associative, graded\/ $R$-algebra}. These algebras are {\it with
identity\/} if $Y$ is compact without boundary, and {\it without
identity\/} otherwise. Pullbacks $h^*$ induce {\it algebra
morphisms\/} on both cochains and cohomology. Kuranishi chains
$KC_*(Y;R)$ are a {\it graded module\/} over $KC^*(Y;R)$, and
Kuranishi homology $KH_*(Y;R)$ is a {\it graded module\/}
over~$KH^*(Y;R)$.
\label{ug4def2}
\end{dfn}
\subsection{Poincar\'e duality, and isomorphism with $H^*_{\rm cs}(Y;R)$}
\label{ug43}
Suppose $Y$ is an oriented manifold, of dimension $n$, without
boundary, and not necessarily compact, and $R$ is a commutative
ring. Then as in Bredon \cite[\S VI.9]{Bred} there are {\it
Poincar\'e duality isomorphisms}
\e
\mathop{\rm Pd}\nolimits:H^k_{\rm cs}(Y;R)\longrightarrow H_{n-k}^{\rm si}(Y;R)
\label{ug4eq6}
\e
between compactly-supported cohomology, and singular homology. If
$Y$ is also {\it compact\/} then it has a {\it fundamental class}
$[Y]\in H_n(Y;R)$, and we can write the Poincar\'e duality map $\mathop{\rm Pd}\nolimits$
of \eq{ug4eq6} in terms of the cap product by $\mathop{\rm Pd}\nolimits(\alpha)=[Y]\cap\alpha$
for $\alpha\in H^k_{\rm cs}(Y;R)$. Satake \cite[Th.~3]{Sata} showed that
Poincar\'e duality isomorphisms \eq{ug4eq6} exist when $Y$ is an
oriented orbifold without boundary and $R$ is a ${\mathbin{\mathbb Q}}$-{\it algebra}.
Let $Y$ be an orbifold of dimension $n$ without boundary, and $R$ a
${\mathbin{\mathbb Q}}$-algebra. We wish to construct an isomorphism $\Pi_{\rm cs}^{\rm Kch}:
H^*_{\rm cs}(Y;R)\rightarrow KH^*(Y;R)$ from compactly-supported cohomology to
Kuranishi cohomology. In the case in which $Y$ is oriented we will
define $\Pi_{\rm cs}^{\rm Kch}$ to be the composition
\e
\smash{\xymatrix@C=25pt{ H^k_{\rm cs}(Y;R) \ar[r]^(0.45){\mathop{\rm Pd}\nolimits} &
H_{n-k}^{\rm si}(Y;R) \ar[r]^(0.45){\Pi_{\rm si}^{\rm Kh}} & KH_{n-k}(Y;R)
\ar[r]^{\Pi^{\rm Kch}_{\rm Kh}} & KH^k(Y;R), }}
\label{ug4eq7}
\e
where the isomorphism $\mathop{\rm Pd}\nolimits$ is as in \eq{ug4eq6}, and $\Pi_{\rm si}^{\rm Kh}$
is as in \eq{ug3eq2} and is an isomorphism by Theorem \ref{ug3thm2},
and $\Pi^{\rm Kch}_{\rm Kh}$ is an isomorphism between Kuranishi (co)homology,
with inverse~$\Pi_{\rm Kch}^{\rm Kh}:KH^k(Y;R)\rightarrow KH_{n-k}(Y;R)$.
These $\Pi_{\rm Kch}^{\rm Kh},\Pi^{\rm Kch}_{\rm Kh}$ are defined in
\cite[Def.~4.14]{Joyc1}. At the (co)chain level, we define
$\Pi_{\rm Kch}^{\rm Kh}:KC^k(Y;R)\rightarrow KC_{n-k}(Y;R)$ by $\Pi_{\rm Kch}^{\rm Kh}:[X,\boldsymbol
f,\boldsymbol C]\mapsto [X,\boldsymbol f,\boldsymbol G_{\boldsymbol C}]$, where $\boldsymbol G_{\boldsymbol C}$ is
gauge-fixing data for $(X,\boldsymbol f)$ constructed from the
co-gauge-fixing data $\boldsymbol C$ is a functorial way, and as $\boldsymbol f:X\rightarrow
Y$ is cooriented and $Y$ is oriented, we obtain an orientation for
$X$ as in \S\ref{ug27}. Then $\partial\circ
\Pi_{\rm Kch}^{\rm Kh}=\Pi_{\rm Kch}^{\rm Kh}\circ{\rm d}$, so they induce morphisms
$\Pi_{\rm Kch}^{\rm Kh}:KH^k(Y;R)\rightarrow KH_{n-k}(Y;R)$ in (co)homology.
For $\Pi^{\rm Kch}_{\rm Kh}$ the story is more complicated. To define
$\Pi^{\rm Kch}_{\rm Kh}:KC_{n-k}(Y;R)\rightarrow KC^k(Y;R)$ we cannot simply map
$[X,\boldsymbol f,\boldsymbol G]\mapsto[X,\boldsymbol f,\boldsymbol C_{\boldsymbol G}]$ for some
co-gauge-fixing data $\boldsymbol C_{\boldsymbol G}$ constructed from $\boldsymbol G$, since
$\boldsymbol f$ need only be strongly smooth for $[X,\boldsymbol f,\boldsymbol G]\in
KC_{n-k}(Y;R)$, but $\boldsymbol f$ must be a strong submersion for $[X,\boldsymbol
f,\boldsymbol C_{\boldsymbol G}]\in KC^k(Y;R)$. Instead, we define $\Pi^{\rm Kch}_{\rm Kh}:
KC_{n-k}(Y;R)\rightarrow KC^k(Y;R)$ by $\Pi^{\rm Kch}_{\rm Kh}:[X,\boldsymbol f,\boldsymbol G]\mapsto
[X^Y,\boldsymbol f{}^Y,\boldsymbol C_{\boldsymbol G}^Y]$. Here $X^Y$ is $X$ equipped with an
{\it alternative Kuranishi structure}, which roughly speaking adds
copies of $\boldsymbol f^*(TY)$ to both the tangent bundle and obstruction
bundle of $X$. Also $\boldsymbol f{}^Y:X^Y\rightarrow Y$ is a lift of $\boldsymbol f$ to
$X^Y$, which is a strong submersion, and $\boldsymbol C_{\boldsymbol G}^Y$ is
co-gauge-fixing data for $(X^Y,\boldsymbol f{}^Y)$ constructed in a
functorial way from $\boldsymbol G$. Then ${\rm d}\circ\Pi^{\rm Kch}_{\rm Kh}=
\Pi^{\rm Kch}_{\rm Kh}\circ\partial$, so they induce morphisms
$\Pi^{\rm Kch}_{\rm Kh}:KH_{n-k}(Y;R)\rightarrow KH^k(Y;R)$ on (co)homology.
In \cite[Th.~4.15]{Joyc1} we show that these morphisms
$\Pi_{\rm Kch}^{\rm Kh},\Pi^{\rm Kch}_{\rm Kh}$ on Kuranishi (co)homology are inverse.
Thus the third morphism $\Pi^{\rm Kch}_{\rm Kh}$ in \eq{ug4eq7} is an
isomorphism, so the composition $\Pi_{\rm cs}^{\rm Kch}:H^k_{\rm cs}(Y;R)\rightarrow
KH^k(Y;R)$ is an isomorphism. Changing the orientation of $Y$
changes the sign of $\mathop{\rm Pd}\nolimits,\Pi^{\rm Kch}_{\rm Kh}$, and so does not change
$\Pi_{\rm cs}^{\rm Kch}$. If $Y$ is not orientable we can make a similar
argument using homology groups $H_{n-k}^{\rm si}(Y;O\times_{\{\pm 1\}}R)$,
$KH_{n-k}(Y;O\times_{\{\pm 1\}}R)$ twisted by the principal
${\mathbin{\mathbb Z}}_2$-bundle $O$ of orientations on $Y$. Thus we
prove~\cite[Cor.~4.17]{Joyc1}:
\begin{thm} Let\/ $Y$ be an orbifold without boundary, and\/ $R$ a
${\mathbin{\mathbb Q}}$-algebra. Then there are natural isomorphisms\/ $\Pi_{\rm cs}^{\rm Kch}:
H^k_{\rm cs}(Y;R)\rightarrow KH^k(Y;R)$ for $k\geqslant 0,$ and\/ $KH^k(Y;R)=0$
when~$k<0$.
\label{ug4thm2}
\end{thm}
In \cite[\S 4.5]{Joyc1} the theorem is extended to $Y$ with
boundary, going via relative homology $H_*^{\rm si}(Y,\partial Y;R),
KH_*(Y,\partial Y;R)$. In \cite[Th.~4.34]{Joyc1} we show that the
isomorphisms $\Pi_{\rm cs}^{\rm Kch}:H^*_{\rm cs}(Y;R)\rightarrow KH^*(Y;R)$ and
$\Pi_{\rm si}^{\rm Kh}:H_*^{\rm si}(Y;R)\rightarrow KH_*(Y;R)$ in Theorems \ref{ug3thm2}
and \ref{ug4thm2} identify the cup and cap products $\cup,\cap$ on
$H^*_{\rm cs}(Y;R),H_*^{\rm si}(Y;R)$ with those on~$KH^*(Y;R),KH_*(Y;R)$.
\section{Kuranishi bordism and cobordism}
\label{ug5}
We now summarize parts of \cite[\S 5]{Joyc1} on Kuranishi
(co)bordism. They are based on the classical bordism theory
introduced by Atiyah \cite{Atiy}. In fact \cite[\S 5]{Joyc1} studies
five different kinds of Kuranishi (co)bordism, but we discuss only
one.
\subsection{Classical bordism and cobordism groups}
\label{ug51}
Bordism groups were introduced by Atiyah \cite{Atiy}, and Connor
\cite[\S I]{Conn} gives a good introduction. Our definition is not
standard, but fits in with~\S\ref{ug52}.
\begin{dfn} Let $Y$ be an orbifold without boundary. Consider pairs
$(X,f)$, where $X$ is a compact, oriented manifold without boundary
or corners, not necessarily connected, and $f:X\rightarrow Y$ is a smooth
map. An {\it isomorphism\/} between two such pairs $(X,f),(\tilde X,\tilde
f)$ is an orientation-preserving diffeomorphism $i:X\rightarrow\tilde X$ with
$f=\tilde f\circ i$. Write $[X,f]$ for the isomorphism class of~$(X,f)$.
Let $R$ be a commutative ring. For each $k\geqslant 0$, define the $k^{\it
th}$ {\it bordism group\/} $B_k(Y;R)$ of $Y$ with coefficients in
$R$ to be the $R$-module of finite $R$-linear combinations of
isomorphism classes $[X,f]$ for which $\mathop{\rm dim}\nolimits X=k$, with the
relations:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(i)] $[X,f]+[X',f']=[X\amalg X',f\amalg f']$ for all
classes $[X,f],[X',f']$; and
\item[(ii)] Suppose $Z$ is a compact, oriented $(k\!+\!1)$-manifold
with boundary but without (g-)corners, and $g:Z\rightarrow Y$ is smooth.
Then~$[\partial Z,g\vert_{\partial Z}]=0$.
\end{itemize}
\label{ug5def1}
\end{dfn}
Here is how this definition relates to those in \cite{Atiy,Conn}.
When $Y$ is a manifold and $R={\mathbin{\mathbb Z}}$, our $B_k(Y;{\mathbin{\mathbb Z}})$ is equivalent to
Connor's {\it differential bordism group\/} $D_k(Y)$, \cite[\S
I.9]{Conn}. Atiyah \cite[\S 2]{Atiy} and Connor \cite[\S I.4]{Conn}
also define {\it bordism groups\/} $MSO_k(Y)$ as for $B_k(Y;{\mathbin{\mathbb Z}})$
above, but only requiring $f:X\rightarrow Y$ to be continuous, not smooth.
Connor \cite[Th.~I.9.1]{Conn} shows that when $Y$ is a manifold, the
natural projection $D_k(Y)\rightarrow MSO_k(Y)$ is an isomorphism.
As in \cite[\S I.5]{Conn}, bordism is a {\it generalized homology
theory}, that is, it satisfies all the Eilenberg--Steenrod axioms
for a homology theory except the dimension axiom. The bordism groups
of a point $MSO_*({\rm pt})$ are known, \cite[\S I.2]{Conn}. This
gives some information on bordism groups of general spaces $Y$: for
any generalized homology theory $GH_*(Y)$, there is a spectral
sequence from the singular homology $H^{\rm si}_* \bigl(Y;GH_*({\rm
pt})\bigr)$ of $Y$ with coefficients in $GH_*({\rm pt})$ converging
to $GH_*(Y)$, so that $GH_*({\cal S}^n)\cong H^{\rm si}_*\bigl({\cal
S}^n;GH_*({\rm pt})\bigr)$, for instance.
Atiyah \cite{Atiy} and Connor \cite[\S 13]{Conn} also define {\it
cobordism groups} $MSO^k(Y)$ for $k\in{\mathbin{\mathbb Z}}$, which are a {\it
generalized cohomology theory\/} dual to bordism $MSO_k(Y)$. There
is a natural product $\cup$ on $MSO^*(Y)$, making it into a
supercommutative ring. If $Y$ is a compact, oriented $n$-manifold
without boundary then \cite[Th.~3.6]{Atiy}, \cite[Th.~13.4]{Conn}
there are canonical Poincar\'e duality isomorphisms
\e
MSO^k(Y)\cong MSO_{n-k}(Y)\quad\text{for $k\in{\mathbin{\mathbb Z}}$.}
\label{ug5eq1}
\e
The definition of $MSO^*(Y)$ uses homotopy theory, direct limits of
$k$-fold suspensions, and classifying spaces. There does not seem to
be a satisfactory differential-geometric definition of cobordism
groups parallel to Definition~\ref{ug5def1}.
\subsection{Kuranishi bordism and cobordism groups}
\label{ug52}
Motivated by \S\ref{ug51}, following \cite[\S 5.2]{Joyc1} we define:
\begin{dfn} Let $Y$ be an orbifold. Consider pairs $(X,\boldsymbol f)$,
where $X$ is a compact, oriented Kuranishi space without boundary,
and $\boldsymbol f:X\rightarrow Y$ is strongly smooth. An {\it isomorphism\/}
between two pairs $(X,\boldsymbol f),(\tilde X,\boldsymbol{\tilde f})$ is an
orientation-preserving strong diffeomorphism $\boldsymbol i:X\rightarrow\tilde X$ with
$\boldsymbol f=\boldsymbol{\tilde f}\circ\boldsymbol i$. Write $[X,\boldsymbol f]$ for the isomorphism
class of~$(X,\boldsymbol f)$.
Let $R$ be a commutative ring. For each $k\in{\mathbin{\mathbb Z}}$, define the $k^{\it
th}$ {\it Kuranishi bordism group\/} $KB_k(Y;R)$ of $Y$ with
coefficients in $R$ to be the $R$-module of finite $R$-linear
combinations of isomorphism classes $[X,\boldsymbol f]$ for which $\mathop{\rm vdim}\nolimits
X=k$, with the relations:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(i)] $[X,\boldsymbol f]+[X',\boldsymbol f']=[X\amalg X',\boldsymbol f\amalg
\boldsymbol f']$ for all classes $[X,\boldsymbol f],[X',\boldsymbol f']$; and
\item[(ii)] Suppose $W$ is a compact, oriented Kuranishi space with
boundary but without (g-)corners, with $\mathop{\rm vdim}\nolimits W=k+1$, and $\boldsymbol
e:W\rightarrow Y$ is strongly smooth. Then~$[\partial W,\boldsymbol e\vert_{\partial W}]=0$.
\end{itemize}
Elements of $KB_k(Y;R)$ will be called {\it Kuranishi bordism
classes}.
Let $h:Y\rightarrow Z$ be a smooth map of orbifolds. Define the {\it
pushforward\/} $h_*:KB_k(Y;R)\rightarrow KB_k(Z;R)$ by $h_*:\sum_{a\in
A}\rho_a[X_a,\boldsymbol f_a]\mapsto\sum_{a\in A}\rho_a[X_a,h\circ\boldsymbol f_a]$.
This takes relations (i),(ii) in $KB_k(Y;R)$ to (i),(ii) in
$KB_k(Z;R)$, and so is well-defined. Pushforward is functorial, that
is,~$(g\circ h)_*=g_*\circ h_*$.
\label{ug5def2}
\end{dfn}
Now Kuranishi bordism $KB_*(Y;R)$ is like Kuranishi homology
$KH_*(Y;R)$ in \S\ref{ug32}, but using Kuranishi spaces $X$ without
boundary, and omitting gauge-fixing data $\boldsymbol G$. Thus it seems
natural to define Kuranishi cobordism $KB^*(Y;R)$ by modifying the
definition of Kuranishi cohomology $KH^*(Y;R)$ in \S\ref{ug42} in
the same way, following~\cite[\S 5.4--\S 5.5]{Joyc1}.
\begin{dfn} Let $Y$ be an orbifold without boundary. Consider pairs
$(X,\boldsymbol f)$, where $X$ is a compact Kuranishi space without
boundary, and $\boldsymbol f:X\rightarrow Y$ is a cooriented strong submersion. An
{\it isomorphism\/} between two pairs $(X,\boldsymbol f),(\tilde X,\boldsymbol{\tilde f})$
is a coorientation-preserving strong diffeomorphism $\boldsymbol i:X\rightarrow\tilde
X$ with $\boldsymbol f=\boldsymbol{\tilde f}\circ\boldsymbol i$. Write $[X,\boldsymbol f]$ for the
isomorphism class of~$(X,\boldsymbol f)$.
Let $R$ be a commutative ring. For each $k\in{\mathbin{\mathbb Z}}$, define the $k^{\it
th}$ {\it Kuranishi cobordism group\/} $KB^k(Y;R)$ of $Y$ with
coefficients in $R$ to be the $R$-module of finite $R$-linear
combinations of isomorphism classes $[X,\boldsymbol f]$ for which $\mathop{\rm vdim}\nolimits
X=\mathop{\rm dim}\nolimits Y-k$, with the relations:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(i)] $[X,\boldsymbol f]+[X',\boldsymbol f']=[X\amalg X',\boldsymbol f\amalg
\boldsymbol f']$ for all classes $[X,\boldsymbol f],[X',\boldsymbol f']$; and
\item[(ii)] Suppose $W$ is a compact Kuranishi space with
boundary but without \hbox{(g-)}\allowbreak corners, with $\mathop{\rm vdim}\nolimits W=\mathop{\rm dim}\nolimits
Y-k+1$, and $\boldsymbol e:W\rightarrow Y$ is a cooriented strong submersion. Then
$\boldsymbol e\vert_{\partial W}:\partial W\rightarrow Y$ is a cooriented strong submersion,
and we impose the relation $[\partial W,\boldsymbol e\vert_{\partial W}]=0$
in~$KB^k(Y;R)$.
\end{itemize}
Elements of $KB^k(Y;R)$ will be called {\it Kuranishi cobordism
classes}.
Define the {\it cup product\/} $\cup:KB^k(Y;R)\times KB^l(Y;R)\rightarrow
KB^{k+l}(Y;R)$ by
\e
\raisebox{-4pt}{\begin{Large}$\displaystyle\Bigl[$\end{Large}}
\sum_{a\in A}\rho_a\bigl[X_a,\boldsymbol f_a\bigr]
\raisebox{-4pt}{\begin{Large}$\displaystyle\Bigr]$\end{Large}}\!\cup\!
\raisebox{-4pt}{\begin{Large}$\displaystyle\Bigl[$\end{Large}}
\sum_{b\in B}\sigma_b\bigl[\tilde X_b,\boldsymbol{\tilde f}_b\bigr]
\raisebox{-4pt}{\begin{Large}$\displaystyle\Bigr]$\end{Large}}\!=\!\!
\sum_{a\in A,\; b\in B\!\!\!\!\!\!}\!\!\rho_a\sigma_b
\bigl[X_a\times_{\boldsymbol f_a,Y,\boldsymbol{\tilde f}_b}\tilde X_b,\boldsymbol\pi_Y\bigr],
\label{ug5eq2}
\e
for $A,B$ finite and $\rho_a,\sigma_b\in R$. The coorientations on
$(X_a,\boldsymbol f_a)$ and $(\tilde X_b,\boldsymbol{\tilde f}_b)$ induce a coorientation
on $(X_a\times_Y\tilde X_b,\boldsymbol\pi_Y)$ as in \S\ref{ug27}. Similarly, define
the {\it cap product\/} $\cap:KB_k(Y;R)\times KB^l(Y;R)\rightarrow
KB_{k-l}(Y;R)$ by the same formula \eq{ug5eq2}, where now $[X_a,\boldsymbol
f_a]\in KB_k(Y;R)$ so that $\boldsymbol f_a$ is strongly smooth and $X_a$
oriented, and the orientation on $X_a$ and coorientation for
$\boldsymbol{\tilde f}_b$ combine to give an orientation for~$X_a\times_Y\tilde X_b$.
One can show that $\cup,\cap$ are well-defined, that $\cup$ is
associative and supercommutative, and that
$(\gamma\cap\delta)\cap\epsilon=\gamma\cap(\delta\cup\epsilon)$ for $\gamma\in KB_*(Y;R)$ and
$\delta,\epsilon\in KB^*(Y;R)$. If $Y$ is also {\it compact\/} then using
the trivial coorientation for $\mathop{\rm id}\nolimits_Y:Y\rightarrow Y$, we have $[Y,\mathop{\rm id}\nolimits_Y]\in
KB^0(Y;R)$, which is the {\it identity\/} for $\cup$ and $\cap$.
Thus, $KB^*(Y;R)$ is a {\it graded, supercommutative, associative
$R$-algebra, with identity\/} if $Y$ is compact, and {\it without
identity\/} otherwise, and $\cap$ makes $KB_*(Y;R)$ into a module
over~$KB^*(Y;R)$.
Let $Y,Z$ be orbifolds without boundary, and $h:Y\rightarrow Z$ a smooth,
proper map. Motivated by \eq{ug4eq2}, define the {\it pullback\/}
$h^*:KB^k(Z;R)\rightarrow KB^k(Y;R)$ by $h^*:\sum_{a\in A}\rho_a[X_a,\boldsymbol
f_a]\mapsto\sum_{a\in A}\rho_a [Y\times_{h,Z,\boldsymbol f_a}X_a,\boldsymbol\pi_Y]$.
This takes relations (i),(ii) in $KB^k(Z;R)$ to (i),(ii) in
$KB^k(Y;R)$, and so is well-defined. Pullbacks are functorial,
$(g\circ h)^*=h^*\circ g^*$. The cup and cap products are compatible
with pullbacks and pushforwards, as in~\eq{ug4eq5}.
\label{ug5def3}
\end{dfn}
\subsection{Morphisms to and from Kuranishi (co)bordism}
\label{ug53}
In \cite[\S 5.3--\S 5.4]{Joyc1} we define morphisms between these
groups.
\begin{dfn} Let $Y$ be an orbifold, and $R$ a commutative ring.
Define morphisms $\Pi_{\rm bo}^{\rm Kb}:B_k(Y;R)\rightarrow KB_k(Y;R)$ for $k\geqslant 0$ by
$\Pi_{\rm bo}^{\rm Kb}:\sum_{a\in A}\rho_a[X_a,f_a]\mapsto
\sum_{a\in A}\rho_a[X_a,f_a]$, interpreting the manifold $X_a$ as a
Kuranishi space, and the smooth map $f_a:X_a\rightarrow Y$ as strongly
smooth.
Define morphisms $\Pi_{\rm Kb}^{\rm Kh}:KB_k(Y;R)\rightarrow KH_k(Y;R\otimes_{\mathbin{\mathbb Z}}{\mathbin{\mathbb Q}})$ for
$k\in{\mathbin{\mathbb Z}}$ by $\Pi_{\rm Kb}^{\rm Kh}:\sum_{a\in A}\rho_a\bigl[X_a,\boldsymbol
f_a\bigr]\mapsto\bigl[\textstyle\sum_{a\in A}\pi(\rho_a)[X_a,\allowbreak\boldsymbol f_a,\boldsymbol
G_a]\bigr]$, where $\boldsymbol G_a$ is some choice of gauge-fixing data for
$(X_a,\boldsymbol f_a)$, which exists by Theorem \ref{ug3thm1}(a), and
$\pi:R\rightarrow R\otimes_{\mathbin{\mathbb Z}}{\mathbin{\mathbb Q}}$ is the natural morphism. Using Theorem
\ref{ug3thm1}(e) over $[0,1]\times X_a$ one can show that $\Pi_{\rm Kb}^{\rm Kh}$
is independent of the choice of $\boldsymbol G_a$, and is well-defined.
Similarly, define morphisms $\Pi_{\rm Kcb}^{\rm Kch}:KB^k(Y;R)\rightarrow
KH^k(Y;R\otimes_{\mathbin{\mathbb Z}}{\mathbin{\mathbb Q}})$ for $k\in{\mathbin{\mathbb Z}}$ by $\Pi_{\rm Kb}^{\rm Kh}:\sum_{a\in
A}\rho_a\bigl[X_a,\boldsymbol f_a\bigr]\mapsto\bigl[\textstyle\sum_{a\in
A}\pi(\rho_a)[X_a,\allowbreak\boldsymbol f_a,\boldsymbol C_a]\bigr]$, where $\boldsymbol C_a$ is
some choice of co-gauge-fixing data for $(X_a,\boldsymbol f_a)$.
These $\Pi_{\rm Kcb}^{\rm Kch},\Pi_{\rm Kb}^{\rm Kh}$ take cup and cap products
$\cup,\cap$ on $KB^*,KB_*(Y;R)$ to $\cup,\cap$ on $KH^*,KH_*
(Y;R\otimes_{\mathbin{\mathbb Z}}{\mathbin{\mathbb Q}})$, and if $Y$ is compact, they take the identity
$[Y,\mathop{\rm id}\nolimits_Y]\in KB^0(Y;R)$ to the identity $\bigl[[Y,\mathop{\rm id}\nolimits_Y,\boldsymbol
C_Y]\bigr]\in KH^0(Y;R\otimes_{\mathbin{\mathbb Z}}{\mathbin{\mathbb Q}})$.
\label{ug5def4}
\end{dfn}
Consider the sequence of morphisms
\begin{equation*}
\smash{\xymatrix@C=25pt{ B_k(Y;R) \ar[r]^(0.45){\Pi_{\rm bo}^{\rm Kb}} &
KB_k(Y;R) \ar[r]^(0.4){\Pi_{\rm Kb}^{\rm Kh}} & KH_k(Y;R\otimes_{\mathbin{\mathbb Z}}{\mathbin{\mathbb Q}})
\ar[r]^(0.55){(\Pi_{\rm si}^{\rm Kh})^{\smash{-1}}} &
H^{\rm si}_k(Y;R\otimes_{\mathbin{\mathbb Z}}{\mathbin{\mathbb Q}}),}}
\end{equation*}
where $(\Pi_{\rm si}^{\rm Kh})^{-1}$ exists by Theorem \ref{ug3thm2}. The
composition is the natural map $B_k(Y;R)\rightarrow H^{\rm si}_k(Y;R\otimes_{\mathbin{\mathbb Z}}{\mathbin{\mathbb Q}})$
taking $[X,f]\mapsto f_*([X])$. Thus we find:
\begin{cor} Let\/ $Y$ be an orbifold, and\/ $R$ a commutative ring.
Then $KB_k(Y;\allowbreak R)$ is at least as large as the image of\/
$B_k(Y;R)$ in\/~$H_k^{\rm si}(Y;R\otimes_{\mathbin{\mathbb Z}}{\mathbin{\mathbb Q}})$.
\label{ug5cor}
\end{cor}
We will see in \S\ref{ug54} that $KB_*(Y;R)$ is actually very large.
The Poincar\'e duality story for Kuranishi (co)homology in
\S\ref{ug43} has an analogue for Kuranishi (co)bordism, as in
\cite[\S 5.4]{Joyc1}. Let $Y$ be an oriented $n$-orbifold without
boundary, and $R$ a commutative ring. Define $R$-module morphisms
$\Pi_{\rm Kcb}^{\rm Kb}:KB^k(Y;R)\rightarrow KB_{n-k}(Y;R)$ for $k\in{\mathbin{\mathbb Z}}$ by
$\Pi_{\rm Kcb}^{\rm Kb}:\sum_{a\in A}\rho_a[X_a,\boldsymbol f_a]\mapsto\sum_{a\in
A}\rho_a[X_a,\boldsymbol f_a]$, using the coorientation for $\boldsymbol f_a$ from
$[X_a,\boldsymbol f_a]\in KB^k(Y;R)$ and the orientation on $Y$ to determine
the orientation on $X_a$ for $[X_a,\boldsymbol f_a]\in KB_{n-k}(Y;R)$.
Define $\Pi^{\rm Kcb}_{\rm Kb}:KB_{n-k}(Y;R)\rightarrow KB^k(Y;R)$ by
$\Pi^{\rm Kcb}_{\rm Kb}:\sum_{a\in A}\rho_a[X_a,\allowbreak\boldsymbol f_a]\mapsto
\sum_{a\in A}\rho_a[X_a^Y,\boldsymbol f{}_a^Y]$, where $X_a^Y$ is $X_a$ with
an {\it alternative Kuranishi structure\/} as in \S\ref{ug43}, and
$\boldsymbol f_a{}^Y:X_a^Y\rightarrow Y$ is a lift of $\boldsymbol f_a$ to $X_a^Y$, which is
a strong submersion. Then \cite[Th.~5.11]{Joyc1} shows that
$\Pi^{\rm Kcb}_{\rm Kb}$ and $\Pi_{\rm Kcb}^{\rm Kb}$ are inverses, so they are both
isomorphisms. Using this and ideas in \S\ref{ug51} including
\eq{ug5eq1}, if $Y$ is a compact manifold we can define a natural
morphism $\Pi_{\rm cb}^{\rm Kcb}:MSO^*(Y)\rightarrow KB^*(Y;{\mathbin{\mathbb Z}})$, so Kuranishi
cobordism is a generalization of classical cobordism.
\subsection{How large are Kuranishi (co)bordism groups?}
\label{ug54}
Theorems \ref{ug3thm2} and \ref{ug4thm2} showed that Kuranishi
(co)homology are isomorphic to classical (compactly-supported)
(co)homology, so they are not new topological invariants. In
contrast, Kuranishi (co)bordism are not isomorphic to classical
(co)bordism, they are genuinely new topological invariants, so it is
interesting to ask what we can say about them. We now summarize the
ideas of \cite[\S 5.6--\S 5.7]{Joyc1}, which show that $KB_*(Y;R)$
and $KB^*(Y;R)$ are {\it very large\/} for any orbifold $Y$ and
commutative ring $R$ with $Y\ne\emptyset$ and~$R\otimes_{\mathbin{\mathbb Z}}{\mathbin{\mathbb Q}}\ne 0$.
One reason for this is that in a class $\sum_{a\in A}\rho_a[X_a,\boldsymbol
f_a]$ in $KB_k(Y;R)$ there is a lot of information stored in the
{\it orbifold strata\/} of $X_a$ for $a\in A$. We define these for
orbifolds,~\cite[Def.~5.15]{Joyc1}.
\begin{dfn} Let $\Gamma$ be a finite group, and consider
(finite-dimensional) real representations $(W,\omega)$ of $\Gamma$, that
is, $W$ is a finite-dimensional real vector space and
$\omega:\Gamma\rightarrow\mathop{\rm Aut}(W)$ is a group morphism. Call $(W,\omega)$ a {\it
trivial representation} if $\omega\equiv\mathop{\rm id}\nolimits_W$, and a {\it nontrivial
representation} if $\mathop{\rm Fix}(\omega(\Gamma))=\{0\}$. Then every
$\Gamma$-representation $(W,\omega)$ has a unique decomposition $W=W^{\rm
t}\oplus W^{\rm nt}$ as the direct sum of a trivial representation
$(W^{\rm t},\omega^{\rm t})$ and a nontrivial representation $(W^{\rm
nt},\omega^{\rm nt})$, where~$W^{\rm t}=\mathop{\rm Fix}(\omega(\Gamma))$.
Now let $X$ be an $n$-orbifold, $\Gamma$ be a finite group, and $\rho$
be an {\it isomorphism class of nontrivial\/ $\Gamma$-representations}.
Each $p\in X$ has a {\it stabilizer group\/} $\mathop{\rm Stab}\nolimits_X(p)$. The
tangent space $T_pX$ is an $n$-dimensional vector space with a
representation $\tau_p$ of $\mathop{\rm Stab}\nolimits_X(p)$. Let $\lambda:\Gamma\rightarrow\mathop{\rm Stab}\nolimits_X(p)$
be an injective group morphism, so that $\lambda(\Gamma)$ is a subgroup of
$\mathop{\rm Stab}\nolimits_X(p)$ isomorphic to $\Gamma$. Hence
$\tau_p\circ\lambda:\Gamma\rightarrow\mathop{\rm Aut}(T_pX)$ is a $\Gamma$-representation, and we
can split $T_pX=(T_pX)^{\rm t}\oplus(T_pX)^{\rm nt}$ into trivial and
nontrivial $\Gamma$-representations, and form the isomorphism class
$\bigl[(T_pX)^{\rm nt},(\tau_p\circ\lambda)^{\rm nt}\bigr]$. As a set,
define the {\it orbifold stratum} $X^{\Gamma,\rho}$ to be
\begin{align*}
X^{\Gamma,\rho}=\bigl\{\mathop{\rm Stab}\nolimits_X(p)\cdot(p,\lambda):\text{$p\in X$,
$\lambda:\Gamma\rightarrow\mathop{\rm Stab}\nolimits_X(p)$ is an injective}& \\
\text{group morphism, $\bigl[(T_pX)^{\rm nt},(\tau_p\circ\lambda)^{\rm
nt}\bigr]=\rho$}&\bigr\},
\end{align*}
where $\mathop{\rm Stab}\nolimits_X(p)$ acts on pairs $(p,\lambda)$ by $\sigma:(p,\lambda)\mapsto
(p,\lambda^\sigma)$, where $\lambda^\sigma:\Gamma\rightarrow\mathop{\rm Stab}\nolimits_X(p)$ is given by
$\lambda^\sigma(\gamma)=\sigma\lambda(\gamma)\sigma^{-1}$. Define a map $\iota^{\Gamma,\rho}:
X^{\Gamma,\rho}\rightarrow X$ by~$\iota^{\Gamma,\rho}:\mathop{\rm Stab}\nolimits_X(p)\cdot(p,\lambda)\mapsto
p$. Then \cite[Prop.~5.16]{Joyc1} shows that $X^{\Gamma,\rho}$ is an
orbifold of dimension $n-\mathop{\rm dim}\nolimits\rho$, and $\iota^{\Gamma,\rho}$ lifts to a
proper, finite immersion.
\label{ug5def5}
\end{dfn}
If $X$ is a Kuranishi space, there is a parallel definition
\cite[Def.~5.18]{Joyc1} of orbifold strata $X^{\Gamma,\rho}$, which we
will not give. The most important difference is that $\rho$ is now a
{\it virtual nontrivial representation\/} of $\Gamma$, that is, a
formal difference of nontrivial representations, so that
$\mathop{\rm dim}\nolimits\rho\in{\mathbin{\mathbb Z}}$ rather than $\mathop{\rm dim}\nolimits\rho\in{\mathbin{\mathbb N}}$. We find
\cite[Prop.~5.19]{Joyc1} that $X^{\Gamma,\rho}$ is a Kuranishi space
with $\mathop{\rm vdim}\nolimits X^{\Gamma,\rho}=\mathop{\rm vdim}\nolimits X-\mathop{\rm dim}\nolimits\rho$, equipped with a proper,
finite, strongly smooth map~$\boldsymbol\iota^{\Gamma,\rho}:X^{\Gamma,\rho}\rightarrow X$.
We would like to define projections $\Pi^{\Gamma,\rho}:KB_k(Y;R)\rightarrow
KB_{k-\mathop{\rm dim}\nolimits\rho}(Y;R)$ mapping $\Pi^{\Gamma,\rho}:[X_a,\boldsymbol f_a]\rightarrow
[X_a^{\Gamma,\rho},\boldsymbol f_a\vert_{X_a^{\Gamma,\rho}}]$. But there is a
problem: we need to define an {\it orientation\/} on
$X_a^{\Gamma,\rho}$ from the orientation on $X_a$, and for general
$\Gamma,\rho$ this may not be possible. To overcome this we suppose
$\md{\Gamma}$ is odd, which implies that $\mathop{\rm dim}\nolimits\rho$ is even for all
$\rho$, and there is then a consistent way to define orientations on
$X_a^{\Gamma,\rho}$, and $\Pi^{\Gamma,\rho}$ is well-defined.
Let $Y$ be a nonempty, connected orbifold. In \cite[\S 5.7]{Joyc1},
for each finite group $\Gamma$ with $\md{\Gamma}$ odd and all isomorphism
classes $\rho$ of virtual nontrivial representations of $\Gamma$, we
construct a class $C^{\Gamma,\rho}\in KB_{\mathop{\rm dim}\nolimits\rho}(Y;{\mathbin{\mathbb Z}})$, such that
$\Pi_{\rm Kb}^{\rm Kh}\circ\Pi^{\Gamma,\rho}(C^{\Gamma,\rho})$ is nonzero in
$KH_0(Y;{\mathbin{\mathbb Q}})\cong H^{\rm si}_0(Y;{\mathbin{\mathbb Q}})\cong{\mathbin{\mathbb Q}}$, and
$\Pi^{\Delta,\sigma}(C^{\Gamma,\rho})=0$ if either $\md{\Delta}\geqslant\md{\Gamma}$ and
$\Delta\not\cong\Gamma$, or if $\Delta=\Gamma$ and $\rho\ne\sigma$. It follows that
taken over all isomorphism classes of pairs $\Gamma,\rho$, the classes
$C^{\Gamma,\rho}\in KB_*(Y;{\mathbin{\mathbb Z}})$ are {\it linearly independent over\/}
${\mathbin{\mathbb Z}}$. Extending to an arbitrary commutative ring $R$, and using the
Poincar\'e duality ideas of \S\ref{ug53}, we deduce:
\begin{thm} Let\/ $Y$ be a nonempty orbifold, and\/ $R$ a
commutative ring with\/ $R\otimes_{\mathbin{\mathbb Z}}{\mathbin{\mathbb Q}}\ne 0$. Then $KB_{2k}(Y;R)$ is
infinitely generated over $R$ for all\/ $k\in{\mathbin{\mathbb Z}}$. If also $Y$ is
oriented of dimension $n$ then $KB^{n-2k}(Y;R)$ is infinitely
generated over $R$ for all\/~$k\in{\mathbin{\mathbb Z}}$.
\label{ug5thm}
\end{thm}
Theorem \ref{ug5thm} supports the idea that Kuranishi bordism, (or
better, {\it almost complex Kuranishi bordism}, as in
\cite[Ch.~5]{Joyc1}) may be a useful tool for studying (closed)
Gromov--Witten invariants. In \cite[\S 6.2]{Joyc1} we define new
Gromov--Witten type invariants $[{\mathbin{\smash{\,\,\overline{\!\!\mathcal M\!}\,}}}_{g,m}(M,J,\beta),\prod_i{\bf
ev}_i]$ in Kuranishi bordism $KB_*(M^m;{\mathbin{\mathbb Z}})$. Theorem \ref{ug5thm}
indicates that $KB_*(M^m;{\mathbin{\mathbb Z}})$ is very large, so that these new
invariants {\it contain a lot of information}, and that much of this
information has to do with the {\it orbifold strata\/} of the moduli
spaces ${\mathbin{\smash{\,\,\overline{\!\!\mathcal M\!}\,}}}_{g,m}(M,J,\beta)$.
Also, these new invariants are defined in groups $KB_*(M^m;{\mathbin{\mathbb Z}})$ {\it
over ${\mathbin{\mathbb Z}}$, not\/} ${\mathbin{\mathbb Q}}$. When we project to Kuranishi homology or
singular homology to get conventional Gromov--Witten invariants, we
must work in homology over ${\mathbin{\mathbb Q}}$. The reason we cannot work over ${\mathbin{\mathbb Z}}$
is because of rational contributions from the orbifold strata of
${\mathbin{\smash{\,\,\overline{\!\!\mathcal M\!}\,}}}_{g,m}(M,J,\beta)$. Kuranishi bordism looks like a good framework
for describing these contributions, and so for understanding the
integrality properties of Gromov--Witten invariants, such as the
Gopakumar--Vafa Integrality Conjecture for Gromov--Witten invariants
of Calabi--Yau 3-folds. This is discussed in \cite[\S 6.3]{Joyc1},
and the author hopes to take it further in~\cite{Joyc4}.
|
1,314,259,996,533 | arxiv | \section{Introduction}
Proving \emph{termination} of \emph{term rewrite systems (TRS\REV{s}{})}
is one of the most important task\REV{s}{} in
program verification and automated theorem proving, where
\emph{reduction orders} play a fundamental role.
The classic use of reduction orders in termination proving is
illustrated in the following example
\REV
\example\label{ex:fact}
Consider the following TRS $\RR_\m{fact}$:
\[
\RR_\m{fact} \DefEq
\left\{
\begin{array}{rcl}
\m{fact}(\m{0}) &\to& \s(\m{0})\\
\m{fact}(\s(x)) &\to& \s(x) \ttimes \m{fact}(x)
\end{array}
\right.
\]
which defines the factorial function,
provided the binary symbol $\ttimes$ is defined as multiplication.
}
\[
\RR_\m{half} \DefEq
\begin{EqSet}
\p(\s(x)) &\to x
\\
\m{half}(0) &\to 0
\\
\m{half}(\s(x)) &\to \s(\m{half}(\p(x)))
\end{EqSet}
\]
The TRS $\RR_\m{half}$ defines the function $\m{half}$ that halves an input natural number.
We can prove termination of $\RR_{\REV{\m{fact}}{\m{half}}}$ by finding a reduction order $\GT$
that satisfies the following constraints
\REV
\begin{align*}
\m{fact}(\m{0}) &\GT \s(\m{0})\\
\m{fact}(\s(x)) &\GT \s(x) \ttimes \m{fact}(x)
\end{align*}
\endexample
}
\begin{align*}
\p(\s(x)) &\GT x
\\
\m{half}(0) &\GT 0
\\
\m{half}(\s(x)) &\GT \s(\m{half}(\p(x)))
\end{align*}
A number of reduction orders have been proposed,
and their efficient
\REV
implementation is demonstrated by several
}
implementations are proposed during
the recent developments of
automatic termination provers such as \AProVE \cite{GST06} or \TTTT \cite{KSZM09}.
One of the most well-known reduction orders is
the \emph{lexicographic path order (LPO)} of Kamin and L\'evy \cite{KL80},
a variant of the \emph{recursive path order (RPO)} of Dershowitz \cite{D82}.
LPO is unified with RPO using \emph{status} \cite{L83}.
Recently,
Codish \etal \cite{CGST12} proposed
an efficient implementation using a SAT solver for
termination proving by
RPO with status.
The \emph{Knuth-Bendix order (KBO)} \cite{KB70} is
the \REV{oldest}{most historical} reduction order.
KBO has become a practical alternative in automatic termination checking since
Korovin and Voronkov \cite{KV03} discovered a polynomial-time algorithm
for termination proof\REV{s}{} with KBO.
Zankl \etal \cite{ZHM09} proposed another implementation method via SAT/SMT encoding,
and verified a significant improvement
in efficiency over dedicated implementations of
the polynomial-time algorithm.
However,
KBO is disadvantageous compared to LPO when \emph{duplicating} rules
(where a variable occurs more often in the right\REV{-}{ }hand side than in the left\REV{-}{ }hand side)
are considered.
Actually, no duplicating rule can be oriented by KBO.
To overcome this disadvantage,
Middeldorp and Zantema \cite{MZ97} proposed the \emph{generalized KBO (GKBO)},
which generalizes weights over
algebras that are weakly monotone and \emph{strictly simple}:
$f(\dots,x,\dots) > x$.
Ludwig and Waldmann proposed another extension of KBO called
the \emph{transfinite KBO (TKBO)} \cite{LW07,KMV11,WZM12},
which extends the weight function to allow linear polynomials over ordinals.
However, proving termination with TKBO involves \REV{solving the }{}satisfiability problem of
non-linear arithmetic which is undecidable in general.
Moreover, TKBO still does not \REV{subsume}{encompass} LPO.
The \emph{polynomial order (POLO)} of Lankford \cite{L75} interprets
each function symbol by a strictly monotone polynomial.
Zantema \cite{Z01} extended the method \REV{to}{over} algebras
\REV
}
that are weakly monotone and \emph{weakly} simple:
$f(\dots,x,\dots) \ge x$,
and suggested combining the ``max'' operator with polynomial interpretations
(\emph{max-polynomials} in terms of \cite{FGMSTZ08}).
Fuhs \etal proposed an efficient SAT encoding of POLO in \cite{FGMSTZ07}, and
a general version of POLO with max in \cite{FGMSTZ08}.
\REV
The \emph{dependency pair (DP) method} of \citet{AG00}
}
The \emph{dependency pair (DP) framework} \cite{AG00,HM05,GTSF06}
significantly enhances the classic approach of reduction orders by
analyzing cyclic dependencies between rewrite rules.
In the DP \REV{method}{framework},
reduction orders are \REV{extended}{relaxed} to \emph{reduction pairs} $\Tp{\GS,\GT}$,
and it suffices if one rule in a \REV{recursive}{cyclic} dependency is strictly oriented,
and other rules are only weakly \REV{oriented}{so}.
\REV
\example\label{ex:DP}\mbox{
}{
Consider again the TRS $\RR_{\REV{\m{fact}}{\m{half}}}$.
There is one cyclic dependency in $\RR_{\REV{\m{fact}}{\m{half}}}$,
that is represented by the \emph{dependency pair}
$\REV{\m{fact}}{\m{half}}^\sharp(\s(x)) \to \REV{\m{fact}}{\m{half}}^\sharp(\p(x))
\REV{, where $\REV{\m{fact}}{\m{half}}^\sharp$ is a fresh symbol. We
}{. Hence we
can prove termination of $\RR_{\REV{\m{fact}}{\m{half}}}$ by finding
a reduction pair $\Tp{\GS,\GT}$ that satisfies the following constraints
\footnote
The last two constraints can be removed by considering \emph{usable rules}
\cite{AG00}.
\REV
\begin{align*}
\m{fact}^\sharp(\s(x)) &\GT \m{fact}^\sharp(x)
\\
\m{fact}(\m0) &\GS \s(\m0)
\\
\m{fact}(\s(x)) &\GS \s(x) \ttimes \m{fact}(x)
\end{align*}
\endexample
}
\begin{align*}
\m{half}^\sharp(\s(x)) &\GT \m{half}^\sharp(\p(x))
\\
\p(\s(x)) &\GS x
\\
\m{half}(0) &\GS 0
\\
\m{half}(\s(x)) &\GS \s(\m{half}(\p(x)))
\end{align*}
One of the typical methods for designing reduction pairs is
\emph{argument filtering} \cite{AG00},
which generates reduction pairs from arbitrary reduction orders.
Hence, \REV{reduction orders are}{a reduction order is} still an important subject to study
in modern termination proving.
\REV{}{\par
Another typical technique is generalizing interpretation methods
to \emph{weakly} monotone ones,
\eg allowing $0$ coefficients for polynomial interpretations \cite{AG00}.
Endrullis \etal \cite{EWZ08} extended polynomial interpretations
to \emph{matrix interpretations},
and presented \REV{their}{its} implementation via SAT encoding.
\REV{}{\par
More recently, Bofill \etal \cite{BBRR13} proposed a reduction pair called \emph{RPOLO},
which unifies standard POLO and RPO by choosing either
\emph{RPO-like} or \emph{POLO-like} comparison
depending on function symbols.
These reduction orders and reduction pairs
require different correctness proofs and different implementations.
In this paper, we extract the underlying essence of these reduction orders and
introduce a general reduction order
called the \emph{weighted path order (WPO)}.
Technically, WPO is a further generalization of GKBO that
relaxes the strict simplicity condition of weights to \emph{weak simplicity}.
This relaxation become\REV{s}{} possible
by combining \REV{the }{}recursive checks of LPO with GKBO.
While strict simplicity is so restrictive that
GKBO does not even subsume the standard KBO,
weak simplicity is so \REV{general}{generous} that
WPO \REV{subsumes}{encompasses} not only KBO but also most of the reduction orders
described above (LPO, TKBO, POLO and so on), except for matrix interpretation
\REV{
which are not weakly simple in genera
}{}.
\REV
There exist several earlier works on generalizing existing reduction orders.
The \emph{semantic path order (SPO)} of \citet{KL80} is a generalization of
RPO where precedence comparison is generalized to
an arbitrary well-founded order on terms.
However, to prove termination by SPO
users have to ensure monotonicity by themselves,
even if the underlying well-founded order is monotone (\cf \cite{BFR00}).
On the other hand,
monotonicity of WPO is guaranteed.
\citet{BFR00} propose a variant of SPO
that ensures monotonicity by using an external monotonic order.
As well as LPO or POLO,
also WPO can be used as such an external order.
The \emph{general path order (GPO)} \cite{DH95,G96} is
a very general framework that many reduction orders are subsumed.
Due to the generality, however,
implementing GPO seems to be quite challenging.
Indeed, we are not aware of any tool that implements GPO.
}{
Instances of WPO are characterized by how weights are computed.
In particular, we introduce the following instances of WPO
and investigate their relationships with existing reduction orders:
\begin{itemize}
\item
$\WPOsum$ which uses summations for weight computation.
KBO can be obtained as a restricted case of $\WPOsum$,
where the \emph{admissibility} condition is enforced, and
weights of constants must be greater than $0$.
$\WPOsum$ is free from these restrictions, and
we verify that each extension strictly increases the power of the order.
\item
$\WPOpol$ which uses monotone polynomial interpretations for weight computation.
As a reduction order,
POLO is subsumed by $\WPOpol$.
TKBO can be obtained as a restricted case of $\WPOpol$,
where interpretations are linear polynomials,
\REV{}{the }admissibility is enforced, and
interpretations of constants are greater than $0$.
\item
$\WPOmax$ which uses maximums for weight computation.
LPO can be obtained as a restricted case of $\WPOmax$,
where the weights of all symbols are fixed to $0$.
In order to keep the presentation simple, we omit \emph{multiset status}
and only consider \emph{LPO with status}.
Nonetheless, it is easy to extend this result \REV{to}{for} \emph{RPO with status}.
\item
$\WPOmp$ which combines polynomial\REV{s}{} and maximum for interpretation,
and its variant $\WPOms$ whose coefficients are fixed to $1$.
WPO($\Ams$) generalizes KBO and LPO, and
$\WPOmp$ moreover subsumes POLO (with max)
as a reduction order.
\end{itemize}
Note that all the instances described above use weakly simple algebras
\REV{which cannot be used for GKBO}{and hence not possible by GKBO}.
Next we extend WPO \REV{to}{as} a reduction pair by incorporating
\emph{partial statuses} \cite{YKS13b}.
This extension further relaxes the weak simplicity condition,
and arbitrary weakly monotone interpretations can be
used for weight computation. Hence as
a reduction pair, WPO also \REV{subsumes}{encompasses the} matrix interpretations,
as well as KBO, TKBO, LPO and POLO.
Though RPOLO also unifies RPO and POLO,
we show that WPO and RPOLO are incomparable in \REV{general}{theory}
\footnote
It is possible to consider an instance of WPO that uses RPOLO for weight computation.
}
Moreover in practice,
WPO \REV{brings}{shows} significant benefit \REV{on}{in} the problems
from the \emph{Termination Problem Data Base (TPDB)} \cite{TPDB13},
while (the first-order version of) RPOLO does not,
as reported in \cite{BBRR13}.
Finally, we present an efficient implementation using
\REV{}{the }state-of-the-art SMT solvers.
By extending \cite{ZHM09},
we present SMT encoding techniques for the instances of WPO introduced so far.
In particular, \REV{the }{}orientability problem\REV{s}{} of $\WPOsum$, $\WPOmax$ and $\WPOms$
are reduced to a satisfiability problem of linear arithmetic, which is known to be decidable.
Through experiments in TPDB problems,
we also verify the efficiency of our implementation
and significance of WPO in practice.
The remainder of this paper is organized as follows:
In \prettyref{sec:preliminaries}
we recall some basic notions of term rewriting and
the definitions of existing orders.
\prettyref{sec:WPO order} is devoted \REV{to}{for} WPO as a reduction order.
There we present the definition of WPO, and then
investigate several instances of WPO and
show their relationships with existing orders.
In \prettyref{sec:pair}
we extend WPO \REV{to}{as} a reduction pair.
We present the definition of the reduction pair
and a soundness proof in \prettyref{sec:pair definition}.
Then the definition is refined in \prettyref{sec:pair refinements} and
\REV{the relationship to}{relationships with} existing reduction pairs
\REV{is}{are} shown in \prettyref{sec:pair instances}.
\prettyref{sec:encodings} presents SMT encodings for
the instances of WPO introduced so far,
and some implementation issues are discussed in \prettyref{sec:optimizations}.
In \prettyref{sec:experiments} we verify the significance of our work
through experiments and we conclude in \prettyref{sec:conclusion}.
A preliminary version of this paper appeared in \cite{YKS13}.
The results for \REV{the reduction orders of}{reduction order in}
\prettyref{sec:WPO order} are basically the same as in \cite{YKS13}\REV{}{,
though the \REV{the presentation is}{presentations are} refined}.
The definition of WPO as a reduction pair in \prettyref{sec:pair} is new, and
hence most of the results in \prettyref{sec:pair} are new.
The \REV{revised}{new} version of WPO further subsumes POLO and matrix interpretations
as a reduction pair\REV{}{, as shown \REV{by}{as}
Corollaries \ref{cor:WPO>=POLO pair} and \ref{cor:WPO>=MAT pair}, \resp,
in contrast to \cite{YKS13}}.
We also conclude that WPO does not subsume RPOLO by \prettyref{ex:RPOLO-WPO},
which was left open in \cite{YKS13}.
\REV{The}
Moreover, the SMT encodings in \prettyref{sec:encodings} are revised to fit
\REV{}{to }the new definition, and
the new} experimental results in
Sections \ref{sec:experiments pair} and \ref{sec:experiments combination} show
a significant improvement due to
the new definition of WPO as a reduction pair.
\section{Preliminaries}\label{sec:preliminaries}
\emph{Term rewrite systems (TRSs)} model first-order functional programs.
We refer the readers to \cite{BN98,Terese} for details \REV{on}{of} rewriting,
and only briefly recall some important notions needed in this paper.
A \Def{signature} $\Sig$ is a finite set of function symbols associated with \REV{an }{}arity.
The set of $n$-ary symbols is denoted by $\Sig_n$.
A \Def{term} is either a variable $x \in \Vars$ or \REV{of the }{in }form
$f(s_1,\dots,s_n)$ where $f \in \Sig_n$ and each $s_i$ is a term.
Throughout the paper, we abbreviate a sequence $a_1,\dots,a_n$ by $\Seq{a_n}$.
The set of terms constructed from $\Sig$ and $\Vars$ is denoted by
$\Terms(\Sig,\Vars)$.
The set of variables occurring in a term $s$ is denoted by $\Var(s)$, and
the number of occurrences of a variable $x$ in $s$ is denoted by $|s|_x$.
\REV{}{\par
A TRS is a set $\RR$ of pairs of terms called \Def{rewrite rules}.
A rewrite rule, written $l \to r$ where
$l \notin \Vars$ and $\Var(l) \supseteq \Var(r)$,
indicates that an instance of $l$ should be rewritten to
\REV{the }{}corresponding instance of $r$.
The \Def{rewrite relation} $\to_\RR$ induced by $\RR$ is
\REV
the least relation which includes $\RR$ and is
monotonic and stable.
Here,
}
the monotonic stable closure of $\RR$, where
a relation $\REL$ on terms is
\begin{itemize}
\item
\Def{monotonic} iff $s \REL t$ implies $f(\dots,s,\dots) \REL f(\dots,t,\dots)$
for every context $f(\dots,\Box,\dots)$, and
\item
\Def{stable} iff $s \REL t$ implies $s\theta \REL t\theta$
for every substitution $\theta$.
\end{itemize}
A TRS $\RR$ is \Def{terminating} iff
no infinite rewrite sequence $s_1 \to_\RR s_2 \to_\RR \dots$ exists.
\subsection{Reduction Orders}
A classic method for proving termination is to find a \emph{reduction order}:
A \Def{reduction order} is a well-founded order which is monotonic and stable.
We say an order $\GT$ \Def{orients} a TRS $\RR$ iff
$l \GT r$ for every rule $l \to r \in \RR$; in other words, $\RR \subseteq {\GT}$.
It is easy to see the following:
\begin{theorem}\label{thm:order}\REV{\cite{Z94}}{}
A TRS is terminating iff
it is oriented by a reduction order.\qed
\end{theorem}
Ensuring well-foundedness of a reduction order is often a non-trivial task.
\REV{The following}{Following}
is a well-known technique of Dershowitz \cite{D82} for ensuring well-foundedness
based on \emph{Kruskal's tree theorem}\REV{.}{:}
A \Def{simplification order} is a strict order $\GT$ on terms, which is
monotonic and stable and satisfies \REV{the }{}\Def{subterm property}:
$f(\dots,s,\dots) \GT s$.
\begin{theorem}\cite{D82}
For a finite signature,
a simplification order is a reduction order.\qed
\end{theorem}
\REV
In the latter, we only consider finite signatures.
}{
In the remainder of this section, we recall several existing reduction orders.
\subsubsection{Lexicographic Path Order}
We consider LPO \cite{KL80} with quasi-precedence and status;
a \Def{quasi-precedence} $\PGS$ is a quasi-order
\REV{(\ie, a reflexive and transitive relation)}{} on $\Sig$,
whose strict part, denoted by $\PGT$, is well-founded.
The equivalence part of $\PGS$ is denoted by $\PSIM$.
A \Def{status function} $\sigma$ assigns
\REV{to }{}each function symbol $f \in \Sig_n$
a permutation $[\Seq{i_n}]$ of positions
in $\SetOf{1,\dots,n}$.
We denote the list $[s_{i_1},\dots,s_{i_n}]$ by
$\AppPerm{\sigma(f)}{s}{n}$ for $\sigma(f) = [\Seq{i_n}]$.
\REV
A strict order $\GT$ (associated with a quasi-order $\GS$)
is lifted on lists as follows:
$[\Seq{s_n}] \GT^\Lex [\Seq{y_m}]$ iff
there exists $k < n$ \st
$x_i \GS y_i$ for each $i \in \SetOf{1,\dots,k}$ and
either $k = m$ or $k < m$ and $x_{k+1} \GT y_{k+1}$.
}{
\begin{definition}\label{def:LPO}
For a quasi-precedence $\PGS$,
the \emph{lexicographic path order} $\GT_\LPO$
with status $\sigma$ is recursively defined as follows:
$s = f(\Seq{s_n}) \GT_\LPO t$ iff
\makeatletter
\renewcommand{\p@enumii}{\theenumi--}
\makeatother
\renewcommand\theenumi{\alph{enumi}}
\renewcommand\labelenumi{(\theenumi)}
\renewcommand\theenumii{\roman{enumii}}
\renewcommand\labelenumii{\theenumii.}
\begin{enumerate}
\item\label{item:LPO-simp}
$\ForSome{i \in \{1,\dots,n \}} s_i\GE_\LPO t$, or
\item\label{item:LPO-args}
$t=g(\Seq{t_m})$, $\ForAll{j\in \{ 1, \dots, m \}} s\GT_\LPO t_j$ and either
\begin{enumerate}
\item\label{item:LPO-prec}
$f\PGT g$, or
\item
$f\PSIM g$ and
$\AppPerm{\sigma(f)}{s}{n} \GT_\LPO^\Lex
\AppPerm{\sigma(g)}{t}{m}$.
\end{enumerate}
\end{enumerate}
\end{definition}
\begin{theorem}\cite{KL80}
$\GT_\LPO$ is a simplification order and hence a reduction order.\qed
\end{theorem}
\unskip
\REV
\begin{example}\label{ex:LPO}
Termination of $\RR_\m{fact}$ from \prettyref{ex:fact} can be shown by LPO.
Both rules are oriented by case \prettyref{item:LPO-prec}
with a precedence \st $\m{fact} \PGT \s$ for the first rule and
$\m{fact} \PGT {\ttimes}$ for the second rule.
\end{example}\unskip
}{
\subsubsection{Polynomial Interpretations}
We basically follow the abstract definitions of \cite{Z01,EWZ08}.
A \Def{well-founded $\Sig$-algebra} $\A$
is a quadruple $\Tp{A,\gs,>,\cdot_\A}$
of a \Def{carrier set} $A$,
a quasi-order $\gs$ on $A$,
a well-founded order $>$ on $A$ which is \emph{compatible} with $\gs$,
\REV{\ie,}{\ie}
${\gs}\circ{>}\circ{\gs} \subseteq {>}$, and
an \Def{interpretation} $f_\A : A^n \to A$ for each $f\in\Sig_n$.
$\A$ is \Def{strictly} (\Def{weakly}) \Def{monotone} iff
$a \REV{\gsopt}{>} b$ implies $f_\A(\dots,a,\dots) \gsopt f_\A(\dots,b,\dots)$, and
\Def{strictly} (\Def{weakly}) \Def{simple} iff
$f_\A(\dots,a,\dots) \gsopt a$ for every $f \in \Sig$.
The relations $\gs$ and $>$ are extended \REV{to}{on} terms as follows:
$s \gsopt_\A t$ iff
$\widehat\alpha(s) \gsopt \widehat\alpha(t)$ holds
for \REV{every}{all} assignment\REV{}{s} $\alpha : \Vars \to A$\REV{,
where $\widehat\alpha : \Terms \to A$ is the homomorphic extension of $\alpha$.
}{and its homomorphic extension $\widehat\alpha$.
A \Def{polynomial interpretation} $\Apol$ interprets
every function symbol $f \in \Sig$ as a polynomial $f_\Apol$.
The carrier set of $\Apol$ is
$\{ a \in \Nat \mid a \ge \Wzero \}$ for some $\Wzero \in \Nat
\REV{, and the}{. The}
orderings are the standard $\ge$ and $>$ on $\Nat$.
$\Apol$ induces a reduction order if it is strictly monotone;
in other words, all arguments have coefficients at least $1$.
\begin{theorem}\cite{MN70,L75}
If $\Apol$ is strictly monotone, then
$>_\Apol$ is a reduction order.\qed
\end{theorem}
\unskip
\REV
\begin{example}\label{ex:POLO}
Termination of $\RR_\m{fact}$ of \prettyref{ex:fact} can be
shown by the polynomial interpretation $\Apol$ defined as follows:
\begin{align*}
\m{fact}_\Apol(x) &= 2 x + 2&
\m{0}_\Apol &= 0\\
\s_\Apol(x) &= 2 x + 1&
x \ttimes_\Apol y &= x + y
\end{align*}
The left- and right-hand sides of
the rule $\m{fact}(\m{0}) \to \s(\m{0})$
are interpreted as $2$ and $1$, \resp, and
those of the rule
$\m{fact}(\s(x)) \to \s(x) \ttimes \m{fact}(x)$
are interpreted as $4x + 4$ and $4x + 3$, \resp.
\end{example}\unskip
}{
\subsubsection{Knuth-Bendix Order}
$\KBO$ \cite{KB70} is induced by a \REV{quasi-}{}precedence and
a \Def{weight function} $\Tp{\Weight,\Wzero}$, where
$\Weight : \Sig \to \Nat$ and $\Wzero \in \Nat$ \st
$\WeightOf c \ge \Wzero$ for every constant $c \in \Sig_0$.
The weight $\WeightOf{s}$ of a term $s$ is defined as follows:
\[
\WeightOf{s} \DefEq
\begin{Cases}
\Wzero
&\text{ if } s \in \Vars \\
\WeightOf{f} + \displaystyle\sum_{i=1}^{n}\WeightOf{s_i}
&\text{ if } s = f(\Seq{s_n})
\end{Cases}
\]
The weight function $\Weight$ is said
\REV{to be }{}\Def{admissible} for $\PGS$ iff
every unary symbol $f \in \Sig_1$ with $\WeightOf{f} = 0$ is
\REV{\emph{greatest} \wrt $\PGS$, \ie, $f \PGS g$ for every $g \in \Sig$}{maximum \wrt $\PGS$}.
In this paper we also consider status for KBO \cite{S89}.
\begin{definition}\label{def:KBO}
For a quasi-precedence $\PGS$
and a weight function $\Tp{\Weight,\Wzero}$,
the \Def{Knuth-Bendix order}
$\GT_\KBO$ with status $\sigma$ is
recursively defined as follows:
$s = f(\Seq{s_n}) \GT_\KBO t$ iff
$|s|_x \ge |t|_x$ for all $x \in \Vars$ and either
\begin{enumerate}
\item\label{item:KBO-gt}
$\WeightOf{s} > \WeightOf{t}$, or
\item\label{item:KBO-ge}
$\WeightOf{s} = \WeightOf{t}$ and either
\begin{enumerate}
\item\label{item:KBO-simp}
$s = f^k(t)$ and $t \in \Vars$ for some $k > 0$, or
\item\label{item:KBO-args}
$t = g(\Seq{t_m})$ and either
\begin{enumerate}
\item\label{item:KBO-prec}
$f\PGT g$, or
\item\label{item:KBO-mono}
$f\PSIM g$ and
$\AppPerm{\sigma(f)}{s}{n} \GT_\KBO^\Lex
\AppPerm{\sigma(g)}{t}{m}$.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{definition}
Here we follow \cite{ZHM09}, and the range of $\Weight$ is restricted to $\Nat$.
According to \cite{KV03}, this does not decrease the power of $\KBO$ for finite TRSs.
Note that we do not assume $\Wzero > 0$ in the definition.
This assumption, together with \REV{}{the }admissibility is required for $\KBO$ to be a simplification order.
For details of the following result, we refer \eg \REV{to }{}\cite[Theorem~5.4.20]{BN98}.
\begin{theorem}
If $\Wzero > 0$ and $\Weight$ is admissible for $\PGS$, then
$\GT_\KBO$ induced by $\Tp{\Weight,\Wzero}$ and $\PGS$ is a simplification order,
and hence a reduction order.
\qed
\end{theorem}
\REV
The \emph{variable condition}
``$|s|_x \ge |t|_x$ for all $x \in \Vars$''
is often said to be a major disadvantage of KBO.
Due to this condition, no duplicating rule can be oriented by KBO.
\begin{example}
Termination of $\RR_\m{fact}$ of \prettyref{ex:fact} cannot be shown by KBO,
since the variable $x$ of the rule
$\m{fact}(\s(x)) \to \s(x) \ttimes \m{fact}(x)$
violates the variable condition.
\end{example}\unskip
}{
\subsubsection{Transfinite KBO}
TKBO \cite{LW07,KMV11,WZM12} extends KBO by introducing
a \Def{subterm coefficient function} $\Coef$,
that assigns a positive intege
\footnote{We do not use \Def{transfinite} coefficients, since
they do not add power when finite TRSs are considered \cite{WZM12}.
\REV
We will still use the acronym TKBO to denote
the finite variant.
}{
}
$\CoefOf{f,i}$ to each $f \in \Sig_n$ and $i \in \{ 1,\dots,n \}$.
For a weight function $\Tp{\Weight,\Wzero}$ and a subterm coefficient function $\Coef$,
\REV{the refined weight }{}$\WeightOf{s}$ is \REV{defined}{refined} as follows:
\[
\WeightOf{s} \DefEq
\begin{Cases}
\Wzero
&\text{ if } s \in \Vars
\\
\WeightOf{f} +
\displaystyle\sum_{i = 1}^{n}\CoefOf{f,i} \cdot \WeightOf{s_i}
&\text{ if } s = f(\Seq{s_n})
\end{Cases}
\]
The \Def{variable coefficient}
$\VCoefOf{x,s}$ of $x$ in $s$ is defined recursively as follows:
\[
\VCoefOf{x,s} \DefEq
\begin{Cases}
1 &\text{if } x = s
\\
0 &\text{if } x \neq \REV{s}{y} \in \Vars
\\
\displaystyle\sum_{i=1}^{n}\CoefOf{f,i} \cdot \VCoefOf{x,s_i}
&\text{if } s = f(\Seq{s_n})
\end{Cases}
\]
Then the order $\GT_\TKBO$ is obtained from \prettyref{def:KBO} by replacing
$|\cdot|_x$ by $\VCoefOf{x,\cdot}$ and $w(\cdot)$ by
\REV{its refined version above}{refined ones}.
\begin{theorem}\cite{LW07}
If $w_0 > 0$ and $w$ is admissible for $\PGS$, then
$\GT_\TKBO$ is a simplification order and hence a reduction order.\qed
\end{theorem}
\unskip
\REV
\begin{example}\label{ex:TKBO}
Consider again the TRS $\RR_\m{fact}$ of \prettyref{ex:fact}.
Consider the weight function $\Weight$ defined as follows:
\begin{align*}
\WeightOf{\m{0}} &= 1&
\WeightOf{\m{s}} &= 0&
\WeightOf{\m{fact}} &= 1&
\WeightOf{\ttimes} &= 0
\end{align*}
and the subterm coefficient function $\Coef$ defined as follows:
\begin{align*}
\CoefOf{\s,1} = \CoefOf{\m{fact},1} &= 2&
\CoefOf{\ttimes,1} = \CoefOf{\ttimes,2} &= 1
\end{align*}
Finally, consider an admissible precedence \st
$\m{s} \PGT \m{fact} \PGT \ttimes$.
The first rule $\m{fact}(\m{0}) \to \s(\m{0})$ of $\RR_\m{fact}$ is
oriented by case \prettyref{item:KBO-gt}.
For both sides of the second rule $\m{fact}(\s(x)) \to \s(x) \ttimes \m{fact}(x)$,
the variable coefficient of $x$ is $4$ and the weight is $3$.
Hence, the rule is oriented by case \prettyref{item:KBO-prec}.
\end{example}\unskip
}{
\subsubsection{Generalized Knuth-Bendix Order}
GKBO \cite{MZ97} uses a weakly monotone and \emph{strictly} simple algebra for weight computation.
In the following version of GKBO, we
extend \cite{MZ97} with quasi-order $\AGS$ and quasi-precedence $\PGS$,
and omit \emph{multiset status}.
\begin{definition}
For a \REV{quasi-}{}precedence \REV{$\PGS$}{$\PGT$} and a well-founded $\Sig$-algebra $\A$,
the \Def{generalized Knuth-Bendix order} $\GT_\GKBO$
is recursively defined as follows: $s =f(\Seq{s_n}) \GT_\GKBO t$ iff
\renewcommand\theenumii{\roman{enumii}
\renewcommand\labelenumii{\theenumii.
\begin{enumerate}
\item\label{item:GKBO-gt}
$s \AGT t$, or
\item\label{item:GKBO-ge}
$s \AGS t = g(\Seq{t_m})$ and either
\begin{enumerate}
\item\label{item:GKBO-prec}
$f \PGT g$, or
\item
$f \PSIM g$ and
$\AppPerm{\sigma(f)}{s}{n} \GT_\GKBO^\Lex
\AppPerm{\sigma(g)}{t}{m}$.
\end{enumerate}
\end{enumerate}
\end{definition}
\begin{theorem}
\cite{MZ97}
If $\A$ is weakly monotone and strictly simple, then
$\GT_\GKBO$ is a simplification order and hence a reduction order.\qed
\end{theorem}
\REV
One of the most important advantage of GKBO is
that it admits weakly monotone interpretations such as $\max$.
\begin{example}
Termination of $\RR_\m{fact}$ of \prettyref{ex:fact} can be shown
by GKBO induced by an algebra $\A$ on $\Nat$ with interpretation \st
\begin{align*}
\m{fact}_\A(x) &= x + 2&
\m{0}_\A &= 0\\
\s_\A(x) &= x + 1&
x \ttimes_\A y &= \max\{x,y\} + 1
\end{align*}
and a precedence \st $\m{fact} \PGT {\ttimes}$.
The first rule $\m{fact}(\m{0}) \to \s(\m{0})$ is oriented by case \prettyref{item:GKBO-gt}.
The second rule $\m{fact}(\s(x)) \to \s(x) \ttimes \m{fact}(x)$ is oriented by
case \prettyref{item:GKBO-prec},
since $x + 3 \ge \max\{x + 1, x + 2 \} + 1$.
\end{example}
Note that the strict simplicity condition is crucial for GKBO
to be well-founded.
If we modify the interpretation of the above example
by $x \ttimes_\A y = \max\{x,y\}$,
then GKBO will admit the following infinite sequence:
}{
\[
\REV
\m{fact}(\m{0}) \GT_\GKBO
\m{0} \ttimes \m{fact}(\m{0}) \GT_\GKBO
\m{0} \ttimes (\m{0} \ttimes \m{fact}(\m{0})) \GT_\GKBO \dots
}{
\]
\subsection{The Dependency Pair Framework and Reduction Pairs}
The
\REV
\emph{dependency pair (DP) method} \cite{AG00}
}
\emph{dependency pair (DP) framework} \cite{AG00,HM05,GTS04,GTSF06}
significantly enhances the classical method of reduction orders
by analyzing dependencies between rewrite rules.
We briefly recall the essential notions for
\REV
its successor, the \emph{DP framework} \cite{HM05,GTS04,GTSF06}.
}
the DP framework.
Let $\RR$ be a TRS over a signature $\Sig$.
The \emph{root symbol} of a term $s = f(\Seq{s_n})$ is $f$ and denoted by
$\Root(s)$.
The set of \emph{defined symbols} \wrt $\RR$ is defined as
$\SigD \DefEq \{ \Root(l) \mid l \to r \in \RR \}$.
For each $f \in \SigD$, the signature $\Sig$ is extended by
a fresh \emph{marked symbol} $f^\sharp$ \REV{having the same arity}{whose arity is the same} as $f$.
For $s = f(\Seq{s_n})$ with $f \in \SigD$, the term
$f^\sharp(\Seq{s_n})$ is denoted by $s^\sharp$.
The set of \emph{dependency pairs} for $\RR$ is defined as
\(
\DP(\RR) \DefEq
\{
l^\sharp \to t^\sharp \mid l \to r \in \RR,
t\text{ is a subterm of }r, \Root(t) \in \SigD
\}
\).
A \emph{DP problem} is a pair $\Tp{\PP,\RR}$
of a TRS $\RR$ and a set $\PP$ of dependency pairs for $\RR$.
A DP problem $\Tp{\PP,\RR}$ is \emph{finite} iff $\to_\PP\cdot\to_\RR^*$ is well-founded,
where $\PP$ is viewed as a TRS.
The main result of the DP framework is the following:
\begin{theorem}\cite{AG00,GTSF06}
A TRS $\RR$ is terminating
\REV{iff}{if} the DP problem $\Tp{\DP(\RR),\RR}$ is finite.
\qed
\end{theorem}
Finiteness of a DP problem is proved by \emph{DP processors}:
A sound \emph{DP processor}
\REV{gets a DP problem as input}{inputs a DP problem}
and outputs a set of (hopefully simpler) DP problems
\st the input problem is finite if all the output problems are finite.
Among other DP processors for transforming or simplifying DP problems
(\cf \cite{GTSF06} for a summary),
we recall the most important one:
A \emph{reduction pair} $\Tp{\GS,\GT}$ is a pair of relations on terms \st
$\GS$ is a monotonic and stable quasi-order, and
$\GT$ is a well-founded stable order which is \emph{compatible} with $\GS$,
\REV{\ie,}{\ie}
${\GS} \circ {\GT} \circ {\GS} \subseteq {\GT}$.
\begin{theorem}\cite{AG00,GTS04,HM05,GTSF06}\label{thm:reduction pair}
Let $\Tp{\GS,\GT}$ be a reduction pair \st
$\PP\cup\RR \subseteq {\GS}$ and $\PP' \subseteq {\GT}$.
Then the DP processor that maps
$\Tp{\PP,\RR}$ to $\{ \Tp{\PP\setminus\PP',\RR} \}$
is sound.
\qed
\end{theorem}
\subsubsection{Weakly Monotone Interpretations}
To define a reduction pair by polynomial
interpretations, an interpretation
$\Apol$ need not be strictly monotone but only weakly \REV{monotone}{so};
in other words, $0$ coefficients are allowed.
\begin{theorem}\cite{AG00}
If $\Apol$ is weakly monotone, then
$\Tp{\ge_\Apol,>_\Apol}$ forms a reduction pair.\qed
\end{theorem}
Endrullis \etal \cite{EWZ08} extend\REV{}{s} linear polynomial interpretations
to \emph{matrix interpretations}.
\begin{definition}
Given a fixed \emph{dimension} $d \in \Nat$,
the well-founded algebra $\Amat$ consists of
the carrier set $\Nat^d$ and
the strict and quasi\REV{-}{ }orders on $\Nat^d$ defined as follows:
\[
\left(\begin{matrix}v_1\\\vdots\\v_d\end{matrix}\right)
\gsopt
\left(\begin{matrix}u_1\\\vdots\\u_d\end{matrix}\right)
\DefIff
v_1 \geopt u_1 \AND v_j \ge u_j \FORALL j \in \SetOf{2,\dots,d}
\]
The interpretation in $\Amat$ is induced by
a function $\VecWeight$ that assigns
a $d$-dimension vector $\VecWeightOf{f}$ to each $f \in \Sig$, and
a function $\Cmat$ that assigns
a $d \ttimes d$ matrix $\CmatOf{f,i}$ to each $f \in \Sig_n$ and
$i \in \SetOf{1,\dots,n}$\REV{. It is}{, and} defined as follows:
\[
f_\Amat(\Seq{{\Vec{x}}_n}) =
\VecWeightOf{f} + \sum_{i=1}^n \CmatOf{f,i}\cdot\Vec{x}_i
\]
The $i$-th row and $j$-th column element of a matrix $M$
is denoted by $M^{i,j}$.
\end{definition}
\begin{theorem}\cite{EWZ08}
For a matrix interpretation $\Amat$,
$\Tp{\gs_\Amat,>_\Amat}$ forms a reduction pair.\qed
\end{theorem}
\subsubsection{Argument Filtering}
\emph{Argument filtering} \cite{AG00,KNT00} is a typical technique to design
a reduction pair from a reduction order:
An \emph{argument filter} $\pi$ maps each $f \in \Sig_n$ to
either a position $i \in \{ 1,\dots, n \}$ or
a list $[\Seq{i_m}]$ of positions \st $1 \le i_1 < \dots < i_m \le n$.
The signature $\Sig^\pi$ consists of
every $f \in \Sig$ \st $\pi(f) = [\Seq{i_m}]$,
and \REV{the }{}arity of $f$ is $m$ in $\Sig^\pi$.
An argument filter $\pi$ induces a mapping
$\pi : \Terms(\Sig,\Vars) \to \Terms(\Sig^\pi,\Vars)$ as follows:
\[
\pi(s) \DefEq
\begin{Cases}
s &\text{if } s \in \Vars
\\
\pi(s_i) &\text{if } s = f(\Seq{s_n})\text{, }\pi(f) = i
\\
f(\pi(s_{i_1}),\dots,\pi(s_{i_m}))
&\text{if }s = f(\Seq{s_n})\text{, }\pi(f) = [\Seq{i_m}]
\end{Cases}
\]
For an argument filter $\pi$ and a reduction order $\GT$ on $\Terms(\Sig^\pi,\Vars)$,
the relations $\GS^\pi$ and $\GT^\pi$ on $\Terms(\Sig,\Vars)$ are defined as follows:
$s \GS^\pi t$ iff $\pi(s) \GE \pi(t)$, and
$s \GT^\pi t$ iff $\pi(s) \GT \pi(t)$.
\begin{theorem}\cite{AG00}
For a reduction order $\GT$ and an argument filter $\pi$,
$\Tp{{\GS}^\pi,{\GT}^\pi}$ forms a reduction pair.
\qed
\end{theorem}
\REV
The effect of argument filtering is especially apparent for KBO;
it relaxes the variable condition.
\begin{example}
By applying an argument filter $\pi$ \st
$\pi(\ttimes) = 1$ for the constraints in \prettyref{ex:DP},
we obtain the following constraints:
\[
\m{fact}(\m{0}) \GS \s(\m{0})\qquad \m{fact}(\s(x)) \GS \s(x) \qquad
\m{fact}^\sharp(\s(x)) \GT \m{fact}^\sharp(x)
\]
The first constraint can be satisfied by KBO with
\eg $\WeightOf{\m{fact}} > \WeightOf{\s}$.
The other constraints are satisfied by any instance of KBO.
\end{example}\unskip
}{
\section{WPO as a Reduction Order}\label{sec:WPO order}
In this section, we introduce a reduction order called
the \emph{weighted path order (WPO)} that further generalizes GKBO
by weakening the simplicity condition on algebras.
After showing some properties for the order,
we then introduce several instances of WPO by fixing algebras.
We investigate relationships between these instances of WPO and
existing reduction orders and show the potential of WPO.
We first introduce the definition of WPO.
In order to admit algebras that are not strictly but only weakly simple,
we employ the recursive checks
\REV{which ensure that LPO is}{that ensures LPO to be} a simplification order.
\def\WPOAS{{\WPO(\A,\sigma)}}
\begin{definition}[WPO]\label{def:WPO}
For a quasi-precedence $\PGS$,
a well-founded algebra $\A$ and a status $\sigma$,
the \emph{weighted path order}
$\GT_\WPOAS$
is defined as follows:
$s = f(\Seq{s_n}) \GT_\WPOAS t$ iff
\begin{enumerate}
\item
$s \AGT t$, or
\item
$s \AGS t$ and
\begin{enumerate}
\item
$\ForSome{i \in \SetOf{1,\dots,n}}s_i \GE_\WPOAS t$, or
\item
$t=g(\Seq{t_m})$,
$\ForAll{j \in \SetOf{1,\dots,m}}s \GT_\WPOAS t_j$ and either
\begin{enumerate}
\item
$f\PGT g$ or
\item
$f\PSIM g$ and
$\AppPerm{\sigma(f)}{s}{n} \GT_\WPOAS^\Lex
\AppPerm{\sigma(g)}{t}{m}$.
\end{enumerate}
\end{enumerate}
\end{enumerate}
We abbreviate $\WPOAS$ by $\WPOA$ and $\WPO$ when no confusion arises.
\end{definition}
Case \prettyref{item:WPO-gt} and the
\REV{precondition of case}{ in} \prettyref{item:WPO-ge} are
the same as $\GKBO$.
Case \prettyref{item:WPO-simp} and the
\REV{precondition of case}{condition in} \prettyref{item:WPO-args}
are the recursive checks that
correspond to \prettyref{item:LPO-simp} and \prettyref{item:LPO-args} of $\LPO$.
Note that here we may restrict \REV{}{\eg }$i$ in \prettyref{item:WPO-simp}
\REV{and $j$ in \prettyref{item:WPO-args}}{} to positions
\st $f(\dots,x_i,\dots) \ANGT x_i$, since otherwise we have $s \AGT t$,
which is \REV{covered by}{considered in} \prettyref{item:WPO-gt}.
Cases \prettyref{item:WPO-prec} and \prettyref{item:WPO-mono} are common among
$\WPO$, $\GKBO$ and $\LPO$.
\REV
In the appendix we prove the following soundness result:
\begin{theorem}\label{thm:WPO simple}
If $\A$ is weakly monotone and weakly simple, then
$\GT_\WPO$ is a simplification order and hence a reduction order.
\end{theorem}
}{
Note that \prettyref{thm:WPO simple} gives an alternative proof for
the following result of Zantema \cite{Z01}:
\begin{theorem}\cite{Z01}
If a TRS $\RR$ is oriented by $\AGT$ for
a weakly monotone and weakly simple algebra $\A$,
then $\RR$ is simply terminating,
\REV{\ie,}{\ie} its termination is shown by a simplification order.
\end{theorem}
\begin{proof}
Since ${\AGT} \subseteq {\GT_\WPO}$, $\RR$ is oriented by
the simplification order $\GT_\WPO$.
\qedhere
\end{proof}
Moreover,
we can verify that $\WPO$ is a \REV{}{further }generalization of $\GKBO$.
\begin{theorem}\label{thm:WPO>=GKBO}
If $\A$ is strictly simple, then
${\GT_\GKBO} = {\GT_\WPO}$.
\end{theorem}
\begin{proof}
The condition $s_i \GE_\WPO t$ of case \prettyref{item:WPO-simp}
\REV{may be dropped}{is ignorable},
since in that case we have $s \AGT s_i \AGS t$ by the assumption.
Analogously,
the condition $s \GT_\WPO t_j$ of case \prettyref{item:WPO-args} always holds,
since $s \AGS t \AGT t_j$.
Hence, case \prettyref{item:WPO-simp} and the condition in
\prettyref{item:WPO-args} can be ignored, and
the definition of $\WPO$ becomes equivalent to that of $\GKBO$.\qedhere
\end{proof}
\REV
The relaxation of strict simplicity to weak simplicity is an important step.
While GKBO does not even subsume the standard KBO,
WPO subsumes not only KBO but also many other reduction orders.
}{
In the remainder of this section,
we investigate several instances of $\WPO$.
\subsection{WPO$(\Asum)$}
The first instance $\WPOsum$ is induced by an algebra $\Asum$, which
interprets function symbols as the summation operator $\sum$.
We obtain KBO as a restricted case of $\WPOsum$.
We design the algebra $\Asum$ from a weight function $\Tp{\Weight,\Wzero}$,
so that ${\GT_\WPOsum} = {\GT_\KBO}$
when $\Wzero > 0$ and \REV{}{the }admissibility is satisfied.
\begin{definition}
The $\Sig$-algebra $\Asum$ induced by a weight function $\Tp{\Weight,\Wzero}$
consists of the carrier set $\{ a \in \Nat \mid a \ge \Wzero \}$ and
the interpretation which is defined as follows:
\[
f_\Asum(\Seq{a_n}) = \WeightOf{f} + \sum_{i=1}^{n}a_i
\]
If $\Wzero > 0$ is satisfied, we also write $\AsumP$ for $\Asum$.
\end{definition}
Obviously, $\Asum$ is strictly (and hence weakly) monotone and weakly simple.
We obtain the following as a corollary of \prettyref{thm:WPO simple}:
\begin{corollary}\label{cor:WPOsum}
$\GT_\WPOsum$ is a reduction order.\qed
\end{corollary}
Now let us prove that $\GT_\KBO$ is obtained as a special case of $\GT_\WPOsumP$.
The following lemma verifies that $\Asum$ indeed works as the weight of KBO.
\begin{lemma}\label{lem:weight}
$s \geopt_\Asum t$ iff
$|s|_x \ge |t|_x$ for all $x \in \Vars$ and
$\WeightOf{s} \geopt \WeightOf{t}$.
\end{lemma}
\begin{proof}
The ``if'' direction is easy. For the ``only-if'' direction,
suppose $s \geopt_\Asum t$.
Define the assignment $\alpha_0$ which maps all variables to $\Wzero$.
We have $\widehat\alpha_0(s) \geopt \widehat\alpha_0(t)$, that is
$\WeightOf{s} \geopt \WeightOf{t}$.
Furthermore, define the assignment $\alpha_x$ which maps
$x$ to $\WeightOf{s} + \Wzero$ and \REV{other variables}{others} to $\Wzero$.
We have $\widehat\alpha_x(s) \geopt \widehat\alpha_x(t)$, which implies
$\WeightOf{s} + |s|_x \cdot \WeightOf{s} \geopt \WeightOf{t} + |t|_x \cdot \WeightOf{s}$.
\REV
Here, $|s|_x < |t|_x$ cannot hold since $\WeightOf{t} \ge 0$.
We conclude
}
Hence, we get
$|s|_x \ge |t|_x$.
\qedhere
\end{proof}
\begin{theorem}\label{thm:WPO>=KBO}
If $\Wzero > 0$ and $\Weight$ is admissible for $\PGS$, then
${\GT_\WPOsum} = {\GT_\KBO}$.
\end{theorem}
\begin{proof}
For arbitrary terms $s = f(\Seq{s_n})$ and $t$,
we show $s \GT_\WPOsum t$ iff $s \GT_\KBO t$ by induction on $|s| + |t|$.
\REV
Because of the admissibility assumption,
we may assume that $\GT_\KBO$ is a simplification order.
}{
\begin{itemize}
\item
Suppose $s \GT_\KBO t$.
If $\WeightOf{s} > \WeightOf{t}$, then we have
$s >_\Asum t$ by \prettyref{lem:weight} and
$s \GT_\WPOsum t$ by \prettyref{item:WPO-gt} of \prettyref{def:WPO}.
Let us consider that
$\WeightOf{s} = \WeightOf{t}$.
\begin{itemize}
\item
Suppose $s = f^k(t)$ and $t \in \Vars$ for some $k > 0$.
Since $\WeightOf{s} = \WeightOf{t}$, $\WeightOf{f} = 0$.
If $k = 1$, then we are done by case \prettyref{item:WPO-simp}.
Otherwise $f^{k-1}(t) \GT_\KBO t$ by case
\prettyref{item:KBO-simp} of \prettyref{def:KBO}.
By the induction hypothesis we get $f^{k-1}(t) \GT_\WPOsum t$, and hence
case \prettyref{item:WPO-simp} of \prettyref{def:WPO} applies.
\item
Suppose $t = g(\Seq{t_m})$ and case \prettyref{item:KBO-prec} or
\prettyref{item:KBO-mono} applies.
For all $j \in \{ 1, \dots, m \}$,
we have $t \GT_\KBO t_j$ by the subterm property of $\GT_\KBO$,
and we get $s \GT_\KBO t_j$ by the transitivity.
By the induction hypothesis, $s \GT_\WPOsum t_j$.
Hence, the side condition in \prettyref{item:WPO-args} of
\prettyref{def:WPO} is satisfied, and subcase
\prettyref{item:WPO-prec} or \prettyref{item:WPO-mono} applies.
\end{itemize}
\item
Suppose $s \GT_\WPOsum t$.
If $s >_\Asum t$, then $\WeightOf{s} > \WeightOf{t}$ by \prettyref{lem:weight} and
$s \GT_\KBO t$ by \prettyref{item:KBO-gt} of \prettyref{def:KBO}.
Otherwise we get $\WeightOf{s} = \WeightOf{t}$ by \prettyref{lem:weight}.
\begin{itemize}
\item
Suppose $s_i \GE_\WPOsum t$ for some $i \in \{1,\dots,n\}$.
By the induction hypothesis, we have $s_i \GE_\KBO t$.
The subterm property of $\GT_\KBO$ ensures $s \GT_\KBO s_i$.
Hence by \REV{}{the }transitivity, we get $s \GT_\KBO t$.
\item
Suppose $t = g(\Seq{t_m})$.
If $f \PGT g$, then case \prettyref{item:KBO-prec} of \prettyref{def:KBO} applies.
If $f = g$ and $[\Seq{s_n}] \GT_\WPOsum^\Lex [\Seq{t_m}]$,
then by the induction hypothesis we get
$[\Seq{s_n}] \GT_\KBO^\Lex [\Seq{t_m}]$,
and hence case \prettyref{item:KBO-mono} applies.
\qedhere
\end{itemize}
\end{itemize}
\end{proof}
Note that we need neither admissibility nor $\Wzero > 0$ in \prettyref{cor:WPOsum}.
Let us see that removal of these conditions \REV{is}{are} indeed advantageous.
The following example illustrates that $\WPOsumP$ properly
\REV{extends}{enhances} KBO because \REV{}{the }admissibility is relaxed.
\begin{example}\label{ex:WPO>KBO+LPO}
Consider the following TRS $\RR_1$:
\[
\RR_1 \DefEq
\begin{EqSet}
\f(\g(x))&\to\g(\f(\f(x)))\\
\f(\h(x))&\to\h(\h(\f(x)))\\
\end{EqSet}
\]
The first rule
\REV
is oriented from right to left
}
cannot be oriented
by $\LPO$ in any precedence.
The second rule
\REV
is oriented from right to left by KBO, since
$\WeightOf{\h} > 0$ implies increase in weights and
$\WeightOf{\h} = 0$ implies $\h \PGS \f$ by the admissibility.
}
cannot be oriented by $\KBO$,
since it requires that $\f \PGT \h$ and $\WeightOf{\h} = 0$
which is not admissible.
On the other hand, $\WPOsumP$ with precedence $\f \PGT \g$, $\f \PGT \h$ and
$\WeightOf{\g} > \WeightOf{\f} = \WeightOf{\h} = 0$ orients all the rules.
Hence, $\RR_1$ is orientable by $\WPOsumP$, but not by $\KBO$ or $\LPO$.
Note that there is no need to consider a status for $\RR_1$,
since all symbols are unary.
\end{example}
Moreover, allowing $\Wzero = 0$ is also a proper enhancement.
\begin{example}\label{ex:WPOsum}
Consider the following TRS $\RR_2$:
\[
\RR_2 \DefEq
\begin{EqSet}
\f(\ca,\cb) &\to \f(\cb,\f(\cb,\ca))\\
\f(\ca,\f(\cb,x)) &\to \f(x,\f(\cb,\cb))
\end{EqSet}
\]
The first rule cannot be oriented by $\KBO$ or $\WPOsumP$, since
$\WeightOf{\cb} = 0$ is required.
The second rule is not orientable by $\LPO$
no matter \REV{how one chooses}{the choice of} $\sigma$.
On the other hand, $\WPOsum$ with
$\WeightOf{\ca} > \WeightOf{\cb} = \WeightOf{\f} = 0$,
$\ca \PGT \cb$ and $\sigma(f) = [1,2]$ orients \REV{}{the }both rules.
Hence, $\RR_2$ is orientable by $\WPOsum$ with $\Wzero = 0$,
but not by $\LPO$, $\KBO$, or $\WPOsumP$.
\end{example}
\subsection{WPO$(\Apol)$}
Next we consider generalizing $\WPOsum$ using monotone polynomial interpretations.
According to Zantema \cite[Proposition 4]{Z01},
every monotone interpretation on a totally ordered set is weakly simple.
Hence a monotone polynomial interpretation
$\Apol$ is weakly simple and we obtain the following:
\begin{corollary}
If $\Apol$ is strictly monotone, then
$\GT_\WPOpol$ is a reduction order.\qed
\end{corollary}
Trivially, POLO is subsumed by $\WPOpol$ as a reduction order.
More precisely, the following relation holds:
\begin{theorem}\label{thm:WPO>=POLO}
${>_\Apol} \subseteq {\GT_\WPOpol}$.\qed
\end{theorem}
In the remainder of this paper, we consider
\REV{an algebra $\Apol$ that}{$\Apol$}
consists of linear polynomial interpretations induced by
a weight function $\Tp{\Weight,\Wzero}$ and a subterm coefficient function $\Coef$,
which is defined as follows:
\[
f_\Apol(\Seq{a_n}) \DefEq
\WeightOf{f} + \sum_{i=1}^{n} \CoefOf{f,i} \cdot a_i
\]
Analogous to \prettyref{thm:WPO>=KBO}, we also obtain the following:
\begin{theorem}\label{thm:WPO>=TKBO}
If $\Wzero > 0$ and $\Weight$ is admissible for $\PGS$, then
${\GT_\WPOpol} = {\GT_\TKBO}$.\qed
\end{theorem}
Moreover, we can verify that $\WPOpol$ strictly enhances both POLO and TKBO.
\REV
More precisely, we show that both POLO and TKBO do not subsume even
WPO($\Asum$).
}
\begin{example}\label{ex:POLO+TKBO<WPO}
POLO cannot orient the first rule of $\RR_1$:
\[
l_1 = \f(\g(x)) \to \g(\f(\f(x))) = r_1
\]
since it is not $\omega$-terminating \cite{Z94}.
Suppose that $\RR_1$ is oriented by TKBO. For the first rule, we need
\[
\VCoefOf{x,l_1} = \CoefOf{\f,1}\cdot\CoefOf{\g,1} \ge
\CoefOf{\g,1}\cdot\CoefOf{\f,1}^2 = \VCoefOf{x,r_1}
\]
Hence $\CoefOf{\f,1} = 1$.
Moreover,
\begin{align*}
\WeightOf{l_1}
&= \WeightOf{\f} + \WeightOf{\g} + \CoefOf{\g,1} \cdot \Wzero
\\
&\ge
\WeightOf{\g} + \CoefOf{\g,1} \cdot (2 \cdot \WeightOf{\f} + \Wzero) =
\WeightOf{r_1}
\end{align*}
Hence $\WeightOf{\f} = 0$.
Analogously, for the second rule of $\RR_1$:
\[
l_2 = \f(\h(x)) \to \h(\h(\f(x))) = r_2
\]
we need
$\CoefOf{\h,1} = 1$ and $\WeightOf{\h} = 0$.
Hence $\WeightOf{l_2} = \WeightOf{r_2}$.
\REV
The admissibility imposes $\f \PSIM \h$,
and thus the rule is oriented only from right to left.
Note also that it is not possible to
orient one of the rules in $\RR_1$ by $>_\Apol$ and
the other by $\ge_\Apol$.
Thus, togather with the discussion in \prettyref{ex:WPO>KBO+LPO},
we conclude that
$\RR_1$ cannot be oriented by any lexicographic composition of
POLO, (T)KBO, and LPO.
}
By the admissibility,
$\f \PGT \h$ cannot hold and this rule
cannot be oriented by TKBO.
\end{example}
\subsection{WPO$(\Amax)$}
Note that $\Apol$ is strictly monotone.
$\WPO$ also admits \emph{weakly} monotone interpretations;
a typical example is $\max$
\footnote{
\REV
Note that weakly monotone polynomials with $0$ coefficients are not weakly simple,
and hence cannot be applied for WPO of this section.
We consider such polynomials in \prettyref{sec:pair}.
}{
}
Let us consider an instance of $\WPO$ using $\max$ for interpretation.
\begin{definition}
A \emph{subterm penalty function} $\Pen$ is a mapping \st
$\PenOf{f,i} \in \Nat$ is defined for each $f \in \Sig_n$ and
$i \in \{ 1, \dots, n \}$.
A weight function $\Tp{\Weight,\Wzero}$ and $\Pen$ induce
the $\Sig$-algebra $\Amax$, which
consists of the carrier set $\{ a \in \Nat \mid a \ge \Wzero \}$ and
interpretations given by:
\[
f_\Amax(\Seq{a_n}) \DefEq
\max
\Big(\WeightOf{f}, \max_{i=1}^n \big(\PenOf{f,i} + a_i\big)
\Big)
\]
\end{definition}
\begin{lemma}
$\Amax$ is weakly monotone and weakly simple.
\end{lemma}
\begin{proof}
Weak simplicity is obvious from the fact that $\max( \dots, a, \dots ) \ge a$.
For weak monotonicity, suppose $a > b$ and let us show
\[
a' = f_\Amax(\Seq{c_{k}}, a, \Seq{d_l})\ge
f_\Amax(\Seq{c_{k}}, b, \Seq{d_l}) = b'
\]
To this end, let $c = f_\Amax(\Seq{c_{k}}, 0, \Seq{d_l} )$.
If $c \ge \PenOf{f,k+1} + a$, then $a' = b' = c$.
Otherwise, we have $a' = \PenOf{f,k+1} + a$ and either
$a' > \PenOf{f,k+1} + b = b' > c$ or $a' > b' = c$.\qedhere
\end{proof}
Note that $\Amax$ can be considered as the \REV{1-dimensional}{dimension-1} variant of
\emph{arctic interpretations} \cite{KW09}.
The weak monotonicity of $\Amax$ is also shown there.
\begin{corollary}
$\GT_\WPOmax$ is a reduction order.\qed
\end{corollary}
Now we show that LPO is obtained as a restricted case of $\WPOmax$.
\begin{theorem}\label{thm:WPO>=LPO}
If $\Wzero = 0$, $\WeightOf{f} = 0$ and
$\PenOf{f,i} = 0$ for all $f \in \Sig_n$ and $i \in \{ 1, \dots, n \}$, then
${\GT_\LPO} = {\GT_\WPOmax}$.
\end{theorem}
\begin{proof}
From the assumptions, $s >_\Amax t$ never holds.
Hence, case \prettyref{item:WPO-gt} of \prettyref{def:WPO} can be ignored.
Moreover,
$s \ge_\Amax t$ is equivalent to $\Var(s) \supseteq \Var(t)$.
One can easily verify the latter holds whenever $s \GT_\LPO t$,
using the fact that $s \NGE_\LPO x$ for $x \notin \Var(s)$.
Hence, the condition of
\REV{case }{}\prettyref{item:WPO-ge} can be ignored and
\prettyref{def:LPO} and \prettyref{def:WPO} become equivalent.\qedhere
\end{proof}
The following example illustrates that $\WPOmax$ properly enhances $\LPO$.
\begin{example}
Consider the following TRS $\RR_3$:
\[
\RR_3 \DefEq
\begin{EqSet}
\f(x,y) &\to \g(x)\\
\f(\g(x),y)&\to \f(x,\g(x))\\
\f(x,\g(y)) &\to \f(y,y)
\end{EqSet}
\]
To orient the first two rules by $\LPO$,
we need $\f \PGT \g$ and $\sigma(\f) = [1,2]$.
$\LPO$ cannot orient the third rule by this precedence and status,
while $\PenOf{\g,1} > \PenOf{\f,1} = 0$ suffices for $\WPOmax$.
Since the last two rules are duplicating,
$\KBO$ or $\WPOsum$ cannot apply for $\RR_3$.
\end{example}
However, $\WPOmax$
\REV
covers neither $\WPOsum$ nor even KBO.
}
does not cover $\WPOsum$, \REV{and}{or} not even $\KBO$.
In the next section, we consider unifying $\WPOsum$ and $\WPOmax$
to cover both KBO and LPO.
\subsection{WPO$(\Amp)$ and WPO$(\Ams)$}
Now we consider unifying $\WPOmax$ and $\WPOpol$.
To this end, we introduce the \emph{weight status} to choose a polynomial or $\max$
for each function symbol.
\begin{definition}
A \emph{weight status function} is a mapping $\Wstatus$ which maps
each function symbol $f$
\REV
either to the symbol $\Pol$ or to the symbol $\Max$.
}
to either symbol $\Pol$ or $\Max$.
The $\Sig$-algebra $\Amp$ consists of the carrier set
$\{ a \in \Nat \mid a \ge \Wzero \}$ and the interpretation which is defined as follows:
\[
f_\Amp(\Seq{a_n}) \DefEq
\begin{Cases}
\WeightOf{f} + \displaystyle\sum_{i = 1}^{n} \CoefOf{f,i} \cdot a_i
&\text{if } \WstatusOf{f} = \Pol
\smallskip\\
\max
\Big( \WeightOf{f}, \displaystyle\max_{i = 1}^{n} \big(\PenOf{f,i} + a_i\big)
\Big)
&\text{if } \WstatusOf{f} = \Max
\end{Cases}
\]
We denote $\Amp$ by $\Ams$ if coefficients are at most $1$.
\end{definition}
\begin{corollary}
$\WPOmp$ is a reduction order.\qed
\end{corollary}
Trivially, $\WPOms$ \REV{subsumes}{encompasses} both $\WPOsum$ and $\WPOmax$.
Hence, we obtain the following more \REV{interesting}{influential} result:
\begin{theorem}\label{thm:WPO>=LPO+KBO}
$\WPOms$ \REV{subsumes}{encompasses} both LPO and KBO.\qed
\end{theorem}
As far as we know, this is the first reduction order
that \REV{unifies}{unify} LPO and KBO.
The following example illustrates that
$\WPOms$ is strictly stronger than the union of $\WPOsum$ and $\WPOmax$.
\begin{example}
Consider the following TRS $\RR_4$:
\[
\RR_4 \DefEq
\begin{EqSet}
\f(\f(x,y),z) &\to \f(x,\f(y,z))\\
\g(\f(\ca,x),\cb) &\to \g(\f(x,\cb),x)
\end{EqSet}
\]
If $\WstatusOf{\f} = \Max$, then the first rule requires $\PenOf{\f,2} = 0$.
Under this restriction the second rule cannot be oriented.
If $\WstatusOf{\f} = \Pol$, then the first rule is
oriented iff $\CoefOf{f,1} = \CoefOf{f,2} = 1$ and $\sigma(f) = [1,2]$.
On the other hand, the duplicating variable $x$ in the second rule requires
$\WstatusOf{\g} = \Max$.
Hence, $\RR_4$ is orientable by $\WPOms$ only if $\WstatusOf{\f} = \Pol$ and
$\WstatusOf{\g} = \Max$.
\end{example}
\REV
The following example
}
Let us close this section with an example that
suggests \REV{that }{}$\WPOms$
advances the state-of-the-art of automated termination proving.
\begin{example}
The most powerful termination provers including
\AProVEver{2013} and \TTTTver{1.11}
fail to prove termination of the following TRS $\RR_5$:
\[
\RR_5 \DefEq
\begin{EqSet}
\f(\g(\g(x,\ca),\g(\cb,y))) &\to \f(\g(\g(\h(x,x),\cb),\g(y,\ca)))
\\
\g(x,y) &\to x
\\
\h(x,\h(y,z)) &\to y
\end{EqSet}
\]
We show that $\WPOms$ with
$\WstatusOf{\g} = \Pol$, $\WstatusOf{\h} = \Max$,
$\WeightOf{\ca} > \WeightOf{\cb}$,
$\WeightOf{\h} = \PenOf{\h,1} = \PenOf{\h,2} = 0$
and $\sigma(\f) = \sigma(\g) = [1,2]$
orients all the rules.
For the first rule, applying case \prettyref{item:WPO-mono} twice
it yields orienting $\g(x,\ca) \GT_\WPOms \g(\h(x,x),\cb)$ where
case \prettyref{item:WPO-gt} applies.
The other rules are trivially oriented.
\end{example}
\REV
\newcommand\mat[1]{
\begin{pmatrix}
#1
\end{pmatrix}
In general,
matrix interpretations cannot be combined with WPO,
since a matrix interpretation is often not weakly simple.
Consider the following TRS from \cite{EWZ08}:
\[
\RR_6 \DefEq \left\{ \f(\f(x)) \to \f(\g(\f(x)))\right\}
\]
which is shown terminating by the following matrix interpretation $\Amat$ \st
\begin{align*}
\f_\Amat(\vec{x}) &= \mat{1&1\\0&0} \cdot \vec{x} + \mat{0\\1}&
\g_\Amat(\vec{x}) &= \mat{1&0\\0&0} \cdot \vec{x}
\end{align*}
However, $\g_\Amat$ is not weakly simple. For example,
\[
\g_\Amat(\mat{0\\1}) = \mat{1&0\\0&0} \cdot \mat{0\\1} = \mat{0\\0}
\ngeq \mat{0\\1}
\]
Hence to unify the matrix interpretation with WPO,
we have to further relax the weak simplicity condition.
This is achieved in the next section by extending WPO to a reduction pair.
}{
\section{WPO as a Reduction Pair}\label{sec:pair}
In this section, we extend WPO to a reduction pair.
First we introduce the basic definition and prove its soundness,
and then \REV{we }{}present two refinements.
\REV{Afterwards}{Then} we introduce some instances of WPO as reduction pairs
and investigate relationships with existing reduction pairs.
In particular, matrix interpretations are
also subsumed by WPO as a reduction pair.
\subsection{WPO with Partial Status}\label{sec:pair definition}
In the preliminary version of this paper \cite{YKS13},
we simply applied argument filtering to obtain the reduction pair
$\Tp{{\GS_\WPO^\pi},{\GT_\WPO^\pi}}$ from WPO.
In this paper, we fully revise this approach and directly define a
reduction pair by incorporating
\emph{partial statuses} \cite{YKS13b} into WPO.
A partial status is a generalization of status
that admits \emph{non-permutations}.
\begin{definition}
A \Def{partial status function} $\sigma$ is a mapping
that assigns \REV{to }{}each $n$-ary symbol $f$ a list $\ListOf{\Seq{i_m}}$
of (distinct) positions in $\{ 1, \dots, n \}$.
We also view $\sigma(f)$ as the set $\SetOf{\Seq{i_m}}$
for $\sigma(f) = \ListOf{\Seq{i_m}}$.
A well-founded algebra $\A$ is \Def{(weakly) simple}
\wrt $\sigma$ iff
$f_\A$ is (weakly) simple in \REV{its }{}$i$-th argument
for every $f \in \Sig$ and $i \in \sigma(f)$.
\end{definition}
Note that here $\sigma(f)$ need not be a permutation,
\REV{since}{and}
some positions may be ignored.
If every $\sigma(f)$ is a permutation,
then we say \REV{that }{}$\sigma$ is \Def{total}.
Conversely if $\sigma(f) = \List\Empty$ for every $f$, then
we call $\sigma$ the \Def{empty} status.
\begin{definition}[WPO with Partial Status]\label{def:WPOpS}
Let $\A$ be a well-founded algebra and $\sigma$ a partial status.
The pair $\Tp{\GS_\WPOAS,\GT_\WPOAS}$ of relations
is defined mutually recursively as follows:
$x \GS_\WPOAS x$, and
$s = f(\Seq{s_n}) \GSopt_\WPOAS t$ iff
\begin{enumerate}
\item\label{item:WPO-gt}
$s \AGT t$, or
\item\label{item:WPO-ge}
$s \AGS t$ and
\begin{enumerate}
\item\label{item:WPO-simp}
$\ForSome{i \in \sigma(f)} s_i \GS_\WPOAS t$, or
\item\label{item:WPO-args}
$t = g(\Seq{t_m})$,
$\ForAll{j \in \sigma(g)} s \GT_\WPOAS t_j$ and either
\begin{enumerate}
\item\label{item:WPO-prec}
$f\PGT g$ or
\item\label{item:WPO-mono}
$f\PSIM g$ and
\(
\AppPerm{\sigma(f)}{s}{n} \GSopt_\WPOAS^\Lex
\AppPerm{\sigma(g)}{t}{m}
\).
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{definition}
\REV
In the appendix we prove that
the pair $\Tp{\GS_\WPO,\GT_\WPO}$ is indeed a reduction pair.
}{
The effect of a partial status has similarity with that of
combining argument filtering and a standard total status.
Indeed, WPO with partial status subsumes
WPO with total status and
\REV{certain form of}{\emph{non-collapsing}} argument filtering.
\REV
An argument filter $\pi$ is said to be \Def{non-collapsing} iff
$\pi(f)$ is a list for every $f \in \Sig$.
}{
\begin{proposition}\label{prop:partial status >= argument filter}
Let $\pi$ be a non-collapsing argument filter\REV{.}{, \REV{\ie,}{\ie}
$\pi(f)$ is a list for every $f \in \Sig$.}
For every $\Sig^\pi$-algebra $\A$ and total status $\sigma$ on $\Sig^\pi$,
there exists an $\Sig$-algebra $\A'$ and a partial status $\sigma'$ on $\Sig$ \st
\[
\Tp{{\GS_\WPOAS^\pi},{\GT_\WPOAS^\pi}} =
\Tp{{\GS_{\WPO(\A',\sigma')}},{\GT_{\WPO(\A',\sigma')}}}
\]
\end{proposition}
\begin{proof}
Let us define the interpretation of each $f \in \Sig_n$ in $\A'$ by
$f_{\A'}(\Seq{x_n}) \DefEq f_\A(\Seq{x_{\pi(f)}})$.
Then obviously, $\pi(s) \gsopt_\A \pi(t)$ iff $s \gsopt_{\A'} t$.
Moreover, we define $\sigma'(f) $ by $\pi(f) \star \sigma(f)$,
where $\star$ is a left-associative operator defined by
\[
[a_1,\dots,a_n] \star [i_1,\dots,i_{n'}] \DefEq [a_{i_1},\dots,a_{i_{n'}}]
\]
Now we verify that
$s \GSopt_\WPOAS^\pi t$ implies $s \GSopt_{\WPO(\A',\sigma')} t$
by induction on $|s| + |t|$.
If $s \in \Vars$, then
$s = \pi(t) = t$ since $\pi$ is non-collapsing, and hence
$s \GS_{\WPO(\A',\sigma')} t$.
Suppose $s = f(\Seq{s_n})$.
We proceed \REV{by}{to} case
\REV{analysis on the}{splitting for} derivation of $\pi(s) \GSopt_\WPOAS \pi(t)$.
\begin{enumerate}
\item
Suppose $\pi(s) \AGT \pi(t)$.
We obviously have $s >_{\A'} t$ and hence $s \GT_{\WPO(\A',\sigma')} t$.
\item
Suppose $\pi(s) \AGS \pi(t)$.
We obviously have $s \gs_{\A'} t$.
\begin{enumerate}
\item
Suppose that $\pi(s) \GSopt_\WPOAS \pi(t)$ is derived by case \prettyref{item:WPO-simp}.
Then we have $s_i \GS_\WPOAS^\pi t$
for some $i \in [1,\dots,n] \star \pi(f)$.
By the induction hypothesis we have $s_i \GS_{\WPO(\A',\sigma')} t$,
and since $i \in \sigma'(f)$,
$s \GT_{\WPO(A',\sigma')} t$ by case \prettyref{item:WPO-simp}.
\item
Suppose that $\pi(s) \GSopt_\WPOAS \pi(t)$ is derived by case \prettyref{item:WPO-args}.
Then we have $t = g(\Seq{t_m})$ and $s \GT_\WPOAS^\pi t_j$
for every $j \in [1,\dots,m] \star \pi(g)$.
By the induction hypothesis we have $s \GT_{\WPO(\A',\sigma')} t_j$,
for all $j \in \sigma'(f)$.
If furthermore $f \PGT g$, then immediately $s \GT_{\WPO(\A',\sigma')} t$
by case \prettyref{item:WPO-prec}.
If case \prettyref{item:WPO-mono} applies,
then we obtain
\[
[\Seq{s_n}] \star \pi(f) \star \sigma(f) \GSopt_\WPOAS^{\pi\;\Lex}
[\Seq{t_m}] \star \pi(g) \star \sigma(g)
\]
By the induction hypothesis and definition of $\sigma'$ we obtain
\[
\AppPerm{\sigma'(f)}{s}{n} \GSopt_{\WPO(\A',\sigma')} \AppPerm{\sigma'(g)}{t}{m}
\tag*{\qed}
\]
\end{enumerate}
\end{enumerate}
\end{proof}
The advantage of partial status over argument filtering
is due to the \emph{weights} of ignored arguments.
This is illustrated by the following example.
\begin{example}\label{ex:predecessor}\cite{YKS13b}
Consider
a DP problem that induces the following constrains:
\begin{align*}
\F(\s(x)) &\GT \F(\p(\s(x)))&
\p(\s(x)) &\GS x
\end{align*}
In order to
\REV{satisfy the first constraint}{strictly orient the rule in $\PP$}
by any simplification order,
the argument of $\p$ must be filtered.
However,
\REV{the second constraint cannot be satisfied}{the rule in $\RR$ cannot be weakly oriented}
under such an argument filtering.
On the other hand, the DP problem can be shown finite
using $\WPO$ with partial status \st
$\s_\A(x) > x$, $\F_\A(x) = \p_\A(x) = x$,
$\sigma(\F) = \sigma(\s) = [1]$, $\sigma(\p) = \List\Empty$, and
$\s \PGT \p$.
We have
$\F(\s(x)) \GT_\WPO \F(\p(\s(x)))$ because of cases
\prettyref{item:WPO-mono} and \prettyref{item:WPO-prec}, and
$\p(\s(x)) \GS_\WPO x$ because of case \prettyref{item:WPO-gt}.
\end{example}
\subsection{Refinements}\label{sec:pair refinements}
As we will see in \prettyref{sec:experiments pair},
the reduction pair processor induced by \prettyref{def:WPOpS} is
already powerful in practice.
However in theory,
\REV{the }{}WPO reduction pair is not a proper extension of the
underlying interpretation, \eg,
$x \GS_\A g(x)$ if $g_\A(x) = x$,
but $x \GS_\WPO g(x)$ cannot hold.
Hence we refine the definition of $\GS_\WPO$ to properly subsume
the underlying interpretation.
\REV
Note that in the above case,
assuming $x \GS_\WPO g(x)$ does not cause a problem
if $g$ is \emph{least} (\ie, $f \PGS g$ for every $f \in \Sig$) and
$\sigma(g) = \List\Empty$.
}{
\begin{proposition}\label{prop:least}
Let $g \in \Sig$ \st $f \PGS g$ for every $f \in \Sig$ and $\sigma(g) = \List\Empty$.
Then $x \AGS t = g(\Seq{t_m})$ implies
$s \GS_\WPO t\Subst{x \TO s}$ for arbitrary non-variable
\REV{terms }{}$s = f(\Seq{s_n})$.
\end{proposition}
\begin{proof}
By the definition of $\AGS$, we have $s \AGS t\Subst{x \TO s}$.
Since $g$ is \REV{least}{minimal} \wrt $\PGS$, we have $f \PGS g$.
Moreover, since $\sigma(g) = \List\Empty$, we have
$\AppPerm{\sigma(f)}{s}{n} \GS_\WPO^\Lex \AppPerm{\sigma(g)}{t}{m} = \List\Empty$.
\qedhere
\end{proof}
\prettyref{prop:least} suggests a refined definition of
$s \GS_\WPO t$ by adding
the following subcase in case \prettyref{item:WPO-ge} of \prettyref{def:WPOpS}
(note that $s \AGS t$ is ensured in this case):
\begin{enumerate}
\setcounter{enumi}{2}
\item[]
\begin{enumerate}[(2a)]
\setcounter{enumii}{2}
\item\label{item:WPO-min}
$s \in \Vars$ and $t = g(\Seq{t_m})$ \st
$\sigma(g) = \List\Empty$ and $g$ is \REV{least}{minimal} \wrt $\PGS$.
\end{enumerate}
\end{enumerate}
Similar refinements are proposed for KBO
\REV{\cite{KV03,ST13}}{\cite{KV03}} and for
RPO \cite{TAN12}, when $t$ is a \REV{least}{minimal} constant.
Our version is more general since $t$ need not be a constant.
\begin{example}\cite{YKS13b}
Consider a DP problem that induces the following constraints:
\begin{align*}
\F(\s(x),y) &\GT \F(\p(\s(x)),\p(y))
\\
\F(x,\s(y)) &\GS \F(\p(x),\p(\s(y)))
\\
\p(\s(x)) &\GS x
\end{align*}
Let
$\sigma(\p) = \List\Empty$, $\sigma(\F) = [1]$,
$\s_\A(x) = x + 1$, $\p_\A(x) = x$, $\F_\A(x,y) = x$ and
$\p$ be \REV{least}{minimal} \wrt $\PGS$.
The first constraint is strictly oriented by case \prettyref{item:WPO-prec}.
For the second constraint, it yields $x \GS_\WPO \p(x)$,
for which case (2c) applies.
Note that the argument of $\p$ cannot be filtered by an argument filter,
because of the third constraint.
Hence the refinements of \REV{\cite{KV03,ST13}}{\cite{KV03}} or \cite{TAN12} do not work for this example.
\end{example}
Moreover, a further refinement is also possible for WPO,
when the right\REV{-}{ }hand side is a variable.
\REV
Note that $f(x) \AGS x$ does not imply $f(x) \GS_\WPO x$ if $\sigma(f) = \List\Empty$.
Nonetheless,
$f(x) \GS_\WPO x$ can be assumed if
$f$ is greatest, and
moreover $f \PSIM g$ implies $\sigma(g) = \List\Empty$.
The latter condition is crucial,
since $g(x) \GT_\WPO f(g(x))$ if $f \PSIM g$ and $g = [1]$.
}{
\begin{proposition}\label{prop:greatest}
Suppose that $\A$ is strictly simple \wrt $\sigma$, and
$f \in \Sig$ \st
either $f \PGT g$\REV{,}{} or $f \PSIM g$ and $\sigma(g) = \List\Empty$ for every $g \in \Sig$.
Then $s = f(\Seq{s_n}) \AGS y$ implies
$s\Subst{y \TO t} \GS_\WPO t$ for arbitrary non-variable
\REV{term }{}$t = g(\Seq{t_m})$.
\end{proposition}
\begin{proof}
By the definition of $\AGS$, we have $s\Subst{y \TO t} \AGS t$.
Moreover by the strict simplicity,
$s\Subst{y \TO t} \AGS t \AGT t_j$ for all $j \in \sigma(g)$.
Hence we get $s \GT_\WPO t_j$.
If $f \PGT g$, then $s\Subst{y \TO t} \GT_\WPO t$ by case \prettyref{item:WPO-prec}.
If $f \PSIM g$ and $\sigma(g) = \List\Empty$, then
$s \Subst{y \TO t} \GS_\WPO t$ by case \prettyref{item:WPO-mono}.
\qedhere
\end{proof}
Provided $\A$ is strictly simple \wrt $\sigma$,
\prettyref{prop:greatest} suggests a refinement of $s \GS_\WPO t$
by adding the following subcase in case \prettyref{item:WPO-ge}:
\begin{enumerate}
\setcounter{enumi}{2}
\item[]
\begin{enumerate}[(2a)]
\setcounter{enumii}{3}
\item\label{item:WPO-max}
$s = f(\Seq{s_n})$ and $t \in \Vars$ \st for every $g \in \Sig$,
either $f \PGT g$ or $f \PGS g$ and $\sigma(g) = \List\Empty$.
\end{enumerate}
\end{enumerate}
Note that an arbitrary algebra is strictly simple \wrt the empty status.
Hence
\REV
WPO with the refinements can subsume even matrix interpretations. We
}
we
obtain the following result:
\begin{theorem}\label{thm:WPO refine}
\REV
Consider an instance of WPO that is induced by
\begin{itemize}
\item
a well-founded algebra $\A$ that is \emph{non-trivial},
\ie, there exist $a,b \in A$ \st $a \ngs b$,
\item
the empty status function $\sigma$, and
\item
the quasi-precedence $\PGS$ \st
$f \PGS g$ for arbitrary $f, g \in \Sig$.
\end{itemize}\noindent
}
Let $\A$ be a non-trivial well-founded algebra,
$\sigma$ the empty status function and $\PGS$ the quasi\REV{-}{ }precedence
$\Sig^2$.
Then, $\Tp{{\AGS},{\AGT}} = \Tp{{\GS_\WPO},{\GT_\WPO}}$ after the refinements.
\end{theorem}
\begin{proof}
From the definition, it is obvious that ${\AGT} \subseteq {\GT_\WPO}$ and
${\GS_\WPO} \subseteq {\AGS}$.
By the assumptions,
cases \prettyref{item:WPO-prec} and \prettyref{item:WPO-mono} of \prettyref{def:WPOpS}
cannot apply for $\GT_\WPO$. Hence we easily obtain ${\GT_\WPO} \subseteq {\AGT}$.
Now suppose $s \gs_\A t$ and let us show $s \GS_\WPO t$.
The proof proceeds \REV{by}{to} case \REV{analysis on}{splitting of}
the structure of $s$ and $t$.
\begin{itemize}
\item
If $s, t \in \Vars$, then from non-triviality
we have $s = t$, and hence $s \GS_\WPO t$.
\item
Suppose $s = f(\Seq{s_n})$ and $t = f(\Seq{t_m})$.
Since $f \PGT g$ never hold\REV{s}{},
case \prettyref{item:WPO-prec} of \prettyref{def:WPOpS} can be ignored.
Moreover, since $f \PGS g$ and $\List\Empty \GS_\WPO^\Lex \List\Empty$,
$s \AGS t$ implies $s \GS_\WPO t$.
\item
If either $s$ or $t$ is a variable, then
\REV{}{either }refinement \prettyref{item:WPO-min} or \prettyref{item:WPO-max}
is satisfied. Hence $s \GS_\WPO t$.
\qedhere
\end{itemize}
\end{proof}
\REV
In the appendix, we prove soundness of WPO with the refinements
\prettyref{item:WPO-min} and \prettyref{item:WPO-max}.
\begin{theorem}[Soundness]\label{thm:WPO pair}
If $\A$ is weakly monotone and
weakly simple \wrt $\sigma$, then
$\Tp{\GS_\WPO,\GT_\WPO}$ forms a reduction pair.
\end{theorem}\unskip
}
\subsection{Comparison with Other Reduction Pairs}
\label{sec:pair instances}
In this section,
we investigate some relationships between
instances of WPO and existing reduction pairs.
In \prettyref{def:WPOpS}, it is obvious that
\REV{the }{}induced $\GT_\WPO$ is identical to that induced by \prettyref{def:WPO},
if we choose a \emph{total} status $\sigma$.
Hence
Theorems \ref{thm:WPO>=KBO}, \ref{thm:WPO>=TKBO} and \ref{thm:WPO>=LPO}
imply that WPO \REV{subsumes}{encompasses} KBO, TKBO and LPO \resp also as a reduction pair.
On the other hand,
\prettyref{thm:WPO>=POLO} does not imply
that $\WPOpol$ subsumes POLO as a reduction pair,
since the ``weak-part'' $\ge_\Apol$
is not considered in the theorem.
Nonetheless, after the refinements in \prettyref{sec:pair refinements},
we obtain the following result from \prettyref{thm:WPO refine}:
\begin{corollary}\label{cor:WPO>=POLO pair}
\REV{}{As a reduction pair,
$\WPOpol$
\REV
with the refinements \prettyref{item:WPO-min} and \prettyref{item:WPO-max}
subsumes
}
encompasses
POLO.\qed
\end{corollary}
It is now easy to obtain the following result:
\begin{corollary}
$\WPOmp$
\REV
with the refinements \prettyref{item:WPO-min} and \prettyref{item:WPO-max}
subsumes
}
encompasses
POLO, KBO, TKBO and LPO.\qed
\end{corollary}
Moreover, WPO also subsumes
the \emph{matrix interpretation method} \cite{EWZ08}, when
weights are computed by a matrix interpretation $\Amat$.
Note that a matrix interpretation is not always weakly simple.
Hence as a \emph{reduction order},
\prettyref{def:WPO} cannot be applied for $\Amat$ in general.
The situation is relaxed for reduction pairs, and
from \prettyref{thm:WPO refine} we obtain the following:
\begin{corollary}\label{cor:WPO>=MAT pair}
\REV
}
The reduction pair induced by
$\WPOmat$ \REV{with the refinements \prettyref{item:WPO-min} and \prettyref{item:WPO-max} subsumes}{encompasses}
the reduction pair induced by the matrix interpretation $\Amat$.\qed
\end{corollary}
Finally, we compare $\WPOmp$ and
\emph{RPOLO} of Bofill \etal \cite{BBRR13},
another approach of unifying LPO and POLO.
It turns out that RPOLO is incomparable with $\WPOmp$.
First we verify that $\WPOmp$ is not subsumed by $\RPOLO$;
more precisely, RPOLO does not subsume KBO.
\begin{example}\label{ex:WPO-RPOLO}
Let us show that the constraint
$\f(\g(x)) \GT \g(\f(\f(x)))$ cannot be
satisfied by RPOLO
\footnote{
To simplify the discussion, we do not consider \REV{the }{}possibility for
\emph{argument filterings} or
\emph{usable rules} \cite{AG00} in the following examples.
\REV
It is easy to exclude these techniques;
for example, by adding the constraint $\g(x) \GS x$ we can enforce
the argument of $\g$ not to be filtered.
}
Nonetheless, it is easy to exclude these techniques by adding rules \eg $\g(x) \to x$.
}
Note that this constraint is satisfied
by KBO with $\WeightOf{\f}=0$ and $\f \PGT \g$.
\begin{itemize}
\item
Suppose $f \in \Sig_\POLO$. Since this
constraint cannot be satisfied by POLO,
$\g$ must be in $\Sig_\RPO$.
Hence we need
$\f_\Apol(v_{\g(x)}) >_{C(\f(\g(x)))} v_{\g(\f(\f(x)))}$.
This requires either
\begin{itemize}
\item $\f_\Apol(x) = x$ and $\g(x) \GT_\RPOLO \g(\f(\f(x)))$, or
\item $\f_\Apol(x) > x$ and $\g(x) \GE_\RPOLO \g(\f(\f(x)))$.
\end{itemize}
In either case, we obtain $\g(x) \GT_\RPOLO \g(x)$,
which is a contradiction.
\item
Suppose $\f \in \Sig_\RPO$. Since this
constraint cannot be satisfied by RPO,
$\g$ must be in $\Sig_\POLO$.
Hence we need
\begin{itemize}
\item $\g(x) \GE_\RPOLO \g(\f(\f(x)))$, or
\item $\f(\g(x)) \GT_\RPOLO \f(\f(x))$.
\end{itemize}
The first case contradicts \REV{}{with }$\f(x) \GT_\RPOLO x$.
The second case contradicts \REV{}{with }the fact that $\f(x) \GT_\RPOLO \g(x)$.
\end{itemize}
\end{example}
On the other hand, $\RPOLO$ is also not subsumed by $\WPOmp$,
as the following example illustrates.
\begin{example}\label{ex:RPOLO-WPO}
Consider a DP problem that induces the following constraints:
\begin{align}
\label{eq:g}
\F(\ci(x,\ci(y,\g(z)))) &\GT \F(\ci(y,\ci(z,x))))
\\\label{eq:Zantema}
\f(\g(\h(x))) &\GS \g(\f(\h(\g(x))))
\\\label{eq:i}
\ci(y,\ci(z,x)) &\GS \ci(x,\ci(y,z))
\end{align}
where constraint \eqref{eq:Zantema} is from \cite[Proposition 10]{Z94}.
\begin{itemize}
\item
First, let us show that
the set of constraints cannot be satisfied
by $\WPOmp$. Since $\f$, $\g$ and $\h$ are unary,
we only consider $\Wstatus(\f) = \Wstatus(\g) = \Wstatus(\h) = \Pol$.
It is easy to adjust \cite[Proposition 10]{Z94} to show that
$\g_\Apol(x) = x$ whenever
\eqref{eq:Zantema} is satisfied.
Together with \eqref{eq:i}, we obtain
\[
\ci(y,\ci(z,x)) \ge_\Amp \ci(x,\ci(y,z)) =_\Amp \ci(x,\ci(y,\g(z)))
\]
Hence case \prettyref{item:WPO-gt} of
\prettyref{def:WPOpS} cannot be applied
for constraint \eqref{eq:g}.
Moreover, by any choice of $\sigma(\ci)$,
case \prettyref{item:WPO-mono} cannot apply, either.
\item
Second, let us show that the
set of constraints can be satisfied by RPOLO.
Consider $\f,\g,\h \in \Sig_\RPO$, $\ci,\F \in \Sig_\POLO$,
$\f \PGT \g \PGT \h$,
$\ci_\Apol(x,y) = x + y$ and $\F_\Apol(x) = x$.
Then
constraint \eqref{eq:Zantema} is strictly oriented
and \eqref{eq:i} is weakly \REV{oriented}{so}.
Since $\g(z) \GT_\RPOLO z$ and $v_{\g(z)} > z$ implies $x+y+v_{\g(z)} > y + z + x$,
constraint \eqref{eq:g} is also satisfied.
\qedhere
\end{itemize}
\end{example}
\section{SMT Encodings}\label{sec:encodings}
In the preceding sections, we have concentrated on theoretical aspects.
In this section, we consider how to implement the instances of $\WPO$ using
SMT solvers.
We extend the corresponding approach for KBO \cite{ZHM09} to WPO.
In particular, $\WPOsum$, $\WPOmax$ and $\WPOms$ are reduced to
SMT problems of linear arithmetic, and as a consequence,
decidability is ensured for orientability problems of these orders.
An \emph{expression} $e$ is
built from (non-negative integer) variables, constants and
the binary symbols $\cdot$ and $+$ denoting multiplication and addition, \resp.
A \emph{formula} is built from
atoms of the form $e_1 > e_2$ and $e_1 \ge e_2$,
negation $\Not$,
and the binary symbols $\And$, $\Or$ and $\Then$ denoting
conjunction, disjunction and implication, \resp.
The precedence of these symbols \REV{is}{are} in the order we listed above.
The main interest of the SMT encoding approach is to employ SMT solvers for
finding a concrete algebra that proves finiteness of a given DP problem
(or termination of a given TRS).
Hence we assume that algebras are
parameterized by a set of expression variables.
\begin{definition}
An algebra $\A$ \Def{parameterized} by a set $V$ of variables
is a mapping that induces a concrete algebra $\A^\alpha$ from an assignment $\alpha$
whose domain contains $V$.
An \Def{encoding} of the relation $\gsopt_\A$ is a function that assigns
for two terms $s$ and $t$ a formula
$\Encode{s \gsopt_\A t}$ over variables from $V$ \st
$\alpha \models \Encode{s \gsopt_\A t}$ iff $s \gsopt_{\A^\alpha} t$.
\end{definition}
In the encodings presented in the rest of this section,
we consider $\A$ \REV{to be}{is} parameterized by at most the following variables:
\begin{itemize}
\item
integer variables $\WeiVarOf{f}$ and $\WeiVarZero$ denoting $\WeightOf{f}$ and $\Wzero$, \resp, and
\item
integer variables $\CoefVarOf{f,i}$ and $\PenVarOf{f,i}$
denoting $\CoefOf{f,i}$ and $\PenOf{f,i}$, \resp.
\end{itemize}
\subsection{The Common Structure}\label{sec:encode WPO}
\def\ST{\mathsf{ST}}
\def\SIMP{\mathsf{SIMP}}
To optimize the presentation,
we present an encoding of the common structure of $\WPO$
independent from the shape of $\A$.
Hence, we assume encodings for
$\AGT$ and $\AGS$ are given.
\REV{Following \cite{ZHM09}, first}{First}
we represent a \REV{quasi-}{}precedence $\PGS$ by integer variables $\PrecVarOf{f}$.
For an assignment $\alpha$,
we define the \REV{quasi-}{}precedence $\PGS^\alpha$ as follows:
$f \PGS^\alpha g$ iff $\alpha \models \PrecVarOf{f} \ge \PrecVarOf{g}$.
Next we consider representing \REV{a partial status
by imitating the encoding of a \emph{filtered permutation}
proposed in \cite{CGST12}.
We introduce
}{
statuses by
the boolean variables
$\PermedVarOf{f,i}$ and $\PermVarOf{f,i,j}$,
so that an assignment $\alpha$ induces a status $\sigma^\alpha$ as follows:
$\alpha \models \PermedVarOf{f,i}$ iff $i \in \sigma^\alpha(f)$, and
$\alpha \models \PermVarOf{f,i,j}$ iff $i$ is the $j$-th element in $\sigma^\alpha(f)$.
\REV
In order for $\sigma^\alpha$ to be well-defined,
every $i \in \sigma^\alpha(f)$ must occur exactly once in $\sigma^\alpha(f)$
and any $i \notin \sigma^\alpha(f)$ must not occur in $\sigma^\alpha(f)$.
These conditions are represented by
}
In order to ensure $\sigma^\alpha$ to be well-defined,
we introduce
the following formula:
\[
\ST \DefEq
\BigAnd_{f\in\Sig_n}\BigAnd_{i=1}^n
\Bigl(
\Bigl( \PermedVarOf{f,i} \Then \sum_{j=1}^n \PermVarOf{f,i,j} = 1 \Bigr)\And
\Bigl( \Not\PermedVarOf{f,i} \Then \sum_{j=1}^n \PermVarOf{f,i,j} = 0 \Bigr)
\Bigr)
\]
It is easy to verify that $\sigma^\alpha$ is well-defined
if $\alpha \models \ST$.
\REV
In contrast to the previous works \cite{ZHM09,CGST12}, we moreover
}
Moreover we
need the following formula to ensure
\REV{}{the }weak simplicity of $\A$ \wrt $\sigma$:
\[
\SIMP \DefEq
\BigAnd_{f \in \Sig_n} \BigAnd_{i = 1}^{n}
\Bigl(\PermedVarOf{f,i} \Then \Encode{f(\Seq{x_n}) \AGS x_i}\Bigr)
\]
Note that in the formula $\SIMP$,
the condition $f(\Seq{x_n}) \AGS x_i$ can often be encoded in a more efficient way.
For linear $\Apol$, for example, this condition is equivalent to $\CoefOf{f,i} \ge 1$.
\begin{lemma}
$\alpha \models \ST \And \SIMP$ iff
$\A^\alpha$ is weakly simple \wrt a partial status $\sigma^\alpha$.\qed
\end{lemma}
Now we present the encodings for $\WPO$.
\begin{definition}
The encodings of $\GT_\WPO$ and $\GS_\WPO$ are defined as follows:
\[
\Encode{s \GSopt_\WPO t} \DefEq
\Encode{s \AGT t} \Or
\bigl( \Encode{s \AGS t} \And s \GSopt_1 t \bigr)
\]
where the formula $s \GSopt_1 t$ is defined as follows:
\begin{eqnarray*}
x \GS_1 t &\DefEq&
\begin{Cases}
\True &\text{if } x = t
\\
\False &\text{otherwise}
\end{Cases}
\\
x \GT_1 t &\DefEq& \False
\\
f(\Seq{s_n}) \GSopt_1 t &\DefEq&
\bigvee_{i=1}^{n}
\Bigl(\PermedVarOf{f,i} \And \Encode{s_i \GS_\WPO t} \Bigr) \Or f(\Seq{s_n}) \GSopt_2 t
\end{eqnarray*}
The formula $f(\Seq{s_n}) \GSopt_2 t$ is defined as follows:
\begin{eqnarray*}
f(\Seq{s_n}) \GSopt_2 y &\DefEq& \False
\\
f(\Seq{s_n}) \GSopt_2 g(\Seq{t_m}) &\DefEq&
\bigwedge_{j=1}^{m}
\Bigl(\PermedVarOf{g,j} \Then \Encode{s \GT_\WPO t_j} \Bigr)
\ \And\\
&&
\Bigl(
\PrecVarOf{f} > \PrecVarOf{g}
\Or
\PrecVarOf{f} = \PrecVarOf{g} \And
\Encode{\ListOf{\Seq{s_{\sigma(f)}}} \GSopt_\WPO^\Lex \ListOf{\Seq{t_{\sigma(g)}}}}
\Bigr)
\end{eqnarray*}
\end{definition}
Here\REV{,
$s \GSopt_1 t$ indicates that $s \GSopt_\WPO t$ is derived by
case \prettyref{item:WPO-simp} or \prettyref{item:WPO-args}, and
$s \GSopt_2 t$ indicates that $s \GSopt_\WPO t$ is derived by
case \prettyref{item:WPO-prec} or \prettyref{item:WPO-mono}.
We
}{ we
do not present the encoding for the lexicographic extension \wrt permutation,
which can be found in \cite{CGST12}.
\REV{\par}{
For an assignment $\alpha$, we write $\GSopt_\WPO^\alpha$ to denote
the instance of WPO corresponding to $\alpha$;
\ie,
$\GSopt_\WPO^\alpha$ is induced by the algebra $\A^\alpha$,
the \REV{quasi-}{}precedence $\PGS^\alpha$ and the partial status $\sigma^\alpha$.
\begin{lemma}\label{lem:encode}
For any assignment $\alpha$ \st
$\alpha \models \ST \And \SIMP$,
$\alpha \models \Encode{s \GSopt_\WPO t}$ iff
$s \GSopt_\WPO^\alpha t$.\qed
\end{lemma}
\begin{theorem}\label{thm:encode DP WPO}
If the following formula is satisfiable:
\begin{equation}\label{eq:encode DP}
\ST \And \SIMP \And
\bigwedge_{l \to r \in \RR \cup \PP}\Encode{l \GS_\WPO r}\ \And
\bigvee_{l \to r \in \PP'}\Encode{l \GT_\WPO r}
\end{equation}
then the DP processor that maps $\Tp{\PP,\RR}$ to $\{ \Tp{\PP\setminus\PP',\RR} \}$
is sound.
\end{theorem}
\begin{proof}
Let $\alpha$ be the assignment that satisfies \eqref{eq:encode DP}.
By \prettyref{lem:encode}, we obtain
$l \GS_\WPO^\alpha r$ for all $l \to r \in \RR \cup \PP$ and
$l \GT_\WPO^\alpha r$ for all $l \to r \in \PP'$.
Moreover by \prettyref{thm:WPO pair}, $\Tp{\GS_\WPO^\alpha, \GT_\WPO^\alpha}$
forms a reduction pair.
Hence \prettyref{thm:reduction pair} concludes the soundness of
this DP processor.
\qedhere
\end{proof}
In the following sections,
we give encodings depending on the choice of $\A$ for each instance of $\WPO$.
\subsection{Encoding WPO$(\Apol)$ and WPO$(\Asum)$}
First we present \REV{an encoding of a}{encodings for} linear polynomial interpretation $\Apol$.
The encodings for $\Asum$ is obtained by fixing \REV{all }{}coefficients to $1$.
The weight of a term $s$ and the variable coefficient of $x$ in $s$
are encoded as follows:
\begin{align*}
\WsumOf{s} &\DefEq
\begin{Cases}
\WeiVarZero &\text{ if } s \in \Vars \\
\WeiVarOf{f} +
\displaystyle\sum_{i = 1}^{n}\CoefVarOf{f,i} \cdot \WsumOf{s_i} &\text{ if } s = f(\Seq{s_n})
\end{Cases}
\\
\VCoefOf{x,s} &\DefEq
\begin{Cases}
1 &\text{if } x = s
\\
0 &\text{if } x \neq s \in \Vars
\\
\displaystyle\sum_{i=1}^{n}\CoefVarOf{f,i} \cdot \VCoefOf{x,s_i}
&\text{if } s = f(\Seq{s_n})
\end{Cases}
\end{align*}
\def\COEF{\mathsf{COEF}}
\def\WMIN{\mathsf{WMIN}}
\REV
We have to ensure $\WeiVarZero$ to be the lower bound of weights of terms.
To ensure $\WsumOf{f(\Seq{s_n})} \ge \WeiVarZero$
for every term $f(\Seq{s_n})$,
we need either $\WeiVarOf{f} \ge \WeiVarZero$ or
one of the arguments to have a positive coefficient
(note that the weight of this argument is at least $\WeiVarZero$).
This is represented by
}
In order to ensure $\Wzero$ to be the lower bound,
we introduce
the following constraint:
\[
\WMIN \DefEq \BigAnd_{f \in \Sig_n}
\Bigl( \WeiVarOf{f} \ge \WeiVarZero \Or \BigOr_{i=1}^n \CoefVarOf{f,i} \ge 1
\Bigr)
\]
Now the relations $>_\Apol$ and $\ge_\Apol$ are encoded as follows:
\[
\Encode{s \geopt_\Apol t} \DefEq
\WsumOf{s} \geopt \WsumOf{t} \And
\bigwedge_{x \in \Var(t)} \VCoefOf{x,s} \geq \VCoefOf{x,t}
\]
\begin{corollary}
If the following formula is satisfiable:
\[
\ST \And \SIMP \And \WMIN \And
\bigwedge_{l \to r \in \RR \cup \PP}\Encode{l \GS_\WPOpol r}\ \And
\bigvee_{l \to r \in \PP'}\Encode{l \GT_\WPOpol r}
\]
then the DP processor that maps $\Tp{\PP,\RR}$ to $\{ \Tp{\PP\setminus\PP',\RR} \}$
is sound.
\qed
\end{corollary}
\subsection{Encoding WPO$(\Amax)$}
In this section, we consider encoding $\WPOmax$.
Unfortunately, we are aware of no SMT solver which supports a built-in $\max$ operator.
Hence we consider encoding the constraint $s >_\Amax t$ into
both quantified and quantifier-free formulas.
First, we present an encoding to a quantified formula.
A straightforward encoding would involve
\[
\WmaxOf{s}\DefEq
\begin{Cases}
s &\text{if } s \in \Vars\\
v &\text{if } s = f(\Seq{s_n})
\end{Cases}
\]
where $v$ is a fresh integer variable
\REV
representing $\max\{ \WeiVarOf{f}, \WmaxOf{s_1},\dots,\WmaxOf{s_n} \}$
}
with the following constraint $\phi$ added into the context:
\begin{align*}
\phi \DefEq{}&
v \ge \WeiVarOf{f} \And \bigwedge_{i=1}^{n} v \ge \WmaxOf{s_i}
\And
\Big(
v = \WeiVarOf{f} \Or \bigvee_{i=1}^{n} v = \WmaxOf{s_i}
\Big)
\end{align*}
Then the constraint $s \geopt_\Amax t$ can be encoded as follows:
\[
\Encode{s \geopt_\Amax t}\DefEq
\ForAll{\Seq{x_k}, \Seq{v_m}}
\phi_1 \And \dots \And \phi_m \Then
\WmaxOf{s} \geopt \WmaxOf{t}
\]
where $\{ x_1, \dots, x_k \} = \Var(s) \cup \Var(t)$ and
each $\Tp{\phi_j,v_j}$ is
the pair of the constraint and the fresh variable
introduced during the encoding.
Although quantified linear integer arithmetic is known to be decidable,
the SMT solvers we have tested could not solve the problems
generated by the above straightforward encoding efficiently, if at all.
Fuhs \etal \cite{FGMSTZ08} propose\REV{}{s} a sound elimination of quantifiers
by introducing new template polynomials.
Here we propose another encoding to quantifier-free formulas
that \REV{does not introduce extra polynomials and
}{}is sound and complete for linear polynomials with max.
\begin{definition}
A \emph{generalized weight} \cite{KV03b} is a pair $\Tp{n,N}$
where $n \in \Nat$ and $N$ is a finite multise
\footnote{In the encoding for $\Amax$, $N$ need not contain more than one variable.
This generality is reserved for \REV{the }{}encoding of $\Amp$.}
over $\Vars$.
We define the following operations:
\begin{align*}
\Tp{n,N} + \Tp{m,M} &\DefEq \Tp{n + m, N \uplus M}
\\
n \cdot \Tp{m,M} &\DefEq \Tp{n \cdot m, n \cdot M}
\end{align*}
where $n \cdot M$ denotes the multiset that maps
$x$ to $n \cdot M(x)$ for every $x \in \Vars$.
We encode a generalized weight as a pair of an expression and
a mapping $N$ from $\Vars$ to expressions \st
the \emph{domain} $\Dom(N) \DefEq \{ x \mid N(x) \neq 0 \}$ of $N$ is finite.
Notations for generalized weights are naturally extended for encoded ones.
The relation $\supseteq$ on multisets is encoded as follows:
\[
N \supseteq M \DefEq \bigwedge_{x \in \Dom(M)} N(x) \ge M(x)
\]
\end{definition}
A generalized weight $\Tp{n,N}$ represents the expression
$n + \sum_{x\in N}x$.
Now we consider removing $\max$.
\begin{definition}\label{def:Wmax}
The \emph{expanded weight} $\XWof{s}$ of a term $s$ induced by
a weight function $\Tp{\Weight,\Wzero}$ and a subterm penalty function
$\Pen$ is a set of generalized weights, which is defined as follows:
\[
\XWof{s} \DefEq
\begin{Cases}
\{ \Tp{\WeiVarZero, \SetOf{s}} \} &\text{if } s \in \Vars
\\
\{ \Tp{\WeiVarOf{f}, \Set\Empty} \} \cup
\{ \PenVarOf{f,i} + p \mid p \in \XWof{s_i} \text{, } 1 \le i \le n \}
&\text{if } s = f(\Seq{s_n})
\end{Cases}
\]
\end{definition}
The expanded weight $\XWof{s} = \{ \Seq{p_n} \}$ represents
the expression $\max \{ \Seq{e_n} \}$,
where each generalized weight $p_i$ represents the expression $e_i$.
Using expanded weights, we can encode $>_\Amax$ and $\ge_\Amax$
in a way similar to the \emph{max set ordering} presented in \cite{BC08}:
\[
\Encode{s \geopt_\Amax t} \DefEq
\bigwedge_{\Tp{m,M} \in \XWof{t}}
\bigvee_{\Tp{n,N} \in \XWof{s}}
(n \geopt m \And N \supseteq M)
\]
Using the quantified or quantifier-free encodings,
we obtain the following corollary of \prettyref{thm:WPO pair}:
\begin{corollary}
If the following formula is satisfiable:
\[
\ST \And
\bigwedge_{l \to r \in \RR \cup \PP}\Encode{l \GS_\WPOmax r}\ \And
\bigvee_{l \to r \in \PP'}\Encode{l \GT_\WPOmax r}
\]
then the DP processor that maps $\Tp{\PP,\RR}$ to $\{ \Tp{\PP\setminus\PP',\RR} \}$
is sound.
\qed
\end{corollary}
\subsection{Encoding WPO$(\Amp)$ and WPO$(\Ams)$}
In this section, we consider encoding linear polynomials with max into SMT formulas.
First we extend \prettyref{def:Wmax} for weight statuses.
\begin{definition}
For a weight status $\Wstatus$,
the \emph{expanded weight} $\XWsmOf{s}$ of a term $s$ is the set of
generalized weight, which is recursively defined as follows:
\begin{align*}
\XWsmOf{s} &\DefEq
\begin{Cases}
\{ ( \WeiVarZero, \{ s \} ) \}&\text{if } s \in \Vars
\\
S&\text{if } s = f(\Seq{s_n}),\ \WstatusOf{f} = \Pol
\\
T&\text{if } s = f(\Seq{s_n}),\ \WstatusOf{f} = \Max
\end{Cases}
\end{align*}
where
\begin{align*}
S &= \Bigl\{ \WeiVarOf{f} + \sum_{i=1}^{n} \CoefVarOf{f,i} \cdot p_i\ \big|\
p_1 \in \XWsmOf{s_1}, \dots, p_n \in \XWsmOf{s_n} \Bigr\}
\\
T &= \{ \WeiVarOf{f} \} \cup
\{
\PenVarOf{f,i} + \CoefVarOf{f,i} \cdot p \mid p \in \XWsmOf{s_i} \text{, }
i \in \SetOf{1,\dots,n}
\}
\end{align*}
\end{definition}
Now the encoding of $>_\Amp$ and $\ge_\Amp$ are given as follows:
\[
\Encode{s \geopt_\Amp t} \DefEq
\bigwedge_{\Tp{m,M} \in \XWsmOf{t}}
\bigvee_{\Tp{n,N} \in \XWsmOf{s}}
\bigl( n \geopt m \And N \supseteq M \bigr)
\]
\begin{corollary}
If the following formula is satisfiable:
\[
\ST \And \SIMP \And \WMIN \And
\bigwedge_{l \to r \in \RR \cup \PP}\Encode{l \GS_\WPOmp r}\ \And
\bigvee_{l \to r \in \PP'}\Encode{l \GT_\WPOmp r}
\]
then the DP processor that maps $\Tp{\PP,\RR}$ to $\{ \Tp{\PP\setminus\PP',\RR} \}$
is sound.
\qed
\end{corollary}
\subsection{Encoding WPO$(\Amat)$}
We omit presenting an encoding of the matrix interpretation method,
which can be found in \cite{EWZ08}.
In order to use a matrix interpretation in WPO,
however, \REV{}{a }small care is needed; one \REV{has}{have} to ensure
weak simplicity of $\Amat$ \wrt $\sigma$.
This can be done as follows:
\begin{lemma}
If
$\CmatOf{f,i}^{j,j} \ge 1$ for all
$f \in \Sig_n$, $i \in \sigma(f)$ and
$j \in \SetOf{1, \dots, d}$,
then $\Amat$ is weakly simple \wrt $\sigma$.\qed
\end{lemma}
\subsection{Encoding for Reduction Orders}\label{sec:encode order}
\def\TOTAL{\mathsf{TOTAL}}
In case one wants an encoding for the reduction order $>_\WPO$ defined in
\prettyref{def:WPO}, then the status $\sigma$ must be total.
This can be ensured by
\REV
enforcing $i \in \sigma(f)$ for all $i \in \{ 1, \dots, n \}$ and $f \in \Sig$,
which is represented by
}
the following formula:
\[
\TOTAL \DefEq \BigAnd_{f\in\Sig_n}\BigAnd_{i=1}^{n}\PermedVarOf{f,i}
\]
or equivalently by replacing all $\PermedVarOf{f,i}$ by $\True$.
Note that $\TOTAL \And \SIMP$
enforces all the subterm coefficients to be greater than
\REV
or equal to
}{
$1$.
\begin{theorem}\label{thm:encode order}
If the following formula is satisfiable:
\[ \TOTAL \And \ST \And \SIMP \And \WMIN \And \BigAnd_{l \to r \in \RR} \Encode{l \GT_\WPOmp r}
\]
then $\RR$ is orientable by $\WPOmp$.\qed
\end{theorem}
\section{Optimizations}\label{sec:optimizations}
In our implementation, some
optimizations are performed during the encoding.
For example,
formulas like $\False \And \phi$ are reduced in advance
to avoid generating meaningless formulas, and
temporary variables are inserted to avoid multiple occurrences of
an expression or a formula.
Moreover, we apply several optimizations that we discuss below.
\subsection{Fixing $\Wzero$}\label{sec:fixing w0}
We can simplify the encoded formulas by fixing $\Wzero$.
For KBO, Winkler \etal \cite{WZM12} show\REV{}{s} that
$\Wzero$ can be fixed to \REV{an }{}arbitrary $k > 0$ \REV{(\eg, $1$)}{\eg $1$}
without \REV{losing any power}{loosing the power of the order}.
\REV
By adapting the proof of \cite[Lemma 3]{WZM12},
it can be shown that $\Wzero$ can be fixed to $0$ for $\WPOsum$.
}
Applying their technique,
it can be shown that for $\WPOsum$, $\Wzero$ can be fixed to $0$.
On the contrary to KBO, however,
\REV
fixing $\Wzero > 0$ will affect the power,
since the transformation of \cite{WZM12} may assign
negative weights to some symbols
when applied to the case $\Wzero = 0$.
}
$\Wzero$ cannot be fixed to $k > 0$
since transforming a weight function $\Tp{\Weight,0}$ into $\Tp{\Weight^k,k}$
may assign negative weights to some symbols.
\subsection{Fixing Weight Status}
For POLO and WPO using algebras $\Ams$ and $\Amp$,
it may not be practical to consider all possible weight statuses.
Hence, we introduce a heuristic for fixing $\Wstatus$.
In case of $\WPOms$,
$\Wstatus$ should at least satisfy the following condition
for all $l \to r \in \RR \cup \PP$:
\[
\BigAnd_{\Tp{m,M} \in \XWsmOf{r}}
\BigOr_{\Tp{n,N} \in \XWsmOf{l}}
N \supseteq M
\]
since otherwise the formula
$\BigAnd_{l \to r \in \RR} \Encode{l \GS_\WPOms r}$
is trivially unsatisfiable.
Hence in our implementation,
we \REV{require that}{consider} $\Ams$ and $\Amp$ are induced by
the weight status which minimizes
the number of $f$ with $\WstatusOf{f} = \Max$,
while satisfying the above condition.
\subsection{Reducing Recursive Checks}\label{sec:reduce recursion}
Encoding KBO as a reduction order \cite{ZHM09}
is notably efficient,
because KBO does not have \REV{}{a }recursive checks like LPO or WPO.
For WPO, we can reduce formulas for recursive checks
by restricting $w_0 > 0$, since under \REV{this}{the} restriction\REV{,}{}
$f(\Seq{s_n}) >_\Amp s_i$ \REV{holds }{}whenever $n \ge 2$ and $\WstatusOf{f} = \Pol$.
Hence if $n \ge 2$ and $\WstatusOf{f} = \Pol$,
we reduce the formula $f(\Seq{s_n}) \GSopt_1 t$ to $f(\Seq{s_n}) \GSopt_2 t$.
Analogously if $m \ge 2$ and $\WstatusOf{g} = \Pol$,
we reduce the formula $f(\Seq{s_n}) \GSopt_2 g(\Seq{t_m})$ to
the following:
\[
\PrecVarOf{f} > \PrecVarOf{g} \Or
\PrecVarOf{f} = \PrecVarOf{g} \And
\Encode{\AppPerm{\sigma(f)}{s}{n} \GSopt_\WPOAS^\Lex
\AppPerm{\sigma(g)}{t}{m}
}
\]
without generating formulas for recursive checks corresponding to
cases \prettyref{item:WPO-simp} and \prettyref{item:WPO-args}.
Note \REV{however that}{that however,} this simplification does not apply
when encoding reduction pairs using argument filtering,
as we will see in \prettyref{sec:experiments pair}.
\section{Experiments}\label{sec:experiments}
In this section we examine the performance of WPO
both as a reduction order and in the DP framework.
\REV
We implemented a simple form of the DP framework as the
\emph{Nagoya Termination Tool} (\NaTT)
and incorporated WPO as a DP processor \cite{YKS14b}.
The
}{We implemented the
encodings presented in \prettyref{sec:encodings}
and optimizations presented in \prettyref{sec:optimizations
\REV{ are implemented}{}.
In the encodings of $\Apol$ and $\Amp$,
we choose $3$ \REV{as upper bound}{for upper bounds} of weights and coefficients\REV{,
in order to achieve a practical runtime}{}.
For comparison, KBO, TKBO, LPO,
polynomial interpretations with or without max
and matrix interpretations are
implemented in the same manner.
For the DP framework,
we implemented
the estimation of \emph{dependency graphs} in \cite{GTS05},
and \emph{strongly connected components} are sequentially processed
in order of size where smaller ones \REV{come first}{are precedent}.
Moreover,
\emph{usable rules} \wrt argument filters are also implemented by
following the encoding proposed in \cite{CSLTG06}.
The test set of termination problems are
the 1463 TRSs from the TRS Standard category of TPDB 8.0.6 \cite{TPDB13}.
The experiments are run on a server equipped with
two quad-core Intel Xeon W5590 processors running at a clock rate of 3.33GHz
and 48GB of main memory, though only one thread of SMT solver runs at once.
As the SMT solver, we choose \Zthreevar{4.3.1}
\footnote{\url{http://z3.codeplex.com/}}
Timeout is set to 60s, as in the \emph{Termination Competition} \cite{TC13}.
Details of the experimental results are available at
\url{http://www.trs.cm.is.nagoya-u.ac.jp/papers/SCP2014/}.
\subsection{Results for Reduction Orders}
First we evaluated WPO as a reduction order
by directly testing orientability for input TRSs.
The results are listed in \prettyref{tab:order}.
Since KBO, POLO($\Asum$), WPO($\Asum$) are only applicable for
non-duplicating TRSs,
the test set is split into
non-duplicating ones (consisting of 439 TRSs) and
duplicating ones (consisting of 1024 TRSs).
In the table,
\REV{the }{}`yes' column indicates the number of successful termination proofs,
`T.O.' indicates the number of timeouts, and `time' indicates the total time.
\REV{
To emphasize the benefit of WPO,
we also compare it with arbitrary lexicographic compositions of
POLO, KBO, and LPO (`POLO+KBO+LPO' row).
}{
\begin{table}[tb]
\caption{Results for Reduction Orders\label{tab:order}}
\centering
\begin{tabular}{cc@{\quad}rc@{\ }rl@{\quad}rc@{\ }r}
\hline
&&\multicolumn{3}{c}{non-dup. TRSs}
&&\multicolumn{3}{c}{dup. TRSs}\\
\cline{3-5}\cline{7-9}
order&algebra &yes&{T.O.}&time&&yes&{T.O.}&time\\
\hline
POLO&$\Asum$ &41 &0 &4.45 &&-- &-- &-- \\
POLO&$\Ams$ &60 &0 &4.46 &&19 &0 & \\
LPO & &90 &0 &31.64 &&90 &0 &35.39\\
KBO & &115&0 &6.20 &&-- &-- &--\\
\hdashline
WPO &$\Asum^+$ &126&0 &6.70 &&-- &-- &-- \\
WPO &$\Asum$ &\bf 135&0 &43.16 &&-- &-- &-- \\
WPO &$\Amax$ &109&0 &53.77 &&125 &0 &49.49 \\
WPO &$\Ams$ &\bf 135&0 &42.72 &&\bf 138 &0 &66.15\\
\hline
POLO&$\Apol$ &104&3 &203.37 &&21 &10 &1065.34\\
POLO&$\Amp$ &104&3 &203.07 &&39 &8 &608.76 \\
TKBO& &132&3 &226.33 &&27 &12 &1414.62\\
\hdashline
WPO &$\Apol$ &\bf 149&3 &280.92 &&29 &12 &1495.55\\
WPO &$\Amp$ &\bf 149&3 &280.86 &&\bf 138 &9 &1008.67\\
\hline
\multicolumn{2}{c}{\REV{POLO+KBO+LPO}{}}
&\REV{130}{}&\REV{0}{} &\REV{35.35}&&\REV{92}{}&\REV{0}{}&\REV{51.35}{}\\
\hline
\end{tabular}
\end{table}
We point \REV{out }{}that $\WPO(\Ams)$ is a balanced choice;
it is significantly stronger than existing orders,
while the runtime is much better than
involving non-linear SMT solving (last 5 columns).
Note that we directly solve non-linear problems using \Zthree;
\REV{it}{It} may be possible to improve efficiency by
\REV{\eg, a}{\eg} SAT encoding like \cite{FGMSTZ07}.
In that case, we expect $\WPO(\Amp)$ to become a practical choice.
If \REV{}{the }efficiency is the main concern, then
$\WPO(\Asum^+)$, a variant of $\WPOsum$ with $\Wzero > 0$,
is a reasonable substitute for KBO.
This efficiency is due to the reduction of recursive checks
proposed in \prettyref{sec:reduce recursion}.
\subsection{Results for Reduction Pairs}\label{sec:experiments pair}
Second,
we evaluated WPO as a reduction pair.
\prettyref{tab:pair} compares the power of the reduction pair processors.
In `total status' column,
we apply standard total statuses and argument filtering to
obtain a reduction pair from a reduction order, as in
\cite{YKS13}.
Because of argument filtering,
\REV{the }{}existence of duplicating rules \REV{is}{are} not an issue in this setting.
For POLO, statuses and argument filtering are ignored
(the latter is considered as $0$-coefficient).
\begin{table}[tb]
\caption{Results for Reduction Pairs\label{tab:pair}}
\centering
\begin{tabular}{cccc@{\quad}c@{\ }rlc@{\quad}c@{\ }r}
\hline
&&\qquad&\multicolumn{3}{c}{total status}
&\quad&\multicolumn{3}{c}{partial status}
\\\cline{4-6}\cline{8-10}
order&algebra&&yes&T.O.&\multicolumn{1}{c}{time}
&&yes&T.O.&\multicolumn{1}{c}{time}
\\\hline
POLO&$\Asum$ &&512&0 &150.99 &&-- &-- &--\\
POLO&$\Ams$ &&522&0 &300.65 &&-- &-- &--\\
LPO & &&502&0 &435.62 &&-- &-- &--\\
KBO & &&497&3 &1001.55&&520 &4 &1238.67\\
\hdashline
WPO &$\Asum$ &&514&3 &907.27 &&560 &5 &1244.04\\
WPO &$\Amax$ &&548&7 &1269.48&&637 &13 &1846.06\\
WPO &$\Ams$ &&\bf 578&5 &1261.63&&\bf 675 &12 &1827.01\\
\hline
POLO&$\Apol$ &&544&19 &1958.44 &&-- &-- &--\\
POLO&$\Amp$ &&540&18 &1889.86 &&-- &-- &--\\
POLO&$\Amat$ &&\bf 645&480 &32367.26 &&-- &-- &--\\
TKBO& &&516&187 &15665.26 &&539 &178&15799.28\\
\hdashline
WPO &$\Apol$ &&527&172 &14535.24 &&579 &153&13579.95\\
WPO &$\Amp$ &&560&88 &7678.43 &&\bf 672 &94 &9269.36\\
WPO &$\Amat$ &&-- &-- &-- &&538 &640&42067.45\\
\hline
\REV{\THOR}{}& &&\REV{418}{}&\REV{261}{}&\REV{18550.62}{}&&-- &-- &--\\
\hline
\end{tabular}
\end{table}
The power of WPO is still measurable here. On
the contrary to the reduction order case,
$\WPOsum$ outperforms KBO both in power and efficiency.
This is because
KBO needs formulas for recursive comparison that resembles WPO,
when argument filters are considered.
Moreover,
encodings of weights are more complex in KBO, since $w_0$ cannot
be fixed to $0$ as discussed in \prettyref{sec:fixing w0}.
Finally, KBO needs extra constraints that correspond to \REV{}{the }admissibility.
In `partial status' column, we moreover admit partial statuses
of \prettyref{def:WPOpS}.
We also apply partial status for KBO as in \cite{YKS13b} but
not for LPO,
since LPO does not benefit from partial statuses because
weights are not considered.
The power of WPO is much more significant in this setting, and
$\WPOms$ is about 30\% stronger than any other existing techniques.
Though the efficiency is sacrificed for partial statuses,
this is not a severe problem in the DP framework,
as we will see in the next section.
On the other hand, our implementation of
the instances of WPO that require non-linear SMT solving are extremely time-consuming.
Especially, $\WPOmat$ \REV{loses}{looses} 107 problems by timeout compared to the
standard matrix interpretation method.
We conjecture that the situation can be improved by SAT encoding or
by using other non-linear SMT solvers such as \cite{ZM10,BLNRR12}.
\REV
In order to estimate the power of RPOLO,
we also ran an experiment with \THOR
\footnote{\url{http://www.lsi.upc.edu/~albert/term.html}}
the only termination prover having RPOLO implemented, as far as we know.
Note however that it might be unfair to compare the results directly;
\THOR is specialized to higher-order case and is based on MSPO,
while our implementation is based on the DP framework.
}{
\subsection{Combining DP Processors}\label{sec:experiments combination}
\REV{}{Finally, we evaluate WPO in a more practical use for termination provers.
In \REV{}{the }modern termination provers,
DP processors are combined in the DP framework and
weak but efficient ones are applied first.
\REV
The default strategy of \NaTT sequentially applies
the \emph{rule removal processor} \cite{GTS04},
the \emph{(generalized) uncurrying} \cite{HMZ13,ST11},
reduction pair processors including
standard POLO, LPO, POLO with max \cite{FGMSTZ08} and
WPO($\Ams$) with partial status,
and then a simple variant of the matrix interpretation method \cite{EWZ08}.
When all reduction pair processors fail,
a naive loop detection is performed to conclude nontermination.
In \prettyref{tab:combination},
we compare the following settings:
`\NaTT' (the default strategy described above),
`\NaTT w/o WPO' (WPO is replaced by KBO)
\footnote{However, KBO does not contribute in this strategy.}
\AProVEver{2014}, and \TTTTver{1.15}.
The `no' column indicates the number of successful nontermination proofs.
}
In \prettyref{tab:combination},
we compare two strategies for combining DP processors.
For `existing' strategy, we sequentially apply
the \emph{rule removal processor} \cite{GTS04},
the \emph{(generalized) uncurrying} \cite{HMZ13,ST11},
and then reduction pair processors including
standard POLO, LPO,
POLO with max \cite{FGMSTZ08},
KBO,\footnote{However, KBO does not contribute in this strategy.}
and then a simpler variant of the matrix interpretation method \cite{EWZ08}.
In `new' strategy, we replace KBO by $\WPOms$ with partial status.
\begin{table}[tb]
\caption{Results for Combination\label{tab:combination}}
\centering
\begin{tabular}{cccccr}
\hline
\REV{tools}{strategy}
&yes &\REV{no}{} &\REV{maybe}{}&T.O. &\multicolumn{1}{c}{time}\\
\hline
\REV{\NaTT}{new}
&848 &\REV{173}{}&\REV{429}{}&\REV{13}{10} &18\REV{65.50}{36.30}\\
\REV{\NaTT w/o WPO}{existing}
&810 &\REV{173}{}&\REV{467}{}&\REV{13}{10} &\REV{2023.18}{1714.07}\\
\REV{\AProVE}
&\REV{1020}{}&\REV{271}{}&\REV{0}{}&\REV{173}{} &\REV{15123.48}{}\\
\REV{\TTTT} &\REV{788}{}&\REV{193}{}&\REV{417}{}&\REV{65}{} &\REV{13784.43}{}\\
\hline
\end{tabular}
\end{table
In this setting, \REV{\NaTT}{our tool} discovered
termination proofs for \REV{36}{40} of \REV{159}{161} problems
whose termination
\REV
could not be proved by any other tools participated
}
are unknown
in the full-run of the Termination Competition
\REV
2013 \cite{TC13}. For 29 of these problems,
}
2011 \cite{TC13}, and for 29 of them,
WPO is essential
\footnote{Due to the efficiency of our implementation,
our tool proves \REV{7}{11} open problems in TPDB without using WPO.}
\REV
In the competition,
\NaTT finished in the remarkable second place in the TRS standard category.
}{
\section{Conclusion}\label{sec:conclusion}
We introduced the weighted path order both as a reduction order
and as a reduction pair.
We presented several instances of WPO as reduction orders:
$\WPOsum$ that subsumes KBO,
$\WPOpol$ that subsumes POLO and TKBO,
$\WPOmax$ that subsumes LPO,
$\WPOms$ that unifies KBO and LPO, and
$\WPOmp$ that unifies all of them.
Moreover, we applied partial status for WPO to obtain a reduction pair,
and presented further refinements.
We show that as a reduction pair, WPO subsumes
KBO, LPO and TKBO with argument filters, POLO, and matrix interpretations.
We also presented SMT encodings for these techniques.
The orientability problems of $\WPOsum$, $\WPOmax$ and $\WPOms$ are decidable,
since they are reduced to satisfiability problems of linear integer arithmetic
which is known to be decidable.
Finally, we verified through experiments the significance of our work
both as a reduction order and as a reduction pair.
In order to keep the presentation simple,
we did not present $\WPO$ with \emph{multiset status}.
Nonetheless, it is easy to define $\WPO$ with multiset status and verify that
$\WPOmax$ with multiset status \REV{subsumes}{encompasses} RPO.
We only considered a straightforward method for combining $\WPOpol$ and $\WPOmax$
using `weight statuses', and moreover heuristically fixed the weight status.
We leave it for future work to
search for other possible weight statuses, or to
find more sophisticated combination\REV{s}{} of max-polynomials
such as $f_\A(x,y,z) = x + \max(y,z)$, or
even trying other algebras including
\emph{ordinal interpretations} \cite{WZM12,WZM13}.
For efficiency, real arithmetic is also attractive to consider, since
SMT for \REV{}{the }real arithmetic is often more efficient than for
\REV{}{the }integer arithmetic.
To this end, we will have to reconstruct the proof \REV{of}{for}
well-foundedness of WPO, since our current proof relies on
well-foundedness of the underlying order,
which does not hold anymore for real numbers.
Another obvious future work is to extend WPO for higher\REV{-}{ }order case.
Since RPOLO has strength in its higher\REV{-}{ }order version \cite{BBRR13},
we expect their technique can be extended for WPO.
\paragraph*{Acknowledgments}
\REV
We are grateful to the anonymous reviewers for their careful inspections
and comments that significantly improved the presentation of this paper.
We thank Sarah Winkler and Aart Middeldorp for discussions at
the early stages of this work.
We thank Florian Frohn and J\"urgen Giesl for
their helps in experiments with \AProVE, and
Albert Rubio, Miquel Bofill and Cristina Borralleras for
their helps in experiments with \THOR.
}
We thank Sarah Winkler and Aart Middeldorp for discussions
and the anonymous reviewers of previous versions of this paper
for helpful comments.
This work was supported by JSPS KAKENHI \#24500012
\REV
|
1,314,259,996,534 | arxiv | \section{Introduction}
The chiral Potts model was originally introduced
\cite{Ostlund,Kardar,Huse,Huse2,Caflisch} to model the full phase
diagram of krypton monolayers, including the epitaxial and
incommensurate ordered phases. In addition to being useful in the
analysis of surface layers, the chiral Potts model has become an
important model of phase transitions and critical phenomena. We have
studied the chiral spin-glass Potts system with $q=3$ states in
$d=2$ and 3 spatial dimensions by renormalization-group theory and
calculated the global phase diagrams (Fig. 1) in temperature,
chirality concentration $p$, and chirality-breaking concentration
$c$, also quantitatively determining phase chaos and phase-boundary
chaos. In $d = 3$, the system has ferromagnetic, left-chiral,
right-chiral, chiral spin-glass, and disordered phases. The phase
boundaries to the ferromagnetic, left- and right-chiral phases show,
differently, an unusual, fibrous patchwork (microreentrances) of all
four (ferromagnetic, left-chiral, right-chiral, chiral spin-glass)
ordered phases, especially in the multicritical region. The chaotic
behavior of the interactions, under scale change, is determined in
the chiral spin-glass phase and on the boundary between the chiral
spin-glass and disordered phases, showing Lyapunov exponents in
magnitudes reversed from the usual ferromagnetic-antiferromagnetic
spin-glass systems. At low temperatures, the boundaries of the
left- and right-chiral phases become thresholded in $p$ and $c$. In
the $d=2$, the chiral spin-glass Potts system does not have a
spin-glass phase, consistently with the lower-critical dimension of
ferromagnetic-antiferromagnetic spin glasses. The left- and
right-chirally ordered phases show reentrance in chirality
concentration $p$.
\section{The Chiral Potts Spin-Glass System}
The chiral Potts model is defined by the Hamiltonian
\begin{equation}
\centering
- \beta {\cal H} = \sum_{\left<ij\right>}[J_0\delta(s_i,s_j) + J_\pm\delta(s_i, s_j\pm1)],
\end{equation}
where $\beta=1/k_{B}T$, at site $i$ the spin $s_{i}=1,2,,...,q$ can
be in $q$ different states with implicit periodic labeling, e.g.
$s_i=q+n$ implying $s_i=n$, the delta function $\delta(s_i,
s_j)=1(0)$ for $s_i=s_j (s_i\neq s_j)$, and $\langle ij \rangle$
denotes summation over all nearest-neighbor pairs of sites. The
upper and lower subscripts of $J_\pm>0$ give left-handed and
right-handed chirality (corresponding to heavy and superheavy domain
walls in the krypton-on-graphite incommensurate ordering
\cite{Kardar, Caflisch}), whereas $J_\pm=0$ gives the non-chiral
Potts model (relevant to the krypton-on-graphite epitaxial ordering
\cite{BerkerPLG}).
In the chiral Potts spin-glass model studied here, the chirality of
each nearest-neighbor interaction is randomly either left-handed, or
right-handed, or zero. This randomness is frozen (quenched) into the
system and the overall fraction of left-, right-, and non-chirality
is controlled by the quenched densities $p$ and $c$ as described
below. Thus, the Hamiltonian of the chiral Potts spin-glass model is
\begin{multline}
- \beta {\cal H} = \sum_{\left<ij\right>}J \,[(1-\eta_{ij})\delta(s_i, s_j)\\
+\eta_{ij}\,[\phi_{ij}\delta(s_i,s_j+1)+(1-\phi_{ij})\delta(s_i,s_j-1)],
\end{multline}
where, for each pair of nearest-neighbor sites $<ij>,$ $\eta_{ij}=0$
(non-chiral) or 1 (chiral). In the latter case, $\phi_{ij}=1$
(left-handed) or 0 (right-handed). Thus, non-chiral, left-chiral,
and right-chiral nearest-neighbor interactions are frozen randomly
distributed in the entire system. For the entire system, the
overall concentration of chiral interactions is given by $p$, with
$0\leq p\leq1$. Among the chiral interactions, the overall
concentrations of left- and right-chiral interactions are
respectively given by $c$ and $1-c$, with $0\leq c\leq1$. Thus, the
model is chiral for $p>0$ and chiral-symmetric $c=0.5$,
chiral-symmetry broken for $c\neq 0.5$. The global phase diagram is
given in terms of temperature $J^{-1}$, chirality concentration $p$,
and chirality-breaking concentration $c$.(Figs. 1-3)
Under the renormalization-group transformations described below, the
Hamiltonian given in Eq.(2) maps onto the more general form
\begin{multline}
- \beta {\cal H} = \sum_{\left<ij\right>} \, [J_0(ij)\delta(s_i,
s_j)+J_+(ij)\delta(s_i,s_j+1)\\
+J_-(ij)\delta(s_i,s_j-1)],
\end{multline}
where for each pair of nearest-neighbor sites $<ij>$, the largest of
the interaction constants $(J_0,J_+,J_-)$ is set to zero, by
subtracting a constant G from each of $(J_0,J_+,J_-)$, with no
effect to the physics.
\section{Renormalization-Group Transformation: Migdal-Kadanoff Approximation / Exact Hierarchical Lattice Solution}
We solve the chiral Potts spin-glass model with $q=3$ states by
renormalization-group theory, in $d=3$ spatial dimension and with
the length rescaling factor $b=2$. Our solution is, simultaneously,
the Migdal-Kadanoff approximation \cite{Migdal,Kadanoff} for the
cubic lattices and exact
\cite{BerkerOstlund,Kaufman1,Kaufman2,McKay,Hinczewski1} for the
$d=3$ hierarchical lattice based on the leftmost graph of Fig. 4.
Exact calculations on hierarchical lattices
\cite{BerkerOstlund,Kaufman1,Kaufman2,McKay,Hinczewski1} are also
currently widely used on a variety of statistical mechanics problems
\cite{Kaufman,Kotorowicz,Barre,Monthus,Zhang,Shrock,Xu,Hwang2013,Herrmann1,
Herrmann2,Garel,Hartmann,Fortin,Wu,Timonin,Derrida,Thorpe,Efrat,Monthus2,
Hasegawa,Lyra,Singh,Xu2014,Hirose1,Silva,Hotta,Boettcher1,Boettcher2,Hirose2,Boettcher3,Nandy}.
This approximation for the cubic lattice is an uncontrolled
approximation, as in fact are all renormalization-group theory
calculations in $d=3$ and all mean-field theory calculations.
However, as noted before \cite{Yunus}, the local summation in
position-space technique used here has been qualitatively,
near-quantitatively, and predictively successful in a large variety
of problems, such as arbitrary spin-$s$ Ising models
\cite{BerkerSpinS}, global Blume-Emery-Griffiths model
\cite{BerkerWortis}, first- and second-order Potts transitions
\cite{NienhuisPotts,AndelmanBerker}, antiferromagnetic Potts
critical phases \cite{BerkerKadanoff1,BerkerKadanoff2}, ordering
\cite{BerkerPLG} and superfluidity \cite{BerkerNelson} on surfaces,
multiply reentrant liquid crystal phases \cite{Indekeu,Garland},
chaotic spin glasses \cite{McKayChaos}, random-field
\cite{Machta,FalicovRField} and random-temperature
\cite{HuiBerker,HuiBerkerE} magnets including the remarkably small
$d=3$ magnetization critical exponent $\beta$ of the random-field
Ising model, and high-temperature superconductors
\cite{HincewskiSuperc}. Thus, this renormalization-group
approximation continues to be widely used
\cite{Gingras1,Migliorini,Gingras2,Hinczewski,Heisenberg,Guven,Ohzeki,Ozcelik,Gulpinar,Kaplan,Ilker1,Ilker2,Ilker3,Demirtas}.
\begin{figure*}[ht!]
\centering
\includegraphics[scale=1.0]{phase_ptE.eps}
\caption{(Color online) Cross-sections, in temperature $J^{-1}$ and
chirality concentration $p$, of the global phase diagram shown in
Fig. 1. The chirality-breaking concentration $c$ is given on each
cross-section. The ferromagnetically ordered phase (F), the chiral
spin-glass phase (S), the left- and right-chirally ordered phases (L
and R), and the disordered phase (D) are marked. Note that, as soon
as the chiral symmetry of the model is broken by $c \neq 0.5$, a
narrow fibrous patchwork (microreentrances) of all four
(ferromagnetic, left-chiral, right-chiral, chiral spin-glass)
ordered phases intervenes at boundaries of the ferromagnetically
ordered phase F. This intervening region is more pronounced close to
the multicritical region where the ferromagnetic, spin-glass, and
disordered phases meet. The interlacing phase transitions inside
this region are more clearly seen in the right-hand side panels of
the figure, where only the phase boundaries are drawn in black. This
intervening region gains importance as $c$ moves away from 0.5. But
it is only at higher values of the chirality-breaking concentration
$c$, such as $c=0.8$ on the figure, that the chirally ordered phase
appears as a compact region at $c,p\lesssim 1$. In this case, again
all four (ferromagnetic, left-chiral, right-chiral, chiral
spin-glass) ordered phases intervene in a narrow fibrous patchwork
at the boundaries of the chirally ordered phase L and R, the latter
mirror-symmetric and not shown here. For $c=1$, for which all
interactions of the system are, with respective concentrations $1-p$
and $p$, either ferromagnetic, or left-chiral, the phase diagram
becomes symmetric with respect to $p=0.5$ as in standard
ferromagnetic-antiferromagnetic spin-glass systems, except that the
chirally ordered phases dominate the fibrous patchwork on both sides
of the phase diagram.}
\end{figure*}
\begin{figure*}[ht!]
\centering
\includegraphics[scale=1.02]{phase_pcE.eps}
\caption{(Color online) Cross-sections, in chirality concentration
$p$ and chirality-breaking concentration $c$, of the global phase
diagram shown in Fig. 1. The temperature $J^{-1}$ is given on each
cross-section. The ferromagnetically ordered phase (F), the chiral
spin-glass phase (S), the left-chirally ordered phase (L), the
right-chirally ordered phase (R), and the disordered phase (D) are
marked. Note the narrow fibrous patches (microreentrances) of all
four (ferromagnetic, left-chiral, right-chiral, chiral spin-glass)
ordered phases intervening at the boundaries of the
ferromagnetically ordered phase F and at the boundaries of the
chirally ordered phases L and R. It is seen here that, within these
regions, the chirally ordered phases L and R form elongated lamellar
patterns. These intervening phase transitions are more clearly seen
in the right-hand side panels of the figure, where only the phase
boundaries are drawn in black. Also note the temperature-independent
square shape, at low temperatures, of the phase boundary of the
chirally ordered phases, creating thresholds of $p = 0.84$ and $c =
0.84$ or 0.16 into L or R, respectively. This is also visible in the
three-dimensional Fig. 1}
\end{figure*}
\begin{figure}[ht!]
\centering
\includegraphics[scale=1.0]{rg_scheme.eps}
\caption{Renormalization-group transformation consisting of
decimation followed by bond moving. The resulting recursion
relations are approximate for the cubic lattice. The corresponding
hierarchical lattice is obtained by the repeated self-imbedding of
the leftmost graph. The recursion relations are exact for this $d =
3$ hierarchical lattice. For the $d = 2$, the number of parallel
strands is 2 instead of 4 shown here.}
\end{figure}
The local renormalization-group transformation is achieved by a
sequence, shown in Fig. 4, of decimations
\begin{multline}
\begin{split}
e^{\widetilde{J}_0(13)-\widetilde{G}}= x_0(12)x_0(23) &+ x_+(12)x_-(23)\\
&+ x_-(12)x_+(23),\\
e^{\widetilde{J}_+(13)-\widetilde{G}}= x_0(12)x_+(23) &+ x_+(12)x_0(23)\\
&+ x_-(12)x_-(23),\\
e^{\widetilde{J}_-(13)-\widetilde{G}}= x_0(12)x_-(23) &+ x_-(12)x_0(23)\\
&+x_+(12)x_+(23),\\
\end{split}
\end{multline}
where $x_0(12) \equiv e^{J_0(12)}$, etc., and $\widetilde{G}$ is the
subtractive constant mentioned in the previous section, and bond
movings
\begin{multline}
\begin {split}
J_0'(13) =&
\widetilde{J}_0^{(1)}(13)+\widetilde{J}_0^{(2)}(13)+\widetilde{J}_0^{(3)}(13)+\widetilde{J}_0^{(4)}(13),\\
J_+'(13) =&
\widetilde{J}_+^{(1)}(13)+\widetilde{J}_+^{(2)}(13)+\widetilde{J}_+^{(3)}(13)+\widetilde{J}_+^{(4)}(13),\\
J_-'(13) =&
\widetilde{J}_-^{(1)}(13)+\widetilde{J}_-^{(2)}(13)+\widetilde{J}_-^{(3)}(13)+\widetilde{J}_-^{(4)}(13),
\end {split}
\end{multline}
where primes mark the interactions of the renormalized system.
\begin{figure}[ht!]
\centering
\includegraphics[scale=1.0]{HistColorC.eps}
\caption{(Color online) The fixed probability distribution of the
quenched random interactions $P_0(J_+,J_-)$ to which all of the
points in the chiral spin-glass phase are attracted under
renormalization-group transformations, namely the sink of the chiral
spin-glass phase. The average interactions $<J_\pm>$ diverge to
negative infinity as $<J_\pm> \sim b^{y_R n}$, where $n$ is the
number of renormalization-group iterations and $y_R = 0.32$ is the
runaway exponent, while $J_0=0$ (See Sec. II). The other two
distributions $P_+(J_0,J_-)$ and $P_-(J_0,J_+)$ have the same sink
distribution. Thus, in the chiral spin-glass phase, chiral symmetry
is broken by local order, but not globally.}
\end{figure}
The starting trimodal quenched probability distribution of the
interactions, characterized by $p$ and $c$ as described above, is
not conserved under rescaling. The renormalized quenched probability
distribution of the interactions is obtained by the convolution
\cite{Andelman}
\begin{multline}
P'(\textbf{J}(i'j')) = \\
\int{\left[\prod_{ij}^{i'j'}d\textbf{J}(ij)P(\textbf{J}(ij))\right]}
\delta(\textbf{J}(i'j')-\textbf{R}(\left\{\textbf{J}(ij)\right\})),
\end{multline}
where $\textbf{J}\equiv (J_0',J_+',J_-')$ and
$\textbf{R}(\left\{\textbf{J}(ij)\right\})$ represents the bond
decimation and bond moving given in Eqs.(4) and (5). Similar
previous studies, on other spin-glass systems, are in Refs.
\cite{Gingras1,Migliorini,Gingras2,Hinczewski,Heisenberg,Guven,Ohzeki,Ozcelik,Gulpinar,Kaplan,Ilker1,Ilker2,Ilker3,Demirtas}.
For numerical practicality, the bond moving of Eq. (5) is achieved
by two sequential pairwise combination of interactions, each
pairwise combination leading to an intermediate probability
distribution resulting from a pairwise convolution as in Eq.(6).
Furthermore, due to our convention of zeroing the largest
interaction constant in each local triplet of interactions, the
quenched probability distribution of three interactions
$P(\textbf{J}(ij))$ is conveniently just composed of the three
probability distributions of two interactions,
$P_0(J_+,J_-),P_+(J_0,J_-),P_-(J_+,J_-)$, where $P_0(J_+,J_-)$ has
the (largest) interaction $J_0 =0,$ etc., which also considerably
simplifies the numerical calculation. We effect this procedure
numerically, by representing each probability distribution by
histograms, as in previous studies
\cite{Migliorini,Hinczewski,Heisenberg,Guven,Ozcelik,Gulpinar,Ilker2,Demirtas}.
The probability distributions of two interactions $P_0(J_+,J_-)$,
$P_+(J_0,J_-)$, and $P_-(J_+,J_-)$ are represented via bivariate
histograms with two-dimensional vectors $(J_+,J_-)$ for $P_0$, etc.
The number of histograms grow rapidly with each
renormalization-group transformation, so that for calculational
purposes, the histograms are binned when the number of histograms
outgrow $40,000$ bins. In the calculation of chiral spin-glass
phase-sink fixed distribution of Fig. 5, the histograms are binned
after $10^8$ histograms.
The different thermodynamic phases of the model are identified by
the different asymptotic renormalization-group flows of the quenched
probability distribution. For all renormalization-group flows,
originating inside the phases and on the phase boundaries, Eq.(6) is
iterated until asymptotic behavior is reached. Thus, we are able to
calculate the global phase diagram of the chiral Potts spin-glass
model.
\section{Chiral Potts Spin Glass: Calculated Global Phase Diagram}
The calculated global phase diagram of the $d=3$ chiral Potts
spin-glass system, in temperature $J^{-1}$, chirality concentration
$p$, and chirality-breaking concentration $c$, is given in Fig. 1.
The ferromagnetically ordered (F) phase occurs at low temperature
and low chirality $p$. The chiral spin-glass ordered (S) phase
occurs at intermediate chirality $p$ for all $c$ and at high
chirality $p$ for intermediate $c$. The left- and right-chirally
ordered phases L and R occur at high chirality $p$ and values of
chirality-breaking $c$ away from 0.5. The disordered phase (D)
occurs at high temperature. The global phase diagram is
mirror-symmetric with respect to the chirality-breaking
concentration $c=0.5$, so that only $1\leq c \leq 0.5$ is shown in
Fig. 1. In the (not shown) mirror-symmetric $0.5\leq c \leq 0$
portion of the global phase diagram, the right-chirally ordered
phase (R) occurs in the place of the left-chirally ordered phase (L)
seen in Fig. 1. Different cross-sections of the global phase diagram
are shown in Figs. 2 and 3.
Under renormalization-group transformations, all points in the
spin-glass phase are attracted to a fixed probability distribution
of the quenched random interactions $P(J_0,J_+,J_-)$, namely to the
sink of the chiral spin-glass phase. As explained in Sec. III,
$P(J_0,J_+,J_-)$ is composed of three distributions, $P_0(J_+,J_-)$,
$P_+(J_0,J_-)$, and $P_-(J_0,J_+)$. Of these, $P_0(J_+,J_-)$ gives
the quenched probability distribution of nearest-neighbor
interactions in which the ferromagnetic interaction $J_0$ is
dominant. Similarly, $P_+(J_0,J_-)$ and $P_-(J_0,J_+)$ give the
quenched probability distributions of nearest-neighbor interactions
in which, respectively, the left-chiral interaction $J_+$ and the
right-chiral interaction $J_-$ are dominant. (As explained in Sec.
II, by subtraction of an overall constant, the dominant interaction
is set to zero and the other two, subdominant interactions are
therefore negative, with no loss of generality.) The sink fixed
distribution for $P_0(J_+,J_-)$ is given in Fig. 5, where the
average interactions $<J_\pm>$ diverge to negative infinity as
$b^{y_R n}$, where $n$ is the number of renormalization-group
iterations and $y_R = 0.32$ is the runaway exponent, while
conserving the shape of the distribution shown in Fig. 5. The other
two distribution $P_+(J_0,J_-)$ and $P_-(J_0,J_+)$ have the same
sink distribution. Thus, in the chiral spin-glass phase, chiral
symmetry is broken by local order, but not globally.
In spin-glass phases, at a specific location in the lattice, the
consecutive interactions, encountered under consecutive
renormalization-group transformations, behave chaotically
\cite{McKayChaos, McKayChaos2,BerkerMcKay}. This chaotic behavior
was found \cite{McKayChaos, McKayChaos2,BerkerMcKay} and
subsequently well established \cite {Bray,Hartford,
Nifle1,Nifle2,Banavar,Frzakala1,Frzakala2,Sasaki,Lukic,Ledoussal,Rizzo,
Katzgraber,Yoshino,Pixley,Aspelmeier1,Aspelmeier2,Mora,Aral,Chen,Jorg,Lima,Katzgraber2,MMayor,ZZhu,Katzgraber3,Fernandez,Ilker1,Ilker2}
in spin-glass systems with competing ferromagnetic and
antiferromagnetic interactions. We find here that the chaotic
rescaling behavior also occurs in our current spin-glass system with
competing left- and right-chiral interactions, as shown in Fig. 6.
In fact, the chaotic rescaling behavior occurs not only within the
spin-glass phase, but also, quantitatively distinctly, at the phase
boundary between the spin-glass and disordered phases \cite{Ilker1}.
This chaotic behavior at the phase boundary is also seen in the
chiral system here and also shown in Fig. 6. It has been shown that
chaos in the interaction as a function of rescaling implies chaos in
the spin-spin correlation function as a function of distance
\cite{Aral}. Chaos in the spin-glass phase and at its phase boundary
are identified and distinguished by different Lyapunov exponents
\cite{Aral,Ilker1,Ilker2}. We have calculated the Lyapunov exponent
\cite{Collet, Hilborn}
\begin{equation}
\lambda = \lim _{n\rightarrow\infty} \frac{1}{n} \sum_{k=0}^{n-1}
\ln \Big|\frac {dx_{k+1}}{dx_k}\Big|
\end{equation}
where $x_k = J(ij)/<J>$ at step $k$ of the renormalization-group
trajectory. The sum in Eq.(7) is to be taken within the asymptotic
chaotic band, which is renormalization-group stable or unstable for
the phase or its boundary, respectively. Thus, we throw out the
first 100 renormalization-group iterations to eliminate the
transient points outside of, but leading to the chaotic band.
Subsequently, typically using 1,000 renormalization-group iterations
in the sum in Eq.(7) assures the convergence of the Lyapunov
exponent value. Thus, the Lyapunov exponents that we obtain are
numerically exact, to the number of digits given. We have calculated
the Lyapunov exponents $\lambda = 1.77$ and 1.94 respectively for
the chiral spin-glass phase and for the boundary between the chiral
spin-glass and disordered phases. At the chiral spin-glass
phase-sink fixed distribution, the average interaction diverges to
negative infinity as $<J> \sim b^{ny_R}$, where $n$ is the number of
renormalization-group iterations and $y_R = 0.32$ is the runaway
exponent. At the fixed distribution of the phase boundary between
the chiral spin-glass and disordered phases, the average interaction
remains fixed at $<J> = -2.53$. Interestingly, chaos is stronger at
the boundary (larger Lyapunov exponent) than inside the chiral
spin-glass phase. The opposite is seen in the usually studied $\pm
J$ ferromagnetic-antiferromagnetic spin glass \cite{Ilker1}.
\begin{figure}[ht!]
\centering
\includegraphics[scale=1.0]{ChaoticTrajD.eps}
\caption{Chaotic renormalization-group trajectory: The three
interactions at a given location, under consecutive
renormalization-group transformations, are shown. Bottom panel:
Inside the chiral spin-glass phase. The corresponding Lyapunov
exponent is $\lambda = 1.77$ and the average interaction diverges as
$<J> \sim b^{y_R n}$, where $n$ is the number of
renormalization-group iterations and $y_R = 0.32$ is the runaway
exponent. Top panel: At the phase boundary between the chiral
spin-glass and disordered phases. The corresponding Lyapunov
exponent is $\lambda = 1.94$ and the average non-zero interaction is
fixed at $<J> = -2.53$. The relative value of the Lyapunov exponents
is unusual for spin-glass systems.}
\end{figure}
By contrast, in each of the ferromagnetic (F), left-chiral (L), and
right-chiral (R) ordered phases, under consecutive
renormalization-group transformations, the quenched probability
distribution of the interactions sharpens to a delta function around
a single value receding to negative infinity, for the respective
pairs of interactions, namely ($J_+,J_-), (J_0, J_+)$, and $(J_0,
J_-)$. There is no asymptotic chaotic behavior under
renormalization-group in these phases F, L, and R.
\begin{figure*}[ht!]
\centering
\includegraphics[scale=1.0]{phase_d2_ptE.eps}
\caption{Representative cross-sections of the $d=2$ chiral Potts
spin-glass system, in temperature $J^{-1}$ and chirality
concentration $p$. The chirality-breaking concentration $c$ is given
on each cross-section. The ferromagnetically ordered phase (F), the
left-chirally ordered phase (L), and the disordered phase (D) are
marked. No chiral spin-glass phase occurs in $d=2$ and no fibrous
patchwork is seen at the phase boundaries. The chirally ordered
phase appears for very high chirality-breaking concentration $c$
(seen here for $c = 0.934$, but not seen for $c = 0.930$) and shows
reentrance in chirality concentration $p$. This reentrance
disappears as $c = 1$ is approached. For $c=1$, for which all
interactions of the system are, with respective concentrations $1-p$
and $p$, either ferromagnetic, or left-chiral, the phase diagram
becomes symmetric with respect to $p=0.5$ as in standard
ferromagnetic-antiferromagnetic spin-glass systems.}
\end{figure*}
Cross-sections of the global phase diagram, in temperature $J^{-1}$
and chirality concentration $p$, are given in Fig. 2. The
chirality-breaking concentration $c$ is indicated for each
cross-section. Note that, as soon as the chiral symmetry of the
model is broken by $c \neq 0.5$, a narrow fibrous patchwork
(microreentrances) of all four (ferromagnetic, left-chiral,
right-chiral, chiral spin-glass) ordered phases intervenes at the
boundaries between the ferromagnetically ordered phase F and the
spin-glass phase S or the disordered phase D. This intervening
region is more pronounced close to the multicritical region where
the ferromagnetic, spin-glass, and disordered phases meet. The
interlacing phase transitions inside this region are more clearly
seen in the right-hand side panels of Fig. 2, where only the phase
boundaries are drawn in black. This intervening region gains
importance as $c$ moves away from 0.5. But it is only at higher
values of the chirality-breaking concentration $c$, such as $c=0.8$
on the figure, that the chirally ordered phase appears as a compact
region at $c,p\lesssim 1$. In this case, again all four
(ferromagnetic, left-chiral, right-chiral, chiral spin-glass)
ordered phases intervene in a narrow fibrous patchwork at the
boundaries of the chirally ordered phases L and R, the latter mirror
symmetric and not shown here. For $c=1$, for which all interactions
of the system are, with respective concentrations $1-p$ and $p$,
either ferromagnetic, or left-chiral, the phase diagram becomes
symmetric with respect to $p=0.5$ as in standard
ferromagnetic-antiferromagnetic spin-glass systems
\cite{NishimoriBook}, except that the chirally ordered phases
dominate the fibrous patchwork on both sides of the phase diagram.
Cross-sections, in chirality concentration $p$ and
chirality-breaking concentration $c$, of the global phase diagram
are given in Fig. 3. The temperature $J^{-1}$ is given on each
cross-section. Note the narrow fibrous patches of all four
(ferromagnetic, left-chiral, right-chiral, chiral spin-glass) phases
intervening at the boundaries of the ferromagnetically ordered phase
F and at the boundaries of the chirally ordered phases L and R. It
is seen here that, within these regions, the chirally ordered phases
L and R form elongated lamellar patterns. The interlacing phase
transitions inside this region are more clearly seen in the
right-hand side panels of the figure, where only the phase
boundaries are drawn in black. It is again seen that the symmetry
around $p=0.5$ at the upper horizontal frame $(c=1)$ of each panel
is broken inside the panel $(c<1)$. Also note the
temperature-independent square shape, at low temperatures, of the
phase boundary of the chirally ordered phases L and R, creating the
threshold value of $p = 0.84$ and $c = 0.84$ or 0.16 into L or R,
respectively. This is also visible in the three-dimensional Fig. 1.
\section{Chiral Reentrance in $d=2$}
The global phase diagram of the $d=2$ chiral Potts spin-glass system
is given in Fig. 7. Representative cross-sections in temperature
$J^{-1}$ and chirality concentration $p$ are shown. The
chirality-breaking concentration $c$ is given on each cross-section.
The ferromagnetically ordered phase (F), the left-chirally ordered
phase (L), and the disordered phase (D) are marked. No chiral
spin-glass phase occurs in $d=2$ and no fibrous patchwork is seen at
the phase boundaries. The chirally ordered phase appears for very
high chirality-breaking concentration $c$ (seen here for $c =
0.934$, but not seen for $c = 0.930$) and shows reentrance
\cite{Cladis, Hardouin, Indekeu, Garland, Netz, Kumari, Caflisch} in
chirality concentration $p$. This reentrance disappears as $c = 1$
is approached. For $c=1$, for which all interactions of the system
are, with respective concentrations $1-p$ and $p$, either
ferromagnetic, or left-chiral, the phase diagram becomes symmetric
with respect to $p=0.5$ as in standard
ferromagnetic-antiferromagnetic spin-glass systems \cite{Ilker2}.
The absence of the chiral spin-glass phase in $d = 2$ is consistent
with standard ferromagnetic-antiferromagnetic Ising spin-glass
systems, where the lower-critical dimension for the spin-glass phase
is found around 2.5
\cite{Parisi,Boettcher,Amoruso,Bouchaud,Demirtas}. Below this
dimension, no spin-glass phase appears (unless some
nano-restructuring is done to the system \cite{Ilker2}).
\section{Conclusion}
We have thus obtained the global phase diagram of the chiral
spin-glass Potts system with $q=3$ states in $d=3$ and 2 spatial
dimensions by renormalization-group theory that is approximate for
the cubic lattice and exact for the hierarchical lattice. Unusual
features have been revealed in $d=3$. The phase boundaries to the
ferromagnetic, left- and right-chiral phases show, differently, an
unusual, fibrous patchwork (microreentrances) of all four
(ferromagnetic, left-chiral, right-chiral, chiral spin-glass)
ordered phases, especially in the multicritical region. In $d=3$,
there is a chiral spin-glass phase. Quite unusually, the phase
boundary between the chiral spin-glass and disordered phases is more
chaotic than the chiral spin-glass phase itself, as judged by the
magnitudes of the respective Lyapunov exponents. At low
temperatures, the boundaries of the left- and right-chiral phases
become temperature-independent and thresholded in chirality
concentration $p$ and chirality-breaking concentration $c$. In the
$d=2$, the chiral spin-glass system does not have a spin-glass
phase, consistently with the lower-critical dimension of
ferromagnetic-antiferromagnetic spin glasses. The left- and
right-chirally ordered phases show reentrance in chirality
concentration $p$.
\begin{acknowledgments}
Support by the Academy of Sciences of Turkey (T\"UBA) is gratefully
acknowledged.
\end{acknowledgments}
|
1,314,259,996,535 | arxiv | \section{Introduction}
Cold nuclear matter at subsaturation density as $\alpha $ matter
has been subjected to a critical study for some time
\cite{cla,car}. The aim is to understand the $\alpha $-clustering
near the surface of heavy nuclei or the putative dilute
alpha-condensate in light 4n nuclei. In astrophysical context,
in following the evolution of the core-collapse supernovae,
these studies have been extended to the case of warm nuclear
matter \cite{lamb}. The homogeneous low-density
nuclear matter stabilizes as a mixture of nucleons and
nucleon-clusters. It has lower free energy compared to
that for nucleonic matter. The cluster
composition is temperature and density dependent, with increasing
temperature or decreasing density, the population of heavier clusters
tends to diminish leading to a mixture of nucleons and light clusters
like d, t, $^{3}$He and $\alpha $ \cite{fri,pei}.
The properties of the clusterized matter undergo a major change,
{\it e.g.},
the incompressibility of clusterized nuclear matter is quite
smaller compared to that for homogeneous nucleon matter
\cite{sam}. This directly influences the collapse and bounce phase
of the supernova matter. The symmetry energies of nuclear matter are
also affected significantly when matter gets clusterized \cite{hor,de}.
This has an important role in a better understanding of neutrino-driven
energy transfer in supernova matter \cite{jan}. The symmetry energy also
influences the cluster composition in the crust
of neutron stars and is thus
instrumental in shaping the details of their mass, cooling and structure
\cite{fuc}.
The equation of state (EOS) of warm dilute nuclear matter with only
light clusters upto $\alpha $ has recently been investigated in the
virial approach \cite{hor,con}; inclusion of heavier clusters has
also been made in the $S$-matrix (SM)
framework \cite{mal}. These
methods relate the calculations directly to the experimental observables
like the binding energies and the phase-shifts and thus, as such, are
model-independent. They are usually taken as {\it bench-mark } calculations
in the domain of low-density and high-temperature; they are understood
to exhaust all the dynamical information concerning the strong
interactions in the medium.
For an
interacting quantum gas, the virial expansion, however,
virtually ends at the second
order. Formulation of higher order virial coefficients are very
involved even at the formal level \cite{pai}, making it difficult to estimate
the domain of validity of the virial series truncated at the second
order. It may further be noted that
the density should be dilute enough so that
the concept of asymptotic wave functions as inherent in the virial
expansion should be meaningful.
An alternate avenue could be to bypass the virial
expansion altogether and take recourse to nucleation in the
framework of the mean-field model with a suitably chosen effective
two-nucleon interaction that inherently takes an inclusive account
of the scattering effects. Unlike the virial ($S$-matrix) approach
which has direct contact with the experimental data,
this method has indirect contact but it can
be applied to relatively higher densities.
With increasing density, a large number of
different fragment species would, however, be formed that makes the numerical
calculation very lengthy. Before attempting any full-blown calculation,
it may then be worthwhile as a first step,
to take only $\alpha $-clustering in the nuclear matter and to
examine whether the model works in the low-density region where
the bench-mark calculations exist. The present work aims
towards that end.
For the study of the so-mentioned $\alpha $-nucleon ($\alpha $N) matter,
we have chosen the Thomas-Fermi prescription for the mean-field model
and the finite range, momentum and density dependent modified
Seyler-Blanchard (SBM) effective interaction \cite{de1}. The properties
we explore include the EOS of the $\alpha $N matter, its symmetry energy,
incompressibility, $\alpha $ concentration etc. In Sec. II, the theoretical
framework for the mean-field and the $S$-matrix approach
is presented. Sec. III contains the results and discussions.
Concluding remarks are given in Sec. IV.
\section{Theoretical framework}
Given an effective two-nucleon interaction, the properties of the
$\alpha $N matter can be evaluated by exploiting the occupation
functions of the n, p and alphas obtained from minimization
of the thermodynamic potential of the system. In Sec. II~A, some details
of the effective interaction used are given. In Sec II~B,
theoretical formulation for obtaining the occupation functions from
the Thomas-Fermi (TF) approximation
is presented. In Sec. II~C, expressions for various observables
explored are given. In Sec.~II~D, a brief outline of the $S$-matrix approach
is made.
\begin{table}
\caption{ The parameters of the effective interaction (in MeV fm units)}
\begin{ruledtabular}
\begin{tabular}{cccccc}
$C_l$& $C_u$& $a$& $b$& $d$& $\kappa $\\
\hline
291.7& 910.6& 0.6199& 928.2& 0.879& 1/6\\
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{The effective interaction}
The form of the SBM effective interaction $v$ is
\begin{eqnarray}
v(r,p,\rho )&=&C_{l,u}\left [v_1(r,p)+v_2(r,\rho )\right ], \nonumber \\
v_1&=&-(1-\frac{p^2}{b^2})f({\bf r_1, r_2}), \nonumber \\
v_2&=&d^2\left [ \rho (r_1) + \rho (r_2) \right ]^\kappa f({\bf r_1, r_2}),
\end{eqnarray}
with
\begin{eqnarray}
f({\bf r_1, r_2})&=& \frac{e^{-|{\bf r_1-r_2 }|/a}}{|{\bf r_1-r_2 }/a}.
\end{eqnarray}
The subscripts $l$ and $u$ to the interaction strength $C$ refer to
like-pair (nn or pp) and unlike-pair (np) interactions, respectively.
The range of the effective interaction is given by $a, ~b$ is the measure
of the strength of the momentum dependence of the interaction. The
relative separation of the interacting nucleons is ${\bf r=r_1-r_2 }$
and the relative momentum is ${\bf p=p_1-p_2 }$; $d$ and $\kappa$ are
the two parameters governing the strength of the density dependence
and $\rho (r_1)$ and $\rho (r_2)$ are the nucleon densities at the sites
of the two interacting nucleons. The potential parameters are given
in Table~I; the details for the determination of these
parameters are given in \cite{de1}. \\
The incompressibility $K_\infty$ of
symmetric nucleon matter is mostly governed by the
parameter $\kappa$; for the potential set we have chosen, the value
of $K_\infty$=238 MeV.
The equation of state of symmetric nuclear matter calculated with the
SBM interaction is seen to agree extremely well \cite{uma} with
that obtained in a variational approach by Friedman and Pandharipande
\cite{fri} with $v_{14}$+TNI interaction. This interaction
also reproduces quite well the binding energies,
rms charge radii, charge distributions
and the giant monopole resonance energies for a host of even-even nuclei
ranging from $^{16}$O to very heavy systems \cite{de1}. Interactions
of this type has been used with great success by Myers and Swiatecki
\cite{mye} in the context of nuclear mass formula.
\subsection{The occupation functions}
The self-consistent occupation probabilities of nucleons and alphas
in $\alpha $N matter at temperature $T$ are
obtained in the TF approximation
by minimizing the thermodynamic potential of the system
\begin{eqnarray}
\Omega = E-TS-\sum_\tau \mu_\tau N_\tau -\mu_\alpha N_\alpha .
\end{eqnarray}
Here $\tau $ represents the isospin index (n,p). The quantities $E,S,
\mu_\tau, \mu_\alpha, N_\tau $ and $N_\alpha $ are the total internal
energy, entropy, nucleon chemical potentials, $\alpha $ chemical potential,
free nucleon numbers and the number
of $\alpha $ particles, respectively, in
the system. Chemical equilibrium in the system ensures
\begin{eqnarray}
\mu_\alpha = 2(\mu_n +\mu_p ).
\end{eqnarray}
The total energy of the $\alpha $N matter in TF approximation is
\begin{eqnarray}
&&E= \sum_\tau\Biggl \{\int d{\bf r_1} d{\bf p_1} \frac{p_1^2}{2m_\tau }
\tilde n_\tau ({\bf p_1}) +\frac{1}{2} \int d{\bf r_1}d{\bf p_1}
d{\bf r_2}d{\bf p_2} \nonumber \\
&&\times [v_1(|{\bf r_1}-{\bf r_2}|, |{\bf p_1}
-{\bf p_2}|)+v_2(|{\bf r_1}-{\bf r_2}|,2\rho)]
[C_l\tilde n_\tau ({\bf p_2}) \nonumber \\
&&+C_u \tilde n_{-\tau }({\bf p_2})] \tilde n_\tau ({\bf p_1})
+\frac{1}{2}(C_l +C_u )\int d{\bf r_1} d{\bf p_1} d{\bf r_2}
d{\bf p_2} \nonumber \\
&&\times \tilde n_\tau ({\bf p_1}) \tilde n_\alpha ({\bf p_2})
\int_{V_\alpha} d{\bf r}\int d{\bf p_i^\alpha }\tilde n_i^\alpha
({\bf p_i^\alpha}) [v_1({\bf R^{\prime }},|{\bf p_1}-({\bf p_i^\alpha}
\nonumber \\
&&+{\bf p_2})|) + v_2({\bf R^{\prime }}, (\sum_{\tau^{\prime }}
\rho_{\tau^{\prime}}+\rho_i^\alpha))] \Biggr \}
\nonumber \\
&&+\int d{\bf r_1}d{\bf p_1}
\frac{p_1^2}{2m_\alpha}\tilde n_\alpha ({\bf p_1}) \nonumber \\
&& +(C_l+C_u)\int d{\bf r_1}d{\bf p_1} d{\bf r_2} d{\bf p_2}
\tilde n_\alpha ({\bf p_1})\tilde n_\alpha ({\bf p_2}) \nonumber \\
&& \times \int_{V_\alpha } d{\bf r} d{\bf r^\prime }
\int d{\bf p_i^\alpha}
d{\bf p_i^{\alpha ^{\prime}}}\tilde n_i^\alpha ({\bf p_i^\alpha})
\tilde n_i^{\alpha^{\prime}}({\bf p_i^{\alpha^{\prime}}}) \nonumber \\
&& \times [v_1({\bf R^{\prime}}, |({\bf p_1}+{\bf p_i^\alpha })
-({\bf p_2}+{\bf p_i^{\alpha^{\prime}}}|)+v_2({\bf R^{\prime}},
2\rho_i^\alpha)] \nonumber \\
&& -N_\alpha B_\alpha
\end{eqnarray}
In Eq.~(5), $m_\tau $ and $m_\alpha $ are the nucleon and
$\alpha $ masses, the first and fourth terms correspond to the kinetic energy
of the nucleons and alphas, the second and the fifth terms refer
to the interaction energy among nucleons and among alphas,
respectively and the third term is the interaction energy
between nucleons and alphas.
The various space coordinates occurring in the third and fifth terms are shown
in Figs.~1 and 2, respectively.
\begin{figure}
\includegraphics[width=1.0\columnwidth,angle=0,clip=true]{fig1.eps}
\caption{ Space coordinates shown
for nucleon (located
at $A$) and alpha (with center at $B$) configuration. The origin
of the coordinate system is at $O$
and $P$ is any arbitrary point within alpha.}
\end{figure}
\begin{figure}
\includegraphics[width=1.0\columnwidth,angle=0,clip=true]{fig2.eps}
\caption{ Space coordinates shown for alpha-alpha configuration with
$O$ as the origin
of the coordinate system. $P$ and
$P^{\prime}$ are arbitrary points within the alphas
with $A$ and $A^{\prime}$ as their centers.}
\end{figure}
These terms are evaluated in the single-folding
and double-folding models. The last term is the binding
energy contribution from the $\alpha $ particles. Here $\tilde n_\tau =
\frac{2}{h^3}n_\tau, \tilde n_\alpha =\frac{1}{h^3}n_\alpha $ where
$n_\tau $ and $n_\alpha $ are the occupation probabilities for
nucleons and alphas, respectively. Similarly,
$\tilde n_i^{\alpha }({\bf p_i^{\alpha }})=\frac{2}{h^3}
n_i^{\alpha }({\bf p_i^{\alpha }})$ represents the occupation
probability of the constituent nucleons in the $\alpha $ particle
and ${\bf p_i^{\alpha }}$ is their intrinsic momentum inside
the $\alpha $. The space coordinates do not enter in the occupation
functions $\tilde n_\tau $ and $\tilde n_\alpha $ as the system
is infinite. For simplicity, the $\alpha $ particles are taken to
be uniform nuclear drops with a sharp surface and hence the space
coordinates do not also occur in $\tilde n_i^{\alpha }$. The
notation $\int_{V_{\alpha }}$ refers to configuration integral
over the volume of $\alpha $. The integral over $\tilde {\bf p_i^\alpha }$
is over the Fermi sphere of the nucleon momenta inside the $\alpha $
particles. Since alphas are difficult to excite (the first excited
state in $\alpha $ is $\sim $ 20 MeV), they are
taken to be in their ground states.
All the other integrals are over the
entire configuration or momentum space unless otherwise specified.
It then follows that
\begin{eqnarray}
\int \tilde n_\tau ({\bf p})d{\bf p}= N_\tau/V=\rho_\tau , \nonumber \\
\int \tilde n_\alpha ({\bf p})d{\bf p} =N_\alpha /V=\rho_\alpha ,
\nonumber \\
\int \tilde n_i^{\alpha }({\bf p}) d{\bf p}=4/V_\alpha =\rho_i^{\alpha } ,
\end{eqnarray}
where $V$ is the volume of the $\alpha$N system and
$V_\alpha =\frac{4}{3}\pi R_\alpha ^3 $, with $R_\alpha $ as
the sharp-surface radius of the $\alpha $ drop taken to be 2.16 fm
obtained from experimental rms charge-radius of $\alpha $;
$\rho_i^\alpha $ is the density of the constituent nucleons of
the $\alpha $ particles.
The total
baryon density $\rho_b $ is given by $\rho_b=\rho +4\rho_\alpha $
where $\rho =\sum_\tau \rho_\tau $ is the density of the free
nucleons and $\rho_\alpha $ is the $\alpha $-particle density.
The total entropy of the $\alpha $N system is
\begin{eqnarray}
S=\sum_\tau S_\tau +S_\alpha ,
\end{eqnarray}
where in the Landau quasi-particle approximation,
\begin{eqnarray}
S_\tau &=&\frac{2}{h^3} \int \bigl [n_\tau ({\bf p}) \ln n_\tau ({\bf p})
\nonumber \\
&&+(1-n_\tau ({\bf p}))\ln (1-n_\tau ({\bf p})) \bigr ] d{\bf r}d{\bf p},
\end{eqnarray}
and
\begin{eqnarray}
S_\alpha & =& \frac{1}{h^3} \int \bigl
[n_\alpha ({\bf p}) \ln n_\alpha ({\bf p}) \nonumber \\
&& -(1+n_\alpha ({\bf p}))\ln (1+n_\alpha ({\bf p})) \bigr ] d{\bf r}d{\bf p}.
\end{eqnarray}
Minimization of $\Omega $ with respect to $n_\tau $ and $n_\alpha $,
remembering that $\delta n_\tau ({\bf p})$ and $\delta n_\alpha
({\bf p})$ are separately arbitrary over the whole phase space,
at the end yields
\begin{eqnarray}
&&\frac{p_1^2}{2m_\tau }+\int d{\bf r_2}d{\bf p_2}\Bigl \{v_1(|{\bf r_1}
-{\bf r_2}|,|{\bf p_1}-{\bf p_2}|) \nonumber \\
&&+v_2(|{\bf r_1}-{\bf r_2}|,2\rho )\Bigr \}
[C_l\tilde n_\tau ({\bf p_2})+C_u\tilde n_{-\tau }({\bf p_2})] \nonumber \\
&& +\kappa d^2 (2\rho ) ^{\kappa -1}\sum_{\tau ^{\prime }}\int
d{\bf p_1^{\prime }}d{\bf r_2}d{\bf p_2} \nonumber \\
&&\times [C_l \tilde n_{\tau^{\prime }}
({\bf p_2})+C_u\tilde n_{-\tau^{\prime }}
({\bf p_2})]\tilde n_{\tau^{\prime }}
({\bf p_1}^{\prime })f({\bf r_1},{\bf r_2}) \nonumber \\
&&+\frac{1}{2}
(C_l+C_u)\int d{\bf r_2}d{\bf p_2}\tilde n_\alpha ({\bf p_2}) \nonumber \\
&& \times \int d{\bf r}d{\bf p_i^\alpha }
\tilde n_i^{\alpha }({\bf p_i^{\alpha }})
\bigl \{v_1({\bf R^{\prime}},|{\bf p_1}-({\bf p_i^{\alpha }}+
{\bf p_2})|) \nonumber \\
&& +v_2({\bf R^{\prime }},(\rho +\rho_i^{\alpha } )) \bigr \}
+\frac{1}{4}(C_l+C_u)\kappa d^2(\rho +\rho_i^\alpha )^{\kappa -1} \nonumber \\
&& \times \sum_{\tau^{\prime }}\int d{\bf p_1^\prime }d{\bf p_2}
\tilde n_{\tau^{\prime }}({\bf p_1^\prime })\tilde n_{\alpha }({\bf
p_2 })\rho_i^\alpha \nonumber \\
&&\times \int d{\bf r_2}\int_{V_{\alpha }}d{\bf r}
\frac{e^{-|{\bf R ^{\prime }}|/a }}{|{\bf R ^{\prime }}|/a } \nonumber \\
&& +T\bigl [\ln n_{\tau }({\bf p_1 })
-\ln (1-n_{\tau }({\bf p_1 }))\bigr ]
-\mu_{\tau}=0 ,
\end{eqnarray}
and
\begin{eqnarray}
&& \frac{p_1^2}{2m_{\alpha }}+2(C_l+C_u)\int d{\bf r_2}d{\bf p_2}
\tilde n_{\alpha}({\bf p_2}) \nonumber \\
&&\times \int d{\bf r}d{\bf p_i^{\alpha }}d{\bf r^{\prime }}d{\bf p_i^
{\alpha \prime }}
\tilde n_i^\alpha ({\bf p_i^\alpha })\tilde n_i^{\alpha \prime }
({\bf p_i^{\alpha \prime }}) \nonumber \\
&&\times \Bigl \{v_1({\bf R^\prime },|({\bf p_1+p_i^\alpha })-({\bf p_2+p_i^{\alpha
\prime }})|)+v_2({\bf R^\prime },2\rho_i^\alpha ) \Bigr \} \nonumber \\
&& +\frac{1}{2}(C_l+C_u)\sum_\tau \int d{\bf p_2}
d{\bf p_i^\alpha }\tilde n_\tau
({\bf p_2}) \tilde n_i^\alpha ({\bf p_i^\alpha }) \nonumber \\
&& \times \int d{\bf r_2}\int_{V_\alpha }d{\bf r}
\Bigl \{v_1({\bf R^\prime },|{\bf p_2}-
({\bf p_1}+{\bf p_i^\alpha })|) \nonumber \\
&& +v_2({\bf R^\prime },
\rho +\rho_i^\alpha ) \Bigr \} +T \bigl [\ln n_\alpha ({\bf p_1})-\ln (1+
n_\alpha ({\bf p_1 })) \bigr ] \nonumber \\
&& -(\mu_\alpha +B_\alpha )=0.
\end{eqnarray}
Without any loss of generality, ${\bf r_1}$ can be set equal to
zero in Eqs.~(10) and (11).
The single-particle occupation functions $n_\tau (p)$ and $n_\alpha (p)$
for nucleons and alphas are determined from Eqs.~ (10) and (11),
respectively.
Eq.~(10), after some algebraic manipulations can be written as
\begin{eqnarray}
&& \frac{p_1^2}{2m_\tau}+V_\tau^0+p_1^2V_\tau^1+V_\tau^2 \nonumber \\
&& + T \bigl [\ln n_\tau ({\bf p_1})
-\ln (1-n_\tau ({\bf p_1})) \bigr ]-\mu_\tau =0.
\end{eqnarray}
The momentum-dependent nucleon single-particle potential
$V_\tau (p) $ is given by
\begin{eqnarray}
V_\tau (p)=V_\tau^0+p^2V_\tau^1 ,
\end{eqnarray}
where $V_\tau^0 $ is the momentum-independent part. Eq.~(12) leads
to
\begin{eqnarray}
n_\tau (p)=\left [1+exp \left \{ \left ( \frac{p^2}{2m_\tau^*}
+V_\tau^0+V_\tau^2-\mu_\tau \right )/T \right \} \right ]^{-1},
\end{eqnarray}
where $m_\tau^* $ is the nucleon effective mass,
\begin{eqnarray}
m_\tau^*=\left [\frac{1}{m_\tau}+2V_\tau^1 \right ]^{-1},
\end{eqnarray}
and $V_\tau^2 $ is the rearrangement potential coming from the
density dependence of the interaction. Similarly Eq.~(11) can
be written as
\begin{eqnarray}
&& \frac{p_1^2}{2m_\alpha}+V_\alpha^0+p_1^2V_\alpha^1
+ T \bigl [\ln n_\alpha ({\bf p_1})
-\ln (1+n_\alpha ({\bf p_1})) \bigr ] \nonumber \\
&& -(\mu_\alpha +B_\alpha ) =0,
\end{eqnarray}
which yields
\begin{eqnarray}
n_\alpha (p)=\left [exp \left ( \left \{ \frac{p^2}{2m_\alpha^*}
+V_\alpha^0-(\mu_\alpha +B_\alpha )\right \}/T \right )
-1 \right ]^{-1}
\end{eqnarray}
where
\begin{eqnarray}
m_\alpha^*=\left [\frac{1}{m_\alpha}+2V_\alpha^1 \right ]^{-1},
\end{eqnarray}
is the $\alpha $ effective mass. $V_\alpha ^0 $ is the momentum-independent
part of the $\alpha $-single particle potential $V_\alpha
(=V_\alpha ^0 +p^2V_\alpha ^1) $ in the system.
The nucleon and $\alpha $ masses
are renormalized due to the momentum dependence in the interaction.
The expressions for $V_\tau^0$
can be arrived at as,
\begin{eqnarray}
&& V_\tau^0= -4\pi a^3\left \{ 1-d^2(2\rho )^\kappa \right \}
(C_l\rho_\tau +C_u\rho_{-\tau }) \nonumber \\
&& +\frac{16\pi^2a^3}{b^2h^3} \biggl [C_l(2m_\tau^*T)^{5/2}J_{3/2}
(\eta_\tau )+C_u(2m_{-\tau }^*T)^{5/2} \nonumber \\
&& \times J_{3/2}(\eta_{-\tau }) \biggr ]
+\frac{1}{4}I(C_l+C_u)\rho_\alpha \rho_\alpha^i
\biggl [ \frac{<p_\alpha^2>+<(p_i^\alpha )^2>}{b^2} \nonumber \\
&& +d^2(\rho
+\rho_i^\alpha )^\kappa -1 \biggr ].
\end{eqnarray}
The first two terms come from the interaction
between free nucleons, the last
term originates from the presence of alphas.
In Eq.~(19), $I$ is the six-dimensional integral (see Fig.~1)
\begin{eqnarray}
I=\int_{V_\alpha} d{\bf r} \int d{\bf R} \frac{e^{-|{\bf r}+{\bf R}|/a}}
{|{\bf r}+{\bf R}|/a}.
\end{eqnarray}
This integral can be evaluated analytically.
The quantity $<p_\alpha^2>$ is the mean squared value of the
$\alpha$ momentum in $\alpha$N matter and $<(p_i^\alpha)^2>$
is the mean squared value of the constituent nucleon momentum inside
the $\alpha$. The value of $<p_\alpha^2>$ is
\begin{eqnarray}
<p_\alpha^2>&=&(2m_\alpha^*T)B_{3/2}(\eta_\alpha)/B_{1/2}(\eta_\alpha)
\nonumber \\
&&\simeq 3m_\alpha^*T,
\end{eqnarray}
and
\begin{eqnarray}
<(p_i^\alpha)^2>\simeq \frac{3}{5}(P_F^\alpha)^2
\end{eqnarray}
where $P_F^\alpha$ is the value
of the zero-temperature nucleon Fermi momentum inside
$\alpha$, taken to be 220.5 MeV/c, consistent with the $\alpha$
sharp surface radius. The $J_k(\eta )$ and $B_k(\eta )$ are the
Fermi and Bose integrals,
\begin{eqnarray}
J_k(\eta )=\int_0^\infty \frac{x^k~dx}{e^{(x-\eta)}+1},
\end{eqnarray}
and
\begin{eqnarray}
B_k(\eta )=\int_0^\infty \frac{x^k~dx}{e^{(x-\eta)}-1},
\end{eqnarray}
with
\begin{eqnarray}
&& \eta_\tau =(\mu_\tau -V_\tau^0 -V_\tau^2 )/T, \nonumber \\
&& \eta_\alpha =(\mu_\alpha +B_\alpha -V_\alpha^0 )/T.
\end{eqnarray}
The expressions for $V_\tau^1 ,
V_\tau^2$, $V_\alpha^0$ and $V_\alpha^1 $
are given as
\begin{eqnarray}
V_\tau^1=\frac{4\pi a^3}{b^2}[C_l\rho_\tau +C_u \rho_{-\tau }]
+\frac{1}{4}I(C_l+C_u) \frac{\rho_\alpha \rho_\alpha^i }{b^2},
\end{eqnarray}
\begin{eqnarray}
&& V_\tau^2= 4 \pi a^3 \kappa d^2(2\rho )^{\kappa -1} \sum_{\tau \prime }
[C_l\rho_{\tau ^\prime }+C_u \rho_{-\tau ^\prime }]\rho_{\tau ^\prime }
\nonumber \\
&& +\frac{1}{4}I(C_l+C_u)\kappa d^2(\rho+\rho_i^\alpha )^{\kappa -1}
\rho_i^\alpha \rho_\alpha \rho,
\end{eqnarray}
\begin{eqnarray}
&& V_\alpha^0 = \frac{1}{4}(C_l+C_u)\rho_i^\alpha \Biggl \{2\rho_i^\alpha
\rho_\alpha I_\alpha \bigl [d^2(2\rho_i^\alpha )^\kappa -1 \nonumber \\
&& +\frac{3m_\alpha^*T+\frac{6}{5}(P_F^\alpha )^2}{b^2} \bigr ]
+I \bigl [\rho \bigl \{ d^2(\rho+\rho_i^\alpha )^\kappa -1 \nonumber \\
&& +\frac{3}{5}\frac{(P_F^\alpha )^2 }{b^2} \bigr \}
+\sum_\tau \frac{ 4\pi (2m_\tau^*T)^{5/2} J_{3/2}(\eta_\tau )}
{h^3b^2} \bigr ] \Biggr \},
\end{eqnarray}
and
\begin{eqnarray}
V_\alpha^1=\frac{1}{4}(C_l+C_u)\rho_i^\alpha \bigl \{2\rho_i^\alpha
\rho_\alpha I_\alpha +I\rho \bigr \}/b^2.
\end{eqnarray}
In both $V_\tau^1 $ and $V_\tau^2 $, the last term stems from
the $\alpha $-N interaction. The effective nucleon mass in pure
nucleonic matter thus gets modified due to clusterization.
The integral $I_\alpha $ occurring in Eqs.~ (28) and
(29) is a nine-dimensional
integral (see Fig.~2),
\begin{eqnarray}
I_\alpha=\int_{V_\alpha }d{\bf r }\int_{V_\alpha }d{\bf r^\prime }
\int d{\bf R} \frac{e^{-|{\bf R}+{\bf r}-{\bf r^\prime }|/a}}
{|{\bf R}+{\bf r}-{\bf r^\prime }|/a},
\end{eqnarray}
which can be evaluated numerically.
If the alphas do not interpenetrate, the integral over ${\bf R }$
excludes the $\alpha $ volumes.
\subsection{Expressions for observables in TF approximation}
{\bf i) Energy per baryon :}
The energy per baryon $e_b$ of the $\alpha$N matter can be calculated
from Eq.~(5). It can be split into the following form,
\begin{eqnarray}
e_b=e_{NN}+e_{\alpha N}+e_{\alpha \alpha }.
\end{eqnarray}
Here $e_{NN} $ comes from the kinetic energy of the free nucleons
and the interactions among them, $e_{\alpha N }$ arises from the
interaction among the free nucleons and the alphas and
$e_{\alpha \alpha }$ stems from the kinetic energy of the
alphas and the interaction among themselves. The expressions for
them are
\begin{eqnarray}
e_{NN}&=&\frac{1}{\rho_b}\sum_\tau \rho_\tau \bigl [TJ_{3/2}(\eta_\tau)/
J_{1/2}(\eta_\tau ) \{ 1-m_\tau^*V_\tau^1 \} \nonumber \\
&&+\frac{1}{2}V_\tau^0 \bigr ],
\end{eqnarray}
\begin{eqnarray}
&& e_{\alpha N}= \frac{1}{4\rho_b}(C_l+C_u)I\rho_\alpha \rho_\alpha^i
\Bigl [\Bigl \{\frac{3m_\alpha^*T+3/5(P_F^\alpha)^2}{b^2} \nonumber \\
&& -1+d^2(\rho +\rho_i^\alpha )^\kappa \Bigr \} \rho \nonumber \\
&& +\frac{1}{b^2}\sum_\tau \bigl (\frac{4\pi }{h^3}(2m_\tau^*T)^{5/2}
J_{3/2}(\eta_\tau ) \bigr ) \Bigr ],
\end{eqnarray}
and
\begin{eqnarray}
&& e_{\alpha \alpha }=\frac{1}{\rho_b} \Bigl [\frac{\pi }{m_\alpha h^3}
(2m_\alpha^*T)^{5/2}B_{3/2}(\eta_\alpha ) \nonumber \\
&& + \frac{1}{4}(C_l+C_u)I_\alpha \rho_\alpha^2 (\rho_i^\alpha )^2
\Bigl \{ d^2(2\rho_i^\alpha )^\kappa -1
\nonumber \\
&& +\frac{6m_\alpha^*T}{b^2}
+\frac{6}{5}\frac {(P_F^\alpha )^2}{b^2} \Bigr \} \Bigr ].
\end{eqnarray}
In the above equations, as stated earlier, $\rho_b$
(=$\rho +4\rho_\alpha $) corresponds to the total baryon density,
$\rho $ and $\rho_\alpha $ are the free nucleon and $\alpha $
densities, respectively, in the $\alpha$N system. \\
{\bf ii) Entropy per baryon:}
The entropy per baryon $s_b$ can be evaluated using Eqs.~(8) and (9). It is
additive and can be written as
\begin{eqnarray}
s_b=s_N+s_\alpha ,
\end{eqnarray}
where $s_N$ and $s_\alpha $ are the contributions to entropy
from free nucleons and alphas respectively. Their expressions
reduce to
\begin{eqnarray}
s_N=\frac{1}{\rho_b }\sum_\tau \rho_\tau \Bigl
[\frac{5}{3} J_{3/2}(\eta_\tau )
/J_{1/2}(\eta_\tau )-\eta_\tau \Bigr ],
\end{eqnarray}
and
\begin{eqnarray}
s_\alpha =\frac{\rho_\alpha }{\rho_b }
\Bigl [\frac{5}{3} B_{3/2}(\eta_\alpha )
/B_{1/2}(\eta_\alpha )-\eta_\alpha \Bigr ].
\end{eqnarray}
\vskip 0.5cm
{\bf iii) Pressure of $\alpha $N matter:}
Once the energy and entropy of the composite system are known,
the pressure can be calculated from the Gibbs-Duhem thermodynamic
identity,
\begin{eqnarray}
P=\sum_\tau \rho_\tau \mu_\tau +\rho_\alpha \mu_\alpha -f_b\rho_b,
\end{eqnarray}
where $f_b$ is the free energy per baryon, $f_b=e_b-Ts_b $. \\
{\bf iv) Incompressibility and the symmetry coefficients:}
The incompressibility
$K$ can be computed from the derivative of pressure
\begin{eqnarray}
K=9\frac{dP}{d\rho}.
\end{eqnarray}
The symmetry free energy and symmetry energy coefficients $C_F$ and
$C_E$ are calculated from
\begin{eqnarray}
C_F=\frac{1}{2}\Bigl ( \frac{\partial^2 f_b}{\partial X^2} \Bigr )_{X=0},
\end{eqnarray}
\begin{eqnarray}
C_E=\frac{1}{2}\Bigl ( \frac{\partial^2 e_b}{\partial X^2} \Bigr )_{X=0},
\end{eqnarray}
where $X$ is the neutron-proton asymmetry of the $\alpha $N system.
It is given as $X=(\rho_b^n-\rho_b^p)/\rho_b $, where $\rho_b^n$
and $\rho_b^p $ are the total neutron and proton
density, respectively.
\vskip 1.0cm
\subsection {The $S$-matrix approach}
The relevant key elements of the $S$-matrix framework \cite{das}
as applied in the context of dilute nuclear matter \cite{mal,de}
are outlined in brief below.
The grand partition function of an interacting infinite system of
neutrons and protons can be written as
\begin{equation}
{\cal Z} = \sum_{Z,N=0}^\infty (\zeta_p)^Z (\zeta_n)^N
\, {\rm Tr}_{Z,N}\, e^{-\beta H}~.
\end{equation}
where $\zeta_p =e^{\beta \mu_p}$ and $\zeta_n=e^{\beta \mu_n}$
are the elementary fugacities with $\beta =1/T$ and $\mu $'s are
the nucleonic chemical potentials. Here $H$ is the total Hamiltonian
of the system and the trace $Tr_{Z,N}$ is taken over states of
$Z$ protons and $N$ neutrons. The partition function can be split
into two types of terms \cite{das}
\begin{equation}
\ln {\cal Z} =\ln {{\cal Z}}_{part}^{(0)} + \ln {{\cal Z}}_{scat}~.
\end{equation}
The first term on the right hand side corresponds to contributions
from stable single-particle states of clusters of different sizes
including free nucleons formed in the system; the second term refers
to all possible scattering states. The superscript (0) indicates that
the clusters behave as an ideal quantum gas. In general,
$\ln {{\cal Z}}_{part}^{(0)} $ contains contributions from the ground states
as well as the particle-stable excited states of all the clusters.
The scattering term $\ln {{\cal Z}}_{scat}$ may be written as a sum
of scattering contributions from a set of channels,
each set having total proton
number $Z_t$ and neutron number $N_t$. Since our interest in the
present work is focused on $\alpha $N matter, in $\ln {{\cal Z}}_{part}^{(0)}$,
we include only the nucleons and the ground state of $\alpha $; similarly
in $\ln {{\cal Z}}_{scat}$, only the scattering channels $NN, \alpha N $ and
$\alpha \alpha $ are considered, so that
\begin{equation}
\ln {{\cal Z}}_{scat}~=~ \ln {{\cal Z}}_{NN}
+\ln {{\cal Z}}_{\alpha N}+\ln {{\cal Z}}_{\alpha \alpha}.
\end{equation}
Each of the terms in Eq.~(44) can be expanded in the respective virial
coefficients. Expansion upto the second-order coefficients are only
considered. They are written as energy integrals of the relevant
phase-shifts \cite{hor,sam}.
The partition function can then be written explicitly as
\begin{eqnarray}
&&\ln {{\cal Z}}=V \Bigl \{\frac{2}{\lambda_N^3} [ \zeta_n +\zeta_p+
\frac{b_{nn}}{2}\zeta_n^2+\frac{b_{pp}}{2}\zeta_p^2
+\frac{1}{2}b_{np}\zeta_n\zeta_p \nonumber \\
&& +8\zeta_\alpha +8b_{\alpha \alpha }\zeta_\alpha^2
+8b_{\alpha n}\zeta_\alpha (\zeta_n+\zeta_p)] \Bigr \},
\end{eqnarray}
where $\lambda_N$=$\frac{h}{\sqrt{2\pi mT}} $ is the nucleon thermal
wavelength, $\zeta_\alpha $=$ e^{\beta (\mu_\alpha +B_\alpha )}$, $B_\alpha $
being the binding energy of $\alpha $ and $\mu_\alpha =2(\mu_n +\mu_p )$.
The $b_{nn}$, $b_{np}$, etc., are the temperature dependent virial coefficients
\cite{hor,mal}. The value of the virial coefficient $b_{np}$ has
been adjusted so as to exclude the resonance formation of deuteron
from n-p scattering to be consistent with our choice of the
$\alpha $N matter.
The knowledge of the partition function allows all the relevant observables
to be calculated. The pressure is given by
\begin{eqnarray}
P=T \ln {{\cal Z}}/V \,.
\end{eqnarray}
The number density $\rho_i$ is calculated from
\begin{eqnarray}
\rho_i=\zeta_i \left (\frac{\partial}{\partial \zeta_i}
\frac{\ln {{\cal Z}}}{V} \right )_{V,T},
\end{eqnarray}
where $i$ stands for n,p, or $\alpha$. Once the pressure,
densities and chemical potentials are known, the free
energy can be obtained from the Gibbs-Duhem relation. The
entropy per baryon is calculated from
\begin{eqnarray}
s_b=\frac{1}{\rho_b}\Bigl (\frac{\partial P}{\partial T} \Bigr )_\mu,
\end{eqnarray}
which yields the energy per baryon as $e_b=f_b+Ts_b$. The explicit
expression for the entropy per baryon is
\begin{eqnarray}
&& s_b=\frac{1}{\rho_b}\Biggl \{\frac{5}{2}\frac{P}{T}
-\sum_i \rho_i \ln \zeta_i \nonumber \\
+&& \frac{T}{\lambda_N^3}\bigl [\zeta_n\zeta_p b_{np}^{\prime}+(\zeta_n^2+
\zeta_p^2)b_{nn}^{\prime}
\nonumber \\
&& +8\zeta_{\alpha}^2b_{\alpha \alpha}^
{\prime}+8\zeta_{\alpha}(\zeta_n+\zeta_p)b_{\alpha n}^{\prime}
\bigr ]\Biggr \}.
\end{eqnarray}
The prime on the virial coefficients denotes their temperature derivatives.
\section{Results and Discussions}
In the mean-field framework, the momentum and density-dependent
finite-range modified Seyler-Blanchard force as scripted in
Eqs.~(1) and (2) has been chosen as the effective two-nucleon interaction
in our calculations.
To start with, we take baryon matter at a given density $\rho_b $
at a temperature $T$ with an isospin asymmetry $X$. The unknowns
are the free nucleon densities $\rho_n$, $\rho_p$ and the $\alpha $
concentration in the matter. The three constraints are the conservation
of the total baryon number, the total isospin and the condition of
chemical equilibrium between the nucleons and alphas.
Starting from a guess value for the
$\alpha $ concentration, the unknowns are determined iteratively
using the Newton-Raphson method. For our calculations, the masses
of neutron and proton are taken to be the same, for $\alpha $
binding energy, the experimental value of 28.3 MeV is used. For
the evaluation of the $\alpha \alpha $-potential, the
$\alpha $-particles are assumed to be nuclear droplets with
sharp boundary and that they do not interpenetrate.
\begin{figure}
\includegraphics[width=1.0\columnwidth,angle=0,clip=true]{fig3.eps}
\caption{(color online) The $\alpha$ fraction $Y_{\alpha}=
4\rho_\alpha /\rho_b $ shown as a function of baryon density
$\rho_b $ in TF and SM approaches at $T$=3, 5 and 10 MeV for
symmetric matter ($X$=0.0) and asymmetric matter ($X$=0.2)
in panels (a) and (b), respectively.}
\end{figure}
\begin{figure}
\includegraphics[width=1.0\columnwidth,angle=0,clip=true]{fig4.eps}
\caption{(color online) The $\alpha $ fraction $Y_\alpha $
displayed as a function of asymmetry $X$ at baryon density $\rho_b$
=0.001 (upper panel) and at 0.01 fm$^{-3}$ (lower panel) at
$T$= 3, 5 and 10 MeV in TF and SM approaches.}
\end{figure}
\begin{figure}
\includegraphics[width=1.0\columnwidth,angle=0,clip=true]{fig5.eps}
\caption{(color online) Free energy per baryon $F/A$ shown as a
function of $\rho_b $ at $T$=3, 5 and 10 MeV in the TF framework
for homogeneous nucleonic matter (blue lines) and $\alpha$N
matter (black lines). The red lines represent results from
the SM approach.}
\end{figure}
\begin{figure}
\includegraphics[width=1.0\columnwidth,angle=0,clip=true]{fig6.eps}
\caption{(color online) Pressure $P$ as a function of $\rho_b$.
The notations are the same as in Fig.~5.}
\end{figure}
\begin{figure}
\includegraphics[width=1.0\columnwidth,angle=0,clip=true]{fig7.eps}
\caption{(color online) The nucleon (full black lines) and
$\alpha $ (dashed black lines) effective masses shown as a
function of $\rho_b $ at $T$=3, 5 and 10 MeV in the TF framework
for $\alpha $N matter. The blue lines refer to the corresponding
nucleon effective masses for homogeneous nucleonic matter.}
\end{figure}
The calculations are done upto a baryon density $\rho_b$=0.01 fm$^{-3}$.
To show the effect of temperature on different properties of the dilute
matter, results are reported for temperatures $T$ = 3, 5, and
10 MeV. In Fig.~3, the baryon fraction in $\alpha $,
$Y_\alpha $ =$4\rho_\alpha /\rho_b $ (hereafter referred to as
$\alpha $ fraction) in $\alpha $N matter
as a function of density at
the three temperatures mentioned are shown for symmetric ($X$=0) and
asymmetric ($X$ =0.2) nuclear matter in panels (a) and (b),
respectively. The black lines correspond to results obtained
in the TF approximation [$\alpha $N (TF)], the red lines refer to
those in the SM approach [$\alpha $N (SM)] with consideration of
only n, p and $\alpha $ as the constituents of the baryonic matter.
At low temperatures and higher densities, it is seen
that alphas are the major constituents of the matter, with
increasing temperature, the free nucleon fraction increases at the
cost of $\alpha $ density. At moderate asymmetry $X$=0.2,
the $\alpha $ population is somewhat lower compared to that for
symmetric nuclear matter. In the temperature and density domain
that we explore, the results from both
the SM and TF approach are found to be quite close.
The asymmetry dependence of $\alpha $ fraction $Y_\alpha $ is displayed
in Fig.~4 at two representative densities $\rho_b$=0.001 and
0.01 fm$^{-3}$ at the three temperatures. With increasing
asymmetry, the $\alpha $ concentration decreases, the decrease is
more prominent at lower temperature. At the lower density (Fig.~4(a)),
results for $T$=10 MeV are not shown as $Y_\alpha $ is close to zero.
In Fig.~5, the free energy per baryon for the homogeneous nucleonic
matter (denoted by N(TF)) and the $\alpha $N matter in the TF
approximation are presented in panels (a), (b), and (c) at $T$=
3, 5, and 10 MeV, respectively. The calculations presented refer
to symmetric nuclear matter. The blue and black lines represent
results for N(TF) and $\alpha $N(TF). It is clearly
seen that the clusterized matter has lower free energy compared to
homogeneous nucleonic matter. This is more prominent at lower temperatures,
higher temperature tends to melt away the clusters. For comparison,
results from the $S$-matrix approach are also presented. They are
shown by the red lines, nearly indistinguishable from those
from $\alpha $N(TF). Fig.~6
displays the pressure of the baryonic matter.
At lower temperatures ($T$=3 and 5 MeV), the nucleonic matter
shows the rise and fall of the pressure with density leading to
unphysical region. For $\alpha $N matter, however, no such
unphysical region is observed in the density region
we have studied.
Both the TF and the SM approaches
yield nearly the same value of pressure. At high temperature
the $\alpha $ concentration becomes very less,
the pressure in all the three approaches are then nearly the same
in this density region.
In Fig.~7, the effective masses of nucleon and $\alpha $
are shown as a function of density at the temperatures mentioned.
The nucleon effective mass
is calculated for both nucleonic matter (blue line) and $\alpha $N
matter (full black line) in the TF approximation.
The nucleon effective mass at a given $\rho_b $ in homogeneous
nucleonic matter is always lower compared to that in clusterized
matter. It is independent of temperature. In $\alpha $N
matter it nominally decreases with temperature.
At high temperature,
the nucleon effective masses calculated in the homogeneous and clusterized
matter are nearly degenerate, with lowering of temperature, the
degeneracy is lifted due to the increase in the $\alpha $ concentration.
The effective $\alpha $ mass is shown by the dashed black lines.
With increasing temperature, the medium effect on the $\alpha $ mass
gets strikingly enhanced. This is due to the interplay of the
temperature-dependent contributions from the $\alpha \alpha $ interactions
and $\alpha $N interactions corresponding to the first and the second
term within the braces in Eq.~(29).
\begin{figure}
\includegraphics[width=1.0\columnwidth,angle=0,clip=true]{fig8.eps}
\caption{(color online) The incompressibility $K$ for baryonic matter
shown as a function of $\rho_b$ at $T$=3, 5 and 10 MeV. The
notations are the same as in Fig.~5.}
\end{figure}
The incompressibility of the baryonic matter as a function of density
is displayed in Fig.~8 at the three temperatures.
At very low density and higher temperature, the matter is mostly nucleonic
in all the three approaches, so the incompressibility $K$ is $\sim $
9$T$; this one sees at the lower densities considered at
$T$ =10 MeV in panel (c) of this figure.
Even at this very high temperature, however, the nucleonic
interactions have their role as the density increases;
this results in the reduction of the incompressibility from the
ideal gas value.
At the lower temperatures (panels (a) and (b)),
clusterization softens the matter towards compression compared to
homogeneous matter (shown in the lower density region); increasing
density, however, pushes the homogeneous matter towards
the unphysical region leading to negative
incompressibility.
\begin{figure}
\includegraphics[width=1.0\columnwidth,angle=0,clip=true]{fig9.eps}
\caption{(color online) The symmetry energy $C_E$ (left panels)
and symmetry free energy coefficients $C_F$ (right panels) shown
as a function of $\rho_b$ at temperatures $T$=3, 5 and 10 MeV.
The notations are the same as in Fig.~5.}
\end{figure}
\begin{figure}
\includegraphics[width=1.0\columnwidth,angle=0,clip=true]{fig10.eps}
\caption{(color online) The free energy per particle, pressure
and $\alpha $ fraction shown as a function of $\rho_b$ at
$T$=3 MeV in panels (a), (b) and (c), respectively, for $\alpha$
drops with no overlap (full black lines) and with at best 5$\% $
overlap (dashed-dot black lines) in the TF approximation. The
same observables are also shown in the SM approach (red lines).}
\end{figure}
The symmetry energy coefficients $C_E$ and $C_F$ of the baryonic
matter as a function of density are displayed in the left and
right panels, respectively, of Fig.~9 at the three temperatures
studied. The blue lines refer to calculations for the homogeneous
matter, the black and red lines represent results for $\alpha $N(TF)
and $\alpha $N(SM). Clusterized matter displays a marked increase
in the symmetry coefficients noticed already earlier \cite{hor,sam}.
The two approaches to clusterization
lead to the same values of the symmetry
coefficients at lower densities, with increase in density
the difference widens, more so at lower temperatures.
The results presented so far in the $\alpha $N(TF) approach have been
calculated with the assumption that the alphas do not overlap, they are
mutually impenetrable spherical drops. This assumption
relies on the fact that the alphas are very tightly bound and very hard
to excite.
To explore the effect of overlap in alphas, we consider a possibility
of penetration with at best a 5 $\% $ overlap in volume
(the value of $I_\alpha $
in Eq.~(30) then changes accordingly). Calculations have been repeated
with this changed condition. The so-calculated free energy per baryon,
pressure and the $\alpha $ fraction $Y_\alpha $ in the baryonic matter
are presented in panels (a), (b) and (c), respectively,
of Fig.~10 at $T$ =3 MeV
(the dot-dashed black lines) and compared with those calculated with the
no-overlap condition (the full black lines) and also those from the
$\alpha $N(SM) approach (the red lines). There is no significant change
in the free energy or in $\alpha $ fraction, but the pressure changes
perceptibly, particularly at higher density. The good agreement
between the no-overlap $\alpha $N(TF) calculations with those from the
bench-mark $\alpha $N(SM) shows the viability
of the approximation of the impenetrability
of the alphas.
\section{Concluding remarks}
Clusterization in warm dilute nuclear matter has been treated
earlier in the virial approach or in the $S$-matrix framework. These
are model-independent parameter-free calculations.
As explained in the introduction, these methods may have limitations
at relatively high densities and low temperatures. An alternate
avenue for dealing with clusterized matter in a broadened density
and temperature domain is suggested in the mean-field framework
in the present paper. The suggested method may be lengthy at
relatively higher densities where many different fragment species
are formed, but it is straightforward. To explore its applicability
in a wider domain, as a first step,
we consider only
n, p and $\alpha $ as the constituents of the matter
at low densities and see how
the results compare with those from the model-independent virial
approach.
We have chosen the SBM interaction that
nicely reproduces the bulk properties
of nuclear matter and of finite nuclei. We have calculated
the $\alpha $ fraction, free energy, pressure, incompressibility and
the symmetry coefficients of this $\alpha $N matter in this mean-field
framework and find that all these results compare extremely well with
those obtained from the $S$-matrix method, particularly in the
low-density high-temperature regime.
This gives one confidence
in the applicability of this mean-field approach in dealing with the
EOS of warm dilute baryonic matter and the possibility of extending
this method to higher densities.
The price, however, is consideration of a larger number of fragment
species and a numerically involved calculation.
\begin{acknowledgments}
S.K.S. and J.N.D acknowledge support of DST, Government of India.
\end{acknowledgments}
|
1,314,259,996,536 | arxiv | \subsection{I. Introduction}
Wafer-scale methods for the epitaxial growth of thin films of Aluminum (Al) on Indium Arsenide (InAs) heterostructures have recently been developed which yield uniform and atomically flat interfaces \cite{Kaushini2018,Shabani2016, JoonSue, Pauka2019}. Josephson junctions fabricated on these materials yield a gate-controllable supercurrent with highly transparent contacts between the Al top layer and an InAs quantum well (QW) directly below the surface \cite{MortenPRA2017,Henri17,Billy2019,Lee2019,Fabrizio17}. Tuning of the semiconductor properties will affect supercurrent and other superconducting properties due to the wavefunction overlap at the epitaxial interface. Josephson junctions made out of Al-InAs have been used for tunable superconducting qubits, the so-called ``gatemon'' where the Josephson energy can be tuned in-situ with an applied electric field \cite{Larsen15, Casparis2018}. Furthermore, since InAs has large spin-orbit coupling, they can host topological superconductivity and Majorana bound states \cite{2019Mayer_Mat, mayer2019anom, Ren2019, FornieriNature2019}. The key feature in these structures is that the two-dimensional electron gases (2DEG) is confined near the surface, in close proximity to the superconductor. While the epitaxial interface creates high contact transparency, it is expected that electron mobility of the 2DEG deteriorates due to increased rates of surface scattering as compared to isolated 2DEGs buried beneath the surface \cite{Kaushini2018,hatke,Shayegan2017}. The myriad of possible applications with this platform implores a deeper study of the characteristics and material properties for near surface InAs QWs. In this work, the transport experiments investigate the isolated semiconductor with the superconducting layer removed and the optical measurements are conducted on the semiconductor samples which did not have a superconducting layer to begin with.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{fig1J.pdf}
\caption{(Color online) (a) Measured longitudinal resistance $R_{xx}$ vs magnetic field over a range of densities from $3.9 \times 10^{11} cm^{-2}$ to $3.1 \times 10^{12} cm^{-2}$. Dashed lines indicate the traces that are shown in (b-d). Integer quantum Hall states are labeled from complementary $R_{xy}$ data. (b-d) Longitudinal $R_{xx}$ and transverse $R_{xy}$ magnetotransport data at particular densities. The various integer quantum Hall states are labeled. The left axis (blue trace) shows the longitudinal resistance $R_{xx}$ and the right axis (red trace) shows the transverse resistance $R_{xy}$.}
\label{fig:Rxx}
\end{figure*}
Two important material parameters of a 2DEG are the effective mass, $m^*$, and the effective $g$ factor, $g^*$. These parameters dictate the response of a material to external electric and magnetic fields. Their effect on device performance should be accounted for in the design of mesoscopic devices and realistic theoretical modeling. Both $m^*$ and $g^*$ have been measured and calculated for bulk InAs \cite{InAsBulk} and for InAs QWs \cite{InAsQW,InAsQWSim}. It is of particular interest that confinement of the electron wave function can strongly affect these values. Confinement becomes relevant when the 2DEG is placed near the surface, as is required for epitaxial contacts. In addition, narrow gap semiconductors can lead to strong non-parabolicity of the bands modifying the $m^*$ and $g^*$. However, to date, very few experimental studies have been performed to quantify the $m^*$ and $g^*$ in near surface InAs quantum wells. Here we report on these properties using Shubnikov-de Haas (SdH) oscillations and cyclotron resonance (CR) technique.
\subsection{II. Sample Growth and Preparation}
The samples were grown on a semi-insulating InP (100) substrate, using a modified Gen II molecular beam epitaxy system. The In$_{x}$Al$_{1-x}$As buffer is grown at low temperature to help mitigate formation of dislocations originating from the lattice mismatch between the InP substrate and higher levels of the heterostructure \cite{Wallart05, ShabaniAPL2014, ShabaniMIT}. The indium content of In$_{x}$Al$_{1-x}$As is step-graded from $x =$ 0.52 to 0.81. Next, a delta-doped Si layer of $\sim 7.5 \times 10^{11}$ cm$^{-2}$ density is placed here followed by 6 nm of In$_{.81}$Al$_{.19}$As. The quantum well is grown next, consisting of a 4 nm thick layer of In$_{0.81}$Ga$_{0.19}$As layer, a 4 nm thick layer of InAs, and finally a 10 nm thick top layer of In$_{0.81}$Ga$_{0.19}$As. A thin film of Al can be epitaxially grown on the final InGaAs layer. For the transport studies of the InAs quantum wells, Al films were selectively etched by Transene type-D solution while for optical studies Al was not grown from the beginning.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{fig2.pdf}
\caption{(Color online) (a) Lifting of $\nu=2$ integer quantum Hall state longitudinal resistance as a function of gate voltage (density) at various temperatures between 1.5~K and 12~K. (b) The natural logarithm of the minima in longitudinal resistance traces shown in (a). The higher temperature range data is linearly fitted and the gap is extracted from the slope. (c) The gap energy shown on a logarithmic scale. The $\nu=2$ gap is plotted for various magnetic fields. This scale is used to highlight the large difference in expected range for the gap versus the measured gap. (d) The gap energy shown for various quantum hall states $\nu = {2,3,4,6,8,10}$. The gaps are extracted in the manner exemplified in (a) and (b). These gap energies when fit to the usual linear field dependance yield values for $m^*\sim0.2-2.1$ which are at least one order of magnitude higher than electron effective masses in general and landau level broadening of 10~K or less which does not represent the strong disorder expected from a two dimensional electron gas near the surface.}
\label{fig:TempGap}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{fig3.pdf}
\caption{(Color online) (a) The amplitude of Shubnikov-de Haas (SdH) oscillations obtained by subtracting the polynomial background from the longitudinal resistance. Traces with largest amplitude (blue) were taken at temperature of 1.5~K and traces with lowest amplitude (red) were taken at a temperature of 30~K. Traces of intermediate amplitude (and color) span the temperature range from 1.5 K to 30 K in steps of approximately 2~K. Labeled quantum Hall states are extracted from Hall resistance. (b) The normalized amplitude of SdH oscillations at B = 4.2~T. The points are data and the dashed line is the fit. The energy gap is extracted from the fits and used to calculate the effective-mass, m*. A value of m* = 0.04 is found for this oscillation extrema near B = 4.2 T. (c) $m^*$ values extracted from all reasonable oscillations. (d) The Landau level broadening $\Gamma$ calculated from the quantum lifetime $\tau_q$ extracted for each temperature where an exponential envelope is fitted to the oscillations.}
\label{fig:SDH}
\end{figure*}
\subsection{III. Device Fabrication and Measurement Setup}
The samples used for our transport measurements were patterned using photolithography. The pattern used was an L-shaped Hall bar geometry allowing simultaneous measurement of longitudinal resistances ($R_{xx}$ and $R_{yy}$) and transverse resistance ($R_{xy}$). Chemical wet etching was performed after lithographic patterning leaving a 900~nm tall mesa. A 50~nm thick aluminum oxide (Al$_2$O$_3$) gate dielectric was then deposited on top of the Hall bar via atomic layer deposition. Gate electrodes were realized by subsequent deposition of 5~nm of titanium and 70~nm of gold. All measurements were performed inside a cryogen-free refrigerator with base temperature of 1.5~K with maximum magnetic field of 12~T. Carrier densities are determined based on the slope of Hall data.
\subsection{IV. Measurement Results}
\subsection{A. Magnetotransport Measurements}
Figure~\ref{fig:Rxx}a shows the color-scale plot of longitudinal magnetotransport, $R_{xx}$, as a function of top gate voltage, $V_{G}$. The Landau level fan diagram is evident from the plot with crossings observed at near $n = 1.3 \times 10^{12}$cm$^{-2}$ and 8~T and another near $n = 2.2 \times 10^{12}$cm$^{-2}$ and 12~T. At lowest densities we only observe well developed integer quantum Hall states up to $n = 1.3 \times 10^{12}$cm$^{-2}$ ($V_{G}<$ -3~V). The first Landau level crossing appears near $V_{G}\sim$ -3~V where it signals occupation of the second electric subband. This is most evident as $\nu$ = 6 stays the same before and after the crossing in Fig.~\ref{fig:Rxx}a. Similar Landau level crossings have been studied extensively in GaAs 2DEGs \cite{Muraki2001,Gossard2006,Zhang2005,LiuPRB2011}. Three magnetotransport traces are shown in Fig. \ref{fig:Rxx}b-d. Longitudinal and Hall resistance as a function of magnetic field are plotted for $n = 2.2$, $1.3$, and $0.68 \times 10^{12}$ cm$^{-2}$. The beating in SdH oscillations clearly suggest occupation of two subbands at $n = 2.2 \times 10^{12}$cm$^{-2}$ where below the crossing clear quantum Hall states develop with vanishing longitudinal resistance at $n = 0.68 \times 10^{12} cm^{-2}$.
In a non-interacting quantum Hall system, the Landau level spacing increases with magnetic field as $\hbar\omega_c$ with $\omega_c=eB/(m^*m_e)$ where B is the magnetic field, and $m_e$ is the bare electron mass. Hence, measurements of energy gaps of integer quantum Hall states should be related to electron mass. Figure \ref{fig:TempGap}a shows the temperature dependence of longitudinal resistance as a function of gate voltage near the filling factor $\nu$ = 2 and at the magnetic field $B =$ 9.5~T. The natural logarithm of the minimum in resistance in a system with parabolic bands has a linear dependence on inverse temperature as shown in Fig. \ref{fig:TempGap}b \cite{lnR}. The energy gap is directly proportional to the magnitude of the slope. We repeated these measurements as we varied the density and hence the position of $\nu$ = 2 in magnetic field. The results are shown in Fig.~\ref{fig:TempGap}c where extracted energy gaps are plotted as a function of magnetic field. For comparison, we also plot the energy gap expected from $\hbar\omega_c$ as a black dashed line. There is a large discrepancy between the measured and expected energy gap. If we allow electron mass to be a fitting parameter we obtain unrealistically high values of $m^* > 0.2$ for electrons. We have also studied the energy gaps of filling factors $\nu$ = 3, 4, 6, 8, 10. Figure~\ref{fig:TempGap}d shows the energy gaps are between 0-10~K. All these values are much smaller than their corresponding $\hbar\omega_c$. The energy gaps for each filling factor first increase with magnetic field, then decrease, and eventually disappear near the Landau level crossings. For odd integer quantum Hall states, the Landau levels are split by the Zeeman energy $g^{*}\mu B$. Our data indicates that odd integers are mainly absent and only begin to develop at higher magnetic field ($\nu$ = 3 near 12 T) as shown in Fig.~\ref{fig:Rxx}a. Given the bulk g-factor in InAs (g = -14), the odd integers should have large enough energy gaps to be clearly observed. Their very weak presence is due to either modified $g^*$ or Landau level broadening due to disorder. To address this and the discrepancy of energy scales for gaps in even integer quantum Hall states we next measure the temperature dependence of the low magnetic field SdH oscillations where only free electrons contribute to the transport.
The SdH oscillation amplitude can be isolated by subtracting the background trend of the longitudinal resistance $R_{xx}$. Figure \ref{fig:SDH}a displays the amplitude of SdH, $A_{SdH}$ for a carrier density of $n = 1.22 \times 10^{12} cm^{-2}$. Taking the points for a single minimum or maximum, normalized by our lowest temperature value, we can fit them to the formula $x/\text{sinh}(x)$ with $x = 2\pi^2 T/\Delta E$, where T is the temperature and $\Delta$E is the gap. This allows us to calculate $m^* = \hbar e B / (m_e \Delta E)$. Figure \ref{fig:SDH}b shows the data and fit for the oscillation near B~=~4.2~T from \ref{fig:SDH}a. We have repeated these measurements for various filling factors to extract $m^*$ as shown in \ref{fig:SDH}c. The experimental values range between 0.035 - 0.05 with an average value near $m^*$ = 0.04. This is slightly higher than bulk values of our quantum well consisting of InAs and In$_{0.81}$Ga$_{0.19}$As with $m^*$ = 0.023 and 0.03 respectively. From the exponential envelope of the SdH oscillations we can also obtain the quantum lifetime and calculate the Landau level broadening, $\Gamma = \hbar/\tau_q$. Figure~\ref{fig:SDH}d shows $\Gamma$ for carrier density n = 1.22$ \times 10^{12}$ cm$^{-2}$. The Landau level broadening range is around 200~K for $n = 1.2 \times 10^{12}$ cm$^{-2}$. The broadening in the near surface InAs 2DEG is significantly larger than in buried InAs 2DEGs where $\Gamma$ is measured to be 5~K \cite{ShabaniMIT}. Here the surface scattering clearly dominates the other scattering mechanisms \cite{Kaushini2018}. Thankfully, the smaller electron mass in InAs enhances the energy scales and therefore enables us to resolve quantum Hall states. Our measured Landau level broadening could qualitatively describe the large discrepancy between energy gap measurements in the quantum Hall states and $\hbar\omega_c$.
\begin{figure*}[htb]
\centering
\includegraphics[width=\textwidth]{fig4.pdf}
\caption{(a) The normalized transmission of 10.6 $\mu$m excitation showing cyclotron resonance (CR) taken at T = 300~K (electron-active). The sample in this measurement has a density of $n=3.6 \times 10^{11}$ cm$^{-2}$. The transitions indicated by arrows are attributed to the spin resolved CR transitions. (b) The CR measurement displays a sharper transitions at T = 20.5~K. Unlike the measurements at 300 K, the spin resolved CR can not be resolved but the broader resonance at 55 T (the Landau level transition n = 1 to n = 2), observed at 300 K, shifts to lower fields and narrows down.
(c) The effective mass $m^*$ as a function of magnetic field at T = 300~K and T = 20.5~K, demonstrate the non-parabolicity. (d) The absolute value of effective g-factor $g^*$ as a function of magnetic field at 20.5~K.
}
\label{new-CR}
\end{figure*}
\subsection{B. Cyclotron Resonance Measurements}
A more direct way to measure $m^*$ is through infrared CR measurements using pulsed ultrahigh magnetic fields ($<$ 150 Tesla) generated by the single-turn coil technique \cite{Mag0,Mag1,Mag2}. The external pulsed magnetic field was applied along the growth direction and measured by a pick-up coil around the sample. The sample and the pick-up coil were placed inside a continuous flow helium cryostat. In this study, we employed infrared radiations from a CO$_2$ laser with wavelengths ranging from 9.2-10.6 $\mu$m. The sample in this measurement has a density of $n=3.6 \times 10^{11}$ cm$^{-2}$. The changes in transmission through the sample were collected using a fast liquid-nitrogen-cooled HgCdTe detector. A multi-channel digitizer placed in a shielded room recorded the signals from the detector and pick-up coil.
The spin resolved CR at 10.6 $\mu$m indicated by the two arrows in Fig. \ref{new-CR}a, separated by $\sim$ 4 Tesla, was observed at T = 300~K. This fact can be expected, as the Landau levels above the Fermi level can be occupied at T = 300~K, allowing the transitions between n = 0 and n = 1 for two different spins. In addition, in Fig. \ref{new-CR}a the broad resonance at $\sim$ 55 T represents a transition between n = 1 and n = 2 which is possible when the carrier lifetime allows time for a finite population of Landau level n = 1. This transition is not predicted from the fixed Fermi energy, but can be attributed to the non-equilibrium electron distribution \cite{CR1,CR2}.
In Fig. \ref{new-CR}b, we present the CR measurements at 20.5~K with an excitation of 10.6~$\mu$m. The spin resolved CR was not observed indicating the states above the Fermi energy are no longer occupied. On the other hand, the broad resonance observed at $\sim$ 55~T and T = 300~K, which is due to the transition from n = 1 to n = 2, remained and narrowed.
Figure~\ref{new-CR}c summarizes our measurements for $m^*$ as a function of magnetic field at T = 300~K (crosses) and T = 20.5~K (filled circles). We note that although the single-turn coil is destroyed in each shot, the sample and pick-up coil remain intact, making it possible to carry out temperature and wavelength dependence measurements on the same sample. Figure~\ref{new-CR}c shows that the $m^*$ varied and increased monotonically with magnetic field. We measured $m^*$ = 0.04 near B = 40~T and $m^*$ = 0.061 near 70~T. Correspondingly we can estimate $g^*$ as a function of magnetic field using appropriate Landau level index using Eq.~1. In Fig.~\ref{new-CR}d we present absolute effective g-factor at 20.5 K as a function of magnetic field.
\subsection{V. Landau Level Modeling}
Next we provide a simple theoretical model to understand $m^*$ and the Landau level fan diagram in InAs which has a non-parabolic conduction band. Unlike the wide gap semiconductors such as GaAs, CR $m^*$ and $g^*$ may vary with subband index, Landau Level index, and external magnetic field. Beginning with expectations from the bulk and introducing confinement we can arrive at expressions for $m^*$ and $g^*$ (the details are presented in the Appendix):
\begin{equation}
\label{12}
{g^{*}_{j,n}} = \frac{{\left( {\varepsilon _{j,n}^ + - \varepsilon _{j,n}^ - } \right)}}{{{\mu _B}B}}
\end{equation}
where $\varepsilon _{j,n}$ is the energy of the $n^{th}$ Landau level, for the $j^{th}$ subband index, and at magnetic field $B$. Plus and minus superscripts represent higher and lower Zeeman split energy bands respectively. As shown in Fig. \ref{new-theory}a, $g^*$ depends on the subband index $j$, the Landau level $n$ as well as the magnetic field $B$. At zero magnetic field, the absolute value of $g^*$ = 12 is reduced from bulk value of $g^*$ = 14 due to confinement and monotonically decreases as magnetic field is increased. The rate depends on the Landau level index.
Similarly one can define $m^*$ obtained by CR as:
\begin{equation}
\label{13}
m_{j,n}^{*, \pm } = \frac{{\hbar eB}/m_e}{{\left( {\varepsilon _{j,n + 1}^ \pm - \varepsilon _{j,n}^ \pm } \right)}}
\end{equation}
We find that $m^*$, as shown in Fig. \ref{new-theory}b also depends on the $n^{th}$ Landau level, the $j^th$ subband index, and the magnetic field $B$ (we plot only the ($-$) solution for clarity). At zero magnetic field we see $m^{*}$ = 0.027 is larger than the bulk value of $m^*$ = 0.023 and increases monotonically as magnetic field is increased. These values are in close agreement with values derived from magnetotransport (over a small region 3~T to 5~T) and CR (40~T $<$ B $<$ 70~T).
\begin{figure}[htb]
\centering
\includegraphics [width = 0.45\textwidth] {fig5.pdf}
\caption{ (a) Absolute values for the effective g-factor $g^*$ for the n = 0, ...5 Landau levels for the lowest subband for an InAs infinite square well with effective well width of 20 nm. One can see the sensitivity of the $g^*$ to the magnetic field and the Landau level index. (b) The effective mass $m^*$(in units of the bare electron mass) for the n = 0, ...5 Landau levels for the lowest subband for an InAs infinite square well with 20 nm effective well width. Similar to $g^*$, $m^*$ varies as a function of the magnetic field and the Landau level index.}
\label {new-theory}
\end{figure}
\subsection{VI. Conclusion}
We have done magnetotransport and ultra high field cyclotron resonance characterization of surface InAs Quantum wells. The density of these structures can be tuned and our magnetotransport measurement provides insight into the Landau level broadening and the quantum Hall energy gaps. By combining magnetotransport and cyclotron resonance measurements we can obtain conduction band effective mass $m^*$ at both low and high magnetic fields respectively. A band structure model which includes the effects of strong non-parabolicity and quantum confinement can describe the extracted $m^*$ from magnetotransport and cyclotron resonance measurements. We used our experimental CR $m^*$ values to determine the effective g-factor $g^*$ as a function of magnetic fields and Landau level index and these values are in a good agreement with the model presented here.
\textbf {Acknowledgment:}
The NYU team acknowledges partial support from U.S. Army Research Office agreements W911NF1810067 and W911NF1810115. and NSF Grants No. NSF-MRSEC 1420073 and No. NSF DMR - 1702594. G.A.K. and C.J.S. acknowledge support from the Air Force Office of Scientific Research under Award No. FA9550-17-1-0341. G.A.K. and B.A.M. acknowledge support from the Japanese visiting program of The Institute for Solid State Physics, The University of Tokyo. J.Y. acknowledges funding from the ARO/LPS QuaCGR fellowship reference W911NF1810067.
\section{Appendix: Simple model for electron mass and \MakeLowercase{g}-factor in a non-parabolic semiconductor.}
The derivation of the theoretical model accounting for non-parabolicity is described in this section. In the absence of external magnetic field (and quantum confinement) a narrow gap semiconductor such as InAs has a conduction band energy, $\varepsilon$ vs. wavevector $k$ given by the dispersion relationship is given by:
\begin{equation}
\label{1}
\varepsilon (1 + \alpha \varepsilon ) = \frac{{{\hbar ^2}{k^2}}}{{2m_o^*}} = \frac{{{\hbar ^2}\left( {k_x^2 + k_y^2 + k_z^2} \right)}}{{2m_o^*}}
\end{equation}
Here, $\alpha$ is the {\it non-parabolicity factor} given by
\begin{equation}
\label{2}
\alpha = 1/{\varepsilon _g}
\end{equation}
with $\varepsilon_g$ being the band-gap, and $m_o^*$ is the CR $m^*m_e$ {\it at the band edge} ($k=0$). For small $\alpha\varepsilon$, the energy depends quadratically on $k$ while for large
$\alpha\varepsilon$, the energy depends linearly on $k$.
In the presence of a magnetic field in the $z$ direction, it can be shown \cite{mavroides1972magneto, lax1960cyclotron, bowers1959magnetic} that one can write:
\begin{equation}
\label{3}
\varepsilon (1 + \alpha \varepsilon ) = \frac{{{\hbar ^2}k_z^2}}{{2m_o^*}} + \left( {n + \frac{1}{2}} \right)\hbar {\omega _{c0}} \pm \frac{1}{2}{\mu _B}g_o^*B
\end{equation}
Here, $n$ is the Landau level index which can take on values [0,1,2,...]. $\omega_{c0}$ is the \textit{band-edge} CR frequency, given by:
\begin{equation}
\label{4}
{\omega _{c0}} = \frac{{eB}}{{m_o^*}}
\end{equation}
and
\begin{equation}
\label{5}
g_o^* = 2\left[ {1 + \,\,\left( {1 - \frac{1}{m^*}} \right)\,\,\frac{\Delta }{{3{\varepsilon _g} + 2\Delta }}} \right]
\end{equation}
is the {\it band-edge} $g^*$. $\Delta$ is the valence band spin-orbit splitting, and $\mu_B$ is the Bohr-magneton given by:
\begin{equation}
\label{6}
{\mu _B} = {\frac{{e\hbar }}{2m_e}}.
\end{equation}
Note that in the Bohr magneton, as opposed to the band-edge CR frequency, it is the bare electron mass that enters the expression.
To simplify, we set the RHS of Eq. \ref{3} to $K$
\begin{equation}
\begin {split}
\label{7}
\varepsilon (1 + \alpha \varepsilon ) = \frac{{{\hbar ^2}k_z^2}}{{2m_o^*}} +\left( {n + \frac{1}{2}} \right)\hbar {\omega _{c0}}\\
\pm \frac{1}{2}{\mu _B}g_o^*B = K
\end {split}
\end{equation}
and then solve for the energy $\varepsilon$.
\begin{equation}
\label{8}
\varepsilon = \frac{{ - 1 \pm \sqrt {1 + 4\alpha K} }}{{2\alpha }}.
\end{equation}
The plus sign corresponds to the conduction band while the minus sign corresponds to the light hole in the valence bands. Quantum confinement will also affect both $g^*$ and $m^*$ for narrow gap materials. To take into account quantum confinement, one quantizes $k_z$ as:
\begin{equation}
\label{9}
{k_z} = \frac{{2\pi }}{\lambda } = \frac{{j\pi }}{L}
\end{equation}
with $j$ a positive integer and $L$ the {\it effective} width of the quantum well. Substituting into equation \ref{3} yields:
\begin{equation}
\begin {split}
\label{11}
\varepsilon _{j,n}^ \pm (1 + \alpha \varepsilon _{j,n}^ \pm ) =\frac{{{\hbar ^2}{j^2}{\pi ^2}}}{{2m_o^*{L^2}}}
+\left( {n + \frac{1}{2}} \right)\hbar {\omega _{c0}}\\
\pm \frac{1}{2}{\mu _B}g_o^*B = K_{j,n}^ \pm
\end {split}
\end{equation}
We assume an {\it effective} well width of 20 $nm$. The gap at low temperatures is given by $\varepsilon_g = 0.4180$ while the spin orbit splitting is $\Delta = 0.38$ eV and the low temperature, band-edge effective mass is: $m_0^* = 0.023 m$. From Eq. \ref{5}, we see this yields a band-edge $g_o^* = -14$.
\begin{figure}[htb]
\centering
\includegraphics[width = 0.5\textwidth]{fig6.pdf}
\caption{
Calculated Landau Levels (n=0,...,5) for the lowest subband for a 20 nm InAs infinite square well in the simple model.
}
\label {landau}
\end{figure}
The Landau fan energies in Eq.~\ref{11} can lead us to calculate and define $g^*$ for different Landau levels by:
\begin{equation}
\label{12}
{g_{j,n}} = \frac{{\left( {\varepsilon _{j,n}^ + - \varepsilon _{j,n}^ - } \right)}}{{{\mu _B}B}}
\end{equation}
We can see that $g^*$ depends on the subband index $j$, the Landau level $n$ as well as the magnetic field $B$.
Similarly one can define $m^*$ by:
\begin{equation}
\label{13}
m_{j,n}^{*, \pm } = \frac{{\hbar eB}/m_e}{{\left( {\varepsilon _{j,n + 1}^ \pm - \varepsilon _{j,n}^ \pm } \right)}}
\end{equation}
Figure \ref{new-theory}(a,b) plots the $m^*$ and $g^*$ as a function of magnetic field and the Landau level index. We plot $m^*$ only for the lowest (-) solution. Since $g^*$ will differ between Landau levels for a non-parabolic system. The + and - effective masses will differ slightly and will lead to spin-split cyclotron resonance peaks under certain conditions. The calculation shows that in presence of non-parabolicity both of these parameters depend on the subband index $j$, the Landau level $n$, and the magnetic field $B$. We note that assuming a smaller effective quantum well width (e.g. 12~nm) will shift $m^*$ to larger values (e.g. $\sim$ 0.035 at B = 0 T) and $g^*$ will shift smaller values ($\sim$ -9.5 at B = 0 T).
As shown in Fig. \ref{landau}, we have also calculated the Landau levels for the 1st subband. With the effective g-factor being negative, Red lines are spin down, Blacks are spin up. The solid green arrows indicate the predicted CR transitions at 10.6 $\mu m$ and are in close agreement with experimental observations indicated by dashed green arrows. While in the theory presented here, we considered the infinite potential well, the agreement between the theory and experiment is better at lower magnetic fields. We should note that the Fermi level can be occupied at T = 300~K, allowing transitions between n = 0 and n = 1 for two different spins. The spin resolved CR was not allowed at lower temperatures and the resonances above 50~T, in Fig.\ref{new-CR}a and Fig.\ref{new-CR}b are attributed to the transitions between n = 1 and n = 2 . These transitions are possible where the photo-excited carrier lifetime is long enough to populate the Landau level n = 1, even though the position of the Fermi level would not predict the transitions.
\begin{center}
{\bf References}
\end{center}
|
1,314,259,996,537 | arxiv | \section{INTRODUCTION}
High-temperature plasmas exist in the clusters of galaxies (Arnaud et al. 1994; Markevitch et al. 1994; Markevitch et al. 1996; Holzapfel et al. 1997). Some clusters have extremely high-temperature electrons, $k_{B} T_{e}$ = $10 \sim 15$keV. Relativistic expressions for the thermal bremsstrahlung emissivity have been discussed by many authors (Gould 1980; Rephaeli \& Yankovitch 1997). However, the relativistic expressions have been so far derived by power-series expansions.
Very recently the present authors (Nozawa, Itoh, \& Kohyama 1998) have calculated the relativistic thermal bremsstrahlung Gaunt factor for the intracluster plasma by using the Bethe--Heitler (1934) cross section corrected by the Elwert (1939) factor for the following cases: $Z$ = 1 (H), 2 (He), 6 (C), 7 (N), 8 (O). They have also calculated the bremsstrahlung Gaunt factor by using the Coulomb--distorted wave functions for nonrelativistic electrons following the method of Karzas \& Latter (1961). They have thereby assessed the accuracy of the calculations. Their method of the calculation closely followed the work on the calculation of the inverse thermal bremsstrahlung by two of the present authors and their collaborators (Itoh, Nakagawa, \& Kohyama 1985; Nakagawa, Kohyama, \& Itoh 1987; Itoh, Kojo, \& Nakagawa 1990; Itoh et al. 1991; Itoh et al. 1997).
In the present paper we will extend the calculation to heavier elements: $Z$ = 10 (Ne), 12 (Mg), 14 (Si), 16 (S), 26 (Fe). For the sake of completeness, we will restate the formalism in this paper.
The present paper is organized as follows. We will give formulations for the calculation of the relativistic thermal bremsstrahlung Gaunt factor in $\S$ 2. The numerical results will be presented in $\S$ 3. We will discuss the results and give concluding remarks in $\S$ 4.
\section{FORMULATION}
In this paper we are concerned with the calculation of the relativistic thermal bremsstrahlung Gaunt factor for a high-temperature, low-density plasma which is relevant to the hot gas in the clusters of galaxies. We will use the accurate relativistic cross section. We will neglect the effects of screening and ionic correlation, which are expected to be small for a high-temperature, low-density plasma which is under investigation.
The cross section for the bremsstrahlung is related to (and can be easily obtained by the principle of detailed balancing from) the inverse bremsstrahlung cross section which is shown in Itoh, Nakagawa, \& Kohyama (1985). In the last three decades there appeared a great deal of theoretical work concerning the accurate relativistic calculation of the bremsstrahlung cross section (Elwert \& Haug 1969; Tseng \& Pratt 1971; Pratt \& Tseng 1975; Lee et al. 1976). Elwert \& Haug (1969) and Pratt \& Tseng (1975) have confirmed that the Bethe-Heitler (1934) cross section corrected by the Elwert (1939) factor gives excellent results for ions with small atomic number $Z_{j}$.
The relativistic cross section for the bremsstrahlung is written (following the notation of the original Bethe-Heitler paper as closely as possible) as
\begin{eqnarray}
\sigma & = & \alpha \, Z_{j}^{2} \, r_{0}^{2} \, \frac{p_{f}}{p_{i}} \, \frac{d \omega}{\omega} \, \frac{a_{f}}{a_{i}} \, \frac{1 - {\rm exp} ( - 2 \pi a_{i})}{1 - {\rm exp} ( - 2 \pi a_{f})} \nonumber \\
& \times & \left\{ \, \frac{4}{3} \, - \, 2 E_{f} E_{i} \frac{ p_{f}^{2} \, + \, p_{i}^{2}}{p_{f}^{2} p_{i}^{2} c^{2}} \, + \, m^{2}c^{2} \left[ \,
\frac{\beta_{f} E_{i}}{p_{f}^{3}c} \, + \, \frac{\beta_{i} E_{f}}{p_{i}^{3}c} \, - \, \frac{\beta_{f} \beta_{i}}{p_{f} p_{i}} \right] \right. \nonumber \\
& & \, + \, L \left[ \, \frac{8}{3} \, \frac{E_{f} E_{i}}{p_{f} p_{i} c^{2}} \, + \, \frac{\hbar^{2} \omega^{2}}{p_{f}^{3} p_{i}^{3} c^{6}} \, \left( E_{f}^{2} E_{i}^{2} \, + \, p_{f}^{2} p_{i}^{2} c^{4} \right) \right. \nonumber \\
& & \, + \, \left. \left. \frac{m^{2}c^{2} \hbar \omega}{2 p_{f} p_{i}} \left( \frac{E_{f} E_{i} + p_{i}^{2}c^{2}}{p_{i}^{3} c^{3}} \beta_{i} \, - \, \frac{E_{f} E_{i} + p_{f}^{2}c^{2}}{p_{f}^{3} c^{3}} \beta_{f} \, + \, \frac{2 \hbar \omega E_{f} E_{i}}{p_{f}^{2} p_{i}^{2} c^{4}} \right) \, \right] \, \right\} \, ,
\end{eqnarray}
\begin{eqnarray}
a_{f} & \equiv & \frac{\alpha \, Z_{j} \, E_{f}}{p_{f}c} \, , \, \, \, \, \,
a_{i} \, \equiv \, \frac{\alpha \, Z_{j} \, E_{i}}{p_{i}c} \, , \\
\beta_{f} & \equiv & 2 \, \, {\rm ln} \, \frac{E_{f} \, + \, p_{f} c}{mc^{2}} \, , \, \, \, \, \, \beta_{i} \, \equiv \, 2 \, \, {\rm ln} \, \frac{E_{i} \, + \, p_{i} c}{mc^{2}} \, , \\
L & \equiv & 2 \, \, {\rm ln} \, \frac{E_{f} E_{i} \, + \, p_{f} p_{i} c^{2} \, - \, m^{2}c^{4}}{mc^{2} \hbar \omega} \, , \\
E_{i} & = & E_{f} \, + \, \hbar \omega \, .
\end{eqnarray}
In the above, $\alpha$ is the fine-structure constant, $r_{0}$ is the classical electron radius, $\omega$ is the angular frequency of the emitted photon, $p_{i}$ is the initial momentum of the electron, $p_{f}$ is the final momentum of the electron, $E_{i}$ is the initial energy of the electron, $E_{f}$ is the final energy of the electron.
We will calculate the bremsstrahlung emissivity
\begin{equation}
W(\omega) \, d \omega \, = \, \hbar \omega \, n_{e} \, n_{j} \, v_{i} \, \sigma \, = \, \hbar \omega \, n_{e} \, \frac{p_{i}c^{2}}{E_{i}} \, n_{j} \, \sigma \, ,
\end{equation}
where $n_{e}$ is the number density of electrons and $n_{j}$ is the number density of ions of charge $Z_{j}$. Then we will average the bremsstrahlung emissivity over a distribution of electrons taking into account the Pauli blocking of the final electron state
\begin{eqnarray}
< W(\omega) > d \omega & = & \frac{ \displaystyle{ \int W(\omega) \, d \omega \, f(E_{i}) [ 1 - f(E_{f})] \, d^{3} p_{i}}}{ \displaystyle{\int f(E_{i}) \, d^{3} p_{i}}} \, , \\
f(E_{i}) & \equiv & \left\{ {\rm exp} \left[ (E_{i} - \mu)/k_{B} T \right] \, + \, 1 \right\}^{-1} \, , \\
\int f(E_{i}) \, d^{3} p_{i} & = & 4 \pi m^{3} c^{3} \, G_{0}^{-}(\lambda, \nu) \, , \\
G_{0}^{-}(\lambda, \nu) & \equiv & \lambda^{3} \, \int_{\lambda^{-1}}^{\infty} \frac{ x (x^{2} - \lambda^{-2})^{1/2}}{1 \, + \, e^{x - \nu}} \, d x \, , \\
\lambda & \equiv & \frac{k_{B}T}{m c^{2}} \, = \, \frac{T}{5.930 \times 10^{9} {\rm K}} \, , \\
\nu & = & \frac{\mu}{k_{B} T} \, ,
\end{eqnarray}
$\mu$ being the electron chemical potential including the rest mass. For the extreme non-degeneracy ($-\eta \gg 1$) and the nonrelativistic temperature ($\lambda \ll 1$), the chemical potential $\mu$ is related to the electron number density $n_{e}$ and the temperature $T$ through the relationship (relativistic Maxwellian distribution)
\begin{eqnarray}
\eta & \equiv & \frac{\mu \, - \, m c^{2}}{k_{B} T} \nonumber \\
& = & {\rm ln} \, \left\{ \frac{1}{2} n_{e} \left(\frac{2 \pi \hbar^{2}}{m k_{B} T} \right)^{3/2} \, \left[ 1 \, + \, \frac{15}{8} \frac{k_{B} T}{m c^{2}} \, + \, \frac{105}{128} \left( \frac{k_{B} T}{m c^{2}} \right)^{2} \right]^{-1} \, \right\} \nonumber \\
& = & {\rm ln} \, \left\{ 4.535 \times 10^{-31} \left[n_{e}({\rm cm^{-3}}) \right] \lambda^{-3/2} \, \left(1 \, + \, \frac{15}{8} \lambda \, + \, \frac{105}{128} \lambda^{2} \right)^{-1} \right\} \, .
\end{eqnarray}
The degeneracy parameter $\eta$ is generally related to the mass density and temperature through the relationship
\begin{eqnarray}
\frac{\rho}{2} \left( 1 \, + \, \frac{0.992 X}{1.008} \right) & = &
\frac{n_{e}}{N_{A}} \, = \, \frac{1}{ N_{A}} \frac{2}{( 2\pi \hbar)^{3}} \int f(E_{i}) \, d^{3} p_{i} \, \nonumber \\
& = & 2.922 \times 10^{6} \, G_{0}^{-}( \lambda, \eta + \lambda^{-1} ) \, .
\end{eqnarray}
In equation (14) $\rho$ is measured in units of gcm$^{-3}$, $X$ is the mass fraction of hydrogen, $n_{e}$ is the electron number density in units of cm$^{-3}$, and $N_{A}$ is Avogadro's number. The reader might wonder the reason why we retain the general Fermi-Dirac distribution function rather than we use the relativisitic Maxwellian distribution function from the outset. The reason is twofold. The first one is very simple: we wish to make strong connections with our previous papers in which we have considered degeneracy of the electrons. The second reason is that we consider it most appropriate to use the relationships such as equation (9) when we deal with relativistic electrons. This choice reflects upon the mathematical preference of the authors who wish to use expressions as general as possible.
We obtain
\begin{equation}
< W(\omega) > d \omega \, = \, \frac{ \displaystyle{n_{e} n_{j} Z_{j}^{2} \, \alpha \, r_{0}^{2} \, \hbar c \lambda^{3} \, J^{-}(\lambda, \nu, u, Z_{j})}}{ \displaystyle{G_{0}^{-}(\lambda, \nu)}} \, d \omega \, ,
\end{equation}
\begin{eqnarray}
& & J^{-}(\lambda, \nu, u, Z_{j}) \nonumber \\
& = & \int_{\lambda^{-1}+u}^{\infty} dx \, \frac{x^{2} - \lambda^{-2}}{ e^{x - \nu} \, + \, 1} \, \frac{x - u}{x} \left(1 - \frac{1}{e^{x-u-\nu}+1} \right) \nonumber \\
& \times & \frac{1 - {\rm exp} \left[-2 \pi \alpha Z_{j} x \left(x^{2} - \lambda^{-2} \right)^{-1/2} \right] }{1 - {\rm exp} \left\{ -2 \pi \alpha Z_{j} (x - u) \left[ (x - u )^{2} - \lambda^{-2} \right]^{-1/2} \right\} } \nonumber \\
& \times & \left( \frac{4}{3} \, - \, 2 (x - u) x \, \frac{ [ (x - u)^{2} - \lambda^{-2} ] \, + \, ( x^{2} - \lambda^{-2})}{ [ (x - u)^{2} - \lambda^{-2} ] \, ( x^{2} - \lambda^{-2})} \right. \nonumber \\
& + & \lambda^{-2} \left\{ \frac{ \beta_{f} \, x}{ [ (x - u)^{2} - \lambda^{-2} ]^{3/2}} + \frac{ \beta_{i} \, (x - u)}{(x^{2} - \lambda^{-2} )^{3/2}} - \frac{ \beta_{f} \, \beta_{i}}{ [(x - u)^{2} - \lambda^{-2} ]^{1/2} (x^{2} - \lambda^{-2} )^{1/2}} \right\}
\nonumber \\
& + & L \, \left[ \frac{8}{3} \, \frac{(x - u) x}{ [(x - u)^{2} - \lambda^{-2} ]^{1/2} (x^{2} - \lambda^{-2} )^{1/2}} \right. \nonumber \\
& & + \, \frac{u^{2}}{ [(x - u)^{2} - \lambda^{-2} ]^{3/2} (x^{2} - \lambda^{-2} )^{3/2}} \left\{ (x - u)^{2}x^{2} \, + \, [(x - u)^{2} - \lambda^{-2} ] (x^{2} - \lambda^{-2} ) \right\} \nonumber \\
& & + \, \frac{\lambda^{-2} u}{ 2 [(x - u)^{2} - \lambda^{-2} ]^{1/2} (x^{2} - \lambda^{-2} )^{1/2} } \times \left\{ \frac{(x-u)x + (x^{2} - \lambda^{-2})}{ (x^{2} - \lambda^{-2} )^{3/2}} \beta_{i} \right. \nonumber \\
& & \left. \, \left. \, \left. \, - \, \frac{(x-u)x + [(x-u)^{2} - \lambda^{-2}]}{ [(x-u)^{2} - \lambda^{-2} ]^{3/2}} \beta_{f} \, + \, \frac{2u(x-u)x}{ [(x-u)^{2} - \lambda^{-2} ] \, (x^{2}-\lambda^{-2})} \, \right\} \, \, \, \, \right] \, \, \, \, \right) \, ,
\end{eqnarray}
\begin{eqnarray}
\beta_{f} & = & 2 \, \, {\rm ln} \frac{(x-u) + [ (x-u)^{2} - \lambda^{-2} ]^{1/2}}{\lambda^{-1}} \, , \\
\beta_{i} & = & 2 \, \, {\rm ln} \frac{x + (x^{2} - \lambda^{-2})^{1/2}} {\lambda^{-1}} \, , \\
L & = & 2 \, \, {\rm ln} \frac{(x-u)x + [ (x-u)^{2} - \lambda^{-2}]^{1/2} (x^{2}-\lambda^{-2})^{1/2} - \lambda^{-2}}{\lambda^{-1} u} \, , \\
u & = & \frac{\hbar \omega}{k_{B}T} \, .
\end{eqnarray}
By doing a transformation $x^{\prime} = x - u$, one finds that the formula (16) is related to $I^{-}(\lambda, \nu, u, Z_{j})$ of Itoh, Nakagawa, \& Kohyama (1985) (their equation (12)) by
\begin{equation}
J^{-}(\lambda, \nu, u, Z_{j}) \, = \, e^{-u} \, I^{-}(\lambda, \nu, u, Z_{j}) \, .
\end{equation}
We define the temperature-averaged relativistic Gaunt factor $g_{Z_{j}}$ for the thermal bremsstrahlung emissivity by
\begin{equation}
< W(\omega) > d \omega \, = \, g_{Z_{j}} \, < W(\omega) >_{K} d \omega \, ,
\end{equation}
where
\begin{eqnarray}
< W(\omega) >_{K} d \omega & = & \frac{2^{5} \pi e^{6}}{3 h m c^{3}} \, n_{e} n_{j} Z_{j}^{2} \, \left( \frac{2 \pi k_{B} T}{3 m} \right)^{1/2} \, e^{-u} \, \frac{\hbar}{k_{B} T} \, d \omega \nonumber \\
& = & 1.426 \times 10^{-27} \left[ n_{e}({\rm cm^{-3}}) \right] \left[n_{j}({\rm cm^{-3}}) \right] Z_{j}^{2} \left[ T({\rm K}) \right]^{1/2} \nonumber \\
& \times & e^{-u} \, du \, \hspace{1.0cm} \, {\rm erg \, \, s^{-1} \, cm^{-3}} \, .
\end{eqnarray}
In the above we have followed the expression of the original paper by Karzas \& Latter (1961) as closely as possible.
Therefore we obtain
\begin{equation}
g_{Z_{j}} \, = \, \frac{3 \sqrt{6}}{32 \sqrt{\pi}} \, \lambda^{7/2} \, e^{u} \, \frac{ J^{-}(\lambda, \nu, u, Z_{j})}{G_{0}^{-}(\lambda, \nu)} \, .
\end{equation}
We note that the formula (24) is identical to equation (21) of Itoh, Kohyama, \& Nakagawa (1985) with the positron contributions omitted, because one has $e^{u} \, J^{-}(\lambda, \nu, u, Z_{j})$ = $I^{-}(\lambda, \nu, u, Z_{j})$.
In this paper we will also calculate the exact nonrelativistic energy-dependent Gaunt factor for bremsstrahlung, and then use this for the calculation of the thermally-averaged Gaunt factor. The formulation is similar to the one presented in the paper by Nakagawa, Kohyama, \& Itoh (1987). However, for the sake of completeness we will present it here. According to Karzas \& Latter (1961), the exact nonrelativistic inverse bremsstrahlung Gaunt factor is given by
\begin{eqnarray}
g & = & \frac{2 \sqrt{3}}{\pi \eta_{i} \eta_{f}} \left[ \left( \eta_{i}^{2} + \eta_{f}^{2} + 2 \eta_{i}^{2} \eta_{f}^{2} \right) \, I_{0} - 2 \eta_{i} \eta_{f} \left(1 + \eta_{i}^{2} \right)^{1/2}
\left(1 + \eta_{f}^{2} \right)^{1/2} \, I_{1} \, \right] \, I_{0} \, , \\
\eta_{i}^{2} & = & \frac{Z_{j}^{2} {\rm R_{y}}}{\epsilon_{i}} \, \, , \hspace{0.5cm} \eta_{f}^{2} \, \, \, = \, \, \, \frac{Z_{j}^{2} {\rm R_{y}}}{\epsilon_{f}} \, \, , \\
I_{l} & = & \frac{1}{4} \left[ \frac{4 k_{i} k_{f}}{(k_{i} - k_{f})^{2}} \right]^{l+1} {\rm exp} \left( \frac{ \pi \mid \eta_{i} - \eta_{f} \mid }{2} \right) \frac{ \mid \Gamma (l+1+i \eta_{i}) \Gamma(l+1+i \eta_{f}) \mid}{\Gamma(2 l + 2)} \, G_{l} \, , \\
G_{l} & = & \left| \frac{k_{f} - k_{i}}{k_{f} + k_{i}} \right|^{i \eta_{i} + i \eta_{f}} \, _{2}F_{1} \left[ l+1- i \eta_{f}, l+1-i \eta_{i}; 2l+2; - \frac{4 k_{i} k_{f}}{(k_{i} - k_{f})^{2}} \right] \, ,
\end{eqnarray}
\begin{equation}
_{2}F_{1} (a,b;c;x) \, \equiv \, \sum_{m=0}^{ \infty} \frac{ \Gamma(a+m) \Gamma(b+m) \Gamma(c)}{ \Gamma(c+m) \Gamma(a) \Gamma(b) } \,
\frac{x^{m}}{m! } \, .
\end{equation}
In the above $\epsilon_{i}$ and $\epsilon_{f}$ are the kinetic energies of the initial electron and final electron for the inverse bremsstrahlung, $k_{i}$ and $k_{f}$ are their wave numbers, and $_{2}F_{1}(a,b;c;x)$ is the hypergeometric function. Karzas \& Latter (1961) have given a power series expansion method for the practical evaluation of $G_{l}$. We have employed their method in the present calculation.
The thermally-averaged Gaunt factor for the inverse bremsstrahlung by nonrelativistic electrons is written as
\begin{eqnarray}
g_{NR}(u, \gamma^{2}) & = & \frac{ \pi^{1/2}}{2} \frac{ \displaystyle{ \int_{0}^{ \infty} d \epsilon_{i} \, g \, f(\epsilon_{i}) \left[ 1 - f(\epsilon_{f}) \right]}}
{ \displaystyle{\beta^{1/2} \int_{0}^{ \infty} d \epsilon_{i} \, \epsilon_{i}^{1/2} f(\epsilon_{i})} } \, , \\
f( \epsilon_{i} ) & = & \left[ {\rm exp} ( \beta \epsilon_{i} - \eta) + 1 \right]^{-1} \, , \, \, \, \beta \, \equiv \, (k_{B}T)^{-1} \, , \\
\gamma^{2} & = & \frac{Z_{j}^{2} {\rm R_{y}}}{k_{B}T} \, = \, Z_{j}^{2} \, \frac{1.579 \times 10^{5} {\rm K}}{T} \, .
\end{eqnarray}
Therefore, the nonrelativistic thermal bremsstrahlung emissivity is
given by
\begin{eqnarray}
< W(\omega) >_{NR} d \omega & = & 1.426 \times 10^{-27} \, g_{NR}(u, \gamma^{2}) \, \left[ n_{e}({\rm cm^{-3}}) \right] \left[n_{j}({\rm cm^{-3}}) \right] Z_{j}^{2} \, \\ \nonumber
& \times & \left[ T({\rm K}) \right]^{1/2} e^{-u} \, du \, \hspace{1.0cm} \, {\rm erg \, \, s^{-1} \, cm^{-3}} \, .
\end{eqnarray}
The thermally-averaged nonrelativistic Gaunt factor $g_{NR}(u, \gamma^{2})$ for the inverse bremsstrahlung has been calculated and tabulated by Nakagawa, Kohyama, \& Itoh (1987), Itoh, Kojo, \& Nakagawa (1990), and by Itoh et al. (1997), for high-temperature, high-density plasmas. Carson (1988) calculated the thermally-averaged nonrelativistic Gaunt factor for the inverse bremsstrahlung for non-degenerate electrons following the method of Karzas \& Latter (1961).
In order to assess the accuracy of the various approximations, we have also calculated the nonrelativistic Gaunt factors with the Born approximation corrected by the Elwert factor. It is given as
\begin{equation}
g_{NRE} \, = \, \frac{\sqrt{3}}{\pi} {\rm ln} \left| \frac{p_{f} + p_{i}}{p_{f} - p_{i}} \right| \, \frac{\eta_{f}}{\eta_{i}} \frac{1 - {\rm exp} ( - 2 \pi \eta_{i})}{1 - {\rm exp} ( - 2 \pi \eta_{f})} \, .
\end{equation}
The thermally-averaged nonrelativistic Gaunt factor in the Elwert approximation is obtained by inserting eq.\ (34) into eq.\ (30).
\section{NUMERICAL RESULTS}
We have carried out the numerical calculations of the thermally averaged relativistic and nonrelativistic Gaunt factors for the thermal bremsstrahlung. In the present paper we are interested in the high-temperature and low-density regime which is relevant to the hot gas in the clusters of galaxies. The electron plasma in this regime is extremely nondegenerate, namely $\eta \approx - \infty$. It is clear from Fig.\ 1 of Nozawa, Itoh, \& Kohyama (1998) that the relativistic Gaunt factor is independent of $\eta$ for negatively large values ($\eta \le -10$). Therefore we adopt $\eta=-70$ as the case of $\eta \approx -\infty$ throughtout this paper.
In Fig.\ 1 we have plotted the thermally averaged relativistic Gaunt factor as a function of the photon energy $u$ for $Z_{j}$=26, $\eta=-\infty$ and the electron temperatures $T=10^{5}$K, 10$^{6}$K, 10$^{7}$K, 10$^{8}$K, 10$^{8.5}$K and 10$^{9}$K. The relativistic effect becomes very important in the high $T$, large-$u$ region. For the typical plasmas in the clusters of galaxies of $T=10^{8}$K the effect is large for the photon energies log$_{10}u \ge 2$.
In Figs.\ 2 and 3 we have plotted the thermally averaged relativistic and nonrelativistic Gaunt factors for $\eta=-\infty$, log$_{10}u$=0 and 1 as a function of the temperature parameter $\gamma^{2}$ defined by eq.\ (32). The dashed curve is the nonrelativistic exact calculation of eq.\ (30), where the thermally averaged Gaunt factor does not depend on $Z_{j}$ and $T$ separately, but on the combination $Z_{j}^{2}/T$. The solid curves from left to right are the relativistic cases for $Z_{j}$=10, 12, 14, 16, and 26, respectively.
The numerical results are also presented in Tables 1 and 2 for $\eta=-\infty$, $10^{-3.0} \leq \gamma^{2} \leq 10^{2.5}$, and $10^{-4.0} \leq u \leq 10^{3.0}$. The first, second, third, fourth, fifth, and sixth entries correspond to the thermally averaged relativistic Gaunt factors of neon, magnesium, silicon, sulphur, iron, and the thermally averaged exact nonrelativistic Gaunt factor. We wish to point out the following fact: for 0.0 $\leq$ log$_{10}u \leq 1.0$ and for 10 $\leq Z_{j} \leq 26$ there exists a certain range of $\gamma^{2}$ where the relativistic Elwert result almost coincides with the nonrelativistic exact result. Let us consider the case of $Z_{j}=26$. From Table 1 we find that for log$_{10}u$=0.0 both results show an excellent agreement (0.1\% accuracy) for log$_{10}\gamma^{2}=-0.5$. For log$_{10}u$=1.0 both results show a very good agreement (0.4\% accuracy) for log$_{10}\gamma^{2}$=1.0. This fact guarantees the accuracy of the relativistic Elwert calculation. At high temperatures (small values of $\gamma^{2}$) the discrepancy is caused by the insufficiency of the nonrelativistic approximation. The discrepancy becomes larger as $Z_{j}$ increases. At low temperatures (large values of $\gamma^{2}$) the Coulomb distortion of the wave function becomes very large and the Elwert approximation becomes less accurate. The accuracy of the numerical calculation of the exact nonrelativistic Gaunt factor for low--temperature cases is about 0.1\%.
In order to assess the accuracy of the various approximations, we have also calculated the thermally averaged nonrelativistic Gaunt factors using the Elwert approximation. We have found that the nonrelativistic Gaunt factor with the Elwert approximation coincides with the nonrelativistic exact Gaunt factor for $\gamma^{2} \leq 10^{-1.0}$ with an accuracy of better than 0.2\% for the photons of moderate energy (say log$_{10}u \sim 0$), thereby proving the excellence of the Elwert approximation at high temperatures. It is also found that the nonrelativistic Gaunt factor with the Elwert approximation coincides with the relativistic Gaunt factor with the Elwert approximation at low temperatures, as it should. For the cases of heavy elements, the Elwert approximation has a lower accuracy compared with the cases of hydrogen and helium. Agreement of the results of the calculations with different methods proves the accuracy of the present calculations.
For clusters of galaxies, the relativistic correction is not too large. Take an example of $Z_{j}=26$ and $\gamma^{2}=1$ ($T=1.067 \times 10^{8}$K). From Table 1 one finds $g_{R}/g_{NR}$=1.019 for log$_{10}u$=0.0 and $g_{R}/g_{NR}$=0.977 for log$_{10}u$=1.0. Therefore, the relativistic correction in the case of Fe is on the order of 2\% for clusters of galaxies.
\section{CONCLUDING REMARKS}
We have calculated the Gaunt factor for the thermal bremsstrahlung in high-temperature plasmas for the cases of $Z$ = 10, 12, 14, 16, 26 by using the accurate relativistic cross section, and have compared the result with the Gaunt factor derived by using Sommerfeld's exact nonrelativistic cross section. Significant deviations from the nonrelativistic results have been found for high temperature cases.
We have presented the results in the form of extensive tables. The accuracy of the Elwert approximation is better than 0.2\% for $\gamma^{2} \leq 10^{-1}$, log$_{10}u \sim 0$. The overall accuracy of the present calculation is generally about 0.4\%. The present paper will be useful to analyze the relativistic effects for the thermal bremsstrahlung in the high-temperature plasmas which exist in the clusters of galaxies. The present paper covers a much wider temperature range than that of the intracluster plasma for the sake of completeness.
\newpage
\references{}
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\newpage
\centerline{\bf \large Figure Captions}
\begin{itemize}
\item Fig.\ 1. Thermally averaged relativistic Gaunt factor for $Z_{j}=26$, $\eta=-\infty$. The dotted curve corresponds to $T=10^{5}$K. The dashed curve corresponds to $T=10^{6}$K. The dash-dotted curve corresponds to $T=10^{7}$K. The solid curves from right to left correspond to $T=10^{8}$K, $10^{8.5}$K and $10^{9}$K, respectively.
\item Fig.\ 2. Thermally averaged Gaunt factors for $\eta=-\infty$, log$_{10}u=0$. The dashed curve corresponds to the nonrelativistic Gaunt factor (N.R. exact). The solid curves from left to right correspond to the relativistic Gaunt factors for $Z_{j}$=10,12,14,16, and 26, respectively. The nonrelativistic Gaunt factor in the Elwert approximation is also displayed as the dotted curve, but it is indistinguishable from other curves.
\item Fig.\ 3. Same as for Fig.\ 2 but for $\eta=-\infty$, log$_{10}u=1$.
\end{itemize}
\end{document}
|
1,314,259,996,538 | arxiv | \section{Introduction}\label{sec:intro}
Emerging software-defined communication networks
provide \emph{direct} and ``\emph{algorithmic}'' control over the
forwarding rules of nodes (i.e., routers and switches)
and hence the network \emph{routes}.
The resulting routes are not restricted to follow
only shortest paths and moreover,
they can be flexibly \emph{adapted over time}, e.g., depending
on certain events in the dataplane.
Indeed, there are many reasons why flows may need to be
\emph{rerouted}~\cite{update-survey}, including
security and policy changes (e.g., suspicious traffic
is rerouted via a firewall),
traffic engineering optimizations, reactions to changes in the demand,
maintenance work, failures, etc.
Implementing route changes however is challenging,
since updating a route usually involves the distribution of
new (forwarding) rules across the asynchronous communication
network, and since even \emph{during} such route changes,
it is important to maintain certain safety properties.
In particular, the routes of flows should be changed without
causing any congestion or introducing temporary forwarding loops.
For example, in a Software Defined Network (SDN),
rules are communicated by the remote software controller.
Therefore, updates have to be distributed in \emph{rounds},
in which switches acknowledge the next batch of updates~\cite{roger,sigmetrics16,ludwig2015scheduling}.
This introduces a \emph{scheduling problem}: In which order
to update the different forwarding rules for the different flows and switches
over time, such that these safety properties are maintained at any time?
And how to schedule these updates such that the rerouting time
(and number of controller interactions) is minimized?
\subsection{A Simple Example}
Figure~\ref{fig:ex} gives an example
of the flow rerouting problem.
We want to schedule
the rerouting of $2$ flows in a $5$-node network,
connecting nodes $\{s,u,v,w,t\}$
with $7$ edges $\{\{s,u\},\{s,w\},\{u,w\},
\{u,v\},\{v,w\},\{v,t\},\{w,t\}\}$.
In this example, both flows originate at $s$
and end at $t$: The first flow is indicated
in \textcolor{red}{red} and the second
flow in \textcolor{blue}{blue}.
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale=0.9]
\tikzset{>=latex}
\node (o) [] {};
\node (c-1-1) [position=0:0mm from o] {};
\foreach\i in {1} {
\foreach\j in {1} {
\node (s-\i-\j) [fill,inner sep=0pt,minimum size=7pt,draw,circle,thick,scale=0.7,position=162:17mm from c-\i-\j] {};
\node (s-\i-\j-1) [inner sep=0pt,position=330:0.72mm from s-\i-\j] {};
\node (s-\i-\j-2) [inner sep=0pt,scale=0.75,position=270:0.72mm from s-\i-\j] {};
\node (s-\i-\j-3) [inner sep=0pt,scale=0.75,position=90:0.72mm from s-\i-\j] {};
\node (s-\i-\j-4) [inner sep=0pt,scale=0.75,position=30:0.72mm from s-\i-\j] {};
\node (t-\i-\j) [fill,inner sep=0pt,minimum size=7pt,draw,circle,thick,scale=0.7,position=18:17mm from c-\i-\j] {};
\node (t-\i-\j-1) [inner sep=0pt,scale=0.75,position=270:0.72mm from t-\i-\j] {};
\node (t-\i-\j-2) [inner sep=0pt,scale=0.75,position=210:0.72mm from t-\i-\j] {};
\node (t-\i-\j-3) [inner sep=0pt,scale=0.75,position=160:0.72mm from t-\i-\j] {};
\node (t-\i-\j-4) [inner sep=0pt,scale=0.75,position=90:0.72mm from t-\i-\j] {};
\node (u-\i-\j) [fill,inner sep=0pt,minimum size=7pt,draw,circle,thick,scale=0.5,position=224:10mm from c-\i-\j] {};
\node (u-\i-\j-1) [inner sep=0pt,scale=0.75,position=90:0.72mm from u-\i-\j] {};
\node (u-\i-\j-2) [inner sep=0pt,scale=0.75,position=180:0.72mm from u-\i-\j] {};
\node (u-\i-\j-3) [inner sep=0pt,scale=0.75,position=120:0.72mm from u-\i-\j] {};
\node (u-\i-\j-4) [inner sep=0pt,scale=0.75,position=0:0.72mm from u-\i-\j] {};
\node (v-\i-\j) [fill,inner sep=0pt,minimum size=7pt,draw,circle,thick,scale=0.5,position=316:10mm from c-\i-\j] {};
\node (v-\i-\j-1) [inner sep=0pt,scale=0.75,position=90:0.72mm from v-\i-\j] {};
\node (v-\i-\j-2) [inner sep=0pt,scale=0.75,position=180:0.72mm from v-\i-\j] {};
\node (v-\i-\j-3) [inner sep=0pt,scale=0.75,position=60:0.72mm from v-\i-\j] {};
\node (v-\i-\j-4) [inner sep=0pt,scale=0.75,position=0:0.72mm from v-\i-\j] {};
\node (w-\i-\j) [fill,inner sep=0pt,minimum size=7pt,draw,circle,thick,scale=0.5,position=90:15mm from c-\i-\j] {};
\node (w-\i-\j-1) [inner sep=0pt,scale=0.75,position=320:0.72mm from w-\i-\j] {};
\node (w-\i-\j-2) [inner sep=0pt,scale=0.75,position=240:0.72mm from w-\i-\j] {};
\node (w-\i-\j-3) [inner sep=0pt,scale=0.75,position=180:0.72mm from w-\i-\j] {};
\node (w-\i-\j-4) [inner sep=0pt,scale=0.75,position=30:0.72mm from w-\i-\j] {};
\node (w-\i-\j-5) [inner sep=0pt,scale=0.75,position=270:0.72mm from w-\i-\j] {};
\node (w-\i-\j-6) [inner sep=0pt,scale=0.75,position=270:0.72mm from w-\i-\j] {};
\pgfmathtruncatemacro{\ilabel}{\j+3*(\i-1)};
\node (labels-\i-\j) [position=150:4mm from s-\i-\j] {$s$};
\node (labelt-\i-\j) [position=30:4mm from t-\i-\j] {$t$};
\node (labelt-\i-\j) [position=270:4mm from u-\i-\j] {$u$};
\node (labelt-\i-\j) [position=270:4mm from v-\i-\j] {$v$};
\node (labelt-\i-\j) [position=90:4mm from w-\i-\j] {$w$};
}
}
\begin{pgfonlayer}{bg}
\draw [cyan,line width=6.5pt,line cap=round,opacity=0.2] (s-1-1) to (w-1-1);
\draw [cyan,line width=6.5pt,line cap=round,opacity=0.2] (w-1-1) to (t-1-1);
\draw [Magenta,line width=6.5pt,line cap=round,opacity=0.2] (s-1-1) to (u-1-1);
\draw [Magenta,line width=6.5pt,line cap=round,opacity=0.2] (u-1-1) to (v-1-1);
\draw [Magenta,line width=6.5pt,line cap=round,opacity=0.2] (v-1-1) to (t-1-1);
\draw [blue,line width=1.3pt,->] (s-1-1-4) to (w-1-1-2);
\draw [blue,line width=1.3pt,->] (w-1-1-1) to (t-1-1-3);
\draw [red,line width=1.3pt,->] (s-1-1-1) to (u-1-1-3);
\draw [red,line width=1.3pt,->] (u-1-1-4) to (v-1-1-2);
\draw [red,line width=1.3pt,->] (v-1-1-3) to (t-1-1-2);
\draw [blue,line width=1.3pt,->,dashed] (s-1-1-2) to (u-1-1-2);
\draw [blue,line width=1.3pt,->,dashed] (u-1-1-1) to (w-1-1-6);
\draw [blue,line width=1.3pt,->,dashed] (w-1-1-5) to (v-1-1-1);
\draw [blue,line width=1.3pt,->,dashed] (v-1-1-4) to (t-1-1-1);
\draw [red,line width=1.3pt,->,dashed] (s-1-1-3) to (w-1-1-3);
\draw [red,line width=1.3pt,->,dashed] (w-1-1-4) to (t-1-1-4);
\node (l-sw-1-1) [position=130:17.5mm from c-1-1] {$\frac{1}{1}$};
\node (l-wt-1-1) [position=50:17.5mm from c-1-1] {$\frac{1}{2}$};
\node (l-uw-1-1) [position=160:7mm from c-1-1] {$\frac{0}{1}$};
\node (l-wv-1-1) [position=20:7mm from c-1-1] {$\frac{0}{1}$};
\node (l-uv-1-1) [position=270:12mm from c-1-1] {$\frac{1}{1}$};
\node (l-su-1-1) [position=200:15mm from c-1-1] {$\frac{1}{2}$};
\node (l-vt-1-1) [position=340:15mm from c-1-1] {$\frac{1}{1}$};
\end{pgfonlayer}
\end{tikzpicture}
\end{center}
\vspace{-1em}
\caption{The flow rerouting problem: Example.}
\label{fig:ex}
\end{figure}
Each of the two flows has an \emph{original (``old'') route} and a \emph{new route},
which it should be updated to.
We indicate the original route
with a \emph{solid}
line and the new route with a \emph{dotted} line.
For example, the original route of the red flow is
$(s,u,v,t)$ and needs to be updated to $(s,w,t)$.
The original route of the blue flow is
$(s,w,t)$ and needs to be updated to $(s,u,w,v,t)$.
In other words, each flow defines an
\emph{update pair}, consisting of two routes (the original and the new one):
Acordingly, updates are denoted using tuples, i.e.,
$(v,\textcolor{blue}{B})$ means that
we activate all
inactive (dotted)
outgoing blue edges (the new \emph{forwarding rules}) of vertex $v$ and deactivate all of its active
(solid) outgoing edges (the old \emph{forwarding rules}).
In this example, we assume that both flows consume
$1$ unit of bandwidth on each link they traverse.
Both flows are unsplittable.
Accordingly, we annotate the network edges in the figure
with two numbers,
$\frac{x}{y}$, where $x$ denotes the bandwidth
consumed by the two flows on the corresponding edge
\emph{before} rerouting and $y$ denotes the edge capacity.
How to reroute the two flows from their old paths to their
new paths in a congestion-free manner?
In this example, initially,
we cannot perform the update $(s,\textcolor{red}{R})$,
since the edge $(s,w)$ has a capacity of $1$ which is currently
used by the blue flow. So
the first part of an update schedule could look like this,
where the updates in this sequence are performed one-by-one:
$$
(u,\textcolor{blue}{B}) , (s,\textcolor{blue}{B}) , (s,\textcolor{red}{R}) , \dots
$$
In this case the red flow would be
routed along the edge $(s,w)$.
However, from there, it could not reach $t$ anymore,
after performing the update $(s,\textcolor{red}{R})$:
the schedule is invalid. In fact,
no valid update sequence can start like
the example above.
One valid sequence is the following:
$$
(u,\textcolor{blue}{B}) , (s,\textcolor{blue}{B}) , (w,\textcolor{red}{R}) ,
(s,\textcolor{red}{R}) , (u,\textcolor{red}{R}) , (v,\textcolor{red}{R}) ,
(v,\textcolor{blue}{B}) , (w,\textcolor{blue}{B})
$$
But this schedule requires $8$ rounds,
updating only one vertex for one flow at a time.
A faster update sequence
schedules multiple updates in a single round, if
possible without introducing congestion: Updates that are
scheduled for the same round are asynchronous and
can occur in any order,
and hence, need to be performed carefully.
The following schedule requires 4 rounds and
is the shortest valid congestion-free
flow rerouting solution for our example:
$$
(u,\textcolor{blue}{B}) , \{(s,\textcolor{blue}{B}) ,
(v,\textcolor{blue}{B}) , (w,\textcolor{red}{R})\} ,
(s,\textcolor{red}{R}) , \{(u,\textcolor{red}{R}) , (v,\textcolor{red}{R}) , (w,\textcolor{blue}{B})\}
$$
A rigorous formal model for this
problem will be given later in this paper.
\subsection{Our Contributions}
This paper initiates the study of polynomial-time
scheduling algoritms to reroute flows in a congestion-free manner
and \emph{fast}. In particular, we contribute the,
to the best of our knowledge \emph{first},
polynomial-time algorithm to compute
shortest rerouting schedules \emph{for two flows}.
In fact, our algorithm runs in (deterministic) \emph{linear time};
its runtime is hence asymptotically optimal.
Moreover, our algorithm is elegant.
We show that this is almost as good as one can hope for
when investing only polynomial time algorithms:
we rigorously prove that even \emph{deciding}
whether a congestion-free reroute schedule exists
is NP-hard, already for 6 flows.
In other words, we provide an almost tight characterization
of the polynomial-time solvability of the problem.
In addition to our formal results, we also show empirically
that the schedules produced by our algorithm are significantly
shorter than the state-of-the-art algorithms focusing on \emph{feasibility}~\cite{icalp18}.
\subsection{Paper Organization}
The remainder of this paper is organized
as follows. Section~\ref{sec:model} presents
a formal model for the problem studied in this paper.
Section~\ref{2flows} describes and analyzes
a polynomial-time update scheduling algorithm
for two flows and Section~\ref{sec:np-hard}
presents the hardness proof for six flows.
We present a nonpolynomial-time algorithm to compute
optimal schedules in Section~\ref{sec:mip}
and present simulation results in Section~\ref{sec:sims}.
After reviewing related work in Section~\ref{sec:relwork},
we conclude our contribution in Section~\ref{sec:conclusion}.
\section{A Rigorous Formal Model}\label{sec:model}
This section presents a rigorous formal model for the
fast congestion-free flow rerouting problem
introduced intuitively in Figure~\ref{fig:ex}.
The problem can be described in terms of edge capacitated directed graphs.
In what follows, we will assume basic familiarity with directed graphs and we refer the
reader to~\cite{digraphs} for more details. We denote a directed
edge $e$ with head $v$ and tail $u$ by $e=(u,v)$. For an undirected
edge $e$ between vertices $u,v$, we write $e=\{u,v\}$;
$u,v$ are called
endpoints of $e$.
For ease of presentation and without loss of generality, we consider directed graphs
with only one source vertex (where flows will originate) and one terminal
vertex (the flows' sink). We call this graph a \emph{flow network}.
The forwarding rules that define the paths considered in
our problem, are best seen as flows in a network.
We will be interested in rerouting flows such that natural
notions of consistency are preserved, such as loop-freedom
and congestion-freedom.
In particular, we will say that a set of
flows is valid if the edge capacities
of the underlying network are respected.
\begin{definition}[Flow Network, Flow, Valid Flow Sets]
A \textbf{flow network} is a directed capacitated graph $G=(V,E,s,t,c)$,
where $s$ is the \emph{source}, $t$ the \emph{terminal}, $V$ is the set
of vertices with $s,t\in V$, $E\subseteq V\times V$ is a set of ordered
pairs known as edges, and $c\colon E\rightarrow\mathbb{N}$ a
capacity function assigning a capacity $c(e)$ to every edge $e\in E$.
An \emph{$(s,t)$-flow} $F$ of capacity $d\in\mathbb{N}$ is a
\emph{directed path} from $s$ to $t$ in a flow network such that
$d\leq c(e)$ for all $e\in E(F)$. Given a
$\mathcal{F}$ of $(s,t)$-flows $F_1,\dots,F_k$ with demands
$d_1,\dots,d_k$ respectively, we call $\mathcal{F}$ a \textbf{valid flow
set}, or simply \textbf{valid}, if $c(e)\geq\sum_{i\colon e\in E(F_i)}d_i$.
\end{definition}
Recall that we consider the problem of how to reroute a current (old) flow to a
new flow, and hence we will consider such flows in ``update pairs'':
\begin{definition}[Update Flow Pair]
An \textbf{update flow pair} $P=(F^o,F^u)$ consists
of two $(s,t)$-flows $F^o$, the \emph{old flow}, and $F^u$,
the \emph{update (or new) flow}, each of demand $d$.
\end{definition}
The update flow network is a flow network (the underlying edge capacitated graph)
together with a \textbf{valid} family of flow pairs. For an illustration, recall
the initial network in Figure~\ref{fig:ex}:
The old flows are presented as the directed paths made of solid edges
and the new ones are represented by the dashed edges.
A flow can be rerouted by updating the outgoing edges of the vertices along its path
(the forwarding rules), i.e.,
by blocking the outgoing edge of the old flow and by allowing
traffic along the outgoing edge of the new flow (if either of them
exists). If these two edges coincide, there are no changes.
In order to ensure transient consistency, the updates of these
outgoing edges need to be scheduled
over time: this results in a sequence which can be partitioned into
update rounds.
\begin{definition}[Resolving Updates, Update Sequence]
Given $G=(V,E,\mathcal{P},s,t,c)$ and an update flow pair $P=(F^o,F^u)\in\mathcal{P}$
of demand $d$, we consider the \textbf{activation label} $\alpha_P\colon E(F^o\cup F^u)\times 2^{V\times\mathcal{P}}\rightarrow \left\{ \operatorname{active},\operatorname{inactive} \right\}$.
For an edge $(u,v)\in E(F^o\cup F^u)$ and a set of updates $U\subseteq V\times\mathcal{P}$, $\alpha_P$
is defined as follows:
\[
\alpha_P((u,v),U)=\left\{\begin{array}{ll} \operatorname{active}, &
\text{if}~(u,P)\notin
U, (u,v)\in
E(F^o), \\\operatorname{active}, &
\text{if}~(u,P)\in
U, (u,v)\in
E(F^u),\\
\operatorname{inactive}, & \text{otherwise.}\end{array}\right.
\]
The graph:
$$\alpha(U,G)=(V,\left\{ e\in E \left|\exists i\in[k]~\text{s.t.}~\alpha_{P_i}(e,U)=\operatorname{active}
\right.\right\})$$ is called the \textbf{$U$-state} of $G$ and we
call any update in $U$ \textbf{resolved}.
An \textbf{update sequence} $\mathfrak{R}=(\mathfrak{r}_1,\dots,\mathfrak{r}_{\ell})$ is an ordered
partition of $V\times\mathcal{P}$. For every such $i$ we define $U_i=\bigcup_{j=1}^i\mathfrak{r}_i$
and consider the activation label $\alpha_P^i(e)=\alpha_P(e,U_i)$ for
every update flow pair $P=(F^o,F^u)\in\mathcal{P}$ of demand $d$ and edge $e\in E(F^o\cup F^u)$.
\end{definition}
Let $(u,P)$ be some update. When we say that we want to \emph{resolve}
$(u,P)$, we mean that we target a state of $G$ in which
$(u,P)$ is resolved. In most cases this will mean to add $(u,P)$ to
the set of already resolved updates.
With a slight abuse of notation, let define $\alpha_P(U,G) = (V(F^o)\cup V(F^u), (E(F^o) \cup E(F^u))$.
In the definition of an update sequence, $\mathfrak{r}_i$ for $i\in[\ell]$ is a \textbf{\emph{round}}.
We define the \emph{initial round} $\mathfrak{r}_0=\emptyset$.
Recall that we consider unsplittable flows which travel along a single path.
The following will clarify
how active edges are to be used.
\begin{definition}[Transient Flow, Transient Family]
The flow pair $P$ is called \textbf{transient} for some set of updates $U\subseteq V\times\mathcal{P}$,
if $\alpha_P(U,G)$ contains a unique valid $(s,t)$-flow $T_{P,U}$.
If there is a \textbf{valid} family $\mathcal{P}=\left\{ P_1,\dots P_k \right\}$ of update flow pairs with
demands $d_1,\dots,d_k$ respectively, we call $\mathcal{P}$ a
\textbf{transient family} for a set of updates $U\subseteq
V\times\mathcal{P}$, if and only if every $P\in\mathcal{P}$ is transient for $U$.
\end{definition}
In short, the transient flows look
like a path of active edges for flow $F$, which starts at the source
vertex and ends at the terminal vertex. Note that there may be some
active edges connected to this path, but they cannot be used to route
the flow since $T_{P,U}$ is unique after resolving
$U$.
The collection of the transient flows corresponding to the transient family
is a snapshot of a valid
updating scenario.
Whenever we say a path $p$ ``routes'' a flow $F$,
we mean that
all edges of path $p$ are active for flow $F$.
In each round $\mathfrak{r}_i$, any subset of updates of $\mathfrak{r}_i$ resolved without
considering the remaining updates of $\mathfrak{r}_i$ should allow a
transient flow for every flow pair. This models the asynchronous
nature of the implementation of the update commands in each round.
\begin{definition}[Consistency Rule]
Let $\mathfrak{R}=(\mathfrak{r}_1,\ldots,\mathfrak{r}_\ell)$ be an update sequence and $i\in[\ell]$.
We require that for any $S\subseteq\mathfrak{r}_i,\mathcal{U}_i^S\coloneqq S\cup \bigcup_{i-1} \mathfrak{r}_j$,
there is a family of transient flow pairs
\end{definition}
\begin{definition}[Valid Update]
An update sequence $\mathfrak{R}$ is \textbf{valid}, or \textbf{feasible},
if every round $\mathfrak{r}_i\in\mathfrak{R}$ obeys the consistency rule.
\end{definition}
Note that we do not forbid any edge $e\in E(F^o_i\cap F^u_i)$ and
we never activate or deactivate such an edge. Starting with an initial update
flow network, these edges will be active and remain so until all updates are resolved.
Hence there are vertices $v\in V$ with either no outgoing edge for a given flow pair $F$ at all;
or $v$ has an outgoing edge, but this edge is used by both the old and the update flow of $F$.
We will call such updates $(v,P)$ \emph{empty}.
Empty updates do not have any impact on the actual problem since they
never change any transient flow. Hence they can always be scheduled in
the first round and thus w.l.o.g.~we can ignore them in the
following. Let us now define the main problem which we
consider in this paper.
\begin{definition}[\textsc{$k$-Network Flow Update Problem}]
Given an update flow network $G$ with $k$ update flow pairs,
is there a feasible update sequence $\mathfrak{R}$? The corresponding optimization
problem is: What is the minimum $\ell$ such that there exists a valid update sequence
$\mathfrak{R}$ using exactly $\ell$ rounds?
\end{definition}
Finally, we introduce some \textbf{preliminaries}.
Let $G=(V,E,\mathcal{P},s,t,c)$ be an update flow network consisting
of two flow pairs $P^1,P^2$, such that
each flow pair is an acyclic graph. For a flow pair $P^i$ ($i\in \{1,2\}$),
let $\prec^i$ be a topological
order on its vertices $V=\left\{ v_1,\dots,v_n \right\}$. We may
write $\prec$ for $\prec^i$ whenever $i$ is clear from the context.
The following applies to both feasible and shortest schedules,
and hence, we use the same terminology as Amiri et
al.~\cite{icalp18}. We only slightly modify the terminology
as unlike Amiri et al., we do not require that the sum of updates forms a DAG
(but only the pairs).
Let $P_i=(F^o_i,F^u_i)$ be an update flow
pair of demand $d$.
We define a topological order $v_1^i,\dots,v_{\ell_i^o}^i$
on the vertices of $F^o_i$ w.r.t.~$\prec^i$ (recall
that we $P_i$ forms an acyclic graph); analogously,
let $u_1^i,\dots,v_{\ell^u_i}^i$
be the order on $F^u_i$. Furthermore, let
$V(F^o_i)\cap V(F^u_i)=\left\{ z_1^i,\dots,z^i_{k_i} \right\}$
be ordered by $\prec^i$ as well.
The subgraph of $F_i^o\cup F_i^u$ induced by the set $\left\{ v\in
V(F_i^o\cup F_i^u) ~|~ z_j^i \prec v \prec z^i_{j+1} \right\}$,
$j\in[k_i-1]$, is called the $j$th \emph{block} of the
update flow
pair $F_i$, or simply the $j$th \emph{$i$-block}. We
will denote this block by $b^i_j$.
For a block $b$, we define $\Start{b}$ to be
the \emph{start of the block}, i.e., the smallest vertex
w.r.t.~$\prec^i$; similarly, $\End{b}$ is the \emph{end of the block}:
the largest vertex w.r.t.~$\prec^i$.
Let $G=(V,E,\mathcal{P},s,t,c)$ be an update flow network with
$\mathcal{P}=\left\{ P_1,\dots,P_k \right\}$ and let
$\mathcal{B}$ be the set of its
blocks. We define a binary relation $<$ between two blocks as follows.
For two blocks $b_1,b_2\in \mathcal{B}$, where $b_1$ is an $i$-block and $b_2$ a
$j$-block, $i,j\in[k]$, we say $b_1<b_2$ ($b_1$ \emph{is smaller than}
$b_2$) if one of the following holds.
\begin{enumerate}[i]
\item $\Start{b_1} \prec \Start{b_2}$,
\item if $\Start{b_1}=\Start{b_2}$ then $b_1<b_2$, if $\End{b_1} \prec \End{b_2}$,
\item if $\Start{b_1}=\Start{b_2}$ and $\End{b_1}=\End{b_2}$ then $b_1<b_2$, if $i<j$.
\end{enumerate}
Let $b$ be an $i$-block and $P_i$ the corresponding update flow pair.
For a feasible update sequence $\mathfrak{R}$, we will denote the round
$\mathfrak{R}(\Start{b},P_i)$ by $\mathfrak{R}(b)$. We say that $i$-block $b$ is
\emph{updated}, if all edges in $b\cap F^u_i$ are active and all edges
in $b\cap F_i^o\setminus F_i^u$ are inactive.
\section{A Fast Scheduling Algorithm}\label{2flows}
This section presents an elegant, \emph{linear-time} and deterministic
and deterministic algorithm to compute
shortest update schedules for two flows.
Let $G=(V,E,\mathcal{P},s,t,c)$ be an update
flow network where $(V,E)$ is the union of the
DAGs implied by the flow pairs.
Let $\mathcal{P}=\left\{ B,R \right\}$ be the two update flow pairs
with $B=(B^o,B^u)$ and $R=(R^o,R^u)$ of demands $d_B$ and $d_R$.
As in the previous section, we identify $B$ with blue and $R$ with red.
We say that an $I$-block $b_1$ is \emph{dependent}
on a $J$-block $b_2$, $I,J\in\left\{ B,R \right\}$, $I\neq J$, if
there is an edge $e\in (E(b_1)\cap E(I^u))\cap(E(b_2)\cap E(J^o))$,
but $c(e) < d_I+d_J$. In fact, to update $b_1$, we either
violate capacity constraints, or we update $b_2$ first in order to
prevent congestion. In this case, we write $b_1\rightarrow b_2$
and say that $b_1$ \emph{requires} $b_2$. A block that does not depend on any other block is called \emph{free}.
We say a block $b$ is a \emph{free block}, if it is not dependent on any
other block. A \emph{dependency graph} of $G$ is a graph
$D=(V_D,E_D)$ for which there exists a bijective
mapping $\mu\colon V(D)\leftrightarrow B(G)$, and there is an
edge $(v_b,v_{b'})$ in $D$ if $b\rightarrow b'$. Clearly, a block $b$
is free if and only if it corresponds to a sink in $D$.
\smallskip
We propose the following algorithm to check the feasibility
of the flow rerouting problem.
\begin{algorithm}\textbf{Feasible $2$-Flow DAG Update}\label[algorithm]{alg:main}
\begin{enumerate}
\item [] \textbf{Input: Update Flow Network $G$}
\item Compute the dependency graph $D$ of $G$.
\item If there is a cycle in $D$, return \emph{impossible to update}.
\item While $D\neq \emptyset$ repeat:
\begin{enumerate}[i]
\item \label{lbl:step} Update all blocks which correspond to the sink
vertices of $D$.
\item Delete all of the current sink vertices from $D$.
\end{enumerate}
\end{enumerate}
\end{algorithm}
Recall that empty updates can always be scheduled in the first round,
even for infeasible problem instances. So
for \Cref{alg:main} and all following algorithms, we simply
assume these updates to be scheduled together with the non-empty updates of round $1$.
Figure~\ref{fig:exwithblocks} gives an example of an update flow
network on a DAG and illustrates the block decomposition and
its value to finding a feasible update sequence.
\begin{figure*}[t]
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\end{pgfonlayer}
\end{tikzpicture}
\end{center}
\vspace{-1em}
\caption{\emph{Example} for \Cref{alg:main}. The $2$ update flow pairs are \textcolor{red}{red} and
\textcolor{blue}{blue},
each of demand $1$.
The active edges of the respective colors are indicated as \emph{solid lines} and the inactive edges are \emph{dashed}.
Each edge in the flow graph is annotated with its current
load (\emph{top}) and its capacity (\emph{bottom}).
We start by identifying the \textcolor{blue}{blue} and \textcolor{red}{red} blocks. For \textcolor{red}{red} there is exactly one such block \textcolor{red}{$r_1$}, since \textcolor{red}{$R^o$} and \textcolor{red}{$R^u$} only coincide in $s$ and $t$. The \textcolor{blue}{blue} flow pair on the other hand omits two blocks \textcolor{blue}{$b_1$} and \textcolor{blue}{$b_2$}: \textcolor{blue}{$B^o$} and \textcolor{blue}{$B^u$} meet again at $w$ and at $t$.
We observe that $\textcolor{blue}{b_2}$
can only be updated after $\textcolor{red}{r_1}$
has been updated; similarly, $\textcolor{red}{r_1}$
can only be updated after $\textcolor{blue}{b_1}$
has been updated.
An update sequence respecting these dependencies
can be constructed as follows.
We can first prepare the blocks by updating
the following two out-edges which currently
do not carry any flow:
$(w,\textcolor{red}{red})$,
$(u,\textcolor{blue}{blue})$, and $(v,\textcolor{blue}{blue})$.
Subsequently, the three blocks can be updated in a congestion-free
manner in the following order:
Prepare the update for all blocks in the first round. Then, update
$\textcolor{blue}{b_1}$ in the second round,
$\textcolor{red}{r_1}$ in the third round,
$\textcolor{blue}{b_2}$ in the fourth round.
}
\label{fig:exwithblocks}
\end{figure*}
Suppose $\mathfrak{R}$ is a feasible update sequence for $G$. We say
that a $c$-block $b$
w.r.t.~$\mathfrak{R}=(\mathfrak{r}_1,\ldots,\mathfrak{r}_\ell)$ is \emph{updated in
consecutive rounds}, if the following holds:
if some of the edges of $b$
are activated/deactivated in
round $i$ and some others in round $j$,
then for every $i<k<j$, there is
an edge of $b$ which is activated/deactivated.
\begin{lemma}\label[lemma]{lem:updateblockstart}
Let $b$ be a $c$-block. Then in
a feasible update sequence $\mathfrak{R}$, all vertices
(resp.~their outgoing $c$ flow edges)
in $F^u_c\cap b - \Start{b}$ are
updated strictly before $\Start{b}$. Moreover, all vertices in $b-F^u_c$ are updated strictly
after $\Start{b}$ is updated.
\end{lemma}
\begin{proof}
In the following, we will implicitly assume
flow $c$, and will not mention
it explicitly everywhere.
We will write $F^u_b$ for $F^u_c\cap b$ and
$F^o_b$
for $F^o_c\cap b$.
For the sake of contradiction, let $U=\{v\in V(G)\mid v\in F^u_b-F^o_b-\Start{b},\mathfrak{R}(v,c) > \mathfrak{R}(\Start{b},c)\}$.
Moreover, let $v$ be the
vertex of $U$ which is updated the latest
and $\mathfrak{R}(v,c) = \max_{u\in U}\mathfrak{R}(u,c)$.
By our condition, the update of $v$ enables
a transient flow along
edges in $F^u_c\cap b$. Hence,
there now exists an $(s,t)$-flow through $b$ using only update edges.
No vertex in $F_1\coloneqq F^o_b-(F^u_b-\Start{b})$
could have been updated before, or
simultaneously with $v$:
otherwise, between the time $u$ has been updated
and before the update
of $v$, there would not exist a transient flow.
But once we update $v$, there is a $c$-flow which
traverses the
vertices in $F^o_b-F^u_b$, and another $c$-flow which
traverses
$v\not\in F_1$: a contradiction. Note that $F_1\neq \emptyset$.
The other direction is obvious: updating
any vertex in $(F^o_c\cap b)-F^u_c$ inhibits any transient flow.
\end{proof}
\begin{lemma}\label[lemma]{lem:updatewholeblock}
Given any feasible (not necessarily shortest)
update sequence $\mathfrak{R}$, there is a feasible
update sequence $\mathfrak{R}'$ which updates every block in at most
$3$ consecutive rounds.
\end{lemma}
\begin{proof}
Consider the following approach to
update free blocks
(for the $i$-block $b$):
first resolve $(v,P_i)$ for all $v\in P^u_i \cap b-\Start{b}$;
then resolve $(\Start{b},P_i)$; finally
resolve $(v,P_i)$ for all $v\in (b-P^u_i)$.
Now let $\mathfrak{R}$ be a feasible update sequence with a minimum number of
blocks which are not updated in~$3$ consecutive rounds.
Furthermore let $b$ be such a $c$-block. Let $i$
be the round in which $\Start{b}$ is updated. Then
by~\Cref{lem:updateblockstart},
all other vertices of
$F^u_c\cap b$ have been updated
in the previous rounds. Moreover, since they do not carry
any flow during these rounds, the edges can all be updated in round $i-1$.
By our assumption, we can
update $\Start{b}$ in round $i$, and hence now
this is still possible.
As $\Start{b}$ is updated in round~$i$, the edges of
$F^o_c\cap b$ do not carry any active $c$-flow in
round~$i+1$ and thus we can deactivate all remaining
such edges in this round. This is a contradiction to the
choice of $\mathfrak{R}$, and hence there is always a feasible
sequence $\mathfrak{R}$ satisfying the requirements of the
lemma.
In particular, the above algorithm is correct.
\end{proof}
From the above lemmas, we immediately
derive a corollary regarding the optimality in terms
of the number of rounds:
the~$3$ rounds feasible update sequence.
\begin{corollary}\label{cor:blockroundoptimal}
Let $b$ be any $c$-block with
$\left| E(b\cap F_c^o)\right|\geq 2$ and
$\left| E(b\cap F_c^u)\right|\geq 2$. Then it is
not possible to update $b$ in less than~$3$
rounds: otherwise it is not possible to
update~$b$ in less than~$2$ rounds.
\end{corollary}
Next we show that if there is a cycle in the
dependency graph, then
it is impossible to update any flow.
\begin{lemma}\label[lemma]{lem:nocycle}
If there is a cycle in the dependency graph, then
there is no feasible update sequence.
\end{lemma}
\begin{proof}
Suppose that there exists a cycle in the dependency graph.
Without loss of generality, we can assume that this is the only
cycle in the dependency graph as
we can always remove vertices without creating new dependencies.
Then it is not possible to update the cycle.
For the sake of contradiction, suppose
that there is a feasible update order; then there is a feasible
update order in which blocks are updated in consecutive (distinct)
rounds. But in this order, one of the vertices in
the dependency graph (a block)
should be earlier than the others.
This is impossible due to dependency
on other vertices.
\end{proof}
We will now slightly modify \Cref{alg:main} to create a new algorithm
which not only computes
a feasible sequence $\mathfrak{R}$ for a given update flow network in polynomial time,
whenever it exists, but which also ensures that $\mathfrak{R}$ is as short as possible
(in terms of number of rounds). For any block $b$, let $c(b)$ denote its corresponding flow pair.
\begin{algorithm}\textbf{Optimal $2$-Flow DAG Update}\label[algorithm]{alg:mainopt}
\begin{enumerate}
\item [] \textbf{Input: Update Flow Network $G$}
\item Compute the dependency graph $D$ of $G$. \label{line:1}
\item If there is a cycle in $D$, return \emph{impossible to update}.
\item If there is any block $b$ corresponding to a
sink vertex of $D$ with $(b\cap
F^u_{c(b)})-\Start{b}\neq\emptyset$ set
$i\colon\!\!\!=2$, otherwise set $i\colon\!\!\!=1.$
\item While $D\neq \emptyset$ repeat:
\begin{enumerate}[i]
\item \label{lbl:stepopt} Schedule the update of all blocks $b$ which correspond to the sink
vertices of $D$
for the
rounds $i-1$, $i$, $i+1$, such that
$\Start{b}$ is updated in round $i$.
\item Delete all of the current sink vertices from $D$.
\item Set $i\colon\!\!\!= i+1$.
\end{enumerate}
\end{enumerate}
\end{algorithm}
\begin{theorem}
An optimal (feasible) update sequence
on acyclic update flow networks with exactly~$2$ update flow pairs can be found in linear time.
\end{theorem}
\begin{proof}
Let $G$ denote the given update flow network.
In the following, for ease of presentation, we will slightly abuse
terminology and say that ``a block is updated in some round'',
meaning that the block is updated in the corresponding consecutive rounds
as in the proof of
\Cref{lem:updatewholeblock}.
We proceed as follows. First, we find a block decomposition and
create the dependency graph of the input instance.
This takes linear time only.
If there is a cycle in that graph, we output \emph{impossible}
(cf~\Cref{lem:nocycle}).
Otherwise, we apply \Cref{alg:mainopt}.
As there is no cycle in the dependency
graph (a property that stays invariant),
in each round, either there exists a free block which is not
processed yet, or everything is already updated or is in the process of being updated. Hence, if there is a
feasible solution (it may not be unique), we can find one in time~$O(\sizeof{G})$.
For the optimality in terms of the number of rounds,
consider two feasible update sequences. Let $\mathfrak{R}_{\text{\sc Alg}}$ be the update
sequence produced by \Cref{alg:mainopt} and let $\mathfrak{R}_{\text{\sc
Opt}}$ be a feasible update sequence that realizes the minimum
number of rounds. According to~\Cref{lem:updateblockstart},
any block $b$ is processed only in
round $\Start{b}$.
Suppose there is a block $b'$ such that $\mathfrak{r}_{\text{\sc
Opt}}(b')<\mathfrak{r}_{\text{\sc Alg}}(b')$. Then let $b$ be the block
with the smallest such $\mathfrak{r}_{\text{\sc Opt}}(b)$. Hence, for every
block $b''$ with $\mathfrak{r}_{\text{\sc Opt}}(b'')\leq\mathfrak{r}_{\text{\sc
Opt}}(b)$, $\mathfrak{r}_{\text{\sc Opt}}(b'')\geq\mathfrak{r}_{\text{\sc
Alg}}(b'')$ holds. Since $\Start{b}$ is updated in round
$\mathfrak{r}_{\text{\sc Opt}}(b)$, there are no dependencies for $b$ that are
still in place in this round. Thus,
according to the
sequence $\mathfrak{R}_{\text{\sc Opt}}$,
$b$ is a sink vertex of the
dependency graph after round $\mathfrak{r}_{\text{\sc Opt}}(b)-1$.
Furthermore, by our previous
observation, every start of some block has been updated up to this
round in the optimal sequence, and hence it is also already updated in the same
round in $\mathfrak{R}_{\text{\sc Alg}}$. This means that after round
$\mathfrak{r}_{\text{\sc Opt}}(b)-1<\mathfrak{r}_{\text{\sc Alg}}(b)-1$, $b$ is a sink
vertex of the dependency graph of $\mathfrak{R}_{\text{\sc Alg}}$ as well.
Thus, \Cref{alg:mainopt} would have scheduled the update of
block $b$ in the rounds $\mathfrak{r}_{\text{\sc Opt}}(b)-1$, $\mathfrak{r}_{\text{\sc
Opt}}(b)$ and $\mathfrak{r}_{\text{\sc Opt}}(b)+1$. Contradiction.
Thus $\mathfrak{r}_{\text{\sc Alg}}(b)\leq \mathfrak{r}_{\text{\sc Opt}}(b)$
for all blocks $b$. Now let $b_1,\dots,b_{\ell}$ be the last
blocks whose starts are updated the latest under
$\mathfrak{R}_{\text{\sc Alg}}$. If there is some $i\in[\ell]$
such that $\left| E_{b_i}^o\right|\geq 2$ and $\left|
E_{b_i}^u\right|\geq 2$, $\mathfrak{R}_{\text{\sc Alg}}$ uses
exactly $\mathfrak{r}_{\text{\sc Alg}}(b_i)+1$ rounds;
otherwise it is one round less, by Corollary~\ref{cor:blockroundoptimal}.
By our previous observation, none of these blocks can start later than
$\mathfrak{r}_{\text{\sc Alg}}(b_i)$ and thus $\mathfrak{r}_{\text{\sc Opt}}$ uses at
least as many rounds as \Cref{alg:mainopt}. Hence the algorithm is
optimal in the number of rounds.
\end{proof}
\section{NP-hardness for More Flows}\label{sec:np-hard}
This section shows that the polynomial-time result derived
above cannot be generalized much further:
it is NP-hard to compute a shortest schedule already for six flows,
and even if the pair of old and new path \emph{forms a DAG}.
In fact, we show that already the decision problem, i.e., whether a feasible
schedule exists, is NP-hard.
\begin{theorem}
\label{thm:6flow_hardness}
Deciding whether a feasible network update schedule exists for
a given update flow network in which each flow pair
forms a DAG is $NP$-hard for 6 flows.
\end{theorem}
We use a reduction from 3-SAT. Let $C$ be any 3-SAT formula
with $n$ variables $x_1,\dots, x_n$ and
$m$ clauses $C_1,\dots, C_m$. The resulting update flow
network is denoted as $G(C)$.
We will create $6$ flow pairs: $X$, $\overline{X}$, $D_1$,
$D_2$, $D_3$ and $B$, each having demand $1$. $B$is
the blocking
pair: it can by updated only if all clauses are satisfied.
Flows $X$ and $\overline{X}$ contain gadgets
for all literals, $X$ for positive ones and $\overline{X}$ for negative
ones. Updating a variable gadget in $X$
corresponds to assigning the variable value $1$ in $C$.
Flow $B$ prevents the variable gadget to be updated
in both $X$ and $\overline{X}$, unless all clauses are satisfied.
Flows $D_1$, $D_2$ and $D_3$ encode clauses of $C$.
Each of these flows contains a clause gadget linking
a clause to one of its literals. This gadget can be updated only if the literal is
satisfied. Updating a clause gadget
in one of those flows will allow $B$ to be updated.
Now we proceed with the detailed description of the reduction.
\begin{enumerate}
\item \textbf{Clause gadgets:} For every clause $i\in[m]$ we
introduce eight vertices: $u^i$, $v^i$
and for $j\in \{1,2,3\}$ $u^i_j$ and $v^i_j$. For $j\in \{1,2,3\}$
we add edge $(u^i, v^i)$ to $D^o_j$
and edges $(u^i, u^i_j)$, $(u^i_j, v^i_j)$ and $(v^i_j, v^i)$ to $D^u_j$.
\item \textbf{Variable Gadgets:} For every $j\in[n]$, we introduce two
vertices: $w_1^j$ and $w_2^j$. Let $P_j=\left\{
p_1^j,\dots,p_{k_j}^j \right\}$ denote the set of indices of the
clauses containing the literal $x_j$ and $\overline{P}_j=\left\{
\overline{p}^j_1,\dots,\overline{p}^j_{k'_j} \right\}$ the set of
indices of the clauses containing the literal
$\overline{x}_j$. Furthermore, let $\pi(i,j)$ denote the position of
$x_j$ in the clause $C_i$, $i\in P_j$. Similarly,
$\overline{\pi}(i',j)$ denotes the position of $\overline{x_j}$ in $C_{i'}$ where
$i'\in\overline{P}_j$.
To $X^u$ and $\overline{X}^u$ we add edge $(w_1^j, w_2^j)$.
To $X^o$ we add the following edges:
\begin{itemize}
\item $(u^{p_i^j}_{\pi(p_i^j,j)}, v^{p_i^j}_{\pi(p_i^j,j)})$ for all $i\in \{1,\dots,k_j\}$,
\item $(v^{p_i^j}_{\pi(p_i^j,j)}, u^{p_{i+1}^j}_{\pi(p_{i+1}^j,j)})$ for all $i\in \{1,\dots,k_j-1\}$,
\item $(w_1^j, u^{p_{1}^j}_{\pi(p_{1}^j,j)})$ and $(v^{p_k^j}_{\pi(p_k^j,j)},w_2^j)$.
\end{itemize}
We proceed similarly with $\overline{X}^o$ and
clauses containing $\overline{x}_j$.
\item \textbf{Blocking flow:} The goal of flow $B$ is to
block update of $w_1^j$, for any $j\in[n]$, in both $X$ and $\overline{X}$.
To do that we add to $B^o$ the following edges:
\begin{itemize}
\item $(w_1^j, w_2^j)$, for all $j\in[n]$,
\item $(w_2^j, w_1^{j+1})$, for all $j\in[n-1]$.
\end{itemize}
We also add the following edges to $B^u$:
\begin{itemize}
\item $(u^i, v^i)$, for all $i\in[m]$,
\item $(v^i, u^{i+1})$, for all $i\in[m-1]$.
\end{itemize}
\item \textbf{Source and Terminal:} Now we need to
connect all the gadgets in the flows.
The source and the terminal of all flows will be $s$ and $t$.
To $D_j^o$ and $D_j^u$, for $j\in\{1,2,3\}$, we add the following edges:
\begin{itemize}
\item $(v^i,u^{i+1})$, for all $i\in[m-1]$,
\item $(s, u^1)$ and $(v^m, t)$.
\end{itemize}
To $X^o$, $X^u$, $\overline{X}^o$ and $\overline{X}^u$
we add the following edges:
\begin{itemize}
\item $(w^j_2, w^{j+1}_1)$, for all $j\in[n-1]$
\item $(s, w^1_1)$ and $(w^n_2, t)$
\end{itemize}
We also add edges $(s, w^1_1)$ and $(w^n_2, t)$ to $B^o$ and
edges $(s, u^1)$ and $(v^m, t)$ to $B^u$.
\item \textbf{Edges capacity:} For all $j\in[n]$ we set the capacity of
edge $(w^j_1, w^j_2)$ to be $2$.
Also for all $i\in[m]$ we set capacity of edge $(u^i,v^i)$ to be $3$
and capacity of edge $(u^i_j,v^i_j)$, for $j\in\{1,2,3\}$, to be $1$.
For all the other edges, we set their capacity to be $6$, that is, to the
number of flows. Therefore they cannot violate any capacity constraint.
\end{enumerate}
\begin{lemma} Given any valid update sequence $\mathfrak{R}$
for the above constructed update flow network $G(C)$,
the following conditions hold.
\begin{enumerate}
\item For every $r<\mathfrak{R}(s,B)$ and $j\in[n]$ $\mathfrak{R}(w^j_1, X) > r$ or $\mathfrak{R}(w^j_1, \overline{X}) > r$.
\label{con:nphardvar}
\item For every $r\geq\mathfrak{R}(s,B)$ and $i\in[m]$ $\mathfrak{R}(u^i,D_1) < r$, $\mathfrak{R}(u^i,D_2) < r$ or $\mathfrak{R}(u^i,D_3) < r$.
\label{con:nphardclause}
\end{enumerate}
\label{lem:6flowhardaux}
\end{lemma}
\begin{proof}
Note that $B^o$ and $B^u$ have no common nodes apart from
$s$ and $t$. That means that for any $r$ either $T_{B,U_r} = B^o$
or $T_{B,U_r} = B^u$. Now we prove both conditions.
\begin{enumerate}
\item Let us consider any $j\in[n]$.
As $r<\mathfrak{R}(s,B)$, then $T_{B,U_r} = B^o$.
The capacity of edge $(w^j_1, w^j_2)$ is $2$ and it belongs
to $B^o$. Therefore it can be in at most
one other transient flow, so the condition holds.
\item Let us consider any $i\in[m]$. As $r\geq\mathfrak{R}(s,B)$,
then $T_{B,U_r} = B^u$.
The capacity of edge $(u^i, v^i)$ is $3$ and it belongs to $B^u$.
Therefore it can be in at most two other transient flows,
so the condition holds.
\end{enumerate}
\end{proof}
\begin{proof}
Now we are ready to prove Theorem \ref{thm:6flow_hardness}.
First, let us assume that $C$ is satisfiable and
we will construct valid update sequence for $G(C)$.
Let $\sigma$ be an assignment satisfying $C$. Then the
update sequence for $G(C)$ is as follows.
\begin{enumerate}
\item For every $j\in[n]$, if $\sigma(x_j) = 1$ then resolve $(w^j_1,X)$,
otherwise resolve $(w^j_1,\overline{X})$.
\item For every clause $C_i$ at least one of
the edges $(u^i_1,v^i_1)$, $(u^i_2,v^i_2)$
and $(u^i_3,v^i_3)$ is neither in $T_{X,r_2^f-1}$ nor $T_{\overline{X},r_2^f-1}$. So the update of $u^i$ can be resolved in
the corresponding flow $D_1$, $D_2$ or $D_3$ (this follows
from $\sigma$ being satisfying assignment).
\item As every $i\in[m]$ edge $(u^i,v^i)$ is used
by at most $2$ flows,
we can resolve every update in $B^u$, resolving $(s,B)$ as the last one.
\item For every $j\in[n]$, resolve either $(w^j_1,X)$ or $(w^j_1,\overline{X})$,
depending on which one was not resolved in step 1.
\item For every $i\in[m]$ resolve updates of $u^i$
in flows $D_1$, $D_2$ and $D_3$
(those that have not been resolved in step 2).
\item Resolve the remaining updates in all flows.
\end{enumerate}
Now let us assume that there is a valid update
sequence $\sigma$ for $G(C)$.
We will show that $C$ is satisfiable by
constructing satisfying assignment $\sigma$.
Let us consider round $r=\sigma(s,B)$. We assign values
in the following way. For $j\in[n]$, if
$\sigma(w_1,X) < r$ then $\sigma(x_j) := 1$
and if $\sigma(w_1,\overline{X}) < r$ then $\sigma(x_j) := 0$. If both
$\sigma(w_1,X) > r$ and $\sigma(w_1,\overline{X}) > r$
we assign to $x_j$ a random
value. By Condition \ref{con:nphardvar} of Lemma \ref{lem:6flowhardaux}
this is a correct assignment,
that is no variable is assigned two values.
We want to prove that this assignment satisfies $\sigma$.
Let us consider any clause $C_i$.
By Condition \ref{con:nphardclause} of Lemma \ref{lem:6flowhardaux}
at least one of $(u^i,D_1)$, $(u_i,D_2)$ or $(u_i, D_3)$
is updated before round $r$. That m0eans that at least for one of variables $x_j$
in $C_i$ $\sigma(w^j_1,X) < r$, if $C_i$ contains literal $x_j$, or $\sigma(w^j_1,\overline{X}) < r$, if $C_i$
contains literal $\overline(x)_j$. This means that
$C_i$ is satisfied by $x_j$ in $\sigma$.
\end{proof}
\section{Optimal Scheduling of Arbitrary Problems}\label{sec:mip}
Since the problem is generally NP-hard, for completeness and in order to
investigate the runtime of such an approach, we in the following
describe an optimal scheduling algorithm for a general model
and arbitrary number of flows, which runs in super-polynomial time.
The algorithm is based on mixed integer linear programming.
The formulation first reserves variables for all possible rounds during which
a node can update a flow (Constraints (\ref{line:1})).
Henceforth, we refer to them as \textit{schedule variables}.
Schedule variables are constrained so that a node updates each of its flows only once.
The remaining constraints ensure the following feasibility criteria:
\begin{enumerate}
\item
Constraints (\ref{LP:init_oldedges}) to (\ref{LP:forknode}) prepare the variables
used in the consistency checks.
\item
Assuming a value assignment to the schedule variables,
Constraints (\ref{LP:transientgraph1}) to (\ref{LP:transientflow_check}) emulate the update
with respect to the schedule and ensure no flow is interrupted during any update round $r$.
That is, for any link that receives a new flow during $r$, the incident node
must have been already updated in an earlier round ($<r$).
Also, any node that removes a flow from an outgoing link
must postpone this to a later round ($>r$).
However, there is an exception for nodes that are incident
to both old and new flow links (denoted by fork nodes).
This criteria accounts for the fact that node updates occur
asynchronously and the flows must not be interrupted in any case.
\item
With the last set of constraints, we ensure that during the emulation all capacities are respected.
\end{enumerate}
\begin{figure}[!h]
\begin{IEEEeqnarray}{ll}
\textbf{Minimize}\;R \label{LP:objective}
\\
\textit{\small{ROUNDS}} = \{1,..,(|V|-1).|P|\} \nonumber
\\
\sum_{r\in \textit{\scriptsize{ROUNDS}}} x^r_{v,i} = 1 & \forall i\in |P|, v\in P_i \label{LP:schedule} \IEEEeqnarraynumspace
\\
y^0_{(u,v),i} = 1 & \forall (u,v)\in F_i^o \label{LP:init_oldedges}
\\
y^0_{(u,v),i} = 0 & \forall (u,v)\in F_i^u \label{LP:init_newedges}
\\
\forall i\in [|P|], r\in \textit{\small{ROUNDS}} \;
\boldsymbol{\{} & \label{LP:repeat}
\\
x^r_{v,i},\textit{fork}^r_{v,i},\textit{join}^r_{v,i} \in \{0,1\} & \forall v\in P_i
\\
y^r_{(u,v),i},f^r_{(u,v),i} \in [0,1]
& \forall (u,v)\in P_i\label{LP:transientgraph1}
\\
R \geq r.x^r_{v,i} & \forall v\in P_i \label{LP:rounds}
\\
y^r_{(u,v),i} = \sum_{r'\leq r} x^{r'}_{u,i} & \forall (u,v)\in F_i^u \label{LP:activeflag1}
\\
y^r_{(u,v),i} = 1 - \sum_{r'\leq r} x^{r'}_{u,i} & \forall (u,v)\in F_i^o \label{LP:activeflag2}
\\
\textit{fork}^r_{v,i} =
\begin{cases}
x^r_{v,i} & \hspace{-0.5em} \exists w,w'\in P_i:
\begin{cases}
(v,w)\in F_i^o \\
(v,w')\in F_i^u
\end{cases} \\
0 & \text{else}
\end{cases}
& \forall v\in P_i \label{LP:forknode}
\\
f^r_{(u,v),i} \leq y^{r-1}_{(u,v),i} + \textit{fork}^r_{u,i}
& \forall (u,v)\in P_i\label{LP:transientgraph1}
\\
f^r_{(u,v),i} \leq y^{r}_{(u,v),i} + \textit{fork}^r_{u,i}
& \forall (u,v)\in P_i \label{LP:transientgraph2}
\\
\textit{join}^r_{v,i} \leq
f^{r}_{(u,v),i},f^{r}_{(u',v),i}
& \forall v,u,u' \! \in \! P_i:\!
\begin{cases}
(u,v) \! \in \! F_i^o \\
(u',v) \! \in \! F_i^u
\end{cases} \label{LP:joinnode}
\\
\sum_{(v,w)\in P_i} f^{r}_{(v,w),i} - \sum_{(u,v)\in P_i} f^{r}_{(u,v),i} =
\begin{cases}
1 + \textit{fork}^r_{s,i} & v=s\\
-(1 + \textit{join}^r_{t,i}) & v=t\\
\textit{fork}^r_{v,i} - \textit{join}^r_{v,i} & \text{else}
\end{cases}
& \forall v\in P_i \label{LP:transientflow_check}
\\
\boldsymbol{\}} \nonumber
\\
\sum_{i\in [|P|]} f^r_{(u,v),i} \leq C_{(u,v)} & \hspace{-5em} \forall r \in \textit{\small{ROUNDS}}, (u,v)\in E \label{LP:capacitycheck}
\end{IEEEeqnarray}
\caption{Mixed Integer Program for $k$ flow pairs}
\label{MIP}
\end{figure}
Next, we describe the formulation in detail.
\begin{itemize}
\item (\ref{LP:schedule}): Each schedule variable $x^r_{v,i}$ indicates whether a node $v$ is scheduled to update flow $i$ in round $r$.
\item (\ref{LP:repeat}): Repeat the embraced lines for every pairs $P_i$ and each round $r \in \textit{ROUNDS}$.
\item (\ref{LP:activeflag1}),(\ref{LP:activeflag2}): $y^r_{(u,v),i}$ indicates whether the link $(u,v)$ is active for pair $i$ immediately after round $r$ (i.e. active graph).
\item (\ref{LP:forknode}): \textit{fork nodes} are the nodes at which old and update paths split.
A fork node $v$ acts as a source, doubling its incoming transient flow $i$, when it updates the flow during round $r$ (i.e.~if $\textit{fork}^r_{v,i}$ is 1).
\item (\ref{LP:joinnode}): \textit{join nodes} are nodes at which the old and update paths meet once again.
A join node acts as a sink (if $\textit{join}^r_{v,i}$ is 1) when the two in-links both carry the transient flow $i$ in the transient state of round $r$.
\item (\ref{LP:transientgraph1}),(\ref{LP:transientgraph2}): $f^r_{u,v,i}$ specifies the transient flow $i$ on a link $(u,v)$.
The first terms on the r.h.s.~constrains together state that the link is allowed to be utilized in the transient state of round $r$,
if it is active before round $r$ and it remains active during the round.
Alternatively, if the link is deactivating in round $r$ due to the updating fork node $u$, then the second term allows the link to be usable in the transient state.
(This, along with Constraint (\ref{LP:transientflow_check}) guarantees there will be no loops on the old out-branch of any updating fork node.)
\item (\ref{LP:transientflow_check}): Runs a variable-size transient flow $i$ from $s$ to $t$
in order to impose $st$-connectivity in the worst-case transient state.
The flow produced at $s$ is of size 1 and it arrives at $t$ with the same size.
In the meanwhile, any active fork node (including possibly $s$) adds one unit to this flow and
splits it into two unit-size flows along both its out-links.
Later, a join node consumes this extra flow by taking away the 1 unit.
\item (\ref{LP:capacitycheck}): The capacity constraints.
\end{itemize}
Because of a possible cleanup round after a fork node updates,
it is necessary to maintain $st$-connectivity via both (old and update) out-links of the fork node,
which is ensured by (15). In other words, no cleanup (i.e.~removal of old flow rules) should occur
on the old branch of the fork node in the same round it reroutes to the new branch.
\section{Empirical Results}\label{sec:sims}
In order to gain insights into the actual number of rounds
needed to reroute flows in real networks, we conducted
a simulation study on real network topologies. In particular,
we want to compare the length of the schedules produced by our
algorithm (which provably provides \emph{shortest}
schedules) to the state-of-the-art algorithm presented in~\cite{icalp18}
(which only computes \emph{feasible} schedules).
In order to study the need for fast algorithms, we
compare the runtime of our algorithm to the mathematical
programming approach, as it is frequently used in the literature~\cite{update-survey}.
We implemented \Cref{alg:mainopt} using standard C++ libraries and performed
a exhaustive evaluation on over 100 topologies provided by
\cite{topologyZoo} and $\approx 136$ million records in total.
The graphs were chosen so that the runtime would be practical.
In another, but similar implementation we evaluated \Cref{alg:main}
on the same input cases and
compared the number of rounds obtained from these algorithms in \Cref{fig:charts_rounds}.
We observe that our Algorithm~\ref{alg:main} is a simpler
feasibility algorithm than~\cite{icalp18}, for two flows:
it employs basic batching, which leads to shorter schedules
compared to ~\cite{icalp18}. In the following,
we hence use Algorithm~\ref{alg:main}
as a baseline and lower bound on the number of rounds needed
by the more complex algorithm in~\cite{icalp18}.
The input data does not provide any capacity on the links.
Capacities determine the block dependency and an insufficient capacity
allocation can lead to a cyclic $D$
which renders an instance infeasible. On the other hand,
examining all possible allocations is not practical.
Hence, in order to capture the maximum rounds in each graph efficiently,
we take into account also the infeasible instances.
Later, we explain how infeasible instances are handled.
In a preprocessing step, the evaluation takes the raw graph and
allocates minimal capacities:
set the capacity to 2 for links that carry the old/update flow paths of both pairs,
otherwise set the capacity to 1.
The program, for every pair of source and destination $(s,t)$, first computes all the paths from $s$ to $t$.
Next, it iterates over all possible path pairs (i.e.~old and update paths) chosen independently for each of the two flows (dismissing identical path pairs).
Each iteration does the following.
\begin{enumerate}
\item
Perform Line~\ref{line:1} on the path pairs and generates a block dependency graph $D$.
\item
Enumerate all paths in $D$ and each path $P$ is weighted as follows.
\begin{enumerate}
\item Initialize $w(P) = |P|$.
\item \label{item:preparation_round}
Let $b_1$ be the block that corresponds to the last vertex of $P$.
Set $w(P) = w(P)+1$ if $|E[b_1 \cap F^u(b_1)]| > 1$.
\item \label{item:cleanup_round}
Let $b_2$ be the block that corresponds to the first vertex of $P$.
Set $w(P) = w(P)+1$ if $|E[b_1 \cap F^o(b_1)]| > 1$.
\end{enumerate}
\item
Find the path $P_{max} = \max_{P'} w(P')$ (ties broken arbitrarily). Let $\ell = w(P_{max})$.
\item \label{itm:singleblocks}
For each block $b$ corresponding to a vertex in $D$ apply
$\ell = \max(\ell, |E[b \cap F^o(b)]| + |E[b \cap F^u(b)]| + 1)$.
\end{enumerate}
At the end, $\ell$ will hold the actual number of rounds it takes in the optimal
schedule produced by \Cref{alg:mainopt}.
The case \ref{item:preparation_round} accounts for the preparation (i.e.~adding new flow rules)
round of the block scheduled earliest in a chain of dependent blocks (i.e.~ current path $P$).
Similarly, \ref{item:cleanup_round} accounts for the cleanup round (i.e.~removal of old flow rules) of the block
scheduled the latest in that chain.
Eventually, $\ell$ is determined either by the chain of dependent blocks that
corresponds to the longest weighted path in $D$,
or by some single block at Line \ref{itm:singleblocks}
due to extra preparation/cleanup rounds consumed by that block.
Any infeasible instance, i.e.~with cyclic block dependency,
can be turned feasible by increasing the capacity of
some link from 1 to 2, hence breaking the cycle.
Therefore, starting from minimal capacity allocation
is always sufficient to preserve the worst case in any topology.
The results show that the optimal number of rounds on the
subject networks vary between 2 and 6 (see \Cref{fig:roundschart_opt}).
Table~\ref{tbl:zoo} lists the numbers obtained from some of
the examined graphs (numbers are rounded to the nearest integer, except those close to 0).
In the second implementation (i.e.~feasibility only),
we evaluated \Cref{alg:main} on the same
input data and obtained feasible schedules under
worst case capacity allocations.
The number of rounds spread between 2 and 14 (see \Cref{fig:roundschart_feas}).
We also implemented the MIP in \Cref{MIP}.
The runtime even for 2 flows are usually in few seconds,
much longer compared to the results from the first implementation
($\approx100$ microseconds, see \Cref{fig:times_plot} and \Cref{fig:example_mip}).
\begin{figure}[!h]
\begin{subfigure}{.33\columnwidth}
\centering
\includegraphics[width=0.99\columnwidth]{rounds_freq.pdf}
\caption{}
\label{fig:roundschart_opt}
\end{subfigure}%
\begin{subfigure}{.33\columnwidth}
\centering
\includegraphics[width=0.99\columnwidth]{rounds_freq_feas.pdf}
\caption{}
\label{fig:roundschart_feas}
\end{subfigure}%
\begin{subfigure}{.33\columnwidth}
\tabcolsep=0.06cm
\begin{tabular}{l c c c c c}
\hline\hline
Graph & $2$ & $3$ & $4$ & $5$ & $6$ \\ [0.5ex]
\hline
AARNET & 46 & 50 & 4 & 0 & 0 \\
Abilene & 37 & 57 & 6 & 0 & 0 \\
Belnet(2003) & 13 & 79 & 8 & 0.01 & 0 \\
GARR(2011) & 21 & 68 & 10 & 1 & 6e-4 \\
SWITCH & 10 & 77 & 13 & 0.03 & 0 \\
\hline
\end{tabular}
\caption{}
\label{tbl:zoo}
\end{subfigure}
\caption{
(\subref{fig:roundschart_opt}) The frequency of each possible number of rounds (in \%) in optimal schedules from \Cref{alg:mainopt},
(\subref{fig:roundschart_feas}) in arbitrary feasible schedules from \Cref{alg:main},
and (\subref{tbl:zoo}) in optimal schedules for a sample of the examined graphs.
}
\label{fig:charts_rounds}
\end{figure}
\begin{figure}[!h]
\begin{subfigure}{.5\columnwidth}
\includegraphics[width=0.99\columnwidth]{times_boxplot.pdf}
\caption{}
\label{fig:times_plot}
\end{subfigure}%
\begin{subfigure}{.5\columnwidth}
\includegraphics[width=0.99\columnwidth]{_DialtelecomCz.pdf}
\caption{}
\label{fig:example_mip}
\end{subfigure}
\caption{
(\subref{fig:times_plot}) Distribution of runtime (in microsecond) over all the
problem instances from some of the evaluated graphs.
(\subref{fig:example_mip}) The graph (Dial Telecom) with a long runtime in our MIP implementation
(the worst case takes $\approx26$ seconds)
}
\end{figure}
\section{Related Work}\label{sec:relwork}
The fundamental problem of how to reroute flows
has recently received much attention in the networking
community and we refer the reader to the recent survey by
Foerster et al.~\cite{update-survey} for an overview
of the field.
Yet, today, and in contrast to the classic problem of how
to \emph{route} flows~\cite{AmiriKMR16,shimon-flows,KawarabayashiKK14,ufkleinberg,leighton1999multicommodity,ufskutella},
we still know surprisingly little about useful
\emph{algorithmic} techniques for efficient
flow \emph{rerouting}.
There exist several empirical studies
motivating our model~\cite{dionysus,kuzniar2015you},
however, this literature is orthogonal to ours.
Moreover, many existing consistent network update algorithms
such as~\cite{infocom15,dionysus,zupdate,roger,abstractions}
require packet tagging and additional forwarding rules, which
render the problem different in nature.
Mahajan and Wattenhofer~\cite{roger}
initiated the study of flow rerouting algorithms which schedule
updates over time. The authors also presented first algorithms
to quickly updates routes in a transiently \emph{loop-free} manner~\cite{sirocco16update,Forster2016Consistent,Forster2016Power},
by maximizing \emph{the number of updates per round}.
A second line of research focuses on \emph{minimizing the number of rounds}
of loop-free updates~\cite{dsn16,sigmetrics16,ludwig2015scheduling,hotnets14update}.
As congestion is known to
negatively affect application performance and
user experience, it has also been studied
intensively in the context of flow rerouting problems.
The seminal work by Hongqiang et al.~\cite{zupdate}
on congestion-free rerouting
has already been extended in several papers,
using static~\cite{roger-infocom,swan,jaq4,icnp-jiaqi},
dynamic~\cite{jaq1}, or time-based~\cite{jaq8,jaq10}
approaches.
Vissicchio et al.~presented FLIP~\cite{vissicchio2016flip},
which combines
per-packet consistent updates with order-based rule replacements,
in order to reduce memory overhead:
additional rules are used only when necessary.
Moreover, Hua et al.~\cite{huafoum}
recently initiated the study of
adversarial settings,
and presented FOUM,
a flow-ordered
update mechanism that is robust to packet-tampering and packet dropping
attacks.
However, none of these papers present polynomial-time
algorithms for rerouting flows without requiring packet tagging.
Our work on polynomial-time algorithms is motivated in particular by
the negative result by Ludwig et al.~\cite{ludwig2015scheduling}
who showed that deciding whether a loop-free 3-round
update schedule exists is NP-hard, even in the absence of
capacity constraints.
Given this negative result, much prior work typically resorts
to heuristics~\cite{icdcs17}, which however do not come with any
formal guarantees on the quality of the computed
schedule, or to algorithms which have a super-polynomial
runtime~\cite{sigmetrics16}.
The only exception is the polynomial-time algorithm
by Amiri et al.~\cite{icalp18} for acyclic flow graphs,
which however is limited to
computing \emph{feasible} (possibly very long)
update schedules.
There are various differences between this paper and the recent work of
Amiri et al.:
(1) In contrast to our work where only flow pairs
need to form a DAG, \cite{icalp18} considers a much more restricted
model where the union of all flows must be acyclic.
This restriction allows the authors to design an FPT
algorithm for $k$ flows, whereas in our model
the problem is NP-complete already for six flows.
Hence, different techniques are required to show hardness.
(2) Similarly to~\cite{icalp18}, our algorithm relies on
a dependency graph that
explains the relation between flows. However,
since we aim to compute schedules only for two flows,
we do not require the big machinery introduced for
$k$ flows instead we provide a more elegant algorithm.
(3) At the same time, since in contrast to prior work, we focus on an optimal solution,
our model is more chalelnging and requires new algorithmic ideas.
Finally, our problem is situated in the larger context of
combinatorial reconfiguration theory, which has recently
received much attention, e.g., in the context of games~\cite{van2013complexity}.
In this respect,
the reconfiguration model closest to ours is by
Bonsma~\cite{bonsma2013complexity}
who studied
how to perform rerouting such that transient paths are always \emph{shortest}.
However, the corresponding techniques and results are not applicable
in our model where we consider flows of certain \emph{demands},
and where different flows may \emph{interfere} due to capacity constraints
in the underlying network.
\section{Conclusion and Future Work}\label{sec:conclusion}
This paper presented the first polynomial-time and optimal scheduling algorithm to
quickly reroute two flows in a congestion-free manner.
In particular, the algorithm can be used to minimize the number of required (asynchronous) interactions between switches and controller in a software-defined network.
We also prove that our result cannot be generalized much further
as the problem becomes NP-hard
already for six flows.
One of the main open question of our work concerns the polynomial-time
tractability for $2<k<6$ flows. These cases
might be very challenging, and currently, we do not have any insights
on how to deal even with three flows.
In this paper we assumed that every pair of flows
forms a DAG, and we did not constrain the underlying network topology
which can be general.
However, many real-world networks are sparse, and feature
nice topological or combinatorial structures. The study of such
networks introduces another interesting direction for future research.
We currently do not know what is the complexity of the problem even on planar
graphs, one of the most interesting classes of sparse graphs.
\bigskip
\noindent \textbf{Acknowledgements.} We would like
to thank Stephan Kreutzer, Arne Ludwig,
and Roman Rabinovich for discussions on this
problem.
The research of Saeed Akhoondian Amiri and Sebastian Wiederrecht has been supported by
the European Research Council (ERC) under the European Union's Horizon
2020 research and innovation programme (ERC consolidator grant DISTRUCT,
agreement No 648527).
\balance
{
|
1,314,259,996,539 | arxiv | \section{Introduction}
Let $\mathcal{F}$ be a tiling of $\Re^2$ with convex polygons. Following \cite{DomLan}, we call a point $p \in \Re^2$ an \emph{irregular vertex} of $\mathcal{F}$ if it is a vertex of a tile, and a relative interior point of an edge of another one; compare Figure \ref{fig:final_tiling}: the edge labelled $E_3$ contains an irregular vertex. A tiling of $\Re^2$ having no irregular vertices is called \emph{edge-to-edge}. A tiling is \emph{normal} if for some positive $r$ and $R$, there is a disk of radius $r$ inside each tile and each tile is contained in a disk of radius $R$ \cite{GS, Schulte}. A tiling is a \emph{unit area tiling} if the area of every tile is one, and it has \emph{bounded perimeter} if perimeters of all tiles are bounded from above and from below by positive numbers. Clearly, a unit area tiling with bounded perimeter is normal, and a tiling with finitely many prototiles is both normal and has bounded perimeter.
In \cite{Nandakumar} Nandakumar asked whether there is a tiling of the plane by pairwise non-congruent triangles of equal area and equal perimeter. Kupavskii, Pach and Tardos proved that there is no such tiling \cite{KPT18_2}.
Nandakumar also asked \cite{Nandakumar} if there is a tiling of the plane with pairwise non-congruent triangles with equal area and bounded perimeters. This was answered affirmatively in \cite{KPT18_1} and in \cite{Frettloh}. The tilings constructed in these papers have many irregular vertices.
Later in \cite{F-R} it was shown that there is such a tiling without any irregular vertices. In the same paper it was shown that there is a tiling of the plane with pairwise non-congruent hexagons with equal area and bounded perimeters such that the tiling \emph{has} irregular vertices.
It is worth noting that while in general the requirement of having no irregular vertices is more restrictive than the requirement of having some, for tilings with convex hexagons as tiles it is the other way around: it is difficult to find hexagon tilings with many irregular vertices.
This observation is illustrated by the fact that the hexagonal tiling in \cite{F-R} has only two irregular vertices.
Motivated by these results, the main goal of this paper is to investigate the number $k$ of irregular vertices in a unit area and bounded perimeter tiling of the Euclidean plane by convex hexagons. Our main result is the following.
\begin{thm}\label{thm:main}
In any normal tiling of the Euclidean plane with convex hexagons there are finitely many irregular vertices. Furthermore, for any integer $k \geq 3$, there is a unit area tiling of the plane with bounded perimeter convex hexagons such that the number of irregular vertices of the tiling is equal to $k$.
\end{thm}
In Section~\ref{sec:proof} we prove Theorem~\ref{thm:main}. Our proof is based on results of Stehling \cite{ste88} and Akopyan \cite{ako18}, which provide an upper bound on the number of irregular vertices in a hexagon tiling (cf. Theorems~\ref{thm:finite} and \ref{thm:indexupperbound}, respectively). In Section \ref{sec:asymptotic} we show that the upper bound of Akopyan in Theorem~\ref{thm:nonhexlowerbound} is asymptotically tight, using a simplified version of the construction in Section~\ref{sec:proof}. We summarize our corresponding results in Theorems~\ref{thm:nonhexlowerbound} and \ref{thm:asymptotic}. Section~\ref{sec:ques} lists some additional questions.
In the sequel we denote the Euclidean plane by $\Re^2$. Furthermore, for any points $p,q \in \Re^2$ we denote the closed segment connecting $p$ and $q$ by $[p,q]$, the convex hull of a set $S \subset \Re^2$ by $\conv S$, and the Euclidean norm of a point $p \in \Re^2$ by $| p |$.
\section{Proof of Theorem~\ref{thm:main}} \label{sec:proof}
In his thesis \cite{ste88}, Stehling proved the following result.
\begin{thm}[Stehling]\label{thm:finite}
In a normal tiling of a plane with convex polygons such that all of them have at least six edges, all but a finite number of tiles are hexagons.
\end{thm}
Akopyan \cite{ako18} found a short proof of this theorem with concrete bounds on the number of non-hexagonal tiles. To formulate his result, we define the \emph{index} of a polygon as the number of its edges minus $6$.
\begin{thm}[Akopyan]\label{thm:indexupperbound}
If the plane is tiled with convex polygons, each having at least six edges, area at least $A$ and diameter not greater than $D$, then the sum of the indices of the tiles is not greater than $\frac{2\pi D^2}{A} - 6$.
\end{thm}
This theorem immediately implies the finiteness of the number of irregular vertices in a normal tiling by hexagons.
\begin{cor}\label{cor:upperbound}
In a normal tiling by hexagons, there is a finite number of irregular vertices.
\end{cor}
\begin{proof}
For each hexagon with an irregular vertex in the interior of one of its edges, we can subdivide this edge into several segments by all irregular vertices it contains and obtain a polygon with a positive index. By Theorem \ref{thm:indexupperbound}, the number of polygons of positive index is finite and, therefore, the number of irregular vertices in the initial tiling is finite, too.
\end{proof}
Now we prove the second part of Theorem~\ref{thm:main}.
\begin{proof}
Consider a unit area regular hexagonal tiling $\mathcal{F}$. Let $H$ be a tile of $\mathcal{F}$, and let $E$ be an edge of $H$. Without loss of generality, we may assume
that the centre of $H$ is the origin $o$. Let $R$ denote the angular region with the apex $o$ and with the boundary consisting of the two half-lines starting at $o$ and passing through the endpoints of $E$ (cf. Figure~\ref{fig:hexagonal_lattice}). Let $T$ be the regular triangle $H \cap R$ of area $1/6$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=.5\textwidth]{hexagonal_lattice.pdf}
\caption{The regular hexagonal lattice tiling $\mathcal{F}$}
\label{fig:hexagonal_lattice}
\end{center}
\end{figure}
Let $\alpha$ be an arbitrary angle measure from $(0,\frac{\pi}{3}]$ and let $f : \Re^2 \to \Re^2$ be an area-preserving linear transformation that maps $T$ to an isosceles triangle $T'$ with the apex $o$ and angle $\alpha$ at $o$. Then $R'=f(R)$ is an angular region with angle $\alpha$ at its apex. We denote the two half lines bounding $R'$ by $L_1$ and $L_2$. Applying $f$ also to the restriction of $\mathcal{F}$ to $R$ we obtain a tiling $\mathcal{F}'$ of $R'$.
Let the vertices of $\mathcal{F}'$ on $L_1$ be denoted by $p_1, p_2, \ldots$, in the consecutive order, and the vertices on $L_2$ be denoted by $q_1, q_2, \ldots$, where $q_i$ are symmetric to $p_i$ with respect to the symmetry axis of $R'$.
Then the tiles in $\mathcal{F}'$ are the following:
\begin{itemize}
\item the triangle $T'$ of area $\frac{1}{6}$;
\item affinely regular hexagons of unit area;
\item trapezoids of area $\frac{1}{2}$, obtained by dissecting an affinely regular hexagon into two congruent parts by a diagonal connecting two opposite vertices.
\end{itemize}
Note that $\mathcal{F}'$ is symmetric with respect to the symmetry axis of $R'$, the edge of each trapezoid in $\mathcal{F}'$ is either $[p_{2m},p_{2m+1}]$ or $[q_{2m},q_{2m+1}]$ for some positive integer $m$ (cf. Figure~\ref{fig:affine_copy}), and the two angles of the trapezoids on these edges are acute.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=.6\textwidth]{affine_copy.pdf}
\caption{Modifying an affine copy $\mathcal{F}'$ of $\mathcal{F} \cap R$}
\label{fig:affine_copy}
\end{center}
\end{figure}
In the next part we replace all points $p_i$ and $q_i$ by points $p_i'$ and $q_i'$, respectively, to obtain a tiling $\mathcal{G}$ of $R'$, combinatorially equivalent to $\mathcal{F}'$, which has the following properties:
\begin{itemize}
\item $\mathcal{G}$ is axially symmetric to the symmetry axis of $R'$,
\item the triangle $T''$ corresponding to $T'$ is an isosceles triangle of area $\frac{1}{3}$, with $o$ as its apex;
\item the hexagons corresponding to the hexagons in $\mathcal{F}'$ have unit area;
\item the quadrangles corresponding to the trapezoids in $\mathcal{F}'$ are also trapezoids of area $\frac{1}{2}$, and the two angles on the edges $[p'_{2m},p'_{2m+1}]$ or $[q'_{2m},q'_{2m+1}]$ ($m \in \mathbb{N}$) are acute.
\end{itemize}
Let $H'$ be the hexagon cell of $\mathcal{F}'$ with $p_1, p_2, q_1, q_2 \in H'$. Let the remaining two vertices of $H'$ be $z_1$ and $z_2$ such that $z_1$ is closer to $L_1$. For $i=1,2$, let $x_i$ be the orthogonal projection of $z_i$ onto $L_i$. It is an elementary computation to show that if $\alpha = \frac{\pi}{3}$, then the area $A$ of the convex pentagon $\conv \{ o, x_1, z_1, z_2, x_2\}$ is $\frac{4}{3}$, and if $0 < \alpha < \frac{\pi}{3}$, then $A > \frac{4}{3}$.
Now we apply modifications as follows. Replace the segment $[p_1,q_1]$ by the parallel segment $[p'_1,q'_1]$ such that $p'_1 \in L_1$, $q'_1 \in L_2$, and the isosceles triangle $T'' = \conv \{ o, p'_1, q'_1 \}$ has area $\frac{1}{3}$. Then the area of the hexagon $\conv \{ p'_1, x_1, z_1, z_2, x_2, p'_2\}$ is greater than $1$. Thus, there are points $p'_2 \in [p'_1,x_1]$, $q'_2 \in [q'_1,x_2]$ such that the area of $\conv \{ p'_1,p'_2, z_1, z_2, q'_2, q'_1\}$ is one, and this hexagon is axially symmetric to the symmetry axis of $R'$; i.e. $|p'_2-p'_1| = |q'_2-q'_1|$. Until now, we have modified the construction to satisfy the conditions in the previous list apart from the fact that the quadrangles with $[p'_2,p_3]$ and $[q'_2,q_3]$ as edges have area $\frac{5}{12}$.
Observe that if $[a,b]$ and $[c,d]$ are parallel edges of a trapezoid, then moving $[c,d]$ parallel to $[a,b]$ does not change the area of the trapezoid.
Using this observation, we set $v_1=p'_2-p_2$, $v_2 = q'_2-q_2$, and $p'_i = p_i+v_1$ and $q'_i = q_i+v_2$ for all integers $i \geq 2$.
Then, in the resulting tiling $\mathcal{G}$ any quadrangle having $[p'_{2m},p'_{2m+1}]$ or $[q'_{2m},q'_{2m+1}]$ ($m \in \mathbb{N}$) as an edge has area $\frac{1}{2}$, and the two angles lying on this edge are acute. Using the same observation, we may derive the analogous statement for the hexagons containing $[p'_{2m-1},p'_{2m}]$ or $[q'_{2m-1},q'_{2m}]$ ($m \in \mathbb{N}$, $m \geq 2$) as an edge. Clearly, the tiling $\mathcal{G}$ obtained in this way satisfies all the required conditions.
Observe that if we rotate $\mathcal{G}$ around $o$ by angle $\alpha$, and glue the pairs of quadrangles meeting in an edge on $L_1$ or $L_2$, we obtain a tiling of an angular region of angle $2\alpha$, in which any tile contained entirely in the interior of the region is a hexagon of unit area.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=.6\textwidth]{final_tiling.pdf}
\caption{The tiling $\mathcal{G}_4$ constructed in the proof of Theorem~\ref{thm:main}}
\label{fig:final_tiling}
\end{center}
\end{figure}
Now we prove Theorem~\ref{thm:main}.
Let $k \geq 3$ be fixed, and let $\mathcal{G}$ be the tiling constructed in the previous part of the proof with angle $\alpha = \frac{2\pi}{3k}$.
Let us rotate $\mathcal{G}$ around $o$ in counterclockwise direction and glue the pairs of quadrangles meeting on $L_1$ or $L_2$. Continuing this process we obtain a tiling of $\Re^2$, in which all tiles are unit area hexagons, outside a regular $(3k)$-gon $P$ centred at $o$, with area $k$. This regular $(3k)$-gon is obtained as the union of all the rotated copies of the triangle $T''$. Let $E_i$, $i=1,2,\ldots, 3k$ denote the edges of $P$. Then, dissecting $P$ with $k$ segments, connecting $o$ to the midpoint of an edge $E_{3j}$, $j=1,2,\ldots,k$, yields a tiling of $P$ with $k$ unit area hexagons (cf. Figure~\ref{fig:final_tiling}). The $k$ midpoints, used as vertices of the tiles in $P$ are not vertices of any other tiles, and thus they are irregular. On the other hand, all other vertices of the obtained tiling, which we denote by $\mathcal{G}_k$, are clearly regular. Finally, $\mathcal{G}_k$ consists of five prototiles, and hence, it has bounded perimeter and is normal.
\end{proof}
\section{Tightness of the upper bound} \label{sec:asymptotic}
Using the construction from Theorem \ref{thm:main} and subdividing the edges of polygons by irregular vertices, as in Corollary \ref{cor:upperbound}, we can find the counterpart of Stehling's theorem, i.e. we show the existence of normal tilings in which all tiles have at least six edges and the number of non-hexagon tiles is an arbitrarily large finite number. However, a simplified version of the construction from Theorem \ref{thm:main} provides also unit area and bounded perimeter tilings with the same property.
\begin{thm}\label{thm:nonhexlowerbound}
For any integer $k \geq 2$, there is a unit area tiling of the plane by convex polygons with at least six edges such that the set of the perimeters of the tiles is bounded, and the number of non-hexagon tiles is equal to $k$.
\end{thm}
\begin{proof}
We follow the proof of Theorem \ref{thm:main} and obtain a tiling $\mathcal{F}'$ of the angular region with the angle measure $\alpha=\frac {\pi} {3k}$. Instead of modifying this tiling we just glue $6k$ copies of $\mathcal{F}'$ to obtain a tiling of the plane (cf. Figure~\ref{fig:nonhex_tiling}).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=\textwidth]{non-vtv8.pdf}
\caption{The tiling used in the proofs of Theorems~\ref{thm:nonhexlowerbound} and
\ref{thm:asymptotic} for $k=2$}
\label{fig:nonhex_tiling}
\end{center}
\end{figure}
This is a tiling of the plane by hexagons of unit area with two prototiles, and a regular $(6k)$-gon of area $k$ in the centre. For $k=2$, we connect the opposite vertices of the central $12$-gon to divide it into two heptagons of unit area. For $k\geq 3$, we connect the centre of the $(6k)$-gon with every sixth vertex to subdivide it into $k$ congruent octagons. Overall, we obtain an edge-to-edge tiling $\mathcal{F}_k$ by unit area polygons with just three prototiles.
\end{proof}
We use the construction above to show that the bound in Theorem \ref{thm:indexupperbound} is asymptotically tight.
To state our result, for any normal tiling $\mathcal{F}$ of the plane by convex polygons having at least six edges, we denote the sum of the indices of the tiles in $\mathcal{F}$ by $i(\mathcal{F})$, and call it the \emph{total index} of $\mathcal{F}$.
\begin{thm} \label{thm:asymptotic}
There is a sequence of normal tilings $\mathcal{F}_k$ by convex polygons having at least six edges such that
\[ \lim_{k \to \infty} \frac{\frac{2 \pi D_k^2}{A_k} -6}{i(\mathcal{F}_k)} = \frac{4}{3}, \]
where, analogously to Theorem \ref{thm:indexupperbound}, $D_k$ denotes the maximal diameter, and
$A_k$ denotes the minimal area of the tiles in $\mathcal{F}_k$, respectively.
\end{thm}
\begin{proof}
We use the construction from the proof of Theorem~\ref{thm:nonhexlowerbound} without
partitioning the central $(6k)$-gon into octagons. Let us denote the resulting tiling
by $\mathcal{F}_k$. All tiles in $\mathcal{F}_k$ but one are unit area hexagons, of two distinct
shapes. The central tile in $\mathcal{F}_k$ is a $(6k)$-gon, hence the
sum of indices in $\mathcal{F}_k$ is $i(\mathcal{F}_k) = 6k-6$. Let us compare this value
with the bound $\frac{2 \pi D_k^2}{A_k}-6$ from Theorem \ref{thm:indexupperbound}.
Obviously the smallest area $A_k$ of the tiles in $\mathcal{F}_k$ is one.
We show that all three prototiles have the same diameter $D_k= \frac{2}{\sqrt{3 \sin(\pi/3k)}}$.
Let us call the area preserving linear transformation that
maps the cone with angle $\frac{\pi}{3}$ to the cone with angle
$\alpha$ by $f$. The image of a regular hexagon with diameter $2r_1$ under
$f$ is an affinely regular hexagon with diameter $2r_k$, see Figure \ref{fig:affine-6gons}.
\begin{figure}
\[ \includegraphics[width=60mm]{affine-6gons} \]
\caption{Diameter of deformed hexagons is $2r_k$
\label{fig:affine-6gons}}
\end{figure}
Note that $f$ distorts the basic regular triangles in the regular hexagons of edge
length $r_1$ into isosceles triangles with longer edge lengths $r_k$.
Furthermore, as the central $(6k)$-gon can be dissected into $6k$ isosceles triangles intersecting in $o$ with longest
edge lengths equal to $r_k$, the diameter of this $(6k)$-gon is also $2r_k$.
Since area is preserved by $f$, we get
$\frac{1}{2}r_k^2 \sin(\frac{\pi}{3k})=\frac{1}{6}$. This yields
\[ r_k = \frac{1}{\sqrt{3 \sin(\pi/3k)}}, \; \mbox{ which implies } D_k = 2 r_k =
\frac{2}{\sqrt{3 \sin(\pi/3k)}}. \]
Let us plug this into the formula for the upper bound on $i(\mathcal{F})$ in Theorem \ref{thm:indexupperbound}, and compute the
ratio of the obtained expression with the total index $i(\mathcal{F}_k) = 6k-6$ of our construction. Then we have
\[
\lim_{k \to \infty} \frac{1}{6k-6} \Big( \frac{2 \pi D_k^2}{A_k} -6 \Big) =
\lim_{k \to \infty} \frac{2 \pi \frac{4}{3 \sin\left(\frac{\pi}{3k} \right)}-6}{6k-6}
= \lim_{k \to \infty} \frac{4 \frac{\frac{\pi}{3k}}{\sin\left(\frac{\pi}{3k}\right)} - \frac{3}{k}} {3-\frac{3}{k}} = \frac{4}{3}, \]
where in the last step we used the well-known limit $\lim\limits_{x \to 0} \frac{\sin (x)}{x} = 1$.
\end{proof}
\section{Open questions}\label{sec:ques}
Our first question is motivated by the result in \cite{KPT18_2}.
\begin{ques}
Let $k \geq 1$ be an arbitrary integer. Prove or disprove that there is a unit area tiling of $\Re^2$ with equal perimeter convex hexagons such that the number of irregular vertices of the tiling is at least $k$. Similarly, are there equal area and equal perimeter tilings of $\Re^2$ in which every tile has at least six edges, and at least $k$ tiles have more?
\end{ques}
\begin{ques}
Let $k \geq 1$ be an arbitrary integer. What is the smallest value of $n$ such that for any $k \geq 3$, there is a tiling $\mathcal{F}$ of $\Re^2$ with convex (or possibly non-convex) hexagons, and with $n$ prototiles, such that the number of irregular vertices in $\mathcal{F}$ is at least $k$? Our proof of Theorem \ref{thm:main} shows $n \le 5$. In particular we ask whether there are monohedral ($n=1$) tilings satisfying this property?
\end{ques}
\begin{ques}
Can we improve the ratio $\frac{4}{3}$ in Theorem \ref{thm:asymptotic}?
That is, are there sequences of normal tilings by convex polygons with at least six edges for which the ratio in Theorem \ref{thm:asymptotic}
tends to less than $\frac{4}{3}$? Are there such sequences of unit area tilings with bounded perimeters?
\end{ques}
\begin{ques}
It is also tempting to ask about an analogue of Theorem \ref{thm:main} in higher dimensions.
The question becomes much harder, because we do not even know what the equivalents of the
hexagons in dimension 3 are. In other words: let $n$ be the minimal number of faces of a tile
in a normal tiling by convex tiles in $\Re^3$. What is the maximal value of $n$? Engel showed
\cite{engel81} that it is at least 38 by constructing a convex polyhedron with 38 faces that
tiles $\Re^3$ in a face-to-face manner.
\end{ques}
\section*{Acknowledgements}
We thank the BIRS and the organizers of the meeting ``Soft Packings, Nested Clusters, and
Condensed Matter'' at Casa Matem\'atica Oaxaca where we obtained the results in this paper.
The first author is supported by the Research Centre for Mathematical Modelling (RCM$^2$) of
Bielefeld University.
The third author is partially supported by the NKFIH Hungarian Research Fund grant 119245, the J\'anos Bolyai Research Scholarship of the Hungarian Academy of Sciences, and grants BME FIKP-V\'IZ and \'UNKP-19-4 New National Excellence Program by the Ministry of Innovation and Technology.
|
1,314,259,996,540 | arxiv | \section{Introduction}
Suppose that $\mathcal{H}$ is a random Hermitian matrix of size $N\times N$ taken from the Gaussian Unitary Ensemble (GUE), with ensemble distribution given by the measure
\begin{equation}
\label{dens}
Const. \exp \left[-2N\mathrm{Tr}(\mathcal{H}^{2})\right]\, \prod_{j=1}^N d\mathcal{H}_{jj} \prod_{1\le j<k\le N} d \mathop{\mathrm{Re}} \mathcal{H}_{jk} d \mathop{\mathrm{Im}} \mathcal{H}_{jk}.
\end{equation}
It is well known that in the limit of infinite matrix dimensions $N\to\infty$, the distribution of the eigenvalues of $\mathcal{H}$ is supported on the interval $[-1,1]$ and has density $\frac{2}{\pi}\sqrt{1-x^2}$ there. This is known as Wigner's semicircle law, see e.g. \cite {PS11} and \cite{AGZ09} for precise statements. In this paper we are concerned with the \emph{random process} in $x$ defined by the logarithm
\begin{equation}
\label{krasov}
D_{N}(x) =-\log |\det({\cal H} - xI)|
\end{equation}
of the characteristic polynomial of $\mathcal{H}$ in the limit $N\to\infty$, with $x$ varying in $(-1,1)$. The quantity $D_N(x)$ is a particular case of linear eigenvalue statistics $X_{N}(f) = \sum_{k=1}^{N} f(x_k)$, where $x_1, \dots, x_N$ are the eigenvalues of $\mathcal{H}$. It is well known that for suitably regular test functions $f$, $X_{N}(f)$ is asymptotically normal as $N \to \infty$ with variance $\sigma^{2}(f) = \frac{1}{4}\sum_{k=1}^{\infty}kc_{k}(f)^{2}$, where $c_{k}(f)$ are the \textit{Chebyshev-Fourier coefficients}:
\begin{equation} \label{chebshev}
c_{k}(f) = \frac{2}{\pi}\int_{-1}^{1}\frac{f(u)T_{k}(u)}{\sqrt{1-u^{2}}}\,du, \qquad T_{k}(u) = \cos(k\,\mathrm{arccos}(u)).
\end{equation}
In fact, the asymptotic normality of $X_{N}(f)$ for regular $f$ has been established for a variety of random matrix ensembles, see for example \cite{Joh98,LP09,PS11} and references therein.
Since $x$ lies in the bulk of the eigenvalue distribution, our test function, $f(u)=\log |u-x|$ is unbounded. Its Chebyshev-Fourier coefficients are proportional to $1/k$, so that $\sigma^{2}(f) =\infty$ and it is then natural to consider normalizing $D_N(x)$ before taking the limit $N \to \infty$. Indeed, for any fixed $x\in (-1,1)$ the variance of $D_N(x)$ grows with $N$ like $\frac{1}{2}\log N$, and for any finite number of distinct points $x_{1}, \ldots ,x_{m} $ in $(-1,1)$ the random vector $(D_{N}(x_{1}),\ldots,D_{N}(x_{m}))/(\frac{1}{2} \log N)^{1/2}$ converges in distribution, after centering, to a collection of $m$ independent standard Gaussians as $N\to\infty$. This can be inferred from the asymptotic identity due to Krasovsky \cite{K07}:
\begin{equation}\label{krasovasympt}
\begin{split}
\mathbb{E}\left\{e^{-\sum_{k=1}^{m}\alpha_{k}D_{N}(x_{k})}\right\} = &\prod_{k=1}^{m}\left[C\left(\frac{\alpha_{k}}{2}\right)(1-x_{k}^{2})^{\alpha_{k}^{2}/8}N^{\alpha_{k}^{2}/4}e^{\alpha_{k}N(2x_{k}^{2}-1-2\log(2))/2}\right]\\
&\times \prod_{1 \leq \nu < \mu \leq m}(2|x_{\nu}-x_{\mu}|)^{-\alpha_{\nu}\alpha_{\mu}/2}\left(1+O\left(\frac{\log N}{N}\right)\right),
\end{split}
\end{equation}
where $C(\alpha) = 2^{2\alpha^{2}}{G(\alpha+1)^{2}}/{G(2\alpha+1)}$ and $G(z)$ is the Barnes G-function.
The most salient feature of the asymptotics in (\ref{krasovasympt}) is the product of differences on the second line which, when rewritten in the form
\begin{equation}
\exp \Big[ -\sum_{1 \leq \nu < \mu \leq m}\frac{\alpha_{\nu}\alpha_{\mu}}{2}\,\log|2(x_{\nu}-x_{\mu})|\Big] , \label{kracovstruct}
\end{equation}
is suggestive of the existence of a logarithmic covariance structure in the Gaussian process $D_N(x)$. However, this term is of sub-leading order to the variance term. Clearly then, the normalization of the process \eqref{krasov} comes at a price, because the non-trivial covariance structure implied by \eqref{kracovstruct} is too small to survive the limit $N \to \infty$.
This motivates the following question. How can we `regularize' the process \eqref{krasov} so that it has a well-defined limit that `feels' the covariance structure implied by \eqref{kracovstruct}? Hughes, Keating and O'Connell \cite{HKOC01} answered this question in the context of the Circular Unitary Ensemble (Haar unitary matrices). Employing convergence in functional spaces instead of point-wise convergence, they proved that the logarithm $V_N(\theta)=-2\log{|p_N(\theta)|}$ of the characteristic polynomial $p_N(\theta)=\det{\left(I-U\,e^{-i\theta}\right)}$ of Haar unitary matrices $U$ converges as $N\to\infty$ to the stochastic process represented by the Fourier series
\begin{equation} \label{1/f}
V(\theta)=\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}} \left(v_n e^{i n \theta}+\overline{v_n} e^{-i n \theta}\right)\,.
\end{equation}
Here, the coefficients $v_n,\overline{v}_n$ are independent standard \emph{complex} Gaussians, $\mathbb{E} \{v_n \overline{v}_n \}=1$, and the convergence of the series is understood in the sense of distributions in a suitable Sobolev space. This process has a logarithmic singularity in the covariance structure:
$
\mathbb{E} \{V(\theta_1)V(\theta_2)\}=-2\log {|e^{i\theta_1}-e^{i\theta_2}|}\, .
$
At this point it is appropriate to mention that random processes and fields with logarithmic covariance structure appear with astonishing regularity in physics and also engineering applications, see e.g. \cite{CLD01} and more recently \cite{FlDR12}. Those objects are intimately related to multifractal cascades emerging in turbulence, and from that angle attracted considerable mathematical interest within the last decade, see, e.g., \cite{BacryMuzy2002} and \cite{BarralMandelbrot2004}. In fact, closely related
mathematical objects appear in the so-called "multiplicative chaos" construction going back to Kahane's work \cite{Kah85}, also see \cite{RV13} and references therein for recent research in that direction which was motivated, in particular, by Quantum Gravity applications. In two spatial dimensions, the most famous example of the random field of that type is the two-dimensional Gaussian Free Field \cite{GFF}. A regularized version of this field appeared in a non-trivial way in the work of Rider and Vir\'ag \cite{VR97}, who showed that it describes the limiting law of the log-modulus of characteristic polynomials in the Ginibre ensemble. The Gaussian Free Field also appeared more recently as the limiting distribution of the eigenvalue counting function in general $\beta$-Jacobi ensembles and their principal subminors \cite{BG13}. As for the one-dimensional processes with logarithmic correlations, they are known in natural sciences under the general name of \textit{1/f noises} (see Section 2 in \cite{FlDR12} for some general references) since, in the spectral representation, the Fourier transform of the covariance or structure function, interpreted as a ''power" of the signal, is inversely proportional to the Fourier variable (i.e. the ''frequency" $f$). The random process $V(\theta)$ is, arguably, the simplest time-periodic stationary version of $1/f$ noise. It was found to play an important role in the construction of conformally invariant planar random curves \cite{AJKS11} and statistical mechanics of disordered systems \cite{FB08}. We note in passing that from a different angle, discrete sequences with $1/f$ properties were considered heuristically in the physics literature, see e.g. \cite{Rel04} and \cite{MLD07}.
The motivation for the work in \cite{HKOC01} came from number theory, as for large $N$, $p_N(\theta)$ provides a good model for describing statistics of the values of the Riemann-zeta function high up the critical line \cite{KS00}. The established relation of $p_N(\theta)$ to
$V(\theta)$ turned out to be fruitful. It allowed one to put forward nontrivial conjectures about statistics of extreme and high values of characteristic polynomials of Haar unitary matrices emerging as $N\to \infty$, and eventually for the Riemann-zeta function \cite{FHK12,FK12}.
The main goal of this paper is to investigate further the relation between $1/f$-noises and the characteristic polynomials of random matrices in the limit $N\to \infty$. Significantly extending the picture found in \cite{HKOC01} we will show that the limiting process depends on the {\it spectral scale} at which one allows the argument $x$ of the characteristic polynomial $\det ({\cal H}-xI)$ to vary. To this end, let us remind the reader that, as is well known in random matrix theory, see e.g., \cite{PS11}, there exist three natural scales in the spectra of large random matrices. One, known as the global, or {\it macroscopic} scale is set for the GUE by the width of the support of the semicircle law and, in the normalization chosen in the present paper, see (\ref{dens}), remains of the order of unity as $N\to \infty$. Second, known as the local, or {\it microscopic} scale is set by the typical separation between neighbouring eigenvalues and is, in the chosen normalization, of order $1/N$ for large $N$. Finally, the third scale which is called {\it mesoscopic} can be defined as intermediate between those two.
Deferring precise statements to the next section, now we will outline the two instances of $1/f$ noise that emerge in the limit $N\to\infty$ for the GUE matrices. On the macroscopic scale, by adapting the arguments of \cite{HKOC01} to our setting, we prove that, as $N\to\infty$, the process $\{D_N(x): x\in (-1,1)\}$ converges, after centering, to the (aperiodic) $1/f$ noise given by the random Chebyshev-Fourier series
\begin{equation}\label{1/fch}
F(x) = \sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}\, a_{n}\, T_{n}(x), \qquad x \in (-1,1),
\end{equation}
where $a_n$, $n=1,2 \ldots$ is a sequence of independent standard real Gaussians. As with the Fourier series in (\ref{1/f}), the convergence in (\ref{1/fch}) has to be understood in the sense of distributions in a suitable Sobolev space. The covariance structure associated with the generalized process \eqref{1/fch} is given by an integral operator with kernel
$ \mathbb{E}\{F(x)F(y)\} = -\frac{1}{2}\log(2|x-y|)$.
The problem of finding a suitable model to describe the statistical properties of the characteristic polynomials of random matrices on the \emph{mesoscopic} rather than macroscopic scale turned out to be much more challenging and is the main focus of the present paper. Our main finding is the emergence of fractional Brownian motion with Hurst index $H=0$ in this context. To describe the latter, we recall that the conventional {\it fractional Brownian motion} (fBm) is a zero-mean Gaussian process $B_H(t)$, $B_H(0)=0$, with stationary increments and the covariance structure given by
\begin{equation}\label{intr1}
\mathbb{E}\left\{[B_H(t_1)-B_H(t_2)]^2\right\}=\sigma^2\ |t_1-t_2|^{2H}\, ,
\end{equation}
where $H\in(0,1)$ and $\sigma^2>0$ are two parameters. Although first introduced by Kolmogorov in 1940, fBm became very popular after the seminal work of Mandelbrot and van Ness \cite{ManvNess68} and proved to be a very rich mathematical object of high utility, see e.g. articles by M. Taqqu and by G. Molchan in the book \cite{Taq2003} for an introduction and further references and applications. The utility of fBm is related to its properties of being {\it self-similar}, i.e. $\{ B_H(at): t\in \mathbb{R}\}\overset{d}=a^H \{ B_H(t): t\in \mathbb{R}\}$ for any $a>0$, and having {\it stationary increments}. These two properties characterize the corresponding Gaussian process uniquely, see, e.g., \cite{Taq2003}. In the context of self-similarity parameter $H$ is also known as the Hurst index $H$ or the scaling exponent.
For $H=1/2$, the fBm $B_{1/2}(t)$ is proportional to the usual Brownian motion (Wiener process). We will denote the latter simply as $B(t)$, with $B(dt)$ being the corresponding white noise measure, $\mathbb{E}\left\{B(dt)\right\}=0$ and $\mathbb{E}\left\{B(dt)B(dt')\right\}=\delta(t-t')dtdt'$, where we have chosen the normalization corresponding to the choice of $\sigma=1$ in (\ref{intr1}).
It is apparent from (\ref{intr1}) that the naive limit $H= 0$ of $B_H(t)$ is not well-defined. To overcome this problem, the first author proposed some time ago to regularize the fBm in the limit $H\to 0$ as follows. Consider the stochastic Fourier integral
\begin{equation}\label{intr3}
B^{(\eta)}_H(t) = \frac{1}{2\sqrt{2}}\int_0^{\infty}\frac{e^{-\eta s}}{s^{1/2+H}}\left[\left(e^{-its}-1\right)B_c(ds)+
\left(e^{its}-1\right)\overline{B_c(ds)}\right], \quad \eta \ge 0\, ,
\end{equation}
where $B_c(t)=B_R(t)+iB_I(t)$ and $B_R(t)$ and $B_I(t)$ are two independent copies of the Brownian motion. For $H\in (0,1)$ the integral in (\ref{intr3}) is well defined for all $\eta \ge 0$ and represents a zero-mean Gaussian process with stationary increments and covariance
$\mathbb{E}\left\{[B^{(\eta)}_H(t_1)-B^{(\eta)}_H(t_2)]^2\right\}=2\phi^{(\eta)}_H(t_1-t_2) $, where
\begin{equation}
\begin{split}
\phi^{(\eta)}_H(t)&=\frac{1}{2}\int_0^{\infty}\frac{e^{-2\eta s}}{s^{1+2H}}\left(1-\cos{(ts)}\right)\, ds\\
&=\frac{1}{4H}\Gamma(1-2H)\left[(4\eta^2+t^2)^H\cos{\left(2 H \arctan{\frac{t}{2\eta}}\right)}-(2\eta)^{2H}\right]. \label{intr5}
\end{split}
\end{equation}
For fixed $H\in (0,1)$, $\lim_{\eta \to 0} \phi^{(\eta)}_H(t) =\frac{1}{4H} \Gamma(1-2H)\cos(\pi H) t^{2H}$, where $\Gamma(z)$ is the Euler gamma-function. Hence $B^{(0)}_H(t)$ is fBm. This also follows from the so-called {\it harmonizable representation} of the fBm, which is precisely the integral on r.h.s. in (\ref{intr3}) when $\eta=0$, see Proposition 9.2 in \cite{Taq2003}, or Eq. (7.16) in \cite{Sam06}.
On the other hand, for any fixed $\eta>0$, the limit of $H=0$ in (\ref{intr3}) is well defined, and
\begin{equation}
\label{covphi}
\lim_{H\downarrow 0} \phi^{(\eta)}_H(t) =
\frac{1}{4}\log\left(\frac{t^{2}}{4\eta^{2}}+1\right)\, .
\end{equation}
We consider the resulting limiting process,
\begin{equation}\label{intr3_h_0}
B^{(\eta)}_{0}(\tau) = \frac{1}{2\sqrt{2}}\int_{0}^{\infty}\frac{e^{-\eta s}}{\sqrt{s}}\left\{[e^{-i\tau s}-1]\,B_{c}(ds)+[e^{i\tau s}-1]\,\overline{B_{c}(ds)}\right\}
\end{equation}
as the most natural extension of the standard fBm to the case of zero Hurst index $H=0$. This process can also be defined axiomatically.
\begin{Def*} The regularized fBm with Hurst index $H=0$ is a real-valued stochastic process $\{B_{0}^{(\eta)}(\tau), \tau \in \mathbb{R} \}$ with the following properties
\begin{itemize}
\item[(i)] $B_{0}^{(\eta)}(t)$ is a Gaussian process with mean 0 and $B_{0}^{(\eta)}(0)=0$,
\item[(ii)] $\var \{B_{0}^{(\eta)}(t)\}= \frac{1}{2}\log\left(\frac{t^{2}}{4\eta^{2}}+1\right)$ for some $\eta >0$,
\item[(iii)] $B_{0}^{(\eta)}(t)$ has stationary increments.
\end{itemize}
\end{Def*}
The increment structure of $B_{0}^{(\eta)}(t)$ depends logarithmically on the time separation:
\begin{equation}\label{logcov}
\mathbb{E}\{[B^{(\eta)}_{0}(t_1)-B^{(\eta)}_{0}(t_2)]^{2}\}= \frac{1}{2}\log\left[\frac{(t_1-t_2)^{2}}{4\eta^{2}}+1\right]\, ,
\end{equation}
and, hence the regularized fBm with $H=0$ defines a {\it bona fide} version of the $1/f$ noise with stationary increments\footnote{Compare (\ref{intr3_h_0}) with a stationary version of fBm with $H=0$ proposed in Eq. (16) of \cite{Schmitt2003}}. Therefore, the stochastic process $B^{(\eta)}_{0}(\tau)$ is of interest in its own right and deserves further study. We do not pursue this direction in the present paper except for noting for future reference that the regularized fBm has continuous sample paths.
\medskip
\textit{Note.} After posting the initial version of this paper to the arXiv, we learnt of the work \cite{U09}, where a regularization of fBm essentially equivalent to our $B_{H}^{(\eta)}(t)$ was introduced for $H>0$. Note that neither the limit $H \to 0$ nor the connection with random matrices were identified or investigated there.
\section{Main results}
\label{se:mainresults}
\subsection{Macroscopic regime}\label{sec2.2}
We start with the simpler case of the macroscopic scale where we extend the analogous construction of \cite{HKOC01} from unitary to Hermitian matrices. The relation between characteristic polynomials of Haar unitary matrices and the random Fourier series in (\ref{1/f}) can be understood by expanding $\log{|p_N(\theta)|}$ into the Fourier series
\begin{equation}
\label{diashah}
V_N(\theta)= -2\log |\det(I-Ue^{-i\theta})| = \sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}\left(v_{n,N}e^{i n\theta}+\overline{v_{n,N}}e^{-in\theta}\right),
\end{equation}
where $v_{n,N} = \frac{1}{\sqrt{n}}\mathop{\mathrm{Tr}} \, (U^{-n})$. Now, the coefficients $v_{n,N}$ converge in distribution as $N\to\infty$ to independent standard complex Gaussians. This is a result due to Diaconis and Shahshahani \cite{DS94} from which it can be inferred \cite{HKOC01} that \eqref{1/f} represents the limit of $V_{N}(\theta)$ in a suitable functional space.
An analogue of the Diaconis-Shahshahani result for the $N \times N$ GUE matrices $\mathcal{H}$ was obtained by Johansson \cite{Joh98}. He proved that for any fixed $m$ the vector $\big(\frac{2}{\sqrt{n}}\mathop{\mathrm{Tr}} T_{n}(\mathcal{H})\big)_{n=1}^m$, with $T_{n}(x) = \cos(n\arccos(x))$ being Chebyshev polynomials, converges, after centering, to a collection of independent standard Gaussians in the limit $N\to \infty$. In view of the handy identity
\begin{equation}\label{haag1}
-\log (2|x-y|) = \sum_{n=1}^{\infty} \frac{2}{n}\ T_n(x)T_n(y), \quad x,y\in [-1,1], \quad x\not=y\, ,
\end{equation}
the desired analogue of Fourier expansion is an expansion in terms of Chebyshev polynomials,
\begin{equation}
\label{logdetH}
D_N(x) =-\log|\det(\mathcal{H} - xI)|
= \sum_{n=1}^{\infty}\frac{a_{n,N}}{\sqrt{n}}T_{n}(x)+N\log 2+ R_{N}(x), \quad a_{n,N} = \frac{2}{\sqrt{n}}\mathop{\mathrm{Tr}} T_{n}(\mathcal{H}),
\end{equation}
where the error term $R_N(x)$ is due to the eigenvalues of $\cal H$ outside the support $[-1,1]$ of the semicircle law. Since the probability of finding such an eigenvalue vanishes fast as $N\to\infty$ it can be shown that the error term does not contribute in the limit (see the proof of Proposition \ref{prop:findim} for a more precise statement).
One then concludes that the natural limit of $D_N(x)$, after centering,
is given by the random Chebyshev-Fourier series (\ref{1/fch}).
We will make this picture mathematically rigorous by working in a suitable functional space. First, let us assign a formal meaning to the series in (\ref{1/fch}) and the corresponding stochastic process. Consider the space $L^2=L^2 ((-1,1), \mu (dx))$ with $ \mu (dx) = dx/\sqrt{1-x^2}$. The Chebyshev polynomials form an orthogonal basis in this space, with $c_n(f)$ (\ref{chebshev}) being the coefficients of the corresponding Chebyshev-Fourier series. For $a>0$, consider the space $V^{(a)}$ of functions $f$ in $L^2$ such that $\sum_{n=0}^{\infty} |c_n(f)|^2 (1+n^2)^a <\infty $. This is a Hilbert space with the inner product
\[
\langle f,g \rangle_{a}=\sum_{n=0}^{\infty} c_n(f)c_n(g)(1+n^{2})^{a}\, .
\]
Its dual, $V^{(-a)}$, is the Hilbert space of generalised functions $F(x)=\sum_{n=0}^{\infty} c_n T_n(x) $ with $ || F ||_{-a}^2=\sum_{n=0}^{\infty} |c_n|^2(1+n^2)^{-a}<\infty $. Setting here $c_0=0$ and $c_n=a_n/\sqrt{n}$ with $a_n$, $n\ge 1$, being independent standard Gaussians, one obtains $F(x)$ of (\ref{1/fch}). In such case $|| F ||_{-a}^2$ is finite with probability one. This defines $F(x)$ in (\ref{1/fch}) as a generalised random function (stochastic process) which acts on a test function $f\in V^{(a)}$ in the usual way,
\[
F[f]=\sum_{n=1}^{\infty} \frac{a_n}{\sqrt{n}}\, c_n(f)=\langle f,F \rangle_{0}\, .
\]
This process is Gaussian with zero mean. Its covariance, $\mathbb{E}\{F[f]F[g]\}$, is given by
\begin{equation}\label{haag}
\mathbb{E}\{F[f]F[g]\}
= \sum_{n=1}^{\infty} \frac{1}{n} \int_{-1}^1 \int_{-1}^1 f(x)g(y) T_n(x)T_n(y)\ \mu (dx)\mu (dy)\, .
\end{equation}
It can be shown, see e.g. Lemma 3.1 in \cite{GP13}, that the order of summation and integration in (\ref{haag}) can be interchanged, and, in view of (\ref{haag1}), one obtains the covariance operator in closed form:
\[
\mathbb{E}\{F[f]F[g]\}= -\int_{-1}^1 \int_{-1}^1 \frac{1}{2}\log(2|x-y|) f(x)g(y) \ \mu (dx)\mu (dy), \quad f,g\in V^{(a)}\, .
\]
We are now in a position to formulate our result. Consider the centered process
\begin{equation}\label{logdet}
\tilde D_{N}(x )=-\log|\det(\mathcal{H}-xI)|+\mathbb{E}\{\log|\det(\mathcal{H}-xI)|\}, \quad x\in (-1,1)\, .
\end{equation}
Since $\log |x|$ is locally integrable, $\tilde D_{N}\in V^{(-a)}$ for every $N$.
\smallskip
\begin{theorem}
\label{th:global}
For every $a>1/2$, $\tilde{D}_N (x) \Rightarrow F(x)$ in $V^{(-a)}$ as $N\to\infty$, where $F(x)$ given by (\ref{1/fch}).
\end{theorem}
Our proof of this theorem in Section \ref{se:weakconv} involves solving at least two technical problems that did not arise in \cite{HKOC01}. First, when proving convergence of the finite-dimensional distributions of $\tilde D_{N}(x)$, we are faced with a test function possessing square-root singularities at the edges of the spectrum, arising from the Chebyshev-Fourier coefficients of the logarithm outside $[-1,1]$, see Lemma \ref{le:chebycoeffs}. Most bounds and concentration inequalities for linear statistics rely on the test function having at least $C^{1}(\mathbb{R})$ regularity, see \textit{e.g.} \cite{PS11, LP09,AGZ09}, while ours is only $C^{1/2}(\mathbb{R})$ (even the recent extension \cite{SW13} of such bounds to test functions from the $C^{1/2+\epsilon}(\mathbb{R})$ class does not suffice here). Making use of fine asymptotics of orthogonal polynomials and Airy functions, we prove that this linear statistic converges to zero, a problem that did not appear in \cite{HKOC01}.
Secondly, when proving tightness of $(\tilde D_{N}(x))_{N=1}^{\infty}$ we need additional control over the variance of $ \mathop{\mathrm{Tr}} (T_{n}(\mathcal{H}))$ for both large $N$ \textit{and} large $n$. In \cite{HKOC01}, the analogous quantity, namely $\var \{ \mathop{\mathrm{Tr}}(U^{-n}) \}$, was known explicitly due to exact results for the unitary group obtained by Diaconis and Shashahani \cite{DS94}. In contrast, for the GUE case, $\var \{ \mathop{\mathrm{Tr}} (T_{n}(\mathcal{H}))\}$ and related quantities need to be estimated asymptotically as $N \to \infty$, \textit{uniformly} in the degree $n$ of the Chebyshev polynomial.
\subsection{Mesoscopic regime}
Now we proceed to our next task of extending the relation between characteristic polynomials of random matrices and $1/f$-noises to the mesoscopic scale. In this case, instead of working directly with a generalised stochastic process, we find it more convenient to work with their \textit{regularized} versions.
To formulate our results more precisely, fix a parameter $\eta>0$ and consider the following sequence of stochastic processes $\{W^{(\eta)}_{N}(\tau): \tau \in \mathbb{R}\}$, $N=1,2, \ldots$:
\begin{equation}
\label{wntau}
W^{(\eta)}_{N}(\tau) = -\log\bigg{|}\det
\left[
\mathcal{H}- \left(x_{0}- \frac{\tau }{d_{N}} \right)I- \frac{i\eta}{d_{N}} I
\right]\bigg{|}
+\log\bigg{|}\det \left[
\mathcal{H}-x_{0}I-\frac{i\eta}{d_{N}}I
\right]\bigg{|}\, .
\end{equation}
Note that $ W^{(\eta)}_{N}(\tau)$ also depends implicitly on three additional parameters: $\eta >0$, $x_{0}\in (-1,1)$ and $d_{N}>0$; their importance is explained below, though for ease of notation we will not emphasize the dependence on $x_0$ when referring to $ W^{(\eta)}_{N}(\tau)$. We use the parameter $d_{N}>0$ to zoom into the appropriate spectral scale of $\mathcal{H}$ centered around a point $x_{0}$ inside the bulk of the limiting spectrum of the GUE matrices ${\cal H}$. On the macroscopic scale $d_N=1$, on the microscopic scale $d_N=N$ whilst on the mesoscopic scale $d_N$ is in between these two extremes, $1 \ll d_N \ll N $. The parameter $\eta$ is an arbitrary but fixed positive real number, introduced to regularize the logarithmic singularity at zero.
Our main result shows that in the \textit{mesoscopic limiting regime} where
\begin{equation}\label{M2}
\text{
\hspace{18ex} $d_{N} \to \infty$ and $ d_N=o(N/\log N)$ as $N \to \infty$
}\hspace{15ex}
\end{equation}
the stochastic process $ W^{(\eta)}_{N}(\tau)$ converges, after centering, to $B_{0}^{(\eta)}(\tau)$; the regularized fractional Brownian motion with Hurst index $H=0$. For finite-dimensional distributions this is the content of the following Theorem. Let
\[
\tilde{W}^{(\eta)}_{N}(\tau)= W^{(\eta)}_{N}(\tau)-\mathbb{E}\{ W^{(\eta)}_{N}(\tau)\}\, .
\]
\begin{theorem}
\label{th:maintheorem} Consider GUE random matrices ${\cal H}$ in (\ref{dens}).
Assume that the reference point $x_0$ is in the bulk of the limiting spectrum of ${\cal H}$, $x_0\in (-1,1)$, and the scaling factor $d_N$ satisfies (\ref{M2}). Then for any fixed $\eta>0$ and any finite number of times $(\tau_{1},\ldots,\tau_{m}) \in \mathbb{R}^{m}$ we have the convergence in distribution
\begin{equation}
\label{mesoconv}
(\tilde{W}^{(\eta)}_{N}(\tau_{1}),\ldots,\tilde{W}^{(\eta)}_{N}(\tau_{m})) \overset{d}{\Longrightarrow} (B^{(\eta)}_{0}(\tau_{1}),\ldots,B^{(\eta)}_{0}(\tau_{m})), \qquad \text{as $N \to \infty$}.
\end{equation}
\end{theorem}
We prove this theorem in Section \ref{se:mesoproof} by adopting Krasovsky's derivation of identity (\ref{krasovasympt}) to the mesoscopic scale. The characteristic function of the random vector on the l.h.s. in (\ref{mesoconv}) is given by a Hankel determinant whose symbol possesses Fisher-Hartwig singularities. The Riemann-Hilbert problem provides a powerful tool to obtain asymptotics of such Hankel determinants \cite{De99,KV03,KMVaV04,K07}. On the mesoscopic scale the Fisher-Hartwig singularities (these are located at points $x_0+(\tau_k+i\eta)/d_N$) are all at distance of order $1/d_N$ from the point $x_0 \in (-1,1)$. Because of this, the system of contours defining the Riemann-Hilbert problem (inside of which the symbol is analytic) close onto the real line as $N\to\infty$. In this regime, the estimates become more delicate. In contrast, in the macroscopic regime the Fisher-Hartwig singularities are real and spaced out and one does not need to consider the case of shrinking contours.
Here it is appropriate to mention that linear eigenvalue statistics on the mesoscopic scale are more challenging to study compared to the macroscopic scale. Known results are sparse and mostly limited to regular test functions with compact support, see \cite{BdMK99a,BdMK99b,S00} and also more recent works \cite{EK13a,EK13b,DJ13,Bou14,Dui14}. One reason is that the majority of concentration inequalities involving derivatives, such as \textit{e.g.} Lipschitz norm \cite{AGZ09} or the Poincar\'e inequality \cite{AGZ09, PS11}, that proved to be so useful on the macroscopic scale, get a factor of $d_N$ in the mesoscopic case and, hence, no longer apply without appropriate modification. In this context, the Riemann-Hilbert problem proves to be a powerful tool for estimating the error terms down to very small scales (\ref{M2}).
One can extend Theorem \ref{th:maintheorem} to an infinite-dimensional setting with a little bit more work. Let $L^{2}[a,b]$ denote the Hilbert space of square integrable functions on $[a,b]$ with the inner product
\begin{equation}
\langle f, g\rangle_{2} = \int_{a}^{b}f(\tau)\overline{g(\tau)}\,d\tau.
\end{equation}
Since the sample paths of $\tilde{W}^{(\eta)}_{N}$ are continuous, $\norm{\tilde{W}^{(\eta)}_{N}}_{2} < \infty$. Therefore, both $W^{(\eta)}_{N}$ and its $N \to \infty$ limit $B^{(\eta)}_{0}$ can be viewed as random elements in the space $L^{2}[a,b]$. We have,
\begin{theorem}
\label{th:compactconv}
Let $-\infty < a < b < \infty$. Then on mesoscopic scales \eqref{M2}, the process $\tilde{W}^{(\eta)}_{N}$ converges weakly (in the sense of probability law) to $B^{(\eta)}_{0}$ in $L^{2}[a,b]$ as $N \to \infty$. Furthermore, for every $h \in L^{2}[a,b]$, we have the convergence in distribution
\begin{equation}
\label{hl2}
\int_{a}^{b}h(\tau)\tilde{W}^{(\eta)}_{N}(\tau)\,d\tau \overset{d}{\Longrightarrow} \int_{a}^{b}h(\tau)B^{(\eta)}_{0}(\tau)\,d\tau, \qquad N \to \infty \, .
\end{equation}
\end{theorem}
This result follows from Theorem $3$ in \cite{G76}, which allows one to deduce weak convergence for general processes in $L^{2}[a,b]$ under the hypothesis that
\begin{enumerate}[(i)]
\item The finite-dimensional distributions of $\tilde{W}^{(\eta)}_{N}$ converge to those of $B^{(\eta)}_{0}$ as $N \to \infty$.
\item For some $C>0$, the bound $\mathbb{E}\{|\tilde{W}^{(\eta)}_{N}(\tau)|^{2}\} \leq C$ holds for all $N$ and $\tau \in [a,b]$ and
\begin{equation}
\lim_{N \to \infty}\mathbb{E}\{|\tilde{W}^{(\eta)}_{N}(\tau)|^{2}\} = \mathbb{E}\{|B^{(\eta)}_{0}(\tau)|^{2}\}.
\end{equation}
\end{enumerate}
Note that item $(\mathrm{i})$ is a restatement of Theorem \ref{th:maintheorem}, while item $(\mathrm{ii})$ will be shown to follow from our proof of Theorem \ref{th:maintheorem}.
Having established the relation between characteristic polynomials of GUE matrices and $1/f$ noise on the mesoscopic scale, let us revisit the series expansions of the macroscopic scale discussed at length in Sec. \ref{sec2.2}. Instead of expanding the process $W^{(\eta)}_{N}(\tau)$ in a Chebyshev-Fourier series and applying the Diaconis-Shahshahani result, in the mesoscopic regime it comes in handy to expand $W^{(\eta)}_{N}(\tau)$ as a Fourier \textit{integral}.
To this end, we now provide a suitable Fourier-integral representation for $W^{(\eta)}_{N}(\tau)$. Such a representation can be derived by making use of the identity (see, e.g., Eq. 7.89 in \cite{De99})
\begin{equation}
\label{deiftint}
\frac{1}{2}\log\left(\frac{t^{2}}{\epsilon^{2}}+1\right) = \int_{0}^{\infty}\frac{e^{-\epsilon s}}{s}[1-\cos(ts)]\,ds, \quad \epsilon>0\, .
\end{equation}
It follows from (\ref{deiftint}) that
\begin{equation}
\label{detforident}
W^{(\eta )}_{N}(\tau) = \frac{1}{2}\int_{0}^{\infty}\frac{e^{-\eta s}}{\sqrt{s}}\left\{[e^{-i\tau s}-1]\,b_{N}(s)+[e^{i\tau s}-1]\,\overline{b_{N}(s)}\right\}\,ds
\end{equation}
where
\begin{equation}
\label{bee}
b_{N}(s) = \frac{1}{\sqrt{s}}\, \mathop{\mathrm{Tr}} e^{-isd_{N}(\mathcal{H}-x_{0}I)}\, .
\end{equation}
The identity \eqref{detforident} can be thought of as the Fourier integral version of the Fourier series (\ref{diashah}). Furthermore, comparison of the harmonizable representation \eqref{intr3_h_0} for $B_0^{(\eta)}(t)$ (which can be thought as a natural integral analogue of the series expansions in \eqref{1/f}) and \eqref{detforident}), suggests that the Fourier coefficients $b_{N}(s)$ converge in the mesoscopic regime to Gaussian white noise. Such a statement may be interpreted as a continuous analogue of the Diaconis-Shahshahani result \cite{DS94} and is the content of our next theorem.
Let $C^{\infty}_{0}(\mathbb{R}_{+})$ be the space of infinitely many times differentiable functions with compact support on $\mathbb{R}_{+}=\{x\in \mathbb{R}: x>0 \}$. Denote
\begin{equation}
\label{cn}
c_{N}(\xi) = \int_{0}^{\infty}\xi(s)\,b_{N}(s)\, ds.
\end{equation}
\begin{theorem}
\label{th:fourierconv}
Consider the mesoscopic regime where $d_N=N^{\alpha}$ with any $\alpha \in (0,1)$. Then for every $\xi \in C^{\infty}_0 (\mathbb{R}_{+})$
\begin{equation}
\label{whitenoiselimit}
\lim_{N \to \infty}\mathbb{E}
\{
e^{-i\mathop{\mathrm{Re}} c_{N}(\xi)}
\} = \exp \left(-\frac{1}{4}\int_{0}^{\infty}|\xi(s)|^{2}\,ds\right).
\end{equation}
Furthermore, for any finite number of $\xi_j \in C^{\infty}_0 (\mathbb{R}_{+})$, the vector $(c_{N}(\xi_{1}),\ldots,c_{N}(\xi_{m}))$ converges in distribution, as $N \to \infty$, to the centered complex Gaussian vector $Z \in \mathbb{R}^{m}$ having relation matrix $\mathbb{E}(ZZ^{\mathrm{T}})=0$ and covariance matrix $\Gamma = \mathbb{E}(ZZ^{\dagger})$ given by
\begin{equation}
\Gamma_{j,k} = \int_{0}^{\infty}\xi_{j}(s)\overline{\xi_{k}(s)}\,ds, \qquad j,k=1,\ldots,m.
\end{equation}
\end{theorem}
\begin{proof}
See Section \ref{se:fouriercoeff}.
\end{proof}
\begin{remark}
As is often the case in random matrix theory, linear eigenvalue statistics such as \eqref{cn} have variance of the order of unity due to strong correlations between the eigenvalues and converge to a Gaussian random variable after centering. One would typically expect that $\mathbb{E}\{c_{N}(\xi)\} = O(N/d_{N})$ as $N \to \infty$. Instead, we find, see Section \ref{se:fouriercoeff}, that the smoothness of $\xi$ and the rapid oscillations in \eqref{bee} imply $\mathbb{E}\{c_{N}(\xi)\} = O(d_{N}^{-1})$ as $N \to \infty$ and, thus, centering is not really needed.
\end{remark}
The rest of the paper is organized as follows. Section \ref{se:mesoproof} is devoted to the proof of Theorem \ref{th:maintheorem}. To do this, we begin by adapting the differential identity used in \cite{K07} and then outline the relevant asymptotic analysis of the Riemann-Hilbert problem, leaving estimation of all error terms to Appendix \ref{ap:riemann}.
Section \ref{se:fouriercoeff} is devoted to proving the convergence of the Fourier coefficients $b_{N}(s)$ to the white noise. In the final section we focus on the macroscopic scale and prove Theorem \ref{th:global}.
\\\\
\section{Mesoscopic regime}
\label{se:mesoproof}
In this section we prove Theorem \ref{th:maintheorem}. Let us fix $m-1$ distinct times $\tau_1, \ldots , \tau_{m-1}$, $m\ge 2$, and consider the characteristic function
\[
\varphi_N ( \alpha_1, \ldots , \alpha_{m-1})= \mathbb{E}\left\{\exp \left(\sum\limits_{k=1}^{m-1}\alpha_{k} W^{(\eta)}_{N}(\tau_{k})\right)\right\}
\]
of the random vector $(W^{(\eta)}_{N}(\tau_1), \ldots, W^{(\eta)}_{N}(\tau_{m-1}))$. Our strategy will be to prove that $\varphi_N$ converges to the characteristic function of the multivariate Gaussian distribution in the limit $N\to\infty$. Theorem \ref{th:maintheorem} will then follow by inspection of the quadratic form in the exponential.
To begin with, we will write the characteristic function $\varphi_N$ as the partition function of a matrix model with Gaussian weight, modified by the singularities
\begin{equation}
\mu_{k}= \sqrt{2N}\left(x_{0}+\frac{\tau_{k}+i\eta}{d_{N}}\right), \qquad \eta>0,
\end{equation}
where $k=1,\ldots,m$ and $\tau_{m}\equiv 0$. A standard calculation (changing variables of integration from ${\cal H}$ to the eigenvalues and eigenvectors of ${\cal H}$ and integrating out the eigenvectors, see e.g. \cite{PS11}) yields
\begin{equation}
\varphi_N ( \alpha_1, \ldots , \alpha_{m-1}) =\frac{1}{C}\int_{\mathbb{R}^{N}}\prod_{j=1}^{N}\,w(x_{j})\prod_{1 \leq i < j \leq N}(x_{i}-x_{j})^{2}\,dx_{1}\ldots dx_{N} \label{multint}
\end{equation}
where the weight function is given by
\begin{equation}
\label{weight}
w(x) = e^{-x^{2}}\prod_{k=1}^{m}|x-\mu_{k}|^{\alpha_{k}}, \qquad \mathrm{Im}(\mu_{k}) \neq 0, \qquad k=1,\ldots,m
\end{equation}
and $\alpha_m=-\alpha_{1} - \ldots - \alpha_{m-1}$. Note the discrepancy with the measure \eqref{dens}; for convenience we have changed variables $x_{j} \to x_{j}/\sqrt{2N}$, the resulting multiplicative constants cancelling each other out.
Our calculation will be guided by that of Krasovsky \cite{K07} who treated a similar partition function, but only for the macroscopic regime $d_{N}=1$ and $\eta=0$. In that case the weight function acquires Fisher-Hartwig singularities inside the spectral interval $(-1,1)$. In contrast, our weight \eqref{weight} posesses singularities in the complex plane that merge towards the point $x_{0}$ on the spectral axis at rate $d_{N}$ as $N\to\infty$. Since this merging process occurs sufficiently slowly (i.e. $d_{N} = o(N)$), these singularities will not play a crucial role in the calculation.
A special feature of the weight function \eqref{weight} is the \textit{cyclic condition}
\begin{equation}
\label{cyclic}
\sum_{k=1}^{m}\alpha_{k}=0.
\end{equation}
This holds because the second term in \eqref{wntau} is independent of $\tau$. Our first step is to express the partition function \eqref{multint} in a form suitable for the computation of asymptotics.
\subsection{Orthogonal polynomials and differential identity}
The multiple integral in \eqref{multint} is intimately connected to the theory of orthogonal polynomials. Let
\[
p_{n}(x) = \chi_{n}(x^{n}+\beta_{n}x^{n-1}+\gamma_{n}x^{n-2}+\ldots), \quad n=0,1,2, \ldots , \]
be orthogonal polynomials with respect to weight function $w(x)$:
$\int_{-\infty}^{\infty}p_{m}(x)p_{n}(x)w(x)\,dx=\delta_{m,n}$. When the $\alpha_{j}$'s are real and each $\alpha_{j}>-1/2$ we have $w(x)\geq0$ and the existence of the polynomials $p_{n}(x)$ is well known \cite{De99}. Then, as in \cite{K07}, the coefficients $\chi_{n},\beta_{n}$ and $\gamma_{n}$ and the polynomials $p_{n}(x)$ are defined for any $\{\alpha_{j}\}_{j=1}^{m} \in \mathbb{C}^{m}$ via analytic continuation, provided each $\mathop{\mathrm{Re}}(\alpha_{j})>-1/2$.
Now, the partition function \eqref{multint} can be written in terms of the coefficients $\{\chi_{j}\}_{j=1}^{N}$ (see \textit{e.g.} \cite{Meh04})
\begin{equation}
\label{prodchi}
\varphi_N (\alpha_{1},\ldots,\alpha_{m-1}) = \frac{N!}{C}\prod_{j=0}^{N-1}\chi_{j}^{-2}.
\end{equation}
Thus, in principle, our problem is reduced to computing the asymptotics of the orthogonal polynomials and related quantities with respect to the weight $w(x)$.
The crucial point observed in \cite{K07} is that by taking the logarithmic derivative on both sides of \eqref{prodchi} with respect to any of the $\alpha_{j}$'s, the right-hand side can be written as a sum involving only $O(m)$ terms, rather than $N$. To state the resulting \textit{differential identity} we also need the following $2 \times 2$ matrix involving the orthogonal polynomials and their Cauchy transforms:
\begin{equation}
Y(z) = \begin{pmatrix} \chi_{N}^{-1}p_{N}(z) & \displaystyle{\chi_{N}^{-1}\int_{-\infty}^{\infty}\frac{p_{N}(x)}{x-z}\frac{w(x)dx}{2\pi i}}\\[3ex]
\displaystyle{-2\pi i\chi_{N-1}p_{N-1}(z)} &\displaystyle{ -\chi_{N-1}\int_{-\infty}^{\infty}\frac{p_{N-1}(x)}{x-z}w(x)dx} \end{pmatrix}. \label{ymatrix}
\end{equation}
\begin{lemma}
For each $k=1,\ldots,m$, let $\mu_{k}$ in \eqref{weight} be any complex parameters satisfying $\mathrm{Im}(\mu_{k}) \neq 0$ and define $\alpha_{m+k}=\alpha_{k}$, $\mu_{m+k} = \overline{\mu_{k}}$. Denoting by $'$ differentiation with respect to $\alpha_{j}$, the following formula holds for any $j=1,\ldots,m$.
\begin{equation}
\label{diffid}
\begin{split}
&(\log \varphi_N)' = -N(\log \chi_{N}\chi_{N-1})'-2\left(\frac{\chi_{N-1}}{\chi_{N}}\right)^{2}\left(\log \frac{\chi_{N-1}}{\chi_{N}}\right)^{'}+2(\gamma_{N}'-\beta_{N}\beta_{N}')\\
&+\frac{1}{2}\sum_{k=1}^{2m}\alpha_{k}(Y_{11}(\mu_{k})'Y_{22}(\mu_{k})-Y_{21}(\mu_{k})'Y_{12}(\mu_{k})+(\log \chi_{N}\chi_{N-1})'Y_{11}(\mu_{k})Y_{22}(\mu_{k})).
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
The proof follows from simple modifications of the arguments given in Sec. 3 of \cite{K07}. In fact, further simplifications occur due to the cyclic condition $\sum_{k=1}^{m}\alpha_{k}=0$ and the fact that the singularities $\mu_{k}$ have non-zero imaginary part ($k=1,\ldots,m$).
\end{proof}
Note that $\chi_{N}$ and the coefficients $\beta_{N}$ and $\gamma_{N}$ can be computed from the relations:
\begin{equation}
\label{coeffid}
\begin{split}
Y_{11}(z) &= z^{N}+\beta_{N}z^{N-1}+\gamma_{N}z^{N-2}+\ldots\\
\chi_{N-1}^{2} &= \lim_{z \to \infty}\frac{iY_{21}(z)}{2\pi z^{N-1}}
\end{split}
\end{equation}
Therefore, our plan will be to compute the asymptotics of $Y(z)$ and then, by making use of identities \eqref{coeffid}, evaluate the right-hand side of \eqref{diffid} to the desired accuracy in the limit as $N \to \infty$. We will find that the error terms in the asymptotics are uniform in the variables $\{\alpha_{k}\}_{k=1}^{m-1}$ belonging to a compact subset of
\begin{equation}
\Omega = \{(\alpha_{1},\ldots,\alpha_{m-1}) \mid \mathop{\mathrm{Re}}(\alpha_{k}) > -1/2, \qquad k=1,\ldots,m-1\}. \label{omset}
\end{equation}
This uniformity property then allows us to integrate the identity \eqref{diffid} recursively with respect to $\{\alpha_{k}\}_{k=1}^{m-1}$ and obtain asymptotics for the characteristic function \eqref{multint}. The asymptotics of $Y(z)$ in the limit $N\to\infty$ can be obtained by using an appropriate Riemann-Hilbert problem. Although this technique is nowadays standard, for the reader's convenience we will briefly summarise the necessary ingredients of the corresponding calculation.
\subsection{The Riemann-Hilbert problem for $Y(z)$}
The relationship between orthogonal polynomials and Riemann-Hilbert problems was established for general weights in \cite{FIK92} where it was shown that $Y(z)$ solves the following problem:\\
\begin{enumerate}
\item $Y(z)$ is analytic in $\mathbb{C} \setminus \mathbb{R}$.\\
\item On the real line there is a jump discontinuity
\begin{equation}
Y_{+}(x) = Y_{-}(x)
\begin{pmatrix} 1 & w(x)\\
0 & 1
\end{pmatrix}, \quad x \in \mathbb{R},
\end{equation}
where $Y_{+}(x)$ and $Y_{-}(x)$ denote the limiting values of $Y(z)$ as $z$ approaches the point $x \in \mathbb{R}$ from above $(+)$ or below $(-)$.\\
\item Near $z=\infty$, we have the following asymptotic behaviour
\begin{equation}
Y(z) = \left(I+O\left(\frac{1}{z}\right)\right)z^{N\sigma_{3}}. \label{Yasympt}
\end{equation}
\end{enumerate}
Here $\sigma_{3}$ is the third Pauli matrix and serves as a convenient notational tool. By definition of the matrix exponential, the notation in \eqref{Yasympt} has the meaning
\begin{equation}
z^{N\sigma_{3}} = \begin{pmatrix} z^{N} & 0\\0 & z^{-N} \end{pmatrix}.
\end{equation}
One can verify directly that $Y(z)$ of \eqref{ymatrix} does indeed solve this Riemann-Hilbert problem, while the uniqueness of this solution can be deduced from the observation that $\det Y(z) \equiv 1$, in conjunction with the Liouville theorem. Further details regarding existence and uniqueness of the problem can be found in \cite{De99}.
In order to obtain asymptotics as $N \to \infty$, we will perform a sequence of transformations to our initial Riemann-Hilbert problem known as the \textit{Deift-Zhou steepest descent} (see e.g. \cite{De99} and \cite{DKMVZ99}). The purpose of these transformations is to identify a `limiting' problem that can be solved with elementary functions, giving the leading order asymptotics to $Y(z)$. For the reader's convenience, we briefly describe the key points underlying these transformations:\\
\begin{enumerate}
\item The first transformation $Y \to T$ normalizes the unsatisfactory asymptotic behaviour in the third condition, equation \eqref{Yasympt}. This comes with the cost that the entries of the jump matrix for $T(z)$ on the interval $(-1,1)$ are now oscillating in $N$ and do not have a limit as $N \to \infty$.\\
\item The second transformation $T \to S$ aims to remove these oscillations by splitting the contour $(-1,1)$ into lens shaped contours where now the jump matrices are exponentially close to the identity. For our particular \textit{mesoscopic} problem, we need the lenses to pass below the singularities for each $k=1,\ldots,m$, so that their distance from $(-1,1)$ is of order $O(d_{N}^{-1})$ (see Figure \ref{fig:contour}).\\
\item Now it turns out that the jump matrices for $S$ tend to the identity as $N \to \infty$, except on the contour $(-1,1)$. But the jump across $(-1,1)$ is of a special form that can be solved exactly in terms of elementary functions. This solution, denoted $P_{\infty}(z)$, gives the leading order contribution to the asymptotics in the required regions of the complex plane.\\
\end{enumerate}
In Sec. \ref{se:asymptotics} we will show that the asymptotics obtained in this way lead directly to Theorem \ref{th:maintheorem}. However, to complete the proof, one has to show that the conclusion of $(3)$, namely that $S(z) \sim P_{\infty}(z)$ as $N \to \infty$, is really correct. This may be regarded as the most technical part of the Deift-Zhou method. The main problem is that although the jump matrix for $S(z)$ converges to that of $P_{\infty}(z)$, this convergence is not uniform near the edges $z=\pm 1$. To remedy this, local solutions known as \textit{parametrices} have to be constructed near these points, and then matched to leading order with the so-called outer parametrix $P_{\infty}(z)$. These final technical issues will be addressed in Appendix A.
\subsection{$T$ and $S$ transformations of the Riemann-Hilbert problem}
\label{se:smatrix}
The $T$ transformation is performed in the usual way. First we define the $g$-function:
\begin{equation}
\label{gfun}
g(z) = \int_{-1}^{1}\log(z-s)\rho(s)\,ds, \qquad z \in \mathbb{C}\setminus(-\infty,1],
\end{equation}
where throughout we take the principal branch of the logarithm. Here and below $\rho(s) = (2/\pi)\sqrt{1-s^{2}}$ denotes the limiting density of eigenvalues. The $Y \to T$ transformation is then given by the formula
\begin{equation}
Y(z\sqrt{2N}) = (2N)^{N\sigma_{3}}e^{Nl\sigma_{3}/2}T(z)e^{N(g(z)-l/2)\sigma_{3}}
\end{equation}
where $l=-1-2\log(2)$. Notice that we have rescaled the Riemann-Hilbert problem so that the singularities of the corresponding weight function are of order $O(1)$ as $N \to \infty$, so that from now on we deal with singularities of the form
\begin{equation}
z_{k} = \frac{\mu_{k}}{\sqrt{2N}} = x_{0}+\frac{\tau_{k}+i\eta}{d_{N}}. \label{sings}
\end{equation}
The resulting jump matrix for $T(z)$ can now be computed from the standard properties of the $g$-function:
\begin{equation}
\label{gprops}
\begin{split}
&g_{+}(x) + g_{-}(x)-2x^{2}-l=0, \qquad x \in (-1,1),\\
&g_{+}(x) + g_{-}(x)-2x^{2}-l<0, \qquad x \in \mathbb{R}\setminus [-1,1],\\
&g_{+}(x)-g_{-}(x) = \begin{cases} 2\pi i & x\leq-1\\ 2\pi i \displaystyle{\int_{x}^{1}\rho(s)ds }& x \in [-1,1]\\0 & x \geq 1. \end{cases}
\end{split}
\end{equation}
In addition, since $g(z) \sim \log(z)$ as $z \to \infty$, we have $e^{Ng(z)\sigma_{3}} \sim z^{N\sigma_{3}}$. Thus one easily verifies that $T(z)$ is normalized at $z=\infty$. We now have the following Riemann-Hilbert problem for $T(z)$:
\begin{enumerate}
\item $T(z)$ is analytic in $\mathbb{C}\setminus \mathbb{R}$.\\
\item We have the jump condition
\begin{align}
T_{+}(x) &= T_{-}(x)\begin{pmatrix} e^{-N(g_{+}(x)-g_{-}(x))} & \displaystyle{\prod_{k=1}^{m}|x-z_{k}|^{\alpha_{k}}}\\[4ex]
0 & e^{N(g_{+}(x)-g_{-}(x))} \end{pmatrix}, \quad x \in (-1,1), \label{tjump1}\\[2ex]
T_{+}(x) &= T_{-}(x)\begin{pmatrix}\quad 1 & \displaystyle{\prod_{k=1}^{m}|x-z_{k}|^{\alpha_{k}}e^{N(g_{+}(x)+g_{-}(x)-2x^{2}-l)}}\\[4ex]
\quad 0 & 1 \end{pmatrix}, \quad x \in \mathbb{R}\setminus[-1,1]. \label{tjump2}
\end{align}
\item $T(z) = I+O(z^{-1})$ as $z \to \infty$.
\end{enumerate}
We see that although the problem for $T(z)$ is normalized at $\infty$, the jump matrix \eqref{tjump1} on $(-1,1)$ has oscillatory diagonal entries that not have a limit as $N \to \infty$. The Deift-Zhou steepest descent procedure remedies this situation by splitting the contour $(-1,1)$ into `lenses' in the complex plane (see Figure \ref{fig:contour}), transforming the unwanted oscillations into exponentially decaying matrix elements.
This procedure is facilitated by the factorization of the jump matrix on $(-1,1)$:
\begin{equation*}
\begin{pmatrix} e^{-Nh(x)} & \omega(x)\\ 0 & e^{Nh(x)} \end{pmatrix} = \begin{pmatrix} 1 & 0\\ \omega(x)^{-1}e^{Nh(x)} & 1\end{pmatrix} \begin{pmatrix} 0 & \omega(x)^{-1}\\ -\omega(x)^{-1} & 0\end{pmatrix} \begin{pmatrix} 1 & 0\\ \omega(x)^{-1}e^{-Nh(x)} & 1\end{pmatrix}
\end{equation*}
where
\begin{align}
\omega(x) &= \prod_{k=1}^{m}|x-z_{k}|^{\alpha_{k}} \label{omdef}\\
h(x) &= g_{+}(x)-g_{-}(x) = -2\pi i \int_{1}^{x}\rho(y)dy \label{hdef}
\end{align}
The latter objects \eqref{omdef} and \eqref{hdef} possess analytic continuations into the lens shaped regions depicted in Figure \ref{fig:contour}. For the weight $\omega(x)$ we have
\begin{equation}\label{omegaweight}
\omega(z) =
\prod_{k=1}^{m-1}\left[\frac{(z-x_{0}-\tau_{k}/d_{N})^{2}+(\eta/d_{N})^{2}}{(z-x_{0})^{2}+(\eta/d_{N})^{2}}\right]^{\alpha_{k}/2},
\end{equation}
where throughout we take the principal branch of the roots. This function is analytic for all $z$ such that the inequality
\begin{equation}
(\mathop{\mathrm{Re}} (z)-\mathop{\mathrm{Re}} (z_{k}))^{2} > (\mathop{\mathrm{Im}}(z_{k}))^{2}- (\mathop{\mathrm{Im}}(z))^{2} \label{ineqanalytic}
\end{equation}
is satisfied for every $k=1,\ldots,m$. One easily verifies that for $x_{0} \in (-1+\delta,1-\delta)$, the inequality \eqref{ineqanalytic} holds for any $z$ chosen from the interior region bounded by the lips $\Sigma_{\pm 1}$ and the discs $z \in \partial B_{\pm 1}(\delta)$ of sufficiently small radius (see Figure \ref{fig:contour}). Finally let $h(z)$ denote the analytic continuation of $\eqref{hdef}$ to $\mathbb{C}\setminus((-\infty,-1]\cup[1,\infty))$. We are now ready to define the $T \to S$ transformation. Let
\begin{equation}
S(z) = \begin{cases} T(z), & \textrm{for $z$ outside the lenses},\\[2ex]
T(z)\begin{pmatrix} 1 & 0\\
-\omega(z)^{-1}e^{-Nh(z)} & 1 \end{pmatrix}, & \textrm{for $z$ in the upper part of the lenses},\\[4ex]
T(z)\begin{pmatrix} 1 & 0\\ \omega(z)^{-1}e^{Nh(z)} & 1 \end{pmatrix}, & \textrm{for $z$ in the lower part of the lenses.}
\end{cases}
\end{equation}
\begin{figure}
\includegraphics[width=16cm]{contourdnnew.pdf}
\caption{The contour $\Sigma$ for the $S$ Riemann-Hilbert problem with $m=3$. The crosses depict the $3$ singularities and their complex conjugates, of distance $O(d_{N}^{-1})$ from the point $x_{0} \in (-1,1)$. The lenses $\Sigma_{\pm}$ pass between the real line and the singularities into the points $\pm 1$.}
\label{fig:contour}
\end{figure}
Now we get the following Riemann-Hilbert problem for $S(z)$:\\
\begin{enumerate}
\item $S(z)$ is analytic in $\mathbb{C}\setminus \Sigma$ where $\Sigma = \Sigma_{+}\cup\mathbb{R}\cup\Sigma_{-}$.\\
\item $S(z)$ has the following jumps on $\Sigma$
\begin{align}\nonumber
S_{+}(x) &= S_{-}(x)\begin{pmatrix} 1 & 0\\ \omega(x)^{-1}e^{\mp Nh(x)} & 1 \end{pmatrix}, \qquad x \in \Sigma_{\pm},\\[2ex]
\nonumber
S_{+}(x) &= S_{-}(x)\begin{pmatrix} 0 & \omega(x)\\-\omega(x)^{-1} & 0 \end{pmatrix}, \qquad x \in (-1,1),\\[2ex]
\nonumber
S_{+}(x) &= S_{-}(x)\begin{pmatrix} 1 & \omega(x)e^{N(g_{+}(x)+g_{-}(x)-2x^{2}-l)}\\ 0 & 1 \end{pmatrix}, \qquad x \in \mathbb{R}\setminus [-1,1].
\end{align}
\item $S(z) = I+O(z^{-1})$ as $z \to \infty$.\\
\end{enumerate}
At this point in the asymptotic analysis, it becomes clear that the mesoscopic regime under consideration becomes important. In order to obtain asymptotics, it is essential that the jump matrix for $S(z)$ approaches the identity as $N \to \infty$ for $z\in\Sigma_{\pm}$. In the Appendix (see Prop. \ref{prop:deltasigr}) we will see that $|e^{\mp Nh(z)}| = O(e^{-c_{1}\frac{N}{d_{N}}})$ as $N \to \infty$ uniformly on $\Sigma_{\pm}\setminus(B_{1}(\delta)\cup B_{-1}(\delta))$. Notice that such a bound fails when one approaches the critical situation $d_{N}=N$ corresponding to the \textit{local} or \textit{microscopic} regime. It is precisely at this scale that one would \textit{not} expect the appearance of a Gaussian process in the limit $N \to \infty$.
Therefore, in the \textit{mesoscopic} regime it is reasonable to expect that in the limit $N \to \infty$ we may neglect the jumps on $\Sigma_{\pm}\cup(\mathbb{R}\setminus[-1,1])$ and approximate $S(z)$ by a Riemann-Hilbert problem with jumps only on the interval $(-1,1)$. This approximation will be valid only in the region $U_{\infty}=\mathbb{C}\setminus (B_{1}(\delta)\cup B_{-1}(\delta))$ and will give rise to an error that is quantified in Appendix \ref{ap:riemann}.
\subsection{Limiting Riemann-Hilbert problem: Parametrix in $U_{\infty}$}
\label{se:outer}
Before we perform the final transformation $S \to R$ of the Riemann-Hilbert problem, we must construct parametrices in the appropriate regions of the complex plane. We saw in the last section how the jump matrices for $S(z)$ converge to the identity as $N \to \infty$, except on $[-1,1]$. Therefore, outside the lenses and the discs, we expect the solution to the following problem to give a good approximation to $S(z)$ for large $N$.
\begin{enumerate}
\item $P_{\infty}(z)$ is analytic in $\mathbb{C}\setminus[-1,1]$.\\
\item We have the jump condition
\begin{equation}
P_{\infty,+}(x)=P_{\infty,-}(x)\begin{pmatrix} 0 & \omega(x)\\ -\omega(x)^{-1} & 0 \end{pmatrix}, \qquad x \in (-1,1).
\end{equation}
\item $P_{\infty}(z) = I+O(z^{-1})$ as $z \to \infty$.\\
\end{enumerate}
This problem has the advantage that it has a completely explicit solution. The solution, as obtained in \cite{KMVaV04}, is given by
\begin{equation}
\label{pinf}
P_{\infty}(z) = \frac{1}{2}(\mathcal{D}_{\infty})^{\sigma_{3}}\begin{pmatrix} a+a^{-1} & -i(a-a^{-1})\\ i(a-a^{-1}) & a+a^{-1}\end{pmatrix}\mathcal{D}(z)^{-\sigma_{3}}, \qquad a(z) = \frac{(z-1)^{1/4}}{(z+1)^{1/4}},
\end{equation}
where $\mathcal{D}(z)$ is the Szeg\"o function
\begin{equation}
\label{sz}
\mathcal{D}(z) = \mathrm{exp}\left(\frac{\sqrt{z+1}\sqrt{z-1}}{2\pi}\int_{-1}^{1}\frac{\log \omega(x)}{\sqrt{1-x^{2}}}\frac{dx}{z-x}\right)
\end{equation}
and
\begin{equation}
\mathcal{D}_{\infty} = \lim_{z \to \infty}\mathcal{D}(z) = \exp\left(\frac{1}{2\pi}\int_{-1}^{1}\frac{\log\omega(x)}{\sqrt{1-x^{2}}}\,dx\right). \label{dinf}
\end{equation}
Recalling the definition of the weight $\omega(x)$ in \eqref{omdef}, the integrals in \eqref{sz} can be calculated explicitly by extending the procedure outlined in \cite{K07} to the case of complex singularities.
As we shall see in the next subsection, the Szeg\"o function $\mathcal{D}(z)$ will turn out to be the key ingredient in deriving the logarithmic covariance structure in \eqref{logcov}.
\subsection{Asymptotics of the polynomials and proof of Theorem \ref{th:maintheorem}}
\label{se:asymptotics}
We are now ready to present the leading order asymptotics $N \to \infty$ of the $Y$-matrix in \eqref{ymatrix}, leaving the technical matters of estimation of errors and the final transformation of the Riemann-Hilbert problem to Appendix \ref{ap:riemann}. Our aim in this subsection is to prove Theorem \ref{th:maintheorem} using these asymptotics.
Tracing back the transformations $S \to T \to Y$, we find that
\begin{equation}
Y(z\sqrt{2N}) = (2N)^{N\sigma_{3}/2}e^{Nl\sigma_{3}/2}S(z)e^{N(g(z)-l/2)\sigma_{3}} \label{yas}
\end{equation}
According to \eqref{diffid}, we need the asymptotics for $Y(z)$ in two different regions of the complex plane, near $z=\infty$ in the first line of \eqref{diffid} and at $z=z_{k}$ in the second line. In the following Proposition, let $\mathcal{A}$ denote the bounded subset of $\mathbb{C}$ enclosed by the lenses $\Sigma_{\pm}$ and the discs $\partial B_{\pm 1}(\delta)$.
\begin{proposition}
\label{prop:match}
Consider the Riemann-Hilbert problems $S(z)$ and $P_{\infty}(z)$ from Sections \ref{se:smatrix} and \ref{se:outer} respectively. Then the following asymptotics hold as $N \to \infty$
\begin{equation}
S(z) = \left(I+\frac{\tilde{R}_{1}(z)}{N}+O\left(\frac{1}{Nd_{N}}\right)+O\left(\log(d_{N})\,e^{-c_{1}\frac{N}{d_{N}}}\right)\right)P_{\infty}(z), \label{zkas}
\end{equation}
uniformly for all $z \in \mathbb{C}\setminus \mathcal{A}$. The function $\tilde{R}_{1}(z)$ has an asymptotic expansion of the form $\tilde{R}_{1}(z) = (A/z+B/z^{2}+O(z^{-3}))$ as $z \to \infty$ where $c_{1}$ is a positive constant depending only on $\delta$ and $\eta$ and
\begin{equation}
A = \begin{pmatrix} 0 & i/24\\ i/24 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} -1/48 & 0\\0 & 1/48 \end{pmatrix}.
\end{equation}
\end{proposition}
\begin{proof}
See Appendix \ref{ap:riemann}.
\end{proof}
\begin{remark}
\label{re:uniformity}
The error terms in \eqref{zkas} are uniform in the parameters $\{\alpha_{k}\}_{k=1}^{m-1}$ belonging to $\Omega$ (cf. \eqref{omset}), $\{\tau_{k}\}_{k=1}^{m-1}$ belonging to a compact subset of $\mathbb{R}$ and $x_{0}$ belonging to a compact subset of $(-1+\delta,1-\delta)$. Furthermore, every such error term is an analytic function in the variables $\{\alpha_{k}\}_{k=1}^{m-1}$ whose derivatives with respect to $\alpha_{j}$ have the same order in $N$ and have the same uniformity property described above. Hence, in the remainder of this Section it will be implicit that the error terms involved are of this form.
\end{remark}
Now inserting the above asymptotics \eqref{zkas} into the differential identity \eqref{diffid}, we obtain
\begin{proposition}
Let $\varphi_{N}$ denote the characteristic function of the stochastic process $W^{(\eta)}_{N}(\tau)$ defined in \eqref{multint}. Then in the limit $N \to \infty$, we have
\begin{equation}
\begin{split}
\label{charasymptotics}
&\varphi_{N}(\alpha_{1},\ldots,\alpha_{m-1}) = \mathrm{exp}\left(N\sum_{k=1}^{m-1}\alpha_{k}(\mathop{\mathrm{Re}}(g(z_{k}))-\mathop{\mathrm{Re}}(g(z_{m})))\right.\\
&\left.+\sum_{k,j=1}^{m-1}\frac{\alpha_{k}\alpha_{j}}{2}\left(\phi^{(\eta)}_{0}(\tau_{k}) +\phi^{(\eta)}_{0}(\tau_{j})-\phi^{(\eta)}_{0}(\tau_{k}-\tau_{j})\right)\right.\\
&\left.+O(d_{N}^{-1})+ O\left(N\log(d_{N})\,\mathrm{exp}\left(-c_{1}\frac{N}{d_{N}}\right)\right)\right),
\end{split}
\end{equation}
where $g(z)$ is defined in \eqref{gfun} and $\phi^{(\eta)}_{0}(\tau)$ in \eqref{covphi}. The asympotics in \eqref{charasymptotics} hold uniformly in the same sense described in Remark \ref{re:uniformity}.
\end{proposition}
\begin{remark}
Notice that the asymptotics in \eqref{charasymptotics} consist of both \textit{global} error terms, which become large when $d_{N} \sim 1$ and \textit{local} error terms, which become large when $d_{N} \sim N$. Throughout the following proof, we will write $e_{N}$ for the local error term of order
\begin{equation}
e_{N} = \log(d_{N})\,\mathrm{exp}\left(-c_{1}\frac{N}{d_{N}}\right).
\end{equation}
\end{remark}
\begin{proof}
We remind the reader that the prime $'$ always denotes differentiation with respect to $\alpha_{j}$. We begin by considering the second line of \eqref{diffid}. Taking into account $\alpha_{m}=-(\alpha_{1}+\ldots+\alpha_{m-1})$, we insert \eqref{zkas} into \eqref{yas} and make use of the explicit formula \eqref{pinf} for $P_{\infty}(z)$. Straightforward calculation then gives
\begin{align}
&Y_{11}(\sqrt{2N}z_{k})'Y_{22}(\sqrt{2N}z_{k})-Y_{21}(\sqrt{2N}z_{k})'Y_{12}(\sqrt{2N}z_{k}) \notag\\
&=(P_{\infty}(z_{k}))'_{11}(P_{\infty}(z_{k}))_{22}-(P_{\infty}(z_{k}))'_{21}(P_{\infty}(z_{k}))_{12}+O(N^{-1})+O(e_{N}).\label{diffofpinfs}\\
&=C(z_{m},z_{k})-C(z_{j},z_{k})+O(d_{N}^{-1})+O(e_{N})\label{diffofys}
\end{align}
where we introduced
\begin{equation}
\label{phifn}
C(\mu,z) = \frac{\sqrt{z+1}\sqrt{z-1}}{2\pi}\int_{-1}^{1}\frac{\log|x-\mu|}{\sqrt{1-x^{2}}}\frac{dx}{z-x},
\end{equation}
and \eqref{diffofys} was obtained from \eqref{diffofpinfs} using the estimate $\mathcal{D}_{\infty}=1+O(d_{N}^{-1})$. Since $C(z_{j},\overline{z_{k}}) = \overline{C(z_{j},z_{k})}$, we find from \eqref{diffofys} that
\begin{align}
&\frac{1}{2}\sum_{k=1}^{2m}\alpha_{k}\left(Y_{11}(\sqrt{2N}z_{k})'Y_{22}(\sqrt{2N}z_{k})-Y_{21}(\sqrt{2N}z_{k})'Y_{12}(\sqrt{2N}z_{k})\right) \label{ysum}\\
&=\sum_{k=1}^{m}\alpha_{k}\,\left(\mathop{\mathrm{Re}}(C(z_{m},z_{k}))-\mathop{\mathrm{Re}}(C(z_{j},z_{k}))\right)+O(d_{N}^{-1})+O(e_{N})\label{ysum2}\\
&=\sum_{k=1}^{m-1}\alpha_{k}\left(\phi^{(\eta)}_{0}(\tau_{k})+\phi^{(\eta)}_{0}(\tau_{j})-\phi^{(\eta)}_{0}(\tau_{k}-\tau_{j})\right)+O(d_{N}^{-1})+O(e_{N}) \label{ysum3}
\end{align}
To obtain \eqref{ysum3} from \eqref{ysum2}, we used the formula \eqref{szasy2} to compute the asymptotics of $\mathop{\mathrm{Re}}(C(z_{j},z_{k}))$ and used that $\alpha_{m}=-(\alpha_{1}+\ldots+\alpha_{m-1})$.\\
Now let us compute the asymptotics of the coefficients $\beta_{N}$, $\gamma_{N}$ and $\chi_{N-1}$ defined in \eqref{coeffid} and appearing in the first line of \eqref{diffid}. As usual, these quantities are all obtained by expanding all $z$-dependent quantities appearing in \eqref{yas} in powers of $1/z$. Firstly, the Szeg\"o function \eqref{sz} satisfies $\mathcal{D}(z) = \mathcal{D}_{\infty}(1+\mathcal{D}_{1}/z+(\mathcal{D}_{1}^{2}/2+\mathcal{D}_{2})/z^{2}+O(z^{-3}))$ as $z \to \infty$, where
\begin{equation}
\begin{split}
\mathcal{D}_{1} &= -\frac{1}{2}\sum_{k=1}^{m}\alpha_{k}\mathop{\mathrm{Re}}\left(\frac{1}{z_{k}+\sqrt{z_{k}+1}\sqrt{z_{k}-1}}\right),\\
\mathcal{D}_{2} &= -\frac{1}{8}\sum_{k=1}^{m}\alpha_{k}\mathop{\mathrm{Re}}\left(\frac{1}{(z_{k}+\sqrt{z_{k}+1}\sqrt{z_{k}-1})^{2}}\right), \label{d2}
\end{split}
\end{equation}
and secondly, use of the definitions \eqref{pinf} and \eqref{gfun} shows that for $z \to \infty$
\begin{equation}
g(z) = \log(z)-\frac{1}{8z^{2}}+O(z^{-4}), \qquad a(z)=1-\frac{1}{2z}+\frac{1}{8z^{2}}+O(z^{-3})
\end{equation}
Then expanding \eqref{zkas} at $z=\infty$, we can compare with \eqref{coeffid} and obtain
\begin{align*}
\beta_{N}&=\sqrt{2N}\left(-\mathcal{D}_{1}+\frac{A_{11}}{N}+O\left(\frac{1}{Nd_{N}}\right)+O(e_{N})\right)\\
\gamma_{N}&= 2N\left(1/8-N/8+\mathcal{D}_{1}^{2}/2-\mathcal{D}_{2}+\frac{B_{11}-A_{11}\mathcal{D}_{1}-iA_{12}/2}{N}+O\left(\frac{1}{Nd_{N}}\right)+O(e_{N})\right)\\
\chi_{N-1}^{2}&=\frac{2^{N-1}}{\sqrt{\pi}(N-1)!}\left(\frac{1}{\mathcal{D}_{\infty}^{2}}+\frac{1}{N}\left(\frac{1}{12\mathcal{D}_{\infty}^{2}}+2iA_{21}\right)+O\left(\frac{1}{Nd_{N}}\right)+O(e_{N})\right)
\end{align*}
A similar computation shows that the asymptotics of $\chi^{2}_{N}$ are given by
\begin{equation}
\chi^{2}_{N} = \frac{2^{N}}{\sqrt{\pi}N!}\left(\frac{1}{\tilde{\mathcal{D}}^{2}_{\infty}}+\frac{1}{N}\left(\frac{1}{12\tilde{\mathcal{D}}^{2}_{\infty}}+2iA_{12}\right)+O\left(\frac{1}{Nd_{N}}\right)+O(e_{N})\right)
\end{equation}
where $\tilde{\mathcal{D}}_{\infty}$ denotes the quantity \eqref{dinf} with rescaled singularities $\tilde{z}_{k} = \sqrt{2N/(2N+2)}z_{k}$. This rescaling is necessary when estimating $\chi^{2}_{N}$, because without it one obtains asymptotics with respect to the weight $w(x)=\prod_{j}|x-\sqrt{2N+2}z_{k}|^{\alpha_{k}}$. Cumbersome though routine manipulations with the above asymptotics yield
\begin{equation}
\label{meanterm}
\begin{split}
-N(\log \chi_{N}\chi_{N-1})^{'}&=2N(C(z_{j},\infty)-C(z_{m},\infty))+O(d_{N}^{-1})+O(Ne_{N}),\\
2(\gamma_{N}'-\beta_{N}\beta_{N}')&=-4N\mathcal{D}_{2}'+O(d_{N}^{-1})+O(Ne_{N}),
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
(\log \chi_{N}\chi_{N-1})'Y_{11}(\sqrt{2N}z_{k})Y_{22}(\sqrt{2N}z_{k}) &= O(d_{N}^{-1})+O(e_{N}),\\
2\left(\frac{\chi_{N-1}}{\chi_{N}}\right)^{2}\left(\log\frac{\chi_{N-1}}{\chi_{N}}\right)^{'} &= O(d_{N}^{-1})+O(e_{N}), \label{smallterms}
\end{split}
\end{equation}
where we introduced
\begin{align}
C(\mu,\infty) &= \lim_{z \to \infty}C(\mu,z) = \frac{1}{2\pi}\int_{-1}^{1}\frac{\log|x-\mu|}{\sqrt{1-x^{2}}}\,dx \\
&= \frac{1}{2}\log|z+\sqrt{z+1}\sqrt{z-1}|-\frac{1}{2}\log(2)\label{phiinf}.
\end{align}
Using the explicit formulae \eqref{phiinf} and \eqref{d2}, we get
\begin{align}
2(C(z_{j},\infty)-C(z_{m},\infty))-4\mathcal{D}'_{2} = \mathop{\mathrm{Re}}(g(z_{j}))-\mathop{\mathrm{Re}}(g(z_{m}))
\end{align}
where we exploited the convenient identity (see \textit{e.g.} the derivation of Eq. 7.89 in \cite{De99})
\begin{equation}
\log|z+\sqrt{z+1}\sqrt{z-1}|+\frac{1}{2}\mathop{\mathrm{Re}}\left(\frac{1}{(z+\sqrt{z+1}\sqrt{z-1})^{2}}\right) = \mathop{\mathrm{Re}}(g(z)).
\end{equation}
Now inserting \eqref{meanterm}, \eqref{ysum3} and \eqref{smallterms} into \eqref{diffid}, we obtain
\begin{equation}
\label{logasy}
\begin{split}
&\frac{\partial}{\partial \alpha_{j}}\log\varphi_{N}(\alpha_{1},\ldots,\alpha_{m-1}) = N(\mathop{\mathrm{Re}}(g(z_{j}))-\mathop{\mathrm{Re}}(g(z_{m})))\\
&+\sum_{k=1}^{m-1}\alpha_{k}\left(\phi^{(\eta)}_{0}(\tau_{k})+\phi^{(\eta)}_{0}(\tau_{j})-\phi^{(\eta)}_{0}(\tau_{k}-\tau_{j})\right)+O(d_{N}^{-1})+O(Ne_{N}).
\end{split}
\end{equation}
Note that the error terms in \eqref{logasy} hold \textit{uniformly} in the parameters $(\alpha_{k})_{k=1}^{m-1}$ (see Remark \ref{re:uniformity}), so that we may integrate both sides of \eqref{logasy} according to the procedure discussed in Sect. $5$ of \cite{K07}, arriving at the asymptotics \eqref{charasymptotics}.
\end{proof}
\begin{proof}[Proof of Theorems \ref{th:maintheorem} and \ref{th:compactconv}]
Bearing in mind Remark \ref{re:uniformity}, we differentiate \eqref{charasymptotics} with respect to the parameters $(\alpha_{k})_{k=1}^{m-1}$ and evaluate near the origin, leading to
\begin{align}
\mathbb{E}\{ W^{(\eta)}_{N}(\tau)\} &= N(\mathop{\mathrm{Re}}(g(z_{k}))-\mathop{\mathrm{Re}}(g(z_{m})))+O(d_{N}^{-1}) + O(Ne_{N})\label{mean}\\
\mathrm{Cov}\{W^{(\eta)}_{N}(\tau), W^{(\eta)}_{N}(\upsilon)\} &= \phi^{(\eta)}_{0}(\tau)+\phi^{(\eta)}_{0}(\upsilon)-\phi^{(\eta)}_{0}(\tau-\upsilon)+O(d_{N}^{-1}) +O(Ne_{N}) \label{covar}
\end{align}
where the error terms are uniform in $\tau$ and $\upsilon$ varying in a compact subset of $\mathbb{R}$. Then defining the centered process $\tilde{W}^{(\eta)}_{N}(\tau) = W^{(\eta)}_{N}(\tau)-\mathbb{E}\{W^{(\eta)}_{N}(\tau)\}$ we immediately find from \eqref{mean} and \eqref{charasymptotics} that in the mesoscopic regime \eqref{M2}, we have
\begin{equation}
\lim_{N \to \infty}\mathbb{E}\left\{e^{i\sum_{k=1}^{m}s_{k}\tilde{W}^{(\eta)}_{N}(\tau_{k})}\right\} = \mathrm{exp}\left(-\frac{1}{2}\sum_{k=1}^{m}\sum_{j=1}^{m}s_{k}s_{j}\left(\phi_{0}(\tau_{k})+\phi_{0}(\tau_{j})-\phi_{0}(\tau_{k}-\tau_{j})\right)\right)
\end{equation}
where $(s_{k})_{k=1}^{m} \in \mathbb{R}^{m}$. Theorem \ref{th:maintheorem} follows immediately. To complete the proof of Theorem \ref{th:compactconv}, it suffices to note that the error terms in \eqref{covar} are uniform, so that the sequence $(\mathbb{E}\{(\tilde{W}_{N}(\tau))^{2}\})_{N=1}^{\infty}$ is uniformly bounded.
\end{proof}
\section{Convergence to white noise in the spectral representation}
\label{se:fouriercoeff}
The main achievement of the previous section was to prove that for any mesoscopic scales of the form \eqref{M2}, the process $\tilde{W}^{(\eta)}_{N}(\tau)$ converges in the sense of finite-dimensional distributions to the regularized fractional Brownian motion $B^{(\eta)}_{0}(\tau)$. We also proved Theorem \ref{th:compactconv} which extends this convergence to an appropriate function space.
In this section we will study $\tilde{W}^{(\eta)}_{N}(\tau)$ from a different point of view, namely by means of the Fourier coefficients $b_{N}(s)$ appearing in the spectral decomposition \eqref{detforident}. We remind the reader of the definition
\begin{equation}
b_{N}(s) = \frac{1}{\sqrt{s}}\mathrm{Tr}\left(e^{-isd_{N}(\mathcal{H}-x_{0}I)}\right), \qquad s>0. \label{bee2}
\end{equation}
A useful and interesting feature of the integral representations \eqref{detforident} and its $N \to \infty$ limit \eqref{intr3} is that they are suggestive of a corresponding limiting law satisfied by the coefficients $b_{N}(s)$. Namely, we expect that $b_{N}(s)$ should `converge' to the white noise measure $B_{c}(ds)/\sqrt{2}$. The precise mode of the convergence we consider is described in Theorem \ref{th:fourierconv} and it is our goal in this Section to prove this result.
By its very definition, the white noise measure $B_{c}(ds)$ cannot be understood in a pointwise sense and must be regularized by integrating against a test function. We will consider test functions $\xi \in C^{\infty}_{0}(\mathbb{R}_{+})$, i.e. $\xi$ is a smooth function with compact support on $\mathbb{R}_{+}$. Then we have the correspondence
\begin{equation}
c_{N}(\xi)=\int_{0}^{\infty}\xi(s)b_{N}(s)\,ds = \sum_{j=1}^{N}f(d_{N}(x_{j}-x_{0})) =: X_{N}(f) \label{cn2}
\end{equation}
where
\begin{equation}
\label{gxi}
f(x) = \int_{0}^{\infty}\frac{\xi(s)}{\sqrt{s}}e^{-isx}\,ds.
\end{equation}
By our assumptions on $\xi$, it follows that $f$ belongs to the Schwartz space of rapidly decaying smooth functions, i.e. $f \in S(\mathbb{R})$ where
\begin{equation}
S(\mathbb{R}) = \bigg\{f \in C^{\infty}(\mathbb{R}): \, \sup_{x \in \mathbb{R}}\bigg{|}x^{\gamma}\frac{d^{\beta}f(x)}{dx^{\beta}}\bigg{|}<\infty \quad \gamma,\beta=0,1,2,\ldots \bigg\}. \label{schwartz}
\end{equation}
In the following three subsections we will obtain results for the mean, variance and distribution of the random variable \eqref{cn2} as $N \to \infty$.
\subsection{Mean}
We begin by proving that centering is not required in Theorem \ref{th:fourierconv}.
\begin{proposition}
On any mesoscopic scales of the form $d_{N} = N^{\alpha}$ with any $\alpha \in (0,1)$, we have
\begin{equation}
\mathbb{E}\{c_{N}(\xi)\} = O(d_{N}^{-1}), \qquad N \to \infty.
\end{equation}
\end{proposition}
\begin{proof}
We write the expectation above as an integral over the normalized density of states $\rho_{N}(x)$,
\begin{equation}
\label{fullint}
\mathbb{E}\{c_{N}(\xi)\}= N\int_{-\infty}^{\infty}f(d_{N}(x-x_{0}))\rho_{N}(x)\,dx
\end{equation}
where
\begin{equation}
\rho_{N}(x) = \frac{1}{N}\mathbb{E}\left\{\sum_{j=1}^{N}\delta(x-x_{j})\right\}. \label{dos}
\end{equation}
Firstly, note that the tails of the integral \eqref{fullint} can be removed using the rapid decay of $f$. For any $\epsilon>0$, we have
\begin{equation}
\label{inst}
\mathbb{E}\{c_{N}(\xi)\}= N\int_{x_{0}-\epsilon}^{x_{0}+\epsilon}f(d_{N}(x-x_{0}))\rho_{N}(x)\,dx + O(Nd_{N}^{-\infty}),
\end{equation}
where here and elsewhere, the notation $O(Nd_{N}^{-\infty})$ refers to a quantity that is $O(Nd_{N}^{-\gamma})$ for any $\gamma>0$. Such a contribution tends to zero for the power law scales $d_{N} = N^{\alpha}$ with any $\alpha \in (0,1)$. Then for small enough $\epsilon$ we have the uniform estimate (see \cite{PS11}, Chapter $5.2$)
\begin{equation}
\label{denslim}
\rho_{N}(x) = \frac{2}{\pi}\sqrt{1-x^{2}}+O(N^{-1}), \qquad x \in (x_{0}-\epsilon,x_{0}+\epsilon)
\end{equation}
After inserting \eqref{denslim} into \eqref{inst} we find that
\begin{equation}
\label{climit}
\mathbb{E}\{c_{N}(\xi)\} = \frac{2N}{\pi}\int_{x_{0}-\epsilon}^{x_{0}+\epsilon}f(d_{N} (x-x_{0}))\sqrt{1-x^{2}}\,dx+E_{N}+O(Nd_{N}^{-\infty})
\end{equation}
where the error term $E_{N} = O(d_{N}^{-1})$, since
\begin{align}
|E_{N}| &\leq C\bigg{|}\int_{x_{0}-\epsilon}^{x_{0}+\epsilon}f(d_{N}(x-x_{0}))\,dx\bigg{|}\leq\frac{C}{d_{N}}\int_{-\infty}^{\infty}|f(x)|\,dx.
\end{align}
Similarly, we can replace the integration limits in \eqref{climit} with $\pm1$ using the Schwartz property of $f$. We have
\begin{equation}
\mathbb{E}\{c_{N}(\xi)\} = \frac{2N}{\pi}\int_{-1}^{1}f(d_{N}(x-x_{0}))\sqrt{1-x^{2}}\,dx + O(d_{N}^{-1}) \label{semiasympt}
\end{equation}
Next we substitute $f$ with the definition \eqref{gxi} and interchange the order of integration (justified by the rapid decay of $\xi(s)$) so that,
\begin{align}
\mathbb{E}\{c_{N}(\xi)\} &= \frac{2N}{\pi}\int_{0}^{\infty}\xi(s)s^{-1/2}e^{isd_{N}x_{0}}\int_{-1}^{1}e^{-isd_{N}x}\sqrt{1-x^{2}}\,dx\,ds+O(d_{N}^{-1}) \notag \\
&= 2N\int_{0}^{\infty}\xi(s)s^{-3/2}J_{1}(d_{N}s)e^{isd_{N}x_{0}}\,ds + O(d_{N}^{-1}) \label{bessel}
\end{align}
where $J_{1}(z)$ is the Bessel function of index $1$. To finish the proof, note that $J_{1}(d_{N}s)$ has an asymptotic expansion (for any fixed $\gamma\in \mathbb{N}$ and $s>0$) as $N \to \infty$,
\begin{align}
\sqrt{\frac{\pi}{2}}J_{1}(d_{N}s) &= \cos(d_{N}s-3\pi/4)\sum_{k=0}^{\gamma-1}\frac{C_{k}}{d_{N}^{2k+1/2}s^{2k+1/2}} \notag\\
&+\sin(d_{N}s-3\pi/4)\sum_{k=0}^{\gamma-1}\frac{D_{k}}{d_{N}^{2k+3/2}s^{2k+3/2}}+E_{N}(s) \label{sumerrors}
\end{align}
where the error term satisfies the bound $|E_{N}(s)| \leq |C_{\gamma}d_{N}^{-2\gamma-1/2}s^{-2\gamma-1/2}|$ and $C_{k},D_{k}$ are constants depending only on $k$. Such asymptotics can be found in e.g. \cite{NIST} or \cite{K11}.
Inserting \eqref{sumerrors} into \eqref{bessel} we see that the contribution from each term in the sum in \eqref{sumerrors} is an oscillatory integral of order $O(Nd_{N}^{-\infty})$, as follows from repeated integration by parts. The final error term $E_{N}(s)$ is integrable with respect to $\xi(s)$ and gives rise to an error of order $O(Nd_{N}^{-2\gamma})$. Since $\gamma>0$ was arbitrary, we conclude that the term proportional to $N$ in \eqref{semiasympt} is in fact asymptotically smaller than the error term. This completes the proof of the proposition.
\end{proof}
\subsection{Covariance}
\label{se:cov}
Having studied the expectation of $b_{N}(s)$ in the previous subsection, we now consider the fluctuations. In the introduction it was remarked, in accordance with the expected white noise limit for $b_{N}(s)$, that we should have $\lim_{N \to \infty}\mathbb{E}\{b_{N}(s_{1})\overline{b_{N}(s_{2})}\} = \delta(s_{1}-s_{2})$. In this subsection we will make this assertion precise by proving that
\begin{equation}
\label{deltacors}
\lim_{N \to \infty}\mathbb{E}\{c_{N}(\xi_{1})\overline{c_{N}(\xi_{2})}\} = \int_{0}^{\infty}\xi_{1}(s)\overline{\xi_{2}(s)}\,ds
\end{equation}
for all smooth functions $\xi_{1},\xi_{2}$ with compact support on $\mathbb{R}_{+}$.
It turns out that there is an exact finite-$N$ formula for the covariance (see Eq. (4.2.38) in \cite{PS11}):
\begin{equation}
\mathbb{E}\{\tilde{X}_{N}(f_{1})\tilde{X}_{N}(f_{2})\}=\frac{1}{8}\int_{\mathbb{R}^{2}}\Delta f_{1}(d_{N}x)\Delta f_{2}(d_{N}x)K_{N}^{2}(x_{1},x_{2})\,dx_{1}\,dx_{2} \label{orig}
\end{equation}
where $f_{1}$ and $f_{2}$ are defined in terms of $\xi_{1}$ and $\xi_{2}$ as in formula \eqref{gxi} and we introduced the notation $\Delta f(x) = f(x_{1})-f(x_{2})$ for any $f$. The function $K_{N}(x_{1},x_{2})$ is the kernel of the GUE ensemble (see \textit{e.g.} \cite{Meh04}, \cite{PS11}) having the explicit formula
\begin{equation}
K_{N}(x,y) = \frac{\psi^{(N)}_{N}(x_{1})\psi^{(N)}_{N-1}(x_{2})-\psi^{(N)}_{N}(x_{2})\psi^{(N)}_{N-1}(x_{1})}{x_{1}-x_{2}} \label{guekernel}
\end{equation}
where
\begin{equation}
\label{wavefns}
\psi^{(N)}_{l}(x) = e^{-Nx^{2}}P^{(N)}_{l}(x),
\end{equation}
and $P^{(N)}_{l}(x)$ are (rescaled) Hermite polynomials, normalized by the condition that $\{\psi^{(N)}_{l}\}_{l=1}^{\infty}$ forms an orthonormal family on $\mathbb{R}$. By making use of the known Plancherel-Rotach asymptotics for the functions $\psi^{(N)}_{l}(x)$, we deduce the following covariance formula. After noting the correspondence \eqref{gxi}, we immediately derive from it the $\delta$-correlations \eqref{deltacors}.
\begin{proposition}
Let the test functions $f_{1}$ and $f_{2}$ belong to the Schwartz space $S(\mathbb{R})$ defined in \eqref{schwartz} and consider the mesoscopic regime $d_{N} = N^{\alpha}$ with any $\alpha \in (0,1)$. We have
\begin{equation}
\lim_{N \to \infty}\mathbb{E}\big\{\tilde{X}_{N}(f_{1})\tilde{X}_{N}(f_{2})\big\} = \frac{1}{2\pi}\int_{-\infty}^{\infty}|s|\hat{f_{1}}(s)\hat{f_{2}}(-s)\,ds. \label{eqnfour}
\end{equation}
where $\hat{f}(s) = (2\pi)^{-1/2}\int_{-\infty}^{\infty}f(x)e^{-isx}\,dx$.
\end{proposition}
\begin{remark}
Formula \eqref{eqnfour} is already known for $C^{1}$ functions with compact support, as in Theorem 5.2.7 (iii) of \cite{PS11}. It was also proved recently in \cite{EK13a} for a class of Wigner matrices with $f$ a Schwartz test function, but only up to scales $d_{N} = N^{\alpha}$ with any $0<\alpha<1/3$. Our main contribution in this subsection is to adapt the argument given in \cite{PS11} to our test functions $f$ in \eqref{gxi}, which cannot be compactly supported due to our assumptions on $\xi$. We note that our proof holds on the full range $0 < \alpha < 1$ and that the smoothness hypothesis can be relaxed to $C^{1}$ functions with rapid decay at $\pm \infty$.
\end{remark}
\begin{proof}
Here we only consider the contribution to integral \eqref{orig} coming from the square $I_{\delta}^{2} = [-(1-\delta),(1-\delta)]^{2}$ for some small $\delta>0$. In Appendix \ref{ap:intcomp} we will show that the complement of this region can be neglected for small enough $\delta$. We will need the following asymptotic formula for the functions $\psi^{(N)}_{N+k}$ defined in \eqref{wavefns}. Uniformly for $|x|<(1-\delta)$ and $k=O(1)$, we have
\begin{equation}
\label{planch2}
\psi^{(N)}_{N+k}(x) = \left(\frac{2}{\pi\sqrt{1-x^{2}}}\right)^{1/2}\cos(N\alpha(x)+(k+1/2)\cos^{-1}(x)-\pi/4)+O(N^{-1})
\end{equation}
where $\alpha(x) = 2\int_{-1}^{x}dt\, \sqrt{1-t^{2}}$. Formula \eqref{planch2} follows immediately from the classical asymptotic results of Plancherel and Rotach (see Sections $5$ in \cite{PS11} and $8$ in \cite{Sze39}).
Now, using the symmetry about the line $x_{1}=x_{2}$, we see that the integral \eqref{orig} restricted to $I_{\delta}^{2}$ can be written in the convenient form,
\begin{equation}
\label{intsquare}
\frac{1}{4}\int_{I_{\delta}^{2}}\frac{\Delta f_{1}(d_{N}x)}{\Delta x}\frac{\Delta f_{2}(d_{N}x)}{\Delta x}\mathcal{F}_{N}(x_{1},x_{2})\,dx_{1}\,dx_{2}
\end{equation}
where
\begin{equation}
\mathcal{F}_{N}(x_{1},x_{2}) = \psi^{(N)}_{N}(x_{1})^{2}\psi^{(N)}_{N-1}(x_{2})^{2}-\psi^{(N)}_{N}(x_{1})\psi^{(N)}_{N-1}(x_{1})\psi^{(N)}_{N}(x_{2})\psi^{(N)}_{N-1}(x_{2}). \label{cfn}
\end{equation}
We insert the Plancherel-Rotach formula \eqref{planch2} into \eqref{intsquare} and denote $\theta(x) = \cos^{-1}(x)$. Using the double angle formula for the cosine, we find that the contribution of $\eqref{planch2}$ to the product of squares in \eqref{cfn} is
\begin{align}
&\frac{1+\cos(2N\alpha(x_{1})+\theta(x_{1})/2-\pi/4)+\cos(2N\alpha(x_{2})-\theta(x_{2})/2-\pi/4)}{\pi^{2}\sqrt{1-x_{2}^{2}}\sqrt{1-x_{1}^{2}}} \label{costerms1}\\
&+\frac{\cos(2N\alpha(x_{1})+\theta(x_{1})/2-\pi/4)\cos(2N\alpha(x_{2})-\theta(x_{2})/2-\pi/4)}{\pi^{2}\sqrt{1-x_{2}^{2}}\sqrt{1-x_{1}^{2}}}+O(N^{-1}). \label{costerms2}
\end{align}
Inserting the oscillatory terms in lines \eqref{costerms1} and \eqref{costerms2} into \eqref{intsquare} gives rise to error terms that are $O((N/d_{N})^{-\infty})$ as $N \to \infty$ for every $\delta>0$. This can be shown by repeated integration by parts, using the fact that $\alpha(x)$ is smooth and increasing on the interval $I_{\delta}$. Combined with a similar calculation applied to the second term in \eqref{cfn}, we see that the integral \eqref{intsquare} is equal to
\begin{align}
&\frac{1}{4\pi^{2}}\int_{I_{\delta}^{2}} \frac{\Delta f_{1}(d_{N}x)}{\Delta x}\frac{\Delta f_{2}(d_{N}x)}{\Delta x}\frac{1-x_{1}x_{2}}{\sqrt{1-x_{1}^{2}}\sqrt{1-x_{2}^{2}}}\,dx_{1}\,dx_{2}+O((N/d_{N})^{-\infty})=\notag\\
&\frac{1}{4\pi^{2}}\int_{\mathbb{R}^{2}}\frac{\Delta f_{1}(x)}{\Delta x}\frac{\Delta f_{2}(x)}{\Delta x}\frac{1-x_{1}x_{2}/d_{N}^{2}}{\sqrt{1-x_{1}^{2}/d_{N}^{2}}\sqrt{1-x_{2}^{2}/d_{N}^{2}}}\chi_{I_{N}}(x_{1})\chi_{I_{N}}(x_{2})\,dx_{1}\,dx_{2}+O((N/d_{N})^{-\infty}) \label{underint}
\end{align}
where $\chi_{I_{N}}(x_{1})$ is the indicator function on the set $I_{N}=(-(1-\delta)d_{N},(1-\delta)d_{N})$.
Now Lebesgue's dominated convergence theorem can be applied to take the limit under the integral in \eqref{underint}. Indeed, it is easy to see that the integrand in \eqref{underint} is bounded by the integrable function
\begin{equation}
\left(\frac{2}{\delta^{2}}-1\right)\bigg{|}\frac{\Delta f_{1}(x)}{\Delta x}\bigg{|}\bigg{|}\frac{\Delta f_{2}(x)}{\Delta x}\bigg{|}
\end{equation}
for any $N \in \mathbb{N}$, $(x_{1},x_{2})\in\mathbb{R}^{2}$ and $0<\delta<1$. We finally see that for all $0<\delta<1$, we have
\begin{equation}
\lim_{N \to \infty}\frac{1}{4}\int_{I_{\delta}^{2}}\frac{\Delta f_{1}(d_{N} x)}{\Delta x}\frac{\Delta f_{2}(d_{N} x)}{\Delta x}\mathcal{F}_{N}(x_{1},x_{2})\,dx_{1}\,dx_{2} = \frac{1}{4\pi^{2}}\int_{\mathbb{R}^{2}}\frac{\Delta f_{1}(x)}{\Delta x}\frac{\Delta f_{2}(x)}{\Delta x}\,dx_{1}\,dx_{2}. \label{prefour}
\end{equation}
Rewriting $f_{1}$ and $f_{2}$ in terms of their Fourier transforms and applying the Plancherel theorem gives the identity
\begin{equation}
\frac{1}{4\pi^{2}}\int_{\mathbb{R}^{2}}\frac{f_{1}(x_{1})-f_{1}(x_{2})}{x_{1}-x_{2}}\frac{f_{2}(x_{1})-f_{2}(x_{2})}{x_{1}-x_{2}}\,dx_{1}\,dx_{2} = \frac{1}{2\pi}\int_{\mathbb{R}}|s|\hat{f_{1}}(s)\hat{f_{2}}(-s)\,ds,
\end{equation}
which is precisely the right-hand side of \eqref{eqnfour}. To complete the proof, we just need to show that the integral \eqref{orig} restricted to the complement of the square $I_{\delta}^{2}$ can be neglected in the limit $N \to \infty$. Namely, we prove in the Appendix that
\begin{equation}
\label{eqn432}
\lim_{N \to \infty}\int_{(I_{\delta}^{2})^{\mathrm{c}}}\Delta f_{1}(d_{N}x)\Delta f_{2}(d_{N}x)K_{N}^{2}(x_{1},x_{2})\,dx_{1}\,dx_{2}=O(\delta), \qquad \delta \to 0,
\end{equation}
and so complete the proof of the Proposition by choosing $\delta>0$ sufficiently small.
\end{proof}
\subsection{Convergence in distribution}
\label{se:fulldist}
The aim of this subsection is to study the full distribution of the coefficients $b_{N}(s)$ and ultimately to prove Theorem \ref{th:fourierconv}. First we need a preliminary result regarding the stochastic process $\tilde{W}^{(\eta)}_{N}(\tau)$. It will be convenient to consider the \textit{increments}
\begin{equation}
\begin{split}
\Delta_{p}[\tilde{W}_{N}^{(\eta)}](\tau) :&= \tilde{W}_{N}^{(\eta)}(\tau)-\tilde{W}_{N}^{(\eta)}(\tau+p)\\
&= \frac{1}{2}\int_{0}^{\infty}\frac{e^{-\eta s}}{\sqrt{s}}\left\{[1-e^{-ips}]e^{-i\tau s}\tilde{b}_{N}(s)+[1-e^{ips}]e^{i\tau s}\overline{\tilde{b}_{N}(s)}\right\}\,ds, \label{detforident2}
\end{split}
\end{equation}
where $\tilde{b}_{N}(s) = b_{N}(s)-\mathbb{E}\{b_{N}(s)\}$.
Similarly, the corresponding limiting object is given by the following stationary Gaussian process
\begin{equation}
\begin{split}
\Delta_{p}[B^{(\eta)}_{0}](\tau) :&= B^{(\eta)}_{0}(\tau)-B^{(\eta)}_{0}(\tau+p)\\
&=\frac{1}{2\sqrt{2}}\int_{0}^{\infty}\frac{e^{-\eta s}}{\sqrt{s}}\left\{[1-e^{-ips}]e^{-i\tau s}B_{c}(ds)+[1-e^{ips}]e^{i\tau s}\overline{B_{c}(ds)}\right\}. \label{b0inc}
\end{split}
\end{equation}
\begin{proposition}
\label{prop:tailsh}
Let $p \in \mathbb{R}$. For any $h \in S(\mathbb{R})$ and on any power law scales $d_{N} = N^{\alpha}$ with $\alpha \in (0,1)$, we have the convergence in distribution
\begin{equation}
\int_{-\infty}^{\infty}h(\tau)\Delta_{p}[\tilde{W}^{(\eta)}_{N}](\tau)\,d\tau \overset{d}{\Longrightarrow} \int_{-\infty}^{\infty}h(\tau)\Delta_{p}[B^{(\eta)}_{0}](\tau)\,d\tau, \qquad N \to \infty. \label{convhschwartz}
\end{equation}
\end{proposition}
\begin{proof}
The proof will be analogous to our proof of Theorem \ref{th:compactconv}, the main difference being we must have good enough control of the tails in the above integrals. This will be taken care of by the rapid decay of $h$. To proceed, we fix some (arbitrary) $M \in \mathbb{R}$ and $\delta_{0}>0$ and decompose the left-hand side of \eqref{convhschwartz} as
\begin{equation}
\begin{split}
&\int_{-M}^{M}h(\tau)\Delta_{p}[\tilde{W}^{(\eta)}_{N}](\tau)\,d\tau\\ &+\int_{|\tau|\in[M,\delta_{0}d_{N}]}h(\tau)\Delta_{p}[\tilde{W}^{(\eta)}_{N}](\tau)\,d\tau+\int_{|\tau| \in [\delta_{0}d_{N},\infty)}h(\tau)\Delta_{p}[\tilde{W}^{(\eta)}_{N}](\tau)\,d\tau \label{decomp}
\end{split}
\end{equation}
and label each of the integrals in \eqref{decomp} with $\mathcal{I}_{1}, \mathcal{I}_{2}$ and $\mathcal{I}_{3}$. Let us begin with the first integral, $\mathcal{I}_{1}$. By Theorem \ref{th:maintheorem} and the Cram\'er-Wold device, the finite-dimensional distributions of $\Delta_{p}[\tilde{W}^{(\eta)}_{N}](\tau)$ converge in law to those of $\Delta_{p}[B^{(\eta)}_{0}](\tau)$. Furthermore, by the uniform estimate \eqref{covar} we have that there is a constant $C>0$ such that $\mathbb{E}\{(\Delta_{p}[B^{(\eta)}_{0}(\tau)])^{2}\}\leq C$ for all $\tau \in [-M,M]$ and for all $N$. Therefore the hypotheses of Theorem 3 in \cite{G76} are satisfied and we conclude that the first integral in \eqref{decomp} converges in distribution to the right-hand side of \eqref{convhschwartz} in the limit $N \to \infty$ followed by $M \to \infty$. To complete the proof, it suffices to show that the second and third integrals in \eqref{decomp} converge in probability to $0$ in the same limit.
For notational convenience we just consider the contributions to $\mathcal{I}_{2}$ and $\mathcal{I}_{3}$ where $\tau>0$ as the situation $\tau<0$ is almost identical. By Chebyshev's inequality and Cauchy-Schwarz, we have
\begin{equation}
\mathbb{P}\{|\mathcal{I}_{2}|>\epsilon\} \leq \epsilon^{-2}\int_{M}^{\delta_{0}d_{N}}|h(\tau)|d\tau\int_{M}^{\delta_{0}d_{N}}|h(\tau)|\,\mathbb{E}\{\Delta_{p}[\tilde{W}^{(\eta)}_{N}](\tau)^{2}\} \, d\tau \label{varrhp}
\end{equation}
We will now argue that the variance term in \eqref{varrhp} is uniformly bounded. Since $|\tau|\leq \delta_{0}d_{N}$, by choosing $\delta_{0}$ small enough we see that $|x_{0}+\tau/d_{N}| < 1-\delta$ for some $\delta>0$ independent of $N$. Hence the singularities of the logarithm in \eqref{wntau} remain inside the bulk region $(-1+\delta,1-\delta)$ for all $N$ and we may apply the methods of Section $3$ with $m=2$ and weight (cf. \eqref{omdef})
\begin{equation}
\omega(z)=\left[\frac{(z-x_{0}(\tau,N)-p/d_{N})^{2}+(\eta/d_{N})^{2}}{(z-x_{0}(\tau,N))^{2}+(\eta/d_{N})^{2}}\right]^{\alpha/2}, \qquad x_{0}(\tau,N)=x_{0}+\tau/d_{N}.
\end{equation}
The only difference in the analysis of the Riemann-Hilbert problem with this weight is that the new reference point $x_{0}(\tau,N)$ can vary with $N$ in the small fixed neighbourhood $[x_{0}-\delta_{0},x_{0}+\delta_{0}]$. However, all the estimates we obtain are uniform for $x_{0}$ varying in compact subsets of $(-1+\delta,1-\delta)$ so that the variance bound \eqref{covar} (with $\upsilon = \tau$) remains valid. This implies that for some $N$-indepedent $C>0$,
\begin{equation}
\mathbb{P}\{|\mathcal{I}_{2}|>\epsilon\} \leq \epsilon^{-2}C\left(\int_{M}^{\delta_{0}d_{N}}|h(\tau)|\,d\tau\right)^{2} \to 0,
\end{equation}
in the limit $N \to \infty$ followed by $M \to \infty$.
To bound the integral $\mathcal{I}_{3}$ we again apply Chebyshev's inequality and exploit the rapid decay of $h$. We have
\begin{align}
&\mathbb{P}\{|\mathcal{I}_{3}|>\epsilon\} \leq \epsilon^{-2}\int_{\delta_{0}d_{N}}^{\infty}\int_{\delta_{0} d_{N}}^{\infty}\mathbb{E}
\{
h(\tau_{1})\Delta_{p}[\tilde{W}^{(\eta)}_{N}](\tau_{1})\overline{h(\tau_{2})}\Delta_{p}[\tilde{W}^{(\eta)}_{N}](\tau_{2})\}\,d\tau_{1}\,d\tau_{2}\\
&=\epsilon^{-2}\int_{\delta_{0}d_{N}}^{\infty}\int_{\delta_{0}d_{N}}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}h(\tau_{1})\overline{h(\tau_{2})}\prod_{j=1}^{2}(q(x_{1},\tau_{j})-q(x_{2},\tau_{j}))\,K_{N}^{2}(x_{1},x_{2})\,dx_{1}dx_{2}d\tau_{1}d\tau_{2}
\end{align}
where we computed the expectation using the identity \eqref{orig} and
\begin{equation}
q(x,\tau) = -\log\bigg{|}x-x_{0}-\frac{\tau+i\eta}{d_{N}}\bigg{|}+\log\bigg{|}x-x_{0}-\frac{\tau+p+i\eta}{d_{N}}\bigg{|}.
\end{equation}
Now, since $h$ is a Schwartz test function, we know that for any $\gamma>0$ and $u>0$, we have $|h(ud_{N})| \leq (d_{N}u)^{-\gamma}$ for $N$ large enough. Then using the inequalities $|q(x,\tau)|\leq C_{p,\eta}$ for some finite constant depending only on $p$ and $\eta$, $K_{N}^{2}(x_{1},x_{2}) \leq N^{2}\rho_{N}(x_{1})\rho_{N}(x_{2})$ and substituting $\tau_{j}=ud_{N}$ we obtain
\begin{equation}
\label{tailineq}
\mathbb{P}(|\mathcal{I}_{3}|>\epsilon) \leq 4\epsilon^{-2} C_{p,\eta}^{2}N^{2}d_{N}^{-2\gamma+2}\left(\int_{\delta_{0}}^{\infty}u^{-\gamma}du\right)^{2}.
\end{equation}
Then provided $d_{N}$ takes the form $d_{N} = N^{\alpha}$ with $\alpha \in (0,1)$ we can always choose $\gamma>0$ large enough such that the right-hand side of \eqref{tailineq} tends to $0$ as $N \to \infty$.
\end{proof}
We can now translate the result \eqref{convhschwartz} into a statement about the Fourier coefficients $b_{N}(s)$, allowing us to prove Theorem \ref{th:fourierconv}. For the convenience of the reader, we repeat the statement of the latter result here.
\begin{theorem}
\label{prop:convcomp}Let $\xi_{1},\ldots,\xi_{m}$ be smooth functions compactly supported on $\mathbb{R}_{+}$. Then the vector $(c_{N}(\xi_{1}),\ldots,c_{N}(\xi_{m}))$ converges in distribution to a centered complex Gaussian vector $Z$ with relation matrix $C=\mathbb{E}\{ZZ^{\mathrm{T}}\}=0$ and covariance matrix $\Gamma = \mathbb{E}\{ZZ^{\dagger}\}$ given by
\begin{equation}
\Gamma_{j,k} = \int_{0}^{\infty}\xi_{j}(s)\overline{\xi_{k}(s)}\,ds, \qquad j,k=1,\ldots,m.
\end{equation}
\end{theorem}
\begin{proof}
Define functions $h_{k}$ in terms of their Fourier transform as
\begin{equation}
\label{hdef2}
\int_{-\infty}^{\infty}h_{k}(\tau)e^{-i\tau s}\,d\tau = \frac{\sqrt{s}}{1-e^{-ips}}e^{\eta s}\xi_{k}(s) \qquad k=1,\ldots,m.
\end{equation}
Then for sufficiently small $p$, the right-hand side of \eqref{hdef2} is smooth and compactly supported. Therefore, its Fourier transform $h_{k}$ is a Schwartz function, i.e. $h_{k} \in S(\mathbb{R})$. Next, note that with $c_{N}(\xi)$ as in \eqref{cn2}, we have the identity
\begin{equation}
c_{N}(\xi_{k})-\mathbb{E}(c_{N}(\xi_{k})) = 2\int_{-\infty}^{\infty}h_{k}(\tau)\Delta_{p}[\tilde{W}^{(\eta)}_{N}](\tau)\,d\tau
\end{equation}
which holds almost surely and follows after inserting the representation \eqref{detforident2} and interchanging the order of integration, justified by the rapid decay of $\xi_{k}$ and $h_{k}$. Now we apply Proposition \ref{prop:tailsh} with $h(\tau)=\sum_{k=1}^{m}\alpha_{k}h_{k}(\tau)$ where $\alpha_{k} \in \mathbb{C}$. Since $\mathbb{E}(c_{N}(\xi_{k}))=O(d_{N}^{-1})$, we get the convergence in distribution
\begin{equation}
\sum_{k=1}^{m}\alpha_{k}c_{N}(\xi_{k}) \overset{d}{\Longrightarrow} 2\sum_{k=1}^{m}\alpha_{k}\int_{-\infty}^{\infty}h_{k}(\tau)\Delta_{p}[B^{(\eta)}_{0}](\tau)\,d\tau, \qquad N \to \infty. \label{limitcm}
\end{equation}
By the Cram\'er-Wold device, this implies the convergence in distribution
\begin{equation}
(c_{N}(\xi_{1}),\ldots,c_{N}(\xi_{k})) \overset{d}{\Longrightarrow} (Z(h_{1}),\ldots,Z(h_{m}))
\end{equation}
where
\begin{equation}
Z(h_{k}) = 2\int_{-\infty}^{\infty}h_{k}(\tau)\Delta_{p}[B^{(\eta)}_{0}](\tau)\,d\tau.
\end{equation}
Since $\Delta_{p}[B^{(\eta)}_{0}](\tau)$ is a Gaussian process, one easily sees that $(Z(h_{1}),\ldots,Z(h_{m}))$ is a mean zero complex Gaussian vector. Then by a simple computation using the integral representation \eqref{b0inc} and basic properties of the white noise measure $B_{c}(ds)$, we find the covariance structure
\begin{equation}
\Gamma_{j,k}=\mathbb{E}\{Z(h_{j})\overline{Z(h_{k})}\} = \int_{0}^{\infty}\xi_{j}(s)\overline{\xi_{k}(s)}\,ds,
\end{equation}
and $C_{j,k}=\mathbb{E}\{Z(h_{j})Z(h_{k})\}=0$ for all $j,k=1,\ldots,m$.
\end{proof}
\section{Macroscopic regime}
\label{se:weakconv}
The main goal of this section is to prove Theorem \ref{th:global}. Namely, we will show that the process $\tilde{D}_{N}(x)$ (\ref{logdet}) converges in probability law as $N \to \infty$ to the generalized Gaussian process $F(x)$ given by (\ref{1/fch}). The convergence is interpreted in the Sobolev space $V^{(-a)}$, \textit{i.e.} the assertion of Theorem \ref{th:global} is that for any bounded continuous functional $q$ on $V^{(-a)}$, we have
\begin{equation} \label{eq:weakconveq}
\lim_{N\to\infty} \mathbb{E}\{ q(\tilde{D}_{N} )\}= \mathbb{E} \{ q(F) \}. \end{equation}
Our proof is an adaptation for the GUE matrices ${\cal H}$ of the proof of a similar result for the CUE matrices given in \cite{HKOC01}. First, we will prove that the finite-dimensional distributions of $\tilde D_N(x)$ converge to those of $F(x)$ and then establish that the sequence $\tilde D_N(x)$ is tight in $V^{(-a)}$. This will imply the convergence in probability law in $V^{(-a)}$ as in (\ref{eq:weakconveq}). As explained in section \ref{sec2.2}, for the GUE matrices there are additional analytical complications compared with the case of CUE matrices.
We start with a deterministic result, writing down the Chebyshev-Fourier series for $\tilde D_N(x)$.
\begin{lemma}
\label{le:chebycoeffs} Let $\cal H$ be a Hermitian matrix of size $N\times N$ with eigenvalues $x_1, \ldots, x_N$. Then
\[
-\log |\det(\mathcal{H}-xI)| = N\log 2+ \sum_{k=0}^{\infty} c_k (D_N) T_k(x)
\]
where the convergence is pointwise for any $x\in [-1,1]\backslash \{x_1, \ldots, x_N \}$ and the Chebyshev-Fourier coefficients $c_{k}({D}_{N})$ are given for any $k > 0$ by the formula
\begin{equation}
\label{insertcoeffs}
c_{k}(D_{N}) = \sum_{j=1}^{N}\frac{2}{k}T_{k}(x_j)+ \sum_{j=1}^{N} r^{+}_{k}(x_{j})+\sum_{j=1}^{N}r^{-}_{k}(x_{j})
\end{equation}
and
\begin{equation}
\label{insertcoeffs2}
c_{0}(D_{N}) = -\sum_{j=1}^{N}r^{+}_{0}(x_{j}) -\sum_{j=1}^{N}r^{-}_{0}(x_{j})\end{equation}
where for $k > 0$
\begin{equation}\label{testfns}
r^{\pm}_{k}(x) =\left[(2/k)(-T_{k}(x) + (x \mp \sqrt{x^{2}-1})^{k}\right]\chi_{(\pm 1, \pm \infty)}(x)
\end{equation}
and
\begin{equation}
r^{\pm}_{0}(x) = \log|x \mp \sqrt{x^{2}-1}| \chi_{(\pm 1, \pm \infty)}(x)
\end{equation}
In the above formulae, $\chi_{J}(x)$ is the indicator function on the set $J$.
\end{lemma}
\begin{proof}
This follows immediately from Lemma 3.1 in \cite{GP13}.
\end{proof}
It follows from this Lemma that for our random matrices $\cal H$, with probability one,
\[
\tilde{D}_{N}(x)= \sum_{k=0}^{\infty} c_k (\tilde D_N) T_k(x), \quad \mathrm{where} \quad c_k (\tilde D_N) =c_{k}({D}_{N})- \mathbb{E}\{ c_{k}({D}_{N})\}. \label{DNexpansion}
\]
\subsection{Convergence of finite-dimensional distributions}
The main goal of this subsection is to establish the following:
\begin{proposition}
\label{prop:findim}
Fix $M \in \mathbb{N}$ and let $X_{1},\ldots,X_{M}$ be independent Gaussian random variables with mean zero and variance one. Then for any
$(t_{k})_{k=1}^{M} \in \mathbb{R}^{M}$ we have the convergence in distribution
\begin{equation}
\sum_{k=0}^{M}c_{k}(\tilde{D}_{N})t_{k} \overset{d}{\Longrightarrow} \sum_{k=1}^{M}\frac{X_{k}}{\sqrt{k}}t_{k} \label{chebydist}, \qquad N \to \infty.
\end{equation}
\end{proposition}
\begin{proof}
We begin by inserting Eq. \eqref{insertcoeffs} into the left-hand side of \eqref{chebydist}. Then from \cite{Joh98} or \cite{PS11}, we know that the sum
\begin{equation}
\sum_{k=1}^{M}t_{k}\left(\sum_{j=1}^{N}\frac{2}{k}T_{k}(x_{j})-\mathbb{E}\left\{\sum_{j=1}^{N}\frac{2}{k}T_{k}(x_{j})\right\}\right)
\end{equation}
converges in distribution to the right-hand side of \eqref{chebydist} as $N \to \infty$. The main technical part of our proof of \eqref{chebydist} consists in showing that the other terms appearing in \eqref{insertcoeffs} and \eqref{insertcoeffs2} do not contribute in the limit $N \to \infty$. All such terms that appear are of the form
\begin{equation}
\label{linstatsing}
A^{\pm}_{k,N} = \sum_{j=1}^{N}r^{\pm}_{k}(x_{j})
\end{equation}
and by definition of the test function $r^{\pm}_{k}(x)$, they are non-zero only when an eigenvalue $x_{j}$ lies outside the bulk of the limiting spectrum $[-1,1]$. Intuitively this is a rare event and we show below that in fact $\mathbb{E}|A^{\pm}_{k,N}| \to 0$ as $N \to \infty$. We note in passing that the regularity of the test functions $r^{\pm}_{k}(x)$ lies outside the best known $C^{1/2+\epsilon}$ threshold in \cite{SW13}, due to the singularities at the spectral edges.
Let us focus our attention on the case $\mathbb{E}\{|A^{+}_{k,N}|\}$, since the estimation of $\mathbb{E}\{|A^{-}_{k,N}|\}$ follows exactly the same pattern. First, one sees from the explicit formula \eqref{testfns} and the elementary inequality $(x-\sqrt{x^{2}-1})^{k} \leq T_{k}(x) \leq (x+\sqrt{x^{2}-1})^{k}$, $x\geq1$ that $-r^{+}_{k}(x)$ is non-negative for all $x \in \mathbb{R}$. Therefore $\mathbb{E}\{|A^{+}_{k,N}|\} = -\mathbb{E}\{A^{+}_{k,N}\}$.
In terms of the normalized eigenvalue density, we have
\begin{equation}
\mathbb{E}\{A^{+}_{k,N}\} = N\int_{1}^{\infty}r^{+}_{k}(x)\rho_{N}(x)dx.
\end{equation}
To proceed, we split the integral as
\begin{equation}
\label{edgeandouter}
\mathbb{E}\{A^{+}_{k,N}\} = N\int_{1}^{1+\delta_{N}}r^{+}_{k}(x)\rho_{N}(x)\,dx+N\int_{1+\delta_{N}}^{\infty}r^{+}_{k}(x)\rho_{N}(x)\,dx
\end{equation}
where we choose $\delta_{N}=N^{-7/12}$. The first integral in \eqref{edgeandouter} is over a shrinking neighbourhood of the spectral edge $x=1$. An estimate that holds uniformly in this region can be given in terms of the Airy function $\mathrm{Ai}(x)$ and its derivatives. In particular, Eq. 4.4 of \cite{EM03} (see also the \textit{Proof of Lemma 2.2} in \cite{G05}) shows that as $N \to \infty$
\begin{equation}
\begin{split}
N\rho_{N}(x) &= \left(\frac{\Phi'(x)}{4\Phi(x)}-\frac{\gamma'(x)}{\gamma(x)}\right)[2\mathrm{Ai}(N^{2/3}\Phi(x))\mathrm{Ai}'(N^{2/3}\Phi(x))]\\
&+N^{2/3}\Phi'(x)[(\mathrm{Ai}'(N^{2/3}\Phi(x)))^{2}-N^{2/3}\Phi(x)(\mathrm{Ai}(N^{2/3}\Phi(x)))^{2}]+O\left(\frac{1}{N(\sqrt{x-1})}\right) \label{airyasy}
\end{split}
\end{equation}
where
\begin{equation}
\gamma(x) = \left(\frac{x-1}{x+1}\right)^{1/4}
\end{equation}
and
\begin{equation}
\Phi(x) = \begin{cases}
&-\left(3\int_{x}^{1}\sqrt{1-y^{2}}\,dy\right)^{2/3}, \qquad |x|\leq 1\\
&\left(3\int_{1}^{x}\sqrt{y^{2}-1}\,dy\right)^{2/3}, \hspace{11pt}\qquad |x|>1
\end{cases}
\end{equation}
Since $\Phi(x)\geq0$ for $x\geq1$, the functions $\mathrm{Ai}(N^{2/3}\Phi(x))$ and $\mathrm{Ai}'(N^{2/3}\Phi(x))$ are uniformly bounded on $[1,\infty)$. Furthermore, $\left(\frac{\Phi'(x)}{4\Phi(x)}-\frac{\gamma'(x)}{\gamma(x)}\right)$ and $\Phi'(x)$ are bounded near $x=1$. Inserting \eqref{airyasy} into the first integral in \eqref{edgeandouter}, we obtain the bound
\begin{equation}
\label{twoints}
N\int_{1}^{1+\delta_{N}}r^{+}_{k}(x)\rho_{N}(x)\,dx = c_{1}N^{2/3}\int_{1}^{1+\delta_{N}}r^{+}_{k}(x)\,dx+O\left(\frac{1}{N}\right),
\end{equation}
where $c_{1}$ is an $N$-independent constant. In \eqref{twoints} we used that $r^{+}_{k}(x)(x-1)^{-1/2}$ is bounded near $x=1$ to estimate the contribution of the error term in \eqref{airyasy}. A simple computation shows that $\int_{1}^{1+\delta_{N}}r^{+}_{k}(x)\,dx=O(\delta_{N}^{3/2})$ as $N \to \infty$ for $k \geq 0$. Inserting the latter into \eqref{twoints} yields the bound
\begin{equation}
N\int_{1}^{1+\delta_{N}}r^{+}_{k}(x)\rho_{N}(x)\,dx = O(N^{2/3}\delta_{N}^{3/2}) = O(N^{-5/24}).
\end{equation}
Now consider the second integral in \eqref{edgeandouter}. We will prove below that it is exponentially small as $N \to \infty$. Using the fact that (for $k\geq 1$) $-r^{+}_{k}(x) \leq T_{k}(x)$ and applying Lemma \ref{le:expdecaylem}, we obtain
\begin{align}
&-N\int_{1+\delta_{N}}^{\infty}r^{+}_{k}(x)\rho_{N}(x)\,dx \label{cint1}\\
&\leq N\delta_{N}\int_{1}^{\infty}T_{k}(1+u\delta_{N})\rho_{N}(1+u\delta_{N})\,du\\
&\leq B^{-1}\int_{1}^{\infty}u^{-1}T_{k}(1+u\delta_{N})e^{-buN^{1/8}}\,du \label{cint}
\end{align}
where $B,b>0$ are absolute constants. Then \textit{e.g.} expanding $T_{k}(1+u\delta_{N})$ in powers of $(u\delta_{N})$ and integrating \eqref{cint} term by term, we can apply the standard Laplace method and find that \eqref{cint} is $O(e^{-cN^{1/8}})$ for some $c>0$. If $k=0$ in the integral \eqref{cint1}, one can use the inequality $|r^{+}_{0}(1+x)| \leq \sqrt{2x}$, $x>0$ and then apply the Laplace method as before yielding a similar error bound. This completes the proof of the Proposition.
\end{proof}
\subsection{Tightness}
The final ingredient required for proving the weak convergence in \eqref{eq:weakconveq} is to show that the sequence $\tilde{D}_{N}$ is tight in $V^{(-a)}$. In direct analogy to the proof given in Theorem 2.5 of \cite{HKOC01} for the Circular Unitary Ensemble, we will exploit the convenient fact that for $-\infty < a < b < \infty$, the closed unit ball in $V^{(b)}$ is compact in $V^{(a)}$. Then by Chebyshev's inequality, tightness follows if we can bound the variance
\begin{equation}
\mathbb{E}\norm{\tilde{D}_{N}}^{2}_{(-b)} = \sum_{k=0}^{\infty}\mathbb{E}\{c_{k}(\tilde{D}_{N})^{2}\}(1+k^{2})^{-b}
\end{equation}
uniformly in $N$. Such a uniform bound will follow for any $b > 1/2$ provided we show that $\mathbb{E}\{c_{k}(\tilde{D}_{N})^{2})\} \leq C$ for some constant $C$ independent of $k$ and $N$. We begin by writing the Chebyshev-Fourier coefficient as
\begin{equation}
c_{k}(\tilde{D}_{N}) = \sum_{j=1}^{N}h_{k}(x_{j})-\mathbb{E}\left\{\sum_{j=1}^{N}h_{k}(x_{j})\right\}
\end{equation}
where
\begin{equation}
\begin{split}
h_{k}(x) &= (2/k)T_{k}(x)\chi_{[-1,1]}(x) - (2/k)(x-\sqrt{x^{2}-1})^{k}\chi_{(1,\infty)}(x)\\
&- (2/k)(x+\sqrt{x^{2}-1})^{k}\chi_{(-1,-\infty)}(x)
\end{split}
\end{equation}
Then by formula \eqref{orig} we have
\begin{equation}
\mathbb{E}\{c_{k}(\tilde{D}_{N})^{2}\} = \frac{1}{8}\int_{\mathbb{R}^{2}}(h_{k}(x_{1})-h_{k}(x_{2}))^{2}K_{N}(x_{1},x_{2})^{2}\,dx_{1}\,dx_{2} \label{varform}
\end{equation}
where $K_{N}(x,y)$ is the GUE kernel defined in Eq. \eqref{guekernel}.
First we consider the contribution to the integral \eqref{varform} coming from the region $[-1,1]^{2}$, namely the integral
\begin{equation}
\frac{1}{2k^{2}}\int_{[-1,1]^{2}}\left(\frac{\Delta T_{k}(x)}{\Delta x}\right)^{2} \mathcal{F}_{N}(x_{1},x_{2})\,dx_{1}\,dx_{2}\label{varform2}
\end{equation}
where $\mathcal{F}_{N}(x_{1},x_{2})$ is defined by \eqref{cfn} and, as in Section 4, for a function $f$, we denote by $\Delta f$ the difference $\Delta f(x) = f(x_{1})-f(x_{2})$. By the Plancherel-Rotach asymptotics of Hermite polynomials, we have the bound (as follows from \textit{e.g.} parts (iii) and (v) of Theorem 2.2 in \cite{DKMVZ99})
\begin{equation}
|\mathcal{F}_{N}(x_{1},x_{2})| \leq \frac{K_{1}}{\sqrt{1-x_{1}^{2}}\sqrt{1-x_{2}^{2}}}
\end{equation}
uniformly for $(x_{1},x_{2}) \in [-1,1]^{2}$. This implies that the modulus of \eqref{varform2} is bounded by
\begin{equation}
\frac{K_{1}}{2k^{2}}\int_{[-1,1]^{2}}\left(\frac{\Delta T_{k}(x)}{\Delta x}\right)^{2}\frac{1}{\sqrt{1-x_{1}^{2}}\sqrt{1-x_{2}^{2}}}\,dx_{1}\,dx_{2}=K_{1}\pi^{2}/8. \label{explicitint}
\end{equation}
The equality in \eqref{explicitint} is a simple exercise involving standard properties of Chebyshev polynomials and we omit the derivation.
Finally consider the contribution to the integral \eqref{varform} from outside the square $[-1,1]^{2}$. For simplicity, consider just the region $1 < x_{1} < \infty$ and $-1<x_{2}<1$, all others being analogous. Since $h_{k}(x)$ is uniformly bounded in $k$ and $x$ on the whole real line, we have
\begin{align}
&\int_{-1}^{1}\int_{1}^{\infty}(h_{k}(x_{1})-h_{k}(x_{2}))^{2}K_{N}(x_{1},x_{2})^{2}\,dx_{1}\,dx_{2}\\
&\leq \int_{-\infty}^{\infty}\int_{1}^{\infty}K_{N}(x_{1},x_{2})^{2}\,dx_{1}\,dx_{2}\\
&=\int_{1}^{\infty}N\rho_{N}(x_{1})\,dx_{1}= \int_{1}^{1+\delta}N\rho_{N}(x_{1})\,dx_{1}+O(Ne^{-c_{\delta}N}) \label{lastint}
\end{align}
where $\delta>0$ is a constant and $c_{\delta}>0$. The last equality in \eqref{lastint} follows from Theorem 5.2.3 (iii) in \cite{PS11}. Now we can insert the formula \eqref{airyasy} which holds uniformly on $[1,1+\delta]$. The first term in \eqref{airyasy} is bounded in $N$ and $x_{1}$ and so its integral over $[1,1+\delta]$ is bounded in $N$. The third term gives an error of order $1/N$. The contribution from the middle term can be explicitly integrated using the substitution $u=N^{2/3}\Phi(x_{2})$:
\begin{align}
&\int_{1}^{1+\delta}N^{2/3}\Phi'(x_{2})\left(\mathrm{Ai}'^{2}(N^{2/3}\Phi(x_{2}))-N^{2/3}\Phi(x_{2})\mathrm{Ai}^{2}(N^{2/3}\Phi(x_{2}))\right)\,dx_{2}\\
&=\int_{0}^{N^{2/3}\Phi(1+\delta)}[\mathrm{Ai}'^{2}(u)-u\mathrm{Ai}^{2}(u)]\,du\\
&=-\left[\frac{2}{3}(u^{2}\mathrm{Ai}^{2}(u)-u\mathrm{Ai}'^{2}(u))-\frac{1}{3}\mathrm{Ai}(u)\mathrm{Ai}'(u)\right]_{0}^{N^{2/3}\Phi(1+\delta)}\\
&=\mathrm{Ai}(0)\mathrm{Ai}'(0)/3+O(e^{-d_{\delta}N})
\end{align}
where $d_{\delta}>0$. A completely analogous argument proves that the integral over the region $\{1<x_{1}<\infty, 1<x_{2}<\infty\}$ is also uniformly bounded in $k$ and $N$, in addition to the remaining $6$ regions that make up $B^{c}$. This completes the proof that $\tilde{D}_{N}$ is tight in $V^{(-a)}$ for any $a > 1/2$ and hence completes the proof of Theorem \ref{th:global}.
|
1,314,259,996,541 | arxiv | \section{Introduction}\label{Sec:Intro}
Consider the MILP problem
\begin{equation}\label{eq:MILPAlg}
\begin{split}
z^\text{IP}:= \min\limits_{ \mb{x}_1, \cdots \mb{x}_N}& \sum_{\nu \in {\cal P} } \mb{c}^\top_\nu \mb{x}_\nu \\
\text{s.t. } & \mb{x}_\nu \in X_\nu, \, \forall \nu\in {\cal P}, \\
& \sum_{\nu\in {\cal P} } \mb{A}_\nu \mb{x}_\nu =\mb{b},
\end{split}
\end{equation}
where ${\cal P}=\{1, \cdots, N\}$ is the set of blocks. In reality, there are cases where each of these blocks are governed by a different agent or owner. Each block $\nu$ has its own $n_\nu$ dimensional vector $\mb{x}_\nu$ of (discrete and continuous) decision variables, and local linear constraints
\begin{equation}\label{eq:Local-X}
\mb{x}_\nu \in X_\nu,
\end{equation}
where $ X_\nu$ is a linear mixed integer set. Different blocks of the problem \eqref{eq:MILPAlg} are linked to each other via the following linear \emph{coupling constraints}:
\begin{equation}\label{eq:MILPAlg-Link}
\sum_{\nu\in {\cal P} } \mb{A}_\nu \mb{x}_\nu =\mb{b}.
\end{equation}
Each $\mb{A}_\nu$ is a $m\times n_\nu$ matrix, for all $ \nu\in {\cal P}$, $\mb{b}$ is a $m$ dimensional vector, where $m$ is the number of coupling constraints \eqref{eq:MILPAlg-Link}.
If $\mb{A}_\nu$s are sparse matrices and the number of coupling constraints \eqref{eq:MILPAlg-Link} is relatively small comparing to the total number of local constraints of type \eqref{eq:Local-X}, then we call the problem \eqref{eq:MILPAlg} a \emph{loosely coupled} MILP. In general, relaxing these coupling constraints makes the remaining problem separable and easier.
In the Lagrangian relaxation (LR), the coupling constraints can be replaced by a linear penalty term in the objective function. Therefore, the LR of MILP \eqref{eq:MILPAlg} will become a separable MILP problem which can be solved in a distributed manner. In contrast to the convex setting, for nonconvex optimization problems such as MILPs, a nonzero duality gap may exist when the coupling constraints are relaxed by using classical Lagrangian dual (LD). In addition to a possible nonzero duality gap, it is not obvious how to obtain optimal Lagrange multipliers and a primal feasible solution by applying LD for MILPs.
Augmented Lagrangian dual (ALD) modifies classical LD by appending a nonlinear penalty on the violation of the dualized constraints. For MILP \eqref{eq:MILPAlg} under some mild assumptions, \cite{Feizollahi:2017Augmented} showed asymptotic zero duality gap property of ALD for MILPs when the penalty coefficient is allowed to go to infinity. They also proved that using any norm as the augmenting function with a sufficiently large but finite penalty coefficient closes the duality gap for general MILPs.
The main drawback of ALD is that the resulting subproblems are not separable because of the nonlinear augmenting functions. To overcome this issue, the alternating direction method of multipliers (ADMM) \citep{Boyd:2011} and related schemes have been developed for convex optimization problems . However, it is not at all clear how to decompose ALD for MILP problems and utilize parallel computation.
Based on ADMM, a heuristic decomposition method was developed in \citep{Feizollahi:2015Large} to solve MILPs arising from electric power network unit commitment problems.
\cite{Bixby:1995} presented a parallel implementation of a branch-and-bound algorithm for mixed 0-1 integer programming problems.
\cite{Ahmed:2013scenario} and \cite{Deng:2017} developed scenario decomposition approaches for 0-1 stochastic programs.
\cite{Munguia:2018} presented a parallel large neighborhood search framework for finding high quality primal solutions for generic MILPs. The approach simultaneously solved a large number of sub-MILPs with the dual objective of reducing infeasibility and optimizing with respect to the original objective.
\cite{Oliveira:2017} proposed a decomposition approach for mixed-integer stochastic programming (SMILP) problems that is inspired by the combination of penalty-based Lagrangian and block Gauss-Seidel methods.
A key challenge is that, because of the non-convex nature of MILPs, classical distributed and decentralized optimization approaches cannot be applied directly to find their optimal solutions.
In this paper, we propose a distributed approach to solve loosely coupled MILP problems. where each block solves its own modified LR subproblem iteratively. This approach provides valid lower and upper bounds for the original MILP problem at each iteration. Based on this distributed approach, we develop two exact algorithms which are able to close the gap between lower and upper bounds, and obtain a feasible and optimal solution to the original MILP problem in a finite number of iterations. The proposed exact algorithms are based on adding primal cuts and restricting the Lagrangian relaxation of the original MILP problem. Note that these cuts are not distributable in general.
We test the proposed algorithms on the unit commitment problem and discuss its pros and cons comparing to the central MILP approach.
This paper is organized as follows. Details of the assumptions and notations are provided in Section \ref{Sec:Prelim}. In Section \ref{Sec:AlgLiterature}, scheme of the dual decomposition and ADMM as two well known distributed optimization technique are presented. Our distributed MILP approach with two exact algorithms are discussed in Section \ref{Sec:ExactAlgs}.
Experimental results are discussed in Section \ref{Sec:Computations} and conclusions are presented in Section \ref{Sec:Conclusion}.
\section{Preliminaries}\label{Sec:Prelim}
Let $\mathbb{R}$, $\mathbb{Z}$, and $\mathbb{Q}$ denote the sets of real, integer and rational numbers, respectively. For a finite dimensional vector $\mb{a}$, denote its transpose by $\mb{a}^\top$. For a set ${\cal S}$, denote its cardinality by $|{\cal S}|$.
In this paper, we consider MILP problem \eqref{eq:MILPAlg} which satisfies the following assumptions.
\begin{assumption}\label{Assump:MILPAlg}
For the MILP \eqref{eq:MILPAlg} we have the following:
\begin{enumerate}[label=(\alph*)]
\item For each block $\nu\in {\cal P}$, $X_\nu$ is a linear mixed integer set defined by
\begin{equation}\label{eq:DefineX}
X_\nu:=\{ (\mb{u}_\nu^\top,\mb{y}_\nu^\top)^\top : \mb{u}_\nu \in U_\nu, ~ \mb{y}_\nu\in Y_\nu(\mb{u}_\nu) \},
\end{equation}
where $\mb{u}_\nu\in \{0,1\}^{n_\nu^1}$ and $\mb{y}_\nu \in \mathbb{R}^{n_\nu^2}$ are the subvectors of $n_\nu^1$ binary and $n_\nu^2$ continuous decision variables, respectively, with $n_\nu=n_\nu^1+n_\nu^2$.
\item In description \eqref{eq:DefineX} of $X_\nu$, $U_\nu$ and $Y_\nu(\mb{u}_\nu)$ are subsets of $\{0,1\}^{n_\nu^1}$ and $\mathbb{R}^{n_\nu^2}$, respectively. Because $U_\nu$ is a finite set, it can be represented by a set of linear inequalities and integrality constraints. For a given $\mb{u}_\nu\in U_\nu$, we assume $Y_\nu(\mb{u}_\nu)$ is a (possibly empty) polyhedron.
In particular, let $Y_\nu(\mb{u}_\nu)=\{ \mb{y}_\nu: \mathbb{R}^{n_\nu^2}: \mb{E}_\nu \mb{u}_\nu +\mb{F}_\nu \mb{y}_\nu \le \mb{g}_\nu\}$, where $\mb{E}_\nu $ and $\mb{F}_\nu$ are matrices and $ \mb{g}_\nu$ is a vector of appropriate finite dimensions, independent of the value of $\mb{u}_\nu$.
\item $\mb{c}_\nu$, $\mb{A}_\nu$, $\mb{E}_\nu $, $\mb{F}_\nu$ and $ \mb{g}_\nu$, for all $\nu\in {\cal P}$, and $\mb{b}$ have rational entries.
\item Problem \eqref{eq:MILPAlg} is feasible and its optimal value is bounded.
\end{enumerate}
\end{assumption}
Let $n^1:=\sum_{\nu \in {\cal P} } n_\nu^1$ and $n^2:=\sum_{\nu \in {\cal P} } n_\nu^2$ denote total number of binary and continuous variables, respectively, and $n=n^1+n^2$. For convenience, let
\begin{equation}\nonumber
\begin{split}
& \mb{c}:= \left[ \begin{array}{c}
\mb{c}_1 \\
\vdots \\
\mb{c}_N
\end{array} \right],~
\mb{x} := \left[ \begin{array}{c}
\mb{x}_1 \\
\vdots \\
\mb{x}_N
\end{array} \right],~
\mb{u}:= \left[ \begin{array}{c}
\mb{u}_1 \\
\vdots \\
\mb{u}_N
\end{array} \right], ~
\mb{y}:= \left[ \begin{array}{c}
\mb{y}_1 \\
\vdots \\
\mb{y}_N
\end{array} \right],\\
& \mb{A}:= [\mb{A}_1,\cdots, \mb{A}_N],~
X:= X_1\times \cdots \times X_N,~
U:= U_1\times \cdots \times U_N,\\
& Y(\mb{u}):= Y_1(\mb{u}_1)\times \cdots Y_N(\mb{u}_N).
\end{split}
\end{equation}
Then, problem \eqref{eq:MILPAlg} can be recast as $z^\text{IP}= \min\limits_{ \mb{x} } \{ \mb{c}^\top \mb{x} : \mb{x}\in X, \mb{A} \mb{x} =\mb{b}\}$.
By Assumption \ref{Assump:MILPAlg}-d), there exists a solution $\mb{x}^\ast$ which satisfies constraints \eqref{eq:Local-X} and \eqref{eq:MILPAlg-Link}, and $\mb{c}^\top \mb{x}^\ast =z^\text{IP}$.
Therefore, by data rationality assumption in part (c), the value of the linear programming (LP) relaxation ($z^\text{LP}$) of \eqref{eq:MILPAlg} is bounded \citep{Blair:1979}, i.e. $-\infty < z^\text{LP} \le z^\text{IP}< \infty$.
\begin{example} \label{Ex:MILPAlgExample}
Following is an example for problem \eqref{eq:MILPAlg} with two blocks.
\begin{equation}\label{eq:MILPAlgExample}
\begin{split}
\min ~~& 70u_{11}+70u_{12}+110u_{13}+2y_{11}+2y_{12}+48u_{21}+48u_{22}+52u_{23}+3y_{21}+3y_{22}\\
\text{s.t. } &
\left. \begin{array}{c}
u_{12}-u_{11} - u_{13}\le 0,\\
30u_{11} \le y_{11} \le 100 u_{11},\\
30u_{12} \le y_{12} \le 100 u_{12},\\
-35 \le y_{12}-y_{11} \le 35,\\
u_{11}, u_{12}, u_{13}\in \{0,1\},
\end{array} \right\} \text{Local constraints for block 1}\\
& \left. \begin{array}{c}
u_{22}-u_{21} - u_{23}\le 0,\\
20u_{11} \le y_{21} \le 80 u_{21},\\
20u_{12} \le y_{22} \le 80 u_{22},\\
-30 \le y_{22}-y_{21} \le 30,\\
u_{21}, u_{22}, u_{23}\in \{0,1\},\\
\end{array} \right\} \text{Local constraints for block 2}\\
& \left. \begin{array}{c}
y_{11}+y_{21}= 90,\\
y_{12}+y_{22}= 120.
\end{array}~~~~~~~~~ \right\} \text{Coupling constraints}
\end{split}
\end{equation}
Recalling the notations described in Sections \ref{Sec:Intro} and \ref{Sec:Prelim},
$\mb{u}_1=(u_{11}, u_{12}, u_{13})^\top$ and $\mb{u}_2=(u_{21}, u_{22}, u_{23})^\top$ are the vectors of binary variables for blocks 1 and 2, respectively. Similarly, $\mb{y}_1=(y_{11}, y_{12})^\top$ and $\mb{y}_2=(y_{21}, y_{22})^\top$ are the vectors of continuous variables for blocks 1 and 2, respectively.
Then, $\mb{x}_1=(\mb{u}_1^\top,\mb{y}_1^\top)^\top$ and $\mb{x}_2=(\mb{u}_2^\top,\mb{y}_2^\top)^\top$ are the vectors of decision variables for blocks 1 and 2, respectively.
Moreover, $\mb{u}=(\mb{u}_1^\top,\mb{u}_2^\top)=(u_{11}, u_{12}, u_{13},u_{21}, u_{22}, u_{23})^\top$ and $\mb{y}=(\mb{y}_1^\top,\mb{y}_2^\top)=(y_{11}, y_{12},y_{21}, y_{22})^\top$ are the overall vectors of binary and continuous variables. In this example, we have
$$ \mb{c}_1=\left[
\begin{array}{c}
70\\ 70\\ 110\\ 2\\ 2
\end{array} \right],~
\mb{c}_2=
\left[ \begin{array}{c}
48 \\ 48 \\ 52 \\ 3 \\ 3
\end{array} \right], \text { and }
\mb{A}_1= \mb{A}_2=\left[
\begin{array}{c c c c c}
0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 1
\end{array} \right].
$$
Moreover,
$$U_1=\{\mb{u}_1\in \{0,1\}^3: u_{12}-u_{11} - u_{13}\le 0 \},$$
$$U_2=\{\mb{u}_2\in \{0,1\}^3: u_{22}-u_{21} - u_{23}\le 0 \},$$
$$U=U_1\times U_2=\left\{\mb{u}\in \{0,1\}^6: \begin{array}{c} u_{12}-u_{11} - u_{13}\le 0,\\ u_{22}-u_{21} - u_{23}\le 0 \end{array}\right\},$$
$$Y_1(\mb{u}_1)=\left\{\mb{y}_1\in \mathbb{R}^2:
\begin{array}{c}
30u_{11} \le y_{11} \le 100 u_{11},\\
30u_{12} \le y_{12} \le 100 u_{12},\\
-35 \le y_{12}-y_{11} \le 35
\end{array}\right \},$$
$$Y_2(\mb{u}_2)=\left\{\mb{y}_2\in \mathbb{R}^2:
\begin{array}{c}
20u_{11} \le y_{21} \le 80 u_{21}\\
20u_{12} \le y_{22} \le 80 u_{22}\\
-30 \le y_{22}-y_{21} \le 30
\end{array}\right \},$$
$$X_1=\left\{\mb{x}_1=(\mb{u}_1^\top,\mb{y}_1^\top)^\top\in \{0,1\}^3\times \mathbb{R}^2:
\begin{array}{c}
u_{12}-u_{11} - u_{13}\le 0,\\
30u_{11} \le y_{11} \le 100 u_{11},\\
30u_{12} \le y_{12} \le 100 u_{12},\\
-35 \le y_{12}-y_{11} \le 35,
\end{array}\right \},$$
$$X_2=\left\{\mb{x}_2=(\mb{u}_2^\top,\mb{y}_2^\top)^\top\in \{0,1\}^3\times \mathbb{R}^2:
\begin{array}{c}
u_{22}-u_{21} - u_{23}\le 0,\\
20u_{11} \le y_{21} \le 80 u_{21},\\
20u_{12} \le y_{22} \le 80 u_{22},\\
-30 \le y_{22}-y_{21} \le 30,
\end{array}\right \}.$$
\end{example}
For a given vector of the dual (Lagrange) variables, $\mb{\mu}\in \mathbb{R}^m$, the standard LR for MILP \eqref{eq:MILPAlg} is
\begin{equation}\label{eq:MILP1-LR}
\begin{split}
z^\text{LR}(\mb{\mu}):=\mb{\mu}^\top \mb{b}+ \min\limits_{ \mb{x}_1,\cdots, \mb{x}_N } & \sum_{ \nu\in {\cal P} } {\cal L}_\nu(\mb{x}_\nu,\mb{\mu})
\\
\text{s.t. } &\mb{x}_\nu \in X_\nu, \, \forall \nu\in {\cal P},
\end{split}
\end{equation}
where
\begin{equation}\nonumber
{\cal L}_\nu(\mb{x}_\nu,\mb{\mu}):=(\mb{c}^\top_\nu -\mb{\mu}^\top \mb{A}_\nu ) \mb{x}_\nu, \, \forall \nu\in {\cal P},
\end{equation}
and the corresponding LD value is
\begin{equation}\label{eq:MILP1-LD}
z^\text{LD}:= \sup\limits_{\mb{\mu}\in \mathbb{R}^m} z^\text{LR}(\mb{\mu}).
\end{equation}
Since \eqref{eq:MILP1-LR} is a relaxation of \eqref{eq:MILPAlg}, $z^\text{LR}(\mb{\mu}) \le z^\text{LD} \le z^\text{IP}$ holds, for any $\mb{\mu}\in \mathbb{R}^m$. Due to the presence of binary variables, a nonzero duality gap may exists \citep{Wolsey:1999}, i.e. $z^\text{LD} < z^\text{IP}$ is possible. Let $\mb{\mu}^\ast$ be a maximizer in \eqref{eq:MILP1-LD}, which exists under Assumption \ref{Assump:MILPAlg}.
Obtaining $\mb{\mu}^\ast$ and $z^{LD}$ are not straight forward in practice. A popular and easy approach to solve \eqref{eq:MILP1-LD} is the subgradient decent method, where the problem \eqref{eq:MILP1-LR} is solved iteratively and the dual multipliers are updated at each iteration. Note that problem \eqref{eq:MILP1-LR} is separable and it can be solved by computing
$$\min\limits_{\mb{x}_\nu} \{ {\cal L}_\nu(\mb{x}_\nu,\mb{\mu}): \mb{x}_\nu \in X_\nu\}$$
for each block $\nu$.
Even with $\mb{\mu}^\ast$ at hand, a {\em primal feasible solution}, one that satisfies all constraints in model \eqref{eq:MILPAlg}, is not readily available.
In other words, an optimal solution of LR \eqref{eq:MILP1-LR} for $\mb{\mu}^\ast$
does not necessarily satisfy the coupling constraints \eqref{eq:MILPAlg-Link} in problem \eqref{eq:MILPAlg}.
For a given $\hat{\mb{u}} \in U$, the best corresponding primal feasible solution, if there exists one, and its objective value, $z(\mb{\hat u})$, can be computed by solving the following LP:
\begin{equation}\label{eq:LP1}
\begin{split}
z(\mb{\hat u}):= \min\limits_{ \mb{y}_1,\cdots,\mb{y}_N } & ~\sum_{\nu \in {\cal P}} \mb{c}_\nu^\top \left[ \def0.5} \begin{array}{l{0.5} \begin{array}{c}
\mb{\hat u}_\nu \\
\mb{y}_\nu
\end{array}
\right] \\
\text{s.t. } & \mb{y}_\nu\in Y_\nu(\mb{\hat u}_\nu), \, \forall \nu \in {\cal P},\\
& \sum_{\nu \in {\cal P}} \mb{A}_\nu \left[ \def0.5} \begin{array}{l{0.5} \begin{array}{l}
\mb{\hat u}_\nu \\
\mb{y}_\nu
\end{array}
\right] =\mb{b} .
\end{split}
\end{equation}
Problem \eqref{eq:LP1} is an LP and can be solved with a distributed algorithm \citep{Boyd:2011}.
Denote the upper and lower bounds on $z^\text{IP}$ by $ub$ and $lb$, respectively. Then, $z(\mb{u})$ and $z^\text{LR}(\mb{\mu})$ are valid $ub$ and $lb$, respectively, for all $\mb{u}\in U$ and $\mb{\mu}\in \mathbb{R}^m$, i.e.
$$z^\text{LR}(\mb{\mu}) \le z^\text{IP} \le z(\mb{u}),~~ \forall \mb{u}\in U, \mb{\mu}\in \mathbb{R}^m.$$
In fact,
\begin{equation}\label{eq:z-U}
z^\text{IP}= \min\limits_{\mb{u}\in U} ~ z(\mb{u}).
\end{equation}
\section{Dual Decomposition and ADMM for MILPs} \label{Sec:AlgLiterature}
Dual decomposition and ADMM are two well known distributed optimization technique in the context of convex optimization. Our distributed MILP algorithms in this paper are based on extensions of these two techniques. Next, we present these schemes and discuss challenges in applying them to MILPs.
\subsection{Dual Decomposition}
Dual decomposition is a well known technique to solve large scale optimization problems. Early works on application of dual decomposition for
large scale linear programming can be found in \citep{Benders:1962,Dantzig:1960,Dantzig:1963,Everett:1963}.
Let $\rho_\mu^k>0$ be the step size for updating the dual vector $\mb{\mu}$ at iteration $k$. Algorithm \ref{Alg:BasicDualD} represents an overall scheme of a dual decomposition method to solve \eqref{eq:MILPAlg}. Each iteration of this method requires a ``broadcast'' and a ``gather'' operation. Dual update step (line \ref{Line:DualUpdate} in Algorithm \ref{Alg:BasicDualD}) requires $\mb{A}_\nu \mb{x}_\nu^k$ values from all blocks. Once $\mb{\mu}^{k}$ is computed, it must be broadcasted to all blocks.
A lower bound for $z^\text{IP}$ can be obtained from Algorithm \ref{Alg:BasicDualD}. If $\sum\limits_{\nu\in {\cal P} } \mb{A}_\nu \mb{x}_\nu^k =\mb{b}$ in some iteration $k$ of this algorithm, $\mb{x}^k$ is a feasible and optimal solution of \eqref{eq:MILPAlg}. But, this case is not likely in practice and there is no hope to find a feasible solution for \eqref{eq:MILPAlg} by running only Algorithm \ref{Alg:BasicDualD}. Therefore, in general we cannot expect to get an upper bound for $z^\text{IP}$ from this algorithm. A modified version of dual decomposition technique is presented in Algorithm \ref{Alg:DualD} which is able to provide upper bounds for $z^\text{IP}$.
\begin{algorithm}
\caption{Basic Dual Decomposition}\label{Alg:BasicDualD}
\begin{algorithmic}[1]
\State $lb \leftarrow -\infty$, $\mb{\mu}^0 \leftarrow \mb{0}$, and $k \leftarrow 0$.
\While{ some termination criteria is not met}
\State $k \leftarrow k+1$
\For{$\nu:=1$ to $N$}
\State solve $\min\limits_{\mb{x}_\nu} \{ {\cal L}_\nu(\mb{x}_\nu,\mb{\mu}^{k-1}): \mb{x}_\nu \in X_\nu\}$
\State let $v_\nu^k$ be the optimal value and $\mb{x}_\nu^k$ be an optimal solution
\EndFor
\If{$lb< \mb{\mu}^\top \mb{b}+\sum\limits_{\nu\in {\cal P}} v_\nu^k$}
\State $lb\leftarrow \mb{\mu}^\top \mb{b}+\sum\limits_{\nu\in {\cal P}} v_\nu^k$
\EndIf
\State $\mb{\mu}^{k} \leftarrow \mb{\mu}^{k-1}+\rho_\mu^{k}\left(\mb{b}-
\sum\limits_{\nu\in {\cal P} } \mb{A}_\nu \mb{x}_\nu^k \right) $ \label{Line:DualUpdate}
\EndWhile
\end{algorithmic}
\end{algorithm}
\subsection{Alternating Direction Method of Multipliers (ADMM)}\label{Sec:ADMM}
ADMM is an algorithm that is intended to blend the separability of dual decomposition with the superior convergence properties of the method of multipliers \citep{Boyd:2011}. For $\rho>0$ and ${\mb \mu}\in \mathbb{R}^m$, the augmented Lagrangian with squared Euclidean norm has the following form.
\begin{equation} \label{eq:ALMILP}
{\cal L}_\rho^+(\mb{x}_1,\cdots,\mb{x}_N,{\mb \mu})= \sum_{\nu\in {\cal P} } \mb{c}^\top_\nu \mb{x}_\nu + {\mb \mu}^\top \left(\mb{b}-
\sum_{\nu\in {\cal P} } \mb{A}_\nu \mb{x}_\nu \right)
+\frac{\rho}{2}\left\|\mb{b} -\sum_{\nu\in {\cal P} } \mb{A}_\nu \mb{x}_\nu \right\|_2^2.
\end{equation}
A robust relaxation for MILP \eqref{eq:MILPAlg} is the augmented Lagrangian relaxation (ALR) which has the following form:
\begin{equation} \label{eq:ALR}
\begin{split}
z^\text{LR+}_\rho({\mb \mu}):= \min\limits_{ \mb{x}_1, \cdots \mb{x}_N}& {\cal L}_\rho^+(\mb{x}_1,\cdots,\mb{x}_N,{\mb \mu}) \\
\text{s.t. } & \mb{x}_\nu \in X_\nu, \, \forall \nu\in {\cal P},
\end{split}
\end{equation}
and the corresponding ALD value is
\begin{equation}\label{eq:MILP1-ALD}
z^\text{LD+}:= \sup\limits_{\mb{\mu}\in \mathbb{R}^m} z^\text{LR+}(\mb{\mu}).
\end{equation}
Since \eqref{eq:ALR} is a relaxation of \eqref{eq:MILPAlg}, $z^\text{LR+}(\mb{\mu}) \le z^\text{LD+} \le z^\text{IP}$ holds, for any $\mb{\mu}\in \mathbb{R}^m$.
For MILP \eqref{eq:MILPAlg} under Assumption \ref{Assump:MILPAlg}, \cite{Feizollahi:2017Augmented} showed that
using ALD with any norm as the augmenting function is able to
close the duality gap with a finite penalty coefficient $\rho$.
It is obvious that ${\cal L}_\rho^+ (\mb{x}_1,\cdots,\mb{x}_N,{\mb \mu})$ in \eqref{eq:ALMILP} is not separable between different blocks, because the nonlinear (quadratic) terms are coupling different block to each other. For convex optimization problems, a decomposable algorithm to solve \eqref{eq:ALR} is ADMM \citep{Boyd:2011}.
\subsubsection{ADMM with two blocks}
Algorithm \ref{Alg:ADMM2} presents an ADMM approach for an optimization problem with two blocks. In $k$th iteration of this algorithm, ${\cal L}_\rho^+ (\mb{x}_1, \mb{x}_2^{k-1},\mb{\mu}^{k-1})$ is first minimized with respect to $\mb{x}_1$, assuming that $\mb{x}_2$ is fixed at its previous value $\mb{x}_2^{k-1}$. Then, ${\cal L}_\rho^+ ( \mb{x}_1^{k}, \mb{x}_2,\mb{\mu}^{k-1})$ is minimized with respected to $\mb{x}_2$, assuming that $\mb{x}_1$ is fixed at its previous value $\mb{x}_1^{k}$. Finally, the vector of dual variables $\mb{\mu}^{k}$ is updated. Note that $\rho>0$ is a given and fixed penalty factor.
\begin{algorithm}
\caption{ADMM procedure for two blocks}\label{Alg:ADMM2}
\begin{algorithmic}[1]
\State $\mb{x}_2^0 \leftarrow \mb{0} $, $\mb{\mu}^0 \leftarrow \mb{0} $, and $k\leftarrow 0$
\While{some termination criteria is not met }
\State $k\leftarrow k+1$
\State $\mb{x}_1^{k} \leftarrow \arg\min\limits_{\mb{x}_1 \in X_1} {\cal L}_\rho^+ (\mb{x}_1, \mb{x}_2^{k-1},\mb{\mu}^{k-1})$
\State $\mb{x}_2^{k} \leftarrow \arg\min\limits_{\mb{x}_2 \in X_2} {\cal L}_\rho^+ ( \mb{x}_1^{k}, \mb{x}_2,\mb{\mu}^{k-1})$
\State Update $\mb{\mu}^{k} \leftarrow \mb{\mu}^{k-1}+ \rho\times [\mb{b}- (\mb{A}_1 \mb{x}_1^{k}+ \mb{A}_2 \mb{x}_2^{k})]$
\EndWhile
\end{algorithmic}
\end{algorithm}
Let $\mb{\alpha}^k$ and $\mb{\beta}^k$ denote vectors of primal and dual residuals at iteration $k$. Then,
\begin{equation}\nonumber
\begin{split}
\mb{\alpha}^k=\mb{b}- (\mb{A}_1 \mb{x}_1^{k}+ \mb{A}_2 \mb{x}_2^{k}) \text { and }
\mb{\beta}^k=\rho \mb{A}_1^\top \mb{A}_2 (\mb{x}_2^k-\mb{x}_2^{k-1}).
\end{split}
\end{equation}
If problem \eqref{eq:MILPAlg} is solvable and the sets $X_1$ and $X_2$ are convex, closed, and non-empty, Algorithm \ref{Alg:ADMM2} can solve \eqref{eq:MILPAlg} in a distributed framework \citep{Boyd:2011}. In this case, primal residuals ($\mb{\alpha}^k$) converge to zero. Moreover, dual variables ($\mb{\mu}^{k}$) and objective value converge to their optimal values \citep{Boyd:2011}. Note that discrete variables destroy the nice convergence properties of ADMM for MILP problems \citep{Feizollahi:2015Large}.
In practice, ADMM converges to modest accuracy --sufficient for many applications-- within a few tens of iterations \citep{Boyd:2011}. However, direct extension of ADMM for multi-block convex
minimization problems is not necessarily convergent \citep{Chen:2016direct}.
\subsubsection{Global Variable Consensus Problem with ADMM}
To extend ADMM for multi-block minimization problems, a global variable consensus problem can be constructed. An equivalent optimization problem for \eqref{eq:MILPAlg} is as follows.
\begin{subequations}\label{eq:consensus}
\begin{eqnarray}
z^\text{IP}:= \min\limits_{ \mb{x}_1, \cdots \mb{x}_N, \bar{\mb x}_1, \cdots, \bar{\mb x}_N} && \sum_{\nu \in {\cal P} } \mb{c}^\top_\nu \mb{x}_\nu \nonumber \\
\text{s.t. } & & \mb{x}_\nu \in X_\nu , \, \forall \nu\in {\cal P}, \nonumber \\
& & \sum_{\nu\in {\cal P} } \mb{A}_\nu \bar{\mb x}_\nu =\mb{b}, \label{eq:LinkCons1}\\
&& \bar{\mb x}_\nu=\mb{x}_\nu, \, \forall \nu\in {\cal P}.\label{eq:LinkCons2}
\end{eqnarray}
\end{subequations}
\begin{algorithm}
\caption{Consensus ADMM} \label{Alg:ADMM}
\begin{algorithmic}[1]
\State $\bar{\mb x}^0 \leftarrow \mb{0}$, $\mb{\mu}^0 \leftarrow \mb{0} $, and $k\leftarrow 0$
\While{some termination criteria is not met }
\State $k\leftarrow k+1$
\For{$\nu:=1$ to $N$}
\State $\mb{x}_\nu^{k} \leftarrow \arg\min\limits_{\mb{x}_\nu \in X_\nu} {\cal L}_{\rho,\nu}^+(\mb{x}_\nu,\bar{\mb x}_\nu^{k-1},\mb{\mu}_\nu^{k-1})$
\EndFor
\State $\bar{\mb x}^{k} \leftarrow \arg\min\limits_{\bar{\mb x}} \left\{ {\cal L}_{\rho}^+(\mb{x}^{k},\bar{\mb x},\mb{\mu}^{k-1}): \sum_{\nu\in {\cal P} } \mb{A}_\nu \bar{\mb x}_\nu =\mb{b} \right\}$ by using \eqref{eq:Update-xbar}
\For{$\nu:=1$ to $N$}
\State $\mb{\mu}_\nu^{k} \leftarrow \mb{\mu}_\nu^{k-1}+ \rho\times (\mb{x}_\nu^{k}-\bar{\mb x}_\nu^{k})$
\EndFor
\EndWhile
\end{algorithmic}
\end{algorithm}
Formulation \eqref{eq:consensus} can be decomposed into two parts, where one part includes variable vectors $\mb{x}_1, \cdots \mb{x}_N$, constraints $\mb{x}_\nu \in X_\nu $, for all $\nu\in {\cal P}$ and the objective function, and the other part contains variable vectors $\bar{\mb x}_1, \cdots, \bar{\mb x}_N$ and constraints \eqref{eq:LinkCons1}. In this case, constraints \eqref{eq:LinkCons2} are coupling these two parts and Algorithm \ref{Alg:ADMM2}, ADMM with two blocks, can be adjusted to solve problem \eqref{eq:consensus} in a distributed manner. Algorithm \ref{Alg:ADMM}, consensus ADMM, represents this process.
Let
\begin{equation}\nonumber
{\cal L}_{\rho}^+(\mb{x},\bar{\mb x},\mb{\mu}):=\sum_{\nu \in {\cal P} } {\cal L}_{\rho,\nu}^+(\mb{x}_\nu,\bar{\mb x}_\nu,\mb{\mu}_\nu),
\end{equation}
where
$
{\cal L}_{\rho,\nu}^+(\mb{x}_\nu,\bar{\mb x}_\nu,\mb{\mu}_\nu):=\mb{c}^\top_\nu \mb{x}_\nu +\mb{\mu}_\nu^\top(\mb{x}_\nu-\bar{\mb x}_\nu)
+ \frac{\rho}{2} \|\mb{x}_\nu-\bar{\mb x}_\nu\|_2^2.
$
Then, the subproblem for part one is
$
\min\limits_{ \mb{x}} \{ {\cal L}_{\rho}^+(\mb{x},\bar{\mb x},\mb{\mu}) :\mb{x}_\nu \in X_\nu , \, \forall \nu\in {\cal P} \},
$
which is separable between blocks and can be solved in parallel. Moreover, the subproblem for part two is
\begin{equation}\nonumber
\min\limits_{ \bar{\mb x}} \{ {\cal L}_{\rho}^+(\mb{x},\bar{\mb x},\mb{\mu}) :\sum_{\nu\in {\cal P} } \mb{A}_\nu \bar{\mb x}_\nu =\mb{b} \}
\end{equation}
which has a closed form solution as follows (assuming $\mb{A}$ has full row rank):
\begin{equation}\label{eq:Update-xbar}
\begin{split}
\arg\min\limits_{\bar{\mb x}} & \left\{ {\cal L}_{\rho}^+(\mb{x},\bar{\mb x},\mb{\mu}): \sum_{\nu\in {\cal P} } \mb{A}_\nu \bar{\mb x}_\nu =\mb{b} \right\}
=\arg\min\limits_{\bar{\mb x}} \left\{\|\mb{x}+\frac{\mb{\mu}}{\rho}-\bar{\mb x}\|_2^2 : \mb{A}\bar{\mb x}={\mb b} \right\}\\
&= [I-\mb{A}^\top (\mb{A}\mb{A}^\top)^{-1}\mb{A}](\mb{x}+\frac{\mb{\mu}}{\rho})+\mb{A}^\top (\mb{A} \mb{A}^\top)^{-1}\mb{b}
\end{split}
\end{equation}
where the second equality is well known in linear algebra for finding the orthogonal projection of a point onto an affine subspace \citep[e.g.][]{Meyer:2000,Plesnik:2007finding}. In general, to compute inverse matrices is not easy \citep{Higham:2002}, but it can be done efficiently for sparse matrices with specific structures.
In distributed consensus optimization, ADMM has a linear convergence rate \citep{Shi:2014linear}. Consensus ADMM can be interpreted as a method for solving problems in which the objective and constraints are distributed across multiple processors. Each processor only has to handle its own objective and constraint term, plus a quadratic term which is updated each iteration. The linear parts of the quadratic terms are updated in such a way that the variables converge to a common value, which is the solution of the full problem \citep{Boyd:2011}.
In our context of MILP \eqref{eq:MILPAlg}, consensus ADMM (Algorithm \ref{Alg:ADMM}) can be used for upper bounding $z^{IP}$. For a given set $\hat{{\cal S}} \subset U$, an upper bounding method is as Algorithm \ref{Alg:UpperBounding}.
\begin{algorithm}
\caption{Upper Bounding Algorithm}\label{Alg:UpperBounding}
\begin{algorithmic}[1]
\For{$\hat{\mb{u}}\in \hat{{\cal S}}$}
\State compute $z(\mb{\hat u})$ by solving LP \eqref{eq:LP1} with consensus ADMM, Algorithm \ref{Alg:ADMM}
\If{$z(\mb{\hat u})<ub$ }
\State $ub \leftarrow z(\mb{\hat u})$
\State $\mb{u}^\ast \leftarrow \mb{\hat u}$
\EndIf
\EndFor
\end{algorithmic}
\end{algorithm}
\subsection{Combination of Dual Decomposition and Consensus ADMM}
A combination of Algorithm \ref{Alg:BasicDualD} (dual decomposition) and Algorithm \ref{Alg:ADMM} (consensus ADMM) can be used to generate lower and upper bounds for $z^\text{IP}$. Algorithm \ref{Alg:DualD} presents a modified version of Algorithm \ref{Alg:BasicDualD}. In this algorithm, for a given binary vector $\hat{\mb u}$, Algorithm \ref{Alg:ADMM} (consensus ADMM) is used to refine continuous variables $\mb{y}$, and obtain an upper bound for $z^\text{IP}$.
Besides the issues related to the non zero duality gap and the challenges in finding the the best dual vector $\mb{\mu}^\ast$, which is a maximizer in \eqref{eq:MILP1-LD}, it is possible for Algorithms \ref{Alg:BasicDualD} and \ref{Alg:DualD} to cycle between non-optimal solutions forever.
\begin{algorithm}
\caption{Modified Dual Decomposition for MILPs}\label{Alg:DualD}
\begin{algorithmic}[1]
\State $ub \leftarrow +\infty$, ${\cal S} \leftarrow \emptyset$, $\mb{u}^\ast \leftarrow \emptyset$, and $k \leftarrow 0$.
\State Solve LP relaxation of \eqref{eq:MILPAlg} with ADMM, Algorithm \ref{Alg:ADMM}.
Let $z^\text{LP}$ be its optimal value, and $\mb{\mu}^0$ be the dual values for the coupling constraints \eqref{eq:MILPAlg-Link}.
\State $lb\leftarrow z^\text{LP}$
\While{ some termination criteria is not met}
\State $k \leftarrow k+1$
\For{$\nu:=1$ to $N$}
\State solve $ \min\limits_{\mb{x}_\nu} \{ {\cal L}_\nu(\mb{x}_\nu,\mb{\mu}^{k-1}): \mb{x}_\nu \in X_\nu\}$
\State let $v_\nu^k$ be the optimal value and $\mb{x}_\nu^{k}= (\mb{u}_\nu^{k},\mb{y}_\nu^{k})$ be an optimal solution
\EndFor
\If{$lb< \mb{\mu}^\top \mb{b}+\sum\limits_{\nu\in {\cal P}} v_\nu^k$}
\State $lb\leftarrow \mb{\mu}^\top \mb{b}+\sum\limits_{\nu\in {\cal P}} v_\nu^k$
\EndIf
\State $\mb{\mu}^{k} \leftarrow \mb{\mu}^{k-1}+\rho_\mu^{k}\left(\mb{b}-
\sum\limits_{\nu\in {\cal P} } \mb{A}_\nu \mb{x}_\nu^k \right) $
\If{$\mb{u}_\nu^{k} \notin {\cal S}$}
\State ${\cal S} \leftarrow {\cal S}\cup \{ \mb{u}_\nu^{k} \}$
\State compute $z(\mb{u}_\nu^{k})$ by solving \eqref{eq:LP1} with ADMM, Algorithm \ref{Alg:ADMM}
\If{$z(\mb{u}_\nu^{k+1})<ub$ }
\State $ub \leftarrow z(\mb{u}_\nu^{k})$
\State $\mb{u}^\ast \leftarrow \mb{u}_\nu^{k}$
\EndIf
\EndIf
\EndWhile
\end{algorithmic}
\end{algorithm}
\subsection{Release-and-Fix Heuristic}
\begin{algorithm}
\caption{Release-and-Fix Heuristic for MILPs \citep{Feizollahi:2015Large} }\label{Alg:RaF}
\begin{algorithmic}[1]
\State $ub \leftarrow +\infty$, $\mb{u}^\ast \leftarrow \emptyset$, and $k \leftarrow 0$.
\State {\bf ADMM-CR}: Solve LP relaxation of \eqref{eq:MILPAlg} with ADMM, Algorithm \ref{Alg:ADMM}.
Let $z^\text{LP}$ be its optimal value, and $\mb{\mu}^0$ be the dual values for the coupling constraints \eqref{eq:MILPAlg-Link}.
\State $lb\leftarrow z^\text{LP}$
\While{ time or iteration limits are not met}
\State $k \leftarrow k+1$
\State {\bf ADMM-Bin+}: Continue ADMM, Algorithm \ref{Alg:ADMM}, for the original MILP \eqref{eq:MILPAlg} until some criteria are not met. In this phase, binary variables are restricted to take only 0 or 1 values. Let $\hat{\mb{u}}$ be the binary subvector of the current solution at the end of this phase.
\State {\bf ADMM-Bin-}: Fix the binary variables at their level of $\hat{\mb{u}}$. Continue ADMM, Algorithm \ref{Alg:ADMM}, to compute $z(\mb{\hat u})$ by solving LP \eqref{eq:LP1}
\If{$z(\mb{\hat u})<ub$ }
\State $ub \leftarrow z(\mb{\hat u})$
\State $\mb{u}^\ast \leftarrow \mb{\hat u}$
\EndIf
\EndWhile
\end{algorithmic}
\end{algorithm}
\cite{Feizollahi:2015Large} have developed an ADMM based a heuristic decomposition method, which was called release-and-fix to solve MILPs arising from electric power network unit commitment problems. Algorithm \ref{Alg:RaF} presents a high level scheme of the release-and-fix method. This algorithm along with some refinements were able to mitigate oscillations and traps in local optimality. This method was able to find very good solutions with relatively small optimality gap for large scale unit commitment problems \citep{Feizollahi:2015Large}. But, it was not able to get the exact solution of MILP \eqref{eq:MILPAlg}.
\section{Exact Distributed Algorithms} \label{Sec:ExactAlgs}
In this section, we propose a distributed MILP approach where each block solves its own modified LR subproblem iteratively.
The approach evaluates the cost of binary solutions as candidate partial solutions and refines them to get a primal feasible solutions to the overall problem. To improve the lower bound and prevent cycling in Algorithm \ref{Alg:DualD}, the explored binary solutions are then cut-off from future consideration in all subproblems.
This idea is similar to the scenario decomposition algorithm for two-stage 0-1 stochastic MILP problems proposed in \citep{Ahmed:2013scenario}. In the two-stage 0-1 stochastic MILP model at \citep{Ahmed:2013scenario}, each scenario is assumed to be a block and nonanticipativity constraints are coupling different scenarios. In that model, binary variables are only present in the first stage and they are the same for different scenarios. Therefore, it is straightforward to cutoff explored binary solutions from the feasible regions of all subproblems. On the contrary, in our loosely coupled MILP model \eqref{eq:MILPAlg}, binary variables are not the same for different blocks. Then, it is not clear how to cutoff a global binary solution from the feasible regions of subproblems. For instance, in Example \ref{Ex:MILPAlgExample}, $\mb{u}_1=(u_{11}, u_{12}, u_{13})^\top$ and $\mb{u}_2=(u_{21}, u_{22}, u_{23})^\top$ are completely different binary vectors for blocks 1 and 2, respectively. In Example \ref{Ex:MILPAlgExample}, consider $\hat{\mb u}=(\hat{\mb u}_1^\top,\hat{\mb u}_2^\top)^\top$ where $\hat{\mb u}_1=(1,1,0)^\top\in U_1$ and $\hat{\mb u}_2=(0,0,0)^\top\in U_2$. Then, it is a challenge to cutoff $\hat{\mb u}=(\hat{\mb u}_1^\top,\hat{\mb u}_2^\top)^\top=(1,1,0,0,0,0)^\top \in U_1\times U_2$ from the local feasible regions of blocks 1 and 2 in a distributed and parallel fashion.
In this section, we propose two exact algorithms to handle this process in a distributed framework.
For given $\mb{\mu}\in \mathbb{R}^m$ and ${\cal S}\subset U$, we define the \emph{restricted Lagrangian relaxation} (RLR)
\begin{equation}\label{eq:MILP1-RLR}
\begin{split}
z^\text{RLR}(\mb{\mu}, {\cal S}):=\mb{\mu}^\top \mb{b}+ \min\limits_{ \mb{x}_1, \cdots , \mb{x}_N } & \sum_{ \nu\in {\cal P} } {\cal L}_\nu(\mb{x}_\nu,\mb{\mu})
\\
\text{s.t. } &\mb{x}_\nu \in X_\nu,\, \forall \nu\in {\cal P}, \\
& \mb{u}\notin {\cal S}.
\end{split}
\end{equation}
Recall from Assumption \ref{Assump:MILPAlg}, $\mb{x}$ consists of the binary variables' subvector $\mb{u}$ and the continuous variables' subvector $\mb{y}$. Note that, $ub({\cal S}) :=\min\limits_{\mb{\hat u} \in {\cal S}} \{ z(\mb{\hat u}) \}$
and $lb(\mb{\mu}, {\cal S}):=\min \{ z^\text{RLR}(\mb{\mu}, {\cal S}), ub( {\cal S}) \}$ are valid upper and lower bounds for $z^\text{IP}$, respectively.
\begin{proposition}\label{prop:finite-iter1}
Consider MILP \eqref{eq:MILPAlg} under Assumption \ref{Assump:MILPAlg}. For any $\mb{\mu}\in \mathbb{R}^m$, there exists a set ${\cal S}\subset U$ such that $ub({\cal S}) =lb(\mb{\mu}, {\cal S})=z^\text{IP}$.
\end{proposition}
\proof{Proof}
By \eqref{eq:z-U}, we know that $ub(U) :=\min\limits_{\mb{\hat u} \in U} \{ z(\mb{\hat u}) \}=z^\text{IP}$.
Clearly, $z^\text{RLR}(\mb{\mu}, U)=+\infty$ and consequently $lb(\mb{\mu}, U):=\min \{ z^\text{RLR}(\mb{\mu}, U), ub(U) \}=ub(U)$. $\Box$
\endproof
Note that for any $\mb{\mu}\in \mathbb{R}^m$, $z^\text{RLR}(\mb{\mu}, {\cal S})$ and $ub({\cal S}) $ are non-decreasing and non-increasing, respectively, functions of ${\cal S}$,
i.e.
$z^\text{RLR}(\mb{\mu}, {\cal S}) \le z^\text{RLR}(\mb{\mu}, {\cal T})$ and $ub( {\cal S}) \ge ub({\cal T})$ for any pair of sets ${\cal S}$ and ${\cal T}$ such that ${\cal S} \subset {\cal T} \subset U$.
Therefore, for any $\mb{\mu}\in \mathbb{R}^m$, there exists a set ${\cal S}(\mb{\mu} )\subset U$ such that $z^\text{RLR}(\mb{\mu}, {\cal S}(\mb{\mu} ) )\ge ub({\cal S}(\mb{\mu} ))$ and consequently $lb(\mb{\mu}, {\cal S}(\mb{\mu} ))=z^\text{IP}=ub({\cal S}(\mb{\mu} ))$. In other words, it is possible to close the duality gap for MILP \eqref{eq:MILPAlg} by cutting off some finite number of binary solutions in \eqref{eq:MILP1-RLR} via constraints $ \mb{u}\notin {\cal S}$.
For a given binary vector $\hat{\mb{u}}\in \{0,1\}^{n^1}$ let us define the simple binary cut (SBC) of $\hat{\mb{u}}$ in terms of binary decision vector $\mb{u}\in \{0,1\}^{n^1}$ as follows:
\begin{equation}\label{eq:SBC}
\text{SBC} (\mb{u},\mb{\hat u}):~~ \sum_{k:\hat{u}_k=0} u_k + \sum_{k:\hat{u}_k=1} (1-u_k) \ge 1.
\end{equation}
Then, $\text{SBC} (\mb{u},\mb{\hat u})$ for $\hat{\mb u}=(1,1,0,0,0,0)^\top $ in Example \ref{Ex:MILPAlgExample} is the following inequality:
\begin{equation}\label{eq:SBC-Ex}
-u_{11}-u_{12}+u_{13}+u_{21}+u_{22}+u_{23} \ge -1.
\end{equation}
To cutoff multiple solutions, stronger cuts can be used as described in \citep{Angulo:2015}.
Using the concept of SBC, the constraint $\mb{u}\notin {\cal S}$ in \eqref{eq:MILP1-RLR} can be represented as $\text{SBC} (\mb{u},\mb{\hat u})$, for all $\mb{\hat u} \in {\cal S}$. However this constraint couples different blocks to each other and defeats the goal of problem decomposition.
For example, in constraint \eqref{eq:SBC-Ex}, all binary variables from blocks 1 and 2 are present.
Next, we propose different techniques to overcome this issue by introducing equivalent formulations of \eqref{eq:MILP1-RLR} which are decomposable.
\subsection{Binary Variables Duplication}
In our first approach of decoupling the constraint $\mb{u}\notin {\cal S}$ in \eqref{eq:MILP1-RLR}, we propose to duplicate the whole vector of binary variables and give a copy of it to each block.
For each pair of $\nu,\nu'\in {\cal P}$, let $\tilde{\mb u}_{\nu,\nu'}\in \tilde{U}_{\nu,\nu'} \subset \{0,1\}^{n_{\nu'}^1}$ be block $\nu$'s perception of $\mb{u}_{\nu'}$, where $U_{\nu,\nu'}$ is the set of all possible values for $\tilde{\mb u}_{\nu,\nu'}$.
For convenience, let $\tilde{\mb u}_\nu$ and $\tilde{U}_{\nu}$ be block $\nu$'s perception of ${\mb u}$ and $U$.
Note that $\tilde{\mb u}_\nu \in \{0,1\}^{n^1}$ and $\tilde{U}_{\nu}\subset \{0,1\}^{n^1}$, for all $\nu\in {\cal P}$.
It can be assumed $U_{\nu'} \subset \tilde{U}_{\nu,\nu'} $ for all $\nu\ne \nu'$ where it is possible that $U_{\nu'} \ne \tilde{U}_{\nu,\nu'}$.
For example one may assume $\tilde{U}_{\nu,\nu'}= \{0,1\}^{n_{\nu'}^1}$. Therefore, it may happen $\tilde{U}_{\nu,\nu'} \backslash U_{\nu'} \ne \emptyset$; i.e. block $\nu$ may not know any explicit or implicit descriptions of $U_{\nu'}$ and consequently its perception of $\mb{u}_{\nu}$ can be infeasible. But, block $\nu$ should receive an infeasibility alert from block $\nu'$, if $\mb{\hat u}_{\nu,\nu'}\notin U_{\nu'}$.
Then, $\mb{\hat u}_{\nu,\nu'}$ can be cut off from $\tilde{U}_{\nu,\nu'}$ using SBC($\mb{u}_{\nu,\nu'},\mb{\hat u}_{\nu,\nu'}$) as defined in \eqref{eq:SBC}.
In this algorithm, we assume $\tilde{U}_\nu=U$, for the sake of simplicity. Later, we will present other algorithms where the blocks do not need to know anything about the feasibility regions of the other blocks.
For Example \ref{Ex:MILPAlgExample}, blocks 1 and 2 perceptions of the overall binary vector $\mb{u}$ are
$\tilde{\mb u}_1=(\tilde{u}_{111},\tilde{u}_{112},\tilde{u}_{113},\tilde{u}_{121},\tilde{u}_{122},\tilde{u}_{123})^\top$ and
$\tilde{\mb u}_2=(\tilde{u}_{211},\tilde{u}_{212},\tilde{u}_{213},\tilde{u}_{221},\tilde{u}_{222},\tilde{u}_{223})^\top$, respectively. In this case,
$\tilde{\mb u}_{11}=(\tilde{u}_{111},\tilde{u}_{112},\tilde{u}_{113})^\top \in U_1$,
$\tilde{\mb u}_{12}=(\tilde{u}_{121},\tilde{u}_{122},\tilde{u}_{123})^\top\in U_1$,
$\tilde{\mb u}_{21}=(\tilde{u}_{211},\tilde{u}_{212},\tilde{u}_{213})^\top\in U_1$, and
$\tilde{\mb u}_{22}=(\tilde{u}_{221},\tilde{u}_{222},\tilde{u}_{223})^\top\in U_2$. Then, $\text{SBC} (\mb{u},\mb{\hat u})$ cut \eqref{eq:SBC-Ex} for $\hat{\mb u}=(1,1,0,0,0,0)^\top $ can be reformulated as
\begin{equation}\label{eq:SBC-Ex1}
-\tilde{\mb u}_{111}-\tilde{\mb u}_{112}+\tilde{\mb u}_{113}+\tilde{\mb u}_{121}+\tilde{\mb u}_{122}+\tilde{\mb u}_{123} \ge 1,
\end{equation}
and
\begin{equation}\label{eq:SBC-Ex2}
-\tilde{\mb u}_{211}-\tilde{\mb u}_{212}+\tilde{\mb u}_{213}+\tilde{\mb u}_{221}+\tilde{\mb u}_{222}+\tilde{\mb u}_{223} \ge 1.
\end{equation}
for blocks 1 and 2, respectively. Note that in inequality \eqref{eq:SBC-Ex1}, only (perception) binary variables from block 1 are present. Similarly, in inequality \eqref{eq:SBC-Ex2}, only (perception) binary variables from block 2 are present.
An equivalent formulation for \eqref{eq:MILP1-RLR} can be constructed by using the binary vectors $\tilde{\mb u}_1, \cdots, \tilde{\mb u}_N$, where all the blocks have the same perceptions of $\mb{u}$, i.e.
\begin{equation}\label{eq:MILP3-Consensus}
\tilde{\mb u}_1= \cdots= \tilde{\mb u}_N,
\end{equation}
and the $\mb{u}\notin {\cal S}$ is replaced by
\begin{equation}\label{eq:MILP3-XS}
\tilde{\mb u}_\nu \in U \backslash {\cal S}.
\end{equation}
In Example \ref{Ex:MILPAlgExample}, constraint \eqref{eq:MILP3-Consensus} has the following form
\begin{equation}\nonumber
\begin{split}
\tilde{u}_{111}&=\tilde{u}_{211},\\
\tilde{u}_{112}&=\tilde{u}_{212},\\
\tilde{u}_{113}&=\tilde{u}_{213},\\
\tilde{u}_{121}&=\tilde{u}_{221},\\
\tilde{u}_{122}&=\tilde{u}_{222},\\
\tilde{u}_{123}&=\tilde{u}_{223}.
\end{split}
\end{equation}
Note that for all $\nu'\ne \nu$, binary vectors $\tilde{\mb u}_{\nu, \nu'}$ are redundant. But, they make it possible to cut a global binary solution $\hat{\mb{u}}$ from the feasible region of all blocks. In other words, we use $\tilde{\mb u}_{\nu, \nu'}$ for $\nu' \ne \nu$ to handle constraint \eqref{eq:MILP3-XS}.
Let $\tilde{\mb x}_\nu:=(\tilde{\mb u}_\nu,\mb{y}_\nu)\in \{0,1\}^{n^1}\times \mathbb{R}^{n^2_\nu}$. Note that for all $\nu\in {\cal P}$, ${\mb u}_\nu$ is a subvector of $\tilde{\mb u}_\nu$ and consequently, $\mb{x}_\nu=({\mb u}_\nu,\mb{y}_\nu)$ is a subvector of $\tilde{\mb x}_\nu$.
Then, problem \eqref{eq:MILP1-RLR} can be reformulated as follows:
\begin{equation}\label{eq:MILP1-RLR2}
\begin{split}
z^\text{RLR}(\mb{\mu}, {\cal S})=\mb{\mu}^\top \mb{b}+ \min\limits_{ \tilde{\mb x}_1,\cdots,\tilde{\mb x}_N } & \sum_{ \nu\in {\cal P} } {\cal L}_\nu(\mb{x}_\nu,\mb{\mu})
\\
\text{s.t. } & \mb{x}_\nu \in X_\nu \text{ and } \tilde{\mb u}_\nu \in U \backslash {\cal S}, \, \forall \nu\in {\cal P}\\
& \tilde{\mb u}_1= \cdots= \tilde{\mb u}_N.
\end{split}
\end{equation}
In the model \eqref{eq:MILP1-RLR2}, the consensus constraints \eqref{eq:MILP3-Consensus} are joint between different blocks.
To decouple these constraints, we use vectors of dual variables $\mb{\lambda}_\nu \in \mathbb{R}^{n^1}$, for all $\nu$ such that $\sum_{\nu\in{\cal P} } \mb{\lambda}_\nu= \mb{0}$. Then, the new restricted Lagrangian relaxation for the model \eqref{eq:MILPAlg} is
\begin{equation}\label{eq:MILP3-LR}
\begin{split}
z^{\text{RLR}'}(\mb{\mu},\mb{\lambda},{\cal S}):=\mb{\mu}^\top \mb{b}+ \min\limits_{ \tilde{\mb x}_1,\cdots,\tilde{\mb x}_N } & \sum_{\nu\in {\cal P}} {\cal L}'_\nu(\tilde{\mb x}_\nu,\mb{\mu},\mb{\lambda}_\nu)\\
\text{s.t. } & \mb{x}_\nu \in X_\nu \text{ and } \tilde{\mb u}_\nu \in U \backslash {\cal S}, \, \forall \nu \in {\cal P},
\end{split}
\end{equation}
where $\mb{\lambda}=(\mb{\lambda}_1^\top,\cdots, \mb{\lambda}_N^\top)^\top$ and
$
{\cal L}'_\nu(\tilde{\mb x}_\nu,\mb{\mu},\mb{\lambda}_\nu):= (\mb{c}^\top_\nu-\mb{\mu}^\top \mb{A}_\nu) \mb{x}_\nu +\mb{\lambda}_\nu^\top \tilde{\mb u}_\nu.
$
To solve problem \eqref{eq:MILP3-LR}, it is sufficient for each block $\nu$ to solve its subproblem of
$\min\limits_{ \tilde{\mb x}_\nu } \{ {\cal L}'_\nu(\tilde{\mb x}_\nu,\mb{\mu},\mb{\lambda}_\nu): \mb{x}_\nu \in X_\nu \text{ and } \tilde{\mb u}_\nu \in U \backslash {\cal S}\}$.
Note that $z^{\text{RLR}'}(\mb{\mu},\mb{\lambda},{\cal S}) \le z^{\text{RLR}}(\mb{\mu},{\cal S})$, for all ${\cal S}\subset U$, $\mb{\mu}\in \mathbb{R}^m$ and $\mb{\lambda}_\nu\in \mathbb{R}^{n^1}$, $\forall \nu\in {\cal P}$ such that $\sum\limits_{\nu\in{\cal P} } \mb{\lambda}_\nu= \mb{0}$. Moreover, $z^{\text{RLR}'}(\mb{\mu},\mb{\lambda},{\cal S})$ is a non-decreasing function of ${\cal S}$.
\begin{algorithm}
\caption{Distributed MILP with Binary Variables Duplication}\label{Alg:DMILP1}
\begin{algorithmic}[1]
\State Run Algorithm \ref{Alg:DualD} to initialize $ub$, $lb$, $\mb{u}^\ast$, $\mb{\mu}^0$ and ${\cal S}$.
\State $\mb{\lambda}^0 \leftarrow \mb{0}$ and $k \leftarrow 0$.
\While{$ub>lb$}
\State Lower bounding:
\While{ some termination criteria is not met}\label{Alg1:LB-start}
\State $k \leftarrow k+1$
\For{$\nu:=1$ to $N$}
\State solve $ \min\limits_{ \tilde{\mb x}_\nu } \{ {\cal L}'_\nu(\tilde{\mb x}_\nu,\mb{\mu}^{k-1},\mb{\lambda}_\nu^{k-1}): \mb{x}_\nu \in X_\nu \text{ and } \tilde{\mb u}_\nu \in U \backslash {\cal S}\}$.
\State let $v_\nu^k$ be the optimal value and $\tilde{\mb x}_\nu^{k}=(\tilde{\mb u}_\nu^{k},\mb{y}_\nu^{k})$ be an optimal solution
\EndFor
\If{$lb< \mb{\mu}^\top \mb{b}+\sum\limits_{ \nu \in {\cal P}} v_\nu^k$}
\State $lb\leftarrow \min\left\{ub, \mb{\mu}^\top \mb{b}+\sum\limits_{ \nu \in {\cal P}} v_\nu^k \right\}$
\EndIf
\State $\bar{\mb u}^{k}\leftarrow \frac{1}{|{\cal P}|} \sum\limits_{\nu\in {\cal P} } \tilde{\mb u}^{k}_{\nu}$
\State $\mb{\mu}^{k} \leftarrow \mb{\mu}^{k-1}+\rho^{k}_\mu \left(\mb{b}-
\sum\limits_{\nu\in {\cal P} } \mb{A}_\nu \mb{x}_\nu^k \right) $
and $\mb{\lambda}^{k}_\nu \leftarrow \mb{\lambda}^{k-1}_\nu+\rho^{k}_\lambda \left(\tilde{\mb u}^{k}_\nu-\bar{\mb u}^{k} \right) $
\EndWhile \label{Alg1:LB-finish}
\State Let $\hat{{\cal S}}^k=\cup_{\nu \in {\cal P}} \{\tilde{\mb u}_\nu^{k}\}$.
\State Upper bounding: run Algorithm \ref{Alg:UpperBounding} for set $\hat{{\cal S}}$ to update $ub$ and $\mb{u}^\ast$.
\State $ {\cal S} \leftarrow {\cal S} \cup \hat{{\cal S}}^k $
\EndWhile
\end{algorithmic}
\end{algorithm}
Let $\rho_\mu^k, \rho_\lambda^k>0$ be the step size for updating the dual vectors $\mb{\mu}$ and $\mb{\lambda}$ at iteration $k$.
Then, our first exact distributed MILP method is as Algorithm \ref{Alg:DMILP1}. This algorithm is initialized by running ADMM to solve the LP relaxation and then switches to dual decomposition. In fact, this step initializes upper and lower bounds as well as dual vectors.
In the lower bounding loop (lines \ref{Alg1:LB-start}-\ref{Alg1:LB-finish}) of Algorithm \ref{Alg:DMILP1}, problem \eqref{eq:MILP3-LR} is solved in parallel by each block and the dual vectors $\mb{\mu}$ and $\mb{\lambda}$ are updated as well as the lower bound and candidate binary subvectors. Then, each candidate binary subvector is evaluated by solving an LP with ADMM method. In this step, the upper bound is updated. Finally, the candidate binary subvectors are added to set ${\cal S}$ and consequently are cutoff from feasible regions of all blocks. The algorithm continues until the lower bound hits the upper bound.
\begin{proposition}\label{prop:finite-DMILP1}
Algorithm \ref{Alg:DMILP1} can find an optimal solution of MILP \eqref{eq:MILPAlg} under Assumption \ref{Assump:MILPAlg} in a finite number of iterations.
\end{proposition}
\proof{Proof}
In the worst case, Algorithm \ref{Alg:DMILP1} needs to be run until cutting off all binary solutions in $U$, which are finite. But for any feasible dual vectors $\mb{\mu}$ and $\mb{\lambda}$, we know that $z^{\text{RLR}'}(\mb{\mu},\mb{\lambda},U)=+\infty> z^\text{IP}$ which implies $ub \ngtr lb$ and the algorithm terminates. $\Box$
\endproof
\subsection{Auxiliary Binary Variables}
In Algorithm \ref{Alg:DMILP1}, each block has as many binary variables as $n^1$, the number of overall binaries in the original MILP problem \eqref{eq:MILPAlg}. Moreover, each block $\nu$ needs to know the constraints defining the set $U_{\nu'}$, for all $\nu'\ne \nu$ or to be able to check the feasibility of $\tilde{\mb u}_{\nu,\nu'}$. Next, we propose another algorithm by introducing some auxiliary binary variables, in which different blocks do not need to know about other blocks' binary variables or feasible regions.
For a given ${\cal S}\subset U$, let ${\cal S}_\nu$, for all $ \nu\in {\cal P}$, be the minimal sets such that ${\cal S}_\nu \subset U_\nu$ and ${\cal S}\subset {\cal S}_1 \times \cdots \times {\cal S}_N$. That is for all $\mb{\hat u}_\nu\in {\cal S}_\nu$ and $\nu \in {\cal P}$, there exists a $\mb{\hat u}\in {\cal S}$ such that the $\nu$th block of $\mb{\hat u}$ is $\mb{\hat u}_\nu$.
Let $S_\nu:=\{1,\cdots, |{\mathcal S}_\nu|\}$ and denote the $l$th solution of ${\cal S}_\nu$ by $\hat{\mb u}_\nu (l)$.
\begin{example} \label{Ex:MILPAlgExample2}
Consider Example \ref{Ex:MILPAlgExample} with ${\cal S}=\{(1,1,0,0,0,0),(1,1,0,0,1,1)\}$. Then, it holds ${\cal S}_1=\{(1,1,0)\}$ and ${\cal S}_2=\{(0,0,0),(0,1,1)\}$.
\end{example}
For $\nu,\nu'\in {\cal P}$ and $l\in S_{\nu'}$, let $w_{\nu,\nu',l}$ be a binary variable which is $1$, if block $\nu$'s perception of $\mb{u}_{\nu'}$ is $\hat{\mb u}_{\nu'} (l)$, and $0$ otherwise.
For convenience, let $w_{\nu,\nu',0}$ be a binary variable which is $1$, if block $\nu$'s perception of $\mb{u}_{\nu'}$ is not in ${\cal S}_{\nu'}$, and $0$ otherwise. Then,
\begin{equation}\label{eq:binaryu}
w_{\nu,\nu',l}\in \{0,1\}, \, \forall \nu'\in N,\, l\in S_{\nu'}\cup \{0\}.
\end{equation}
Then, for Example \ref{Ex:MILPAlgExample2}, block 1 has auxiliary binary variables $w_{1,1,0}$, $w_{1,1,1}$, $w_{1,2,0}$, $ w_{1,2,1}$, $w_{1,2,2}$.
Binary variable $w_{111}$ is 1 if and only if block 1 perception of $\mb{u}_1$ are $(1,1,0)$. Binary variables $w_{121}$ and $w_{122}$ are 1 if and only if blocks 1 perceptions of $\mb{u}_2$ are $(0,0,0)$ and $(0,1,1)$, respectively. Similarly, $w_{110}$ and $w_{120}$ are 1 if and only if blocks 1 perceptions of $\mb{u}_1$ and $\mb{u}_2$ do not exist in ${\cal S}_1$ and ${\cal S}_1$, respectively. Likewise, block 2 has auxiliary binary variables $w_{2,1,0}$, $w_{2,1,1}$, $w_{2,2,0}$, $w_{2,2,1}$, $w_{2,2,2}$.
Note that block $\nu$ does not know the length of $\mb{u}_{\nu'}$ or the values in the $\hat{\mb u}_\nu (l)$, unless $\nu=\nu'$. Therefore, $\mb{u}_\nu= \hat{\mb u}_\nu (l)$ if and only if $w_{\nu,\nu,l}=1$. This relation between the binary vector $\mb{u}_\nu$ and the binary variable $w_{\nu,\nu,l}$ can be imposed by constraints \eqref{eq:XeqXhat} and \eqref{eq:XnoteqXhat}.
\begin{equation}\label{eq:XeqXhat}
\left\{ \begin{array}{l l}
u_{\nu k} \ge w_{\nu,\nu,l}, & \text{ if } \hat{u}_{\nu k}(l)=1 \\
u_{\nu k} \le 1-w_{\nu,\nu,l}, & \text{ Otherwise}
\end{array} \right.~~ \forall l\in S_{\nu}, k=1,\cdots,n_\nu^1,
\end{equation}
\begin{equation}\label{eq:XnoteqXhat}
\sum_{k:\hat{u}_{\nu k}(l)=0} u_{\nu k} + \sum_{k:\hat{u}_{\nu k}(l)=1} (1-u_{\nu k}) \ge w_{\nu, \nu,0}, \, \forall l\in S_\nu.
\end{equation}
Each block $\nu$ should consider exactly one of the binary solutions $\hat{\mb u}_{\nu'}$ in ${\cal S}_{\nu'}$, for all $\nu' \in {\cal P}$, i.e.
\begin{equation}\label{eq:ExactlyOne}
\sum_{l\in S_{\nu}\cup \{0\} }w_{\nu,\nu',l}=1, \, \forall \nu'\in {\cal P}.
\end{equation}
Inequality \eqref{eq:GlobalCut0} cuts the explored binary solutions to prevent cycling.
\begin{equation} \label{eq:GlobalCut0}
\sum_{\nu' \in {\cal P}} \left[\sum_{l:\mb{\hat u}_{\nu'}(l)\ne \mb{\hat u}_{\nu'}(s) } w_{\nu,\nu',l} + \sum_{l:\mb{\hat u}_{\nu'}(l)= \mb{\hat u}_{\nu'}(s)} (1-w_{\nu,\nu',l}) \right] \ge 1, \, \forall s\in {\mathcal S},
\end{equation}
Because of the constraints \eqref{eq:binaryu} and \eqref{eq:ExactlyOne}, constraint \eqref{eq:GlobalCut0} can be strengthened as follows:
\begin{equation} \label{eq:GlobalCut}
\sum_{
\substack{
\nu'\in{\cal P}\\ l\in S_{\nu'}:\mb{\hat u}_{\nu'}(l)\ne \mb{\hat u}_{\nu'}(s) }} w_{\nu,\nu',l} \le N-1, \, \forall s\in {\mathcal S}.
\end{equation}
Constraints \eqref{eq:XeqXhat}-\eqref{eq:ExactlyOne}, and \eqref{eq:GlobalCut} for block 2 in Example \ref{Ex:MILPAlgExample2} have the following form:
\begin{equation}\nonumber
\begin{split}
& \left.\begin{array}{l}
u_{21}\le 1- w_{221}, ~ u_{22}\le 1- w_{221}, ~u_{23}\le 1- w_{221},\\
u_{21}\le 1- w_{222}, ~ u_{22}\ge w_{222}, ~ u_{23}\ge w_{222},
\end{array} \right\} \text{Constraint \eqref{eq:XeqXhat}}\\
\end{split}
\end{equation}
\begin{equation}\nonumber
\begin{split}
& \left.\begin{array}{l}
u_{21}+u_{22}+u_{22} \ge w_{220},\\
u_{21}+1-u_{22}+1-u_{22} \ge w_{220},\\
\end{array} \right\} \text{Constraint \eqref{eq:XnoteqXhat}}\\
\end{split}
\end{equation}
\begin{equation}\nonumber
\begin{split}
& \left.\begin{array}{l}
w_{210}+w_{211}=1,\\
w_{220}+w_{221}+w_{223}=1,
\end{array} \right\} \text{Constraint \eqref{eq:ExactlyOne}}\\
\end{split}
\end{equation}
\begin{equation}\nonumber
\begin{split}
& \left.\begin{array}{l}
w_{211}+w_{221}\le 1,\\
w_{211}+w_{222}\le 1.
\end{array} \right\} \text{Constraint \eqref{eq:GlobalCut}}\\
\end{split}
\end{equation}
Let $\mb{w}_\nu$ be the vector of all binary variables $w_{\nu,\nu',l}$, for all $\nu'\in {\cal P}$ and all $l\in S_{\nu'}$
In the second distributed MILP algorithm, we use the auxiliary binary vector $\mb{w}_\nu\in \{0,1\}^{|{\cal P}|+\sum\limits_{\nu' \in {\cal P}} |{\cal S}_{\nu'}| }$, for all $\nu \in {\cal P}$, to develop another equivalent model for \eqref{eq:MILP1-RLR}. Considering the consensus constraints
\begin{equation}\label{eq:ConsensusU}
\mb{w}_1=\cdots= \mb{w}_N,
\end{equation}
problem \eqref{eq:MILP1-RLR} can be reformulated as follows.
\begin{equation}\label{eq:MILP-Alg2}
\begin{split}
z^\text{RLR}(\mb{\mu},{\cal S})=\mb{\mu}^\top \mb{b}+ \min\limits_{\mb{x}, \mb{w}_1,\cdots,\mb{w}_N } & \sum_{ \nu\in {\cal P} } {\cal L}_\nu(\mb{x}_\nu,\mb{\mu})\\
\text{s.t. } & \mb{x}_\nu \in X_\nu \text{ and } \eqref{eq:binaryu}-\eqref{eq:ExactlyOne},\eqref{eq:GlobalCut}, \, \forall \nu\in {\cal P}, \\
& \mb{w}_1=\cdots= \mb{w}_N.
\end{split}
\end{equation}
Consensus constraints \eqref{eq:ConsensusU} are coupling different block in the problem \eqref{eq:MILP-Alg2}.
To decouple these constraints, we use the feasible dual variable vectors $\mb{\gamma}_\nu \in \mathbb{R}^{{|{\cal P}|+\sum\limits_{\nu' \in {\cal P}} |{\cal S}_{\nu'}| }}$, for all $ \nu \in {\cal P}$ such that $\sum_{\nu\in{\cal P} } \mb{\gamma}_\nu= \mb{0}$. Then, the new restricted Lagrangian relaxation for the model \eqref{eq:MILPAlg} is
\begin{equation}\label{eq:MILP-RLR3}
\begin{split}
z^{\text{RLR}''} (\mb{\mu},\mb{\gamma},{\cal S}):=\mb{\mu}^\top \mb{b}+ & \min\limits_{\mb{x}, \mb{w}_1,\cdots,\mb{w}_N } \sum_{\nu\in {\cal P}} {\cal L}''_\nu(\mb{x}_\nu,\mb{w}_\nu,\mb{\mu},\mb{\gamma}_\nu)\\
&~~ \text{s.t. } ~~ \mb{x}_\nu \in X_\nu, \text{ and } \eqref{eq:binaryu}-\eqref{eq:ExactlyOne},\eqref{eq:GlobalCut}, \, \forall i\in {\cal N},
\end{split}
\end{equation}
where $\mb{\gamma}=(\mb{\gamma}_1, \cdots, \mb{\gamma}_N)$ and ${\cal L}''_\nu(\mb{x}_\nu,\mb{w}_\nu,\mb{\mu},\mb{\gamma}_\nu):= (\mb{c}^\top_\nu-\mb{\mu}^\top \mb{A}_\nu) \mb{x}_\nu +\mb{\gamma}_\nu \mb{w}_\nu$.
Note that $z^{\text{RLR}''} (\mb{\mu},\mb{\gamma},{\cal S}) \le z^{\text{RLR}}(\mb{\mu},{\cal S})$, for all ${\cal S}\subset U$, and feasible dual variable vectors $\mb{\mu}$ and $\mb{\gamma}$. Moreover, $z^{\text{RLR}''} (\mb{\mu},\mb{\gamma},{\cal S})$ is a non-decreasing function of ${\cal S}$.
Let $\rho_\gamma^k>0$ be the step size for updating the dual vector $\mb{\gamma}$ at iteration $k$. Then, our second exact distributed MILP approach is as Algorithm \ref{Alg:DMILP2}. The overall scheme of Algorithm \ref{Alg:DMILP2} is similar to Algorithm \ref{Alg:DMILP1}. The main difference is that instead of problem \eqref{eq:MILP3-LR}, problem \eqref{eq:MILP-RLR3} is solved in parallel in
the lower bounding loop (lines \ref{Alg2:LB-start}-\ref{Alg2:LB-finish}) of Algorithm \ref{Alg:DMILP2}. Different blocks do not need to know about other blocks' vector $\mb{u}_\nu$ of binary variables or feasible regions $U_\nu$ to solve problem \eqref{eq:MILP-RLR3} in parallel. Moreover, in line \ref{Step:1} of Algorithm \ref{Alg:DMILP2}, a new binary solution is added to ${\cal S}_\nu$ which results in adding a new corresponding binary variable $w$ and a new dual variable $\gamma$ to all blocks.
\begin{algorithm}
\caption{Distributed MILP with Auxiliary Binary Variables}\label{Alg:DMILP2}
\begin{algorithmic}[1]
\State Run Algorithm \ref{Alg:DualD} to initialize $ub$, $lb$, $\mb{u}^\ast$, $\mb{\mu}^0$ and ${\cal S}$.
\State Based on ${\cal S}$, set up the sets ${\cal S}_\nu$, for all $\nu\in {\cal P}$.
\State $\mb{\gamma}^0 \leftarrow \mb{0}$ and $k \leftarrow 0$.
\While{$ub>lb$}
\State Lower bounding:
\While{ some termination criteria is not met} \label{Alg2:LB-start}
\State $k \leftarrow k+1$.
\For{$\nu:=1$ to $N$}
\State solve $ \min\limits_{\mb{x}_\nu, \mb{w}_\nu } \{{\cal L}''_\nu(\mb{x}_\nu,\mb{w}_\nu,\mb{\mu}^{k-1},\mb{\gamma}_\nu^{k-1})
: \mb{x}_\nu \in X_\nu, \eqref{eq:binaryu}-\eqref{eq:ExactlyOne}, \eqref{eq:GlobalCut}\}$
\State let $v_\nu^k$ be the optimal value and $(\mb{x}_\nu^{k}, \mb{w}_\nu^{k})$ be an optimal solution
\EndFor
\If{$lb< \mb{\mu}^\top \mb{b}+\sum\limits_{\nu \in {\cal P}} v_\nu^k$}
\State $lb\leftarrow \min\left\{ub, \mb{\mu}^\top \mb{b}+\sum\limits_{\nu \in {\cal P}} v_\nu^k \right\}$
\EndIf
\State $\bar{\mb w}^{k}\leftarrow \frac{1}{|{\cal P}|} \sum\limits_{\nu\in {\cal P} } {\mb w}^{k}_{\nu}$
\State $\mb{\mu}^{k} \leftarrow \mb{\mu}^{k-1}+\rho^{k}_\mu \left(\mb{b}-\sum\limits_{\nu\in {\cal P} } \mb{A}_\nu \mb{x}_\nu^k \right) $ and $\mb{\gamma}^{k}_\nu \leftarrow \mb{\gamma}^{k-1}_\nu+\rho^{k}_\gamma \left({\mb w}^{k}_\nu-\bar{\mb w}^{k} \right) $
\EndWhile\label{Alg2:LB-finish}
\For{$\nu:=1$ to $N$}
\If{$\sum_{\nu' \in {\cal P}} w_{\nu,\nu',0} \ge 1$}
\State ${\cal S}_\nu \leftarrow {\cal S}_\nu \cup \{{\mb u}_\nu (0)\}$ \label{Step:1}
\EndIf
\EndFor
\State Let $\tilde{\mb u}_\nu^{k}$ be the corresponding $\tilde{\mb u}_\nu \in U$ to $\mb{w}_\nu^{k}$
\State $\hat{{\cal S}}\leftarrow \cup_{\nu \in {\cal P}} \{\tilde{\mb u}_\nu^{k}\}$.
\State Upper bounding: run Algorithm \ref{Alg:UpperBounding} for set $\hat{{\cal S}}$ to update $ub$ and $\mb{u}^\ast$.
\State $ {\cal S} \leftarrow {\cal S} \cup \hat{{\cal S}}^k $
\EndWhile
\end{algorithmic}
\end{algorithm}
\begin{proposition}\label{prop:finite-DMILP2}
Algorithm \ref{Alg:DMILP2} can find an optimal solution of MILP \eqref{eq:MILPAlg} under Assumption \ref{Assump:MILPAlg} in a finite number of iterations.
\end{proposition}
\proof{Proof}
In the worst case, Algorithm \ref{Alg:DMILP2} needs to be run until cutting off all binary solutions in $U$, which are finite. But for any feasible dual vectors $\mb{\mu}$ and $\mb{\gamma}$, we know that $z^{\text{RLR}''}(\mb{\mu},\mb{\gamma},U)=+\infty> z^\text{IP}$ which implies $ub \ngtr lb$ and the algorithm terminates. $\Box$
\endproof
\section{Illustrative Computations}\label{Sec:Computations}
In this section, we present numerical results testing the exact distributed MILP Algorithms \ref{Alg:DMILP1} and \ref{Alg:DMILP2} presented in Section \ref{Sec:ExactAlgs}, on small UC instances. We used 6 small unit commitment (UC) instances with 3, 4 and 5 generators for $T$=12 and 24 hours of planning. For details of UC formulation which is a MILP problem the reader can see \citep{Carrion:2006, Feizollahi:2015Large, Costley:2017}.
Table \ref{table:ExactTestCases} presents details of these instances. In Table \ref{table:ExactTestCases}, ``\# Gen'' and ``Gen. types'' denote the number and types of generator in each instance (see Table \ref{Table:GeneratorData} for details of each generator type). The total system demand at each hour is determined as given in Table \ref{Table:TotalDemand}.
The labels ``\# Bin. Vars.'', ``\# Cont. Vars.'', and ``\# Constr.'' denote the number of binary variables, continuous variables, and constraints, respectively, for each test case.
Moreover, the columns
$z^\text{LP}$, $z^\text{IP}$, ``Duality Gap'', and $t_\text{C}$ represent optimal objective value of LP relaxation and MILP formulation for UC, relative duality gap in percentage (between $z^\text{LP}$ and $z^\text{IP}$), and the solution time (in seconds) in central approach, respectively. An estimation for Lagrangian dual, which is obtained as the best lower bound in 100 iterations of the dual decomposition method, is denoted by $\tilde{z}^\text{LD}$. Note that finding an optimal vector of dual variables in the dual decomposition algorithm is not guaranteed. Then, $\tilde{z}^\text{LD}$ is not necessarily equal or close to the value of Lagrangian dual.
\begin{table}
\caption{Generator Data \citep{Carrion:2006}}
\label{Table:GeneratorData}
\center
\setlength\extrarowheight{2pt}
\scalebox{1}{
\begin{tabular}{|@{\hspace{0.2mm}}c@{\hspace{0.2mm}} |@{\hspace{0.2mm}}c@{\hspace{0.2mm}} |@{\hspace{0.2mm}}c@{\hspace{0.2mm}} |@{\hspace{0.2mm}}c@{\hspace{0.2mm}} |@{\hspace{0.2mm}}c@{\hspace{0.2mm}} |@{\hspace{0.2mm}}c@{\hspace{0.2mm}} |@{\hspace{0.2mm}}c@{\hspace{0.2mm}} |@{\hspace{0.2mm}}c@{\hspace{0.2mm}} |@{\hspace{0.2mm}}c@{\hspace{0.2mm}} |@{\hspace{0.2mm}}c@{\hspace{0.2mm}} |@{\hspace{0.2mm}}c@{\hspace{0.2mm}} |@{\hspace{0.2mm}}c@{\hspace{0.2mm}} |}
\hline
\multirow{3}{*}{Gen} & \multicolumn{6}{c|}{Technical Information} & \multicolumn{5}{c|}{Cost Coefficients}\\ \cline{2-12}
& \multicolumn{1}{c|}{$\overline{P}$} & \multicolumn{1}{c|}{$\underline{P}$} & \scriptsize{ TU/TD }& \scriptsize{RU/RD} & $T^{\text{Init}}$ & $T^{\text{cold}}$ & $C^{\text{NL}}$ & $C^{\text{LV}}$ & $C^{\text{Q}}$ & $C^{\text{HS}}$ & $C^{\text{CS}}$ \\
& (MW) & (MW) & (h) & (MW/h) & (h) & (h) & (\$/h) & (\$/MWh) & (\$/MW$^2$h) & (\$) & (\$) \\
\hhline{|============|}
1 & 455 & 150 & 8 & 225 & +8 & 5 & 1000 & 16.19 & 0.00048 & 4500 & 9000 \\[1pt] \hline
2 & 455 & 150 & 8 & 225 & +8 & 5 & 970 & 17.26 & 0.00031 & 5000 & 10000 \\[1pt] \hline
3 & 130 & 20 & 5 & 50 & -5 & 4 & 700 & 16.60 & 0.00200 & 550 & 1100 \\[1pt] \hline
4 & 130 & 20 & 5 & 50 & -5 & 4 & 680 & 16.50 & 0.00211 & 560 & 1120 \\[1pt] \hline
5 & 162 & 25 & 6 & 60 & -5 & 4 & 450 & 19.70 & 0.00398 & 900 & 1800 \\[1pt] \hline
6 & 80 & 20 & 3 & 60 & -3 & 2 & 370 & 22.26 & 0.00712 & 170 & 340 \\[1pt] \hline
7 & 85 & 25 & 3 & 60 & -3 & 2 & 480 & 27.74 & 0.00079 & 260 & 520 \\[1pt] \hline
8 & 55 & 10 & 1 & 135 & -1 & 0 & 660 & 25.92 & 0.00413 & 30 & 60 \\[1pt] \hline
\end{tabular}
}
\end{table}
\begin{table}
\caption{Total Demand (\% of Total Capacity)}
\label{Table:TotalDemand}
\center
\scalebox{.9}{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Time & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
Demand & 71\% & 65\% & 62\% & 60\% & 58\% & 58\% & 60\% & 64\% & 73\% & 80\% & 82\% & 83\% \\ \hline
Time & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\
Demand & 82\% & 80\% & 79\% & 79\% & 83\% & 91\% & 90\% & 88\% & 85\% & 84\% & 79\% & 74\% \\
\hline
\end{tabular}
}
\end{table}
\begin{table}
\center
\caption{Test case details for exact algorithms}\label{table:ExactTestCases}
\scalebox{0.85}{
\begin{tabular}{|c|p{1.5cm}|c|p{1.2cm}|p{1.4cm}|c|c|c|p{1.5cm}|p{.8cm}|c|}
\hline
\# Gen & Gen. types & T & \# Bin. Vars. & \# Cont. Vars. & \# Constr. & $z^\text{LP}$ & $z^\text{IP}$ & Duality Gap (\%) & $t_\text{C}$ (Sec) & $\tilde{z}^\text{LD}$\\ \hline \hline
\multirow{2}{*}{3}& \multirow{2}{*}{6,7,8} & 24 & 216 & 144 & 891 & 139896 & 146403 & 4.44 & 0.03 & 139933\\ \cline{3-11}
& & 12 & 108 & 72 & 435 & 68212 & 70945 & 3.85 & 0.09 & 68226\\ \hline \hline
\multirow{2}{*}{4}& \multirow{2}{*}{3,5,6,8} & 24 & 288 & 192 & 1244 & 207068 & 212771 & 2.68 & 0.14 & 207100\\ \cline{3-11}
& & 12 & 144 & 96 & 596 & 101676 & 104381 & 2.59 & 0.09 & 101686\\ \hline \hline
\multirow{2}{*}{5}& \multirow{2}{*}{1,5,6,7,8} & 24 & 360 & 240 & 1514 & 354684 & 359197 & 1.26 & 0.22 & 354705\\ \cline{3-11}
& & 12 & 180 & 120 & 722 & 171099 & 172994 & 1.10 & 0.11 & 171110\\ \hline
\end{tabular}
}
\end{table}
All algorithms were coded in C++ using CPLEX 12.6 through the Concert API. Central UC instances were solved using internal CPLEX multi-threading with four cores.
The step sizes $\rho_\mu$, $\rho_\lambda$ and $\rho_\gamma$ were set to be 0.01, 10 and 50, respectively. The algorithms start with running ADMM to solve the LP relaxations of the UC instances to initialize the vector of dual variables $\mb{\mu}$ and the lower bound $lb$. Then, they do 100 iterations of the dual decomposition algorithm to improve the lower bound. Then, the main body of Algorithms \ref{Alg:DMILP1} and \ref{Alg:DMILP2} starts with 200 iterations limit where the first 10 iterations are spent on updating dual vectors $\mb{\lambda}$ and $\mb{\gamma}$ without adding cuts. In each iteration, the lower bounding phase does 10 sub-iterations. Then, new candidate binary vectors are explored by the upper bounding procedure and cutoff from the feasible regions of all blocks.
\begin{table}
\center
\caption{Summary of the results for the exact Algorithm \ref{Alg:DMILP1}}\label{table:ResultsExactAlg1}
\begin{tabular}{|c|c|r|r
|r|r|r|r|r|r|r|}
\hline
\# Gen & T & $t_\text{0}$ & $t_\text{1}$ & $t^\ast$ & $t_\text{all}$ & iter$_1$ & iter$^\ast$ & iter$_\text{all}$ & \# Feas. & \# Cut\\ \hline \hline
\multirow{2}{*}{3} & 24 & 3.85 & 4.02 & 4.67 & 4.69 & 1 & 5 & 5 & 12 & 16\\ \cline{2-11}
& 12 & 2.16 & 2.23 & 2.59 & 3 & 1 & 4 & 7 & 12 & 19\\ \hline \hline
\multirow{2}{*}{4} & 24 & 4.54 & 4.7 & 7.4 & 193.1 & 1 & 11 & 118 & 90 & 530\\ \cline{2-11}
& 12 & 2.23 & 2.31 & 5.36 & 34.9 & 1 & 16 & 61 & 38 & 252\\ \hline \hline
\multirow{2}{*}{5} & 24 & 5.07 & 5.29 & 97.08 & 1621.72 & 1 & 42 & 190$^\ast$ & 303 & 1004\\ \cline{2-11}
& 12 & 2.18 & 2.29 & 3.01 & 715.88 & 1 & 5 & 190$^\ast$ & 290 & 962\\ \hline
\end{tabular}
\end{table}
\begin{table}
\center
\caption{Summary of the results for the exact Algorithm \ref{Alg:DMILP2}}\label{table:ResultsExactAlg2}
\begin{tabular}{|c|c|r|r
|r|r|r|r|r|r|r|}
\hline
\# Gen & T & $t_\text{0}$ & $t_\text{1}$ & $t^\ast$ & $t_\text{all}$ & iter$_1$ & iter$^\ast$ & iter$_\text{all}$ & \# Feas. & \# Cut\\ \hline \hline
\multirow{2}{*}{3} & 24 & 4.07 & 4.39 & 4.39 & 13.47 & 2 & 2 & 45 & 4 & 9\\ \cline{2-11}
& 12 & 2.35 & 2.49 & 2.49 & 3.97 & 1 & 1 & 11 & 4 & 7\\ \hline \hline
\multirow{2}{*}{4} & 24 & 4.56 & 4.81 & 4.81 & 6.81 & 2 & 2 & 10 & 12 & 23\\ \cline{2-11}
& 12 & 2.07 & 2.17 & 2.17 & 2.87 & 1 & 1 & 6 & 6 & 11\\ \hline \hline
\multirow{2}{*}{5} & 24 & 4.45 & 5.05 & 5.05 & 601.12 & 4 & 4 & 190$^\ast$ & 84 & 373\\ \cline{2-11}
& 12 & 1.98 & 2.19 & 2.19 & 24.06 & 3 & 3 & 38 & 54 & 122\\ \hline
\end{tabular}
\end{table}
Summary of the results for exact Algorithms \ref{Alg:DMILP1} and \ref{Alg:DMILP2} are presented in Tables \ref{table:ResultsExactAlg1} and \ref{table:ResultsExactAlg2}, respectively.
In Tables \ref{table:ResultsExactAlg1} and \ref{table:ResultsExactAlg2}, $t_\text{0}$, $t_\text{1}$, $t^\ast$, and $t_\text{all}$ are the estimated parallel times spent to initialize the algorithm, to find the first and best feasible solution, and to terminate the algorithm, respectively.
The exact algorithms were initialized by running ADMM for the LP relaxation and 100 iterations of the dual decomposition.
``iter$_1$'', ``iter$^\ast$'', ``iter$_\text{all}$'' are the corresponding number of iteration to $t_\text{1}$, $t^\ast$, and $t_\text{all}$, respectively.
``\# Feas.'', ``\# Cut'' are the number of feasible explored solutions and cuts (all explored binary solution), respectively.
For the 5 generator cases with $T$=24 and 12, Algorithm \ref{Alg:DMILP1} terminated with \% 1.078 and \%0.911 optimality gaps after 190 iterations. For the 5 generator case with $T$=24, Algorithm \ref{Alg:DMILP2} terminated with \%0.671 optimality gap after 190 iterations.
All other cases were solved to optimality. Based on the results in Tables \ref{table:ResultsExactAlg1} and \ref{table:ResultsExactAlg2}, for most cases, Algorithm \ref{Alg:DMILP2} outperforms Algorithm \ref{Alg:DMILP1}, in the sense that it requires less solution time ($t_\text{all}$), total number of iterations (iter$_\text{all}$) and cuts.
\section{Conclusions and Future Work}\label{Sec:Conclusion}
In this paper, we proposed exact distributed algorithms to solve MILP problems. A key challenge is that, because of
the non-convex nature of MILPs, classical distributed and decentralized optimization
approaches cannot be applied directly to find their optimal solutions.
The main contributions of the paper are as follows:
\begin{enumerate}
\item two exact distributed MILP algorithms which are able to optimally solve MILP problems in a distributed manner and output primal feasible solutions
\item primal cuts were added to restrict the Lagrangian relaxation
and improve the lower bound on the objective function of the original MILP problem.
\item illustrative computation on unit commitment problem.
\end{enumerate}
The main conclusions are as follows:
\begin{enumerate}
\item The proposed exact algorithms are proof-of-concept implementations to verify possibility of obtaining the global optimal solutions of MILPs in a distributed manner. Hence, the focus is not on computational times or number of iterations.
\item Algorithm \ref{Alg:DMILP2} requires less information exchange between block than Algorithm \ref{Alg:DMILP1}.
\item Based on the results in Tables \ref{table:ExactTestCases}-\ref{table:ResultsExactAlg2}, these exact distributed algorithms take much more time than the central approach. In particular, the solution times for Algorithms \ref{Alg:DMILP1} and \ref{Alg:DMILP2} are 3 seconds to 30 minutes while the central problems are solved in less than a second.
\item In general, Algorithm \ref{Alg:DMILP2} outperforms Algorithm \ref{Alg:DMILP1} with respect to solution time, number of iterations and number of cuts.
\item With the current implementation and numerical results, the main advantage of Algorithms \ref{Alg:DMILP1} and \ref{Alg:DMILP2} is that they preserve data privacy for different blocks.
\end{enumerate}
Finally, we note that distributed and decentralized optimization are dynamic and evolving area. Data privacy, distributed databases, and computational gains motivate to adapt distributed optimization in many industries such as electric power systems, supply chain, health care systems and etc. Therefore, developing fast and robust distributed exact and heuristic methods for MILPs.
A possible direction for future research is to blend the speed of R\&F and precision of the exact methods. Another topic for future work is investigating stronger primal cuts to speed up the proposed exact methods. Moreover, the proposed methods can be improved for specific applications by exploiting the problem structures.
\bibliographystyle{spmpsci}
|
1,314,259,996,542 | arxiv | \section{Introduction}\label{sec:intro}
Our goal in this paper is to prove various refinements of the following assertion:
\begin{theorem}[Main Theorem]\label{theorem:intro}
If $M$ and $N$ are modules over a commutative ring $R$ with $N$ Noetherian, then the existence of an injection in $\Hom_R(N,M)$ is a local property.
\end{theorem}
There are two reasons why we find this result worthy of report. One is the spectacular failure of the analogous statement for surjections, and the other is the voluminous response to this failure that has, at least to our knowledge, excluded consideration of the problem that we present here. We open with a survey of the related literature on epimorphisms and proceed to illustrate that, inasmuch as a comparison can be made, the monic case affords stronger conclusions with weaker hypotheses.
Theorems on factor modules relevant to our discussion come in two types. Findings of the first type deliver values $t$ for which $\Hom_R(M,N^{\oplus t})$ harbors a surjection, where $M$ and $N$ are modules over a ring~$R$. For example, in~\cite{Bai1}, the first author shows that, if $R$ is a $d$-dimensional commutative Noetherian ring and $M$ and $N$ are finitely generated $R$-modules such that $N^{\oplus (t+d)}$ is locally a factor of $M$, then $N^{\oplus t}$ is globally a factor of~$M$~\cite[Theorem~1.1(1)]{Bai1}. The same statement holds if we replace \textit{factor} with \textit{direct summand}~\cite[Theorem~1.1(2)]{Bai1}. Specializations of these results are due to several authors: Serre~\cite[\textit{Th\'eor\`eme~1}]{Ser} addresses the setting in which $M$ is projective and $N=R$; Bass~\cite[Theorem~8.2]{Bas} removes Serre's projective condition assuming that $t=1$; and De~Stefani--Polstra--Yao~\cite[Theorem~3.13]{DSPY} extends Bass by allowing $t$ to be an arbitrary positive integer. Additional variations on Serre's Splitting Theorem, all of which involve some notion of dimension, appear in Coquand--Lombardi--Quitt\'e~\cite[Corollary~3.2]{CLQ}, Heitmann~\cite[Corollary~2.6]{Heit}, and Stafford~\cite[Proposition~5.5]{Sta}. These results are sharp to the extent that the coordinate ring of the real $d$-sphere admits an indecomposable rank-$d$ projective module for every positive even integer~$d$~\cite[Theorem~3]{Swa2}. So, the existence of an epimorphism between two fixed modules, though not a local property, does follow from, and generally requires, certain dimensional restrictions, given that some finiteness conditions are in place.
This fact naturally affects module generation, the subject of the second type of result that concerns us here. For modules $M$ and $N$ over a ring~$R$, we say that \textit{$u$ copies of $M$ generate $N$} if $\Hom_R(M^{\oplus u},N)$ contains an onto map. The local existence of such integers $u$ does not ensure global existence: If $R=M$ is a direct product of infinitely many fields and $N$ is the direct sum of the local rings of~$R$, then $N$ is finitely generated locally but not globally. In order to avoid this sort of situation, theorists traditionally assume global existence when applying local data to the issue of module generation. Exploiting this concession, the first author certifies in a forthcoming paper that, if $M$ and $N$ are finitely generated modules over a $d$-dimensional commutative Noetherian ring $R$ such that $t-d$ copies of $M$ generate~$N$ locally, then $t$ copies of $M$ generate~$N$ globally~\cite{Bai4}. In the case that $M=R$, the preceding assertion is a theorem of Forster~\cite[{\textit{Satz~2}}]{For}. A refinement of Forster achieved by Swan~\cite[Theorem~2]{Swa} is the celebrated theorem now bearing the names of both men. Further elaborations on Forster's Theorem are due to Coutinho~\cite[Corollary~8.5]{Cou2}~\cite[Theorem~5.4]{Cou}, Eisenbud--Evans~\cite[Theorem~B]{EE}, Lyubeznik~\cite[Theorem~2]{Lyu}, and Warfield~\cite[Theorem~2]{War2}. Despite the many ways that Forster's Theorem has been extended, Forster's local condition cannot be improved in general: For all nonnegative integers $d$ and $t$ with $d<t$, there is a $d$-dimensional commutative ring $R$ admitting a projective module of rank $t-d$ that is not a factor of~$R^{\oplus (t-1)}$~\cite[Theorem~4]{Swa2}. Thus, once again, we are confronted with a question on epimorphisms that resists reduction to the local realm and exhibits strong ties to dimension, even with the provision of several finiteness hypotheses.
In contrast, our solution to the problem of embeddability boils down to the local case and obviates dimensional considerations. Moreover, the Noetherian condition on the source of our maps, which constitutes our only finiteness assumption, cannot be removed: Adapting a prior example, we observe that, if $R=N$ is a direct product of infinitely many fields and $M$ is the direct sum of the local rings of~$R$, then $N$ embeds into~$M$ locally but not globally. Assuming, on the other hand, that $M$ and $N$ are modules over a commutative ring $R$ with $N$ Noetherian, we may marshal the fact (Lemma~\ref{lemma:reduction} below) that a map $h\in\Hom_R(N,M)$ is monic if and only if $h_{\p}$, when restricted to the socle of~$N_{\p}$, is monic for every $\p$ in the finite set $\Ass_R(N)$. This characterization of injectivity allows us to use a ``general position" argument in the style of~\cite[Sections~6 and~7]{Bai1}, which would suffice to prove our main theorem. To complement the existing literature on direct summands of factor modules and on factor modules of direct sums, we could then replace $\Hom_R(N,M)$ in our main theorem with $\Hom_R(N^{\oplus t},M)$ or $\Hom_R(N,M^{\oplus t})$ for every nonnegative integer~$t$.
Instead, we take the following route: We begin by laying down the technical foundations for our argument in Section~\ref{sec:tech}, introducing two invariants of the triple $(M,N,R)$: the \textit{injective capacity of $M$ with respect to $N$ over~$R$} and the \textit{number of cogenerators of $N$ with respect to $M$ over~$R$}. In Section~\ref{sec:root-rank}, we establish results on roots of polynomials and ranks of matrices to which we appeal later in the paper. Section~\ref{sec:main} offers enhancements of our main theorem using injective capacities and numbers of cogenerators. Our main theorem follows as a corollary of these statements. Finally, in Section~\ref{sec:inj-graded}, we initiate an investigation on embeddings of graded modules.
\section{Conventions and definitions}\label{sec:tech}
In this section, we flesh out our conventions for the rest of the paper and cover definitions that will streamline our discussion.
Throughout this paper, the letter $R$ refers to a commutative ring with unity. The set of all maximal ideals of~$R$, called the \textit{maximal spectrum of~$R$}, is denoted $\Max(R)$. The set of all prime ideals of~$R$, referred to as the \textit{prime spectrum of~$R$}, is written as $\Spec(R)$. For every $\p\in\Spec(R)$, the symbol $\kappa(\p)$ represents the \textit{residue field of $R$ at $\p$} or, in other words, the ring $R_{\p}/\p_{\p}$.
Every $R$-module in this paper is standard. The letter $M$ refers to an arbitrary $R$-module, and $N$ stands for a Noetherian $R$-module. The \textit{annihilator of $N$ in $R$}, written as $\Ann_R(N)$, is the largest ideal $I$ of $R$ satisfying $IN=0$. We signify with $\Supp_R(N)$ the set of all $\p\in\Spec(R)$ such that $N_{\p}\neq 0$, and we call this set the \textit{support of $N$ over $R$}. For every $\p\in\Spec(R)$, the \textit{socle of $N_{\p}$ over $R_{\p}$}, denoted $\Soc_{R_{\p}}(N_{\p})$, is the $\kappa(\p)$-module consisting of all elements of $N_{\p}$ annihilated by~$\p_{\p}$. An \textit{associated prime of $N$ in $R$} is a prime $\p$ of $R$ such that $\Soc_{R_{\p}}(N_{\p})\neq 0$. The set $\Ass_R(N)$ is the finite set of all associated primes of $N$ in $R$. For every $R$-module~$L$ and nonnegative integer~$t$, the notation $L^{\oplus t}$ serves as shorthand for the direct sum of $t$ copies of~$L$, with $L^{\oplus 0}$ signifying the zero $R$-module. $\Hom_R(N,M)$ refers to the $R$-module of all $R$-linear maps from $N$ to~$M$. For every nonnegative integer~$t$, we view a member of $\Hom_R(N^{\oplus t},M)$ as a row $(h_1,\ldots,h_t)$, and we view a member of $\Hom_R(N,M^{\oplus t})$ as a column
\[
\begin{pmatrix}
h_1 \\
\vdots \\
h_t \\
\end{pmatrix},
\]
where $h_1,\ldots,h_t\in\Hom_R(N,M)$. Often, we write $(h_1,\ldots,h_t)^{\top}$ to denote the transpose of a row. For an $R$-submodule $F$ of $\Hom_R(N,M)$, context will determine the meaning of the symbol $F^{\oplus t}$: If we write $F^{\oplus t}\subseteq\Hom_R(N^{\oplus t},M)$, then $F^{\oplus t}$ refers to the $R$-module of all rows $(f_1,\ldots,f_t)$, where $f_1,\ldots,f_t\in F$; if we write $F^{\oplus t}\subseteq\Hom_R(N,M^{\oplus t})$, then $F^{\oplus t}$ represents the $R$-module of all columns of length $t$ with entries in $F$.
We now expand on two terms first mentioned in our introduction:
\begin{definition}\label{definition:inj}
We let $\inj_R^F(M,N)$ denote the supremum of the nonnegative integers $t$ for which an injection exists in $F^{\oplus t}\subseteq\Hom_R(N^{\oplus t},M)$. We call $\inj^F_R(M,N)$ the \textit{global injective capacity of $M$ with respect to $N$ over~$R$ when restricted to $F$}. For every $\p\in\Spec(R)$, we call $\inj^{F_{\p}}_{R_{\p}}(M_{\p},N_{\p})$ the \textit{local injective capacity of $M$ with respect to $N$ over $R$ when restricted to $F$ at~$\p$}. We omit the superscript $F$ and the phrase \textit{when restricted to $F$} in the case that $F=\Hom_R(N,M)$. We set $\sup(\varnothing)=0$ and $\sup(\{0,1,2,\ldots\})=\infty$.
\end{definition}
\begin{definition}\label{definition:cog}
The \textit{global number of cogenerators of $N$ with respect to $M$ over~$R$ when restricted to $F$}, denoted $\cog_R^F(N,M)$, refers to the infimum of the nonnegative integers $t$ such that there is an injection in $F^{\oplus t}\subseteq\Hom_R(N,M^{\oplus t})$. For every $\p\in\Spec(R)$, the \textit{local number of cogenerators of $N$ with respect to $M$ over $R$ when restricted to $F$ at~$\p$} is the number $\cog^{F_{\p}}_{R_{\p}}(N_{\p},M_{\p})$. If $F=\Hom_R(N,M)$, we withhold the superscript $F$ and all references to restriction. We set $\inf(\varnothing)=\infty$ and $\inf(\{0,1,2,\ldots\})=0$.
\end{definition}
To recapitulate our main conventions, $R$ denotes a commutative ring; $M$ refers to an arbitrary $R$-module; and $N$ signifies a Noetherian $R$-module.
\section{Roots and ranks}\label{sec:root-rank}
In preparation for our ``general position" arguments, we prove three lemmas below that may attract independent interest. Lemma~\ref{lemma:multi-poly-roots} describes a way to ensure, for example, that a Cartesian product of finite subsets of a field is not contained in (a fixed embedding of) a given affine algebraic set over that field. Building on this result, Lemma~\ref{lemma:multi-rank-block} specifies an instance in which two matrices, one of which has large rank, can be combined to produce a matrix that still has large rank. Lemma~\ref{lemma:multi-rank} is a variant of Lemma~\ref{lemma:multi-rank-block} that we use in the last section of our paper when discussing graded modules.
\begin{lemma}\label{lemma:multi-poly-roots}
Let $k_1,\ldots,k_s$ be fields and $x_1,\ldots,x_t$ indeterminates. For every $i\in\{1,\ldots,s\}$, let $f_i$ be a nonzero polynomial in $k_i[x_1,\dotsc, x_t]$ such that every monomial in its support divides~$x_1^{n_{i1}}\dotsb x_t^{n_{it}}$. For every $j\in\{1,\ldots,t\}$, let $C_j$ be a set with $|C_j| > \sum_{i=1}^s n_{ij}$, and suppose that there are embeddings $\phi_{j1},\ldots,\phi_{js}$ of $C_j$ into $k_1,\ldots,k_s$, respectively. Then there exists $(c_1, \dotsc, c_t) \in C_1 \times \dotsb \times C_t$ such that $f_i(\phi_{1i}(c_1),\dotsc,\phi_{ti}(c_t))\neq 0$ for every $i\in\{1,\ldots,s\}$.
\end{lemma}
\begin{proof}
Since $k[x_1,\dotsc,x_t]=k[x_1,\dotsc,x_{t-1}][x_t]$ for every field~$k$, induction on~$t$ suffices.
\end{proof}
\begin{lemma}\label{lemma:multi-rank-block}
For all $i\in\{1,\ldots,s\}$ and $j\in\{1,\ldots,t\}$, let $A_i$ and $B_i := (B_{i1}\hspace{1mm}\vert\dotsb\vert\hspace{1mm} B_{it})$ be $m_i \times n_i$ matrices with entries in a field $k_i$; suppose that $\rank(B_i) \ge r_i$ and that $B_{ij}$ has size $m_i\times n_{ij}$; and let $C_j$ be a set of size exceeding $\sum_{i=1}^s \min\{r_i,n_{ij}\}$ that can be embedded into $k_1,\ldots,k_s$ via maps $\phi_{j1},\ldots,\phi_{js}$, respectively. Then there exists $(c_1, \dotsc, c_t)\in C_1 \times \dotsb \times C_t$ such that $\rank(A_i + (\phi_{1i}(c_1)B_{i1}\hspace{1mm}\vert\dotsb\vert\hspace{1mm} \phi_{ti}(c_t)B_{it})) \ge r_i$ for every $i\in\{1,\ldots,s\}$. An analogous statement holds for the transposes of $A_1,B_1,\ldots,A_s,B_s$.
\end{lemma}
\begin{proof}
For every $i\in\{1,\ldots,s\}$, we may assume, without loss of generality, that $A_i$ and $B_i$ are both $r_i \times r_i$ matrices and that $\det(B_i) \neq 0$. (So, now, $r_i = n_{i1}+ \dotsb + n_{it}$.) For every $i\in\{1,\ldots,s\}$, let $f_i:=\det(A_i+(x_1B_{i1}\hspace{1mm}\vert\dotsb\vert\hspace{1mm} x_tB_{it})) = \det(A_i) + \dotsb + \det(B_i)(x_1^{n_{i1}}\dotsm x_t^{n_{it}})$ so that $f_i$ is a nonzero polynomial in $k_i[x_1,\dotsc, x_t]$ with every monomial in its support a factor of $x_1^{n_{i1}}\dotsm x_t^{n_{it}}$. Lemma~\ref{lemma:multi-poly-roots} now finishes the proof of the first claim of the lemma. The second claim can be verified in a similar manner.
\end{proof}
\begin{lemma}\label{lemma:multi-rank}
For all $i\in\{1,\ldots,s\}$ and $j\in\{1,\ldots,t\}$, let $A_i$ and $B_{ij}$ be $m_i \times n_i$ matrices with entries in a field $k_i$. Suppose that, for every $i\in\{1,\ldots,s\}$, there exists $j_i\in\{1,\ldots,t\}$ such that $\rank(B_{i,j_i})\geqslant r_i$. For every $j\in\{1,\ldots,t\}$, let $C_j$ be a set containing at least $1+\sum_{i=1}^s r_i$ elements but at most $\min\{|k_1|,\ldots,|k_s|\}$ elements so that there exist injections $\phi_{j1},\ldots,\phi_{js}$ from $C_j$ to $k_1,\ldots,k_s$, respectively. Then there exists $(c_1, \dotsc, c_t)\in C_1 \times \dotsb \times C_t$ such that $\rank(A_i + \phi_{1i}(c_1)B_{i1}+\cdots + \phi_{ti}(c_t)B_{it}) \ge r_i$ for every $i\in\{1,\ldots,s\}$.
\end{lemma}
\begin{proof}
For every $i\in\{1,\ldots,s\}$, we reduce to the case in which $m_i=n_i=r_i$ and $\det(B_{i,j_i})\neq 0$ and define $f_i:=\det(A_i + x_1B_{i1}+\cdots + x_tB_{it}) = \det(A_i) + \dotsb + \det(B_{i,j_i})x_{j_i}^{r_i} + \dotsb$, a nonzero polynomial in $k_i[x_1,\dotsc, x_t]$ with every monomial in its support dividing $x_1^{r_i}\dotsm x_t^{r_i}$. Applying Lemma~\ref{lemma:multi-poly-roots} is all that remains to be done.
\end{proof}
\section{Main results}\label{sec:main}
Our purpose in this section is to achieve several stronger versions of our main theorem. Our primary message is that neither all of $\Hom_R(N,M)$ nor the entirety of $\Supp_R(N)$ necessarily needs to be studied when estimating $\inj_R(M,N)$ or $\cog_R(N,M)$. Indeed, it suffices to work with an $R$-submodule of $\Hom_R(N,M)$ admitting large local injective capacities or small local numbers of cogenerators on the finite set $\Ass_R(N)$.
Each observation in this section also contributes something unique to our discussion. Theorem~\ref{theorem:inj} produces a finite list of maps in $\Hom_R(N,M)$ whose order is linked to local injective capacities on $\Ass_R(N)$, and Theorem~\ref{theorem:cog} serves as its dual for local numbers of cogenerators. Corollary~\ref{corollary:inj} delineates a local criterion for an $R$-submodule $L$ of~$M$ to intersect trivially with a isomorphic copy of~$N^{\oplus t}$ in~$M$, and Corollary~\ref{corollary:cog} replaces~$L$,~$M$, and $N^{\oplus t}$ in the last statement with~$L^{\oplus t}$,~$M^{\oplus t}$, and~$N$, respectively. This section ends with Theorem~\ref{theorem:summary}, which summarizes our findings: A global injective capacity is the infimum of its local analogues on~$\Ass_R(N)$; a global number of cogenerators is the supremum of the corresponding local invariants on~$\Ass_R(N)$; and an $R$-submodule of $\Hom_R(N,M)$ contains an injection if and only if its localizations at the associated primes of $N$ contain injections.
\begin{theorem}\label{theorem:inj}
Let $M$ and $N$ be modules over a commutative ring~$R$ with $N$ nonzero Noetherian, and let $F$ be an $R$-submodule of $\Hom_R(N,M)$. For every $\p\in\Ass_R(N)$, let $t(\p)$ be a positive integer, and suppose that $t(\p)\leqslant\inj^{F_{\p}}_{R_{\p}}(M_{\p},N_{\p})$. Next, for every multiplicatively closed subset $S$ of~$R$ avoiding~$\Ann_R(N)$, define $u(S):=\min\{t(\p):\p\in\Ass_R(N),\p\cap S=\varnothing\}$. Also, let $v:=\max\{t(\p):\p\in\Ass_R(N)\}$. Then there exist $h_1,\ldots,h_v\in F$ such that the map $S^{-1}(h_1,\ldots,h_{u(S)})$ is injective for every multiplicatively closed subset $S$ of~$R$ avoiding~$\Ann_R(N)$. Hence, letting $u:=u(\{1\})$, we find that $(h_1,\ldots,h_u)$ is injective.
\end{theorem}
To prove this theorem, it suffices to compute $h_1,\ldots,h_v\in F$ such that $(h_1,\ldots,h_{t(\p)})_{\p}$, when restricted to $\Soc_{R_{\p}}(N_{\p}^{\oplus t(\p)})$, is injective for every $\p\in\Ass_R(N)$, as Lemma~\ref{lemma:reduction} indicates below. To find such maps $h_1,\ldots,h_v$, we use Lemma~\ref{lemma:multi-rank-block} as part of a ``general position" argument somewhat reminiscent of~\cite[Sections~6 and~7]{Bai1}.
\begin{lemma}\label{lemma:reduction}
Let $M$ and $N$ be modules over a commutative ring~$R$ with $N$ Noetherian. Let $h\in\Hom_R(N,M)$. Then $h$ is injective if and only if $h_{\p}$, when restricted to $\Soc_{R_{\p}}(N_{\p})$, is injective for every $\p\in\Ass_R(N)$.
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lemma:reduction}]
The forward direction is obvious. For the reverse direction, let $K:=\ker(h)$. Then $\Ass_R(K)\subseteq\Ass_R(N)$. Since $\Soc_{R_{\p}}(K_{\p})=K_{\p}\cap\Soc_{R_{\p}}(N_{\p})=0$ for every $\p\in\Ass_R(N)$, we see that $\Ass_R(K)=\varnothing$. Hence $K=0$, and so $h$ is injective.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theorem:inj}]
The main portion of our proof works only for $\p\in\Ass_R(N)$ such that $|\kappa(\p)|>\dim_{\kappa(\p)}(\Soc_{R_{\p}}(N_{\p}))$. This condition always holds for the members of $\Ass_R(N)$ not in $\Max(R)$. To account for the remaining members of $\Ass_R(N)$, we take inspiration from the ``general position" argument in~\cite[Section~6]{Bai1}: For every $\m\in\Ass_R(N)\cap\Max(R)$, let $s(\m)$ be an element of $R$ that avoids $\m$ but belongs to every other member of $\Ass_R(N)\cap\Max(R)$, and let $d(\m)$ be a member of $F^{\oplus v}\subseteq\Hom_R(N^{\oplus v},M)$ whose first $t(\m)$ components form a map that becomes injective after localizing at $\m$ and whose last $v-t(\m)$ components are zero. Let
\[
(e_1,\ldots,e_v):=\sum_{\m\in\Ass_R(N)\cap\Max(R)} s(\m)d(\m).
\]
Then, for every $\m\in\Ass_R(N)\cap\Max(R)$, the map $(e_1,\ldots,e_{t(\m)})_{\m}$ becomes injective when restricted to $\Soc_{R_{\m}}(N_{\m}^{\oplus t(\m)})$. Hence, if $\Ass_R(N)\subseteq\Max(R)$, then we may take $(h_1,\ldots,h_v):=(e_1,\ldots,e_v)$ and apply Lemma~\ref{lemma:reduction} to close out the proof.
Otherwise, we may continue by revising the ``general position" argument from~\cite[Section~7]{Bai1} in the following way: Let $\q_1,\ldots,\q_m$ be the distinct members of $\Ass_R(N)\setminus\Max(R)$, and suppose that, for every $\ell\in\{1,\ldots,m\}$, the ideal $\q_1\cap\cdots\cap\q_{\ell-1}$ is not contained in~$\q_{\ell}$. Fix an $\ell\in\{1,\ldots,m\}$, and choose a map $(f_1,\ldots,f_v)\in F^{\oplus v}$ as follows: If $\ell=1$, then take $(f_1,\ldots,f_v):=(e_1,\ldots,e_v)$; if $\ell\geqslant 2$, then suppose inductively that, for every $\p\in\Ass_R(N)$ with $\p\not\in\{\q_{\ell},\ldots,\q_m\}$, the map $(f_1,\ldots,f_{t(\p)})_{\p}$ becomes injective when restricted to $\Soc_{R_{\p}}(N_{\p}^{\oplus t(\p)})$. Let $J$ denote the intersection of the members of $\Ass_R(N)\setminus\{\q_{\ell},\ldots,\q_m\}$, and let $\q:=\q_{\ell}$ and $t:=t(\q)$. Now let $(g_1,\ldots,g_t)\in F^{\oplus t}$ be a map that becomes injective after localizing at~$\q$. Since $|(J + \q)/\q| = \infty > \dim_{\kappa(\q)}(\Soc_{R_{\q}}(N_\q))$, we may use Lemma~\ref{lemma:multi-rank-block} to conclude that there exist $c_1,\ldots,c_t \in J$ such that $(f_1, \dotsc, f_t) + (c_1g_1, \dotsc, c_tg_t)$ becomes injective after localizing at $\q$ and restricting to $\Soc_{R_{\q}}(N_{\q}^{\oplus t})$. We may thus take $(f_1+c_1g_1, \dotsc, f_t+c_tg_t,f_{t+1},\ldots,f_v)$ to complete our inductive step, and we can conclude the proof with an application of Lemma~\ref{lemma:reduction}.
\end{proof}
\begin{corollary}\label{corollary:inj}
Let $A \le C$ and $B \le D$ be modules over a commutative ring~$R$ with $A$ nonzero Noetherian, and let $G$ be an $R$-submodule of $\Hom_R(C,D)$. For every $\p\in\Ass_R(A)$, let $t(\p)$ be a positive integer, and suppose that there exists a map in $G_{\p}^{\oplus t(\p)}\subseteq\Hom_{R_{\p}}(C_{\p}^{\oplus t(\p)},D_{\p})$ under which the preimage of $B_{\p}$ has trivial intersection with $A_{\p}^{\oplus t(\p)}$. Next, for every multiplicatively closed subset $S$ of~$R$ avoiding~$\Ann_R(A)$, define $u(S):=\min\{t(\p):\p\in\Ass_R(A),\p\cap S=\varnothing\}$. Also, let $v:=\max\{t(\p):\p\in\Ass_R(A)\}$. Then there exist $h_1,\ldots,h_v\in G$ such that the preimage of $S^{-1}B$ under $S^{-1}(h_1,\ldots,h_{u(S)})$ has trivial intersection with $S^{-1}A^{\oplus u(S)}$ for every multiplicatively closed subset $S$ of~$R$ avoiding~$\Ann_R(A)$. Hence, if $u:=u(\{1\})$, then the preimage of $B$ under $(h_1,\ldots,h_u)$ has trivial intersection with $A^{\oplus u}$.
\end{corollary}
\begin{proof}
This is an application of Theorem~\ref{theorem:inj} to $\inj_R^F(M,N)$ with $N=A$, $M = D/B$, and $F$ being the image of $G$ under the natural map $\Hom_R(C,D) \to \Hom_R(A,D/B)$.
\end{proof}
\begin{theorem}\label{theorem:cog}
Let $M$ and $N$ be modules over a commutative ring~$R$ with $N$ nonzero Noetherian, and let $F$ be an $R$-submodule of $\Hom_R(N,M)$. For every $\p\in\Ass_R(N)$, suppose that $\cog^{F_{\p}}_{R_{\p}}(N_{\p},M_{\p})$ is finite, and let $t(\p)$ be an integer $\geqslant\cog^{F_{\p}}_{R_{\p}}(N_{\p},M_{\p})$. For every multiplicatively closed subset $S$ of~$R$ avoiding~$\Ann_R(N)$, let $v(S):=\max\{t(\p):\p\in\Ass_R(N),\hspace{1mm}\p\cap S=\varnothing\}$. Also, let $v:=v(\{1\})$. Then there exist $h_1,\ldots,h_v\in F$ such that $S^{-1}(h_1,\ldots,h_{v(S)})^{\top}$ is injective for every multiplicatively closed subset $S$ of~$R$ avoiding~$\Ann_R(N)$. Hence $(h_1,\ldots,h_v)^{\top}$ is injective.
\end{theorem}
\begin{proof}
We may proceed as in the proof of Theorem~\ref{theorem:inj} except that we must take the transpose of every matrix involved and consider $N_{\p}$ and $M_{\p}^{\oplus t(\p)}$ instead of $N_{\p}^{\oplus t(\p)}$ and $M_{\p}$ for every $\p\in\Ass_R(N)$. Note that this proof uses the second claim of Lemma~\ref{lemma:multi-rank-block} rather than the first.
\end{proof}
\begin{corollary}\label{corollary:cog}
Let $A \le C$ and $B \le D$ be modules over a commutative ring~$R$ with $A$ nonzero Noetherian, and let $G$ be an $R$-submodule of $\Hom_R(C,D)$. For every $\p\in\Ass_R(A)$, let $t(\p)$ be an integer, and suppose that there is a map in $G_{\p}^{\oplus t(\p)}\subseteq\Hom_{R_{\p}}(C_{\p},D_{\p}^{\oplus t(\p)})$ under which the preimage of $B_{\p}^{\oplus t(\p)}$ has trivial intersection with $A_{\p}$. Next, for every multiplicatively closed subset $S$ of~$R$ avoiding~$\Ann_R(A)$, define $v(S):=\max\{t(\p):\p\in\Ass_R(A),\p\cap S=\varnothing\}$. Also, let $v:=v(\{1\})$. Then there exist $h_1,\ldots,h_v\in G$ such that the preimage of $S^{-1}B^{\oplus v(S)}$ under $S^{-1}(h_1,\ldots,h_{v(S)})^{\top}$ has trivial intersection with $S^{-1}A$ for every multiplicatively closed subset $S$ of~$R$ avoiding~$\Ann_R(A)$. Hence, if $v:=v(\{1\})$, then the preimage of $B^{\oplus v}$ under $(h_1,\ldots,h_v)^{\top}$ has trivial intersection with $A$.
\end{corollary}
\begin{proof}
This is an application of Theorem~\ref{theorem:cog} to $\cog_R^F(N,M)$ with $N=A$, $M = D/B$, and $F$ being the image of $G$ under the natural map $\Hom_R(C,D) \to \Hom_R(A,D/B)$.
\end{proof}
\begin{theorem}\label{theorem:summary}
Let $M$ and $N$ be modules over a commutative ring~$R$ with $N$ Noetherian, and let $F$ be an $R$-submodule of $\Hom_R(N,M)$. Then the following statements hold:
\begin{enumerate}
\item $\inj^F_R(M,N)=\inf\{\inj^{F_{\p}}_{R_{\p}}(M_{\p},N_{\p}):\p\in\Ass_R(N)\}$.
\item $\cog^F_R(N,M)=\sup\{\cog^{F_{\p}}_{R_{\p}}(N_{\p},M_{\p}):\p\in\Ass_R(N)\}$.
\item $F$ contains an injection if and only if $F_{\p}$ contains an injection for every $\p\in\Ass_R(N)$.
\end{enumerate}
In fact, in each of the statements above, we can replace $\Ass_R(N)$ with the set consisting of just the maximal members of~$\Ass_R(N)$.
\end{theorem}
\begin{proof}
(1) Let $t$ and $u$ denote the left and right sides of the asserted equation, respectively. Since an injective map remains injective upon localization, $t\leqslant u$. To prove the reverse inequality, we note that either $u=0$, in which case $u\leqslant t$ automatically, or else Theorem~\ref{theorem:inj} shows that $u\leqslant t$, including the case in which $u = \infty$.
(2) Label the left and right sides of the proposed equality as $t$ and~$v$, respectively. If $v$ is infinite, then $t\leqslant v$ at once; otherwise, Theorem~\ref{theorem:cog} suffices. We may also consider two cases to prove that $v\leqslant t$: If $t$ is infinite, then $v\leqslant t$ immediately; otherwise, we may appeal to the exactness of localization.
(3) We may apply either Part~(1) or Part~(2) of the present theorem since $F$ contains an injection if and only if $\inj^F_R(M,N)\geqslant 1$ if and only if $\cog^F_R(N,M)\leqslant 1$.
Having proved Parts (1), (2), and (3), we yield the last claim of the theorem by recalling once again that localization preserves injectivity.
\end{proof}
\section{A graded embedding theorem}\label{sec:inj-graded}
We close this paper with some thoughts on the existence of a homogeneous monomorphism between two $\ZZ$-graded modules. We refer the reader to~\cite[Section~1.5]{BH} for an introduction to $\ZZ$-graded rings and modules but summarize some standard notation and results here for the reader's convenience. To begin with, a \textit{$\ZZ$-grading} on a commutative ring $R$ is a decomposition of $R$ into a direct sum of $\ZZ$-modules $\ldots,R_{-1},R_0,R_1,\ldots$ such that, for all $i,j\in\ZZ$ and $a\in R_i$ and $b\in R_j$, the element $ab$ belongs to $R_{i+j}$. In this section, $R$ denotes a commutative ring with a fixed $\ZZ$-grading. A \textit{$\ZZ$-grading} on an $R$-module $M$ is a decomposition of $M$ into a direct sum of $\ZZ$-modules $\ldots,M_{-1},M_0,M_1,\ldots$ such that, for all $i,j\in\ZZ$ and $a\in R_i$ and $x\in M_j$, the element $ax$ belongs to $M_{i+j}$. Note that $R_0$ is a $\ZZ$-graded commutative ring and that every $\ZZ$-graded $R$-module is naturally a $\ZZ$-graded $R_0$-module. Henceforth, $M$ and $N$ signify $R$-modules with fixed $\ZZ$-gradings. For every $i\in\ZZ$, a member of $M_i$ is called a \textit{homogeneous element of $M$ of degree~$i$}, and $M_i$ is referred to as the \textit{$i$th homogeneous component of~$M$}. Also, for every~$i\in\ZZ$, the symbol $M[i]$ stands for the $\ZZ$-graded $R$-module such that, for every~$j\in\ZZ$, we have $(M[i])_j=M_{i+j}$. If $L$ is a $\ZZ$-graded $R$-submodule of~$M$, then $M/L$ is naturally a $\ZZ$-graded $R$-module such that, for every $i\in\ZZ$, we have $(M/L)_i\cong M_i/L_i$ as $R_0$-modules. We assume that $N$ is Noetherian, which is equivalent to assuming that every ascending chain of $\ZZ$-graded $R$-submodules of $N$ stabilizes. As a consequence, every member of $\Ass_R(N)$ is a $\ZZ$-graded ideal of $R$, and $\Hom_R(N,M)$ is a $\ZZ$-graded $R$-module such that, for every $i\in\ZZ$, the $i$th homogeneous component of $\Hom_R(N,M)$ is the set of all $f$ such that, for every~$j\in\ZZ$, we have $f(N_j)\subseteq M_{i+j}$. For every $\p\in\Spec(R)$, if $S$ is the set of all homogeneous elements of $R$ avoiding~$\p$, then the \textit{homogeneous localization $M_{(\p)}$ of $M$ at $\p$} refers to the $\ZZ$-graded $S^{-1}R$-module~$S^{-1}M$ such that, for every~$i\in\ZZ$, the $i$th homogeneous component of $S^{-1}M$ is the set of all $x/s$ such that $x\in M_{i+j}$ and $s\in R_j\setminus\p$ for some~$j\in\ZZ$. For every $\ZZ$-graded $\p\in\Spec(R)$, the \textit{socle of $N_{(\p)}$ over $R_{(\p)}$}, denoted $\Soc_{R_{(\p)}}(N_{(\p)})$, is the $\ZZ$-graded module over $R_{(\p)}/\p_{(\p)}$ consisting of all elements of $N_{(\p)}$ annihilated by~$\p_{(\p)}$. A prime $\p$ of $R$ is associated to $N$ if and only if $\p$ is $\ZZ$-graded and $\Soc_{R_{(\p)}}(N_{(\p)})\neq 0$.
We now offer a $\ZZ$-graded version of Theorem~\ref{theorem:summary} in the case that $R_0/\p_0$ is sufficiently large for every $\p\in\Ass_R(N)$.
\begin{theorem}\label{theorem:inj-graded}
Let $M$ and $N$ be $\ZZ$-graded modules over a $\ZZ$-graded commutative ring~$R$ with $N$ Noetherian. For every $\p\in\Ass_R(N)$, let $r(\p):=\rank_{R_{(\p)}/\p_{(\p)}}(\Soc_{R_{(\p)}}(N_{(\p)}))$; let $P(\p):=\{\p'\in\Ass_R(N):\p'_0=\p_0\}$; and assume that $|R_0/\p_0|>\sum_{\p'\in P(\p)} r(\p')$. Let $F$ be a $\ZZ$-graded $R$-submodule of $\Hom_R(N,M)$. Assume that there exists a degree $i$ such that, for every $\p \in \Ass_R(N)$, the $i$th homogeneous component $F_i$ of $F$ contains a map that becomes injective after homogeneous localization at~$\p$. Then $F_i$ contains an injection.
\end{theorem}
The following example shows that we cannot remove the uniformity condition on degree in Theorem~\ref{theorem:inj-graded}: Suppose that $R=R_0$ and that $R$ has two incomparable prime ideals $\p$ and~$\q$. Let $N: = (R/\p) \oplus (R/\q)$, and let $M := (R/\p)[-1] \oplus (R/\q)$. Then $\Hom_R(N,M)$ contains homogeneous maps $f$ and $g$ such that $f_{(\p)}$ and $g_{(\q)}$ are injective, but $\Hom_R(N,M)$ does not itself contain a homogeneous injection.
Since the uniformity condition in Theorem~\ref{theorem:inj-graded} cannot be removed, we would like to mention one way to guarantee that this condition is satisfied: First assume that, for every $\p\in\Ass_R(N)$, there exists a homogeneous map $f(\p)$ in $\Hom_R(N,M)$ of degree $i(\p)$ that becomes injective after homogeneous localization at~$\p$. Next, suppose that there exists $i \in \ZZ$ such that, for every $\p\in\Ass_R(N)$, we have $R_{i-i(\p)}\not\subseteq \p$. Then, for every $\p\in\Ass_R(N)$, we can choose $s(\p) \in R_{i - i(\p)}\setminus \p$ and then replace $f(\p)$ with $s(\p)f(\p) \in F_i$.
We can reduce the proof of Theorem~\ref{theorem:inj-graded} to finding a degree-$i$ homogeneous map in $\Hom_R(N,M)$ that becomes injective after homogeneously localizing at any associated prime of~$N$ and restricting to the socle, as the following lemma indicates. We omit the proof of the following lemma since it is similar to the one for Lemma~\ref{lemma:reduction}.
\begin{lemma}\label{lemma:reduction-graded}
Let $M$ and $N$ be $\ZZ$-graded modules over a $\ZZ$-graded commutative ring~$R$ with $N$ Noetherian. Let $h$ be a homogeneous map in $\Hom_R(N,M)$. Then $h$ is injective if and only if $h_{(\p)}$, when restricted to $\Soc_{R_{(\p)}}(N_{(\p)})$, is injective for every $\p\in\Ass_R(N)$.
\end{lemma}
\begin{proof}[Proof of Theorem~\ref{theorem:inj-graded}]
List the members of the set $Q:=\{\p_0 : \p \in \Ass_R(N)\}$ so that no member contains any of its predecessors. Let $\q\in Q$, and suppose inductively that there exists $f\in F_i$ such that, for every $\p\in\Ass_R(N)$ with $\p_0$ a predecessor of $\q$, the map $f_{(\p)}$ becomes injective when restricted to $\Soc_{R_{(\p)}}(N_{(\p)})$. Let $J$ denote the intersection of the predecessors of~$\q$, and let $P$ denote the fiber of $\q$ in $\Ass_R(N)$. For every $\p\in P$, let $g(\p)$ be a map in $F_i$ that becomes injective after homogeneously localizing at~$\p$. Note that $|(J+\q)/\q|>\sum_{\p\in P} r(\p)$. Hence, by Lemma~\ref{lemma:multi-rank}, there exists a function $c:P\rightarrow J$ such that, for every $\p\in P$, the map $f + \sum_{\p'\in P} c(\p')g(\p')$ becomes injective after homogeneously localizing at $\p$ and restricting to $\Soc_{R_{(\p)}}(N_{(\p)})$. Induction, followed by an application of Lemma~\ref{lemma:reduction-graded}, brings the proof to a close.
\end{proof}
\section*{Acknowledgements}\label{sec:acknowledgements}
The first author would like to thank Ela and Olgur Celikbas, Guantao Chen, Florian Enescu, Neil Epstein, Karl Schwede, Alexandra Smirnova, and Janet Vassilev for providing opportunities to speak about the results of this paper. The first author would also like to thank Craig Huneke for offering helpful feedback on the content of the paper. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
\bibliographystyle{amsplain}
|
1,314,259,996,543 | arxiv | \section{Introduction}\label{sec:Intro}
Coordinated networks of mobile robots are already in use for environmental monitoring
and warehouse logistics.
In the near future, autonomous robotic teams will revolutionize transportation of passengers and goods,
search and rescue operations, and other applications.
These tasks share a common feature: the robots are asked to provide service
over a space.
One question which arises is: when a group of robots is waiting for a task request to come in, how can they best position themselves to be ready to respond?
The distributed {\em environment partitioning problem} for robotic
networks consists of designing individual control and communication laws such
that the team divides a large space into regions. Typically, partitioning is
done so as to optimize a cost function which measures the quality of service
provided over all of the regions. {\em Coverage control} additionally optimizes the positioning
of robots inside a region as shown in Fig.~\ref{fig:cover_example}.
This paper describes a distributed partitioning and coverage control algorithm for a network of robots
to minimize the expected distance between the closest robot and spatially distributed events which will appear at discrete points in a non-convex environment.
Optimality is defined with reference to a relevant ``multicenter'' cost
function. As with all multirobot coordination
applications, the challenge comes from reducing the communication
requirements: the proposed algorithm requires only short-range ``gossip"
communication, i.e., asynchronous and unreliable
communication between nearby robots.
\begin{figure}[t]
\centering
\includegraphics[width=0.99\columnwidth]{sim_campus_2.eps}
\caption{Example of a team of robots providing efficient coverage of a non-convex environment, as measured by an appropriate multicenter cost function.}
\label{fig:cover_example}
\end{figure}
\subsection*{Literature Review}
Territory partitioning and coverage control have applications in many fields.
In cyber-physical systems, applications include automated environmental monitoring~\cite{RS-JD-GS:10}, fetching and delivery~\cite{PRW-RdA-MM:08}, construction~\cite{SY-MS-DR:09}, and other vehicle routing scenarios~\cite{FB-EF-MP-KS-SLS:10k}.
More generally, coverage of discrete sets is also closely related to the literature on data clustering and $k$-means~\cite{AKJ-MNM-PJF:99}, as well as the facility location or $k$-center problem~\cite{VVVa:01}.
Partitioning of graphs is its own field of research, see \cite{POF:98} for a survey.
Territory partitioning through local interactions is also studied for animal groups, see for example~\cite{FRA-DMG:03}.
A broad discussion of algorithms for partitioning and coverage control in robotic networks is presented in~\cite{FB-JC-SM:09} which builds on the classic work of Lloyd~\cite{SPL:82} on optimal quantizer selection through ``centering and partitioning."
The Lloyd approach was first adapted for distributed coverage control in~\cite{JC-SM-TK-FB:02j}.
Since this beginning, similar algorithms have been applied to non-convex environments~\cite{MZ-CGC:08,LCAP-VK-RCM-GASP:08}, unknown density functions~\cite{MS-DR-JJS:08,RC-HT-RL:09}, equitable partitioning~\cite{OB-OB-DK-QW:07}, and construction of truss-like objects~\cite{SY-MS-DR:09}.
There are also multi-agent partitioning algorithms built on market principles or auctions, see~\cite{MBD-RZ-NK-AS:06} for a survey.
While Lloyd iterative optimization algorithms are popular and work well in simulation,
they require synchronous and reliable communication among neighboring robots. As robots with adjacent regions may be arbitrarily far apart, these communication requirements are burdensome and unrealistic for deployed robotic networks.
In response to this issue, in~\cite{FB-RC-PF:08u-web} the authors have shown how a group of robotic agents can optimize the partition of a convex bounded set using a Lloyd algorithm with gossip communication.
A Lloyd algorithm with gossip communication has also been applied to optimizing partitions of non-convex environments in~\cite{JWD-RC-PF-FB:08z}, the key idea being to transform the coverage problem in Euclidean space into a coverage problem on a graph with geodesic distances.
Distributed Lloyd methods are built around separate partitioning and centering steps, and they are attractive because there are known ways to characterize their equilibrium sets (the so-called centroidal Voronoi partitions) and prove convergence. Unfortunately, even for very simple environments (both continuous and discrete) the set of centroidal Voronoi partitions may contain several sub-optimal configurations.
We are thus interested in studying (discrete) gossip coverage algorithms for two reasons: (1) they apply to more realistic robot network models featuring very limited communication in large non-convex environments, and (2) they are more flexible than typical Lloyd algorithms meaning they can avoid poor suboptimal configurations and improve performance.
\subsection*{Statement of Contributions}
There are three main contributions in this paper.
First, we present a discrete partitioning and coverage optimization algorithm for mobile robots with unreliable, asynchronous, and short-range communication.
Our algorithm has two components: a \emph{motion protocol} which drives the robots to meet their neighbors, and a \emph{pairwise partitioning rule} to update territories when two robots meet.
The partitioning rule optimizes coverage of a set of points connected by edges to form a graph. The flexibility of graphs allows the algorithm to operate in non-convex, non-polygonal environments with holes. Our graph partition optimization approach can also be applied to non-planar problems, existing transportation or logistics networks, or more general data sets.
Second, we provide an analysis of both the convergence properties and computational requirements of the algorithm. By studying a dynamical system of partitions of the graph's vertices, we prove that almost surely the algorithm converges to a pairwise-optimal partition in finite time. The set of pairwise-optimal partitions is shown to be a proper subset of the well-studied set of centroidal Voronoi partitions.
We further describe how our pairwise partitioning rule can be implemented to run in anytime and how the computational requirements of the algorithm can scale up for large domains and large teams.
Third, we detail experimental results from our implementation of the algorithm in the Player/Stage robot control system. We present a simulation of 30 robots providing coverage of a portion of a college campus to demonstrate that our algorithm can handle large robot teams, and a hardware-in-the-loop experiment conducted in our lab which incorporates sensor noise and uncertainty in robot position. Through numerical analysis we also show how our new approach to partitioning represents a significant performance improvement over both common Lloyd-type methods and the recent results in~\cite{FB-RC-PF:08u-web}.
The present work differs from the gossip Lloyd method~\cite{FB-RC-PF:08u-web} in three respects.
First, while~\cite{FB-RC-PF:08u-web} focuses on territory partitioning in a convex continuous domain, here we operate on a graph which allows our approach to consider geodesic distances, work in non-convex environments, and maintain connected territories.
Second, instead of a pairwise Lloyd-like update, we use an iterative optimal two-partitioning approach which yields better final solutions. Third, we also present a motion protocol to produce the sporadic pairwise communications required for our gossip algorithm and characterize the computational complexity of our proposal.
Preliminary versions of this paper appeared in~\cite{JWD-RC-PF-FB:08z} and \cite{JWD-RC-FB:10h}. Compared to these, the new content here includes: (1) a motion protocol; (2) a simplified and improved pairwise partitioning rule; (3) proofs of the convergence results; and (4) a description of our implementation and a hardware-in-the-loop experiment.
\subsection*{Paper Structure and Notation}
In Section~\ref{sec:prelim} we review and adapt coverage and geometric concepts (e.g., centroids, Voronoi partitions) to a discrete environment like a graph. We formally describe our robot network model and the discrete partitioning problem in Section~\ref{sec:algorithm}, and then state our coverage algorithm and its properties.
Section~\ref{sec:convergence} contains proofs of the main convergence results. In Section~\ref{sec:results} we detail our implementation of the algorithm and present experiments and comparative analysis. Some conclusions are given in Section~\ref{sec:conclusion}.
In our notation, $\R_{\geq 0}$ denotes the set of non-negative real numbers
and $\ensuremath{\mathbb{Z}}_{\ge 0}$ the set of non-negative integers. Given a
set $A$, $\card{A}$ denotes the number of elements in $A$.
Given sets $A,B$, their difference is $A\setminus B=\setdef{a\in A}{a\notin B}$. A set-valued map, denoted by $\setmap{T}{A}{B}$, associates to an
element of $A$ a subset of $B$.
\section{Preliminaries}
\label{sec:prelim}
We are given a team of $N$ robots tasked with providing coverage of a finite set of points in a non-convex and non-polygonal environment. In this Section we translate concepts used in coverage of continuous environments to graphs.
\subsection{Non-convex Environment as a Graph}
Let $Q$ be a finite set of points in a continuous environment. These points
represent locations of interest, and are assumed to be connected by weighted edges.
Let $G(Q)=(Q,E,w)$ be an (undirected) weighted graph with edge set
$E\subset Q\times Q$ and weight map $\map{w}{E}{\ensuremath{\mathbb{R}}_{>0}}$; we let
$w_{e}>0$ be the weight of edge $e$.
We assume that $G(Q)$ is connected and think of the edge weights as
distances between locations.
\begin{remark}[Discretization of an Environment]\label{rem:discretization}
For the examples in this paper we will use a coarse {\em occupancy grid map}
as a representation of a continuous environment. In an occupancy grid~\cite{HM:88}, each grid cell is
either free space or an obstacle (occupied). To form a weighted graph,
each free cell becomes a vertex and free cells are connected with edges
if they border each other in the grid.
Edge weights are the distances between the centers of the cells, i.e., the grid resolution.
There are many other methods to discretize a space, including triangularization and other approaches
from computational geometry~\cite{MdB-MvK-MO-OS:00}, which could also be used.
\end{remark}
In any weighted graph $G(Q)$ there is a standard notion of distance between
vertices defined as follows. A \emph{path} in $G$ is an ordered sequence of
vertices such that any consecutive pair of vertices is an edge of
$G$. The \emph{weight of a path} is the sum of the weights of the
edges in the path. Given vertices $h$ and $k$ in $G$, the
\emph{distance} between $h$ and $k$, denoted $d_G(h,k)$, is the weight of
the lowest weight path between them, or $+\infty$ if there is no path. If $G$ is connected,
then the distance between any two vertices in $G$ is finite.
By convention, $d_G(h,k)=0$ if $h=k$. Note
that $d_G(h,k)=d_G(k,h)$, for any $h,k \in Q$.
\subsection{Partitions of Graphs}
We will be partitioning $Q$ into $N$ connected subsets or
regions which will each be covered by an individual robot. To do so we need to
define distances on induced subgraphs of $G(Q)$.
Given $I\subset Q$, the \emph{subgraph induced by the
restriction of $G$ to $I$}, denoted by $G\ensuremath{\operatorname{\cap}}{}I$, is the graph
with vertex set equal to $I$ and edge set containing
all weighted edges of $G$ where both vertices belong to $I$. In other
words, we set
$(Q,E,w)\ensuremath{\operatorname{\cap}}{}I=(Q\ensuremath{\operatorname{\cap}}{}I,E\ensuremath{\operatorname{\cap}}{}(I\times{I}),w|_{I\times{I}})$.
The induced subgraph is a weighted graph with a notion of
distance between vertices: given $h,k\in I$, we write
$
d_I(h,k) := d_{G\ensuremath{\operatorname{\cap}}{}I}(h,k).
$
Note that $d_I(h,k)\ge d_G(h,k).$
We define a {\em connected subset of $Q$} as a subset
$A \subset Q$ such that $A\neq\emptyset$ and $G \ensuremath{\operatorname{\cap}} A$ is
connected.
We can then partition $Q$ into connected subsets as follows.
\begin{definition}[Connected Partitions]
\label{def:ConPartitions}
Given the graph $G(Q)=(Q,E,w),$ we define a {\em connected $N-$partition
of $Q$} as a collection $P=\{P_i\}_{i=1}^{N}$ of $N$ subsets of $Q$ such that
\begin{enumerate}
\item ${\bigcup_{i=1}^{N}P_i=Q}$;
\item $P_i\cap P_j=\emptyset$ if $i\neq j$;
\item $P_i\neq\emptyset$ for all $i\in \until{N}$; and
\item $P_i$ is connected for all $i\in \until{N}$.
\end{enumerate}
Let $\ConnPart$ to be the set of connected $N-$partitions
of $Q$.
\end{definition}
Property (ii) implies that each element of $Q$ belongs to just one $P_i$, i.e., each location in the environment is covered by just one robot. Notice that each $P_i \in P$ induces a connected subgraph in $G(Q)$.
In subsequent references to $P_i$ we will often mean $G \ensuremath{\operatorname{\cap}}{} P_i$, and in fact we refer to $P_i(t)$ as the \emph{dominance subgraph} or \emph{region} of the $i$-th robot at time $t$.
Among the ways of partitioning $Q$, there are some which are worth special attention. Given a vector of distinct points $c\in Q^N$, the partition $P \in \ConnPart$ is said to be a \emph{Voronoi partition of Q generated by c} if, for each $P_i$ and all $k \in P_i$, we have
$c_i\in P_i$
and
$d_G(k,c_i) \le d_G(k,c_j)$, $\forall j\neq i$.
Note that the Voronoi partition generated by $c$ is not unique since how to apportion tied vertices is unspecified.
\subsection{Adjacency of Partitions}
For our gossip algorithms we need to introduce the notion of adjacent subgraphs. Two distinct connected subgraphs $P_i$, $P_j$ are said to be \emph{adjacent} if there are two vertices $q_i$, $q_j$ belonging, respectively, to $P_i$ and $P_j$ such that $(q_i, q_j) \in E$. Observe that if $P_i$ and $P_j$ are adjacent then $P_i \operatorname{\cup} P_j$ is connected.
Similarly, we say that robots $i$ and $j$ are adjacent or are neighbors if their subgraphs $P_i$ and $P_j$ are adjacent.
Accordingly, we introduce the following useful notion.
\begin{definition}[Adjacency Graph]
For $P\in \ConnPart$, we define
the {\em adjacency graph} between regions of partition $P$ as
$\AdjG(P)=(\{1,\ldots,N\},\E(P))$, where $(i,j)\in \E(P)$ if $P_i$ and $P_j$ are adjacent.
\end{definition}
Note that $\AdjG(P)$ is always connected since $G(Q)$ is.
\subsection{Cost Functions}
We define three coverage cost functions for graphs: $\subscr{\mathcal{H}}{one}$, $\subscr{\mathcal{H}}{multicenter}$, and $\Hexp$.
Let the \emph{weight function} $\map{\phi}{Q}{\ensuremath{\mathbb{R}}_{>0}}$ assign a relative weight to each element of $Q$.
The {\em one-center function} $\subscr{\mathcal{H}}{one}$ gives the cost for a robot to cover a connected subset $A \subset Q$ from a vertex $h\in A$ with relative prioritization set by $\phi$:
$$\subscr{\mathcal{H}}{one}(h; A)={\sum_{k\in A} {d_{A}(h,k)\phi(k)}}.$$
A technical assumption is needed to solve the problem of minimizing $\subscr{\mathcal{H}}{one}(\cdot, A)$: we assume from now on that a {\em total order} relation, $<$, is defined on $Q$, i.e., that $Q=\until{\card{Q}}$. With this assumption we can deterministically pick a vertex in $A$ which minimizes $\subscr{\mathcal{H}}{one}$ as follows.
\begin{definition}[Centroid]\label{def:Centroid}
Let $Q$ be a totally ordered set, and let $A \subset Q$. We define the set of generalized centroids of $A$ as the set of vertices in $A$ which minimize $\subscr{\mathcal{H}}{one}$, i.e.,
\begin{align*}
\Centroids(A):=\mathop{\operatorname{argmin}}_{h\in A} \subscr{\mathcal{H}}{one}(h;A).
\end{align*}
Further, we define the map $\Cd$ as $\Cd(A) := \min\{ c\in \Centroids(A) \}$. We call $\Cd(A)$ the \emph{generalized centroid} of $A$.
\end{definition}
In subsequent use we drop the word ``generalized" for brevity. Note that with this definition the centroid is well-defined, and also that the centroid of a region always belongs to the region.
With a slight notational abuse, we define $\map{\Cd}{\ConnPart}{Q^N}$ as the map which associates to a partition the vector of the centroids of its elements.
We define the \emph{multicenter function}
$\subscr{\mathcal{H}}{multicenter}$ to measure the cost for $N$ robots to cover a connected $N$-partition $P$ from the vertex set $c \in Q^N$:
$$\subscr{\mathcal{H}}{multicenter}(c,P)=\frac{1}{\sum_{k\in Q} \phi(k)}\sum_{i=1}^{N} \subscr{\mathcal{H}}{one}(c_i; P_i).$$
We aim to minimize the performance function $\subscr{\mathcal{H}}{multicenter}$ with respect to both the vertices $c$ and the partition $P$.
We can now state the coverage cost function we will be concerned with for the rest of this paper. Let $\map{\Hexp}{\ConnPart}{\R_{\geq 0}}$
be defined by
\begin{align*}
\Hexp(P) &= \subscr{\mathcal{H}}{multicenter}(\Cd(P),P).
\end{align*}
In the motivational scenario we are considering, each robot will periodically be asked to perform a task somewhere in its region with tasks appearing according to distribution $\phi$. When idle, the robots would position themselves at the centroid of their region. By partitioning $G$ so as to minimize $\Hexp$, the robot team would minimize the expected distance between a task and the robot which will service it.
\subsection{Optimal Partitions}
We introduce two notions of optimal partitions: centroidal Voronoi and pairwise-optimal.
Our discussion starts with the following simple result about the multicenter cost function.
\begin{proposition}[Properties of Multicenter Function]\label{prop:optimal-for-Hgeneric}
Let $P\in \ConnPart$ and $c\in Q^N$. If $P'$ is a Voronoi partition generated by $c$ and
$c' \in Q^n$ is such that $c'_i \in \Centroids(P_i) ~\forall~ i$, then
\begin{align*}
\subscr{\mathcal{H}}{multicenter}(c,P') & \le \subscr{\mathcal{H}}{multicenter}(c,P), \text{and}\\
\subscr{\mathcal{H}}{multicenter}(c',P) &\le \subscr{\mathcal{H}}{multicenter}(c,P).
\end{align*}
The second inequality is strict if any $c_i \notin \Centroids(P_i)$.
\end{proposition}
Proposition~\ref{prop:optimal-for-Hgeneric} implies the following necessary condition:
if $(c,P)$ minimizes $\subscr{\mathcal{H}}{multicenter}$, then $c_i \in \Centroids(P_i) ~ \forall i$ and $P$ must be a Voronoi partition generated by~$c$.
Thus, $\Hexp$ has the following property as an immediate consequence of Proposition~\ref{prop:optimal-for-Hgeneric}: given $P\in \ConnPart$, if $P^*$ is a Voronoi partition generated by $\Cd(P)$ then
$$
\Hexp(P^*)\leq \Hexp(P).
$$
This fact motivates the following definition.
\begin{definition}[Centroidal Voronoi Partition]
$P\in \ConnPart$ is a \emph{centroidal Voronoi partition} of $Q$ if there exists a $c\in Q^n$ such that $P$ is a Voronoi partition generated by $c$ and $c_i \in \Centroids(P_i) ~\forall~ i$.
\end{definition}
\smallskip
The set of \emph{pairwise-optimal partitions} provides an alternative definition for the optimality of a partition: a partition is pairwise-optimal if, for every pair of adjacent regions, one can not find a better two-partition of the union of the two regions.
This condition is formally stated as follows.
\begin{definition}[Pairwise-optimal Partition]
$P\in \ConnPart$ is a \emph{pairwise-optimal partition} if for every $(i,j)\in \E(P)$,
\begin{align*}
&\quad \subscr{\mathcal{H}}{one}(\Cd(P_i); P_i)+\subscr{\mathcal{H}}{one}(\Cd(P_j); P_j)=\\
&~\min_{a, b \in P_i \cup P_j} \biggl\{\sum_{k\in P_i \cup P_j} \min \left\{d_{P_i \cup P_j} (a,k), d_{P_i \cup P_j} (b,k)\right\} \phi(k) \biggr\}.
\end{align*}
\end{definition}
The following Proposition states that the set pairwise-optimal partitions is in fact a subset of the set of centroidal Voronoi partitions. The proof is involved and is deferred to Appendix~\ref{sec:appendix_C}. See Fig.~\ref{fig:voronoi} for an example which demonstrates that the inclusion is strict.
\begin{proposition}[Pairwise-optimal Implies Voronoi]\label{prop:OptPair}
Let $P \in \ConnPart$ be a \emph{pairwise-optimal partition}. Then $P$ is also a \emph{centroidal Voronoi partition}.
\end{proposition}
For a given environment $Q$, a pair made of a centroidal Voronoi partition $P$ and the corresponding vector of centroids $c$ is locally optimal in the following sense: $\Hexp$ cannot be reduced by changing either $P$ or $c$ independently.
A pairwise-optimal partition achieves this property and adds that
for every pair of neighboring robots $(i,j)$, there does not exist a two-partition of $P_i \cup P_j$ with a lower coverage cost.
In other words, positioning the robots at the centroids of a centroidal Voronoi partition (locally) minimizes the expected distance between a task
appearing randomly in $Q$ according to relative weights
$\phi$ and the robot who owns the vertex where the task appears. Positioning at the centroids of a pairwise-optimal partition improves performance by reducing the number of sub-optimal solutions which the team might converge to.
\begin{figure}[tbp]
\centering
\subfigure[]
{
\includegraphics[angle=0,width=0.28\columnwidth]{centroidal_voronoi_1.eps}
}
\hfill
\subfigure[]
{
\includegraphics[angle=0,width=0.28\columnwidth]{centroidal_voronoi_3.eps}
}
\hfill
\subfigure[]
{
\includegraphics[angle=0,width=0.28\columnwidth]{centroidal_voronoi_2.eps}
}
\caption{All possible centroidal Voronoi partitions of a uniform $2 \times 5$ grid. Assuming all edge weights are $w$ and all vertices have priority $1$, then (a) has a cost of $1.2 w$, (b) has a cost of $1.1 w$, and (c) has a cost of $1.0 w$. Only (c) is pairwise-optimal by definition.}
\label{fig:voronoi}
\end{figure}
\section{Models, Problem Formulation, and Proposed Solution}
\label{sec:algorithm}
We aim to partition $Q$ among $N$ robotic agents using only asynchronous, unreliable, short-range communication.
In Section~\ref{sec:Model} we describe the computation, motion, and communication capabilities required of the team of robots, and in Section~\ref{sec:ProblemFormulation} we formally state the problem we are addressing. In Section~\ref{sec:Algorithm} we propose our solution, the \emph{Discrete Gossip Coverage Algorithm}, and in \ref{sec:illustrative} we provide an illustration. In Sections~\ref{sec:ConvProp} and \ref{sec:computation} we state the algorithm's convergence and complexity properties.
\subsection{Robot Network Model with Gossip Communication}\label{sec:Model}
Our Discrete Gossip Coverage Algorithm requires a team of $N$ robotic agents where each agent $i\in \until{N}$ has the following basic computation and motion capabilities:
\begin{enumerate}
\item [(C1)] agent $i$ knows its unique identifier $i$;
\item [(C2)] agent $i$ has a processor with the ability to store $G(Q)$ and perform operations on subgraphs of $G(Q)$; and
\item [(C3)] agent $i$ can determine which vertex in $Q$ it occupies and can move at speed $\speed$ along the edges of $G(Q)$ to any other vertex in $Q$.
\end{enumerate}
\begin{remark}[Localization]
The localization requirement in (C3) is actually quite loose. Localization is only used for navigation and not for updating partitions, thus limited duration localization errors are not a problem.
\end{remark}
The robotic agents are assumed to be able to communicate with each other according to the \emph{range-limited gossip communication model} which is described as follows:
\begin{enumerate}
\item [(C4)] given a communication range $\subscr{r}{comm} > \max_{e \in E} w_e$, when any two agents reside for some positive duration at a distance $r < \subscr{r}{comm}$, they communicate at the sample times of a Poisson process with intensity $\lambda_{\text{comm}} > 0$.
\end{enumerate}
Recall that an homogeneous Poisson process is a widely-used stochastic model for events which occur randomly and independently in time, where the expected number of events in a period $\Delta$ is $\Delta \lambda_{\text{comm}}$.
\begin{remark}[Communication Model]\label{rem:comm}
\noindent (1)~~This communication capability is the minimum necessary for our algorithm, any additional capability can only reduce the time required for convergence.
For example, it would be acceptable to have
intensity $\lambda(r)$ depend upon the pairwise robot distance
in such a way that $\lambda(r) \geq \lambda_{\text{comm}}$ for $r < \subscr{r}{comm}$.
\noindent (2)\quad We use distances in the graph to model limited range communication. These graph distances are assumed to approximate geodesic distances in the underlying continuous environment and thus path distances for a diffracting wave or moving robot.
\end{remark}
\subsection{Problem Statement}\label{sec:ProblemFormulation}
Assume that, for all $t\in\R_{\geq 0}$, each agent $i\in \until{N}$ maintains in memory a connected subset $P_i(t)$ of environment $Q$.
Our goal is to design a distributed algorithm that iteratively updates the partition $P(t)=\{P_i(t)\}_{i=1}^{N}$ while solving the following optimization problem:
\begin{equation}\label{eq:min}
\min_{P\in \ConnPart} \Hexp(P),
\end{equation}
subject to the constraints imposed by the robot network model with range-limited gossip communication from Section~\ref{sec:Model}.
\begin{figure*}[tbp]
\centering
\subfigure
{
\includegraphics[width=0.185\textwidth]{sim_nonconvex_1.eps}
\label{fig:sim_four_initial_centroids}
}
\hspace{-0.1in}
\subfigure
{
\includegraphics[width=0.185\textwidth]{sim_nonconvex_3.eps}
\label{fig:sim_four_swap}
}
\hspace{-0.1in}
\subfigure
{
\includegraphics[width=0.185\textwidth]{sim_nonconvex_4.eps}
\label{fig:sim_four_swap2}
}
\hspace{-0.1in}
\subfigure
{
\includegraphics[width=0.185\textwidth]{sim_nonconvex_6.eps}
\label{fig:sim_four_swap3}
}
\hspace{-0.1in}
\subfigure
{
\includegraphics[width=0.185\textwidth]{sim_nonconvex_7.eps}
\label{fig:sim_four_final}
}
\caption{Simulation of four robots dividing a square environment with obstacles. The boundary of each robots territory is drawn in a different color, the centroid of a territory is drawn with an X, and pairwise communication is drawn with a solid red line. On the left is the initial partition assigned to the robots. The middle frames show two pairwise territory exchanges, with updated territories highlighted with solid colors. The final partition is shown at right.}
\label{fig:sim_four}
\end{figure*}
\subsection{The Discrete Gossip Coverage Algorithm}\label{sec:Algorithm}
In the design of an algorithm for the minimization problem~\eqref{eq:min}
there are two main questions which must be addressed.
First, given the limited communication capabilities in (C4), how should the robots move inside $Q$
to guarantee frequent enough meetings between pairs of robots?
Second, when two robots are communicating, what information should they exchange and how should they update their regions?
In this section we introduce the \emph{Discrete Gossip Coverage Algorithm} which, following these two questions, consists of two components:
\begin{enumerate}[(1)]
\item the \emph{Random Destination \& Wait Motion Protocol}; and
\item the \emph{Pairwise Partitioning Rule}.
\end{enumerate}
The concurrent implementation of the Random Destination \& Wait Motion Protocol and the Pairwise Partitioning Rule determines the evolution of the positions and dominance subgraphs of the agents as we now formally describe. We start with the Random Destination \& Wait Motion Protocol.
\renewcommand\footnoterule{\hrule width \linewidth height .4pt}
\medskip\footnoterule\vspace{.75\smallskipamount}
\noindent\hfill\textbf{Random Destination \& Wait Motion Protocol}\hfill\vspace{.75\smallskipamount}
\footnoterule\vspace{.75\smallskipamount}
\noindent Each agent $i \in \until{N}$ determines its motion by repeatedly performing the following actions:
\begin{algorithmic}[1]
\STATE agent $i$ samples a \emph{destination vertex} $q_i$ from a uniform distribution over its dominance subgraph $P_i$;
\STATE agent $i$ moves to vertex $q_i$ through the shortest path in $P_i$ connecting the vertex it currently occupies and $q_i$; and
\STATE agent $i$ waits at $q_i$ for a duration $\tau>0$.
\end{algorithmic}
\vspace{.5\smallskipamount}\footnoterule\smallskip
If agent $i$ is moving from one vertex to another we say that agent $i$ is in the \emph{moving} state while if agent $i$ is waiting at some vertex we say that it is in the \emph{waiting} state.
\begin{remark}[Motion Protocol]
The motion protocol is designed to ensure frequent enough communication between pairs of robots. In general, any motion protocol can be used which meets this requirement, so $i$ could select $q_i$ from the boundary of $P_i$ or use some heuristic non-uniform distribution over $P_i$.
\end{remark}
If any two agents $i$ and $j$ reside in two vertices at a graphical distance smaller that $\subscr{r}{comm}$ for some positive duration, then at the sample times of the corresponding communication Poisson process the two agents exchange sufficient information to update their respective dominance subgraphs $P_i$ and $P_j$ via the Pairwise Partitioning Rule.
\renewcommand\footnoterule{\hrule width \linewidth height .4pt}
\medskip\footnoterule\vspace{.75\smallskipamount}
\noindent\hfill\textbf{Pairwise Partitioning Rule}\hfill\vspace{.75\smallskipamount}
\footnoterule\vspace{.75\smallskipamount}
\noindent
Assume that at time $t\in\R_{\geq 0}$, agent $i$ and agent $j$ communicate. Without loss of generality assume that $i<j$. Let $P_i(t)$ and $P_j(t)$ denote the current dominance subgraphs of $i$ and $j$, respectively. Moreover, let $t^+$ denote the time instant just after $t$. Then, agents $i$ and $j$ perform the following tasks:
\begin{algorithmic}[1]
\STATE agent $i$ transmits $P_i(t)$ to agent $j$ and vice-versa
\STATE initialize $W_{a^*} := P_i(t)$, $W_{b^*} := P_j(t)$, $a^* := \Cd(P_i(t))$, $b^* := \Cd(P_j(t))$
\STATE compute $U:=P_i(t)\operatorname{\cup} P_j(t)$ and an ordered list $S$ of all pairs of vertices in $U$
\FOR{each $(a,b) \in S$}
\STATE compute the sets\\
\qquad $W_{a} :=\left\{x\in U : d_{U}(x, a) \leq d_{U}(x, b)\right\}$ \\
\qquad $W_{b} :=\left\{x\in U : d_{U}(x, a) > d_{U}(x, b)\right\}$
\STATE \textbf{if~} $\subscr{\mathcal{H}}{one}(a; W_{a}) ~+~ \subscr{\mathcal{H}}{one}(b;W_{b}) < $ \\
$\quad~ \subscr{\mathcal{H}}{one}(a^*; W_{a^*}) + \subscr{\mathcal{H}}{one}(b^*;W_{b^*})$ \textbf{~then}
\STATE $\qquad W_{a^*}:=W_{a}, W_{b^*}:=W_{b}, a^* := a, b^* := b$
\ENDFOR
\STATE $P_i(t^+):=W_{a^*}, \quad P_j(t^+):=W_{b^*}$
\end{algorithmic}
\vspace{.5\smallskipamount}\footnoterule\smallskip
Some remarks are now in order.
\begin{remark}[Partitioning Rule]
\noindent (1)~~
The Pairwise Partitioning Rule is designed to find a minimum cost two-partition of $U$. More formally, if list $S$ and sets $W_{a^*}$ and $W_{b^*}$ for $(a^*,b^*)\in S$ are defined as in the Pairwise Partitioning Rule, then $W_{a^*}$ and $W_{b^*}$ are an optimal two-partition of $U$.
\noindent (2)\quad
While the loop in steps 4-7 must run to completion to guarantee that $W_{a^*}$ and $W_{b^*}$ are an optimal two-partition of $U$, the loop is designed to return an intermediate sub-optimal result if need be. If $P_i$ and $P_j$ change, then $\Hexp$ will decrease and this is enough to ensure eventual convergence.
\noindent (3)\quad We make a simplifying assumption in the Pairwise Partitioning Rule that, once two agents communicate, the application of the partitioning rule is instantaneous. We discuss the actual computation time required in Section~\ref{sec:computation} and some implementation details in Section~\ref{sec:results}.
\noindent (4)\quad Notice that simply assigning $W_{a^*}$ to $i$ and $W_{b^*}$ to $j$ can cause the robots to ``switch sides'' in $U$. While convergence is guaranteed regardless, switching may be undesirable in some applications. In that case, any smart matching of $W_{a^*}$ and $W_{b^*}$ to $i$ and $j$ may be inserted.
\noindent (5)\quad Agents who are not adjacent may communicate but the partitioning rule will not change their regions. Indeed, in this case $W_{a^*}$ and $W_{b^*}$ will not change from $P_i(t)$ and $P_j(t)$.
\end{remark}
Some possible modifications and extensions to the algorithm are worth mentioning.
\begin{remark}[Heterogeneous Robotic Networks]
In case the robots have heterogeneous dynamics, line 5 can be modified to consider per-robot travel times between vertices. For example, $d_U(x,a)$ could be replaced by the expected time for robot $i$ to travel from $a$ to $x$ while $d_U(x,b)$ would consider robot $j$.
\end{remark}
\begin{remark}[Coverage and Task Servicing]
Here we focus on partitioning territory, but this algorithm can easily be combined with methods to provide a service in $Q$ as in~\cite{FB-EF-MP-KS-SLS:10k}. The agents could split their time between moving to meet their neighbors and update territory, and performing requested tasks in their region.
\end{remark}
\subsection{Illustrative Simulation}
\label{sec:illustrative}
The simulation in Fig.~\ref{fig:sim_four} shows four
robots partitioning a square environment with obstacles where the free space is represented by a $12 \times 12$ grid. In the initial partition shown in the left panel, the robot in the top
right controls most of the environment while the robot in the bottom
left controls very little. The robots then move according to
the Random Destination \& Wait Motion Protocol, and communicate
according to range-limited gossip communication model with $r_{comm} = 2.5m$
(four edges in the graph).
The first pairwise territory exchange is shown in the second panel, where the bottom left robot claims some territory from the robot on the top left.
A later exchange between the two robots on the top is shown in the next two panels. Notice that the cyan robot in the top right gives away the vertex it currently occupies. In such a scenario, we direct the robot to follow the shortest path in $G(Q)$ to its updated territory before continuing on to a random destination.
After 9 pairwise territory exchanges, the robots reach the pairwise-optimal partition shown at right in Fig.~\ref{fig:sim_four}.
The expected distance between a random vertex and the closest robot decreases from $2.34m$ down to $1.74m$.
\subsection{Convergence Property}
\label{sec:ConvProp}
The strength of the Discrete Gossip Coverage Algorithm is the possibility of enforcing that a
partition will converge to a pairwise-optimal partition through pairwise territory exchange.
In Theorem~\ref{th:main} we summarize this convergence property, with proofs given in Section~\ref{sec:convergence}.
\begin{theorem}[Convergence Property]\label{th:main}
Consider a network of $N$ robotic agents endowed with computation and motion capacities (C1), (C2), (C3), and communication capacities (C4). Assume the agents implement the \emph{Discrete Gossip Coverage Algorithm} consisting of the concurrent implementation of the \emph{Random Destination \& Wait Motion Protocol} and the \emph{Pairwise Partitioning Rule}. Then,
\begin{enumerate}[(i)]
\item\label{item:well-posedness} the partition $P(t)$ remains connected and is described by $P:\R_{\geq 0} \to \ConnPart,$ and
\item\label{item:convergence} $P(t)$ converges almost surely in finite time to a pairwise-optimal partition.
\end{enumerate}
\end{theorem}
\begin{remark}[Optimality of Solutions]
By definition, a pairwise-optimal partition is optimal in that $\Hexp$ can not be improved by changing only two regions in the partition.
\end{remark}
\begin{remark}[Generalizations]
For simplicity we assume uniform robot speeds, communication processes, and waiting times. An extension to non-uniform processes would be straightforward.
\end{remark}
\subsection{Complexity Properties and Discussion}
\label{sec:computation}
In this subsection we explore the computational requirements
of the Discrete Gossip Coverage Algorithm, and make some comments on implementation.
Cost function $\subscr{\mathcal{H}}{one}(h; P_i)$ is the sum of the distances between $h$ and all other vertices in $P_i$. This computation of one-to-all distances is the core computation of the
algorithm.
For most graphs of interest the total number of edges $\card{E}$ is proportional to $\card{Q}$, so we will state bounds on this computation in terms of $\card{P_i}$.
Computing one-to-all distances requires one of the following:
\begin{itemize}
\item if all edge weights in $G(Q)$ are the same (e.g., for a graph from an occupancy grid), a breadth-first search approach can be used which requires \bigO{\card{P_i}} in time
and memory;
\item otherwise, Dijkstra's algorithm must be used which requires \bigO{\card{P_i} \log\left(\card{P_i}\right)} in time and \bigO{\card{P_i}} in memory.
\end{itemize}
Let $\search(P_i)$ be the time to compute one-to-all distances in $P_i$, then computing $\subscr{\mathcal{H}}{one}(h; P_i)$ requires \bigO{\search(P_i)} in time.
\begin{proposition}[Complexity Properties]
\label{prop:computation}
The motion protocol requires
\bigO{\card{P_i}} in memory, and \bigO{\search(P_i)} in computation time.
The partitioning rule requires
\bigO{\card{P_i} + \card{P_j}} in communication bandwidth between robots $i$ and $j$, \bigO{\card{P_i} + \card{P_j}} in memory, and can run in any time.
\end{proposition}
\begin{IEEEproof}
We first prove the claims for the motion protocol. Step 2 is the only non-trivial step and requires finding a shortest path in $P_i$, which is equivalent to computing one-to-all distances from the robot's current vertex. Hence, it requires \bigO{\search(P_i)} in time and \bigO{P_i} in memory.
We now prove the claims for the partitioning rule.
In step 1, robots $i$ and $j$ transmit their subgraphs to each other, which requires \bigO{\card{P_i} + \card{P_j}} in communication bandwidth. For step 3, the robots determine $U := P_i \cup P_j$, which requires \bigO{\card{P_i} + \card{P_j}} in memory to store.
Step 4 is the start of a loop which executes \bigO{\card{U}^2} times, affecting the time complexity of steps 5, 6 and 7.
Step 5 requires two computations of one-to-all distances in $U$ which each take \bigO{\search(U)}.
Step 6 involves four computations of $\subscr{\mathcal{H}}{one}$ over different subsets of $U$, however those for $W_{a^*}$ and $W_{b^*}$ can be stored from previous computation. Since $W_a$ and $W_b$ are strict subsets of $U$, step 5 takes longer than step 6.
Step 7 is trivial, as is step 8.
The total time complexity of the loop is thus \bigO{\card{U}^2 \search(U)}.
However, the loop in steps 4-7 can be truncated after any number of iterations. While it must run to completion to guarantee that $W_{a^*}$ and $W_{b^*}$ are an optimal two-partition of $U$, the loop is designed to return an intermediate sub-optimal result if need be. If $P_i$ and $P_j$ change, then $\Hexp$ will decrease. Our convergence result will hold provided that all elements of $S$ are eventually checked if $P_i$ and $P_j$ do not change.
Thus, the partitioning rule can run in any time with each iteration requiring \bigO{\search(U)}.
\end{IEEEproof}
All of the computation and communication requirements in Proposition~\ref{prop:computation} are independent of the number of robots and scale with the size of a robot's partition, meaning the Discrete Gossip Coverage Algorithm can easily scale up for large teams of robots in large environments.
\section{Convergence Proofs}
\label{sec:convergence}
This section is devoted to proving the two statements in Theorem~\ref{th:main}.
The proof that the Pairwise Partitioning Rule maps a connected $N$-partition into a connected $N$-partition is straightforward. The proof of convergence is more involved and is based on the application of Lemma~\ref{lem:finite-LaSalle} in Appendix~\ref{sec:appendix_A} to the Discrete Gossip Coverage Algorithm. Lemma~\ref{lem:finite-LaSalle} establishes strong convergence properties for a particular class of set valued maps (set-valued maps are briefly reviewed in Appendix~\ref{sec:appendix_A}).
We start by proving that the Pairwise Partitioning Rule is well-posed in the sense that it maintains a connected partition.
\begin{IEEEproof}[Proof of Theorem~\ref{th:main} statement (\ref{item:well-posedness})]
To prove the statement we need to show that $P(t^+)$ satisfies points (i) through (iv) of Definition~\ref{def:ConPartitions}. From the definition of the Pairwise Partitioning Rule, we have that $P_i(t^+) \cup P_j(t^+)=P_i(t) \cup P_j(t)$ and $P_i(t^+) \cap P_j(t^+) = \emptyset$. Moreover, since $a^*\in P_i(t^+)$ and $b^*\in P_j(t^+)$, it follows that $P_i(t^+)\neq \emptyset$ and $P_j(t^+)\neq \emptyset$. These observations imply the validity of points (i), (ii), and (iii) for $P(t^+)$. Finally,
we must show that $P_i(t^+)$ and $P_j(t^+)$ are connected, i.e., $P(t^+)$ also satisfies point (iv).
To do so we show that, given $x\in W_{a^*}$, any shortest path in $P_i(t) \cup P_j(t)$ connecting $x$ to $a^*$ completely belongs to $W_{a^*}$. We proceed by contradiction. Let $s_{x,a^*}$ denote a shortest path in $P_i(t)\cup P_j(t)$ connecting $x$ to $a^*$ and let us assume that there exists $m\in s_{x,a^*}$ such that $m\in W_{b^*}$. For $m$ to be in $W_{b^*}$ means that $d_{P_i(t)\cup P_j(t)}(m, b^*) < d_{P_i(t)\cup P_j(t)}(m, a^*)$. This implies that
\begin{align*}
d_{P_i\cup P_j}(x, b^*)&\leq d_{P_i\cup P_j}(m,b^*)+d_{P_i \cup P_j}(x,m)\\
&< d_{P_i\cup P_j}(m,a^*)+d_{P_i\cup P_j}(x,m)\\
&= d_{P_i \cup P_j}(x,a^*).
\end{align*}
This is a contradiction for $x\in W_{a^*}$. Similar considerations hold for $W_{b^*}$.
\end{IEEEproof}
The rest of this section is dedicated to proving convergence.
Our first step is to show that the evolution determined by the
Discrete Gossip Coverage Algorithm can be seen as a set-valued map.
To this end, for any pair of robots $(i,j)\in\until{N}^2$, $i\not=j$, we define the map
$\map{T_{ij}}{ \ConnPart}{ \ConnPart}$ by
\begin{align*}
T_{ij}(P) =
(P_1,\dots,\widehat{P}_i,\dots,\widehat{P}_j, \dots,P_N),
\end{align*}
where $\widehat{P}_i = W_{a^*}$ and $\widehat{P}_j = W_{b^*}$.
If at time $t\in \R_{\geq 0}$ the pair $(i,j)$ and no other pair of robots perform an iteration of the Pairwise Partitioning Rule, then the dynamical system on the space of partitions is described by
\begin{equation}\label{eq:Tij}
P(t^+)=T_{ij}\left( P(t) \right).
\end{equation}
We define the set-valued map $\setmap{T}{\ConnPart}{\ConnPart}$ as
\begin{equation}\label{eq:AlgoT}
T(P)=\setdef{T_{ij}(P)}{(i,j)\in\until{N}^2, i\not=j}.
\end{equation}
Observe that~\eqref{eq:Tij} can then be rewritten as $P(t^+)\in T(P(t))$.
The next two Propositions state facts whose validity is ensured by
Lemma~\ref{lemma:OnMotionProtocol} of Appendix~\ref{sec:appendix_B} which states a key property of the Random Destination \& Wait Motion Protocol.
\begin{proposition}[Persistence of Exchanges]\label{prop:tk}
Consider~$N$ robots implementing the Discrete Gossip Coverage Algorithm. Then, there almost surely exists an increasing sequence of time instants $\left\{t_k\right\}_{k \in \ensuremath{\mathbb{Z}}_{\ge 0}}$ such that
$
P(t_k^+)=T_{ij}(P(t_k))
$
for some $(i,j)\in \mathcal{E}(P(t_k))$.
\end{proposition}
\begin{IEEEproof}
The proof follows directly from Lemma~\ref{lemma:OnMotionProtocol} which implies that the time between two consecutive pairwise communications is almost surely finite.
\end{IEEEproof}
\vspace{0.1cm}
The existence of time sequence $\left\{t_k\right\}_{k \in \ensuremath{\mathbb{Z}}_{\ge 0}}$ allows us to to express the evolution generate by the Discrete Gossip Coverage Algorithm as a discrete time process. Let $P(k):=P(t_k)$ and $P(k+1):=P(t_k^+)$, then
$$
P(k+1)\in T\left( P(k) \right)
$$
where $\setmap{T}{\ConnPart}{\ConnPart}$ is defined as in~\eqref{eq:AlgoT}.
Given $k \in \ensuremath{\mathbb{Z}}_{\ge 0}$, let $\mathcal{I}_k$ denote the information which completely characterizes the state of Discrete Gossip Coverage Algorithm just after the $k$-th iteration of the partitioning rule, i.e., at time $t_{k-1}^+$. Specifically, $\mathcal{I}_k$ contains the information related to the partition $P(k)$, the positions of the robots at $t_{k-1}^+$, and whether each robot is in the \emph{waiting} or \emph{moving} state at $t_{k-1}^+$.
The following result characterizes the probability that, given $\mathcal{I}_k$, the $(k+1)$-th iteration of the partitioning rule is governed by any of the maps $T_{ij}$, $(i,j)\in \mathcal{E}(P(k))$.
\begin{proposition}[Probability of Communication]\label{prop:pi}
Consider a team of $N$ robots with capacities (C1), (C2), (C3), and (C4) implementing the Discrete Gossip Coverage Algorithm. Then, there exists a real number $\bar{\pi} \in (0,1)$, such that, for any $k\in \ensuremath{\mathbb{Z}}_{\ge 0}$ and $(i,j)\in \mathcal{E}(P(k))$
\begin{equation*
\Prob\left[P(k+1)=T_{ij}(P(k)) \;|\; \mathcal{I}_k\right]\geq \bar{\pi}.
\end{equation*}
\end{proposition}
\begin{IEEEproof}
Assume that at time $\bar t$ one pair of robots communicates. Given a pair $(\bar i,\bar j)\in \E(P(\bar t))$, we must find a lower bound for the probability that $(\bar i,\bar j)$ is the communicating pair.
Since all the Poisson communication processes have the same intensity, the distribution of the chance of communication is uniform over the pairs which are ``able to communicate,'' i.e., closer than $\subscr{r}{comm}$ to each other. Thus, we must only show that $(\bar i,\bar j)$ has a positive probability of being able to communicate at time $\bar t$, which is equivalent to showing that $(\bar i,\bar j)$ is able to communicate for a positive fraction of time with positive probability. The proof of Lemma~\ref{lemma:OnMotionProtocol} implies that with probability at least $\alpha/(1-e^{-\subscr{\lambda}{comm}\tau})$ any pair in $\E(P(\bar t))$ is able to communicate for a fraction of time not smaller than $\frac{\tau}{\Delta},$ where $\alpha$ and $\Delta$ are defined in the proof of Lemma~\ref{lemma:OnMotionProtocol}. Hence the result follows.
\end{IEEEproof}
\vspace{0.1cm}
The property in Proposition~\ref{prop:pi} can also be formulated as follows.
Let
$\map{\sigma}{\ensuremath{\mathbb{Z}}_{\ge 0}}{\left\{(i,j)\in\until{N}^2, i\not=j\right\}}$ be the stochastic process such that $\sigma(k)$ is the communicating pair at time $k$.
Then, the sequence of pairs of robots performing the partitioning rule at time instants $\left\{t_k\right\}_{k \in \ensuremath{\mathbb{Z}}_{\ge 0}}$ can be seen as a realization of the process $\sigma$, which satisfies
\begin{equation}\label{eq:RanPer2}
\Prob\big[\sigma(k+1)=(i,j) \;|\; \sigma(k)\big] \geq \bar{\pi}
\end{equation}
for all $(i,j)\in \mathcal{E}(P(k))$.
\smallskip
Next we show that the cost function decreases whenever the application of $T$ from \eqref{eq:AlgoT} changes the territory partition. This fact is a key ingredient to apply Lemma~\ref{lem:finite-LaSalle}.
\begin{lemma}[Decreasing Cost Function]\label{lemma:Tdecr}
Let $P\in\ConnPart$ and let $P^+\in T(P)$.
If $P^+ \neq P$, then $\Hexp(P^+) < \Hexp(P)$.
\end{lemma}
\begin{IEEEproof}
Without loss of generality assume that $(i,j)$ is the pair executing the Pairwise Partitioning Rule.
Then
\begin{align*}
&\Hexp(P^+)- \Hexp(P)\\
&\qquad\qquad\qquad= \subscr{\mathcal{H}}{one}(\Cd(P_i^+); P_i^+)+ \subscr{\mathcal{H}}{one}(\Cd(P_j^+); P_j^+)\\
&\qquad\qquad\qquad\quad-\subscr{\mathcal{H}}{one}(\Cd(P_i); P_i)- \subscr{\mathcal{H}}{one}(\Cd(P_j); P_j).
\end{align*}
According to the definition of the Pairwise Partitioning Rule we have that if $P_i^+\neq P_i$, $P_j^+\neq P_j$, then
\begin{align*}
&\subscr{\mathcal{H}}{one}(\Cd(P_i^+); P_i^+)+ \subscr{\mathcal{H}}{one}(\Cd(P_j^+); P_j^+)\\
&\qquad\qquad \leq \subscr{\mathcal{H}}{one}(a^*; P_i^+)+ \subscr{\mathcal{H}}{one}(b^*; P_j^+)\\
&\qquad\qquad < \subscr{\mathcal{H}}{one}(\Cd(P_i); P_i)+ \subscr{\mathcal{H}}{one}(\Cd(P_j); P_j)
\end{align*}
from which the statement follows.
\end{IEEEproof}
\smallskip
We now complete the proof of the main result, Theorem~\ref{th:main}.
\begin{IEEEproof}[Proof of Theorem~\ref{th:main} statement (\ref{item:convergence})]
Note that the algorithm evolves in a finite space of partitions, and by Theorem~\ref{th:main} statement (\ref{item:well-posedness}), the set $\ConnPart$ is strongly positively
invariant. This fact implies that assumption~(i) of Lemma~\ref{lem:finite-LaSalle} is satisfied.
From Lemma~\ref{lemma:Tdecr} it follows that assumption (ii) is also satisfied, with $\Hexp$ playing the role of the function $U$.
Finally, the property in~\eqref{eq:RanPer2} is equivalent to the property of \emph{persistent random switches} stated in Assumption~(iii) of Lemma~\ref{lem:finite-LaSalle}, for the special case $h=1$.
Hence, we are in the position to apply Lemma~\ref{lem:finite-LaSalle} and conclude convergence in finite-time to an element of the intersection of the equilibria of the maps $T_{ij}$, which by definition is the set of the pairwise-optimal partitions.
\end{IEEEproof}
\section{Experimental Methods \& Results}
\label{sec:results}
To demonstrate the utility and study practical issues of the Discrete Gossip Coverage Algorithm, we implemented it using the open-source Player/Stage robot control system \cite{BG-RTV-AH:03} and the Boost Graph Library (BGL) \cite{JBS-LQL-AL:07}. All results presented here were generated using Player 2.1.1, Stage 2.1.1, and BGL 1.34.1. To compute distances in uniform edge weight graphs we extended the BGL breadth-first search routine with a distance recorder event visitor.
\begin{figure*}[t]
\centering
\subfigure
{
\includegraphics[width=0.98\columnwidth]{sim_campus_1.eps}
}
\hspace{0.0in}
\subfigure
{
\includegraphics[width=0.98\columnwidth]{sim_campus_2.eps}
}
\caption{Images of starting and final partitions for a simulation with 30 robots providing coverage of a portion of campus at UCSB.}
\label{fig:large_sim}
\end{figure*}
\subsection{Large-scale Simulation}
To evaluate the performance of our gossip coverage algorithm with larger teams, we tested 30 simulated robots partitioning a map representing a $350m \times 225m$ portion of campus at the University of California at Santa Barbara. As shown in Fig.~\ref{fig:large_sim}, the robots are tasked with providing coverage of the open space around some of the buildings on campus, a space which includes a couple open quads, some narrower passages between buildings, and a few dead-end spurs.
For this large environment the simulated robots are $2m$ on a side and can move at $3.0\tfrac{m}{s}$. Each territory cell is $3m \times 3m$.
In this simulation we handle communication and partitioning as follows. The communication range is set to $30m$ (10 edges in the graph) with $\subscr{\lambda}{comm} = 0.3\frac{\text{comm}}{s}$.
The robots wait at their destination vertices for $\tau = 3.5s$. This value for $\tau$ was chosen so that on average one quarter of the robots are waiting at any moment. Lower values of $\tau$ mean the robots are moving more of the time and as a result more frequently miss connections, while for higher $\tau$ the robots spend more time stationary which also reduces the rate of convergence.
With the goal of improving communication, we implemented a minor modification to the motion protocol: each robot picks its random destination from the cells forming the open boundary\footnote{The open boundary of $P_i$ is the set of vertices in $P_i$ which are adjacent to at least one vertex owned by another agent.} of its territory.
In our implementation, the full partitioning loop may take $5$ seconds for the largest initial territories in Fig.~\ref{fig:large_sim}. We chose to stop the loop after a quarter second for this simulation to verify the anytime computation claim.
The 30 robots start clustered in the center of the map between Engineering II and Broida Hall, and an initial Voronoi partition is generated from these starting positions.
This initial partition is shown on the left in Fig.~\ref{fig:large_sim} with the robots positioned at the centroids of their starting regions. The initial partition has a cost of $37.1m$.
The team spends about 27 minutes moving and communicating according to the Discrete Gossip Coverage Algorithm before settling on the final partition on the right of Fig.~\ref{fig:large_sim}. The coverage cost of the final equilibrium improved by $54\%$ to $17.1m$. Visually, the final partition is also dramatically more uniform than the initial condition.
This result demonstrates that the algorithm is effective for large teams in large non-convex environments.
\begin{figure}[t]
\centering
\psfrag{x_label}{\small{Time (minutes)}}
\psfrag{y_label}{\small{Total cost $\Hexp$}}
\includegraphics[height=3.32cm]{sim_campus_cost_2}
\caption{Graph of the cost $\Hexp$ over time for the simulation in Fig.~\ref{fig:large_sim}.}
\label{fig:large_sim_cost}
\end{figure}
Fig.~\ref{fig:large_sim_cost} shows the evolution of $\Hexp$ during the simulation.
The largest cost improvements happen early when the robots that own the large territories on the left and right of the map communicate with neighbors with much smaller territories. These big territory changes then propagate through the network as the robots meet and are pushed and pulled towards a lower cost partition.
\subsection{Implementation Details}
\label{sec:implementation}
We conducted an experiment to test the algorithm using three physical robots in our lab, augmented by six simulated robots in a synthetic environment extending beyond the lab.
Our lab space is $11.3m$ on a side and is represented by the upper left portion of the territory maps in Fig.~\ref{fig:experiment}. The territory graph loops around a center island of desks. We extended the lab space through three connections into a simulated environment around the lab, producing a $15.9m \times 15.9m$ environment.
The map of the environment was specified with a $0.15m$ bitmap which we overlayed with a $0.6m$ resolution occupancy grid representing the free territory for the robots to cover. The result is a lattice-like graph with all edge weights equal to $0.6m$. The $0.6m$ resolution was chosen so that our physical robots would fit easily inside a cell.
Additional details of our implementation are as follows.
\subsubsection*{Robot hardware}
We use Erratic mobile robots from Videre Design, as shown in Fig.~\ref{fig:robot}.
The vehicle platform has a roughly square footprint $(40 cm \times 37cm)$, with two differential
drive wheels and a single rear caster.
Each robot carries an onboard computer with a 1.8Ghz Core 2 Duo processor, 1 GB
of memory, and 802.11g wireless communication.
For navigation and localization, each robot is equipped with a Hokuyo URG-04LX laser rangefinder.
The rangefinder scans $683$ points over $240\ensuremath{^\circ}$ at $10Hz$ with a range of $5.6$ meters.
\begin{figure}[t]
\centering
\psfrag{rangefinder}{\small Rangefinder}
\psfrag{drive wheel}{\small Drive wheel}
\psfrag{Computer}{\small Computer}
\psfrag{Rear caster}{\small Rear caster}
\includegraphics[height=3.32cm]{robot}
\caption{Erratic mobile robot with URG-04LX laser rangefinder.}
\label{fig:robot}
\end{figure}
\begin{figure*}[t]
\centering
\begin{minipage}{0.325\linewidth}
\includegraphics[width=\linewidth]{exp_terr_1.eps}
\vspace{0.03in}
\includegraphics[width=\linewidth]{exp_photo_1.eps}
\end{minipage}
\begin{minipage}{0.325\linewidth}
\includegraphics[width=\linewidth]{exp_terr_2.eps}
\vspace{0.03in}
\includegraphics[width=\linewidth]{exp_photo_2.eps}
\end{minipage}
\begin{minipage}{0.325\linewidth}
\includegraphics[width=\linewidth]{exp_terr_3.eps}
\vspace{0.03in}
\includegraphics[width=\linewidth]{exp_photo_3.eps}
\end{minipage}
\caption{Each column contains a territory map and the
corresponding overhead camera image for a step of the
hardware-in-the-loop simulation. The position of the camera in the
environment is shown with a camera icon in the territory map. The
physical robots are numbered 1, 2, and 3 and have the orange, blue, and lime green partitions. Their
positions in each territory map are
indicated with numbered circles.}
\label{fig:experiment}
\end{figure*}
\subsubsection*{Experiment setup}
Our mixed physical and virtual robot experiments are run from a central computer which is
attached to a wireless router so it can communicate with the physical robots.
The central computer creates a simulated world using Stage which mirrors and extends the
real space in which the physical robots operate. The central computer also simulates the virtual
members of the robot team. These virtual robots are modeled off of our hardware: they are differential drive
with the same geometry as the Erratic platform and use simulated Hokuyo URG-04LX rangefinders.
\subsubsection*{Localization}
We use the \texttt{amcl} driver in Player which implements Adaptive Monte-Carlo Localization~\cite{DF-WB-FD-ST:01}. The physical robots are provided with a map of our lab with a $15cm$ resolution and told their starting pose within the map. We set an initial pose standard deviation of $0.9m$ in position and $12\ensuremath{^\circ}$ in orientation, and request localization updates using $50$ of the sensor's range measurements for each change of $2cm$ in position or $2\ensuremath{^\circ}$ in orientation reported by the robot's odometry system. We then use the most likely pose estimate output by \texttt{amcl} as the location of the robot. For simplicity and reduced computational demand, we allow the virtual robots access to perfect localization information.
\subsubsection*{Motion Protocol}
Each robot continuously executes the Random Destination \& Wait Motion Protocol, with navigation handled by the \texttt{snd} driver in Player which implements Smooth Nearness Diagram navigation~\cite{JWD-FB:08a}.
For \texttt{snd} we set the robot radius parameter to $22cm$, obstacle avoidance distance to $0.7m$, and maximum speeds
to $0.4 \frac{m}{s}$ and $40 \tfrac{\ensuremath{^\circ}}{s}$.
The \texttt{snd} driver
is a local obstacle avoidance planner, so we feed it a series of waypoints every couple meters
along paths found in $G(Q)$.
We consider a robot to have achieved its target location when it is within $20cm$ and it will then wait for $\tau = 3.5s$.
For the physical robots the motion protocol and navigation processes run on board, while
there are separate threads for each virtual robot on the central computer.
\subsubsection*{Communication and Partitioning}
As the robots move, a central process monitors their positions and simulates the range-limited gossip communication model between both real and virtual robots.
We set $r_{comm} = 2.5m$ and $\lambda_{comm} = 0.3\frac{\text{comm}}{s}$. These parameters were chosen so that the robots would be likely to communicate when separated by at most four edges, but would also sometimes not connect despite being close.
When this process determines two robots
should communicate, it informs the robots who then perform the
Pairwise Partitioning Rule. Our pairwise communication implementation is blocking: if robot $i$ is
exchanging territory with $j$, then it informs the match making process
that it is unavailable until the exchange is complete.
\subsection{Hardware-in-the-Loop Simulation}
The results of our experiment with three physical robots and six simulated robots are shown in Figs.~\ref{fig:experiment} and \ref{fig:exp_cost}. The left column in Fig.~\ref{fig:experiment} shows the starting positions of the team of robots, with the physical robots, labeled 1, 2, and 3, lined up in a corner of the lab and the simulated robots arrayed around them. The starting positions are used to generate the initial Voronoi partition of the environment. The physical robots own the orange, blue, and lime green territories in the upper left quadrant. We chose this initial configuration to have a high coverage cost, while ensuring that the physical robots will remain in the lab as the partition evolves.
In the middle column, robots 1 and 2 have met along their shared border and are exchanging territory. In the territory map, the solid red line indicates 1 and 2 are communicating and their updated territories are drawn with solid orange and blue, respectively. The camera view confirms that the two robots have met on the near side of the center island of desks.
The final partition at right in Fig.~\ref{fig:experiment} is reached after $9 \frac{1}{2}$ minutes. All of the robots are positioned at the centroids of their final territories. The three physical robots have gone from a cluster in one corner of the lab to a more even spread around the space.
\begin{figure}[t]
\centering
\psfrag{x_label}{Time (minutes)}
\psfrag{y_label_1}{Robot costs (m)}
\psfrag{y_label_2}{$\Hexp$ (m)}
\includegraphics[width=0.99\columnwidth]{exp_cost_2}
\caption{Evolution of cost functions during the experiment in Fig.~\ref{fig:experiment}. The total cost $\Hexp$ is shown above in black, while $\subscr{\mathcal{H}}{one}$ for each robot is shown below in the robot's color.}
\label{fig:exp_cost}
\end{figure}
Fig.~\ref{fig:exp_cost} shows the evolution of the cost function $\Hexp$ as the experiment progresses, including the costs for each robot. As expected, the total cost never increases and the disparity of costs for the individual robots shrinks over time until settling at a pairwise-optimal partition.
In this experiment the hardware challenges of sensor noise, navigation, and uncertainty in position were efficiently handled by the \texttt{amcl} and \texttt{snd} drivers. The coverage algorithm assumed the role of a higher-level planner, taking in position data from \texttt{amcl} and directing \texttt{snd}. By far the most computationally demanding component was \texttt{amcl}, but the position hypotheses from \texttt{amcl} are actually unnecessary: our coverage algorithm only requires knowledge of the vertex a robot occupies. If a less intensive localization method is available, the algorithm could run on robots with significantly lower compute power.
\subsection{Comparative analysis}
\label{sec:analysis}
In this subsection we present a numerical comparison of the performance of the Discrete Gossip Coverage Algorithm and the following two Lloyd-type algorithms.
\subsubsection*{Decentralized Lloyd Algorithm}
This method is from \cite{JC-SM-TK-FB:02j} and \cite{FB-JC-SM:09}, we describe it here for convenience.
At each discrete time instant $t \in \ensuremath{\mathbb{Z}}_{\ge 0}$, each robot $i$ performs the following tasks: (1) $i$ transmits its position and receives the positions of all adjacent robots; (2) $i$ computes its Voronoi region $P_i$ based on the information received; and (3) $i$ moves to $\Cd(P_i)$.
\subsubsection*{Gossip Lloyd Algorithm}
This method is from \cite{JWD-RC-PF-FB:08z}.
It is a gossip algorithm, and so we have used the same communication model and the Random Destination \& Wait Motion Protocol to create meetings between robots. Say robots $i$ and $j$ meet at time $t$, then the pairwise Lloyd partitioning rule works as follows: (1) robot $i$ transmits $P_i(t)$ to $j$ and vice versa; (2) both robots determine $U = P_i(t) \cup P_j(t)$; (3) robot $i$ sets $P_i(t^+)$ to be its Voronoi region of $U$ based on $\Cd(P_i(t))$ and $\Cd(P_j(t))$, and $j$ does the equivalent.
For both Lloyd algorithms we use the same tie breaking rule when creating Voronoi regions as is present in the Pairwise Partitioning Rule: ties go to the robot with the lowest index.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\columnwidth]{analysis_bad_start}
\vspace{0.1cm}
\psfrag{xlabel}{Final cost (m)}
\psfrag{ylabel}{Simulation count}
\includegraphics[width=0.99\columnwidth]{bad_start_cost2}
\caption{Initial partition and histogram of final costs for a Monte Carlo test comparing the Discrete Gossip Coverage Algorithm (black bars), Gossip Lloyd Algorithm (gray bars), and Decentralized Lloyd Algorithm (red dashed line). For the gossip algorithms, 116 simulations were performed with different sequences of pairwise communications. The Decentralized Lloyd Algorithm is deterministic given an initial condition so only one final cost is shown.}
\label{fig:bad_start}
\end{figure}
Our first numerical result uses a Monte Carlo probability estimation
method from~\cite{RT-GC-FD:05} to place probabilistic bounds on the
performance of the two gossip algorithms. Recall that the Chernoff bound
describes the minimum number of random samples $K$ required to reach a
certain level of accuracy in a probability estimate from independent
Bernoulli tests. For an accuracy $\epsilon \in (0,1)$ and confidence $1
- \eta \in (0,1)$, the number of samples is given by
$K \geq \tfrac{1}{2\epsilon^2} \log \tfrac{2}{\eta}.$
For $\eta = 0.01$ and $\epsilon = 0.1$, at least 116 samples are required.
Figure~\ref{fig:bad_start} shows both the initial territory partition of the extended laboratory environment used and also a histogram of the final results for the following Monte Carlo test. The environment and robot motion models used are described in Section~\ref{sec:implementation}.
Starting from the indicated initial condition, we ran 116 simulations of both gossip algorithms. The randomness in the test comes from the sequence of pairwise communications. These sequences were generated using:
(1) the Random Destination \& Wait Motion Protocol with $q_i$ sampled uniformly from the open boundary of $P_i$ and $\tau = 3.5s$; and
(2) the range-limited gossip communication model with $r_{comm} = 2.5m$ and $\lambda_{comm} = 0.3\frac{\text{comm}}{s}$.
The cost of the initial partition in Fig.~\ref{fig:bad_start} is $5.48m$, while the best known partition for this environment has a cost of just under $2.18m$. The histogram in Fig.~\ref{fig:bad_start} shows the final equilibrium costs for 116 simulations of the Discrete Gossip Coverage Algorithm (black) and the Gossip Lloyd Algorithm (gray). It also shows the final cost using the Decentralized Lloyd Algorithm (red dashed line), which is deterministic from a given initial condition.
The histogram bins have a width of $0.10m$ and start from $2.17m$. For the Discrete Gossip Coverage Algorithm, $105$ out of $116$ trials reach the bin containing the best known partition and the mean final cost is $2.23m$. The Gossip Lloyd Algorithm reaches the lowest bin in only $5$ of $116$ trials and has a mean final cost of $2.51m$. The Decentralized Lloyd Algorithm settles at $2.48m$. Our new gossip algorithm requires an average of $96$ pairwise communications to reach an equilibrium, whereas gossip Lloyd requires $126$.
Based on these results, we can conclude with $99\%$ confidence that there is at least an $80\%$ probability that 9 robots executing the Discrete Gossip Coverage Algorithm starting from the initial partition shown in Fig.~\ref{fig:bad_start} will reach a pairwise-optimal partition which has a cost within $4\%$ of the best known cost. We can further conclude with $99\%$ confidence that the Gossip Lloyd Algorithm will settle more than $4\%$ above the best known cost at least $86\%$ of the time starting from this initial condition.
\begin{figure}[t]
\centering
\psfrag{xlabel}{Final cost (m)}
\psfrag{ylabel}{Simulation count}
\includegraphics[width=0.99\columnwidth]{multi_compare2}
\caption{Histograms of final costs from 10 Monte Carlo tests using random initial conditions in the environment shown in Fig.~\ref{fig:bad_start} comparing Discrete Gossip Coverage Algorithm (black bars), Gossip Lloyd Algorithm (gray bars), and Decentralized Lloyd Algorithm (red dashed line). For the gossip algorithms, 116 simulations were performed with different sequences of pairwise communications. The Decentralized Lloyd Algorithm is deterministic given an initial condition so only one final cost is shown. The initial cost for each test is drawn with the green dashed line.}
\label{fig:multi_compare}
\end{figure}
Figure~\ref{fig:multi_compare} compares final cost
histograms for $10$ different initial conditions for the same environment
and parameters as described above.
Each initial condition was created by selecting unique starting locations
for the robots uniformly at random and using these locations to generate
an initial Voronoi partition. The initial cost for each test is shown with
the green dashed line. In 9 out of 10 tests the Discrete Gossip Coverage Algorithm reaches the histogram bin with the best known partition in at least $112$ of $116$ trials.
The two Lloyd methods get stuck in sub-optimal centroidal Voronoi partitions more than $4\%$ away from the best known partition in more than half the trials in 7 of 10 tests.
\section{Conclusion}
\label{sec:conclusion}
We have presented a novel distributed partitioning and coverage control algorithm which requires
only unreliable short-range communication between pairs of robots and works in non-convex environments.
The classic Lloyd approach to coverage optimization involves iteration
of separate centering and Voronoi partitioning steps.
For gossip algorithms, however, this separation
is unnecessary computationally and we have
shown that improved performance can be achieved without it.
Our new Discrete Gossip Coverage Algorithm provably
converges to a subset of the set of centroidal Voronoi partitions
which we labeled pairwise-optimal partitions.
Through numerical comparisons we demonstrated that this new subset of solutions
avoids many of the local minima in which Lloyd-type algorithms
can get stuck.
Our vision is that this partitioning and coverage algorithm will form the foundation of a distributed task servicing setup for teams of mobile robots. The robots would split their time between servicing tasks in their territory and moving to contact their neighbors and improve the coverage of the space. Our convergence results only require sporadic improvements to the cost function, affording flexibility in robot behaviors and capacities, and offering the ability to handle heterogeneous robotic networks. In the bigger picture, this paper demonstrates the potential of gossip communication in distributed coordination algorithms. There appear to be many other problems where this realistic and minimal communication model could be fruitfully applied.
\begin{appendices}
\section{}
\label{sec:appendix_A}
For completeness we present a convergence result for set-valued algorithms on finite state spaces, which can be recovered as a direct consequence of~\cite[Theorem~4.5]{FB-RC-PF:08u-web}.
Given a set $X$, a set-valued map $\setmap{T}{X}{X}$ is a map which associates to an element $x\in X$ a subset $Z\subset X.$ A set-valued map is non-empty if
$T(x)\neq \emptyset$ for all $x\in{X}$.
Given a non-empty set-valued map $T$, an evolution of the dynamical system associated to $T$ is a sequence
$\seqdef{x_n}{n\in\ensuremath{\mathbb{Z}}_{\ge 0}}\subset X$ where
$x_{n+1}\in T(x_n)$ for all $n\in\ensuremath{\mathbb{Z}}_{\ge 0}.$
A set $W\subset X$ is
\emph{strongly positively invariant} for $T$ if $T(w)\subset{W}$ for all
$w\in{W}$.
\begin{lemma}[Persistent random switches imply convergence]
\label{lem:finite-LaSalle}
Let $(X,d)$ be a finite metric space. Given a collection of maps
$\map{T_1,\ldots, T_m}{X}{X}$, define the set-valued map
$\setmap{T}{X}{X}$ by $T(x)=\left\{T_1(x),\ldots, T_m(x)\right\}$. Given
a stochastic process $\map{\sigma}{\ensuremath{\mathbb{Z}}_{\ge 0}}{\until{m}}$, consider
an evolution $\seqdef{x_n}{n\in\ensuremath{\mathbb{Z}}_{\ge 0}}$ of $T$ satisfying
$
x_{n+1} = T_{\sigma(n)}(x_n).
$
Assume that:
\begin{enumerate}
\item there exists a set $W\subseteq X$ that is strongly positively
invariant for $T$;
\item there exists a function $\map{U}{W}{\R}$ such that $U(w')< U(w)$,
for all $w\in W$ and $w'\in T(w)\setminus\{w\}$; and
\item there exist $p\in{(0,1)}$ and $k\in\mathbb{N}$ such that, for all
$i\in\until{m}$ and $n\in\ensuremath{\mathbb{Z}}_{\ge 0}$, there exists
$h\in\until{k}$ such that
$
\Prob\big[\sigma(n+h)=i \,|\, \sigma(n),\dots,\sigma(1)\big] \geq p.
$
\end{enumerate}
For $i\in \until{m}$, let $F_i$ be the set of fixed points of $T_i$
in $W$, i.e., $F_i=\setdef{w\in W}{T_i(w)=w}$.
If $x_0\in{W}$, then the evolution $\seqdef{x_n}{n\in\ensuremath{\mathbb{Z}}_{\ge 0}}$ converges almost surely in finite time to an element of the set $(F_1\ensuremath{\operatorname{\cap}} \cdots \ensuremath{\operatorname{\cap}} F_m)$, i.e.,
there exists almost surely $\tau\in\mathbb{N}$ such that, for some $\bar{x} \in (F_1\ensuremath{\operatorname{\cap}} \cdots \ensuremath{\operatorname{\cap}} F_m)$, $ x_n=\bar{x}$ for $n\geq \tau.$
\end{lemma}
\section{}
\label{sec:appendix_B}
This Appendix proves a property of the Random Destination \& Wait Motion Protocol which is needed to show the persistence of pairwise exchanges.
\begin{lemma} \label{lemma:OnMotionProtocol}
Consider $N$ robots implementing the Discrete Gossip Coverage Algorithm starting from an arbitrary $P \in \ConnPart$. Consider $t\in \R_{\geq 0}$ and let $P(t)$ denote the partition at time $t$. Assume that at time $t$ no two robots are communicating. Then, there exist $\Delta>0$ and $\alpha\in(0,1)$, independent of $P(t)$ and the positions and states of the robots at time $t$, such that, for every $(i,j)\in\E(P(t))$,
$\Prob\left[(i,j) \text{ communicate within}\,\, (t,t+\Delta) \right]\ge \alpha.$
\end{lemma}
\begin{IEEEproof}
To begin, we define two useful quantities. Let
$\displaystyle\mathcal{S}(Q):= \max_{P\in \ConnPart}\max_{P_i\in P}\max_{h,k\in P_i} d_{P_i}(h,k)$
be a pseudo-diameter for $Q$,
and then choose $\Delta:=2\frac{\mathcal{S}(Q)}{v} + 2\tau$.
We fix a pair $(i,j)\in\E(P)$, and pick adjacent vertices $a\in P_i$, $b\in P_j$.
Our goal is to lower bound the probability that $i$ and $j$ will communicate within the interval $\left(t,t+\Delta \right)$. To do so we construct {\em one} sequence of events of positive probability which enables such communication.
Consider the following situation: $i$ is in the \emph{moving} state and needs time $t_i$ to reach its destination $q_i$, whereas robot $j$ is in the \emph{waiting} state at vertex $q_j$ and must wait there for time $\tau_j\leq \tau$.
We denote by $t(a)$ (resp. $t(b)$) the time needed for $i$ (resp. $j$) to travel from $q_i$ (resp. $q_j$) to $a$ (resp. $b$).
Let $E_{i}$ be the event such that $i$ performs the following actions in $(t,t+\Delta)$ without communicating with any robot $k \ne j$:
\begin{enumerate}
\item $i$ reaches $q_i$ and waits at $q_i$ for the duration $\tau$; and
\item $i$ chooses vertex $a$ as its next destination and then stays at $a$ for at least $\Delta-t(a)-t_i-\tau$.
\end{enumerate}
Let $E_{j}$ be the event such that $j$ performs the following actions in $(t,t+\Delta)$ without communicating with any $k \ne i$:
\begin{enumerate}
\item $j$ waits at $q_j$ for the duration $\tau_j$; and
\item $j$ chooses vertex $b$ as its next destination and then stays at $b$ for at least $\Delta-t(b)-\tau_j$.
\end{enumerate}
Let $E_{ij}=E_i \cap E_j$.
\newcommand{\subscr{\lambda}{comm}}{\subscr{\lambda}{comm}}
Next, we lower bound the probability that event $E_i$ occurs. Recall the definition of $\lambda_{\text{comm}}$ from Sec.~\ref{sec:Model}.
Since a robot can have at most $N-1$ neighbors, the probability that (i) of $E_i$ happens is lower bounded by
$
e^{-\subscr{\lambda}{comm} \tau N}.
$
For (ii), the probability that $i$ chooses $a$ is $1/\card{P_i}$, which is lower bounded by $1/\card{Q}$. Then, in order to spend at least $(\Delta-t(a)-t_i-\tau)$ at $a$, $i$ must choose $a$ for $\lceil \frac{\Delta-t(a)-t_i-\tau}{\tau} \rceil$ consecutive times. Finally, the probability that during this interval $i$ will not communicate with any robot other than $j$ is lower bounded by
$
e^{-\subscr{\lambda}{comm} \Delta (N-2)}.
$
The probability that (ii) occurs is thus lower bounded by
$
\left(1 / \card{Q}\right)^{\lceil \frac{\Delta}{\tau} \rceil} e^{-\subscr{\lambda}{comm} \Delta N}.
$
Combining the bounds for (i) and (ii), it follows that
$$
\Prob[E_i]\geq \bigl(\tfrac{1}{\card{Q}}\bigr)^{\lceil \frac{\Delta}{\tau} \rceil} e^{-\subscr{\lambda}{comm} (\Delta+\tau) N}.
$$
The same lower bound holds for $\Prob[E_j]$, meaning that
\begin{align*}
\Prob\left[E_{ij}\right]&=\Prob\left[E_{i}\right]\, \Prob\left[E_{j}\right]
\geq \bigl(\tfrac{1}{\card{Q}}\bigr)^{2 \lceil \frac{\Delta}{\tau} \rceil} e^{-2 \subscr{\lambda}{comm} (\Delta+\tau) N}.
\end{align*}
If event $E_{ij}$ occurs, then robots $i$ and $j$ will be at adjacent vertices for an amount of time during the interval $(t,t+\Delta)$ equal to
$
\min \left\{\Delta-t(a)-t_i-\tau, \Delta-t(b)-\tau_j\right\}.
$
Since $t(a)$ and $t(b)$ are no more than $\frac{\mathcal{S}(Q)}{v}$, we can conclude that $i$ and $j$ will be within $\subscr{r}{comm}$ for at least $\tau$.
Conditioned on $E_{ij}$ occurring, the probability that $i$ and $j$ communicate in $(t,t+\Delta)$ is lower bounded by $1-e^{-\subscr{\lambda}{comm} \tau}$. A suitable choice for $\alpha$ from the statement of the Lemma is thus
$$
\alpha= \bigl(\tfrac{1}{|Q|}\bigr)^{2 \lceil \frac{\Delta}{\tau} \rceil} e^{-2 \subscr{\lambda}{comm} (\Delta+\tau) N} \left(1-e^{-\subscr{\lambda}{comm}\tau}\right).
$$
It can be shown that this also constitutes a lower bound for the other possible combinations of initial states:
robot $i$ is \emph{waiting} and robot $j$ is \emph{moving};
robots $i$ and $j$ are both \emph{moving}; and
robots $i$ and $j$ are both \emph{waiting}.
\end{IEEEproof}
\section{}
\label{sec:appendix_C}
In this appendix we provide the proof of Proposition~\ref{prop:OptPair} which states that any pairwise-optimal partition is also a centroidal Voronoi partition.
\begin{IEEEproof}[Proof of Proposition~\ref{prop:OptPair}]
To create a contradiction, assume that $P \in \ConnPart$ is a pairwise-optimal partition but not a centroidal Voronoi partition. In other words, there exist components $P_i$ and $P_j$ in $P$ and an element $x$ of one component, say $x\in P_i$, such that
\begin{equation}\label{eq:not_voronoi}
d_G\left(x, \Cd(P_i)\right)> d_{G}\left(x, \Cd(P_j)\right).
\end{equation}
Choose $P_j$ such that for all $k \neq j$
\begin{equation}\label{eq:lowest_j}
d_G\left(x, \Cd(P_k)\right) \geq d_{G}\left(x, \Cd(P_j)\right).
\end{equation}
Let $\short{a}{b}{G}$ be a shortest path in $G$ connecting $a$ to $b$ and let $m \in \short{x}{\Cd(P_j)}{G}$ be the first element of the path starting from $\Cd(P_j)$ which is not in $P_j$. Let $\ell$ be such that $m \in P_\ell$.
If $m = x$, then from \eqref{eq:not_voronoi} and the definition of $\short{x}{\Cd(P_j)}{G}$ we have that
\begin{align*}
&d_{P_i}\left(x, \Cd(P_i)\right) \geq d_{G}\left(x, \Cd(P_i)\right)\\
&\qquad\qquad > d_{G}\left(x, \Cd(P_i)\right) = d_{P_i \cup P_j}\left(x, \Cd(P_j)\right)
\end{align*}
which, since $x \in P_i$, creates a contradiction of the fact that $P$ is pairwise-optimal.
If $m \neq x$, then, given \eqref{eq:lowest_j}, one of these two conditions holds:
\begin{enumerate}
\item $d_G\left(m, \Cd(P_\ell)\right) > d_{G}\left(m, \Cd(P_j)\right)$, or
\item $d_G\left(m, \Cd(P_\ell)\right) = d_{G}\left(m, \Cd(P_j)\right)$.
\end{enumerate}
In the first case, we again have a contradiction using the same logic above with $m$ in place of $x$. In the second case, we must further consider whether there exists a $\short{m}{\Cd(P_\ell)}{G}$ such that every vertex in $\short{m}{\Cd(P_\ell)}{G}$ is also in $P_\ell$. If there is not such a path, then
$$d_{P_\ell}\left(m, \Cd(P_\ell)\right) > d_G\left(m, \Cd(P_\ell)\right) = d_{P_\ell \cup P_j}\left(m, \Cd(P_j)\right)$$
and we again have a contradiction as above. If there is such a path, then we can instead repeat this analysis using using $\ell$ in place of $j$ and considering the path formed by this $\short{m}{\Cd(P_\ell)}{G}$ and the vertices in $\short{x}{\Cd(P_j)}{G}$ after $m$. Since the next vertex playing the role of $m$ must be closer to $x$, we will eventually find a vertex which creates a contradiction.
\end{IEEEproof}
\end{appendices}
\bibliographystyle{ieeetr}
|
1,314,259,996,544 | arxiv | \section{Introduction}
Understanding the physical processes involved in the evolution of
galaxies is a key goal of extragalactic astronomy. Although stellar
(or halo) mass is emerging as playing a fundamental role for galaxy
evolution, environmental influences may also have an impact
\citep{kauf03a,kauf04,bald06,cucc10,peng10}. This is particularly
relevant in dense environments, in which galaxies may experience a
wide range of externally driven processes such as mergers, tidal
interactions, and gas stripping due to interactions with ambient
gas. All of these processes may act to remove a substantial fraction
of the interstellar medium (ISM), or rapidly consume or eject it
through interaction--induced star formation and nuclear activity,
eventually leading the galaxy to transition from blue and star-forming
to red and quiescent \citep{miho96,quil00,verd01,chun07,kawa08}
The relative importance of these various processes should itself be a
function of environment, with mergers and tidal interactions
dominating in small galaxy groups (e.g., \citealt{barn89,mamo07}), and
with galaxy harassment, ram pressure stripping, and starvation (the
cut-off of the supply of cold gas for star formation from warm/hot gas
in the halo) becoming more efficient in massive clusters exhibiting
high galaxy velocities \citep{moor99,quil00}. {\em Chandra}
observations do show that a lower fraction of early-type galaxies in
clusters contain hot halos than their counterparts in groups
\citep{jelt08}, consistent with expectations of the hot halo stripping
efficiency being higher in more massive systems. Evidence for ram
pressure stripping of the {\em cold} ISM component in rich clusters is
also well-established through many H{\sc i} studies
\citep{cham80,caya94,schr01,sola01,chun07,levy07,chun09,cort11},
suggesting that interactions between galaxies and the intracluster
medium may generally be an important route to removing galactic gas in
the most massive systems.
The situation is less clear in smaller groups, despite these
representing much more typical galaxy environments. Naively,
galaxy--galaxy interactions and mergers should be relatively more
important, given the lower galaxy velocities and intergalactic medium
densities in groups. Indeed, the results of \citet{jelt08} and
\citet{mulc10} show that hot X-ray halos are retained around the
majority of $L_K>L^\ast$ early-types in the central regions of groups,
and these halos are not strongly X-ray underluminous compared to those
of field galaxies. Hence, removal of hot halo gas by ram pressure must
be very modest in groups, at least for massive galaxies. Nevertheless,
numerical simulations suggest that starvation may still occur for
moderate-luminosity galaxies on their first passage through even
fairly small groups \citep{kawa08}. Viscous stripping of galactic gas
through Kelvin--Helmholtz instabilities, the efficiency of which
depends only linearly on galaxy velocity \citep{nuls82}, could also
play a role in groups even when brute-force ram pressure is
unimportant \citep{rasm08}.
For the cold gas, a number of group galaxies show evidence for
extended H{\sc i} that has been stripped from their host (e.g.,
\citealt{verd01,kant05,kilb06}), and many are deficient in H{\sc i}
compared to similar field galaxies \citep{verd01,seng07,kilb09}. Some
of these objects represent strong candidates for ram pressure
stripping of both cold and hot ISM
\citep{bure02,rasm06,seng07,mcco07,bail07}. However, the H{\sc i}
properties of spirals in these environments can often be equally well
explained by tidal encounters \citep{kern08,rasm08,kilb09}, and so it
is still not fully clear which process, if any, dominates the gas
removal from typical group galaxies.
Quantifying the importance of the various mechanisms acting on group
galaxies requires detailed observations of the ISM in these galaxies,
both within individual groups and across systems displaying a range of
global properties. To identify signatures of ongoing gas removal and
how these may depend on local group environment, the ISM must be
probed on spatial scales of individual galaxies across the full
system, with a sensitivity extending down to moderate-luminosity
galaxies. Exploring the role of hot gas stripping, whether induced by
ram pressure or galaxy--galaxy interactions, is currently largely
limited to early-type galaxies or starbursting spirals, however, since
only these galaxy types generally contain X-ray detectable halo gas
\citep{rasm09}. In more quiescent late-type spirals, gas stripping can
be far more efficiently explored using H{\sc i} data. Combining X-ray
and H{\sc i} data therefore allows one to study the ISM over the full
galaxy morphological range.
Here we combine X-ray mosaicing observations from {\em Chandra} with
analogous H{\sc i} observations from the Very Large
Array\footnote{The Very Large Array is operated by the National
Radio Astronomy Observatory, which is a facility of the National
Science Foundation (NSF), operated under cooperative agreement by
Associated Universities, Inc.} (VLA) to study the hot and cold ISM
in the nearby, X-ray bright NGC\,2563 group. Our goal is to search
for evidence of ongoing or recent gas stripping (we here use this term
to mean any externally induced removal of gas from galaxies, including
via tidal stripping), explore possible mechanisms involved, and
understand their environmental dependence. While X-ray
\citep{fabb92,mulc98,osmo04,gast07} and optical \citep{zabl98a,zabl00}
observations of this group already exist, a unique aspect of the
present study is that both our X-ray and H{\sc i} observations cover
the entire group (out to a projected radius of $R=1.15$~Mpc, well
beyond the estimated virial radius), while still benefiting from the
sub-kpc spatial resolution of {\em Chandra} at the target
redshift. Combined with the sensitivity of the VLA to low--surface
brightness diffuse H{\sc i} emission (with a synthesized beam width of
$\approx 45''$ in the most compact ``D-array'' conguration employed
here), this enables a detailed investigation of the spatial variation
in global ISM properties across the full group environment.
We describe the overall properties of the group in
Section~\ref{sec,group}. The X-ray and H{\sc i} analyses are detailed
in Section~\ref{sec,analysis}, and results are presented in
Section~\ref{sec,results}. Section~\ref{sec,discuss} considers the
evidence for recent and past ISM stripping in the group, and our
results and conclusions are summarized in Section~\ref{sec,summary}.
We assume $H_0=70$~km~s$^{-1}$~Mpc$^{-1}$, $\Omega_{m}=0.27$, and
$\Omega_{\Lambda}=0.73$. The target redshift of $z=0.0157$ then
corresponds to a luminosity distance of 68.1~Mpc, and $1'$ on the sky
to a projected distance of 19.2~kpc. Uncertainties are quoted at the
$1\sigma$ level unless otherwise specified.
\vspace{4mm}
\section{Group Membership and Properties}\label{sec,group}
Extensive optical spectroscopy of the NGC\,2563 field exists from the
studies of \citet{zabl98a,zabl00}. The field is also covered by the
Sloan Digital Sky Survey (SDSS), providing images and magnitudes for
all group members, and flux-calibrated spectra for most of them. From
the measured redshifts, we determined group membership using the
ROSTAT package \citep{beer90}, considering all galaxies with known
recessional velocities within $\pm 3000$~km~s$^{-1}$ of the group
mean. We then calculated the biweight estimators of velocity mean and
dispersion. Objects with velocities beyond $\pm 3\sigma$$_{\rm biwt}$
from the mean were discarded, and the process repeated iteratively
until convergence. This technique identified 64 group members within a
projected radius of $R=60'$ ($R = 1.15$~Mpc) from the group center,
including two newly discovered members from our H{\sc i} observations
(see Section~\ref{sec,HI}). The resulting mean redshift and radial
velocity dispersion of the group based on these 64 galaxies are
$z=0.0157 \pm 0.0001$ and $\sigma_{\rm biwt} =
364^{+36}_{-33}$~km~s$^{-1}$, respectively. Based on objects
classified as galaxies in the SDSS survey, the group membership is
100\% (98\%) spectroscopically complete down to $M_r=-18$ ($-17$)
within $60'$ of the group center.
Group members were morphologically classified independently by JSM and
AIZ using SDSS $r$-band images. The two authors' classifications
agreed within one Hubble type in all cases, and our adopted
morphologies represent the average of the two results. We use SDSS
$g$-band magnitudes as a measure of the blue luminosity, and, for
comparison to previous studies, $K_s$ magnitudes from the 2-Micron All
Sky Survey (2MASS) as a rough estimate of galaxy stellar mass. The
latter magnitudes are available for most of the group members, except
for a few galaxies in close pairs. In these cases, the $K_s$ magnitude
was estimated from the SDSS $r$-band magnitude using $r$--$K_s=2.90$
for early-type galaxies, and $r$--$K_s=2.30$ for late-types, based on
the average values for the group members with both magnitudes
available. The scatter in each case is $\sim$\,0.20. As none of the
galaxies without $K_s$ magnitudes is detected in X-rays or H{\sc i},
adopting these relationships has minimal impact on our
conclusions. Magnitudes were converted to solar luminosities assuming
$g_{\bigodot}=5.12$ and $K_{\bigodot}=3.39$ after correcting for
Galactic extinction. We further adopt $L_g^\ast = 10^{10.17} L_\odot$
\citep{mont09} and $L_K^\ast =10^{11.08} L_\odot$ \citep{sun07}.
While a detailed X-ray mass analysis of the group is beyond the scope
of this paper, having a rough estimate of the group virial radius
$R_{\rm vir}$ is instructive for the discussion to follow. This can be
obtained from the hot gas temperature of $kT \approx 1.1$~keV derived
by \citet{mulc03}, combined with the scaling relation of
\citet{fino07}. This would suggest $R_{500} \sim 420$~kpc, and hence
$R_{\rm vir}\approx R_{100}\approx 2R_{500} \approx 850$~kpc in the
adopted cosmology, assuming a Navarro--Frenk--White potential
\citep{nava97} with a typical concentration parameter of
$c=5$--10. Here $R_\Delta$ is the radius enclosing a mean density of
$\Delta$ times the critical value.
\vspace{3mm}
\section{Observations and Analysis}\label{sec,analysis}
\vspace{1mm}
\subsection{{\em Chandra} Imaging and Spectroscopy}\label{sec,spec}
The distribution of group members in NGC\,2563 allows most of them to
be observed with 14 ACIS-I pointings: a $3\times3$ central mosaic to
cover the inner $45'\times45'$ of the group and five outer
pointings. A total of 54 of the 64 confirmed group members were
observed by {\em Chandra}. The majority of the remaining galaxies are
optically faint and therefore not likely to be strong X-ray
sources. The 14 pointings were done between June 2007 and March 2008
for 30~ks each, except for the central pointing which was observed for
50~ks to achieve a similar contrast for galaxies in the X-ray bright
central region. All observations were conducted in VFAINT telemetry
mode. The level~1 event files were processed with {\sc
ciao}\footnote{http://cxc.harvard.edu/ciao/} v4.0 and
CALDB\footnote{http://cxc.harvard.edu/caldb/} v3.4.3, filtering out
bad pixels and applying the latest gain files. Events with {\em ASCA}
grades\footnote{see
http://cxc.harvard.edu/ciao/ahelp/acis\_process\_events.html} 1, 5,
and 7, and a non-zero status flag were discarded. Finally, background
flares were screened for using the routine 'lc\_clean.sl' in {\sc
ciao}, excluding time periods with background count rates greater
than $20\%$ of the quiescent rate. Flaring was not a problem for any
of our observations.
Source detection was performed using the ``Mexican Hat'' wavelet
source detection algorithm 'wavdetect' in {\sc ciao}, with scales of
1, 2, 4, 8, 16 and 32 pixels, using both soft (0.3--2.0~keV), hard
(2.0--7.0~keV), and ``full'' band (0.3--7~keV) event lists. The
detection threshold was set to limit the number of false detections to
$\approx 4$ per CCD. As summarized in Table~\ref{tab:X}, a total of 17
confirmed group members were detected, including the largest
elliptical, NGC\,2563 itself. This galaxy is coincident with the group
center as defined by the peak of the diffuse group X-ray emission. As
it is not straightforward to cleanly separate the emission of this
galaxy from that of the intragroup medium, the central galaxy is
treated as a special case for the remainder of the paper.
\begin{deluxetable*}{llrrrrrrcccll}
\tabletypesize{\scriptsize}
\tablecaption{Group Members Detected With Chandra\label{tab:X}}
\tablewidth{0pt}
\tablehead{ \colhead{Galaxy} & \colhead{Morph.} & \colhead{$R$} &
\colhead{log\,$L_K$} & \colhead{log\,$L_g$} &
\colhead{Cts.} & \colhead{$L_{\rm X,th}$} &
\colhead{$L_{\rm X,pl}$} & \colhead{$kT$} & \colhead{$\Gamma$} &
\colhead{$H\!R$} & \colhead{Spec.} & \colhead{Notes} \\
\colhead{ } & \colhead{ } & \colhead{(kpc)} & \colhead{($L_\odot$)} &
\colhead{($L_\odot$)} & \colhead{ } & \colhead{ } & \colhead{ } &
\colhead{(keV)} & \colhead{ }& \colhead{ }& \colhead{ }& \colhead{ }\\
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{(5)} &
\colhead{(6)} & \colhead{(7)} & \colhead{(8)} & \colhead{(9)} & \colhead{(10)}&
\colhead{(11)} & \colhead{(12)} & \colhead{(13)}
}
\startdata
NGC\,2563 & E & 0 & 11.43 & 10.54 & 1547 & $74.2^{+5.0}_{-4.8}$ &
$22.6^{+5.6}_{-5.7}$ & $0.97^{+0.03}_{-0.04}$ & $2.16^{+0.21}_{-0.24}$ &
$0.11\!\pm\!0.01$ & 1, TP & E\\
NGC\,2562 & S0/a & 91 & 11.21 & 10.35 & 50 & $<1.87$ &
$2.51^{+0.58}_{-0.45}$ & 0.7$^\ast$ & $1.84^{+0.35}_{-0.30}$ & $0.56\!\pm\!0.17$ &
2, P & E\\
NGC\,2560 & S0/a & 218 & 11.11 & 10.21 & 139 & $<4.49$ &
$8.45^{+1.65}_{-1.56}$ & 0.7$^\ast$ & $1.38^{+0.29}_{-0.27}$ & $0.59\!\pm\!0.10$ &
2, P & E\\
CGCG119-069 & E & 238 & 10.13 & 9.49 & 9 & $<5.06$ &
$<1.39$ & 0.7$^\ast$ & 1.7$^\ast$ & $0.63\!\pm\!0.42$ & 3, U & P?\\
UGC\,04344 & Sc & 238 & 10.14 & 9.91 & 17 & $<2.07$ &
$<1.86$ & 0.7$^\ast$ & 1.7$^\ast$ & $0.38\!\pm\!0.21$ & 3, U & E?, SF\\
UGC\,04332 & Sapec& 264 & 10.94 & 10.03 & 117 & $<3.82$ &
$6.45^{+1.18}_{-1.06}$ & 0.7$^\ast$ & $1.85^{+0.27}_{-0.27}$ & $1.32\!\pm\!0.25$
& 2, P & E, Sy2\\
NGC\,2569 & E & 307 & 10.71 & 9.99 & 18 & $<1.99$ &
$<2.09$ & 0.7$^\ast$ & 1.7$^\ast$ & $0.26\!\pm\!0.15$ & 3, UT & E?\\
UGC\,04329 & Scpec& 439 & 10.72 & 10.06 & 57 & $<4.03$ &
$5.93^{+1.92}_{-1.46}$ & 0.7$^\ast$ & $2.39^{+1.16}_{-0.68}$ & $0.62\!\pm\!0.17$
& 2, P & E, SF\\
IC\,2293 & SBbc & 470 & 10.37 & 9.89 & 9 & $<1.29$ &
$<1.00$ & 0.7$^\ast$ & 1.7$^\ast$ & $0.20\!\pm\!0.18$ &3, UT & P?, SF\\
2MJ082236 & S0/a & 542 & 9.99 & 9.32 & 13 & $<1.57$ &
$<1.53$ & 0.7$^\ast$ & 1.7$^\ast$ & $0.49\!\pm\!0.28$ & 3, U & E?, SF\\
NGC\,2557 & SB0 & 567 & 11.15& 10.29 & 162 & $11.2^{+3.0}_{-2.3}$ &
$10.4^{+1.4}_{-1.2}$ & $0.34^{+0.10}_{-0.07}$ & 1.7$^\ast$ & $0.27\!\pm\!0.05$ &
1, TP & E, LI\\
UGC\,04324 & Sab & 664 & 10.71 & 9.92 & 35 & $<3.63$ &
$<3.96$ & 0.7$^\ast$ & 1.7$^\ast$ & $0.42\!\pm\!0.16$ & 3, U & E, SF\\
NGC\,2558 & Sb & 739 & 11.03 & 10.26 & 45 & $<2.45$ &
$3.90^{+0.86}_{-0.75}$ & 0.7$^\ast$ & $1.53^{+0.30}_{-0.33}$ & $0.45\!\pm\!0.15$
& 2, P & E, LI\\
IC\,2338 & SBcd & 848 & 10.27 & 9.87 & 59 & $<3.09$ &
$4.75^{+1.02}_{-0.95}$ & 0.7$^\ast$ & 1.7$^\ast$ & $0.35\!\pm\!0.10$ &
2, TP& E, SF\\
IC\,2339&SBcpec& 858& 10.48& 10.02& 32 & $1.85^{+0.91}_{-0.84}$ &
$1.69^{+0.67}_{-0.60}$ & 0.7$^\ast$ & 1.7$^\ast$ & $0.21\!\pm\!0.10$ &
1, TP& E, SF\\
CGCG119-047 & Sab & 923 & 10.62 & 10.04 & 26 & $<2.80$ &
$<2.68$ & 0.7$^\ast$ & 1.7$^\ast$ & $0.36\!\pm\!0.16$ & 3, U & P?, SF\\
IC\,2341 & E/S0 & 929 & 10.91 & 10.19 & 32 & $<3.54$ &
$<2.89$ & 0.7$^\ast$ & 1.7$^\ast$ & $0.88\!\pm\!0.31$ & 3, U & E, LI
\enddata
\tablecomments{Col.~(3): Projected distance from peak of the
intragroup X-ray emission. Col.~(6): Net counts in the ''full''
0.3--7~keV band. Col.~(7): 0.5--2~keV thermal luminosities
($10^{39}$~erg~s$^{-1}$). Col.~(8): 0.5--2~keV power-law
luminosities ($10^{39}$~erg~s$^{-1}$). Col.~(9): Best-fit X-ray
temperature; asterisk indicates a fixed value. Col.~(10): Best-fit
power-law index; asterisk indicates a fixed value. Col.~(11):
(2--7~keV)/(0.3--2~keV) hardness ratio. Col.~(12): Classification
of the X-ray spectrum according to statistical quality (1--3, as described
in Section~\ref{sec,spec}), and according to consistency with a power-law only
(P), both thermal and power-law components present (TP), or spectral
composition unknown (U) but hardness ratio suggests thermal
component present (T). Col.~(13): Specification of
whether the X-ray emission is extended (E), point-like (P), or
uncertain (``?''), and whether the galaxy is star-forming (SF), a
LINER (LI), or Seyfert~II (Sy2).}
\end{deluxetable*}
For all detected group members, source spectra and associated response
files were extracted within circular regions extending to where the
galaxy surface brightness becomes consistent with the local background
level. Background spectra were extracted in surrounding annuli, and
results were fitted in the 0.3--7.0~keV band using {\sc xspec}
v.\,12.3 \citep{arna96}. As many of the sources have few counts, the
maximum likelihood--based C-statistic \citep{cash79} was used to
determine best-fit parameters. The goodness-of-fit was verified using
the $\chi^2$ statistic for all sources with sufficient photon
statistics to allow meaningful constraints from spectra accummulated
in bins of at least 20~net~counts.
The X-ray emission in luminous early-types is generally dominated by
that of hot gas and low-mass X-ray binaries (e.g., \citealt{fabb06}).
We modeled any hot gas emission in the group members using the {\em
mekal} optically-thin thermal plasma model \citep{mewe85,lied95} in
{\sc xspec}, assuming solar abundance ratios. Any X-ray binary/active
galactic nucleus (AGN) component was modeled with a power law. In all
fits, the absorbing hydrogen column density was fixed at the Galactic
value ($N_{\rm H}=4.0\times10^{20}$~cm$^{-2}$; \citealt{kalb05}),
since the limited photon statistics generally precluded useful
constraints on $N_{\rm H}$ from spectral fitting itself. Following
\citet{sun07} and \citet{jelt08}, the metallicity in the {\em mekal}
model was fixed at $0.8~Z_\odot$, the mean value found for galactic
hot gas in the large cluster galaxy sample of \citet{sun07}. This left
four free model parameters: Gas temperature $T$, power-law index
$\Gamma$, and two normalizations. However, due to limited number of
counts, the $3\sigma$ uncertainties on $T$ and $\Gamma$ were
unconstrained for most of the galaxies. In such cases we first fixed
$\Gamma$ at 1.7, as is typical for X-ray binary and AGN spectra, and,
if necessary, also $T$ at 0.7~keV. Based on the photon statistics and
spectral results, the detected galaxies were then divided into three
categories, as summarized in Col.~(12) of Table~\ref{tab:X}:
(1) Galaxies for which the normalization of both components was
well-constrained at the $2\sigma$ level (three galaxies in total;
well-constrained here means that the $2\sigma$ uncertainties are
finite, and that the normalization is non-zero within those
uncertainties). For the central galaxy NGC\,2563, all of the four
parameters are well constrained, but its emission is not easily
deblended from that of the intragroup medium. The other two galaxies
are NGC\,2557, where $\Gamma$ was fixed, and IC\,2339, where also $T$
was fixed. Best-fit values and $1\sigma$ errors for all fit parameters
were determined using Markov Chain Monte Carlo (MCMC) simulations in
{\sc xspec}. The values reported were obtained from the median value
of the chain. We note for completeness that the standard deviation
leading to a parameter being here considered well-constrained (which
is based on the change in the adopted fit statistic when the parameter
is varied, using {\sc xspec}'s standard ``error'' command) is not
necessarily identical to that resulting from the MCMC simulations.
(2) Galaxies with $\ga 30$~counts but for which the normalization of
one or both components remained unconstrained within the $2\sigma$
uncertainties (six galaxies in total). In the cases where the
$2\sigma$ lower limit on the thermal luminosity $L_{\rm X,th}$ was
consistent with zero, whereas that on the power-law component was not,
the spectrum was assumed to be consistent with a power-law
only. IC\,2338 is consistent with displaying both a thermal and a
power-law component, but, unlike galaxies in category~(1), its
$2\sigma$ upper limit on $L_{\rm X,th}$ remained unconstrained. Using
MCMC, we report the corresponding $1\sigma$ upper limit. For the
remaining galaxies in this category, the best-fit $\Gamma$ was
consistent with $\approx 1.7$ as expected from X-ray binaries or an
AGN, whereas the best-fit temperature was much higher ($T >4$~keV)
than expected for hot galactic gas. Hence, a power-law component
likely dominates their X-ray output. For these sources, we fixed the
best-fit power-law model and added a $T=0.7$~keV thermal
component. The limit on $L_{\rm X,th}$ was determined from the
$1\sigma$ upper limit of the normalization of the {\em mekal} model.
(3) Galaxies detected with $\la 30$~net counts (eight in total). In
these cases, statistics are insufficient to allow a robust
determination of the nature of their X-ray emission. A $1\sigma$
upper limit to their $L_{\rm X,th}$ was estimated assuming a
$T=0.7$~keV {\em mekal} model, but these galaxies are labeled with
``U" (for unknown spectral composition) in Col.~(12) of
Table~\ref{tab:X}. To help clarify the nature of these sources, we
also considered their hardness ratios $H\!R$ (here the
2--7~keV/0.3--2~keV flux ratio). This ratio is 0.02--0.05 for a {\em
mekal} model with $T=0.7$--1.0~keV, and 0.35--0.47 for a power-law
spectrum with $\Gamma=1.7$--2.0. For $0.05\!<\!H\!R\!<\!0.35$, the
dominant component is ambiguous, but some contribution from a thermal
component is allowed. Although these classifications are only
tentative, given the Poisson uncertainties on $H\!R$, we do note that
all group members spectroscopically confirmed to require only a
power-law show $H\!R>0.4$, whereas $H\!R\lesssim0.35$ for the galaxies
with a spectroscopically identified thermal component. In
Table~\ref{tab:X}, a ``T" is listed after ``U" in Col.~(12) if
$H\!R<0.35$, to indicate that the galaxy may contain thermal emission.
The remaining galaxies are labeled with either ``TP", meaning that
both thermal and power-law components are likely present
($0.05\!<\!H\!R\!<\!0.35$), or ``P", indicating consistency with a
power-law only ($H\!R>0.35$).
To further aid in identifying the nature of galactic X-ray emission,
we used its spatial extent to help discriminate between nuclear (AGN)
and galaxy-wide emission (e.g., hot gas halos, X-ray binaries). For
each detected group member, we fitted the 0.5--2~keV brightness
profile with a Gaussian, and classified it as extended if the
$1\sigma$ lower bound of its full width at half-maximum (FWHM) was
$\geq 10$\% larger than that of the local point spread function (PSF)
extracted at peak source energy. Overall, most of the detected group
members appear extended in X-rays, and these are labeled with an ``E''
in Col.~(13) of Table~\ref{tab:X}, or with a ``P'' (point-like)
otherwise. For most sources with $\la 30$~counts the extent remains
poorly constrained, however, and Col.~(13) then includes a question
mark.
For the X-ray undetected group members, we followed the procedure of
\citet{sun07} and \citet{jelt08} to estimate luminosity upper
limits. Count rates were measured within a circular aperture of $R=
3$~kpc from the optical galaxy center, in a few cases expanded to
include the $90\%$ encircled energy radius of the local PSF at
$E=1$~keV to account for PSF smearing. The upper limit on $L_{\rm
X,th}$ was derived from the Poisson $3\sigma$ upper limit on the
count rate \citep{gehr86} assuming a $T=0.7$~keV {\em mekal} model.
\vspace{2mm}
\subsection{VLA Observations and Analysis}\label{sec,hiobs}
The H{\sc i} analysis was based on two different observing runs at the
VLA in its most compact, 1~km (D-array), configuration. The first run,
done in 1999, covered the entire group with a $6\times 6$ point
mosaic. Individual pointings were separated by $15'$, fully sampling
the primary beam with its FWHM of $30\arcmin$. To probe as wide a
velocity range as possible with sufficient velocity resolution, we
used a total bandwidth of 3.125~MHz and four Intermediate Frequency
(IF) channels with two frequency settings and slightly overlapping
bands. The total velocity range covered was $\sim$\,1100~km~s$^{-1}$
with a resolution of 20~km~s$^{-1}$. The data were calibrated using
standard AIPS\footnote{http://www.aips.nrao.edu/} procedures, and then
imported into
MIRIAD\footnote{http://www.atnf.csiro.au/computing/software/miriad/}
for mosaicing and joint deconvolution, turning the individual
pointings into one mosaiced cube for each IF. The continuum was then
subtracted from the $u$--$v$ data using a fit through the line-free
channels identified within the cube. However, inspection of the final
cube showed that our velocity coverage was insufficient, with several
group members detected close to the edge of the band and their H{\sc
i} velocity range only partly probed.
To remedy this, NGC\,2563 was reobserved in 2007, using a 14-point
mosaic that covered an area identical to that of our {\em Chandra}
mosaic. To also cover a larger velocity range, we used a total
bandwidth of 6.25~MHz with two IFs and no online Hanning
smoothing. Each pointing was observed at two separate frequencies,
resulting in a total velocity coverage of 2100~km~s$^{-1}$. Each
pointing and velocity setting was calibrated separately, and cubes
were made using AIPS. Overlapping frequencies were then combined in
the image plane, resulting in a 101--channel cube per pointing. After
identifying channels containing H{\sc i}, the continuum was
iteratively subtracted from each cube using a linear fit in the image
plane through the line--free channels. The resulting cubes were then
CLEANed, combined into one, and the output corrected for the primary
beam response. Where cubes overlapped, they were averaged with
weighting that accounted for the location within the pointing and for
the distance from the center of each field. Although interference
generated by the VLA rendered data unusable within a narrow velocity
range (4331--4374~km~s$^{-1}$), the only group member affected is an
Sm dwarf with $L_K \approx 1\times 10^9 L_\odot$, so this has little
impact on our results.
Instrumental parameters and specifics of the cubes for both data sets
are summarized in Table~\ref{tab,VLA}, including their velocity ranges
(heliocentric, optical definition). Since complete velocity coverage
is important here, the 36-point mosaic was used exclusively to search
for H{\sc i} emission, combining it with the central part of the
14-point mosaic. All remaining H{\sc i} analysis employed the 14-point
mosaic only. This was searched for neutral hydrogen using three
different methods. First the 101--channel cube was divided into groups
of 25, smoothed spatially to half resolution, and Hanning--smoothed in
velocity. This was then used as a mask for the full-resolution cube,
blanking all pixels below $2\sigma$ in the smoothed cube and summing
over 25~channels. Next we searched the cube by eye, stepping through
the velocity channels at different speeds and comparing our detections
with the first method. Finally, we plotted spectra integrated over a
few beams at the location of the optically identified group
members. This did not give any additional detections, so our search
was essentially an optically blind one.
\begin{table}
\begin{center}
\caption{Parameters of the VLA D-array Observations\label{tab,VLA}}
\begin{tabular}{lcc} \tableline \hline
Parameter & 36-Point Mosaic & 14-Point Mosaic \\ \hline
RA (J2000) & 08 20 23.7 & 08 20 23.7 \\
Dec (J2000) & 21 05 00.5 & 21 05 00.5 \\
Vel.~range (km~s$^{-1}$) & 4364--5449 & 3834--5942 \\
Vel.~resolution (km~s$^{-1}$) & 21 & 21 \\
Synthesized beam ($\arcsec$) & $70\times 59$ & $64\times 53$ \\
{\em rms} noise (mJy beam$^{-1}$) & 0.5 & 0.6 \\ \hline
\end{tabular}
\end{center}
\end{table}
Figure~\ref{fig:HIchannel} shows a grey-scale image of a typical
channel in the cube, to illustrate the spatial noise distribution and
the position of the group members. In the central square degree ($R
\la 600$~kpc~$\approx 0.7 R_{\rm vir}$) the {\em rms} noise (using
robust weighting) is remarkably uniform at 0.6~mJy~beam$^{-1}$,
equivalent to $N_{\rm H} \simeq 4 \times 10^{18}$~cm$^{-2}$. Assuming
$3\sigma$ over five~channels, the corresponding sensitivity is $M_{\rm
HI} = 2 \times 10^8$~M$_{\odot}$. Along the $9\arcmin$ wide outer
edges, the noise and sensitivity rise to 3.0~mJy~beam$^{-1}$ and
$1\times 10^9$~M$_{\odot}$, respectively, due to the correction for
the primary beam response. This only affects five out of the 64 group
members and none of our conclusions. A total noise--free H{\sc i}
image was produced by again using the smoothed cube as a mask and
summing each detected galaxy over the narrow velocity range in which
H{\sc i} was detected. The total H{\sc i} mass for each galaxy was
then determined by summing the flux in individual channels in a box
centered on the galaxy. The resulting typical uncertainty is 10\% for
the galaxies with $M_{\rm HI}$ above a few times $10^9$~M$_{\odot}$,
and $2\times 10^8$~M$_{\odot}$ for the remainder.
\begin{figure*}
\begin{center}
\epsscale{.8}
\plotone{f1.eps}
\end{center}
\figcaption{Grey-scale image of a typical channel in the H{\sc i} data
cube for the 14 point VLA mosaic. Crosses mark the position of the
group members. The one group member not covered in the 14 point VLA
mosaic, IC\,2253, was observed but not detected in the 36 point
mosaic.
\label{fig:HIchannel}}
\end{figure*}
\vspace{3mm}
\section{Results}\label{sec,results}
\subsection{X-ray Results}\label{sec,X}
Table~\ref{tab:X} lists all 17 X-ray detected group members (of the 54
observed), and whether these show evidence for star formation or AGN
activity. This is based on their location in a BPT diagram
\citep{bald81,kewl01,kauf03b}, as inferred from an emission-line
analysis of their SDSS spectra, with optical line fluxes measured
using the code described in \citet{trem04}.
Figure \ref{fig:mosaic} shows a 0.5--2~keV mosaic image of the group
from our 14 {\em Chandra} pointings, adaptively smoothed using the
'csmooth' algorithm, with signal significance set between
3$\sigma$--5$\sigma$. To produce this, spectrally weighted exposure
maps were smoothed to the same spatial scales, and the
exposure-corrected images combined into one. The impact of edge
effects was reduced by removing regions with (smoothed) exposure
values below 3\% of the maximum, but small enhancements at the
boundaries of adjacent pointings still remain. Detected group galaxies
are marked with squares in the figure, with interlopers accounting for
the remaining X-ray sources.
\begin{figure*}
\begin{center}
\epsscale{.95}
\plotone{f2.ps}
\end{center}
\figcaption{Adaptively smoothed 0.5--2.0~keV image of the NGC\,2563
group from our 14 {\em Chandra} ACIS-I pointings, with intensity
plotted on a log scale. All group members detected in the unsmoothed
{\em Chandra} data are marked with squares.
\label{fig:mosaic}}
\end{figure*}
Due to the spatial variation in {\em Chandra} effective area, our
X-ray detection limits are not uniform across the field, but we reach
a $3\sigma$ upper limit in the 0.5--2.0~keV band of $L_{\rm X} \approx
3\times10^{39}$~erg~s$^{-1}$ or better for all of the observed group
members. These span almost three orders of magnitude in $L_K$, with
nearly all galaxies above $0.5 L^\ast$ in the $g$-- or $K$--band
detected in X-rays (7/8 in $K$, 15/16 in $g$), whereas the
corresponding fraction is less than half (4/9) for those fainter than
$0.2L_K^\ast$. This is qualitatively consistent with other studies
which show a strong correlation between galaxy optical and X-ray
luminosity \citep{fabb92,read01,sull01,sull03}. As noted earlier, the
ten members not observed by {\em Chandra} are mostly optically faint,
with seven below $0.1L_{K}^\ast$ and only one above $L_{K}^\ast$.
Among the nine galaxies for which X-ray spectral fitting provides
useful constraints (cases~1 and 2 in Section~\ref{sec,spec}), four
show evidence for a thermal component, including the central NGC\,2563
itself. From the hardness ratios of the remaining (case~3) galaxies,
two additional objects show evidence for thermal emission, for a total
of six such galaxies. However, recalling from Section~\ref{sec,spec}
that $H\!R < 0.05$ for a thermal--only model, we note that no source
besides NGC\,2563 itself has a hardness ratio $H\!R<0.2$, so a
power-law component is likely present (if not dominant) in all the
X-ray detected galaxies. In all group members with more than
40~counts, the emission is inferred to be spatially extended, and
their smoothed 0.5--2~keV contours overlayed on SDSS $r$-band images
are shown in Figure~\ref{fig:2}. Note the offset between the X-ray
peak and the optical center in NGC\,2560, possibly indicating the
presence of an ultra-luminous X-ray source as seen in other nearby
galaxies \citep{mill04,robe07}.
\begin{figure}
\begin{center}
\epsscale{1.17}
\plotone{f3.ps}
\end{center}
\figcaption{Smoothed 0.5--2 keV X-ray contours overlayed on SDSS
$r$-band images for the eight X-ray brightest group members. Each
image is $2\arcmin \times 2\arcmin$ ($\sim 40 \times 40$~kpc).
\label{fig:2}}
\end{figure}
Previous studies have revealed a correlation between $L_K$ and thermal
X-ray luminosity for early-type galaxies in all environments
\citep{sull01,sun07,jelt08,mulc10}. Taking early-types to be Sa or
earlier for consistency with these works, then roughly half of the
members in NGC\,2563 belong to this category.
Figure~\ref{fig,scaling}(a) compares $L_{\rm X,th}$ for our optically
bright group members to the scaling relation derived by \citet{jelt08}
for early-types in groups. The central galaxy NGC\,2563 itself is here
excluded for reasons discussed earlier. Our two other early-types with
evidence for a thermal component (NGC\,2557 and NGC\,2569) are
consistent with the \citet{jelt08} relation, while the majority of the
X-ray undetected galaxies are too optically faint to offer much
further insight with the present X-ray detection limits. With the
limited statistics, we simply conclude that the detected early-types
in NGC\,2563 do not deviate significantly from the $L_{\rm X}$--$L_K$
relation found for other group ellipticals.
\begin{figure}
\begin{center}
\epsscale{1.17}
\plotone{f4.eps}
\end{center}
\figcaption{(a) Thermal X-ray luminosities of all optically bright
group members covered by {\em Chandra} (excluding the central
NGC\,2563 itself), and the corresponding scaling with galaxy $L_K$
for early-types in groups from \citet{jelt08} (dashed) and its
$3\sigma$ errors (dotted). Black symbols represent galaxies with
evidence for a thermal component, grey symbols the remaining X-ray
detected galaxies, and empty symbols the X-ray undetected
objects. Early-type galaxies are shown by squares, late-types by
triangles. (b) 0.3--8~keV $L_{\rm X}$ for all X-ray detected
members, again excluding NGC\,2563 itself, along with the scaling
relation (and $1\sigma$ errors) for total $L_{\rm X}$ from low-mass
X-ray binaries in early-types from \citet{kim04}. Symbols are as
above.
\label{fig,scaling}}
\end{figure}
To further elucidate the potential hot gas contribution to the total
$L_X$ of our early-types, Figure~\ref{fig,scaling}(b) compares the
0.3--8~keV $L_X$ of the X-ray detected group members to that expected
from low-mass X-ray binaries using the scaling relation of
\citet{kim04}. NGC\,2563 itself is again excluded. None of of our
early-types shows evidence of a significant X-ray excess relative to
this expectation, suggesting that the majority of their X-ray output
is of stellar origin. A few of the more luminous galaxies even lie
below this relation, including NGC\,2557 which does contain a thermal
component.
Of the eight X-ray detected late-type members, three show evidence for
a thermal component. All eight galaxies have $L_g>0.5L_{g}^\ast$ but
span a relatively larger range in $L_{K}$, suggesting their total
X-ray output is more directly related to the blue stellar light than
to stellar mass. We note the similarity to other studies of late-type
galaxies that find their diffuse $L_{\rm X}$ to correlate more
strongly with $L_B$ (and hence star formation rate) than with $L_K$ or
stellar mass \citep{tull06,sun07}. Our SDSS spectra generally support
this, with seven of our eight X-ray detected late-type members showing
optical line emission consistent with star formation. The eighth
galaxy (NGC\,2558) has line ratios consistent with a LINER. Based on
the BPT diagram, there are 12 other actively star-forming late-types
in the group, but all have $L_g < 0.5L_g^{\ast}$ and remain X-ray
undetected.
\subsection{HI Results}\label{sec,HI}
We detect H{\sc i} in 20 of the 64 group members down to our H{\sc i}
mass limit of $\sim$ $2\times 10^8$~M$_{\odot}$ (recall that our
limits are higher for galaxies at the very edge of the VLA
mosaic). The results for these 20 objects are summarized in
Table~\ref{tab:HI}, along with results for three large spirals that
remain undetected in H{\sc i} but whose H{\sc i} deficiencies can
still be estimated (see below). Quoted H{\sc i} velocity widths
correspond to the velocity range of channels with signal at the $\geq
2\sigma$ level; note that for UGC\,04332 the H{\sc i} profile extends
to the edge of our velocity coverage, so the H{\sc i} width and mass
provided in Table~\ref{tab:HI} are lower limits. Two of the 20 H{\sc
i} detections are associated with galaxies not previously identified
as group members from optical spectroscopy (SDSSJ082044.60+210715.0
and SDSSJ081931.17+203843.9). Both of these represent optically faint
irregular galaxies.
\begin{table*}
\begin{center}
\caption{Parameters of the VLA D-array Observations\label{tab:HI}}
\begin{tabular}{llrrccrr} \tableline \hline
Galaxy & Morph. & $R$ & H{\sc i} width & H{\sc i} vel. &
Optical vel. & Log\,(M$_{\rm HI}$) & Def$_{\rm HI}$ \\
& & (kpc) & (km~s$^{-1}$) & (km~s$^{-1}$) & (km~s$^{-1}$) & (M$_{\odot}$) \\
\hline
SDSSJ082044.60+210715.0& Irr& 73 & 42 & 4406\tablenotemark{a} & -- & 8.28 & -- \\
CGCG119-061& Sab & 112 & -- & -- & 5235 & $<$8.60\tablenotemark{b} & $>$$+0.56$ \\
CGCG119-059&Sc & 165 & 170 & 4215 &4211 & 8.73 & $+0.39$ \\
UGC\,04344&Sc & 238 & 149 & 5033 & 5041 & 9.89 & $-0.10$ \\
SDSSJ081938.81+210353.0 &Irr & 254 & 43 & 4236 & 4240 & 8.48 & -- \\
UGC\,04332 & Sa pec & 264 & $>$449\tablenotemark{c} & 5481 & 5514 &
$>$9.41\tablenotemark{c} & $<$$+0.04$\tablenotemark{c} \\
CGCG119-053 & Sa pec & 339 & 171 & 4852 & 4877 & 9.26 & $-0.17$ \\
UGC\,04329 & Sc pec & 439 & 254 & 4109 & 4099 & 9.98 & $+0.10$ \\
SDSSJ081905.46+211448.0 & Sm? & 451 & 106 & 4842 & 4844 & 8.51 & -- \\
IC\,2293& SBbc & 470 & 212 & 4088 & 4094 & 8.91 & $+0.55$ \\
CGCG119-043& Sc pec & 506 & 170 & 4470 & 4458 & 8.65 & $+0.47$ \\
CGCG119-051& Sb\tablenotemark{d} & 513 & 191 & 5033 & 5028 & 9.32 & $-0.02$ \\
SDSSJ081931.17+203843.9& Irr & 564 & 85 & 4980\tablenotemark{a} & -- & 8.81 & -- \\
CGCG119-040& Sa & 663 & 21 & 4841 & 4816 & 7.78 & $+1.23$ \\
UGC04324& Sab\tablenotemark{d} & 664 & 340 & 4831 & 4814 & 9.20 & $+0.32$ \\
SDSSJ081904.25+213521.0& Irr & 724 & 106 & 4863 & 4865 & 8.95 & -- \\
NGC\,2558& Sb & 739 & 405 & 4990 & 4998 & 10.08 & $-0.23$ \\
IC\,2338 & SBcd\tablenotemark{d} & 848 & 313\tablenotemark{e}
& 5413\tablenotemark{e} & 5400 & 9.76\tablenotemark{e} & $-0.34$\tablenotemark{e} \\
IC\,2339 & SBc pec & 858 & 313\tablenotemark{e} & 5413\tablenotemark{e} &5420 & 9.76\tablenotemark{e} & $-0.34$\tablenotemark{e} \\
CGCG119-082&SBa & 898 & -- & -- & 4783 & $<$9.00\tablenotemark{b} & $>$$+0.09$ \\
UGC\,04386& Sab pec & 921 & -- & -- & 4640& $<$9.00\tablenotemark{b} & $>$$+0.61$ \\
CGCG119-047&Sab\tablenotemark{d} & 923 & 276 & 4502 & 4506 & 9.73 & $-0.45$\\
SDSSJ082352.25+212507.2 &Sc & 964 & 171 & 5086 & 5095 & 8.85 &$+0.45$ \\ \hline
\end{tabular}
\end{center}
\tablecomments{(a) New H{\sc i} group member. (b) H{\sc i}
undetected. (c) Incomplete H{\sc i} velocity coverage. (d) H{\sc i}
tail. (e) Closely interacting pair. Values apply for the entire
system.}
\end{table*}
H{\sc i} is detected only in galaxies of types Sa and later, including
in all 15 such galaxies with $L_K >0.1L_K^\ast$.
Figure~\ref{fig,hi_mosaic} shows contours from our noise--free H{\sc
i} mosaic overlayed on the {\em Chandra} mosaic of the full group,
and Figure~\ref{fig:SDSS_HI} displays an H{\sc i}/optical overlay for
each H{\sc i} detected galaxy. The latter figure reveals four
significant H{\sc i} extensions in the group associated with six
galaxies (IC\,2238/IC\,2339, UGC\,04324/CGCG119-040, CGCG119-047 and
CGCG119-051, all labeled with an asterisk in the Figure). This implies
that $\sim 30$\% of our H{\sc i} detected group members display H{\sc
i} evidence for being involved in an ongoing interaction. The
galaxies IC\,2338 and IC\,2339 represent a particularly close pair,
and the H{\sc i} quantities quoted in Table~\ref{tab:HI} therefore
apply for the entire system. The absence of detectable H{\sc i} among
the early-type group members is consistent with previous studies
finding H{\sc i} above our detection limits in only a small fraction
of early-types \citep{burs87,morg06,dise07}. Since we have performed
an optically blind search for H{\sc i} across the entire group, we can
furthermore rule out the existence of any optically dark H{\sc i}
clouds not associated with any galaxy down to our H{\sc i} mass limit.
\begin{figure*}
\begin{center}
\epsscale{.85}
\plotone{f5.eps}
\end{center}
\figcaption{0.5--2.0~keV grey-scale image of the group, with total
H{\sc i} contours overlayed at 0.04, 0.4, 1.2, and 2.4
Jy~beam$^{-1}$~km~s$^{-1}$ (corresponding to $N_{\rm H} = 1.27$,
12.7, 38.2, and $76.4\times 10^{19}$~cm$^{-2}$). Individual group
members are marked by crosses; IC\,2253, not covered by the 14-point
VLA mosaic, is not included.
\label{fig,hi_mosaic}}
\end{figure*}
\begin{figure*}
\begin{center}
\epsscale{.92}
\plotone{f6.ps}
\end{center}
\figcaption{H{\sc i} contours overlayed on SDSS $r$-band images for
the 20 H{\sc i} detected group members, in order of increasing
distance from the group center. Each image is $7' \times 7'$ ($\sim
130\times 130$~kpc), with contours at 0.04, 0.08, 0.16, 0.32, 0.64,
1.28, and 2.50~Jy~beam$^{-1}$. Arrows indicate the direction to the
group centre. Objects with significant H{\sc i} tails/extensions are
marked with an asterisk in the lower left corner.
\label{fig:SDSS_HI}}
\end{figure*}
To investigate to what extent the group members may be deficient in
H{\sc i}, we estimated H{\sc i} deficiencies Def$_{\rm HI}$ following
the usual prescription,
\begin{equation}
\mbox{Def}_{\rm HI} = \mbox{log}\,(M_{\rm HI})_{\rm
expected} - \mbox{log}\,(M_{\rm HI})_{\rm observed}.
\end{equation}
The expected H{\sc i} masses were computed from the best-fit
relationship between total H{\sc i} mass and galaxy optical diameter
$D$ derived for galaxies in the field \citep{sola96}. $D$ is here
defined as the galaxy major axis measured from the Palomar Observatory
Sky Survey blue prints. Where not already available from the Uppsala
General Catalog of Galaxies \citep{nils73}, we measured $D$ directly
from these prints. As the \citet{sola96} field sample was restricted
to rather large spirals of types Sa through Sc, only group members of
those types and with $D>10$~kpc were considered here. Note, as
mentioned above, that this includes three galaxies not detected in
H{\sc i}, and these are also listed in Table~\ref{tab:HI}. For the
IC\,2338/2339 pair, a single deficiency was calculated from the total
observed and expected H{\sc i}~masses given the optical sizes of both
galaxies.
The resulting deficiencies listed in Table~\ref{tab:HI} span a wide
range, but given the uncertainties in $D$ and optical morphology, as
well as the dispersion in the H{\sc i} masses of field galaxies,
objects with observed H{\sc i} masses within a factor of a few of the
expected values cannot be considered deficient. However, there are six
galaxies with Def$_{\rm HI} \ga 0.45$, consistent with an H{\sc i}
deficiency of at least a factor of three. One of these is UGC\,04386,
but Def$_{\rm HI}$ for this object is not well-established, in part
because it resides at the eastern edge of the H{\sc i} mosaic where
the data are noisier, and in part because it is viewed edge-on and is
of uncertain morphological type.
\subsection{The Intragroup Medium}
Although our main goal is to study the ISM in individual group
members, it is also relevant to establish which galaxies are embedded
within detectable intragroup gas. To do so, a radial surface
brightness profile was extracted from the unsmoothed
exposure-corrected {\em Chandra} mosaic, centered on NGC\,2563 itself,
and with all other bright sources masked out. The background level was
estimated from the four outermost pointings, which are all centered at
$R=40\arcmin$--$45\arcmin$ from the X-ray peak and show consistent
full-chip count rates. The resulting background-subtracted profile,
shown in Figure~\ref{fig:surfbright}, reveals diffuse emission out to
at least $R \approx 21\arcmin$ ($\approx 400$~kpc in projection). This
is a few arcminutes further than the maximum extent determined from
{\em ROSAT} observations \citep{mulc03,osmo04}, and roughly
corresponds to our estimate of $R_{500} \approx 420$~kpc. Modeling the
profile as a sum of two $\beta$--models yields $\beta_1 =
0.61^{+0.09}_{-0.03}$, $r_{c1} = 1.0\pm 0.1$~kpc, and $\beta_2 =
0.32\pm 0.01$, $r_{c2} = 1.1^{+1.3}_{-1.0}$~kpc, for the inner and
outer component, respectively. With a reduced $\chi^2=3.0$, the fit is
poor however, in part because the profile steepens beyond that of a
$\beta$--model at large radii, as also seen in other groups and
clusters \citep{vikh06,sun09}.
\begin{figure}
\begin{center}
\epsscale{1.2}
\mbox{
\hspace{1mm}
\plotone{f7.eps}}
\end{center}
\figcaption{Background-subtracted radial profile of the 0.5--2~keV
intragroup emission, extracted in $2\arcsec$ pixels. Solid line
shows the best-fit double $\beta$--model, with the individual model
components shown as dot-dashed lines. Error bars include
uncertainties on the background level.
\label{fig:surfbright}}
\end{figure}
The relatively high central surface brightness could potentially
compromise our ability to detect individual galaxies in the group
core. We have attempted to circumvent this issue by requiring a longer
central {\em Chandra} exposure (50~ks). In addition, the surface
brightness has dropped from its central value by three orders of
magnitude already at $R=5\arcmin \approx 100$~kpc. Apart from the
central galaxy, there are no optically luminous group members within
the central few arcmin, so our X-ray detection limits are impacted
minimally by the presence of intragroup gas. This is corroborated by
the fact that we find no systematic radial variation in limiting
$L_{\rm X,th}$ for the 54 group members covered by {\em Chandra}. A
detailed analysis of the intragroup medium in NGC\,2563 will be the
subject of future work.
\section{Gas Stripping in NGC\,2563}\label{sec,discuss}
With our H{\sc i} and X-ray results extending well beyond the group
virial radius, we can now explore how the ISM properties of individual
galaxies vary with position across the entire group, and in particular
to what extent there is evidence for ongoing or recent ISM stripping
from the group members.
\subsection{Evidence for Ram Pressure Stripping}
The detection of X-ray tails with {\em Chandra} has provided direct
evidence for ram pressure stripping in both groups
\citep{rasm06,jelt08} and clusters \citep{sun07}, and similar
inferences have been made from the presence of H{\sc i} tails within
these systems \citep{kenn04,crow05,chun07}. In X-rays, however, the
detection rate of these features is generally very small ($<10\%$),
suggesting that strong stripping of hot ISM is either very rare or
proceeds very rapidly. No evidence for such activity in the form of
X-ray tails is seen within NGC\,2563, but this is not surprising given
the low detection rate of thermal coronae within the sample.
In H{\sc i}, we identify two tails associated with the relatively
isolated CGCG119-047 and CGCG119-051, both with H{\sc i} extending
beyond the optical disk in the direction opposite of the group center
(Figure~\ref{fig:SDSS_HI}). Neither of these objects is H{\sc i}
deficient, so any stripping activity must have recently
commenced. Their respective radial velocities relative to the group
mean of 200 and 320~km~s$^{-1}$ imply Mach numbers of ${\cal M} \ga
0.4$ and $\ga 0.6$ for an ambient gas temperature of $T\approx
1$~keV. Although both galaxies are currently beyond the radius to
which intragroup gas is detected ($R\approx 500$ and $\approx
900$~kpc), their H{\sc i} morphologies are suggestive of ongoing ram
pressure (or viscous) stripping. Multi-wavelength observations and
higher-resolution H{\sc i} data would be required to confirm this
(e.g., \citealt{murp09}). Another relatively isolated spiral, IC\,2293
at $R\sim 470$~kpc, displays no H{\sc i} tail but is highly deficient
(Def$_{\rm HI} = 0.55$), and is also a candidate for recent ram
pressure stripping.
To test for the {\em global} importance of ram pressure stripping
within the group, we next consider the radial distribution of ISM
detections. To put all galaxies on an equal footing, their thermal
X-ray luminosities and H{\sc i} masses were normalized by the galaxy
$L_K$. Excluding again the central galaxy NGC\,2563, evidence for hot
gas is seen in only five of the group members, and
Figure~\ref{fig,lxlk} shows that all of these reside relatively far
from the group center, with four out of five located beyond the radius
$R\sim 400$~kpc to which intragroup gas is detected. In contrast, no
hot ISM component is found within $R\sim 300$~kpc, where several of
the optically luminous early-types reside. This includes NGC\,2562 at
$R\sim 90$~kpc, the closest bright galaxy to the group center, with an
upper limit to $L_{\rm X,th}$ which is an order of magnitude below the
value of log\,$(L_{\rm X,th}/L_K) \simeq -4.6$ suggested by the
$L_K$--$L_{\rm X,th}$ relation for early-types in groups
\citep{jelt08}.
\begin{figure}
\begin{center}
\epsscale{1.17}
\hspace{0mm}
\plotone{f8.eps}
\end{center}
\figcaption{$K$-band normalized thermal X-ray luminosity as a function
of projected group radius $R$, excluding the central galaxy
NGC\,2563. Symbols are as in Figure~\ref{fig,scaling}, with symbol
sizes scaling with galaxy $L_K$. Bottom panel shows the cumulative
fraction of optically bright ($L_K>10^{10} L_\odot$) group members
within a given $R$ that are X-ray detected (empty symbols; scaled
down by a factor of 3 for ease of comparison), the fraction
containing a thermal component (grey), and that of the early-types
doing so (black).
\label{fig,lxlk}}
\end{figure}
To further illustrate this point, we note that, given our X-ray
detection limits (Section~\ref{sec,X}), the \citet{kim04} relation
would suggest that we should detect all early-type members brighter
than log\,$(L_K/L_\odot) \approx 10.0$, irrespective of their hot gas
content. In practice, only 8/14 ($57\pm 25$\%) are detected, even if
including NGC\,2563 itself and excluding Sa galaxies. The bottom panel
in Figure~\ref{fig,lxlk} shows the cumulative fraction of group
members covered by {\em Chandra} above this $L_K$ that is X-ray
detected, and the fraction that also contains evidence for a thermal
component. While the former remains constant with $R$, the thermal
fraction drops toward the group core. Although subject to large
Poisson errors, this result is consistent with hot ISM having been
stripped within the dense group core, while the more distant galaxies
retaining a thermal component have yet to experience peak ram pressure
during their orbit (but see also \citealt{balo00}).
In contrast, galaxies with a detectable {\em cold} ISM component are
distributed more evenly with $R$. This is illustrated in
Figure~\ref{fig,mhi_lk}(a), which compares our measured H{\sc i}
masses to those of similar galaxies in the field. A handful of
galaxies show evidence of a significant shortfall in H{\sc i} content
(i.e.\ Def$_{\rm HI}\ga 0.5$), but the values scatter broadly around
Def$_{\rm HI}=0$ and show no systematic dependence on $R$. A
Kolmogorov test was performed to test whether the inferred
deficiencies are, in fact, consistent with being drawn from a Gaussian
parent distribution centered at Def$_{\rm HI} = 0$ with some width
$\sigma$. We find that for $\sigma$ in the range 0.3--0.5, this
probability is at least $85$\%, regardless of whether the upper/lower
limits and the highly deficient CGCG119-040 are included. There is
thus no strong evidence from this that the large Sa--Sc group members
are globally H{\sc i} deficient. However, we present evidence in
Section~\ref{sec,glob} that the late-type population as a whole {\em
is} deficient relative to the field.
\begin{figure}
\begin{center}
\epsscale{1.}
\hspace{0mm}
\plotone{f9.eps}
\end{center}
\figcaption{(a) H{\sc i} deficiencies Def$_{\rm HI}$ of all large
group spirals of types Sa through Sc, along with a histogram of
Def$_{\rm HI}$. Galaxies above the dashed line are deficient by at
least a factor of three. Upper limit represents UGC\,4332, which has
incomplete H{\sc i} velocity coverage and so only a lower limit to
its H{\sc i} mass (cf.\ Table~\ref{tab:HI}). (b) $K$--band
normalized H{\sc i} masses for all H{\sc i} detected group members,
color-coded according to galaxy $L_K$. Lower limit marks
UGC\,4332. In both plots, symbol sizes scale with $L_K$, and
asterisk represents the IC\,2338/2339 pair.
\label{fig,mhi_lk}}
\end{figure}
The estimated H{\sc i} deficiencies depend on assumed galaxy
morphology and the somewhat uncertain comparison values for field
galaxies. A more robust measure of the relative H{\sc i} content
within the group members might be provided by $M_{\rm HI}/L_K$. The
dependence of this quantity on $R$ is shown in
Figure~\ref{fig,mhi_lk}(b). Note that at fixed $R$, lower-mass
galaxies would be more easily ram pressure stripped, but they are also
likely to be more H{\sc i}--rich to begin with. To take this into
account, the galaxies were divided into three bins of comparable
$L_K$. Figure~\ref{fig,mhi_lk}(b) confirms that even at ``fixed''
$L_K$, there is no systematic radial dependence of H{\sc i} content
within the group, and hence that ram pressure stripping of cold ISM is
unlikely to be globally important in this system.
\subsection{Evidence for Tidal Gas Stripping}
The presence of H{\sc i} tails associated with close galaxy pairs is
commonly taken as a sign of tidal encounters \citep{deme08,koop08}.
NGC\,2563 contains two galaxy pairs with clear H{\sc i} extensions
indicating such encounters. The most prominent of these is associated
with IC\,2338 and IC\,2339, a spiral pair separated by only $\sim
15$~kpc in projection and $\sim 20$~km~s$^{-1}$ in velocity. The SDSS
images show evidence for a stellar bridge between the galaxies,
supporting a tidal origin for the H{\sc i} tails. An H{\sc i}
extension is also connecting UGC\,04324 with CGCG119-040; the latter
represents the most H{\sc i} deficient group member (deficient by a
factor of $\sim 15$), with all its H{\sc i} associated with the H{\sc
i} bridge. While no corresponding stellar feature is seen in the
relatively shallow SDSS images, the H{\sc i} morphology is strongly
suggestive of a tidal encounter.
To explore whether recent removal of cold ISM through tidal
interactions can generally explain the observed H{\sc i} deficiencies
within the group, we show in Figure~\ref{fig:HIdef}(a) the H{\sc i}
deficiencies as a function of projected distance $R_1$ to the nearest
neighbor. If excluding the closely interacting IC\,2338/2339 pair and
the upper/lower limits, a Kendall correlation test suggests an
anticorrelation at the $1.0\sigma$ level. Including the upper/lower
limits at their nominal values strengthens the correlation
significance to $2.2\sigma$. The correlation with $R_1$ remains
significant at the $2\sigma$ level when instead considering the
normalized H{\sc i} masses in Figure~\ref{fig:HIdef}(b). While only
indicative, this result is consistent with some H{\sc i} having been
removed in galaxy--galaxy interactions within the group.
\begin{figure*}
\begin{center}
\epsscale{0.9}
\plotone{f10.eps}
\end{center}
\figcaption{(a) H{\sc i} deficiencies Def$_{\rm HI}$ of all large
group spirals of types Sa through Sc as a function of projected
distance $R_1$ to the nearest neighbor. Dashed line is as in
Figure~\ref{fig,mhi_lk}. (b) $K$-band normalized H{\sc i} masses for
the H{\sc i} detected objects against $R_1$. Lower limit marks UGC\,4332.
(c) As (a), but as a
function of $\xi$ from equation~(\ref{eq,xi}). (d) As (b), but as a
function of $\xi$. In all plots, symbol sizes scale with $L_K$, and
asterisk represents the IC\,2338/2339 pair.
\label{fig:HIdef}}
\end{figure*}
Since strong tidal interactions require proximity of two galaxies in
both position and velocity space, a plot of H{\sc i} content against
$R_1$ may be subject to large scatter due to some galaxies being close
in projection only. For each galaxy, we therefore also consider the
minimum value of
\begin{equation}
\xi = \sqrt{ (\Delta R/\Delta R_{\rm max})^2 + (\Delta v_r/\Delta v_{\rm
max})^2} ,
\label{eq,xi}
\end{equation}
where $\Delta R$ and $\Delta v_r$ are the separations in projection
and radial velocity between {\em any} two group members, and $\Delta
R_{\rm max} \approx 2.1$~Mpc and $\Delta v_{\rm max} \approx
1600$~km~s$^{-1}$ are the corresponding maximum values between all
galaxies in NGC\,2563. Strong interactions would require $\xi \ll 1$,
and, if generally prominent in removing H{\sc i}, would imply a
correlation between H{\sc i} content and $\xi$ for small values of the
latter. For NGC\,2563, one might expect interactions to be important
up to at least $\xi \approx 0.1$, as this would correspond to two
galaxies at the same $v_r$ separated by $\Delta R\la 200$~kpc, or to a
physically ``overlapping'' pair with $\Delta v \la
160$~km~s$^{-1}$. As suggested by Figure~\ref{fig:HIdef}(c) and (d),
there is some evidence of such a correlation for $\xi \la 0.1$ among
the individually H{\sc i} detected galaxies. However, this is
significant at less than $2\sigma$, so we can only conclude that our
results are at least consistent with some H{\sc i} removal due to
galaxy--galaxy interactions. Repeating this analysis for the X-ray
detected group members reveals no indication of a systematic trend in
$L_{\rm X,th}/L_K$ with either $R_1$ or $\xi$. However, many of these
galaxies have X-ray luminosities only slightly above our completeness
limit, so deeper observations of a larger sample would be required to
confirm this result within groups in general.
\subsection{Global HI Properties of the Group}\label{sec,glob}
Our comprehensive VLA coverage of NGC\,2563 enables a census of the
global amount and distribution of H{\sc i} in the group, and provides
further evidence of recent H{\sc i} mass loss from the group members.
Based on measurements from the H{\sc i} Parkes All Sky Survey
(HIPASS), \citet{evol11} have quantified the typical H{\sc i} mass for
late-type galaxies (Sb and later) in the general ``field'' with
$M_\ast > 10^8 M_\odot$:
\begin{equation}
M_{\rm HI} = M_1 \left(\frac{M_\ast}{M_2}\right)^{0.19}
\left[1+\left(\frac{M_\ast}{M_2}\right)^{0.76}\right],
\label{eq,HI}
\end{equation}
where $M_1=3.36\times 10^9 M_\odot$ and $M_2=3.3\times 10^{10}
M_\odot$. To compare this to results for NGC\,2563, we used estimates
of $M_\ast$ for the group members from
SDSS\footnote{http://www.mpa-garching.mpg.de/SDSS/DR7/}, available for
all but six of the 64 members; for those six, we assume a $K$--band
stellar mass-to-light ratio of 0.55, which is the average for the
members with both $L_K$ and $M_\ast$ available.
Equation~(\ref{eq,HI}) would then suggest a total H{\sc i} mass of the
relevant group members of $6.7 \pm 2.0 \times 10^{10} M_\odot$,
whereas the value inferred from Table~\ref{tab:HI} is $M_{\rm HI} <
4.4\times 10^{10} M_\odot$, when including the average $3\sigma$ upper
limit of $2\times 10^8M_\odot$ for the undetected members. This
suggests that globally, the $M_\ast > 10^8 M_\sun$ late-types in the
group have suffered mild H{\sc i} mass loss with respect to similar
field galaxies. Compared to estimates of Def$_{\rm HI}$, this result
may represent a more robust estimate of global H{\sc i} deficiency
within the group, as it includes also smaller galaxies, depends less
sensitively on the inferred morphology and optical size of individual
group members, and takes the dispersion seen for field galaxies into
account.
We also derive a total stellar mass in the group of $M_\ast \approx
1.1\times 10^{12} M_\odot$ for all the 64 spectroscopically identified
members, and a corresponding total range of $M_{\rm HI} = (5.7-
6.5)\times 10^{10} M_\odot$. This implies $M_{\rm HI}/M_\ast
=0.054$--0.062 for the full group (down to our 98\% spectroscopic
completeness limit of $M_r = -17$), under the plausible assumption
that our VLA observations are not missing any large-scale diffuse
H{\sc i}. Future comparison of this result for an X-ray bright group
to other groups and clusters could provide further information on the
nature of gas removal in these environments.
On the largest scales, the distribution of H{\sc i}--detected galaxies
in Figure~\ref{fig,hi_mosaic} reveals a peculiar lopsidedness, with 16
of the 20 detections occurring on the western side of the
group. This is not simply due to asymmetric spatial VLA coverage, as
demonstrated in Figure~\ref{fig:HIchannel}, nor to a larger fraction
of H{\sc i} deficient galaxies in the east. The wide velocity coverage
of our VLA data ($\Delta v_r \approx \pm 3\sigma_{\rm biwt}$) further
renders it unlikely that a significant fraction of potential group
members have been missed on the eastern side of the system. The
lopsided H{\sc i} distribution thus reflects a real asymmetry in the
distribution of late-type members, with nearly all the large spirals
located in the western half of the group.
The origin of this asymmetry remains unclear. One possibility is that
the western spirals are part of an infalling subgroup, but echoing an
earlier conclusion by \citet{zabl98b}, we find no evidence for
kinematic substructure in the group using the technique pioneered by
\citet{dres88}. Another possible scenario is that galaxy morphologies
have been modified by strong interactions with a similarly asymmetric
intragroup medium. This can be ruled out, however, because the
intragroup gas distribution in NGC\,2563 appears fairly symmetric on
large scales and, if anything, seems more strongly extended to the
west (see Figure~\ref{fig:mosaic} and \citealt{mulc98}). Finally,
exploiting the extensive SDSS coverage in this field, we have also
examined the overall galaxy distribution within a projected distance
of $R=5$~Mpc of the group center, and find no evidence for a general
large-scale enhancement in galaxy density on the western side of the
group. Further studies of other groups on very large scales may help
establish how common such morphological segregations are in these
systems.
\section{Summary and Conclusions}\label{sec,summary}
Our extensive {\em Chandra} and VLA coverage of the NGC\,2563 galaxy
group has enabled us to probe both the hot and cold ISM within a
nearby group out to $\sim 1.4$ times the estimated virial radius. The
main aim has been to characterize the evidence for galactic gas
removal in this system, and understand the mechanisms involved.
Although the limited number of galaxies preclude strong conclusions
based on a single group, our results suggest that both hot and cold
gas have been stripped from some of the group members, and that both
ram pressure stripping and gas removal via galaxy--galaxy interactions
are all occurring simultaneously within this one system. This is based
on the following evidence:
(i) The thermal X-ray deficiency of optically luminous early-type
galaxies in the group core suggests recent ram pressure stripping of
their hot gas. The central group galaxy aside, no hot ISM is detected
within $R\sim 300$~kpc from the group core, whereas four out of the
five non-central galaxies with such a component reside beyond the
radius to which intragroup gas is detected. The thermal luminosity of
the bright early-type closest to the group core (NGC\,2562) is at
least an order of magnitude below that expected for typical group
galaxies of the relevant $L_K$ \citep{jelt08}.
(ii) Comparison of the H{\sc i} content of the late-type group members
to that of similar galaxies in the field suggests that the former are,
on average, mildly deficient in H{\sc i}. This points to one or more
mechanisms removing cold ISM within the group.
(iii) Ram pressure (or viscous) stripping of cold gas is suspected in
a few cases. The H{\sc i} data reveal one relatively isolated spiral
(IC\,2293) which is H{\sc i} deficient by more than a factor of
three. Another two such spirals show H{\sc i} morphologies suggestive
of an ongoing ram pressure interaction, with H{\sc i} tails pointing
away from the group core (CGCG\,119-047 and 119-051 in
Figure~\ref{fig:SDSS_HI}). However, these are not (yet) H{\sc i}
deficient by the usual definition, indicating that significant
interactions with the intragroup medium may have only recently
commenced.
(iv) Ongoing galaxy--galaxy interactions removing H{\sc i} are also
strongly suggested in at least two cases. The two most prominent H{\sc
i} tails/extensions in the group occur within close galaxy pairs,
and the most H{\sc i} deficient group member (CGCG\,119-040, deficient
by a factor of $\sim 15$) is a member of one of these. Suggestive
evidence is further seen for galaxies with close neighbors in
position--velocity space to show relatively low H{\sc i} content,
consistent with tidal stripping of H{\sc i}.
The inference that ram pressure stripping of hot galactic gas may have
occurred in the central group regions would be in line with other {\em
Chandra} studies of group and cluster galaxies \citep{rasm06,sun07},
and with simulations which suggest that such stripping can be
efficient even in small galaxy groups \citep{kawa08}. However, it is
worth noting that a large fraction of early-type galaxies generally do
retain halos even in X-ray bright groups, including some in NGC\,2563,
and that such halos are not generally underluminous compared to those
of galaxies in the field \citep{jelt08,mulc10}. Despite the detection
of a hot intragroup medium within NGC\,2563 out to $R\approx R_{500}$,
there is also no global evidence for ram pressure stripping of the
{\em cold} ISM within the group. Specifically, no radial trends are
seen in the stellar mass--normalized H{\sc i} content among the 20
H{\sc i} detected group members.
The indicative result that ram pressure may affect only the hot ISM in
typical group galaxies is consistent with existing numerical studies
\citep{kawa08,rasm08}. Galaxy--galaxy interactions remain the main
candidate for removing cold ISM within NGC\,2563 and explaining the
global H{\sc i} deficiency inferred for the late-type group members.
Other studies have also implied that tidal encounters have an
important impact on the H{\sc i} properties of galaxies in groups
\citep{kern08,rasm08}. In addition, such encounters may work to
enhance the susceptibility of a galaxy to ram pressure stripping, by
perturbing the distribution of the cold gas
\citep{davi97,maye06}. Nevertheless, despite the detection of two new
group members from our optically blind H{\sc i} search, we find no
evidence within the NGC\,2563 group for isolated, optically dark H{\sc
i} clouds that might represent previously removed material.
Given the small number of luminous galaxies in a single group, similar
studies of a larger group sample will be required to better understand
how the group environment impacts galaxy evolution. For example, we
note that the inferred H{\sc i}-to-stellar mass ratio of $\approx
0.06$ inferred for this X-ray bright group may provide a useful
benchmark for the H{\sc i} content of dynamically evolved
groups. Future comparison of this result to those of X-ray faint
systems, richer clusters, and groups at higher redshift (of which the
upcoming Square Kilometre Array should detect many thousands) could
further improve our understanding of the nature and epoch of gas
removal in dense environments. Such studies would also help to address
the commonality of the highly lopsided distribution of H{\sc i} seen
in NGC\,2563, which reflects a puzzling galaxy morphological
segregation within the group.
\acknowledgments
We are grateful to the referee for a careful and constructive report
which significantly improved the presentation of our results. We thank
Christy Tremonti for providing the emission-line ratios for the SDSS
spectra. This research has made use of the NASA/IPAC Extragalactic
Database (NED). JR acknowledges support by the Carlsberg
Foundation. XB thanks Prof.\ Zhang, S.~Nan, and R.~Fengyun from
Tsinghua Center for Astrophysics (THCA) for discussions on {\em
Chandra} analysis. Support for this work was provided by the
National Science Foundation under grant number 0607643 to Columbia
University, and by the National Aeronautics and Space Administration
through Chandra Award Number G07-8134X issued by the Chandra X-ray
Observatory Center, which is operated by the Smithsonian Astrophysical
Observatory for and on behalf of the National Aeronautics Space
Administration under contract NAS8-03060.
|
1,314,259,996,545 | arxiv | \section{Introduction}
\label{1}
During disruptive events, like the COVID-19 pandemic, it becomes important for policy makers to quickly estimate the impact of the event on various parameters like GDP (Gross Domestic Product), unemployment rate, industrial output, etc. Existing methods to collect or estimate these parameters are time and labour intensive as they require extensive surveys and data collection. Therefore, computation and publication of these data takes time, and policy making is often done in hindsight. While the existing models are time tested and work well in normal times, they fall short during disruptive events like the COVID-19 pandemic, when policy makers need methods with quick turnaround time to estimate economic parameters and subsequently make policy decisions.
In this paper we present \textit{ReGNL} (Regional GDP-NightLight), a neural networks based technique to rapidly predict the GDP of a given geography using remote sensing. The remote sensing data we use is \textit{Nightlights}, processed from data collected by sensors on board earth-orbiting satellites. Nightlights refer to the visible lights being emitted by activities on earth's surface, remotely sensed by sensors on-board various satellites. With advances in data storage and data processing techniques, and Machine Learning algorithms, it is possible to leverage the large amounts of remote sensing datasets for estimation of macroeconomic parameters like GDP.
Off-late, nightlight has emerged as a convenient proxy used to ascertain economic activity of a region \cite{10.1257/jep.30.4.171, 10.1145/3394486.3403347}. We develop ReGNL to estimate the GDP of a given geography using the nightlights data from the same geography. We present our technique on datasets from different states in the USA, and also demonstrate its applicability on the provinces of Germany, because of the availability of regular data from these countries. However, our techniques are general and can be extended to other geographies (cities, states and countries) as well, given that granular and regular data is available. Our model (ReGNL) is trained on the quarterly nightlights and GDP data from 50 US states from 2014 to 2018. We test ReGNL for a normal (economy wise) timeframe by predicting the GDP for the year 2019 (weighted error: 0.7066, weighted error is explained in Section \ref{4.2}). Next, we test ReGNL for resilience against disruptive events by considering Quarter 2 (April to June) of 2020 as the disruption timeframe. ReGNL is able to predict the GDP for the 50 US states for Quarter 2 of 2020 with a weighted error of 0.7243 from the actual GDP values. In comparison, when we use a time series analysis using ARIMA as the baseline prediction model, we obtain a weighted error of 1.9488 for the same quarter. ReGNL therefore outperforms the timeseries ARIMA model for a timeframe with a disruptive event.
The novelty of our technique is in the curation of a custom dataset comprising of around 1000 datapoints which combines the nightlight values, GDP values, along with the geographical coordinates in the form of latitude and longitude (centroid) of the states. The inclusion of geographical coordinates in the dataset is able to introduce an aspect of location proximity to our model which is not captured by the nightlights data alone.
The salient contributions of our work are:
\begin{enumerate}
\item We curate a custom dataset combining the quarterly nightlights values, quarterly GDP values and latitude and longitude of 50 US states from 2014-2020. We also curate a second dataset with nightlights and GDP values from Germany, India, China and Spain.
\item We develop the ReGNL (Regional GDP-NightLight Forecasting) model, a neural network which predicts the GDP of a geography using the nightlights and latitude and longitude as features. ReGNL is able to estimate the GDP with a consistent weighted error of just 0.6981, on an average.
\item Proof that with the enough data, nightlights can be used to make estimations agnostic of disruptive events.
\end{enumerate}
Our results validate previous claims that nightlight can be used as a proxy for GDP, and further asserts that it is robust even during disruptive events, when conventional data collection methods may not be possible.
The rest of the paper is structured as follows - Section \ref{2} contexualizes our work within existing research in the domain. Section \ref{3} gives a detailed description of the dataset curation process. Section \ref{4} provides a detailed description of the approach and model we develop as well as the details of the results obtained. Finally conclusions are drawn and future research directions are given in section \ref{5}.
\section{Related Work}
\label{2}
Several researchers have explored using remote sensing data in various applications in Economics. Bansal et al. \cite{compass-1, compass-2} curate custom datasets from India which can be used for various policy decisions. Robinson et al. build a model to understand changes in building patterns longitudinally \cite{compass-3}. Donaldson, et al. \cite{10.1257/jep.30.4.171} summarise some existing work in the domain of economics using satellite imagery. They talk about the scope that exists for further research in the field of Economics using remote sensing. The paper also discusses the various shortcomings of using remote sensing for research in the economic context, such as spatial dependence, privacy concerns, dataset size and measurement error. This helps researchers understand the progress that has been made so far in this domain and about the gap that exists. Xie, et al. \cite{10.5555/3016387.3016457} highlight the problems faced by policymakers due to lack of reliable data to base their decisions on. The paper suggests the use of satellite image data to gauge the socio-economic status of different regions. However, training Deep Learning models on image data requires a huge amount of data and the training data available is very limited. Hence, the authors use a Transfer Learning framework which involves using a model trained for a separate task/dataset to be reused in a different context. In this case the nightlight data is used as a proxy. The accuracy of the model was found to be close to that of actual field surveys. Liu, et al. \cite{liu2021nightlight} also use a transfer learning framework on the premise that nightlights are an effective proxy for the economic activity of a region (Mainland China in this case). The authors use satellite images along with nightlight intensity levels as the input set of features to train their model. Eventually, they were able to predict the 2018 GDP with a reasonably high R-squared value of 0.71. Our work is different in the sense that (i) we do not use a transfer learning framework, and, (ii) we do not use satellite ``image'' data, instead we collect and compile remote sensing nightlight data for a region and train our model (ReGNL) on the same. Our motivation for doing this is because it is computationally cheaper to train a model on the nightlight (numeric) values instead of training it on images. Furthermore, Computer Vision tasks become susceptible to issues regarding image quality, saturation, blurring, etc.
Han, et al. \cite{10.1145/3394486.3403347} in their work present a three step approach for judging the economic status of a region without having access to ground truth figures by using high resolution satellite images: (i) the man-made and natural objects in a satellite image are classified (segregated) into different collections by using a clustering framework, (ii) partial order graph of the collections identified in the previous step are developed, and, finally, (iii) a Convolutional Neural Network (CNN) based framework is used to sort each of the satellite images (grids) on the basis of the relative positioning of the collections. Eventually a score is assigned to each of the locations (satellite images), this score is indicative of the near real-time economic development of that corresponding location. The robustness of the methodology proposed in this paper is demonstrated by applying the same methodology to different economies such as Vietnam (a developing economy), South Korea (a developed economy) and Malawi. Similar to the previous paper, this paper is also based on satellite ``image'' data, whereas our work here is based on processed (numeric) nighlight data, in order to avoid the computational complexity associated with processing satellite images. Otchia, et al. \cite{otchia2020industrial} use Machine Learning methodologies on nighttime data to study the industrial progress in Africa. This research further confirms the validity of using nighttime data for gauging the economic progress for regions where there is a gap in the data or the quality of data is poor.
Asher, et al. \cite{10.1093/wber/lhab003} found a statistically significant relationship between nightlight data and economic variables such as, per-capita income and consumption, electricity coverage, population growth and density. The analysis done in this paper is based on about 6 lakh locations across India. This validates our assumption of using nightlight data for determining, or using as a proxy for, the economic parameters of a region. Finally, the authors identify the poverty distribution across India and point out that poverty alleviation programmes will have more impact if implemented at the (much more granular) village level as opposed to the district level. Dasgupta, et al. \cite{PPR:PPR260387} use Nightlight Data in combination with electricity consumption data to predict the impact of COVID-19 on the Indian economy. This analysis was done at a national level and predicted a contraction of 24 percent in in the 1st Quarter of the Financial Year 2020, which was quite close to the official figure of -23.9 percent. However, the model performed below par on the state level. This is one of the gaps that our work aims to fill. Our work focuses on much more specific, state or provincial levels so as to help policy-making decisions with the predictions being reported at a much higher resolution (granularity) about each state or province, rather than it being reported just at the national level. Ustaoglu, et al. \cite{ustaoglu2020spatial} develop a pixel-level agricultural and non-agricultural GDP map for Turkey. The authors use various datasets, combining the Terra MODIS-Enhanced Vegetation Index, the VIIRS-NPP nighttime image data and land use/cover data from CORINE. Including the vegetation information and land cover data leads to a marked improvement in the GDP estimates since agriculture is a major driving force in Turkey's economic output. The work done by the authors in this paper outline the significant impact the remotely sensed information can make in GDP estimation and any other economic indicator.
Kulkarni et al. \cite{kulkarni2011revisiting} use satellite image data, to obtain nightlight information, to make estimates of the economic activity at the sub-regional levels for the nations: India, USA and China. These estimates were then compared with the sub-regional economic indicators. This analysis was done for China, USA and India for the years 2001 to 2007 and the results were quite encouraging for USA and India but not for China. The unfavourable results obtained for China, were attributed to various factors, such as, the saturation level of satellite sensors, the internal migration in China, etc. Skoufias et al. \cite{skoufias2021can} use VIIRS data (which is a format of nightlight data) to gauge the impact that natural disasters such as floods, earthquakes etc. have on the nightlight intensity of a region and how do these correspond to the short-term damage caused there. The authors conclude by apprising researchers to be careful while using the VIIRS data to critique the consequences of natural hazards.
Gallup, et al. \cite{gallup1999geography} examined the role that the geography of a region plays in its economic progress. In their work they highlight how the location and climate of a region impacts its economy through different facets such as agricultural productivity, transportation costs and the burden of disease. In several locations where the economic growth has been slow, the problem has been further compounded by an increased rate of growth in population. Yet another unfavourable situation for countries are when they are landlocked and located in a region with lack of access to coasts, this leads to a reduced amount of international trade which is one of the important drivers for economic growth. These disadvantaged regions are the ones that are predicted to witness the majority of the global population growth. The hypotheses proposed in this paper acts as the basis for us to include geographical coordinates using longitude and latitude as one of the input features in our analysis so as to enhance the working of our model, ReGNL. Maskell, et al. \cite{doi:10.1080/00291959608542835} talk about majority of a region's economic activity being driven by organisations embedded in that region's structure and that their economic growth is reliant on competencies specific to them, despite the increased levels of globalization, worldwide. Bickenbach, et al. \cite{bickenbach2016night} in their work highlight that the relationship between the growth in nightlight and the growth in GDP varies significantly, economically as well as statistically, across regions. This was established using regional data from Brazil and India. Therefore, trying to draw a relationship between the two without accounting for the positional information (such as longitude and latitude) may lead to poor results. These papers outline the importance of including geographical location in our analysis.
Another possible use of this analysis can be to verify GDP estimates released by various government bodies which have been known to inflate their findings. Martinez, Luis R. \cite{martinez2018much} in his work finds that most authoritarian regimes inflate their growth rate findings by a factor of about 1.15 to 1.3. It was found that the misrepresentation of GDP statistics in authoritarian or autocratic regimes is higher when there is a higher incentive to exaggerate economic well being of a region. Some of these could be just before elections or to hide a sustained dip in economic growth. Using our work and other similar methodologies these GDP findings can be scrutinised with a reasonable level of accuracy.
All of the work cited in this section points towards the feasibility of using remotely sensed nightlight data to reasonably estimate the economic output of a region. We hence believe that nightlights can be used for overcoming the lack of real-time data for assessing the economic health of a region and proceed towards extending this idea to disruptive events.
\section{Dataset}
\label{3}
In this section we describe the data collection and processing pipeline. There are 3 components to the dataset: nightlight values, GDP values and geographical coordinates in the form of latitudes and longitudes.
\subsection{Nightlights Data and Geographic Coordinates}\label{sec:nightlights}
Several remote sensing datasets and tools such as the SHRUG open data tool \cite{almn2021}, USGS Earth Explorer Tool \cite{usgs} and the Google Earth Engine \cite{earthengine} were explored. The Google Earth Engine tool was chosen as it had some specific advantages over the others: Google Earth Engine has a rich repository of datasets collected from different verified sources; the datasets are preprocessed and ready for use in different research contexts; it comes with an in-built code editor and library which enables researchers to write code in JavaScript and easily use the predefined functions in the library.
The Google Earth Engine, as outlined in the previous paragraph, is just a tool and not the dataset itself. The dataset that we use in this work for obtaining the remote-sensing data is provided by ``Earth Observation Group, Payne Institute for Public Policy, Colorado School of Mines'' \cite{doi:10.1080/01431161.2017.1342050}. The dataset contains nightlight data in the form of Average Day Night Band (DNB) radiance. The average DNB radiance band, part of the VIIRS dataset, is a metric that ranges from -1.5 to 193565 units and is reported in $\frac{nanoWatts}{cm^{2}/sr}$. This data is from the Visible and Infrared Imaging Suite (VIIRS) Day Night Band (DNB) on board the JPSS (Joint Polar Satellite System) satellites. The VIIRS-DNB has shown a significant improvement in capturing low light images as compared to the Defense Meteorological Satellite Program (DMSP) satellites which were mounted with the Operational Linescan Sensors (OLS). Elvidge, et al. \cite{articleVIIRS} conclude in their work the superiority of VIIRS data as compared to DMSP data based on these parameters: dynamic range, spatial resolution, calibrations, quantization and the
availability of spectral bands suitable for discrimination of thermal sources of light emissions. Similarly Gibson, et al. \cite{GIBSON2021102602} show the superiority of VIIRS data over DMSP data by comparing the two data sources (VIIRS and DMSP) by using them for GDP predictions. The predictive performance of DMSP data was found to be significantly lower than the VIIRS data particularly for rural areas which have lower population densities and at lower levels of spatial hierarchy like at the country level. The relationship between GDP and city lights was found to be twice as noisy in the DMSP data than the VIIRS data. The VIIRS data was found to be superior to the DMSP data in several aspects including lack of proper calibration, top-coding and blurring. This furthers our reasoning for using the newer and better VIIRS dataset over the DMSP dataset.
Shapefiles contain the geometric information about the boundaries of a region like country or state. When overlaid on a map, shapefiles can be used to identify the states or provinces we want to study. The shapefiles for our analysis were obtained from DIVA-GIS \cite{diva-gis} and the US National Weather Service \cite{weatherShapeFile}. Using the VIIRS dataset and the shapefiles we obtain the nightlight information mapped to each of corresponding state along with the state's geographic coordinates in the form of latitudes and longitudes.
\begin{figure*}[hbt!]
\centering
\includegraphics[width=\textwidth]{flowchart.pdf}
\caption{The proposed methodology.}
\label{fig: 2}
\end{figure*}
\subsection{GDP Data}
The dataset obtained Section \ref{sec:nightlights} has the State or Province Name, Mean Nightlight Value, Latitude, Longitude with one observation for every month. These features need to be mapped with the GDP values for each of the corresponding provinces or states. The quarterly GDP values for the United States of America were obtained from Bureau of Economic Analysis \cite{bea}. Subsequently, we averaged the monthly nightlight values into quarters and mapped them to their corresponding GDP values. Furthermore, GDP Data is reported with respect to a particular base year. This to allow for purchasing power comparisons and to be able to report growth of a region while adjusting for inflation. These base years are updated from time to time which is an issue for our analysis because we need the entire dataset to have a common base year so as to have a uniformity in the dataset, and to ensure that the economic activity for a given region is measured on a common scale for all the regions in our dataset. To address this issue, the base year was chosen to be 2011 and adjusted the GDP values of all the regions accordingly.
\begin{table*
\caption{Structure of our dataset}
\label{table: 6}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
State/Province & Year & Quarter & Latitude & Longitude & Mean Nightlight & GDP \\
\hline
California & 2014 & Q1 & 37.2453 & -119.6081 & 1.4385 & 2252602.814 \\
Alabama & 2014 & Q1 & 32.7935 & -.826786 & 1.4918 & 183844.57 \\
Virginia & 2018 & Q4 & 37.5164 & -78.829 6 & 2.0349 & 475154.764 \\
\hline
\end{tabular}
\end{table*}
A snapshot of the final dataset is in Table \ref{table: 6}. While the main focus of our study was the United States, we were also able to collect annual GDP data for Spain \cite{countryeconomy}, Germany \cite{Statisticsportal}, India \cite{mspi} and China \cite{nbs-china}. Since these countries follow different cycles for their financial year GDP reporting, we averaged out the nightlight data as per their respective financial years. We performed similar experiments on the data from Germany, Spain, India and China. Out of these countries, at the time of our analysis, the annual GDP data for 2020 was only available for Germany, to verify our findings. We report the results of applying our model on Germany in Section \ref{4.4}. We show the pipeline for dataset creation and modelling in Figure \ref{fig: 2}.
\section{Experiments, Model and Results}
\label{4}
We aim to answer the following with regards to ReGNL:
\begin{enumerate}
\item Does it work for both regular (economy-wise) years and years with disruptive events?
\item Does the inclusion of geographical coordinates in the form of latitude and longitude result in improvement in its performance?
\end{enumerate}
\subsection{Timeframe with Disruptive Event}
The US GDP for the years 2019 and 2020 is plotted in Figure \ref{fig: 1}. The United States experienced a significant drop in the national GDP during the 2nd quarter of 2020, which can be attributed to the COVID - 19 pandemic. To test our method across regular timeframes and timeframes with disruptive events, we trained ReGNL on data from 2014-2018 and used it to make predictions for 2019 (regular year) and 2020 (year with a disruptive event).
\begin{figure}[hbt!]
\centering
\includegraphics[width=\linewidth]{usa_gdp.pdf}
\caption{USA GDP 2019-20 (indicating the economic dip in 2020 Q2)}
\label{fig: 1}
\end{figure}
\subsection{Evaluating the Incorporation of Geographic Coordinates}
\label{4.2}
To evaluate the value added by incorporating geographical coordinates in the dataset, ReGNL was trained under 2 different scenarios - one containing just the mean nightlight of a region and the other consisting of the mean nightlight as well as the latitude and longitude of the region. The metric used to compare different predictions was a weighted error, in which the error of each state was weighted by its contribution to the total GDP of the country. The reason behind using a weighted error was to ensure that adequate importance is given to states in accordance with how high or low their GDP contribution is to the country. The error is computed using Equation \ref{eqn:1}.
\begin{equation}\label{eqn:1}
error = \sum_{states} \Big(\frac{\left | GDP_{actual} - GDP_{predicted} \right |}{GDP_{actual}}*\frac{GDP_{state}}{GDP_{national}}*10\Big)
\end{equation}
The weight was multiplied by a constant (10) to ensure floating point errors did not occur and the numbers were of a comparable size. The results are shown in Table \ref{table: 1}.
\begin{table}[hbt!]
\centering
\caption{Reduction in weighted error of predicted GDP values after adding latitude and logitude (2019)}
\label{table: 1}
\begin{tabular}{|c|c|c|}
\hline
Quarter & Only Nightlights & ReGNL (Nightlights + Lat, Long)\\
\hline
2019 Q1 & 7.3477 & 0.7697\\
2019 Q2 & 5.7446 & 0.5034\\
2019 Q3 & 5.8081 & 0.8681\\
2019 Q4 & 5.9619 & 0.6852\\ \hline \hline
Average & 6.2156 & 0.7066\\ \hline
\end{tabular}
\end{table}
As is observed in Table \ref{table: 1} we were able to obtain a significant increase in ReGNL's performance by adding the latitude and longitude to our feature set. This is inline with previous works that have highlighted the role geography plays in the economy of a region. The model is able to take advantage of this information and use it to help determine which states are economically better off.
\subsection{Model}
\label{4.3}
This section elaborates further on our model, ReGNL. Several Machine Learning algorithms such as Support Vector Regressors \cite{10.5555/2998981.2999003}, Linear (and Polynomial) Regression, XGBoost \cite{Chen_2016} as well as Neural Networks \cite{deeplearning-book, feedforward-nn} were tested and it was found that the Neural Networks provided better and more consistent results when compared to the others. The results obtained using the different models are presented in Table \ref{table: 5}. Given their superior performance, a Neural Network architecture was adopted for ReGNL.
\begin{table}[hbt!]
\centering
\caption{Comparisions of different algorithms in terms of their average weighted error of 2019-20 GDP predictions}
\label{table: 5}
\begin{tabular}{|l|c|}
\hline
Method & Error\\
\hline
SVR & 6.8221\\
Linear Regression & 5.1546\\
Neural Network & 0.7066\\
XGBoost & 5.1436\\ \hline
\end{tabular}
\end{table}
After running various tests with different architectures, based on train and test loss, the final model developed and trained was a feed forward neural network consisting of 8 hidden layers. The ReLU activation function was used after each layer (barring the last one) to introduce non-linearity. Regularization techniques such as dropout and weight decay were used to avoid over-fitting on the training data. The model was trained for $5x10^{6}$ epochs with a learning rate of $10^{-6}$. A schematic structure of the neural network is shown in Figure \ref{fig: 5}.
\begin{figure}
\centering
\includegraphics[scale=0.85]{nn.pdf}
\caption{The Neural Network}
\label{fig: 5}
\end{figure}
Predictions for the year 2020 using all 3 features (nightlight, latitude and longitude) were made using ReGNL and the results of the same can be visualised in Figures \ref{fig: 3} and \ref{fig: 4} and their respective errors are shown in Table \ref{table: 2}.
\begin{table}[hbt!]
\centering
\caption{Weighted error of predicted GDP values (2020)}
\label{table: 2}
\begin{tabular}{|c|c|}
\hline
Quarter & ReGNL (Nightlight, Lat, Long)\\
\hline
2020 Q1 & 0.7243\\
2020 Q2 & 0.6879\\
2020 Q3 & 0.6983\\
2020 Q4 & 0.6478\\ \hline \hline
Average & 0.6895\\ \hline
\end{tabular}
\end{table}
\begin{figure}
\centering
\includegraphics[width=\linewidth,height=\textheight,keepaspectratio]{20_q1.pdf}
\caption{The predicted GDP (ARIMA and ReGNL model) values vs actual values for USA Q1 2020.}\label{fig: 3}
\end{figure}
\begin{figure}
\label{fig: 4}
\centering
\includegraphics[width=\linewidth,height=\textheight,keepaspectratio]{20_q2.pdf}
\caption{The predicted GDP (ARIMA and ReGNL model) values vs actual values for USA Q2 2020.}\label{fig: 4}
\end{figure}
\begin{figure*}[hbt!]
\centering
\includegraphics[width=\textwidth]{germany_19-20_new.pdf}
\caption{The predicted GDP values vs actual values for Germany (2019-20).}
\label{fig: germany}
\end{figure*}
As can be seen from the table, the error obtained across the year was fairly consistent. The biggest takeaway is the low error from 2020 Q2. As shown in Figure \ref{fig: 3}, despite the GDP being hit drastically during this period, our model did not produce any anomalous result and continued to perform as hypothesized, in a similar fashion to as it was before the pandemic struck. These results provide evidence to support our claim that nightlight along with geographical data can act as a reasonable proxy to determine the GDP for the United States, even during disruptive events such as the COVID-19 pandemic.
ARIMA is a technique used for time-series forecasting and has been used to forecast GDP values in previous works \cite{articlearima}. We employ ARIMA to predict the GDP in USA states for the years 2019 and 2020 as a baseline prediction estimate. We can also use this information to gain insight into whether the nightlight and geographical features of a region were able to serve as proxies to estimate the GDP more accurately.
The results obtained through the ARIMA model are reported in Table \ref{table: 4}. A comparison of The actual and predicted GDP values using ARIMA and our model (ReGNL) is shown in Figure \ref{fig: 3} and Figure \ref{fig: 4}. It is clear that the ARIMA model overestimated the GDP values in 2020 Q2 whereas the estimations from the ReGNL model were closer to the ground truth.
\begin{table}
\centering
\caption{Weighted error of predicted GDP values using ARIMA (2020) vs error of predicted GDP values using ReGNL}
\label{table: 4}
\begin{tabular}{|c|c|c|}
\hline
Quarter & ARIMA Error & ReGNL Error\\
\hline
2020 Q1 & 0.1235 & 0.7243\\
2020 Q2 & 1.4976 & 0.6879\\ \hline
\end{tabular}
\end{table}
As can be observed, while the ARIMA model performs well in the first quarter, it is unable to take into account the impact of the pandemic and gives us predictions that are quite far off in the second quarter. Whereas ReGNL is able to perform much better during this time period and provides more consistent results even in the quarters affected by the disruptive event of the pandemic. ReGNL takes into account the nightlight along with geographical information and hence is able to make reasonably accurate predictions even during a disruptive event such as the COVID-19 pandemic, unlike ARIMA, which is a time-series based algorithm.
\subsection{Performance in other countries}
\label{4.4}
We also created a dataset based on the annual GDP for other countries, namely, China, India, Spain and Germany. Out of them, the GDP data for 2020 to evaluate the model was available only for Germany. An identical approach was employed to predict the 2020 GDP for this country to understand if our method could be extended beyond the United States. However, only annual GDP data is available for Germany (while quarterly data is unavailable). In order to improve the number of data points, we combined the data from all the four countries in this experiment for the training phase. We were able to establish reasonable predictions for the pre-pandemic timeframe, without any disruptive event. The results for the same can be seen in Figure \ref{fig: germany} and Table \ref{table: 3}. We note that the results obtained here are not as accurate as those obtained for the United States. Possible reasons for the same are lack of granularity in the available GDP data for Germany. Only annual GDP data was available for Germany, as opposed to the quarterly data available in the United States, which reduced the number of data points on which the model could be trained for Germany. Moreover, we pooled data from all these different countries i.e., China, India, Spain and Germany, to improve the number of data points, which might have affected the model. With sufficient and regular granular data, similar results as that of the USA can be obtained for these countries as well.
\begin{table}[hbt!]
\centering
\caption{Weighted error of predicted GDP values for Germany}
\label{table: 3}
\begin{tabular}{|c|c|}
\hline
Year & ReGNL (Nightlight, Lat, Long)\\
\hline
2019 & 1.8649\\
2020 & 1.9488\\ \hline
\end{tabular}
\end{table}
\section{Conclusions and Future Work}
\label{5}
Prediction of economic parameters, like GDP, becomes important in the case of disruptive events like the COVID-19 pandemic. However the economic parameters are not immediately available for the policymakers as the traditional methods to compute the parameters are time and resource intensive. In this paper we propose ReGNL, a neural networks based method, to be able to estimate the GDP of a geography using the nightlights data (obtained via remote sensing) in combination with geographical coordinates in the form of latitude and longitude. We are able to demonstrate that adding latitude and longitude as features for our predictive model improves the performance. Using USA as an example, we have also shown that the model is agnostic to the COVID-19 disruptive event. We have used ARIMA as a baseline for comparison and found that our model outperforms ARIMA for the timeframe when the disruptive event affected the economy. Additionally, we have presented a short extension of our work on Germany, in an attempt to build a model with a wider scope.
We believe one of the major bottlenecks in this space is the lack of available data. First, the VIIRS methodology for remotely sensing nightlight data has been deployed recently. Consequently, the VIIRS data is not available in abundance, putting a limit on the amount of training data we could use. The use of transfer learning could also be explored, potentially being able to fine tune for \emph{country A} when there is a pretained model for an economically similar \emph{country B}. Secondly, not all countries release their GDP statistics in the same format or with the same frequency, once again limiting the amount of data that could be used to construct a unified dataset. More research can also be done to understand the potential benefits of adding other remotely sensed indicators such as Carbon Monoxide emissions, NDVI (Normalized Difference Vegetation Index), etc. to the feature set to see if these can be used to predict economic parameters of different regions with improved accuracy. Finally, different model architectures can also be tried for the same task. The authors plan on releasing the dataset and source code publicly for the benefit of the scientific community. We want to make a case through this paper that countries should build robust data collection and publication frameworks which can help researchers and policy makers build models to understand the economic impact during disruptive events.
\bibliographystyle{ACM-Reference-Format}
|
1,314,259,996,546 | arxiv | \section{Introduction}
In \cite{bmmpcr2008}, given
$f:\mathbb{D}\mathrel{\mathop:}= \{z\in \mathbb{C}:|z|<1 \}\rightarrow \mathbb{C}$ analytic, $f (0)=0$, we considered different ways
of measuring $f (r\mathbb{D})$ for $0<r<1$.
\begin{itemize}
\item Maximum modulus: $\operatorname{Rad} f (r\mathbb{D})=\max_{|z|\leq r}|f (z)|$ (Schwarz)
\item Diameter: $\operatorname{Diam} f (r\mathbb{D})$ (Landau-Toeplitz)
\item $n$-Diameter: $\operatorname{n-Diam} f (r\mathbb{D})$
\item Capacity: $\operatorname{Cap} f (r\mathbb{D})$
\item Area: $\operatorname{Area} f (r\mathbb{D})$
\item What else? Perimeter, eigenvalues of the Laplacian, etc\dots
\end{itemize}
Note that if $f^{\prime} (0)\neq 0$, then $f$ is univalent on
$|z|<r_{0}$ for some $r_{0}>0$.
Let $M (f (r\mathbb{D}))$ be a measurement as above. Define:
\[
\phi_{M} (r)\mathrel{\mathop:}= \frac{M (f (r\mathbb{D}))}{M (r\mathbb{D})}.
\]
\begin{theoremA}[\cite{bmmpcr2008}] Let $M$ be Radius, $n$-Diam,
or Capacity. Then $\phi_{M} (r)$ is increasing and its $\log$ is a convex
function of $\log r$. Actually, it is
strictly increasing unless $f$ is linear. In particular,
\[
M (f (r\mathbb{D}))\leq \phi_{M} (R) M(r\mathbb{D})\qquad 0<r<R
\]
When $M$ is Area ($f$ not univalent) ``logconvexity'' might fail but
strict monotonicity persists.
\end{theoremA}
Given a Jordan curve $J$ let $G_{1}$ be its interior and
$G_{2}$ be its exterior in $\mathbb{C} \cup\{\infty \}$, and assume that $0\in G_{1}$.
Compute the reduced modulus (defined below) $M_{1}$ of $J$ with
respect to $0$ in $G_{1}$ and the reduced modulus $M_{2}$ of $J$ with
respect to $\infty$ in $G_{2}$. Then $M_{1}+M_{2}\leq 0$ with equality
if and only if $J$ is a circle of the form $\{|z|=r \}$. In fact, the sum $M_{1}+M_{2}$
can be thought of as a measure of how far $J$ is from being a circle.
Teichm\"uller's famous {\em Modulsatz} says that if $-\delta \leq
M_{1}+M_{2}\leq 0$, for $\delta$ sufficiently small, then the
oscillation of $J$ is controlled in the sense that, for some finite
constant $C$ (independent of $J$)
\[
\frac{\sup_J|z|}{\inf_{J}|z|}\leq 1+C\sqrt{\delta \log \frac{1}{\delta}}.
\]
See Chap. V.4 in \cite{garnett-marshall2005}.
Even though a definition of reduced modulus can be given using modulus of path families,
it turns out that
\[
M_{1}=\frac{1}{2\pi}\log |g_{1}^{\prime} (0)|, \qquad M_{2}=\frac{1}{2\pi}\log \left|\left(\frac{1}{g_{2}} \right)^{\prime} (0)
\right|
\]
where $g_{1}$ is a conformal map of $\mathbb{D}$ onto $G_{1}$ with $g_{1} (0)=0$
and $g_{2}$ is a conformal map of $\mathbb{D}$ onto $G_{2}$ with $g_{2}(0)=\infty$.
Note also that the conformal map $\psi \mathrel{\mathop:}= 1/g_{2}^{-1}$ sends
$G_{2}$ to $\{|z|>1 \}\cup\{\infty \}$ and $\psi (\infty)=\infty$, so
$M_{2}$ is related to the usual concept of logarithmic capacity of $J$:
\begin{equation}\label{eq:m2}
M_{2}=-\frac{1}{2\pi}\log \operatorname{Cap} (J).
\end{equation}
In fact, letting $1/J=\{z:1/z\in J \}$, we find that
\begin{equation}\label{eq:formula}
- (M_{1}+M_{2})=\frac{1}{2\pi}\log \left(\operatorname{Cap} (J)\operatorname{Cap} (1/J) \right)\geq 0.
\end{equation}
Now consider a conformal map $f$ on $\mathbb{D}$ with $f (0)=0$ and let $J
(r)=f (r\bd\mathbb{D})$. Let $T (r)=- (M_{1} (r)+M_{2} (r))$, as above, measure how
different $J (r)$ is from a circle centered at the origin. Note that
$g_{1}(z)\mathrel{\mathop:}= f (zr)$ maps $\mathbb{D}$ conformally onto the interior of $J
(r)$, so $M_{1} (r)= (1/ (2\pi))\log r|f^{\prime} (0)|$. On the other
hand, see for instance Example III.1.1 in \cite{garnett-marshall2005},
$M_{2} (r)= -(1/ (2\pi))\log \operatorname{Cap} (f (r\mathbb{D}))$. Thus we have
\[
T (r)=\frac{1}{2\pi}\log \frac{\operatorname{Cap} f (r\mathbb{D})}{|f^{\prime} (0)|\operatorname{Cap}
r\mathbb{D}}.
\]
By Theorem~A, we now know that $T (r)$, as defined above, is
increasing and convex, in fact strictly increasing unless $f$ is linear.
The proof of this Theorem~A consisted in establishing the ``increasing'' and
``log-convexity'' part first and then deducing the ``Moreover part'' from
the behavior of $\operatorname{Area} f (r\mathbb{D})$ and P\'olya's inequality relating area
and capacity. More specifically, P\'olya's inequality gives the
following relation:
\[
\phi_{\operatorname{Area}} (r)\leq \phi_{\operatorname{Cap}}^{2} (r).
\]
To show that $\phi_{\operatorname{Cap}}$ is strictly increasing, by log
convexity it's enough to show that $\phi_{\operatorname{Area}} (r)$ is strictly
increasing at $0$.
After \cite{bmmpcr2008} appeared in print, C.~Pommerenke pointed out that in
\cite{pom1961} he had already shown the following result:
\begin{theoremB}
If f is one-to-one and analytic in the annulus $\{a < |z| < b\}$,
then $\log\operatorname{Cap}\{ f(r\bd\mathbb{D}) \}$ when expressed as a function of $\log
r$ is a convex function for $a<r<b$.
\end{theoremB}
For the reader's convenience we replicate Pommerenke's proof here.
\begin{proof}[Proof of Theorem~B]
Fix $r$ as above and find $w_{1},\dotsc ,w_{n}\in \bd\mathbb{D}$ so that
\[
\operatorname{n-Diam} f (r\bd\mathbb{D})=\prod_{j<k} |f (w_{j}r)-f (w_{k}r)|^{2/ (n (n-1))}.
\]
Then consider the function
\[
H (z)\mathrel{\mathop:}= \prod_{j<k} (f (w_{j}z)-f (w_{k} (z)))
\]
which is analytic in the annulus $\{a<|z|<b \}$.
Fix $a<r_{1}<r<r_{2}<b$. Then by Hadamard's three-circles theorem,
\[
\log \max_{|z|=r}|H (z)|\leq \frac{\log (r_{2}/r)}{\log
(r_{2}/r_{1})}\log \max_{|z|=r_{1}}|H (z)| +\frac{\log (r/r_{1})}{\log
(r_{2}/r_{1})}\log \max_{|z|=r_{2}}|H (z)|,
\]
i.e.,
\[
\log \operatorname{n-Diam} f (r\bd\mathbb{D})\leq \frac{\log (r_{2}/r)}{\log
(r_{2}/r_{1})}\log \operatorname{n-Diam} f (r_{1}\bd\mathbb{D}) +\frac{\log (r/r_{1})}{\log
(r_{2}/r_{1})}\log \operatorname{n-Diam} f (r_{2}\bd\mathbb{D}).
\]
Now let $n$ tend to infinity.
\end{proof}
In this note we study the case of analytic functions defined on an annulus.
For $R>1$, consider the family $\mathcal{S} (R)$ of analytic and one-to-one
functions $f$ that map the annulus $A (1,R)=\{1<|z|<R \}$ onto a
topological annulus $\mathcal{A}$ such that the bounded component of $\mathbb{C}
\setminus \mathcal{A}$ coincides with the unit disk $\mathbb{D}$, and so that as
$|z|\downarrow 1$ we have $|f (z)|\rightarrow 1$. By the Schwarz
Reflection Principle, the conformal map $f$ extends analytically across
$|z|=1$, so the family $\mathcal{S} (R)$ can be defined more compactly as the
set of analytic and one-to-one maps on $A (1,R)$ such that $|f
(z)|>1$ for all $z\in A(1,R)$ and $|f (z)|=1$ for $|z|=1$.
We prove the following monotonicity result.
\begin{theorem}\label{thm:monotone}
Let $f\in \mathcal{S} (R)$ and let $T (r)$ be defined as above for $1\leq
r<R$. Then, $T (r)$ is a convex function of $\log r$ and is strictly
increasing, unless $f$ is the identity.
\end{theorem}
Also, in analogy with the local case, for a function $f\in \mathcal{S} (R)$ we
define the ratios:
\[
\psi_{\operatorname{n-Diam}}(r)\mathrel{\mathop:}= \frac{\operatorname{n-Diam} (f (\mathcal{A}
(1,r))\cup\overline{\mathbb{D}})}{\operatorname{n-Diam}(r\mathbb{D})}
\qquad\mbox{ and }\qquad
\psi_{\operatorname{Cap}}(r)\mathrel{\mathop:}=\frac{\operatorname{Cap}(f(\mathcal{A} (1,r))\cup\overline{\mathbb{D}})}{\operatorname{Cap}
(r\mathbb{D})}.
\]
Then by Theorem~B, both $\log \psi_{\operatorname{n-Diam}}$ and $\log \psi_{\operatorname{Cap}}$ are
convex functions of $\log r$. It turns out that they are also strictly
increasing unless
$f$ is the identity.
\begin{theorem}\label{thm:monotone2}
Let $f\in \mathcal{S} (R)$ and let $\psi_{\operatorname{n-Diam}} (r)$, $\psi_{\operatorname{Cap}} (r)$ be
defined as above for $1\leq r<R$. Then, $\log\psi_{\operatorname{Cap}}$ is a convex
function of $\log r$ and $\psi_{\operatorname{Cap}}$ is strictly
increasing, unless $f$ is the identity.
In particular,
\[
\operatorname{Cap} (f (\mathcal{A} (1,r))\cup\overline{\mathbb{D}})\leq r\frac{\operatorname{Cap} (f (\mathcal{A}
(1,R))\cup\overline{\mathbb{D}})}{R}.
\]
\end{theorem}
One line of proof is similar to the disk case in
that looking at the area turns out to be the crucial ingredient.
However, we show that Theorem \ref{thm:monotone2} can also be deduced
from Theorem \ref{thm:monotone}.
\section{Proof of Theorem \ref{thm:monotone}}\label{sec:proof}
By Theorem~B and (\ref{eq:formula}) we see that $T (r)$ is a convex
function of $\log r$, for $1<r<R$.
Therefore, since $T (r)\geq 0$ and $T (0)=0$, we get that
$T (r)$ is an increasing function of $r$.
Assume that $f$ is not the identity, then $f (\{|z|=r \})$ is not a circle
for $1<r<R$. For if it were, then $f$ would be a conformal map between
circular annuli and hence would be linear and hence the
identity. Therefore, by Teichm\"uller's Modulsatz, $T (r)>0$ for $1<r<R$.
Suppose $T (r)$ fails to be strictly
increasing. Then by monotonicity it would have to be constant on an
interval $[s,t]$ with $1<s<t<R$. By convexity, it would then have
to be constant and equal to $0$ on the interval $[1,t]$, but this
would yield a contradiction. So Theorem \ref{thm:monotone} is proved.
\begin{remark}\label{rem:logconv}
Note that if $F (r)=G (\log r)$ for
some convex function $G$ and $F^{\prime} (1)\geq 0$, then
$G^{\prime} (0)\geq 0$ and by convexity $G^{\prime} (t)\geq 0$ for all
$t\geq 0$, i.e., $F^{\prime} (r)\geq 0$ for all $r\geq 1$.
\end{remark}
We now turn to Theorem \ref{thm:monotone2}. First we show how it can
be deduced from Theorem \ref{thm:monotone}.
\section{Consequences of the serial rule}\label{sec:serial}
On one hand by (\ref{eq:m2}) we have
\[
\operatorname{Cap} \left(f (\mathcal{A} (1,r)\cup\overline{\mathbb{D}}) \right)=e^{-2\pi M_{2} (r)}.
\]
On the other hand, by the serial rule, see (V.4.1) of
\cite{garnett-marshall2005},
\[
M_{1} (r)\geq M_{1} (1)+\operatorname{Mod} (f (\mathcal{A} (1,r))).
\]
However, $M_{1} (1)=0$ and by conformal invariance $\operatorname{Mod} (f (\mathcal{A}
(1,r)))=\frac{1}{2\pi}\log r$. So
\[
\frac{1}{r}\leq e^{-2\pi M_{1} (r)}.
\]
Putting this together, we get
\[
\frac{\operatorname{Cap} \left(f (\mathcal{A} (1,r)\cup\overline{\mathbb{D}}) \right)}{r}\geq
e^{-2\pi (M_{1} (r)+M_{2} (r))}=e^{2\pi T (r)}.
\]
i.e.,
\begin{equation}\label{eq:lowerbd}
T (r)\leq \frac{1}{2\pi}\log (\psi_{\operatorname{Cap}} (r)).
\end{equation}
Now assume that $f$ is not linear. By Teichm\"uller's Modulsatz, $T
(r)>0$ for $1<r<R$. So by (\ref{eq:lowerbd}), $\psi_{\operatorname{Cap}} (r)>1$ and
by Theorem~B, $\psi_{\operatorname{Cap}} (r)$ is a convex function of $\log r$.
Therefore, we can
conclude as above that $\psi_{\operatorname{Cap}} (r)$ is
strictly increasing.
Teichm\"uller's Modulsatz is based on the so-called
Area-Theorem. Alternatively, Theorem \ref{thm:monotone2} can be proved
using ``area'' and P\'olya's inequality, in the spirit of
\cite{bmmpcr2008}, as we will show next.
\section{From area to capacity}\label{sec:fromatoc}
Recall P\'olya's inequality:
\[
\operatorname{Area} E\leq \pi (\operatorname{Cap} E)^{2}.
\]
It implies that
\begin{equation}\label{eq:alc}
\psi_{\operatorname{Area}} (r)\leq \psi_{\operatorname{Cap}}^{2} (r)
\end{equation}
for all $1\leq r<R$.
Lemma \ref{lem:area} below will establish that
\begin{equation}\label{eq:lemar}
\psi_{\operatorname{Area}} (\rho)>1
\end{equation}
for $1<\rho<R$, unless $f$ is linear.
Moreover $\psi_{\operatorname{Cap}} (1)=1$. So the derivative
\[
\frac{d}{dr}_{\mid r=1}\log \psi_{\operatorname{Cap}} (r)\geq 0.
\]
Hence, by ``convexity'', $\psi_{\operatorname{Cap}} (r)$ is an increasing function
of $r$.
In fact, suppose $\psi_{\operatorname{Cap}} (r)$ fails to be strictly
increasing. Then by monotonicity it would have to be constant on an
interval $[s,t]$ with $1<s<t<R$. By ``convexity'', it would then have
to be constant and equal to $1$ on the interval $[1,t]$, but this
would yield a contradiction in view of (\ref{eq:alc}) and (\ref{eq:lemar}).
So Theorem \ref{thm:monotone2} will be proved if we can establish
(\ref{eq:lemar}).
\section{Area considerations}\label{sec:areaconsid}
Each map $f \in \mathcal{S} (R)$ can be expanded in a Laurent series
\[
f (z)=a_{0}+\sum_{n\neq 0}a_{n}z^{n}.
\]
The key now is to study the area function
$h (\rho )\mathrel{\mathop:}= \operatorname{Area} f (A(1,\rho))$. We use Green's theorem to
compute the area enclosed by the Jordan curve $\gamma_{\rho}(t)=f (\rho
e^{it})$, $t\in [0,2\pi]$. Thus
\[h (\rho
)+\pi=-\frac{i}{2}\int_{\gamma _{\rho}}\bar{w}dw=\frac{-i}{2}\int_{0}^{2\pi}\bar{f}
(\rho e^{it})f_{\theta} (\rho e^{it})dt=
\pi\sum_{n\neq 0}n|a_{n}|^{2}\rho^{2n}.
\]
In particular, when $\rho =1$, $h (\rho)=0$, so
\begin{equation}\label{eq:prelim}
\sum_{n\neq 0}n|a_{n}|^{2}=1
\end{equation}
The following lemma can be deduced from problem 83
in \cite{polya-szego1972}.
\begin{lemma}\label{lem:area}
For all $f\in\mathcal{S} (R)$, except the identity, we have for $1<\rho<R$,
\[
\operatorname{Area} f (A (1,\rho))> \operatorname{Area} A(1,\rho).
\]
\end{lemma}
\begin{proof}
Let $1<\rho<R$. Then, by (\ref{eq:prelim}),
\begin{eqnarray*}
h (\rho) & = & -\pi +\pi \sum_{n\neq 0}n|a_{n}|^{2} \rho^{2n}\\
& = & \pi (\rho^{2}-1)+\pi \sum_{n\neq 0}n|a_{n}|^{2} (\rho^{2n}-\rho^{2})\\
& = & \operatorname{Area} A (1,\rho)+\pi\rho^{2}\sum_{n\neq 0}n|a_{n}|^{2} (\rho^{2n-2}-1)
\end{eqnarray*}
But $n (\rho^{2n-2}-1)\geq 0$ for all integers.
\end{proof}
This concludes the proof of Theorem \ref{thm:monotone2}.
\section{Principal frequency}\label{sec:evlapl}
Another measure for $f (r\mathbb{D})$ is to consider:
\[
M_{0} (f (r\mathbb{D}))\mathrel{\mathop:}= \frac{1}{\Lambda_{1}(f (r\mathbb{D}))}.
\]
Recall that given a bounded domain $\Omega \subset\mathbb{C}$,
\[
\Lambda_{1}^{2} (\Omega )=\inf \frac{\int_{\Omega }|\nabla u|^{2}dA}{\int_{\Omega }
u^{2}dA}
\]
where the infimum ranges over all functions $u\in C^{1}
(\overline{\Omega })$ vanishing on $\bd\Omega $ and is attained by a function $w\in C^{2}
(\overline{\Omega })$ which is characterized as being the unique solution
to
\[
\Delta w +\Lambda^{2}w=0 , w>0 \mbox{ on $\Omega $}, w=0 \mbox{ on $\bd\Omega $}.
\]
It follows from \cite{polya-szego1951} p.~98 (5.8.5) that
\[
\phi_{M_{0}} (r)\left(=\frac{\Lambda_{1} (r\mathbb{D})}{\Lambda_{1} (f (r\mathbb{D}))}
\right)>|f^{\prime} (0)|.
\]
{\bf Problem:} Show that $\phi_{M_{0}} (r)$ is strictly
increasing when $f$ is not linear.
This problem turns out to have been solved already by work of Laugesen
and Morpurgo \cite{laugesen-morpurgo1998}. Although, I'm not sure if
essentially different techniques are required in the case of the annulus.
\def$'${$'$}
|
1,314,259,996,547 | arxiv | \section{Introduction}
The computer package {\tt DIZET} was created as
electroweak and QCD library of
the {\tt ZFITTER} program~\cite{Bardin:1999yd,Arbuzov:2005ma} which was one of the
main tools for the high-precision verification of the Standard Model at LEP~\cite{ALEPH:2005ab}.
{\tt DIZET} can also be linked as
a library by other projects, e.g., it is used by the {\tt HECTOR} program~\cite{Arbuzov:1995id}
and by the {\tt KKMC} Monte Carlo event generator~\cite{Arbuzov:2020coe}.
{\tt DIZET} can be used for fitting EWPOs, for instance $\sin^2\vartheta_{eff}$ at LHC as discussed at the LHC EW precision workshop \cite{dizetws}.
In each new version of {\tt DIZET}, the compatibility with all previous versions has been preserved.
Thus, the numerics of a previous version can be fully reproduced,
except for changes caused by the correction of bugs. The latter is documented
in the header of the code.
We remind, that between {\tt DIZET} versions 6.21~\cite{Bardin:1999yd} and
6.42~\cite{Arbuzov:2005ma}, there are changes affecting the $W$ boson width and the running of
the electromagnetic coupling $\alpha_{QED}$.
In addition, the treatment of so-called box-like diagram contributions
(controlled by the IBOXF flag of the {\tt ROKANC} subroutine in {\tt ZFITTER})
and of the $b$ quark production channel (IBFLA flag in the same subroutine)
have been changed.
Links to publications and the public versions of {\tt DIZET} can be found at the {\tt ZFITTER} project webpage
\href{http://sanc.jinr.ru/users/zfitter/}{http://sanc.jinr.ru/users/zfitter/}.
The last documented version of {\tt DIZET} is
6.42~\cite{Arbuzov:2005ma}.
In the present paper, we describe the actual {\tt DIZET} version 6.45~\cite{dizetv645}.
We present the transition from {\tt DIZET} v.~6.42 to {\tt DIZET} v.~6.45 and show the numerical impact of the newly introduced modifications, controlled by the corresponding options and flags on pseudo-observables (EWPOs).
The contributions added in {\tt DIZET} v.~6.45 are connected with the completion of the 2-loop EW radiative corrections given in \cite{Dubovyk:2016aqv,Dubovyk:2019szj}
and which complement earlier works on radiative corrections, namely: the complete fermionic two-loop corrections to the $W$ boson mass \cite{Freitas:2002ja}; the leading ${\cal O}(\alpha \alpha_s)$
\cite{
Djouadi:1993ss}
and next-to-leading ${\cal O}(\alpha\alpha_s^2)$
\cite{Avdeev:1994db,Chetyrkin:1995ix,Chetyrkin:1995js}
QCD corrections, as well as leading three-loop corrections in an expansion in $m_t^2$ of order ${\cal O}(\alpha^3)$ and ${\cal O}(\alpha^2 \alpha_s)$
\cite{Faisst:2003px}.
These modifications are relevant for future precision HL-LHC studies and the LHC electroweak Working Group activities.
They are also needed as a first step towards high precision predictions of the Standard Model electroweak effects
at future high energy colliders. In the context of the future circular electron-positron collider (FCC-ee)~\cite{Abada:2019zxq},
the anticipated experimental accuracy on EWPOs has to be matched with theory predictions of at least the same level of accuracy to achieve maximum usage of experimental data.
For the present situation concerning EWPOs determination and their future estimate, see Tab.~\ref{tab:th} and references \cite{Blondel:2018mad,Freitas:2019bre} for more details. In Tab.~\ref{tab:th}, we put experimental predictions for EWPOs at FCC-ee as the most stringent among future experimental setups, particularly in the Z-resonance region.
Other widely considered future $e^+e^-$ collider projects are CEPC~\cite{CEPCStudyGroup:2018ghi}, ILC~\cite{Baer:2013cma,Bambade:2019fyw}, and CLIC~\cite{Linssen:2012hp,Charles:2018vfv}.
Through their high integrated luminosities of several ab$^{-1}$ (practically for all relevant Z-resonance, $HZ$, $WW$, and $t\bar{t}$ modes \cite{Abada:2019zxq,CEPCStudyGroup:2018ghi}) these machines will be sensitive to very small deviations between the measured value and the SM expectation
for a given observable.
To account correctly for such slight deviations, dedicated programs like here discussed
{\tt DIZET} will be highly needed.
Table~\ref{tab:th} shows the comparison between the estimated FCC-ee experimental precision, the current theoretical uncertainty,
and the so-called projected one for
representative EWPOs, see Chapter~B in~\cite{Blondel:2018mad} and~\cite{Freitas:2019bre}.
By the projected theoretical uncertainty we mean an estimate of the future theoretical uncertainty when the leading 3-loop $\mathcal{O}(\alpha^3, \alpha^2\alpha_s, \alpha\alpha_s^2)$
corrections will become available.
\begin{table}[ht]
\centering
\begin{tabular}{|l|ccc|}
\hline
Quantity & FCC-ee & Current theory
& Projected theory \\
& & uncertainty &
uncertainty \\ \hline
$m_\mathrm{W}$ (MeV) & $0.5-1$ & 4
& 1 \\
$\sin^2 \vartheta^\ell_{\rm{eff}}$ ($10^{-5}$) & 0.6 & 4.5
& 1.5 \\
$\Gamma_\mathrm{Z}$ (MeV) & 0.1 & 0.4
& 0.15 \\
$R_b$ ($10^{-5}$) & 6 & 11
& 5 \\
$R_\ell$ ($10^{-3}$) & 1 & 6
& 1.5 \\
\hline
\end{tabular}
\caption{Estimated precision for the direct determination of representative EWPOs at FCC-ee (column 2), current theory uncertainties
for the SM prediction of these quantities (column 3), and the projected theoretical uncertainty (column 4).}
\label{tab:th}
\end{table}
Indeed, as {\tt DIZET} includes Standard Model higher order radiative corrections, it can be used for comparisons with experimental results in search for models which go beyond the Standard Model. We discuss here the impact of newly implemented corrections in {\tt DIZET}
on EWPOs and form-factors at the $e^+e^-$ resonance.
\section{Release {\tt DIZET} from v.~6.42 to v.~6.45}
The Fortran code {\tt DIZET} is a library for the calculation of
electroweak radiative corrections and it is part of the {\tt ZFITTER} distribution package.
It can also be used in a stand-alone mode.
On default, {\tt DIZET} performs the following calculations:
\begin{itemize}
\item by call of subroutine {\tt ROKANC}: four weak neutral-current (NC)
form factors, running electromagnetic
and strong couplings needed for the calculation of effective NC Born
cross sections for the production of massless fermions (however, the mass
of the top quark appearing in the virtual state of loop diagrams
for the process $e^+ e^- \to f \bar{f}$ is not ignored);
\item
by call of subroutine {\tt RHOCC}: the corresponding form factors and running
strong coupling for the calculation of effective CC Born cross sections;
\item
by call of subroutine {\tt ZU\_APV}: $Q_W(Z,A)$ -- the weak charge used for
the description of parity violation in heavy atoms.
\end{itemize}
If needed, the form factors of cross sections may be made to contain
the contributions from
$WW$ and $ZZ$ box diagrams thus ensuring the correct kinematic
behaviour over a larger energy range compared to the Z pole.
Between the {\tt DIZET} versions 6.42 and 6.45 there are changes affecting the running
of the QED coupling $\alpha(s)$ and the QCD corrections of
order
$\alpha\alpha_S$ to the $Z$ boson partial widths.
Also, starting from v.6.44, {\tt DIZET} uses the complete $\alpha_s^4$ QCD corrections to hadronic $Z$-decays \cite{Baikov:2012er} by default. In order to reproduce the old behavior, the IBAIKOV flag
must be set to 2014 in the code of {\tt DIZET} v.~6.45.
The largest contribution of the electroweak (EW) corrections
comes from the $s$ channel QED running of $\alpha(s)$, and
the main load in it is due to the hadronic component
$\Delta\alpha^{(5)}_{had}(M_{\sss Z})$~\cite{Jegerlehner:2003ip}.
\subsection{New options in {\tt DIZET} v.~6.45}
In this section, we give the descriptions
of flags and
added options implemented in {\tt DIZET} v.~6.45.
\vspace*{0.4cm}
$\bullet$ flag {\bf IAMT4:}
\begin{description}
\item{\underline{\bf IAMT4:}
two-loop $\alpha^2$ bosonic and/or fermionic radiative corrections: }
\begin{description}
\item[IVALUE =~~I]
\item[\phantom{ AVALUE =} 6] --- fermionic two-loop corrections to $\seff{l}$~\cite{Awramik:2004qv};
\item[\phantom{ AVALUE =} 7] --- the complete two-loop corrections to $\seff{b}$ and $\seff{l}$ according to
Refs.~\cite{Dubovyk:2016aqv,Awramik:2006uz}.
\item[\phantom{ AVALUE =} 8] --- the complete electroweak two-loop radiative corrections to all the
relevant electroweak precision pseudo-observables related to the Z-boson,
according to Ref.~\cite{Dubovyk:2019szj}.
\end{description}
\end{description}
The complete set of EWPOs related to the $Z$-boson for
IAMT4 = 8 includes:
the leptonic and bottom-quark effective weak mixing angles
$\seff{\ell}$, $\seff{b}$,
the $Z$-boson partial
decay widths $\Gamma_f$, where $f$ indicates any charged lepton, neutrino and quark flavor (except for the top quark), the total $Z$ decay width $\Gamma_Z$, the
branching ratios $R_\ell$, $R_c$, $R_b$, and the hadronic cross section $\sigma_{\rm had}^0$.
\vspace*{0.4cm}
$\bullet$ flag {\bf IHVP:}
\begin{description}
\item{\underline{\bf IHVP} --- choice of hadronic vacuum polarization $\Delta \alpha^{(5)}_{had}(M_{\sss Z})$
using public versions of the {\tt AlphaQED} code by F.~Jegerlehner:}
\begin{description}
\item[IVALUE =~~I]
\item[\phantom{ AVALUE =} 1] --- realization of the fit given in \cite{Eidelman:1995ny},
\item[\phantom{ AVALUE =} 4] --- realization
of the fit by \cite{Jegerlehner:2015stw},
\item[\phantom{ AVALUE =} 5] --- realization of the fit by \cite{Jegerlehner:2017zsb}.
\end{description}
\end{description}
Details on hadronic vacuum polarization effects
can be found in~\cite{Jegerlehner:2017zsb}.
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
IHVP & 1 & 4 & 5 \\
\hline
version & FJ-1995 & FJ-2016 & FJ-2017 \\
\hline
$\Delta \alpha^{(5)}_{had}(M_{\sss Z}) $ & 2.8039e-2 & 2.7586e-2 & 2.7576e-2\\
\hline
\end{tabular}
\caption{Results of the fit for hadronic vacuum polarization
$\Delta \alpha^{(5)}_{had}(M_{\sss Z})$
for different versions
(1995-\cite{Eidelman:1995ny},
2016-\cite{Jegerlehner:2015stw},
2017-\cite{Jegerlehner:2017zsb})}
\end{table}
Note that the {\tt AlphaQED} (2017) code provides an estimation of statistical and systematic errors.
To estimate the resulting uncertainties of a {\tt DIZET} output one has to run the code in a cycle with
variation of the input parameters such as the top quark and Higgs boson masses within their error bars,
see, e.g., Ref.~\cite{Bardin:1999gt}. In addition one has to estimate the missing contributions of not
yet computed higher order corrections.
\section{Numerical results}
All numbers presented below are obtained with the following set of Input Parameters (IPS)
and their variations within experimental errors taken from PDG Summary Tables, ~\cite{ParticleDataGroup:2020ssz}:
$\alpha^{-1}(0) = 137.035999084$,
$\alpha_s(M_Z) = 0.1179$,
$M_Z = 91.1876 \; \mathrm{GeV}$,
$M_H = 125.25 \; \mathrm{GeV}$,
$m_t = 172.76 \; \mathrm{GeV}$.
The masses of the five light quarks are chosen in the usual way to
reproduce the hadronic contribution to the photon vacuum polarization (relevant only for $IHVP=2$)
$\Delta \alpha^{(5)}_{had}(M_{\sss Z})$.
Here numerical calculations were carried out at the fixed value IBAIKOV=2014.
The numerical results presented here are slightly different from those of our
report~\cite{dizetws} due to the change of input parameters.
The present values of pseudo-observables (EW boson widths and the weak mixing angle) are~\cite{ParticleDataGroup:2020ssz}:
$\Gamma_Z = 2495.200 \pm 2.300$~MeV,
$G_Z(\mu\mu) = 83.99 \pm 0.16$~MeV,
$\Gamma_W =2085 \pm 42$~MeV,
$G_W(l\nu) = 226.4 \pm 1.9$~MeV,
$\sin^2 \vartheta_{eff}\times 10^6= 231480 \pm 160$.
\subsection{Parametric uncertainties}
{\tt DIZET} can calculate pseudo-observables and EW form-factors in a wide range of input parameters:
$M_{\sss H}$, $M_{\sss Z}$, $m_t$, $\alpha_s$.
Tables~\ref{tablemt}
- \ref{tableas} present the dependence of pseudo-observables on the experimental uncertainty of input parameters: ($m_t=172.76(0.30) \unskip\,\mathrm{GeV}$, $M_{\sss H}=125.25(0.17) \unskip\,\mathrm{GeV}$, $M_{\sss Z}=91.1876(0.0021) \unskip\,\mathrm{GeV}$, $\alpha_s=0.1179(0.0009)$).
The first type of theoretical uncertainties are due to variation of input parameters within
experimental errors.
We consider first the
parametric uncertainties due to variation of the masses $m_t$, $M_{\sss H}$ and $M_{\sss Z}$
and in addition the $\alpha_s$-dependence.
\begin{table}[ht]
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
$m_t,~\unskip\,\mathrm{GeV}$ & $172.76-0.30$ & 172.76 & $172.76+0.30$ & Diff.\\
\hline
$G_Z(\mu\mu),~\unskip\,\mathrm{MeV}$& 83.982 & 83.985 & 83.987 & 0.005\\
\hline
$\Gamma_Z,~\unskip\,\mathrm{MeV}$ & 2494.746 & 2494.814 & 2494.883 & 0.137\\
\hline
$G_W(l\nu),~\unskip\,\mathrm{MeV}$ & 678.935 & 678.981 & 679.027 & 0.092\\
\hline
$\Gamma_W,~\unskip\,\mathrm{MeV}$ & 2089.825 & 2089.967 & 2090.109 & 0.284\\
\hline
$\sin^2 \vartheta^l_{eff}\times 10^6$ & 231508 & 231500 & 231491 & 17\\
\hline
\end{tabular}
\caption{The effect of the parametric uncertainty in $m_t$
on the magnitudes of pseudo-observables.}
\label{tablemt}
\end{table}
As one can see, the parametric uncertainties for the listed pseudo-observables
are less than the current experimental errors~\cite{ParticleDataGroup:2020ssz}.
\begin{table}[h!]
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
$M_{\sss H},~\unskip\,\mathrm{GeV}$ & $125.25-0.17$ & 125.25 & $125.25+0.17$ & Diff.\\
\hline
$G_{\sss Z}(\mu\mu),~\unskip\,\mathrm{MeV}$ & 83.985 & 83.985 & 83.985 & 0\\
\hline
$\Gamma_{\sss Z},~\unskip\,\mathrm{MeV}$ & 2494.818 & 2494.814 & 2494.811 & 0.007\\
\hline
$G_{\sss W}(l\nu),~\unskip\,\mathrm{MeV}$ & 678.983 & 678.981 & 678.979 & 0.004\\
\hline
$\Gamma_{\sss W},~\unskip\,\mathrm{MeV}$& 2089.973 & 2089.967 & 2089.961 & 0.012\\
\hline
$\sin^2 \vartheta^l_{eff}\times 10^6$ & 231499 & 231500 & 231500 & 1\\
\hline
\end{tabular}
\caption{
The effect of the parametric uncertainty in $M_{\sss H}$
on the magnitudes of pseudo-observables.}
\label{tablemh}
\end{table}
Tables~\ref{tablemt} and~\ref{tablemh} show that the effect of experimental uncertainty of $m_t$ and $M_{\sss H}$
changes the partial widths by an interval not exceeding their experimental errors.
\begin{table}[ht]
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
$M_{\sss Z},~\unskip\,\mathrm{GeV}$ & $91.1876-0.0021$ & 91.1876 & $91.1876+0.0021$ & Diff.\\
\hline
$G_{\sss Z}(\mu\mu),~\unskip\,\mathrm{MeV}$ & 83.978 & 83.985 & 83.991 & 0.013\\
\hline
$\Gamma_{\sss Z},~\unskip\,\mathrm{MeV}$ & 2494.602 & 2494.814 & 2495.027 & 0.425\\
\hline
$G_{\sss W}(l\nu),~\unskip\,\mathrm{MeV}$ & 678.914 & 678.981 & 679.048 & 0.287\\
\hline
$\Gamma_{\sss W},~\unskip\,\mathrm{MeV}$ & 2089.761 & 2089.967 & 2090.173 & 0.412\\
\hline
$\sin^2
\vartheta^l_{eff}\times 10^6$ & 231515 & 231500 & 231485 & 30\\
\hline
\end{tabular}
\caption{The effect of the parametric uncertainty in $M_{\sss Z}$
on the magnitudes of pseudo-observables.}
\label{tablemz}
\end{table}
\begin{table}[!ht]
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
$\alpha_s$ & $0.1179-0.0009$ & 0.1179 & $0.1179+0.0009$ & Diff.\\
\hline
$G_{\sss Z}(\mu\mu),~\unskip\,\mathrm{MeV}$ & 83.985 & 83.985 & 83.984 & 0.001\\
\hline
$\Gamma_{\sss Z},~\unskip\,\mathrm{MeV}$ & 2494.338 & 2494.814 & 2495.290 & 0.952\\
\hline
$G_{\sss W}(l\nu),~\unskip\,\mathrm{MeV}$ & 678.995 & 678.981 & 678.967 & 0.028\\
\hline
$\Gamma_{\sss W},~\unskip\,\mathrm{MeV}$ & 2089.607 & 2089.967 & 2090.326 & 0.719\\
\hline
$\sin^2 \vartheta^l_{eff}\times 10^6$ & 231497 & 231500 & 231503 & 6\\
\hline
\end{tabular}
\caption{The effect of the parametric uncertainty in $\alpha_s$
on the magnitudes of pseudo-observables.}
\label{tableas}
\end{table}
As seen from Tables \ref{tablemt} $-$ \ref{tablemz}, the largest uncertainty comes to $\Gamma_{\sss W}$ and $\Gamma_{\sss Z}$ due to errors in $M_{\sss Z}$.
These parametric uncertainties remain, however, well below the corresponding experimental errors.
\subsection{Impact of new options}
Numerical results for the comparison of versions
are conveniently presented as
difference in values
for a given observable at different
sets of flags IHVP and IAMT4.
\subsubsection{Partial $G_{ij}$ and total $\Gamma_{tot}$ decay widths of the Z-boson}
\begin{table}[!h]
\centering
\begin{tabular}{|l|ccc|c|c|}
\hline
IHVP,~IAMT4 & 1,8 & 5,6 & 5,8
& $|\delta_{(1,8)-(5,8)}|$&$|\delta_{(5,6)-(5,8)}|$ \\
\hline
channel & & & & & \\
\hline
$G_{\nu,\bar\nu},~\unskip\,\mathrm{MeV}$ & 167.202 & 167.202 & 167.202 & 0 & 0 \\
$G_{e^+,e^-},~\unskip\,\mathrm{MeV}$ & 83.977 & 83.984 & 83.985 & 0.008 & 0.001 \\
$G_{\mu^+,\mu^-},~\unskip\,\mathrm{MeV}$ & 83.977 & 83.983 & 83.985 & 0.008 & 0.002 \\
$G_{\tau^+,\tau^-},~\unskip\,\mathrm{MeV}$ & 83.787 & 83.794 & 83.795 & 0.008 & 0.001 \\
$G_{u,\bar u},~\unskip\,\mathrm{MeV}$ & 299.832 & 299.902 & 299.918 & 0.086 & 0.016 \\
$G_{d,\bar d},~\unskip\,\mathrm{MeV}$ & 382.783 & 382.846 & 382.861 & 0.078 & 0.015 \\
$G_{c,\bar c},~\unskip\,\mathrm{MeV}$ & 299.766 & 299.836 & 299.852 & 0.086 & 0.016 \\
$G_{s,\bar s},~\unskip\,\mathrm{MeV}$ & 382.783 & 382.846 & 382.861 & 0.078 & 0.015 \\
$G_{b,\bar b},~\unskip\,\mathrm{MeV}$ & 375.874 & 375.839 & 375.951 & 0.077 & 0.112 \\
$G_{hadron},~\unskip\,\mathrm{MeV}$ & 1741.039 & 1741.268 & 1741.442 & 0.403 & 0.174 \\
$\Gamma_Z,~\unskip\,\mathrm{MeV}$ & 2494.387 & 2494.636 & 2494.814 & 0.427 & 0.178 \\
\hline
\end{tabular}
\caption{
Partial $G_{ij}$ and total $\Gamma_Z$ decay widths of the Z-boson
for sets of flags (IHVP,~IAMT4): (1,8) in comparison with (5,6) and (1,8) in comparison with (5,8).}
\label{widths}
\end{table}
In
Table
\ref{widths}
we present the relevant numbers obtained with the latest update
of {\tt DIZET} options
and previous actual options of these flags, i.e.
(IHVP,~IAMT4): (1,8) in comparison with (5,6) and (1,8) in comparison with (5,8).
The main improvement comes from accounting previously missing bosonic $O(\alpha^2)$ corrections to the $Z\to b\bar{b}$ decay~\cite{Dubovyk:2016aqv}.
\subsubsection{The effective weak mixing angle $\sin^2\vartheta_{eff}$}
In Table~\ref{sineff} we illustrate
various options for flag AMT4 in
{\tt DIZET} v.~6.45
to estimate
$\sin^2\vartheta_{eff}$ in different channels.
\begin{table}[ht]
\centering
\begin{tabular}{|l|ccc|c|c|}
\hline
(IHVP,~IAMT4) & (1,8) & (5,6) & (5,8) & $|\delta_{(1,8)-(5,8)}|\cdot 10^{3}$& $|\delta_{(5,6)-(5,8)}|\cdot 10^{3}$ \\
\hline
$\sin^2\vartheta_{eff}$ channel & & & & & \\
\hline
$\nu,\bar\nu$ &0.231280 & 0.231149 & 0.231118 & 0.162 & 0.031 \\
$e^+,e^-$ &0.231661 & 0.231530 & 0.231500 & 0.161 & 0.030 \\
$\mu^+,\mu^-$ &0.231661 & 0.231530 & 0.231500 & 0.161 & 0.030 \\
$\tau^+,\tau^-$ &0.231661 & 0.231530 & 0.231500 & 0.161 & 0.030 \\
$u,\bar u$ &0.231555 & 0.231424 & 0.231393 & 0.162 & 0.031 \\
$d,\bar d$ &0.231428 & 0.231297 & 0.231266 & 0.162 & 0.031 \\
$c,\bar c$ &0.231555 & 0.231424 & 0.231393 & 0.162 & 0.031 \\
$s,\bar s$ &0.231428 & 0.231297 & 0.231266 & 0.162 & 0.031 \\
$b,\bar b$ &0.232895 & 0.232970 & 0.232732 & 0.163 & 0.238 \\
\hline
\end{tabular}
\caption{
The effective weak mixing angle $\sin^2\vartheta_{eff}$ for all channels
calculated for
sets of flags (IHVP,~IAMT4):
(1,8) in comparison with (5,6) and (1,8) in comparison with (5,8).}
\label{sineff}
\end{table}
The results for the channels are similar. The largest shift comes from accounting previously missing bosonic $O(\alpha^2)$ corrections to $\sin^2\theta^b_{eff}$~\cite{Dubovyk:2016aqv}.
\clearpage
\subsubsection{Cross Sections}
\begin{figure}[ht]
\centering
\label{fig_sigma}
\includegraphics[width=12cm]{figs/plot_sigma_u.pdf}
\caption{Differences $\delta$ defined in Eq.~(\ref{eq:diffxsection}) for cross sections of $e^+e^-\to u \bar{u}$ for sets of flags (IHVP,~IAMT4): {(1,8), (4,8), (5,6), (5,7)} relative to the current best set {(5,8)}.}
\label{xs-uu}
\end{figure}
Using the example of the channel $e^+e^-\to u \bar{u}$,
Figure~\ref{xs-uu} shows the cross section differences for sets of the flags (IHVP, IAMT) relative to the current best set (IHVP, IAMT) =(5,8);
\begin{equation}
\delta = \displaystyle{\frac{\sigma (\rm{IHVP, IAMT}) - \sigma (5,8)}{\sigma (5,8)}}\cdot1000[\permil] .\nonumber \label{eq:diffxsection}
\end{equation}
The main influence on the result is the use of a modern parametrization for the
hadronic vacuum polarization. The effect of bosonic corrections is weaker. The results close to the $Z$ peak ($\pm$ 1 GeV) show that the relative shift is below $5\cdot10^{-5}$.
\subsubsection{
Left-Right and Forward-Backward Asymmetries}
The channel $e^+e^-\to u \bar{u}$ is used to show in
Figures~\ref{fig_ALR} and \ref{fig_AFB} the differences for the left-right asymmetries and for the forward-backward asymmetries for sets of the flags (IHVP, IAMT) compared with the current best set (IHVP, IAMT) = (5,8);
\begin{equation}
\Delta \rm{A} = \rm{A}(\rm{IHVP, IAMT})- \rm{A}(5,8). \nonumber
\end{equation}
\begin{figure}[ht]
\centering
\includegraphics[width=12cm]{figs/plot_ALR_u.pdf}
\caption{Differences for left-right asymmetries of $e^+e^-\to u \bar{u}$ for sets of flags (IHVP,~IAMT4): {(1,8), (4,8), (5,6), (5,7)} from the current best set {(5,8)}.}
\label{fig_ALR}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=12cm]{figs/plot_AFB_u.pdf}
\caption{Differences for forward-backward asymmetries of $e^+e^-\to u \bar{u}$ for sets of flags (IHVP,~IAMT4): {(1,8), (4,8), (5,6), (5,7)} from the current best set {(5,8)}.}
\label{fig_AFB}
\end{figure}
As in the case of corrections to the cross section the main effect is due to changes in the hadronic vacuum polarization treatment. Bosonic corrections of the $O(\alpha^2)$ order taken into account change the left-right asymmetry by no more than $3\cdot10^{-4}$ and the forward-backward asymmetry by no more than $2\cdot10^{-4}$.
\clearpage
\section{Benchmarks {\tt DIZET} v.~6.45}
Here, we provide a benchmark for the default set of parameters and flags.
\begin{verbatim}
DIZET flags, see routine DIZET for explanation:
IHVP = 5 IAMT4 = 8
Iqcd = 3 Imoms = 1
Imass = 0 Iscre = 0
Ialem = 3 Imask = 0
Iscal = 0 Ibarb = 2
IFtjr = 1 Ifacr = 0
IFact = 0 Ihigs = 0
Iafmt = 3 Iewlc = 1
Iczak = 1 Ihig2 = 0
Iale2 = 3 Igfer = 2
Iddzz = 1 Iamw2 = 0
Isfsr = 1 Idmww = 0
Idsww = 0
IBAIKOV = 2012
IBAIKOV = 2012
DIZET input parameters:
ZMASS 91.187600000000003 TMASS 172.75999999999999
HMASS 125.25000000000000 WMASS 0.0000000000000000
DAL5H 0.0000000000000000 ALQED5 137.03599908400000
ALFAS 0.11790000000000000
DIZET results:
SIN2TW 0.22340388419691781
WMASSsin 80.358790700232220
WMASS 80.358790700232220
DAL5H 2.7576193213462830E-002
ALQED5 128.95030472145015
ALST 0.10755034917841029
ALPAS 0.11790000000000000
CHANNEL WIDTH RHO_F_R RHO_F_T SIN2_EFF
------- ------- -------- -------- --------
nu,nubar 167.202 1.007963 1.007963 0.231118
e+,e- 83.985 1.005219 1.005062 0.231500
mu+,mu- 83.985 1.005219 1.005062 0.231500
tau+,tau- 83.795 1.005219 1.005062 0.231500
u,ubar 299.918 1.005812 1.005757 0.231393
d,dbar 382.861 1.006733 1.006723 0.231266
c,cbar 299.852 1.005812 1.005757 0.231393
s,sbar 382.861 1.006733 1.006723 0.231266
t,tbar 0.000 0.000000 0.000000 0.000000
b,bbar 375.839 0.994198 0.994198 0.232732
hadron 1741.442
total 2494.814
W-widths
lept,nubar 678.981
down,ubar 1410.986
total 2089.967
FF:
RHO (0.99876990486335659,-4.73600008701756652E-003)
RHO (0.99876990 -.00473600)
KAPPA.I (1.03606482 0.01353154)
KAPPA.J (1.04380408 0.01353154)
KAPPA.IJ (1.08147655 0.02706307)
AL_I(s) (129.37048577139927,1.9810219926255057)
AL_5_I(s) (129.36196765366330,1.9807610678870036)
****************************************************
\end{verbatim}
\section{Conclusions}
The new version 6.45 of the {\tt DIZET} electroweak library is described.
In this work, we benchmark the novel implementation
of two-loop $\alpha^2$ bosonic and fermionic radiative corrections
and several fits of the hadronic vacuum polarization
$\Delta \alpha^{(5)}_{had}(M_{\sss Z})$ using public versions of the {\tt AlphaQED} code.
The presented numerical results show the impact of the new options
for cross sections,
left-right and forward-backward asymmetries.
We can see that the updates
are relevant for high-precision experiments at future electron-positron colliders.
Compatibility with previous versions of the code is supported.
Predictions for observables and pseudobservables have been produced with the {\tt ZFITTER} program \cite{Bardin:1999yd,Arbuzov:2005ma}.
The code is available directly with the link:
\href{http://sanc.jinr.ru/users/zfitter/DIZET_v6.45.tgz} {DIZET6.45}.
Recently the new electroweak library {\tt GRIFFIN} was created \cite{Chen:2022dow} in which comparisons with {\tt DIZET} are given. The new version of the discussed here {\tt DIZET} program can serve for further tuned comparisons in future high-precision studies.
\\
{\bf{Note:}} The work done in this paper is motivated by the HL-LHC studies and the LHC electroweak Working Group activity and is based on the implementation of the new version of the DIZET code released on December 2019 \cite{dizetv645} and public presentations of results during EWG meetings in 2019 \cite{dizetws,dizetws2} and 2020 \cite{dizetws3}.
\section*{Acknowledgments}
This work has been supported in part by the Polish National Science Center (NCN) under grant 2017/25/B/ST2/01987. A.A., L.K. and V.Ye. are grateful for the support to RFBR grant N~20-02-00441.
\bibliographystyle{elsarticle-num-ID}
|
1,314,259,996,548 | arxiv | \section{Introduction}
Action principles for systems with boundaries are a fundamental tool
in studying quantum effects in diffeomorphism invariant systems.
They have been used to derive black hole entropy in the
semiclassical approximation
\cite{HI} and to discuss the pair
creation and annihilation of magnetically charged black holes
\cite{pairs}. Recently, Carlip \cite{steve} has even suggested
that the entropy of the 2+1 dimensional
Ban\~ados, Teitelboim, and Zanelli black hole \cite{BTZ} can be
derived by counting the states produced by the degrees of freedom
that arise through a ``restriction of the diffeomorphism invariance''
by the presence of a boundary; in that case, the horizon of a
black hole. In addition, asymptotically flat and other
noncompact spacetimes may be described as systems with a boundary
\cite{roger}, and spacelike singularities form a natural
past or future boundary for many classical solutions of
general relativity and low energy string theory (see, e.g., \cite{sing}).
Finally, to render the action of {\it any} system finite, it is generally
necessary to consider the system only between two times or between
two spacelike hypersurfaces.
Thus, there is ample motivation to understand any subtleties that
arise in the use of variational principles for bounded generally covariant
systems.
The actions for such systems are typically of the local form
\begin{equation}
\label{s0}
S^{{\cal M}}_0 = \int_{\cal M} {\cal L} \ d^nx
\end{equation}
where ${\cal M}$ is an $n$-manifold and ${\cal L}$ is a scalar density on ${\cal M}$.
Such an action is invariant under any diffeomorphism
$\psi: {\cal M}\rightarrow {\cal M}$. Note that such a map
induces a diffeomorphism of the boundary $\partial{\cal M}$ of ${\cal M}$ as well. If
the map did not preserve $\partial{\cal M}$, it would
correspond to enlarging (or shrinking)
the system considered and would in general change the action.
While the addition of the proper boundary
term to \ref{s0} can enlarge the gauge invariance of $S_0^{\cal M}$
\cite{HTV,conf},
this will in general require the full solution of
the equations of motion. As
a result, if the action $S_0^{\cal M}$
defines the notion of gauge equivalence,
an infinitesimal map $\delta \psi$ is a gauge transformation only if
it preserves $\partial{\cal M}$.
Suppose now that ${\cal M}$ may be embedded in some larger manifold
${\cal M}'$. We would like to understand how the notions of gauge invariance
defined by $S_0^{{\cal M}}$ and $S_0^{{\cal M}'}$ relate. Because
$S_0^{{\cal M}'}$ is invariant under diffeomorphisms that move the
boundary of ${\cal M}$, $S_0^{{\cal M}'}$ provides a larger set of infinitesimal
gauge transformations than does $S_0^{{\cal M}}$, even within the
image of ${\cal M}$. Thus, the class of gauge invariants defined by
$S_0^{{\cal M}}$ is larger than that defined by $S_0^{{\cal M}'}$. In particular,
if the theory contains a metric $g_{\mu \nu}$ and ${\cal M}$ is compact,
the quantity
\begin{equation}
\label{vol}
\int_{{\cal M}} \sqrt{-g} d^nx
\end{equation}
is invariant with respect to gauge transformations defined by
$S_0^{{\cal M}}$, but not those defined by $S_0^{{\cal M}'}$.
When the boundaries of ${\cal M}$ are not specified
by a physical condition, the action $S_0^{{\cal M}}$ and $S_0^{{\cal M}'}$
seem to describe quite different physics.
If, however, the above embedding is chosen in a field dependent but
diffeomorphism invariant manner (such as by mapping timelike
boundaries to sheets of steel and spacelike boundaries to
hypersurfaces defined by the reading of some clock), this
picture is physically reasonable as \ref{vol} may be interpreted as the
spacetime volume of the region bounded by the steel sheets for the
appropriate clock readings. Such a quantity is gauge invariant as defined by
$S_0^{{\cal M}'}$ as well. We would like to make this
connection explicit by describing the part of the system within
such boundaries in a way that
is invariant under all diffeomorphisms of ${\cal M}'$, even those that
move $\partial {\cal M}$.
The purpose of this letter is to use this physical picture to
provide an action principle which achieves these goals {\it without}
first solving the equations of motion. In
particular, given any action principle of the form \ref{s0} and
any (not necessarily local!) scalar field $f$, we show that
variation of the action
\begin{equation}
\label{newS}
S^{{\cal M}} = \int_{{\cal M}} \theta(f) {\cal L} \ d^nx
\end{equation}
where $\theta$ is the Heaviside step function,
yields the same Euler-Lagrange equations as $S_0^{{\cal M}}$ in the region where
$f>0$ {\it but provides no other restrictions on the dynamics}
when varied within a diffeomorphism invariant class of field
histories. This property is nontrivial only on the surface $f=0$,
but, as should be expected, follows on this surface
only if the variations preserve appropriate
`boundary conditions' on the field histories.
Note that, provided $f < 0$ on the boundary of ${\cal M}$, the
action $S^{\cal M}$ is invariant under a larger class of gauge transformations
than $S_0^{\cal M}$. Effectively, it {\it is} invariant under
transformations that
move the boundary of ${\cal M}$. These results will be derived in the next
section.
\section{The Variational Principle for $S^{\cal M}$}
We now consider a general coordinate invariant
action principle of the form \ref{s0}
where the Lagrangian density ${\cal L}$ is a function of some
collection of fields and their first derivatives. An action
of this form yields a well-defined variational principle whenever
all fields whose derivatives appear in ${\cal L}$ are
fixed on the boundary. We refer to such fields as type I; fields
whose derivatives do not appear in ${\cal L}$ will be referred to
as type II. However, for some first order systems such as spinor fields
or Chern-Simons fields \cite{ex1,ex2} it is only appropriate to specify
certain parts of these fields and to do so in a generally covariant
manner typically involves complicated nonlocal constructions\footnote{
Thanks to Steve Carlip for bringing this to the author's attention.}.
We do not treat such cases, but we expect that they may be
addressed by an analysis similar to
what follows. We point out that our analysis {\it is}
appropriate to standard first order formulations of systems, like
gravity in any number of dimensions, which also have a second order
formulation. In addition, although
not manifestly so, the usual action
\begin{equation}
{1 \over {16 \pi}} \int_{{\cal M}} \sqrt{-g} R + {1 \over {8\pi}} \int_{\partial {\cal M}}
K
\end{equation}
for general relativity {\it is} of this form, as the boundary
term exactly cancels a total divergence which contains the higher
derivatives of $g_{\mu \nu}$.
More general variational principles, where
various momenta are fixed on the boundary, can be obtained from
actions of the above form by adding a total divergence to ${\cal L}$. If the
Lagrangian is in first order form (see, for example, \cite{KK}),
the addition of such a divergence
leaves ${\cal L}$ a function only of the fields and their first
derivatives. Thus, by passing through the first order
formulation, we see that actions of this type are
quite general.
Suppose that the Lagrangian depends on a set of
fields (labeled by $a$) of tensor type $i = (i_1,i_2)$ and
density weight $j$ which we write as $\phi^{ija\{\nu\}}_{\{\mu\}}$.
Here, $\{\nu\}$ and $\{\mu\}$ denote the appropriate collection
of abstract tensor indices as specified by the value of $i$.
${\cal L}$ will generally depend on fields of more than one
tensor type so that both $i$ and $j$ will take multiple values.
We employ the summation convention on the indices $i,j,a$ as
well as $\{\nu\}$ and $\{\mu\}$ so that the expression
\begin{equation}
j {{\partial {\cal L}} \over {\partial \phi^{ija\{\nu\}}_{\{\mu\}}}}
\phi^{ija\{\nu\}}_{\{\mu\}} \end{equation}
represents a sum over fully contracted fields of all tensor types
and all density weights with the contribution of each field
multiplied by its density weight.
It will be convenient to introduce an arbitrary background
spacetime connection $\Gamma^{\sigma}_{\alpha \beta}$ together
with the
associated covariant derivative operator ${\cal D}_{\alpha}$ and
to write ${\cal L}$ as a function of the
$\phi^{ijk\{\nu\}}_{\{\mu\}}$
and the ${\cal D}_{\alpha} \phi^{ijk\{\nu\}}_{\{\mu\}}$. Since this
connection was introduced by hand, the fields and derivatives must
enter the Lagrangian in such a way that ${\cal L}$
is independent of $\Gamma^{\sigma}_{\alpha \beta}$. We may thus
vary $\Gamma^{\sigma}_{\alpha \beta}$ as well in our action
principle and, while the resulting equation of motion will be
identically satisfied, this provides a convenient way
to keep track of the relationships that conspire to make
${\cal L}$ independent of $\Gamma^{\sigma}_{\alpha \beta}$ and thus
make $S_0^{\cal M}$ diffeomorphism invariant.
We will need the explicit form of the change of ${\cal L}$
under an infinitesimal coordinate transformation $x^\mu
\rightarrow x^\mu + \delta x^\mu$:
\begin{eqnarray}
\label{dx}
\delta{\cal L} = &-& { {\partial {\cal L}} \over {\partial \phi^{ija\{\nu\}}_{\{\mu\}}}}
[ \partial_\alpha \phi^{ija\{\nu\}}_{\{\mu\}} \ \delta x^\alpha + j \phi^{ija\{\nu\}}_{\{\mu\}} \partial_\alpha \ \delta x^\alpha
+ \sum_k \phi^{ija\{\nu\}}_{\{\mu\}_1 \alpha \{\mu\}_2} \ \partial_{\mu_k} \delta x^\alpha - \sum_k \phi^{ija\{\nu\}_1\beta \{\nu\}_2}_{\{\mu\}} \ \partial_\beta \delta x^{\nu_k}]
\cr
&-& {{\partial {\cal L}} \over {\partial ({\cal D}_\rho \phi^{ija\{\nu\}}_{\{\mu\}})}} [ \partial_\alpha ({\cal D}_\rho \phi^{ija\{\nu\}}_{\{\mu\}}) \ \delta x^\alpha
+ j {\cal D}_\rho \phi^{ija\{\nu\}}_{\{\mu\}} \ \partial_\alpha \delta x^\alpha + {\cal D}_\alpha \phi^{ija\{\nu\}}_{\{\mu\}} \ \partial_\rho \delta x^\alpha
\cr && \ \ \ \ \ \ \ \ \ \ + \sum_k {\cal D}_\rho \phi^{ija\{\nu\}}_{\{\mu\}_1 \alpha \{\mu\}_2} \ \partial_{\mu_k} \delta x^\alpha
- \sum_k {\cal D}_\rho \phi^{ija\{\nu\}_1\beta \{\nu\}_2}_{\{\mu\}} \ \partial_{\beta} \delta x^{\nu_k}]
\end{eqnarray}
where the terms in square brackets are the explicit forms of
$\delta \phi^{ija\{\nu\}}_{\{\mu\}}$ and $\delta {\cal D}_\rho \phi^{ija\{\nu\}}_{\{\mu\}}$ under a change of
coordinates. The notation $\sum_k \phi^{ija\{\nu\}}_{\{\mu\}_1 \alpha \{\mu\}_2} \partial_{\mu_k} \delta x^\alpha$ represents
a sum whose $k$th term has the $k$th covariant index in the
set $\{\mu\}$ replaced by $\alpha$ and is contracted on that index
with $\partial_{\mu_k}\delta x^\alpha$,
where $\mu_k$ is just this missing index. The corresponding notation
is used for the contravariant case as well.
Equation \ref{dx}
may be written in the form $\delta {\cal L} = - (\partial_\alpha {\cal L} \ \delta x^\alpha
+ Q_\alpha^\beta \ \partial_\beta \delta x^\alpha)$, from which will follow the
identity that captures the coordinate invariance
of $S_0^{\cal M}$.
Since $S_0^{\cal M}$ is unchanged by an arbitrary infinitesimal
coordinate transformation, we have that
\begin{eqnarray}
\label{currents}
0 = \delta S_0^{\cal M} &= & -
\int_{\cal M} [Q^\beta_\alpha \ \partial_\beta \delta x^a + \partial_\alpha{\cal L} \ \delta x^\alpha]
+ \int_{\partial {\cal M}} {\cal L} n_\alpha \delta x^\alpha \cr
&=& - \int_{\cal M} \partial_\beta [{\cal L} \delta^\beta_\alpha - Q^\beta_\alpha ] \ \delta x^\alpha
+ \int_{\partial {\cal M}} [{\cal L} \delta^\beta_\alpha - Q^\beta_\alpha ] n_b \delta x^\alpha
\end{eqnarray}
where $n_\beta$ is the outward pointing normal vector field to $\partial {\cal M}$.
Thus, we may conclude that $\partial_\beta[{\cal L} \delta^\beta_\alpha - Q^\beta_\alpha]$
vanishes in the interior and that
$n_\beta[{\cal L} \delta{}^\beta_\alpha - Q^\beta_\alpha]$ vanishes on the boundary.
Since the fields themselves are unrestricted on $\partial {\cal M}$ (only
the variations of the fields are constrained), we must in fact
have
\begin{equation}
\label{cc}
{\cal L} \delta^\beta_\alpha - Q^\beta_a = 0
\end{equation}
{\it identically} everywhere. This is just the generalization of the
familiar statement that, on a $0+1$ dimensional spacetime, the
Hamiltonian constructed from a diffeomorphism invariant
action for a system of scalar fields vanishes identically.
The result \ref{cc} may be used to show that the action \ref{newS}
provides an acceptable variational principle when
$S^{\cal M}$ is varied within a class of
histories for which the fields are fixed on
the $f=0$ surface. Specifically, fix some $n-1$
manifold $\Sigma$ and an embedding $\eta: \Sigma \rightarrow {\cal M}$ such that
${\cal M} - \Sigma$ has exactly two connected components (which we arbitrarily
call the inside and the outside). Note that $\Sigma$ may have
a boundary $\partial \Sigma$.
Consider the class of histories for which the surface defined by
$f=0$ gives the above embedding of $\Sigma$ into ${\cal M}$ up to
a diffeomorphism and for which $f>0$ inside and $f<0$ outside. The
inside may still contain part of $\partial {\cal M}$, although the
most interesting case is when $\partial {\cal M}$ lies completely outside. Furthermore,
we will vary the histories only within the class for which all type
I fields are fixed (up to a diffeomorphism of ${\cal M}$)
on the part of $\partial {\cal M}$ inside of $f=0$
and on the $f=0$ surface. This may be done, for example, by
choosing histories for which some set of scalar fields may be
used as a coordinate system near $\partial {\cal M}$ and $f=0$ and for which the
components of the tensor fields have some fixed relationship with
these scalars and their gradients.
A direct variation of $S_0$, together with the usual integrations
by parts yields:
\begin{eqnarray}
\label{EL}
\delta S^{\cal M} &= \int_{\cal M} \Biggl( \theta(f) &\Bigl[ \
\Bigl( {{\partial {\cal L}} \over {\partial \phi^{ija\{\nu\}}_{\{\mu\}}}} -
{\cal D}_\rho {{\partial {\cal L}} \over {\partial ({\cal D}_\rho \phi^{ija\{\nu\}}_{\{\mu\}})}} \Bigr) \delta \phi^{ija\{\nu\}}_{\{\mu\}} \cr
&& + \Bigl( j \phi^{ija\{\nu\}}_{\{\mu\}} \delta^\beta_\alpha + \sum_k \phi^{ija\{\nu\}}_{\{\mu\}_1 \alpha \{\mu\}_2} \delta^\beta_{\mu_k}
- \sum_k \phi^{ija\{\nu\}_1\beta \{\nu\}_2}_{\{\mu\}} \delta^{\nu_k}_\alpha \Bigr) {{\partial {\cal L}} \over {\partial ({\cal D}_\rho \phi^{ija\{\nu\}}_{\{\mu\}})}}
\delta \Gamma^\alpha_{\rho \beta } \Bigr] \cr
&& + \delta(f) \Bigl[ \delta f {\cal L} - {{\partial {\cal L}} \over {{\cal D}_\rho \phi^{ija\{\nu\}}_{\{\mu\}}}} \partial_\rho f \
\delta \phi^{ija\{\nu\}}_{\{\mu\}} \Bigr]
\Biggr) \cr
&+& \int_{\partial {\cal M}} \theta(f) n_\rho {{\partial {\cal L}} \over {\partial ({\cal D}_\rho \phi^{ija\{\nu\}}_{\{\mu\}})}} \delta \phi^{ija\{\nu\}}_{\{\mu\}}
\end{eqnarray}
where $\delta(f)$ is the Dirac delta-function and is not to be
confused with the variation $\delta f$.
The first line in \ref{EL} contains just the usual Euler-Lagrange
equations inside the surface $f=0$ and the second line explicitly
displays the conspiracies that make ${\cal L}$ independent of the background
connection. Line four contains the usual boundary terms on the
inside part of $\partial {\cal M}$, which vanish for our class of variations.
The third line contains the terms of interest; in order
that our new variational principle not restrict the dynamics excessively,
we must show that these terms vanish when the variations satisfy the
boundary conditions given above and when the Euler-Lagrange equations
are satisfied on the $f=0$ surface\footnote{As may be seen from
a brief study of the
free relativistic particle, an attempt to leave these variations
arbitrary (as in a naive application of \ref{newS})
and use their coefficients as additional equations of
motion leads to nonsense.}.
To do so, note that the variation $\delta f$ will move the $f=0$
boundary surface. Since all fields, including $f$ itself, are
specified on the $f=0$ surface up to diffeomorphism,
the variations $\delta \phi^{ija\{\nu\}}_{\{\mu\}}$ must be just those
induced by some diffeomorphism
$x^\alpha \rightarrow x^\alpha +\delta x^\alpha$, which were explicitly
displayed in equation \ref{dx}. Similarly, $\delta f =
- \partial_\alpha \delta x^\alpha$. Thus these variations, when evaluated on the
$f=0$ surface, are not independent. The relationships between the
$\delta \phi^{ija\{\nu\}}_{\{\mu\}}$ will conspire, together with the general covariance of
$S_0^{\cal M}$, to make the term proportional to $\delta(f)$ vanish when the
usual Euler-Lagrange equations of $S_0^{\cal M}$ are imposed on the $f=0$
surface.
Now, a comparison of \ref{dx} and \ref{EL} shows that use of the Euler-Lagrange
equations for $\Gamma^{\sigma}_{\alpha \beta}$ greatly
simplifies the term proportional to $\delta(f)$ since, when
contracted with ${{\partial {\cal L}} \over {\partial ({\cal D}_\rho \phi^{ija\{\nu\}}_{\{\mu\}})}}$, $\delta \phi^{ija\{\nu\}}_{\{\mu\}}$
may be replaced by $\partial_\alpha \phi^{ija\{\nu\}}_{\{\mu\}} \ \delta x^\alpha$. Thus, we find
\begin{equation}
\label{diff}
\delta S^{\cal M} - \theta(f) \delta S_0^{\cal M} = \int_{\cal M} \delta(f)
[{\cal L} \delta^\beta_\alpha - {{\partial {\cal L}} \over {\partial ({\cal D}_\beta \phi^{ija\{\nu\}}_{\{\mu\}})}} \partial_\alpha \phi^{ija\{\nu\}}_{\{\mu\}}] \partial_\beta f
\delta x^\alpha
\end{equation}
which looks suspiciously like \ref{cc}.
Returning to the definition of $Q_\alpha^\beta$ in \ref{dx},
the Euler-Lagrange equations for $\phi^{ija\{\nu\}}_{\{\mu\}}$ show that we have
\begin{equation}
\label{almost}
Q^\beta_\alpha = {\cal D}_\partial \Bigl[ {{\partial {\cal L}} \over {\partial ({\cal D}_\rho \phi^{ija\{\nu\}}_{\{\mu\}})}}
{{\partial (\delta\phi^{ija\{\nu\}}_{\{\mu\}})} \over {\partial (\partial_\beta \delta x^a)}}\Bigr]
+ {{\partial {\cal L}} \over { \partial ({\cal D}_\rho \phi^{ija\{\nu\}}_{\{\mu\}})}} {\cal D}_\alpha \phi^{ija\{\nu\}}_{\{\mu\}} \delta^\beta_\rho.
\end{equation}
where we have written the transformation $\delta \phi^{ija\{\nu\}}_{\{\mu\}}$ displayed in
\ref{dx} as a function of $\delta x^{\sigma}$ and $\partial_\beta \delta x^\alpha$.
As before, the term in brackets vanishes by the Euler-Lagrange
equations for $\Gamma^{\sigma}_{\alpha \beta}$. In the second
term, the contraction of these same Euler-Lagrange equations with
$\Gamma^{\sigma}_{\alpha \gamma}$ on the indices $\sigma$ and $\gamma$
shows that ${\cal D}_\rho \phi^{ija\{\nu\}}_{\{\mu\}}$ may be replaced with $\partial_\rho \phi^{ija\{\nu\}}_{\{\mu\}}$ in \ref{almost}.
Equation \ref{cc} then reads
\begin{equation}
{\cal L} \delta^\beta_\alpha - {{\partial {\cal L}} \over {\partial ({\cal D}_\rho \phi^{ija\{\nu\}}_{\{\mu\}})}}
\partial_\alpha \phi^{ija\{\nu\}}_{\{\mu\}} \delta^\beta_\rho = 0
\end{equation}
so that $\delta S_0^{\cal M}$ vanishes when the
Euler-Lagrange equations hold inside and on the surface $f=0$.
Thus, the action \ref{newS} provides a perfectly valid variational
principle for our partial system.
This leads to the interesting question of how path integrals
based on \ref{newS} differ from those based on \ref{s0}. Note that
in a canonical framework, and as opposed to the traditional approach
of \cite{Claudio,JJ}, use of an action of the form \ref{newS} allows the
lapse $N$ to be {\it completely} fixed along with the gauge freedom.
It seems likely that path integrals of both types are appropriate
to the study of diffeomorphism invariant systems,
though with different interpretations
which are yet to be fully understood.
\acknowledgments
The author would like to express his thanks to Steve Carlip,
Fay Dowker, Jim Hartle,
Gary Horowitz, and Jorma Louko for sharpening his thinking on this subject
and for suggesting important referecnes.
This work was supported by NSF grant PHY-908502.
|
1,314,259,996,549 | arxiv | \section{Introduction}
\medskip
\medskip
Let $N\geq 1$ be an integer and let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of
$SL_2(\mathbb Z)$. In \cite{CM1}, \cite{CM2}, Connes and Moscovici have introduced
the ``modular Hecke algebra'' $\mathcal A(\Gamma)$ that combines the pointwise product on
modular forms with the action of Hecke operators. Further, Connes and Moscovici have shown that
the modular Hecke algebra $\mathcal A(\Gamma)$ carries an action of ``the Hopf algebra
$\mathcal H_1$ of codimension $1$ foliations''. The Hopf algebra $\mathcal H_1$ is part of a larger
family of Hopf algebras $\{\mathcal H_n|n\geq 1\}$ defined in \cite{CM0}. The objective of this paper
is to introduce and study quasimodular Hecke algebras $\mathcal Q(\Gamma)$ that similarly combine
the pointwise product on quasimodular forms with the action of Hecke operators. We will see that the quasimodular
Hecke algebra $\mathcal Q(\Gamma)$ carries several other operators in addition to an action of
$\mathcal H_1$. Further, we will also
study the collection $\mathcal Q_\sigma(\Gamma)$ of quasimodular Hecke operators twisted
by some $\sigma\in SL_2(\mathbb Z)$. The latter is a generalization of our theory of twisted modular Hecke
operators introduced in \cite{AB1}.
\medskip
We now describe the paper in detail. In Section 2, we briefly recall the notion of modular Hecke algebras
of Connes and Moscovici \cite{CM1}, \cite{CM2}. We let $\mathcal{QM}$ be the ``quasimodular tower'', i.e.,
$\mathcal{QM}$ is the colimit
over all $N$ of the spaces $\mathcal{QM}(\Gamma(N))$ of quasimodular forms of level $\Gamma(N)$ (see \eqref{2.8}). We define a quasimodular Hecke operator
of level $\Gamma$ to be a function of finite support from $\Gamma\backslash GL_2^+(\mathbb Q)$
to the quasimodular tower $\mathcal{QM}$ satisfying a certain covariance condition (see Definition \ref{maindef}).
We then show that the collection $\mathcal Q(\Gamma)$ of quasimodular Hecke operators
of level $\Gamma$ carries an algebra structure $(\mathcal Q(\Gamma),\ast)$ by considering a convolution product over cosets of $\Gamma$ in $GL_2^+(\mathbb Q)$.
Further, the modular Hecke algebra of Connes and Moscovici embeds naturally as a subalgebra of $
\mathcal Q(\Gamma)$. We also show that the quasimodular Hecke operators of level $\Gamma$ act on
quasimodular forms of level $\Gamma$, i.e., $\mathcal{QM}(\Gamma)$ is a left $\mathcal Q(\Gamma)$-module.
In this section, we will also define a second algebra structure $(\mathcal Q(\Gamma),\ast^r)$ on $\mathcal Q(\Gamma)$ by considering the
convolution product over cosets of $\Gamma$ in $SL_2(\mathbb Z)$. When we consider $\mathcal Q(\Gamma)$
as an algebra equipped with this latter product $\ast^r$, it will be denoted by $\mathcal Q^r(\Gamma)=(\mathcal Q(\Gamma),\ast^r)$.
\medskip
In Section 3, we define Lie algebra and Hopf algebra actions on $\mathcal Q(\Gamma)$. Given a quasimodular form
$f\in \mathcal{QM}(\Gamma)$ of level $\Gamma$, it is well known that we can write $f$ as a sum
\begin{equation}\label{1.0ner}
f=\sum_{i=0}^sa_i(f)\cdot G_2^i
\end{equation} where the coefficients $a_i(f)$ are modular forms of level $\Gamma$ and $G_2$ is the classical
Eisenstein series of weight $2$. Therefore, we can consider two different sets of operators on the quasimodular tower
$\mathcal{QM}$: those which
act on the powers of $G_2$ appearing in the expression for $f$ and those which act on the modular coefficients $a_i(f)$. The collection
of operators acting on the modular coefficients $a_i(f)$ are studied in Section 3.2. These induce on $\mathcal Q(\Gamma)$ analogues of operators acting on
the modular Hecke algebra $\mathcal A(\Gamma)$ of Connes and Moscovici and we show that
$\mathcal Q(\Gamma)$ carries an action of the same Hopf algebra $\mathcal H_1$ of codimension $1$ foliations
that acts on $\mathcal A(\Gamma)$. On the other hand, by considering operators on $\mathcal{QM}$ that act on the powers
of $G_2$ appearing in \eqref{1.0ner}, we are able to define additional operators $D$, $\{T_k^l\}_{k\geq 1,l\geq 0}$ and
$\{\phi^{(m)}\}_{m\geq 1}$ on $\mathcal Q(\Gamma)$ (see Section 3.1). Further, we show that these operators satisfy the following commutator
relations:
\begin{equation}\label{1.2ner}
\begin{array}{c}
[T_k^l,T_{k'}^{l'}]=(k'-k)T_{k+k'-2}^{l+l'}\\
\mbox{$[D,\phi^{(m)}]=0 \qquad [T_k^l,\phi^{(m)}]=0 \qquad [\phi^{(m)},
\phi^{(m')}]=0$} \\
\mbox{$[T_k^l,D]=\frac{5}{24}(k-1)T^{l+1}_{k-1}-\frac{1}{2}(k-3)T^l_{k+1}$}\\
\end{array}
\end{equation} We then consider the Lie algebra $\mathcal L$ generated by the symbols
$D$, $\{T_k^l\}_{k\geq 1,l\geq 0}$,
$\{\phi^{(m)}\}_{m\geq 1}$ satisfying the commutator relations in \eqref{1.2ner}. Then, there
is a Lie action of $\mathcal L$ on $\mathcal Q(\Gamma)$. Finally, let $\mathcal H$ be the Hopf algebra given by the
universal enveloping algebra $\mathcal U(\mathcal L)$ of $\mathcal L$. Then, we show that $\mathcal H$ has a Hopf action with
respect to the product $\ast^r$ on $\mathcal Q(\Gamma)$ and this action captures the operators $D$, $\{T_k^l\}_{k\geq 1,l\geq 0}$ and
$\{\phi^{(m)}\}_{m\geq 1}$ on $\mathcal Q(\Gamma)$. In other words, $\mathcal H$ acts on $\mathcal Q(\Gamma)$
such that:
\begin{equation}\label{1.3ner}
h(F^1\ast^r F^2)=\sum h_{(1)}(F^1)\ast^r h_{(2)}(F^2) \qquad\forall \textrm{ } h\in \mathcal H, \textrm{ }F^1,F^2\in \mathcal Q(\Gamma)
\end{equation} where the coproduct $\Delta:\mathcal H\longrightarrow \mathcal H\otimes \mathcal H$ is given by
$\Delta(h)=\sum h_{(1)}\otimes h_{(2)}$ for any $h\in \mathcal H$.
\medskip
In Section 4, we develop the theory of twisted quasimodular Hecke operators. For any
$\sigma\in SL_2(\mathbb Z)$, we define in Section 4.1 the collection $\mathcal Q_\sigma(\Gamma)$
of quasimodular Hecke operators of level $\Gamma$ twisted by $\sigma$. When $\sigma=1$, this reduces
to the original definition of $\mathcal Q(\Gamma)$. In general, $\mathcal Q_\sigma(\Gamma)$ is not
an algebra but we show that $\mathcal Q_\sigma(\Gamma)$ carries a pairing:
\begin{equation}\label{1.7m}
(\_\_,\_\_):\mathcal Q_\sigma(\Gamma)\otimes \mathcal Q_\sigma(\Gamma)\longrightarrow \mathcal Q_\sigma(\Gamma)
\end{equation} Further, we show that $\mathcal Q_\sigma(\Gamma)$ may be equipped with the structure
of a right $\mathcal Q(\Gamma)$-module. We can also extend the action of the Hopf algebra
$\mathcal H_1$ of codimension $1$ foliations to $\mathcal Q_\sigma(\Gamma)$. In fact, we show that
$\mathcal H_1$ has a ``Hopf action'' on the right $\mathcal Q(\Gamma)$ module
$\mathcal Q_\sigma(\Gamma)$, i.e.,
\begin{equation}
h(F^1\ast F^2)=\sum h_{(1)}(F^1)\ast h_{(2)}(F^2) \qquad\forall \textrm{ } h\in \mathcal H_1, \textrm{ }F^1\in \mathcal Q_\sigma(\Gamma), \textrm{ }
F^2\in \mathcal Q(\Gamma)
\end{equation} where the coproduct $\Delta:\mathcal H_1\longrightarrow \mathcal H_1\otimes \mathcal H_1$ is given by
$\Delta(h)=\sum h_{(1)}\otimes h_{(2)}$ for any $h\in \mathcal H_1$. We recall from \cite{CM1} that
$\mathcal H_1$ is equal as an algebra to the universal enveloping algebra of the Lie algebra $\mathcal L_1$ with
generators $X$, $Y$, $\{\delta_n\}_{n\geq 1}$ satisfying the following relations:
\begin{equation}
[Y,X]=X \quad [X,\delta_n]=\delta_{n+1}\quad [Y,\delta_n]=n\delta_n\quad [\delta_k,\delta_l]=0\qquad\forall\textrm{ }k,l,n\geq 1
\end{equation} Then, we can consider the smaller Lie algebra $\mathfrak{l}_1\subseteq \mathcal L_1$ with two generators
$X$, $Y$ satisfying $[Y,X]=X$. If we let $\mathfrak{h}_1$ be the Hopf algebra that is the universal enveloping algebra
of $\mathfrak{l}_1$, we show that the pairing in \eqref{1.7m} on $\mathcal Q_\sigma(\Gamma)$ carries a
``Hopf action'' of $\mathfrak{h}_1$. In other words, we have:
\begin{equation}\label{1.10m}
h(F^1,F^2)=\sum (h_{(1)}(F^1),h_{(2)}(F^2))\qquad\forall \textrm{ } h\in \mathfrak{h}_1, \textrm{ }F^1,F^2\in \mathcal Q_\sigma(\Gamma)
\end{equation} where the coproduct $\Delta:\mathfrak{h}_1\longrightarrow \mathfrak{h}_1\otimes \mathfrak{h}_1$ is given by
$\Delta(h)=\sum h_{(1)}\otimes h_{(2)}$ for any $h\in \mathfrak{h}_1$. In Section 4.2, we consider operators
between the modules $\mathcal Q_\sigma(\Gamma)$ as $\sigma$ varies over $SL_2(\mathbb Z)$. More precisely, for any
$\tau, \sigma\in SL_2(\mathbb Z)$, we define a morphism:
\begin{equation}
X_\tau:\mathcal Q_\sigma(\Gamma)\longrightarrow \mathcal Q_{\tau\sigma}(\Gamma)
\end{equation} In particular, this gives us operators acting between the levels of the graded module
\begin{equation}
\mathbb Q_\sigma(\Gamma)=\bigoplus_{m\in \mathbb Z}\mathcal Q_{\sigma(m)}(\Gamma)
\end{equation} where for any $\sigma \in SL_2(\mathbb Z)$, we set $\sigma(m)
=\begin{pmatrix} 1 & m \\ 0 & 1 \\ \end{pmatrix}\cdot \sigma$. Further, we generalize the pairing
on $\mathcal Q_\sigma(\Gamma)$ in \eqref{1.7m} to a pairing:
\begin{equation}\label{1.13m}
(\_\_,\_\_):\mathcal Q_{\tau_1\sigma}(\Gamma)\otimes \mathcal Q_{\tau_2\sigma}(\Gamma)\longrightarrow \mathcal Q_{\tau_1\tau_2\sigma}(\Gamma)
\end{equation} where $\tau_1$, $\tau_2$ are commuting matrices in $SL_2(\mathbb Z)$. In particular, \eqref{1.13m} gives us a pairing
$\mathcal Q_{\sigma(m)}(\Gamma)\otimes \mathcal Q_{\sigma(n)}(\Gamma)\longrightarrow \mathcal Q_{\sigma(m+n)}(\Gamma)$, $\forall$
$m$, $n\in \mathbb Z$ and hence a pairing on the tower $\mathbb Q_\sigma(\Gamma)$. Finally, we consider
the Lie algebra $\mathfrak{l}_{\mathbb Z}\supseteq \mathfrak{l}_1$ with generators $\{Z,X_n|n\in \mathbb Z\}$ satisfying the following
commutator relations:
\begin{equation}
[Z,X_n]=(n+1)X_n\qquad [X_n,X_{n'}]=0\qquad \forall\textrm{ }n,n'\in \mathbb Z
\end{equation} Then, if we let $\mathfrak{h}_{\mathbb Z}$ be the Hopf algebra that is the universal enveloping
algebra of $\mathfrak{l}_{\mathbb Z}$, we show that $\mathfrak{h}_{\mathbb Z}$ has a Hopf action on the
pairing on $\mathbb Q_\sigma(\Gamma)$. In other words, for any $F^1$, $F^2\in \mathbb Q_{\sigma}(\Gamma)$, we have
\begin{equation}\label{last4.48}
h(F^1,F^2)=\sum (h_{(1)}(F^1),h_{(2)}(F^2))\qquad \forall\textrm{ }h\in \mathfrak{h}_{\mathbb Z}
\end{equation} where the coproduct $\Delta:\mathfrak h_{\mathbb Z}
\longrightarrow \mathfrak h_{\mathbb Z}\otimes \mathfrak h_{\mathbb Z}$ is defined
by setting $\Delta(h):=\sum h_{(1)}\otimes h_{(2)}$ for each $h\in \mathfrak h_{\mathbb Z}$.
\medskip
\medskip
\section{The Quasimodular Hecke algebra}
\medskip
\medskip
We begin this section by briefly recalling the notion of quasimodular forms. The notion of quasimodular forms is due to Kaneko and Zagier \cite{KZ}. The theory has been further developed in Zagier \cite{Zag}. For an introduction to the basic theory of quasimodular forms, we refer the reader to the exposition of Royer \cite{Royer}.
\medskip
Throughout, let $\mathbb H\subseteq \mathbb C$ be the upper half plane. Then, there is a well known
action of $SL_2(\mathbb Z)$ on $\mathbb H$:
\begin{equation}
z\mapsto \frac{az+b}{cz+d}\qquad \forall \textrm{ }z\in \mathbb H, \begin{pmatrix} a & b \\
c & d \\ \end{pmatrix} \in SL_2(\mathbb Z)
\end{equation} For any $N\geq 1$, we denote by $\Gamma(N)$ the following principal congruence
subgroup of $SL_2(\mathbb Z)$:
\begin{equation}
\Gamma(N):=\left\{ \left. \begin{pmatrix} a & b \\
c & d \\ \end{pmatrix}\in SL_2(\mathbb Z) \right | \begin{pmatrix} a & b \\
c & d \\ \end{pmatrix} \equiv \begin{pmatrix} 1 & 0 \\
0 & 1 \\ \end{pmatrix} \mbox{($mod$ $N$)} \right\}
\end{equation} In particular, $\Gamma(1)=SL_2(\mathbb Z)$. We are now ready to define
quasimodular forms.
\medskip
\begin{defn}\label{Def2.1} Let $f:\mathbb H\longrightarrow \mathbb C$ be a holomorphic function and let
$N\geq 1$, $k$, $s\geq 0$ be integers. Then, the function $f$ is a quasimodular form of
level $N$, weight $k$ and depth $s$ if there exist holomorphic functions $f_0$, $f_1$, ..., $f_s:
\mathbb H\longrightarrow \mathbb C$ with $f_s\ne 0$ such that:
\begin{equation}\label{2.3}
(cz+d)^{-k}f\left(\frac{az+b}{cz+d}\right)=\sum_{j=0}^s f_j(z)\left(\frac{c}{cz+d}\right)^j
\end{equation} for any matrix $\begin{pmatrix} a & b \\
c & d \\ \end{pmatrix} \in \Gamma(N)$. The collection of quasimodular forms of level $N$, weight $k$ and depth $s$ will be denoted by $\mathcal{QM}_k^s(\Gamma(N))$. By convention, we let the zero function $0\in \mathcal{QM}_k^0(\Gamma(N))$ for every $k\geq 0$, $N\geq 1$.
\end{defn}
\medskip
More generally, for any holomorphic function $f:\mathbb H\longrightarrow \mathbb C$ and any matrix
$\alpha = \begin{pmatrix} a & b \\
c & d \\ \end{pmatrix}\in GL_2^+(\mathbb Q)$, we define:
\begin{equation}
(f\vert_k\alpha)(z):=(cz+d)^{-k}f\left(\frac{az+b}{cz+d}\right) \qquad \forall \textrm{ }k\geq 0
\end{equation} Then, we can say that $f$ is quasimodular of level $N$, weight $k$ and depth $s$
if there exist holomorphic functions $f_0$, $f_1$, ..., $f_s:
\mathbb H\longrightarrow \mathbb C$ with $f_s\ne 0$ such that:
\begin{equation}
(f\vert_k\gamma)(z) = \sum_{j=0}^s f_j(z)\left(\frac{c}{cz+d}\right)^j \qquad\forall\textrm{ }
\gamma = \begin{pmatrix} a & b \\
c & d \\ \end{pmatrix} \in \Gamma(N)
\end{equation} When the integer $k$ is clear from context, we write $f\vert_k\alpha$ simply
as $f\vert \alpha$ for any $\alpha\in GL_2^+(\mathbb Q)$. Also, it is clear that we have a product:
\begin{equation}
\mathcal{QM}^s_k(\Gamma(N))\otimes \mathcal{QM}^t_l(\Gamma(N))\longrightarrow \mathcal{QM}^{s+t}_{k+l}
(\Gamma(N))
\end{equation} on quasi-modular forms. For any $N\geq 1$, we now define:
\begin{equation}
\mathcal{QM}(\Gamma(N)):=\bigoplus_{s=0}^\infty\bigoplus_{k=0}^\infty
\mathcal{QM}_k^s(\Gamma(N))
\end{equation} We now consider the direct limit:
\begin{equation}\label{2.8}
\mathcal{QM}:=\underset{N\geq 1}{\varinjlim} \textrm{ }\mathcal{QM}(\Gamma(N))
\end{equation} which we will refer to as the quasimodular tower. Additionally, for any $k\geq 0$ and $N\geq 1$, we let $\mathcal M_k(\Gamma(N))$ denote the collection of usual modular forms
of weight $k$ and level $N$. Then, we can define the modular tower $\mathcal M$:
\begin{equation}\label{mtower}
\mathcal{M}:=\underset{N\geq 1}{\varinjlim} \textrm{ }\mathcal{M}(\Gamma(N))\qquad
\mathcal{M}(\Gamma(N)):=\bigoplus_{k=0}^\infty
\mathcal{M}_k(\Gamma(N))
\end{equation} We now recall the modular Hecke algebra of Connes and Moscovici \cite{CM1}.
\medskip
\begin{defn}\label{CMdef} (see \cite[$\S$ 1]{CM1}) Let $\Gamma=\Gamma(N)$ be a principal congruence
subgroup of $SL_2(\mathbb Z)$. A modular Hecke operator of level $\Gamma$ is a function
of finite support
\begin{equation}
F:\Gamma\backslash GL_2^+(\mathbb Q)\longrightarrow \mathcal{M} \qquad \Gamma\alpha
\mapsto F_\alpha
\end{equation} such that for any $\gamma\in \Gamma$, we have:
\begin{equation}\label{tt2.11}
F_{\alpha\gamma}=F_\alpha|\gamma
\end{equation} The collection of all modular Hecke operators of level $\Gamma$ will be denoted
by $\mathcal A(\Gamma)$.
\end{defn}
\medskip
Our first aim is to define a quasimodular Hecke algebra $\mathcal Q(\Gamma)$ analogous to the modular Hecke algebra $\mathcal A(\Gamma)$ of Connes and Moscovici. For this, we recall the structure
theorem for quasimodular forms, proved by Kaneko and Zagier \cite{KZ}.
\medskip
\begin{Thm}\label{Th2.1} (see \cite[$\S$ 1, Proposition 1.]{KZ}) Let $\Gamma=\Gamma(N)$ be a principal congruence
subgroup of $SL_2(\mathbb Z)$. For any even number $K\geq 2$, let $G_K$ denote the classical Eisenstein series of weight $K$:
\begin{equation}\label{2.12ov}
G_K(z):=-\frac{B_K}{2K}+\sum_{n=1}^\infty\left(\sum_{d|n}d^{K-1}\right)e^{2\pi inz}
\end{equation} where $B_K$ is the $K$-th Bernoulli number and $z\in \mathbb H$.
Then, every quasimodular form in $\mathcal{QM}(\Gamma)$ can be written uniquely as a polynomial in $G_2$ with coefficients in
$\mathcal M(\Gamma)$. More precisely, for any quasimodular form $f\in \mathcal{QM}^s_k(\Gamma)$, there exist functions $a_0(f)$, $a_1(f)$, ..., $a_s(f)$ such that:
\begin{equation}
f=\underset{i=0}{\overset{s}{\sum}} a_i(f)G_2^i
\end{equation} where $a_i(f)\in \mathcal M_{k-2i}(\Gamma)$ is a modular form of weight $k-2i$ and level $\Gamma$ for each
$0\leq i\leq s$.
\end{Thm}
\medskip
We now consider a quasimodular form $f\in \mathcal{QM}$. For sake of definiteness, we may assume
that $f\in \mathcal{QM}^s_k(\Gamma(N))$, i.e. $f$ is a quasimodular form
of level $N$, weight $k$ and depth $s$. We now define an operation on $\mathcal{QM}$ by setting:
\begin{equation}\label{t2.11}
f||\alpha = \underset{i=0}{\overset{s}{\sum}} (a_i(f)|_{k-2i}\alpha ) G_2^i \qquad \forall \textrm{ }\alpha\in GL_2^+(\mathbb Q)
\end{equation} where $\{a_i(f)\in \mathcal M_{k-2i}(\Gamma(N))\}_{0\leq i\leq s}$ is the collection of modular forms determining
$f=\sum_{i=0}^s a_i(f)G_2^i$ as in Theorem \ref{Th2.1}. We know that for any $\alpha
\in GL_2^+(\mathbb Q)$, each $(a_i(f)|_{k-2i}\alpha)$ is an element of the modular tower $\mathcal M$. This shows that $f||\alpha = \underset{i=0}{\overset{s}{\sum}} (a_i(f)|_{k-2i}\alpha ) G_2^i\in \mathcal{QM}$. However, we note that for arbitrary $\alpha\in GL_2^+(\mathbb Q)$ and
$a_i(f)\in \mathcal M_{k-2i}(\Gamma(N))$, it is not necessary that $(a_i(f)|_{k-2i}\alpha )
\in \mathcal M_{k-2i}(\Gamma(N))$. In other words, the operation defined in \eqref{t2.11} on the quasimodular tower $\mathcal{QM}$ does not descend to an endomorphism on each
$\mathcal{QM}^s_k(\Gamma(N))$. From the expression in \eqref{t2.11}, it is also clear that:
\begin{equation}\label{tv2.11}
(f\cdot g)||\alpha = (f||\alpha)\cdot (g||\alpha) \qquad f||(\alpha\cdot \beta)=(f||\alpha)||\beta \qquad
\forall\textrm{ }f,g\in \mathcal{QM}, \textrm{ }\alpha,\beta\in GL_2^+(\mathbb Q)
\end{equation} We are now ready to define
the quasimodular Hecke operators.
\medskip
\begin{defn}\label{maindef} Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup. A quasimodular Hecke
operator of level $\Gamma$ is a function of finite support:
\begin{equation}
F:\Gamma\backslash GL_2^+(\mathbb Q)\longrightarrow \mathcal{QM} \qquad \Gamma\alpha
\mapsto F_\alpha
\end{equation} such that for any $\gamma\in \Gamma$, we have:
\begin{equation}\label{tt2.16}
F_{\alpha\gamma}=F_\alpha||\gamma
\end{equation} The collection of all quasimodular Hecke operators of level $\Gamma$ will be denoted
by $\mathcal Q(\Gamma)$.
\end{defn}
\medskip
We will now introduce the product structure on $\mathcal Q(\Gamma)$. In fact, we will introduce
two separate product structures $(\mathcal Q(\Gamma),\ast)$ and $(\mathcal Q(\Gamma),\ast^r)$ on
$\mathcal Q(\Gamma)$.
\medskip
\begin{thm}(a) Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup and let $\mathcal Q(\Gamma)$
be the collection of quasimodular Hecke operators of level $\Gamma$. Then, the product defined
by:
\begin{equation}\label{2.11}
(F\ast G)_\alpha:=\sum_{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}F_{\beta}\cdot (G_{\alpha\beta^{-1}}||\beta)\qquad \forall\textrm{ }\alpha\in GL_2^+(\mathbb Q)
\end{equation} for all $F$, $G\in \mathcal Q(\Gamma)$ makes $\mathcal Q(\Gamma)$ into an associative algebra.
\medskip
(b) Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup and let $\mathcal Q(\Gamma)$
be the collection of quasimodular Hecke operators of level $\Gamma$. Then, the product defined
by:
\begin{equation}\label{2.11vv}
(F\ast^r G)_\alpha:=\sum_{\beta\in \Gamma\backslash SL_2(\mathbb Z)}F_{\beta}\cdot (G_{\alpha\beta^{-1}}||\beta)\qquad \forall\textrm{ }\alpha\in GL_2^+(\mathbb Q)
\end{equation} for all $F$, $G\in \mathcal Q(\Gamma)$ makes $\mathcal Q(\Gamma)$ into an associative algebra which we denote by $\mathcal Q^r(\Gamma)$.
\end{thm}
\begin{proof} (a) We need to check that the product in \eqref{2.11} is associative. First of all, we note
that the expression in \eqref{2.11} can be rewritten as:
\begin{equation}\label{2.12}
(F\ast G)_\alpha = \sum_{\alpha_2\alpha_1=\alpha} F_{\alpha_1}\cdot G_{\alpha_2}||\alpha_1
\qquad\forall\textrm{ }\alpha\in GL_2^+(\mathbb Q)
\end{equation} where the sum in \eqref{2.12} is taken over all pairs $(\alpha_1,\alpha_2)$ with $\alpha_2\alpha_1=\alpha$ modulo the following equivalence relation:
\begin{equation}
(\alpha_1,\alpha_2)\sim (\gamma\alpha_1,\alpha_2\gamma^{-1}) \qquad\forall\textrm{ }
\gamma\in \Gamma
\end{equation} Hence, for $F$, $G$, $H\in \mathcal Q(\Gamma)$, we can write:
\begin{equation}\label{2.14}
\begin{array}{ll}
(F\ast (G\ast H))_\alpha & =\sum_{\alpha'_2\alpha_1=\alpha}F_{\alpha_1}\cdot (G\ast H)_{\alpha'_2}||\alpha_1 \\
& =\sum_{\alpha'_2\alpha_1=\alpha}F_{\alpha_1}\cdot (\sum_{\alpha_3\alpha_2=\alpha'_2}
G_{\alpha_2}\cdot H_{\alpha_3}||\alpha_2)||\alpha_1 \\
& =\sum_{\alpha_3\alpha_2\alpha_1=\alpha}F_{\alpha_1}\cdot (G_{\alpha_2}||\alpha_1)\cdot
(H_{\alpha_3}||\alpha_2\alpha_1)\\
\end{array}
\end{equation} where the sum in \eqref{2.14} is taken over all triples $(\alpha_1,\alpha_2,\alpha_3)$ with $\alpha_3\alpha_2\alpha_1=\alpha$ modulo the following equivalence relation:
\begin{equation}\label{2.15}
(\alpha_1,\alpha_2,\alpha_3)\sim (\gamma\alpha_1,\gamma'\alpha_2\gamma^{-1},\alpha_3\gamma'^{-1})\qquad \forall\textrm{ }\gamma,\gamma'\in \Gamma
\end{equation} On the other hand, we have
\begin{equation}\label{2.16}
\begin{array}{ll}
((F\ast G)\ast H)_\alpha & =\sum_{\alpha_3\alpha''_2=\alpha}(F\ast G)_{\alpha''_2}\cdot
H_{\alpha_3}||\alpha''_2 \\
& =\sum_{\alpha_3\alpha''_2=\alpha} (\sum_{\alpha_2\alpha_1=\alpha''_2} F_{\alpha_1}
\cdot G_{\alpha_2}||\alpha_1)\cdot H_{\alpha_3}||\alpha''_2 \\
&=\sum_{\alpha_3\alpha_2\alpha_1=\alpha}F_{\alpha_1}\cdot (G_{\alpha_2}||\alpha_1)
\cdot (H_{\alpha_3}||\alpha_2\alpha_1) \\
\end{array}
\end{equation} where the sum in \eqref{2.16} is taken over all triples $(\alpha_1,\alpha_2,\alpha_3)$ with $\alpha_3\alpha_2\alpha_1=\alpha$ modulo the equivalence relation in \eqref{2.15}. From
\eqref{2.14} and \eqref{2.16} the result follows.
\end{proof}
\medskip
We know that modular forms are quasimodular forms of depth $0$, i.e., for any $k\geq 0$, $N\geq 1$,
we have $\mathcal M_k(\Gamma(N))=\mathcal{QM}_k^0(\Gamma(N))$. It follows that the modular
tower $\mathcal M$ defined in \eqref{mtower} embeds into the quasimodular tower $\mathcal{QM}$
defined in \eqref{2.8}. We are now ready to show that the modular Hecke algebra
$\mathcal A(\Gamma)$ of Connes and
Moscovici embeds into the quasimodular Hecke algebra $\mathcal Q(\Gamma)$ for
any congruence subgroup $\Gamma=\Gamma(N)$.
\medskip
\begin{thm} Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of $SL_2(\mathbb Z)$.
Let $\mathcal A(\Gamma)$ be the modular Hecke algebra of level $\Gamma$ as defined in
Definition \ref{CMdef} and let $\mathcal Q(\Gamma)$ be the quasimodular Hecke algebra
of level $\Gamma$ as defined in Definition \ref{maindef}.
Then, there is a natural embedding of algebras $\mathcal A(\Gamma)\hookrightarrow
\mathcal Q(\Gamma)$.
\end{thm}
\begin{proof} For any $\alpha\in GL_2^+(\mathbb Q)$ and any $f\in \mathcal{QM}^s_k(\Gamma)$, we
consider the operation $f\mapsto f||\alpha$ as defined in \eqref{t2.11}:
\begin{equation}\label{2.24}
f||\alpha=\underset{i=0}{\overset{s}{\sum}}(a_i(f)|_{k-2i}\alpha)G_2^i \in \mathcal{QM}
\end{equation} In particular, if $f\in \mathcal M_k(\Gamma)=\mathcal{QM}^0_k(\Gamma)$ is a modular form, it follows
from \eqref{2.24} that:
\begin{equation}\label{2.25}
f||\alpha = a_0(f)|_k\alpha = f|_k\alpha = f|\alpha \in \mathcal M
\end{equation} Hence, using the embedding of $\mathcal M$ in $\mathcal{QM}$, it follows from
\eqref{tt2.11} in the definition of $\mathcal A(\Gamma)$ and from \eqref{tt2.16} in the definition
of $\mathcal Q(\Gamma)$ that we have an embedding $\mathcal A(\Gamma)\hookrightarrow
\mathcal Q(\Gamma)$ of modules. Further, we recall from \cite[$\S$ 1]{CM1} that the product on
$\mathcal A(\Gamma)$ is given by:
\begin{equation}\label{kittyS}
(F\ast G)_\alpha:=\sum_{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}F_{\beta}\cdot (G_{\alpha\beta^{-1}}|\beta)\qquad \forall\textrm{ }\alpha\in GL_2^+(\mathbb Q), \textrm{ }F,G\in \mathcal A(\Gamma)
\end{equation} Comparing \eqref{kittyS} with the product on $\mathcal Q(\Gamma)$ described in \eqref{2.11}
and using \eqref{2.25} it follows that $\mathcal A(\Gamma)\hookrightarrow
\mathcal Q(\Gamma)$ is an embedding of algebras.
\end{proof}
\medskip
We end this section by describing the action of the algebra $\mathcal Q(\Gamma)$ on
$\mathcal{QM}(\Gamma)$.
\begin{thm} Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup and let $\mathcal Q(\Gamma)$
be the algebra of quasimodular Hecke operators of level $\Gamma$. Then, for any element
$f\in \mathcal{QM}(\Gamma)$ the action of $\mathcal Q(\Gamma)$ defined by:
\begin{equation}\label{2.28}
F\ast f:=\sum_{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}F_\beta \cdot f||\beta\qquad
\forall\textrm{ }F\in \mathcal Q(\Gamma)
\end{equation} makes $\mathcal{QM}(\Gamma)$ into a left module over $\mathcal Q(\Gamma)$.
\end{thm}
\begin{proof} It is easy to check that the right hand side of \eqref{2.28} is independent of the choice
of coset representatives. Further, since $F\in \mathcal Q(\Gamma)$ is a function of finite support, we can choose finitely many coset representatives $\{\beta_1,\beta_2,...,\beta_n\}$ such that
\begin{equation}\label{tv2.29}
F\ast f=\sum_{j=1}^n F_{\beta_j}\cdot f||\beta_j
\end{equation} It suffices to consider the case $f\in \mathcal{QM}_k^s(\Gamma)$ for some
weight $k$ and depth $s$. Then, we can express $f$ as a sum:
\begin{equation}
f=\sum_{i=0}^s a_i(f)G_2^i
\end{equation} where each $a_i(f)\in \mathcal M_{k-2i}(\Gamma)$. Similarly, for any $\beta\in GL_2^+(\mathbb Q)$, we can express $F_{\beta}$ as a finite sum:
\begin{equation}
F_{\beta}=\sum_{r=0}^{t_\beta}a_{\beta r}(F_{\beta})\cdot G_2^r
\end{equation} with each $a_{\beta r}(F_{\beta})\in \mathcal M$. In particular, we let $t=max\{t_{\beta_1},t_{\beta_2},...,t_{\beta_n}\}$ and we can now write:
\begin{equation}
F_{\beta_j}=\sum_{r=0}^ta_{\beta_j r}(F_{\beta_j})\cdot G_2^r
\end{equation}by adding appropriately many terms with zero coefficients in the expression for each
$F_{\beta_j}$. Further, for any $\gamma\in \Gamma$, we know that $F_{\beta_j\gamma}
=F_{\beta_j}||\gamma=\sum_{r=0}^t(a_{\beta_jr}(F_{\beta_j})|\gamma)\cdot G_2^r$. In other words, we have, for each $j$:
\begin{equation}\label{2.33cv}
F_{\beta_j\gamma}=\sum_{r=0}^ta_{\beta_j\gamma r}(F_{\beta_j\gamma})\cdot G_2^r \qquad
a_{\beta_j\gamma r}(F_{\beta_j\gamma})=(a_{\beta_jr}(F_{\beta_j})|\gamma)
\end{equation} The sum in \eqref{tv2.29} can now be expressed as:
\begin{equation}
F\ast f:=\sum_{j=1}^nF_{\beta_j} \cdot f||\beta_j
=\sum_{i=0}^s\sum_{r=0}^{t}\sum_{j=1}^na_{\beta_jr}(F_{\beta_j})\cdot (a_i(f)|\beta_j)\cdot G_2^{r+i}\end{equation} For any $i$, $r$, we now set:
\begin{equation}\label{2.35cv}
A_{ir}(F,f):=\sum_{j=1}^na_{\beta_jr}(F_{\beta_j})\cdot (a_i(f)|\beta_j)\end{equation} Again, it is easy to see that the sum $A_{ir}(F,f)$ in \eqref{2.35cv} does not depend on the choice of the coset representatives $\{\beta_1,\beta_2,...,\beta_n\}$. Then, for any $\gamma\in \Gamma$, we have:
\begin{equation}\label{2.36}
A_{ir}(F,f)|\gamma=\sum_{j=1}^n (a_{\beta_jr}(F_{\beta_j})|\gamma)\cdot (a_i(f)|\beta_j\gamma)
=\sum_{j=1}^n a_{\beta_j\gamma r}(F_{\beta_j\gamma}) \cdot (a_i(f)|\beta_j\gamma)=A_{ir}(F,f)
\end{equation} where the last equality in \eqref{2.36} follows from the fact that
$\{\beta_1\gamma,\beta_2\gamma,...,\beta_n\gamma\}$ is another collection of distinct cosets reprsentatives of
$\Gamma$ in $GL_2^+(\mathbb Q)$. From \eqref{2.36}, we note that each $A_{ir}(F,f)\in \mathcal M(\Gamma)$. Then, the sum:
\begin{equation}
F\ast f=\sum_{i=0}^s\sum_{r=0}^tA_{ir}(F,f)\cdot G_2^{i+r}
\end{equation} is an element of $\mathcal{QM}(\Gamma)$. Hence, $\mathcal{QM}(\Gamma)$ is a left module over $\mathcal Q(\Gamma)$.
\end{proof}
\medskip
\medskip
\section{The Lie algebra and Hopf algebra actions on $\bf \mathcal Q(\Gamma)$}
\medskip
\medskip
Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of $SL_2(\mathbb Z)$.
In this section, we will describe two different sets of operators on the collection $\mathcal Q(\Gamma)$ of quasimodular Hecke operators
of level $\Gamma$. Given a quasimodular form $f\in \mathcal{QM}(\Gamma)$ of level $\Gamma$, we have mentioned
in the last section that $f$ can be expressed as a finite sum:
\begin{equation}\label{3.1ner}
f=\sum_{i=0}^sa_i(f)\cdot G_2^i
\end{equation} where $G_2$ is the classical Eisenstein series of weight $2$ and each $a_i(f)$ is a modular
form of level $\Gamma$. Then in Section 3.1, we consider operators on the quasimodular tower that
act on the powers of $G_2$ appearing in \eqref{3.1ner}. These induce operators $D$, $\{T_k^l\}_{k\geq 1,l\geq 0}$
on the collection $\mathcal Q(\Gamma)$ of quasimodular Hecke operators
of level $\Gamma$. In order to understand the action of these operators on products of elements
in $\mathcal Q(\Gamma)$, we also need to define extra operators $\{\phi^{(m)}\}_{m\geq 1}$. Finally, we show
that these operators may all be described in terms of a Hopf algebra $\mathcal H$ with a ``Hopf action''
on $\mathcal Q^r(\Gamma)$, i.e.,
\begin{equation}
h(F^1\ast^r F^2)=\sum h_{(1)}(F^1)\ast^r h_{(2)}(F^2) \qquad\forall \textrm{ } h\in \mathcal H, \textrm{ }F^1,F^2\in \mathcal Q^r(\Gamma)
\end{equation} where the coproduct $\Delta:\mathcal H\longrightarrow \mathcal H\otimes \mathcal H$ is given by
$\Delta(h)=\sum h_{(1)}\otimes h_{(2)}$ for any $h\in \mathcal H$. In Section 3.2, we consider operators on the quasimodular tower
$\mathcal{QM}$ that act on the modular coefficients $a_i(f)$ appearing in \eqref{3.1ner}. These induce on $\mathcal Q(\Gamma)$
analogues of operators acting on the modular Hecke algebra $\mathcal A(\Gamma)$ of Connes and Moscovici \cite{CM1}. Then, we
show that $\mathcal Q(\Gamma)$ carries a Hopf action of the same Hopf algebra $\mathcal H_1$ of codimension
$1$ foliations that acts on $\mathcal A(\Gamma)$.
\medskip
\medskip
\subsection{The operators $\bf D$, $\bf \{T^l_k\}$ and $\bf \{\phi^{(m)}\}$ on $\mathcal Q(\Gamma)$}
\medskip
\medskip
For any even number $K\geq 2$, let $G_K$ be the classical Eisenstein series of weight $K$ as in
\eqref{2.12ov}. Since $G_2$ is a quasimodular form, i.e., $G_2\in \mathcal{QM}$, its derivative $G_2'\in \mathcal{QM}$. Further, it is well known that:
\begin{equation}\label{3.1}
G_2'=\frac{5\pi i}{3}G_4 - 4\pi iG_2^2
\end{equation} where $G_4$ is the Eisenstein series of weight $4$ (which is a modular form). For our
purposes, it will be convenient to write:
\begin{equation}
G_2'=\sum_{j=0}^2g_jG_2^j
\end{equation} with each $g_j$ a modular form. From \eqref{3.1}, it follows that:
\begin{equation}\label{3.3}
g_0=\frac{5\pi i}{3}G_4 \qquad g_1=0 \qquad g_2=-4\pi i
\end{equation} We are now ready to define the operators $D$ and $\{W_k\}_{k\geq 1}$ on $\mathcal{QM}$. The first operator $D$ differentiates
the powers of $G_2$: \begin{equation}\label{3.4}
\begin{array}{c}
D:\mathcal{QM}\longrightarrow \mathcal{QM} \\
\begin{array}{ll}
f=\underset{i=0}{\overset{s}{\sum}}a_i(f)G_2^i \mapsto & -\frac{1}{8\pi i}\left( \underset{i=0}{\overset{s}{\sum}}ia_i(f)G_2^{i-1}\cdot G_2' \right)\\
& = -\frac{1}{8\pi i}\underset{i=0}{\overset{s}{\sum}}\underset{j=0}{\overset{2}{\sum}} ia_i(f)g_jG_2^{i+j-1}
\end{array}
\end{array}
\end{equation} The operators $\{W_k\}_{k\geq 1}$ are ``weight operators'' and $W_k$ also steps up the power of $G_2$ by $k-2$. We set:
\begin{equation}\label{3.5x}
W_k:\mathcal{QM}\longrightarrow \mathcal{QM} \qquad
f=\underset{i=0}{\overset{s}{\sum}}a_i(f)G_2^i \mapsto \underset{i=0}{\overset{s}{\sum}}ia_i(f)G_2^{i+k-2}
\end{equation} From the definitions in \eqref{3.4} and \eqref{3.5x}, we can easily check that $D$ and
$W_k$ are derivations on $\mathcal{QM}$.
Finally, for any $\alpha\in GL_2^+(\mathbb Q)$ and any integer $m\geq 1$, we set
\begin{equation}\label{nu}
\nu_\alpha^{(m)}=-\frac{5}{24}\left( G_4^m|\alpha - G_4^m\right)
\end{equation}
\medskip
\begin{lem}\label{L3.1} (a) Let $f\in \mathcal{QM}$ be an element of the quasimodular tower and
$\alpha\in GL_2^+(\mathbb Q)$. Then, the operator $D$ satisfies:
\begin{equation}\label{3.6}
D(f)||\alpha=D(f||\alpha) + \nu_\alpha^{(1)}\cdot (W_1(f)||\alpha)
\end{equation} where, using \eqref{nu}, we know that $\nu_\alpha^{(1)}$ is given by:
\begin{equation}\label{3.7}
\nu_\alpha^{(1)}:= -\frac{1}{8\pi i}(g_0|\alpha - g_0)=-\frac{5}{24}\left( G_4|\alpha - G_4\right)\qquad \forall\textrm{ }
\alpha\in GL_2^+(\mathbb Q)
\end{equation}
\medskip
(b) For $f\in\mathcal{QM}$ and $\alpha\in GL_2^+(\mathbb Q)$, each operator $W_k$, $k\geq 1$ satisfies:
\begin{equation}\label{3.8}
W_k(f)||\alpha=W_k(f||\alpha)
\end{equation}
\end{lem}
\begin{proof} We start by proving part (a). For the sake of definiteness, we assume that $ f=\underset{i=0}{\overset{s}{\sum}}a_i(f)G_2^i$
with each $a_i(f)\in \mathcal M$. For $\alpha\in GL_2^+(\mathbb Q)$, it follows from \eqref{3.4} that:
\begin{equation}\label{3.8z}
\begin{array}{lllll}
D(f)||\alpha & = -\frac{1}{8\pi i}\left(\underset{i}{\sum}\underset{j}{\sum} ia_i(f)g_jG_2^{i+j-1}\right)||\alpha & \textrm{ }& D(f||\alpha) & = D\left(\underset{i}{\sum} (a_i(f)|\alpha)G_2^i \right)\\
& = -\frac{1}{8\pi i} \underset{i}{\sum}\underset{j}{\sum}\textrm{ } i(a_i(f)|\alpha)(g_j|\alpha)G_2^{i+j-1} & & & = -\frac{1}{8\pi i}\underset{i}{\sum}
\underset{j}{\sum}\textrm{ }i(a_i(f)|\alpha)g_jG_2^{i+j-1} \end{array}
\end{equation} From \eqref{3.8z} it follows that:
\begin{equation}\label{3.9}
D(f)||\alpha - D(f||\alpha)= -\frac{1}{8\pi i}\sum_{i=0}^s\sum_{j=0}^2 \textrm{ }i(a_i(f)|\alpha)(g_j|\alpha - g_j)G_2^{i+j-1}
\end{equation} From \eqref{3.3}, it is clear that $g_j|\alpha - g_j=0$ for $j=1$ and $j=2$. It follows that:
\begin{equation*}
D(f)||\alpha - D(f||\alpha)= -\frac{1}{8\pi i}\sum_{i=0}^s\textrm{ }i(a_i(f)|\alpha)(g_0|\alpha - g_0)G_2^{i-1}= -\frac{1}{8\pi i}(g_0|\alpha
- g_0)\cdot \left (\sum_{i=0}^s\textrm{ }i(a_i(f)|\alpha) G_2^{i-1}\right)
\end{equation*} This proves the result of (a). The result of part (b) is clear from the definition in \eqref{3.5x}.
\end{proof}
\medskip
We note here that it follows from \eqref{nu} that for any $\alpha$, $\beta\in GL_2^+(\mathbb Q)$, we have:
\begin{equation}\label{3.12}
\nu_{\alpha\beta}^{(m)}=\nu_{\alpha}^{(m)}|\beta + \nu_\beta^{(m)} \qquad \forall \textrm{ }m\geq 1
\end{equation} Additionally, since each $G_4^m$ is a modular form, we know that when $\alpha\in SL_2(\mathbb Z)$:
\begin{equation}\label{3.125}
\nu_\alpha^{(m)}=-\frac{5}{24}(G_4^m|\alpha - G_4^m) = 0 \qquad \forall\textrm{ }
\alpha\in SL_2(\mathbb Z), m\geq 1
\end{equation} Moreover, from the definitions in \eqref{3.4} and \eqref{3.5x} respectively, it is easily verified that $D$ and $\{W_k\}_{k\geq 1}$
are derivations on the quasimodular tower $\mathcal{QM}$. We now proceed to define operators
on the quasimodular Hecke algebra $\mathcal Q(\Gamma)$ for some principal congruence subgroup
$\Gamma=\Gamma(N)$. Choose $F\in \mathcal Q(\Gamma)$. We set:
\begin{equation}\label{3.13}
\begin{array}{c}
D,W_k,\phi^{(m)}: \mathcal Q(\Gamma)\longrightarrow \mathcal Q(\Gamma) \quad k\geq 1, m\geq 1\\
D(F)_\alpha: = D(F_\alpha) \quad W_k(F)_\alpha :=W_k(F_\alpha) \quad \phi^{(m)}(F)_\alpha:=\nu_\alpha^{(m)}
\cdot F_\alpha \qquad \forall \textrm{ }\alpha\in GL_2^+(\mathbb Q) \\
\end{array}
\end{equation} From Lemma \ref{L3.1} and the properties of $\nu_\alpha^{(m)}$ described in
\eqref{3.12} and \eqref{3.125}, it may be easily verified that the operators
$D$, $W_k$ and $\phi^{(m)}$ in \eqref{3.13} are well defined on $\mathcal Q(\Gamma)$.
We will now compute the commutators of the operators $D$, $\{W_k\}_{k\geq 1}$ and $\{\phi^{(m)}\}_{m\geq 1}$ on
$\mathcal Q(\Gamma)$. In order to describe these commutators, we need one more operator $E$:
\begin{equation}\label{3.19T}
E:\mathcal{QM}\longrightarrow \mathcal{QM}\qquad f\mapsto G_4\cdot f
\end{equation} Since $G_4$ is a modular form of level $\Gamma(1)=SL_2(\mathbb Z)$, i.e.,
$G_4|\gamma=G_4$ for any $\gamma\in SL_2(\mathbb Z)$, it is clear that $E$ induces a
well defined operator on $\mathcal Q(\Gamma)$:
\begin{equation}
E:\mathcal Q(\Gamma)\longrightarrow \mathcal Q(\Gamma) \qquad E(F)_\alpha:=E(F_\alpha)=G_4\cdot F_\alpha\quad \forall\textrm{ }F\in
\mathcal Q(\Gamma), \alpha\in GL_2^+(\mathbb Q)
\end{equation} We will now describe the commutator relations between the operators $D$, $E$, $\{E^lW_k\}_{k\geq 1,l\geq 0}$ and $\{\phi^{(m)}\}_{m\geq 1}$ on $\mathcal Q(\Gamma)$.
\medskip
\begin{thm}\label{P3.3} Let $\Gamma =\Gamma(N)$ be a principal congruence subgroup and let $\mathcal Q(\Gamma)$ be the algebra of quasimodular Hecke operators of level $\Gamma$. The operators $D$, $E$, $\{E^lW_k\}_{k\geq 1,l\geq 0}$ and $\{\phi^{(m)}\}_{m\geq 1}$ on $\mathcal Q(\Gamma)$ satisfy the following relations:
\begin{equation}\label{3.19}
\begin{array}{c}
[E,E^lW_k]=0 \quad [E,D]=0 \quad [E,\phi^{(m)}]=0 \quad [D,\phi^{(m)}]=0 \quad [W_k,\phi^{(m)}]=0 \quad [\phi^{(m)},
\phi^{(m')}]=0 \\
\mbox{$[E^lW_k,D]=\frac{5}{24}(k-1)(E^{l+1}W_{k-1})- \frac{1}{2}(k-3)E^lW_{k+1}$}\\
\end{array}
\end{equation}
\end{thm}
\begin{proof} For any $F\in \mathcal Q(\Gamma)$ and any $\alpha\in GL_2^+(\mathbb Q)$, by definition, we know that $D(F)_\alpha=D(F_\alpha)$, $W_k(F)_\alpha=W_k(F_\alpha)$ and
$E(F)_\alpha=E(F_\alpha)$. Hence, in order to prove that $[E,W_k]=0$ and
$[E,D]=0$, it suffices to show that $[E,W_k](f)=0$ and $[E,D](f)=0$ respectively for any element $f\in
\mathcal{QM}$. Both of these are easily verified from the definitions of $D$ and $W_k$ in
\eqref{3.4} and \eqref{3.5x} respectively. Further, since $[E,W_k]=0$, it is clear that
$[E,E^lW_k]=0$.
\medskip Similarly, in order to prove the expression for $[E^lW_k,D]$, it suffices to prove that:
\begin{equation}
[E^lW_k,D](f) = \frac{5}{24}(k-1)(E^{l+1}W_{k-1})(f)- \frac{1}{2}(k-3)E^lW_{k+1}(f)
\end{equation} for any $f\in \mathcal{QM}$. Further, it suffices to consider the case where $f=\sum_{i=0}^sa_i(f)G_2^i$ where the
$a_i(f)\in \mathcal M$. We now have:
\begin{equation}\label{3.20}
\begin{array}{c}
W_kD(f) = -\frac{1}{8\pi i}W_k\left(\underset{i=0}{\overset{s}{\sum}}\underset{j=0}{\overset{2}{\sum}} ia_i(f)g_jG_2^{i+j-1} \right) = -\frac{1}{8\pi i}\underset{i=0}{\overset{s}{\sum}}\underset{j=0}{\overset{2}{\sum}} i(i+j-1)a_i(f)g_jG_2^{i+j+k-3}\\
DW_k(f)=D\left( \underset{i=0}{\overset{s}{\sum}}ia_i(f)G_2^{i+k-2}\right)= -\frac{1}{8\pi i}\underset{i=0}{\overset{s}{\sum}}\underset{j=0}{\overset{2}{\sum}} i(i+k-2)a_i(f)g_jG_2^{i+j+k-3}\\
\end{array}
\end{equation} It follows from \eqref{3.20} that:
\begin{equation*}\label{3.21}
\begin{array}{ll}
[W_k,D](f)&=-\frac{1}{8\pi i}\underset{i=0}{\overset{s}{\sum}}\underset{j=0}{\overset{2}{\sum}} ija_i(f)g_jG_2^{i+j+k-3} +\frac{1}{8\pi i}\underset{i=0}{\overset{s}{\sum}}\underset{j=0}{\overset{2}{\sum}} i(k-1)a_i(f)g_jG_2^{i+j+k-3} \\ & =-\frac{2g_2}{8\pi i}\underset{i=0}{\overset{s}{\sum}}ia_i(f)G_2^{i+k-1}+(k-1)\frac{1}{8\pi i}\underset{i=0}{\overset{s}{\sum}}ia_i(f)g_0G_2^{i+k-3}
+(k-1)\frac{g_2}{8\pi i}\underset{i=0}{\overset{s}{\sum}}ia_i(f)G_2^{i+k-1} \\
\end{array}
\end{equation*} where the second equality uses the fact that $g_1=0$. Further, since $g_0=\frac{5\pi i}{3}G_4$ and $g_2=-4\pi i$, it follows from \eqref{3.21} that we have:
\begin{equation}\label{3.24cv}
\begin{array}{l}
[W_k,D](f)=\frac{5}{24}(k-1)\underset{i=0}{\overset{s}{\sum}}iG_4a_i(f)G_2^{i+k-3}
-\frac{1}{2}(k-3)\underset{i=0}{\overset{s}{\sum}}ia_i(f)G_2^{i+k-1} \\
=\frac{5}{24}(k-1)(EW_{k-1})(f) - \frac{1}{2}(k-3)W_{k+1}(f)\\
\end{array}
\end{equation} Finally, since $E$ commutes with $\{W_k\}_{k\geq 1}$ and $D$, it follows
from \eqref{3.24cv} that:
\begin{equation}
[E^lW_k,D]=\frac{5}{24}(k-1)(E^{l+1}W_{k-1})- \frac{1}{2}(k-3)E^lW_{k+1} \qquad \forall\textrm{ }k\geq 1,l\geq 0
\end{equation} as operators on $\mathcal Q(\Gamma)$. Finally, it may be easily verified from the definitions that
$[E,\phi^{(m)}]= [D,\phi^{(m)}]= [W_k,\phi^{(m)}]=0$.
\end{proof}
\medskip The operators $\{E^lW_k\}_{k\geq 1,l\geq 0}$ appearing in Proposition \ref{P3.3} above can be described
more succintly as:
\begin{equation}
T^l_k:\mathcal{QM}\longrightarrow \mathcal{QM} \qquad T^l_k:=E^lW_k \qquad \forall\textrm{ }k\geq 1, l\geq 0
\end{equation} and
\begin{equation}
T^l_k:\mathcal{Q}(\Gamma)\longrightarrow \mathcal{Q}(\Gamma) \qquad T^l_k(F)_\alpha:=T^l_k(F_\alpha)=E^lW_k(F_\alpha)
\qquad \forall\textrm{ }F\in \mathcal Q(\Gamma),\alpha\in GL_2^+(\mathbb Q)
\end{equation}
We are now ready to describe the Lie algebra action on $\mathcal Q(\Gamma)$.
\medskip
\begin{thm}\label{excp} Let $\mathcal L$ be the Lie algebra generated by the symbols $D$, $\{T^l_k\}_{k\geq 1, l\geq 0}$,
$\{\phi^{(m)}\}_{m\geq 1}$ along with the following relations between the commutators: \begin{equation}\label{3.27}
\begin{array}{c}
[T_k^l,T_{k'}^{l'}]=(k'-k)T_{k+k'-2}^{l+l'}\\
\mbox{$[D,\phi^{(m)}]=0$} \qquad [T^l_k,\phi^{(m)}]=0 \qquad [\phi^{(m)},
\phi^{(m')}]=0 \\
\mbox{$[T^l_k,D]$}= \frac{5}{24}(k-1)T_{k-1}^{l+1}- \frac{1}{2}(k-3)T^l_{k+1}\\
\end{array}
\end{equation} Then, for any principal congruence subgroup $\Gamma=\Gamma(N)$, we have a Lie action of $\mathcal L$ on
the algebra of quasimodular Hecke operators $\mathcal Q(\Gamma)$ of level $\Gamma$.
\end{thm}
\begin{proof} For any $k\geq 1$ and $l\geq 0$, $T^l_k$ has been defined to be the operator $E^lW_k$ on $\mathcal Q(\Gamma)$.
We want to verify that:
\begin{equation}\label{3.27EX}
[T_k^l,T_{k'}^{l'}]=(k-k')T_{k+k'-2}^{l+l'}\qquad\forall\textrm{ }k,k'\geq 1,\textrm{ }l,l'\geq 0
\end{equation} As in the proof of Proposition \ref{P3.3}, it suffices to show that the relation in \eqref{3.27EX}
holds for any $f\in \mathcal{QM}$. As before, we let $f=\sum_{i=0}^sa_i(f)G_2^i$ where each $a_i(f)\in \mathcal M$. We now have:
\begin{equation}\label{3.28EX}
\begin{array}{c}
T_k^lT_{k'}^{l'}(f)=T_k^l\left(\underset{i=0}{\overset{s}{\sum}}ia_i(f)G_4^{l'}\cdot G_2^{i+k'-2}\right)=\underset{i=0}{\overset{s}{\sum}}i(i+k'-2)a_i(f)G_4^{l+l'}\cdot G_2^{i+k'+k-4}\\
T_{k'}^{l'}T_k^l(f)=T_{k'}^{l'}\left(\underset{i=0}{\overset{s}{\sum}}ia_i(f)G_4^{l}\cdot G_2^{i+k-2}\right)=\underset{i=0}{\overset{s}{\sum}}i(i+k-2)a_i(f)G_4^{l+l'}\cdot G_2^{i+k'+k-4}\\
\end{array}
\end{equation} From \eqref{3.28EX} it follows that:
\begin{equation}
[T_k^l,T_{k'}^{l'}](f)=(k'-k)\underset{i=0}{\overset{s}{\sum}}ia_i(f)G_4^{l+l'}\cdot G_2^{i+k'+k-4}=(k'-k)T_{k+k'-2}^{l+l'}
\end{equation} Hence, the relation \eqref{3.27EX} holds for the operators $T_k^l$, $T_{k'}^{l'}$ acting on
$\mathcal Q(\Gamma)$.
The remaining relations in \eqref{3.27} for the Lie action of $\mathcal L$ on $\mathcal Q(\Gamma)$ follow from \eqref{3.19}.
\end{proof}
\medskip
\begin{lem}\label{L3.5} Let $f\in \mathcal{QM}$ be an element of the quasimodular tower and let $\alpha\in GL_2^+(\mathbb Q)$. Then, for
any $k\geq1 $, $l\geq 0$, the operator $T^l_k:\mathcal{QM}
\longrightarrow \mathcal{QM}$ satisfies:
\begin{equation}
T^l_k(f)||\alpha = T^l_k(f||\alpha)-\frac{24}{5}\nu_{\alpha}^{(l)}\cdot (T^0_k(f)||\alpha)
\end{equation}
\end{lem}
\begin{proof} For the sake of definiteness, we assume that $f=\sum_{i=0}^sa_i(f)\cdot G_2^i$ with each $a_i(f)\in \mathcal M$. We now compute:
\begin{equation}
\begin{array}{lll}
T^l_k(f)||\alpha = (E^lW_k)(f)||\alpha& \qquad \qquad & T^l_k(f||\alpha) = (E^lW_k)(f||\alpha) \\
= \left(\underset{i=0}{\overset{s}{\sum}}iG_4^l\cdot a_i(f)G_2^{i+k-2}\right)||\alpha & &
=(E^lW_k)\left(\underset{i=0}{\overset{s}{\sum}}(a_i(f)|\alpha)G_2^i\right)\\
= \underset{i=0}{\overset{s}{\sum}}i(G_4^l|\alpha)\cdot (a_i(f)|\alpha)G_2^{i+k-2} & &
=\underset{i=0}{\overset{s}{\sum}}i(G_4^l)\cdot (a_i(f)|\alpha)G_2^{i+k-2}\\
\end{array}
\end{equation}
Subtracting, it follows that:
\begin{equation}
T^l_k(f)||\alpha - T^l_k(f||\alpha) = (G_4^l|\alpha - G_4^l)\cdot \left( \underset{i=0}{\overset{s}{\sum}}i(a_i(f)|\alpha)
G_2^{i+k-2}\right)=-\frac{24}{5}\nu_{\alpha}^{(l)}\cdot (W_k(f)||\alpha)
\end{equation} Putting $T^0_k=E^0W_k=W_k$, we have the result.
\end{proof}
\medskip
\begin{thm}\label{Prp3.6} Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup and let $\mathcal Q(\Gamma)$ be the algebra
of quasimodular Hecke operators of level $\Gamma$. Then, for any $k\geq 1$, $l\geq 0$, the operator
$T_k^l$ satisfies:
\begin{equation}\label{3.32}
T_k^l(F^1\ast F^2)=T_k^l(F^1)\ast F^2 + F^1\ast T_k^l(F^2)+\frac{24}{5} (\phi^{(l)}(F^1)\ast T_k^0(F^2))_\alpha \qquad \forall \textrm{ } F^1,F^2 \in \mathcal Q(\Gamma)
\end{equation} Further, the operators $\{T_k^l\}_{k\geq 1,l\geq 0}$ are all derivations on the algebra
$\mathcal Q^r(\Gamma)=(\mathcal Q(\Gamma),\ast^r)$.
\end{thm}
\begin{proof} We know that $T_k^l= E^lW_k$ and that $W_k$ is a derivation on $\mathcal{QM}$. We choose quasimodular
Hecke operators $F^1$, $F^2\in \mathcal Q(\Gamma)$. Then, for any $\alpha\in GL_2^+(\mathbb Q)$, we know that:
\begin{equation*}
\begin{array}{l}
T_k^l(F^1\ast F^2)_\alpha =E^lW_k\left(\underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta
\cdot (F^2_{\alpha\beta^{-1}}||\beta)\right) \\
= \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} E^lW_k\left(F^1_\beta
\cdot (F^2_{\alpha\beta^{-1}}||\beta) \right) \\
= \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} G_4^l\cdot W_k(F^1_\beta)\cdot (F^2_{\alpha\beta^{-1}}||\beta)+\underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta\cdot G_4^l\cdot W_k(F^2_{\alpha\beta^{-1}}||\beta)\\
= \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} G_4^l\cdot W_k(F^1_\beta)\cdot (F^2_{\alpha\beta^{-1}}||\beta)+\underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta\cdot G_4^l\cdot (W_k(F^2_{\alpha\beta^{-1}})||\beta)\\
=(T_k^l(F^1)\ast F^2)_\alpha + \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta\cdot (G_4^l|\beta)\cdot (W_k(F^2_{\alpha\beta^{-1}})||\beta)- \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta\cdot (G_4^l|\beta-G_4^l) \cdot (W_k(F^2_{\alpha\beta^{-1}})||\beta)\\
= (T_k^l(F^1)\ast F^2)_\alpha + (F^1\ast T_k^l(F^2))_\alpha +\frac{24}{5} \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta\cdot \nu_\beta^{(l)} \cdot (W_k(F^2_{\alpha\beta^{-1}})||\beta) \\
= (T_k^l(F^1)\ast F^2)_\alpha + (F^1\ast T_k^l(F^2))_\alpha +\frac{24}{5} (\phi^{(l)}(F^1)\ast T_k^0(F^2))_\alpha\\
\end{array}
\end{equation*} where it is understood that $\phi^{(0)}=0$. This proves \eqref{3.32}. Further, since $\nu_\beta^{(l)}=0$ for any $\beta\in SL_2(\mathbb Z)$,
when we consider the product $\ast^r$ defined in \eqref{2.11vv} on the algebra $\mathcal Q^r(\Gamma)$, the calculation above reduces to
\begin{equation}
T_k^l(F^1\ast^r F^2)=T_k^l(F^1)\ast^r F^2 + F^1\ast^r T_k^l(F^2)
\end{equation} Hence, each $T_k^l$ is a derivation on $\mathcal Q^r(\Gamma)$.
\end{proof}
\medskip
\begin{thm}\label{Prp3.2} Let $\Gamma =\Gamma(N)$ be a principal congruence subgroup and let $\mathcal Q(\Gamma)$ be the algebra of quasimodular Hecke operators of level $\Gamma$.
\medskip
(a) The operator $D:\mathcal Q(\Gamma)\longrightarrow \mathcal Q(\Gamma)$ on the algebra
$(\mathcal Q(\Gamma),\ast)$ satisfies:
\begin{equation}\label{3.15}
D(F^1\ast F^2)=D(F^1)\ast F^2 + F^1\ast D(F^2) -\phi^{(1)}(F^1)\ast T_1^0(F^2) \qquad\forall\textrm{ }F^1,F^2\in \mathcal Q(\Gamma)
\end{equation} When we consider the product $\ast^r$, the operator $D$ becomes a derivation
on the algebra $\mathcal Q^r(\Gamma)=(\mathcal Q(\Gamma),\ast^r)$, i.e.:
\begin{equation}\label{3.151z}
D(F^1\ast^r F^2)=D(F^1)\ast^r F^2 + F^1\ast^r D(F^2) \qquad\forall\textrm{ }F^1,F^2\in \mathcal Q^r(\Gamma)
\end{equation}
\medskip
(b) The operators $\{W_k\}_{k\geq 1}$ and $\{\phi^{(m)}\}_{m\geq 1}$ are derivations on $\mathcal Q(\Gamma)$, i.e.,
\begin{equation}\label{3.152z}
\begin{array}{c}
W_k(F^1\ast F^2)=W_k(F^1)\ast F^2 + F^1\ast W_k(F^2) \\
\phi^{(m)}(F^1\ast F^2)=\phi^{(m)}(F^1)\ast F^2 + F^1\ast \phi^{(m)}(F^2) \\
\end{array}
\end{equation} for any $F^1$, $F^2\in \mathcal Q(\Gamma)$. Additionally, $\{\phi^{(m)}\}_{m\geq 1}$ and
$\{W_k\}_{k\geq 1}$ are also derivations on the algebra $\mathcal Q^r(\Gamma)=(\mathcal Q(\Gamma),\ast^r)$.
\end{thm}
\begin{proof} (a) We choose quasimodular Hecke operators $F^1$, $F^2\in \mathcal Q(\Gamma)$. We have mentioned before that $D$ is a derivation on $\mathcal{QM}$. Then,
for any $\alpha\in GL_2^+(\mathbb Q)$, we have:
\begin{equation*}
\begin{array}{ll}
D(F^1\ast F^2)_\alpha& =D\left(\underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta
\cdot (F^2_{\alpha\beta^{-1}}||\beta)\right) \\
& = \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} D\left(F^1_\beta
\cdot (F^2_{\alpha\beta^{-1}}||\beta) \right) \\
& = \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} D(F^1_\beta)\cdot (F^2_{\alpha\beta^{-1}}||\beta)+\underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta\cdot D(F^2_{\alpha\beta^{-1}}||\beta)\\
& =(D(F^1)\ast F^2)_\alpha + \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta\cdot (D(F^2_{\alpha\beta^{-1}})||\beta)- \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta\cdot \nu_\beta^{(1)} \cdot (W_1(F^2_{\alpha\beta^{-1}})||\beta)\\
& = (D(F^1)\ast F^2)_\alpha + (F^1\ast D(F^2))_\alpha - (\phi^{(1)}(F^1)\ast T_1^0(F^2))_\alpha\\
\end{array}
\end{equation*} This proves \eqref{3.15}. In order to prove \eqref{3.151z}, we note that
$\nu_\beta^{(1)}=0$ for any $\beta\in SL_2(\mathbb Z)$ (see \eqref{3.125}). Hence, when we use
the product $\ast^r$ defined in \eqref{2.11vv}, the calculation above
reduces to
\begin{equation}
D(F^1\ast^r F^2)=D(F^1)\ast^r F^2 + F^1\ast^r D(F^2)
\end{equation} for any $F^1$, $F^2\in \mathcal Q^r(\Gamma)$.
\medskip
(b) For any $F^1$, $F^2\in \mathcal Q(\Gamma)$ and knowing from \eqref{3.12} that
$\nu_\alpha^{(m)}=\nu_\beta^{(m)}+\nu_{\alpha\beta^{-1}}^{(m)}|\beta$, we have:
\begin{equation}
\begin{array}{ll}
\phi^{(m)}(F^1\ast F^2)_\alpha& =\nu_\alpha^{(m)}\cdot \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta
\cdot (F^2_{\alpha\beta^{-1}}||\beta) \\
& = \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} (\nu_\beta^{(m)} \cdot F^1_\beta)
\cdot (F^2_{\alpha\beta^{-1}}||\beta) + \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta
\cdot (\nu_{\alpha\beta^{-1}}^{(m)}|\beta)\cdot (F^2_{\alpha\beta^{-1}}||\beta) \\
& = \phi^{(m)}(F^1)\ast F^2 + \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta
\cdot ((\nu_{\alpha\beta^{-1}}^{(m)}\cdot F^2_{\alpha\beta^{-1}})||\beta)\\
& = \phi^{(m)}(F^1)\ast F^2 + \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta
\cdot (\phi^{(m)}(F^2)_{\alpha\beta^{-1}}||\beta)\\
& = \phi^{(m)}(F^1)\ast F^2 + F^1\ast \phi^{(m)}(F^2) \\
\end{array}
\end{equation} The fact that each $W_k$ is also a derivation on $\mathcal Q(\Gamma)$ now follows from a similar calculation using the fact that $W_k$ is a derivation on the quasimodular tower $\mathcal{QM}$ and that $W_k(f)||\alpha=W_k(f||\alpha)$ for any $f\in \mathcal{QM}$, $\alpha\in GL_2^+(\mathbb Q)$ (from
\eqref{3.8}). Finally, a similar calculation may be used to verify that $\{W_k\}_{k\geq 1}$ and $\{\phi^{(m)}\}_{m\geq 1}$ are all derivations
on $\mathcal Q^r(\Gamma)$.
\end{proof}
\medskip
We now introduce the Hopf algebra $\mathcal H$ that acts on $\mathcal Q^r(\Gamma)$. The Hopf algebra $\mathcal H$ is the
universal enveloping algebra $\mathcal U(\mathcal L)$ of the Lie algebra $\mathcal L$ defined by generators
$D$, $\{T_k^l\}_{k\geq 1,l\geq 0}$, $\{\phi^{(m)}\}_{m\geq 1}$ satisfying the following relations:
\begin{equation}
\begin{array}{c}
[T_k^l,T_{k'}^{l'}]=(k'-k)T_{k+k'-2}^{l+l'}
\qquad
\mbox{$[T^l_k,D]$}= \frac{5}{24}(k-1)T_{k-1}^{l+1}- \frac{1}{2}(k-3)T^l_{k+1}\\
\mbox{$[D,\phi^{(m)}]=0$} \qquad [T^l_k,\phi^{(m)}]=0 \qquad [\phi^{(m)},
\phi^{(m')}]=0 \\
\end{array}
\end{equation}
As such, the coproduct $\Delta:
\mathcal H\longrightarrow \mathcal H\otimes \mathcal H$ is defined by:
\begin{equation}\label{3.34}
\Delta(D)=D\otimes 1+1\otimes D \qquad
\Delta(T_k^l)=T_k^l\otimes 1+1\otimes T_k^l \qquad
\Delta(\phi^{(m)})=\phi^{(m)}\otimes 1 +1\otimes \phi^{(m)}
\end{equation}
We will now show that $\mathcal H$ has a Hopf action on the algebra $\mathcal Q^r(\Gamma)$.
\medskip
\begin{thm} Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of $SL_2(\mathbb Z)$. Then, there is a Hopf action of $\mathcal H$ on
the algebra $\mathcal Q^r(\Gamma)$, i.e.,
\begin{equation}\label{3.44}
h(F^1\ast^r F^2)=\sum h_{(1)}(F^1)\ast^r h_{(2)}(F^2) \qquad\forall\textrm{ }F^1, F^2\in \mathcal Q^r(\Gamma),
\textrm{ }h\in \mathcal H
\end{equation} where $\Delta(h)=\sum h_{(1)}\otimes h_{(2)}$ for any $h\in \mathcal H$.
\end{thm}
\begin{proof} In order to prove \eqref{3.44}, it suffices to verify the relation for $D$ and each of $\{T_k^l\}_{k\geq 1,l\geq 0}$,
$\{\phi^{(m)}\}_{m\geq 1}$. From Proposition \ref{Prp3.6} and Proposition \ref{Prp3.2},
we know that for $F^1$, $F^2\in \mathcal Q^r(\Gamma)$ and any $k\geq 1$, $l\geq 0$, $m\geq 1$:
\begin{equation}
\begin{array}{c}
D(F^1\ast^r F^2)=D(F^1)\ast^r F^2 +F^1\ast^r D(F^2)\\
T_k^l(F^1\ast^r F^2)=T_k^l(F^1)\ast^r F^2+F^1\ast^r T_k^l(F^2) \\
\phi^{(m)}(F^1\ast^r F^2)=\phi^{(m)}(F^1)\ast^r F^2+F^1\ast^r \phi^{(m)}(F^2) \\
\end{array}
\end{equation} Comparing with the expressions for the coproduct in \eqref{3.34}, it is clear that \eqref{3.44}
holds for each $h\in \mathcal H$.
\end{proof}
\medskip
\medskip
\subsection{The operators $\bf X$, $\bf Y$ and $\bf \{\delta_n\}$ of Connes and Moscovici}
\medskip
\medskip
Let $\Gamma=\Gamma(N)$ be a congruence subgroup. In this subsection, we will show that the
algebra $\mathcal Q(\Gamma)$ carries an action of the Hopf algebra $\mathcal H_1$ of Connes and Moscovici \cite{CM0}. The Hopf algebra $\mathcal H_1$ is part of a larger family
$\{\mathcal H_n\}_{n\geq 1}$ of Hopf algebras
defined in \cite{CM0} and $\mathcal H_1$ is the Hopf algebra corresponding to ``codimension
1 foliations''. As an algebra, $\mathcal H_1$ is identical to the universal enveloping algebra
$\mathcal U(\mathcal L_1)$ of the Lie algebra $\mathcal L_1$ generated by $X$, $Y$, $\{\delta_n\}_{n\geq 1}$ satisfying the commutator relations:
\begin{equation}\label{3.2.46}
[Y,X]=X \quad [X,\delta_n]=\delta_{n+1} \quad [Y,\delta_n]=n\delta_n\quad [\delta_k,\delta_l]=0\quad\forall \textrm{ }k,l,n\geq 1
\end{equation} Further, the coproduct $\Delta:\mathcal H_1\longrightarrow
\mathcal H_1\otimes \mathcal H_1$ on $\mathcal H_1$ is determined by:
\begin{equation}\label{3.2.47qz}
\begin{array}{c}
\Delta(X)=X\otimes 1+1\otimes X+\delta_1\otimes Y \\
\Delta(Y)=Y\otimes 1+1\otimes Y\qquad \Delta(\delta_1)=\delta_1\otimes 1+1\otimes\delta_1 \\
\end{array}
\end{equation} Finally, the antipode $S:\mathcal H_1\longrightarrow \mathcal H_1$ is given by:
\begin{equation}\label{3.2.48qz}
S(X)=-X+\delta_1Y\qquad S(Y)=-Y \qquad S(\delta_1)=-\delta_1
\end{equation} Following Connes and Moscovici \cite{CM1}, we define the operators $X$ and
$Y$ on the modular tower: for any congruence subgroup $\Gamma=\Gamma(N)$, we set:
\begin{equation}
Y:\mathcal M_k(\Gamma)\longrightarrow \mathcal M_k(\Gamma)\qquad Y(f):=\frac{k}{2}f \qquad \forall
\textrm{ }f\in \mathcal M_k(\Gamma)
\end{equation} Further, the operator $X:
\mathcal M_k(\Gamma)\longrightarrow \mathcal M_{k+2}(\Gamma)$ is the Ramanujan differential operator on modular forms:
\begin{equation}
X(f):=\frac{1}{2\pi i}\frac{d}{dz}(f)-\frac{1}{12\pi i}\frac{d}{dz}(\log \Delta)\cdot Y(f) \qquad \forall
\textrm{ }f\in \mathcal M_k(\Gamma)
\end{equation} where $\Delta(z)$ is the well known modular form of weight $12$ given by:
\begin{equation}
\Delta(z)=(2\pi )^{12}q\prod_{n=1}^\infty (1-q^n)^{24}, \textrm{ }q=e^{2\pi iz}
\end{equation} We start by extending these operators to the quasimodular tower
$\mathcal{QM}$. Let $f\in \mathcal{QM}^s_k(\Gamma)$ be a quasimodular form. Then, we can
express $f=\underset{i=0}{\overset{s}{\sum}}a_i(f)G_2^i$ where $a_i(f)\in \mathcal M_{k-2i}(\Gamma)$. We set:
\begin{equation}\label{3.52}
X(f)=\sum_{i=0}^sX(a_i(f))\cdot G_2^i\qquad Y(f)=\sum_{i=0}^sY(a_i(f))\cdot G_2^i
\end{equation} From \eqref{3.52}, it is clear that $X$ and $Y$ are derivations on $\mathcal{QM}$.
\medskip
\begin{lem}\label{L3.2.1} Let $f\in \mathcal{QM}$ be an element of the quasimodular tower. Then, for any $\alpha\in GL_2^+(\mathbb Q)$, we have:
\begin{equation}\label{3.2.53}
X(f)||\alpha = X(f||\alpha)+(\mu_{\alpha^{-1}}\cdot Y(f))||\alpha
\end{equation} where, for any $\delta\in GL_2^+(\mathbb Q)$, we set:
\begin{equation}\label{3.2.54}
\mu_\delta:=\frac{1}{12\pi i}\frac{d}{dz}\log \frac{\Delta |\delta}{\Delta}
\end{equation} Further, we have $Y(f||\alpha)=Y(f)||\alpha$.
\end{lem}
\begin{proof} Following \cite[Lemma 5]{CM1}, we know that for any $g\in \mathcal M$, we have:
\begin{equation}
X(g)|\alpha = X(g|\alpha)+(\mu_{\alpha^{-1}}\cdot Y(g))|\alpha\qquad\forall\textrm{ }\alpha\in GL_2^+(
\mathbb Q)
\end{equation} It suffices to consider the case $f\in \mathcal{QM}^s_k(\Gamma)$ for some congruence
subgroup $\Gamma$. If we express $f\in \mathcal{QM}^s_k(\Gamma)$ as $f=\underset{i=0}{\overset{s}{\sum}}a_i(f)G_2^i$ with $a_i(f)\in
\mathcal M_{k-2i}(\Gamma)$, it follows that:
\begin{equation}\label{3.2.56}
X(a_i(f))|\alpha = X(a_i(f)|\alpha)+(\mu_{\alpha^{-1}}\cdot Y(a_i(f)))|\alpha\qquad\forall\textrm{ }\alpha\in GL_2^+(
\mathbb Q)
\end{equation} for each $0\leq i\leq s$. Combining \eqref{3.2.56} with the definitions of $X$
and $Y$ on the quasimodular tower in \eqref{3.52}, we can easily prove \eqref{3.2.53}. Finally, it
is clear from the definition of $Y$ that $Y(f||\alpha)=Y(f)||\alpha$.
\end{proof}
\medskip
From the definition of $\mu_{\delta}$ in \eqref{3.2.54}, one may verify that (see \cite[$\S$ 3)]{CM1}):
\begin{equation}\label{3.2.57X}
\mu_{\delta_1\delta_2}=\mu_{\delta_1}|\delta_2 +\mu_{\delta_2}
\qquad \forall\textrm{ }\delta_1,\delta_2\in GL_2^+(\mathbb Q)
\end{equation} and that $\mu_\delta=0$ for any $\delta\in SL_2(\mathbb Z)$. We now define operators $X$, $Y$ and $\{\delta_n\}_{n\geq 1}$ on the quasimodular Hecke algebra $\mathcal Q(\Gamma)$
for some congruence subgroup $\Gamma=\Gamma(N)$. Let $F\in \mathcal Q(\Gamma)$ be a
quasimodular Hecke operator of level $\Gamma$; then we define operators:
\begin{equation}\label{3.2.58}
\begin{array}{c}
X, Y , \delta_n:\mathcal Q(\Gamma)\longrightarrow \mathcal Q(\Gamma) \\
X(F)_\alpha:=X(F_\alpha)\qquad Y(F)_\alpha:=Y(F_\alpha)\qquad
\delta_n(F)_\alpha=X^{n-1}(\mu_{\alpha})\cdot F_\alpha \qquad \forall\textrm{ }\alpha\in
GL_2^+(\mathbb Q) \\
\end{array}
\end{equation} We will now show that the Lie algebra
$\mathcal L_1$ with generators $X$, $Y$, $\{\delta_n\}_{n\geq 1}$ satisfying the commutator
relations in \eqref{3.2.46} acts on the algebra $\mathcal Q(\Gamma)$. Additionally, in order to give a Lie action on
the algebra $\mathcal Q^r(\Gamma)=(\mathcal Q(\Gamma),\ast^r)$, we define at this juncture the smaller Lie algebra
$\mathfrak l_1\subseteq \mathcal L_1$ with generators $X$ and $Y$ satisfying the relation
\begin{equation}\label{3.59uu}
[Y,X]=X
\end{equation} Further, we consider the Hopf algebra $\mathfrak h_1$ that arises as the universal
enveloping algebra $\mathcal U(\mathfrak l_1)$ of the Lie algebra $\mathfrak l_1$. We will
show that $\mathcal H_1$ (resp. $\mathfrak h_1$) has a Hopf action on the algebra
$\mathcal Q(\Gamma)$ (resp. $\mathcal Q^r(\Gamma))$. We start by describing the Lie actions.
\medskip
\begin{thm}\label{P3.10} Let $\mathcal L_1$ be the Lie algebra with generators $X$, $Y$ and
$\{\delta_n\}_{n\geq 1}$ satisfying the following commutator relations:
\begin{equation}\label{3.2.60}
[Y,X]=X \quad [X,\delta_n]=\delta_{n+1} \quad [Y,\delta_n]=n\delta_n\quad [\delta_k,\delta_l]=0\quad\forall \textrm{ }k,l,n\geq 1
\end{equation}
Then, for any given congruence subgroup $\Gamma=\Gamma(N)$ of
$SL_2(\mathbb Z)$, we have a Lie action of $\mathcal L_1$ on the module $\mathcal Q(\Gamma)$.
\end{thm}
\begin{proof} From \cite[$\S$ 3]{CM1}, we know that for any element $g\in \mathcal M$ of the modular tower, we have $[Y,X](g)=X(g)$. Since the action of $X$ and $Y$ on the quasimodular tower
$\mathcal{QM}$ (see \eqref{3.52}) is naturally extended from their action on $\mathcal M$,
it follows that $[Y,X]=X$ on the quasimodular tower $\mathcal{QM}$. In particular, given any
quasimodular Hecke operator $F\in \mathcal Q(\Gamma)$ and any $\alpha\in GL_2^+(\mathbb Q)$, we have $[Y,X](F_\alpha)=X(F_\alpha)$ for the element $F_\alpha\in\mathcal{QM}$. By definition,
$X(F)_\alpha=X(F_\alpha)$ and $Y(F_\alpha)=Y(F)_\alpha$ and hence $[Y,X]=X$ holds
for the action of $X$ and $Y$ on $\mathcal Q(\Gamma)$.
\medskip
Further, since $X$ is a derivation on $\mathcal{QM}$ and $\delta_n(F)_\alpha=X^{n-1}(\mu_\alpha)\cdot F_\alpha$,
we have
\begin{equation}
\begin{array}{ll}
[X,\delta_n](F)_\alpha & = X(X^{n-1}(\mu_{\alpha})\cdot F_\alpha) - X^{n-1}(\mu_{\alpha})\cdot X(F_\alpha)\\
& = X(X^{n-1}(\mu_{\alpha}))\cdot F_\alpha =X^n(\mu_{\alpha})\cdot F_\alpha=\delta_{n+1}(F)_\alpha \\
\end{array}
\end{equation} Similarly, since $\mu_\alpha
\in \mathcal M \subseteq \mathcal{QM}$ is of weight $2$ and $Y$ is a derivation on $\mathcal{QM}$, we have:
\begin{equation}
\begin{array}{ll}
[Y,\delta_n](F)_\alpha &=Y(X^{n-1}(\mu_{\alpha})\cdot F_\alpha) - X^{n-1}(\mu_{\alpha})\cdot Y(F_\alpha) \\
&=Y(X^{n-1}(\mu_{\alpha}))\cdot F_\alpha=nX^{n-1}(\mu_{\alpha})\cdot F_\alpha=n\delta_n(F)_\alpha \\
\end{array}
\end{equation} Finally, we can verify easily that $[\delta_k,\delta_l]=0$ for any $k$, $l\geq 1$.
\end{proof}
\medskip
From Proposition \ref{P3.10}, it is also clear that the smaller Lie algebra $\mathfrak l_1\subseteq
\mathcal L_1$ has a Lie action on the module $\mathcal Q(\Gamma)$.
\medskip
\begin{lem}\label{L3.11} Let $\Gamma=\Gamma(N)$ be a congruence subgroup of $SL_2(\mathbb Z)$ and let $\mathcal Q(\Gamma)$ be the algebra of quasimodular Hecke operators of level $\Gamma$. Then, the operator
$X:\mathcal Q(\Gamma)\longrightarrow \mathcal Q(\Gamma)$ on the algebra
$(\mathcal Q(\Gamma),\ast)$ satisfies: \begin{equation} \label{3.2.63}
X(F^1\ast F^2)=X(F^1)\ast F^2 + F^1\ast X(F^2) +\delta_{1}(F^1)\ast Y(F^2) \qquad\forall\textrm{ }F^1,F^2\in \mathcal Q(\Gamma)
\end{equation} When we consider the product $\ast^r$, the operator $X$ becomes a derivation
on the algebra $\mathcal Q^r(\Gamma)=(\mathcal Q(\Gamma),\ast^r)$, i.e.:
\begin{equation} \label{3.2.64}
X(F^1\ast^r F^2)=X(F^1)\ast^r F^2 + F^1\ast^r X(F^2) \qquad\forall\textrm{ }F^1,F^2\in \mathcal Q^r(\Gamma)
\end{equation}
\end{lem}
\begin{proof} We choose quasimodular Hecke operators $F^1$, $F^2\in \mathcal Q(\Gamma)$. Using \eqref{3.2.57X}, we also note that
\begin{equation}
0=\mu_1=\mu_{\beta^{-1}}|\beta +\mu_\beta \qquad \forall \textrm{ }\beta\in GL_2^+(\mathbb Q)
\end{equation}
We have mentioned before that $X$ is a derivation on $\mathcal{QM}$. Then,
for any $\alpha\in GL_2^+(\mathbb Q)$, we have:
\begin{equation*}
\begin{array}{ll}
X(F^1\ast F^2)_\alpha& =X\left(\underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta
\cdot (F^2_{\alpha\beta^{-1}}||\beta)\right) \\
& = \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} X\left(F^1_\beta
\cdot (F^2_{\alpha\beta^{-1}}||\beta) \right) \\
& = \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} X(F^1_\beta)\cdot (F^2_{\alpha\beta^{-1}}||\beta)+\underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta\cdot X(F^2_{\alpha\beta^{-1}}||\beta)\\
& =(X(F^1)\ast F^2)_\alpha + \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta\cdot (X(F^2_{\alpha\beta^{-1}})||\beta)- \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} F^1_\beta\cdot ((\mu_{\beta^{-1}} \cdot Y(F^2_{\alpha\beta^{-1}}))||\beta)\\
& =(X(F^1)\ast F^2)_\alpha + (F^1\ast X(F^2))_\alpha + \underset{\beta\in \Gamma \backslash GL_2^+(\mathbb Q)}{\sum} (F^1_\beta\cdot \mu_{\beta})\cdot (Y(F^2_{\alpha\beta^{-1}})||\beta)\\
& = (X(F^1)\ast F^2)_\alpha + (F^1\ast X(F^2))_\alpha + (\delta_1(F^1)\ast Y(F^2))_\alpha\\
\end{array}
\end{equation*} This proves \eqref{3.2.63}. In order to prove \eqref{3.2.64}, we note that
$\mu_\beta=0$ for any $\beta\in SL_2(\mathbb Z)$. Hence, if we use the product $\ast^r$, the calculation above
reduces to
\begin{equation}\label{3.2.65}
X(F^1\ast^r F^2)=X(F^1)\ast^r F^2 + F^1\ast^r X(F^2)
\end{equation} for any $F^1$, $F^2\in \mathcal Q^r(\Gamma)$.
\end{proof}
\medskip
Finally, we describe the Hopf action of $\mathcal H_1$ on the algebra $(\mathcal Q(\Gamma),
\ast)$ as well as the Hopf action of $\mathfrak h_1$ on the algebra
$\mathcal Q^r(\Gamma)=(\mathcal Q(\Gamma),\ast^r)$.
\medskip
\begin{thm} Let $\Gamma=\Gamma(N)$ be a congruence subgroup of $SL_2(\mathbb Z)$. Then, the Hopf algebra $\mathcal H_1$ has a Hopf action on the quasimodular Hecke algebra
$(\mathcal Q(\Gamma),\ast)$; in other words, we have:
\begin{equation}\label{3.2.66}
h(F^1\ast F^2)=\sum h_{(1)}(F^1) \otimes h_{(2)}(F^2)\qquad \forall\textrm{ }h\in \mathcal H_1,
F^1,F^2\in \mathcal Q(\Gamma)
\end{equation} where the coproduct $\Delta:\mathcal H_1\longrightarrow \mathcal H_1
\otimes \mathcal H_1$ is given by $\Delta(h)=\sum h_{(1)}\otimes h_{(2)}$ for any
$h\in \mathcal H_1$. Similarly, there exists a Hopf action of the Hopf algebra
$\mathfrak h_1$ on the algebra $\mathcal Q^r(\Gamma)=(\mathcal Q(\Gamma),\ast^r)$.
\end{thm}
\begin{proof} In order to prove \eqref{3.2.66}, it suffices to check the relation for
$X$, $Y$ and $\delta_1\in \mathcal H_1$. For the element $X\in \mathcal H_1$, this is already the result
of Lemma \ref{L3.11}. Now, for any $F^1$, $F^2\in \mathcal Q(\Gamma)$ and $\alpha\in GL_2^+(\mathbb Q)$, we have:
\begin{equation}\label{3.2.67}
\begin{array}{ll}
\delta_1(F^1\ast F^2)_\alpha& = \mu_\alpha\cdot\left(\underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum}F^1_\beta
\cdot (F^2_{\alpha\beta^{-1}}||\beta)\right)\\
&=\underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum}(\mu_\beta\cdot F^1_\beta)\cdot (F^2_{\alpha\beta^{-1}}||\beta) + \underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum}
F^1_\beta\cdot ((\mu_{\alpha\beta^{-1}}\cdot F^2_{\alpha\beta^{-1}})||\beta)\\
& = (\delta_1(F^1)\ast F^2)_\alpha + (F^1\ast \delta_1(F^2))_\alpha \\
\end{array}
\end{equation} Further, using the fact that $Y$ is a derivation on $\mathcal{QM}$ and
$Y(f||\alpha)=Y(f)||\alpha$ for any $f\in \mathcal{QM}$, $\alpha\in GL_2^+(\mathbb Q)$, we can
easily verify the relation \eqref{3.2.66} for the element $Y\in \mathcal H_1$. This proves
\eqref{3.2.66} for all $h\in \mathcal H_1$.
\medskip
Finally, in order to demonstrate the Hopf action of $\mathfrak h_1$ on $\mathcal Q^r(\Gamma)$,
we need to check that:
\begin{equation}
X(F^1\ast^r F^2)=X(F^1)\ast^r F^2 + F^1\ast^r X(F^2)\qquad Y(F^1\ast^r F^2)=Y(F^1)\ast^r F^2 + F^1\ast^r Y(F^2)
\end{equation} for any $F^1$, $F^2\in \mathcal Q^r(\Gamma)$. The relation for $X$ has already been proved in \eqref{3.2.65}. The relation for $Y$
is again an easy consequence of the fact that $Y$ is a derivation on $\mathcal{QM}$ and
$Y(f||\alpha)=Y(f)||\alpha$ for any $f\in \mathcal{QM}$, $\alpha\in GL_2^+(\mathbb Q)$.
\end{proof}
\medskip
\medskip
\section{Twisted Quasimodular Hecke operators}
\medskip
\medskip
Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of $SL_2(\mathbb Z)$. For any
$\sigma\in SL_2(\mathbb Z)$, we have developed the theory of $\sigma$-twisted modular Hecke operators in \cite{AB1}.
In this section, we introduce and study the collection $\mathcal Q_\sigma(\Gamma)$ of quasimodular Hecke operators
of level $\Gamma$ twisted by $\sigma$. When $\sigma=1$, $\mathcal Q_\sigma(\Gamma)$ coincides with the algebra
$\mathcal Q(\Gamma)$ of quasimodular Hecke operators. In general, we will show that $\mathcal Q_\sigma(\Gamma)$ is a right
$\mathcal Q(\Gamma)$-module and carries a pairing:
\begin{equation}\label{4.1pb}
(\_\_,\_\_):\mathcal Q_\sigma(\Gamma)\otimes \mathcal Q_\sigma(\Gamma)\longrightarrow
\mathcal Q_\sigma(\Gamma)
\end{equation} We recall from Section 3 the Lie algebra $\mathfrak{l}_1$ with two generators $Y$, $X$ satisfying
$[Y,X]=X$. If we let $\mathfrak{h}_1$ be the Hopf algebra that is the universal enveloping algebra
of $\mathfrak{l}_1$, we show in Section 4.1 that the pairing in \eqref{4.1pb} on $\mathcal Q_\sigma(\Gamma)$ carries a
``Hopf action'' of $\mathfrak{h}_1$. In other words, we have:
\begin{equation}
h(F^1,F^2)=\sum (h_{(1)}(F^1),h_{(2)}(F^2))\qquad\forall \textrm{ } h\in \mathfrak{h}_1, \textrm{ }F^1,F^2\in \mathcal Q_\sigma(\Gamma)
\end{equation} where the coproduct $\Delta:\mathfrak{h}_1\longrightarrow \mathfrak{h}_1\otimes \mathfrak{h}_1$ is given by
$\Delta(h)=\sum h_{(1)}\otimes h_{(2)}$ for any $h\in \mathfrak{h}_1$. In Section 4.2, we consider operators
$X_\tau:\mathcal Q_\sigma(\Gamma)\longrightarrow \mathcal Q_{\tau\sigma}(\Gamma)$ for any $\tau$, $\sigma\in SL_2(\mathbb Z)$. In particular, we consider operators
acting between the levels of the graded module:
\begin{equation}
\mathbb Q_\sigma(\Gamma)=\bigoplus_{m\in \mathbb Z}\mathcal Q_{\sigma(m)}(\Gamma)
\end{equation} where for any $\sigma \in SL_2(\mathbb Z)$, we set $\sigma(m)
=\begin{pmatrix} 1 & m \\ 0 & 1 \\ \end{pmatrix}\cdot \sigma$. Further, we generalize the pairing
on $\mathcal Q_\sigma(\Gamma)$ in \eqref{4.1pb} to a pairing:
\begin{equation}\label{4.4pb}
(\_\_,\_\_):\mathcal Q_{\sigma(m)}(\Gamma)\otimes \mathcal Q_{\sigma(n)}(\Gamma)\longrightarrow \mathcal Q_{\sigma(m+n)}(\Gamma)
\qquad \forall\textrm{ }m,n\in \mathbb Z
\end{equation} We show that the pairing in \eqref{4.4pb} is a special case of a more general pairing
\begin{equation}
(\_\_,\_\_):\mathcal Q_{\tau_1\sigma}(\Gamma)\otimes \mathcal Q_{\tau_2\sigma}(\Gamma)\longrightarrow \mathcal Q_{\tau_1\tau_2\sigma}(\Gamma)
\end{equation} where $\tau_1$, $\tau_2$ are commuting matrices in $SL_2(\mathbb Z)$. From \eqref{4.4pb}, it is clear
that we have a graded pairing on
$\mathbb Q_{\sigma}(\Gamma)$ that extends the pairing on $\mathcal Q_{\sigma}(\Gamma)$.
Finally, we consider
the Lie algebra $\mathfrak{l}_{\mathbb Z}$ with generators $\{Z,X_n|n\in \mathbb Z\}$ satisfying the
commutator relations:
\begin{equation}
[Z,X_n]=(n+1)X_n\qquad [X_n,X_{n'}]=0\qquad \forall\textrm{ }n,n'\in \mathbb Z
\end{equation} Then, for $n=0$, we have $[Z,X_0]=X_0$ and hence the Lie algebra $\mathfrak{l}_{\mathbb Z}$ contains
the Lie algebra $\mathfrak{l}_1$ acting on $\mathcal Q_\sigma(\Gamma)$. Then, if we let $\mathfrak{h}_{\mathbb Z}$ be the Hopf algebra that is the universal enveloping
algebra of $\mathfrak{l}_{\mathbb Z}$, we show that $\mathfrak{h}_{\mathbb Z}$ has a Hopf action on the
pairing on $\mathbb Q_\sigma(\Gamma)$. In other words, for any $F^1$, $F^2\in \mathbb Q_{\sigma}(\Gamma)$, we have
\begin{equation}
h(F^1,F^2)=\sum (h_{(1)}(F^1),h_{(2)}(F^2))\qquad \forall\textrm{ }h\in \mathfrak{h}_{\mathbb Z}
\end{equation} where the coproduct $\Delta:\mathfrak h_{\mathbb Z}
\longrightarrow \mathfrak h_{\mathbb Z}\otimes \mathfrak h_{\mathbb Z}$ is defined
by setting $\Delta(h):=\sum h_{(1)}\otimes h_{(2)}$ for each $h\in \mathfrak h_{\mathbb Z}$.
\medskip
\medskip
\subsection{The pairing on $\mathcal Q_\sigma(\Gamma)$ and Hopf action}
\medskip
\medskip
Let $\sigma\in SL_2(\mathbb Z)$ and let $\Gamma=\Gamma(N)$ be a principal congruence subgroup
of $SL_2(\mathbb Z)$. We start by defining the collection
$\mathcal Q_\sigma(\Gamma)$ of quasimodular Hecke operators of level $\Gamma$
twisted by $\sigma$. When $\sigma=1$, this reduces to the definition of
$\mathcal Q(\Gamma)$.
\medskip
\begin{defn} Choose $\sigma\in SL_2(\mathbb Z)$ and let $\Gamma=\Gamma(N)$ be a principal congruence
subgroup of $SL_2(\mathbb Z)$. A $\sigma$-twisted quasimodular Hecke operator
$F$ of level $\Gamma$ is a function of finite support:
\begin{equation}\label{4.1}
F:\Gamma\backslash GL_2^+(\mathbb Q)\longrightarrow \mathcal{QM}\qquad
\Gamma\alpha\mapsto F_\alpha\in \mathcal{QM}
\end{equation} such that:
\begin{equation}\label{4.2}
F_{\alpha\gamma}=F_\alpha||\sigma\gamma\sigma^{-1}\qquad\forall\textrm{ }\gamma\in\Gamma
\end{equation} We denote by $\mathcal Q_\sigma(\Gamma)$ the collection of $\sigma$-twisted
quasimodular Hecke operators of level $\Gamma$.
\end{defn}
\medskip
\begin{thm} Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of $SL_2(\mathbb Z)$
and choose some $\sigma\in SL_2(\mathbb Z)$. Then there exists a pairing:
\begin{equation}\label{4.3}
(\_\_,\_\_):\mathcal Q_\sigma(\Gamma)\otimes \mathcal Q_\sigma(\Gamma)
\longrightarrow \mathcal Q_\sigma(\Gamma)
\end{equation} defined as follows:
\begin{equation}\label{4.4}
(F^1,F^2)_{\alpha}:=\underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
F_{\beta\sigma}^1\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}||\sigma\beta)
\qquad \forall\textrm{ }F^1, F^2\in \mathcal Q_\sigma(\Gamma),\alpha\in GL_2^+(\mathbb Q)
\end{equation}
\end{thm}
\begin{proof} We choose $\gamma\in \Gamma$. Then, for any $\beta\in SL_2(\mathbb Z)$, we have:
\begin{equation}
F^1_{\gamma\beta\sigma}=F^1_{\beta\sigma}\qquad
F^2_{\alpha\sigma^{-1}\beta^{-1}\gamma^{-1}}||\sigma\gamma\beta = F^2_{\alpha\sigma^{-1}\beta^{-1}}||\sigma\gamma^{-1}\sigma^{-1}\sigma\gamma\beta=
F^2_{\alpha\sigma^{-1}\beta^{-1}}||\sigma\beta
\end{equation} and hence the sum in \eqref{4.4} is well defined, i.e., it does not depend
on the choice of coset representatives. We have to show that $(F^1,F^2)\in \mathcal Q_\sigma(\Gamma)$. For this, we first note that $F^2_{\gamma\alpha\sigma^{-1}\beta^{-1}}= F^2_{\alpha\sigma^{-1}\beta^{-1}}$ for any $\gamma\in \Gamma$ and hence from the expression
in \eqref{4.4}, it follows that $(F^1,F^2)_{\gamma\alpha}=(F^1,F^2)_\alpha$. On the other hand, for any $\gamma\in \Gamma$,
we can write:
\begin{equation}\label{4.6}
(F^1,F^2)_{\alpha\gamma}=\underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
F^1_{\beta\sigma}\cdot (F^2_{\alpha\gamma\sigma^{-1}\beta^{-1}}||\sigma\beta)
\end{equation} We put $\delta=\beta\sigma\gamma^{-1}\sigma^{-1}$. It is clear that as $\beta$ runs through all the coset representatives of $\Gamma$ in $SL_2(\mathbb Z)$, so does $\delta$. From \eqref{4.2}, we know that $F^1_{\delta\sigma\gamma}=F^1_{\delta\sigma}||\sigma\gamma\sigma^{-1}$. Then, we can rewrite \eqref{4.6} as:
\begin{equation}
\begin{array}{ll}
(F^1,F^2)_{\alpha\gamma} & = \underset{\delta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
F^1_{\delta\sigma\gamma}\cdot (F^2_{\alpha\sigma^{-1}\delta^{-1}}||\sigma\delta\sigma\gamma\sigma^{-1})\\
& = \underset{\delta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(F^1_{\delta\sigma}||\sigma\gamma\sigma^{-1})\cdot ((F^2_{\alpha\sigma^{-1}\delta^{-1}}||\sigma\delta)||\sigma\gamma\sigma^{-1})\\
& = \left(\underset{\delta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
F_{\delta\sigma}^1\cdot (F^2_{\alpha\sigma^{-1}\delta^{-1}}||\sigma\delta)\right)||(\sigma\gamma\sigma^{-1})\\
&=(F^1,F^2)_\alpha||\sigma\gamma\sigma^{-1}\\
\end{array}
\end{equation} It follows that $(F^1,F^2)\in \mathcal Q_\sigma(\Gamma)$ and hence we have a well defined pairing $(\_\_,\_\_):\mathcal Q_\sigma(\Gamma)\otimes \mathcal Q_\sigma(\Gamma)
\longrightarrow \mathcal Q_\sigma(\Gamma)$.
\end{proof}
\medskip
We now consider the Hopf algebra $\mathfrak h_1$ defined in Section 3.2. By definition, $\mathfrak h_1$ is the universal enveloping algebra of the Lie algebra $\mathfrak l_1$ with two generators
$X$ and $Y$ satisfying $[Y,X]=X$. We will now show that $\mathfrak l_1$ has a Lie action
on $\mathcal Q_\sigma(\Gamma)$ and that $\mathfrak h_1$ has a ``Hopf action'' with respect to the pairing on $\mathcal Q_\sigma(\Gamma)$.
\medskip
\begin{thm}\label{Prop4.3} Let $\sigma\in SL_2(\mathbb Z)$ and let $\Gamma=\Gamma(N)$ be a principal
congruence subgroup of $SL_2(\mathbb Z)$.
\medskip
(a) The Lie algebra $\mathfrak l_1$ has a Lie action on $\mathcal Q_\sigma(\Gamma)$ defined
by:
\begin{equation}\label{4.8}
X(F)_\alpha:=X(F_\alpha) \qquad Y(F)_\alpha:=Y(F_\alpha)\qquad\forall\textrm{ }F\in \mathcal Q_\sigma(\Gamma), \alpha\in GL_2^+(\mathbb Q)
\end{equation}
\medskip
(b) The universal enveloping algebra $\mathfrak h_1$ of the Lie algebra $\mathfrak l_1$ has a ``Hopf action'' with respect to the pairing on $\mathcal Q_\sigma(\Gamma)$; in other words, we have:
\begin{equation}\label{4.9}
h(F^1,F^2)=\sum (h_{(1)}(F^1),h_{(2)}(F^2))\qquad\forall\textrm{ }F^1,F^2\in \mathcal Q_\sigma(\Gamma), h\in \mathfrak h_1
\end{equation} where the coproduct $\Delta:\mathfrak h_1\longrightarrow
\mathfrak h_1\otimes \mathfrak h_1$ is given by $\Delta(h)=\sum h_{(1)}\otimes
h_{(2)}$ for any $h\in \mathfrak h_1$.
\end{thm}
\begin{proof} (a) We need to verify that for any $F\in \mathcal Q_\sigma(\Gamma)$ and
any $\alpha\in GL_2^+(\mathbb Q)$, we have $([Y,X](F))_\alpha=X(F)_\alpha$. We know that for any element $g\in \mathcal{QM}$ and hence in particular for the element $F_\alpha\in \mathcal{QM}$,
we have $[Y,X](g)=X(g)$. The result now follows from the definition of the action of $X$
and $Y$ in \eqref{4.8}.
\medskip
(b) The Lie action of $\mathfrak l_1$ on $\mathcal Q_\sigma(\Gamma)$ from part (a) induces an action
of the universal enveloping algebra $\mathfrak h_1$ on $\mathcal Q_\sigma(\Gamma)$. In order
to prove \eqref{4.9}, it suffices to prove the result for the generators $X$ and $Y$. We have:
\begin{equation}\label{4.10}
\begin{array}{l}
(X(F^1,F^2))_\alpha=X((F^1,F^2)_\alpha)\\
= X\left( \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}F^1_{\beta\sigma}
\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}||\sigma\beta)\right)\\
= \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum} X(F^1_{\beta\sigma})
\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}||\sigma\beta)+ \underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum}F^1_{\beta\sigma}
\cdot X(F^2_{\alpha\sigma^{-1}\beta^{-1}}||\sigma\beta)\\
= \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum} X(F^1_{\beta\sigma})
\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}||\sigma\beta)+ \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}F^1_{\beta\sigma}
\cdot (X(F^2_{\alpha\sigma^{-1}\beta^{-1}})||\sigma\beta)\\
= (X(F^1),F^2))_\alpha + (F^1,X(F^2))_\alpha\\
\end{array}
\end{equation} In \eqref{4.10}, we have used the fact that $\sigma\beta\in SL_2(\mathbb Z)$ and
hence $X(F^2_{\alpha\sigma^{-1}\beta^{-1}}||\sigma\beta)=X(F^2_{\alpha\sigma^{-1}\beta^{-1}})||\sigma\beta$. We can similarly verify the relation \eqref{4.9} for $Y\in \mathfrak h_1$. This proves the result.
\end{proof}
\medskip
Our next aim is to show that $\mathcal Q_\sigma(\Gamma)$ is a right $\mathcal Q(\Gamma)$-module. Thereafter, we will consider
the Hopf algebra $\mathcal H_1$ defined in Section 3.2 and show that there is a ``Hopf action'' of
$\mathcal H_1$ on the right $\mathcal Q(\Gamma)$-module $\mathcal Q_\sigma(\Gamma)$.
\medskip
\begin{thm} Let $\sigma\in SL_2(\mathbb Z)$ and let $\Gamma=\Gamma(N)$ be a principal congruence
subgroup of $SL_2(\mathbb Z)$. Then, $\mathcal Q_\sigma(\Gamma)$ carries a right
$\mathcal Q(\Gamma)$-module structure defined by:
\begin{equation}\label{4.11}
(F^1\ast F^2)_\alpha :=\underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum}
F^1_{\beta\sigma}\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}|\beta)
\end{equation} for any $F^1\in \mathcal Q_\sigma(\Gamma)$ and any
$F^2\in \mathcal Q(\Gamma)$.
\end{thm}
\begin{proof} We take $\gamma\in \Gamma$. Then, since $F^1\in \mathcal Q_\sigma(\Gamma)$ and
$F^2\in \mathcal Q(\Gamma)$, we have:
\begin{equation}
F^1_{\gamma\beta\sigma}=F^1_{\beta\sigma}\qquad
F^2_{\alpha\sigma^{-1}\beta^{-1}\gamma^{-1}}|\gamma\beta=F^2_{\alpha\sigma^{-1}\beta^{-1}}|\gamma^{-1}\gamma\beta=F^2_{\alpha\sigma^{-1}\beta^{-1}}|\beta
\end{equation} It follows that the sum in \eqref{4.11} is well defined, i.e., it does not depend
on the choice of coset representatives for $\Gamma$ in $GL_2^+(\mathbb Q)$. Further, it is clear
that $(F^1\ast F^2)_{\gamma\alpha}=(F^1\ast F^2)_\alpha$. In order to show that
$F^1\ast F^2\in \mathcal Q_\sigma(\Gamma)$, it remains to show that
$(F^1\ast F^2)_{\alpha\gamma}=(F^1\ast F_2)_\alpha||\sigma\gamma\sigma^{-1}$. By definition,
we know that:
\begin{equation}\label{4.13}
\begin{array}{ll}
(F^1\ast F^2)_{\alpha\gamma} & = \underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum}
F^1_{\beta\sigma}\cdot (F^2_{\alpha\gamma\sigma^{-1}\beta^{-1}}|\beta)\\
\end{array}
\end{equation} We now set $\delta=\beta\sigma\gamma^{-1}\sigma^{-1}$. This allows us to rewrite
\eqref{4.13} as follows:
\begin{equation}\label{4.14}
\begin{array}{ll}
(F^1\ast F^2)_{\alpha\gamma}& = \underset{\delta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum}
F^1_{\delta\sigma\gamma}\cdot (F^2_{\alpha\sigma^{-1}\delta^{-1}}|\delta\sigma\gamma\sigma^{-1})\\
& = \underset{\delta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum}
(F^1_{\delta\sigma}||\sigma\gamma\sigma^{-1})\cdot ((F^2_{\alpha\sigma^{-1}\delta^{-1}}|\delta)|\sigma\gamma\sigma^{-1}))\\
& =\left(\underset{\delta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum}
F^1_{\delta\sigma}\cdot (F^2_{\alpha\sigma^{-1}\delta^{-1}}|\delta)\right) ||\sigma\gamma\sigma^{-1}\\
& =(F^1\ast F^2)_\alpha ||\sigma\gamma\sigma^{-1}\\
\end{array}
\end{equation} Hence, $(F^1\ast F^2)\in \mathcal Q_\sigma(\Gamma)$. In order to show that
$\mathcal Q_\sigma(\Gamma)$ is a right $\mathcal Q(\Gamma)$-module, we need to check that
$F^1\ast (F^2\ast F^3)=(F^1\ast F^2)\ast F^3$ for any $F^1\in \mathcal Q_\sigma(\Gamma)$ and
any $F^2,F^3\in \mathcal Q(\Gamma)$. For this, we note that:
\begin{equation}\label{4.15}
(F^1\ast F^2)_\alpha=\underset{\alpha_2\alpha_1=\alpha}
{\sum} F^1_{\alpha_1}\cdot (F^2_{\alpha_2}|\alpha_1\sigma^{-1})\qquad \forall\textrm{ }\alpha\in GL_2^+(\mathbb Q)
\end{equation} where the sum in \eqref{4.15} is taken over all pairs $(\alpha_1,\alpha_2)$ such that
$\alpha_2\alpha_1=\alpha$ modulo the the following equivalence relation:
\begin{equation}
(\alpha_1,\alpha_2)\sim (\gamma\alpha_1,\alpha_2\gamma^{-1})\qquad\forall\textrm{ }\gamma\in \Gamma
\end{equation} It follows that for any $\alpha\in GL_2^+(\mathbb Q)$, we have:
\begin{equation}\label{4.17}
((F^1\ast F^2)\ast F^3)_\alpha=\underset{\alpha_3\alpha_2\alpha_1=\alpha}{\sum}F^1_{\alpha_1}\cdot
(F^2_{\alpha_2}|\alpha_1\sigma^{-1})\cdot (F^3_{\alpha_3}|\alpha_2\alpha_1\sigma^{-1})
\end{equation} where the sum in \eqref{4.17} is taken over all triples $(\alpha_1,\alpha_2,\alpha_3)$ such
that $\alpha_3\alpha_2\alpha_1=\alpha$ modulo the following equivalence relation:
\begin{equation}\label{4.18}
(\alpha_1,\alpha_2,\alpha_3)\sim (\gamma\alpha_1,\gamma'\alpha_2\gamma^{-1},\alpha_3\gamma'^{-1})
\qquad\forall\textrm{ }\gamma,\gamma'\in \Gamma
\end{equation} On the other hand, we have:
\begin{equation}\label{4.19}
\begin{array}{ll}
(F^1\ast (F^2\ast F^3))_\alpha & =\underset{\alpha_2'\alpha_1=\alpha}{\sum} F^1_{\alpha_1}\cdot ((F^2\ast F^3)_{\alpha'_2}|\alpha_1\sigma^{-1})\\
& = \underset{\alpha_3\alpha_2\alpha_1=\alpha}{\sum} F^1_{\alpha_1}\cdot (F^2_{\alpha_2}|\alpha_1\sigma^{-1})\cdot (F^3_{\alpha_3}|\alpha_2\alpha_1\sigma^{-1})\\
\end{array}
\end{equation} Again, we see that the sum in \eqref{4.19} is taken over all triples $(\alpha_1,\alpha_2,\alpha_3)$ such
that $\alpha_3\alpha_2\alpha_1=\alpha$ modulo the equivalence relation in \eqref{4.18}. From \eqref{4.17} and \eqref{4.19}, it follows that $(F^1\ast (F^2\ast F^3))_\alpha=((F^1\ast F^2)\ast F^3)_\alpha$. This proves the
result.
\end{proof}
\medskip
We are now ready to describe the action of the Hopf algebra $\mathcal H_1$ on $\mathcal Q_\sigma(\Gamma)$. From Section 3.2, we know that
$\mathcal H_1$ is generated by $X$, $Y$, $\{\delta_n\}_{n\geq 1}$ which satisfy the relations \eqref{3.2.46}, \eqref{3.2.47qz}, \eqref{3.2.48qz}.
\medskip
\begin{thm} Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of $SL_2(\mathbb Z)$
and choose some $\sigma\in SL_2(\mathbb Z)$.
\medskip
(a) The collection of $\sigma$-twisted quasimodular Hecke operators of level $\Gamma$ can be made into an $\mathcal H_1$-module as follows; for any $F\in \mathcal Q_\sigma(\Gamma)$ and $\alpha\in GL_2^+(\mathbb Q)$:
\begin{equation}\label{4.20}
X(F)_\alpha:=X(F_\alpha) \qquad Y(F)_\alpha:=Y(F_\alpha)\qquad \delta_n(F)_\alpha:=X^{n-1}(\mu_{\alpha\sigma^{-1}})\cdot F_\alpha\qquad\forall\textrm{ }n\geq 1
\end{equation}
\medskip
(b) The Hopf algebra $\mathcal H_1$ has a ``Hopf action'' on the right $\mathcal Q(\Gamma)$-module
$\mathcal Q_\sigma(\Gamma)$; in other words, for any $F^1\in \mathcal Q_\sigma(\Gamma)$ and
any $F^2\in \mathcal Q(\Gamma)$, we have:
\begin{equation}\label{4.21z}
h(F^1\ast F^2)=\sum h_{(1)}(F^1)\ast h_{(2)}(F^2)\qquad \forall\textrm{ }h\in \mathcal H_1
\end{equation} where the coproduct $\Delta:\mathcal H_1\longrightarrow \mathcal H_1\otimes
\mathcal H_1$ is given by $\Delta(h)=\sum h_{(1)}\otimes h_{(2)}$ for each $h\in \mathcal H_1$.
\end{thm}
\begin{proof} (a) For any $F\in \mathcal Q_\sigma(\Gamma)$, we have already checked in the proof
of Proposition \ref{Prop4.3} that
$X(F)$, $Y(F)\in \mathcal Q_\sigma(\Gamma)$. Further, from \eqref{3.2.57X}, we know that for any
$\alpha\in GL_2^+(\mathbb Q)$ and $\gamma\in \Gamma$, we have:
\begin{equation}\label{4.22}
\begin{array}{c}
\mu_{\gamma\alpha\sigma^{-1}}=\mu_{\gamma}|\alpha\sigma^{-1}+\mu_{\alpha\sigma^{-1}}=\mu_{\alpha\sigma^{-1}}\\
\mu_{\alpha\gamma\sigma^{-1}}=\mu_{\alpha\sigma^{-1}}|\sigma\gamma\sigma^{-1}+
\mu_{\sigma\gamma\sigma^{-1}}=\mu_{\alpha\sigma^{-1}}|\sigma\gamma\sigma^{-1}\\
\end{array}
\end{equation} Hence, for any $F\in \mathcal Q_\sigma(\Gamma)$, we have:
\begin{equation}
\begin{array}{c}
\delta_n(F)_{\gamma\alpha}=X^{n-1}(\mu_{\gamma\alpha\sigma^{-1}})\cdot F_{\gamma\alpha}=
X^{n-1}(\mu_{\alpha\sigma^{-1}})\cdot F_{\alpha}=\delta_n(F)_\alpha\\
\delta_n(F)_{\alpha\gamma}=X^{n-1}(\mu_{\alpha\gamma\sigma^{-1}})\cdot F_{\alpha\gamma}
=X^{n-1}(\mu_{\alpha\sigma^{-1}}|\sigma\gamma\sigma^{-1})\cdot (F_\alpha||\sigma\gamma\sigma^{-1})
=\delta_n(F)_\alpha||\sigma\gamma\sigma^{-1}\\
\end{array}
\end{equation} Hence, $\delta_n(F)\in \mathcal Q_\sigma(\Gamma)$. In order to show that there is an action
of the Lie algebra $\mathcal L_1$ (and hence of its universal eneveloping algebra $\mathcal H_1$) on
$\mathcal Q_\sigma(\Gamma)$, it remains to check the commutator relations \eqref{3.2.46} between the operators
$X$, $Y$ and $\delta_n$ acting on $\mathcal Q_\sigma(\Gamma)$. We have already checked that
$[Y,X]=X$ in the proof of Proposition \ref{Prop4.3}. Since $X$ is a derivation on $\mathcal{QM}$ and
$\delta_n(F)_\alpha=X^{n-1}(\mu_{\alpha\sigma^{-1}})\cdot F_\alpha$, we have:
\begin{equation}
\begin{array}{ll}
[X,\delta_n](F)_\alpha & = X(X^{n-1}(\mu_{\alpha\sigma^{-1}})\cdot F_\alpha) - X^{n-1}(\mu_{\alpha\sigma^{-1}})\cdot X(F_\alpha)\\
& = X(X^{n-1}(\mu_{\alpha\sigma^{-1}}))\cdot F_\alpha =X^n(\mu_{\alpha\sigma^{-1}})\cdot F_\alpha=\delta_{n+1}(F)_\alpha \\
\end{array}
\end{equation} Similarly, since $\mu_{\alpha\sigma^{-1}}
\in \mathcal M\subseteq \mathcal{QM}$ is of weight $2$ and $Y$ is a derivation on $\mathcal{QM}$, we have:
\begin{equation}
\begin{array}{ll}
[Y,\delta_n](F)_\alpha &=Y(X^{n-1}(\mu_{\alpha\sigma^{-1}})\cdot F_\alpha) - X^{n-1}(\mu_{\alpha\sigma^{-1}})\cdot Y(F_\alpha) \\
&=Y(X^{n-1}(\mu_{\alpha\sigma^{-1}}))\cdot F_\alpha=nX^{n-1}(\mu_{\alpha\sigma^{-1}})\cdot F_\alpha=n\delta_n(F)_\alpha \\
\end{array}
\end{equation} Finally, we can verify easily that $[\delta_k,\delta_l]=0$ for any $k$, $l\geq 1$.
\medskip
(b) In order to prove \eqref{4.21z}, it is enough to check this equality for the generators
$X$, $Y$ and $\delta_1\in \mathcal H_1$. For $F^1\in \mathcal Q_\sigma(\Gamma)$, $F^2\in \mathcal Q(\Gamma)$ and $\alpha\in GL_2^+(\mathbb Q)$, we have:
\begin{equation}\label{4.26rty}
\begin{array}{l}
(X(F^1\ast F^2))_\alpha=X((F^1\ast F^2)_\alpha)\\
= \underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum}X(F^1_{\beta\sigma}\cdot (F^2_{
\alpha\sigma^{-1}\beta^{-1}}|\beta)) \\
=\underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum} X(F^1_{\beta\sigma})\cdot (F^2_{
\alpha\sigma^{-1}\beta^{-1}}|\beta) + \underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum} F^1_{\beta\sigma}\cdot X(F^2_{
\alpha\sigma^{-1}\beta^{-1}}|\beta) \\
=(X(F^1)\ast F^2)_\alpha + \underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum} F^1_{\beta\sigma}\cdot X(F^2_{
\alpha\sigma^{-1}\beta^{-1}})|\beta - \underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum} F^1_{\beta\sigma}\cdot (\mu_{\beta^{-1}}|\beta)\cdot Y(F^2_{\alpha\sigma^{-1}\beta^{-1}})|\beta \\
=(X(F^1)\ast F^2)_\alpha + \underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum} F^1_{\beta\sigma}\cdot X(F^2_{
\alpha\sigma^{-1}\beta^{-1}})|\beta + \underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum} F^1_{\beta\sigma}\cdot \mu_{\beta}\cdot Y(F^2_{\alpha\sigma^{-1}\beta^{-1}})|\beta \\
=(X(F^1)\ast F^2)_\alpha + (F^1\ast X(F^2))_\alpha + \sum_{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}
\delta_1(F)_{\beta\sigma}\cdot Y(F^2)_{\alpha\sigma^{-1}\beta^{-1}}|\beta \\
=(X(F^1)\ast F^2)_\alpha + (F^1\ast X(F^2))_\alpha + (\delta_1(F^1)\ast Y(F^2))_\alpha\\
\end{array}
\end{equation} In \eqref{4.26rty} above, we have used the fact that $0=\mu_{\beta^{-1}\beta}
=\mu_{\beta^{-1}}|\beta +\mu_\beta$. For $\alpha$, $\beta\in GL_2^+(\mathbb Q)$, it follows from \eqref{3.2.57X} that
\begin{equation}\label{216XXY}
\mu_{\alpha\sigma^{-1}}=\mu_{\alpha\sigma^{-1}\beta^{-1}\beta}=\mu_{\alpha\sigma^{-1}
\beta^{-1}}|\beta + \mu_{\beta}
\end{equation} Since $F^2\in \mathcal Q(\Gamma)$ we know from \eqref{3.2.58} that
$\delta_1(F^2)_{\alpha\sigma^{-1}\beta^{-1}}=
\mu_{\alpha\sigma^{-1}\beta^{-1}}\cdot F^2_{\alpha\sigma^{-1}\beta^{-1}}$. Combining with \eqref{216XXY}, we have:
\begin{equation}
\begin{array}{l}
\delta_1((F^1\ast F^2))_\alpha=\mu_{\alpha\sigma^{-1}}\cdot (F^1\ast F^2)_\alpha= \underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum} \mu_{\alpha\sigma^{-1}}\cdot (F^1_{\beta\sigma}\cdot (F^2_{
\alpha\sigma^{-1}\beta^{-1}}|\beta))\\
= \underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum}
(\mu_\beta\cdot F^1_{\beta\sigma})\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}|\beta)
+ \underset{\beta\in \Gamma\backslash GL_2^+(\mathbb Q)}{\sum} F^1_{\beta\sigma}
\cdot (\mu_{\alpha\sigma^{-1}\beta^{-1}}\cdot F^2_{\alpha\sigma^{-1}\beta^{-1}})|\beta\\
=(\delta_1(F^1)\ast F^2)_\alpha + (F^1\ast\delta_1(F^2))_\alpha
\end{array}
\end{equation} Finally, from the definition of $Y$, it is easy to show that $(Y(F^1\ast F^2))_\alpha
=(Y(F^1)\ast F^2)_\alpha + (F^1\ast Y(F^2))_\alpha$.
\end{proof}
\medskip
\medskip
\subsection{The operators $X_\tau:\mathcal Q_\sigma(\Gamma)\longrightarrow \mathcal Q_{\tau\sigma}(\Gamma)$ and
Hopf action}
\medskip
\medskip
Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup and choose some $\sigma\in
SL_2(\mathbb Z)$. In Section 4.1, we have only considered operators $X$, $Y$ and $\{\delta_n\}_{n\geq 1}$ that are endomorphisms of $\mathcal Q_\sigma(\Gamma)$. In this section, we will define an operator
\begin{equation}\label{4.2.28}
X_\tau:\mathcal Q_\sigma(\Gamma)\longrightarrow \mathcal Q_{\tau\sigma}(\Gamma)
\end{equation} for $\tau\in SL_2(\mathbb Z)$. In particular, we consider the commuting family $\left\{\rho_n:=\begin{pmatrix}
1 & n \\ 0 & 1 \\ \end{pmatrix}\right\}_{n\in \mathbb Z}$ of matrices in $SL_2(\mathbb Z)$ and write $\sigma(n):=
\rho_n\cdot \sigma$. Then, we have operators:
\begin{equation}\label{4.2.29}
X_{\rho_n}:\mathcal Q_{\sigma(m)}(\Gamma)\longrightarrow \mathcal Q_{\sigma(m+n)}(\Gamma) \qquad \forall\textrm{ }m,n\in \mathbb Z
\end{equation} acting ``between the levels'' of the graded module $\mathbb Q_\sigma(\Gamma):=
\underset{m\in \mathbb Z}{\bigoplus}\mathcal Q_{\sigma(m)}(\Gamma)$. We already know that $\mathcal Q_\sigma(\Gamma)$ carries an action of the Hopf algebra $\mathfrak h_1$. Further, $\mathfrak h_1$ has a Hopf action on the pairing on $\mathcal Q_\sigma(\Gamma)$ in the sense of Proposition \ref{Prop4.3}. We will now show that $\mathfrak h_1$ can be naturally embedded into a larger Hopf algebra $\mathfrak h_{\mathbb Z}$ acting on
$\mathbb Q_\sigma(\Gamma)$ that incorporates the operators $X_{\rho_n}$ in
\eqref{4.2.29}. Finally, we will show that the pairing
on $\mathcal Q_\sigma(\Gamma)$ can be extended to a pairing:
\begin{equation}
(\_\_,\_\_):\mathcal Q_{\sigma(m)}(\Gamma)\otimes \mathcal Q_{\sigma(n)}(\Gamma)
\longrightarrow \mathcal Q_{\sigma(m+n)}(\Gamma)\qquad \forall\textrm{ }m,n\in \mathbb Z
\end{equation} This gives us a pairing on $\mathbb Q_\sigma(\Gamma)$ and we prove that this pairing carries a Hopf action of
$\mathfrak h_{\mathbb Z}$. We start by defining the operators $X_\tau$ mentioned in \eqref{4.2.28}.
\medskip
\begin{thm}\label{Prop4.6} (a) Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of $SL_2(\mathbb Z)$
and choose $\sigma\in SL_2(\mathbb Z)$.
\medskip
(a) For each $\tau\in SL_2(\mathbb Z)$, we have a morphism:
\begin{equation}\label{4.2.31}
X_\tau:\mathcal Q_\sigma(\Gamma)\longrightarrow \mathcal Q_{\tau\sigma}(\Gamma)\qquad X_\tau(F)_\alpha
:=X(F_\alpha)||\tau^{-1}\qquad \forall\textrm{ }F\in \mathcal Q_\sigma(\Gamma),\textrm{ }\alpha\in GL_2^+(\mathbb Q)
\end{equation}
\medskip
(b) Let $\tau_1$, $\tau_2\in SL_2(\mathbb Z)$ be two matrices such that $\tau_1\tau_2=\tau_2\tau_1$. Then, the commutator $[X_{\tau_1},X_{\tau_2}]=0$.
\end{thm}
\begin{proof} (a) We choose any $F\in \mathcal Q_\sigma(\Gamma)$. From \eqref{4.2.31}, it is clear that $X_\tau(F)_{\gamma\alpha}=X_\tau(F)_\alpha$ for
any $\gamma\in \Gamma$ and $\alpha\in GL_2^+(\mathbb Q)$. Further, we note that:
\begin{equation}\label{4.33xp}
\begin{array}{ll}
X_{\tau}(F)_{\alpha\gamma}=X (F_{\alpha\gamma})||\tau^{-1}&= X(F_\alpha ||\sigma\gamma\sigma^{-1})||\tau^{-1} \\
& =X(F_\alpha||\tau^{-1})||\tau\sigma\gamma\sigma^{-1}\tau^{-1} \\
& = X_\tau(F_\alpha)||((\tau\sigma)\gamma(\sigma^{-1}\tau^{-1}))\\
\end{array}
\end{equation} It follows from \eqref{4.33xp} that $X_\tau(F)\in \mathcal Q_{\tau\sigma}(\Gamma)$ for any $F\in \mathcal Q_\sigma(\Gamma)$.
\medskip
(b) Since $\tau_1$ and $\tau_2$ commute, both $X_{\tau_1}X_{\tau_2}$ and
$X_{\tau_2}X_{\tau_1}$ are operators from $\mathcal Q_{\sigma}(\Gamma)$
to $\mathcal Q_{\tau_1\tau_2\sigma}(\Gamma)=\mathcal Q_{\tau_2\tau_1\sigma}(\Gamma)$. For any
$F\in \mathcal Q_\sigma(\Gamma)$, we have ($\forall$ $\alpha\in GL_2^+(\mathbb Q)$):
\begin{equation}
(X_{\tau_1}X_{\tau_2}(F))_\alpha=X(X_{\tau_2}(F)_\alpha)||\tau_1^{-1}=X^2(F_\alpha)||\tau_2^{-1}\tau_1^{-1}
=X^2(F_\alpha)||\tau_1^{-1}\tau_2^{-1} = (X_{\tau_2}X_{\tau_1}(F))_\alpha
\end{equation} This proves the result.
\end{proof}
\medskip
As mentioned before, we now consider the commuting family $\left\{\rho_n:=\begin{pmatrix}
1 & n \\ 0 & 1 \\ \end{pmatrix}\right\}_{n\in \mathbb Z}$ of matrices in $SL_2(\mathbb Z)$ and set $\sigma(n):=
\rho_n\cdot \sigma$ for any $\sigma\in SL_2(\mathbb Z)$. We want to define a pairing on the graded module
$\mathbb Q_\sigma(\Gamma)=\underset{m\in \mathbb Z}{\bigoplus}\mathcal Q_{\sigma(m)}(\Gamma)$ that extends the pairing on $\mathcal Q_\sigma(\Gamma)$. In fact, we will prove a more general result.
\medskip
\begin{thm} Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of $SL_2(\mathbb Z)$
and choose $\sigma\in SL_2(\mathbb Z)$. Let $\tau_1$, $\tau_2\in SL_2(\mathbb Z)$ be two matrices such that $\tau_1\tau_2=\tau_2\tau_1$. Then, there exists a pairining
\begin{equation}\label{4.34.2}
(\_\_,\_\_):\mathcal Q_{\tau_1\sigma}(\Gamma)\otimes \mathcal Q_{\tau_2\sigma}(\Gamma)
\longrightarrow \mathcal Q_{\tau_1\tau_2\sigma}(\Gamma)
\end{equation} defined as follows: for any $F^1\in \mathcal Q_{\tau_1\sigma}(\Gamma)$ and
any $F^2\in \mathcal Q_{\tau_2\sigma}(\Gamma)$, we set:
\begin{equation}\label{4.2.35}
(F^1,F^2)_\alpha:=\underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum} (F^1_{\beta\sigma}||\tau_2^{-1})\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}||\tau_2\sigma\beta\tau_1^{-1}\tau_2^{-1})
\qquad \forall\textrm{ }\alpha\in GL_2^+(\mathbb Q)
\end{equation} In particular, when $\tau_1=\tau_2=1$, the pairing in \eqref{4.2.35} reduces
to the pairing on $\mathcal Q_\sigma(\Gamma)$ defined in \eqref{4.4}.
\end{thm}
\begin{proof} We choose some $\gamma\in \Gamma$. Then, for any $\alpha\in GL_2^+(\mathbb Q)$, $\beta\in SL_2(\mathbb Z)$,
we have $F^1_{\gamma\beta\sigma}=F^1_{\beta\sigma}$ and:
\begin{equation*}
(F^2_{\alpha\sigma^{-1}\beta^{-1}
\gamma^{-1}}||\tau_2\sigma\gamma\beta\tau_1^{-1}\tau_2^{-1}) =(F^2_{\alpha\sigma^{-1}\beta^{-1}}||\tau_2\sigma\gamma^{-1}\sigma^{-1}\tau_2^{-1}\tau_2\sigma\gamma\beta
\tau_1^{-1}\tau_2^{-1})= (F^2_{\alpha\sigma^{-1}\beta^{-1}}||\tau_2\sigma\beta
\tau_1^{-1}\tau_2^{-1})
\end{equation*} It follows that the sum in \eqref{4.2.35} is well defined, i.e., independent of the choice
of coset representatives of $\Gamma$ in $SL_2(\mathbb Z)$. Additionally, we have:
\begin{equation}\label{4.2.36y}
(F^1,F^2)_{\alpha\gamma} :=\underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(F^1_{\beta\sigma}||\tau_2^{-1})\cdot (F^2_{\alpha\gamma\sigma^{-1}\beta^{-1}}||\tau_2
\sigma\beta
\tau_1^{-1}\tau_2^{-1})
\end{equation} We now set $\delta=\beta\sigma\gamma^{-1}
\sigma^{-1}$. Since $F^1\in \mathcal Q_{\tau_1\sigma}(\Gamma)$,
we know that $F^1_{\delta\sigma\gamma}=F^1_{\delta\sigma}||\tau_1\sigma\gamma\sigma^{-1}\tau_1^{-1}$. Then, we can rewrite the expression in \eqref{4.2.36y} as follows:
\begin{equation}\label{4.2.37}
\begin{array}{ll}
(F^1,F^2)_{\alpha\gamma} &=\underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(F^1_{\delta\sigma\gamma}||\tau_2^{-1})\cdot (F^2_{\alpha\sigma^{-1}\delta^{-1}}||\tau_2\sigma\delta\sigma\gamma\sigma^{-1}
\tau_1^{-1}\tau_2^{-1}) \\
&=\underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(F^1_{\delta\sigma}||\tau_1\sigma\gamma\sigma^{-1}\tau_1^{-1}\tau_2^{-1})\cdot (F^2_{\alpha\sigma^{-1}\delta^{-1}}||\tau_2\sigma\delta\sigma\gamma\sigma^{-1}
\tau_1^{-1}\tau_2^{-1}) \\
&=\left(\underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(F^1_{\delta\sigma}||\tau_2^{-1})\cdot (F^2_{\alpha\sigma^{-1}\delta^{-1}}||\tau_2\sigma\delta\tau_1^{-1}\tau_2^{-1})\right){||} \tau_1\tau_2\sigma\gamma\sigma^{-1}
\tau_1^{-1}\tau_2^{-1}\\
&=(F^1,F^2)_\alpha || \tau_1\tau_2\sigma\gamma\sigma^{-1}
\tau_1^{-1}\tau_2^{-1}\\
\end{array}
\end{equation} From \eqref{4.2.37} it follows that $(F^1,F^2)\in \mathcal Q_{\tau_1\tau_2\sigma}(\Gamma)$.
\end{proof}
\medskip
In particular, it follows from the pairing in \eqref{4.34.2} that for any $m$, $n\in \mathbb Z$, we have a pairing
\begin{equation}\label{4.38.2}
(\_\_,\_\_):\mathcal Q_{\sigma(m)}(\Gamma)\otimes
\mathcal Q_{\sigma(n)}(\Gamma)\longrightarrow \mathcal Q_{\sigma(m+n)}(\Gamma)
\end{equation} It is clear that \eqref{4.38.2} induces a pairing on $\mathbb Q_\sigma(\Gamma)=\underset{m\in \mathbb Z}{\bigoplus}\mathcal Q_{\sigma(m)}(\Gamma)$ for
each $\sigma\in SL_2(\mathbb Z)$. We will now define operators $\{X_n\}_{n\in \mathbb Z}$ and
$Z$ on $\mathbb Q_\sigma(\Gamma)$. For each $n\in \mathbb Z$, the operator
$X_n:\mathbb Q_\sigma(\Gamma)\longrightarrow \mathbb Q_\sigma(\Gamma)$ is induced
by the collection of operators:
\begin{equation}\label{4.2.39}
X_n^m:=X_{\rho_n}:\mathcal Q_{\sigma(m)}(\Gamma)\longrightarrow \mathcal Q_{\sigma(m+n)}(\Gamma)
\qquad \forall\textrm{ }m\in \mathbb Z
\end{equation} where, as mentioned before, $\rho_n=\begin{pmatrix} 1 & n \\ 0 & 1 \\ \end{pmatrix}
$. Then, $X_n:\mathbb Q_\sigma(\Gamma)\longrightarrow
\mathbb Q_\sigma(\Gamma)$ is an operator of homogeneous degree $n$ on the graded module
$\mathbb Q_\sigma(\Gamma)$. We also consider:
\begin{equation}\label{4.2.40}
Z:\mathcal Q_{\sigma(m)}(\Gamma)\longrightarrow \mathcal Q_{\sigma(m)}(\Gamma)
\qquad Z(F)_\alpha:=mF_\alpha +Y(F_\alpha)\qquad \forall\textrm{ }F\in \mathcal Q_{\sigma(m)}(
\Gamma), \alpha\in GL_2^+(\mathbb Q)
\end{equation} This induces an operator $Z:\mathbb Q_\sigma(\Gamma)
\longrightarrow \mathbb Q_\sigma(\Gamma)$ of homogeneous degree $0$ on
the graded module $\mathbb Q_\sigma(\Gamma)$. We will now show that $\mathbb Q_\sigma(\Gamma)$
is acted upon by a certain Lie algebra $\mathfrak l_{\mathbb Z}$ such that the Lie action incorporates
the operators $\{X_n\}_{n\in \mathbb Z}$ and $Z$ mentioned above. We define $\mathfrak l_{\mathbb Z}$
to be the Lie algebra with generators $\{Z,X_n|n\in \mathbb Z\}$ satisfying the following commutator
relations:
\begin{equation}\label{4.2.41}
[Z,X_n]=(n+1)X_n \qquad [X_n,X_{n'}]=0 \qquad \forall\textrm{ }n,n'\in \mathbb Z
\end{equation} In particular, we note that $[Z,X_0]=X_0$. It follows that the Lie algebra
$\mathfrak l_{\mathbb Z}$ contains the Lie algebra $\mathfrak l_1$ defined in \eqref{3.59uu}. We now
describe the action of $\mathfrak l_{\mathbb Z}$ on $\mathbb Q_\sigma(\Gamma)$.
\medskip
\begin{thm} Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of $SL_2(\mathbb Z)$
and let $\sigma\in SL_2(\mathbb Z)$. Then, the Lie algebra $\mathfrak l_{\mathbb Z}$ has a Lie action
on $\mathbb Q_\sigma(\Gamma)$.
\end{thm}
\begin{proof} We need to check that $[Z,X_n]=(n+1)X_n$ and $[X_n,X_{n'}]=0$, $\forall$ $n$, $n'\in \mathbb Z$
for the operators $\{Z, X_n | n\in\mathbb Z\}$ on $\mathbb Q_\sigma(\Gamma)$. From part (b) of
Proposition \ref{Prop4.6}, we know that $[X_n,X_{n'}]=0$. From \eqref{4.2.39} and \eqref{4.2.40}, it is clear that in order to show that $[Z,X_n]=(n+1)X_n$, we need to check that
$[Z,X_n^m]=(n+1)X_n^m:\mathcal Q_{\sigma(m)}(\Gamma)\longrightarrow
\mathcal Q_{\sigma(m+n)}(\Gamma)$ for any given $m\in \mathbb Z$. For any $F\in \mathcal Q_{\sigma(m)}(\Gamma)$ and any $\alpha\in GL_2^+(\mathbb Q)$, we now check that:
\begin{equation}\label{4.2.42}
\begin{array}{c}
(ZX_n^m(F))_\alpha=(n+m)X_n^m(F)_\alpha+Y(X_n^m(F)_\alpha)=
(n+m)X(F_\alpha)||\rho_n^{-1}+YX(F_\alpha)||\rho_n^{-1}\\\
(X_n^mZ(F))_\alpha = X(Z(F)_\alpha)||\rho_n^{-1}= mX(F_\alpha)||\rho_n^{-1}+XY(F_\alpha)||\rho_n^{-1} \\
\end{array}
\end{equation} Combining \eqref{4.2.42} with the fact that $[Y,X]=X$, it follows that
$[Z,X_n^m]=(n+1)X_n^m$ for each $m\in \mathbb Z$. Hence, the result follows.
\end{proof}
\medskip
We now consider the Hopf algebra $\mathfrak h_{\mathbb Z}$ that is the universal enveloping algebra
of the Lie algebra $\mathfrak l_{\mathbb Z}$. Accordingly, the coproduct $\Delta$ on $\mathfrak h_{\mathbb Z}$
is given by:
\begin{equation}\label{4.2.43}
\Delta(X_n)=X_n\otimes 1+1\otimes X_n \qquad \Delta(Z)=Z\otimes 1+1\otimes Z\qquad
\forall\textrm{ }n\in \mathbb Z
\end{equation} It is clear that $\mathfrak h_{\mathbb Z}$ contains the Hopf algebra $\mathfrak h_1$, the
universal enveloping algebra of $\mathfrak l_1$. From Proposition \ref{Prop4.3}, we know that
$\mathfrak h_1$ has a Hopf action on the pairing on $\mathcal Q_\sigma(\Gamma)$. We want to show
that $\mathfrak h_{\mathbb Z}$ has a Hopf action on the pairing on $\mathbb Q_\sigma(\Gamma)$. For this, we prove the following Lemma.
\medskip
\begin{lem}\label{lem49x} Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of $SL_2(\mathbb Z)$ and let $\sigma
\in SL_2(\mathbb Z)$. Let $\tau_1$, $\tau_2$,
$\tau_3\in SL_2(\mathbb Z)$ be three matrices such that $\tau_i\tau_j=\tau_j\tau_i$,
$\forall$ $i$, $j\in \{1,2,3\}$. Then, for any $F^1\in \mathcal Q_{\tau_1\sigma}
(\Gamma)$, $F^2\in \mathcal Q_{\tau_2\sigma}(\Gamma)$, we have:
\begin{equation}\label{4.2.44}
X_{\tau_3}(F^1,F^2)=(X_{\tau_3}(F^1),F^2)+ (F^1,X_{\tau_3}(F^2))
\end{equation}
\end{lem}
\begin{proof} Consider any $\alpha\in GL_2^+(\mathbb Q)$. Then, from the definition of
$X_{\tau_3}$, it follows that
\begin{equation}\label{4.2.45}
\begin{array}{l}
X_{\tau_3}(F^1, F^2)_\alpha = \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
X((F^1_{\beta\sigma}||\tau_2^{-1})\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}||\tau_2
\sigma\beta
\tau_1^{-1}\tau_2^{-1}))||\tau_3^{-1} \\
= \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(X(F^1_{\beta\sigma})||\tau_2^{-1}\tau_3^{-1})\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}||\tau_2\sigma\beta
\tau_1^{-1}\tau_2^{-1}\tau_3^{-1}) \\ \quad \quad + \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(F^1_{\beta\sigma}||\tau_2^{-1}\tau_3^{-1})\cdot (X(F^2_{\alpha\sigma^{-1}\beta^{-1}})||\tau_2\sigma\beta
\tau_1^{-1}\tau_2^{-1}\tau_3^{-1}) \\
\end{array}
\end{equation} Since $F^1\in \mathcal Q_{\tau_1\sigma}(\Gamma)$, it follows that
$X_{\tau_3}(F^1)\in \mathcal Q_{\tau_1\tau_3\sigma}(\Gamma)$. Similarly, we see that
$X_{\tau_3}(F^2)\in \mathcal Q_{\tau_2\tau_3\sigma}(\Gamma)$. It follows that:
\begin{equation}\label{4.2.46}
\begin{array}{ll}
(X_{\tau_3}(F^1),F^2)_\alpha &= \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(X_{\tau_3}(F^1)_{\beta\sigma}||\tau_2^{-1})\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}||\tau_2\sigma\beta
\tau_1^{-1}\tau_2^{-1}\tau_3^{-1})\\
& = \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(X(F^1_{\beta\sigma})||\tau_2^{-1}\tau_3^{-1})\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}||\tau_2\sigma\beta
\tau_1^{-1}\tau_2^{-1}\tau_3^{-1})\\
(F^1,X_{\tau_3}(F^2))_\alpha & = \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(F^1_{\beta\sigma}||\tau_2^{-1}\tau_3^{-1})\cdot (X_{\tau_3}(F^2)_{\alpha\sigma^{-1}\beta^{-1}}||\tau_2\tau_3\sigma\beta
\tau_1^{-1}\tau_2^{-1}\tau_3^{-1}) \\
& = \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(F^1_{\beta\sigma}||\tau_2^{-1}\tau_3^{-1})\cdot (X(F^2_{\alpha\sigma^{-1}\beta^{-1}})||\tau_3^{-1}\tau_2\tau_3\sigma\beta
\tau_1^{-1}\tau_2^{-1}\tau_3^{-1}) \\
& = \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(F^1_{\beta\sigma}||\tau_2^{-1}\tau_3^{-1})\cdot (X(F^2_{\alpha\sigma^{-1}\beta^{-1}})||\tau_2\sigma\beta
\tau_1^{-1}\tau_2^{-1}\tau_3^{-1}) \\
\end{array}
\end{equation} Comparing \eqref{4.2.45} and \eqref{4.2.46}, the result of
\eqref{4.2.44} follows.
\end{proof}
\medskip
As a special case of Lemma \ref{lem49x}, it follows that for any
$F^1\in \mathcal Q_{\sigma(m)}(\Gamma)$ and $F^2\in \mathcal Q_{\sigma(m')}(\Gamma)$,
we have:
\begin{equation}\label{4.2.47}
X_{\rho_n}(F^1,F^2)=X_n(F^1,F^2)=(X_n(F^1),F^2)+(F^1,X_n(F^2))\qquad \forall \textrm{ }n\in \mathbb Z
\end{equation}
We conclude by showing that $\mathfrak h_{\mathbb Z}$ has a Hopf action
on the pairing on $\mathbb Q_\sigma(\Gamma)$.
\medskip
\begin{thm} Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of $SL_2(\mathbb Z)$ and let $\sigma
\in SL_2(\mathbb Z)$. Then, the Hopf algebra
$\mathfrak h_{\mathbb Z}$ has a Hopf action on the pairing on $\mathbb Q_\sigma(\Gamma)$.
In other words, for $F^1$, $F^2\in \mathbb Q_{\sigma}(\Gamma)$, we have
\begin{equation}\label{4.2.48}
h(F^1,F^2)=\sum (h_{(1)}(F^1),h_{(2)}(F^2))
\end{equation} where the coproduct $\Delta:\mathfrak h_{\mathbb Z}
\longrightarrow \mathfrak h_{\mathbb Z}\otimes \mathfrak h_{\mathbb Z}$ is defined
by setting $\Delta(h):=\sum h_{(1)}\otimes h_{(2)}$ for each $h\in \mathfrak h_{\mathbb Z}$.
\end{thm}
\begin{proof} It suffices to prove the result in the case where $F^1\in \mathcal Q_{\sigma(m)}(\Gamma)$,
$F^2\in \mathcal Q_{\sigma(m')}(\Gamma)$ for some $m$, $m'\in \mathbb Z$. Further, it suffices
to prove the relation \eqref{4.2.48} for the generators $\{Z,X_n|n\in \mathbb Z\}$ of the Hopf algebra
$\mathfrak{h}_{\mathbb Z}$. For the generators $X_n$, $n\in \mathbb Z$, this is already the result of
\eqref{4.2.47} which follows from Lemma \ref{lem49x}. Since $\Delta(Z)=Z\otimes 1+1\otimes Z$, it remains
to show that \begin{equation}
Z(F^1,F^2)=(Z(F^1),F^2)+(F^1,Z(F^2)) \qquad \forall\textrm{ }F^1\in \mathcal Q_{\sigma(m)}(\Gamma),
F^2\in \mathcal Q_{\sigma(m')}(\Gamma)
\end{equation} By the definition of the pairing on $\mathbb Q_\sigma(\Gamma)$, we know that $(F^1,F^2)\in
\mathcal Q_{\sigma(m+m')}(\Gamma)$. Then, for any $\alpha\in GL_2^+(\mathbb Q)$, we have:
\begin{equation}
\begin{array}{ll}
Z(F^1,F^2)_\alpha & = (m+m')(F^1,F^2)_\alpha + Y(F^1,F^2)_\alpha \\
& = (m+m') \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
((F^1_{\beta\sigma}||\rho_{m'}^{-1})\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}||
\rho_{m'}\sigma\beta\rho_{m}^{-1}\rho_{m'}^{-1})) \\ & \quad + \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
Y((F^1_{\beta\sigma}||\rho_{m'}^{-1})\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}||
\rho_{m'}\sigma\beta\rho_{m}^{-1}\rho_{m'}^{-1}))\\
& = \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
((mF^1_{\beta\sigma}+Y(F^1_{\beta\sigma}))||\rho_{m'}^{-1})\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}||
\rho_{m'}\sigma\beta\rho_{m}^{-1}\rho_{m'}^{-1}) \\
&\quad + \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(F^1_{\beta\sigma}||\rho_{m'}^{-1})\cdot ((m'F^2_{\alpha\sigma^{-1}\beta^{-1}}
+Y(F^2_{\alpha\sigma^{-1}\beta^{-1}}))||\rho_{m'}\sigma\beta\rho_{m}^{-1}\rho_{m'}^{-1})\\
& = \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(Z(F^1)_{\beta\sigma}||\rho_{m'}^{-1})\cdot (F^2_{\alpha\sigma^{-1}\beta^{-1}}||
\rho_{m'}\sigma\beta\rho_{m}^{-1}\rho_{m'}^{-1}) \\
&\quad + \underset{\beta\in \Gamma\backslash SL_2(\mathbb Z)}{\sum}
(F^1_{\beta\sigma}||\rho_{m'}^{-1})\cdot (Z(F^2)_{\alpha\sigma^{-1}\beta^{-1}}||\rho_{m'}\sigma\beta\rho_{m}^{-1}\rho_{m'}^{-1})\\
&= (Z(F^1),F^2)_\alpha+(F^1,Z(F^2))_\alpha \\
\end{array}
\end{equation}
\end{proof}
\medskip
\medskip
|
1,314,259,996,550 | arxiv | \section{Introduction}
In cluster populations with a power law cluster initial mass function (CIMF) the highest appearing mass in a sample is determined by the number of clusters (besides statistical fluctuations). This is no longer true if there exists a physical upper mass limit for clusters.
Information about the CIMF can be derived from the present day luminosity function (LF). Although there is not a one-to-one relation between the CIMF and the LF, modeling the LF with an artificial cluster population with varying CIMF parameters can aid the interpretation of the LF. According to the models of \cite{gieles06}, a bend in the LF (i.e. a double power law distribution function) may be the signature of a CIMF, truncated at the high mass end. The (filter dependent) location of the bend is intimately connected to the value of the upper mass limit (brighter bends correspond to higher possible cluster masses).
Bends in the LF are observed in NGC 6946, M51 \cite{gieles06, gieles06a} and NGC 4038/4039 \cite{whitmore99}. Here we again examine the cluster population of M51, dividing it in subsamples at different galactocentric radii. By comparing to the artificial populations we draw conclusions on the CIMF and cluster disruption parameters across the disk.
\section{The cluster population of M51}
We use the Hubble Heritage ACS mosaic of M51 in the passbands \textit{F435W}, \textit{F555W} and \textit{F814W}, covering the complete system of M51 and its companion \cite{mutchler05}. Point sources are extracted and qualified as a cluster if (1) the source is detected in all three broadband filters, (2) the radius of the source was fit better with an extended cluster profile than with a delta function (using the \textit{ISHAPE} algorithm \cite{larsen99}) (3) the cluster has a radius of at least 0.5 pc (the resolution of the \textit{ISHAPE} routine), and (4) the nearest neighbouring source is at least 5 pixels away.
After performing aperture photometry on all sources, the LF can be
constructed for the population brighter than the 95\% completeness limit (-6.3 abs mag \textit{F435W} and \textit{F555W} and -6.0 abs mag for \textit{F814W}) and for three galactocentric distance intervals. Results can be seen in Table~\ref{tab:fits}.
\begin{table*}
\caption{Fit results of the complete sample in all three pass bands. (1) is the passband, (2) the number of clusters within the fit range. (3), (4) and (5) contain both slopes and the location of the bend of the double power law fit respectively. The second part contains for the B band the galactocentric distance dependence. Similar results are obtained in the other filters.}
\label{tab:fits}
\centering
\begin{tabular}{l l l l l}
\hline\hline
Passband & N & $\alpha_1$ & $\alpha_2$ & $M_{\textrm{\tiny{bend}}}$ \\
\hline
\textit{F435W} & 3891 & 1.96 $\pm$ 0.04 & 2.52 $\pm$ 0.08 & -8.33 $\pm$ 0.15\\
\textit{F555W} & 4750 & 1.99 $\pm$ 0.04 & 2.56 $\pm$ 0.07 & -8.38 $\pm$ 0.13\\
\textit{F814W} & 8041 & 2.08 $\pm$ 0.02 & 2.54 $\pm$ 0.08 & -8.90 $\pm$ 0.16\\
\hline
\hline
$F435W$: & N & $\alpha_1$ & $\alpha_2$ & $M_{\textrm{\tiny{bend}}}$ \\
\hline
0 $< d <$ 3 kpc & 1267 & 1.67 $\pm$ 0.06 & 2.60 $\pm$ 0.17 & -8.76 $\pm$ 0.17\\
3 $< d <$ 5.5 kpc & 1415 & 2.08 $\pm$ 0.05 & 2.71 $\pm$ 0.22 & -8.42 $\pm$ 0.22\\
5.5 $< d <$ 8.5 kpc & 1209 & 2.17 $\pm$ 0.03 & 2.55 $\pm$ 0.31 & -7.99 $\pm$ 0.31\\
\hline
\end{tabular}
\end{table*}
\section{Conclusions on upper mass limits and disruption parameters}
Although more detailed modeling is necessary for quantitative results, the conclusions following the models of \cite{gieles06} for the galactocentric dependence of the upper mass limit and disruption parameters are:
\begin{enumerate}
\item Subsets closer to the center have their bends at brighter magnitudes, suggesting higher possible cluster masses.
\item Subsets closer to the center have shallower faint-end slopes, indicating faster disruption.
\end{enumerate}
The two effects are not to be seen separately though. Faster disruption alters the location of the bend in the same direction as a higher upper mass limit. In a follow up study the statistical significance of the results will be further investigated and a more quantitative analysis will be given \cite{haas06}.
|
1,314,259,996,551 | arxiv |
\subsection{Dataset}\label{subsec:data}
We evaluate the performance of our method on StudentLife dataset~\cite{wang2014studentlife}
which is collected from 48 Dartmouth students over a 10 week term.
It contains sensor data,
EMA data (Ecological Momentary Assessment, i.e. stress),
pre and post survey responses (i.e. PHQ-9 depression scale) and educational data (GPA).
During the 10 weeks, students carry their phones throughout the day.
Data streams from multiple sensors, including accelerometer, microphone, GPS, Bluetooth, light sensor, phone charge, phone lock, and WiFi are
collected in real time by the mobile phone.
Besides, students are asked to respond to various EMA questions and surveys, which are provided by psychologists to measure their mental health status.
Educational performance data, such as the grades are also collected.
In our experiment,
we use data streams collected by three representative sensors (i.e., accelerometer, microphone, WiFi) as the input, since these three kinds of sensor data
contain most useful information to reflect the individual behaviors.
The reason for collecting location information with wifi instead of GPS is
that most of students' activities are in indoor environment,
where the college's WiFi AP deployment is more effective to accurately infer the location information than the GPS.
For the groundtruth annotations of the multi-source data streams,
we use \textit{photographic affect meter (PAM)}~\cite{pollak2011pam}
values in EMA data.
The PAM value practically represents a score between 1-16 which
represents the Positive and Negative Affect Schedule (PANAS)\cite{watson1988development} and reflects the instantaneous mental health status of users.
The annotations are collected by a mobile application which captures user' feeling according to user' preference of specific photos.
To keep with the conceptualization of PANAS which ranges from low pleasure and low arousal to high pleasure and high arousal,
the PAM score is further divided into 4 quadrants: negative valence and low arousal with score 1-4, negative valence and high arousal with score 5-8, positive valence and low arousal with score 9-12, and positive valence and high arousal with score 13-16.
Following \cite{pollak2011pam},
we map the PAM value into the above 4 classes.
Finally,
we use data streams of 30 students who have valid PAM annotations.
Specifically, we get 912 samples in total, each sample consists of 3-day multi-modal data streams
collected by the sensors of accelerometer, microphone, and wifi.
For each sample, the instantaneous PAM label of the last day is regarded as the groundtruth label of the whole data streams.
The sample number for each student and the sample distribution on 4 classes are shown in Figure~\ref{dataset}.
Up to now, the mental health status prediction task in our experiment is practically a 4-class classification problem.
For the training and test, we split our datasets into 10 splits and build 10 tasks on them. Each task takes 9 splits for training and the remaining one for test.
We also show the average results of all 10 tasks.
\begin{figure}[t]
\centering
\subfigure[Sample number for each student]{
\begin{minipage}[t]{0.5\linewidth}
\centering
\includegraphics[width=6cm, height=4cm]{user_len.png}
\end{minipage}%
}%
\subfigure[Label distribution]{
\begin{minipage}[t]{0.5\linewidth}
\centering
\includegraphics[width=6cm, height=4.5cm]{label.png}
\end{minipage}%
}%
\centering
\caption{{Dataset information. (a) The number of samples for each student. (b) The distribution over 4 classes.}}
\label{dataset}
\end{figure}
\textbf{Discussion on data selection.}
In this paper, our aim is to model the personal behavior and predict the health status based on the multi-source sensor data collected from people's daily life.
To the best of our knowledge,
StudentLife is the only public dataset that satisfies our requirements.
On the one hand, the data should be collected from both the healthy and unhealthy people.
On the other hand, the sensor data we take advantage of should be continuous and long-term,
i.e.,
the data are recorded as long as the people carry their wearable devices, such as mobile phone and smart wristband.
Some work has been done on this task.
For example, \cite{canzian2015trajectories} uses GPS data to predict depression with a SVM classifier.
\cite{burns2011harnessing} predicts depression based on GPS, accelerometer and light sensor data from smartphone.
However, all these work does not release their data.
Although some medical datasets, such as MIMIC-III~\cite{johnson2016mimic} and ANDI~\cite{ridge2018assembly},
have been widely used to predict diseases~\cite{che2017rnn} or other medical related events~\cite{futoma2015comparison},
they do not satisfy the need of our task.
Firstly, these medical datasets pay more attention to disease analysis where the studied people are mainly patients.
By contrast, our work focuses on personal health, where daily behaviors are considered for both healthy and unhealthy people.
Secondly, these medical datasets are discretely collected from patients during the hospital treatment process with professional medical equipment, such as medical imaging, while the long-term sensor data we use is continuously collected from daily life.\\
\subsection{Implementation Details}\label{subsec:impl}
As introduced in Section \ref{sec:method},
to create the behavior graph which reflects the structure information
contained in the multi-source data streams,
we firstly need to detect the behavior-related concepts contained in the data streams.
Here, three backbone models proposed in~\cite{lane2011bewell} are adopted to get the middle-level semantic concepts from raw sensor data.
For the accelerometer,
a decision tree model for \textit{activity} classification is used with the features extracted from the accelerometer stream to infer the concept class (i.e. $stationary$, $walking$, $running$, and $unknown$).
For the microphone,
the $audio$ data are classified into concept classes (i.e. $silence$, $voice$, $noise$, and $other$) with a HMM model.
As for the wifi,
students' wifi scan logs are firstly recorded, then the $location$ concept classes,
such as \textit{in[dana-library]},
are inferred according to the WiFi AP deployment information,
which results in a number of 9037 classes.
The $location$ classes are in a long-tail distribution with many classes appear few times,
hence we choose the top 100 most frequent classes, which cover 93\% of the $location$ data.
After obtaining the three kinds of detected concepts, we cut the sequences in days and build the local context graph and the global temporal graph
to predict mental health status with the method illustrated in Section~\ref{sec:method}.
We use the metrics of accuracy, precision, recall and F1-score to evaluate our model.
Accuracy is the ratio of correctly predicted samples to total predicted samples. Precision, recall and F1-score are firstly calulated in each class and then weighted by the sample number of each class.
\textbf{Discussion on data synchronization.}
In existing health related systems and methods for analyzing wearable sensor data,
such as the risk situation recognition system~\cite{yebda2019multi},
synchronization of different sensors is a very important issue.
Specifically, the system always contains several devices to collect different kinds of sensor data as well as a smartphone to receive the data from sensors.
An algorithm of data synchronisation is necessary since the sensors are on different devices and there exist time differences between sensor data generated by sensors and received by smartphones.
As for our work,
the sensors here we use are all embedded in the same smartphone~\cite{wang2014studentlife},
which have a common reference time naturally and do not have the time error when sending and receiving the data, thus the synchronisation is not essential.
\subsection{Compared Methods}\label{subsec:compare_method}
Since there are no previous work on PAM prediction on the same dataset,
we compare our method
with three popularly used conventional machine learning algorithms (i.e., RF~\cite{598994}, KNN~\cite{Laaksonen2002Classification}, and SVM~\cite{burges1998tutorial})
and two deep learning algorithms (i.e., DNN~\cite{lecun2015deep}, LSTM~\cite{hochreiter1997long}).
To apply these baselines on multi-source data streams collected from wearable devices,
we compute the~\textit{behavior feature}
by extracting a 108-dimensional feature vector
to represent the duration time of three kinds of concept classes in one day.
Specifically, the $activity$ concept takes 4 elements of the vector, which represent the duration time of \textit{stationary, walking, running, unknown} in one day.
The $audio$ concept takes 4 elements, which represent the duration time of \textit{silence, voice, noise, other} in one day.
The $location$ concept takes 100 elements, which represent the duration time of 100 locations student stay in one day.
The~\textit{behavior feature} contains the principle information of the individual life,
such as where did she/he go in that day and how long did she/he communicate with other people.
We also compare our method with a recent GNN-based method (i.e., HAN~\cite{wang2019heterogeneous}) which can capture the structural information of a graph with different kinds of nodes.
Details of the compared baselines and the variants of our method are illustrated as follows.
\begin{itemize}
\item \textbf{RF\cite{598994}:}
This method uses Random Forest to do the classification.
Specifically, we concatenate the~\textit{behavior features} of three days to get a 324-dimensional feature.
%
Then we input it to the Random Forest.
\item \textbf{KNN\cite{Laaksonen2002Classification}:}
This method uses K-nearest neighbor.
We first get the 324-dimensional feature of continuous three days as in~\textbf{RF}.
%
Then we train a K-nearest neighbor based on the feature.
\item \textbf{SVM\cite{burges1998tutorial}:}
%
This method trains SVM to do the classification.
Features are obtained as in~\textbf{RF}, and then a SVM is trained to predict the health status.
\item \textbf{DNN\cite{lecun2015deep}:}
This method uses
a two-layer deep neural network with the input feature computed as in~\textbf{RF}.
%
Each layer is fully-connected and the hidden size is 50, which is determined by cross-validation.
\item \textbf{LSTM\cite{hochreiter1997long}:}
%
This method uses a LSTM with the hidden size of 100 to capture the temporal information of sequences.
Specifically, we transform the 3-day data into a sequence with the length of 3.
%
Each element in the sequence is a 108-dimensional~\textit{behavior feature}.
%
Then the sequence is input into LSTM, and the hidden state at the last step is used to predict the health status.
\item \textbf{HAN\cite{wang2019heterogeneous}:}
%
This method uses the heterogeneous attention network~\cite{wang2019heterogeneous} instead of our local context graph modeling method to learn the node embedding and graph representation.
Specifically, we get the meta-path based neighbors for each kind of nodes according to our local context graph.
%
Then the node-level attention and semantic-level attention are performed as \textbf{HAN\cite{wang2019heterogeneous}} to get the local context graph representation, which is finally input into the self-attention network to predict the health status.
\item \textbf{Ours$\backslash$he:}
%
This variant of the proposed method omits the heterogeneous message passing in Section 3.4.1 while keeps the homogeneous message passing in the heterogeneous graph neural network.
\item \textbf{Ours$\backslash$ho:}
%
This variant omits the homogeneous message passing in Section 3.4.1 while keeps the heterogeneous message passing in the heterogeneous graph neural network.
%
It is used to compare with Ours$\backslash$he to illustrate the impact of the homogeneous and heterogeneous message passing.
%
\item \textbf{Ours$\backslash$s:}
%
In this variant, we omit the semantic representation learned by Equation (4), and only use the structural representation of the local context graph.
The structural representation is then input into the self-attention network to get the global temporal graph representation.
%
\item \textbf{Ours$\backslash$t:}
%
In this variant, we omit the structural representation learned by Equation (6), and only use the semantic representation of the local context graph.
The semantic representation is then input into the self-attention network to get the global temporal graph representation.
\item \textbf{Ours$\backslash$g:}
In this variant,
representations of all local context graphs are directly added to get the final global temporal graph representation without using the self-attention network.\\
\end{itemize}
\renewcommand{\arraystretch}{1.5}
\begin{table}[tp]
\setlength\tabcolsep{1pt}
\scriptsize
\centering
\caption{PAM prediction results.(A: Accuracy, P: Precision, R: Recall, F1: F1 score)}\label{tab:res}
\begin{threeparttable}
\begin{tabular}{ccccccccccccccccccccc}
\toprule
\multirow{2}{*}{Method}&
\multicolumn{4}{c}{T1}&\multicolumn{4}{c}{T2}&\multicolumn{4}{c}{T3}&\multicolumn{4}{c}{T4}&\multicolumn{4}{c}{T5} \cr
\cmidrule{2-5} \cmidrule{6-9} \cmidrule{10-13} \cmidrule{14-17} \cmidrule{18-21}
& A & P & R &F1 & A & P & R &F1 & A & P & R &F1 & A & P & R &F1 & A & P & R &F1 \cr
\midrule
RF\cite{598994}
& 0.34 & 0.31 & 0.34 & 0.32
& 0.33 & 0.30 & 0.33 & 0.31
& 0.33 & 0.29 & 0.33 & 0.31
& 0.34 & 0.30 & 0.34 & 0.32
& 0.34 & 0.30 & 0.34 & 0.32
\cr
KNN\cite{Laaksonen2002Classification}
& 0.36 & 0.32 & 0.36 & 0.34
& 0.36 & 0.31 & 0.36 & 0.34
& 0.35 & 0.31 & 0.35 & 0.33
& 0.34 & 0.30 & 0.34 & 0.32
& 0.35 & 0.31 & 0.35 & 0.33
\cr
SVM\cite{burges1998tutorial}
& 0.37 & 0.37 & 0.37 & 0.37
& 0.37 & 0.33 & 0.37 & 0.35
& 0.36 & 0.32 & 0.36 & 0.34
& 0.36 & 0.31 & 0.36 & 0.34
& 0.37 & 0.33 & 0.37 & 0.35
\cr
\midrule
DNN\cite{lecun2015deep}
& 0.39 & 0.35 & 0.39 & 0.35
& 0.39 & 0.34 & 0.39 & 0.36
& 0.40 & 0.34 & 0.40 & 0.37
& 0.37 & 0.33 & 0.37 & 0.35
& 0.39 & 0.34 & 0.39 & 0.36
\cr
LSTM\cite{hochreiter1997long}
& 0.41 & 0.37 & 0.41 & 0.39
& 0.41 & 0.35 & 0.41 & 0.37
& 0.42 & 0.36 & 0.42 & 0.38
& 0.43 & 0.41 & 0.43 & 0.41
& 0.41 & 0.35 & 0.41 & 0.37
\cr
HAN\cite{wang2019heterogeneous}
& 0.42 & 0.39 & 0.42 & 0.40
& 0.43 & 0.41 & 0.43 & 0.41
& 0.42 & 0.40 & 0.42 & 0.41
& 0.42 & 0.41 & 0.44 & 0.42
& 0.42 & 0.39 & 0.42 & 0.40
\cr
\midrule
Ours
&\textbf{0.47} &\textbf{0.44}&\textbf{0.47}&\textbf{0.43}
&\textbf{0.48} &\textbf{0.42}&\textbf{0.48}&\textbf{0.44}
&\textbf{0.47} &\textbf{0.46}&\textbf{0.47}&\textbf{0.45}
&\textbf{0.46} &\textbf{0.45}&\textbf{0.46}&\textbf{0.44}
&\textbf{0.48} &\textbf{0.47}&\textbf{0.48}&\textbf{0.46}
\cr
\bottomrule
\end{tabular}
\end{threeparttable}
\begin{threeparttable}
\begin{tabular}{ccccccccccccccccccccc}
\toprule
\multirow{2}{*}{Method}&
\multicolumn{4}{c}{T6}&\multicolumn{4}{c}{T7}&\multicolumn{4}{c}{T8}&\multicolumn{4}{c}{T9}&\multicolumn{4}{c}{T10} \cr
\cmidrule{2-5} \cmidrule{6-9} \cmidrule{10-13} \cmidrule{14-17} \cmidrule{18-21}
& A & P & R &F1 & A & P & R &F1 & A & P & R &F1 & A & P & R &F1 & A & P & R & F1 \cr
\midrule
RF\cite{598994}
& 0.33 & 0.34 & 0.33 & 0.33
& 0.33 & 0.34 & 0.33 & 0.33
& 0.33 & 0.29 & 0.33 & 0.31
& 0.34 & 0.30 & 0.34 & 0.32
& 0.33 & 0.29 & 0.33 & 0.31 \cr
KNN\cite{Laaksonen2002Classification}
& 0.35 & 0.31 & 0.35 & 0.33
& 0.36 & 0.31 & 0.36 & 0.33
& 0.36 & 0.34 & 0.36 & 0.35
& 0.35 & 0.31 & 0.35 & 0.33
& 0.35 & 0.34 & 0.35 & 0.34 \cr
SVM\cite{burges1998tutorial}
& 0.35 & 0.31 & 0.35 & 0.33
& 0.36 & 0.31 & 0.36 & 0.34
& 0.37 & 0.33 & 0.37 & 0.35
& 0.37 & 0.33 & 0.37 & 0.35
& 0.37 & 0.37 & 0.37 & 0.37 \cr
\midrule
DNN\cite{lecun2015deep}
& 0.40 & 0.34 & 0.40 & 0.37
& 0.37 & 0.37 & 0.37 & 0.37
& 0.39 & 0.34 & 0.39 & 0.36
& 0.37 & 0.33 & 0.37 & 0.35
& 0.39 & 0.33 & 0.39 & 0.36 \cr
LSTM\cite{hochreiter1997long}
& 0.42 & 0.36 & 0.42 & 0.38
& 0.41 & 0.35 & 0.41 & 0.37
& 0.42 & 0.36 & 0.42 & 0.38
& 0.42 & 0.40 & 0.42 & 0.41
& 0.41 & 0.35 & 0.41 & 0.37
\cr
HAN\cite{wang2019heterogeneous}
& 0.43 & 0.41 & 0.43 & 0.41
& 0.43 & 0.40 & 0.43 & 0.40
& 0.43 & 0.41 & 0.43 & 0.41
& 0.42 & 0.39 & 0.42 & 0.40
& 0.42 & 0.41 & 0.42 & 0.42 \cr
\midrule
Ours
&\textbf{0.47} &\textbf{0.46}&\textbf{0.47}&\textbf{0.45}
&\textbf{0.47} &\textbf{0.46}&\textbf{0.47}&\textbf{0.45}
&\textbf{0.47} &\textbf{0.43}&\textbf{0.47}&\textbf{0.44}
&\textbf{0.48} &\textbf{0.47}&\textbf{0.48}&\textbf{0.46}
&\textbf{0.47} &\textbf{0.46}&\textbf{0.47}&\textbf{0.45}
\cr
\bottomrule
\end{tabular}
\end{threeparttable}
\end{table}
\subsection{Result Analysis}
\subsubsection{Performance Comparison}
Here we show both the results on 10 tasks in Table~\ref{tab:res} and Table~\ref{tab:ablation} and the average results in Figure~\ref{result}.
It can be seen that our model performs better than baselines on all metrics, and most variants of the proposed method also have good results.
Compared with the traditional machine learning methods, all the deep learning based methods perform better. Baseline-LSTM has better results than Baseline-DNN since it takes the temporal information into consideration. The HAN updates the node embedding and graph representation in a meta-path way with several kinds of adjacent matrix. However, this method does not consider the direct connection between heterogeneous nodes, which ignores the semantic interaction between different kinds of nodes thus
it performs worse than our method.
\begin{table}[tp]
\setlength\tabcolsep{1pt}
\scriptsize
\centering
\caption{Ablation study results.(A: Accuracy, P: Precision, R: Recall, F1: F1 score)}\label{tab:ablation}
\begin{threeparttable}
\setlength{\abovecaptionskip}{0.cm}
\begin{tabular}{ccccccccccccccccccccc}
\toprule
\multirow{2}{*}{Method}&
\multicolumn{4}{c}{T1}&\multicolumn{4}{c}{T2}&\multicolumn{4}{c}{T3}&\multicolumn{4}{c}{T4}&\multicolumn{4}{c}{T5} \cr
\cmidrule{2-5} \cmidrule{6-9} \cmidrule{10-13} \cmidrule{14-17} \cmidrule{18-21}
& A & P & R &F1 & A & P & R &F1 & A & P & R &F1 & A & P & R &F1 & A & P & R &F1 \cr
\midrule
Ours$\backslash$he
& 0.42 & 0.37 & 0.42 & 0.39
& 0.42 & 0.38 & 0.42 & 0.38
& 0.40 & 0.34 & 0.40 & 0.36
& 0.44 & 0.41 & 0.44 & 0.39
& 0.43 & 0.42 & 0.43 & 0.39 \cr
Ours$\backslash$ho
& 0.45 & 0.42 &0.45 &0.42
& 0.45 & 0.43 & 0.45 & 0.40
& 0.46 & 0.43 &0.46 &0.43
& 0.45 & 0.42 & 0.45 & 0.42
& 0.44 & 0.41 &0.44 &0.41 \cr
Ours$\backslash$s
& 0.43 & 0.40 & 0.43 & 0.39
& 0.42 & 0.40 & 0.42 & 0.40
& 0.43 & 0.40 & 0.43 & 0.41
& 0.43 & 0.43 & 0.43 & 0.42
& 0.44 & 0.41 & 0.44 & 0.42 \cr
Ours$\backslash$t
& 0.46 & 0.44 & 0.46 & 0.45
& 0.45 & 0.43 & 0.46 & 0.43
& 0.46 & 0.43 & 0.45 & 0.44
& 0.45 & 0.44 & 0.45 & \textbf{0.44}
& 0.46 & 0.44 & 0.46 & 0.45 \cr
Ours$\backslash$g
& 0.44 & 0.41 & 0.44 & 0.40
& 0.43 & \textbf{0.44} & 0.43 & 0.42
& 0.42 & 0.43 & 0.42 & 0.43
& 0.44 & 0.41 & 0.44 & 0.40
& 0.44 & 0.42 & 0.44 & 0.43 \cr
Ours
&\textbf{0.47} &\textbf{0.44}&\textbf{0.47}&\textbf{0.43}
&\textbf{0.48} &0.42 &\textbf{0.48}&\textbf{0.44}
&\textbf{0.47} &\textbf{0.46}&\textbf{0.47}&\textbf{0.45}
&\textbf{0.46} &\textbf{0.45}&\textbf{0.46}&\textbf{0.44}
&\textbf{0.48} &\textbf{0.47}&\textbf{0.48}&\textbf{0.46}
\cr
\bottomrule
\end{tabular}
\begin{tabular}{ccccccccccccccccccccc}
\toprule
\multirow{2}{*}{Method}&
\multicolumn{4}{c}{T6}&\multicolumn{4}{c}{T7}&\multicolumn{4}{c}{T8}&\multicolumn{4}{c}{T9}&\multicolumn{4}{c}{T10} \cr
\cmidrule{2-5} \cmidrule{6-9} \cmidrule{10-13} \cmidrule{14-17} \cmidrule{18-21}
& A & P & R &F1 & A & P & R &F1 & A & P & R &F1 & A & P & R &F1 & A & P & R &F1 \cr
\midrule
Ours$\backslash$he
& 0.41 & 0.36 & 0.41 & 0.37
& 0.42 & 0.40 & 0.42 & 0.38
& 0.40 & 0.35 & 0.40 & 0.37
& 0.41 & 0.36 & 0.41 & 0.37
& 0.42 & 0.37 & 0.42 & 0.39
\cr
Ours$\backslash$ho
& 0.44 & 0.41 & 0.44 & 0.42
& 0.46 & 0.44 &0.46 &0.44
& 0.43 & 0.42 & 0.43 & 0.42
& 0.45 & 0.42 &0.45 &0.43
& 0.45 & 0.43 & 0.45 & 0.44 \cr
Ours$\backslash$s
& 0.43 & 0.40 & 0.43 & 0.39
& 0.43 & 0.44 & 0.43 & 0.42
& 0.42 & \textbf{0.43} & 0.42 & 0.41
& 0.44 & 0.42 & 0.44 & 0.43
& 0.41 & 0.42 & 0.41 & 0.41
\cr
Ours$\backslash$t
& 0.44 & 0.41 & 0.44 & 0.42
& 0.44 & 0.42 & 0.44 & 0.43
& 0.45 & 0.42 & 0.45 & 0.42
& 0.46 & 0.43 & 0.46 & 0.43
& 0.46 & 0.44 & 0.46 & 0.44 \cr
Ours$\backslash$g
& 0.44 & 0.42 & 0.44 & 0.43
& 0.45 & 0.42 & 0.45 & 0.42
& 0.44 & 0.41 & 0.44 & 0.40
& 0.44 & 0.42 & 0.44 & 0.43
& 0.44 & 0.41 & 0.44 & 0.40
\cr
Ours
&\textbf{0.47} &\textbf{0.46}&\textbf{0.47}&\textbf{0.45}
&\textbf{0.47} &\textbf{0.46}&\textbf{0.47}&\textbf{0.45}
&\textbf{0.47} &\textbf{0.43}&\textbf{0.47}&\textbf{0.44}
&\textbf{0.48} &\textbf{0.47}&\textbf{0.48}&\textbf{0.46}
&\textbf{0.47} &\textbf{0.46}&\textbf{0.47}&\textbf{0.45}
\cr
\bottomrule
\end{tabular}
\end{threeparttable}
\end{table}
As for the ablation study, it can be concluded that each module in our framework plays a significant role in the performance improvement.
By comparing the full model and Ours$\backslash$he as well as Ours$\backslash$ho,
we note that the full model performs better than both two variants,
which proves the assumption that the message passing module has positive effects on the node embedding learning, and the homogeneous edges and heterogeneous edges succeed in building the homogeneous and heterogeneous node structures.
When comparing the performances of the homogeneous message passing Ours$\backslash$he
and the heterogeneous message passing Ours$\backslash$ho,
it can be seen that the heterogeneous message passing performs better than the homogeneous message passing which may be because to heterogeneous edges help learn more comprehensive embeddings by getting additional information from other types of nodes.
\begin{figure}[htbp]
\centering
\vspace{-0.3cm}
\setlength{\abovecaptionskip}{-0.05cm}
\setlength{\belowcaptionskip}{-0.1cm}
\subfigure[Accuracy]{
\begin{minipage}[t]{0.5\linewidth}
\centering
\includegraphics[width=5.5cm, height=4cm]{accuracy.png}
\end{minipage}%
}%
\subfigure[Precision]{
\begin{minipage}[t]{0.5\linewidth}
\centering
\includegraphics[width=5.5cm, height=4cm]{precision.png}
\end{minipage}%
}
\subfigure[Recall]{
\begin{minipage}[t]{0.5\linewidth}
\centering
\includegraphics[width=5.5cm, height=4cm]{recall.png}
\end{minipage}%
}%
\subfigure[F1-score]{
\begin{minipage}[t]{0.5\linewidth}
\centering
\includegraphics[width=5.5cm, height=4cm]{F1.png}
\end{minipage}%
}%
\centering
\caption{The average results on the 10 tasks.}
\label{result}
\end{figure}
As for the comparison between Ours$\backslash$t and Ours$\backslash$s,
which use semantic representations and structural representations, respectively,
it can be noted that both the semantic representation and the structural representation benefit the prediction process,
meaning that these two kinds of representations reveal different aspects of the graph information.
The semantic representations perform better than structural representations,
which may be because the semantic representation provides more global information of the graph.
When we omit the global temporal graph while only use the representation of the local context graph to predict PAM in Ours$\backslash$g,
the performance drops a little,
meaning that not only the local context information has an positive effect on individual's health status,
but also the long-term temporal structure of behaviors makes a difference.
In Table~\ref{tab:res}, Table~\ref{tab:ablation} and Figure~\ref{result}, it is worth noting that the results of all methods are below 50\%,
which demonstrates that it is a extremely challenging task to predict mental health status based on personal behaviors in daily life, especially with limited samples.
However, the average accuracy improvement of our method accounts for about $5\%$ of the result
of the second best method HAN as shown in Figure~\ref{result}(a),
which still shows the advantage of the proposed method.
We believe that our model will get better performances on larger-scale datasets.
\subsubsection{Parameter Analysis}
Here we investigate the influence of two import hyper-parameters $m$
and $d_p$ which represent the number of layers in the local context graph
and the dimension of the final local context graph representation, respectively.
We vary $m$ from 1 to 5 and keep other settings fixed,
the results on PAM prediction are shown in Figure~\ref{parameter}.
It can be noted that the performance is improved firstly with the increase of $m$.
However, when the layer number keeps increasing, the performance drops, because too many layers could make the node embedding less discriminative.
As for $d_p$, we vary it from 16 to 256 while keep other settings fixed.
The results are shown in Figure~\ref{parameter}.
We can see that the best performance is achieved in 64,
since low dimension is hard to get useful information,
while the high dimension is difficult to train with limited instances.
\begin{figure}[htb]
\centering
\setlength{\abovecaptionskip}{-0.05cm}
\subfigure[Node message passing module layer $m$]{
\begin{minipage}[t]{0.45\linewidth}
\centering
\includegraphics[width=4cm, height=3.5cm]{layer.png}
\end{minipage}%
}\hspace{10mm}
\subfigure[Local context graph representation dimension $d_p$]{
\begin{minipage}[t]{0.45\linewidth}
\centering
\includegraphics[width=4cm, height=3.5cm]{embed.png}
\end{minipage}%
}%
\centering
\caption{Parameter analysis of $m$, $d_p$}
\label{parameter}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=12cm, height=9cm]{visualization.jpg}
\centering
\caption{Visualization of local context graphs and the global temporal graph.}
\label{visual}
\end{figure}
\subsubsection{Visualization}
Here we analyse the attention in learning representations from local context graphs and the global temporal graph to figure out the factors that influence the mental health.
Figure~\ref{visual} shows the example of an individual's 3-day data streams.
The PAM is predicted with the global representation which is learned based on representations of three local context graphs.
The attention score $\gamma_{ij}$ between two different local context graphs is computed with Eq.~\eqref{eq:day attention}.
We show the attention score between any two local graphs by the darkness of the corresponding line.
The attention of each graph is represented by the darkness of the text box.
We can see that for each local context graph,
the attention to itself tends to play the major role,
though each one pays attention to all other ones.
Besides, it can be seen that the third local context graph
has much influence on all three ones.
We further visualize the concept sequences in the third day, which are used to build the local context graph in that day. In the concept sequences, each text box represents a concept and different concept sequences are shown in different colors.
The importance of each concept is calculated with the attention mechanism in Eq.~\eqref{eq: node attention}, and represented by the darkness of text box.
It is noted that concepts of $stationary$, $silence$ and $classroom$ play the major role in learning the representation from the local context graph.
Homogeneous edges that represent the behavior transfers are shown as arrows, while heterogeneous edges that represent behavior co-occurrences are shown as dotted lines.
The importance of the edge is also indicated by the darkness.
We notice that in general,
heterogeneous edges receive more attention than homogeneous edges although their occurrences are few,
which demonstrates the importance of combination of different concepts and validates that our attention mechanism successfully finds useful patterns from the multi-source data streams.
Besides, the edges, which have nodes from the $location$ type,
tend to get more attention than the ones which have nodes from the $activity$ and $audio$ types, because locations reveal much information of the daily lives.
\subsection{Application on Grade Prediction}
To better illustrate the effectiveness of our model for
learning representations from behavior-related data streams,
we propose to apply our model on the grade prediction task.
The grade annotation is the GPA which indicates a student's overall long term academic performance in a range 0-4.
Firstly,
we take our model, which is well trained on the health status prediction task,
to extract the global representation for each student's data stream and
use KNN to do the grade prediction.
Then we compare our model with a baseline, which takes
the 108-dimensional feature introduced in Section~\ref{subsec:compare_method}
as the representation for each day and add them to get the final feature.
The same KNN is used to do the grade prediction.
We use the mean absolute errors (MAE),
the coefficient of determination ($R^2$) and Pierson correlation as evaluation metrics for the grade prediction.
We adopt the leave-one-out way to evaluate the performance.
The average results are shown in Table~\ref{grade}.
As shown,
our model outperforms the baseline on three metrics,
which demonstrates that our model is effective in learning the structure-aware representation of the individual's long-term behavior.
Moreover, our model has good generalization ability and can be used to extract global
behavior-related features in different tasks without model finetuning.
\begin{table}[t]
\centering
\setlength{\abovecaptionskip}{-0.04cm}
\fontsize{6.5}{8}\selectfont
\begin{threeparttable}
\caption{Grade prediction results.}
\label{grade}
\begin{tabular}{cccccc}
\toprule
\multirow{2}{*}{Method}& \multicolumn{3}{c}{Grade Prediction}\cr
&MAE&$R^2$&Pearson\\
\midrule
Hand-crafted feature + KNN & 0.296 & 0.10 & 0.32 \cr
Graph representation + KNN & \textbf{0.195} & \textbf{0.21} & \textbf{0.51} \cr
\bottomrule
\end{tabular}
\end{threeparttable}
\end{table}
\subsection{Framework Overview}
The individual behavior refers to the way that a person lives.
Our purpose is to model the individual behavior in a period based on multi-modal data streams collected by wearable devices,
and then learn effective representations to predict the health status.
As shown in Figure \ref{fig:framework1}, we take multi-source data streams as input,
and detect behavior-related middle-level concept sequences with pre-trained backbone models.
Then, the behavior-related middle-level concept sequences are used to build the behavior graph
which consists of multiple local context sub-graphs and a global temporal sub-graph.
Specifically, the concepts are regarded as different types of nodes to build local context sub-graphs.
Each local context graph is regarded as a node in the global temporal sub-graph to catch temporal dependency.
The representations of local context sub-graph and global temporal sub-graph are learned by local context modeling and global temporal relation modeling, based on which the final representation of the behavior graph is learned and used to predict the health status.
\subsection{Behavior-related Concept Detection}\label{subsec:backbone}
We take three kinds of data streams (i.e. accelerometer, microphone and wifi) as input,
and detect behavior-related middle-level concept sequences $\{C^1, C^2, C^3\}$ with three pre-trained backbone models (i.e. \textit{activity, audio, location} detectors),
where each concept sequence $C^k = \{c^k_1, c^k_2, \cdots c^k_T\}$
and the $c_k^t$ denotes the specific concept class (e.g. $walking$ detected by the $activity$ detector).
Meanwhile, we also obtain timestamp sequences $\{U^1,U^2,U^3\}$,
where $U^k = \{u^k_1, u^k_2, \cdots u^k_T\}$
and each timestamp $u^k_t$ is a 2-dimensional vector which represents the start and end time of the detected concept class $c^k_t$ in the corresponding data stream.
More details of the pre-trained backbone models are introduced in Section~\ref{subsec:impl}, and the notations and their corresponding explanations are shown in Table ~\ref{tab:notation}.
\begin{table}\footnotesize
\caption{Notations and explanations}
\begin{tabular}{cc}
\toprule
Notation&Explanation\\
\midrule
$c^k_t$& concept class for type $k$ at the $t$ th moment\\
$u^k_t$& a 2-d vector of start and end time of $c^k_i$\\
$a^k_i$& the attribute of $i$-th node from type $k$.\\
$x^k_i$& the embedding of $i$-th node from type $k$.\\
$w_{ij}$& the weight of edge between node pair $(i,j)$,\\
$e_{ij}$& the embedding of edge between node pair $(i,j)$\\
\bottomrule
\end{tabular}
\label{tab:notation}
\end{table}
\subsection{Behavior Graph Building}
To capture the temporal structure of the individual behavior,
we need to build a behavior graph that contains both the local context information and the long-term relationships from the multimodal data stream.
However, a huge densely connected network may increase the computation complexity and impact the performance.
For this reason, we decompose the whole graph into two kinds of sub-graphs: local context graphs to explore the local information of individual behaviors in a short term, and a global temporal graph to capture temporal dependency in the long term.
The local context graphs are regarded as nodes of the global temporal graph.
\subsubsection{Local Context Sub-Graph}\label{subsubsec:localgraph}
The local context graph is built based on the daily data streams, which could reflect the individual behaviors (e.g. activities, audio, locations) from various aspects.
Taking these individual behaviors into consideration, the local context graph is actually a heterogeneous graph.
As illustrated in Section~\ref{subsec:backbone},
we have detected three types of concept sequences $\{C^1, C^2, C^3\}$
from the data streams.
In the following, we explicitly use $activity$, $audio$, and $location$ to denote the type names.
For the $t$-th time step, we use a sliding time window of one day
to crop out three concept sub-sequences $\{\hat{C}^1, \hat{C}^2, \hat{C}^3\}$ from the original concept sequences.
Each concept sub-sequence $\hat{C}^k=\{c^k_{t-h_k/2}, \cdots, c^k_t, \cdots, c^k_{t+h_k/2}\}$ represents
several consecutive concepts detected in one day.
Here the $h_k$ denotes the size of the sliding time window which represents the number of timestamps
contained in one day for the $k$-th type of the concept sequence.
Accordingly, we can obtain the timestamp sub-sequences $\{\hat{U}^1, \hat{U}^2, \hat{U}^3\}$,
where $\hat{U}_k=\{u^k_{t-h_k/2}, \cdots, u^k_t, \cdots, u^k_{t+h_k/2}\}$.
Next, we will introduce how to create the local context graph based on $\{\hat{C}^1, \hat{C}^2, \hat{C}^3\}$ and $\{\hat{U}^1, \hat{U}^2, \hat{U}^3\}$.
For the $t$-th time step,
the local context graph can be formally defined as $\mathcal{G}_t=(\mathcal{V}_t, \mathcal{E}_t)$, where $\mathcal{V}_t = ({\mathcal{N}_1, \mathcal{N}_2,\mathcal{N}_3})$, representing different types of nodes.
Each type corresponds to a specific aspect to describe individual behavior.
$\mathcal{E}_t$ is the set of edges, containing both the homogeneous edges, which connect two nodes in the same type and heterogeneous edges which connect two nodes in different types.
The nodes of the local context graph are comprised of all the concept classes in $\{\hat{C}^1, \hat{C}^2, \hat{C}^3\}$.
Since there are three types of concept classes,
the local context graph has three different node types
and thus is a heterogeneous graph.
Each node has an attribute and an embedding representation,
noted as $a_i^k$ and $x_i^k$ for the $i$-th node of type $k$.
For the attribute $a_i^k$ of the node with concept class $c^k_i$,
we use its corresponding timestamp $u^k_i$ to compute the time interval which represents
duration of the concept-related behavior.
For the node embedding $x_i^k$, we introduce external semantic knowledge to help its learning.
Specifically,
we extract Glove embeddings\cite{pennington2014glove} corresponding to the concept class name of each node,
which are pre-trained on Wikipedia according to the word by word co-occurrence.
The embeddings are proved effective in capturing semantic meanings of words in many NLP tasks.
Therefore, these word embeddings would provide a reasonable representation for nodes at first.
As for edges, homogeneous edges and heterogeneous edges are considered in different ways.
Two nodes with the same type $k$ are connected with homogeneous edges if they are temporal neighborhoods in the concept sequence $\hat{C}^k$.
For example,
a node with concept class ${c}^3_t = dormitory$ from the type $loation$ is connected to
another node with concept class ${c}^3_{t+1} = library$ from the same type if
an individual moves to library from dormitory with continuous timestamps in reality.
The weight of the homogeneous edge is noted as the frequency of two nodes being neighborhood in the concept sequence.
With the edge weight,
the homogeneous edge captures specific patterns of an individual's behavior change information.
While for heterogeneous nodes (e.g. $c^k_i$, $c^{k'}_j$, $k\neq k'$),
we connect them according to their co-occurrences in time, i.e., $u^k_i \cap u^{k'}_j \neq \varnothing$.
For example, a node with concept class ${c}^2_t = silence$ from the type \textit{audio} and a node with concept class ${c}^3_t = library$ from the type \textit{location} are connected when they
are detected from the data streams at the same timestamp.
The co-occurrences in time reflect the interactions of heterogenous nodes,
which could describe the individual behaviors from more aspects.
We do not connect heterogeneous nodes which are temporal neighbouring in the concept sequence, because connecting different types of nodes according to their temporal relations has no practical significance.
The weight of the heterogeneous edge is the frequency of co-occurrences in time.
Note that the weights of both homogeneous edges and heterogeneous edges are written as $w_{ij}$ with $i,j$ as node indexes.
Whether $w_{ij}$ represents a homogeneous edge or a heterogeneous edge depends on specific types of node $i$ and node $j$.
\subsubsection{Global Temporal Sub-Graph}\label{subsubsec:globalgraph}
The global temporal sub-graph models the long-term time dependency of the daily information and gets the global information for the whole period, which is used to predict the health status.
Formally,
the global temporal graph is denoted as $\mathcal{G}=(\mathcal{V}, \mathcal{E})$,
where nodes in $\mathcal{V}$ refer to local context graphs introduced in Section~\ref{subsubsec:localgraph}, and $\mathcal{E}$ represents interactions between any two local context graphs.
\subsection{Local Context Graph Modeling}\label{subsec:localgraphmodel}
As introduced in Section~\ref{subsubsec:localgraph},
we have created a heterogeneous local context graph $\mathcal{G}_t$ at a each time step of multi-source data streams.
Now we will introduce how to capture local context information of the short term individual behavior with the heterogeneous graph neural network.
The network contains $m$ layers of node message passing modules and edge embedding learning modules.
Here we only introduce these two kinds of modules for one layer.
As shown in Figure \ref{fig:framework2}, the node message passing module is used to learn the node embeddings and graph semantic representation, while the edge embedding learning module is used to learn the edge embeddings and graph structural representation.
Then the final representation of the local context graph is obtained with the combination of
the semantic and the structural representation.
It is worth noting that all local context graphs share the same network parameters.
\begin{figure*}[t]
\centering
\setlength{\belowcaptionskip}{-0.2cm}
\includegraphics[width=11cm]{framework2.jpg}
\centering
\caption{Local context graph modeling.}
\label{fig:framework2}
\end{figure*}
\subsubsection{Node Message Passing}\label{subsubsec:nodemessage}
We consider the node message passing process in two ways:
homogeneous message passing through homogeneous edges and heterogeneous message passing through heterogeneous edges.
At first, we multiply each node embedding $x_i^k$ with its attribute $a_i^k$, which reveal the node importance in time.
For simplicity, the representation of each node is still denoted by $x_i^k$.
Homogeneous message passing aims to learn information from the same type of nodes according to their edges.
For the \textit{i}th node of type $k$, its message passing process is done as below:
\begin{center}
\begin{equation}
x^k_i = W^k_x x^k_i + W^k_{\alpha}\sum_{j, j\neq i}\alpha_{ij}x^k_j
\end{equation}
\end{center}
where $W^k_x$ and $W^k_{\alpha}$ are learnable matrices.
The $i$ and $j$ are node indexes, and $\alpha_{ij} = \frac{w_{ij}}{\sum_{j,j\neq i}{w_{ij}}}$ is the normalized value of the homogeneous edge weight between the $i$-th node and the $j$-th node defined in Section~\ref{subsubsec:localgraph}.
The calculations for other types of nodes are in the same way with different projection matrices.
By this means, each node gets information from its homogeneous neighborhoods according to their connections.
Heterogeneous message passing manages to capture additional semantic meanings from other types of nodes,
thus learns a comprehensive representation for individual behaviors.
Since each node has more than one type of heterogeneous neighbours,
we add embeddings learned from all types of heterogeneous neighbors to do the message passing,
shown as follows:
\begin{center}
\begin{equation}
\setlength{\abovedisplayskip}{-3pt}
\setlength{\belowdisplayskip}{-3pt}
x^k_i = W^k_x x^k_i + W^{k'}_{\alpha} \sum_{k'}{\sum_{j}{\alpha_{ij}x^{k'}_j}}
\end{equation}
\end{center}
where $W^k_x$ and $W^{k'}_{\alpha}$ are learnable matrices.
The $k$ and $k'$ are node type indexes and $k' \neq k$.
The $i$ and $j$ are node indexes.
The $\alpha_{ij} = \frac{w_{ij}}{\sum_{j,j\neq i}{w_{ij}}}$ is the normalized value of the heterogeneous edge weight defined in Section~\ref{subsubsec:localgraph}.
\subsubsection{Edge Embedding Learning}
Compared with the node embeddings,
edge embeddings reveal much more structure information of the graph.
For our heterogenous graph,
edges are also in different types since they connect different types of nodes.
Considering that nodes have been embedded in a common semantic space
by the node message passing module,
we directly concatenate them and use a projection to extract the edge embedding:
\begin{center}
\begin{equation}
e_{ij} = W_e[x^k_i \oplus x^{k'}_j]
\end{equation}
\end{center}
where $k$ and $k'$ are node type indexes and $\oplus$ means concatenation.
The $W_e$ is a learnable matrix.
If $k = k'$, $e_{ij}$ is the embedding of homogeneous edge, otherwise heterogeneous edge.
\subsubsection{Local Context Graph Representations}
For each local context graph,
we learn two kinds of representations to capture the short-term behavior information:
a semantic representation that reflects semantic meanings and a structural representation that catches information of the graph structure.
We obtain the semantic representation of the local context graph by combining the embeddings of all types of nodes learned with the node message passing module.
Although the attribute of a node defined in Section~\ref{subsubsec:localgraph} could reflect its importance,
the semantic meaning of the node should also be considered.
Because a concept appearing few times in the concept sequence
may contain important factors for the health status prediction, we take advantage of soft-attention mechanism to determine the importance of different nodes and combine node embeddings to get the semantic representation:
\begin{center}
\begin{equation}
\setlength{\abovedisplayskip}{-3pt}
\setlength{\belowdisplayskip}{-3pt}
g_s = \sum_{k,i} \beta_i^k x^k_i
\end{equation}
\end{center}
\begin{center}
\begin{equation}
\setlength{\abovedisplayskip}{-3pt}
\setlength{\belowdisplayskip}{-3pt}
\beta_i^k = \frac{exp(q \cdot x^k_i)}{\sum_{k,i}exp(q \cdot x^k_i)} \label{eq: node attention}
\end{equation}
\end{center}
where $g_s$ is the semantic representation for the local context graph, $\beta_i^k$ is the relevant importance given to each node when blending all nodes together, and $q$ is a trainable vector used as query.
The reason for using softmax in~\eqref{eq: node attention} mainly lies in two points.
(1)
The softmax is differentiable thus can be easily integrated in the graph neural networks for end-to-end training.
(2)
The output values of softmax function are in the range of
[0,1] with the sum of 1.
With the softmax function, $\beta_i^k$ can be interpreted as the relevant importance given to node $x_i^k$ when blending all nodes together.
For the structural representation,
since different edges play various roles for the graph structure,
we also use attention mechanism to calculate their correlations and then combining them to get the graph structural representation.
Since the semantic representation $g_s$ contains the global information of the local context graph with semantic meanings of all nodes considered,
we treat it as a query vector to help learning more effective attentions for combining edge embeddings:
\begin{center}
\begin{equation}
\setlength{\abovedisplayskip}{-3pt}
\setlength{\belowdisplayskip}{-3pt}
g_e = \sum_{i,j} \beta_{ij} e_{ij}
\end{equation}
\end{center}
\begin{center}
\begin{equation}
\setlength{\abovedisplayskip}{-3pt}
\setlength{\belowdisplayskip}{-3pt}
\beta_{ij} = \frac{exp[(W_{\beta}g_s) \cdot e_{ij}]}{\sum_{i,j}exp[(W_{\beta}g_s) \cdot e_{ij}]}. \label{eq: edge attention}
\end{equation}
\end{center}
where $e_{ij}$ is the embedding for either homogeneous edges or heterogeneous edges.
$\beta_{ij}$ is the relevant importance given to each edge when blending all edges together, and $W_{\beta}$ is a projection matrix for semantic representation $g_s$.
We get the final representation for each local context graph with the concatenation of its semantic and structural representations as $g = [g_e;g_s]$.
\subsection{Global Temporal Relation Modeling}
Self-attention network (SAN) is introduced in Transformer\cite{vaswani2017attention} for the first time,
which has a sequence-to-sequence architecture and is popularly used in neural machine translation.
Taking a token sequence as input,
SAN calculates the attention scores between each token and other tokens with multiple attention heads.
Then the token embeddings are updated with other token embeddings according to their attention scores.
From this perspective,
SAN can be regarded as a graph neural network with token sequence as fully-connected nodes,
while the multi-head attention mechanism is a special message passing method.
Inspired by this,
we implement the global temporal relation modeling with the self-attention network.
Specifically,
we can get a sequence of all local context graph representations
$\{g_1, g_2, \cdots, g_T\}$ as illustrated in Section~\ref{subsec:localgraphmodel}.
For the temporal information among different local graphs,
we adopt position embeddings in \cite{vaswani2017attention} to encode the relative position of each local context graph,
noted as $\{p_1, p_2, \cdots, p_T\}$.
Therefore, the representation for the $i$th graph is the sum of $g_i$ and $p_i$.
Then the correlations between any two local context graphs can be calculated with attention scheme:
\begin{center}
\begin{equation}
\setlength{\abovedisplayskip}{-5pt}
\setlength{\belowdisplayskip}{-3pt}
f(g_i, g_j) = {[W_q(g_i + p_i)]}^{\top}[W_p(g_j+p_j)]
\end{equation}
\end{center}
where $W_q$ and $W_p$ are learnable matrices.
The attention scores are scaled and normalized with a Softmax function,
which are used to get an attended representation for each local context graph $g_i$:
\begin{center}
\begin{equation}
\setlength{\abovedisplayskip}{-3pt}
\setlength{\belowdisplayskip}{-3pt}
\gamma_{ij} = \frac{exp(f(g_i, g_j)/\sqrt{d_p})}{\sum_{j=1}^{T}{exp(f(g_i, g_j)/\sqrt{d_p})}} \label{eq:day attention}
\end{equation}
\end{center}
\begin{center}
\begin{equation}
\setlength{\abovedisplayskip}{-3pt}
\setlength{\belowdisplayskip}{-3pt}
g'_i = \sum_{j=1}^{T}{\gamma_{ij}W_gg_j}
\end{equation}
\end{center}
where $W_g$ is a learnable projection matrix,
and $d_p$ is the dimension of $g_i$.
$T$ is the number of local context graphs.
$g'_i$ is the attended representation for the $i$th local context graph.
Finally, we can get the structure-aware representation $g^*$ of the global temporal graph $\mathcal{G}$ by adding
the attended representations of all local context graphs.
\subsection{Objective Function}
The final loss function is written as the sum of a classification loss and a node variance constraint:
$L = L_c + \lambda L_n$,
where $\lambda$ is a trade-off parameter.
\textbf{Classification Loss:}
With the representation $g^*$ of the global temporal graph,
we predict the health status label $y$ by a fully-connected layer with Softmax activation.
Then we calculate the cross-entropy loss:
\begin{center}
\begin{equation}
\setlength{\abovedisplayskip}{-3pt}
\setlength{\belowdisplayskip}{-3pt}
L_c = \frac{1}{N} \sum_{i=1}^NCrossEntropy(y_i, y'_i)
\end{equation}
\end{center}
where $N$ is the number of instances used in training process
and $y'_i$ is the groundtruth label of the health status.
\textbf{Node Variance Loss:}
It is worth noting that many GNNs will face the problem of node homogenization after several epoches of node message passing since all nodes exchange information with their neighbors.
In the local context graph modeling illustrated in Section~\ref{subsec:localgraphmodel},
the message passing is not only applied on homogeneous nodes but also on heterogeneous nodes,
which may make the representations of nodes similar.
To alleviate this problem,
we add a constraint $L_n$ to the node representations to control the variance of all nodes.
Specifically, we concatenate all the node embeddings into a matrix noted as
$E \in \mathbb{R}^{N \times d}$,
where $N$ is the number of nodes of all types and
$d$ is the dimension of the node embedding.
Then we calculate the variance and get a vector $v \in \mathbb{R}^d$,
where each element represents the variance of the corresponding dimension in $E$.
Finally,
the node variance loss is defined as the average of the elements in the variance vector followed by a sigmoid function:
\begin{center}
\begin{equation}
\setlength{\abovedisplayskip}{-3pt}
\setlength{\belowdisplayskip}{-3pt}
L_n = -sigmoid(\frac{1}{d}\sum_{i=1}^dv_i)
\end{equation}
\end{center}
\subsection{Health Status Prediction}
Our task is to predict the health status based on personal behaviors with the multi-source sensor data collected in daily life.
This task has important medical implications because it can provide early prevention of diseases and complement the clinical treatment in hospital.
As we know, most existing health prediction methods can be divided into two categories: Electronic Health Record(EHR) based methods~\cite{{mcgrath2013toward,mwangi2012prediction}} and mobile sensor data based methods~\cite{canzian2015trajectories,burns2011harnessing}.
The EHR based methods are more relevant to medical studies since they use the record data collected during hospital treatment process with professional medical equipment.
However, people have few EHR data before they are detected with diseases, thus this kind of methods, however good, can only provide a piece of the puzzle.
More kinds of data about early-life experience of patients should also be considered in treatment, which is actually what the second kind of methods do.
The sensor data based methods focus on the personal behavior in daily life and take advantage of mobile devices, which are more convenient to monitor people's health and also provide additional useful information for clinical treatment. Below we will introduce these two categories in detail.
Electronic Health Record (EHR) data are collected during the hospital treatment process, containing nearly all the information of patients, such as the diagnoses, medication prescriptions and clinical notes.
In early stages, expert-defined rules are adopted to identify disease based on the EHR data, such as type 2 diabetes\cite{kho2012use} and cataract \cite{peissig2012importance}.
Much work based on EHR data has been done with deep learning models, with disease classification task most commonly. Cheng \textit{et al.} \cite{cheng2016risk} and Acharya \textit{et al.} \cite{acharya2018deep} train a CNN model to classify the normal, preictal, and seizure EEG signals. Che \textit{et al.} \cite{che2017rnn} propose a multi-class classification task to predict the different stages of Parkinson's disease with Recurrent Neural network(RNN).
Kam \textit{et al.} \cite{kam2017learning} do binary classification of sepsis by regarding the EHR data as input to a long short-term memory network (LSTM).
In addition to the disease classification,
the future event prediction is another task attracting much attention recently,
which aims to predict the future medical events according to the historical records. For example, Joseph \textit{et al.} \cite{futoma2015comparison} and Rajkomar \textit{et al.} \cite{rajkomar1801scalable} use the EHR data from the hospital
to predict events such as mortality, readmission, length of stay and discharge diagnoses
with deep feed forward network and LSTMs.
There has been much work on mental health prediction based on EHR data, among which T1-weighted imaging~\cite{mcgrath2013toward,mwangi2012prediction} and functional MRI (fMRI)~\cite{fu2008pattern,hahn2011integrating} are the most commonly used data to study brain structure, with other physiological signals such as electroencephalogram also playing an important role.
Recently,
machine learning receives
more attention for its effect on improving the management of mental health.
Costafreda \textit{et al.} \cite{costafreda2009prognostic} do a depression classification task with SVM using the smoothed gray matter voxel-based intensity values.
Rosa \textit{et al.} \cite{rosa2015sparse} propose a sparse L1-norm SVM to predict depression with the feature of region-based functional connectivity.
Cai \textit{et al.} \cite{cai2018pervasive} collect the electroencephalogram (EEG) signals of participants and use four classification methods (SVM, KNN, DT, and ANN) to distinguish the depressed participants from normal controls.
Mobile devices provide another way for health status prediction, where diverse sensors can be used to catch various signals of people, thus make it easier to monitor daily behavior and predict health status.
Machado \textit{et al.} \cite{machado2015human} calculate the signal magnitude area of the acceleration signal to recognize activities with several cluster algorithms(eg. K-Means, Mean Shift).
Koenig \textit{et al.} \cite{koenig2016validation} and Banhalmi \textit{et al.} \cite{banhalmi2018analysis} use camera in smartphone to monitor the heart rate(HR) and heart rate variability(HRV), which are vital signs of cardiovascular health.
Stafford \textit{et al.} \cite{stafford2016flappy} and Goel \textit{et al.} \cite{goel2016spirocall} detect the sound of breathing and cough by microphone in smartphone for assessing pulmonary health in a quick and efficient way.
Besides the use of data from only one source,
many researchers are devoted to taking advantage of information from various sources\cite{huang2020knowledge}\cite{zhang2019multimodal}\cite{qi2020emotion}. Asselbergs \textit{et al.} \cite{asselbergs2016mobile} integrate accelerometer data, call history as well as the short message service pattern to predict the mood. Burns \textit{et al.} \cite{burns2011harnessing} predict depression based on GPS, accelerometer and light sensor data from smartphone.
Nag \textit{et al.} \cite{nag2018cross} estimate the heart health status by combining sensor data from wearable devices and other factors, such as inherent genetic traits, circadian rhythm and living environmental risks analysed from cross-modal data, which provides better personalized health insight.
However, all of the above work cannot well explore local and global temporal characteristics of the daily behavior based on multi-source wearable sensors.
\subsection{Graph Neural Network}
Recently, the emergence of structural data, especially the structured graphs promote the development of Graph Neural Networks(GNNs) \cite{gao2019graph}\cite{wu2020comprehensive}\cite{zhang2020deep}.
As the early work of GNNs, recurrent graph neural networks (RecGNNs) \cite{scarselli2008graph}\cite{gallicchio2010graph} apply recurrent architectures to learn the node representation, where message passing is done constantly with nodes' neighborhoods until the node representations are stable.
Inspired by the success of Convolutional Neural Network(CNN),
\textit{convolution} operation is also introduced to graph data in both spectural\cite{henaff2015deep}\cite{defferrard2016convolutional}\cite{kipf2016semi} and spatial ways\cite{atwood2016diffusion}\cite{junyu2019AAAI_TS-GCN}\cite{gao2020learning}.
The spectral approaches adapt the spectral graph theory to design a graph convolution.
The spatial approaches inherit the message passing idea in RecGNNs while have the difference in getting node representations by stacking multiple convolutional layers.
Besides RecGNNs and ConvGNNs, many other graph architectures have been developed to cope with different scenarios.
For example, Graph autoencoders(GAEs)~\cite{cao2016deep}\cite{wang2016structural} are used to learn the graph embedding by reconstructing the structural information such as adjacency matrix of graph.
Spatial-temporal graph neural networks (STGNNs) \cite{seo2018structured}\cite{li2017diffusion}\cite{jain2016structural} aim to model both the spatial and temporal dependency of data and learn the representation of spatial-temporal graph, which have advantages in the related tasks, such as human action recognition.
Most of the existing GNNs focus on homogeneous graphs where nodes are in the same type and can be calculated in the same way.
In comparison, heterogeneous graph contains diverse types of nodes and edges, leading to a more complicated situation in calculation.
On the one hand, different types of nodes may have different semantic meanings and in different feature spaces.
On the other hand, the heterogeneous graph represents both homogeneous and heterogeneous relations of data.
Recently, some work has been done on heterogeneous graphs.
Dong \textit{et al.} \cite{dong2017metapath2vec} propose a path2vec method to learn heterogeneous graph embeddings with a meta-path based random walk. Chen \textit{et al.} \cite{chen2018pme} process different kinds of nodes with several projection matrices used to embed all the nodes into a same space and then do the link prediction.
Wang \textit{et al.} \cite{wang2019heterogeneous} further introduce the hierarchical attention to heterogeneous graph to learn attentions for both nodes and meta-paths.
Until now, the application
of heterogeneous GNNs to individual behavior analysis and health status prediction is yet to be explored.
\begin{figure*}[htbp]
\centering
\setlength{\belowcaptionskip}{-0.2cm}
\includegraphics[width=14cm, height=4.5cm]{framework1.jpg}
\centering
\caption{Overview of our framework. We take multi-source data streams as input, and detect concept sequences to build a behavior graph which consists of local context sub-graph and global temporal sub-graph.}
\label{fig:framework1}
\end{figure*}
\section{Introduction}
\input{intro}
\section{Related Work}
\input{relat}
\section{Methods}\label{sec:method}
\input{method}
\section{Experiments}
\input{expr}
\section{Conclusion}
\input{conc}
\bibliographystyle{ACM-Reference-Format}
|
1,314,259,996,552 | arxiv | \section{Introduction and preliminary results}
\setcounter{equation}{0}
Polyanalytic functions were introduced in 1908 by Kolossov to solve problems in elasticity theory, see \cite{kolossov}. For a general introduction to this topic see \cite{Alpay2015, Balk1991, balk_ency}. In more recent times, this function theory was
studied by several authors from different perspectives, see
\cite{abreu, abreufeicht, agranovsky, begehr, vasilevski} and the references
therein.
Polyanalytic functions are used also to study sampling and interpolation problems on Fock spaces using time frequency analysis techniques such as short-time Fourier transform (STFT) or Gabor transforms, see \cite{A2010}. In the next parts of this introduction, we collect some basic definitions and explain what we mean by polyanalytic functions of infinite order in our setting. Some important facts that will be needed in the sequel will be also revised. Then, we will explain the general construction of the kernels associated to the reproducing kernel Hilbert spaces of polyanalytic functions of infinite order, which will be studied in this paper. We conclude by describing the contents of the paper.
\subsection{Definitions}
A complex valued function $f:\Omega\subset \mathbb{C}\longrightarrow \mathbb{C}$ which belongs to the kernel of a power $n\geq 1$ of the classical Cauchy-Riemann operator $\displaystyle \frac{\partial}{\partial \overline{z}}$, that is $$\displaystyle \frac{\partial^n}{\partial \overline{z}^n}f(z)=0, \quad \forall z\in\Omega,$$
is called a polyanalytic function of order $n$.
\smallskip
An interesting fact regarding these functions is that any polyanalytic function of order $n$ can be decomposed in terms of $n$ analytic functions so that we have a decomposition of the following form
\begin{equation}
f(z)=\displaystyle \sum_{k=0}^{n-1}\overline{z}^kf_k(z),
\end{equation}
for which all $f_k$ are analytic functions on $\Omega$. In particular, expanding each analytic component using the series expansion theorem lead to an expression of this form
\begin{equation}\label{exp1}
f(z)=\displaystyle \sum_{k=0}^{n-1}\sum_{j=0}^{\infty}\overline{z}^kz^ja_{k,j},
\end{equation}
where $(a_{k,j})$ are complex coefficients.
In this paper, we are interested by the case where the expansion \eqref{exp1} is of infinite order, which means that we consider functions of the form
\begin{equation}\label{exp2}
f(z)=\displaystyle \sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\overline{z}^kz^ja_{k,j},
\end{equation}
which will be called polyanalytic functions of infinite order. We note that such functions were discussed in \cite{Balk1991, balk_ency} in which they were mentioned as conjugate analytic functions.
\\ \\
For $n=1,2,...$ we recall that polyanalytic Fock spaces of order $n$ can be defined as follows
$$\mathcal{F}_n(\mathbb{C}):=\left\lbrace g\in H_n(\mathbb{C}), \quad \frac{1}{\pi}\int_{\mathbb{C}}|g(z)|^2e^{-|z|^2}dA(z)<\infty \right\rbrace.$$
The reproducing kernel associated to the space $\mathcal{F}_n(\mathbb{C})$ is given by
\begin{equation}\label{Kn}
K_n(z,w)=e^{z\overline{w}}\displaystyle \sum_{k=0}^{n-1}\frac{(-1)^k}{k!}{n \choose k+1}|z-w|^{2k},
\end{equation}
for every $z,w\in\mathbb{C}.$
\subsection{The kernels construction: general discussion}
Consider a function $F(z_1,z_2)$ in the Hardy space of the bidisk $\mathbb{D}^2$ in $\mathbb{C}^2$. Then, $g(z)=F(z,z)$ belongs to the Bergman space of the disk $\mathbb{D}$, and the map $F\mapsto g$ is onto and contractive, but not one-to-one.
For instance, the polynomials $z_1^nz_2^m$ belong to $\mathbf H^2(\mathbb D^2)$ and have the same image $z^{s}$ with $n+m=s$. On the other hand the map $g(z)=F(z,\overline{z})$ is one-to-one, and its image is the
reproducing kernel Hilbert space with reproducing kernel
\[
\frac{1}{(1-z\overline{w})(1-\overline{z} w)}
\]
The corresponding reproducing kernel Hilbert space consists of polyanalytic functions of infinite order. \\ \\ Motivated by the above discussion
we consider a function $c(\mathbf z, \mathbf w)$ positive definite in some open subset $\Omega$ of $\mathbb C^{2N}$, and analytic in $\mathbf z$ and $\overline{\mathbf w}$.
We assume that
\begin{equation}
\Omega_s=\left\{z\in\mathbb C^N\,:\, (z,\overline{z})\in\Omega\right\}
\label{sym}
\end{equation}
is open and non-empty. The function
\[
k(z,w)=c((z,\overline{z}),(\omega,\overline{\omega}))
\]
is then positive definite in $\Omega_s$. The purpose of the present work is to study the corresponding reproducing kernel Hilbert spaces of polyanalytic functions of infinite
order. For instance in the case of the Fock space with reproducing kernel $c(\overline{z},\omega)=e^{\sum_{n=1}^{2N}z_n\overline{\omega_n}}$ we have the kernel
\begin{equation}
k(z,w)=e^{\sum_{n=1}^N(z_n\overline{w_n}+\overline{z_n}w_n)}
\end{equation}
while in the case of the Drury-Arveson space with reproducing kernel $\frac{1}{1-\sum_{n=1}^{2N}z_n\overline{w_n}}$ which is positive definite in the open unit ball of
$\mathbb C^{2N}$, the corresponding kernel is
\begin{equation}
\frac{1}{1-\sum_{n=1}^N(z_n\overline{w_n}+\overline{z_n}w_n)}
\end{equation}
positive definite in the open ball of $\mathbb C^N$ centered at the origin and with radius $\frac{1}{\sqrt{2}}$. In this paper we will focus on $N=1$.\\
A general family of examples correspond to
\[
k(z,w)=K_1(z,w)\overline{K_2(z,w)}
\]
where $K_1$ and $K_2$ are analytic kernels, or, in the matrix-valued case,
\[
k(z,w)=K_1(z,w)\otimes \overline{K_2(z,w)}.
\]
The structure of the paper is as follows: in Section 2 we introduce the kernel function $K$ associated to the polyanalytic Fock space $\mathcal{SF}(\mathbb C)$ of infinite order and we study various properties. We give a sequential characterization of the space $\mathcal{SF}(\mathbb C)$ and, in particular, we prove that the creation and annihilation operators are adjoint of each other. We also introduce and study two backward shift operators. In Section 3 we prove that by taking the power series of the polyanalytic Fock kernels of finite order $(K_n)_{n\geq 0}$ we obtain the kernel function $K$ multiplied up to an exponential kernel. In Section 4 and 5 we study Segal-Bargmann and Berezin type transforms and some related operators. In Section 6 we present the polyanalytic Hardy space of infinite order and we study the Gleason problem. We also prove some results on the backward shift operator in this setting. Finally, Section 7 is devoted to the case of Drury-Arveson space.
\section{The polyanalytic Fock space of infinite order and associated kernel}
\setcounter{equation}{0}
We denote by $M_z$ and $M_{\overline{z}}$ the multiplication operators by $z$ and $\overline{z}$. Then, we will prove the following main result
\begin{theorem}
The reproducing kernel Hilbert space with reproducing kernel $e^{z\overline{w}+\overline{z} w}$ is, up to a multiplicative positive factor, the only reproducing kernel
Hilbert space of polyanalytic functions of infinite order, regular at the origin, and for which
\begin{eqnarray}
\left(\frac{\partial}{\partial z}\right)^*&=&M_z\\
\left(\frac{\partial}{\partial \overline{z}}\right)^*&=&M_{\overline{z}}.
\end{eqnarray}
\end{theorem}
To this end, we need the following:
\begin{definition}
We consider the kernel function given by
\begin{equation}\label{newFock-kernel}
K(z,w)=e^{z\overline{w}+\overline{z} w}=e^{2Re(z\overline{w})}, \quad \forall (z,w)\in\mathbb{C}^2.
\end{equation}
We denote by $(\mathcal{H}(K),\langle \cdot, \cdot \rangle_{\mathcal{H}(K)})$ the reproducing kernel
Hilbert space associated to the kernel function \eqref{newFock-kernel}.
\end{definition}
\begin{proposition}
The function $K:\mathbb{C}\times \mathbb{C}\longrightarrow \mathbb{C}$ defined by \eqref{newFock-kernel} is a positive definite kernel.
\end{proposition}
\begin{proof}
It is clear that we have
\begin{equation}\label{FFbar}
K(z,w)=F(z,w)\overline{F(z,w)},
\end{equation}
for every $z,w\in\mathbb{C}$ and $F$ denotes the reproducing kernel of the classical Fock space $\mathcal{F}(\mathbb{C})$. Thus, since $\overline{F(z,w)}$ is also a positive definite kernel we can conclude, since $K$ is a product of positive definite kernels.
\end{proof}
We observe that the following integral representation holds
\begin{proposition}
It holds that \begin{equation}
\displaystyle \frac{1}{\pi}\int_{\mathbb{C}}K(z,w)e^{-|w|^2}dA(w)=e^{|z|^2}=\sqrt{K(z,z)}, \text{ for any } z\in\mathbb{C}.
\end{equation}
\end{proposition}
\begin{proof}
We set $w=x+iy$, we identify $\mathbb{C}$ with $\mathbb{R}^2$ and use the classical Gaussian integral
$$\displaystyle\int_{\mathbb{R}} e^{-at^2+bt}dt=\sqrt{\frac{\pi}{a}}e^{\frac{b^2}{4a}}, \quad a>0, b\in\mathbb{C}.$$
We have
\[
\begin{split}
\displaystyle \frac{1}{\pi}\int_{\mathbb{C}}K(z,w)e^{-|w|^2}dA(w) &= \frac{1}{\pi}\int_{\mathbb{C}}e^{z\overline{w}+\overline{z} w}e^{-|w|^2}dA(w) \\
&= \frac{1}{\pi}\int_{\mathbb{C}}e^{z(x-iy)+\overline{z}(x+iy)}e^{-(x^2+y^2)}dxdy \\
&= \frac{1}{\pi}\left(\int_{\mathbb{R}}e^{x(\overline{z}+z)-x^2}dx\right)\left(\int_{\mathbb{R}}e^{y i(\overline{z}-z)-y^2}dy \right)
\\
&= e^{\frac{(z+\overline{z})^2}{4}}e^{-\frac{(\overline{z}-z)^2}{4}}
\\
&= e^{|z|^2}
\\
&=\sqrt{K(z,z)},
\end{split}
\]
as stated.
\end{proof}
Next result is simple but very useful:
\begin{proposition} \label{Fockpptd}
For any $z,w\in \mathbb{C}$ it holds that
\begin{itemize}
\item[i)] $\displaystyle \frac{\partial}{\partial z}K(z,w)=\overline{w}K(z,w)$ and $\displaystyle \frac{\partial}{\partial \overline{z}}K(z,w)=wK(z,w)$.
\item[ii)] $\displaystyle \frac{\partial}{\partial w}K(z,w)=\overline{z}K(z,w)$ and $\displaystyle\frac{\partial}{\partial \overline{w}}K(z,w)=zK(z,w)$.
\end{itemize}
\end{proposition}
\begin{proof}
It is an easy calculation based on the expression of the kernel function $K(z,w)$ given by formula \eqref{newFock-kernel}.
\end{proof}
\begin{corollary}\label{Fockppt}
For any $z,w\in \mathbb{C}$ and $n=1,2,...$ it holds that
\begin{itemize}
\item[i)] $\displaystyle \frac{\partial^n}{\partial z^n}K(z,w)=\overline{w}^nK(z,w)$ and $\displaystyle \frac{\partial^n}{\partial \overline{z}^n}K(z,w)=w^nK(z,w)$.
\item[ii] $\displaystyle \frac{\partial^n}{\partial w^n}K(z,w)=\overline{z}^nK(z,w)$ and $\displaystyle\frac{\partial^n}{\partial \overline{w}^n}K(z,w)=z^nK(z,w)$.
\end{itemize}
\end{corollary}
\begin{proof}
We will prove i), the other statements follow similar arguments. We apply Proposition \ref{Fockpptd} and get $$\displaystyle \frac{\partial}{\partial z}K(z,w)=\overline{w}K(z,w).$$
Thus, we apply a second time the complex derivative with respect to the variable $z$, we use again Proposition \ref{Fockpptd} and get $$\displaystyle \frac{\partial^2}{\partial z^2}K(z,w)=\overline{w}^2K(z,w).$$
Then, we repeat the same calculation $n$-times and obtain $$\displaystyle \frac{\partial^n}{\partial z^n}K(z,w)=\overline{w}^nK(z,w).$$
\end{proof}
Now, let us consider the commutator operators given by \begin{equation}
\displaystyle \left[\frac{\partial}{\partial z}, M_z\right]:=\frac{\partial}{\partial z}M_z-M_z\frac{\partial}{\partial z}
\end{equation}
and
\begin{equation}
\displaystyle \left[\frac{\partial}{\partial \overline{z}}, M_{\overline{z}}\right]:=\frac{\partial}{\partial \overline{z}}M_{\overline{z}}-M_{\overline{z}}\frac{\partial}{\partial \overline{z}}.
\end{equation}
Then, we can prove the following
\begin{proposition}\label{Commutator}
For any $z,w\in\mathbb{C}$ we have
$$\displaystyle \left[\frac{\partial}{\partial z}, M_z\right]K(z,w)=K(z,w)$$
and
$$\displaystyle \left[\frac{\partial}{\partial \overline{z}}, M_{\overline{z}}\right]K(z,w)=K(z,w).$$
\end{proposition}
\begin{proof}
We have
\[
\begin{split}
\displaystyle\frac{\partial}{\partial z}M_z K(z,w)&=\frac{\partial}{\partial z}\left(zK(z,w) \right)\\
&=z\frac{\partial}{\partial z}K(z,w)+K(z,w)\\
&=z\overline{w}K(z,w)+K(z,w).
\end{split}
\]
On the other hand
\[
\begin{split}
\displaystyle M_z\frac{\partial}{\partial z}K(z,w)&=M_z\left(\overline{w}K(z,w)\right)\\
&=z\overline{w}K(z,w),\\
\end{split}
\]
hence, we obtain $$\displaystyle \left[\frac{\partial}{\partial z}, M_z\right]K(z,w)=K(z,w).$$
In a similar way we can prove that
$$\displaystyle \left[\frac{\partial}{\partial \overline{z}}, M_{\overline{z}}\right]K(z,w)=K(z,w).$$
\end{proof}
Thanks to the reproducing kernel property we have this more general result
\begin{theorem}
For any $f\in\mathcal{H}(K)$, the following identities hold
$$\displaystyle \left[\frac{\partial}{\partial z}, M_z\right]f=f$$
and
$$\displaystyle \left[\frac{\partial}{\partial \overline{z}}, M_{\overline{z}}\right]f=f.$$
\end{theorem}
\begin{proof}
Let $f\in\mathcal{H}(K)$, we know that $$f(z)=<f,K_z>_{\mathcal{H}(K)}, \text{ for any } z\in \Omega.$$
Thus, for any $z\in \Omega$ we apply Proposition \ref{Commutator} and get \[
\begin{split}
\displaystyle \left[\frac{\partial}{\partial z}, M_z\right]f(z) &= <f, \left[\frac{\partial}{\partial z}, M_z\right]K_z>_{\mathcal{H}(K)}\\
&= <f,K_z>_{\mathcal{H}(K)}\\
&=f(z).
\end{split}
\]
Hence, it follows that $$\displaystyle \left[\frac{\partial}{\partial z}, M_z\right]f=f.$$
In the same way we can prove that $$\displaystyle \left[\frac{\partial}{\partial \overline{z}}, M_{\overline{z}}\right]f=f.$$
\end{proof}
As a consequence of Proposition \ref{Fockpptd} we can study more properties of the kernel function in \eqref{newFock-kernel}.
\begin{proposition}\label{deltadelta}
For any $z, w\in\mathbb{C}$ we have
\begin{equation}
\displaystyle \Delta_z K(z,w)=4|w|^2K(z,w)
\end{equation}
and \begin{equation}
\Delta_w\Delta_zK(z,w)=16\left(1+w\overline{z}+\overline{w}z+|w|^2|z|^2\right)K(z,w)=16|1+\overline{w}z|^2K(z,w).
\end{equation}
\end{proposition}
\begin{proof}
We note that using \eqref{newFock-kernel} and Proposition \ref{Fockpptd} we have
\[
\begin{split}
\displaystyle \frac{\partial^2}{\partial \overline{z}\partial z}K(z,w) &= \overline{w}\frac{\partial}{\partial \overline{z}}K(z,w) \\
&= \overline{w}wK(z,w)\\
&=|w|^2K(z,w).
\end{split}
\]
Since $\displaystyle \frac{\partial^2}{\partial \overline{z}\partial z}=\frac{1}{4}\Delta_z$ we conclude that
\begin{equation}\label{deltazK}
\displaystyle \Delta_z K(z,w)=4|w|^2K(z,w).
\end{equation}
Furthermore, by taking the derivative with respect to $w$ we get \[
\begin{split}
\displaystyle \frac{\partial}{\partial w}\Delta_z K(z,w) &= 4\overline{w}\frac{\partial}{\partial w}\left(wK(z,w)\right) \\
&= 4\overline{w}(K(z,w)+w\overline{z}K(z,w))\\
&=4(1+w\overline{z})\overline{w}K(z,w).
\end{split}
\]
Then, we apply the derivative with respect to $\overline{w}$, develop the computations and get \[
\begin{split}
\displaystyle \frac{\partial^2}{\partial \overline{w}\partial w}\Delta_z K(z,w) &= 4(1+w\overline{z})\frac{\partial}{\partial \overline{w}}\left(\overline{w} K(z,w)\right) \\
&=4(1+w\overline{z})(1+\overline{w}z)K(z,w) \\
&=4|1+\overline{w}z|^2K(z,w),
\end{split}
\]
from which we conclude that $$\Delta_w\Delta_zK(z,w)=16|1+\overline{w}z|^2K(z,w).$$
\end{proof}
\begin{corollary}
Let $\Omega=\mathbb{D}$ denote the unit disk, we consider the operator $$T_{z,w}:=\frac{\Delta_w\Delta_z}{16|1+\overline{w}z|^2}, \quad \forall (z,w)\in\Omega\times \Omega.$$
Then, we have \begin{equation}
T_{z,w}K(z,w)=K(z,w), \quad \forall (z,w)\in\Omega\times \Omega.
\end{equation}
\end{corollary}
\begin{proof}
It is a direct consequence of Proposition \ref{deltadelta}.
\end{proof}
\begin{remark}
From \eqref{deltazK} we deduce that if $w$ is a fixed parameter, then the kernel function $K(z,w)$ can be seen as an eigenfunction of the Laplace operator $\Delta_z$ with eigenvalue given by $\displaystyle 4|w|^2$.
\end{remark}
\begin{definition}
The polyanalytic Fock space $\mathcal{SF}(\mathbb C)$ of infinite order is the set of functions of the form \begin{equation}
f(z)=\displaystyle \sum_{n=0}^{\infty}\overline{z}^nf_n(z),
\end{equation}
satisfying the conditions
\begin{enumerate}
\item[i)] $f_n\in \mathcal{F}(\mathbb{C})$ for any $n\geq 0$;
\item[ii)]$\displaystyle ||f||^{2}_{\mathcal{SF}(\mathbb{C})}= \sum_{n=0}^{\infty} n! ||f_n||^{2}_{\mathcal{F}(\mathbb{C})}<\infty.$
\end{enumerate}
Then, we consider the scalar product on $\mathcal{SF}(\mathbb{C})$ given by \begin{equation}\displaystyle
\langle f,g \rangle_{\mathcal{SF}(\mathbb{C})}:=\sum_{k=0}^{\infty}k!\langle f_k,g_k \rangle_{\mathcal{F}(\mathbb{C})},
\end{equation}
for any $f=\displaystyle \sum_{k=0}^{\infty}\overline{z}^kf_k$ and $g=\displaystyle \sum_{k=0}^{\infty}\overline{z}^kg_k$ with $f_k,g_k\in\mathcal{F}(\mathbb{C})$ for every $k\geq 0$.
\end{definition}
\begin{proposition}\label{seqNF}
A function $f:\mathbb{C}\longrightarrow \mathbb{C}$ belongs to $\mathcal{SF}(\mathbb{C})$ if and only if $f$ is of the form $$\displaystyle f(z)=\sum_{(m,n)\in\mathbb{N}^2}z^m\overline{z}^n\alpha_{m,n},$$
with $(\alpha_{m,n})\subset\mathbb{C}$ and such that \begin{equation}\displaystyle
||f||^{2}_{\mathcal{SF}(\mathbb{C})}=\sum_{(m,n)\in\mathbb{N}^2}m!n!|\alpha_{m,n}|^2<\infty.
\end{equation}
Moreover, if for any $(m,n)\in\mathbb{N}^2$ we set $\phi_{m,n}(z,\overline{z})=\displaystyle \frac{z^m\bar{z}^n}{\sqrt{m!n!}}$ then, the family of functions $\lbrace \phi_{m,n} \rbrace_{m,n\geq 0}$ form an orthonormal basis of $\mathcal{SF}(\mathbb{C})$.
\end{proposition}
\begin{proof}
Let
$$\displaystyle f(z)=\sum_{(m,n)\in\mathbb{N}^2}\overline{z}^nz^m\alpha_{m,n},$$
with $(\alpha_{n,m})\subset\mathbb{C}$.
Setting $f_n(z)=\displaystyle \sum_{m=0}^\infty z^m\alpha_{m,n}$, it is clear that $\displaystyle f(z)=\sum_{n=0}^{\infty}\overline{z}^nf_n(z)$. Moreover, we have
$$\displaystyle ||f||^{2}_{\mathcal{SF}(\mathbb{C})}=\sum_{n=0}^{\infty}n!||f_n||^{2}_{\mathcal{F}(\mathbb{C})}=\sum_{(m,n)\in\mathbb{N}^2}m!n!|\alpha_{m,n}|^2.$$
Therefore, $f$ belongs to the space $\mathcal{SF}(\mathbb{C})$ if and only if
$$\displaystyle
||f||^{2}_{\mathcal{SF}(\mathbb{C})}=\sum_{(m,n)\in\mathbb{N}^2}m!n!|\alpha_{m,n}|^2<\infty.$$
On the other hand, easy computations lead to $$\langle f,\phi_{m,n}\rangle_{\mathcal{SF}(\mathbb{C})}=\sqrt{n!m!}\alpha_{m,n}, \quad \forall m,n \geq 0.$$
If $\langle f,\phi_{m,n}\rangle_{\mathcal{SF}(\mathbb{C})}=0$ for any $m,n\geq 0$, then we have $\alpha_{m,n}=0$ for any $m,n\geq 0$.
We note also that $$\langle \phi_{m,n},\phi_{m,n}\rangle_{\mathcal{SF}(\mathbb{C})}=1 \text{ and } \langle \phi_{m,n},\phi_{p,q}\rangle_{\mathcal{SF}(\mathbb{C})}=0 \text{ whenever } (m,n)\neq (p,q).$$
In particular, this shows that $\lbrace \phi_{m,n} \rbrace_{m,n\geq 0}$ form an orthonormal basis of $\mathcal{SF}(\mathbb{C})$. This ends the proof.
\end{proof}
\begin{example}
We recall the complex Hermite polynomials introduced in \cite{Ito}
\begin{equation}
H_{m,n}(z,\overline{z}):=\displaystyle \sum_{k=0}^{\min{(m,n)}}(-1)^k k! {m\choose k}{n \choose k}z^{m-k}\overline{z}^{n-k}.
\end{equation}
It is easy to prove that $H_{m,n}$ belong to $\mathcal{SF}(\mathbb{C})$.
\end{example}
We now provide a sequential charachterization of the space $\mathcal{H}(K)$
\begin{theorem}
We have
$$\mathcal{H}(K)=\mathcal{SF}(\mathbb{C}).$$
Moreover, it holds that
\begin{equation}
K(z,w)=\displaystyle \sum_{m,n=0}^{\infty}\phi_{m,n}(z,\overline{z})\overline{\phi_{m,n}(w,\overline{w})}, \quad \text{ for any } z,w\in\mathbb{C}.
\end{equation}
\end{theorem}
\begin{proof}
Since $(\phi_{m,n})_{m,n\geq 0}$ is an orthonormal basis of the space $\mathcal{SF}(\mathbb{C})$, the associated reproducing kernel is given by the convergent series
$$\displaystyle \sum_{m,n=0}^{\infty}\phi_{m,n}(z,\bar{z})\overline{\phi_{m,n}(w,\bar{w})}<\infty, \text{ for any } z,w\in\mathbb{C}.$$
More precisely, for any $(z,w)\in\mathbb{C}^2$ we have the equalities
\[
\begin{split}
\displaystyle \sum_{m,n=0}^{\infty}\phi_{m,n}(z,\bar{z})\overline{\phi_{m,n}(w,\bar{w})}
&= \sum_{m,n=0}^{\infty}\frac{z^m\bar{z}^n \bar{w}^m w^n}{m!n!} \\
&= \left(\sum_{m=0}^{\infty} \frac{z^m\bar{w}^m}{m!}\right) \left(\sum_{n=0}^{\infty} \frac{w^n\bar{z}^n}{n!}\right)
\\
&= e^{z\bar{w}}e^{w\bar{z}}
\\
&=e^{z\bar{w}+w\bar{z}}
\\
&=K(z,w).
\end{split}
\]
\end{proof}
\begin{theorem}\label{Adj}
It holds that
\begin{equation}\displaystyle
\langle \frac{\partial}{\partial z}f,g\rangle_{\mathcal{SF}(\mathbb{C})}=\langle f,M_{z}g\rangle_{\mathcal{SF}(\mathbb{C})},
\end{equation}
moreover
\begin{equation}\displaystyle
\langle \frac{\partial}{\partial \overline{z}}f,g\rangle_{\mathcal{SF}(\mathbb{C})}=\langle f,M_{\overline{z}}g\rangle_{\mathcal{SF}(\mathbb{C})}.
\end{equation}
\end{theorem}
\begin{proof}
Let $\displaystyle f=\sum_{k=0}^{\infty}\overline{z}^kf_k$ and $\displaystyle g=\sum_{k=0}^{\infty}\overline{z}^kg_k$ in $\mathcal{SF}(\mathbb{C})$ that belongs to the domains of the creation and annihilation operators. Firstly, we note that we have $$\displaystyle M_z(g)=\sum_{k=0}^{\infty}\overline{z}^kM_z(g_k).$$ Then, it follows that
\[
\begin{split}
\displaystyle \langle \frac{\partial}{\partial z}f,g\rangle_{\mathcal{SF}(\mathbb{C})}
&= \sum_{k=0}^{\infty}k!\langle \frac{\partial}{\partial z} f_k,g_k \rangle_{\mathcal{F}(\mathbb{C})}\\
&= \sum_{k=0}^{\infty}k!\langle f_k,(\frac{\partial}{\partial z})^*g_k \rangle_{\mathcal{F}(\mathbb{C})}
\\
&= \sum_{k=0}^{\infty}k!\langle f_k,M_z(g_k) \rangle_{\mathcal{F}(\mathbb{C})}
\\
&=\langle f,M_{z}(g)\rangle_{\mathcal{SF}(\mathbb{C})}.
\end{split}
\]
Moreover, since $$\displaystyle \frac{\partial}{\partial\overline{z}}(f)(z)=\sum_{h=0}^{\infty}(h+1)\overline{z}^hf_{h+1},$$
and
$$\displaystyle M_{\overline{z}}(g)(z)=\sum_{h=1}^{\infty}\overline{z}^hg_{h-1},$$
it follows that
\[
\begin{split}
\displaystyle \langle \frac{\partial}{\partial\overline{z}}f,g\rangle_{\mathcal{SF}(\mathbb{C})} &= \sum_{k=0}^{\infty}k!\langle (k+1)f_{k+1},g_k \rangle_{\mathcal{F}(\mathbb{C})}
\\
&= \sum_{k=0}^{\infty}(k+1)!\langle f_{k+1},g_k \rangle_{\mathcal{F}(\mathbb{C})}
\\
&= \sum_{k=1}^{\infty}k!\langle f_{k},g_{k-1} \rangle_{\mathcal{F}(\mathbb{C})}
\\
&=\langle f,M_{\overline{z}}(g)\rangle_{\mathcal{SF}(\mathbb{C})}.
\end{split}
\]
\end{proof}
\begin{remark}
We shall see later that that we have $$\mathcal{F}(\mathbb{C})\subset \mathcal{SF}(\mathbb{C})\subset L^2(\mathbb{C},d\mu_\beta), \quad \beta>2.$$
The previous inclusions are strict. It is well known that the classical Fock space $\mathcal{F}(\mathbb{C})$ is the only space of entire functions on which the creation and annihilation operators are adjoints of each others and satisfy the classical commutation rules. Of course, this is not true anymore on $L^2(\mathbb{C},d\mu)$, see Proposition 7.2 in \cite{Shige}. However, the previous theorem shows that the result still holds in the subspace $\mathcal{SF}(\mathbb{C})$ of the space of polyanalytic functions of infinite order.
\end{remark}
\begin{definition}\label{RinfinityL}
Let $f,g\in\mathcal{SF}(\mathbb{C})$ and let $f(z)=\displaystyle \sum_{n=0}^{\infty}\bar{z}^nf_n(z)$ and $g(z)=\displaystyle \sum_{m=0}^{\infty}z^mg_m(\bar{z})$. Then, we define two backward shift operators $R_\infty$ and $L_\infty$ with respect to the variables $z$ and $\bar{z}$ respectively given by
\begin{equation}
R_{\infty}(f)(z,\bar{z})=\displaystyle \sum_{n=0}^{\infty}\bar{z}^nR_0(f_n)(z)=\frac{1}{z}\left(f(z,\bar{z})-\sum_{n=0}^{\infty}\bar{z}^{n}f_n(0)\right), \quad z\in\mathbb{C}
\end{equation}
and
\begin{equation}
L_{\infty}(g)(z,\bar{z})=\displaystyle \frac{1}{\bar{z}}\left(g(z,\bar{z})-\sum_{m=0}^{\infty}z^{m}g_m(0)\right), \quad z\in\mathbb{C}.
\end{equation}
\end{definition}
It turns out that both the backward shift operators $R_\infty$ and $L_\infty$ define two contractions on the polyanalytic Fock space of infinite order $\mathcal{SF}(\mathbb{C})$. Indeed, the following result holds:
\begin{proposition}
For any $f\in\mathcal{SF}(\mathbb{C}),$ we have
\begin{equation}
||R_{\infty}(f)||_{\mathcal{SF}(\mathbb{C})}^2\leq ||f||_{\mathcal{SF}(\mathbb{C})}^2-\sum_{n=0}^{\infty}n!|f_n(0)|^2 \leq||f||_{\mathcal{SF}(\mathbb{C})}^2.
\end{equation}
and \begin{equation}
||L_{\infty}(g)||_{\mathcal{SF}(\mathbb{C})}^2\leq ||g||_{\mathcal{SF}(\mathbb{C})}^2-\sum_{n=0}^{\infty}n!|g_n(0)|^2 \leq||g||_{\mathcal{SF}(\mathbb{C})}^2.
\end{equation}
\end{proposition}
\begin{proof}
We will prove the result for $R_{\infty}$. Indeed, if we consider
$$f(z)=\displaystyle \sum_{n=0}^{\infty}\bar{z}^n f_n(z), \quad f_n\in\mathcal{F}(\mathbb{C}),$$
we have $$R_{\infty}(f)(z) \displaystyle =\sum_{n=0}^{\infty}\bar{z}^nR_0(f_{n})(z).$$
Thus, using the fact $R_0$ is a contraction on the Fock space (see \cite{ACK}) we deduce
\[
\begin{split}
\displaystyle ||R_{\infty}(f)||_{\mathcal{SF}(\mathbb{C})}^2 &= \sum_{n=0}^{\infty}n!||R_0(f_{n})||_{\mathcal{F}(\mathbb{C})}^2
\\
&\leq \sum_{n=0}^{\infty}n!(||f_{n}||_{\mathcal{F}(\mathbb{C})}^2-|f_n(0)|^2)
\\
&=||f||_{\mathcal{SF}(\mathbb{C})}^2-\sum_{n=0}^{\infty}n!|f_n(0)|^2\leq ||f||_{\mathcal{SF}(\mathbb{C})}^2
\\
&
\end{split}
\]
and this ends the proof. We note that the argument follows in a similar way for $L_{\infty}$ using the fact that we can write $f$ also in the form
$$f(z)=\displaystyle\sum_{m=0}^{\infty}z^mf_m(\overline{z}).$$
\end{proof}
Now, we consider other two operators:
\begin{definition}
Let $f,g\in\mathcal{SF}(\mathbb{C})$ and let $f(z)=\displaystyle \sum_{n=0}^{\infty}\bar{z}^nf_n(z)$ and $g(z)=\displaystyle \sum_{m=0}^\infty z^mg_m(\bar{z})$. Then, we define two operators with respect to the variables $z$ and $\bar{z}$ respectively which are given by
\begin{equation}
I_{\infty}(f)(z,\bar{z})=\displaystyle \sum_{n=0}^{\infty}\bar{z}^nI(f_n)(z), \quad z\in\mathbb{C}
\end{equation}
and
\begin{equation}
J_{\infty}(g)(z,\bar{z})=\displaystyle \sum_{m=0}^{\infty}z^mJ(g_m)(\bar{z}), \quad z\in\mathbb{C}.
\end{equation}
\end{definition}
We point out that the operator $I$ is the integration operator considered in \cite{ACK}, while $J$ is the integration with respect to the conjugate variable.
\begin{remark}
It holds that
\begin{equation}
I_{\infty}(\bar{z}^nz^m)(z)=\displaystyle \frac{\bar{z}^{n}z^{m+1}}{m+1} , \quad z\in\mathbb{C}
\end{equation}
and
\begin{equation}
J_{\infty}(\bar{z}^nz^m)(z)= \displaystyle \frac{\bar{z}^{n+1}z^m}{n+1} , \quad z\in\mathbb{C}.
\end{equation}
\end{remark}
As a consequence, we have the following result:
\begin{theorem}
The adjoints of $R_\infty$ and $L_\infty$ satisfy
\begin{equation}
R_{\infty}^{*}=I_\infty
\end{equation}
and
\begin{equation}
L_{\infty}^{*}=J_\infty.
\end{equation}
\end{theorem}
\begin{proof}
Let $f,g\in\mathcal{SF}(\mathbb{C})$; we will prove that
\begin{equation}
\langle I_{\infty}(f),g \rangle_{\mathcal{SF}(\mathbb{C})}=\langle f, R_{\infty}(g) \rangle_{\mathcal{SF}(\mathbb{C})}.
\end{equation}
Indeed, we write $f(z)=\displaystyle \sum_{n=0}^{\infty}\overline{z}^n f_n(z)$ and $g(w)=\displaystyle \sum_{n=0}^{\infty}\overline{z}^n g_n(z)$ with $f_n,g_n\in\mathcal{F}(\mathbb{C})$ for any $n\geq 0$. We have
$$I_{\infty}(f)(z)=\displaystyle \sum_{n=0}^{\infty} \bar{z}^nI(f_n)(z) $$
and
$$R_{\infty}(g)(z)=\sum_{m=0}^\infty \bar{z}^m R_0(g_m)(z).$$
Then, we use the scalar product on $\mathcal{SF}(\mathbb{C})$ and apply the result on the classical backward shift operator, see \cite{ACK}. Therefore, it follows that
\[
\begin{split}
\displaystyle \langle I_\infty(f),g\rangle_{\mathcal{SF}(\mathbb{C})} &= \sum_{k=0}^{\infty}k!\langle I(f_{k}),g_k \rangle_{\mathcal{F}(\mathbb{C})}
\\
&= \sum_{k=0}^{\infty}k!\langle f_{k},R_0(g_k) \rangle_{\mathcal{F}(\mathbb{C})}
\\
&= \langle f,R_\infty(g) \rangle_{\mathcal{SF}(\mathbb{C})}.
\\
&
\end{split}
\]
The second part of the statement can be proved in a similar way.
\end{proof}
\section{A kernel function relating polyanalytic Fock spaces of finite and infinite order}
\setcounter{equation}{0}
In this section we study how the polyanalytic Fock spaces of finite and infinite order are related between them.
We denote by $\mathcal{F}_n(\mathbb{C})$ the classical polyanalytic Fock space whose kernel is given by the formula \eqref{Kn}.
The relation between the kernels $K$ and $(K_n)_{n\geq 1}$ is described in the next result.
\begin{proposition}[kernel formula]
For any $z,w\in\mathbb{C}$ we set $$G(z,w)=e^{z\overline{w}-(|z|^2+|w|^2)}.$$
Then, it holds that
\begin{equation}
\displaystyle \sum_{n=1}^{\infty} \frac{K_n(z,w)}{2^{n+1}}=G(z,w)K(z,w), \quad \text{ for any } z,w\in\mathbb{C}.
\end{equation}
\end{proposition}
\begin{proof}
We note that the polyanalytic Fock kernels given by \eqref{Kn} can be written in terms of the generalized Laguerre polynomials as follows
\begin{equation}\label{LKn}
K_n(z,w)=e^{z\overline{w}}L_{n-1}^{1}(|z-w|^2), \quad \text{ for any } z,w\in\mathbb{C}.
\end{equation}
Then, taking the series \eqref{LKn} we obtain
\begin{equation}\label{sumKn}
\displaystyle \sum_{n=1}^{\infty}\frac{K_n(z,w)}{2^{n-1}}=e^{z\overline{w}}\sum_{n=1}^{\infty}\frac{L_{n-1}^{1}(|z-w|^2)}{2^{n-1}}=e^{z\overline{w}}\sum_{n=0}^{\infty}\frac{L_{n}^{1}(|z-w|^2)}{2^{n}}.
\end{equation}
Morever, we note that for any $a,\alpha>0$ we have the following expansion, see \cite[Example 2, pp 89]{L}
\begin{equation}\label{ax}
e^{-ax}=(a+1)^{-(\alpha+1)}\sum_{n=0}^{\infty}\left(\frac{a}{a+1}\right)^{n}L_n^\alpha(x),\quad x\geq 0.
\end{equation}
In particular, inserting $\alpha=a=1$ and $x=|z-w|^2$ in \eqref{ax} we obtain
$$e^{-|z-w|^2}=\frac{1}{2^2}\sum_{n=0}^{\infty}\frac{L^1_n(|z-w|^2)}{2^n}, \quad z,w\in\mathbb{C}.$$
Hence, with some computations involving \eqref{sumKn} we obtain that
$$\displaystyle \sum_{n=1}^{\infty} \frac{K_n(z,w)}{2^{n+1}}=G(z,w)K(z,w),$$
\text{ for any } $z,w\in\mathbb{C}$ where $G(z,w)=e^{z\overline{w}-(|z|^2+|w|^2)}.$
\end{proof}
\begin{remark}
We observe that the classical creation and annihilation operators are adjoint of each others on the polyanalytic Fock space of infinite order $\mathcal{SF}(\mathbb{C})$, see Theorem \ref{Adj}.
\end{remark}
\section{A Segal-Bargmann type transform and related operators}
\setcounter{equation}{0}
In this section, we deal with a Segal-Bargmann type transform related to the polyanalytic Fock spaces of infinite order. We discuss also some related operators.
Let ($\psi_n(x))_{n\geq 0}$ denote the normalized Hermite functions and consider the Segal-Bargmann kernel $A(z,x)$ which is given by \begin{equation}
A(z,x):=\displaystyle \sum_{n=0}^{\infty}\frac{z^n}{\sqrt{n!}}\psi_n(x)=e^{-\frac{1}{2}(z^2+x^2)+\sqrt{2}zx}, \quad \text{ for any } (z,x)\in\mathbb{C}\times \mathbb{R}.
\end{equation}
For any $z\in\mathbb{C}$ fixed we use also the notation $A_z(x)=A(z,x)$ for all $x\in\mathbb{R}.$ The kernel \eqref{newFock-kernel} can be factorized as follows:
\begin{theorem} \label{kerFac}
For any $(z,w)\in\mathbb{C}^2$, we have
\begin{equation}
K(z,w)= \langle A_z\otimes A_{\bar{z}},A_w \otimes A_{\bar{w}}\rangle_{L^2(\mathbb{R}^2)}.
\end{equation}
\end{theorem}
\begin{proof}
The proof is based on computations using Fubini's theorem combined with the following well-known fact $$\langle A_z, A_w\rangle_{L^2(\mathbb{R})}=e^{z\bar{w}}.$$
Indeed, for $z,w\in\mathbb{C}$ we have the explicit computations
\[
\begin{split}
\displaystyle \langle A_z\otimes A_{\bar{z}},A_w \otimes A_{\bar{w}}\rangle_{L^2(\mathbb{R}^2)} &= \int_{\mathbb{R}^2} (A_z\otimes A_{\bar{z}})(x,y)\overline{(A_w\otimes A_{\bar{w}})(x,y)} dxdy\\
&=\int_{\mathbb{R}^2}A_z(x)A_{\overline{z}}(y) \overline{A_w(x)}\textbf{ }\overline{A_{\overline{w}}(y)}dxdy\\
&= \left(\int_{\mathbb{R}}A_z(x) \overline{A_w(x)}dx \right) \left(\int_{\mathbb{R}}\overline{A_{z}(y)} A_w(y) dy\right)
\\
&= \langle A_z, A_w\rangle_{L^2(\mathbb{R})} \langle A_w, A_z\rangle_{L^2(\mathbb{R})}
\\
&= e^{z\bar{w}} e^{w\bar{z}}
\\
&=K(z,w).
\end{split}
\]
\end{proof}
\begin{definition}\label{Ttran}
For a given $\varphi\in L^2(\mathbb{R}^2)$, we define the so-called first Segal-Bargmann type transform by
\begin{equation} \displaystyle
T(\varphi)(z,\bar{z})=\langle \varphi, \overline{A_z\otimes A_{\bar{z}}}\rangle_{L^2(\mathbb{R}^2)}= \int_{\mathbb{R}^2} A_z(x)A_{\bar{z}}(y)\varphi(x,y)dxdy.
\end{equation}
\end{definition}
Then, as a consequence, we can write the kernel function $K(z,w)$
as a function in the range of the transform $T$ thanks to the following
\begin{proposition}
For a fixed $w\in\mathbb{C}$, we set $\varphi_w(t_1,t_2)=(A_{\overline{w}}\otimes A_w)(t_1,t_2)$, with $t_1,t_2\in\mathbb{R}$. Then
$$K(z,w)=T(\varphi_w)(z,\overline{z}), \quad \text{ for any } z\in\mathbb{C}.$$
\end{proposition}
\begin{proof}
This result can be obtained as a direct application of Theorem \ref{kerFac} and Definition \ref{Ttran} taking into account that
$$\overline{A_{\overline{w}}\otimes A_w}=A_w\otimes A_{\overline{w}}, \quad w\in\mathbb{C}.$$
Indeed, for any $z,w\in\mathbb{C}$ we have
\[
\begin{split}
\displaystyle T(\varphi_w)(z,\overline{z})&= \langle \varphi_w, \overline{A_z\otimes A_{\bar{z}}}\rangle_{L^2(\mathbb{R}^2)}\\
&= \langle A_{\overline{w}}\otimes A_{w}, \overline{A_z\otimes A_{\bar{z}}}\rangle_{L^2(\mathbb{R}^2)}\\
&= \int_{\mathbb{R}^2}(A_{\overline{w}}\otimes A_{w})(t_1,t_2)(A_z\otimes A_{\bar{z}})(t_1,t_2)dt_1dt_2
\\
&= \langle A_z\otimes A_{\bar{z}}, \overline{A_{\overline{w}} \otimes A_{w}}\rangle_{L^2(\mathbb{R}^2)}
\\
&= \langle A_z\otimes A_{\bar{z}},A_w \otimes A_{\bar{w}}\rangle_{L^2(\mathbb{R}^2)}
\\
&=K(z,w).
\end{split}
\]
\end{proof}
\begin{remark}
As a consequence of the previous result, we observe that for any fixed $w\in\mathbb{C}$, we have
$$T(\varphi_w)(z)=K(z,w)=K_w(z), \quad z\in\mathbb{C}.$$
Then, $T^*=T^{-1}$ since $T$ is a unitary operator; moreover, for any $w\in\mathbb{C}$
$$T^{-1}(K_w)(t_1,t_2)=T^{*}(K_w)(t_1,t_2)=(A_{\overline{w}}\otimes A_w)(t_1,t_2), \quad \text{ for all } (t_1,t_2)\in\mathbb{R}^2.$$
\end{remark}
As a first example, we consider the family of functions given by $$\psi_{m,n}(x,y):=(\psi_m\otimes \psi_n)(x,y)=\psi_m(x)\psi_n(y), \text{ for any } m,n\geq 0.$$ We have
\begin{proposition}\label{Action1}
For every $z\in\mathbb{C}$ we have
\begin{equation}
T(\psi_{m,n})(z,\bar{z})=\frac{z^m\overline{z^n}}{\sqrt{m!n!}}=\phi_{m,n}(z,\overline{z}), \quad m,n=0,1,...
\end{equation}
and
\begin{equation}
\Delta_z \phi_{p,q}(z,\overline{z})=4\sqrt{pq}\phi_{p-1,q-1}(z,\overline{z}), \quad p,q=1,2,....
\end{equation}
\end{proposition}
\begin{proof}
Recalling that $A_{\overline{z}}(y)=\overline{A_z(y)}$ for any $y\in\mathbb{R}$, using the Fubini's theorem we have: \[
\begin{split}
\displaystyle T(\psi_{(m,n)})(z,\bar{z}) &= \int_{\mathbb{R}^2} (A_z\otimes A_{\bar{z}})(x,y)\psi_{m,n}(x,y)dxdy\\
&=\int_{\mathbb{R}^2}A_z(x)A_{\overline{z}}(y) \psi_{m}(x)\psi_n(y) dxdy\\
&= \left(\int_{\mathbb{R}}A_z(x) \psi_m(x) dx \right) \left(\int_{\mathbb{R}}\overline{A_{z}(y)}\psi_n(y) dy\right)
\\
&= B(\psi_m)(z)\overline{B(\psi_n)(z)}
\\
&= \frac{z^m\overline{z}^n}{\sqrt{m!n!}}
\\
&=\phi_{m,n}(z,\overline{z}).
\end{split}
\]
Now, using the fact that $\displaystyle \Delta_z=4\frac{\partial^2}{\partial z \partial \overline{z}}$ we get
$$\displaystyle \Delta_z(\phi_{p,q})(z,\overline{z})=4\frac{pq}{\sqrt{p!q!}}z^{p-1}\overline{z}^{q-1}=4\sqrt{pq}\phi_{p-1,q-1}(z,\overline{z}).$$
\end{proof}
\begin{example}
Let $$f(z,\overline{z})=T(\psi_{n,m})(z,\overline{z}), \text{ for any } z\in\mathbb{C}.$$
Then $f\in\mathcal{SF}(\mathbb{C})$, moreover we have $$||f||_{\mathcal{SF}(\mathbb{C})}=||T(\psi_{n,m})||=1=||\psi_{n,m}||_{L^2(\mathbb{R}^2)}.$$
\end{example}
\begin{theorem}
The first Segal-Bargmann type transform $T$ defines an isometric isomorphism from $L^2(\mathbb{R}^2)$ onto $\mathcal{SF}(\mathbb{C})$.
\end{theorem}
\begin{proof}
The normalized Hermite functions $(\psi_{m,n})_{m,n\geq 0}$ form an orthonormal basis of $L^2(\mathbb{R}^2)$, thus for any $\varphi\in L^2(\mathbb{R}^2,\mathbb{C}),$ there exist unique coefficients $(\beta_{m,n})_{m,n\geq 0}$ in $\mathbb{C}$ such that
$$\varphi(x,y)=\displaystyle \sum_{m,n=0}^{\infty} \psi_{m,n}(x,y)\beta_{m,n}, \quad \text{ and }\quad ||\varphi||_{L^2(\mathbb{R}^2)}^{2}=\sum_{m,n=0}^{\infty}|\beta_{m,n}|^2<\infty. $$
Therefore, inserting $\varphi$ in the definition of the transform $T$ and using some standard arguments we have
\[
\begin{split}
\displaystyle T(\varphi)(z,\bar{z}) &= \int_{\mathbb{R}^2} A_z(x) A_{\bar{z}}(y)\left(\sum_{m,n=0}^{\infty}\psi_{m,n}(x,y)\beta_{m,n}\right)dxdy\\
&=\sum_{m,n=0}^{\infty}\left(\int_{\mathbb{R}^2}A_z(x)A_{\overline{z}}(y) \psi_{m,m}(x,y) dxdy\right) \beta_{m,n}\\
&= \sum_{m,n=0}^{\infty}T(\psi_{m,n})(z,\overline{z})\beta_{m,n}.
\end{split}
\]
Applying the first part of Proposition \ref{Action1} we obtain
\[
\begin{split}
\displaystyle T(\varphi)(z,\bar{z}) &= \sum_{m,n=0}^{\infty}\phi_{m,n}(z,\overline{z})\beta_{m,n}
\\
&= \sum_{m,n=0}^{\infty} \frac{z^m\overline{z}^n}{\sqrt{m!n!}}\beta_{m,n}
\\
&= \sum_{n=0}^{\infty}\overline{z}^n\left(\sum_{m=0}^{\infty}\frac{z^m}{\sqrt{m!n!}}\beta_{m,n}\right).
\end{split}
\]
Then, setting $\displaystyle f_n(z)=\frac{1}{\sqrt{n!}}\sum_{m=0}^{\infty}\frac{z^m}{\sqrt{m!}}\beta_{m,n}$ for any $n\geq 0 $ we have
$$T(\varphi)(z,\overline{z})=\displaystyle \sum_{n=0}^{\infty}\overline{z}^nf_n(z), \quad z\in\mathbb{C}.$$
We observe that $f_n$ are entire functions. Moreover, it is immediate to see that
$$\displaystyle ||f_n||^{2}_{\mathcal{F}(\mathbb{C})}=\frac{1}{n!}\sum_{m=0}^{\infty}|\beta_{m,n}|^2<\infty.$$
Hence, we deduce
\[
\begin{split}
\displaystyle ||T(\varphi)||_{\mathcal{SF}(\mathbb{C})}^{2} &= \sum_{n=0}^{\infty} n!||f_n||^{2}_{\mathcal{F}(\mathbb{C})}\\
&=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}|\beta_{m,n}|^2\\
&=||\varphi||_{L^2(\mathbb{R})}^{2}.
\\
&
\end{split}
\]
This proves that the transform $T$ defines an isometric operator from $L^2(\mathbb{R})$ into the polyanalytic Fock space of infinite order $\mathcal{SF}(\mathbb{C}).$ On the other hand, we note that using Proposition \ref{Action1} we have
$$T(\psi_{m,n})=\frac{z^m\overline{z}^n}{\sqrt{m!n!}}, \quad m,n\geq 0.$$
This allows to justify that if $f\in\mathcal{SF}(\mathbb{C})$ there exists $g\in L^2(\mathbb{R})$ such that $f=T(g)$. In particular, this shows that $T$ is also surjective which ends the proof.
\end{proof}
\begin{remark}
We know that $T$ defines an isometric and surjective transform from $L^2(\mathbb{R}^2)$ onto $\mathcal{SF}(\mathbb{C})$. This shows that $T$ is invertible and the inverse of $T$ exists. However, we do not know how to explicitly calculate this inverse because of the lack of a geometric description of the space $\mathcal{SF}(\mathbb{C})$.
\end{remark}
We recall the position operators given by $$X\varphi(x,y)=x\varphi(x,y)$$ and $$Y\varphi(x,y)=y\varphi(x,y).$$
We denote by $\mathcal{D}(X)=\left\lbrace{\varphi\in L^2(\mathbb{R}), X(\varphi)\in L^2(\mathbb{R})}\right\rbrace$ the domain of the position operator $X$.
\begin{proposition}\label{PositionX}
The following relations hold:
$$\displaystyle T^{-1}\left(\frac{\partial}{\partial z}+M_z\right)T=\sqrt{2}X, \quad \text{ on } \mathcal{D}(X)$$
and $$\displaystyle T^{-1}\left(\frac{\partial }{\partial \overline{z}}+M_{\overline{z}}\right)T=\sqrt{2}Y, \quad \text{ on } \mathcal{D}(Y).$$
\end{proposition}
\begin{proof}
We will make the calculations for the operator $X$ and for $Y$ it can be done in a similar way. As in the classical case, for $\varphi\in \mathcal{D}(X)$ we have
$$\displaystyle \frac{\partial}{\partial z}[T(\varphi)](z)=\displaystyle \int_{\mathbb{R}^2}\frac{\partial}{\partial z}(A_z(x))A_{\overline{z}}(y)\varphi(x,y)dxdy$$.
It is easy to check that
$$\displaystyle \frac{\partial}{\partial z} (A(z,x))=(-z+\sqrt{2}x)A(z,x).$$
Thus, we obtain
\[
\begin{split}
\displaystyle \frac{\partial}{\partial z} T(\varphi)(z) &= \int_{\mathbb{R}^2}(-z+\sqrt{2}x)A_z(x)A_{\overline{z}}(y)\varphi(x,y)dxdy
\\
&= -z \int_{\mathbb{R}^2}A_z(x)A_{\overline{z}}(y)\varphi(x,y)dxdy+ \sqrt{2}\int_{\mathbb{R}^2}A_z(x)A_{\overline{z}}(y)x\varphi(x,y)dxdy
\\
&=-M_zT(\varphi)(z)+\sqrt{2}T(X \varphi)(z).
\\
&
\end{split}
\]
Therefore, it follows that for any $\varphi \in \mathcal{D}(X)$ we have
$$\displaystyle \frac{\partial}{\partial z} T(\varphi)+M_zT(\varphi)=\sqrt{2}TX(\varphi).$$
Hence, we obtain
$$\left(\frac{\partial}{\partial z}+M_z \right)T=\sqrt{2}TX.$$
Finally, this leads to
$$\displaystyle T^{-1}\left(\frac{\partial}{\partial z}+M_z\right)T=\sqrt{2}X, \quad \text{ on } \mathcal{D}(X).$$
\end{proof}
\begin{proposition}
We have
$$\frac{1}{2}T^{-1}\left(\frac{1}{4}\Delta_z+z\frac{\partial}{\partial \overline{z}}+\overline{z}\frac{\partial}{\partial z}+M_{|z|^2}\right)T=YX, \text{ on } \mathcal{D}(YX).$$
\end{proposition}
\begin{proof}
We observe that Proposition \ref{PositionX} yields
$$\displaystyle\frac{1}{2}T^{-1}\left(\frac{\partial}{\partial \overline{z}}+M_{\overline{z}}\right)\left(\frac{\partial}{\partial z}+M_z\right)T=YX.$$
Then, the result holds since we have $$\left(\frac{\partial}{\partial\overline{z}}+M_{\overline{z}}\right)\left(\frac{\partial}{\partial z}+M_z\right)=\frac{1}{4}\Delta_z+z\frac{\partial}{\partial \overline{z}}+\overline{z}\frac{\partial}{\partial z}+M_{|z|^2}.$$
\end{proof}
Let $\mathcal{D}(M_z)=\left\lbrace{f\in \mathcal{SF}(\mathbb{C}), M_z(f)\in \mathcal{SF}(\mathbb{C})}\right\rbrace$ be the domain of the creation operator $M_z$.
\begin{proposition} \label{CreationM}
The following relations hold
$$T\left(X-\frac{\partial}{\partial x}\right)T^{-1}=\sqrt{2}M_z, \quad \text{ on }\mathcal{D}(X)\cap \mathcal{D}(\frac{\partial}{\partial x})$$
and
$$T\left(Y-\frac{\partial}{\partial y}\right)T^{-1}=\sqrt{2}M_{\overline{z}}, \quad \text{ on }\mathcal{D}(Y)\cap \mathcal{D}(\frac{\partial}{\partial y}).$$
\end{proposition}
\begin{proof}
We will prove the first statement of this result; the second one follows with similar arguments. Indeed, let $\displaystyle \varphi\in \mathcal{D}(X)\cap \mathcal{D}(\frac{\partial}{\partial x})$ we have
$$T(X-\frac{\partial}{\partial x})\varphi (z)=T(X\varphi)(z)-T(\frac{\partial}{\partial x}\varphi)(z).$$
By definition we have
$$\displaystyle T(X\varphi)(z)=\displaystyle \int_{\mathbb{R}^2} A_z(x)A_{\bar{z}}(y)x\varphi(x,y)dxdy,$$
so that
$$\displaystyle T(\frac{\partial}{\partial x}\varphi)(z)= \int_{\mathbb{R}^2} A_z(x)A_{\bar{z}}(y)\frac{\partial}{\partial x}\varphi(x,y)dxdy=- \int_{\mathbb{R}^2} \frac{\partial}{\partial x}(A_z(x))A_{\bar{z}}(y)\varphi(x,y)dxdy,$$
since $$\displaystyle \frac{\partial}{\partial x}(A_z(x))=(-x+\sqrt{2}z)A_z(x),$$
we are lead to
$$T\left(X-\frac{\partial}{\partial x}\right)\varphi(z)=\sqrt{2}zT(\varphi)(z).$$
Hence, we obtain
$$T\left(X-\frac{\partial}{\partial x}\right)T^{-1}=\sqrt{2}M_z, \quad \text{ on }\mathcal{D}(X)\cap \mathcal{D}(\frac{\partial}{\partial x}).$$
\end{proof}
As a consequence of the previous result we can prove the following
\begin{corollary}
We have
$$T\left(\frac{\partial^2}{\partial x\partial y}+XY-X\frac{\partial}{\partial y}-Y\frac{\partial}{\partial x}\right)T^{-1}=2M_{|z|^2}.$$
\end{corollary}
\begin{proof}
Indeed, we just need to apply Proposition \ref{CreationM} combined with the relation $$\left(X-\frac{\partial}{\partial x}\right)\left(Y-\frac{\partial}{\partial y}\right)=\frac{\partial^2}{\partial x \partial y}+XY-X\frac{\partial}{\partial y}-Y\frac{\partial}{\partial x}.$$
\end{proof}
\section{A Berezin transform and related operators}
\setcounter{equation}{0}
We now use the kernel function \eqref{newFock-kernel} to study a Berezin integral transform and develop further results on it.
\begin{definition}
Let $f:\mathbb{C}\longrightarrow \mathbb{C}$ and let $d\mu(w)=\frac{1}{\pi}e^{-|w|^2}dA(w)$ be the Gaussian measure. Then, we consider the following integral transform
\begin{equation}\label{Noperator}
\displaystyle \mathcal{B}(f)(z)=\int_{\mathbb{C}} e^{-|z|^2}K(z,w)f(w)d\mu(w), \quad z\in\mathbb{C},
\end{equation}
when the integral exists.
\end{definition}
\begin{remark}
We observe that $$e^{-|z|^2}K(z,w)e^{-|w|^2}=e^{-|z-w|^2},\quad z,w\in \mathbb{C}.$$
Thus, it turns out that the integral transform given by \eqref{Noperator} coincides with the so-called Berezin transform considered in \cite[p. 101]{Zhu}. However, since $\mathcal{B}$ can be expressed in terms of the kernel function $K(z,w)$ we can use various properties of this kernel in order to develop further results. A similar transform in the case of two complex variables was considered in \cite{BG2019}. It is important to note that the Berezin transform was first introduced by Berezin in \cite{Berezin} as a general concept of quantization.
\end{remark}
We start first by observing that the Berezin transform $\mathcal{B}$ fixes all the complex monomials $z^n$, $n\in\mathbb N$ which form an orthogonal basis of the classical Fock $\mathcal{F}(\mathbb{C})$:
\begin{proposition}\label{Bzn}
For $n=0,1,...$ it holds that
\begin{equation}
\mathcal{B}(z^n)=z^n,\quad \forall z\in\mathbb{C}.
\end{equation}
\end{proposition}
\begin{proof}
We set $f_n(z)=z^n$ with $n=0,1,...$.
Then, for every $\alpha\in\mathbb{C}$ we can write
\begin{equation}
\displaystyle \sum_{n=0}^{\infty}\alpha^n \frac{\mathcal{N}(f_n)(z)}{n!}=\frac{1}{\pi}e^{-|z|^2}\int_{\mathbb{C}}K(z,w)e^{\alpha w}e^{-|w|^2}dA(w).
\end{equation}
Let us set $w=t_1+it_2$ and let us replace $K(z,w)$ by its expression to obtain
$$\displaystyle \sum_{n=0}^{\infty}\alpha^n \frac{\mathcal{N}(f_n)(z)}{n!}=\frac{1}{\pi}e^{-|z|^2}\int_{\mathbb{R}^2}e^{z(t_1-it_2)+(t_1+it_2)\overline{z}+\alpha(t_1+it_2)}e^{-(t_1^2+t_2^2)}dt_1dt_2.$$
Therefore, thanks to Fubini's theorem we have
$$\displaystyle \sum_{n=0}^{\infty} \alpha^n \frac{\mathcal{B}(f_n)(z)}{n!}=\frac{1}{\pi}e^{-|z|^2}\left(\int_{\mathbb{R}}e^{-t_1^2+t_1(z+\overline{z}+\alpha)}dt_1\right)\left(\int_{\mathbb{R}}e^{-t_2^2+t_2i(\overline{z}-z+\alpha)}dt_2\right).$$
Now, we recall the classical Gaussian integral
\begin{equation}
\displaystyle \int_{\mathbb{R}}e^{-at^2+bt}dt=\sqrt{\frac{\pi}{a}}e^{\frac{b^2}{4a}},\qquad a>0, b\in\mathbb{C}.
\end{equation}
Thus, setting $b_1(z,\overline{z})=z+\overline{z}+\alpha$ and $b_2(z,\overline{z})=i(\overline{z}-z+\alpha)$ we obtain
$$\displaystyle \sum_{n=0}^{\infty} \alpha^n\frac{\mathcal{B}(f_n)(z)}{n!}=\frac{1}{\pi}e^{-|z|^2}\pi e^{\frac{b_1^2(z,\overline{z})}{4}}e^{\frac{b_2^2(z,\overline{z})}{4}}.$$
Therefore, we have $$\displaystyle \sum_{n=0}^{\infty} \alpha^n\frac{\mathcal{B}(f_n)(z)}{n!}=e^{-|z|^2}e^{\frac{b^2_1(z,\overline{z})+b^2_2(z,\overline{z})}{4}}$$
We have $b_1^2(z,\overline{z})=z^2+\overline{z}^2+2|z|^2+\alpha^2+2\alpha z+2\alpha \overline{z}$ and $b_2^2(z,\overline{z})=-(\overline{z}^2+z^2-2|z|^2+\alpha^2+2\alpha \overline{z}-2\alpha z)$, which leads to
$$b^2_1(z,\overline{z})+b^2_2(z,\overline{z})=4(|z|^2+\alpha z).$$
Hence
$$\displaystyle \sum_{n=0}^{\infty} \alpha^n\frac{\mathcal{B}(f_n)(z)}{n!}=e^{\alpha z}=\sum_{n=0}^{\infty}\alpha^n \frac{z^n}{n!}, \quad \forall \alpha \in\mathbb{C}.$$
Finally, we identify the coefficients with respect to the variable $\alpha$ and get
$$\mathcal{B}(f_n)(z)=z^n, \quad n=0,1,...$$
\end{proof}
\begin{remark}
It is impotant to note that in \cite{Zhu} it was proved that a function $f\in L^\infty(\mathbb{C})$ is a fixed point for the Berezin transform $\mathcal{B}$ if and only if $f$ is constant. Then, Proposition \ref{Bzn} shows that monomials are fixed by the Berezin transform. In particular, this suggests to consider a more general fixed point problem: to find when $\mathcal{B}(f)=f$ for $f$ in some suitable function space.
\end{remark}
\begin{remark}
We observe that
\begin{equation}
\displaystyle \mathcal{B}(f)(z)=\frac{1}{\pi}\int_{\mathbb{C}} e^{\overline{w}z+w\overline{z}-|z|^2}f(w)e^{-|w|^2}dA(w).
\end{equation}
If $z=0$, we have
$$\displaystyle \mathcal{B}(f)(0)=\frac{1}{\pi}\int_{\mathbb{C}} f(w)e^{-|w|^2}dA(w).$$
In particular, using the Cauchy-Schwarz inequality we obtain
$$|\mathcal{B}(f)(0)|\leq ||f||_{L^2(\mathbb{C},\mu)},$$
and by taking the function $f=1$, we get
$$\mathcal{B}(1)(z)=\displaystyle e^{-|z|^2} \frac{1}{\pi}\int_{\mathbb{C}} K(z,w)e^{-|w|^2}dA(w).$$
\end{remark}
\begin{theorem}
For any $\alpha>0$, let $d\mu_{\alpha}(w)=\frac{\alpha}{\pi}e^{-\alpha|w|^2}dA(w)$ be the weighted Gaussian measure. For any $\beta>2$ the operator $\mathcal{B}$ is bounded from $L^2(\mathbb{C},d\mu)$ into $L^2(\mathbb{C},d\mu_\beta)$. In particular, for any $f\in L^2(\mathbb{C},d\mu)$ it holds that
\begin{equation}
||\mathcal{B}(f)||_{L^2(\mathbb{C},d\mu_\beta)}\leq \frac{1}{\sqrt{\beta-2}} ||f||_{L^2(\mathbb{C},d\mu)}.
\end{equation}
\end{theorem}
\begin{proof}
For $z\in \mathbb{C}$ fixed, we have $$\mathcal{B}(f)(z)=e^{-|z|^2}\int_{\mathbb{C}} K(z,w)f(w)d\mu(w).$$
Thus, setting $K_z(w)=K(z,w)$ for any $w\in\mathbb{C}$ and noting that $f,K_z \in L^2(\mathbb{C},d\mu)$ we obtain
$$\displaystyle |\mathcal{B}(f)(z)|\leq e^{-|z|^2}\int_{\mathbb{C}}|K(z,w)||f(w)|d\mu(w),$$
and using the Cauchy Schwarz inequality
$$|\mathcal{B}(f)(z)|\leq e^{-|z|^2}||K_z||_{L^2(\mathbb{C},d\mu)}||f||_{L^2(\mathbb{C},d\mu)}.$$
Moreover, developing some calculations using Gaussian integrals we get
$$ ||K_z||_{L^2(\mathbb{C},d\mu)} =e^{2|z|^2},$$
and so
$$|\mathcal{B}(f)(z)|\leq e^{|z|^2} ||f||_{L^2(\mathbb{C},d\mu)}.$$
As a consequence, for any $\beta>2$ we have $$||\mathcal{B}(f)||^{2}_{L^2(\mathbb{C},d\mu_\beta)}\leq ||f||^{2}_{L^2(\mathbb{C},d\mu)}\left(\frac{1}{\pi}\int_{\mathbb{C}}e^{-(\beta-2)|z|^2}dA(z)\right)=\frac{1}{\beta-2}||f||^{2}_{L^2(\mathbb{C},d\mu)}.$$
Hence, $\mathcal{B}$ is a bounded operator from $L^2(\mathbb{C},d\mu)$ into $L^2(\mathbb{C},d\mu_\beta)$ with $\beta>2$.
\end{proof}
\begin{remark}
In particular, for any $\beta>2$ we have
$$||\mathcal{B}(f)||_{L^2(\mathbb{C},d\mu_\beta)}\leq ||\mathcal{B}(f)||_{\mathcal{SF}(\mathbb{C})}.$$
This explains somehow that $$\mathcal{SF}(\mathbb{C})\subset L^2(\mathbb{C},d\mu_\beta),\quad \beta>2.$$
\end{remark}
Now, we can prove the following
\begin{lemma}\label{Naction}
For any $p,q=0, 1, 2, \ldots$, we have
$$ \mathcal{B}(H_{p,q})(z,\overline{z})=z^p\overline{z}^q.$$
We have also
$$||\mathcal{B}(H_{p,q})||_{\mathcal{SF}(\mathbb{C})}=||H_{p,q}||_{L^2(\mathbb{C},d\mu)}.$$
\end{lemma}
\begin{proof}
For any $u,v\in\mathbb{C}$ we have (see \cite{Ismail, Ito} )
\begin{equation}
\displaystyle \sum_{m,n=0}^{\infty}H_{m,n}(z,\overline{z})\frac{u^mv^n}{m!n!}=e^{uz+v\overline{z}-uv}, \quad \forall z\in\mathbb{C}.
\end{equation}
For $w\in\mathbb{C}$ we set
\begin{equation}
\displaystyle \mathcal{R}(z,w):= \sum_{m,n=0}^{\infty}\frac{\overline{z}^mz^n}{m!n!} H_{m,n}(w,\overline{w})=e^{\overline{z}w+z\overline{w}-|z|^2}, \quad z\in\mathbb{C}.
\end{equation}
It is immediate that
$$\overline{\mathcal{R}(z,w)}=\mathcal{R}(z,w), \quad z,w\in\mathbb{C}.$$
From formula \eqref{Noperator} we obtain
\[
\begin{split}
\displaystyle \mathcal{B}(H_{p,q})(z) &= \int_{\mathbb{C}} e^{-|z|^2} K(z,w)H_{p,q}(w,\overline{w})d\mu(w)\\
&= \int_{\mathbb{C}} e^{-|z|^2+z\overline{w}+w\overline{z}}H_{p,q}(w,\overline{w})d\mu(w)\\
&= \int_{\mathbb{C}}\left(\sum_{m,n=0}^{\infty}\frac{\overline{z}^mz^n}{m!n!} H_{m,n}(w,\overline{w}) \right) H_{p,q}(w,\overline{w})d\mu(w)
\\
&= \int_{\mathbb{C}}\mathcal{R}(z,w)H_{p,q}(w,\overline{w})d\mu(w)
\\
&= \int_{\mathbb{C}}\overline{\mathcal{R}(z,w)}H_{p,q}(w,\overline{w})d\mu(w)
\\
&= \sum_{m,n=0}^{\infty}\frac{z^m\overline{z}^n}{m!n!}\int_{\mathbb{C}}\overline{H_{m,n}(w,\overline{w})}H_{p,q}(w,\overline{w})d\mu(w)
\\
&= \sum_{m,n=0}^{\infty}\frac{z^m\overline{z}^n}{m!n!} \langle H_{p,q},H_{m,n} \rangle_{L^2(\mathbb{C},d\mu)}.
\\
&
\end{split}
\]
However, we know that the complex Hermite polynomials $(H_{m,n})_{m,n\geq 0}$ form an orthonormal basis of $L^2(\mathbb{C},d\mu)$ (see \cite{Ito}) so that we have
$$\langle H_{p,q},H_{m,n} \rangle_{L^2(\mathbb{C},d\mu)}=p!q!\delta_{(p,q);(m,n)}.$$
In particular, this leads to
$$\mathcal{B}(H_{p,q})(z,\overline{z})=z^p\overline{z}^q, \quad z\in\mathbb{C}.$$
As a consequence, it is clear that for any $p,q= 0, 1, 2, \ldots$ we have
$$||\mathcal{B}(H_{p,q})||_{\mathcal{SF}(\mathbb{C})}=||z^p\overline{z}^q||_{\mathcal{SF}(\mathbb{C})}=\sqrt{p!q!}=||H_{p,q}||_{L^2(\mathbb{C},d\mu)}.$$
\end{proof}
\begin{theorem}
The integral transform $\mathcal{B}$ is an isomorphism from $L^2(\mathbb{C},d\mu)$ onto $\mathcal{SF}(\mathbb{C})$ and for any $f\in L^2(\mathbb{C},d\mu)$ we have
\begin{equation}
||\mathcal{B}(f)||_{\mathcal{SF}(\mathbb{C})}=||f||_{L^2(\mathbb{C},d\mu)}.
\end{equation}
\end{theorem}
\begin{proof}
For any $f\in L^2(\mathbb{C},d\mu)$ we can write the following decomposition using Hermite polynomials
$$f(z)=\displaystyle \sum_{p,q=0}^{\infty}H_{p,q}(z,\overline{z})\alpha_{p,q}, \quad z\in\mathbb{C},$$
and $$||f||_{L^2(\mathbb{C},d\mu)}^{2}=\displaystyle \sum_{p,q=0}^{\infty}p!q!|\alpha_{p,q}|^2 .$$
Thus, thanks to Lemma \ref{Naction} we get
\[
\begin{split}
\displaystyle \mathcal{B}(f)(z) &= \sum_{p,q=0}^{\infty}\mathcal{B}(H_{p,q})(z,\overline{z})\alpha_{p,q} \\
&=\sum_{p,q=0}^{\infty}z^p\overline{z}^q\alpha_{p,q}.
\\
&
\end{split}
\]
Then, using also Proposition \ref{seqNF} we obtain
\[
\begin{split}
\displaystyle || \mathcal{B}(f)||_{\mathcal{SF}(\mathbb{C})}^2&= \sum_{p,q=0}^{\infty}p!q!|\alpha_{p,q}|^2 \\
&=||f||^2_{L^2(\mu)}.\\
&
\\
&
\end{split}
\]
We observe that the surjectivity of the transform $\mathcal{B}$ is a direct consequence of Lemma \ref{Naction}. This allows to consider the integral transform $\mathcal{B}:L^2(\mathbb{C},d \mu)\longrightarrow \mathcal{SF}(\mathbb{C})$ which defines an unitary operator.
\end{proof}
\begin{proposition} \label{d^1}
For any $f\in L^2(\mathbb{C},d\mu)$, it holds that $$\displaystyle \frac{\partial}{\partial z}\mathcal{B}(f)(z)=-\overline{z}\mathcal{B}(f)(z)+\mathcal{B}(\overline{w}f)(z).$$
In a similar way, we have also
$$\displaystyle \frac{\partial}{\partial \overline{z}}\mathcal{B}(f)(z)=-z\mathcal{B}(f)(z)+\mathcal{B}(wf)(z).$$
\end{proposition}
\begin{proof}
We observe that thanks to the definition of $\mathcal{B}$ we have
$$\displaystyle \mathcal{B}(f)(z)=\frac{1}{\pi}\int_{\mathbb{C}} e^{-|z|^2}K(z,w)f(w)e^{-|w|^2}dA(w), \quad z\in\mathbb{C}.$$
Thus, we have
\begin{equation}\label{dzn}
\displaystyle \frac{\partial}{\partial z}\mathcal{B}(f)(z)=\frac{1}{\pi}\int_{\mathbb{C}}\frac{\partial}{\partial z} (e^{-|z|^2}K(z,w))f(w)e^{-|w|^2}dA(w), \quad z\in\mathbb{C}.
\end{equation}
Applying the Leibniz rule and Proposition \ref{Fockpptd} we get
\[
\begin{split}
\displaystyle \frac{\partial}{\partial z} (e^{-|z|^2}K(z,w)) &= e^{-|z|^2}\frac{\partial}{\partial z} K(z,w)+\frac{\partial}{\partial z}(e^{-|z|^2})K(z,w) \\
&=e^{-|z|^2}\overline{w}K(z,w)-\overline{z}e^{-|z|^2}K(z,w)\\
&=(\overline{w}-\overline{z})e^{-|z|^2}K(z,w).
\\
&
\end{split}
\]
Hence, we insert the previous computations in the formula \eqref{dzn} and this leads to
\[
\begin{split}
\displaystyle \frac{\partial}{\partial z} \mathcal{B}(f)(z)&= \frac{1}{\pi}\int_{\mathbb{C}} e^{-|z|^2}(\overline{w}-\overline{z})K(z,w)f(w)e^{-|w|^2}dA(w)\\
&=\mathcal{B}(\overline{w}f)(z)-\overline{z}\mathcal{B}(f)(z).\\
&
\end{split}
\]
\end{proof}
As a consequence of the previous result we prove
\begin{proposition}\label{Ndzbarz}
It holds that
\begin{equation}
\mathcal{B}^{-1}\left(\frac{\partial}{\partial z}+\overline{z}\right)\mathcal{B}=M_{\overline{w}},
\end{equation}
and \begin{equation}
\mathcal{B}^{-1}\left(\frac{\partial}{\partial \overline{z}}+z\right)\mathcal{B}=M_{w}.
\end{equation}
\end{proposition}
\begin{proof}
We note that thanks to Proposition \ref{d^1} we can write
$$\displaystyle \frac{\partial}{\partial z}\mathcal{B}(f)(z)+\overline{z}\mathcal{B}(f)(z)=\mathcal{B}(\overline{w}f)(z).$$
Then, using the fact that $\mathcal{B}$ is an isomorphism it is easy to see that
$$\mathcal{B}^{-1}\left(\frac{\partial}{\partial z}+\overline{z}\right)\mathcal{B}=M_{\overline{w}}.$$
In a similar way we can prove the second statement.
\end{proof}
As a consequence, we can prove the following
\begin{corollary}
It holds that
\begin{equation}
\mathcal{B}^{-1}\left(\frac{1}{4}\Delta_z+\frac{\partial}{\partial\overline{z}} \overline{z}+z\frac{\partial}{\partial z}+|z|^2\right)\mathcal{B}=M_{|w|^2}.
\end{equation}
\end{corollary}
\begin{proof}
We observe that using the two expressions proved in Proposition \ref{Ndzbarz} we obtain that
$$\mathcal{B}^{-1}\left(\frac{\partial}{\partial z}+\overline{z}\right)\left(\frac{\partial}{\partial \overline{z}}+z\right)\mathcal{B}=M_{\overline{w}}M_w=M_{|w|^2}.$$
On the other hand, we have
$$\left(\frac{\partial}{\partial z}+\overline{z}\right)\left(\frac{\partial}{\partial \overline{z}}+z\right)=\left(\frac{1}{4}\Delta_z+\frac{\partial}{\partial\overline{z}} \overline{z}+z\frac{\partial}{\partial z}+|z|^2\right),$$
and this ends the proof.
\end{proof}
In the next result, we will calculate $\displaystyle \frac{\partial^n}{\partial \overline{z}^n}\mathcal{B}(f)(z)$ and $\displaystyle \frac{\partial^n}{\partial z^n}\mathcal{B}(f)(z)$:
\begin{proposition}
For any $n= 0, 1,2, \ldots$, it holds that
\begin{equation}\label{d^n}
\displaystyle \frac{\partial^n}{\partial z^n}\mathcal{B}(f)(z)=\sum_{k=0}^n(-1)^k {n \choose k}\overline{z}^k\mathcal{B}(\overline{w}^{n-k}f)(z)=\mathcal{B}((\overline{w}-\overline{z})^nf)(z),
\end{equation}
and
\begin{equation}
\displaystyle \frac{\partial^n}{\partial \overline{z}^n}\mathcal{B}(f)(z)=\sum_{k=0}^n(-1)^k {n \choose k}z^k\mathcal{B}(w^{n-k}f)(z)=\mathcal{B}((w-z)^nf)(z).
\end{equation}
\end{proposition}
\begin{proof}
The proof is done by induction. For $n=0$ the relations are true. We assume that \eqref{d^n} holds for some $n$ and we prove that we have
\begin{equation}\label{d^n+1}
\displaystyle \frac{\partial^{n+1}}{\partial z^{n+1}}\mathcal{B}(f)(z)=\sum_{k=0}^{n+1}(-1)^k {n+1 \choose k}\overline{z}^k\mathcal{B}(\overline{w}^{n+1-k}f)(z).
\end{equation}
Indeed, using Proposition \ref{d^1} we have
$$\displaystyle \frac{\partial}{\partial z}\mathcal{B}(f)(z)=-\overline{z}\mathcal{B}(f)(z)+\mathcal{B}(\overline{w}f)(z).$$
Thus, applying the operator $\displaystyle \frac{\partial^n}{\partial z^n}$ we get
$$\displaystyle \frac{\partial^{n+1}}{\partial z^{n+1}}\mathcal{B}(f)(z)=-\overline{z}\frac{\partial^n}{\partial z^n}\mathcal{B}(f)(z)+\frac{\partial^n}{\partial z^n}\mathcal{B}(\overline{w}f)(z).$$
Applying the induction hypothesis to the functions $f$ and $g=\overline{w}f$ we obtain the chain of equalities
\[
\begin{split}
\displaystyle \frac{\partial^{n+1}}{\partial z^{n+1}}\mathcal{B}(f)(z)
&= -\overline{z}\left(\sum_{k=0}^{n}(-1)^k{n \choose k} \overline{z}^k\mathcal{B}(\overline{w}^{n-k}f)(z)\right)+\sum_{k=0}^{n}(-1)^k{n \choose k} \overline{z}^k\mathcal{B}(\overline{w}^{n+1-k}f)(z)\\
&= \sum_{k=0}^{n}(-1)^{k+1}{n \choose k} \overline{z}^{k+1}\mathcal{B}(\overline{w}^{n-k}f)(z)+\sum_{k=0}^{n}(-1)^k{n \choose k} \overline{z}^k\mathcal{B}(\overline{w}^{n+1-k}f)(z)
\\
&= \sum_{h=1}^{n+1}(-1)^{h}{n \choose h-1} \overline{z}^{h}\mathcal{B}(\overline{w}^{n+1-h}f)(z)+\sum_{h=0}^{n}(-1)^h {n \choose h} \overline{z}^h\mathcal{B}(\overline{w}^{n+1-h}f)(z)
\\
&= {n \choose 0}\mathcal{B}(\overline{w}^{n+1}f)(z)+ \sum_{h=1}^{n}(-1)^{h}\left({n \choose h}+ {n \choose h-1} \right) \overline{z}^{h}\mathcal{B}(\overline{w}^{n+1-h}f)(z)
\\
&+(-1)^{n+1} {n \choose n}\overline{z}^{n+1}\mathcal{B}(f)(z).
\end{split}
\]
By Pascal identity we have
$${n \choose h}+ {n \choose h-1}={n+1 \choose h}, \quad \text{ for any } h\geq 1,$$
so that
$$\displaystyle \frac{\partial^{n+1}}{\partial z^{n+1}}\mathcal{B}(f)(z)=\sum_{k=0}^{n+1}(-1)^k {n+1 \choose k}\overline{z}^k\mathcal{B}(\overline{w}^{n+1-k}f)(z),$$
and this ends the proof. The second statement follows using similar arguments.
\end{proof}
\begin{theorem}\label{Ndwbar'}
For any $f\in L^2(\mathbb{C},d\mu)$, it holds that \begin{equation}
\mathcal{B}(\partial_{\overline{w}}f)(z)=\mathcal{B}(wf)(z)-z\mathcal{B}(f)(z).
\end{equation}
In a similar way, we have
\begin{equation}
\mathcal{B}(\partial_{w}f)(z)=\mathcal{B}(\overline{w}f)(z)-\overline{z}\mathcal{B}(f)(z).
\end{equation}
\end{theorem}
\begin{proof}
This result can be proved using some computations that are based on Proposition 7.2 of \cite{Shige}. Indeed, setting $d\mu(z):=\frac{1}{\pi}e^{-|z|^2}dA(z)$ we get
$$\displaystyle \int_{\mathbb{C}}u(w) \overline{(\partial_{\overline{w}}v(w))}d\mu(w)=\int_{\mathbb{C}}\left(-\partial_{w}+\overline{w}\right)(u(w))\overline{v(w)}d\mu(w).$$
In particular, this means that on $L^2(\mathbb{C},d\mu)$ we have
$$(\partial_w)^{*}=-\partial_w+\overline{w}.$$
Then, using this fact we deduce the following computations
\[
\begin{split}
\displaystyle M_z\mathcal{B}(f)(z) &=z\mathcal{B}(f)(z) \\
&=e^{-|z|^2}\int_{\mathbb{C}} zK(z,w)f(w)d\mu(w)\\
&=e^{-|z|^2}\int_{\mathbb{C}} \frac{\partial}{\partial \overline{w}}(K(z,w))f(w)d\mu(w)
\\
&=e^{-|z|^2}\langle \partial_{\overline{w}}K_z,\overline{f} \rangle_{\mu}
\\
&= e^{-|z|^2}\langle K_z,\partial_{\overline{w}}^{*}(\overline{f}) \rangle_{\mu}
\\
&= e^{-|z|^2}\langle K_z,(-\partial_{w}+\overline{w})(\overline{f}) \rangle_{\mu}
\\
&=e^{-|z|^2} \int_{\mathbb{C}} K(z,w)\overline{(-\partial_{w}+\overline{w})(\overline{f})(w)}d\mu(w)
\\
&= e^{-|z|^2}\int_{\mathbb{C}} K(z,w) (-\partial_{\overline{w}}f+wf)d\mu(w)
\\
&=\mathcal{B}(-\partial_{\overline{w}}f+wf)(z)
\\
&=\mathcal{B}(wf)(z)-\mathcal{B}(\partial_{\overline{w}}f)(z),
\end{split}
\]
where we used
$\overline{\partial_{w}f}=\partial_{\overline{w}}\overline{f},$ which leads to
$$\overline{(-\partial_w+\overline{w})(\overline{f})}=(-\partial_{\overline{w}}f+wf).$$
This ends the proof.
\end{proof}
As a consequence of the previous result, we prove that the operators $w-\partial_{\overline{w}}$, $\overline{w}-\partial_w$ are similar to $M_z$ and $M_{\overline{z}}$ respectively:
\begin{proposition}
It holds that
\begin{equation}
\mathcal{B}\left(w-\partial_{\overline{w}}\right)\mathcal{B}^{-1}=M_z,
\end{equation}
and
\begin{equation}
\mathcal{B}\left(\overline{w}-\partial_{w}\right)\mathcal{B}^{-1}=M_{\overline{z}}.
\end{equation}
\end{proposition}
\begin{proof}
By Theorem \ref{Ndwbar'} we have
$$\mathcal{B}(\partial_{\overline{w}}f)(z)=\mathcal{B}(wf)(z)-z\mathcal{B}(f)(z),$$
from which we deduce
$$\mathcal{B}((w-\partial_{\overline{w}})f)(z)=z\mathcal{B}(f)(z).$$
Hence, using the fact that the integral transform $\mathcal{B}$ is a unitary operator from $L^2(\mathbb{C},d\mu)$ onto $\mathcal{SF}(\mathbb{C})$ we multiply the operator $\mathcal{B}^{-1}$ on the right and obtain $$\mathcal{B}\left(w-\partial_{\overline{w}}\right)\mathcal{B}^{-1}=M_z.$$
The second part of the statement can be proved similarly.
\end{proof}
\begin{proposition}\label{p5.12}
We have
\begin{equation}
\partial_z \mathcal{B}(f)=\mathcal{B}(\partial_w f)
\end{equation}
and
\begin{equation}
\partial_{\overline{z}}N(f)=\mathcal{B}(\partial_{\overline{w}}f).
\end{equation}
\end{proposition}
\begin{proof}
Applying Proposition \ref{d^1} and Theorem \ref{Ndwbar'} we have
$$\displaystyle \partial_z\mathcal{B}(f)(z)=-\overline{z}\mathcal{B}(f)(z)+\mathcal{B}(\overline{w}f)(z)$$
and $$\mathcal{B}(\partial_{w}f)(z)=\mathcal{B}(\overline{w}f)(z)-\overline{z}\mathcal{B}(f)(z).$$
Therefore, from the two previous relations we obtain
$$\displaystyle \partial_z\mathcal{B}(f)(z)=\mathcal{B}(\partial_{w}f)(z).$$
The second statement can be proved in a similar way and this will end the proof.
\end{proof}
In the table below, we summarize different operators that are equivalent to each others thanks to the Berezin transform $\mathcal{B}$.
\begin{table}[ht]
\caption{Equivalent operators using the transform $\mathcal{B}$}
\centering
\large
\begin{tabular}{cc }
\hline
$\mathcal{SF}(\mathbb{C})$ & $L^2(\mathbb{C},d\mu)$ \\ [1ex]
\hline
\vspace{0.2cm}
$\frac{\partial}{\partial z}+\overline{z}$ & $\overline{w}$ \\
\vspace{0.2cm}
$\frac{\partial}{\partial\overline{z}}+z$ & $w$ \\
\vspace{0.2cm}
$\frac{\partial^n}{\partial z^n}$ & $\frac{\partial^n}{\partial w^n}$\\
\vspace{0.2cm}
$\frac{\partial^n}{\partial\overline{z}^n}$ & $\frac{\partial^n}{\partial \overline{w}^n}$ \\
\vspace{0.2cm}
$\frac{1}{4}\Delta_z+\frac{\partial}{\partial\overline{z}} \overline{z}+z\frac{\partial}{\partial z}+|z|^2$ & $|w|^2$ \\
\vspace{0.2cm}
$M_{|z|^2}$ & $\frac{1}{4}\Delta_w-\frac{\partial}{\partial\overline{w}} \overline{w}-w\frac{\partial}{\partial w}+|w|^2$ \\
\vspace{0.2cm}
$z$ & $w-\frac{\partial}{\partial\overline{w}}$ \\
\vspace{0.2cm}
$\overline{z}$ & $\overline{w}-\frac{\partial}{\partial w}$ \\
\hline
\end{tabular}
\label{table:nonlin}
\end{table} \newpage
Inspired by \cite{Zhu2} we consider the following integral operator on the Fock space $\mathcal{F}(\mathbb{C})$
\begin{definition}
Let $\varphi:\mathbb{C}\longrightarrow \mathbb{C}$ and denote by $d\mu(w)=\frac{1}{\pi}e^{|w|^2}dA(w)$ the normalized Gaussian measure. Then, we define the following integral transform when it exists
\begin{equation}
\mathcal{S}_\varphi(f)(z):=\displaystyle \int_{\mathbb{C}}e^{z\overline{w}}f(w)\varphi(w-z)d\mu(w),\quad f\in\mathcal{F}(\mathbb{C}).
\end{equation}
\end{definition}
\begin{remark}
If $\varphi=1$, it is clear by the reproducing kernel property for the Fock space that in this case $$ \mathcal{S}_\varphi(f)(z):=\displaystyle \int_{\mathbb{C}}e^{z\overline{w}}f(w)d\mu(w)=f(z),\quad \forall f\in\mathcal{F}(\mathbb{C}).$$
\end{remark}
\begin{example}
For every $a\in\mathbb{C}$ set $\varphi_a(w)=e^{\overline{a}w}$. Then, we have
$$\mathcal{S}_{\varphi_z}(f)(z)=\displaystyle \int_{\mathbb{C}}e^{z\overline{w}}f(w)\varphi_z(w-z)d\mu(w)=\mathcal{B}(f)(z).$$
It turns out that the transform $\mathcal{B}$ is a particular case of the general integral operator $\mathcal{S}_{\varphi}$.
\end{example}
\section{The polyanalytic Hardy space of infinite order and Gleason problem}
\setcounter{equation}{0}
Let us denote by $\mathbb{D}$ the unit disk and by $\mathcal{H}^2(\mathbb{D})$ the classical Hardy space. In this section, we will prove the following main result:
\begin{theorem}
The reproducing kernel Hilbert space with reproducing kernel $\displaystyle \frac{1}{(1-z\overline{w})(1-\overline{z} w)}$ is, up to a multiplicative positive factor, the only reproducing kernel
Hilbert space of polyanalytic functions of infinite order, regular at the origin, and for which
\begin{eqnarray}
R_\infty^*&=&M_z\\
L_\infty^*&=&M_{\overline{z}}.
\end{eqnarray}
\end{theorem}
To prove this theorem we need a couple of preliminary results, including a sequential characterization of the space $\mathcal{H}(\mathsf{K})$.
\begin{definition}
The polyanalytic Hardy space of infinite order $\mathcal{SH}(\mathbb{D})$
is the space of functions of the form \begin{equation}
f(z)=\displaystyle \sum_{n=0}^{\infty}\overline{z}^nf_n(z),
\end{equation}
satisfying
\begin{enumerate}
\item[i)] $f_n\in \mathcal{H}^2(\mathbb{D})$ for any $n\geq 0$;
\item[ii)]$\displaystyle ||f||^{2}_{\mathcal{SH}(\mathbb{D})}= \sum_{n=0}^{\infty} ||f_n||^{2}_{\mathcal{H}^2(\mathbb{D})}<\infty.$
\end{enumerate}
Then, we consider the scalar product on $\mathcal{SH}(\mathbb{D})$ given by \begin{equation}\displaystyle
\langle f,g \rangle_{\mathcal{SH}(\mathbb{D})}:=\sum_{k=0}^{\infty}\langle f_k,g_k \rangle_{\mathcal{H}^2(\mathbb{D})},
\end{equation}
for any $f=\displaystyle \sum_{k=0}^{\infty}\overline{z}^kf_k$ and $g=\displaystyle \sum_{k=0}^{\infty}\overline{z}^kg_k$ with $f_k,g_k\in\mathcal{H}^2(\mathbb{D})$ for every $k\geq 0$.
\end{definition}
\begin{proposition}
A function $f:\mathbb{D}\longrightarrow \mathbb{C}$ belongs to $\mathcal{SH}(\mathbb{D})$ if and only if $f$ is of the form $$\displaystyle f(z)=\sum_{(m,n)\in\mathbb{N}^2}z^m\overline{z}^n\alpha_{m,n},$$
with $(\alpha_{m,n})\subset\mathbb{C}$ and such that \begin{equation}\displaystyle
||f||^{2}_{\mathcal{SH}(\mathbb{D})}=\sum_{(m,n)\in\mathbb{N}^2}|\alpha_{m,n}|^2<\infty.
\end{equation}
Moreover, if for any $(m,n)\in\mathbb{N}^2$ we set $\upsilon_{m,n}(z,\overline{z})=\displaystyle z^m\bar{z}^n$. Then, the family of functions $\lbrace \upsilon_{m,n} \rbrace_{m,n\geq 0}$ form an orthonormal basis of $\mathcal{SH}(\mathbb{D})$.
\end{proposition}
\begin{proof}
This proof follows the arguments used to prove Proposition \ref{seqNF}.
\end{proof}
\begin{lemma}
We have
$$\mathcal{H}(\mathsf{K})=\mathcal{SH}(\mathbb{D}).$$
Moreover, it holds that
\begin{equation}
\mathsf{K}(z,w)=\displaystyle \sum_{m,n=0}^{\infty}\upsilon_{m,n}(z,\overline{z})\overline{\upsilon_{m,n}(w,\overline{w})},
\end{equation}
for every $z,w\in\mathbb{D}$.
\end{lemma}
\begin{proof}
We note that $(\upsilon_{m,n})_{m,n\geq 0}$ is an orthonormal basis of the space $\mathcal{SH}(\mathbb{D})$. Thus, the associated reproducing kernel is given by the following series which converges uniformly on each compact so that
$$\displaystyle \sum_{m,n=0}^{\infty}\upsilon_{m,n}(z,\bar{z})\overline{\upsilon_{m,n}(w,\bar{w})}<\infty, \text{ for any } z,w\in\mathbb{D}.$$
More precisely, for any $(z,w)\in\mathbb{D}^2$ we have the equalities
\[
\begin{split}
\displaystyle \sum_{m,n=0}^{\infty}\upsilon_{m,n}(z,\bar{z})\overline{\upsilon_{m,n}(w,\bar{w})}
&= \sum_{m,n=0}^{\infty} z^m\bar{z}^n \bar{w}^m w^n \\
&= \left(\sum_{m=0}^{\infty} z^m\bar{w}^m \right) \left(\sum_{n=0}^{\infty} w^n\bar{z}^n\right)
\\
&= \frac{1}{(1-z\overline{w})} \frac{1}{(1-\overline{z} w)}
\\
&=\frac{1}{(1-z\overline{w})(1-\overline{z} w)}
\\
&=\mathsf{K}(z,w).
\end{split}
\]
\end{proof}
\begin{lemma}\label{HardyR_0}
It holds that
\begin{equation}\displaystyle
\langle R_\infty(f),g\rangle_{\mathcal{SH}(\mathbb{D})}=\langle f,M_{z}g\rangle_{\mathcal{SH}(\mathbb{D})},
\end{equation}
and \begin{equation}\displaystyle
\langle L_\infty(f),g\rangle_{\mathcal{SH}(\mathbb{D})}=\langle f,M_{\overline{z}}g\rangle_{\mathcal{SH}(\mathbb{D})}.
\end{equation}
\end{lemma}
\begin{proof}
Let $\displaystyle f=\sum_{k=0}^{\infty}\overline{z}^kf_k$ and $\displaystyle g=\sum_{k=0}^{\infty}\overline{z}^kg_k$ in $\mathcal{SH}(\mathbb{D})$. Since we have $$\displaystyle M_z(g)=\sum_{k=0}^{\infty}\overline{z}^kM_z(g_k),$$ it follows that
\[
\begin{split}
\displaystyle \langle R_\infty(f),g\rangle_{\mathcal{SH}(\mathbb{D})}
&= \sum_{k=0}^{\infty}\langle R_0(f_k),g_k \rangle_{\mathcal{H}^2(\mathbb{D})}\\
&= \sum_{k=0}^{\infty}\langle f_k,(R_0)^*g_k \rangle_{\mathcal{H}^2(\mathbb{D})}
\\
&= \sum_{k=0}^{\infty}\langle f_k,M_z(g_k) \rangle_{\mathcal{H}^2(\mathbb{D})}
\\
&= \sum_{k=0}^{\infty}\langle f_k,zg_k \rangle_{\mathcal{H}^2(\mathbb{D})}
\\
&=\langle f,M_{z}(g)\rangle_{\mathcal{SH}(\mathbb{D})}.
\end{split}
\]
In a similar way, we can prove the second part of the statement.
\end{proof}
\begin{example}
On the bidisc $\mathbb{D}^2$ we consider the inner function defined by $$j(z_1,z_2)=\displaystyle \frac{z_1+z_2+2z_1z_2}{z_1+z_2+2}, \quad \forall (z_1,z_2)\in\mathbb{D}^2.$$
Then, $j(z_1,z_2)$ is a contractive multiplier of the Hardy space $\mathcal{H}^2(\mathbb{D}^2)$ and hence (dimension 2) is in the Schur-Agler class, see \cite{BB2012,BVD2015}. Moreover, if we set
$$\rho_w(z)=1-z\overline{w},$$
we can consider the kernel function
\begin{equation}
\mathsf{K}_j((z_1,z_2);(w_1,w_2)):= \frac{1-j(z_1,z_2)\overline{j(w_1,w_2)}}{\rho_{w_1}(z_1)\rho_{w_2}(z_2)}, \quad \forall (z_1,z_2); (w_1,w_2) \in\mathbb{D}^2.
\end{equation}
We note that
\[
\begin{split}
\displaystyle \mathsf{K}_j((z_1,z_2);(w_1,w_2))
&=\frac{2}{(z_1+z_2+2)(\overline{w_1}+\overline{w_2}+2)}\\
& \cdot \left( \frac{(z_1+1)(\overline{w_1}+1)}{(1-z_1\overline{w_1})}+\frac{(z_2+1)(\overline{w_2}+1)}{(1-z_2\overline{w_2})} \right).
\\
&
\end{split}
\]
Then, by taking $z_1=z, z_2=\overline{z}$ and $w_1=w, w_2=\overline{w}$ we have
\begin{equation}\label{j}
\displaystyle j(z,\overline{z})=\frac{z+\overline{z}+2|z|^2}{z+\overline{z}+2}=\frac{\Re(z)+|z|^2}{1+\Re(z)}, \quad \forall z\in\mathbb{D},
\end{equation}
and we can write
$$j(z,\overline{z})=\frac{P(z,\overline{z})}{Q(z,\overline{z})},$$
where both the polynomials $P$ and $Q$ are polyanalytic of order $2$. We observe that $Q(z,\overline{z})=0$ if and only if $\Re(z)=-1,$ so, $Q(z,\overline{z})\neq 0$ for every $z\in \mathbb{D}$.
On the other hand, we note that $j(z,\overline{z})=1$ on the boundary $\partial\mathbb{D}$.
We have
\begin{equation}
\displaystyle \mathsf{K}_j(z,w)=\frac{1-j(z,\overline{z})\overline{j(w,\overline{w})}}{\rho_w(z)\rho_{\overline{w}}(\overline{z})}, \quad \forall (z,w) \in\mathbb{D}^2,
\end{equation}
and, as a consequence,
\[
\begin{split}
\displaystyle \mathsf{K}_j((z,\overline{z});(w,\overline{w}))
&=\frac{2}{(z+\overline{z}+2)(\overline{w}+w+2)}\\
& \cdot \left( \frac{(z+1)(\overline{w}+1)}{(1-z\overline{w})}+\frac{(\overline{z}+1)(w+1)}{(1-\overline{z}w)} \right).
\\
&
\end{split}
\]
It is important to note also that the function $j$ given by \eqref{j} is polyrational in the sense of \cite[pp 175]{Balk1991}.
\end{example}
\begin{remark}
According to \cite{agler2, BT1998} we observe that \begin{equation}
j(z_1,z_2)=C(I_2-ZA)^{-1}ZB,
\end{equation}
and using the unitary matrix $M$ given by
\[
M=\begin{pmatrix}A & B \\
C & D
\end{pmatrix}=\begin{pmatrix}-\frac{1}{2}&-\frac{1}{2}&\frac{1}{\sqrt{2}}\\
-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{\sqrt{2}}\\
\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&0
\end{pmatrix},
\]
we have
\[
A=\begin{pmatrix}-\frac{1}{2}&-\frac{1}{2}\\
-\frac{1}{2}&-\frac{1}{2}
\end{pmatrix}, C=\begin{pmatrix} \frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}
\end{pmatrix}, B=\begin{pmatrix}\frac{1}{\sqrt{2}}\\
-\frac{1}{\sqrt{2}}
\end{pmatrix}
\]
and $D=0$ where we set
$
Z=\begin{pmatrix}z_1& 0 \\
0& z_2
\end{pmatrix}.
$
\end{remark}
Inspired by \cite{ABDS2001} we can prove the following result concerning the backward shift operators considered in Definition \ref{RinfinityL} in the case of the polyanalytic Hardy space of infinite order $\mathcal{SH}(\mathbb{D})$. Indeed, we have
\begin{theorem}
A function $f\in\mathcal{SH}(\mathbb{D})$ is a common eigenfunction for the backward shift operators $R_\infty$ and $L_\infty$ with corresponding eigenvalues given respectively by $\lambda_1, \lambda_2\in\mathbb{D}$ if and only if
\begin{equation}\label{fexpression}
\displaystyle
f(z,\overline{z})=\frac{f(0,0)}{(1-\lambda_1 z)(1-\lambda_2\overline{z})}, \quad \forall z\in\mathbb{D}.
\end{equation}
\end{theorem}
\begin{proof}
It is easy to check that if $f$ is of the form \eqref{fexpression}, then we have
$R_\infty(f)=\lambda_1f$ and $L_\infty(f)=\lambda_2f.$
For the converse, we assume that $f$ is an eigenfunction for $R_\infty$ and $L_\infty$ with corresponding eigenvalues given by $\lambda_1$ and $\lambda_2$. By writing $$\displaystyle f(z,\overline{z})=\sum_{n=0}^\infty\overline{z}^nf_n(z),$$
we observe that $R_\infty(f)=\lambda_1 f$ if and only if $$\displaystyle \sum_{n=0}^{\infty} \overline{z}^nR_0(f_n)(z)=\sum_{n=0}^\infty \overline{z}^n(\lambda_1f_n)(z).$$
In particular, this shows that $$R_0(f_n)(z)=\lambda_1 f_n(z), \quad \text{ for any } n= 0,1, 2, \ldots.$$
Using the classical result on the backward shift operator $R_0$ for any $n= 0, 1, 2, \ldots$ we get
$$\displaystyle f_n(z,\overline{z})=\frac{f_n(0)}{1-\lambda_1z},\quad z\in\mathbb{D}.$$
Now, we insert the expression of $f_n$ in the $f$ decomposition which leads to the following calculations
\[
\begin{split}
\displaystyle f(z)
&= \sum_{n=0}^{\infty}\overline{z}^nf_n(z)\\
&= \sum_{n=0}^{\infty}\overline{z}^n\frac{f_n(0)}{1-\lambda_1z}
\\
&= \left(\sum_{m=0}^{\infty}z^m \lambda_1^m \right)\left(\sum_{n=0}^{\infty}\overline{z}^nf_n(0)\right)
\\
&=\sum_{m=0}^{\infty}z^mg_m(\overline{z})
\end{split}
\]
where we set $\displaystyle g_m(\overline{z})= \lambda_1^m\left(\sum_{n=0}^{\infty}\overline{z}^nf_n(0)\right)$ for any $z\in\mathbb{D}$. On the other hand, we note that by definition $$\displaystyle L_\infty(f)=\sum_{m=0}^{\infty}z^mL_0(g_m),$$
with $\displaystyle L_0(g_m)(\overline{z})=\frac{g_m(\overline{z})-g_m(0)}{\overline{z}}$. Then, following a similar reasoning as we did in the case of $R_\infty$, using the fact that $L_\infty(f)=\lambda_2 f$ we deduce tha $L_0(g_m)(\overline{z})=\lambda_2 g_m(\overline{z})$. Thus, in particular this allows to write
$$g_m(\overline{z})=\frac{g_m(0)}{(1-\lambda_2\overline{z})}, \quad z\in\mathbb{D}.$$
So, now we insert $g_m$ in the expression of $f$ and get
$$\displaystyle f(z)=\sum_{m=0}^\infty z^m \frac{g_m(0)}{1-\lambda_2\overline{z}}.$$
We note that $g_m(0)=\lambda_1^mf_0(0)=\lambda_1^mf(0)$. Then, we obtain
$$f(z)=\frac{f(0)}{1-\lambda_2\overline{z}}\sum_{m=0}^{\infty}\lambda_1^mz^m, \quad z\in \mathbb{D}.$$
Hence, we conclude that
$$\displaystyle
f(z,\overline{z})=\frac{f(0,0)}{(1-\lambda_1 z)(1-\lambda_2\overline{z})}, \quad \text{ for any } z\in\mathbb{D}.$$
\end{proof}
\begin{lemma}
For any $f\in \mathcal{C}^1$ we have
\[
\frac{\rm d}{\rm dt}f(tx,ty)=z\partial f+\overline{z} \overline{\partial}f
\]
\end{lemma}
\begin{proof}
It follows from
\[
\begin{split}
\frac{\rm d}{\rm dt}f(tx,ty)&=x\frac{\partial f }{\partial x}(tx,ty)+y\frac{\partial f }{\partial y}(tx,ty)\\
&=\frac{1}{2}(x+iy)\left(\frac{\partial f }{\partial x}(tx,ty)--i\frac{\partial f }{\partial y}(tx,ty)\right)+\frac{1}{2}(x-iy)\left(\frac{\partial f }{\partial x}(tx,ty)
+i\frac{\partial f }{\partial y}(tx,ty)\right)\\
&=
z\partial f+\overline{z} \overline{\partial}f.
\end{split}
\]
\end{proof}
Let
\begin{equation}
f(z,\overline{z})=zf_1(z,\overline{z})+\overline{z} f_2(z,\overline{z})
\end{equation}
where $f_1$ and $f_2$ are required to be in the same space as $f$. Then
\begin{eqnarray}
(A_0 f)(z,\overline{z})&=&\int_0^1\frac{\partial}{\partial z}f(tz,t\overline{z})dt\\
(B_0 f)(z,\overline{z})&=&\int_0^1\frac{\partial}{\partial \overline{z}}f(tz,t\overline{z})dt
\end{eqnarray}
We have
\[
\frac{\rm d}{{\rm d}t}f(tz,t\overline{z})=z\frac{\partial}{\partial z}f(tz,t\overline{z})dt\\+
\overline{z} \frac{\partial}{\partial \overline{z}}f(tz,t\overline{z})dt
\]
and so
\begin{equation}
\label{345}
f(z,\overline{z})-f(0,0)=zA_0f(z,\overline{z})+\overline{z} B_0f(z,\overline{z})
\end{equation}
\begin{lemma}
Let $\mathfrak M$ be a finite dimensional space in which \eqref{345} holds. Any $f\in \mathfrak{M}$ which is a common eigenfunction of $A_0$ and $B_0$ can be written in the form
\[
f(z,\overline{z})=\frac{f(0,0)}{1-az-b\overline{z}}.
\]
\end{lemma}
\begin{proof}
Since $A_0$ and $B_0$ commute they can be simultaneously triangularized.
Let
\[
A_0f=a f\quad{\rm and}\quad B_0f=b f
\]
We have
\[
f(z,\overline{z})=f(0,0)+(az+b\overline{z})f
\]
and so
\[
f(z,\overline{z})=\frac{f(0,0)}{1-az-b\overline{z}}
\]
\end{proof}
\begin{lemma}
The following equalities hold:
\begin{eqnarray}
A_0(z^n\overline{z}^m)&=&\frac{n}{n+m}z^{n-1}\overline{z}^m\\
B_0(z^n\overline{z}^m)&=&\frac{m}{m+n}z^n\overline{z}^{m-1}.
\end{eqnarray}
\end{lemma}
\begin{proof}
It follows from
\[
\begin{split}
A_0(z^n\overline{z}^m)&=\int_0^1(nz^{n-1}\overline{z}^m)(tz,t\overline{z})dt\\
&=nz^{n-1}\overline{z}^m\int_0^1nt^{n+m-1}dt\\
&=\frac{n}{n+m}z^{n-1}\overline{z}^m,
\end{split}
\]
and similarly for $B_0$.
\end{proof}
\begin{remark}
We remark that $A_0$ is not $R_0$, in general; it will reduce to $R_0$ when $f$ is analytic. We note that both the operators $R_\infty$ and $A_0$ extend the classical backward shift operator $R_0$ on the Hardy space, but these two operators are different.
\end{remark}
\section{The Drury-Arveson space case}
\setcounter{equation}{0}
Let us consider the kernel function given by
\begin{equation}
\label{arve}
k(z,w)= \frac{1}{1-(z\overline{w}+\overline{z} w)}.
\end{equation}
We denote by $\mathfrak H(k)$ the associated reproducing kernel Hilbert space. Setting $z=x+iy$ and $w=t+iu$, we have
\begin{equation}
\frac{1}{1-(z\overline{w}+\overline{z} w)}=\frac{1}{1-2(xt+yu)}
\end{equation}
We note that \eqref{arve} is a complete Nevanlinna-Pick kernel, meaning that
\[
\frac{1}{k(z,w)}=1-2xt-2yu
\]
has one positive square in $B(0,1/\sqrt{2})$. Such kernels were introduced by Agler, see \cite{agler} and also the paper of Quiggin \cite{PQ}.
\begin{lemma}\label{kresult}
The function \eqref{arve} is positive definite in $|z|<1/\sqrt{2}$ and the functions
\[
\frac{\partial k}{\partial t}\quad and\quad \frac{\partial k}{\partial u}
\]
belong to $\mathfrak H(k)$. Furthermore, it holds that
\begin{eqnarray}
\langle f, \frac{\partial f}{\partial t}\rangle_{\mathfrak H(k)}&=&\frac{\partial f}{\partial x}\\
\langle f, \frac{\partial f}{\partial u}\rangle_{\mathfrak H(k)}&=&\frac{\partial f}{\partial y}.
\end{eqnarray}
\end{lemma}
\begin{proof}
For a fixed choice of $(t,u)$ and for $h\in\mathbb R$ small enough we set
\[
f_h(x,y,t,u)=\frac{k(x,y,t,u+h)-k(x,y,t,u)}{h}.
\]
Then $f_h\in\mathfrak H(k)$ and
\[
\|f_h\|^2_{\mathcal H(k)}=\frac{k(t,u+t,t,u+h)+k(t,u,t,u)-2k(t,u+h,t,u)}{h^2}
\]
uniformly bounded in $h$ for $h$ small. Thus $f_h$ has a weakly convergent subsequence, with limit say $g_{t,u}$.
Since weak convergence implies pointwise convergence we have
\[
\begin{split}
g_{t,u}(x,y)&=\langle g_{t,u},k(\cdot,\cdot, x,y)\rangle_{\mathfrak H(k)}\\
&=\lim_{h\rightarrow 0} \langle f_h,k(\cdot,\cdot, x,y)\rangle_{\mathfrak H(k)}\\
&=\lim_{h\rightarrow 0} \left\langle \frac{k(\cdot,\cdot,t,u+h)-k(\cdot,\cdot,t,u)}{h},k(\cdot,\cdot, x,y)\right\rangle_{\mathfrak H(k)}\\
&=\lim_{h\rightarrow 0} \frac{k(x,y,t,u+h)-k(x,y,t,u)}{h}\\
&= \frac{\partial f}{\partial u}(x,y,t,u).
\end{split}
\]
Furthermore, for $f\in\mathfrak H(k)$, we have:
\[
\begin{split}
\langle f,g_{t,u}\rangle_{\mathfrak H(k)}=&=\lim_{h\rightarrow 0} \langle f,f_h\rangle_{\mathfrak H(k)}\\
&=\lim_{h\rightarrow 0} \left\langle f(\cdot,\cdot), \frac{k(\cdot,\cdot,t,u+h)-k(\cdot,\cdot,t,u)}{h}\right\rangle_{\mathfrak H(k)}\\
&=\lim_{h\rightarrow 0} \frac {f(t,u+h)-f(t,u)}{h}\\
&=\frac{\partial f}{\partial y}(t,u)
\end{split}
\]
The other claims are proved in the same way.
\end{proof}
Iterating the above result we get:
\begin{corollary}
Let $k$ be as in \eqref{arve}, then for $n,m=0,1,2,\ldots$
\begin{equation}
\frac{\partial^{n+m}k}{\partial t^n\partial u^m}(\cdot,\cdot, t,u)\in\mathfrak H(k)
\end{equation}
and
\begin{equation}
\langle f(\cdot,\cdot), \frac{\partial^{n+m}k}{\partial t^n\partial u^m}(\cdot,\cdot, t,u)\rangle_{\mathfrak H(k)}=
\frac{\partial^{n+m}f}{\partial x^n\partial y^m}(t,u).
\end{equation}
\end{corollary}
\begin{proof}
This a direct consequence of Lemma \ref{kresult}.
\end{proof}
\begin{corollary}
For for $n,m=0,1,2,\ldots$ we have
\[
x^ny^m\in\mathfrak H(k).
\]
\end{corollary}
\begin{proof}
It suffices to set $t=u=0$ in the previous corollary.
\end{proof}
We now give a characterization of the space $\mathfrak H(k)$.
\begin{proposition}
The space $\mathfrak H(k)$ consists of the functions of the form
\begin{equation}
f(z,\overline{z})=\sum_{a,b=0}^\infty c_{a,b}z^a\overline{z}^b
\end{equation}
with norm
\begin{equation}
\|f\|^2=\sum_{a,b=0}^\infty |c_{a,b}|^2\frac{a!b!}{(a+b)!}
\end{equation}
\end{proposition}
\begin{proof}
It suffices to observe that we have
\begin{equation}
k(z,w)= \sum_{a,b\in\mathbb N_0}\frac{(a+b)!}{a!b!}z^a\overline{z}^bw^b\overline{w}^{a}.
\end{equation}
\end{proof}
\begin{proposition}
The operators $M_z$ and $M_{\overline{z}}$ are bounded in $\mathfrak H(k)$, with $\|M_{z}\|\le 1$ and $\|M_{\overline{z}}\|\le 1$. Their adjoints are given by
\begin{eqnarray}
M_z^*&=&A_0\\
M_{\overline{z}}^*&=&B_0.
\end{eqnarray}
\end{proposition}
\begin{proof}
The first claim follows from
\[
\frac{1-z\overline{w}}{1-2{\rm Re}\, z\overline{w}}=1+\frac{\overline{z}w}{1-2{\rm Re}\, z\overline{w}}\ge 0
\]
and
\[
\frac{1-\overline{z}w}{1-2{\rm Re}\, z\overline{w}}=1+\frac{z\overline{w}}{1-2{\rm Re}\, z\overline{w}}\ge 0
\]
The second claim follows from
\[
\begin{split}
\langle M_z(z^n\overline{z}^m),z^u\overline{z}^v\rangle&=\langle z^{n+1}\overline{z}^m,z^u\overline{z}^v\rangle
\end{split}
\]
and similarly for $M_{\overline{z}}$.
\end{proof}
Following \cite[Corollary 2.4, p. 7]{AT} we introduce
\begin{eqnarray}
(A_af)(z)&=&\frac{z}{1-2{\rm Re}\, z\overline{a}}f(z)\\
(B_af)(z)&=&\frac{\overline{z}}{1-2{\rm Re}\, \overline{z}a}f(z)
\end{eqnarray}
with $a\in B(0,1/\sqrt{2})$.
\begin{proposition}
Let $a\in B(0,1/\sqrt{2})$. The operators $A_a$ and $B_a$ are bounded and it holds that
\begin{equation}
\label{wer}
f(z)-f(a)=(z-a)(A_a^*f)(z)+(\overline{z}-\overline{a})(B_a^*f)(z),\quad f\in\mathfrak H(k).
\end{equation}
\end{proposition}
\begin{proof}
We have
\begin{eqnarray}
(A_a^*k_b)(z)&=\frac{\overline{b}}{1-2{\rm Re}\, (b\overline{a})}k_b(z)\\
(B_a^*k_b)(z)&=\frac{b}{1-2{\rm Re}\, (b\overline{a})}k_b(z)
\end{eqnarray}
\[
\begin{split}
(z-a)(A_a^*k_b)(z)+(\overline{z}-\overline{a})(B_a^*k_b)(z)&=\frac{(z-a)\overline{b}+(\overline{z}-\overline{a})b}{(1-2{\rm Re}\, b\overline{a})(1-2{\rm Re}\, z\overline{b})}\\
&=\frac{z\overline{b}+\overline{z}b-a\overline{b}-\overline{a}b}{(1-2{\rm Re}\, b\overline{a})(1-2{\rm Re}\, z\overline{b})}\\
&=k_b(z)-k_b(a)
\end{split}
\]
\end{proof}
\begin{example}
Let us consider the coefficients $c_n$ such that
\begin{equation}\label{cn}
\displaystyle 1-\sqrt{1-t}=\sum_{n=1}^{\infty} c_nt^n, \quad t<1.
\end{equation}
Then, the function
\[
f_m(z,\overline{z})=z+\sum_{n=1}^mc_n\overline{z}^{2m}
\]
is bounded by one in modulus in $|z|<\frac{1}{\sqrt{2}}$, but is not a Schur multiplier.
\end{example}
Indeed, recalling that the $c_n>0$ and satisfy $\sum_{n=1}^\infty c_n=1$ we have
\[
\begin{split}
\left|z+\sum_{n=1}^mc_n\overline{z}^{2n}\right|&\le |z|+\sum_{n=1}^m c_n|z|^{2n}\\
&\le |z|+\sum_{n=1}^\infty c_n|z|^{2n}\\
&=|z|+1-\sqrt{1-|z|^2}\\
&\le \frac{1}{\sqrt{2}}+1-\sqrt{\frac{1}{2}}\\
&=1.
\end{split}
\]
But $\|f_m\|^2=1+\sum_{n=1}^mc_n\|\overline{z}^{2n}\|^2>1$.
For $a\in B(0,1/\sqrt{2})$ we set
\begin{equation}
b_a(z)=\frac{(1-2|a|^2)\begin{pmatrix}z-a&\overline{z}-\overline{a}\end{pmatrix}}{1-{\rm Re}\, z\overline{a}}\sqrt{I_2-\begin{pmatrix}\overline{a}\\ a\end{pmatrix}
\begin{pmatrix}a&\overline{a}\end{pmatrix}}
\end{equation}
We note that (with $c_1,c_2,\ldots$ as in \eqref{cn})
\[
\begin{split}
\sqrt{I_2-\begin{pmatrix}\overline{a}\\ a\end{pmatrix}\begin{pmatrix}a&\overline{a}\end{pmatrix}}&=I_2-\sum_{n=1}^\infty c_n
\left(\begin{pmatrix}\overline{a}\\ a\end{pmatrix}
\begin{pmatrix}a&\overline{a}\end{pmatrix}
\right)^n\\
&=I_2-\sum_{n=1}^\infty c_n \left(\begin{pmatrix}\overline{a}\\ a\end{pmatrix}
\begin{pmatrix}a&\overline{a}\end{pmatrix}\right)\left(\begin{pmatrix}\overline{a}& a\end{pmatrix} \begin{pmatrix}a\\ \overline{a}\end{pmatrix}
\right)^{n-1}\\
&=I_2-\frac{\begin{pmatrix}\overline{a}& a\end{pmatrix} \begin{pmatrix}a\\ \overline{a}\end{pmatrix}}{2|a|^2}\sum_{n=1}^\infty c_n(2|a|^2)^n\\
&I_2-\frac{\begin{pmatrix}\overline{a}& a\end{pmatrix} \begin{pmatrix}a\\ \overline{a}\end{pmatrix}}{2|a|^2}\left(1-\sqrt{1-2|a|^2}\right).
\end{split}
\]
\begin{theorem}
The function $f$ belongs to $\mathfrak H(k)$ and $f(a)=0$ if and only if
\[
f(z)=b_a(z)g(z),
\]
with $g\in{\mathfrak H(k)}^2$.
\end{theorem}
\begin{proof} We follow \cite{AT}. One direction is trivial while the converse is a direct consequence of \eqref{wer} with $g(z)=\begin{pmatrix}A_a^*f\\ B_a^*f\end{pmatrix}$
since $f(a)=0$.
\end{proof}
More generally, as in Proposition 4.5 and Section 5 of \cite{AT} we have
\begin{theorem}
Let $z_1,\ldots, z_N\in B(0,1/\sqrt{2})$ and $w_1,\ldots, w_N\in\mathbb C$. There exists a Schur multiplier $s$ such that
\begin{equation}
s(z_n)=w_n,\quad n=1,\ldots, N
\end{equation}
if and only if the $N\times N$ matrix with $(n,m)$ entry
\begin{equation}
\frac{1-w_n\overline{w_m}}{1-2{\rm Re}\, z_n\overline{z_m}}
\end{equation}
is non negative.
\end{theorem}
\begin{proof}
This holds thanks to the fact that the kernel $k(z,w)=\frac{1}{1-2{\rm Re}\, z\overline{w}}$ is a complete Nevanlinna-Pick kernel, and so the Nevanlinna-Pick interpolation problem is solved.
\end{proof}
\bibliographystyle{plain}
|
1,314,259,996,553 | arxiv | \section*{Introduction and Summary}
Classical elasticity theory rests on the hypothesis that a solid may exist in a stress-free state in Euclidean space. The dynamics of the solid is then described by a time-dependent mapping taking the position of each material particle in this relaxed reference state to its actual position at a given time. Now the hypothesis of the existence of a relaxed state is violated when dislocations are present in the solid. These cause internal stresses even in the absence of external forces.
Solids in which a continuous distribution of dislocations is present have been treated in the linear approximation (\cite{K1}, \cite{K2}, \cite{N1}). However, the conceptual framework of this linear theory is inadequate for the formulation of an exact theory. The appropriate conceptual framework was introduced in \cite{C2} and an exact nonlinear theory was proposed. The basic concept introduced in \cite{C2} is that of the material manifold which captures those properties of a crystalline solid which are intrinsic to it, being independent of the way it is embedded in space. The dynamics of the elastic body is then described by a mapping from space-time into this material manifold.
While an exact theory of crystalline solids with a continuous distribution of dislocations was formulated in \cite{C2}, the theory was left undeveloped up to the present time. The aim of the present work is to develop the theory and derive results which may be brought into contact with experimental data. Our focus in the present paper is on the static case, where we have a crystalline solid with a uniform distribution of elementary dislocations in equilibrium in free space. As a basic example we study, in the continuum limit, a two-dimensional crystalline solid with a uniform distribution of edge dislocations, one of the two elementary types of dislocations. In this case the material manifold is the affine group of the real line, the hyperbolic plane. We then study, in the continuum limit, a three-dimensional crystalline solid with a uniform distribution of screw dislocations, the other elementary type of dislocation. In this case the material manifold is the Heisenberg group.
The purpose of the present work is to introduce to the mathematics and theoretical physics community a field where beautiful differential geometric structures, in particular Lie groups, form the basis of a classical physical theory. Moreover, the laws of the theory form a nonlinear system of variational partial differential equations, which in the static case is elliptic and in the dynamical case is of hyperbolic type. Both cases constitute a worthy challenge for the geometric analyst.
\cutpage
\setcounter{page}{2}
\noindent
This paper is organized as follows. In Part I, we give a general introduction to the theory of crystalline solids containing an arbitrary distribution of dislocations, based on the work of Christodoulou \cite{C1}, \cite{C2}. We introduce the basic concepts of material manifold and crystalline structure. In the present work, the fundamental notion is the canonical form, which is used to define the dislocation density. We illustrate the theory by giving two basic examples of solids with a uniform distribution of dislocations. In order to state the laws governing the dynamics, the thermodynamic space is introduced and the energy function defined. The dynamics is described by a mapping from space-time into the material manifold.
The equivalence relations for crystalline structures and for the mechanical properties of a solid are discussed to give the proper physical interpretation of the theory.
Finally, the Eulerian picture is given including the non-relativistic limit. In the Eulerian picture the material manifold is eliminated.
In Part II, we focus our attention on the static case. We derive the boundary value problem from an action principle and give the Legendre-Hadamard conditions for the energy function. We then consider the two examples of uniform distributions of elementary dislocations and motivate the choice of our model energy function. We separately discuss the special cases of uniform distributions of edge and screw dislocations in two and three dimensions, respectively.
In concluding the second part, we derive the scaling properties of the theory.
Part III is devoted to the analysis of equilibrium configurations of a crystalline solid with a uniform distribution of elementary dislocations in two and three dimensions. We solve the problem in two dimensions and give the method for the solution of the anisotropic problem in the case of a uniform distribution of screw dislocations in three dimensions.
\newpage
\part{The General Setting} \label{Setting}
\setcounter{section}{0}
\section{The Material Manifold}
Let $\mathcal N$ be an oriented $n$-dimensional differentiable manifold, called the {\bf material manifold}. It describes a material together with those of its properties which are intrinsic to it being independent of the way in which it is to be embedded in physical space. A point $y \in \mathcal N$ represents a particle. We denote by $\mathcal X (\mathcal N)$ the $C^{\infty}$-vectorfields on the material manifold $\mathcal N$. Let
\begin{equation} \label{evalmap}
\begin{array}{rcl}
\epsilon_y : \mathcal X (\mathcal N) & \to & T_y\mathcal N \\
X & \mapsto & \epsilon_y(X)=X(y)
\end{array}
\end{equation}
be the evaluation map at a point $y \in \mathcal N$.
\begin{definition}
A {\bf crystalline structure} on $\mathcal N$ is a distinguished linear subspace $\mathcal V$ of $\mathcal X (\mathcal N)$ such that the evaluation map $\epsilon_y$ restricted to $\mathcal V$ is an isomorphism for each $y \in \mathcal N$.
\end{definition}
\begin{remark}
The orientation of $\mathcal N$ induces an orientation in $\mathcal V$ such that $\epsilon_y$ is orientation preserving.
$\mathcal N$ admits a crystalline structure if and only if $\mathcal N$ is parallelizable, see \cite{C2}.
\end{remark}
We introduce a $1$-form $\nu$ on $\mathcal N$ with values in $\mathcal V$ defined by
\begin{equation} \label{defnu}
\nu_y(Y_y)=\epsilon_y^{-1}(Y_y) \quad : \forall Y_y \in T_y\mathcal N \, .
\end{equation}
In the case that $(\mathcal N, \mathcal V)$ is a Lie group with corresponding Lie algebra, $\nu$ is the \emph{Maurer-Cartan} form.
\begin{remark} \label{mfo}
The most fundamental object is the $1$-form $\nu$. We may in fact replace $\mathcal V$ by an abstract real vectorspace $W$ of the same dimension, $n$, as the manifold $\mathcal N$. Then we define a canonical form $\nu$ to be a $W$-valued $1$-form on $\mathcal N$ such that
\begin{equation*}
\nu_y := \left. \nu \right|_{T_y\mathcal N} : T_y \mathcal N \to W
\end{equation*}
is an isomorphism for each $y \in \mathcal N$. Given an element $v \in W$, we may then define a vectorfield $Y_v$ on $\mathcal N$ by
\begin{equation*}
Y_v(y)=\nu_y^{-1}(v) \quad : \, \forall y \in \mathcal N
\end{equation*}
We then define the crystalline structure $\mathcal V$ by
\begin{equation*}
\mathcal V = \left\{ X_v : v \in W \right\} \, .
\end{equation*}
Then the canonical form corresponds to the $1$-form $\nu$ defined in (\ref{defnu}).
\end{remark}
We say that the crystalline structure $\mathcal V$ on $\mathcal N$ is {\bf complete} if each $X \in \mathcal V$ is a complete vectorfield on $\mathcal N$. Then each element of $\mathcal V$ generates a $1$-parameter group of diffeomorphisms of $\mathcal N$, which represents (in the continuum limit) a group of translations of the crystal lattice with parameter proportional to the number of atoms traversed.
A complete crystalline structure $\mathcal V$ on the material manifold $\mathcal N$ defines an {\bf exponential map}
\begin{equation*}
\exp : \mathcal N \times \mathcal V \to \mathcal N
\end{equation*}
as follows. Let $\exp(y,X)$ be the point in $\mathcal N$ that is at parameter value $1$ from $y$ along the integral curve of $X$ initiating at $y$. For each $y \in \mathcal N$, let
\begin{equation*} \begin{array}{rcl}
\exp_y : \mathcal V & \to & \mathcal N \\
X & \mapsto & \exp_y(X)=\exp(y, X) \, .
\end{array}
\end{equation*}
We have
\begin{equation*}
\exp_y(0)=y \quad , \quad d\exp_y(0)=\epsilon_y \, .
\end{equation*}
Thus $d\exp_y(0)$ is an isomorphism for each $y \in \mathcal N$. By the implicit function theorem it follows that, for each $y \in \mathcal N$, there is a neighborhood $\mathcal U_y$ of the zero vector in $\mathcal V$ such that $\exp_y$ restricted to $\mathcal U_y$ is a diffeomorphism onto its image in $\mathcal N$.
Now choose a totally antisymmetric $n$-linear form $\omega$ on $\mathcal V$ which is positive when evaluated on a positive basis.
The $n$-form $\omega$ defines a {\bf volume form} $d\mu_{\omega}$, called {\bf mass form} on $\mathcal N$ by:
\begin{equation}\label{volumeform}
\begin{array}{rcl}
d\mu_{\omega}\left(Y_{1,y},\ldots, Y_{n,y}\right) & = & \omega\left(\epsilon_y^{-1}(Y_{1,y}),\ldots,\epsilon_y^{-1}(Y_{n,y})\right) \\
& & : \, \forall y \in \mathcal N, \forall Y_{1,y}, \ldots, Y_{n,y} \in T_y\mathcal N \, .
\end{array}
\end{equation}
The volume assigned by $d\mu_{\omega}$ to a domain $\mathcal D \subset \mathcal N$,
\begin{equation*}
\int_{\mathcal D}d\mu_{\omega} \, ,
\end{equation*}
is the rest mass of $\mathcal D$.
\begin{definition}
Given a crystalline structure $\mathcal V$ on $\mathcal N$ we can define a mapping
\begin{equation*}
\Lambda:\mathcal N \to \mathcal L(\mathcal V \wedge \mathcal V, \mathcal V)
\end{equation*}
by:
\begin{equation} \label{defdisden}
\Lambda(y)(X,Y)=\epsilon_y^{-1}\left([X,Y](y)\right) \in \mathcal V \quad , \quad \forall y \in \mathcal N, X,Y \in \mathcal V \, .
\end{equation}
We call $\Lambda$ {\bf dislocation density}.
\end{definition}
Suppose $X, Y \in \mathcal V$ are complete and generate $1$-parameter groups of diffeomorphisms $\Phi_t$, $\Psi_t$, $t \in \mathbb R$ from $\mathcal N$ to $\mathcal N$. For $y \in \mathcal N$
\begin{equation*}
(t,s) \mapsto \Psi_{-s}\left(\Phi_{-t}\left(\Psi_s\left(\Phi_t(y)\right)\right)\right)
\end{equation*}
coincides with $(t,s)\mapsto \Xi_{ts}$, where $\Xi_t$ is the $1$-parameter group of diffeomorphisms of $\mathcal N$ generated by $\Lambda(y)(X,Y)$.
The dislocation density is a concept that arises in the continuum limit of a distribution of elementary dislocations in a crystal lattice. An elementary dislocation has the property that, if we start at an atom in the crystal lattice and move according to one group of lattice transformations $k$ atoms in one direction, then according to a different group $l$ atoms in a second direction, according to the first $-k$ atoms and finally $-l$ atoms in the second directions, then we arrive at a different atom than we started from, but which, provided the circuit encloses a single elementary dislocation, is reached at in a single step corresponding to a lattice translation. This step is called {\bf Burger's vector}.
If $\Lambda$ is constant on $\mathcal N$ then, for all $X, Y \in \mathcal V$, there is a $Z \in \mathcal V$ such that
\begin{equation*}
[X,Y]=Z \, ,
\end{equation*}
corresponding to a uniform distribution of elementary dislocations of the same kind, see below.
Thus $\mathcal V$ constitutes in this case a {\bf Lie algebra}, i.e.~a vectorspace $\mathcal V$ over $\mathbb R$ with a bracket
\begin{equation*}
[\, .\, ,\, .\, ] \, : \, \mathcal V \wedge \mathcal V \to \mathcal V
\end{equation*}
satisfying the Jacobi identity.
By the fundamental theorems of Lie group theory, upon choosing an identity element $e \in \mathcal N$, the material manifold $\mathcal N$ can then be given the structure of a {\bf Lie group} such that $\mathcal V$ is the space of vectorfields on $\mathcal N$ which generate the right action of the group on itself. $\mathcal V$ is then the space of vectorfields on $\mathcal N$ which are left invariant, i.e.~invariant under left group multiplication.
The dual space $\mathcal V^*$ is then the space of left invariant $1$-forms on $\mathcal N$.
Let us consider $d\nu$, a $2$-form on $\mathcal N$ : for any pair of vectorfields $X, Y$ on $\mathcal N$ we have
\begin{equation} \label{derofnu}
d\nu(X,Y)=X(\nu(Y))-Y(\nu(X))-\nu([X,Y])
\end{equation}
by the formula for the exterior derivative of a $1$-form.
In particular, this holds for $X, Y \in \mathcal V$. Now for $X \in \mathcal V$ we have
\begin{equation*}
\nu(X)(y)=\nu_y(X_y)=\epsilon_y^{-1}(X_y)=X \quad , \quad \forall y \in \mathcal N, X \in \mathcal V \, ,
\end{equation*}
a constant $\mathcal V$-valued function on $\mathcal N$. Similarly with $X$ replaced by $Y$.
Therefore, from (\ref{derofnu}) we have
\begin{equation}\label{dernu}
d\nu (X,Y) = -\nu([X,Y])\quad , \, \forall X,Y \in \mathcal V \, .
\end{equation}
On the other hand,
\begin{equation} \label{Lambdaeqnu}
\Lambda(y)(X,Y)=\epsilon_y^{-1}([X,Y](y))=\nu_y([X,Y](y)) \quad , \, \forall y \in \mathcal N, X,Y \in \mathcal V \, .
\end{equation}
Thus, comparing (\ref{dernu}) and (\ref{Lambdaeqnu}), we obtain
\begin{equation}\label{dnumlam}
\left(d\nu(X,Y)\right)(y)=-\nu_y\left([X,Y](y)\right)=-\Lambda(y)(X,Y) \, .
\end{equation}
Let us then define the $\mathcal V$-valued $2$-form $\lambda$ on $\mathcal N$ by:
\begin{equation*}
\lambda_y(Y_{1,y},Y_{2,y}) = \Lambda(y)\left(\epsilon_y^{-1}(Y_{1,y}),\epsilon_y^{-1}(Y_{2,y})\right) \quad : \, \forall Y_{1,y}, Y_{2,y} \in T_y\mathcal N \, ,
\end{equation*}
at any point $y \in \mathcal N$. We conclude from (\ref{dnumlam}) that
\begin{equation*}
d\nu= -\lambda \, .
\end{equation*}
Let $\Gamma$ be a closed curve in $\mathcal N$ and let $\Sigma$ be any surface spanning $\Gamma$, i.e.~
$\partial \Sigma=\Gamma$. We finally conclude
\begin{equation} \label{sumofBv}
-\int_{\Gamma} \nu = \int_{\Sigma}\lambda \, .
\end{equation}
The right-hand side of (\ref{sumofBv}) is the sum of all Burger's vectors enclosed by the curve $\Gamma$ (or threading $\Sigma$).
\subsection{Uniform Dislocation Distributions and Lie Groups}
At the atomic level, two kinds of elementary dislocations are found. They are called {\bf edge} and {\bf screw} dislocations.
In the following, we consider these two types of dislocations taking them as our model cases. For a uniform distribution of these two types of elementary dislocations, we determine the corresponding Lie groups, the affine group and the Heisenberg group, respectively. For a detailed description see \cite{IK}.
\subsubsection{Edge Dislocations and the Affine Group} \label{edag}
The most basic type of a dislocation in a $2$-dimensional crystal lattice is an edge dislocation. It
appears in a $2$-dimensional lattice in which an extra half-line of atoms has been inserted along the positive $1$st axis. A circuit of translations in the directions of the $1$st and $2$nd axis, alternately, which encloses the origin, ends at an atom which is reached in a single step by a translation in the direction of the $2$nd axis. On the other hand, circuits not enclosing the origin close. Mathematically, this phenomenon is represented by the commutation relation $[E_1,E_2]=E_2$, where $E_1, E_2$ are the vectorfields along the coordinate axis.
We want to show that a uniform distribution of edge dislocations in a $2$-dimensional lattice gives rise in the continuum limit to the {\bf affine group}. This group is characterized by transformations of the real line of the form
\begin{equation*} \begin{array}{rcl}
\mathbb R & \to & \mathbb R \\
x & \mapsto & e^{y^1}x+y^2 \, ,
\end{array}
\end{equation*}
where $(y^1, y^2) \in \mathbb R^2$ are two parameters. The subgroups of the affine group are $t\mapsto e^{y^1}t$ (multiplication) and $s\mapsto s + y^2$ (translation).
We have as the group manifold $\mathbb R^2$ equipped with the following multiplication
\begin{equation*}
(y^1,y^2)(\tilde y^1,\tilde y^2)=(y^1+\tilde y^1,y^2+e^{y^1}\tilde y^2) \, .
\end{equation*}
\begin{equation*}
X=\frac{\partial}{\partial y^1} \quad \textrm{and} \quad Y=e^{y^1}\frac{\partial}{\partial y^2}
\end{equation*}
generate the right action with
\begin{equation*}
[X,Y] = \frac{\partial}{\partial y^1}e^{y^1}\frac{\partial}{\partial y^2}-e^{y^1}\frac{\partial}{\partial y^2}\frac{\partial}{\partial y^1} = e^{y^1}\frac{\partial}{\partial y^2}= Y \, .
\end{equation*}
The Lie algebra of the affine group is thus generated by the vectorfields $X, Y$, which satisfy the commutation relation
\begin{equation*}
[X,Y]=Y \, .
\end{equation*}
Therefore, the affine group is the material manifold $\mathcal N$ endowed with the crystalline structure.
If we take $\{E_1=X, E_2=Y\}$ as a basis of $\mathcal V$, we have the following dual basis $\{\omega^1, \omega^2\}$ for $\mathcal V^*$
\begin{equation*}
\omega^1 = dy^1 \quad , \quad \omega^2=e^{-y^1}dy^2 \, .
\end{equation*}
The corresponding left invariant metric
\begin{equation} \label{hypmetafgr}
\stackrel{\circ}{n}=(\omega^1)^2+(\omega^2)^2=(dy^1)^2+e^{-2y^1}(dy^2)^2
\end{equation}
on $\mathcal N$ makes $\mathcal N$ the {\bf hyperbolic plane}.
The dislocation density $\lambda$ is
\begin{equation*}
\lambda(E_1,E_2)(y)=\Lambda(y)(E_1,E_2) = \epsilon_y^{-1}\left([E_1,E_2](y)\right) = \epsilon_y^{-1}\left(\epsilon_y(E_2)\right) = E_2 \, .
\end{equation*}
Since $\omega_1 \wedge \omega_2=e^{-y_1}\left(dy^1\wedge dy^2\right)$, it follows
\begin{equation*}
\lambda = e^{-y_1}\left(dy^1\wedge dy^2\right)E_2 \, ,
\end{equation*}
and therefore
\begin{equation*}
\int_{\Sigma}\lambda =\int_{\Sigma}e^{-y_1}\left(dy^1\wedge dy^2\right)E_2=A(\Sigma)E_2 \, ,
\end{equation*}
where $A(\Sigma)$ is the area of the surface $\Sigma$, a domain in $\mathcal N$. This makes sense since the sum of the Burger vectors associated to a domain in a uniform distribution of edge dislocations should be proportional to the area of the domain.
\subsubsection{Screw Dislocations and the Heisenberg Group} \label{sdhg}
The second kind of elementary dislocation is called a {\bf screw dislocation}. It appears in a $3$-dimensional lattice in the following way. A circuit of translations along the direction of the $1$st and $2$nd axis, alternately, which encloses the $3$rd, ends at an atom which is reached at a single step by a translation in the direction of the $3$rd axis, while circuits not enclosing the $3$rd axis close.
Mathematically, this phenomenon is represented by the commutation relations $[E_1, E_2]=E_3$, $[E_1, E_3]=[E_2, E_3]=0$, where $E_1, E_2, E_3$ are the vectorfields along the coordinate axis.
We want to show that a uniform distribution of dislocations of the screw type give rise in the continuum limit to the {\bf Heisenberg group}. This group is represented as a group of unitary transformations on the space of square integrable complex valued functions $\Psi$ on $\mathbb R$ as follows
\begin{equation*} \begin{array}{rcl}
L^2(\mathbb R, \mathbb C) & \to & L^2(\mathbb R, \mathbb C) \\
\Psi(x) & \mapsto & \Psi'(x)=e^{i(y^2 x+y^3)}\Psi(x+y^1) \, ,
\end{array}
\end{equation*}
where $(y^1,y^2,y^3) \in \mathbb R^3$ are three parameters. The subgroups of the Heisenberg group are $t \mapsto \Psi(x+t)$ (translation in position), $s \mapsto e^{is x}\Psi (x)$ (translation in momentum), and $u \mapsto e^{iu}\Psi(x)$ (multiplication by a phase).
We have as the group manifold $\mathbb R^3$ equipped with the following multiplication
\begin{equation*}
(y^1,y^2,y^3)(\tilde y^1,\tilde y^2,\tilde y^3)=(y^1+\tilde y^1,y^2+\tilde y^2,y^3+\tilde y^3+y^1\tilde y^2) \, .
\end{equation*}
\begin{equation*}
X=\frac{\partial}{\partial y^1} \quad , \quad Y=\frac{\partial}{\partial y^2}+y^1\frac{\partial}{\partial y^3} \quad \textrm{and} \quad Z=\frac{\partial}{\partial y^3}
\end{equation*}
generate the right action with
\begin{equation*}
\left[X,Y\right] = \left[\frac{\partial}{\partial y^1},\frac{\partial}{\partial y^2}+y^1\frac{\partial}{\partial y^3}\right]=\frac{\partial}{\partial y^3}= Z \quad , \quad \left[X,Z\right] = \left[Y,Z\right] = 0 \, .
\end{equation*}
The Heisenberg group is thus generated by the vectorfields $E_1=X, E_2=Y, E_3=Z$, which fulfill the commutation relations
\begin{equation*}
[E_1, E_2]=E_3 \, , \, [E_1, E_3]=0 \, , \, [E_2, E_3]=0 \, ,
\end{equation*}
and generate the right multiplication. The linear span of $(E_1, E_2, E_3)$ forms a Lie algebra (crystalline structure) corresponding to a uniform distribution of screw dislocations in a three-dimensional crystal lattice. Therefore, we can associate the Heisenberg group, which is the group corresponding to the crystalline structure, with the material manifold $\mathcal N$.
If we take $\{E_1, E_2, E_3\}$ as a basis of $\mathcal V$, we have the following dual basis $\{\omega^1, \omega^2, \omega^3\}$ for $\mathcal V^*$
\begin{equation*}
\omega^1 =dy^1 \quad , \quad \omega^2=dy^2 \quad , \quad \omega^3=dy^3-y^1 dy^2 \, ,
\end{equation*}
and the corresponding metric
\begin{equation}\label{homspacemet}
\stackrel{\circ}{n}=(\omega^1)^2+(\omega^2)^2+(\omega^3)^2=(dy^1)^2+(dy^2)^2+(dy^3-y^1 dy^2)^2 \, ,
\end{equation}
which is a {\bf Bianchi type VII} metric. The manifold $\mathcal N$ endowed with this metric is a {\bf homogeneous space}.
Here, the dislocation density $\lambda$ turns out to be
\begin{equation*}
\lambda(E_1,E_2)(y)=E_3 \, , \, \lambda(E_1,E_3)(y)=\lambda(E_2,E_3)(y)=0 \, ,
\end{equation*}
and therefore
\begin{equation*}
\lambda =\left(\omega_1 \wedge \omega_2\right) E_3 = \left(dy^1\wedge dy^2\right)E_3 \, .
\end{equation*}
The integral of $\lambda$ over a surface $\Sigma$ in $\mathcal N$ is
\begin{equation*}
\int_{\Sigma}\lambda =\int_{\Sigma}\left(dy^1\wedge dy^2\right)E_3=A(\Pi\Sigma)E_3 \, ,
\end{equation*}
where $\Pi$ is the projection map of the line bundle of the homogeneous space (\ref{homspacemet}) over $\mathbb R^2$ with the standard metric on the base (the curvature of the bundle being $-dy^1\wedge dy^2$).
\section{The Thermodynamic State Space}\label{tdss}
Consider the space $S_2^+(\mathcal V)$ of inner products on the crystalline structure $\mathcal V$.
The {\bf thermodynamic state space} is defined as the product
\begin{equation*}
S_2^+(\mathcal V) \times \mathbb R^+
\end{equation*}
and its elements are $(\gamma,\sigma)$, where $\gamma \in S_2^+(\mathcal V)$ is the {\bf configuration} and $\sigma \in \mathbb R^+$ is the {\bf entropy per unit mass}.
Each $\gamma \in S_2^+(\mathcal V)$ defines a totally antisymmetric $n$-linear form $\omega_{\gamma}$ on the crystalline structure $\mathcal V$ by the condition that if $(E_1, \ldots, E_n)$ is a positive basis for $\mathcal V$, orthonormal with respect to $\gamma$, i.e.~$\gamma_{AB}:=\gamma(E_A,E_B)=\delta_{AB}$, then
\begin{equation*}
\omega_{\gamma}(E_1,\ldots,E_n)=1 .
\end{equation*}
It follows that there is a positive function $V$ on $S_2^+(\mathcal V)$ such that
\begin{equation*}
\omega_{\gamma}=V(\gamma)\omega \, .
\end{equation*}
The positive real number $V(\gamma)$ is the {\bf volume per unit mass} corresponding to the configuration $\gamma$.
The {\bf thermodynamic state function} $\kappa$ is a real-valued function on the thermodynamic state space $S_2^+(\mathcal V)\times \mathbb R^+$. The Lagrangian which determines the dynamics will be defined through this function.
The {\bf thermodynamic stress} corresponding to a thermodynamic state $(\gamma,\sigma)$ is the element $\pi(\gamma,\sigma)$ of $(S_2(\mathcal V))^*$ defined by
\begin{equation*}
\frac{\partial\left(\kappa(\gamma,\sigma)V(\gamma)\right)}{\partial\gamma}=-\frac 1 2 \pi(\gamma,\sigma)V(\gamma) \, .
\end{equation*}
The {\bf temperature} corresponding to a thermodynamic state $(\gamma,\sigma)$ is the real number $\vartheta(\gamma,\sigma)$ given by
\begin{equation*}
\vartheta(\gamma,\sigma)=\frac{\partial\left(\kappa(\gamma,\sigma)V(\gamma)\right)}{\partial\sigma} \, ,
\end{equation*}
with the requirement that $\vartheta(\gamma,\sigma)$ is positive and tends to zero as $\sigma$ tends to zero.
\section{The Dynamics}
In the {\bf general theory of relativity} the space-time manifold is an oriented $(n+1)$-dimensional differentiable manifold $\mathcal M$ endowed with a {\bf Lorentzian metric} $g$, that is a continuous assignment of a symmetric bilinear form $g_x$ of index $1$ in $T_x\mathcal M, \, \forall x \in \mathcal M$. The Lorentzian metric divides $T_x\mathcal M$ into three different subsets $I_x, N_x, S_x$, the set of {\bf timelike}, {\bf null} and {\bf spacelike} vectors at $x$ respectively, according to whether $g_x$ restricted to the corresponding subset is negative, zero or positive. The subset $N_x$ is a double cone called the null cone at $x$. The subset $I_x$ is the interior of this cone, an open set of two components, the future and past component. $S_x$ is the subset outside the null cone, a connected open set for $n>1$. A curve $\gamma$ is called {\bf causal} if its tangent vector belongs to $I_x \cup N_x, \, \forall x \in \gamma$, and it is called timelike if its tangent vector belongs to $I_x$, $\forall x \in \gamma$. We assume that $(\mathcal M, g)$ is time oriented, that is a continuous choice of future component $I_x^+$ of $I_x$ can and has been made $\forall x \in \mathcal M$. This choice determines the future component $N_x^+$ of $N_x$ at each $x \in \mathcal M$. A timelike or causal curve is then future or past directed, according to which component its tangent vector belongs to. A hypersurface $\mathcal H$ is called spacelike if at each $x \in \mathcal H$ the restriction of $g_x$ to $T_x\mathcal H$ is positive definite. A spacelike hypersurface in $\mathcal M$ is called a {\bf Cauchy hypersurface} if each causal curve in $\mathcal M$ intersects $\mathcal H$ exactly once. We assume that $(\mathcal M, g)$ possesses such a Cauchy hypersurface \cite{CBG}.
The motion of the material continuum is described by a mapping $f$ from the space-time manifold $\mathcal M$ into the material manifold $\mathcal N$,
\begin{equation} \label{defmapf}
f: \mathcal M \to \mathcal N \, .
\end{equation}
This mapping specifies which material particle is at a given event in space-time.
It is subject to the following conditions:
\begin{itemize}
\item[{\bf i)}] The mapping $f$, restricted to a Cauchy hypersurface $\mathcal H$, $\left. f \right|_{\mathcal H}$, is one to one.
\item[{\bf ii)}] The differential of the mapping $f$, $df(x)$, has a $1$-dimensional kernel contained in $I_x$, $\forall x \in \mathcal M$.
\end{itemize}
Then, for each $y \in f(\mathcal M) \subset \mathcal N$, $f^{-1}(y)$ is a timelike curve in $\mathcal M$.
The {\bf material velocity} $u$ is the future directed unit tangent vectorfield of the timelike curve $f^{-1}$:
\begin{equation*}
\mathrm{span}(u_x)=\ker\left(df(x)\right)=T_xf^{-1}(y) \, , \, f(x)=y \, , \, g(u_x,u_x)=-1 \, , \forall x \in \mathcal M \, .
\end{equation*}
The {\bf simultaneous space} at $x$ is the orthogonal complement of the linear span of $u_x$:
\begin{equation*}
\Sigma_x = \left(\textrm{span}(u_x)\right)^{\perp} \, .
\end{equation*}
Note that $g_{\Sigma_x}=\left.g_x \right|_{\Sigma_x}$ is positive definite. The restriction of the differential $df(x)$ to $\Sigma_x$ is an isomorphism of $\Sigma_x$ onto $T_y\mathcal N$, where $y=f(x)$.
\begin{itemize}
\item[{\bf iii)}] The isomorphism $\left.df(x)\right|_{\Sigma_x}$ is orientation preserving.
\end{itemize}
The {\bf equations of motion}, a second order system of partial differential equations for the mapping $f$, are derived from a {\bf Lagrangian} $L$, a function on $\mathcal M$ which is constructed from $f$. The action $\mathcal A$ in a domain $\mathcal D \subset \mathcal M$ is the integral
\begin{equation*}
\int_{\mathcal D} L \, d\mu_g \, ,
\end{equation*}
where $d\mu_g$ is the volume form of $(\mathcal M,g)$.
Any mapping $f$ fulfilling the three requirements stated above, defines an orientation preserving isomorphism $j_{f,x}$ of $\mathcal V$ onto $\Sigma_x$ by
\begin{equation}\label{isoj}
j_{f,x}=\left(\left.df(x)\right|_{\Sigma_x}\right)^{-1}\circ \epsilon_{f(x)} \, .
\end{equation}
Define
\begin{equation}
j_{f,x}^* g_x =\gamma \, ,
\end{equation}
$\gamma \in S_2^+(\mathcal V)$ and depends only on $f$ and $x$.
Note that
\begin{equation*}
\mu(x) = \frac{1}{V\left(j^*_{f,x}g_x\right)}
\end{equation*}
is the rest mass density at $x$.
To take into account thermal effects in the Lagrangian picture we invoke the adiabatic condition which states that the entropy per unit mass of any element of the material remains unchanged. This allows us to consider the entropy per unit mass $\sigma$ as a given function on the material manifold $\mathcal N$.
The {\bf Lagrangian} function $L$ on $\mathcal M$ is defined by:
\begin{equation}\label{lagrangian}
L(x)=\kappa\left(j^*_{f,x}g_x,\sigma(f(x))\right) \, , \, \forall x \in \mathcal M \, ,
\end{equation}
where $\kappa$ is the thermodynamic state function on $S_2^+(\mathcal V) \times \mathbb R^+$. Note that $L(x)$ depends on $g$ only through $g_x$ and not on derivatives of $g$.
The {\bf energy-momentum-stress tensor} $T_x$ at $x$ is an element of the dual space $\left(S_2(T_x\mathcal M)\right)^*$ defined by
\begin{equation}\label{partiall}
\frac{\partial\left(L(x)d\mu_g(x)\right)}{\partial g_x}=-\frac 1 2 T_x d\mu_g(x) \, .
\end{equation}
From \cite{C2} we have the following
\begin{proposition} \label{ems}
We can write the energy-momentum-stress tensor at $x$ as follows:
\begin{equation} \label{enmomstrtens}
T_x=\rho(x) u_x\otimes u_x +S_x
\end{equation}
with the mass-energy density $\rho$ (since we are in the relativistic framework this includes the rest mass energy) given by
\begin{equation} \label{rhokappav}
\rho(x)=\kappa(j^*_{f,x}g_x,\sigma(f(x))) \, .
\end{equation}
The stress tensor is given by
\begin{equation*}
S_x(\dot g_x)=\pi(j^*_{f,x}g_x,\sigma(f(x)))(j^*_{f,x}\dot g_x) \, .
\end{equation*}
Since $j^*_{f,x}\dot g_x=j^*_{f,x}\dot g_{\Sigma_x}$, $S_x$ can be viewed as an element of $\left(S_2(\Sigma_x)\right)^*$.
\end{proposition}
\begin{remark}
Hence we see from Proposition \ref{ems} that in the case of pure continuum mechanics (in the absence of electromagnetic fields) the energy per unit mass is $e=\kappa V$.
\end{remark}
Define the {\bf principal pressures} $p_i$, $i=1,\ldots,n$ as the eigenvalues of $S_{\flat \flat}$ relative to $\left. g\right|_{\Sigma}$, where the subscript $\flat \flat$ means lowering of the indices with respect to $g$. Note that the principal pressures $p_i : i=1,\ldots,n$ can equivalently be described as the eigenvalues of $\pi_{\flat \flat}$ relative to $\gamma$.
The positivity condition on $T_x$ requires that
\begin{eqnarray}
H_x^+ & \to & T_x \mathcal M \nonumber \\
v^{\mu} & \mapsto & -T_{\nu}^{\mu} v^{\nu} \label{poscond}
\end{eqnarray}
maps into $\overline{I_x^+}$. This is equivalent to the condition that $|p_i|\leq \rho \quad : \, i=1,\ldots,n$. Here, in the case of crystalline solids, we assume the stronger condition that the range of (\ref{poscond}) lies in $I_x^+$ which is in turn equivalent to $|p_i| < \rho \quad : \, i=1,\ldots,n$.
\subsection{Variation of the Lagrangian}
Let $v=df(x) \in U_{(x,y)} \subset \mathcal L(T_x\mathcal M, T_y \mathcal N)$, $y=f(x)$, where $U_{(x,y)}$ is the open subset of the linear space $\mathcal L (T_x \mathcal M , T_y \mathcal N)$ consisting of those $v \in \mathcal L (T_x \mathcal M , T_y \mathcal N)$ which verify the two conditions
\begin{itemize}
\item[1)] $\textrm{ker}\,v$ is a time like line in $T_x \mathcal M$,
\item[2)] with $\Sigma_x$ the $g_x$-orthogonal complement of $\textrm{ker}\,v$ in $T_x \mathcal M$, the isomorphism
\begin{equation*}
\left. v \right|_{\Sigma_x} : \Sigma_x \to T_y \mathcal N
\end{equation*}
is orientation-preserving.
\end{itemize}
These above conditions 1) and 2) correspond to the conditions {\bf ii)} and {\bf iii)}, respectively. Note that {\bf i)} is not a local condition so it does not reduce to a condition on $v$.
Since the isomorphism
\begin{equation*}
j_{f,x}= \left(\left. v\right|_{\Sigma_x}\right)^{-1} \circ \epsilon_y \, ,
\end{equation*}
as defined in (\ref{isoj}), depends only on $v$, we may write
\begin{equation*}
i(v)=j_{f,x} \, ,
\end{equation*}
so the configuration $\gamma_{f,x}$ corresponding to the isomorphism $v \in U_{(x,y)}$ is given by the positive quadratic form on $\mathcal V$
\begin{equation} \label{defgamma}
\gamma (v) = i^*(v) \left.g\right|_{\Sigma_x} \, .
\end{equation}
Now $v \mapsto \gamma(v)$ is a mapping of
\begin{equation} \label{bundleB}
\mathcal B := \bigcup_{(x,y)\in \mathcal M \times \mathcal N} U_{(x,y)} \, ,
\end{equation}
into $S_2^+(\mathcal V)$. The bundle $\mathcal B$, a bundle over $\mathcal M \times \mathcal N$, is the bundle over which the Lagrangian is defined.
The mapping $v \mapsto \gamma(v)$ is described by the functions $\gamma_{AB}(v)$ on $\mathcal B$, where
\begin{equation} \label{gammaofv}
\gamma_{AB}(v)=\gamma(v)\left(E_A,E_B\right) \, ,
\end{equation}
where $E_A: A=1,\ldots,n$ is a basis of $\mathcal V$ such that $\omega(E_1,\ldots,E_n)=1$.
By the properties of $v$ we have a positive basis $(X_A(x) \, : \, A=1,\ldots,n)$ of $\Sigma_x$ defined by:
\begin{equation*}
v \cdot X_A(x) =E_A(y) \, : \, A=1,\ldots,n \, ,
\end{equation*}
where $E_A(y)=\epsilon_y(E_A) \in T_y\mathcal N (A=1,\ldots,n)$, and thus from (\ref{defgamma}) and (\ref{gammaofv})
\begin{equation*}
\gamma_{AB}=\left.g\right|_{\Sigma_x}(X_A,X_B) \, .
\end{equation*}
Note that $(u,X_1,\ldots, X_n)$ is a frame field for $\mathcal M$.
Let $\dot v \in \mathcal L(T_x \mathcal M, T_y \mathcal N)$ be a variation of $v \in U_{(x,y)}$. To describe $\dot v$ we must give $\dot v \cdot u \in T_y\mathcal N$ and $\dot v \cdot X_A \in T_y \mathcal N$,
\begin{eqnarray*}
\dot v \cdot u(x) & = & \dot v_0^A E_A(y) \, ,\\
\dot v \cdot X_A(x) & = & \dot v_A^B E_B(y) \, .
\end{eqnarray*}
This is because $(E_A(y): A=1,\ldots,n)$ is a basis for $T_y\mathcal N$. So the $(\dot v_0^A : A=1, \ldots,n ; \dot v_A^B : A,B=1, \ldots,n)$ can be thought of as the the components of $\dot v$.
In view of (\ref{lagrangian}) and (\ref{rhokappav}) the Lagrangian function $L$ is $L(v)=\rho\left(\gamma(v),\sigma(y)\right)$, where $\rho$ is the relativistic energy-density which includes the rest mass contribution, a function of a thermodynamic state $\left(\gamma(v),\sigma(y)\right) \in S_2^+(\mathcal V)\times \mathbb R^+$. $L$ is a function on the bundle $\mathcal B$ (\ref{bundleB}).
The Lagrangian form, a top degree form on $\mathcal M$, is $L d\mu_g$.
The Lagrangian $L$ depends on $v$ through the configuration $\gamma$, $L(\gamma_{AB})$, and the first variation reads
\begin{equation*}
\dot L=\frac{\partial \rho}{\partial \gamma_{AB}}\dot \gamma_{AB} \, ,
\end{equation*}
where the first variation of $\gamma_{AB}$ is given by
\begin{equation*} \label{firstvargamma}
\dot \gamma_{AB}=-\dot v_A^C\gamma_{BC}-\dot v_B^C \gamma_{AC} \,
\end{equation*}
(see \cite{C1}).
To formulate the hyperbolicity condition (see below), we need to consider the second variation of $L$ with respect to $v$,
\begin{equation} \label{defofh*}
\ddot L=\frac{\partial^2 L}{\partial v^2}\cdot(\dot v,\dot v)=h(\dot v,\dot v) \, ,
\end{equation}
where, in general,
\begin{equation} \label{h*indetail}
h(\dot v,\dot v)=h^{00}_{AB} \dot v_0^A \dot v_0^B+ 2 h^{C0}_{AB} \dot v_C^A \dot v_0^B+ h^{CD}_{AB} \dot v_C^A \dot v_D^B \, .
\end{equation}
For $L=L(\gamma_{ab})$, using the formula for the second variation of $\gamma_{AB}$,
\begin{equation*} \label{secondvargamma}
\ddot \gamma_{AB}=2\left(\gamma_{AC}\gamma_{BD}\dot v_0^C\dot v_0^D+\gamma_{CD}\dot v_A^C\dot v_B^D+\gamma_{AC}\dot v_B^D\dot v_D^C+\gamma_{BC}\dot v_A^D\dot v_D^C \right)
\end{equation*}
(see \cite{C1}), we obtain
\begin{eqnarray*}
\ddot L & = & \frac{\partial \rho}{\partial \gamma_{AB}}\ddot \gamma_{AB}+\frac{\partial^2 \rho}{\partial \gamma_{AB} \partial \gamma_{CD}}\dot \gamma_{AB}\dot \gamma_{CD} \\
& = & 2\frac{\partial \rho}{\partial \gamma_{AB}}\left(\gamma_{AC}\gamma_{BD}\dot v_0^C\dot v_0^D+\gamma_{AC}\dot v_B^D\dot v_D^C+\gamma_{BC}\dot v_A^D\dot v_D^C+\gamma_{CD}\dot v_A^C\dot v_B^D \right) \\
& & + \frac{\partial^2 \rho}{\partial \gamma_{AB} \partial \gamma_{CD}}\big( \dot v_A^E\gamma_{EB}+\dot v_B^E \gamma_{EA} \big)\left( \dot v_C^F\gamma_{FD}+\dot v_D^F \gamma_{FC} \right) \\
& = & 2 \frac{\partial \rho}{\partial \gamma_{CD}} \gamma_{AC}\gamma_{BD}\dot v_0^A\dot v_0^B + 2 \frac{\partial \rho}{\partial \gamma_{CD}} \gamma_{AB}\dot v_C^A\dot v_D^B \\
& & + 2 \left(\frac{\partial \rho}{\partial \gamma_{CE}} \gamma_{BE} \delta_A^D + \frac{\partial \rho}{\partial \gamma_{DE}} \gamma_{AE}\delta_B^C\right) \dot v_C^A\dot v_D^B\\
& & + 4 \frac{\partial^2 \rho}{\partial \gamma_{CE}\partial \gamma_{DF}}\gamma_{AE}\gamma_{BF}\dot v_C^A\dot v_D^B \, .
\end{eqnarray*}
Comparing coefficients with (\ref{h*indetail}) using (\ref{defofh*}), we see that:
\begin{eqnarray*}
h^{00}_{AB} & = & 2 \frac{\partial \rho}{\partial \gamma_{CD}} \gamma_{AC}\gamma_{BD} \, ,\qquad h^{C0}_{AB}=0 \, , \\
h^{CD}_{AB} & = & 4 \frac{\partial^2 \rho}{\partial \gamma_{CE}\partial \gamma_{DF}}\gamma_{AE}\gamma_{BF} \\
& & + 2\left(\frac{\partial \rho}{\partial \gamma_{CD}} \gamma_{AB}+ \frac{\partial \rho}{\partial \gamma_{CE}} \gamma_{BE} \delta_A^D + \frac{\partial \rho}{\partial \gamma_{DE}} \gamma_{AE}\delta_B^C \right) \, ,
\end{eqnarray*}
where the last expression will be made use of in the derivation of the Legendre-Hadamard conditions in Part \ref{static}.
\subsection{Hyperbolicity and Characteristic Speeds}
Set
\begin{equation*}
G_{AB}= -h_{AB}^{00} \quad , \quad M_{AB}^{CD}=h_{AB}^{CD} \, .
\end{equation*}
Then
\begin{equation*}
h(\dot v, \dot v)= -G_{AB}\dot v_0^A \dot v_0^B + M_{AB}^{CD} \dot v_C^A \dot v_D^B \, .
\end{equation*}
The hyperbolicity condition in general is that there is a pair $(T,\theta) \in T_x \mathcal M \times T_x^* \mathcal M$ with $\theta \cdot T > 0$ such that $h$ is \emph{negative-definite} on
\begin{equation} \label{hnegdef}
\left\{ \dot v : \dot v = \theta \otimes Y , Y \in T_y \mathcal N \right\}
\end{equation}
and \emph{positive-definite} on
\begin{equation} \label{hposdef}
\left\{ \dot v : \dot v = \kappa \otimes Y , \kappa \cdot T=0, Y \in T_y \mathcal N \right\}
\end{equation}
(see \cite{C1}).
We stipulate here that the above conditions hold with $(T,\theta)$ defined by the rest frame of the material at $x$, i.e.~$T=u_x$, $\Sigma_x=\textrm{ker}\,\theta$, $\theta\cdot T=1$.
For $\dot v$ in (\ref{hnegdef}) we have:
\begin{equation*}
\dot v \cdot u = Y \quad , \quad \dot v \cdot X_A = 0 \, ,
\end{equation*}
i.e.
\begin{equation*}
\dot v_0^A = Y^A \quad , \quad \dot v_A^B = 0 \, ,
\end{equation*}
and the condition that $h$ is negative definite on (\ref{hnegdef}) is the condition that
\begin{equation} \label{firstcondG}
G_{AB} Y^A Y^B > 0 \quad : \, \forall Y \neq 0 \, ,
\end{equation}
i.e.~that $G$ is positive-definite.
Since
\begin{equation*}
G_{AB}=\pi_{AB} + \rho \gamma_{AB}
\end{equation*}
(we are raising and lowering indices with respect to $\gamma_{AB}$) and the principal pressures $p_1,\ldots, p_n$ are the eigenvalues of $\pi_{AB}$ with respect to $\gamma_{AB}$, this condition is:
\begin{equation*}
\min_i p_i > -\rho \, ,
\end{equation*}
which of course follows from $\max_i |p_i| < \rho$.
For $\dot v$ in (\ref{hposdef}) we have:
\begin{equation*}
\dot v \cdot u = 0 \quad , \quad \dot v \cdot X_A = \kappa_A Y \, ,
\end{equation*}
where $\kappa_A= \kappa \cdot X_A$, i.e.~
\begin{equation*}
\dot v_0^A = 0 \quad , \quad \dot v_A^B = \kappa_A Y^B \, ,
\end{equation*}
and the condition that $h$ is positive definite on (\ref{hposdef}) is the condition that
\begin{equation} \label{hposdeftilde}
M_{AB}^{CD} \kappa_C \kappa_D Y^A Y^B >0 \quad : \, \forall \kappa \neq 0, \forall Y \neq 0 \, .
\end{equation}
Set
\begin{equation*}
H_{AB}(\kappa)=M_{AB}^{CD} \kappa_C \kappa_D \, .
\end{equation*}
Then condition (\ref{hposdeftilde}) is that $H_{AB}(\kappa)$ is positive-definite for all $\kappa \neq 0$.
If the first condition (\ref{firstcondG}) is satisfied, the second condition (\ref{hposdeftilde}) becomes the condition that for $\kappa \neq 0$ the eigenvalues $\lambda_1(\kappa),\ldots, \lambda_n(\kappa)$ of $H_{AB}(\kappa)$ with respect to $G_{AB}$ are all positive. Note that $H_{AB}(\kappa)$ is homogeneous of degree $2$ in $\kappa$, hence so are the $\lambda_1(\kappa), \ldots , \lambda_n(\kappa)$.
Now the characteristic matrix is
\begin{equation*}
\chi_{AB}(\xi)=h_{AB}^{\mu\nu} \xi_{\mu}\xi_{\nu} \, .
\end{equation*}
We use the basis $(u,X_1,\ldots,X_n)$ for $T_x\mathcal M$. We denote by $\omega$ the frequency and by $\kappa_A$, $A=1,\ldots,n$, the wave number components, i.e.~
\begin{equation*}
\omega := \xi_0 := \xi \cdot u \quad , \quad \kappa_A := \xi_A := \xi \cdot X_A \, .
\end{equation*}
Then
\begin{equation*}
\chi_{AB}=-G_{AB} \omega^2 + H_{AB}(\kappa) \, ,
\end{equation*}
so
\begin{equation*}
\det \chi = \det G \cdot \prod_{i=1}^n \left( \lambda_i(\kappa)-\omega^2\right) \, .
\end{equation*}
We see that the second condition is equivalent to the $2n$ roots $\pm \sqrt{\lambda_i(\kappa)}$, $i:1,\ldots,n$, of the characteristic polynomial $\det \chi$ as a polynomial in $\omega$ being real. The characteristic speeds are
\begin{equation} \label{defetai}
\eta_i =\frac{\sqrt{\lambda_i(\kappa)}}{|\kappa|} \quad , \quad \textrm{where} \,\, |\kappa|=\sqrt{\left(\gamma^{-1}\right)^{AB}\kappa_A \kappa_B} \, .
\end{equation}
Note that the $\eta_i$ are homogeneous of degree $0$ in $\kappa$.
The causality condition is that the inner characteristic core in $T_x^* \mathcal M$ contains the null cone of $g$ in $T_x^* \mathcal M$. This reads (in units $c=1$)
\begin{equation*}
\eta_i < 1 \quad , \, \forall i=1,\ldots,n.
\end{equation*}
\section{Properties of the Energy per unit Mass}
{\postulate \label{pmine} We stipulate that the energy per unit mass $e=\rho V$ has a strict minimum at a certain inner product $\stackrel{\circ}{\gamma}$.}
{\remark The above postulate has an intuitive physical interpretation. Note that the set of inner products $\gamma$ is an open positive cone in the linear space of quadratic forms. For large $\gamma$ (large expansion) and also for $\gamma$ near the boundary (large compression) the energy $e$ is physically expected to blow up, so we can restrict ourselves to a compact set of inner products, where $e$ necessarily attains a minimum.}
\vspace{2mm}
We choose $(E_1,\ldots,E_n)$ to be an orthonormal basis relative to $\stackrel{\circ}{\gamma}$, so $\stackrel{\circ}{\gamma}_{AB}=\delta_{AB}$. This is compatible with the previous condition $\omega(E_1,\ldots,E_n)=1$ for a suitable volume form $\omega$ on $\mathcal V$.
This choice of $\omega$ corresponds to a choice of unit of mass so that the mass density associated to the configuration $\stackrel{\circ}{\gamma}$ is equal to $1$.
$\stackrel{\circ}{\gamma}$ defines a metric $\stackrel{\circ}{n}$ on $\mathcal N$ by
\begin{equation*}
\stackrel{\circ}{n}=\sum\limits_{A,B=1}^n\stackrel{\circ}{\gamma}_{AB}\omega^A \otimes \omega^B = \sum\limits_{A=1}^n \omega^A \otimes \omega^A \, ,
\end{equation*}
where $(\omega^1,\ldots,\omega^n)$ is the dual basis to $(E_1,\ldots,E_n)$. If $\mathcal V$ is a Lie algebra, so that $\mathcal N$ is a Lie group, then $\stackrel{\circ}{n}$ is a left-invariant metric, i.e.~invariant under the actions of $\mathcal N$ on itself by left multiplications by elements of $\mathcal N$.
Thus $(\mathcal N, \stackrel{\circ}{n})$ is a homogeneous Riemannian manifold (which in general is not isotropic).
The Riemannian manifold $(\mathcal N, \stackrel{\circ}{n})$ has curvature except when $\mathcal V$ is Abelian, so there are no dislocations.
Note that $d\mu_{\stackrel{\circ}{n}}=\sqrt{\det \stackrel{\circ}{n}} \, d^n y = d\mu_{\omega}$, therefore we have for any domain $\Omega$ in $\mathcal N$
\begin{equation*}
M(\Omega)=\int\limits_{\Omega} d\mu_{\omega} = \int\limits_{\Omega} d\mu_{\stackrel{\circ}{n}} \, ,
\end{equation*}
that is, in our choice of units the mass of $\Omega$ is equal to its volume with respect to $\stackrel{\circ}{n}$.
\subsection{The Isotropic Case} \label{isotropiccase}
Let us define the orthogonal group corresponding to $\stackrel{\circ}{\gamma}$,
\begin{equation*}
O_{\stackrel{\circ}{\gamma}}=\left\{ O \in \mathcal L(\mathcal V, \mathcal V) : \, \stackrel{\circ}{\gamma}(OX,OY)= \, \stackrel{\circ}{\gamma}(X,Y) \, , \, \forall X,Y \in \mathcal V \right\} \, .
\end{equation*}
$O_{\stackrel{\circ}{\gamma}}$ acts on $S_2^+(\mathcal V)$ in the following way: $\gamma \mapsto O \gamma $, where
\begin{equation*}
\left( O\gamma\right)(X,Y) = \gamma(OX,OY) \, .
\end{equation*}
Since $\stackrel{\circ}{\gamma} \, \mapsto O \! \stackrel{\circ}{\gamma} \, =\stackrel{\circ}{\gamma}$, the symmetric bilinear form $\stackrel{\circ}{\gamma}$ is a fixed point of the action of $O_{\stackrel{\circ}{\gamma}}$ on $S_2^+(\mathcal V)$.
If the energy density $e(\gamma)$ is invariant under $O_{\stackrel{\circ}{\gamma}}$ we are in the case of {\bf isotropic elasticity}.
Then $e(\gamma)$ depends only on the eigenvalues $\lambda_1,\ldots ,\lambda_n$ of $\gamma$ relative to $\stackrel{\circ}{\gamma}$,
\begin{equation*}
e(\gamma)=e(\lambda_1, \ldots , \lambda_n) \, ,
\end{equation*}
where $e(\lambda_1, \ldots , \lambda_n)$ is totally symmetric in its arguments. This is the simplest case of an energy density.
Consider
\begin{equation*}
q=f^*\stackrel{\circ}{n} \, .
\end{equation*}
This is a $2$-covariant symmetric tensorfield on $\mathcal M$, i.e.~at each $x \in \mathcal M$ $q_x$ is a quadratic form in $T_x \mathcal M$. The vector $u_x$ belongs to the null space of $q_x$ and the restriction $\left. q_x \right|_{\Sigma_x}$ is positive-definite.
Let $\Lambda_1,\ldots,\Lambda_n$ be the eigenvalues of $\left.q_x\right|_{\Sigma_x}$ relative to $\left.g_x\right|_{\Sigma_x}$. We then have
\begin{proposition}
The eigenvalues $\Lambda_1,\ldots,\Lambda_n$ are the inverses of the eigenvalues $\lambda_1,\ldots,\lambda_n$ of $\gamma=j_{f,x}^* g_x$ relative to $\stackrel{\circ}{\gamma}$. In particular,
\begin{equation*}
\sqrt{\Lambda_1 \cdot \ldots \cdot \Lambda_n}=\frac 1{\sqrt{\lambda_1 \cdot \ldots \cdot \lambda_n}} = \frac 1 v = N \, .
\end{equation*}
Therefore, in the variational principle, the crystalline structure $\mathcal V$ on $\mathcal N$ is eliminated in favor of the Riemannian metric $\stackrel{\circ}{n}$.
\end{proposition}
\section{Equivalences}
\subsection{Equivalence of Crystalline Structures}
{\definition Two crystalline structures $\mathcal V$ and $\mathcal V'$ on $\mathcal N$ are said to be equivalent if there is a diffeomorphism $\psi$ of $\mathcal N$ onto itself such that $\psi_*$, the push-forward of $\psi$, induces an isomorphism of $\mathcal V$ onto $\mathcal V'$.}
Let $\nu$ be the canonical $1$-form (\ref{defnu}) associated to $\mathcal V$ and $\nu'$ the one associated to $\mathcal V'$. We have
\begin{equation} \label{equivcs}
\begin{array}{rcl}
\nu(Y_y) & = & Y \in \mathcal V \quad : \, Y_y \in T_y\mathcal N \, ,\\
\nu'(d\psi \cdot Y_y) & = & \psi_* Y \in \mathcal V' \quad : \, d\psi \cdot Y_y \in T_{\psi(y)}\mathcal N \, .
\end{array}
\end{equation}
We may take the above as the definition of pullback for $\mathcal V$-valued $1$-forms:
\begin{equation*}
\nu = \psi^* \nu' \, .
\end{equation*}
It then follows that:
\begin{equation} \label{equivlambda}
\lambda=\psi^* \lambda' \, ,
\end{equation}
with the natural extension of the above definition of pullback to $\mathcal V$-valued $2$-forms, i.e.~
\begin{equation*}
\lambda'\left(d\psi \cdot X_y, d\psi \cdot Y_y\right) = \psi_*(\lambda(X_y,Y_y)) \quad : \, \forall X,Y \in \mathcal V, \forall y \in \mathcal N \, .
\end{equation*}
In fact, we have for all $X,Y \in \mathcal V$ and $y \in \mathcal N$:
\begin{eqnarray*}
\Lambda'(\psi_*X,\psi_*Y)\left(\psi(y)\right) & = & \nu'\left([\psi_* X,\psi_* Y](\psi(y)) \right) \\
& = & \nu'\left(\psi_* [X,Y](\psi(y))\right) \\
& = & \nu' \left( d\psi \cdot [X,Y](y)\right) \\
& = & \psi_*\left(\nu([X,Y](y))\right) \\
& = & \psi_*\left(\Lambda(X,Y)(y)\right) \, ,
\end{eqnarray*}
where we have made use of the fact that $[\psi_* X, \psi_* Y]=\psi_* [X,Y]$.
\subsection{Equivalence of Mechanical Properties}
We investigate the question of the equivalence of the mechanical properties of a solid. A certain solid phase of a certain substance is described by an equation of state.
Let $A:\mathcal V \to \mathcal V'$ be a linear isomorphism. The group of all such $A$ is homomorphic to $GL_n(\mathbb R)$. Then $A^*: S_2^+(\mathcal V') \to S_2^+(\mathcal V)$ is the induced isomorphism defined by $\gamma = A^* \gamma'$, where
\begin{equation*}
\gamma(X,Y)=\gamma'(A X, A Y) \quad , \, \forall X,Y \in \mathcal V \, .
\end{equation*}
The corresponding energy functions on $S_2^+(\mathcal V) $ and $S_2^+(\mathcal V')$ are denoted by $e$ and $e'$, respectively.
{\definition \label{equivnrg} Two energy functions $e$ on $\mathcal V$ and $e'$ on $\mathcal V$ are equivalent if there exists a linear isomorphism $A: \mathcal V \to \mathcal V'$ such that $e'(\gamma')=e(\gamma)$, where $ \gamma = A^*\gamma'$, for all $\gamma' \in S_2^+(\mathcal V')$.}
We illustrate the above definition of mechanical equivalence by giving an example in the isotropic case. Let $\gamma \in S_2^+(\mathcal V)$ be given, and define $M\in \mathcal L(\mathcal V, \mathcal V)$ by
\begin{equation*}
\stackrel{\circ}{\gamma}(M X, Y) = \gamma(X,Y) \quad , \, \forall X,Y \in \mathcal V \, .
\end{equation*}
Similarly, given an isomorphism $A$ as above (i.e.~$\gamma=A^*\gamma'$), we define $M' \in \mathcal L(\mathcal V', \mathcal V')$ from the corresponding $\gamma' \in S_2^+(\mathcal V')$ as
\begin{equation*}
\stackrel{\circ}{\gamma'}(M' X', Y') = \gamma'(X',Y') \quad , \, \forall X',Y' \in \mathcal V' \, .
\end{equation*}
In the above, $\stackrel{\circ}{\gamma}=A^*\stackrel{\circ}{\gamma'}$.
Then $M'=AMA^{-1}$.
For, given any $X', Y' \in \mathcal V'$ let $X, Y \in \mathcal V$ be $X=A^{-1}X'$, $Y=A^{-1}Y'$. Then
\begin{eqnarray*}
\stackrel{\circ}{\gamma'}(M' X', Y') & = & \gamma'(X',Y')=\gamma(X,Y)=\stackrel{\circ}{\gamma}(M X, Y) \\
& = & \stackrel{\circ}{\gamma'}(AM X, A Y)=\stackrel{\circ}{\gamma'}(AMA^{-1} X', Y') \, .
\end{eqnarray*}
Therefore, the eigenvalues $\lambda_1',\ldots,\lambda_n'$ of $M'$ coincide with the eigenvalues $\lambda_1,\ldots,\lambda_n$ of $M$. In the isotropic case, $e'(\gamma')=e(\gamma)$ means
\begin{equation*}
e'(\lambda_1',\ldots,\lambda_n')=e(\lambda_1,\ldots,\lambda_n) \, ,
\end{equation*}
where both sides of the above equation are symmetric functions, so the energy functions coincide, i.e.~$e'=e$.
The equivalence of the energy functions, however, does not fully capture the equivalence of having two solids of the same substance in the same phase. What is required in addition is to have the same equilibrium mass density of infinitesimal portions.
In fact, it is the triplet $(\mathcal V, \omega, e)$ which defines a solid with its mechanical properties. Two solids of the same substance in the same phase, but with possibly different dislocation structures, to the extent that they can be described by the same manifold $\mathcal N$ (which is true if they are diffeomorphic) are defined by the triplets $(\mathcal V, \omega, e)$ and $(\mathcal V', \omega', e')$, respectively. Additionally, there is an isomorphism $A: \mathcal V \to \mathcal V'$, such that
\begin{equation*}
\omega = A^* \omega' \, ,
\end{equation*}
i.e.~$\omega(X_1, \ldots , X_n)=\omega'(AX_1, \ldots, AX_n) \, \, : \, \forall X_1, \ldots, X_n \in \mathcal V$ as well as
\begin{equation}
e'(\gamma')=e(\gamma) \quad , \quad \gamma = A^* \gamma' \quad : \, \forall \gamma' \in S_2^+(\mathcal V') \,
\end{equation}
in accordance with Definition \ref{equivnrg}. Thus if $(E_1, \ldots, E_n)$ is a positive basis for $\mathcal V$ which is orthonormal relative to $\stackrel{\circ}{\gamma}$, then
\begin{equation*}
\omega_{\stackrel{\circ}{\gamma}}(E_1,\ldots,E_n)=1 \, ,
\end{equation*}
while $\omega(E_1,\ldots,E_n)=\mu_0$. On the other hand, with $E_i'=A E_i \, : i=1, \ldots , n$, $(E_1', \ldots , E_n')$ is orthonormal relative to $\stackrel{\circ}{\gamma'}$, thus
\begin{equation*}
\omega_{\stackrel{\circ}{\gamma'}}(E_1',\ldots,E_n')=1 \, ,
\end{equation*}
while $\omega'(E_1',\ldots,E_n')=\omega(E_1,\ldots,E_n)=\mu_0$. Consequently, the mass density corresponding to $\stackrel{\circ}{\gamma'}$ relative to the triplet $(\mathcal V', \omega', e')$ is the same as the mass density corresponding to $\stackrel{\circ}{\gamma}$ relative to the triplet $(\mathcal V, \omega, e)$.
\begin{remark}
If we have two triplets $(\mathcal V, \omega, e)$ and $(\mathcal V', \omega', e')$ which correspond to the same substance in the same phase and, additionally, the isomorphism $A: \mathcal V \to \mathcal V'$ is of the form $A=\psi_*$, where $\psi$ is a diffeomorphism of $\mathcal N$ onto itself, then the dislocation structures are also equivalent. In this case, if $f: \mathcal M \to \mathcal N$ is a dynamical solution of the problem corresponding to $(\mathcal V, \omega, e)$, then $f':=\psi \circ f : \mathcal M \to \mathcal N$ is a dynamical solution of the problem corresponding to $(\mathcal V', \omega', e')$.
\end{remark}
\subsection{Similarity} \label{similarity}
Let now $\mathcal V$ be fixed, so that $S_2^+(\mathcal V)$ is also fixed. Two different energy functions $e$ and $e'$ may be related as follows. There is a constant $a>0$ and an isomorphism $\mathcal V \to \mathcal V$, defined by
\begin{equation*}
X \mapsto a X \quad : \, \forall X \in \mathcal V \, .
\end{equation*}
This defines an isomorphism $S_2^+(\mathcal V) \to S_2^+(\mathcal V)$ by $\gamma \mapsto \gamma'$, where
\begin{equation*}
\gamma'(X,Y)=\gamma(aX,aY)=a^2 \gamma(X,Y) \quad : \, \forall X,Y \in \mathcal V \, ,
\end{equation*}
i.e.~$\gamma \mapsto \gamma' = a^2 \gamma$.
According to the above discussion of equivalence of mechanical properties, we must have $e(\gamma)=e'(\gamma')$, $\forall \gamma \in S_2^+(\mathcal V)$, \emph{and}, additionally $\omega'=a^n\omega$ for the same material in the same phase, for
\begin{equation*}
\omega'(X_1,\ldots,X_n)=\omega(aX_1,\ldots,X_n)=a^n\omega(X_1,\ldots,X_n) \, .
\end{equation*}
Then the two theories (primed and unprimed) represent the same material in the same phase.
Moreover, since $\mathcal V$ is identical in the two theories, the dislocation structures may seem identical. However, for a given domain $\Omega \subset \subset \mathcal N$, which has the same crystalline structure $\left.\mathcal V\right|_{\Omega}$ (the restriction to $\Omega$ of the vectorfields in $\mathcal V$), the primed theory actually assigns a \emph{physical dislocation density} $a^{-2}$ times that of the unprimed theory. This is due to the fact that $\omega'=a^n\omega$ and the observation that in any dimension, the elementary dislocations are co-dimension two objects. Thus their density refers to the two-dimensional measure of a cross-section.
\section{The Eulerian Picture}
As a prelude to formulating the Eulerian picture, we view the crystalline structure $\mathcal V$ as an abstract $n$-dimensional real vector space, as in Remark \ref{mfo}. We then view the canonical form $\nu$ as a $1$-form on $\mathcal N$ with values in $\mathcal V$ such that $\left. \nu \right|_{T_y\mathcal N}$ is an isomorphism from $T_y \mathcal N$ onto $\mathcal V$ for all $y \in \mathcal N$. Thus, given any $v \in \mathcal V$, we have a tangent vector
\begin{equation*}
\left(\left. \nu \right|_{T_y\mathcal N}\right)^{-1} \cdot v = Y_y(v) \in T_y\mathcal N \quad : \, \forall y \in \mathcal N \, ,
\end{equation*}
that is a vectorfield $Y(v)$ on $\mathcal N$. Then the crystalline structure in the original sense is the space
\begin{equation*}
\{ Y(v) : v \in \mathcal V \}
\end{equation*}
of vectorfields on $\mathcal N$.
To obtain the Eulerian description we must eliminate the material manifold $\mathcal N$. A fundamental variable is the material velocity $u$, a future-directed timelike unit vectorfield on $\mathcal M$. This defines the distribution of local simultaneous spaces
\begin{equation} \label{lss}
\Sigma=\{\Sigma_x : x \in \mathcal M \} \, ,
\end{equation}
where $\Sigma_x$ is the orthogonal complement of $u_x$ in $T_x \mathcal M$.
We then need another entity defined on $\mathcal M$ to play the role of the canonical form $\nu$. Consider
\begin{equation*}
\xi = f^* \nu \, .
\end{equation*}
This is a $1$-form on $\mathcal M$ with values in $\mathcal V$, viewed as an abstract $n$-dimensional vector space.
The $1$-form $\xi$ has the following properties:
\begin{itemize}
\item[{\bf 1)}]
$\xi \cdot u = 0$,
\item[{\bf 2)}]
for each $x \in \mathcal M$, $\left. \xi \right|_{\Sigma_x}$ is an isomorphism from $\Sigma_x$ onto $\mathcal V$, {\it and}
\item[{\bf 3)}]
$\mathcal L_u \xi =0$.
\end{itemize}
We thus introduce ab initio a $1$-form $\xi$ on $\mathcal M$ with values in $\mathcal V$ possessing the above three properties, as another fundamental Eulerian variable besides $u$.
By {\bf 2)}, given any $v \in \mathcal V$, we have a tangent vector
\begin{equation*}
\left(\left.\xi\right|_{\Sigma_x}\right)^{-1} \cdot v = X_x(v) \in \Sigma_x
\end{equation*}
at each $x \in \mathcal M$, that is we obtain a vectorfield $X(v)$ whose value at each point belongs to the distribution (\ref{lss}), i.e.~it is orthogonal to $u$.
A mapping $\mathcal M \to S_2^+(\mathcal V)$, $x \mapsto \gamma_x$, is then defined by:
\begin{equation} \label{defgammax}
\gamma_x(v_1,v_2)=g_x(X_x(v_1),X_x(v_2))=\left.g\right|_{\Sigma_x}(X_x(v_1),X_x(v_2)) \quad : \, \forall v_1, v_2 \in \mathcal V \, .
\end{equation}
The last fundamental variable is the entropy $s$, a positive function on $\mathcal M$.
The volume form $\omega$, the space of thermodynamic configurations $S_2^+(\mathcal V)$, and the volume per unit mass $V$ are defined as in Section \ref{tdss}. The thermodynamic state space is $S_2^+(\mathcal V)\times \mathbb R^+$ and the energy per unit mass $e$ is a function on this space, as before. The thermodynamic stress $\pi$ at each $(\gamma, s) \in S_2^+(\mathcal V)\times \mathbb R^+$, $\pi(\gamma, s) \in \left(S_2(\mathcal V)\right)^*$, is defined by
\begin{equation*}
\frac{\partial e(\gamma, s)}{\partial\gamma}=-\frac 1 2 V(\gamma) \pi(\gamma,s)\, ,
\end{equation*}
as in Section \ref{tdss}. Also, the temperature $\theta$ is defined, as before, by
\begin{equation*}
\theta(\gamma,s)=\frac{\partial e(\gamma, s)}{\partial s} \, .
\end{equation*}
Thus
\begin{equation*}
de = -\frac 1 2 V \pi \cdot d\gamma + \theta ds
\end{equation*}
expresses the first law of thermodynamics in the present framework.
The stress $S$ is a $2$-contravariant symmetric tensorfield on $\mathcal M$, an assignment of an element $S_x \in \left(S_2(\Sigma_x)\right)^*$ at each $x \in \mathcal M$. This is defined as follows. Given any $\dot g_x \in S_2(T_x \mathcal M)$, we define $\dot \gamma_x \in S_2^+(\mathcal V)$ as in (\ref{defgammax}) by:
\begin{equation*}
\dot \gamma_x(v_1,v_2)=\dot g_x(X_x(v_1),X_x(v_2))=\left.\dot g\right|_{\Sigma_x}(X_x(v_1),X_x(v_2)) \quad : \, \forall v_1, v_2 \in \mathcal V \, .
\end{equation*}
Then
\begin{equation*}
S_x(\dot g_x)= \pi(\gamma_x,s(x))(\dot \gamma_x) \, .
\end{equation*}
The mass-energy density $\rho$ is then the positive function on $\mathcal M$ given by
\begin{equation*}
\rho(x)=\frac{e(\gamma_x, s(x))}{V(\gamma_x)} \, ,
\end{equation*}
and the energy-momentum-stress tensor is defined according to (\ref{enmomstrtens}).
The Eulerian equations of motion are a first order system of partial differential equations consisting of
\begin{itemize}
\item[{\bf (a)}] $\mathcal L_u \xi =0$, i.e.~property {\bf 3)} of $\xi$, {\it and}
\item[{\bf (b)}] $\nabla \cdot T=0$.
\end{itemize}
\begin{remark}
In $n$ space dimensions, $\mathrm{dim} \mathcal M=n+1$, there are $n^2+n+1$ dependent variables, the $n^2$ algebraically independent components of $\xi$, the $n$ algebraically independent components of $u$, and also $s$. The Eulerian equations are also $n^2+n+1$ in number, $n^2$ independent equations in {\bf (a)} and $n+1$ independent equations in {\bf (b)}.
\end{remark}
For solutions of the Eulerian equations such that $u$, $\xi$, and $s$ are continuous, {\bf (b)} implies the adiabatic condition on $s$:
\begin{equation*}
u(s)=0 \, .
\end{equation*}
For the proof of this fact see \cite{C2}.
\begin{remark}
The case of absence of dislocations is the case $d \xi =0$.
\end{remark}
Let $(E_A: A=1,\ldots,n)$ be a basis for $\mathcal V$. Given any $\gamma \in S_2^+(\mathcal V)$, we set
\begin{equation*}
\gamma_{AB}=\gamma(E_A,E_B) \, .
\end{equation*}
According to the above, at each $x \in \mathcal M$ there is a unique $X_A \in \Sigma_x$ such that
\begin{equation*}
\xi \cdot X_A = E_A \quad : \, A=1, \ldots, n \, .
\end{equation*}
Thus $X_A$ is a vectorfield on $\mathcal M$ belonging to $\Sigma$.
Then
\begin{equation*}
S = \pi^{AB} X_A \otimes X_B \, ,
\end{equation*}
where $\pi^{AB} \in S_2^+(\mathcal V)^*$ is the thermodynamic stress defined above.
Fix an element $v \in \mathcal V$ and consider the vectorfield $X\subset \Sigma$ such that
\begin{equation*}
\xi \cdot X = v \, .
\end{equation*}
Then
\begin{equation*}
0=u(\xi \cdot X)=\left(\mathcal L_u \xi \right) \cdot X + \xi \cdot \left[ u , X \right] \, ,
\end{equation*}
i.e.~(according to {\bf 3)}) $ \xi \cdot \left[ u , X \right]=0$. It follows that
\begin{equation}\label{projpi}
\Pi\left[u,X\right] = 0 \, ,
\end{equation}
where $\Pi$ is the $g$-orthogonal projection to $\Sigma$.
\begin{proposition}
Define $\mu:=\frac 1 V$. Then the mass current $I=\mu u$ satisfies the equation of continuity:
\begin{equation} \label{continuity}
\nabla \cdot I = 0 \, .
\end{equation}
\end{proposition}
\begin{proof}
$\mathcal V$ is endowed with a volume form $\omega$ and $\omega_{\gamma}=V \omega$.
Choose a basis $(E_1,\ldots,E_n)$ for $\mathcal V$ such that $\omega(E_1,\ldots,E_n)=1$.
Then $V=\sqrt{\det \gamma}$, and, by (\ref{defgammax}),
\begin{equation}\label{gammag}
\gamma_{AB}=\gamma(E_A,E_B)=g(X_A,X_B) \, ,
\end{equation}
hence
\begin{equation} \label{uv}
u (V) = \frac 1 2 V \left(\gamma^{-1}\right)^{AB} u (\gamma_{AB}) \, ,
\end{equation}
and, using (\ref{gammag}),
\begin{eqnarray*}
u(\gamma_{AB}) & = & u(g(X_A,X_B)) \\
& = & g(\nabla_u X_A, X_B) + g(X_A, \nabla_u X_B) \\
& = & g(\nabla_{X_A}u, X_B) + g(X_A, \nabla_{X_B}u) +g([u,X_A],X_B)+g(X_A, [u,X_B]) \, .
\end{eqnarray*}
By (\ref{projpi}), the last two terms in the above equation vanish.
Let us define $\kappa$ by
\begin{eqnarray}\label{defkappa}
\nabla_{X_A}u=\kappa_A^B X_B \, .
\end{eqnarray}
[$(X_A : A=1, \ldots, n)$ is a basis for $\Sigma_x$ at each point.]
Then
\begin{equation*}
g(\nabla_{X_A}u,X_B)= \kappa_A^C g(E_C,E_B)=\gamma_{CB} \kappa_A^C \, .
\end{equation*}
Hence we obtain
\begin{equation*}
u(\gamma_{AB})=\gamma_{CB}\kappa_A^C+\gamma_{AC}\kappa_B^C \, .
\end{equation*}
Substituting in (\ref{uv}) yields
\begin{equation*}
u(V)=\frac 1 2 V \left(\gamma^{-1}\right)^{AB}u(\gamma_{AB})= V \textrm{tr} \, \kappa \, .
\end{equation*}
Then $\mu=V^{-1}$ satisfies
\begin{equation*}
u(\mu)+\mu\textrm{tr} \, \kappa = 0 \, ,
\end{equation*}
and since, from (\ref{defkappa}), $\textrm{tr} \,\kappa = \nabla \cdot u$, we obtain
\begin{equation}
\nabla \cdot I = \nabla (\mu u) = u(\mu) + \mu \nabla \cdot u= 0 \, ,
\end{equation}
which establishes (\ref{continuity}).
\end{proof}
\subsection{The Non-relativistic Limit}
First of all, we restrict ourselves to the case where $(\mathcal M, g)$ is the Minkowski space-time.
Then we consider the non-relativistic limit, where Minkowski space-time is replaced by Galilean space-time. There, we have the hyperplanes $\Sigma_t$ of absolute simultaneity, which are isometric to $n$-dimensional Euclidean space.
Any family of parallel lines transversal to the $\Sigma_t$ represents a Galilean frame, that is, a family of observers in uniform motion and at rest relative to each other. Any such family of parallel lines defines an isometry of the $\Sigma_t$ onto each other.
In terms of a Galilean frame and a rectangular coordinate system $(x^1, \ldots, x^n)$ in Euclidean space, and with $x^0=ct$ ($c$: the speed of light in vacuum), the space-time velocity $u = u^{\mu}\frac{\partial}{\partial x^{\mu}}$ is represented in terms of the space velocity $v=v^i\frac{\partial}{\partial x^i}$ by
\begin{equation*}
u^0=\frac 1{\sqrt{1-|v|^2/c^2}} \quad , \quad u^i = \frac{\frac{v^i}{c}}{\sqrt{1-|v|^2/c^2}} \, .
\end{equation*}
Therefore, in the non-relativistic limit $c \to \infty$
\begin{equation*}
c u = \frac{\partial}{\partial t}+ v^i \frac{\partial}{\partial x^i} \, .
\end{equation*}
Also, the condition
\begin{equation*}
\xi_{\mu} u^{\mu} = 0
\end{equation*}
is simply $\xi_0=0$ and thus $\xi$ becomes a $\mathcal V$-valued $1$-form on each $\Sigma_t$,
\begin{equation*}
\xi=\xi_i dx^i \, .
\end{equation*}
Equations {\bf (a)} become
\begin{equation}
\frac{\partial \xi}{\partial t}+ \mathcal L_v \xi =0 \quad \left( \textrm{or} \quad \frac{\partial \xi_i}{\partial t}+v^j \frac{\partial \xi_i}{\partial x^j}+ \xi_j \frac{\partial v^j}{\partial x^i}=0 \right) \, ,
\end{equation}
and
\begin{equation*}
S^{\mu \nu} u_{\nu} = 0 \quad , \quad u_{\nu}=g_{\mu\nu}u^{\mu} \, ,
\end{equation*}
reads (note that $u_{\nu}=g_{\kappa \nu}u^{\kappa}$, i.e.~$u_0=-u^0$, while $u_i=u^i$)
\begin{eqnarray} \label{sio}
S^{i0}-S^{ij} \frac{v^j}{c} & = & 0 \, , \\
S^{00}-S^{0i} \frac{v^i}{c} & = & 0 \, .
\end{eqnarray}
So, in the limit $c \to \infty$, we have $S^{i0}=S^{00}=0$ and, therefore,
\begin{equation*}
S=S^{ij} \frac{\partial}{\partial x^i}\otimes \frac{\partial}{\partial x^j} \, .
\end{equation*}
Let
\begin{eqnarray*}
e & = & c^2 + e' \, , \\
\rho & = & \mu e = \mu c^2 + \varepsilon \quad , \quad \varepsilon=\mu e' \, .
\end{eqnarray*}
We have for the mass current $I^{\nu}=\mu u^{\nu}$:
\begin{eqnarray*}
I^0 & = & \frac{\mu}{\sqrt{1-|v|^2/c^2}} \, = \, \mu + \frac 1 2 \frac{\mu |v|^2}{c^2} + O(c^{-4}) \, , \\
I^i & = & \frac{\mu\frac{v^i}{c}}{\sqrt{1-|v|^2/c^2}} \, = \, \frac{\mu v^i}{c^2} + \frac 1 2 \frac{\mu|v|^2 v^i}{c^3} + O(c^{-5}) \, .
\end{eqnarray*}
In the non-relativistic limit the equation of continuity $\nabla_{\nu} I^{\nu}=0$ becomes the classical continuity equation:
\begin{equation}
\frac{\partial \mu}{\partial t}+ \frac{\partial \left( \mu v^i \right)}{\partial x^i} = 0 \, .
\end{equation}
Similarly, for the energy-momentum-stress tensor $T_{\kappa\lambda}$, we have
\begin{eqnarray*}
T^{00} & = & \rho (u^0)^2 + S^{00} \, = \, \mu c^2 + (\varepsilon + \mu v^2) + O(c^{-2}) \, , \\
T^{0i} & = & \rho u^0 u^i + S^{0i} \, = \, \mu v^i c + (\varepsilon + \mu v^2)\frac{v^i}{c^2} - S^{ij} \frac{v^j}{c} + O(c^{-2}) \, ,\\
T^{ij} & = & \frac{\rho v^i v^j/c^2}{1-|v|^2/c^2}+ S^{ij} \, = \, \mu v^i v^j + S^{ij} + O(c^{-2}) \, .
\end{eqnarray*}
The equations of motion $\nabla_{\lambda} T^{\kappa\lambda}=0$ reads
\begin{eqnarray*}
\kappa=0: \quad \frac{\partial T^{00}}{\partial x^0} + \frac{\partial T^{0i}}{\partial x^i}= 0 & \stackrel{c\to \infty}{\longrightarrow} & \frac{\partial \mu}{\partial t}+ \frac{\partial \left( \mu v^i \right)}{\partial x^i} = 0 \, , \\
\kappa=i: \quad \frac{\partial T^{i0}}{\partial x^0} + \frac{\partial T^{ij}}{\partial x^j}= 0 & \stackrel{c\to \infty}{\longrightarrow} & \frac{\partial \left(\mu v^i\right)}{\partial t}+ \frac{\partial \left( \mu v^i v^j + S^{ij} \right)}{\partial x^j} = 0 \, .
\end{eqnarray*}
In order to obtain the energy equation, let us consider
\begin{equation*}
F^{\nu}= T^{0\nu}-c^2 I^{\nu} \quad , \, \textrm{where} \quad \nabla_{\nu}F^{\nu}=0 \, .
\end{equation*}
For the components of $F^{\nu}$ we have, using (\ref{sio}), i.e.~$S^{i0}=S^{ij}v^j/c$ and $S^{00}=S^{ij}v^i v^j / c^2$,
\begin{eqnarray*}
F^0 & = & \frac{\mu |v|^2 + \varepsilon}{1-|v|^2/c^2}+ \frac{S^{ij}v^i v^j}{c^2} - \frac{c^2 \mu}{\sqrt{1-|v|^2/c^2}} \, = \, \frac 1 2 \mu |v|^2 + \varepsilon + O(c^{-2}) \\
F^i & = & \frac{\left(\mu |v|^2 + \varepsilon\right)v^i/c}{1-|v|^2/c^2}+ S^{0i} - \frac{c \mu v^i}{\sqrt{1-|v|^2/c^2}} \\
& = & \left(\mu |v|^2 + \varepsilon \right) \frac{v^i}{c} + S^{ij} \frac{v^j}{c} - \frac 1 2 \mu |v|^2 \frac{v^i}{c} + O(c^{-2}) \, .
\end{eqnarray*}
Hence,
\begin{equation*}
c \nabla_{\nu} F^{\nu} = \frac{\partial F^0}{\partial t}+ c \frac{\partial F^i}{\partial x^i}=0
\end{equation*}
is the energy equation
\begin{equation}
\frac{\partial}{\partial t}\left(\frac 1 2 \mu |v|^2 + \varepsilon \right) + \frac{\partial}{\partial x^i}\left[ \left(\frac 1 2 \mu |v|^2 + \varepsilon \right) v^i + S^{ij} v^j \right] = 0 \, .
\end{equation}
The non-relativistic Eulerian equations are
\begin{eqnarray*}
{\bf (a)} & & \frac{\partial \xi}{\partial t}+ \mathcal L_v \xi = 0 \, , \\
{\bf (b0)} & & \frac{\partial \mu}{\partial t}+\frac{\partial}{\partial x^i} \left(\mu v^i\right) = 0 \, , \\
{\bf (b1)} & & \frac{\partial \left(\mu v^i\right)}{\partial t}+ \frac{\partial}{\partial x^j} \left( \mu v^i v^j + S^{ij} \right) = 0 \, , \\
{\bf (b2)} & & \frac{\partial}{\partial t}\left(\frac 1 2 \mu |v|^2 + \varepsilon \right) + \frac{\partial}{\partial x^i}\left[ \left(\frac 1 2 \mu |v|^2 + \varepsilon \right) v^i + S^{ij} v^j \right] = 0 \, .
\end{eqnarray*}
Notice:
\begin{itemize}
\item ${\bf (b0)}$ is the differential mass conservation law, a consequence of ${\bf (a)}$,
\item ${\bf (b1)}$ is the differential momentum conservation law, {\it and}
\item ${\bf (b2)}$ is the differential energy conservation law.
\end{itemize}
\begin{remark}
{\bf (a)} may be thought of as expressing a law of conservation of dislocations. In fact, let $C_0$ be a closed curve on $\Sigma_0$. Let $\phi_t$ be the flow in Galilean space-time generated by the vectorfield
\begin{equation*}
u= \frac{\partial}{\partial t}+ v^i \frac{\partial}{\partial x^i} \, .
\end{equation*}
Then $\left. \phi_t\right|_{\Sigma_0}$ is a diffeomorphism from $\Sigma_0$ onto $\Sigma_t$. Let $C_t=\phi_t(C_0)$. Then, according to {\bf (a)},
\begin{equation*}
\int\limits_{C_t} \xi = \int\limits_{C_0} \xi \, .
\end{equation*}
The left-hand side corresponds, in the continuum limit, to minus the sum of the Burger's vectors of all the dislocation lines enclosed by $C_t$.
\end{remark}
\begin{remark}
For continuous solutions, i.e.~($\xi$, $v$, $s$) continuous, {\bf (b2)} is equivalent to the adiabatic condition
\begin{itemize}
\item[{\bf (c)}] $\frac{\partial s}{\partial t}+v^i \frac{\partial s}{\partial x^i}=0$,
\end{itemize}
modulo the other equations. When discontinuities such as shocks develop, this equivalence no longer holds. However, the Eulerian equations {\bf (a)}-{\bf (b)} still hold, but in a weak or integral sense.
\end{remark}
\subsection{From the Eulerian to the Lagrangian Picture}
In order to go from the Eulerian picture to the Lagrangian formulation we have to extract the canonical form $\nu$ from $\xi$.
First note that
\begin{equation}\label{liuxi}
\mathcal L_u \xi =0 \quad \Leftrightarrow \quad \phi_t^* \xi = \xi \, ,
\end{equation}
where $\phi_t$ is the flow generated by $u$.
Let $\mathcal H$ be a Cauchy hypersurface in $\mathcal M$. We identify $\mathcal H$ with $\mathcal N$ and define $f: \mathcal M \to \mathcal N$ as follows:
\begin{equation*}
f(x)=y \in \mathcal H
\end{equation*}
is the point at which the integral curve of $u$ through intersects $\mathcal H$.
For $X_x \in T_x\mathcal M$ we have from (\ref{liuxi})
\begin{equation*}
\left(\phi^*\xi\right) = \xi\left(d\phi_t \cdot X_x\right)=\xi(X_x) \, .
\end{equation*}
Define $\nu$ to be the $\mathcal V$-valued $1$-form induced by $\xi$ on $\mathcal H$:
\begin{equation*}
\nu(X_x)=\xi(X_x) \quad : \, \forall X_x \in T_x {\mathcal H} , x \in \mathcal H \, .
\end{equation*}
We must show
\begin{proposition}
\begin{equation*}
f^* \nu = \xi \, .
\end{equation*}
\end{proposition}
\begin{proof}
Let $V_p \in T_p \mathcal M$, $p\in \phi_t(\mathcal H)$ for some $t \in \mathbb R$. $V_p$ uniquely decomposes into
\begin{equation*}
V_p= \lambda u + Y_p \quad , \, \textrm{where} \quad Y_p \in T_p \phi_t(\mathcal H) \, .
\end{equation*}
Therefore, it suffices to show that $(f^*\nu)(u_p)=\xi(u_p)=0$, which is obvious, and
\begin{equation*}
\left(f^*\nu\right)(Y_p)=\xi(Y_p) \, .
\end{equation*}
Since $\left.\phi_t \right|_{\mathcal H}$ is a diffeomorphism from $\mathcal H$ onto $\phi_t(\mathcal H)$, $d\phi_t(q)$ is an isomorphism from $T_q \mathcal H$ onto $T_p\phi_t(\mathcal H)$, $\phi_t(q)=p$. Therefore, there exists a unique $X_q \in T_q \mathcal H$, such that
\begin{equation*}
d\phi_t \cdot X_q = Y_p \, .
\end{equation*}
Now,
\begin{equation*}
\xi(Y_p)=\xi(d\phi_t \cdot X_q)=\left(\phi_t^* \xi\right)(X_q)=\xi(X_q)=\nu (X_q)
\end{equation*}
and
\begin{equation*}
\left(f^* \nu\right) = \left(f^*\nu\right)(d\phi_t \cdot X_q) = \left(\phi_t^* f^*\nu \right)(X_q) = \left(f^* \nu \right)(X_q) = \nu (df \cdot X_q) \, .
\end{equation*}
But $\left. f \right|_{\mathcal H}= id$, hence $df \cdot X_q = X_q$ because
\begin{equation*}
\left. df \right|_{T_q\mathcal H}= id \, .
\end{equation*}
\end{proof}
\begin{remark}
The Eulerian picture is the one more related to direct physical experience and also the one which may serve as a basis for extending the theory beyond the domain of elasticity theory, where the dislocations are no longer anchored in the solid. We shall see in the next part, however, that the Lagrangian picture provides the suitable framework for the study of static problems.
\end{remark}
\newpage
\part{The Static Case} \label{static}
\setcounter{section}{0}
\section{Formulation of the Static Problem} \label{staticsetting}
Let $\mathcal N=\Omega \subset \mathbb R^n$ be the material manifold and $\mathcal M=E^n$ Euclidean space ($n=2,3$ being the cases of interest).
In the static case, we adopt the material picture, that is, we consider one to one mappings $\phi$ from the material manifold into space,
\begin{eqnarray}\label{defphi}
\begin{array}{rcl}
\phi : \mathcal N & \to & \mathcal M \\
y & \mapsto & \phi(y)=x \, .
\end{array}
\end{eqnarray}
They correspond to mappings $ f : \mathcal M \times \mathbb R \to \mathcal N$ such that
\begin{equation*}
f(x,t)=\phi^{-1}(x) \quad : \, \forall t \, .
\end{equation*}
In the material picture, the interchange of the roles of the domain and target space transforms a free boundary problem into a fixed boundary problem.
We recall that the \emph{configuration} $\gamma$ is an element of the space of inner product on the crystalline structure $\mathcal V$, $\gamma \in S_2^+(\mathcal V)$. In terms of the mapping $\phi$ it is defined as the pullback of the metric $g$ on $\mathcal M$ by the isomorphism $i_{\phi , y}$
\begin{equation} \label{defthermconfig}
\gamma = i_{\phi, y}^* g \, ,
\end{equation}
where $i_{\phi, y}=d\phi(y)\circ \epsilon_y$ and $\epsilon_y$ is the evaluation map (\ref{evalmap}) from $\mathcal V$ to $T_y\mathcal N$.
The energy per unit mass $e(\gamma)$ defines the {\it thermodynamic stress} $\pi$, an element in $\left(S_2(\mathcal V)\right)^*$, by
\begin{equation} \label{defthermstress}
\frac{\partial e}{\partial \gamma}=-\frac 1 2 \pi V \, ,
\end{equation}
where $V\equiv V(\gamma)$ is the volume per unit mass, related to the mass density $\mu$ by
\begin{equation*}
V(\gamma)=\frac 1{\mu(x)} \quad , \quad x=\phi(y) \, .
\end{equation*}
In the following, we assume the entropy $\sigma$ to be constant, what is called isentropic.
Since we are in the relativistic framework, $e$ includes the contribution $c^2$ of the rest mass-energy, so
\begin{equation*}
e=c^2+e'
\end{equation*}
(in conventional units), where $e'$ is what we would call energy per unit mass in the non-relativistic framework. Note that the additive constant $c^2$ does \emph{not} affect the definition (\ref{defthermstress}) of the thermodynamic stress $\pi$, which may equally well be written in the form
\begin{equation*}
\frac{\partial e'}{\partial \gamma}=-\frac 1 2 \pi V \, .
\end{equation*}
Also, in view of (\ref{energy}) below, we have
\begin{equation*}
E=Mc^2+E' \, ,
\end{equation*}
where
\begin{equation*}
M= \int_{\Omega} d\mu_{\omega} \quad , \quad E'=\int_{\Omega} e'(\gamma) d\mu_{\omega} \, .
\end{equation*}
The additive constant $Mc^2$ does not affect variations of $E$, which are the same as those of $E'$.
For this reason we shall not distinguish in the static theory $e'$, $E'$ from $e$, $E$, and we shall denote the former by the latter.
{\remark The static theory would be formally identical if formulated wholly within the non-relativistic framework. What distinguishes the two theories is that the non-relativistic theory is applicable only when the characteristic speeds $\eta_i$ (\ref{defetai}) are negligible in comparison to $c$, while the relativistic theory holds for any values of the $\eta_i$ less than $c$.}
\section{Boundary Value Problem and Legendre-Hadamard Condition} \label{ELeqs}
\subsection{Euler-Lagrange Equations}
We choose a basis $E_1,\ldots ,E_n$ of $\mathcal V$ such that $\omega (E_1,\ldots ,E_n)=1$, where $\omega$ is the volume form on $\mathcal V$.
We denote by $\gamma_{AB}=\gamma(E_A, E_B)$ the matrix representing the configuration in this frame.
We then have
\begin{equation*}
\omega_{\gamma} = V(\gamma)\omega = \sqrt{\det \gamma} \cdot \omega \, ,
\end{equation*}
and for the total energy of a domain $\Omega$ in the material manifold
\begin{equation}\label{energy}
E=\int\limits_{\Omega} e(\gamma) d\mu_{\omega} \, ,
\end{equation}
where $d \mu_{\omega}$ is the mass element on $\mathcal N$ induced by $\omega$:
\begin{equation} \label{detomega}
d \mu_{\omega}\left(\left.\frac{\partial}{\partial y_1}\right|_y, \ldots , \left.\frac{\partial}{\partial y_n}\right|_y\right) = \omega \left( \epsilon_y^{-1}\left(\left.\frac{\partial}{\partial y_1}\right|_y\right), \ldots , \epsilon_y^{-1}\left(\left.\frac{\partial}{\partial y_n}\right|_y\right)\right)\, .
\end{equation}
Let $(\omega^A : A=1,\ldots,n)$ be the basis for $\mathcal V^*$ which is dual to the basis $(E_A: A=1,\ldots,n)$ for $\mathcal V$. The $\omega^A$ are $1$-forms on $\mathcal N$ such that $\omega^A(E_B)=\delta^A_B$.
Since $(E_A(y): A=1,\ldots,n)$ is a basis for $T_y\mathcal N$, given any $Y \in T_y \mathcal N$, there are coefficients $Y^A : A=1,\ldots,n$ such that $Y=Y^A E_A(y)$. Then:
\begin{equation*}
\omega^A(Y) = \omega^A \left(Y^B E_B(y)\right)=Y^B(\omega^A(E_B))(y)= Y^B \delta^A_B = Y^A \, .
\end{equation*}
Setting $Y=\left. \frac{\partial}{\partial y^a}\right|_y$ we obtain
\begin{equation*}
Y^A=\omega^A \left(\left. \frac{\partial}{\partial y^a}\right|_y\right) =\omega_a^A(y) \, ,
\end{equation*}
and we have $\epsilon_y^{-1}(Y)= Y^A E_A \in \mathcal V$.
Thus
\begin{equation}\label{omegacoeff}
\epsilon_y^{-1}\left(\left.\frac{\partial}{\partial y^a}\right|_y\right)=\omega_a^A(y)E_A \, .
\end{equation}
Let us denote by $\det \omega (y)$ the determinant of the $n$-dimensional matrix with entries $\omega^A_a(y)$.
We conclude from (\ref{detomega}), (\ref{omegacoeff}) that
\begin{equation} \label{dmuomega}
d \mu_{\omega} = \det \omega (y) d^ny \, .
\end{equation}
The first variation of the energy (\ref{energy}) is:
\begin{equation}\label{firstvarE}
\dot E = \left.\frac{\partial}{\partial \lambda}\right|_{\lambda =0} E(\gamma+\lambda \dot\gamma)=\left.\frac{\partial}{\partial \lambda}\right|_{\lambda =0}\int\limits_{\Omega} e(\gamma+\lambda \dot\gamma) d\mu_{\omega}=\int\limits_{\Omega}\frac{\partial e(\gamma)}{\partial \gamma} \cdot \dot \gamma \, d\mu_{\omega} \, ,
\end{equation}
where $\dot \gamma \in S_2^+(\mathcal V)$ is the variation of the configuration $\gamma$.
By definition of the thermodynamic stress (\ref{defthermstress}), we conclude, in view of (\ref{dmuomega}),
\begin{equation*}
\dot E = - \int\limits_{\Omega} \frac 1 2 \pi^{AB}\dot \gamma_{AB} \sqrt{\det \gamma}\det \omega (y) \ d^ny \, .
\end{equation*}
Denoting by $m_{ab}$ the pullback by $\phi$ to $\mathcal N$ of the Euclidean metric $g$ on $\mathcal M$, we have
\begin{eqnarray}
m & = & \phi^*g \label{mpbphi} \\
m_{ab} & = &\frac{\partial x^i}{\partial y^a}\frac{\partial x^j}{\partial y^b} g_{ij} \qquad [x^i=\phi^i(y)] \, , \label{mabpbm} \\
\gamma_{AB} & = & E_A^aE_B^b m_{ab} \, , \label{gammapbm}
\end{eqnarray}
where the $n$-dimensional matrix with entries $E_A^a(y)$ is the reciprocal of the $n$-dimensional matrix with entries $\omega_a^A(y)$. Thus $\det E(y)=\left( \det \omega(y)\right)^{-1}$, and by (\ref{gammapbm}) we have
\begin{eqnarray*}
\det \gamma & = & \det m \left(\det E\right)^2 \, , \\
\sqrt{\det \gamma} \det \omega & = & \sqrt{\det m} \, .
\end{eqnarray*}
For the stresses $S$ on $\mathcal N$ and $T=\phi_* S$ on $\mathcal M$ we have:
\begin{eqnarray}
S^{ab} & = & \pi^{AB} E_A^aE_B^b \, , \label{defSpfpi} \\
T^{ij} & = & S^{ab}\frac{\partial x^i}{\partial y^a}\frac{\partial x^j}{\partial y^b} \, .
\end{eqnarray}
We will now formulate the Euler-Lagrange equations on both the material manifold and on space.
We calculate for the variation of $\gamma_{AB}$ from (\ref{gammapbm}) using linear coordinates in $E^n$ (i.e.~$\partial_k g_{ij}=0$)
\begin{equation*}
\dot \gamma_{AB}=E_A^a E_B^b \dot m_{ab} \quad , \quad \dot m_{ab} = g_{ij}\left(\frac{\partial \dot x^i}{\partial y^a}\frac{\partial x^j}{\partial y^b}+\frac{\partial x^i}{\partial y^a}\frac{\partial \dot x^j}{\partial y^b}\right) \, .
\end{equation*}
The first variation of the energy thus reads
\begin{equation} \label{dotE}
\dot E = -\int\limits_{\Omega} \frac 1 2 S^{ab}\dot m_{ab} \sqrt{\det m} \, d^ny = -\int\limits_{\Omega} S^{ab}g_{ij}\frac{\partial \dot x^i}{\partial y^a}\frac{\partial x^j}{\partial y^b}\sqrt{\det m} \, d^ny \, ,
\end{equation}
where $\dot x^i$ is the variation of $x^i$. Hence (\ref{dotE}) is
\begin{equation} \label{starS}
-\int\limits_{\Omega} S^{ab}g_{ij}\frac{\partial \dot x^i}{\partial y^a}\frac{\partial x^j}{\partial y^b} d\mu_m \, ,
\end{equation}
where $d\mu_m$ is the volume form on $\Omega$ associated to the metric $m$. By the divergence theorem, (\ref{starS}) becomes
\begin{equation} \label{dstarS}
-\int\limits_{\partial \Omega} S^{ab} g_{ij}\frac{\partial x^j}{\partial y^b} M_a \dot x^i\, dA_m + \int\limits_{\Omega} \dot x^i \overset{m}{\nabla}_a Q_i^a d\mu_m \, ,
\end{equation}
where $Q_i$ are the vectorfields with components
\begin{equation} \label{Qia}
Q_i^a = S^{ab}g_{ij}\frac{\partial x^j}{\partial y^b}
\end{equation}
and $\overset{m}{\nabla}$ is the covariant derivative operator on $\mathcal N$ associated to $m$. Thus, in local coordinates on $\mathcal N$,
\begin{equation} \label{covderQ}
\overset{m}{\nabla}_a Q_i^a = \frac{1}{\sqrt{\det m}} \frac{\partial}{\partial y^a} \left(\sqrt{\det m}Q_i^a\right) \, .
\end{equation}
In (\ref{dstarS}), $M^a$ are the components of the outward unit normal to $\partial \Omega$ with respect to $m$, $M_a=m_{ab}M^b$, and $dA_m$ is the area element of $\partial\Omega$ associated to the metric induced by $m$. So $M_a$ are the components of the covectorfield along $\partial \Omega$ whose null space is the tangent plane to $\partial \Omega$ at each point.
The Euler-Lagrange equations are obtained by requiring $\dot E=0$ for variations $\dot x^i$ which are supported in $\Omega$ and vanish on $\partial \Omega$. In view of (\ref{dstarS}), (\ref{Qia}), (\ref{covderQ}), the Euler-Lagrange equations are:
\begin{equation} \label{equivEL}
\frac{\partial}{\partial y^a}\left(Q_i^a\sqrt{\det m}\right)=0 \quad \textrm{on} \, \, \Omega \, .
\end{equation}
Note that with $Q_i^a$ defined in (\ref{Qia}) we have using (\ref{mabpbm})
\begin{equation} \label{QS}
Q_i^a \frac{\partial x^i}{\partial y^b}= g_{ij}S^{ac} \frac{\partial x^j}{\partial y^c} \frac{\partial x^i}{\partial y^b} = m_{bc} S^{ac} = S_b^a \, .
\end{equation}
Consider
\begin{equation} \label{covdermS}
\overset{m}{\nabla}_a S_b^a = \frac{1}{\sqrt{\det m}} \frac{\partial}{\partial y^a} \left(\sqrt{\det m}S_b^a\right)-\overset{m}{\Gamma_{ab}^c} S_c^a \, .
\end{equation}
The first term on the right-hand side of (\ref{covdermS}) is, by (\ref{equivEL}), (\ref{QS}),
\begin{equation*}
\frac{1}{\sqrt{\det m}}\frac{\partial}{\partial y^a} \left(\sqrt{\det m}Q_i^a \frac{\partial x^i}{\partial y^b}\right) = Q_i^a \frac{\partial^2 x^i}{\partial y^a \partial y^b}= S^a_c \frac{\partial y^c}{\partial x^i} \frac{\partial^2 x^i}{\partial y^a \partial y^b} \, .
\end{equation*}
Thus (\ref{equivEL}) is equivalent to:
\begin{equation} \label{nablaS}
\overset{m}{\nabla}S_b^a = \left( \frac{\partial y^c}{\partial x^i} \frac{\partial^2 x^i}{\partial y^a \partial y^b}-\overset{m}{\Gamma_{ab}^c} \right) S_c^a \, .
\end{equation}
We shall show now that
\begin{equation}\label{gammam}
\overset{m}{\Gamma_{ab}^c}=\frac{\partial y^c}{\partial x^i} \frac{\partial^2 x^i}{\partial y^a \partial y^b} \, ,
\end{equation}
so that the factor in parenthesis in (\ref{nablaS}) vanishes. In fact, (\ref{gammam}) is equivalent to:
\begin{equation*}
\overset{m}{\Gamma}_{ab,c}=m_{cd} \frac{\partial y^d}{\partial x^i} \frac{\partial^2 x^i}{\partial y^a \partial y^b} \, ,
\end{equation*}
which, since $m_{cd} \frac{\partial y^d}{\partial x^i}= g_{ij}\frac{\partial x^j}{\partial y^c}$, is in turn equivalent to
\begin{equation} \label{gammaml}
\overset{m}{\Gamma}_{ab,c}= g_{ij}\frac{\partial x^j}{\partial y^c} \frac{\partial^2 x^i}{\partial y^a \partial y^b} \, .
\end{equation}
Now we have by the definition of the Christoffel symbols $\overset{m}{\Gamma}_{ab,c}$ and $m$ from (\ref{mabpbm}):
\begin{equation*}
\overset{m}{\Gamma}_{ab,c} = \frac 1 2 \left(\frac{\partial m_{ac}}{\partial y^b}+ \frac{\partial m_{bc}}{\partial y^a} - \frac{\partial m_{ab}}{\partial y^c}\right) = g_{ij} \frac{\partial x^j}{\partial y^c} \frac{\partial^2 x^i}{\partial y^a \partial y^b} \, .
\end{equation*}
Thus (\ref{gammaml}) is indeed verified. We conclude that the Euler-Lagrange equations (\ref{equivEL}) are equivalent to:
\begin{equation} \label{divstresszeroN}
\overset{m}{\nabla}_aS_b^a=0 \quad , \, \textrm{or} \quad \overset{m}{\nabla}_a S^{ab} = 0 \quad \textrm{on} \, \, \Omega \, .
\end{equation}
{\remark $T^{ij}$ being the push-forward of $S^{ab}$ by $\phi$ and $m_{ab}$ being the pull-back of $g_{ij}$ by $\phi$, the last equations (\ref{divstresszeroN}) are in turn equivalent to
\begin{equation} \label{divstresszeroE}
\overset{g}{\nabla}_i T^{ij} =0 \quad , \, \textrm{or} \quad \partial_i T^{ij} = 0 \quad \textrm{on} \, \, \phi(\Omega)
\end{equation}
in linear coordinates.}
\subsection{Boundary Conditions}
Let now equations (\ref{equivEL}) hold. Requiring again $\dot E=0$, this time for arbitrary variations $\dot x$, we obtain from the first term of (\ref{dstarS})
\begin{equation} \label{bdrycondS}
0 = \int\limits_{\partial\Omega} S^{ab}g_{ij}\frac{\partial x^j}{\partial y^b}M_a \dot x^i \, dA_m \, .
\end{equation}
Here the covectorfield along $\partial \Omega$ with components $M_a$ is defined by
\begin{equation*}
M_aY^a=0 \quad : \, \forall Y \in T_y \partial \Omega \, ,
\end{equation*}
together with the condition $M_aY^a>0$ for all $Y \in T_y \Omega$, $y \in \partial \Omega$, such that $Y$ points to the exterior of $\Omega$, and the normalization condition $\left(m^{-1}\right)^{ab} M_a M_b = 1$.
As (\ref{bdrycondS}) is to hold for arbitrary variations $\dot x^i$, we obtain the boundary conditions
\begin{equation} \label{bdryc}
S^{ab}g_{ij}\frac{\partial x^j}{\partial y^b} M_a = 0 \quad , \, \textrm{or} \quad S^{ab}M_b=0 \quad \textrm{on} \, \, \partial \Omega \, .
\end{equation}
Let $X^i=\frac{\partial x^i}{\partial y^a}Y^a$. Setting $M_a=\frac{\partial x^i}{\partial y^a}N_i$ we have:
\begin{equation} \label{defNi}
N_i X^i =0 \quad : \, \forall X \in T_x \phi(\partial \Omega) \, .
\end{equation}
In fact, the covectorfield along $\phi(\partial \Omega)$ with components $N_i$ is defined by (\ref{defNi}), together with the condition that
\begin{equation*}
N_i X^i > 0 \quad: \, \forall X \in T_x \phi(\Omega) \, ,
\end{equation*}
$x \in \phi(\partial \Omega)=\partial (\phi(\Omega))$, such that $X$ points to the exterior of $\phi(\Omega)$, and the normalization condition $\left(g^{-1}\right)^{ij} N_i N_j = 1$.
In other words, $N^i:=(g^{-1})^{ij} N_j$ are the components of the outward unit normal to $\partial(\phi(\Omega))$ in $E^n$.
In view of the fact that $S^{ab}\frac{\partial x^j}{\partial y^b}=T^{ij} \frac{\partial y^a}{\partial x^i}$, the boundary conditions (\ref{bdryc}) are equivalent to
\begin{equation}\label{bdrycE}
T^{ij}N_i = 0 \quad \textrm{on} \, \,\phi(\partial \Omega)=\partial \phi(\Omega) \, ,
\end{equation}
which means that no forces are acting on the boundary $\partial \phi(\Omega)$.
\subsection{Legendre-Hadamard Condition} \label{SectLegHad}
The \emph{Legendre-Hadamard} condition in the static case is just the second part of the hyperbolicity condition stated in (\ref{hposdef}), see \cite{C1} for details.
Here, it is formulated in terms of $\frac{\partial \phi^i}{\partial y^a}(y)=v^i_a$.
It requires for $\xi_a, \xi_b \in T^*_y\mathcal N, \eta^i, \eta^j \in T_{\phi(y)}\mathcal M$ that
\begin{equation} \label{L-H}
\frac 1 4 \frac{\partial ^2 e}{\partial v^i_a \partial v^j_b} \xi_a \xi_b \eta^i \eta^j \quad : \quad \textrm{positive for}\,\, \xi, \eta \neq 0 \, .
\end{equation}
We differentiate $e$ with respect to $v^j_b$:
\begin{equation*}
\frac{\partial e}{\partial v^j_b}=\frac{\partial e}{\partial \gamma_{AB}}\frac{\partial\gamma_{AB}}{\partial v^j_b} \, ,
\end{equation*}
where $\gamma_{AB}=E_A^a E_B^b g_{ij} v^i_a v^j_b$, hence
\begin{equation*}
\frac{\partial\gamma_{AB}}{\partial v^j_b}=2E_{(A}^a E_{B)}^b g_{ij} v^i_a \, ,
\end{equation*}
($E_{(A}^a E_{B)}^b$ denotes symmetrization in $A$ and $B$) and thus
\begin{equation} \label{firstdere}
\frac 1 2 \frac{\partial e}{\partial v^j_b}=\frac{\partial e}{\partial \gamma_{AB}}E_A^a E_B^b g_{ij} v^i_a \, .
\end{equation}
We differentiate (\ref{firstdere}) once more with respect to $v^i_a$ to get
\begin{equation*}
\frac 1 4 \frac{\partial ^2 e}{\partial v^i_a \partial v^j_b} = \frac{\partial ^2 e}{\partial\gamma_{AB}\partial\gamma_{CD}}g_{li}g_{kj} v^l_c v^k_d E_A^d E_B^b E_C^c E_D^a + \frac 1 2 \frac{\partial e}{\partial\gamma_{AB}}E_A^a E_B^b g_{ij} \, .
\end{equation*}
Therefore,
\begin{equation}
\frac 1 4 \frac{\partial ^2 e}{\partial v^i_a \partial v^j_b} \xi_a \xi_b \eta^i \eta ^j = \frac 1 2 \frac{\partial e}{\partial\gamma_{AB}}|\eta|^2 \xi_A \xi_B + \frac{\partial ^2 e}{\partial\gamma_{AB}\partial\gamma_{CD}} \eta_C \eta_A \xi_B \xi_D \, , \label{LegHadgen}
\end{equation}
where $|\eta|^2=g_{ij}\eta^i\eta^j$, $\xi_A=E_A^a\xi_a$ and $\eta_C= E_C^c v^l_c g_{li} \eta^i$.
Thus the Legendre-Hadamard condition reads:
\begin{equation*}
\frac{\partial^2 e}{\partial\gamma_{AB}\partial\gamma_{CD}} \eta_C \eta_A \xi_B \xi_D + \frac 1 2 \frac{\partial e}{\partial\gamma_{AB}}|\eta|^2 \xi_A \xi_B > 0 \quad (\eta, \xi \neq 0) \, .
\end{equation*}
At $\stackrel{\circ}{\gamma}_{AB}$ the Legendre-Hadamard condition reduces to
\begin{equation*}
\frac{\partial ^2 e}{\partial \gamma_{AB} \partial \gamma_{CD}} \eta_C \eta_A \xi_B \xi_D > 0 \qquad (\eta ,\, \xi \neq 0) \, ,
\end{equation*}
because the first term on the right-hand side of (\ref{LegHadgen}) (containing $\frac{\partial e}{\partial \gamma_{AB}}$) vanishes.
Note that this condition is satisfied by virtue of the Postulate \ref{pmine} that $e$ has a strict minimum at $\stackrel{\circ}{\gamma}$, since this implies
\begin{equation*}
\frac{\partial ^2 e}{\partial \gamma_{AB} \partial \gamma_{CD}} (\stackrel{\circ}{\gamma}) \sigma_{AB} \sigma_{CD} > 0
\end{equation*}
for any non-zero symmetric $\sigma_{AB}$ and, in particular, for
\begin{equation*}
\sigma_{AB}=\xi_A\eta_B+\eta_A\xi_B \, .
\end{equation*}
In the general case, we define $\eta^A$ by $\eta^A E_A^a v^i_a = \eta^i$ and we have
\begin{equation*}
|\eta|^2=g_{ij}\eta^i \eta^j=g_{ij} v^i_a v^j_b E_A^a E_B^b \eta^A \eta^B = \gamma_{AB}\eta^A \eta^B \, .
\end{equation*}
Additionally,
\begin{equation*}
\gamma_{AB} \eta^B= g_{ij} v^i_a v^j_b E_A^a E_B^b \eta^B= g_{ij} v^i_a E_A^a \eta^j = \eta_A \, ,
\end{equation*}
and hence $\eta^A=\left(\gamma^{-1}\right)^{AB}\eta_B$.
We finally have for the inverse $\left(\gamma^{-1}\right)^{AB}$ of $\gamma_{AB}$ with $|\eta|^2=\eta_A\eta^A= \eta_A\left(\gamma^{-1}\right)^{AB}\eta_B$
\begin{equation*}
|\eta|^2=\left(\gamma^{-1}\right)^{AB}\eta_A\eta_B \, .
\end{equation*}
\section{Uniform Distribution of Edge Dislocations in $2$d}
The case of a uniform distribution of 2-dimensional \emph{edge dislocations} is realized by the material manifold $\mathcal N$ being the affine group, see \ref{edag}.
The corresponding metric $\stackrel{\circ}{n}$ from (\ref{hypmetafgr}) is isometric to the metric of the hyperbolic plane.
In general, we consider the total energy $E$ (\ref{energy}) of a domain $\Omega$ in the material manifold $\mathcal N$.
In agreement with Postulate \ref{pmine}, we assume that the energy per unit mass $e$ has a strict minimum at $\stackrel{\circ}{\gamma}$.
The symmetry of the problem motivates the choice of an isotropic energy function $e$ that is a symmetric function of the eigenvalues of $m=\phi^* g$ relative to $\stackrel{\circ}{n}$.
Therefore, in this model case, the crystalline structure is eliminated in favor of the Riemannian manifold $(\mathcal N, \stackrel{\circ}{n})$.
\subsection{Isotropic Energy Function}\label{ToyNrgS}
Concerning our toy energy function, we make the following basic choice:
\begin{equation} \label{toynrg}
e(\gamma)=e(\lambda_1, \ldots , \lambda_n)= \frac 1 2 \sum_{k=1}^n \left(\lambda_k -1\right)^2 \geq 0
\end{equation}
that satisfies $e(\lambda_1=1, \ldots , \lambda_n=1)=0$. Thus we have a strict minimum of the energy density at $\gamma= \, \stackrel{\circ}{\gamma}$. Note that, in (\ref{toynrg}), we are subtracting the rest mass energy, something which does not affect the variation of $E$, see also the discussion at the end of Section \ref{staticsetting}.
Therefore,
\begin{equation} \label{ToyEnergy}
e(\gamma)=\frac 1 2 \sum_{k=1}^n \left(\lambda_k^2-2\lambda_k+1\right)=\frac 1 2 \mathrm{tr}\, \gamma^2 -\mathrm{tr}\, \gamma +\frac n 2 =\frac 1 2 \sum_{A,B=1}^n \gamma_{AB}^2- \sum_{A=1}^n \gamma_{AA} +\frac n 2 \, .
\end{equation}
Here and in the following the basis $(E_A: A=1, \ldots, n)$ for $\mathcal V$ is chosen to be orthonormal relative to $\stackrel{\circ}{\gamma}$, hence $\stackrel{\circ}{\gamma}_{AB}=\delta_{AB}$. For such a choice to be compatible with the condition $\omega(E_1, \ldots, E_n)=1$, we must choose the physical unit of mass so that the mass density corresponding to $\stackrel{\circ}{\gamma}$ is equal to $1$.
In case that the metric of the target space is Euclidean, $g_{ij}=\delta_{ij}$, we find from (\ref{gammapbm})
\begin{equation}\label{gammacomp}
\gamma_{AB}= E_A^a E_B^b \frac{\partial x^i}{\partial y^a}\frac{\partial x^j}{\partial y^b}\delta_{ij}=E_A^a E_B^b \frac{\partial \phi}{\partial y^a}\cdot\frac{\partial \phi}{\partial y^b} \, ,
\end{equation}
where the dot denotes the Euclidean inner product.
For the total energy $E$ (\ref{energy}) of a domain $\Omega \subset \subset \mathcal N$, where $\omega$ coincides with the volume form on $\mathcal V$ induced by $\stackrel{\circ}{\gamma}$, we have $d\mu_{\omega}=d\mu_{\stackrel{\circ}{n}}$ (the volume form on $\mathcal N$ corresponding to the Riemannian metric $\stackrel{\circ}{n}$), and we obtain for the total energy $E$ expressed in terms of the components of $\gamma$,
\begin{equation} \label{nrgcomp}
E=\int\limits_{\Omega}\left( \frac 1 2 \sum_{A,B=1}^n \gamma_{AB}^2 -\sum_{A=1}^n \gamma_{AA} + \frac n 2 \right) d\mu_{\stackrel{\circ}{n}} \, .
\end{equation}
In the case $n=2$, the invariants of a linear mapping are, respectively, the trace and the determinant. Expressed in terms of the eigenvalues $\lambda_1, \lambda_2$, they read:
\begin{eqnarray*}
\tau & = & \lambda_1 + \lambda_2= \textrm{tr}\, m \, , \\
\delta & = & \lambda_1 \lambda_2 = \det m \, .
\end{eqnarray*}
We have:
\begin{equation*}
\tau^2=\lambda_1^2 + \lambda_2^2 + 2 \delta \, .
\end{equation*}
The toy energy (\ref{toynrg}) thus reads
\begin{equation*}
e(\lambda_1, \lambda_2)=\frac 1 2 \left( (\lambda_1 -1)^2+ (\lambda_1 -1)^2\right) = \frac 1 2 \left( \tau^2 -2 \tau - 2 \delta + 2\right) \, .
\end{equation*}
\subsection{Expansion of the Hyperbolic Metric} \label{HyperbolicExpansion}
We shall perform a dilation of the hyperbolic plane in order to be able to establish an existence result for a fixed domain, from which we can deduce an analogous result for energy minimizing mappings from a suitably small domain in the original hyperbolic plane to the Euclidean plane.
We start with the usual metric, of constant curvature $-1$ for the hyperbolic plane $H$, expressed in polar normal coordinates $(R, \theta)$:
\begin{equation*}
ds_{H}^2 = dR^2 + \sinh^2 R d\theta^2 \, .
\end{equation*}
Given a parameter $\varepsilon \in \mathbb R^+$, we dilate $H$ by a factor $\varepsilon^{-1}$ obtaining $H_{\varepsilon}$. The corresponding metric, of curvature $-\varepsilon^2$, is:
\begin{equation*}
ds_{H_{\varepsilon}}^2 = \varepsilon^{-2} \left( dR^2 + \sinh^2 R d\theta^2 \right) =dr^2 + F_{\varepsilon}^2(r) d\theta^2 \, ,
\end{equation*}
where $r=\varepsilon^{-1}R$ and
\begin{equation*}
F_{\varepsilon}(r)= \varepsilon^{-1} \sinh \left(\varepsilon r \right)=\varepsilon^{-1}\left(\varepsilon r + \frac 1 {3!}\left(\varepsilon r\right)^3 +\frac 1 {5!}\left(\varepsilon r\right)^5+ \ldots \right) \, .
\end{equation*}
$(r, \theta)$ are polar normal coordinates in $H_{\varepsilon}$. The expression for $F_{\epsilon}^2(r)$ thus reads:
\begin{equation} \label{epsilonexp}
F_{\varepsilon}^2(r)=r^2\left(1+\frac 1 3 (\varepsilon r)^2 + \frac 2 {45} (\varepsilon r)^4 + \ldots \right) \, .
\end{equation}
The main point in the expansion (\ref{epsilonexp}) is that $F_{\varepsilon}^2(r)$ is a real analytic function converging to $r^2$ for $\varepsilon \to 0$, corresponding to the transition of the negatively curved hyperbolic plane to the flat Euclidean plane.
In order to simplify the calculations, we go back to rectangular coordinates.
We denote rectangular normal coordinates in $H_{\varepsilon}$ by $(y^i: i=1,2)$
\begin{eqnarray*}
y^1 & = & r \cos \theta \, , \\
y^2 & = & r \sin \theta \, .
\end{eqnarray*}
Then
\begin{equation*}
ds_{H_{\varepsilon}}^2 = \left(dy^1\right)^2 + \left(dy^2\right)^2 + \varepsilon^2\left(\frac 1 3 + \frac 2{45}(\varepsilon r)^2+ \ldots \right) r^4 d\theta^2 \, .
\end{equation*}
Since
\begin{equation*}
d\theta = \frac 1{r^2}\left(-y^2 dy^1+ y^1 dy^2 \right) \, ,
\end{equation*}
denoting by $ds_E^2$ the Euclidean metric
\begin{equation*}
ds_E^2= \left(dy^1\right)^2 + \left(dy^2\right)^2 \, ,
\end{equation*}
the metric of $H_{\varepsilon}$ takes the form:
\begin{equation} \label{hypmetrectcoords}
ds_{H_{\varepsilon}}^2 = ds_E^2 + \varepsilon^2 \left(\frac 1 3 + \frac 2 {45} (\varepsilon r)^2+ \ldots \right)\left(-y^2 dy^1 + y^1 dy^2 \right)^2 \, .
\end{equation}
The crucial point in the above expression for the metric $h$ of $H_{\varepsilon}$
is that it takes the form
\begin{equation} \label{hypmetexpeps}
h_{ab}=\delta_{ab}+ \varepsilon^2 f_{ab} \, ,
\end{equation}
where $f_{ab}$ are analytic functions in $(y^1,y^2)$ and $\varepsilon^2$. In fact,
\begin{equation} \label{rectmeth}
\left(-y^2 dy^1+y^1 dy^2 \right)^2 = \left(r^2 \delta_{ab}-y^a y^b \right)dy^a dy^b \, ,
\end{equation}
hence, by (\ref{hypmetrectcoords}), (\ref{hypmetexpeps}) and (\ref{rectmeth}),
\begin{equation}
f_{ab}=\left(\frac 1 3 + \frac 2 {45} (\varepsilon r)^2+ \ldots \right)\left(r^2 \delta_{ab} - y^a y^b \right) \, .
\end{equation}
That is,
\begin{equation*}
f_{ab}= f(\varepsilon^2 r^2)\left(r^2 \delta_{ab} - y^a y^b \right) \quad , \, \textrm{where} \quad f(x)= \frac 1 3 + \frac 2 {45} x + \ldots \, .
\end{equation*}
\section{Uniform Distribution of Screw Dislocations in $3$d} \label{udsd}
The case of a uniform distribution of (3-dimensional) \emph{screw dislocations} is realized when the material manifold $\mathcal N$ is the Heisenberg group, see \ref{sdhg}.
The corresponding metric $\stackrel{\circ}{n}$ from (\ref{homspacemet}) is isometric to the metric of a homogeneous but anisotropic space.
\subsection{Anisotropic Energy Function}
The symmetry of the problem in this case motivates the choice of an anisotropic energy function $e$, since the dislocation lines have a preferred direction.
Therefore, in this model case, the crystalline structure cannot be eliminated in favor of the Riemannian manifold $(\mathcal N, \stackrel{\circ}{n})$, in contrast to the $2$-dimensional case previously discussed.
The question of the right choice of energy per unit mass for the Heisenberg group is investigated in the following.
Consider the orthogonal transformation $O$ of $\mathcal V$,
given by $(X',Y',Z')=O(X,Y,Z)$ with
\begin{eqnarray}
X' & = & \cos \theta \cdot X + \sin \theta \cdot Y \, , \nonumber \\
Y' & = & -\sin \theta \cdot X + \cos \theta \cdot Y \, , \label{orthtrsf} \\
Z' & = & Z \, . \nonumber
\end{eqnarray}
The commutation relations $[X,Y]=Z$, $[X,Z]=[Y,Z]=0$ are preserved by this transformation since
\begin{eqnarray*}
\left[X',Y'\right] & = & \cos^2 \theta [X,Y] - \sin^2 \theta [Y,X]=\left(\cos^2 \theta +\sin^2 \theta\right) [X,Y]=Z=Z' \, , \\
\left[X',Z'\right] & = & [Y',Z'] = 0 \, .
\end{eqnarray*}
Consider the metric $m=\phi^* g$. Since $m$ is symmetric, we have generally six independent components in three dimensions. This also holds for $\gamma$, the corresponding inner product on the crystalline structure:
\begin{equation*}
\gamma = \left( \begin{array}{cc} \bar{\gamma}_{AB} & \theta_A \\ \theta_A^T & \rho \end{array} \right) \, ,
\end{equation*}
where the six components are
\begin{eqnarray*}
\bar{\gamma}_{AB} & = & \gamma_{AB} \qquad A,B=1,2 \, ,\\
\theta_A & = & \gamma_{A3} \qquad A=1,2 \, , \\
\rho & = & \gamma_{33} \, .
\end{eqnarray*}
The energy density $e$ depends on the variables $(\bar{\gamma}, \theta, \rho)$. $e$ must be invariant under orthogonal transformations (\ref{orthtrsf}), that is $e(\bar{\gamma}, \theta, \rho)= e (O \bar{\gamma} \widetilde O, O \theta, \rho)$, for any $O$ given by (\ref{orthtrsf}).
$|\theta|^2$ is such an invariant; together with the invariants $\textrm{tr}\,\bar{\gamma}$ and $\textrm{tr}\,\bar{\gamma}^2$, which are the same as in the two-dimensional case, we have the following four invariants:
\begin{equation} \label{invariants3d}
\textrm{tr}\,\bar{\gamma} \, , \, \textrm{tr}\,\bar{\gamma}^2 \, ,\, \abs{\theta}^2 \, , \, \rho \, .
\end{equation}
Thus, starting with six variables, we have eliminated two and are left with an energy per unit mass of the form
\begin{equation*}
e=e(\mu_1, \mu_2, \abs{\theta}^2, \rho) \, ,
\end{equation*}
where $\mu_{1,2}$ are the eigenvalues of $\bar{\gamma}_{AB}$ with respect to $\overset{\circ}{\bar{\gamma}}_{AB}$ in the $XY$-plane ($A,B=1,2$).
We may choose in particular:
\begin{equation}\label{AnisotropicEnergy}
e=\frac 1 2 \left( (\mu_1 -1)^2 + (\mu_2 -1)^2\right) +\frac{\alpha}{2} \abs{\theta}^2+ \frac{\beta}{2} (\rho-1)^2 \, ,
\end{equation}
which has a minimum at the identity $\gamma_{AB}=\stackrel{\circ}{\gamma}_{AB}=\delta_{AB}$.
It is not surprising that the screw dislocations, which have a distinguished direction, break the isotropy and we are lead to an anisotropic energy of the form (\ref{AnisotropicEnergy}), which is invariant under the transformation (\ref{orthtrsf}).
\subsection{The Heisenberg Group as a Homogeneous Space} \label{HGahs}
Let
\begin{equation} \label{vectfHeis}
X=\frac{\partial}{\partial x} \quad , \quad Y=\frac{\partial}{\partial y}+x \frac{\partial}{\partial z} \quad , \quad Z=\frac{\partial}{\partial z}
\end{equation}
be, as in Section \ref{udsd}, the basis for the Lie algebra $\mathcal V$ of the Heisenberg group. We have the commutation relations
\begin{equation} \label{commutationHeisenberg}
[X, Y]= Z \quad , \quad [X, Z]=[Y, Z]=0 \, .
\end{equation}
The basis of $1$-forms $(\xi, \zeta, \eta)$ dual to $X, Y, Z$ is
\begin{equation*}
\xi=dx \quad , \quad \zeta = dy \quad , \quad \eta= dz - x dy \, .
\end{equation*}
A left-invariant metric on the Heisenberg group is, up to isometries and an overall scale factor, given by:
\begin{equation} \label{HGmetrit}
\stackrel{\circ}{n} = \xi \otimes \xi + \zeta \otimes \zeta + e^{2\beta} \eta \otimes \eta \, ,
\end{equation}
that is:
\begin{equation} \label{Heisenbergmetric}
ds^2= dx^2 + dy^2 + e^{2\beta}(dz- x dy)^2 \, ,
\end{equation}
where $\beta$ is a real constant.
The metric (\ref{Heisenbergmetric}) is called a Bianchi type VII metric (for the classification into Bianchi types, see \cite{LL2}).
It represents a homogeneous space which is not isotropic.
The inner product $\stackrel{\circ}{\gamma}$ on $\mathcal V$ giving rise to the metric $\stackrel{\circ}{n}$ is given in the basis $(X,Y,Z)$ by:
\begin{eqnarray*}
\stackrel{\circ}{\gamma}(X,X) & = & \stackrel{\circ}{\gamma}(Y,Y) \quad , \quad \stackrel{\circ}{\gamma}(Z,Z)=e^{2\beta} \, , \\
\stackrel{\circ}{\gamma}(X,Y) & = & \stackrel{\circ}{\gamma}(X,Z) = \stackrel{\circ}{\gamma} (Y,Z) = 0 \, .
\end{eqnarray*}
Thus defining
\begin{equation} \label{onbHG}
E_1=X \quad , \quad E_2 = Y \quad , \quad E_3=e^{-\beta} Z \, ,
\end{equation}
$(E_A: A=1,2,3)$ is an orthonormal basis for $\stackrel{\circ}{\gamma}$:
\begin{equation} \label{minmetHG}
\stackrel{\circ}{\gamma}_{AB} := \stackrel{\circ}{\gamma}(E_A,E_B) = \delta_{AB} \, .
\end{equation}
The dual basis is:
\begin{equation}\label{dualbHG}
\omega^1 = \xi \quad , \quad \omega^2 = \zeta \quad , \quad \omega^3 = e^{\beta} \eta \, .
\end{equation}
In the following we shall denote the metric $\stackrel{\circ}{n}$ by $h$. Thus, in terms of the basis $(\omega^A: A=1,2,3)$ we have:
\begin{equation*}
h = \sum_A \omega^A \otimes \omega^A \, .
\end{equation*}
Now $(\mathcal N, h)$ being a homogeneous space, any point in $\mathcal N$ may be taken as the origin. Consider a local coordinate system $(y^a: a=1,2,3)$ on $\mathcal N$ with origin a given point. Since in the Euclidean space $E^3$ we may set up a rectangular coordinate system $(x^i: i=1,2,3)$, a mapping $id$ is then defined in the domain of the coordinate system on $\mathcal N$ by:
\begin{equation*}
id(y^1,y^2,y^3)=(y^1,y^2,y^3) \in E^3 \, ,
\end{equation*}
that is, $id$ is expressed in the respective coordinate systems as the identity mapping. Thus $m$, the pullback to $\mathcal N$ by $id$ of the Euclidean metric $\delta_{ij} dx^i \otimes dx^j$ is
\begin{equation*}
m = \delta_{ab} dy^a \otimes dy^b
\end{equation*}
in the local coordinates $(y^a: a=1,2,3)$ on $\mathcal N$. The following problem then arises in connection with the analytic method to be presented below: given any point in $\mathcal N$, find a local coordinate system $(y^a: a=1,2,3)$ on $\mathcal N$ with origin the given point such that $\gamma_{AB}$, the components at $y$ of the corresponding inner product on $\mathcal V$ in the basis $(E_A: A=1,2,3)$, that is:
\begin{equation*}
\gamma_{AB}(y) = m_{ab}(y) E_A^a(y) E_B^b(y) = \delta_{ab} E_A^a(y) E_B^b(y)
\end{equation*}
satisfy:
\begin{equation*}
\gamma_{AB}(y)-\stackrel{\circ}{\gamma}_{AB} = O(|y|^2) \, .
\end{equation*}
(Recall that $\stackrel{\circ}{\gamma}_{AB}=\delta_{AB}$.) The solution is simply to set up Riemannian normal coordinates on $(\mathcal N,h)$ with origin the given point. For, the components of $h$ in such coordinates satisfy:
\begin{equation*}
h_{ab}(y)=\delta_{ab}+ O(|y|^2) \, .
\end{equation*}
Since
\begin{equation*}
h_{ab}(y) E_A^a(y) E_B^b(y) = \delta_{AB}
\end{equation*}
identically, then
\begin{equation*}
\gamma_{AB}(y)-\delta_{AB}= \left(\delta_{ab}-h_{ab}(y)\right) E_A^a(y) E_B^b(y) = O(|y|^2) \, ,
\end{equation*}
as required.
\section{Scaling Properties} \label{scaling}
\subsection{The General Isotropic Case}
Let $\mathcal N=\Omega \subset \mathbb R^n$ and $\phi(\Omega) \subset E^n$ be the domain and the target space, respectively, of the mapping $\phi$ with the same properties as in (\ref{defphi}). We fix coordinates in $\mathcal N$ and also fix an origin in $E^n$.
We may assume that the coordinate origin in $\mathcal N$ is mapped by $\phi$ to the origin in $E^n$. Consider
\begin{equation*}
\tilde \phi(y)=l \phi \left(\frac y l \right) \, ,
\end{equation*}
where $l>0$ is a given positive constant.
If $U$ is a domain in $E^n$, we denote by $lU$ the domain $\{ lx : x \in U\}$. Similarly for a domain in $\mathbb R^n$. The domain of $\tilde \phi$ is $\tilde \Omega= l \Omega$, and
\begin{equation*}
\tilde \phi : \tilde \Omega = l \Omega \to \tilde \phi (\tilde \Omega) =l \phi(\Omega) \, .
\end{equation*}
The change from $\phi$ to $\tilde \phi$ induces a change from $m$ to $\tilde m$ (and similarly for the inverses) as follows
\begin{equation}\label{scalemetric}
\tilde m_{ab} (y) = \sum_{i=1}^n \frac{\partial \tilde \phi^i(y)}{\partial y^a}\frac{\partial \tilde \phi^i(y)}{\partial y^b}= m_{ab}\left(\frac y l\right) \quad , \quad \left(\tilde m^{-1}\right)^{ab}(y)=\left(m^{-1}\right)^{ab}\left(\frac y l \right) \, .
\end{equation}
For, since $\tilde \phi^i(y)=l \phi^i \left( \frac y l \right)$, we have
\begin{equation} \label{scalephi}
\frac{\partial \tilde \phi^i(y)}{\partial y^a}=l\frac{\partial \phi^i(y/l)}{\partial y^a}\frac 1 l = \frac{\partial \phi^i(y/l)}{\partial y^a} \, .
\end{equation}
We consider here the case of an isotropic energy function. We have seen that in this case the crystalline structure $\mathcal V$ can be eliminated in favor of a Riemannian metric $h$.
Let us define a new metric $\tilde h$ on $\mathcal N$ by $\tilde h_{ab}(y)=h_{ab}\left(\frac y l \right)$.
Note that $\tilde h$ is \emph{not} isometric to $h$. In the case of the toy energy function (\ref{ToyEnergy}), we have (see (\ref{stressonN}) below),
\begin{equation*}
\sqrt{\frac{\det \tilde m}{\det \tilde h}}\tilde S^{ab}= 2 \left(\tilde h^{-1}\right)^{ac} \left(\tilde h^{-1}\right)^{bd}\left(\tilde h_{cd}-\tilde m_{cd}\right) \, ,
\end{equation*}
and similarly with $\tilde h$, $\tilde m$, $\tilde S$ replaced by $h$, $m$, $S$.
Using (\ref{scalemetric}) we then obtain:
\begin{equation} \label{scalestress}
\tilde S^{ab}(y) =S^{ab}\left(\frac y l \right) \, .
\end{equation}
Now the equations (\ref{divstresszeroN}) transform as follows. From the definition of the covariant derivative, we have
\begin{equation} \label{defcovderS}
\overset{m}{\nabla}_b S^{ab}=\frac{\partial S^{ab}}{\partial y^b} + \overset{m}{\Gamma_{bc}^a} S^{bc} + \overset{m}{\Gamma_{ac}^b} S^{ac} \, ,
\end{equation}
where $\overset{m}{\Gamma_{bc}^a}$ are the Christoffel symbols with respect to $m$ defined by
\begin{equation} \label{defChristoffel}
\overset{m}{\Gamma_{bc}^a}=\frac 1 2 (m^{-1})^{ad} \left(\frac{\partial m_{bd}}{\partial y^c} + \frac{\partial m_{cd}}{\partial y^b}-\frac{\partial m_{bc}}{\partial y^d} \right) \, ,
\end{equation}
as well as analogous expressions for $\overset{\tilde m}{\nabla}_b \tilde S^{ab}$ and $\overset{\tilde m}{\Gamma_{bc}^a}$.
From (\ref{scalemetric}) and (\ref{scalestress}) we then obtain
\begin{equation} \label{derscalemandS}
\frac{\partial \tilde m_{bc}}{\partial y^d}(y)=\frac 1 l \frac{\partial m_{bc}}{\partial y^d} \left(\frac y l \right) \qquad \textrm{and} \qquad \frac{\partial \tilde S^{ab}}{\partial y^b}(y)=\frac 1 l \frac{\partial S^{ab}}{\partial y^b}\left(\frac y l \right) \, ,
\end{equation}
and, consequently, from (\ref{defChristoffel}),
\begin{equation} \label{scaleChristoffel}
\overset{\tilde m}{\Gamma_{bc}^a} (y)= \frac 1 l \overset{m}{\Gamma_{bc}^a} \left(\frac y l \right) \, .
\end{equation}
We conclude, using (\ref{defcovderS}) with (\ref{derscalemandS}) and (\ref{scaleChristoffel}),
\begin{equation*}
\overset{\tilde m}{\nabla}_b \tilde S^{ab}(y)=\frac 1 l \left(\overset{m}{\nabla}_b S^{ab} \right) \left(\frac y l \right) \, .
\end{equation*}
So, if $\phi$ is a solution relative to $h$ and $\Omega$, $\tilde \phi$ is a solution relative to $\tilde h$ and $\tilde \Omega$, the tangent plane to $\tilde \Omega = l\Omega$ at the point $l y \in \partial \tilde \Omega$ being parallel to the tangent plane to $\partial \Omega$ at $y$ relative to the linear structure of $\mathbb R^n \supset \Omega$.
If $(E_A: A=1,\ldots,n)$ is an orthonormal frame field for $h$ then $(\tilde E_A: A=1,\ldots,n)$, with $\tilde E_A$ defined by
\begin{equation*}
\tilde E_A^a(y)=E_A^a\left(\frac y l\right) \, ,
\end{equation*}
is an orthonormal frame field for $\tilde h$. Also, using a formula similar to (\ref{scaleChristoffel}) with $h$ in the role of $m$, we deduce
\begin{equation*}
\tilde R^a_{bcd}(y)= l^{-2} R^a_{bcd}\left(\frac y l \right) \, .
\end{equation*}
It follows that, if $K_{\Pi}$ is the sectional curvature of $h$ corresponding to the plane $\Pi$ at $y$ and $\tilde K_{\tilde \Pi}$ is the sectional curvature of $\tilde h$ corresponding to the plane $\tilde \Pi$ at $\tilde y=S_l y=ly$, where $\tilde \Pi=dS_l(\Pi)$ ($S_l$ the scaling map), then
\begin{equation*}
\tilde K_{\tilde \Pi} = l^{-2} K_{\Pi} \, .
\end{equation*}
\subsection{The $2$d Case}
Consider the two-dimensional case. The metric of the hyperbolic plane of curvature $-\varepsilon^2$ is given in Riemannian normal coordinates by:
\begin{eqnarray*}
h_{ab} & = & \delta_{ab}+\varepsilon^2 f_{ab}(y) \, , \\
f_{ab}(y) & = & f(\varepsilon^2 r^2) l_{ab}(y) \, ,\\
l_{ab} (y) & = & \delta_{ab} r^2 -y^a y^b \, ,
\end{eqnarray*}
and $f(z)$ is an entire function with $f(0)=\frac 1 3$.
We now consider the metric $\tilde h$, the components of which in the above coordinates are: $\tilde h_{ab}(y)=h_{ab}(y/l)$. We have:
\begin{eqnarray*}
\tilde h_{ab}(y) & = & \delta_{ab}+\varepsilon^2 f_{ab}\left(\frac y l \right) \, ,\\
f_{ab} \left(\frac y l \right) & = & f\left(\varepsilon^2 \frac{r^2}{l^2}\right) \frac{1}{l^2} l_{ab}(y) \, ,
\end{eqnarray*}
therefore,
\begin{equation*}
\tilde h_{ab}(y) = \delta_{ab}+ \tilde \varepsilon^2 f \left( \tilde \varepsilon^2 r^2\right) l_{ab}(y)=\delta_{ab}+ \tilde \varepsilon^2 \tilde f_{ab}(y) \, ,
\end{equation*}
where $\tilde \varepsilon = \varepsilon / l$ and $\tilde f_{ab}(y)$ is $f_{ab}(y)$ with $\varepsilon$ replaced by $\tilde \varepsilon$.
While the curvature $K$ of $h$ is $-\varepsilon^2$, the curvature $\tilde K$ of $\tilde h$ is
\begin{equation*}
-\tilde \varepsilon^2 = -\frac{\varepsilon^2}{l^2} \, .
\end{equation*}
Given a domain $\Omega_1$, we shall show in Part III below that there is a $\varepsilon_1>0$ such that we can solve equations (\ref{divstresszeroN}) for all $0< \varepsilon < \varepsilon_1$.
Let $\phi_{1,\varepsilon}$ be the solution corresponding to $\Omega_1$ and $\varepsilon \in (0,\varepsilon_1)$. $\Omega_1$ is a domain in the hyperbolic plane of curvature $-\varepsilon^2$. We define $\phi_{l,\varepsilon/l}$ by
\begin{equation} \label{defphiscale}
\phi_{l, \varepsilon/l}(y)= l \phi_{1,\varepsilon}\left(\frac y l \right) \, .
\end{equation}
This is the solution corresponding to the domain $\Omega_l=l\Omega_1$ in the hyperbolic plane of curvature $-\varepsilon^2/l^2$.
Choosing then $l=\varepsilon$ we have from (\ref{defphiscale})
\begin{equation*}
\phi_{l,1}(y)= l \phi_{1,l} \left(\frac y l \right) = l \phi_{1,\varepsilon}\left(\frac y l \right)
\end{equation*}
a solution of the problem for the domain $\Omega_l:=l\Omega_1=\varepsilon \Omega_1$ in the hyperbolic plane of curvature $-\varepsilon^2/l^2=-1$, that is, the standard hyperbolic plane. In conclusion, once we have a solution for the domain $\Omega_1$ and curvature $-\varepsilon^2$, we automatically have a solution for the smaller (rescaled) domain $\Omega_l=\varepsilon \Omega_1$ and curvature $-1$.
Consider the stress $T^{ij}$ and its rescaled version $\tilde T^{ij}$ at the respective points in Euclidean space. From (\ref{scalestress}) and (\ref{scalephi}) we conclude:
\begin{eqnarray*}
T^{ij}\left(\phi(y)\right) & = & S^{ab}(y) \frac{\partial \phi^i(y)}{\partial y^a}\frac{\partial \phi^j(y)}{\partial y^b} \, ,\\
\tilde T^{ij}\left(\tilde \phi(y)\right) & = & \tilde S^{ab}(y) \frac{\partial \tilde \phi^i(y)}{\partial y^a}\frac{\partial \tilde \phi^j(y)}{\partial y^b} = S^{ab}\left( \frac y l \right) \frac{\partial \phi^i(y/l)}{\partial y^a}\frac{\partial \phi^j(y/l)}{\partial y^b} \\
& = & T^{ij}\left(\phi\left(\frac y l \right) \right) \, .
\end{eqnarray*}
The rescaled stress $\tilde T$ at the rescaled point $\tilde \phi(y)$ is thus the same as the original stress $T$ at the point $\phi(y/l)$.
\subsection{The $3$d Case}
Let $(E_A: A=1, \ldots , 3)$ be the vectorfields (\ref{onbHG}) that satisfy the commutation relations $[E_1, E_2]=e^{\beta}E_3$, $[E_1, E_3]=[E_2,E_3]=0$ and $(\omega^A :A=1, \ldots , 3)$ the dual $1$-forms (\ref{dualbHG}), so that
\begin{equation}
h=\sum_{A=1}^3 \omega^A \otimes \omega^A \quad , \quad h_{AB}=h(E_A,E_B)=\delta_{AB} \, .
\end{equation}
We introduce Riemannian normal coordinates $(y^a : a=1,2,3)$ for $h$ at a given point in $\mathcal N$, which we take as the origin. The components of the metric $h$ in these coordinates are of the form:
\begin{equation} \label{RNKh}
h_{ab}(y)=\delta_{ab}+\epsilon_{ab}(y) \quad , \quad \epsilon_{ab}(y) = O(\abs{y}^2) \, .
\end{equation}
Let $id$ be the identity map defined in Section \ref{HGahs} and $m$ the pullback by $id$ of the Euclidean metric $\delta_{ij} dx^i \otimes dx^j$. Then the components $m_{ab}$ of $m$ in the coordinates $(y^a: a=1,2,3)$ are simply $m_{ab}=\delta_{ab}$ and the corresponding inner product on $\mathcal V$, which depends on $y$, is given by:
\begin{eqnarray*}
\gamma_{AB}(y) & := & \gamma(y)(E_A,E_B) \\
& = & m_{ab}(y)E_A^a(y) E_B^b(y) \\
& = & \delta_{ab} E_A^a(y) E_B^b(y) \\
& = & \left(h_{ab}(y)-\epsilon_{ab}(y)\right)E_A^a(y) E_B^b(y) \\
& = & \delta_{AB}-\epsilon_{AB}(y) \, ,
\end{eqnarray*}
where $\epsilon_{AB}(y)=\epsilon_{ab}(y) E_A^a(y) E_B^b(y)= O(\abs{y}^2)$.
We now dilate the Heisenberg group metric $h$ by the factor $l>1$, i.e.~we set:
\begin{equation*}
\tilde h = l^2 h \quad , \quad \tilde h(E_A, E_B)= l^2 h(E_A, E_B)=l^2 \delta_{AB} \, .
\end{equation*}
Define now $\tilde E_A=l^{-1} E_A$, so
\begin{equation*}
\tilde h(\tilde E_A, \tilde E_B)= \delta_{AB} \, ,
\end{equation*}
that is $(\tilde E_A: A=1,2,3)$ is an orthonormal frame field relative to $\tilde h$. We denote by $ \overset{\tilde\circ}{\gamma}$ the corresponding inner product on $\mathcal V$, i.e.~$ \overset{\tilde\circ}{\gamma}(\tilde E_A, \tilde E_B)=\delta_{AB}$. Remark that the commutation relations of the frame field $(\tilde E_A : A=1,2,3)$ read
\begin{equation*}
[\tilde E_1, \tilde E_2]=l^{-1}\tilde E_3 \quad , \quad [\tilde E_1, \tilde E_3]=[\tilde E_2, \tilde E_3]=0 \, .
\end{equation*}
Now:
\begin{equation*}
\tilde h = l^2 h = l^2 h_{ab}(y) dy^a \otimes dy^b \, .
\end{equation*}
Set $\tilde y^a= l y^a$, then $\tilde h = \tilde h_{ab}(\tilde y) d\tilde y^a \otimes \tilde y^b$ and hence
\begin{equation} \label{scaleHGmet}
\tilde h_{ab}(\tilde y) = h_{ab}\left(\frac{\tilde y}{l}\right) \, .
\end{equation}
Consider the geodesic ray through the origin of the coordinate system $y^a$:
\begin{equation*}
y^a = \lambda^a t \quad , \quad \tilde y^a = \tilde \lambda^a t \quad \textrm{where} \quad \tilde \lambda^a= l \lambda^a \, .
\end{equation*}
For the Christoffel symbols of the metric $h$ with respect to the coordinates $y^a$ we have:
\begin{equation*}
\Gamma_{bc}^a(\lambda t)\lambda^b \lambda^c = 0 \, .
\end{equation*}
But from (\ref{scaleChristoffel}),
\begin{equation*}
\tilde \Gamma_{bc}^a(\tilde y)= \frac 1 l \Gamma_{bc}^a\left(\frac{\tilde y}{l}\right) \, ,
\end{equation*}
hence:
\begin{equation}
\tilde \Gamma_{bc}^a(\tilde \lambda t) \tilde \lambda^b \tilde \lambda^c= \frac 1 l \Gamma_{bc}^a(\lambda t) l^2 \lambda^b \lambda^c=0 \, .
\end{equation}
Thus $(\tilde y^a: a=1,2,3)$ are Riemannian normal coordinates for $\tilde h$.
We remark that the mapping $id$ defined in Section \ref{HGahs} depends on the choice of local coordinates in $\mathcal N$. Let us denote by $\tilde{id}$ the mapping associated to the coordinates $(\tilde y^a : a=1,2,3)$, reserving the notation $id$ for the mapping associated to the original coordinates $(y^a : a=1,2,3)$. Let $\tilde m$ be the pullback of the Euclidean metric $\delta_{ij} dx^i \otimes dx^j$ by $\tilde{id}$. Then the components $\tilde m_{ab}$ of $\tilde m$ in the coordinates $(\tilde y^a : a=1,2,3)$ are again simply $\tilde m_{ab}=\delta_{ab}$.
Now, by (\ref{RNKh}) and (\ref{scaleHGmet}),
\begin{equation*}
\tilde h_{ab}(\tilde y) = \delta_{ab} + \tilde \epsilon_{ab} (\tilde y) \, ,
\end{equation*}
where
\begin{equation*}
\tilde \epsilon_{ab}(\tilde y) = \epsilon_{ab}\left(\frac{\tilde y}{l}\right) \, .
\end{equation*}
Therefore:
\begin{equation*}
\tilde \epsilon_{ab}(\tilde y) = O\left(l^{-2} |\tilde y|^2\right) \, ,
\end{equation*}
and we obtain, in analogy with the above:
\begin{eqnarray*}
\tilde \gamma_{AB}(\tilde y) & := & \tilde \gamma(\tilde y)(\tilde E_A,\tilde E_B) \\
& = & \tilde m_{ab}(\tilde y)\tilde E_A^a(\tilde y) \tilde E_B^b(\tilde y) \\
& = & \delta_{ab} \tilde E_A^a(\tilde y) \tilde E_B^b(\tilde y) \\
& = & \left(\tilde h_{ab}(\tilde y)-\tilde \epsilon_{ab}(\tilde y)\right)\tilde E_A^a(\tilde y) \tilde E_B^b(\tilde y) \\
& = & \delta_{AB}-\tilde \epsilon_{AB}(\tilde y) \, ,
\end{eqnarray*}
where $\tilde \epsilon_{AB}(\tilde y)=\tilde \epsilon_{ab}(\tilde y) \tilde E_A^a(\tilde y) \tilde E_B^b(\tilde y)= O(l^{-2}\abs{\tilde y}^2)$. Here $\tilde E_A^a$ are the components of the vectorfield $\tilde E_A$ in the coordinate system $(\tilde y^a: a=1,2,3)$.
Let now $\tilde \Omega$ be a fixed domain in the $\tilde y$ coordinates containing the origin. Let $\tilde \phi$ be a solution of the boundary value problem
\begin{equation} \label{tildebvp}
\left\{ \begin{array}{rll} \overset{\tilde m}{\nabla}_b \tilde S^{ab} & = 0 & :\quad \textrm{in} \quad \tilde \Omega \, , \\ \tilde S^{ab} \tilde M_b & = 0 & : \quad \textrm{on} \quad \partial \tilde \Omega \, , \end{array} \right.
\end{equation}
such that $\tilde \phi$ takes the $\tilde y$ coordinate origin in $\mathcal N$ to the $x$ coordinate origin in $E^3$.
In (\ref{tildebvp}) $\tilde S^{ab}= \tilde \pi^{AB} \tilde E_A^a \tilde E_B^b$ and, by definition,
\begin{equation*}
\sqrt{\frac{\det \tilde \gamma}{\det \overset{\tilde \circ}{\gamma}}}\tilde \pi^{AB}= -2\frac{\partial \tilde e}{\partial \tilde \gamma_{AB}} \, .
\end{equation*}
Since $\tilde h=\tilde h_{ab}(\tilde y) d\tilde y^a \otimes d\tilde y^b$ and $y^a=l^{-1} \tilde y^a$ we have
\begin{equation*}
h=l^{-2} \tilde h = l^{-2}\tilde h_{ab}(\tilde y) d\tilde y^a \otimes d\tilde y^b=\tilde h_{ab}(l y) dy^a \otimes d y^b \, ,
\end{equation*}
hence $h_{ab}(y)=\tilde h_{ab}(ly)$.
We now consider the rescaled (smaller) domain $\Omega = \{ y= l^{-1}\tilde y , \tilde y \in \tilde \Omega \}$ and define the mapping
\begin{equation*}
\phi(y)=l^{-1} \tilde \phi(ly) \qquad : \forall y \in \Omega \, .
\end{equation*}
The pullback by $\phi$ of the Euclidean metric is $m$, the components of which in the $y^a$ coordinates are
\begin{equation*}
m_{ab}(y)=\delta_{ij} \frac{\partial \phi^i}{\partial y^a}(y) \frac{\partial \phi^j}{\partial y^b}(y) \, .
\end{equation*}
Since
\begin{equation*}
\frac{\partial \phi^i}{\partial y^a}(y)= \frac{\partial \tilde \phi^i}{\partial \tilde y^a}(\tilde y) \, ,
\end{equation*}
we have $m_{ab}(y)=\tilde m_{ab}(\tilde y)$.
Hence, the corresponding inner product on $\mathcal V$, $\gamma_{AB}(y)=\gamma(y)(E_A,E_B)$ in terms of the original basis $(E_A: A=1,2,3)$, is:
\begin{equation*}
\gamma_{AB}(y)=m_{ab}(y) E_A^a(y) E_B^b(y)= \tilde m_{ab}(\tilde y) \tilde E_A^a(\tilde y) \tilde E_B^b(\tilde y)=\tilde \gamma_{AB}(\tilde y) \, .
\end{equation*}
For,
\begin{equation*}
\tilde E_A^a(\tilde y) \frac{\partial}{\partial \tilde y^a}=l^{-1}E_A^a(y)\frac{\partial}{\partial y^a} \, ,
\end{equation*}
hence $\tilde E_A^a(\tilde y)=E_A^a(y)$.
For the stress tensors in both coordinate systems we have $\pi^{AB}(\gamma(y))=\tilde \pi^{AB}(\tilde \gamma(\tilde y))$ because intrinsically $\tilde e(\tilde \gamma)=e(\gamma)$. Therefore:
\begin{equation*}
\tilde S^{ab}(\tilde y)=\tilde \pi^{AB}(\tilde \gamma(\tilde y)) \tilde E_A^a(\tilde y) \tilde E_B^b(\tilde y)= \pi^{AB}(\gamma(y)) E_A^a(y) E_B^b(y)=S^{ab}(y) \, ,
\end{equation*}
which shows that the stress is scaling invariant, as expected.
\begin{remark}
The energy per unit mass $e$ is a function of the configuration $\gamma$, an inner product on $\mathcal V$. However, when representing $e$ as a function of the components $\gamma_{AB}$ of $\gamma$ in a basis $(E_A: A=1,\ldots,n)$, which is orthonormal relative to $\stackrel{\circ}{\gamma}$, $e$ in this representation depends indirectly on $\stackrel{\circ}{\gamma}$.
\end{remark}
Consider the equations in (\ref{tildebvp}). Starting from $\tilde m_{ab}(\tilde y)=m_{ab}(y)$,
\begin{equation*}
\frac{\partial \tilde m_{ab}}{\partial \tilde y^c}(\tilde y)=l^{-1}\frac{\partial \tilde m_{ab}}{\partial y^c}(\tilde y) = l^{-1} \frac{\partial m_{ab}}{\partial y^c}(y) \, ,
\end{equation*}
the Christoffel symbols $\Gamma_{ab}^c$ transform accordingly
\begin{equation} \label{scalechristoffelH}
\overset{\tilde m}{\Gamma _{ab}^c} (\tilde y)= l^{-1} \overset{m}{\Gamma_{ab}^c}(y) \, .
\end{equation}
Since we have
\begin{equation} \label{scalederSH}
\frac{\partial \tilde S^{ab}}{\partial \tilde y^c}(\tilde y)= l^{-1} \frac{\partial S^{ab}}{\partial y^c}(y) \, ,
\end{equation}
we finally obtain, using (\ref{scalechristoffelH}), (\ref{scalederSH}),
\begin{equation*}
\left(\overset{\tilde m}{\nabla}_b \tilde S^{ab}\right)(\tilde y)= \frac{\partial \tilde S^{ab}}{\partial \tilde y^b}(\tilde y)+ \overset{\tilde m}{\Gamma_{bc}^a}(\tilde y)\tilde S^{bc}(\tilde y) + \overset{\tilde m}{\Gamma_{ac}^b}(\tilde y) \tilde S^{ac}(\tilde y)=l^{-1} \left(\overset{m}{\nabla}_b S^{ab} \right)(y) \, .
\end{equation*}
Therefore, once we have a solution for the equation in the $\tilde y$ coordinates, we immediately obtain the solution for the original equation and the boundary condition is also satisfied since $\tilde M_a(\tilde y)=M_a(y)$.
\newpage
\part{The Analysis of Equilibrium Configurations} \label{analysises}
\setcounter{section}{0}
\section{Stress Tensor}
We will now show that the stress tensor $S$ on the material manifold $\mathcal N$ is given by
\begin{equation} \label{stressonN}
\sqrt{\frac{\det m}{\det \stackrel{\circ}{n}}} S^{ab}=-2 \frac{\partial e}{\partial m_{ab}} \, .
\end{equation}
Recall first the definition (\ref{defthermstress}) of the thermodynamic stress $\pi$ on the crystalline structure $\mathcal V$
\begin{equation} \label{defstresstilde}
\sqrt{\frac{\det \gamma}{\det \stackrel{\circ}{\gamma}}} \pi^{AB} = -2 \frac{\partial e}{\partial \gamma _{AB}} \, ,
\end{equation}
where we have made use of the definition of the volume $V(\gamma)=\sqrt{\frac{\det \gamma}{\det \stackrel{\circ}{\gamma}}}$ in the frame $(E_1,\ldots,E_n)$ satisfying $\omega(E_1,\ldots,E_n)=1$, $\stackrel{\circ}{\gamma}_{AB}=\delta_{AB}$ (so in fact $\det \stackrel{\circ}{\gamma}=1$).
Now, from (\ref{gammapbm}) we have
\begin{equation*}
\frac{\partial e}{\partial m_{ab}}= \frac{\partial e}{\partial \gamma_{AB}}\frac{\partial \gamma_{AB}}{\partial m_{ab}} = \frac{\partial e}{\partial \gamma_{AB}} E_A^a E_B^b \, ,
\end{equation*}
hence, with $E(y)=\omega(y)^{-1}$,
\begin{equation} \label{deregammaderem}
\frac{\partial e}{\partial \gamma_{AB}}=\frac{\partial e}{\partial m_{ab}} \omega_a^A \omega_b^B \, ,
\end{equation}
and finally, using (\ref{defSpfpi}), (\ref{defstresstilde}), (\ref{deregammaderem}) and the equality
\begin{equation} \label{detratio}
\frac{\det m}{\det \stackrel{\circ}{n}}=\frac{\det \gamma}{\det \stackrel{\circ}{\gamma}} \, ,
\end{equation}
we obtain:
\begin{equation*}
\sqrt{\frac{\det m}{\det \stackrel{\circ}{n}}} S^{ab} \omega_a^A \omega_b^B = \sqrt{\frac{\det \gamma}{\det \stackrel{\circ}{\gamma}}} \pi^{AB} = -2 \frac{\partial e}{\partial \gamma _{AB}}= -2 \frac{\partial e}{\partial m_{ab}} \omega_a^A \omega_b^B \, ,
\end{equation*}
which directly implies (\ref{stressonN}). The equality (\ref{detratio}) is a special case of the following proposition which applies to the isotropic case.
{\proposition \label{eigenvalues} The eigenvalues $\lambda_1, \ldots, \lambda_n$ of $\gamma$ with respect to $\stackrel{\circ}{\gamma}$ coincide with the eigenvalues of $m$ with respect to $\stackrel{\circ}{n}$.}
\begin{proof}
We define $A(y) \in \mathcal L(\mathcal V, \mathcal V)$ by
\begin{equation*}
\stackrel{\circ}{\gamma}\left(A(y)Y_1,Y_2\right) = \gamma(y) (Y_1,Y_2) \quad : \, \forall Y_1, Y_2 \in \mathcal V \, .
\end{equation*}
Hence $A(y)E_B=A_B^A(y)E_A$, where
\begin{equation*}
A_B^A(y) = (\stackrel{\circ}{\gamma}^{-1})^{AC}\gamma_{BC}(y) \, .
\end{equation*}
Thus $\lambda_1, \ldots, \lambda_n$ are the eigenvalues of $A$.
Similarly, we define $B(y) \in \mathcal L(T_y \mathcal N, T_y \mathcal N)$ by
\begin{equation*}
\stackrel{\circ}{n}\left(B(y)Y_{1,y},Y_{2,y}\right) = m(y) (Y_{1,y},Y_{2,y}) \quad : \, \forall Y_{1,y}, Y_{2,y} \in T_y\mathcal N \, .
\end{equation*}
and using $B(y)\left.\frac{\partial}{\partial y^a}\right|_y=B_a^b(y)\left.\frac{\partial}{\partial y^b}\right|_y$, we want to express $A(y)$ in terms of $B(y)$.
Since the evaluation map $\epsilon_y$ is an isomorphism from $\mathcal V$ to $T_y \mathcal N$ for each $y \in \mathcal N$, we have
\begin{equation*}
A(y)= \epsilon_y^{-1} \circ B(y) \circ \epsilon_y \, .
\end{equation*}
Hence, using $E_B(y)=E_B^b(y)\left.\frac{\partial}{\partial y^b}\right|_y$,
\begin{eqnarray*}
A_B^A(y) E_A & = & A(y)E_B = \epsilon_y^{-1} \left( B(y) E_B(y) \right) \\
& = & \epsilon_y^{-1} \left(E_B^b(y) B_b^a(y) \left.\frac{\partial}{\partial y^a}\right|_y \right) \\
& = & E_B^b(y) B_b^a(y) \omega_a^A(y) E_A \, ,
\end{eqnarray*}
we conclude
\begin{equation*}
A(y)=E(y)B(y)\omega(y)=\omega(y)^{-1}B(y)\omega(y) \quad , \quad E(y)=\omega(y)^{-1} \, ,
\end{equation*}
i.e.~the linear mappings $A(y)$ and $B(y)$ are conjugate and therefore $\lambda_1, \ldots, \lambda_n$ are also the eigenvalues of $B(y)$. \end{proof}
Let us now calculate the stress tensor $S^{ab}$ corresponding to the toy energy (\ref{toynrg}). Here, we denote $\stackrel{\circ}{n}$ by $h$, the metric of the hyperbolic plane that is the metric of the material manifold for a uniform distribution of elementary edge dislocations. Recall that
\begin{eqnarray*}
e = \frac 1 2 \left( (\lambda_1-1)^2+(\lambda_2-1)^2 \right) & = & \frac 1 2 \left( \lambda_1^2 + \lambda_2^2 -2 (\lambda_1+\lambda_2)+2\right) \\
& = & \frac 1 2 \textrm{tr}_h(m^2)-\textrm{tr}_h m + 1 \\
& = & \frac 1 2 \left(h^{-1}\right)^{ac}\left(h^{-1}\right)^{bd}m_{ab}m_{cd}-\left(h^{-1}\right)^{ab}m_{ab}+1 \, .
\end{eqnarray*}
Then:
\begin{equation}\label{stressedgeup}
\sqrt{\frac{\det m}{\det h}} S^{ab}=-2 \frac{\partial e}{\partial m_{ab}}=-2\left(\left(h^{-1}\right)^{ac}\left(h^{-1}\right)^{bd}m_{cd}-\left(h^{-1}\right)^{ab}\right) \, ,
\end{equation}
or,
\begin{equation}\label{stressedge}
S^{ab}=2 \sqrt{\frac{\det h}{\det m}}\left(h^{-1}\right)^{ac}\left(h^{-1}\right)^{bd}\left(h_{cd}-m_{cd}\right) \, .
\end{equation}
The physical interpretation of (\ref{stressedge}) is the following:
\begin{itemize}
\item[{\bf i)}] If $m$ is smaller than $h$ then the stress is positive,
\item[{\bf ii)}] if $m$ is larger than $h$ then the stress is negative.
\end{itemize}
Recall that a quadratic form $q=h-m$ on $T_y \mathcal N$ is said to be positive (negative) if, for all $v \in T_y\mathcal N$,
\begin{equation*}
q(v,v) > 0 \, (<0) \quad : \, \forall v \neq 0 \, .
\end{equation*}
\section{Setup and Method in $2$d}\label{setupPDE2d}
As was shown in Part I, Section \ref{edag}, the material manifold $\mathcal N$ for the case of a uniform distribution of edge dislocations in two dimensions is given by the affine group, and a left-invariant metric on $\mathcal N$ gives $\mathcal N$ the structure of the hyperbolic plane $H_{\varepsilon}$ of curvature $-\varepsilon^2$.
To solve the problem in this case, we fix an origin in $H_{\varepsilon}$ and set up Riemannian normal coordinates $(y^a: a=1,2)$ as in Section \ref{HGahs} of Part II. Let $\Omega$ be any smooth bounded domain in these coordinates, containing the origin. Note that as $\varepsilon \to 0$, $\left.H_{\varepsilon}\right|_{\Omega}$ tends to $\left.H_0\right|_{\Omega}$, where $H_0=E$ is the Euclidean plane. We also choose an origin and set up rectangular coordinates $(x^i:i=1,2)$ in $E$. An identity mapping
\begin{equation} \label{idmap}
\begin{array}{rcl}
id: H_{\varepsilon} & \to & E \\
(y^1,y^2) & \mapsto & (x^1,x^2)=(y^1,y^2)
\end{array}
\end{equation}
is then defined as in Section \ref{HGahs} of Part II.
We now restrict the allowed mappings $\phi: \Omega \subset H_{\varepsilon} \to E$ by the following two requirements. First, $\phi$ should map the origin in $H_{\varepsilon}$ into the origin in $E$. Second, $d\phi(0)$ should map the vector $\left.\frac{\partial}{\partial y^1}\right|_0$ into a vector of the form $\lambda \left.\frac{\partial}{\partial x^1}\right|_0$ for some $\lambda > 0$. By virtue of this restriction, the identity mapping (\ref{idmap}) (restricted to $\Omega$) is for $\varepsilon = 0$ the unique minimizer of our toy energy $e$ from (\ref{toynrg}).
{\remark The restriction is needed to ensure uniqueness. Otherwise, composition on the left with a rigid motion of $E$ gives another minimizer. The appropriate restriction in the three dimensional case will be stated below in Section \ref{uniqueness}. Analogously, it can be formulated in any number of space dimensions. Thus uniqueness is ensured in general. The argument of Part I, Section \ref{HyperbolicExpansion} applies with the hyperbolic plane $H_{\varepsilon}$ and the Euclidean plane replaced by the $n$-dimensional hyperbolic space $H_{\varepsilon}^n$ and the $n$-dimensional Euclidean space. This is as long as the toy energy (\ref{toynrg}) is considered.}
\vspace{2mm}
From (\ref{divstresszeroN}) and (\ref{bdryc}) the system of partial differential equations and the corresponding boundary conditions for the static problem of a uniform distribution of edge dislocations in $n=2$ dimensions is of the form
\begin{equation*}
F_{\varepsilon}\left[ \phi \right]=0 \, ,
\end{equation*}
where
\begin{equation} \label{Fepspb}
F_{\varepsilon}\left[ \phi \right]=\left( \begin{array}{c} \overset{m}{\nabla}_b S^{ab} \\ S^{ab}M_b \end{array} \right) \, .
\end{equation}
We linearize the equations at the identity mapping, which is a solution for $\varepsilon=0$. First, we solve the linearized problem using the theorem of Lax-Milgram (as in \cite{ABS}). An iteration will then show that there exists a solution to the nonlinear problem for sufficiently small $\varepsilon$, and, therefore, by the scaling argument of Section \ref{scaling} of Part II, that there is a mapping from a rescaled domain $\tilde \Omega$ in the standard hyperbolic plane $H_1=H$ of curvature $-1$ to the Euclidean plane, $\phi: \tilde \Omega \subset H \to E$, satisfying the conditions of the problem.
We set $\phi = id + \psi$, where $\psi$ is a small deviation from the identity mapping. We have:
\begin{equation*}
F_0\left[ id \right]=0 \, .
\end{equation*}
In a neighborhood of the identity $F_{\varepsilon}$ is of the form
\begin{equation*}
F_{\varepsilon}\left[ \phi \right]=F_{\varepsilon}\left[id\right] + D_{id} F_{\varepsilon} \cdot \psi + N_{\varepsilon}[\psi] \, ,
\end{equation*}
where $N_{\varepsilon}[\psi]$ is to leading order quadratic in $\psi$. We denote by $L_{\varepsilon}$ the linearized operator $D_{id}F_{\varepsilon}$. Then $F_{\varepsilon}\left[ \phi \right]=0$ reads
\begin{equation} \label{linearizationF}
L_{\varepsilon} \cdot \psi = -F_{\varepsilon}\left[id\right] -N_{\varepsilon}[\psi] \, .
\end{equation}
We will solve this by an iteration starting at $\psi_0=0$. In the first step of the iteration we have the linearized equations
\begin{equation} \label{stepone}
L_{\varepsilon} \cdot \psi_1 = -F_{\varepsilon}\left[ id \right] \, .
\end{equation}
The term $-F_{\varepsilon}\left[id\right]$ in (\ref{linearizationF}), (\ref{stepone}) can be interpreted as a source term.
A possible approach to the problem is to study the iteration
\begin{equation} \label{firstit}
L_{\varepsilon} \cdot \psi_{n+1} = -F_{\varepsilon}\left[ id \right]-N_{\varepsilon}[\psi_n]
\end{equation}
and show that $\psi_n$ converges to a solution $\psi=\phi-id$ of (\ref{linearizationF}), provided that we choose $\varepsilon$ appropriately small. However, we follow a different approach.
Adding $L_0 \cdot \psi$ on both sides of (\ref{linearizationF}) yields
\begin{equation*}
L_0 \cdot \psi = -\left(L_{\varepsilon}-L_0\right)\cdot \psi -F_{\varepsilon}[id]-N_{\varepsilon}[\psi] \, .
\end{equation*}
What we actually do is to consider the iteration
\begin{equation}\label{iteration}
L_0 \cdot \psi_{n+1} = - \left(L_{\varepsilon} - L_0\right)\cdot \psi_n - F_{\varepsilon}[id] - N_{\varepsilon}[\psi_n] \, .
\end{equation}
Note that while both (\ref{firstit}), (\ref{iteration}) are linear in the next iterate $\psi_{n+1}$, in (\ref{firstit}) the operator $L_{\varepsilon}$ which refers to $H_{\varepsilon}$ acts on $\psi_{n+1}$ whereas in (\ref{iteration}) the operator which refers to $H_0=E$ acts on $\psi_{n+1}$.
In (\ref{iteration}), $L_{\varepsilon}-L_0$ is a pair of linear operators, a second order operator in $\Omega$ and a first order operator on $\partial \Omega$. The coefficients of these operators are of order $\varepsilon^2$, which we may write symbolically in the form:
\begin{equation}\label{linit}
L_{\varepsilon}-L_0 \sim \partial \left(h - \delta\right) = \varepsilon^2 \partial f \, ,
\end{equation}
since, from (\ref{hypmetexpeps}), $h-\delta = \varepsilon^2 f$.
The iteration (\ref{iteration}) starts also with $\psi_0=0$. Then, setting $n=0$ in (\ref{iteration}), we have:
\begin{equation} \label{step0it}
L_0 \cdot \psi_1 = -F_{\varepsilon}[id] \, ,
\end{equation}
thus, taking into account the fact that $\det h=1+O(\varepsilon^2)$, (\ref{stressedge}) implies $S^{ab}(id)=-2\varepsilon^2 \left.f_{ab}\right|_{\varepsilon=0}+O(\varepsilon^4)$. It follows that $\psi_1$ is of order $\varepsilon^2$. This is step one of the iteration, the linear level.
For $n \geq 1$, taking the difference between (\ref{iteration}) and the same with $n$ replaced by $n-1$, we obtain
\begin{equation*}
L_0\cdot\left(\psi_{n+1}-\psi_n\right)= -\left(L_{\varepsilon}-L_0\right)\cdot \left(\psi_n-\psi_{n-1}\right)-\left( N_{\varepsilon}[\psi_n]-N_{\varepsilon}[\psi_{n-1}]\right) \, .
\end{equation*}
In view of (\ref{linit}) and the fact that $N_{\varepsilon}[\psi]$ is to leading order quadratic in $\psi$, for sufficiently small $\varepsilon$, contraction will hold and $\psi_n$ will converge to a solution $\psi$ of (\ref{linearizationF}).
Thus $\phi=id+\psi$ solves the problem (\ref{Fepspb}).
Consider now the equations in (\ref{Fepspb}).
To analyze the problem at the linearized level, we consider the variation $\dot m_{ab}$ of the metric $m_{ab}$ at the identity mapping $id$. Setting
\begin{equation}\label{variationid}
\phi^i= y^i+s \psi^i \, ,
\end{equation}
and recalling that
\begin{equation*}
m_{ab}(y)= \frac{\partial \phi^i(y)}{\partial y^a}\frac{\partial \phi^i(y)}{\partial y^b} \, ,
\end{equation*}
we obtain:
\begin{equation} \label{Deltamab}
\dot m_{ab}=\left. \frac{d}{ds}\right|_{s=0} \left[\left(\delta_a^i+s\frac{\partial \psi^i}{\partial y^a}\right)\cdot \left(\delta_b^i+s\frac{\partial \psi^i}{\partial y^b}\right)\right]=\frac{\partial \psi^i}{\partial y^a}\delta_b^i+\delta_a^i\frac{\partial\psi^i}{\partial y^b}=\frac{\partial \psi^b}{\partial y^a}+\frac{\partial\psi^a}{\partial y^b} \, .
\end{equation}
Hence, $\dot m_{ab}$ is the Lie derivative of the metric $\delta_{ab}$ on $H_{\varepsilon}$ (the pullback by $id$ of the metric $\delta_{ij}$ on $E$) with respect to the vectorfield $\psi^a \frac{\partial}{\partial y^a}$ on $H_{\varepsilon}$ (which the push-forward by $id$ takes to the vectorfield $\psi^i \frac{\partial}{\partial x^i}$ on $E$).
\section{Analogous Geometric Linear Problem} \label{AnalogousP}
In the work on the stability of the Minkowski space-time \cite{CK}, a linear geometric problem was studied analogous to the linear problems in (\ref{firstit}) and (\ref{iteration}).
Let $(M,g)$ be a compact Riemannian manifold with boundary $\partial M$ and $X$ a vectorfield on $M$. We set
\begin{equation} \label{stresspi}
\pi= \mathcal L_X g \quad , \quad \pi_{ij}=\nabla_i X_j + \nabla_j X_i \quad , \quad X_i=g_{ij}X^j \, ,
\end{equation}
the Lie derivative of the metric $g$ along $X$, a symmetric $2$-covariant tensorfield on $M$. The variation of $\pi$ with respect to $X$ reads:
\begin{equation} \label{varofpi}
\dot \pi_{ij}=\nabla_i \dot X_j +\nabla_j\dot X_i \, .
\end{equation}
We consider the problem of free minimization of the action integral
\begin{equation} \label{actionM}
A=\int_{M}\left(\frac 1 4 |\pi|_g^2 +\rho^i X_i \right)d\mu_g-\int_{\partial M} \tau^i X_i \left.d\mu_g\right|_{\partial M} \, .
\end{equation}
Here $\rho$ is a given vectorfield on $M$ and $\tau$ is a given vectorfield along $\partial M$. In mechanical terms $\rho$ is the body force and $\tau$ the boundary force.
The first variation of (\ref{actionM}), using (\ref{varofpi}), is
\begin{equation} \label{firstvarAinh}
\dot A = \int_{M}\left(\pi^{ij}\nabla_j\dot X_i +\rho^i \dot X_i\right) d\mu_g - \int_{\partial M} \tau^i \dot X_i \left.d\mu_g\right|_{\partial M} \, .
\end{equation}
For variations of $X_i$ which vanish near the boundary we have
\begin{equation*}
\dot A = -\int_M \left( \nabla_j \pi^{ij} - \rho^i \right) \dot X_i d\mu_g \, ,
\end{equation*}
and requiring $\dot A=0$ for such variations yields the Euler-Lagrange equations:
\begin{equation} \label{ELinhom}
\nabla_j \pi^{ij} = \rho^i \quad :\textrm{in} \,\, M \, .
\end{equation}
Requiring then $\dot A=0$ for arbitrary variations:
\begin{equation*}
- \int_{\partial M} \tau^i \dot X_i \left.d\mu_g\right|_{\partial M} = \int_{\partial M}\left(\pi^{ij} N_j - \tau^i\right)\dot X_i \left.d\mu_g\right|_{\partial M}=0 \, ,
\end{equation*}
yields the boundary conditions:
\begin{equation} \label{bdryinhom}
\pi^{ij}N_j = \tau^i \quad :\textrm{on} \,\, \partial M \, .
\end{equation}
The equations (\ref{ELinhom}), together with the boundary conditions (\ref{bdryinhom}), correspond to the linearized boundary value problem (\ref{linit}) corresponding to a crystalline solid with a uniform distribution of dislocations in equilibrium if we identify $(\Omega, \dot m, id^*\delta)$ with $(M,\pi,g)$, $\delta$ being the Euclidean metric.
We proceed in showing self adjointness of the above operators in the following sense. Let $Y$ be a vectorfield on $M$, $\sigma=\mathcal L_Y g$, i.e.~
\begin{equation} \label{defsigmacomp}
\sigma_{ij}=\nabla_i Y_j+\nabla_j Y_i=\sigma_{ji} \quad , \quad Y_i=g_{ij}Y^{j} \, .
\end{equation}
Then we have by repeated partial integration using (\ref{stresspi}),
\begin{equation*}
\left<Y, \nabla \cdot \pi \right>_{L^2(M)}-\left<Y, \pi \cdot N\right>_{L^2(\partial M)}=
\left<X, \nabla \cdot \sigma \right>_{L^2(M)}-\left<X, \sigma \cdot N\right>_{L^2(\partial M)} \, .
\end{equation*}
Suppose now that $Y$ is a Killing field, i.e.~$\sigma=\mathcal L_Y g=0$.
Then we have by (\ref{ELinhom}), (\ref{bdryinhom}) and (\ref{defsigmacomp})
\begin{equation*}
\int_M Y_i \rho^i = \int_M Y_i \nabla_j \pi^{ij} = -\frac 1 2 \int_M \sigma_{ij} \pi^{ij} + \int_{\partial M} Y_i \pi^{ij} N_j = \int_{\partial M} Y_i \tau^i \, ,
\end{equation*}
since $\sigma_{ij}=0$ on $M$. Thus the integrability condition reads:
\begin{equation} \label{intcond}
\int_M Y_i \rho^i = \int_{\partial M} Y_i \tau^i \, .
\end{equation}
This condition guarantees the existence of a solution for the boundary value problem (\ref{ELinhom}), (\ref{bdryinhom}) by the theorem of Lax-Milgram.
\subsection{Uniqueness of the Solution} \label{uniqueness}
In fact, the solution is unique up to an additive Killing field. For, if we take two solutions $X_1$ and $X_2$ of (\ref{ELinhom}) with (\ref{bdryinhom}), their difference $X=X_1-X_2$ satisfies the homogeneous equation of (\ref{ELinhom}), i.e.~$\rho = 0$ with zero boundary conditions, (\ref{bdryinhom}) for $\tau=0$.
Therefore, we have, setting $\pi=\mathcal L_X g$, $\nabla_j\left(\pi^{ij}X_i\right)=\pi^{ij} \nabla_j X_i$,
\begin{equation*}
A= \frac 1 4 \int_M |\pi|^2 =\frac 1 2 \int_M \pi^{ij}\nabla_j X_i=\frac 1 2 \int_M \nabla_j \left(\pi^{ij}X_i \right) \, .
\end{equation*}
Using Gauss's theorem we obtain:
\begin{equation*}
A=\frac 1 2 \int_M \nabla_j \left(\pi^{ij}X_i \right) =\frac 1 2 \int_{\partial M} \underbrace{\left(\pi^{ij}N_j \right)}_{=0} X_i =0 \, .
\end{equation*}
It follows that $\pi=0$, and thus $X=X_1-X_2$ is a Killing field, i.e.~the solutions $X_1, X_2$ only differ by a Killing field.
For our problem, where $(M,g)$ is isometric to a domain in the Euclidean plane, recalling the restriction
\begin{eqnarray*}
\phi(0) & = & 0 \, , \\
d\phi(0)\cdot \left. \frac{\partial}{\partial y^1} \right|_0 & = & \lambda \left. \frac{\partial}{\partial x^1} \right|_0 \, ,
\end{eqnarray*}
we obtain, setting
\begin{equation*}
\phi^i(x)=y^i+s X^i(y) \, ,
\end{equation*}
the conditions
\begin{eqnarray*}
s X^i(0) & = & 0 \, , \\
\delta_1^i + s \frac{\partial X^i}{\partial y^1}(0) & = & \lambda(s) \delta_1^i \, .
\end{eqnarray*}
Taking then the derivative with respect to $s$ at $s=0$ yields the linearized conditions:
\begin{eqnarray}
X^i(0) & = & 0 \, , \label{firstcond} \\
\frac{\partial X^i}{\partial y^1}(0) & = & \mu \delta_1^i \, , \label{secondcond}
\end{eqnarray}
where $\mu=\dot \lambda(0)$. The second of the above conditions is
\begin{equation*}
\frac{\partial X^2}{\partial y^1}(0)=0 \, .
\end{equation*}
Substituting the general form of a Killing field,
\begin{equation*}
X^i=\alpha^i_j y^j + \beta^i \quad , \quad \alpha^i_j=-\alpha^j_i \quad ,
\end{equation*}
it follows from the first condition (\ref{firstcond}) that $\beta^i=0$, and from the second condition (\ref{secondcond}) that $\alpha_1^2=0$, hence $\alpha^i_j=0$, i.e.~$X=0$. So the solution of the linearized problem is in fact unique.
We remark that in three dimensions, one must add, for uniqueness, the condition that
\begin{equation*}
d\phi(0)\cdot \left. \frac{\partial}{\partial y^2} \right|_0 \,
\end{equation*}
is a vector at the origin contained in the plane spanned by $\left. \frac{\partial}{\partial x^1} \right|_0 $ and $\left. \frac{\partial}{\partial x^2} \right|_0$. This can always be arranged by a suitable rotation in three-dimensional Euclidean space. Thus
\begin{equation*}
d\phi(0)\cdot \left. \frac{\partial}{\partial y^2} \right|_0 = \lambda_1 \left. \frac{\partial}{\partial x^1} \right|_0 + \lambda_2 \left. \frac{\partial}{\partial x^2} \right|_0 \, .
\end{equation*}
At the linearized level, this additional condition gives:
\begin{equation} \label{addcond}
\frac{\partial X^i}{\partial y^2}(0) = \mu_1 \delta_1^i + \mu_2 \delta_2^i \, ,
\end{equation}
where $\mu_1=\dot \lambda_1(0)$, $\mu_2=\dot \lambda_2(0)$. Setting $i=3$ in (\ref{addcond}), we have:
\begin{equation*}
\frac{\partial X^3}{\partial y^2}(0)=0 \, ,
\end{equation*}
while the conditions from (\ref{secondcond}) read
\begin{equation*}
\frac{\partial X^2}{\partial y^1}(0)=\frac{\partial X^3}{\partial y^1}(0)=0 \, .
\end{equation*}
Hence, if $X$ is a Killing field,
\begin{equation*}
\alpha_1^2=\alpha_1^3=0 \quad \textrm{and} \quad \alpha_2^3=0 \, ,
\end{equation*}
or, taking into account that also $\beta^i=0$ ($i=1,\ldots,3$), we conclude that $X=0$.
\section{The Linear Problem}\label{linearcase}
The unknown of the problem being the mapping $\phi: \Omega \to E$, the pullback metric $m=\phi^* \delta$ depends on $\phi$ according to:
\begin{equation*}
m_{ab}[\phi](y)=\sum_i \frac{\partial \phi^i(y)}{\partial y^a}\frac{\partial \phi^i(y)}{\partial y^b} \, .
\end{equation*}
Since $m_{ab}(id)=\delta_{ab}$, and $\phi= id + \psi$, i.e.~$\phi^i=y^i+\psi^i(y)$, we have:
\begin{equation*}
m_{ab}=\delta_{ab}+\dot m_{ab}+\mu_{ab}[\psi] \quad , \, \textrm{where} \quad \dot m_{ab}=\frac{\partial \psi^b}{\partial y^a}+\frac{\partial \psi^a}{\partial y^b}
\end{equation*}
and $\mu_{ab}[\psi]$ is quadratic in $\psi$.
It follows from (\ref{iteration}), together with (\ref{linit}), that $\psi=O(\varepsilon^2)$ at the linear level, whence $\psi^2=O(\varepsilon^4)$. Furthermore, from the expansion of the hyperbolic metric in rectangular coordinates we have
\begin{equation*}
h_{ab}=\delta_{ab}+\varepsilon^2 l_{ab}+O(\varepsilon^4) \, ,
\end{equation*}
where $l_{ab}= \left. f_{ab} \right|_{\varepsilon = 0}$.
Linearizing the operator $F_{\varepsilon}$ from (\ref{Fepspb}) at the identity mapping, $L_0 \psi =D_{id}F_0\cdot \psi$, we find:
\begin{equation*}
L_0 \psi = \left\{ \begin{array}{rl} -2 \partial_b \dot m_{ab} & : \, \textrm{in} \,\, \Omega \, , \\ -2\dot m_{ab} M_b & : \, \textrm{on} \,\, \partial \Omega \, , \end{array} \right.
\end{equation*}
with $\dot m_{ab}$ as above. Moreover, for $\phi=id$ we have, $m_{ab}=\delta_{ab}$, $\overset{m}\nabla_b =\frac{\partial}{\partial y^b}$. Hence:
\begin{equation} \label{fracstress}
\frac 1 2 S^{ab}(id)=\sqrt{\det h}\left(h^{-1}\right)^{ac}\left(h^{-1}\right)^{bd}\left(h_{cd}-m_{cd}\right)= \varepsilon^2 l_{ab} + O(\varepsilon^4) \, ,
\end{equation}
where we have made use of
\begin{equation*}
\sqrt{\det h}=\sqrt{\det(\delta+\varepsilon^2 l)}=1+O(\varepsilon^2) \, .
\end{equation*}
Therefore,
\begin{equation*}
F_{\varepsilon}(id) = \left\{ \begin{array}{rll} \partial_b S^{ab}(id) \, = & 2 \varepsilon^2 \partial_b l^{ab} + O(\varepsilon^4) & : \, \textrm{in} \,\, \Omega \, , \\ S^{ab}(id)M_b \, = & 2\varepsilon^2 l^{ab}M_b + O(\varepsilon^4) & : \, \textrm{on} \,\, \partial \Omega \, . \end{array} \right.
\end{equation*}
Thus, dropping terms of $O(\varepsilon^4)$ in $F_{\varepsilon}(id)$ which come from the $O(\varepsilon^4)$ terms in (\ref{fracstress}), the linearized problem (\ref{step0it}) reduces to the boundary value problem
\begin{equation}\label{LinearPb}
\left\{ \begin{array}{rll} \frac{\partial}{\partial y^b} \left(\dot m_{ab} - \varepsilon^2 l_{ab} \right) & = 0 & :\quad \textrm{in} \quad \Omega \, , \\ \left(\dot m_{ab} - \varepsilon^2 l_{ab} \right) M_b & = 0 & : \quad \textrm{on} \quad \partial \Omega \, . \end{array} \right.
\end{equation}
We consider the following problem analogous to (\ref{LinearPb}):
\begin{equation} \label{anabvp}
\left\{ \begin{array}{rll} \nabla_j\left(\pi^{ij}-\sigma^{ij} \right) & = 0 & :\quad \textrm{in} \quad M \, , \\ \left(\pi^{ij}-\sigma^{ij}\right)N_j & = 0 & : \quad \textrm{on} \quad \partial M \, , \end{array} \right.
\end{equation}
where $\pi_{ij}=\left(\mathcal L_X g\right)_{ij}$, $\sigma_{ij}$ is a given symmetric $2$-covariant tensorfield on $M$, $N_j$ is a covector whose null space is the tangent plane $T_x M$ at $x\in\partial M$, and $N^i=(g^{-1})^{ij}N_j$ is the corresponding outer unit normal vector to $\partial M$.
To obtain the solution of (\ref{anabvp}), we find a vectorfield $\tilde X$ such that $\tilde \pi=\mathcal L_{\tilde X}g$ satisfies
\begin{equation} \label{bdryvalue}
\left(\tilde \pi^{ij}-\sigma^{ij}\right)N_j=0 \qquad : \quad \textrm{on} \quad \partial M \, .
\end{equation}
We define $X'=X-\tilde X$ and $\pi'=\mathcal L_{X'} g=\mathcal L_X g-\mathcal L_{\tilde X}g=\pi-\tilde \pi$, to obtain from (\ref{bdryvalue}) and the boundary conditions of (\ref{anabvp}):
\begin{equation*}
(\pi')^{ij}N_j=\left(\pi^{ij}-\tilde \pi^{ij} \right)N_j=\left(\pi^{ij}-\sigma^{ij} \right)N_j=0 \qquad : \quad \textrm{on} \quad \partial M \, .
\end{equation*}
Defining the vectorfield $\rho$ in $M$ by $\rho^i=\nabla_j \left(\sigma^{ij}-\tilde \pi^{ij}\right)$, the problem then reduces to:
\begin{equation*}
\left\{ \begin{array}{rll} \nabla_j (\pi')^{ij} & = \rho^i & :\quad \textrm{in} \quad M \, , \\ (\pi')^{ij}N_j & = 0 & : \quad \textrm{on} \quad \partial M \, . \end{array} \right.
\end{equation*}
This is of the same form as (\ref{anabvp}), but with zero boundary conditions. To see whether this problem has a solution, we need to check the integrability condition (\ref{intcond}) (i.e.~orthogonality to the Killing fields in the $L^2$ sense). Here, the boundary terms vanish since we have zero boundary conditions and it remains for us to show that
\begin{equation*}
\int_M \xi_i \rho^i d\mu_g =0 \, ,
\end{equation*}
for all Killing fields $\xi$. We have:
\begin{eqnarray*}
\int_M\xi_i \rho^i d\mu_g & = & \int_M \xi_i\nabla_j\left(\sigma^{ij}-\tilde \pi^{ij} \right) d\mu_g \\
& = & \int_{\partial M} \xi_i\left(\sigma^{ij}-\tilde \pi^{ij} \right)N_j \left. d\mu_g\right|_{\partial M}- \int_M \nabla_j\xi_i \left(\sigma^{ij}-\tilde \pi^{ij} \right) d\mu_g=0 \, ,
\end{eqnarray*}
where the first term is zero due to (\ref{bdryvalue}) (that is, the choice of $\tilde X$), and the second one vanishes by virtue of the fact that $\xi$ is a Killing field,
\begin{equation*}
\nabla_j \xi_i \left(\sigma^{ij}-\tilde \pi^{ij}\right)= \frac 1 2 \left( \nabla_i \xi_j + \nabla_j \xi_i \right) \left(\sigma^{ij}-\tilde \pi^{ij}\right)=0 \, .
\end{equation*}
By applying the Lax-Milgram theorem as in \cite{ABS}, we conclude that there is a solution $X$ to the generalized linear problem which is unique up to an additive Killing field. Thus, also the linear case of the original problem (\ref{LinearPb}), viewed as a special case of the above, has a unique solution up to an additive Euclidean Killing field.
\section{The Nonlinear Case}
We will show that the nonlinear system (\ref{Fepspb}) is solvable in $\Omega$ under a certain smallness assumption on the parameter $\varepsilon$. Let us first state the nonlinear case of the original problem $(P)$, again.
\begin{equation} \label{nonlinpb}
(P) \left\{ \begin{array}{rll} \overset{m}{\nabla}_b S^{ab} & = 0 & :\quad \textrm{in} \quad \Omega \, , \\ S^{ab}M_b & = 0 & : \quad \textrm{on} \quad \partial \Omega \, , \end{array} \right.
\end{equation}
where $S^{ab}$ is given by (\ref{stressedge}) and $M_b$ is a covector whose null space is the tangent plane $T_y \Omega$ at $y \in \partial \Omega$. The orientation may be defined by $M_bX^a>0$, whenever $X^b$ is a vector pointing to the exterior of $\Omega$. $M^a=(m^{-1})^{ab}M_b$ is the outer normal to $\partial \Omega$.
The strategy for solving the problem $(P)$ is the following. We set up an iteration scheme, where the first step is the linearized problem. By the result for problem (\ref{anabvp}), the linearized problem is solvable because the integrability condition is automatically satisfied. The integrability condition of the iteration can then be satisfied by applying a doping technique similar to the one in \cite{Ka}.
\subsection{Iteration} \label{iterationmethod}
We first study the iteration scheme.
For the analogous (generalized) problem $(AP)$ we have from (\ref{iteration}), (\ref{nonlinpb}) (now written in terms of the coordinates $y^a$ on $\Omega$, and $\pi$ denoting the linearized metric $\dot m$):
\begin{equation} \label{AP}
(AP) \left\{ \begin{array}{rll} \frac{\partial \pi_{n+1}^{ab}}{\partial y^b} & = \rho_n^a & :\quad \textrm{in} \quad \Omega \, , \\ \left(\pi_{n+1}^{ab}-\sigma_n^{ab}\right)M_b & = 0 & : \quad \textrm{on} \quad \partial \Omega \, , \end{array} \right.
\end{equation}
where
\begin{equation*}
\pi^{ab}_{n+1}=\frac{\partial \psi^b_{n+1}}{\partial y^a}+\frac{\partial \psi^a_{n+1}}{\partial y^b} \, .
\end{equation*}
Now we set $\psi_{n+1}'=\psi_{n+1}-\tilde \psi_{n+1}$, where the $\tilde \psi$-part satisfies the boundary conditions, i.e.~
\begin{equation} \label{piprimesigmabdry}
\left(\tilde \pi^{ab}_{n+1}-\sigma^{ab}_n\right)M_b=0 \, ,
\end{equation}
and we have a modified problem $(AP')$ with zero boundary conditions
\begin{equation*}
(AP') \left\{ \begin{array}{rll} \frac{\partial \pi_{n+1}^{' \, ab}}{\partial y^b} & = \rho_n^a-\frac{\partial \tilde \pi_{n+1}^{ab}}{\partial y^b} & :\quad \textrm{in} \quad \Omega \, , \\ \pi_{n+1}^{' \, ab}M_b & = 0 & : \quad \textrm{on} \quad \partial \Omega \, . \end{array} \right.
\end{equation*}
The integrability condition, which yields existence of solutions of the problem by the Lax-Milgram theorem \cite{Ev}, now reads:
\begin{equation*}
0=\int_{\Omega}\xi^a \left(\rho_n^a-\frac{\partial \tilde\pi_{n+1}^{ab}}{\partial y^b}\right) = \int_{\Omega}\xi^a \rho_n^a -\int_{\partial \Omega} \xi^a \tilde \pi_{n+1}^{ab} M_b + \int_{\Omega} \tilde\pi_{n+1}^{ab}\frac{\partial \xi^a}{\partial y^b} \, .
\end{equation*}
That is:
\begin{equation} \label{intcondnl}
\int_{\Omega}\xi^a \rho_n^a - \int_{\partial \Omega} \xi^a \sigma_n^{ab} M_b = 0 \, ,
\end{equation}
for every Killing field $\xi$ of a background Euclidean metric on $\Omega$. This metric is $id^*\delta$, where $\delta$ is the Euclidean metric of $E$. The $y^a$ are rectangular coordinates on $\Omega$ relative to this metric. In particular, $M_b$ is a unit covector relative to this Euclidean metric and the integral $\int_{\partial \Omega}$ is taken with respect to the measure on $\partial \Omega$ corresponding to the metric (arc length if $\dim \Omega=2$) induced on $\partial \Omega$ by this Euclidean metric on $\Omega$.
In the above, we have made use of (\ref{piprimesigmabdry}) and the properties of $\xi$ as a Killing field. In particular,
\begin{equation*}
\xi^a=\alpha_b^a y^b + \beta^a \quad , \quad \alpha_b^a = -\alpha_a^b \quad \Rightarrow \quad \frac{\partial \xi^a}{\partial y^b}=\alpha_b^a=\frac 1 2 \left(\alpha_a^b-\alpha_b^a \right) \, ,
\end{equation*}
and thus the contraction of the symmetric tensors $\tilde \pi_{n+1}^{ab}$, respectively $\sigma_n^{ab}$, with $\frac{\partial \xi^a}{\partial y^b}$ vanishes.
In conclusion, the integrability condition at the $n+1$ step of the iteration is (\ref{intcondnl}).
\subsection{Killing Fields and Doping Technique}
In $n$ dimensional Euclidean space we have the following linearly independent Killing fields:
\begin{itemize}
\item[{\bf i)}] $n$ translations,
\item[{\bf ii)}] $\frac{n(n-1)}{2}$ rotations.
\end{itemize}
So the space of Killing fields on $E^n$ is $N$ dimensional, where
\begin{equation*}
N=n + \frac{n(n-1)}{2}=\frac{n(n+1)}{2} \, .
\end{equation*}
Let $\left(\xi_A: A=1,\ldots ,N\right)$ be a basis for the space of Killing fields. In the spirit of \cite{Ka}, we have to modify $\rho$, that is the inhomogeneity on the right-hand side of (\ref{AP}). This technique is called \emph{doping}. We replace $\rho$ by
\begin{equation*}
\rho' =\rho + \sum_A c_A \xi_A \, ,
\end{equation*}
and require in accordance with (\ref{intcondnl}) that
\begin{equation} \label{dopcond}
\int_{\Omega} \xi_A \cdot \rho' = \int_{\partial \Omega} \xi_A \cdot \sigma_M \quad : \quad \forall A=1, \ldots , N \, ,
\end{equation}
where $\sigma_M^a=\sigma^{ab}M_b$. Since
\begin{equation*}
\int_{\Omega} \xi_A \cdot \xi_B = M_{AB}
\end{equation*}
is positive definite, we obtain a linear system of equations for the coefficients $c_A \, : \, A=1, \ldots , N$ as follows:
\begin{equation*}
\int_{\Omega} \xi_A \cdot \rho' =\int_{\Omega} \xi_A \cdot \rho+ \int_{\Omega} \xi_A \left(\sum_B c_B \xi_B\right) = \int_{\partial \Omega} \xi_A \cdot \sigma_M \quad : \quad \forall A=1, \ldots , N \, ,
\end{equation*}
and thus
\begin{equation*}
\sum_B M_{AB} c_B= \int_{\partial \Omega} \xi_A \cdot \sigma_M - \int_{\Omega} \xi_A \cdot \rho =: \sigma_A \quad : \quad \forall A=1, \ldots , N \, .
\end{equation*}
The system $\sum_B M_{AB} c_B= \sigma_A$ can be solved, and there is a solution
\begin{equation} \label{coeffsol}
c_A= \sum_B\left( M^{-1}\right)_{AB}\sigma_B \, ,
\end{equation}
$M_{AB}$ being positive definite, hence non-singular.
We reformulate the problem $(AP)$ as follows:
\begin{equation} \label{APtilde}
(\widetilde{AP}) \left\{ \begin{array}{rll} \frac{\partial \pi_{n+1}^{ab}}{\partial y^b} & = \rho_n^{' \, a} & :\quad \textrm{in} \quad \Omega \, , \\ \left(\pi_{n+1}^{ab}-\sigma_n^{ab}\right)M_b & = 0 & : \quad \textrm{on} \quad \partial \Omega \, . \end{array} \right.
\end{equation}
Writing $\psi_{n+1}^a=X^a$, $\rho_{n}^{' \, a}=\rho^{' \, a}$, $\sigma_n^{ab} M_b = \tau^a$, (\ref{APtilde}) is of the form
\begin{equation} \label{limitAPtilde}
\left\{ \begin{array}{rll} \frac{\partial}{\partial y^b}\left( \frac{\partial X^b}{\partial y^a}+\frac{\partial X^a}{\partial y^b} \right) & = \rho^{' \, a} & :\quad \textrm{in} \quad \Omega \, , \\ \left( \frac{\partial X^b}{\partial y^a}+\frac{\partial X^a}{\partial y^b} \right)M_b & = \tau^a & : \quad \textrm{on} \quad \partial \Omega \, . \end{array} \right.
\end{equation}
Under the restriction discussed above, which if $X$ is a Euclidean Killing field forces $X$ to vanish identically, the linear system (\ref{limitAPtilde}) has no kernel. The following estimate then holds (see \cite{ADN})
\begin{equation} \label{estimate}
|| X ||_{H_{s+2}(\Omega)} \leq C \left\{ || \rho' ||_{H_s(\Omega)}+ || \tau ||_{H_{s+1/2}(\partial \Omega)}\right\} \quad , \quad C=C(\Omega) \, .
\end{equation}
Applying this estimate to (\ref{APtilde}) and taking $\varepsilon$ suitably small we can prove contraction of the sequence $(\psi_n)$ in $H_{s+2}(\Omega)$ for $s> \frac n 2$, $H_s(\Omega)$ and $H_{s-1/2}(\partial \Omega)$ being under this condition Hilbert algebras.
\begin{equation*}
\left. \begin{array}{lrcl} & \sigma_{n,A} & \to & \sigma_A \\ & C_{n,A} & \to & C_A \\ \textrm{in} \, \, H_s(\Omega) & \rho_n^a & \to & \rho^a \\ \textrm{in} \, \, H_{s+1/2}(\partial \Omega) & \sigma_n^{ab} & \to & \sigma^{ab} \\ \textrm{in} \, \, H_{s+1}(\Omega) & \pi_{n+1}^{ab} & \to & \pi^{ab} \\ \textrm{in} \, \, H_s(\Omega) & \rho_n^{' \, a} & \to & \rho^{' \, a} \end{array} \right\} \quad \textrm{for} \quad n \to \infty \, ,
\end{equation*}
and, in the limit $n \to \infty$ we deduce in terms of the modified inhomogeneity $\rho'$
\begin{equation}\label{dopedpb}
\left\{ \begin{array}{rll} \frac{\partial \pi^{ab}}{\partial y^b} & = \rho^{' \, a} & :\quad \textrm{in} \quad \Omega \, , \\ \left(\pi^{ab}-\sigma^{ab}\right)M_b & = 0 & : \quad \textrm{on} \quad \partial \Omega \, . \end{array} \right.
\end{equation}
We have:
\begin{equation} \label{rhoprime}
\rho^{'\,a}=\rho^a + \sum_A c_A \xi_A^a \, ,
\end{equation}
and $c_A$ is given in terms of $\sigma_A$ by (\ref{coeffsol}).
From (\ref{nonlinpb}), (\ref{dopedpb}) and (\ref{rhoprime}), we find
\begin{equation} \label{dagger}
\left\{ \begin{array}{rll}
\overset{m}{\nabla}_b S^{ab}=\frac{\partial \pi^{ab}}{\partial y^b}-\rho^a=\rho'^a-\rho^a & =: X^a & \quad : \, \textrm{in} \quad \Omega \, , \\
S^{ab} M_b & = 0 & \quad : \, \textrm{on} \quad \partial\Omega \, .
\end{array} \right.
\end{equation}
We have to show that
\begin{proposition}
\begin{equation} \label{defX}
X^a = \sum_A c_A \xi_A^a = 0 \quad : \, a=1, \ldots , n \, ,
\end{equation}
if $\varepsilon$ is suitably small.
\end{proposition}
\begin{proof}
Let $\xi^a$ be a Killing field of $\delta_{ab}$. We have:
\begin{equation}\label{differenceKF}
\int_{\Omega} \delta_{ab} \xi^a X^b d^2y = \int_{\Omega} m_{ab} \xi^a X^b d^2\mu_m +O(\varepsilon^2 \abs{X}_{\infty}) \, ,
\end{equation}
where we say that a real-valued function $Q(\varepsilon,X)$ is $O(\varepsilon^2\left|X\right|_{\infty})$, if there is a constant $C$ such that
\begin{equation*}
\left| Q (\varepsilon,X)\right| \leq C \varepsilon^2 \left|X\right|_{\infty} \, .
\end{equation*}
(\ref{differenceKF}) holds because $m_{ab}=\delta_{ab}+O(\varepsilon^2)$. Consider the vectorfield $\zeta$ on $\Omega$, the push-forward of which by $\phi$ to the Euclidean plane $E$ coincides with $\xi$
\begin{equation*}
\zeta^a= \xi^i \frac{\partial y^a}{\partial x^i} \quad , \quad \left[ \, \xi^i = \frac{\partial \phi^i}{\partial y^a} \zeta^a \, \right] \, .
\end{equation*}
Then $\zeta$ is a Killing field of the metric $m=\phi^*\delta$ on $\Omega$. Hence:
\begin{equation} \label{KFzeta}
\int_{\Omega} m_{ab} \zeta^a X^b d\mu_m = \int_{\Omega} \zeta_a \overset{m}{\nabla}_b S^{ab} d\mu_m = - \int_{\Omega} \overset{m}{\nabla}_b \zeta_a S^{ab} d\mu_m + \int_{\partial \Omega} \zeta_a S^{ab} M_b \left. d\mu_m\right|_{\partial \Omega} \, ,
\end{equation}
where $\zeta_a=m_{ab}\zeta^b$. The integral on $\Omega$ in (\ref{KFzeta}) vanishes because
\begin{equation*}
\overset{m}{\nabla}_b \zeta_a S^{ab}=\frac 1 2 \left( \overset{m}{\nabla}_b \zeta_a + \overset{m}{\nabla}_a \zeta_b\right) S^{ab} = 0 \, .
\end{equation*}
The integral on $\partial \Omega$ vanishes by virtue of the boundary condition in (\ref{dagger}). Then, since
\begin{equation*}
\xi^a = \zeta^a + O(\varepsilon^2) \, ,
\end{equation*}
we deduce:
\begin{equation} \label{orderKF}
\int_{\Omega} \delta_{ab} \xi^a X^b d^2y = \int_{\Omega} m_{ab} \zeta^a X^b d\mu_m + O(\varepsilon^2 \abs{X}_{\infty})=O(\varepsilon^2 \abs{X}_{\infty}) \, .
\end{equation}
However,
\begin{equation*}
X^a=\sum_A c_A \xi_A^a \quad : \quad a=1,\ldots,n \, ,
\end{equation*}
and since Killing fields are analytic functions, we have on a bounded domain $\Omega$:
\begin{equation} \label{KFcoeff}
\abs{X}_{\infty}=C \max_A \abs{c_A} \, .
\end{equation}
From (\ref{orderKF}),
\begin{equation}\label{estX}
\left|\int_{\Omega} \xi_A \cdot X \, d^ny \right| \leq C \varepsilon^2 \abs{X}_{\infty} \, .
\end{equation}
But from the definition of X in (\ref{defX}),
\begin{equation*}
\int_{\Omega}\xi_A \cdot X \, d^ny = \sum_{B} \int_{\Omega} \xi_A \xi_B c_B \, d^2y =\sum_{B}M_{AB} c_B = \sigma_A \, ,
\end{equation*}
and hence, from (\ref{estX}),
\begin{equation}\label{estsigmaX}
\abs{\sigma_A} \leq C \varepsilon^2 \abs{X}_{\infty} \, .
\end{equation}
On the other hand, we see from (\ref{coeffsol}) that
\begin{equation} \label{estcsigma}
\max_A \abs{c_A} \leq \tilde C \max_A \abs{\sigma_A} \, ,
\end{equation}
where $\tilde C=||M^{-1}||$. Finally, using (\ref{KFcoeff}), (\ref{estcsigma}) and (\ref{estsigmaX}), we obtain
\begin{equation*}
\abs{X}_{\infty}=C_1 \max_A \abs{c_A} \leq C_2 \max_A \abs{\sigma_A} \leq C_3 \varepsilon^2 \abs{X}_{\infty} \, ,
\end{equation*}
which implies $X \equiv 0$ for $\varepsilon$ sufficiently small ($C_3 \varepsilon^2 < 1 $). This finishes the proof.
\end{proof}
{\remark Concerning the regularity of the solution for $C^{\infty}$ domain $\Omega$ (smooth boundary) the solution is also $C^{\infty}$. For, $\psi \in H_{s+2}$, $s > n/2$ implies $\psi \in H_{s+3}(\Omega)$. Therefore, by induction, $\psi \in H_k(\Omega)$ for every $k$.}
\newpage
\bibliographystyle{my-h-elsevier}
|
1,314,259,996,554 | arxiv | \section{Introduction}
The construction for blowing up points and subspaces which
is a mainstay in algebraic geometry, especially in the resolution
of singularities, is investigated here from a dynamical point of
view. The blowup in this sense of a point $p$ in a smooth manifold
$M^n$ is a map of manifolds, $q\colon V^n \to M^n$, which is a
homeomorphism away from $q^{-1}(\{p\})$, and for which
$q^{-1}(\{p\})$ is a nonempty compact set, classically a
projective space. Real and complex versions of the construction
are both considered here.
Some other notions of blowing up points appear in dynamics,
notably in studies of normal forms for vector fields and in
constructions which delete a fixed point set and manipulate an
open cylinder.
The principal theme presented here is that the algebraic geometers'
form of blowup (Section \ref{Section:Model}) is so natural that it
easily induces continuous homomorphisms on diffeomorphism groups
(Sections \ref{Section:Natural} and \ref{Sec:Maps}), defined by an
explicit model given in Section \ref{Section:Natural}. These lifting
homomorphisms render derivative data at the space level since the
exceptional locus in blowup is a projectivized tangent space. For
complex manifolds and biholomorphic maps this rendering works nicely,
as it does for real $C^\infty$ manifolds and diffeomorphisms.
Dynamical consequences of the construction are laid out in Section
\ref{Sec:Dynamics}.
However, for finitely differentiable diffeomorphisms the
loss of regularity (Example \ref{Example:Regularity})
in blowup
becomes interesting and leads to a second theme: at
the lowest order of differentiability we find that a $C^1$
diffeomorphism fixing a point might lift to many homeomorphisms with
variant dynamics and quotient projections (Section \ref{Section:Variant}).
The argument for this
nonuniqueness claim uses local $C^0$ conjugacy facts for hyperbolic
fixed points of diffeomorphisms and suggests that the dynamical
universality of the classical blowup is much more distinctive than the
spatial or single--map aspects of the construction. This $C^0$
variation of blowup also indicates limitations on neighborhood--based
invariants of dynamics.
The paper's third theme is that while the blowup notion is easy
to generalize in a $C^0$ context (Section \ref{Section:Blowups}),
greatly enlarging on the topological effects of classical
blowup (Section \ref{Sec:Topology}),
when we give up differentiability entirely
it turns out that the only reasonable generalized blowups which allow
every homeomorphism of the base manifold to lift are necessarily
homeomorphic to the base manifold (Theorem \ref{Thm:NoTopLift}).
Topologists are familiar with blowup as a construction tool and
stabilizing device in four--manifold topology. Nash \cite{Nash52}
posed questions, since amplified by others, on the equivalence
relation on manifolds which blowup generates. Major progress on
Nash's space--level question was made in
\cite{AbkulutKing91,BenedettiMarin92}, and especially in the work of
Mikhalkin \cite{Mikhalkin97}. Blowup equivalence of diffeomorphisms
or group actions may be ripe for study after those advances.
We close the introduction with some notational conventions.
$B^n$ denotes the open ball $\{{\mathbf x} \in {\mathbf R}^n :
|{\mathbf x}| < 1\}$, while the closed disk
$D^n = \{{\mathbf x} \in {\mathbf R}^n :
|{\mathbf x}| \leq 1\}$.
The derivative of $h$ at $p$ is written $Dh|_p$.
The projection from a Cartesian product onto its $j$--th factor
is denoted $\operatorname{pr}_j$.
Mapping spaces for pairs appear frequently below. If $M$ is a
manifold and $A \subset M$ then $\operatorname{Aut}_{C^{k}}(M,\ A)$
denotes the space of $C^k$ diffeomorphisms $f$ of $M$ such that $f(A)
= A$; $f$ is not obliged to fix $A$ pointwise. The analogous reading
is used for spaces of homeomorphisms or other self--maps such as
$\operatorname{Homeo}(M, A)$.
\section{Blowups of a Manifold or Map} \label{Section:Blowups}
This section considers nonclassical, merely continuous versions of
the notion of blowing up a manifold or a map between manifolds.
Although these are easy to construct topologically, they do not
ordinarily have the universality properties of the classical
constructions, and it seems likely that homomorphisms such as
those exhibited in Theorem \ref{Theorem:Homom} are a distinguishing
feature of the classical constructions whose description
begins in Section \ref{Section:Model}.
\begin{definition} \label{Def:Blowup}
A \emph{topological blowup} of an $n$--manifold $M^n$ at a point
$p \in M$ is a quotient map $q\colon V^n \to M^n$ such that
\newline
(1) $V^n$ is also an $n$--manifold,
\newline
(2) $\Sigma := q^{-1}(\{p\})$ is a connected, compact,
nonempty subset of $V$, and
\newline
(3) $q|_{V \mysetminus \Sigma}\colon V \mysetminus \Sigma
\to M \mysetminus \{p\}$ is a homeomorphism.
A topological blowup of a self--map $f\colon M^n \to M^n$
of an $n$--manifold $M^n$ at a fixed point $p = f(p) \in M$
of $f$ is a topological blowup $q\colon V^n \to M^n$
of $M$ at $p$ together with a self--map ${\widetilde f}\colon V \to V$
such that this diagram commutes:
\begin{equation*}
\begin{CD}
V @> {\widetilde f} >> V \\
@VV q V @VV q V \\
M @>> f > M,
\end{CD}
\end{equation*}
i.e., $q\circ {\widetilde f} = f \circ q \colon V \to M$.
\end{definition}
$\Sigma = q^{-1}(\{p\})$ is called the \emph{exceptional locus} and
$M$ is sometimes described as a ``blowdown'' of $V$. Our first
examples are constructed top--downwards, by beginning with $V$
and $\Sigma$.
One could very reasonably add to Definition \ref{Def:Blowup} the
requirement that $V \mysetminus \Sigma$ should be dense in $V$. We
shall not do so in this paper, but note in advance the relevance of
this density condition in Theorem \ref{Thm:NoTopLift}.
A subset $\Sigma$ of a manifold $V^n$ is
\emph{cellular} if there are closed sets $S_i \subset V^n$
such that $S_1 \supseteq S_2 \supseteq \dots \supseteq S_i
\supseteq S_{i+1} \dots$,
$\Sigma = \cap_1^\infty S_i$ and
for every $i$, $S_i \cong D^n$ is a disk imbedded with bicollared
boundary.
\begin{example} \label{Example:CE}
If $\Sigma$ is a cellular subset of $V^n$ then
$M = V/\Sigma$ is a manifold and the quotient map
$q\colon V \to V/\Sigma$ is a blowup of $M$ at
the image of $\Sigma$.
\end{example}
See Figure \ref{Figure:CE} for a sketch of this sort of example, in
which $\Sigma$ is a finite polyhedral tree.
\epsfysize1.7in
\begin{figure}[tb]
\epsfbox{CE-A2.eps}
\caption{Blowup with a cellular exceptional locus.}
\label{Figure:CE}
\end{figure}
\begin{example} \label{Example:MapCE}
If $g\colon V^n \to V^n$ is a map which preserves a cellular
subset $\Sigma \subset V$, then $g$ descends to a map
$f\colon V/\Sigma \to V/\Sigma$ and $g$ together with
the quotient map
$q\colon V \to V/\Sigma$ defines a blowup of $f$.
\end{example}
Examples \ref{Example:CE} and \ref{Example:MapCE} are misleading,
inasmuch as $V^n$ and $M^n$ are homeomorphic. This is not usually
the case, and the replacement of $\{p\}$ by $\Sigma$ can affect
the global topology of a manifold in drastic ways.
\begin{example} \label{Example:Skeleton}
If $V^n \cong W^n \# X^n$ is a connected sum and
$\Sigma = X^{(n-1)}$ is the codimension--one skeleton
of a CW structure for $X$ which has one top--dimensional
cell, then the quotient map
$q\colon V \to V/\Sigma \cong W$ is a blowup of $W$
with exceptional locus $\Sigma$.
\end{example}
For instance, if $V^n$ is a compact, connected manifold and
$\Sigma$ is the codimension--one skeleton of a cell structure for
$V$ which has one top cell then $V/\Sigma \cong S^n$.
\begin{example}
Examples of topological blowups for maps as well as spaces are not
hard to produce, and one is sketched in Figure
\ref{Figure:Rotation}. Suppose that $g\colon V^n \to V^n$ is
periodic of period $r$ (so $g^r = \operatorname{Id}_V$),
that $g$ has a fixed point $a$, and that the action of the
cyclic group $C_r$ generated by $g$ is effective on $V$ and
locally linear at $a$. Form a
$g$--invariant polyhedral tree $\Sigma$ with $r$ legs emanating from
$a$, beginning with
a short segment $J$ based at $a$ so that $J \mysetminus \{a\}$
lies in the open dense set of $V$ on which $C_r$ acts freely.
If $J$ is sufficiently short and becomes smooth in a linear model
for the action near $a$, then $\Sigma = \cup_1^r g^i(J)$ is the
desired tree, lying in a Euclidean ball about $a$.
The periodic map $g$ descends to a periodic map $f$ on $V/\Sigma$
and the pair $q\colon V \to V/\Sigma$, $g\colon V \to V$
defines a blowup of $f$.
\end{example}
\epsfysize1.7in
\begin{figure}[tb]
\epsfbox{Rot-A3.eps}
\caption{Blowing up a rotation.}
\label{Figure:Rotation}
\end{figure}
\begin{remark} \label{Remark:Collar}
The distinctive property of the exceptional locus $\Sigma$ in a
blowup $q\colon V^n \to M^n$ arises from the requirement that
$V^n \mysetminus \Sigma \cong M^n \mysetminus \{p\}$ and
concerns deleted neighborhoods: There is an
open neighborhood $U$ of $\Sigma$ in $V$ such that
$U \mysetminus \Sigma \cong B^n \mysetminus \{{\mathbf 0}\}$.
Such a neighborhood provides collared codimension--one spheres
exhibiting a connected sum structure for $V$, so Example
\ref{Example:Skeleton} is more typical than it might appear, although
up to this point we have allowed non--CW compacta to appear as
exceptional loci. The classical blowup construction of algebraic
geometry exploits an instance of this neighborhood structure in
projective space, exactly along the lines of Example
\ref{Example:Skeleton}.
\end{remark}
\section{The Classical Model Construction} \label{Section:Model}
The most classical form of blowing up is performed at the origin in
${\mathbf F}^n$, where ${\mathbf F}$ is ${\mathbf R}$ or ${\mathbf C}$. Our account
mostly follows \cite{McDuffSalamon98}, and a good description of the
construction and the properties which extend it from the affine model
to other varieties is found in \cite{Harris92}.
${\mathbf P}({\mathbf F}^n)$ denotes the projective space of the
vector space ${\mathbf F}^n$, defined as the quotient
${\mathbf P}({\mathbf F}^n) = ({\mathbf F}^n \mysetminus \{{\mathbf 0}\})/\sim$,
where ${\mathbf v} \sim {\mathbf w}$ if and only if there exists
$\lambda \in {\mathbf F} \mysetminus \{0\}$ such that
${\mathbf v} = \lambda{\mathbf w}$. Square brackets denote homogeneous
coordinates on a projective space, so that
the image in ${\mathbf P}({\mathbf F}^n)$
of $(v_1, \dots, v_n) \in {\mathbf F}^n \mysetminus \{{\mathbf 0}\}$
is written $[v_1,\dots,v_n]$.
We will also use $[{\mathbf v}]$ to label the image in
${\mathbf P}({\mathbf F}^n)$ of a nonzero vector $\mathbf v$ in
${\mathbf F}^n$.
Let $X \subset {\mathbf F}^n \times {\mathbf P}({\mathbf F}^n)$ be the
subset
\begin{equation*}
X = \{\left((x_1,\dots,x_n), [y_1,\dots,y_n]\right) :
\ \text{for every $j,k$,}
\ x_j y_k = x_k y_j \}
\end{equation*}
and let
\begin{align*}
q\colon X &\to {\mathbf F}^n \\
({\mathbf x}, [{\mathbf y}]) &\mapsto {\mathbf x}
\end{align*}
be the restriction of first--coordinate projection
$\operatorname{pr}_1\colon {\mathbf F}^n \times {\mathbf P}({\mathbf F}^n)
\to {\mathbf F}^n$.
\begin{lemma}
$X = \{({\mathbf x}, [{\mathbf y}]) : \ \text{there exists}
\ \mu \in {\mathbf F}\ \text{such that}\ {\mathbf x} = \mu{\mathbf y} \}$.
In addition, $X$ is
a subvariety of ${\mathbf F}^n \times {\mathbf P}({\mathbf F}^n)$,
$q$ is an algebraic map, and preimages under $q$,
\begin{equation*}
q^{-1}(\{{\mathbf x}\}) =
\begin{cases}
\{ ({\mathbf x}, [{\mathbf x}]) \},&
\text{if ${\mathbf x} \neq {\mathbf 0}$},\\
\{ ({\mathbf 0}, [{\mathbf y}]) : [{\mathbf y}] \in
{\mathbf P}({\mathbf F}^n)\},&
\text{ if ${\mathbf x} = {\mathbf 0}$},
\end{cases}
\end{equation*}
are such that $q$ is an isomorphism away from the origin
and the fiber of $q$ over the origin is isomorphic to
the projective space ${\mathbf P}({\mathbf F}^n)$.
\qed
\end{lemma}
$\Sigma = q^{-1}(\{{\mathbf 0}\}) \cong {\mathbf P}({\mathbf F}^n)$
is usually called the
\emph{exceptional locus} or \emph{exceptional divisor}.
The quotient map $q$ is sometimes called the \emph{blowdown}
map, since it alters $X$ only by identifying $\Sigma$ to a point
(thus ``blowing down $\Sigma$'').
First--coordinate projection in ${\mathbf F}^n \times
{\mathbf P}({\mathbf F}^n)$ defined the blowdown map $q$, and
second--coordinate projection determines the structure of a
neighborhood of the exceptional locus in classical blowups.
\begin{lemma} \label{Lemma:LineBundle}
Second--coordinate projection restricts
to $X$ as
\begin{align*}
({\mathbf x}, [{\mathbf x}]) &\mapsto
[{\mathbf x}] \\
({\mathbf 0}, [{\mathbf y}]) &\mapsto
[{\mathbf y}]
\end{align*}
and identifies $X$ with the universal
line bundle over ${\mathbf P}({\mathbf F}^n)$, i.e., the ${\mathbf F}^1$--bundle
over this projective space whose fiber at $[{\mathbf y}]$ is the
line $\{\lambda{\mathbf y} : \lambda \in {\mathbf F}\}$
through the origin and $\mathbf y$ in ${\mathbf F}^n$.
$\Sigma$ is identified with the zero section in this bundle, so
the normal bundle of $\Sigma$ in $X$ is identified with
the universal line bundle.
\qed
\end{lemma}
We will return to this bundle structure in Section \ref{Sec:Topology}.
\section{Naturality Properties and Manifold Constructions}
\label{Section:Natural}
Any self--map of $({\mathbf F}^n, \{{\mathbf 0}\})$
such that $h$ is differentiable at the origin and
$Dh|_0$ is a ${\mathbf F}$--linear isomorphism
lifts to a map ${\widehat h}$ of $X$,
where ${\widehat h}$ is defined by:
\begin{align} \label{Equation:HatMapDef}
{\widehat h}\colon ({\mathbf x}, [{\mathbf x}]) &\mapsto
(h({\mathbf x}), [h({\mathbf x})]), \\
\notag
{\widehat h}\colon ({\mathbf 0}, [{\mathbf y}]) &\mapsto
({\mathbf 0}, [Dh|_{\mathbf 0}({\mathbf y})]).
\end{align}
\begin{lemma}
If $h\colon ({\mathbf F}^n, \{{\mathbf 0}\}) \to
({\mathbf F}^n, \{{\mathbf 0}\})$ is a continuous map which is
differentiable at ${\mathbf 0}$, and
if $Dh|_0$ is a ${\mathbf F}$--linear isomorphism, then
the map ${\widehat h}\colon (X, \Sigma) \to (X, \Sigma)$
defined above is continuous and makes this diagram commute
\begin{equation*}
\begin{CD} (X, \Sigma) @> {\widehat h} >> (X, \Sigma) \\
@V q VV @VV q V \\
({\mathbf F}^n, \{{\mathbf 0}\}) @>h >> ({\mathbf F}^n, \{{\mathbf 0}\}),
\end{CD}
\end{equation*}
i.e., $q \circ {\widehat h} = h \circ q\colon
(X, \Sigma) \to ({\mathbf F}^n, \{{\mathbf 0}\})$.
\end{lemma}
\begin{proof}
The claim that
$q \circ {\widehat h} = h \circ q$ follows immediately from
$q = \operatorname{pr}_1|_X$ and ${\widehat h}(\Sigma) = \Sigma$.
Because $q\colon X \mysetminus \Sigma \to {\mathbf F}^n \mysetminus
\{{\mathbf 0}\}$ is a homeomorphism and $h$ is continuous, the
continuity claim only needs to be confirmed at points of $\Sigma$.
The restriction of $\widehat h$ to $\Sigma = \{{\mathbf 0}\} \times
{\mathbf P}({\mathbf F}^n)$ is the projectivization
${\mathbf P}(Dh|_{\mathbf 0})$ of a linear isomorphism, so this
restriction is a $C^\infty$ diffeomorphism on $\Sigma$. Because
$h({\mathbf x}) = Dh|_{\mathbf 0}({\mathbf x}) + R({\mathbf x})$,
where $R({\mathbf x}) = o(|{\mathbf x}|)$ as
$|{\mathbf x}| \to 0$, $\widehat h$ is continuous
on the normal line $\{(\lambda {\mathbf x}, [\lambda {\mathbf x}])\}$
through $({\mathbf 0}, [{\mathbf x}]) \in \Sigma$:
\begin{align*}
{\widehat h}\left(\lambda {\mathbf x}, [\lambda {\mathbf x}]\right) &=
\left(h(\lambda {\mathbf x}), [h(\lambda {\mathbf x})]\right) \\ &=
\left(h(\lambda {\mathbf x}), [Dh|_{\mathbf 0}(\lambda {\mathbf x})
+ R(\lambda {\mathbf x})]\right) \\ &=
\left(h(\lambda {\mathbf x}),
[(\lambda|{\mathbf x}|)^{-1}Dh|_{\mathbf 0}(\lambda {\mathbf x})
+ (\lambda|{\mathbf x}|)^{-1} R(\lambda {\mathbf x})]\right) \\ &=
\left(h(\lambda {\mathbf x}),
[Dh|_{\mathbf 0}(|{\mathbf x}|^{-1}{\mathbf x})
+ (\lambda|{\mathbf x}|)^{-1} R(\lambda {\mathbf x})]\right),
\end{align*}
which tends to $({\mathbf 0}, [Dh|_{\mathbf 0}({\mathbf x})])$
as $\lambda \to 0$, with convergence uniform
in $|{\mathbf x}|^{-1}{\mathbf x}$.
Therefore, by the triangle
inequality, $\widehat h$ is continuous at every point of $\Sigma$.
\end{proof}
The next lemma follows from the Chain Rule. Recall that a map or
homeomorphism of pairs $(X, \Sigma) \to (X, \Sigma)$ is required
to carry $\Sigma$ to itself but need not restrict to the identity
on $\Sigma$.
\begin{lemma} \label{Lemma:Homomorphism}
Let $g,h\colon ({\mathbf F}^n, \{{\mathbf 0}\}) \to
({\mathbf F}^n, \{{\mathbf 0}\})$ be continuous maps which are
differentiable at ${\mathbf 0}$ and have ${\mathbf F}$--linear
isomorphisms as their derivatives at the origin. Then
$\widehat{g \circ h} = {\widehat g} \circ {\widehat h}\colon
(X, \Sigma) \to (X, \Sigma)$.
\qed
\end{lemma}
Since ${\widehat {\operatorname{Id}_{{\mathbf F}^n}}} = \operatorname{Id}_X$,
the Lemma shows that $h \mapsto \widehat h$ defines a homomorphism
\begin{equation*}
\beta\colon\operatorname{Aut}_{C^{1}, {\mathbf F}}({\mathbf F}^{n},\ \{{\mathbf 0}\}) \to
\operatorname{Homeo}(X,\ \Sigma),
\end{equation*}
where the ${\mathbf F}$ decoration indicates that the derivatives are
required to be ${\mathbf F}$--linear.
\begin{prop}
A smooth real or complex manifold $M$ can be blown up at any point $p$
to produce a quotient map from a smooth real or complex manifold
$\widehat M$,
\begin{equation*}
q\colon ({\widehat M}, \Sigma) \to (M, \{p\}),
\end{equation*}
which restricts to an isomorphism
$q|\colon {\widehat M} \mysetminus \Sigma \xrightarrow{\cong}
M \mysetminus \{p\}$.
If $M$ is modelled on ${\mathbf F}^n$ then
$\Sigma \cong {\mathbf P}({\mathbf F}^n)$.
\end{prop}
\begin{proof}
Lemma \ref{Lemma:Homomorphism} shows that origin--preserving
coordinate changes
with ${\mathbf F}$--linear derivatives act as automorphisms of
$q\colon (X, \Sigma) \to ({\mathbf F}^n, \{{\mathbf 0}\})$.
If $\phi_i\colon U \to {\mathbf F}^n$ are local coordinate systems
($i=1,2$)
on a neighborhood $U$ of $p$ in $M$ then
${\widehat {(\phi_1\circ\phi_2^{-1})}} \colon X \to X$
gives a change of coordinates on the model blowup. The
homomorphism properties established in
Lemma \ref{Lemma:Homomorphism} show that blownup coordinate change
maps satisfy the cocycle condition, yielding a consistent pasting
construction for $\widehat M$ from the data defining $M$.
\end{proof}
The same naturality properties used above give diffeomorphism blowups
on smooth manifolds, which are treated in detail in
Theorem \ref{Theorem:Homom}.
\section{Topology} \label{Sec:Topology}
This section describes the effects of the classical blowup
construction on topology. We begin with the model construction at the
origin in ${\mathbf F}^n$, where the space $X$ and the normal bundle of
$\Sigma$ in $X$ are identified with the universal line bundle over the
projective space $\Sigma$.
This bundle description gives a picture of the blowdown map which may
be helpful (see Figure \ref{Figure:Mobius}). Let $S({\mathbf F}^n)$ denote
the unit sphere in ${\mathbf F}^n$; then a tubular neighborhood of $\Sigma$
in $X$ is identified with the mapping cylinder of the Hopf map
$h\colon S({\mathbf F}^n) \to {\mathbf P}({\mathbf F}^n)$ and the blowdown
quotient on this tubular neighborhood is the natural map between the
mapping cylinders for this Hopf map and for the constant map
$c\colon S({\mathbf F}^n) \to \text{point}$, i.e., the map of
pairs $(\operatorname{MapCyl}(h), {\mathbf P}({\mathbf F}^n)) \to
(\operatorname{MapCyl}(c), \{\text{point}\})$.
\epsfysize3in
\begin{figure}[tb]
\epsfbox{mob-d2.eps}
\caption{Normal structure of the exceptional locus.}
\label{Figure:Mobius}
\end{figure}
In the complex case a bit of attention is required to the line bundles
playing roles in this discussion. McDuff and Salamon
\cite{McDuffSalamon98} describe these identifications or computations
carefully:
\newline
(a) $\nu_X(\Sigma)$ is identified with the universal
line bundle $L$ over the projective space
$\Sigma \cong {\mathbf P}({\mathbf C}^n)$;
\newline
(b) the first Chern class $c_1(\nu_X(\Sigma) = -c$, where
$c$ is the positive or canonical generator of
$H^2(\Sigma; {\mathbf Z})$;
\newline
(c) the normal line bundle to the hyperplane section in
${\mathbf P}({\mathbf C}^{n+1})$ has
first Chern class $c_1(\nu_{{\mathbf P}({\mathbf C}^{n+1})}
{\mathbf P}({\mathbf C}^n)) = c$; and
\newline
(d) the normal line bundle to the hyperplane section in
the conjugate complex structure
$\overline{{\mathbf P}({\mathbf C}^{n+1})}$ has
first Chern class
$c_1\left(\nu_{\,\overline{{\mathbf P}({\mathbf C}^{n+1})}\,}
\overline{{\mathbf P}({\mathbf C}^n)}\right) = -c$.
The real blowup of a point in a Riemann surface has
$\Sigma \cong {\mathbf R} P^1 \cong S^1$, where the model space
$X$ is the nonorientable line bundle over $S^1$ whose total space
is a M\"obius band. Thus,
for surfaces the mapping cylinder description of blowing up
and down suggests that blowing up a point has the global topological
effect of sewing in a crosscap. This is true, and in general
the global effect of blowing up a point is a connected sum operation,
as in Example \ref{Example:Skeleton} and Remark \ref{Remark:Collar}.
For real blowups,
\begin{equation*}
{\widehat M}^n \cong M^n \# {\mathbf R}{}P^n,
\end{equation*}
and for complex blowups
\begin{equation*}
{\widehat M}^n \cong M^n \# \overline{{\mathbf C}{}P^n}.
\end{equation*}
A conjugate complex structure appears in the second connected
sum because of the determinations of line bundles in the preceding
paragraph.
\section{Blowing Up Maps at a Fixed Point} \label{Sec:Maps}
The naturality properties of the classical blowup construction
suggest that (\ref{Equation:HatMapDef}) shows how to extend blowup
to a homomorphism of diffeomorphism groups. This is possible,
but a kink develops in the $C^k$ case.
The regularity loss in the theorem below is formally due to a division
when one considers the homogeneous coordinate side of the formula for
$\widehat h$. More geometrically, the blowup construction renders
tangential data for $h$ as spatial data for
$\widehat h$ since $\Sigma$ is the space of
lines in $T_pM$, so the loss of one derivative should be expected.
An example is worked out below to show that the
the loss of a derivative is genuine.
\begin{theorem} \label{Theorem:Homom}
Classical blowup of a point $p \in M$ determines
continuous, injective homomorphisms
\begin{equation*}
\beta\colon\operatorname{Aut}_{C^{k}}(M^{n},\ \{p\}) \to
\operatorname{Aut}_{C^{k-1}}(\widehat{M},\ \Sigma)
\end{equation*}
(in the real case) and
\begin{equation*}
\beta\colon\operatorname{Aut}_{\text{Holo}}(M^{n},\ \{p\}) \to
\operatorname{Aut}_{\text{Holo}}(\widehat{M},\ \Sigma)
\end{equation*}
(in the complex analytic case), where
$\beta(h) = {\widehat h}$ in both cases.
\end{theorem}
\begin{proof}
Away from $\Sigma$ we know that
$\widehat h$ and $h$ may be identified, so the regularity issue
only arises along $\Sigma$. Along $\Sigma$ we have defined
$\widehat h$ to be the projectivization of the linear
map $Dh|_{\mathbf 0}$, so $\widehat h$ is infinitely differentiable
in those directions.
If for ${\mathbf x}$ near ${\mathbf 0}$ we have
$h({\mathbf x})
= \sum_{j=1}^k D^j h|_{\mathbf 0} ({\mathbf x},\dots, {\mathbf x})
+ R({\mathbf x})$, where $R({\mathbf x}) = o(|{\mathbf x}|^k)$
as ${\mathbf x} \to {\mathbf 0}$, then
\begin{align*}
{\widehat h}({\mathbf x}, [{\mathbf x}])
&=
\left( \sum_{j=1}^k D^j h|_{\mathbf 0} ({\mathbf x},\dots, {\mathbf x})
+ R({\mathbf x}),\ %
[\sum_{j=1}^k D^j h|_{\mathbf 0} ({\mathbf x},\dots, {\mathbf x})
+ R({\mathbf x})] \right)\\
&=
\left( \sum_{j=1}^k D^j h|_{\mathbf 0} ({\mathbf x},\dots, {\mathbf x})
+ R({\mathbf x}),\ %
[\sum_{j=1}^k D^j h|_{\mathbf 0} ({\mathbf x/|{\mathbf x}|},{\mathbf x},
\dots, {\mathbf x})
+ R({\mathbf x})/|{\mathbf x}|]
\right).
\end{align*}
The division by $|{\mathbf x}|$ inside the homogeneous coordinates
gives a zero--th order term of
$Dh|_{\mathbf 0}({\mathbf x}/|{\mathbf x}|)$, similarly reduces the
degree of the other homogeneous terms in the Taylor expansion,
and reduces by one the order of vanishing for the remainder term.
The resulting expansion of $\widehat h$ near
$(0,\ [{\mathbf x}]) \in \Sigma$ shows that we lose one
partial derivative
of $\widehat h$ along the fibers of the normal bundle $\nu_X(\Sigma)$,
compared to the degree of smoothness of $h$ at ${\mathbf 0}$.
(Recall from Section \ref{Section:Natural}
that $\widehat h$ is $C^\infty$ along $\Sigma$.)
The partial derivatives of $\widehat h$ along the singular locus
and normal to it are continuous, through order $k-1$, so $\widehat h$
is $C^{k-1}$ at points of $\Sigma$ by the familiar theorem deducing
(Fr\'echet) differentiability from continuous partial derivatives.
Lemma \ref{Lemma:Homomorphism} and surrounding discussion show that
$\beta$ is a homomorphism.
Once we know that $\widehat h$ is continuous, it
follows that $\beta$ is injective,
since $h$ determines $\widehat h$ on the dense subset
${\widehat M} \mysetminus \Sigma$.
$\beta$ is continuous because the $C^r$ distance between
diffeomorphisms on $M^n$ majorizes the $C^{r-1}$ distance between
their blowups on ${\widehat M} \mysetminus \Sigma$ and the $C^{r-1}$
distance between those blowups on $\Sigma$.
\end{proof}
\begin{example} \label{Example:Regularity}
This is a two--dimensional example of
the regularity loss from $C^1$ to $C^0$
indicated in the theorem.
Let $g\colon {\mathbf R} \to {\mathbf R}$ be defined by
$g(x) = x + x|x|$, so
that $g'(x) = 1 + 2|x|$. $g$ is a $C^1$ diffeomorphism,
but not $C^2$, and $g(0) = 0$.
Define $h\colon {\mathbf R}^2 \to {\mathbf R}^2$ to be the $C^1$ diffeomorphism
$h(x,y) = (g(x), y)$. This map preserves the origin and blows up there
to ${\widehat h}\colon \left( (x,y),\ [x,y] \right) \mapsto
\left( (x + x|x|, y),\ [x+ x|x|, y] \right)$. The parametrized
line
$t \mapsto (t, mt)$ of slope $m \neq 0$ in the plane is covered in
the blownup plane by the $C^{\infty}$ parametric
curve $c\colon t \mapsto \left((t, mt),\ [t, mt] \right)$,
and the composite
\begin{align*}
{\widehat h} \circ c\colon t
&\mapsto
\left( (t + t|t|, mt),\ [t+ t|t|, mt] \right) \\
&=
\left( (t + t|t|, mt),\ [m^{-1} + m^{-1}|t|, 1] \right) \\
\end{align*}
is continuous but not differentiable at $t = 0$, since in the
usual local coordinate system about $[0,1] \in {\mathbf P}({\mathbf R}^2)$
the second component becomes $t \mapsto m^{-1} + m^{-1}|t|$:
therefore $\widehat h$ is not differentiable, and
not even G\^ateaux differentiable,
at $c(0) = \left((0,0),\ [1,m]\right) \in \Sigma$.
\end{example}
Similar examples for $C^k$ to $C^{k-1}$ regularity loss are
available for all $k \geq 1$.
Theorem \ref{Thm:NoTopLift}
indicates that $C^0$ to $C^0$ lifting of automorphisms through
blowups is problematic for other
reasons.
\section{Dynamics} \label{Sec:Dynamics}
The dynamics of a classically blown--up diffeomorphism $\widehat h$
off, on, and near the singular locus are described in terms
of basic features of the original diffeomorphism $h$.
If $h\colon (M, \{p\}) \to (M, \{p\})$ is a diffeomorphism
fixing $p$ then the restriction of $q$ gives
\begin{equation*}
{\widehat h}|_{{\widehat M} \smallsetminus \Sigma}
\cong
h|_{ M \smallsetminus \{p\}},
\end{equation*}
so blowup does not modify dynamics far from the exceptional locus.
\begin{lemma} \label{Lemma:BlowFix}
On the exceptional locus $\Sigma = {\mathbf P}(T_pM)$
\begin{equation*}
{\widehat h}|_{\Sigma} \cong
{\mathbf P}(Dh|_p)
\end{equation*}
is a projectivized linear map, with the new fixed point set
given by
\begin{equation} \label{Equation:SigmaFixedSet}
\Sigma\,\cap\,\operatorname{Fix}({\widehat h}) =
\amalg_{\lambda \in {\mathbf F}}
\ {\mathbf P}(E_\lambda),
\end{equation}
where $E_\lambda = \ker(\lambda I - Dh|_p)$
and $\amalg$ denotes a disjoint union.
\end{lemma}
\begin{proof}
Equation \ref{Equation:HatMapDef} defines
${\widehat h}\colon ({\mathbf 0}, [{\mathbf y}]) \mapsto
({\mathbf 0}, [Dh|_{\mathbf 0}({\mathbf y})])$ in the
model case, so the restriction
of $\widehat h$ to $\Sigma$ is the projectivization of
the derivative $Dh|_p$.
Therefore, fixed points of $\widehat h$
in $\Sigma = {\mathbf P}(T_pM)$ are solutions of
$[Dh|_p(v)] = [v]$, i.e. projective
equivalence classes of tangent vectors $v$ for which there
exist scalars $\lambda \in {\mathbf F}$ satisfying
$Dh|_p(v) = \lambda v$, that is, projective equivalence
classes of eigenvectors of $Dh|_p$.
Equation \ref{Equation:SigmaFixedSet} describes the set of
all such projective classes as a disjoint union of projectivized
subspaces of $T_p M$.
\end{proof}
Derivative computations for $\widehat h$ at points of $\Sigma$
and in directions not tangent to $\Sigma$ will
involve the loss of order noted
in Theorem \ref{Theorem:Homom}. These are omitted here -- see the
displayed equation in the proof of that theorem for the appearance of
second derivatives of $h$ in the first derivative of $\widehat h$.
Despite this derivative complication,
a qualitative picture of part of the dynamics of $\widehat h$ normal
to $\Sigma$ is provided by some naturality observations.
First, if $N^k$ is a $C^1$ submanifold of $N^n$ passing through
$p$ and $\widehat M$ is the blowup of $M^n$ at $p$ then there is a
$C^0$ submanifold $V^k$ of $\widehat M$ such that $V$ is homeomorphic
to $\widehat N$ and $q_N \colon ({\widehat N}, \Sigma_N) \to
(N, \{p\})$ is equivalent to $q_M|\colon (V, V\cap \Sigma_M)
\to (N, \{p\})$. The main step in checking this claim is a confirmation that
the closure of $q_M^{-1}(N \mysetminus \{p\})$ in $\widehat M$ meets
$\Sigma_M$ in ${\mathbf P}(T_p N^k)$.
Second, if $h\colon (M^n, N^k, \{p\}) \to (M^n, N^k, \{p\})$ is
a $C^1$ diffeomorphism then $\widehat h$ preserves the submanifold
$V^k$ defined in the preceding paragraph.
In particular, if $p$ is a hyperbolic fixed point
of $h$ then this applies to the stable and unstable manifolds at $p$,
and also to any invariant local submanifolds tangent to
other invariant subspaces of $T_p M$, such as eigenspaces of
$Dh|_p$. Because $\Sigma \cap V = {\mathbf P}(T_pN)$,
in Figure \ref{Figure:Hyperbolic} the blownup one--dimensional
stable submanifold at $p$
meets $\Sigma$ in the point corresponding to the appropriate
eigenspace of $Dh|_p$, while the blownup unstable manifold meets
$\Sigma$ in the point corresponding to another one--dimensional
eigenspace.
\epsfysize3in
\begin{figure}[tb]
\epsfbox{BlMob-A2.eps}
\caption{Blowing up a hyperbolic fixed point.}
\label{Figure:Hyperbolic}
\end{figure}
This low--dimensional example suggests the behavior of blowups at
hyperbolic fixed points, but is a bit simpler than the general case,
which we sketch now.
Suppose that $p$ is a hyperbolic fixed point of $h$, that $E_s$ and
$E_u$ are the stable and unstable subspaces of $T_p M$, and that the
stable and unstable submanifolds at $p$ are $W_s$ and $W_u$. $W_s$
and $W_u$ blow up at $p$ to give invariant submanifolds $V_s$, $V_u$
which meet $\Sigma$ in a submanifold (either ${\mathbf P}(E_s)$ or
${\mathbf P}(E_u)$) which is invariant and forward attracting
(respectively backward attracting).
The dynamics of $\widehat h$ restricted to
${\mathbf P}(E_s)$ or
${\mathbf P}(E_u)$) are those of a projectivized linear map.
\section{Variant Blowups} \label{Section:Variant}
$C^0$ data for $h$ are not enough to determine ${\widehat h}|_\Sigma$
as a homeomorphism. For example, in the hyperbolic case we can apply
local conjugacy results to obtain lots of homeomorphisms blowing up a
given $C^1$ diffeomorphism. A handy reference for these facts on
topological conjugacy is \cite[Sec.~6.3]{KatokHasselblatt95}, which
gives a proof of this result on local equivalence of hyperbolic fixed
points:
\begin{remark} \label{Remark:Conjugacy}
Topological conjugacy classes of hyperbolic diffeomorphisms
with $p$ as an isolated fixed point are
determined by the dimensions and orientations of the stable
and unstable manifolds of these diffeomorphisms at $p$.
\end{remark}
For example, if $M$ is even--dimensional over ${\mathbf R}$, $p$ is a fixed
point for $h_i$ ($i=1,2$), and $D{h_1}|_p$ is diagonalizable with all
eigenvalues lying in the interval $\lambda > 1$, while $D{h_2}|_p$ has
only non--real eigenvalues, all satisfying $|\lambda| > 1$, then $h_1$
and $h_2$ are locally conjugate near $p$. Figure \ref{Figure:Spiral}
suggests how different in appearance such topologically conjugate
diffeomorphisms can be, and indicates that the cause of the phenomenon
is a familiar difficulty: we can unwind a spiral with a continuous
automorphism, but not with a smooth one.
\epsfysize2in
\begin{figure}[tb]
\epsfbox{spi-A1.eps}
\caption{Topologically conjugate hyperbolic fixed points.}
\label{Figure:Spiral}
\end{figure}
A global topological conjugacy $\phi\colon M \to M$ from
$h_1$ to $h_0$ leads to a variant blowup of $h_0$ with
${\widehat h}_1$ as the covering homeomorphism and
$\phi\circ q$ as blowdown map.
\begin{equation*}
\begin{CD}
{\widehat M} @> {\widehat h}_1 >> {\widehat M} \\
@V q VV @VV q V \\
M @> h_1 >> M \\
@V \phi VV @VV \phi V \\
M @> h_0 >> M
\end{CD}
\end{equation*}
The dynamics on $\Sigma$ of these conjugacy--induced blowups can
differ dramatically from those of the classical construction. Fixed
point sets on $\Sigma$ may differ drastically in dimension as we run
over diffeomorphisms $h_1$ which are topologically conjugate to $h_0$,
ranging from empty to discrete to connected and high--dimensional.
We emphasize this variability with a proposition.
\begin{prop}
Let $h\colon M^n \to M^n$ be a diffeomorphism with an isolated
hyperbolic fixed point $p$. If the stable and unstable
subspaces $E^s_p, E^u_p \subseteq T_pM$ are both even--dimensional
then there are conjugacy--induced topological blowups
${\widetilde h}$ of $h$ such that
$\Sigma \cap \operatorname{Fix}({\widetilde h})$
is of any of these sorts:
\newline
(a) empty,
\newline
(b) discrete, containing any even number of points between $2$ and $n$,
or
\newline
(c) positive--dimensional, with any dimension between $1$ and
$-1 + \max(\dim(E^u_p),\,\dim(E^s_p))$.
\end{prop}
\begin{proof}
In each case a $C^0$
conjugacy as described in Remark \ref{Remark:Conjugacy}
between $h$ and another diffeomorphism $g$ with a
hyperbolic fixed point at $p$ yields the topological blowup.
Our job here is to allocate eigenvalues for $Dg|_p$ and apply
Lemma \ref{Lemma:BlowFix} to $\widehat g$.
If every eigenvalue $\lambda \in {\mathbf C} \mysetminus {\mathbf R}$
then $Dg|_p$ has no real eigenvectors and the classical blowup's
$\Sigma \cap \operatorname{Fix}({\widehat g})$ is empty.
If $Dg|_p$ has $k$ distinct real eigenvalues and $n - k$
complex eigenvalues appearing in conjugate pairs, then
$k$ must be even but may otherwise assume any value between
$0$ and $n$. In this case we see $k$ isolated fixed points
for $\widehat g$ on $\Sigma$.
Positive--dimensional fixed point sets arise from repeated real
eigenvalues for $Dg|_p$. These may appear in combinations so that
the multiplicities of real eigenvalues form partitions of some
even numbers $0 \leq e_u \leq \dim(E_p^u)$,
$0 \leq e_s \leq \dim(E_p^s)$:
\begin{align*}
k_1 + k_2 + \dots + k_q &= e_u,\\
l_1 + l_2 + \dots + l_r &= e_s,\\
\end{align*}
where $k_i$, $l_j$ are the multiplicities of unstable, respectively
stable, real eigenvalues
of $Dg|_p$. Each of these real eigenvalues produces a component
of the fixed point set $\Sigma \cap \operatorname{Fix}({\widetilde g})$
which is diffeomorphic to a
projective space: If $\lambda$ is a real eigenvalue of
multiplicity $m$ then ${\mathbf P}(E_{\lambda}) \cong
{\mathbf P}({\mathbf R}^{m}) \cong {\mathbf R} P^{m - 1}$.
The largest dimension arising in this way
$\max(\dim({\mathbf P}(E^u_p)),\,\dim({\mathbf P}(E^s_p)))$.
\end{proof}
Note that $\Sigma \cap \operatorname{Fix}({\widehat g})$ might have
components of different dimensions. The complex case is similar, but
the fixed point set must be nonempty.
The next few results indicate that
topological blowups are necessarily limited in naturality.
\begin{lemma} \label{Lemma:TubeHomeo}
Let $N^k$ be a connected topological manifold without boundary.
If $\{x_i\}$ and $\{y_i\}$ are sequences in $N \times (0,\infty)$
such that $\{\operatorname{pr}_2(x_i)\} \to \infty$,
$\{\operatorname{pr}_2(y_i)\} \to \infty$, and both of these sequences
in ${\mathbf R}$ are strictly increasing, then there is a homeomorphism
$g\colon N \times (0,\infty) \to N \times (0,\infty)$ such that
for every $i$, $g(x_i) = y_i$, and such that for some $\varepsilon > 0$
the restriction
$g|_{(0,\varepsilon]}$ is the identity.
\end{lemma}
\begin{proof}
This is a consequence of the following version of the homogeneity
of manifolds: for any $x,y \in N$ there
exists an isotopy from $\operatorname{Id}_N$ to a homeomorphism
which carries $x$ to $y$.
In more detail, $g$ can be built in segments which are pasted
together.
We may apply a homeomorphism of the form $\rho\colon (x,t) \mapsto
(x, \psi(t))$, where $\psi\colon (0,\infty) \to (0,\infty)$ is
a homeomorphism, to arrange that $\operatorname{pr}_2(\rho(x_i))
= \operatorname{pr}_2(y_i)$ for all $i$. Continue the argument with
the sequence $\{\rho(x_i)\}$ replacing $\{x_i\}$.
Let $0 < s_1 < \operatorname{pr}_2 (x_1)$ and
let $\phi_t\colon N \to N$ be an isotopy over $s_1 \leq t \leq
\operatorname{pr}_2(x_1)$ such that $\phi_{s_1} = \operatorname{Id}_N$
and $\phi_{\operatorname{pr}_2(x_1)}(\operatorname{pr}_1(x_1)) = \operatorname{pr}_1(y_1)$.
Define the first two pieces of
$g\colon N \times (0,\infty) \to N \times (0,\infty)$ by
$g|_{N \times (0,s_1]} = \operatorname{Id}_{N \times (0,s_1]}$ and
$g|_{N \times [s_1, \operatorname{pr}_2(x_1)]}\colon
(x,t) \mapsto (\phi_t(x), t)$.
Subsequent segments are defined by taking an isotopy
$\phi_t$ over $\operatorname{pr}(x_{i-1})
\leq t \leq \operatorname{pr}_2(x_{i})$ which starts with
the $\phi_{\operatorname{pr}(x_{i-1})}$ already selected
and ends at a homeomorphism which carries
$\phi_{\operatorname{pr}(x_{i-1})}(x_i)$ to $y_i$.
\end{proof}
The topological blowups $q\colon (V, \Sigma) \to (M, \{p\})$
of greatest interest will share with
the classical construction the property that
$V \mysetminus \Sigma$ is dense in $V$. This line of argument
shows that in no such case
can we find a lifting construction for homeomorphisms.
\begin{theorem} \label{Thm:NoTopLift}
Let $q\colon (V^n,\Sigma) \to (M^n, \{p\})$ be a topological blowup
of the manifold $M^n$ at $p$ such that at least two points lie on
the frontier of $\Sigma$ in $V$. If $n \geq 2$
then there is a homeomorphism $h\colon (M, \{p\}) \to (M, \{p\})$
which does not lift through $q$.
\end{theorem}
\begin{proof}
Suppose that $x, y \in \Sigma$ are distinct points and that
$\{x_i\}$, $\{y_i\}$ are sequences in $V \mysetminus \Sigma$
such that $x = \lim_{i \to \infty} x_i$ and $y = \lim_{i \to \infty} y_i$.
We may assume that both sequences lie in a neighborhood $U$ of $\Sigma$
which admits a homeomorphism
$f\colon U \mysetminus \Sigma \xrightarrow{\cong}
S^{n-1} \times (0,\infty)$ and that the sequences
$\{\operatorname{pr}_2\circ f(x_i)\}$ and
$\{\operatorname{pr}_2\circ f(y_i)\}$ are strictly increasing and converge
to $\infty$.
Form a third sequence $\{z_i\}$ such that each $z_{2j}$ is one of the
$y_i$, each $z_{2j+1}$ is one of the $x_i$, and the real sequence
$\{\operatorname{pr}_2\circ f(z_i)\}$ is strictly increasing and converges
to $\infty$. $\{z_i\}$ is not convergent in $V$, but all three of
the sequences $\{q(x_i)\}$, $\{q(y_i)\}$, and $\{q(z_i)\}$ converge
to $p$ in $M$.
Since $n \geq 2$, Lemma \ref{Lemma:TubeHomeo}
implies that there is a homeomorphism $h\colon (M, \{p\}) \to
(M, \{p\})$ such that for every $i$, $h(q(x_i)) = q(z_i)$. If $h$
is covered by $\widetilde h\colon V \to V$ then
$\{{\widetilde h}(x_i)\}$ is divergent although $\{x_i\}$ converges
to $x$, so $\widetilde h$ is not continuous.
\end{proof}
\begin{cor}
Suppose that $n \geq 2$ and let
$q\colon (V^n,\Sigma) \to (M^n, \{p\})$ be a topological blowup
of the manifold $M^n$ at $p$ such that $V \mysetminus \Sigma$ is
dense in $V$. If every homeomorphism of $(M, \{p\})$ is
covered by a homeomorphism of $(V, \Sigma)$, then
$\Sigma = \{\text{pt.}\}$ and $q$ is a homeomorphism.
\qed
\end{cor}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
|
1,314,259,996,555 | arxiv | \section{Introduction}
\IEEEPARstart{T}{he} Gut Bacteriome (GB) is an ecosystem of a massive number of bacterial cells which play a vital role in maintaining the stability of the host's metabolism\cite{ursell2012defining}.
The bacterial populations of the GB build complex interaction networks by exchanging metabolites with the host and/or other bacterial populations, resulting in the production of new metabolites or other molecules, such as Short Chain Fatty Acids (SCFA) (essential for the host's health) and proteins \cite{sanna2019causal}. This process occurs inside of each bacterial cell and is supported by specific complex gene regulatory networks, which are modulated depending on the established interaction networks in the human GB.
External factors such as the availability of nutrients, antibiotics and pathogens can affect this interaction network resulting in disruptions to the overall composition and, in turn, behavior of the human GB \cite{wen2017factors}. These factors mainly alter the compositional balance of the human GB, i.e. dysbiosis, disrupting the metabolite production \cite{iljazovic2020perturbation}. In humans, these GB changes have a significant impact on the host’s health and may lead to many diseases including \sm{inflammatory} bowel disease, type-2 diabetes, and obesity. Therefore, several studies have been undertaken to precisely identify the causes for microbial behavioral alterations and their consequent health effects in humans and animals \cite{gupta2020predictive,lynch2019translating,He2018}. For example, Yang et al. \cite{yang2020landscapes} performed a cross-sectional whole-genome shotgun metagenomics analysis of the microbiome and proposed a combinatorial marker panel to demarcate microbiome related major depressive disorders from a healthy microbiome. From a different perspective, Kim et al. introduced a split graph model to denote the composition and interactions of a given human gut microbiome \cite{kim2019novel}. They studied three different sample types (classified as healthy or Crohn's disease microbiome) to analyse the influence of microbial compositions on different behaviors of the host's cells. Inspired by these works, we propose a novel tool to further characterize the interactions among the bacterial populations often found in the human GB.
In this paper we propose a two-layer interaction model supported by the exchange of molecular signals, i.e. metabolites, to model the human GB.
Here, we identify the interactions between bacterial cells as \textbf{Molecular Communications} (MC) systems and their collective behavior as a MC network. MC aims to model the communication between biological components using molecules as information \cite{akyildiz2019moving} and it is fundamental to characterize the exchange of metabolites in our two-layer interaction model.
In the graph network, bacterial populations act as nodes while the edges represent the interactions between them. This interpretation allows to quantify the behaviors of the human GB using graph theoretical incorporating MC analysis to understand impacts from distances between different graph states and variations of node/edge weights. The theoretical quantification of the node behaviors and edge behaviors that will be discussed in subsequent sections explain the quantification of the graph behaviors using Flux Balance Analysis (FBA).
\sm{Moreover, conducting \textit{in-vivo} or \textit{in-vitro} experiments on the human GB to extract data related to each interaction of the network often requires a significant amount of resources.}
Hence, we designed an agent-based simulator (henceforth named as virtual GB) to simulate the human GB which produces data that can be used to quantify the same set of measures by avoiding complex calculations. The main reason for the complexity of theoretical calculations using FBA is the number of parameters and their stochastic nature. The virtual GB performs the behaviors of the human GB considering natural characteristics. Hence, the generated data represents bacterial behaviors that are influenced by the aforementioned stochastic parameters.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.55\textwidth}
\centering
\includegraphics[width=\textwidth]{diagrams/SysModel-1-2.pdf}
\caption{}
\vspace{63pt}
\label{fig:VGB}
\end{subfigure}
\begin{subfigure}[b]{0.43\textwidth}
\centering
\includegraphics[width=\textwidth]{diagrams/SysModel-3.pdf}
\caption{}
\label{fig:TwoLayerModel}
\end{subfigure}
\caption{Illustration of the system model. (a) We recreated the human GB functionalities on virtual GB using voxel architecture and parallel processing dedicating one GPU block for each bacterial cell to produces quantitative data on MC layer, and (b) we propose a two-layer system model to investigate the molecular interactions simulated in the virtual GB.}
\label{fig:SystemModel}
\end{figure*}
Our main contributions are as follows:
\begin{itemize}
\item \textbf{Design of a two-layer interaction model of the human GB}: The gut bacteria
consume, metabolise and secrete metabolites as single cells and use them to form a complex interaction network among the different bacterial populations in the human GB. Hence, in this study we design a layered interaction model to investigate the dynamics of the human GB based on the exchange of metabolites.
\item \textbf {Analysing molecular communication impact on the human GB graph structure:}
Deviations of bacterial populations' metabolism cause alterations in molecular interaction within the human GB, which may impact the graph layer structure. We analyse this relationship between the MC measures and the graph structure of the human GB in terms of graph nodes and edges behaviours.
\item \textbf{Development of a human GB simulator to perform \textit{in-silico} experiments:}
We design and utilize an \textit{in-silico simulation model} of the human GB to investigate the direct and hidden interactions among bacterial populations based on the exchange of metabolites.
\item \textbf {Explaining the dynamics of human GB using graph analyses:}
The bacterial growth and changes in the human GB metabolism can be interpreted as a result of a series of cascading metabolic activities through bacterial interactions.
In this study, we explain the dynamics of bacterial growth and metabolism changes using graph analysis on the bacterial interactions.
\end{itemize}
In the next sections, we detail our approach to model the human GB and assess its network performance. Section \ref{sec:background}, we describe the basics of the human GB, and highlight the existent gaps that this research aims to address. Our proposed model is detailed in Section \ref{sec:two-layer}. Then, in Section \ref{sec:VirtualGB}, we introduce the simulation environment built to utilize metagenomics data and perform \textit{in-silico} experiments with the human GB. The metrics considered in this paper are introduced in Section \ref{sec:SystemDynamics}, and our analysis results are presented in Section \ref{sec:AnalyticalResults}. Finally, our conclusions are shown in Section \ref{sec:conclusions}.
\section{Background on the Human GB Model}\label{sec:background}
The human GB is the bacterial ecosystem residing inside the human digestive system, comprising of approximately 1000 species interacting with each other, and carrying out crucial functions such as nutrient metabolism and immunomodulation of the host \cite{jandhyala2015role}. These bacteria do not manifest their cellular functions as individual entities but exhibit various social behaviors such as commensalism \cite{hooper2001commensal}, amensalism \cite{garcia2017microbial}, mutualism\cite{chassard2006h2}, parasitism and competition by interacting with other populations mainly using molecules (e.g., proteins, metabolites and \textit{quorum} sensing) \cite{hasan2015social}. Moreover, similar bidirectional exchanges of molecules occur between the GB bacterial populations and the human gut cells. These bacteria utilize products of the host metabolism activities or dietary components from the gastrointestinal tract to convert into various products essential for the host, through different metabolic pathways \cite{visconti2019interplay}.
This composition of human GB is a crucial driver for processing of metabolites (i.e., small molecules produced and used in metabolic reactions) in the lower intestine, which significantly impacts the health of the host \cite{singh2017influence}. Hence, any imbalance of the GB may result in negative impacts on the human health ranging from metabolic deficiencies to diseases such as type-2 diabetes, inflammatory bowel diseases and cancers \cite{henson2017microbiota,bjerrum2015metabonomics}. The composition of the human GB differs between individuals, and it depends on various factors including dietary patterns, gut diseases, exercise regimes, antibiotic usage, age, and genetic profiles \cite{rajoka2017interaction}.
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{diagrams/DCgraph.pdf}
\caption{Degree Centrality distribution of the full human GB MC network.}
\label{fig:FullGraph_DC}
\end{figure}
The relationship between the GB and the human gut not only benefits the host, but also the bacterial populations. They utilize the nutrient availability (i.e. the metabolic inputs to the GB) to modulate their growth and improving their survivability \cite{shanahan2017feeding}. Some bacteria can shift their metabolic pathways based on the extracellular signals such as the concentrations of nutrients and the growth condition \cite{shimizu2013metabolic}. For instance, \textit{Roseburia inulinivorans} switches the gene regulation switches between consumption of glucose and fucose in SCFA production according to the nutrient availability\cite{scott2006whole}. In addition to that, some of the bacterial populations will be highly benefited than others depending on the metabolite/nutrient that is being processed by the GB, which will require different signalling pathways, resulting in single or multiple bacterial interactions. For example \textit{Eubacterium rectale} consumes acetate as a growth substance and produces butyrate which is one of the most important SCFA \cite{riviere2016bifidobacteria}\cite{o2016bifidobacteria}. Similarly, the growths of \textit{Roseburia intestinalis} and \textit{Faecalibacterium prausnitzii} are also stimulated by acetate \cite{rowland2018gut}. Hence, it is evident that overall metabolic functionality relies on the composition of the GB and the metabolic inputs have a significant impact on GB composition.
The bacterial population signalling process in the GB is quite similar to routing and relaying information in a conventional network system, and has inspired different network models (including ours) of the human GB interactions. For example \cite{naqvi2010network} used a network-based approach to characterize the human gut microbiome composition and analyzed the healthy vs diseased states using network statistics. Another study focuses on the use of Boolean dynamic models that combines genome-scale metabolic networks, to determine the metabolic deviations between community members, which was used to characterize their metabolic roles of interactions\cite{steinway2015inference}.
\section{Two-layer Human GB Interaction Model}\label{sec:two-layer}
In this paper, we represent the metabolic interactions between bacterial populations in the human GB as a two-layer interaction model.
Figure \ref{fig:SystemModel} explains how we designed the model. First, the compositional and behavioral data on the human GB is extracted from the databases and literature. Then, the extracted data is used in implementing the virtual GB (Figure \ref{fig:VGB}). Finally, the virtual GB simulates the human GB functionalities according to various experimental setups (later explained in section \ref{sec:AnalyticalResults}) and produces data on bacterial, molecular and gut environmental behaviors. Then the produced data is analysed according to the introduced Two-layer interaction model as shown in Figure \ref{fig:TwoLayerModel}.
The upper layer of this model which is the Bacterial population graph layer defines the interconnections and overall structure of the human GB where we model the bacterial populations and host as nodes, and interactions between them as edges. To minimize the complexity of the model, this graph layer considers genera as nodes as species in the same genus share a common ancestral origin and the data availability. Further, the edges of the network represent the direct connections between the nodes that produce a particular metabolite and the nodes that consume the corresponding metabolite (An example graph network of the human GB can be seen in Figure \ref{fig:SubGraph}).
In this layer, we can observe which bacterial genus is predominant in the human GB to measure the compositional changes and its impacts on the host's health. In other words, this model allows us to investigate the network topology of the human GB.
The bottom layer consists of the cascaded molecular communications systems created by the bacterial populations to establish their exchange of metabolites and support their network structure. Here, each node is viewed as a molecular transceiver and the edges are the communications channels interconnecting the nodes. Furthermore, this model extends to the molecular signals that reach the human GB from the environment, as well as, the ones that are output from the human GB and return to the environment. The interactions represented in this layer are dynamic and will depend on several environmental conditions, such as media characteristics, bacterial population sizes, and human GB composition. Please note, this is the layer where we initially observe the impacts of any alterations on the human GB composition (we further model and analyse this effect in Section \ref{sec: InputVsStructure}). The upper layer and the bottom layer are further described in the following sections.
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{diagrams/FullGraph1.png}
\caption{Illustrates a subgraph of the human GB only considering species of nine genera related to SCFA. The nodes are color-coded according the degree ranking, where the darker color indicates higher number of inward and outward interactions while nodes with lesser number of interactions are with lighter color.}
\label{fig:SubGraph}
\end{figure*}
\subsection{Upper Layer - Bacterial Population Graph Layer}\label{sec:upper_layer}
Bacteria display a wide variety of social behaviors, and this can lead to processes such as metabolism of molecules or coordinated biofilm formation \cite{unluturk2016impact}. The bacteria's ability to consume and produce multiple metabolites results in dense interaction patterns that can lead to challenges in the analysis. However, representing the interactions of human GB as a graph network can provide a new avenue towards analyzing the bacterial communication and the impact on the overall bacteriome stability.
Our human GB interaction model aims to provide a better global view of the functionality of the human GB, leading to the understanding of the causes and effects of its imbalance to propose precise alterations to fix such issues. Therefore, we model the human GB as follows. We first consider that all bacterial cells $b^{B_{k}}$ of a bacterial population $B_{k}$ (where $k$ is the bacterial population identifier) perform the same series of metabolic functions to process the metabolites input in the human gut. Each node of the proposed MC network is a bacterial population and comprises the collective metabolic functions of all cells. \sm{Let $\Omega$ be the set of all agents we considered in our human GB interactions study, i.e. host cells and bacterial populations, $\Omega=\{host,B_k\}$. In this case, the molecular intake of population $B_{k}$ from $\Omega$, $C_{(\Omega,B_k)}$. is considered $C_{(\Omega, B_k)}\simeq\sum c_{(\Omega, b^{B_{k}})}$ where $C$ represents population interactions, $c$ represents the intercellular interactions and $c_{(\Omega, b^{B_{k}})}$ is the molecular reception of bacterial cell $b^{B_{k}}$ from a $B_{k}$ source. In the same way, molecular emission of the population is considered the combined molecular emission of all bacterial cells of the particular population, $C_{(B_k, \Omega)}\simeq\sum c_{(b^{B_{k}}, \Omega)}$, where $C_{(B_k,\Omega)}$ is the molecular emission from population
$B_k$ to any receiver (host or other bacterial population) and $c_{(b^{B_{k}}, \Omega)}$ is the molecular emission of single bacterial cell of the population $B_k$ to any receiver. Additionally, the metabolite consumed by the bacterial cell $b^{B_k}$, $M_{Con}({b^{B_{k}}})$ is obtained as $M_{Con}(b^{B_{k}})=c_{(\Omega, b^{B_k})}-c_{(b^{B_k}, \Omega)}$. Hence the metabolite consumption of a bacterial population is defined as $M_{Con}(B_{k}) \simeq \sum M_{Con}(b^{B_{k}})$.}
\sm{Next, we map the interactions between bacterial populations to a directed multi-graph network $\Gamma = (B, C, B^{s}, B^{d}, M)$ where $B$ is the set of all bacterial populations, $C$ is the set of all interaction in the human GB, $B^{s}\in B$ is \sm{the bacterial population} interaction sources, $B^{d}\in B$ \sm{the bacterial population} interaction destinations and $M$ is the set of metabolites. Using this definition and the data from NJS16 database, we can create the full graph network of the human GB $\Gamma_{full}$ where $|B|=532$, $|C|=30,085$ \cite{sung2017global}. Figure \ref{fig:FullGraph_DC} shows the normalized degree distribution of the full network $\Gamma_{full}$ which aligns with a power-law $P_{\text{deg}}(\kappa) \propto \kappa^{-\gamma}$ (where $\kappa$ is the degree and $\gamma$ is the decaying factor) distribution. Hence, we categorize the human GB as a scale-free network. Additionally, we emphasize a dominating quality of scale-free network which is the relative nodes that are common with a high number of interactions. Nodes with a significant number of edges are considered as hubs that perform specific functions. In this work we reduce such complex MC network of the human GB by focusing our model on the bacterial population interactions related to one of the main metabolic interactions, i.e. production of SCFAs. For example, Figure \ref{fig:SubGraph} represents a bacterial population subgraph $\Gamma_{sub}$ of the human GB, where the number of bacterial populations $|B|=50$, number of interactions $|C|=393$ and number of molecules used in interactions $|M|=230$, which will be investigated in the further sections of this work. In this graph, some nodes are involved in a significantly large number of interactions, compared to others. The color scheme used for the nodes represents the degree ranking, where darker colors indicates the higher degree ranking while lighter colors indicates the opposite.}
The examples introduced in this section show that the human GB is a complex network to analyse. Therefore, we reduce the complexity of the human GB interaction model, even more, by considering the bacteria genera instead of the species. By taking this approach, we are able to represent the human GB with a few nodes and edges and, we exemplify this approach in Figure \ref{fig:FBA}. In this graph, we identified three types of interactions. First is the interactions between the host and the bacterial populations, where the bacterial populations consume nutrients (external inputs of the graph) from the host. These interactions are denoted as $C_{(host, B_k)}$, where the interaction starts with the host and ends with the bacterial population $B_k$, which acts as an inward node of the graph. The second type of interaction is identified when the host consumes the metabolite produced by the bacterial populations. These interactions are presented as $C_{(B_{k}, host)}$ where the interactions start with $B_k$ and end with the host where the bacterial population, $B_k$ acts as an outward node of the graph. Finally, when the metabolites produced by one bacterial population and consumed by other bacterial populations in the human GB, we classify them as direct interactions, $C_{(B_k, B_{k'})}$ where $B_{k'}$ is any bacterial population except $B_k$. Here, the bacterial populations $B_k$ and $B_k'$ act as relay nodes of the graph.
The bacterial populations in the human GB are capable of receiving and producing multiple metabolites that are processed through multiple pathways. Therefore, we can safely assume that all bacterial populations in the human GB consume or produce metabolites. Hence, the inward degree and outward degree of a node in our bacterial population graph layer must be $\text{deg}^-(B_{k}), \text{deg}^+(B_{k}) \geq 1$.
We would also like to highlight that the outputs of this metabolism, metabolite production and the bacterial populations' growth play significant roles in the performance and the robustness of the human GB. Therefore, to analyse the contribution and role of each node in the graph, we consider the metabolism output of each bacterial population as the weight of the node. The metabolism output is modeled as the signal processing performance $SPP(B_{k})$, which is further detailed in Section \ref{sec:bottom_layer}.
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{diagrams/DummyGraph.pdf}
\caption{Graph representation of the interaction between the bacterial populations investigated in this paper, which are commonly found in the human GB.}
\label{fig:FBA}
\end{figure}
\subsection{Bottom Layer - MC System}\label{sec:bottom_layer}
As detailed in the previous section, the metabolism of nutrients by the human GB involves the reception, processing, production of metabolites. These activities are fundamental for the maintenance of the human GB, and this is modeled as the MC layer shown in Figure \ref{fig:TwoLayerModel}. Our aim of having the two-layer model is to determine how the changes due to molecular signals of the metabolites will affect the relationship of the bacterial population graph layer. Therefore, any changes in the bottom layer directly affect the upper layer and vice-versa.
Here, we model the bottom layer based on the definitions of the upper layer. Therefore, we define the metabolites as the molecular signals that are exchanged by the nodes, which can assume different functions depending on the MC network structure. For example, when the node receives molecular signals, we model it as a receiver, and when processing and secreting molecular signals we define them as transmitters based on the MC paradigm. The edges of the proposed MC network are represented as the MC channels to model the physical transport of molecular signals between the nodes. Figure \ref{fig:TwoLayerModel} shows a visual representation of the proposed bottom layer and its relationship with the upper layer.
One of the functions that the nodes can execute is the molecular signal transmission and in this case, the emission of metabolites. Once the molecular signal is released to the extracellular space, it is diffused through the fluidic MC channel interconnecting the nodes. This process is governed by Fick's Law of diffusion and is represented as follows
\begin{equation}
\frac{\partial c(x,t)}{\partial t}=-D(x,t)\frac{\partial^2 c(x,t)}{\partial x^2},
\end{equation}
where $c(x,t)$ is the molecular signal concentration, $D$ is the diffusion coefficient, $x$ is the particle position (in metres) and $t$ is the time (in seconds).
The diffused molecular signal is received by the nodes which have the membrane receptors that will allow the metabolites to bind. The performance of this network node function (i.e., molecular reception) relies on many factors such as molecule size \cite{farsad2016comprehensive, chahibi2014molecular}, ligand-receptor maximum attraction length \cite{chahibi2014molecular}, binding noise due to the Brownian motion of molecules near the receptors \cite{kuscu2019transmitter}, ligand-receptor bond equilibrium \cite{chahibi2014molecular}, and the minimum required concentration to be detected \cite{llatser2013detection}. After receiving the molecular signals, the node will process them internally which may result in the production of a new molecular signal to be transmitted to the next node (focus of this paper) or consume the molecular signal.
Upon reception of the molecular signals, the nodes execute a series of actions to internally processes them. These actions are often modelled as signalling pathways and the end result of this process are the metabolites that will be transmitted to the next node or to the host \cite{oliphant2019macronutrient}.
\sm{The signalling process performance of a bacterial cell $b^{B_{k}}$ related to any metabolite $M_j$ is represented as $SPP_{b^{B_{k}}}(M_j)$. Let's assume that the cell $b^{B_{k}}$ produces $M_j$ by consuming another metabolite $M_{j'}$.
Then the signal process performance $SPP_{b^{B_{k}}}(M_j)$ can be modeled by considering the metabolite $M_{j'}$ reception process (defined as $R_{b^{B_{k}}}(M_{j'})$), the encoding/decoding process from metabolite $M_{j'}$ to $M_j$ (defined as $E_{b^{B_{k}}}(M_{j'},M_{j})$), and $M_j$ metabolite secretion process by the cells in the bacterial population $b^{B_{k}}$ (defined as $S_{b^{B_{k}}}(M_{j})$). Hence, we represent the signal process performance as follows,}
\begin{equation}
\label{eq:SPPCalc}
SPP_{b^{B_{k}}}(M_{j})=f(R_{b^{B_{k}}}(M_{j'}),E_{b^{B_{k}}}(M_{j'},M_{j}),S_{b^{B_{k}}}(M_{j})).
\end{equation}
Therefore, the SPP of the populations $B_{k}$ can be modeled as follows,
\begin{equation}
\label{eq:NodeSPP}
SPP_{B_{k}}(M_{j}) = \sum SPP_{b^{B_{k}}}(M_{j}).
\end{equation}
Since, the output of the molecular signal processing is the emission of a particular molecular signal, it is fair to say,
\begin{equation}
\label{eq:SPPvsInteractions}
SPP_{B_{k}}(M_{j}) = C^r_{(B_k, \Omega)}(M_{j})
\end{equation}
where $C^r_{(B_k, \Omega)}(M_{j})$ is the rate of molecule $M_j$ production by the bacterial population $B_k$ to any node (either other bacterial populations or host cells).
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{diagrams/AnalysisModels.pdf}
\caption{Illustration of the analysis structure of this study, where Analysis 1 focuses on the influence of inputs on the graph structure while Analysis 2 examine the behaviors of graph output against the structural deviations. See Sections \ref{sec: InputVsStructure} and \ref{sec:StructureVsSNR} for our analysis methods, and Sections \ref{sec:analysis1} and \ref{sec:StructureVSOutput} for our results.}
\label{fig:analyses_structure}
\end{figure}
\section{System Dynamics}\label{sec:SystemDynamics}
Based on the proposed \sm{two-layer human GB model}, we investigate the system dynamics of the human GB through a series of simulations using a virtual GB model. \sm{To that end, we first recreate the digital form of human GB on the simulator. This recreation process is explained in depth later in Section \ref{sec:VirtualGB}.} Then we perform two main sets of experiments as depicted in Figure \ref{fig:analyses_structure}. In the first set, we analyse the impact of the system's inputs on the connectivity structure of the virtual GB as we understand that the molecular input signals may work as an altering factor of the human GB interaction patterns. In the second set, we manipulate the composition of our virtual GB to investigate the impact on the metabolite production that leaves the outward node of our MC network. Through this second set of experiments, we aim to identify the nodes that have the largest impact on the overall production of the metabolites and can play a pivoting role in the GB imbalances.
In our analyses, first we define a standard graph state $S_{0}$, which represents the functionality of an average healthy human GB with the intention of quantifying structural connectivity changes and behavioral deviations that is relative to the standard structures. The average composition, interactions and metabolite production dynamics were mainly considered in defining the $S_{0}$. Finally, the accuracy of the $S_{0}$ is confirmed using the output metabolite ratios over the exact values as the virtual GB scales down the ecosystem. The average composition and the interactions of $S_{0}$ for the case study of this paper is presented in Section \ref{sec:AnalyticalResults}
\subsection{Molecular input impact on the human GB structure}\label{sec: InputVsStructure}
Due to the variety of bacterial behaviors induced by the exchange of molecules, some of the molecular input signals have a significant impact on the structure of the human GB (our focus), while others are directly converted into output metabolites. In this section, we detail how the molecular input signals impact the structure of our MC network. The structural deviations of the graph is a crucial measurement in understanding the deviation of the human GB behavior from the healthy state. This structural deviation is evaluated in terms of edges and nodes weight variations using the rates of the interaction of the nodes. Hence, we explain how the interaction rates can be calculated theoretically using FBA and are represented as follows
\begin{equation}\label{eq:FBA1}
F_{[k\times q]} \cdot \vec{C} = \vec{M}_{Con}(B_{k})
\end{equation}
where $F_{[k\times q]}$ is the stoichiometric matrix of $k$ number of bacterial populations and $q$ number of interactions based on the flux of metabolites between the nodes in the MC network.
Here $\vec{C} =[C^r_{1}, C^r_{2},..., C^r_{q}]_{1\times q}$ and $C^r_q$ is the rate of interactions for $C_q$. We can solve (\ref{eq:FBA1}) as follows,
\begin{equation}\label{eq:int_rate}
\begin{blockarray}{ccccc}
\begin{block}{c(cccc)}
B_{1} & a_{1,1} & a_{1,2} &... & a_{1,q}\\
B_{2} & a_{2,1} & a_{2,2} &... & a_{2,q} \\
\vdots &\vdots &\vdots &\ddots &\vdots \\
B_{k} & a_{k,1} & a_{k,2} &... &a_{k,q}\\
\end{block}
\end{blockarray}
\cdot
\begin{blockarray}{c}
\begin{block}{(c)}
C^r_{1}\\
C^r_{2}\\
\vdots \\
C^r_{q}\\
\end{block}
\end{blockarray}
=
\begin{blockarray}{c}
\begin{block}{(c)}
\frac{\mathrm{d}M_{Con}(B_{1})}{\mathrm{d}t}\\
\frac{\mathrm{d}M_{Con}(B_{2})}{\mathrm{d}t}\\
\vdots \\
\frac{\mathrm{d}M_{Con}(B_{k})}{\mathrm{d}t}\\
\end{block}
\end{blockarray}
\end{equation}
where, $a_{k,q}$ is the stoichiometry of the interaction $C^r_q$ for bacterial population $B_k$.
\sm{Based on (\ref{eq:int_rate})}, we can extract the relationship between rates of interactions starting from the node $B_{k}$ using Mass Balance Equation (MBE) as, which is based on the following relationship
\begin{equation}
\frac{\mathrm{d}M_{Con}(B_{k})}{\mathrm{d}t}=\sum_{q}a_{(k,q)}C^r_{q}.
\end{equation}
On the other hand, the rate of molecular consumption can be modeled as follows \cite{martins2016using},
\begin{equation}
\frac{\mathrm{d}M_{Con}(B_{k})}{\mathrm{d} t}=-U_{1}\left ( \mu_{k}\frac{M_{Con}(B_{k})}{M_{Con}(B_{k})+K_{S1}} \right )N_{B_{k}}
\end{equation}
where $N_{B_{k}}$ is the bacterial concentration, $\mu_{k}$ is maximum growth rate, $K_{S1}$ is the half-saturation constant of the bacteria, and $U_{1}$ is an utility parameter.
Hence,
\begin{equation}
-U_{1}\left ( \mu_{k}\frac{M_{Con}(B_{k})}{M_{Con}(B_{k})+K_{S1}} \right )N_{B_{k}}=\sum_{q}a_{(k,q)}C^r_{q}.
\end{equation}
By solving the series of MBEs, all the interaction rates can be calculated. This is a highly complex calculation due to the massive number of nodes and the edges of the network. Additionally, there is a large number of parameters associated to the structural connections that influence the MBEs. In order to minimize this complexity, the virtual GB produces data that is related to the rates of interactions, and this will avoid the complex FBA calculations.
The extracted rates of interactions are then used to quantify the graph structural changes in two ways. First, we investigate the graph structural changes considering the behaviors of the node weights. Here, the statistical distances between the weights of the same node in different graph states are measured. The node weights of this system are modeled using the $SPP$. Collective metabolite production of the bacterial population $B_{k}$ in the graph state $S_g$ is considered as the node weight $B_{k}^{w}(S_g)$ and can be evaluated as follows,
\begin{equation}\label{eq:node_weight}
B_{k}^{w}(S_g)=\sum_{j}SPP_{B_{k}}(M_{j}).
\end{equation}
Alternatively, using (\ref{eq:SPPvsInteractions}) we compute the node weight as follows,
\begin{equation}\label{eq:node_weightbyInteraction}
B_{k}^{w}(S_g)=\sum_{j}C^r_{(B_k,\Omega)}(M_{j}).
\end{equation}
Based on this, $d(B^w_{k}:S_{g},S_{0})$ that represents the distance of node $B_{k}$ between the two graph states $S_{0}$ and $S_{q}$ is evaluated as follows,
\begin{equation}
\label{eq:NodeHammingDistance}
d(B^w_{k}:S_{g},S_{0}) = B_{k}^{w}(S_g)-B_{k}^{w}(S_0).
\end{equation}
Next, we quantify the structural deviation of the graph using the interaction changes.
There are several methods from the literature for distance calculation between two graph states. In this study, we consider static snapshots of different graph states that can enable the use of the \emph{Hamming Distance} to evaluate graphical distances for two states \cite{deza2009encyclopedia}. The Hamming distance $d_{h}(S_{0}, S_{g})$ between the graph states $S_g$ and the standard state $S_0$ is defined as the difference of two adjacent matrices corresponding to the two graph states. First, we define the adjacency matrix of the graph state $S_{g}$ as follows,
\begin{equation}
\begin{blockarray}{ccccc}
& B_{1} & B_{2} & ... & B_{k}\\
\begin{block}{c(cccc)}
B_{1} & C^w_{(B_{1},B_{1})} & C^w_{(B_{1},B_{2})} &... & C^w_{(B_{1},B_{k})}\\
B_{2} & C^w_{(B_{2},B_{1})} & C^w_{(B_{2},B_{2})} &... & C^w_{(B_{2},B_{k})} \\
\vdots &\vdots &\vdots &\ddots &\vdots \\
B_{k} & C^w_{(B_{k},B_{1})} & C^w_{(B_{k},B_{2})} &... & C^w_{(B_{k},B_{k})}\\
\end{block}
\end{blockarray}
\end{equation}
where $C^w_{(B_{k},B_{k})}$ is the weight of the interaction $C_{(B_{k},B_{k})}$.
\sm{We define the weight of the interaction $C_{(B_k,B_{k'})}(M_j)$ as follows,}
\begin{equation}
\label{eq:EdgeWeighEmpirical}
C^w_{(B_k,B_{k'})}(M_j)=\frac{C^r_{(B_k, \Omega)}(M_j)}{\sum_{k}C^r_{(B_k, \Omega)}(M_j)}\cdot \frac{C^r_{(\Omega, B_k')}(M_j)}{\sum_{B_k'}C^r_{(\Omega, B_k')}(M_j)}.
\end{equation}
\sm{Moreover, from the number of molecules released by a bacterial population, only a small portion is consumed directly by the other populations and the remaining concentration will get accumulated in the environment.}
This means the most significant portion of molecular consumption by the bacterial populations is from the environment. We define this process with the help of a memory component concept as elaborated in Figure \ref{fig:MemModule}. \sm{Since the metabolites are accumulated in the environment, we consider it as memory, then model the metabolite accumulation as an interaction starting from a bacterial population that releases the metabolites and ending with the memory, $C_{(B_k, Mem)}$. In the same way, the metabolite consumption from the environment is modeled as an interaction starting from the memory and ending with a bacterial population that consumes the particular metabolite, $C_{(Mem, B_k)}$.}
Hence, we modify (\ref{eq:EdgeWeighEmpirical}) by applying the memory and is represented as follows,
\begin{equation}
\label{eq:EdgeWeighWithMem}
\begin{split}
C^w_{(B_k,B_{k'})}(M_j)=\frac{C^r_{(B_k, Mem)}(M_j)}{\sum_{k}C^r_{(B_k, Mem)}(M_j)}\\[1mm]
\cdot \frac{C^r_{(Mem, B_k')}(M_j)}{\sum_{B_k'}C^r_{(Mem, B_k')}(M_j)}.
\end{split}
\end{equation}
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{diagrams/MemModule.pdf}
\caption{Illustration of the environment working as a memory of molecules.}
\label{fig:MemModule}
\end{figure}
\sm{Then, the Hamming distance, $d_{h}(S_{0}, S_{g})$ can be represented as,
\begin{equation}
d_{h}(S_{0}, S_{g})=\sum_{k,k'} |C_{(B_{k},B_{k'})}^{w}(S_g)- C_{(B_{k},B_{k'})}^{w}(S_0)|
\end{equation}
where, $C_{(B_{k},B_{k'})}^{w}(S_g)$ and $C_{(B_{k},B_{k'})}^{w}(S_0)$ are the weights of interaction $C_{(B_{k},B_{k'})}$ in graph states $(S_g)$ and $S_0$ respectively.}
\subsection{Human GB structure impact on the molecular output}
\label{sec:StructureVsSNR}
This analysis explores the impact of interaction variations of the human GB on the output. Here, we keep the inputs at an optimal level and manually alter the graph structure by changing the population sizes which leads to variations in the $SPP$ of the nodes. Then the output of the system is measured in different graph states and the weights of the edges are calculated using (\ref{eq:EdgeWeighEmpirical}) to determine the molecular output of the MC layer using graph theory.
\sm{The ratio between the three SCFAs can be identified as a critical measurement to evaluate the metabolite production accuracy of the bacteriome. We adopt SNR to evaluate the consistency of the output signal ratios. In this analysis, we calculate SNR of any signal $SNR(M_j)$, considering the other output signals, $M_{j'}$ as noise. This SNR value directly indicates the ratio between the molecular signal $M_j$ and other metabolite signals $M_{j'}$. Then $SNR(M_j)$ is calculated as follows,}
\begin{equation}
\label{eq:SNRCalulation}
SNR(M_j)=\displaystyle\sum_{k}\frac{C_{(B_k,host)}(M_{j})}{\sum_{j'} C_{(B_k,host)}(M_{j'})}.
\end{equation}
\sm{Moreover, some bacterial populations do not produce specific SCFAs, but have an indirect influence on them. For example, \textit{Bacteroides} cells do not produce butyrate, but the acetate produced by the \textit{Bacteroides} cells is a substrate for the butyrate production by \textit{Faecalibacterium} and \textit{Roseburia} cells. Hence, the impact of changes in \textit{Bacteroides} population sizes cascades through the interaction network and affects the butyrate production. Considering the above mentioned effect, a correlation matrix is generated for variation of node weights vs the collective SCFA output of the human GB to analyse the impact of various bacterial populations in SCFA production.
Here, we denote the rate of SCFA $M_j$ output by all the bacterial populations as $O^r(M_j)$ where}
\begin{equation}
O^r(M_j) = \sum_{k} C^r_{(B_k, host)}(M_j).
\end{equation}
Then, the correlation coefficient $r(B_k)$ of node weight $B_k^w$ versus the collective output of $M_j$ is calculated based on the following relationship.
\begin{equation}
\begin{split}
r(B_k) =\displaystyle\sum_{g}\frac{\overline{B^w_{k}}-B^w_{k}(S_{g})}{\sqrt{(\overline{B^w_{k}}-B^w_{k}(S_{g}))^2}}\\[1mm]
\cdot\frac{(\overline{O^r}(M_j)-O^r(M_j))}{\sqrt{(\overline{O^r}(M_j)-O^r(M_j))^2}}
\end{split}
\end{equation}
where, $\overline{O^r}(M_j)$ is the standard collective output rate for $M_{j}$ by all the bacterial populations and $\overline{B^w_{k}}$ is the weight of the node $B_k$ in the standard state $S_0$.
\section{Analytical Results}\label{sec:AnalyticalResults}
In this section, we describe the development of the virtual GB and the results from our analysis that is based on the models presented in Section \ref{sec:SystemDynamics}.
\subsection{Virtual GB Design}
\label{sec:VirtualGB}
We developed the virtual GB using the metagenomic data to characterize the bacterial populations signaling interactions and its impact on the network relationships. The virtual GB is inspired by the BSim agent-based cell simulator \cite{Gorochowski2012}.
The virtual GB is written in C++ with CUDA platform for parallel processing to increase the simulation performance and most importantly, mimic the parallel processing typically executed by the bacterial cells. \sm{We dedicate one GPU block for each bacterial cell, and the threads of that block to intracellular functions of the corresponding cell.} To simulate the bacterial interactions, we model the exchange of molecules using metabolic flux in a diffusive media. The simulator is equipped with a 3D environment, which can be divided into chambers to place host or bacterial cells in specific regions of the environment. The 3D environment is further designed using a voxel architecture as shown in Figure \ref{fig:VGB} which provides the ability of extracting data on metabolites interactions and accumulations separately. Moreover, we can introduce any new cell type by creating their internal metabolic pathways and other physiological characteristics such as motility, shape, size, etc. Therefore, the simulator can be used for a range of setups including other metabolic function, microbial ecosystems in different body habitats or targeting specific bacterial behavior like quorum sensing.
\sm{We designed this simulator to extract multi-dimensional data from the MC network of the human GB, and its data extraction capabilities can be divided into two categories.} In the case of data extraction from bacterial cells, the simulator can log data on the metabolite consumption, metabolite production and growth of every bacterial population. The simulator also generates data of the bacterial ecosystem environment. This includes data on consumption of nutrients, metabolite accumulation, and the locations of the bacterial cells and host cell can be easily determined. In this study, we setup the virtual GB to simulate the SCFA production using the metagenomic data obtained in \cite{oliphant2019macronutrient}, KEGG \cite{kanehisa2000a, kanehisa2019a, kanehisa2021a} and MetaCyc databases \cite{caspi-a}
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{diagrams/SCFASubGraph.pdf}
\caption{Representation of the subgraph considered in the case study which contains the nodes and edges related to SCFA production.}
\label{fig:SCFAMetabolism}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{diagrams/SCFADiagram.pdf}
\caption{Combined and simplified SCFA production pathway of converting fucose and glucose, into SCFAs.}
\label{fig:SCFAProduction}
\end{figure}
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{diagrams/BayesianNeworkWithMemory.pdf}
\caption{Representation of the simplified MC network with plots illustrating link behaviors for different compositional changes. Please note that $glu$, $ace$, $lact$ and $bte$ stands for glucose, acetate, lactate and butyrate respectively.}
\label{fig:A2-Network_SimplifiedWithGraphs}
\end{figure*}
Here, we present a series of analyses conducted on the SCFA production within the human GB using the two-layer model of the Bacterial Population Graph layer and the Molecular Communication Layer. SCFA are the main products of fermentation of non-digestible carbohydrates\cite{morrison2016formation}. First, we defined the average composition of the human GB using the relative abundance (RA) data extracted from 352 samples of the MicrobiomeDB \cite{MicrobiomeDB}. \sm{In-depth graphical representation of RA data of all these samples} are shown in the Appendix. Then, we calculate the average RAs for the most abundant genera related to the SCFA production, which is shown in Table \ref{table:RA}.
\sm{Using these RA data along with the extracted interaction data from the databases mentioned earlier,}
we created a graph network for SCFA production, $\Gamma_{SCFA}$ \sm{following the definitions presented in Section \ref{sec:upper_layer}}.
Figure \ref{fig:SCFAMetabolism} illustrates the $\Gamma_{SCFA}$ where nodes sizes of Figure are proportional to the RAs of the respective genera shown in Table \ref{table:RA}. Furthermore, the edges are color coded to highlight the strengths of the interactions which are quantified using (\ref{eq:EdgeWeighEmpirical}). The direct interactions between the host and the first layer nodes represent the GB consumption of glucose and fucose supplied by the host. The interaction from the host to the \textit{Bacteroides} cells through glucose, $C_{(host, Bact)}(M_{glu})$, and \textit{Bacteroides} cells to \textit{Faecalibacterium} cells through acetate, $C_{(Bact, Fae)}(M_{ace})$, are the strongest since the \textit{Bacteroides} cells dominate the metabolism process. The interactions from the first layer of bacterial population nodes to the second layer are composed of interactions through lactate as well as acetate, which are shown later in detail in Figures \ref{fig:A1-GlucoseSubGraph}, \ref{fig:A2-BacteroidesSubGraph}, \ref{fig:A2-FaecalibacteriumSubGraph} and \ref{fig:A2-RuminococcusSubGraph}.
\sm{
For illustration purposes, we combine the metabolic processes executed on different bacterial cells and simplify the SCFA pathway to focus on the most important steps that leads to the production of the three most abundant SCFAs in the human GB, namely acetate, butyrate and propionate (see Figure \ref{fig:SCFAProduction})\cite{rowland2018gut}. Please note that in a typical human GB their abundance ratios range from 3:1:1 to 10:2:1 \cite{den2013role}, and we utilize this metric to establish the normal behavior of the bacterial ecosystem within our virtual GB. This enables us to modify the MC network inside the virtual GB to better investigate the effects of the molecular interactions between nodes. An example of such investigation can be seen in Figure \ref{fig:A2-Network_SimplifiedWithGraphs}, where each link is an interaction between the graph nodes in our virtual GB. Furthermore, each plot in Figure \ref{fig:A2-Network_SimplifiedWithGraphs} shows the behavior of the molecular signal transported in each link and their multiple molecular count values result from the different bacterial population sizes considered for each link. More detailed results for the virtual GB compositional changes are explained in following sections. Please note that the high variation in the molecular count shown in the interactions between \textit{Faecalibacterium}/\textit{Roseburia}, \textit{Eubacterium} and host cells (see Figure \ref{fig:A2-Network_SimplifiedWithGraphs}) are due to diffusion effects and are not treated as noise in this work. Therefore, our results are based on the average MC network behavior after twenty runs of the virtual GB to take this effect into account.}
\begin{table}[t!]
\centering
\caption{Average RAs of bacterial populations}
\label{table:RA}
\begin{tabular}{l|l}
\hline
Genus & Average RA \\ \hline
\textit{Bacteroides} & 0.4899173 \\
\textit{Alistipes} & 0.05960802 \\
\textit{Faecalibacterium} & 0.04329791 \\
\textit{Parabacteroides} & 0.04096428 \\
\textit{Ruminococcus} & 0.03320183 \\
\textit{Roseburia} & 0.01039938 \\
\textit{Eubacterium} & 0.0093219 \\
\textit{Bifidobacterium} & 0.00179366 \\
\textit{Escherichia} & 0.00185639 \\
\hline
\end{tabular}
\end{table}
\begin{figure*}[t!]
\centering
\subfloat[\label{fig:A1-GlucoseSubGraph}]{
\includegraphics[width=0.28\textwidth]{diagrams/A1-GlucoseSubGraph.pdf}}
\subfloat[\label{fig:A1-GlucoseMiddleLayer.pdf}]{
\includegraphics[trim=10 14 10 0,clip,width=0.36\textwidth]{diagrams/A1-GlucoseMiddleLayer.pdf}}
\subfloat[\label{fig:A1-GlucoseMCLayer.pdf}]{
\includegraphics[trim=10 14 10 0,clip,width=0.35\textwidth]{diagrams/A1-GlucoseMCLayer.pdf}}
\caption{Deviation of population sizes of \textit{Faecalibacterium}, \textit{Eubacterium} and Escherichia from the optimal levels due to different input concentrations of glucose: (a) subgraph for the glucose consumption, (b) edge weight behaviors of the intermediate interactions, and (c) population growth behaviors.}
\label{fig:A1-GlucoseVsGrowth}
\end{figure*}
\subsection{Analysis 1: Molecular input effects on the graph structure}\label{sec:analysis1}
\begin{figure}[t!]
\centering
\includegraphics[trim=5 0 0 0,clip,width=\columnwidth]{diagrams/A1-WightsBar.pdf}
\caption{Changes of node weights due to the variations in molecular signal inputs.}
\label{fig:Weights}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{diagrams/A1-GraphDistance.pdf}
\caption{Behaviors of overall graph weights against the changes in inputs and their concentrations.}
\label{fig:InputVsGraphDistance}
\end{figure}
\sm{Here}, we present the results for the analyses mentioned in Section \ref{sec: InputVsStructure}. The analyses are conducted by regulating the inputs glucose rate $C^r_{(host, Mem)}(M_{glu})$ and fucose rate $C^r_{(host, Mem)}(M_{fse})$ \sm{from the host cells} to the system that contains memory of existing metabolites and evaluating the \sm{human GB} compositional changes. The simulation for these experiments only contains growth dynamics of \textit{Faecalibacterium}, \textit{Eubacterium} and \textit{Escherichia} bacteria.
Figure \ref{fig:A1-GlucoseVsGrowth} illustrates the impact of glucose on the three bacterial populations. Figure \ref{fig:A1-GlucoseSubGraph} shows the glucose consumption graph $\Gamma_{glu}$, which is a subgraph of SCFA production graph, $\Gamma_{glu} \subseteq \Gamma_{SCFA}$. The colors used in Figures \ref{fig:A1-GlucoseMiddleLayer.pdf} and \ref{fig:A1-GlucoseMCLayer.pdf} follow the same color scheme as in Figure \ref{fig:A1-GlucoseSubGraph}. Figure \ref{fig:A1-GlucoseMiddleLayer.pdf} shows the behaviors of edge weight deviations of the bacteria in the host due to the changes in the glucose input rates $C^r_{(host, Mem)}(M_{glu})$. Meanwhile, Figure \ref{fig:A1-GlucoseMCLayer.pdf} depicts the variation of population sizes due to the changes in $C^r_{(host, Mem)}(M_{glu})$ as a fraction of the value when the input is at the standard level. The variations of the input rate $C^r_{(host, Mem)}(M_{glu})$ alters the intermediate interaction from any bacterial population $B_k$ to other species $B_{k'}$ through acetate, $C_{(B_k, B_{k'})}(M_{ace})$ and lactate $C_{(B_k, B_{k'})}(M_{lact})$, which are required for the growth of \textit{Faecalibacterium} and \textit{Eubacterium}, respectively. Figure \ref{fig:A1-GlucoseMiddleLayer.pdf} explains the graph theoretical behavior of indirect influence on the growth dynamics of the respective bacterial populations. The growth of \textit{Eubacterium} keeps increasing steadily until the $C^r_{(host, Mem)}(M_{glu})$ is twice of the standard level, while the growths of the other two bacterial populations converges to the static standard level. This phenomenon is manifested by the stoichiometry of the metabolite conversion, where an acetate molecule is produced by one glucose molecule while in the case of the lactate molecule it requires two glucose molecules. The growth of \textit{Escherichia} is directly altered by the variations of glucose inputs and the behaviors is similar to that of the \textit{Faecalibacterium} population. We calculated the Mean Standard Error (MSE) for the experiment by iterating the experiment 20 times and the maximum MSEs recorded for any metabolite is 0.03374.
\begin{figure*}[t!]
\centering
\subfloat[\label{fig:A2-BacteroidesSubGraph}]{
\includegraphics[width=0.32\textwidth]{diagrams/A2-BacteroidesSubGraph.pdf}}
\subfloat[\label{fig:A2-BacteroidesMiddleLayer}]{
\includegraphics[trim=0 10 0 0,clip,width=0.33\textwidth]{diagrams/A2-BacteroidesMiddleLayer.pdf}}
\subfloat[SCFA Output\label{fig:A2-BacteroidesMCLayer}]{
\includegraphics[trim=0 10 0 0,clip,width=0.34\textwidth]{diagrams/A2-BacteroidesMCLayer.pdf}}
\caption{Behaviors of SCFA production for various in \textit{Bacteroides} population sizes: (a) subgraph of Bacteroides population interactions, (b) edge weight behaviors, and (c) SCFA output.}
\label{fig:A2-BacteroideVsSCFA}
\end{figure*}
\begin{figure*}[t!]
\centering
\subfloat[\label{fig:A2-FaecalibacteriumSubGraph}]{
\includegraphics[width=0.34\textwidth]{diagrams/A2-FaecalibacteriumSubGraph.pdf}}
\subfloat[\label{fig:A2-FaecalibacteriumMiddleLaye}]{
\includegraphics[trim=5 10 0 0,clip,width=0.33\textwidth]{diagrams/A2-FaecalibacteriumMiddleLayer.pdf}}
\subfloat[\label{fig:A2-FaecalibacteriumMCLayer}]{
\includegraphics[trim=5 10 0 0,clip,width=0.32\textwidth]{diagrams/A2-FaecalibacteriumMCLayer.pdf}}
\caption{Behaviors of SCFA production for various in \textit{Faecalibacterium} population sizes: (a) subgraph related to the interactions of \textit{Faecalibacterium} population, (b) edge weight behaviors, and (c) SCFA output.}
\label{fig:A2-FaecaliVsSCFA}
\end{figure*}
\begin{figure}
\centering
\begin{subfigure}[b]{\linewidth}
\includegraphics[width=\linewidth]{diagrams/A2-RuminococcusSubGraph.pdf}
\caption{}\label{fig:A2-RuminococcusSubGraph}
\end{subfigure}
\begin{subfigure}[b]{.49\linewidth}
\includegraphics[width=\linewidth]{diagrams/A2-RuminococcusMiddleLayer.pdf}
\caption{}\label{fig:A2-RuminococcusMiddleLayer}
\end{subfigure}
\begin{subfigure}[b]{.49\linewidth}
\includegraphics[width=\linewidth]{diagrams/A2-RuminococcusMCLayer.pdf}
\caption{}\label{fig:A2-RuminococcusMCLayer}
\end{subfigure}
\caption{Behaviors of SCFA production for various in \textit{Ruminococcus} population sizes: (a) subgraph related to the interactions of \textit{Ruminococcus} population, (b) edge weight behavior, and (c) SCFA output.}
\label{fig:A2-RuminococcusVSSCFA}
\end{figure}
\begin{figure*}[t!]
\centering
\subfloat[\label{fig:A2-Bacteroides-SNR}]{
\includegraphics[trim=0 0 0 0,clip,width=0.33\textwidth]{diagrams/A2-Bacteroides-SNR.pdf}}
\subfloat[\label{fig:A2-Faecalibacterium-SNR}]{
\includegraphics[width=0.33\textwidth]{diagrams/A2-Faecalibacterium-SNR.pdf}}
\subfloat[\label{fig:A2-Ruminococcus-SNR}]{
\includegraphics[width=0.33\textwidth]{diagrams/A2-Ruminococcus-SNR.pdf}}
\caption{Simulation results for SNR of three output signal with the changes in population sizes of \textit{Bacteroides}, Feacalibacterium and \textit{Ruminococcus}. The populations sizes are changed as a ratio of the standard level: (a) SNR results for \textit{Bacteroides} population, (b) SNR results for Feacalibacterium population, and (c) SNR results for \textit{Ruminococcus} population.}
\label{fig:A2-BacteriaVsSNR}
\end{figure*}
\begin{figure*}[t!]
\centering
\includegraphics[trim=0 0 80 0,clip,width=0.9\textwidth]{diagrams/A2-Correlation_HeatMap.pdf}
\caption{Pearson correlation heat map of the impact on the three output signals by nine bacterial populations.}
\label{fig:An2-Correlation}
\end{figure*}
Deviations of a bacterial population concentration refer to deviations in node weights according to the (\ref{eq:NodeSPP}) and (\ref{eq:node_weight}). Figure \ref{fig:Weights} represents the node weight deviation compared to standard graph state $S_{0}$ due to the variability in inputs. This analysis reveals the impact of different input conditions on the molecular signal performance $SPP$ of bacterial populations.
While Figure \ref{fig:Weights} explains the node weight variations, Figure \ref{fig:InputVsGraphDistance} focuses on the overall interaction weight behaviors compared to $S_{0}$. This graph provides an insight into how the structure is being modified by the input variability. When the $C^r_{(host, Mem)}(M_{glu})$ is low compared to the standard level, the graph deviates significantly from the standard level and when the $C^r_{(host, Mem)}(M_{glu})$ exceeds the standard level, the graph starts to deviate again from the standard structure, but with a lesser magnitude compared to a weaker signal (the standard level is 1.0). This reveals that the human GB is more sensitive to low glucose concentrations. The experiment is repeated for the fucose input rates $C^r_{(host, Mem)}(M_{fse})$ as well, but the impact is minimal compared to $C^r_{(host, Mem)}(M_{glu})$.
\subsection{Analysis 2: Human GB structure effect on the graph outputs}
\label{sec:StructureVSOutput}
In this section, we analyze the direct and indirect impacts of the human GB compositional changes on the network behaviors. The analyses are conducted by altering the bacterial population sizes manually on the virtual GB and extracting the metabolite production data with respect to each alteration. The resulting behaviors of the MC layer are explained using the graph analyses. Although we conduct similar experiments for all the nine populations, we only show results on \textit{Bacteroides}, \textit{Faecalibacterium} and \textit{Ruminococcus} populations as they provide a better understanding of the metabolite production dynamics of the human GB. Figures \ref{fig:A2-BacteroideVsSCFA}, \ref{fig:A2-FaecaliVsSCFA} and \ref{fig:A2-RuminococcusVSSCFA} illustrate the influence of \textit{Bacteroides}, \textit{Faecalibacterium} and \textit{Roseburia} populations on the SCFA production.
Figure \ref{fig:A2-BacteroideVsSCFA} shows the impact of \textit{Bacteroides} population size variation on the human GB SCFA production. In this experiment, we focus on the graph $\Gamma_{Bct}$ considering only the interactions that are related to the \textit{Bacteroides} population, as shown in Figure \ref{fig:A2-BacteroidesSubGraph}. $\Gamma_{Bct}$ is a subgraph of SCFA production graph, $\Gamma_{Bct} \subseteq \Gamma_{SCFA}$. The color scheme used in Figures \ref{fig:A2-BacteroidesMiddleLayer} and \ref{fig:A2-BacteroidesMCLayer} follow the same color scheme as in Figure \ref{fig:A2-BacteroidesSubGraph}. The metabolite inputs to the graph and the population sizes are maintained fixed at the standard level except for the \textit{Bacteroides} population size. We modify the population size of \textit{Bacteroides} ($|B_{Bct}|$) from zero cells to 2.2 times the standard population size. Figure \ref{fig:A2-BacteroidesMiddleLayer} explains the behaviors of the intermediate links from \textit{Bacteroides} to \textit{Faecalibacterium} node through acetate $C_{(Bact, Fae)}(M_{ace})$, \textit{Bacteroides} to \textit{Eubacterium} populations through lactate $C_{(Bact, Eub)}(M_{lact})$ and \textit{Bacteroides} to \textit{Roseburia} populations through acetate $C_{(Bact, Ros)}(M_{ace})$ while Figure \ref{fig:A2-BacteroidesMCLayer} shows SCFA production behaviors in the MC layer due to changes in the \textit{} population size.
From Figure \ref{fig:A2-BacteroidesMCLayer}, it is evident that all the SCFAs have strong positive relationships with the population size of \textit{Bacteroides}. Acetate and propionate are direct productions of \textit{Bacteroides} cells. As a result of that, acetate and propionate outputs show steady trends against the increment of \textit{Bacteroides} population sizes. Moreover, the edge weight variations shown in Figure \ref{fig:A2-BacteroidesMiddleLayer} justify the butyrate signal behavior in the MC layer shown in Figure \ref{fig:A2-BacteroidesMCLayer}. To be more precise, the butyrate output curve starts to become flatter when the \textit{Bacteroides} population size $|B_{Bct}|$ is greater than 0.8 times the standard value. The graph theoretical quantification of links also shows the same trend in Figure \ref{fig:A2-BacteroidesMiddleLayer}, emphasizing that the graph theoretical measures can be used to explain the metabolite production behaviors.
In the same way, Figure \ref{fig:A2-FaecaliVsSCFA} illustrates the results for a similar experiment on \textit{Faecalibacterium} population. Figures \ref{fig:A2-FaecalibacteriumSubGraph}, \ref{fig:A2-FaecalibacteriumMiddleLaye} and \ref{fig:A2-FaecalibacteriumMCLayer} represent the subgraph $\Gamma_{Fae}$ ($\Gamma_{Fae} \subseteq \Gamma_{SCFA}$), edge and the MC layer behaviors, respectively. Similarly to the previous analysis, we modify the population size of \textit{Faecalibacterium} $|B_{Fae}|$ ranging from zero cells to 2.2 times the standard population size. As the \textit{Faecalibacterium} cells consume acetate and produce butyrate, the rate of acetate consumption from the environment, $C^r_{(Mem, Fae)}(M_{ace})$ increases when the $|B_{Fae}|$ is increased. Hence, the weight of interaction between environment and \textit{Faecalibacterium} population, $C^w_{(Mem, Fae)}(M_{ace})$ increases which can be observed in Figure \ref{fig:A2-FaecalibacteriumMiddleLaye} and the resulting reduction in acetate output is visible in Figure \ref{fig:A2-FaecalibacteriumMCLayer}. Moreover, since \textit{Faecalibacterium} population is one of the key butyrate producers, there is a clear positive relationship evident between $|B_{Fae}|$ and butyrate. Due to the smaller population size of \textit{Roseburia} population, the influence on the metabolite production is relatively low, which can be observed from Figure \ref{fig:A2-FaecalibacteriumMiddleLaye}. For all the graphs the maximum MSEs are calculated below 0.03087.
Figure \ref{fig:A2-RuminococcusVSSCFA} elaborates the analytical results for a similar analysis on \textit{Ruminococcus} population size variation as we conduct on \textit{Bacteroides} and \textit{Faecalibacterium} populations. First, we extract a graph considering the metabolite consumption and production of \textit{Ruminococcus} cells $\Gamma_{Rcc}$ which is a subgraph of the SCFA production graph, $\Gamma_{Rcc} \subseteq \Gamma_{SCFA}$. Figures \ref{fig:A2-RuminococcusSubGraph}, \ref{fig:A2-RuminococcusMiddleLayer} and \ref{fig:A2-RuminococcusMCLayer} represent the subgraph $\Gamma_{Rcc}$ edge behaviors and the metabolite outputs, respectively. Similar to previous analysis, we modify the population size of \textit{Ruminococcus} $|B_{Rcc}|$ ranging from 0 - 2.2 times the standard population size. In this analysis, we considered fucose as the input signal to the \textit{Ruminococcus} population, $C_{(host,Rum)}(M_{fse})$ due to the metabolic switching from converting glucose into acetate to converting fucose into propionate of \textit{Ruminococcus} cells in the presence of fucose. It is clear in Figure \ref{fig:A2-RuminococcusMiddleLayer} the edge weight $C^w_{(host,Rum)}(M_{fse})$ is increased in parallel to the $|B_{Rcc}|$. Due to this behavior, the propionate production is increased, which can be observed in Figure \ref{fig:A2-RuminococcusMCLayer}. Again, these results confirm that the graph theoretical analysis reflects the behaviors of metabolite production in the MC Layer.
The MC layer results presented for the three analyses on \textit{Bacteroides}, \textit{Faecalibacterium} and \textit{Ruminococcus} populations (Figures \ref{fig:A2-BacteroidesMCLayer}, \ref{fig:A2-FaecalibacteriumMCLayer} and \ref{fig:A2-RuminococcusMCLayer}) are then interpreted in terms of SNR in Figure \ref{fig:A2-BacteriaVsSNR}. In the plots of this figure, SNR values are shown as ratios of the SNR value at the standard state of human GB and the bacterial population sizes are increased similar to the previous analyses. Here, we show the three SNRs of acetate, propionate and butyrate of three bacterial populations: \textit{Bacteroides}, \textit{Faecalibacterium} and \textit{Ruminococcus}. Figure \ref{fig:A2-Bacteroides-SNR} shows the SNR behaviors of the three SCFAs against the $|B_{Bct}|$. It is clearly evident that the acetate production is higher compared to the other two SCFAs when the $|B_{Bct}|$ is increased. This means, when the composition of human GB is changed as the $|B_{Bct}|$ increases, the output of the GB also loses the balance and tends to produce more acetate compared to the other two SCFAs. On the contrary, propionate production rate reduces when the $|B_{Bct}|$ increases. When the population size of \textit{Bacteroides} $|B_{Bct}|$ is smaller than the standard level, the system tends to produce molecular signal with higher deviated ratios, but when $|B_{Bct}|$ is greater than the standard level, the deviation is relatively low.
Figure \ref{fig:A2-Faecalibacterium-SNR} shows the SNR behaviors of the three SCFAs against the $|B_{Fae}|$. Since \textit{Faecalibacterium} is the main butyrate producer of this network, the butyrate SNR increases with the $|B_{Fae}|$ increment. Hence, compositional imbalance related to \textit{Faecalibacterium} causes a significant imbalance in output molecular signal ratios. Furthermore, due to the acetate consumption of \textit{Faecalibacterium}, the acetate signal becomes weaker, resulting in the acetate SNR deviating from the standard level. Moreover, Figure \ref{fig:A2-Ruminococcus-SNR} elaborates the SNR behavior due to variations in the \textit{Ruminococcus} population size, $|B_{Rcc}|$. Since, the population size of \textit{Ruminococcus} is small, the impact of it on the output signal ratio is relatively low compared to the other populations. The SNR for propionate is clearly increased with the \textit{Ruminococcus} population size increases as \textit{Ruminococcus} is a propionate producer.
The heat map shown in Figure \ref{fig:An2-Correlation} explains the correlation between each bacterial population and the SCFA abundance in the gut environment. Although the \textit{Bacteroides} is the biggest producer of all the SCFAs, it has a week correlation with SCFAs compared to other producers such as \textit{Alistipes} and \textit{Parabacteroides}. This reveals that the reduction of glucose consumption by \textit{Bacteroides} increases the other bacterial population resulting in boosted SCFA production. Note that, even the SCFA production of the other bacterial population is boosted in the absence of \textit{Bacteroides}, the overall production is lower. Since the \textit{Faecalibacterium} and \textit{Roseburia} consume acetate, the heat map shows a strong negative correlation with acetate. Interestingly, this heat map indicates the metabolic switching for \textit{Escherichia} which is switching from a SCFA producer to consumer of high acetate concentrations. This is same for the \textit{Ruminococcus} when the fucose concentration is not sufficient for the increased population, it switches from fucose consumption to glucose consumption reducing the intermediate metabolite production which causes a reduction in butyrate production.
\section{Conclusion}\label{sec:conclusions}
The gut bacteriome has been largely investigated due to its importance to the human health. We contribute to this research topic by introducing a two-layer GB interaction model to investigate the impacts of bacterial population compositional changes on the overall structure of the human GB utilizing data collected from MicrobiomeDB and NJS16 databases. Our proposed human GB interaction model combines a bacterial population graph layer, which models the structure typically found in the human GB (i.e. bacterial populations genus and sizes), with a molecular communications layer, which models the exchange of metabolites by the bacterial populations in this structure. Supported by these models we also developed a virtual GB to simulate the metabolic interactions that typically occurs in the human GB. These simulations allowed us to study the impacts caused by the metabolite exchanges on the human GB structure (i.e. nodes weight and hamming distance). Through our analyses we found that the molecular input availability affect differently the bacterial populations in the human GB by modifying the nodes and edges weights of our GB interaction model. Our results also show that modifications in the human GB structure, in specific changing the sizes of \textit{Bacteroides}, \textit{Faecalibacterium} and \textit{Ruminococcus} populations can lead to improvement/reduction on the production of SCFA, which may result in metabolic diseases in humans. Based on our results we also infer that there is an intrinsic
relationship between the investigated bacterial populations sizes, the increase/decrease of specific metabolites (acetate, butyrate and propionate) and the overall balance of the human GB. These results can support the development of novel strategies to treat unbalanced human GB and can provide insights on the role of other metabolites and molecules on the maintenance of a healthy gut bacteriome.
|
1,314,259,996,556 | arxiv | \section{Introduction}
\subsection{Motivations}
\IEEEPARstart{T}{he} proliferation of mobile devices is leading to an explosion of global mobile traffic, which is estimated to reach 30.6 exabytes per month by 2020 \cite{Cisco2016}. To accommodate this {{rapidly}} growing mobile traffic, 3GPP has been working on proposals to enable LTE to operate in the unlicensed 5GHz band \cite{3GPP}.{\footnote{{The LTE unlicensed technology can also work in the 3.5GHz band \cite{kim2015design}. However, since the available spectrum resources for the LTE technology in the 5GHz band ($500$MHz) are much more than those in the 3.5GHz band ($80$MHz), we focus on the interaction between the LTE and Wi-Fi in the 5GHz in this paper.}}} By extending LTE to the unlicensed spectrum, the LTE provider can significantly expand its network capacity, and tightly integrate its control over the licensed and unlicensed bands \cite{Senza}. Furthermore, since the LTE technology has an efficient framework of traffic management (\emph{e.g.,} congestion control), it is capable of achieving a much higher spectral efficiency than Wi-Fi networks in the unlicensed spectrum, if there is no competition between these two technologies \cite{forum}.
Key market players, such as AT\&T, Verizon, T-Mobile, Qualcomm, and Ericsson, have already demonstrated the potential of LTE in the unlicensed band through experiments \cite{forum}, and have formed several forums (\emph{e.g.}, LTE-U Forum \cite{LTEforum} and EVOLVE \cite{EVOLVE}) to promote this promising LTE unlicensed technology.
A key technical challenge for LTE working in the unlicensed spectrum is that it can significantly degrade the Wi-Fi network performance if there is no effective co-channel interference avoidance mechanism.
To address this issue, industries have proposed two major mechanisms for LTE/Wi-Fi coexistence:
(a) Qualcomm's carrier-sensing adaptive transmission (CSAT) scheme \cite{Qualcomm}, where the LTE transmission follows a periodic on/off pattern creating interference-free zones for Wi-Fi during certain periods, and (b) Ericsson's ``Listen-Before-Talk'' (LBT) scheme \cite{Ericsson}, where LTE transmits only when it senses the channel being idle for at least certain duration.
{{However, field tests revealed that these solutions often perform below expectations in practice.}}
In particular, a series of experiments by Google revealed that both mechanisms severely affect the performance of Wi-Fi \cite{Google}: for the CSAT mechanism, since Wi-Fi is not designed in anticipation of LTE's activity, it cannot respond well to LTE's on-off cycling, and its transmission is severely affected; for the LBT mechanism, it is challenging to choose the proper backoff time and transmission length for LTE to fairly coexist with Wi-Fi.
{{Therefore, beyond these coexistence mechanisms, there is a need for a novel framework that can effectively explore the potential cooperation opportunity between LTE and Wi-Fi to directly avoid the co-channel interference. This motivates our study in this work.}}
\subsection{Contributions}
{{Unlike previous solely technical coexistence mechanisms that focused on the fair competition between LTE and Wi-Fi, we design a novel coopetition framework. The basic idea is that the two types of networks (LTE and Wi-Fi) should explore the potential benefits of cooperation before deciding whether to enter head-to-head competition.
Under certain conditions (\emph{e.g.}, the co-channel interference heavily reduces the data rates of both LTE and Wi-Fi), it would be more beneficial for both types of networks to reach an agreement on the cooperation;
otherwise, they will compete with each other based on a typical coexistence mechanism (\emph{e.g.}, CSAT or LBT).}}
{{In our coopetition framework,}}
the LTE network works in either the \emph{competition mode} or the \emph{cooperation mode}. For the \emph{competition mode}, the LTE network simply shares the access of a channel with the corresponding Wi-Fi access point.{\footnote{We consider a general coexistence scheme between LTE and Wi-Fi. Hence, our model applies to both the CSAT and the LBT mechanisms.}}
For the \emph{cooperation mode}, the LTE network \emph{exclusively} occupies a Wi-Fi access point's channel and the corresponding Wi-Fi access point does not transmit, which avoids the co-channel interference and hence generates a high LTE data rate. Meanwhile, the Wi-Fi access point \emph{onloads} its users to the LTE network,{\footnote{{With the industrial standardization efforts (\emph{e.g.}, Hotspot 2.0 \cite{4Ginteg}), there is a trend of tightly integrating the Wi-Fi technology with the cellular networks. This enables various forms of cooperations between the cellular and Wi-Fi network providers. A successful example is Wi-Fi data offloading, where the cellular network providers offload their cellular traffic to the third-party Wi-Fi networks to relieve the cellular congestion \cite{iosifidis2015iterative,gao2014bargaining,dong2014ideal,lu2014easybid}. In terms of the practical implementation, one advantage of the data onloading over the Wi-Fi data offloading is that the data onloading can be more secure and better protect the mobile users' privacy. This is because the cellular networks usually provide better security guarantees than the Wi-Fi networks.}}} which serves the Wi-Fi access point's users with some data rates based on the access point's request.
Since LTE usually achieves a much higher spectral efficiency than Wi-Fi \cite{forum,canousing}, such a cooperation {{can potentially}} lead to a \emph{win-win} situation for both networks.
{{In our work, we want to answer the following two questions: (1) \emph{How would LTE and Wi-Fi negotiate over which mode (competition mode or cooperation mode) that LTE would use?} (2) \emph{If the LTE network works in the cooperation mode, how much Wi-Fi traffic should it serve?} Addressing these questions is challenging because of the following reasons: (i) given the increasingly large penetration of Wi-Fi technology, there are usually {{multiple}} Wi-Fi networks in range. {{As we will show in our analysis, the cooperation between the LTE network and one Wi-Fi network imposes a positive externality to other Wi-Fi networks not involved in the cooperation}}; (ii) there is no centralized decision maker in such a system, and different networks have conflicting interests as each of them wants to maximize the total data rate received by its own users; (iii) the throughput of a network (LTE or Wi-Fi) when it exclusively occupies a channel is its private information not known by others, which makes the coordination difficult.}}
{{To address these issues, in Section \ref{sec:model}, we design a mechanism that operates with minimum signaling and computations, and can be implemented in an almost real-time fashion. Specifically, the mechanism is based on}}
a reverse auction where the LTE provider is the auctioneer (buyer) and wants to \emph{exclusively} obtain the channel from one of the Wi-Fi access point owners {{(APOs, sellers)}}.{\footnote{We consider one LTE network and multiple Wi-Fi access points, since the LTE network has a larger coverage than the Wi-Fi access points, and the Wi-Fi access points are already very popular and exist in many areas.}}
{{We define the payoff of a network (LTE or Wi-Fi) as the total data rate received by its users.}}
{{In Stage I of the auction,}} the LTE provider announces the maximum data rate (\emph{i.e.,} reserve rate) that it is willing to allocate for serving users of the winning APO. By optimizing the reserve rate, the LTE provider can affect the APOs' willingness of cooperation, and hence maximize its expected payoff.
{{In Stage II of the auction,}} given the reserve rate, the APOs report whether they are willing to cooperate and what are the data rates that they request from the LTE provider.
{{Different APOs may have different requests, since they can have different data rates when exclusively occupying their channels.}}
If no APO wants to cooperate, the LTE network works in the \emph{competition mode}, and randomly accesses an APO's channel {{(based on a coexistence mechanism like CSAT or LBT)}}; otherwise, it works in the \emph{cooperation mode}, {{and cooperates with the APO that requests the lowest data rate from the LTE provider}}. Such an auction mechanism is particularly challenging to analyze since it induces \emph{positive allocative externalities} \cite{jehiel2000auctions}: the cooperation between the LTE provider and one APO will benefit other APOs not involved in this collaboration, because other APOs can avoid the potential interference generated by the LTE network under the \emph{competition mode}.
{{In Section \ref{sec:stageII:APO}, we analyze the APOs' equilibrium strategies in Stage II of the auction, given the LTE provider's reserve rate in Stage I. We show that an APO always has a unique form of the bidding strategy at the equilibrium under a given reserve rate. However, such a unique form of the bidding strategy may have different closed-form expressions based on different intervals of the reserve rate.}}
Furthermore, our study shows that for some APOs, the data rates they request from the LTE provider are lower than the rates they can obtain by themselves without the LTE's interference. Intuitively, such a low request motivates the LTE network to work in the \emph{cooperation mode} rather than the \emph{competition mode}. In the latter case, the APOs may receive even lower data rates due to the potential co-channel interference from the LTE network.
{{In Section \ref{sec:stageI:LTE}, we analyze the LTE provider's equilibrium choice of reserve rate in Stage I of the auction, by anticipating the APOs' equilibrium strategies in Stage II.}}
The LTE network's expected payoff has different closed-form function forms, over different intervals of the reserve rate. We analyze the optimal reserve rate by jointly considering all the reserve rate intervals. We show that when the LTE network's throughout exceeds a threshold, it will choose a reasonably large reserve rate and cooperate with some APOs; otherwise, it will restrict the reserve rate to a small value, and eventually work in the \emph{competition mode}.
The main contributions of this work are as follows:
\begin{itemize}
\item \emph{Proposal of the LTE/Wi-Fi coopetition framework:}
{{We propose a coopetition framework that explores the cooperation opportunity between LTE and Wi-Fi in order to determine whether they should directly compete with each other. Unlike previously proposed LTE/Wi-Fi coexistence mechanisms, our framework can avoid the data rate reduction when there is a cooperation opportunity between LTE and Wi-Fi.
Furthermore, our framework can be implemented without revealing the private throughput information of the networks.}}
\item \emph{Equilibrium analysis of the auction with allocative externalities:} {{We provide rigorous analysis for an auction mechanism with positive allocative externalities that involves more than two bidders. To the best of our knowledge, this is the first work studying such a mechanism in auction theory. Moreover, our work introduces a methodology for modeling and analyzing the allocative dependencies that arise increasingly often in wireless systems.}}
\item \emph{Characterization of the optimal reserve rate:} We analyze the reserve rate that maximizes the LTE network's payoff, and investigate its relation with the LTE throughput. Through simulation, we show that the optimal reserve rate is non-increasing in the LTE's data rate discounting factor, and non-decreasing in the LTE throughput, the number of APOs, and the APOs' data rate discounting factor.
\item \emph{Performance evaluation of the LTE/Wi-Fi coopetition framework:} Numerical results show that our framework achieves larger LTE's and APOs' payoffs comparing with {{a state-of-the-art benchmark scheme, which only considers the competition between LTE and APOs.}} In particular, our framework increases the LTE's payoff by $70\%$ on average when the LTE has a large throughput and a small data rate discounting factor. Furthermore, our framework leads to a close-to-optimal social welfare for a large LTE throughput.
\end{itemize}
\subsection{Related Work}
{{This paper is an extension of our conference paper \cite{haoran2016wiopt}, where we considered a basic model with two APOs.
In this paper, we generalize the model by considering an arbitrary number of APOs, which substantially extends the scope of the paper and the applicability of the results, but also significantly complicates the analysis.
Furthermore, in this paper, we investigate the impact of the number of APOs on the LTE provider's and the APOs' strategies, and compare our auction-based scheme with a state-of-the-art benchmark scheme through simulation. We also extensively discuss the generalization of our work to more complicated scenarios (\emph{e.g.}, multi-LTE scenario).}}
{{Several recent studies focused on the spectrum sharing problems for the LTE unlicensed technology.}}
{{Cano \emph{et al.} in \cite{canousing} and Zhang \emph{et al.} in \cite{zhanglte} discussed the major challenges for the LTE/Wi-Fi coexistence.}}
{{References \cite{cavalcante2013performance,rupasinghe2014licensed} provided performance evaluations for the LTE/Wi-Fi coexistence.}}
{{Li \emph{et al.} in \cite{li2015modelingJ} applied stochastic geometry to characterize the main performance metrics (\emph{e.g.,} SINR coverage probability) for the neighboring LTE and Wi-Fi networks in the unlicensed spectrum.
Jeon \emph{et al.} in \cite{jeon2014lte} applied a fluid network model to analyze the interference between the LTE and Wi-Fi.
Chen \emph{et al.} in \cite{chen2016cellular} jointly considered the Wi-Fi data offloading and the spectrum sharing between the LTE and Wi-Fi.
Cano \emph{et al.} in \cite{cano2016fair} addressed the fair coexistence problem for general scheduled and random access transmitters that share the same channel.}}
{{Cano \emph{et al.} in \cite{cano2015coexistence} studied the LTE network's channel access probability in the CSAT mechanism to ensure the fairness between LTE and Wi-Fi.}} Zhang \emph{et al.} in \cite{zhangmodeling} proposed a new LBT-based MAC protocol that allows LTE to friendly coexist with Wi-Fi.
{{Guan \emph{et al.} in \cite{guan4cu}} investigated the LTE provider's joint channel selection and fractional spectrum access problem with the consideration of the fairness between LTE and Wi-Fi.}
{{Zhang \emph{et al.} in \cite{zhang2015hierarchical} analyzed the spectrum sharing among multiple LTE providers in the unlicensed spectrum through a hierarchical game.}}
{{However, these studies did not consider the cooperation between LTE and Wi-Fi. We include the existing studies on LTE/Wi-Fi coexistence like \cite{cano2015coexistence,zhangmodeling} as part of our framework (\emph{i.e.}, in the competition mode), and also consider the new possibility of cooperation between LTE and Wi-Fi (\emph{i.e.}, in the cooperation mode).}}
In terms of the auction with \emph{allocative externalities}, the most relevant works are \cite{jehiel2000auctions} and \cite{bagwell2003case}. Jehiel and Moldovanu in \cite{jehiel2000auctions} provided a systematic study of the second-price forward auction with allocative externalities. They characterized the bidders' bidding strategies at the equilibrium for general payoff functions. However, they did not prove the uniqueness of the equilibrium strategies. Bagwell \emph{et al.} in \cite{bagwell2003case} studied a special example in the WTO system, where the retaliation rights were allocated through a first-price forward auction among different countries. The auction involves positive allocative externalities, and the authors showed the uniqueness of the countries' bidding strategies.
Both \cite{jehiel2000auctions} and \cite{bagwell2003case} only studied two bidders in the auction. In contrast, we consider an auction with an arbitrary number of bidders, and show the impact of the number of bidders on the auction outcome.
Furthermore, the bidders' equilibrium strategies have different expressions under different reserve rates announced by the auctioneer, which makes our analysis of the optimal reserve rate {{much more challenging than}} \cite{jehiel2000auctions} and \cite{bagwell2003case}.
\section{System Model}\label{sec:model}
\subsection{Basic Settings}
We consider a time-slotted system, where the length of each time slot corresponds to several minutes. We assume that the system is quasi-static, \emph{i.e.}, the system parameters (which involve mostly time average values) remain constant during each time slot, but can change over time slots. Our analysis focuses on the interaction between LTE and Wi-Fi networks in a single generic time slot.{\footnote{{Since the LTE unlicensed technology (time-division duplex mode) supports both the uplink and downlink transmissions \cite{zhanglte,Senza}, the LTE network is able to onload both the APOs' uplink and downlink traffic. Our framework works for both the uplink scenario (the networks only have uplink traffic) and downlink scenario (the networks only have downlink traffic). For example, in the uplink scenario, all throughputs in our model correspond to the networks' uplink throughputs. For the most general scenario, where the networks serve uplink and downlink traffic simultaneously, each network should choose its strategy by considering both the uplink and downlink transmissions, and we leave the analysis of this scenario as our future work.}}}
We consider one LTE small cell network and a set ${\cal K}\triangleq \left\{1,2,\ldots,K\right\}$ ($K\ge2$) of Wi-Fi access points.
The LTE small cell network is owned by an LTE provider,{\footnote{{In Section \ref{subsec:discussion:multiLTE}, we will discuss the extension to the scenario where there are multiple LTE providers.}}} and the $k$-th ($k\in{\cal K}$) Wi-Fi access point is owned by APO $k$. We assume that the APOs occupy different unlicensed channels so that they do not interfere with each other. We use channel $k$ to represent the channel occupied by APO $k$.
The LTE small cell network has a larger coverage area than the Wi-Fi access points \cite{Qualcomm,forum}. Furthermore, it can work in one of the $K$ channels, and cause interference to the corresponding access point in the channel.{\footnote{For ease of exposition, we use ``LTE provider'' and ``LTE network'' interchangeably. Similarly, we use ``APO'' and ``access point'' interchangeably.}} The assumption that the APOs occupy different channels simplifies the problem and helps us gain key insights into the proposed auction framework. {{In Section \ref{subsec:discussion:APOshare}, we will discuss the extension to the scenario where different APOs can share the same channel.}}
{\bf APOs' Rates:} {{We consider fully loaded APOs,{\footnote{{Since the length of each time slot corresponds to several minutes, we assume that a network has enough traffic to serve during a time slot and will not complete its service within a time slot. This assumption simplifies the problem, and helps us understand the fundamental benefit of organizing an auction to onload the Wi-Fi traffic to the LTE network. Many papers made similar saturation assumptions to analyze the network performance \cite{bianchi2000performance,cali2000dynamic,wu2002performance}. {{In the future work, we will study the scenario where the networks do not have full loads, and can precisely predict their traffic loads in the next few minutes.}}}}} and use $r_k$ to denote the throughput that APO $k\in{\cal K}$ can achieve to serve its users when it \emph{exclusively} occupies channel $k$ (without the interference from the LTE network).}}
{{The value of $r_k$ in the time slot that we are interested in is the private information of APO $k$. The LTE provider and the other $K-1$ APOs only know the probability distribution of $r_k$. Specifically, we assume that $r_k$ is a continuous random variable drawn from interval $\left[r_{\min},r_{\max}\right]$ ($r_{\min},r_{\max}\ge0$), and follows a probability distribution function (PDF) $f\left(\cdot\right)$ and a cumulative distribution function (CDF) $F\left(\cdot\right)$.{\footnote{We assume that all $r_k$ ($k\in{\cal K}$) follow the same distribution, and hence both functions $f\left(\cdot\right)$ and $F\left(\cdot\right)$ are independent of index $k$. We will study problem with the non-identical variable $r_k$ in our future work.}} Moreover, we assume that $f\left(\cdot\right)>0$ for all $r\in\left[r_{\min},r_{\max}\right]$.}}
{\bf LTE's Dual Modes:} {{We consider a fully loaded LTE network,}} and assume that it achieves a channel independent throughput of $R_{\rm LTE}>0$ when it \emph{exclusively} occupies one of the $K$ channels (without the interference from the APOs).{\footnote{{As we will show in the analysis, the APOs make their decisions based on the LTE provider's reserve rate $C$ instead of the throughput $R_{\rm LTE}$. In other words, the APOs do not need to know the value of $R_{\rm LTE}$. Therefore, we do not need to assume a probability distribution of $R_{\rm LTE}$ to model the APOs' knowledge of $R_{\rm LTE}$.}}} The LTE provider can operate its network in one of the following modes
-\emph{competition mode:} the LTE provider randomly chooses each channel $k\in{\cal K}$ with an equal probability and coexists with APO $k$. {{Since our main focus is the design of the auction framework, the LTE provider simply coexists with APO $k$ based on a typical coexistence mechanism (\emph{e.g.}, CSAT or LBT) and setting (\emph{e.g.}, the LTE's backoff time and transmission length in the LBT)}}. The co-channel interference decreases both the data rates of the LTE provider and the corresponding APO. We use ${\delta^{\rm LTE}}\in\left(0,1\right)$ and ${\eta^{\rm APO}}\in\left(0,1\right)$ to denote the LTE's and the APO's data rate discounting factors, respectively;{\footnote{{Based on \cite{Google,cavalcante2013performance,rupasinghe2014licensed}, the data rate reduction of the APO due to the co-channel interference is much heavier than that of the LTE. Hence, factor ${\eta^{\rm APO}}$ is usually smaller than ${\delta^{\rm LTE}}$. The values of ${\eta^{\rm APO}}$ and ${\delta^{\rm LTE}}$ depend on the concrete coexistence mechanisms and settings. For example, the study in \cite{Google} showed that ${\eta^{\rm APO}}$ ranges from $0.1$ to $0.5$ given different LTE off time under the CSAT mechanism.
In this work, we assume that the LTE provider adopts the same mechanism (\emph{e.g.}, CSAT or LBT) and settings (\emph{e.g.}, LTE off time in CSAT) when coexisting with any APO. Hence, the LTE provider has the same discounting factor ${\delta^{\rm LTE}}$ for the coexistence with any APO, and the APOs have the same discounting factor ${\eta^{\rm APO}}$.}}}
-\emph{cooperation mode:} the LTE provider reaches an agreement with APO $k\in{\cal K}$, where APO $k$ stops transmission and the LTE provider exclusively occupies channel $k$. In this case, there is no co-channel interference, and the LTE provider's data rate is simply $R_{\rm LTE}$. As a compensation, the LTE provider will serve APO $k$'s users with a guaranteed data rate $r_{\rm pay}\in\left[0,R_{\rm LTE}\right]$.{\footnote{{{{{According to \cite{Cisco2016}, in 2015, $88$\% of global mobile devices are the mobile phones (including the smartphones, non-smartphones, and phablets), which have the cellular interfaces. Only $12$\% of global mobile devices are the tablets, laptops, and other devices that may not have the cellular interfaces. Therefore, we assume that all the mobile devices served by the APOs (during the considered time slot) have the cellular interfaces and hence can be onloaded to the LTE network if needed. In Section \ref{subsec:discussion:dumb}, we will discuss the extension to the scenario where some mobile devices (\emph{e.g.}, laptops) do not have the cellular interfaces.}}}}}}
The remaining $K-1$ APOs occupy their own channels, and are not interfered by the LTE provider. Which APO the LTE provider chooses to cooperate with and what the value $r_{\rm pay}$ should be will be determined through a reverse auction design in the next subsection.
\subsection{Second-Price Reverse Auction Design}\label{subsec:auctionframe}
We design a second-price reverse auction, where the LTE provider is the auctioneer (buyer) and the APOs are the bidders (sellers).
The auction is held at the beginning of each time slot.
The private type of APO $k$ is $r_k$ (\emph{i.e.}, the data rate when it exclusively occupies channel $k$), {{and APO $k$'s item for sale is the right of onloading APO $k$'s traffic. When the LTE provider obtains the item from APO $k$, the LTE provider can onload APO $k$'s traffic and exclusively occupy channel $k$.}}
Since we assume that the LTE provider cannot occupy more than one channel at the same time, the LTE provider is only interested in obtaining one item from one of the APOs.{\footnote{{Since the LTE unlicensed technology is still in an early stage of development, the existing relevant experiments and studies focused on the situation where the LTE network can only utilize a single unlicensed channel \cite{Google,forum,Qualcomm}. In the future, it is likely that the LTE network can aggregate multiple unlicensed channels through the carrier aggregation technology \cite{zhang2010carrier}.}}} Different from the conventional reverse auction where the auctioneer pays the winner money to obtain the item, here the LTE serves the winning APO's users with the rate $r_{\rm pay}$ as the payment.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{JLTEUReverseAuction11-eps-converted-to.pdf}
\vspace{-0.2cm}
\caption{Illustration of The Reverse Auction.}
\label{fig:auction}
\vspace{-0.7cm}
\end{figure}
{\bf Reserve Rate and Bids:} {{In Stage I of the auction,}} the LTE provider announces its reserve rate $C\in\left[0,\infty\right)$, which corresponds to the maximum data rate that it is willing to accept to serve the winning APO's users. {{In Stage II of the auction,}} after observing the reserve rate $C$, APO $k$ submits a bid $b_k\in\left[0,C\right]\cup\left\{``\rm N \textquotedblright\right\}$: (a) $b_k\in\left[0,C\right]$ indicates the data rate that APO $k$ requests the LTE provider to serve APO $k$'s users; (b) $b_k=``\rm N \textquotedblright$ means that APO $k$ does not want to sell its item (\emph{i.e.}, the right of {{onloading APO $k$'s traffic}}) to the LTE provider.{\footnote{{{If APO $k$ bids any value greater than the reserve rate $C$, the LTE provider will not cooperate with APO $k$ based on the definition of $C$. Hence, any bid greater than $C$ leads to the same result to APO $k$. In order to facilitate the description, we use $``\rm N \textquotedblright$ to represent any bid greater than $C$.}} Intuitively, if the reserve rate $C$ is very small, APO $k$ is more likely to bid $``\rm N \textquotedblright$. In this case, APO $k$ can achieve an expected data rate {{(considering all possible auction results)}} higher than that when onloading the users to the LTE provider.}} We define the vector of APOs' bids as ${\bm b}\triangleq \left(b_k,\forall k\in{\cal K}\right)$.
The auction design is illustrated in Fig. \ref{fig:auction}.
{\bf Auction Outcomes:} Next we discuss the auction outcomes based on the different values of ${\bm b}$ and $C$. For ease of exposition, we define the comparison between $``\rm N \textquotedblright$ and any bid $b_k$ as
\begin{align}
\min\left\{{``\rm N \textquotedblright},b_k\right\}=\left\{\begin{array}{ll}
{b_k,} & {\rm if~}{b_k\in\left[0,C\right],}\\
{{``\rm N \textquotedblright},} & {{\rm if~}{b_k={``\rm N \textquotedblright}}.}
\end{array} \right.
\end{align}
Furthermore, we use ${\cal I}_{\min}$ to denote the set of APOs with the minimum bid, and define it as
\begin{align}
{\cal I}_{\min}\triangleq \left\{i\in{\cal K}:i=\arg\min_{k\in{\cal K}}{b_k} \right\}.
\end{align}
The auction has the following possible outcomes:
(a) When $\left|{\cal I}_{\min}\right|=1$,{\footnote{Condition $\left|{\cal I}_{\min}\right|=1$ implies $\min_{k\in{\cal K}}{b_k} \in \left[0,C\right]$ as we have $K\ge2$ APOs.}} then APO $i= \arg\min_{k\in{\cal K}}{b_k}$ is the winner, and leaves channel $i$ to the LTE provider. The LTE provider works in the \emph{cooperation mode} and exclusively occupies channel $i$. Furthermore, the LTE serves APO $i$'s users with a rate $r_{\rm pay}=\min\left\{C,b_1,\ldots,b_{i-1},b_{i+1},\ldots,b_{K}\right\}$, which is the lowest rate among the reserve rate and all the {\emph{other}} APOs' bids, based on the rule of the second-price auction. {{In this case, the allocated rate $r_{\rm pay}$ is greater than the winning APO's bid (\emph{i.e.}, $\min_{k\in{\cal K}}{b_k}$)}}
(b) When $\min_{k\in{\cal K}}{b_k} \in \left[0,C\right]$ and $\left|{\cal I}_{\min}\right|>1$, the LTE provider works in the \emph{cooperation mode}, randomly chooses an APO from set ${\cal I}_{\min}$ with the probability $\textstyle \frac{1}{\left|{\cal I}_{\min}\right|}$ to exclusively occupy the corresponding channel, and serves the APO's users with a rate $r_{\rm pay}=\min_{k\in{\cal K}}{b_k}$. {{In this case, the allocated rate $r_{\rm pay}$ equals the winning APO's bid}};
(c) When $\min_{k\in{\cal K}}{b_k} =``\rm N \textquotedblright$,{\footnote{In this case, all APOs bid $``\rm N \textquotedblright$.}} the LTE provider works in the \emph{competition mode}, randomly chooses one of the $K$ channels with the probability $\textstyle\frac{1}{K}$, and shares the channel with the corresponding APO.{\footnote{{{Because the LTE provider does not have the private information $r_k$, it cannot differentiate the channels.}} We consider a specific protocol where the LTE provider randomly accesses each channel with an equal probability in the \emph{competition mode}.}}
\vspace{-0.4cm}
\subsection{LTE Provider's Payoff}\label{subsec:LTEpayoff}
Based on the summary of auction outcomes in the last subsection, we can write $r_{\rm pay}$ as a function of $\bm b$ and $C$:
\vspace{-0.2cm}
\begin{align}
\nonumber
& r_{\rm pay}\left({{\bm b},C}\right)= \\
& \left\{\begin{array}{ll}
{\min\!\left\{\!C,\!\min_{k\ne i,k\in{\cal K}}b_k\right\}\!,} & {\rm if~}{\left|{\cal I}_{\min}\right|=1,}\\
{\min_{k\in{\cal K}}{b_k},} & {\rm if~}{\min_{k\in{\cal K}}\!{b_k}\! \in\!\left[0,\!C\right] {~\!\rm and~\!\!}\left|{\cal I}_{\min}\right|\!\!>\!\!1,}\\
{0,} & {\rm if~}{\min_{k\in{\cal K}}{b_k} =``\rm N \textquotedblright.}
\end{array} \right.\label{equ:rpay}
\end{align}
We define the LTE provider's payoff as the data rate that it can allocate to its own users, and compute it as:{\footnote{Notice that $\min_{k\in{\cal K}}{b_k} \in \left[0,C\right]$ contains two possible situations: (i) $\left|{\cal I}_{\min}\right|=1$; (ii) $\min_{k\in{\cal K}}{b_k} \in\left[0,C\right] {~\rm and~}\left|{\cal I}_{\min}\right|>1$.}}
\begin{align}
\Pi^{\rm LTE}\left({\bm b},C\right)=\left\{\begin{array}{ll}
{R_{\rm LTE}-r_{\rm pay}\left({{\bm b},C}\right),} & {\rm if~}{\min_{k\in{\cal K}}{b_k} \in\left[0,C\right],}\\
{{\delta^{\rm LTE}} R_{\rm LTE},} & {\rm if~}{\min_{k\in{\cal K}}{b_k} =``\rm N \textquotedblright.}
\end{array} \right.\label{equ:LTEpayoff}
\end{align}
Equation (\ref{equ:LTEpayoff}) captures two possible situations:
(a) when the minimum bid lies in $\left[0,C\right]$, the LTE provider works in the \emph{cooperation mode}, exclusively occupies a channel, and obtains a total data rate of $R_{\rm LTE}$. Since the LTE provider needs to allocate a rate of $r_{\rm pay}\left({{\bm b},C}\right)$ to the winning APO's users, its payoff is $R_{\rm LTE}-r_{\rm pay}\left({{\bm b},C}\right)$;
(b) when all APOs bid $``{\rm N} \textquotedblright$, the LTE provider works in the \emph{competition mode}, and ${\delta^{\rm LTE}}\in\left(0,1\right)$ captures the discount in the LTE provider's data rate due to the interference from the Wi-Fi APO in the same channel.
\vspace{-0.4cm}
\subsection{APOs' Payoffs and Allocative Externalities}
We define the payoff of APO $k\in{\cal K}$ as the data rate that its users receive: when APO $k$ cooperates with the LTE provider, these users are served by the LTE provider; otherwise, they are served by APO $k$. Based on the summary of auction outcomes in Section \ref{subsec:auctionframe} and the definition of $r_{\rm pay}\left({\bm b},C\right)$ in (\ref{equ:rpay}), we summarize APO $k$'s expected payoff as follows:
\begin{align}
\nonumber
& \Pi^{\rm APO}_{k}\left({\bm b},C\right)=\\
& \left\{\begin{array}{ll}
{r_k,} & {\rm if~}{b_k>\min_{j\in{\cal K}}{b_j},}\\
{\frac{1}{\left|{\cal I}_{\min}\right|}r_{\rm pay}\left({\bm b},C\right)\!+\!\frac{\left|{\cal I}_{\min}\right|-1}{\left|{\cal I}_{\min}\right|}r_k,} & {\rm if~}{b_k\!=\! \min_{j\in{\cal K}}{b_j}\!\in\!\left[0,C\right],}\\
{\frac{K-1+{\eta^{\rm APO}}}{K}r_k,} & {\rm if~}{\min_{j\in{\cal K}}{b_j}=``{\rm N} \textquotedblright.}
\end{array} \right.\label{equ:APOpayoff}
\end{align}
Equation (\ref{equ:APOpayoff}) summarizes three possible situations: (a) when $b_k>\min_{j\in{\cal K}}{b_j}$, the LTE provider exclusively occupies a channel from one of the APOs (other than APO $k$) with the minimum bid. As a result, APO $k$ can exclusively occupy its own channel $k$, and serve its users with rate $r_k$; (b) when $b_k= \min_{j\in{\cal K}}{b_j}\in\left[0,C\right]$, the LTE provider cooperates with APO $k$ and one of the other APOs with the minimum bid with the probability ${\frac{1}{\left|{\cal I}_{\min}\right|}}$ and the probability $1-{\frac{1}{\left|{\cal I}_{\min}\right|}}$ ($1\le {\left|{\cal I}_{\min}\right|}\le K$), respectively. Hence, APO $k$'s users receive rate $r_{\rm pay}\left({\bm b},C\right)$ and rate $r_k$ with the probability ${\frac{1}{\left|{\cal I}_{\min}\right|}}$ and the probability $1-{\frac{1}{\left|{\cal I}_{\min}\right|}}$, respectively. In this case, the expected data rate that APO $k$'s users receive is ${\frac{1}{\left|{\cal I}_{\min}\right|}r_{\rm pay}\left({\bm b},C\right)+\frac{\left|{\cal I}_{\min}\right|-1}{\left|{\cal I}_{\min}\right|}r_k,}$; (c) when $\min_{j\in{\cal K}}{b_j}=``{\rm N} \textquotedblright$, there is no winner in the auction, and the LTE provider randomly chooses one of the $K$ channels to coexist with the corresponding APO. With the probability ${\frac{1}{K}}$, APO $k$ coexists with the LTE provider and has a data rate of ${{\eta^{\rm APO}}}r_k$; with the probability $1-\frac{1}{K}$, APO $k$ has a data rate of $r_k$ by exclusively occupying channel $k$. In this case, the expected data rate that APO $k$'s users receive is $\frac{K-1+{\eta^{\rm APO}}}{K}r_k$.
We note that APO $k$ does not win the auction in either of the following two cases: $b_k>\min_{j\in{\cal K}}{b_j}$ and $\min_{j\in{\cal K}}{b_j}=``{\rm N} \textquotedblright$. However, the APO $k$'s payoff is different in these two cases: it obtains a payoff of $r_k$ when $b_k>\min_{j\in{\cal K}}{b_j}$, and achieves a smaller payoff of $\frac{K-1+{\eta^{\rm APO}}}{K}r_k$ when $\min_{j\in{\cal K}}{b_j}=``{\rm N} \textquotedblright$. That is to say, even if APO $k$ does not win the auction, it is more willing to see the other APOs winning (\emph{i.e.}, $b_k>\min_{j\in{\cal K}}{b_j}$) rather than losing the auction (\emph{i.e.}, $\min_{j\in{\cal K}}{b_j}=``{\rm N} \textquotedblright$). This shows \emph{positive allocative externalities} of the auction, which make our problem substantially different from conventional auction problems. At the equilibrium of the conventional second-price auction, bidders bid truthfully according to their private values, regardless of other bidders' valuations. With allocative externalities in our problem, when APO $k$ evaluates its payoff when losing the auction, it needs to consider whether the other APOs win the auction or not. Hence, the distributions of the other APOs' valuations (types) affect APO $k$'s strategy. As we will show in the following sections, this leads to a special structure of APOs' bidding strategies at the equilibrium, and bidding truthfully is no longer a dominate strategy.
\begin{table}[t]\small
\centering
\caption{Main Notations}\label{table:notation}
\begin{tabular}{|c|p{6cm}|}
\hline
{\minitab[c]{${\cal K},K$}} & {The set of APOs and its cardinality}\\
\hline
{\minitab[c]{$r_k$}} & {APO $k$'s {throughput without interference (private valuation, also called \emph{type})}}\\
\hline
{\minitab[c]{$r_{\min}$, $r_{\max}$}} & {Lower and upper bounds of $r_k$, $k\in{\cal K}$}\\
\hline
{\minitab[c]{$f\left(\cdot\right)$, $F\left(\cdot\right)$}} & {PDF and CDF of $r_k$, $k\in{\cal K}$}\\
\hline
{\minitab[c]{$R_{\rm LTE}$}} & {LTE provider's throughput without interference}\\
\hline
{\minitab[c]{${\eta^{\rm APO}}$}} & {APOs' data rate discounting factor}\\
\hline
{\minitab[c]{${\delta^{\rm LTE}}$}} & {LTE provider's data rate discounting factor}\\
\hline
{\minitab[c]{$C$}} & {LTE provider's reserve rate (\emph{decision variable})}\\
\hline
{\minitab[c]{$b_k$}} & {APO $k$'s bid (\emph{decision variable})}\\
\hline
{\minitab[c]{${\cal I}_{\min}$}} & {The set of APOs with the minimum bid}\\
\hline
{\minitab[c]{$\Pi^{\rm LTE}\left({\bm b},C\right)$}} & {LTE provider's payoff}\\
\hline
{\minitab[c]{$r_{\rm pay}\left({\bm b},C\right)$}} & {Data rate LTE allocates to the winning APO}\\
\hline
{\minitab[c]{$\Pi^{\rm APO}_{k}\left({\bm b},C\right)$}} & {APO $k$'s payoff}\\
\hline
\end{tabular}
\vspace{-0.5cm}
\end{table}
We summarize the main notations in Table \ref{table:notation}. For the parameters and distributions that characterize the APOs, $r_k$ is APO $k$'s private information, and the remaining information, \emph{i.e.}, $K,r_{\min},r_{\max},f\left(\cdot\right),F\left(\cdot\right),$ and ${\eta^{\rm APO}}$, is publicly known to all the APOs and the LTE provider. For the parameters that characterize the LTE provider, \emph{i.e.}, $R_{\rm LTE}$ and ${\delta^{\rm LTE}}$, as we will see in later sections, they will not affect the APOs' strategies. Therefore, they can be either known or unknown to the APOs.
{{Next we analyze the auction by backward induction. In Section \ref{sec:stageII:APO}, we analyze the APOs' equilibrium strategies in Stage II, given the LTE provider's reserve rate $C$ in Stage I. In Section \ref{sec:stageI:LTE}, we analyze the LTE provider's equilibrium reserve rate $C^*$ in Stage I by anticipating the APOs' equilibrium strategies in Stage II.}}
\begin{comment}
{{
\subsection{Two-Stage Game}
We formulate the interaction between the LTE provider and APOs in the auction as a two-stage game. In Stage I, the LTE provider announces the reserve price $C$. In Stage II, each APO $k\in{\cal K}$ submits bid $b_k$. We illustrate the two-stage game in Fig. \ref{fig:stage}, and analyze it by backward induction. In Section \ref{sec:stageII:APO}, we analyze the APOs' equilibrium strategies in Stage II, given the LTE provider's reserve rate $C$ in Stage I. In Section \ref{sec:stageI:LTE}, we analyze the LTE provider's optimal reserve rate $C^*$ in Stage I by anticipating the APOs' equilibrium strategies in Stage II.}}
\begin{figure}[t]
\centering
\begin{tabular}{|p{8.5cm}<{\centering}|}
\hline
{{\bf{Stage I}}}\\
\hline
{The LTE provider announces the reserve rate $C$.}\\
\hline
\end{tabular}
\centerline{$\Downarrow$}
\begin{tabular}{|p{8.5cm}<{\centering}|}
\hline
{\bf{Stage II}}\\
\hline
{Each APO $k\in{\cal K}$ submits bid $b_k$.}\\
\hline
\end{tabular}
\caption{Two-Stage Game.}\label{fig:stage}
\vspace{-6mm}
\end{figure}
\end{comment}
\begin{comment}
In Section \ref{sec:EQtypicalC}, we consider the situation where $C$ is from $\left[r_{\min},r_{\max}\right)$, and show the three-region structure of the bidding strategies: some APO types bid their types, some APO types bid the reserve rate, and some APO types bid $``{\rm N} \textquotedblright$. In Section \ref{sec:EQotherC}, we consider the situation where $C$ is from $C\in\left[0,r_{\min}\right)\cup\left[r_{\max},\infty\right)$. Since in this situation, the reserve rate is either very small or very large, among the three regions of APOs' strategies introduced in Section \ref{sec:EQtypicalC}, only one or two will appear in this case. Hence, the equilibrium analysis in Section \ref{sec:EQotherC} can be treated as the special case of the analysis in Section \ref{sec:EQtypicalC}.
\end{comment}
\vspace{-0.2cm}
\section{{{Stage II: APOs' Equilibrium Bidding Strategies}}}\label{sec:stageII:APO}
{{In this section, we assume that the reserve rate $C$ of the LTE provider in Stage I is given, and analyze the APOs' equilibrium strategies in Stage II.
In Section \ref{subsec:definitionEQ}, we define the equilibrium for the APOs under a given $C$.
In Sections \ref{subsec:stageII:1}, \ref{subsec:lowC}, \ref{subsec:stageII:3}, and \ref{subsec:stageII:4}, we analyze the APOs' equilibrium strategies by considering different intervals of $C$.
In Section \ref{sec:comparison}, we summarize the results for the APOs' equilibrium strategies.}}
\vspace{-0.3cm}
\subsection{{Definition of Symmetric Bayesian Nash Equilibrium}}\label{subsec:definitionEQ}
We focus on the symmetric Bayesian Nash equilibrium (SBNE), which is defined as follows.
\begin{definition}
Under a reserve rate $C$, a bidding strategy function $b^*\left(r,C\right)$, $r\in\left[r_{\min},r_{\max}\right]$, constitutes a symmetric Bayesian Nash equilibrium if
relation (\ref{equ:defineEQ}) holds
for all $s_k\in \left[0,C\right]\cup\left\{{``{\rm N}\textquotedblright}\right\}$, all $r_k\in \left[r_{\min},r_{\max}\right]$, and all $k\in{\cal K}$.
\end{definition}
\begin{figure*}[t]
\begin{align}
\nonumber
& \mathbb{E}_{{\bm r}_{-k}}\left\{\Pi^{\rm APO}_k\left(\left(b^*\left(r_1,C\right),\ldots,b^*\left(r_{k-1},C\right),b^*\left(r_k,C\right),b^*\left(r_{k+1},C\right),\ldots,b^*\left(r_K,C\right)\right),C\right)|r_k\right\}\ge\\
& \mathbb{E}_{{\bm r}_{-k}}\left\{\Pi^{\rm APO}_k\left(\left(b^*\left(r_1,C\right),\ldots,b^*\left(r_{k-1},C\right),s_k,b^*\left(r_{k+1},C\right),\ldots,b^*\left(r_K,C\right)\right),C\right)|r_k\right\},\label{equ:defineEQ}
\end{align}
\hrulefill
\end{figure*}
Since it is the symmetric equilibrium, all the APOs apply the same bidding strategy function $b^*\left(r,C\right)$ at the equilibrium. The left hand side of inequality (\ref{equ:defineEQ}) stands for APO $k$'s expected payoff when it bids $b^*\left(r_k,C\right)$. The expectation is taken with respect to ${\bm r}_{-k}\triangleq \left(r_j,\forall j\ne k,j\in{\cal K}\right)$, which denotes all the other APOs' types and is unknown to APO $k$. Inequality (\ref{equ:defineEQ}) implies that APO $k\in{\cal K}$ cannot improve its expected payoff by unilaterally changing its bid from $b^*\left(r_k,C\right)$ to any $s_k\in \left[0,C\right]\cup\left\{{``{\rm N}\textquotedblright}\right\}$.
\vspace{-0.4cm}
\subsection{{{APOs' Equilibrium When $C\in\left[r_{\min},r_{\max}\right)$}}}\label{subsec:stageII:1}
We assume that the reserve rate $C$ is given from $\left[r_{\min},r_{\max}\right)$,{\footnote{{We first analyze the case where $C\in\left[r_{\min},r_{\max}\right)$, because it has the most complicated equilibrium analysis. We can apply a similar analysis approach in this section to the other cases.}}} and show the unique form of bidding strategy that constitutes an SBNE. We first introduce the following lemma (the proofs of all lemmas and theorems can be found in the appendix{{}}).
\begin{lemma}\label{lemma:rT}
The following equation admits at least one solution $r$ in $\left(C,r_{\max}\right)$:
\begin{multline}
\sum_{n=1}^{K-1}{\binom{K-1}{n} \left(F\left(r\right)-F\left(C\right)\right)^n\left(1-F\left(r\right)\right)^{K-1-n}\frac{C-r}{n+1}} \\
+\left(1-F\left(r\right)\right)^{K-1}\left(C-\frac{K-1+{\eta^{\rm APO}}}{K}r\right)=0,\label{equ:rT}
\end{multline}
where $F\left(\cdot\right)$ is the CDF of random variable $r_k$, $k\in{\cal K}$. We denote the solutions $r$ in $\left(C,r_{\max}\right)$ as $r_1^t\left(C\right), r_2^t\left(C\right),\ldots, r_M^t\left(C\right)$, where $M=1,2,\ldots,$ is the number of solutions.
\end{lemma}
Based on the definition of $r_1^t\left(C\right), r_2^t\left(C\right),\ldots, r_M^t\left(C\right)$ in Lemma \ref{lemma:rT}, we introduce the following theorem.
\begin{theorem}\label{theorem:equilibrium
Consider an $r_T\left(C\right)\in\left(C,r_{\max}\right)$ that belongs to the set of $\left\{r_1^t\left(C\right), r_2^t\left(C\right),\ldots, r_M^t\left(C\right)\right\}$, then the following bidding strategy $b^*$ constitutes an SBNE:
\begin{align}
{b^*}\left(r_k,C\right)=\left\{\begin{array}{ll}
{{\rm any\!~value\!~in} \left[0,\!r_{\min}\right]\!,} & {\rm if~}{r_k=r_{\min},}\\
r_k, & {{\rm if~}{r_k\in\left(r_{\min},C\right]},}\\
C, & {{\rm if~}{ r_k\in\left(C,r_T\left(C\right)\right)},}\\
C{\rm~or~}``{\rm N}\textquotedblright, & {\rm if~}{r_k=r_T\left(C\right),}\\
``{\rm N} \textquotedblright, & {{\rm if~} r_k\in\left(r_T\left(C\right),r_{\max}\right],}
\end{array} \right.\label{equ:equilibrium}
\end{align}
for all $k\in{\cal K}$.
\end{theorem}
We illustrate the structure of strategy $b^*$ in Fig. \ref{fig:S1}, in which we notice that
\vspace{-3mm}
\begin{figure}[h]
\centering
\includegraphics[scale=0.4]{JStructure-eps-converted-to.pdf}
\vspace{-2mm}
\caption{Bidding Strategy Structure at SBNE When $C\in\left[r_{\min},r_{\max}\right)$.}
\label{fig:S1}
\vspace{-4mm}
\end{figure}
(a) For an APO $k$ with type $r_k\in\left(r_{\min},C\right]$, it bids $r_k$. In other words, APO $k$ requests the LTE provider to serve APO $k$'s users with at least the rate that APO $k$ can achieve by exclusively occupying channel $k$;
(b) For an APO $k$ with type $r_k\in\left(C,r_T\left(C\right)\right)$, it bids $C$. Since $C<r_k$, the data rate APO $k$ requests from the LTE provider is smaller than the rate that APO $k$ achieves by exclusively occupying channel $k$. Recall that the feasible bid should be from $\left[0,C\right]\cup \left\{``{\rm N} \textquotedblright\right\}$. If APO $k$ bids $``{\rm N} \textquotedblright$, there is a chance that all the other APOs also bid $``{\rm N} \textquotedblright$, which makes the LTE provider work in the \emph{competition mode} and leads to a payoff of $\frac{K-1+{\eta^{\rm APO}}}{K}r_k$ to APO $k$ based on (\ref{equ:APOpayoff}). In order to avoid such a situation, APO $k$ would bid $C$, and ensure that its payoff is at least $C$;{\footnote{Specifically, based on (\ref{equ:rpay}), if APO $k$ bids $C$ and wins the auction, its payoff will be $C$; if APO $k$ bids $C$ but loses the auction, its payoff will be $r_k>C$.}}
(c) For an APO $k$ with type $r_k\in\left(r_T\left(C\right),r_{\max}\right]$, it bids $``{\rm N} \textquotedblright$. Similar as case (b), there is a chance that all the other APOs also bid $``{\rm N} \textquotedblright$, and APO $k$ obtains a payoff of $\frac{K-1+{\eta^{\rm APO}}}{K}r_k$. However, with $r_k\in\left(r_T\left(C\right),r_{\max}\right]$, the value $\frac{K-1+{\eta^{\rm APO}}}{K}r_k$ is already large enough so that there is no need for APO $k$ to lower its bid from $``{\rm N} \textquotedblright$ to any value from $\left[0,C\right]$.
There are two special points in (\ref{equ:equilibrium}):
(d) For an APO $k$ with $r_k=r_{\min}$, it has the same payoff if it bids any value from $\left[0,r_{\min}\right]$. This is because with probability one, APO $k$ wins the auction.{\footnote{Notice that for any APO $j\ne k, j\in{\cal K}$, the probability that $r_j=r_{\min}$ is zero based on the continuous distribution of $r_j$. In other words, with probability one, $r_j$ is from the interval $\left(r_{\min},r_{\max}\right]$. Based on (\ref{equ:equilibrium}), APO $j\ne k$ bids from $\left(r_{\min},C\right]\cup \left\{``{\rm N} \textquotedblright\right\}$ and APO $k$ wins the auction.}} From (\ref{equ:rpay}) and (\ref{equ:APOpayoff}), APO $k$'s payoff is $\min\left\{C,b_1,\ldots,b_{k-1},b_{k+1},\ldots,b_{K}\right\}$, which does not depend on APO $k$'s bid $b_k$ {and is always no smaller than $r_{\min}$};
(e) For an APO $k$ with $r_k=r_T\left(C\right)$, it has the same expected payoff under bids $C$ and $``{\rm N} \textquotedblright$.
It is easy to show that $b^*\left(r_k,C\right)$ in (\ref{equ:equilibrium}) is not a dominant strategy for the APOs. For example, if APO $k$'s type $r_k\in\left(C,r_T\left(C\right)\right)$ and $\min_{j\in{\cal K},j\ne k}b_j=C$, bidding $``{\rm N} \textquotedblright$ generates a larger payoff to APO $k$ than bidding $b^*\left(r_k,C\right)=C$. This result is different from that of the conventional second-price auction, where bidding the truthful valuation constitutes an equilibrium, and is also the weakly dominant strategy for the bidders.
Notice that equation (\ref{equ:rT}) may admit multiple solutions, \emph{i.e.}, $M>1$. Based on Theorem \ref{theorem:equilibrium}, each solution $r_m^t$, $m=1,2,\ldots,M$, corresponds to a strategy $b^*$ defined in (\ref{equ:equilibrium}).
\begin{comment}
\subsection{Proof of the Unique Form of SBNE}
In this section, we show that (\ref{equ:equilibrium}) in Theorem \ref{theorem:equilibrium} provides the unique form of bidding strategy that constitutes an SBNE. First, we introduce Lemma \ref{lemma:monotonicity} to Lemma \ref{lemma:rBrA}, which provide the necessary conditions for a bidding strategy to constitute an SBNE. Recall that the reserve rate $C$ can be any value from $\left[r_{\min},r_{\max}\right)$.
\begin{lemma}\label{lemma:monotonicity
For a strategy function $\tilde b$ that constitutes an SBNE, if $r_L<r_H$ and $r_L,r_H\in\left[r_{\min},r_{\max}\right]$, we have ${\tilde b}\left(r_L,C\right)\preceq {\tilde b}\left(r_H,C\right)$.
\end{lemma}
Lemma \ref{lemma:monotonicity} shows the monotonicity of the bidding strategy function at the equilibrium. Intuitively, the bid from a lower type APO will not be larger than that from a higher type APO.
\begin{lemma}\label{lemma:activity}
For a strategy function $\tilde b$ that constitutes an SBNE, there is a positive measure of APO types that bid $``{\rm N} \textquotedblright$.
\end{lemma}
Lemma \ref{lemma:activity} implies that it is always possible for the LTE provider to work in the \emph{competition mode}.
\begin{lemma}\label{lemma:combine:rArB}
For a strategy function $\tilde b$ that constitutes an SBNE, there exist $r_A,r_B\in\left[r_{\min},r_{\max}\right)$ with $r_A>r_B$ such that: (1) ${\tilde b}\left(r,C\right)=r$ for all $r\in\left(r_{\min},r_B\right)$; (2) ${\tilde b}\left(r,C\right)=C$ for all $r\in\left[r_B,r_{A}\right)$; (3) ${\tilde b}\left(r,C\right)=``{\rm N} \textquotedblright$ for all $r\in\left(r_A,r_{\max}\right]$.
\end{lemma}
Lemma \ref{lemma:combine:rArB} shows that at the equilibrium: (a) APOs with low types ($r\in\left(r_{\min},r_B\right)$) bid the values of their types; (b) APOs with medium types ($r\in\left[r_B,r_A\right)$) bid the reserve rate $C$; (c) APOs with high types ($r\in\left(r_A,r_{\max}\right]$) bid $``{\rm N} \textquotedblright$. We determine the threshold types $r_B$ and $r_A$ in Lemma \ref{lemma:rBrA}.
\begin{lemma}\label{lemma:rBrA}
For a strategy function $\tilde b$ that constitutes an SBNE, $r_B=C$ and $r_A=r_T\left(C\right)$, where $r_T\left(C\right)$ can be any value from $\left\{r_1^t\left(C\right), r_2^t\left(C\right),\ldots, r_M^t\left(C\right)\right\}$.
\end{lemma}
Based on the necessary conditions summarized in the lemmas above, we conclude the unique form of bidding strategy at SBNE in the following theorem.
\end{comment}
In the following theorem, we show the unique form of bidding strategy under an SBNE.
\begin{theorem}\label{theorem:combine:unique}
The strategy function in (\ref{equ:equilibrium}) is the unique form of bidding strategy that constitutes an SBNE.
\end{theorem}
The sketch of the proof is as follows: first, we show the necessary conditions that a bidding strategy needs to satisfy to constitute an SBNE; second, we show that the function in (\ref{equ:equilibrium}) is the only function that satisfies all these conditions. We leave the detailed proof in Appendices \ref{appendix:sec:preliminary} and \ref{appendix:sec:theorem2}.
\subsection{APOs' Equilibrium When $C\in\left[0,\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}\right]$}\label{subsec:lowC}
We assume that the reserve rate $C$ is given from interval $\left[0,\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}\right]$, and summarize the form of the bidding strategy at the SBNE in the following theorem.
\begin{theorem}\label{theorem:lowC}
When $C\in\left[0,\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}\right)$, there is a unique SBNE, where ${b^*}\left(r_k,C\right)=``{\rm N}\textquotedblright$, $k\in{\cal K}$, for all $r_k\in\left[r_{\min},r_{\max}\right]$; when $C=\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$,
a strategy function constitutes an SBNE if and only if it is in the following form
\begin{align}
{b^*}\left(r_k,C\right)\!=\! \left\{\begin{array}{ll}
{\rm \!any\!~value\!~in} \left[0,\!C\right]{\rm{~\!or~\!}}``{\rm N} \textquotedblright\!, & {{\rm if~}{ r_k=r_{\min}},}\\
``{\rm N}\textquotedblright, & {{\rm if~} r_k\!\in\left(r_{\min},\!r_{\max}\right].}
\end{array} \right.\label{equ:lowCbid}
\end{align}
\end{theorem}
When $C\in\left[0,\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}\right]$, the LTE provider only wants to allocate a limited data rate to the winning APO's users. In this case, the APOs bid $``{\rm N} \textquotedblright$ with probability one.{\footnote{In Theorem \ref{theorem:lowC}, when $C=\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$, the APO with type $r_{\min}$ can bid any value. However, the probability for an APO to have the type $r_{\min}$ is zero due to the continuous distribution of $r$.}}
\vspace{-0.2cm}
\subsection{APOs' Equilibrium When $C\in\left(\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},r_{\min}\right)$}\label{subsec:stageII:3}
We assume that the reserve rate $C$ is given from interval $\left(\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},r_{\min}\right)$, and show that the bidding strategy that constitutes an SBNE has a unique form. First, we introduce the following lemma.
\begin{lemma}\label{lemma:rX}
The following equation admits at least one solution $r$ in $\left(r_{\min},r_{\max}\right)$:
\begin{multline}
\sum_{n=1}^{K-1}{\binom{K-1}{n} F^n\left(r\right)\left(1-F\left(r\right)\right)^{K-1-n}\frac{C-r}{n+1}}\\
+\left(1-F\left(r\right)\right)^{K-1}\left(C-\frac{K-1+{\eta^{\rm APO}}}{K}r\right)=0,\label{equ:rX}
\end{multline}
where $F\left(\cdot\right)$ is the CDF of random variable $r_k$, $k\in{\cal K}$. We denote the solutions $r$ in $\left(r_{\min},r_{\max}\right)$ as $r_1^x\left(C\right),r_2^x\left(C\right),\ldots,r_L^x\left(C\right)$, where $L=1,2,\ldots,$ is the number of solutions.
\end{lemma}
Based on the definition of $r_1^x\left(C\right),r_2^x\left(C\right),\ldots,r_L^x\left(C\right)$ in Lemma \ref{lemma:rX}, we introduce the following theorem.
\begin{theorem}\label{theorem:middleC}
When $C\in\left(\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},r_{\min}\right)$, consider an $r_X\left(C\right)\in\left(r_{\min},r_{\max}\right)$ that belongs to the set of $\left\{r_1^x\left(C\right),r_2^x\left(C\right),\ldots,r_L^x\left(C\right)\right\}$, then the following bidding strategy $b^*$ constitutes an SBNE:
\begin{align}
{b^*}\left(r_k,C\right)=\left\{\begin{array}{ll}
C, & {{\rm if~}{ r_k\in\left[r_{\min},r_X\left(C\right)\right)},}\\
C{\rm~or~}``{\rm N} \textquotedblright, & {\rm if~}{r_k=r_X\left(C\right),}\\
``{\rm N} \textquotedblright, & {{\rm if~} r_k\in\left(r_X\left(C\right),r_{\max}\right],}
\end{array} \right.\label{equ:simplyequilibrium}
\end{align}
where $k\in{\cal K}$. Furthermore, such a bidding strategy $b^*$ is the unique form of bidding strategy that constitutes an SBNE.
\end{theorem}
The bidding strategy in (\ref{equ:simplyequilibrium}) is similar to that in (\ref{equ:equilibrium}), except that here it only has two regions instead of three regions. Specifically, here there are no APOs that bid their types $r_k$. This is because here the reserve rate $C$ is smaller than $r_{\min}$, hence bidding any type $r_k\in\left[r_{\min},r_{\max}\right]$ is not feasible. We illustrate the structure of strategy function $b^*$ in Fig. \ref{fig:S2}.
\begin{figure}[h]
\centering
\includegraphics[scale=0.4]{JStructure2-eps-converted-to.pdf}
\caption{Bidding Strategy Structure at SBNE When $C\in\left(\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},r_{\min}\right)$.}
\label{fig:S2}
\vspace{-4mm}
\end{figure}
Similar as the equilibrium analysis for $C\in\left[r_{\min},r_{\max}\right)$, here equation (\ref{equ:rX}) may admit multiple solutions, \emph{i.e.}, $L>1$, in which case each solution $r_l^x$, $l=1,2,\ldots,L$, corresponds to a strategy $b^*$ defined in (\ref{equ:simplyequilibrium}).
\vspace{-4mm}
\subsection{APOs' Equilibrium When $C\in\left[r_{\max},\infty\right)$}\label{subsec:stageII:4}
We assume that the reserve rate $C$ is given from interval $C\in\left[r_{\max},\infty\right)$, and show the unique form of bidding strategy that constitutes an SBNE in the following theorem.
\begin{theorem}\label{theorem:highC}
When $C\in\left[r_{\max},\infty\right)$, a strategy function constitutes an SBNE if and only if it is in the following form ($k\in{\cal K}$):
\begin{align}
{b^*}\!\left(r_k,\!C\right)\!=\!\!\left\{\begin{array}{ll}
{{\rm \!any~\!value~\!in} \left[0,\!r_{\min}\right]\!,} & {\rm \!if~\!}{r_k=r_{\min},}\\
r_k, & {{\rm \!if~\!}{r_k\!\in\!\left(r_{\min},\!r_{\max}\right)\!},}\\
{\rm \!any~\!value~\!in}\left[r_{\max},\!C\right]\!\cup\!\left\{\!``{\rm N}\textquotedblright\right\}\!, & {\rm \!if~\!}{r_k=r_{\max}.}\\
\end{array} \right.\label{equ:EQhighC}
\end{align}
\end{theorem}
When $C\in\left[r_{\max},\infty\right)$, the LTE provider is willing to allocate a large data rate to the winning APO's users. Based on (\ref{equ:EQhighC}), all APOs bid values from $\left[0,C\right]$ with probability one.{\footnote{Notice that the probability for an APO to have the type $r_{\max}$ is zero due to the continuous distribution of $r$.}}
\vspace{-0.5cm}
\subsection{{{Summary of APOs' Equilibriums}}}\label{sec:comparison}
\vspace{-0.2cm}
\begin{figure}[h]
\centering
\includegraphics[scale=0.38]{LSumJfourC-eps-converted-to.pdf}
\caption{APOs' Strategies under Different $C$.}
\label{fig:fourC}
\end{figure}
{{Based on Sections \ref{subsec:stageII:1}, \ref{subsec:lowC}, \ref{subsec:stageII:3}, and \ref{subsec:stageII:4}, there is always a unique form of APO $k$'s bidding strategy $b^*\left(r_k,C\right)$ at the SBNE for any reserve rate $C\in\left[0,\infty\right)$.}}
{{We summarize the APOs' strategies under different intervals of $C$ in Fig. \ref{fig:fourC}.}}{\footnote{{{When $C\in\left[r_{\min},r_{\max}\right)$, an APO with type $r_k=r_{\min}$ can bid any value from $\left[0,r_{\min}\right]$ at the equilibrium based on (\ref{equ:equilibrium}). Since the probability for an APO to have the type $r_{\min}$ is zero, the strategy of this particular APO type is not shown in Fig. \ref{fig:fourC}.}}}}
We find that some APO types bid the reserve rate $C$ in Fig. \ref{fig:fourC} when $C\in\left(\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},r_{\max}\right)$. This is due to the unique feature of the auction with \emph{allocative externalities}: {{first, if none of the other APOs submits its bid from interval $\left[0,C\right]$, these types of APOs prefer to cooperate with the LTE provider rather than {{to interfere with}} the LTE in the \emph{competition mode}; second, if at least one of the other APOs submits its bid from interval $\left[0,C\right]$, these types of APOs prefer to occupy their own channels rather than to cooperate with the LTE provider, as the LTE will not generate interference to their channels in this case. The first reason motivates these APO types to bid from interval $\left[0,C\right]$, and the second reason motivates these APO types to reduce their chances of winning the auction as much as possible. As a result, these APO types bid the reserve rate $C$ at the equilibrium.}}
\section{{{Stage I: LTE Provider's Reserve Rate}}}\label{sec:stageI:LTE}
{{In this section, we analyze the LTE provider's optimal reserve rate by anticipating APOs' equilibrium strategies in Stage II.
In Section \ref{subsec:stageI:0}, we define the LTE provider's expected payoff.
In Section \ref{subsec:stageI:1}, we compute the LTE provider's expected payoff based on different intervals of $C$. In Section \ref{subsec:stageI:2}, we formulate the LTE provider's payoff maximization problem. In Section \ref{subsec:stageI:3}, we analyze the LTE provider's optimal reserve rate $C^*$.}}
\vspace{-4mm}
\subsection{{{Definition of LTE Provider's Expected Payoff}}}\label{subsec:stageI:0}
We first make the following assumption on the CDF of an APO's type.
\begin{assumption}\label{assumption:unique}
Under the cumulative distribution function $F\left(\cdot\right)$, (a) equation (\ref{equ:rT}) has a unique solution in $\left(C,r_{\max}\right)$, {i.e.}, $M=1$, and (b) equation (\ref{equ:rX}) has a unique solution in $\left(r_{\min},r_{\max}\right)$, {i.e.}, $L=1$.
\end{assumption}
Assumption \ref{assumption:unique} implies that $r_T\left(C\right)$ and $r_X\left(C\right)$ are unique. Such an assumption is mild.
{{When $K=2$, we have proved that Assumption \ref{assumption:unique} holds for the uniform distribution.
For a general $K$, we have run simulation and shown that Assumption \ref{assumption:unique} holds for both the uniform distribution and truncated normal distribution.
The details of the proof and simulation can be found in Appendices \ref{appendix:sec:K2} and \ref{appendix:sec:Kgeneral}, respectively.}}
Based on Theorem \ref{theorem:equilibrium} and Theorem \ref{theorem:combine:unique}, the uniqueness of $r_T\left(C\right)$ implies the unique expression of APOs' bidding strategy $b^*$ for $C\in\left[r_{\min},r_{\max}\right)$. Similarly, from Theorem \ref{theorem:middleC}, the uniqueness of $r_X\left(C\right)$ implies the unique expression of strategy $b^*$ for $C\in\left(\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},r_{\min}\right)$.
We define the LTE provider's expected payoff as
\begin{align}
{\bar \Pi}^{\rm LTE}\!\left(C\right)\! \triangleq \!\mathbb{E}_{\bm r}\left\{{\Pi^{\rm LTE}\!\left(\!\left(b^*\!\left(r_1,C\right),\ldots,b^*\!\left(r_K,C\right)\!\right)\!,\!C\right)\!}\right\}\!,\label{equ:expectedpayoff}
\end{align}
where ${\bm r}\triangleq \left(r_k,\forall k\in{\cal K}\right)$ denotes the types of all APOs, and $b^*\left(r_k,C\right)$, $k\in{\cal K}$, is given in (\ref{equ:equilibrium}), (\ref{equ:lowCbid}), (\ref{equ:simplyequilibrium}), and (\ref{equ:EQhighC}) based on the different intervals of $C$.
\vspace{-4mm}
\subsection{{{Computation of LTE Provider's Expected Payoff}}}\label{subsec:stageI:1}
{{Since $b^*\left(r_k,C\right)$ in (\ref{equ:expectedpayoff}) has different expressions for four different intervals of $C$, we characterize ${\bar \Pi}^{\rm LTE}\left(C\right)$ based on these four intervals of $C$.}}
\subsubsection{$C\in\left[0,\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}\right]$}\label{subsec:LTEpayoffI}
The APOs submit their bids according to strategy $b^*$ in (\ref{equ:lowCbid}). It is easy to find that $b^*\left(r_k,C\right)=``{\rm N}\textquotedblright$ with probability one for all $k\in{\cal K}$, and hence the LTE provider always works in the \emph{competition mode}. Based on (\ref{equ:LTEpayoff}), we can compute ${\bar \Pi}^{\rm LTE}\left(C\right)$ as
\begin{align}
{\bar \Pi}^{\rm LTE}\left(C\right)={\delta^{\rm LTE}} R_{\rm LTE}.\label{equ:LTEpayoff:smallC}
\end{align}
\subsubsection{$C\in\left(\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},r_{\min}\right)$}\label{subsec:LTEpayoffII}
The APOs' bidding strategy is summarized in (\ref{equ:simplyequilibrium}). Hence, the probabilities for an APO with a random type to bid $C$ and $``{\rm N}\textquotedblright$ are $F\left(r_X\left(C\right)\right)$ and $1-F\left(r_X\left(C\right)\right)$, respectively.
Therefore, we can compute ${\bar \Pi}^{\rm LTE}\left(C\right)$ a
\begin{align}
\nonumber
{\bar \Pi}^{\rm LTE}\left(C\right) = &\left(1-F\left(r_X\left(C\right)\right)\right)^{K} {\delta^{\rm LTE}} R_{\rm LTE} + \\
& \left(1-\left(1-F\left(r_X\left(C\right)\right)\right)^K \right) \left(R_{\rm LTE}-C\right).\label{equ:LTEpayoff:relow}
\end{align}
That is to say: (a) when all the APOs bid $``{\rm N}\textquotedblright$, the LTE provider works in the \emph{competition mode}, and obtains a payoff of ${\delta^{\rm LTE}} R_{\rm LTE}$; (b) when at least one APO bids $C$, the LTE provider works in the \emph{cooperation mode}, and allocates a rate of $C$ to the winning APO's users.
\subsubsection{$C\in\left[r_{\min},r_{\max}\right)$}\label{subsec:LTEpayoffIII}
The APOs' strategy is given in (\ref{equ:equilibrium}).
\begin{comment}
Recall that in (\ref{equ:rpay}), we have $r_{\rm pay}\left({{\bm b},C}\right)=0$ if $b_1=b_2=``{\rm N} \textquotedblright$. Hence, we can rearrange (\ref{equ:LTEpayoff}) as
\begin{align}
\Pi^{\rm LTE}\left({\bm b},C\right)=-r_{\rm pay}\left({{\bm b},C}\right)+\left\{\begin{array}{ll}
{{\delta^{\rm LTE}} R_{\rm LTE},} & {\rm if~}{b_1=b_2=``{\rm N} \textquotedblright,}\\
{R_{\rm LTE},} & {\rm otherwise.}
\end{array} \right.
\end{align}
\end{comment}
We can compute ${\bar \Pi}^{\rm LTE}\left(C\right)$ a
\begin{align}
\nonumber
{\bar \Pi}^{\rm LTE}\left(C\right) = & \left(1-F\left(r_T\left(C\right)\right)\right)^K {\delta^{\rm LTE}} R_{\rm LTE} + \\
& \left(1-\left(1-F\left(r_T\left(C\right)\right)\right)^K \right) R_{\rm LTE}- {\bar r}_{\rm pay}\left(C\right).\label{equ:LTEpayoff:typicalC}
\end{align}
Here, ${\bar r}_{\rm pay}\left(C\right)$ is defined as the expected data rate that the LTE provider allocates to the winning APO's users, and is given as (the details of computing ${\bar r}_{\rm pay}\left(C\right)$ can be found in Appendix \ref{appendix:sec:rpay})
\begin{align}
\nonumber
{\bar r}_{\rm pay}\left(C\right)\triangleq & K\left(K-1\right) \int_{r_{\min}}^{C} r f\left(r\right) F\left(r\right) \left(1-F\left(r\right)\right)^{K-2} dr \\
\nonumber
& +KCF\left(C\right)\left(1-F\left(C\right)\right)^{K-1} \\
&+C\left(\left(1-F\left(C\right)\right)^K-\left(1-F\left(r_T\left(C\right)\right)\right)^K\right).\label{equ:rpay:multi}
\end{align}
\subsubsection{$C\in\left[r_{\max},\infty\right)$}\label{subsec:LTEpayoffIV}
Based on (\ref{equ:EQhighC}), the APOs bid values from $\left[0,C\right]$ with probability one, and the LTE provider always works in the \emph{cooperation mode}. Then we can compute ${\bar \Pi}^{\rm LTE}\left(C\right)$ as
\begin{align}
\nonumber
& {\bar \Pi}^{\rm LTE}\left(C\right)=R_{\rm LTE}\\
& -K\left(K-1\right) \int_{r_{\min}}^{r_{\max}} r f\left(r\right) F\left(r\right) \left(1-F\left(r\right)\right)^{K-2} dr.\label{equ:LTEpayoff:largeC}
\end{align}
\subsection{LTE Provider's Payoff Maximization Problem}\label{subsec:stageI:2}
Based on ${\bar \Pi}^{\rm LTE}\left(C\right)$ derived in Section \ref{subsec:stageI:1}, we can verify that ${\bar \Pi}^{\rm LTE}\left(C\right)$ is continuous for $C\in\left[0,\infty\right)$. The LTE provider determines the optimal reserve rate by solving
{{
\begin{align}
\max_{C\in\left[0,\infty\right)} {~\bar \Pi}^{\rm LTE}\left(C\right) {~~~~~~\rm s.t.~~} b_{\max}\left(C\right) \le R_{\rm LTE},\label{equ:optABC}
\end{align}}}
where we define
\begin{align}
b_{\max}\left(C\right) \triangleq \max \left\{ b^*\left(r_k,C\right)\in\left[0,C\right]:{r_k\in\left[r_{\min},r_{\max}\right]} \right\},
\end{align}
which is the maximum possible bid (except $``{\rm N}\textquotedblright$) from the APOs at the SBNE under $C$. Constraint $b_{\max}\left(C\right) \le R_{\rm LTE}$ ensures that the LTE provider has enough capacity to satisfy the bid from the winning APO.
\vspace{-0.3cm}
\subsection{LTE Provider's Optimal Reserve Rate}\label{subsec:stageI:3}
{{In the following theorem, we characterize the optimal reserve rate $C^*$ that solves problem (\ref{equ:optABC}) for a general distribution function $F\left(\cdot\right)$ that satisfies Assumption \ref{assumption:unique}.}}
\begin{comment}
We first show the following lemma.
\begin{lemma}\label{lemma:local:etarmin}
(a) When $R_{\rm LTE}>\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}$, $C=\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$ is a local minimum of ${\bar \Pi}^{\rm LTE}\left(C\right)$; (b) when $R_{\rm LTE}\le \frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}$, $C=\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$ is a global maximum of ${\bar \Pi}^{\rm LTE}\left(C\right)$.
\end{lemma}
When we have $R_{\rm LTE}>\!\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}$, we can show that $\lim_{C\downarrow \frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}} \frac{d {{\bar \Pi}^{\rm LTE}}\left(\!C\right)}{dC}>0$.{\footnote{The downward arrow $\downarrow$ corresponds to the right-sided limit.}} Furthermore, ${\bar \Pi}^{\rm LTE}\left(C\right)$ is a constant for $C\in\left[0,\frac{K-1+{\eta^{\rm APO}}}{K}\!r_{\min}\right]$ based on (\ref{equ:LTEpayoff:smallC}). Hence, it is easy to find that $C=\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$ is a local minimum point, and the LTE provider will not choose $C^*$ from the interval $\left[0,\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}\right]$. To understand the intuition, notice that $R_{\rm LTE}> \frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}$ is equivalent to $\left(1-{\delta^{\rm LTE}}\right)R_{\rm LTE}> \frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$. Here, $\left(1-{\delta^{\rm LTE}}\right)R_{\rm LTE}$ stands for the additional increase in the LTE network's capacity when it works in the \emph{cooperation mode}. Based on (\ref{equ:equilibrium}), (\ref{equ:lowCbid}), (\ref{equ:simplyequilibrium}), and (\ref{equ:EQhighC}), $\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$ is the lower bound of the data rate that any APO with type in $\left(r_{\min},r_{\max}\right]$ may request from the LTE provider. When $\left(1-{\delta^{\rm LTE}}\right)R_{\rm LTE}> \frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$, the LTE provider benefits from cooperating with the APOs that request small data rates. Therefore, it will definitely choose $C^*$ greater than $\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$ so as to accept the bids from these APOs.
When $R_{\rm LTE}\le \frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}$, we can show that ${\bar \Pi}^{\rm LTE}\left(C\right)$ under $C\in\left(\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},\infty\right)$ is smaller than that under $C\!=\!\frac{K-1+{\eta^{\rm APO}}}{K}\! r_{\min}$. Since ${\bar \Pi}^{\rm LTE}\left(C\right)$ is a constant for $C\!\in\!\left[0,\!\frac{K-1+{\eta^{\rm APO}}}{K}\!r_{\min}\right]$, point $C=\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$ is a global maximum. The intuition for this is as follows. When $\left(1-{\delta^{\rm LTE}}\right)R_{\rm LTE}\le \frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$, the additional gain in the LTE network's capacity cannot cover the request from the APOs. Hence, the LTE provider prefers to work in the \emph{competition mode} by choosing $C=\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$.
Based on Lemma \ref{lemma:local:etarmin}, we characterize the optimal $C^*$ that solves problem (\ref{equ:optABC}) in the following theorem.
\end{comment}
\begin{theorem}\label{theorem:optimalCcase}
The LTE provider's optimal reserve rate $C^*$ satisfies the following properties:\\
(1) When $R_{\rm LTE}\le\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}$, {{$C^*$}} can be any value from $\left[0,\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}\right]$;\\
(2) When $\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}<\! R_{\rm LTE}\le \! r_{\max}$, {{$C^*$ can be chosen from}} $\left(\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},\!R_{\rm LTE}\right]$;\\
(3) When $R_{\rm LTE}\!>\!\max\left\{r_{\max},\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}\!\right\}$, {{$C^*$ can be chosen from}} $\left(\!\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},\!r_{\max}\!\right]$.
\end{theorem}
{{
When $R_{\rm LTE}\le\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}$, the LTE provider does not have enough capacity to satisfy any APO's request. Specifically, $R_{\rm LTE}\le \frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}$ is equivalent to $\left(1-{\delta^{\rm LTE}}\right)R_{\rm LTE}\le \frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$. Here, $\left(1-{\delta^{\rm LTE}}\right)R_{\rm LTE}$ stands for the additional increase in the LTE network's capacity when it works in the \emph{cooperation mode}. Based on (\ref{equ:equilibrium}), (\ref{equ:lowCbid}), (\ref{equ:simplyequilibrium}), and (\ref{equ:EQhighC}), $\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$ is the lower bound of the data rate that any APO with type in $\left(r_{\min},r_{\max}\right]$ may request from the LTE provider. Therefore, when $R_{\rm LTE}\le\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}$, the additional gain in the LTE network's capacity under cooperation cannot cover the request from any APO, and the LTE provider sets $C^*\in \left[0,\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}\right]$ to work in the \emph{competition mode}.
When $\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}<R_{\rm LTE}\le r_{\max}$, the LTE network's capacity can cover the requests from the APOs that bid small values. Hence, the LTE provider chooses $C^*$ above $\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}$ to accept these APOs' bids. Meanwhile, the LTE provider has to choose $C^*$ no larger than $R_{\rm LTE}$, otherwise it does not have enough capacity to satisfy the APOs that bid large values.
When $R_{\rm LTE}\!>\!\max\left\{r_{\max},\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}\right\}$, since the maximum possible bid from the APOs is $r_{\max}$, the LTE provider always has enough capacity to satisfy the APOs' requests. In this case, the LTE provider chooses $C^*$ from $\left(\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},r_{\max}\right]$, and $C^*$ is no longer constrained by the LTE throughput $R_{\rm LTE}$.}}
{{Next we discuss the choice of $C^*$ based on Theorem \ref{theorem:optimalCcase}. When $R_{\rm LTE}\!\le\!\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}$, Theorem \ref{theorem:optimalCcase} indicates that any value from interval $\!\left[0,\!\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min}\!\right]$ is the optimal $C^*$ for a general distribution function $F\left(\cdot\right)$.}} However, when $R_{\rm LTE}>\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}$, it is difficult to characterize the closed-form expression of $C^*$ even under a specific function $F\left(\cdot\right)$.
This is because (i) it is difficult to solve equations (\ref{equ:rT}) and (\ref{equ:rX}) and obtain the closed-form expressions of $r_T\left(C\right)$ and $r_X\left(C\right)$, respectively, and (ii) the expression of ${\bar \Pi}^{\rm LTE}\left(C\right)$ in (\ref{equ:LTEpayoff:typicalC}) is complicated and hard to analyze. Therefore, we determine the optimal $C^*$ numerically for $R_{\rm LTE}>\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}$. Specifically, we have the following observation from the simulation.
\begin{observation}\label{observation:0}
${\bar \Pi}^{\rm LTE}\left(C\right)$ is strictly unimodal for $C\in\left(\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},r_{\max}\right]$.
\end{observation}
\begin{figure*}[t]
\centering
\begin{minipage}[t]{.49\linewidth}
\centering
\includegraphics[scale=0.3]{CQsimu0-eps-converted-to.pdf}
\vspace{-2mm}
\caption{Example of Function ${\bar \Pi}^{\rm LTE}\left(C\right)$.}
\label{fig:simu:0}
\vspace{-5mm}
\end{minipage}
\begin{minipage}[t]{.49\linewidth}
\centering
\includegraphics[scale=0.3]{QsimuA-eps-converted-to.pdf}
\vspace{-2mm}
\caption{Impact of $K$ on LTE Provider's and APOs' Strategies.}
\label{fig:simu:A}
\vspace{-5mm}
\end{minipage}
\end{figure*}
We have verified Observation \ref{observation:0} for the uniform distribution function $F\left(\cdot\right)$ and the truncated normal distribution function $F\left(\cdot\right)$. In Fig. \ref{fig:simu:0}, we illustrate an example of ${\bar \Pi}^{\rm LTE}\left(C\right)$, where $K=2$, ${\delta^{\rm LTE}}=0.4$, ${\eta^{\rm APO}}=0.3$, $R_{\rm LTE}=300{\rm~Mbps}$, and $r_k\in\left[50~{\rm Mbps},200~{\rm Mbps}\right]$ follows a truncated normal distribution.{\footnote{We choose $R_{\rm LTE}=300{\rm~Mbps}$ because the peak LTE throughput ranges from $250~{\rm Mbps}$ to $370~{\rm Mbps}$ based on \cite{canousing}. {{Moreover, we choose ${\eta^{\rm APO}}$ smaller than ${\delta^{\rm LTE}}$, because the degeneration of Wi-Fi's data rate due to the co-channel interference is usually heavier than that of the LTE, as we discussed in Section \ref{sec:model}.}}}} Based on Theorem \ref{theorem:optimalCcase} and Observation \ref{observation:0}, when $\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}<R_{\rm LTE}\le r_{\max}$ and $R_{\rm LTE}>\max\left\{r_{\max},\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}\right\}$, we can use the Golden Section method \cite{bertsekas1999nonlinear} on interval $\left(\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},R_{\rm LTE}\right]$ and interval $\left(\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},r_{\max}\right]$, respectively, to determine the optimal $C^*$.
\vspace{-0.3cm}
\section{Numerical Results}\label{sec:numerical}
In this section, we first investigate the impacts of the system parameters on the LTE's optimal reserve rate, the LTE's expected payoff, and the APOs' equilibrium strategies.
{{Then we compare our auction-based spectrum sharing scheme with a state-of-the-art benchmark scheme. Specifically, we randomly pick the APOs, and implement both schemes. We compare several criteria (such as the LTE provider's payoff, the APOs' total payoff, and the social welfare) achieved by our auction-based scheme and the benchmark scheme.}}
\subsection{Influences of System Parameters}
\subsubsection{Influence of $K$} We first study the impact of the number of APOs $K$ on the LTE provider's and APOs' strategies. We choose $R_{\rm LTE}=95{\rm~Mbps}$, ${\delta^{\rm LTE}}=0.4$, and ${\eta^{\rm APO}}=0.3$, and assume that $r_k$, $k\in{\cal K}$, follows a truncated normal distribution. Specifically, we obtain the distribution of $r_k$ by truncating the normal distribution ${\cal N}\left(125~{\rm Mbps},2500~{{\rm Mbps}^2}\right)$ to interval $\left[50~{\rm Mbps},200~{\rm Mbps}\right]$. We change $K$ from $2$ to $7$, and determine the corresponding optimal reserve rate $C^*$ numerically based on the approach discussed in Section \ref{subsec:stageI:3}.
We plot $C^*$ against $K$ in Fig. \ref{fig:simu:A}, and observe that $C^*$ increases with $K$. This is because that the probability of a particular APO being interfered by the LTE in the \emph{competition mode} decreases with the number of APOs. Hence, the APOs are less willing to cooperate with the LTE provider under a larger $K$, and the LTE provider needs to increase $C^*$ to attract the APOs.
In Fig. \ref{fig:simu:A}, we observe that $C^*\in\left(\frac{K-1+{\eta^{\rm APO}}}{K}r_{\min},r_{\min}\right)$ for $2\le K \le 4$. Based on (\ref{equ:simplyequilibrium}), in this case, APOs with types in $\left[r_{\min},r_X\left(C^*\right)\right)$ and $\left(r_X\left(C^*\right),r_{\max}\right]$ bid $C^*$ and $``{\rm N}\textquotedblright$, respectively. To study the impact of $K$ on the APOs' strategies, we plot $r_X\left(C^*\right)$ for $2\le K \le 4$ in Fig. \ref{fig:simu:A}. We observe that $r_X\left(C^*\right)$ decreases with $K$. This means that when $K$ increases from $2$ to $4$, more APOs bid $``{\rm N}\textquotedblright$ instead of $C^*$. On the other hand, we find that $C^*\in\left[r_{\min},r_{\max}\right)$ for $5\le K \le 7$. Based on (\ref{equ:equilibrium}), in this case, APOs with types in $\left[C^*,r_T\left(C^*\right)\right)$ and $\left(r_T\left(C^*\right),r_{\max}\right]$ bid $C^*$ and $``{\rm N}\textquotedblright$, respectively. We plot $r_T\left(C^*\right)$ for $5\le K \le 7$, and observe that $r_T\left(C^*\right)$ decreases with $K$. Since $C^*$ increases with $K$, it is easy to conclude that when $K$ increases from $5$ to $7$, fewer APOs bid $C^*$, and more APOs bid $``{\rm N}\textquotedblright$. Combining the observations for $2\le K \le 4$ and $5\le K \le 7$, we summarize that the increase of $K$ makes more APOs switch from bidding $C^*$ to bidding $``{\rm N}\textquotedblright$. The reason is that each APO has a smaller chance to be interfered by the LTE in the \emph{competition mode} under a larger $K$. Therefore, the APOs with large $r_k$ are less willing to cooperate with the LTE provider, and more APOs bid $``{\rm N}\textquotedblright$ instead of $C^*$.
We summarize the observations for Fig. \ref{fig:simu:A} as follows.
\begin{observation}\label{observation:1}
When the number of APOs increases, (i) the LTE provider's optimal reserve rate $C^*$ increases, and (ii) more APOs switch from bidding $C^*$ to bidding $``{\rm N}\textquotedblright$.
\end{observation}
\begin{figure*}[t]
\centering
\begin{minipage}[t]{.32\linewidth}
\centering
\includegraphics[scale=0.3]{QsimuB-eps-converted-to.pdf}
\vspace{-3mm}
\caption{${\bar \Pi}^{\rm LTE}\left(C^*\right)$ (Large $R_{\rm LTE}$).}
\label{fig:simu:B}
\vspace{-5mm}
\end{minipage}
\begin{minipage}[t]{.32\linewidth}
\includegraphics[scale=0.3]{QsimuC-eps-converted-to.pdf}
\vspace{-3mm}
\caption{${\bar \Pi}^{\rm LTE}\left(C^*\right)$ (Small $R_{\rm LTE}$).}
\label{fig:simu:C}
\vspace{-5mm}
\end{minipage}
\begin{minipage}[t]{.32\linewidth}
\centering
\includegraphics[scale=0.3]{CQsimuD-eps-converted-to.pdf}
\vspace{-3mm}
\caption{Impacts of $R_{\rm LTE}$, ${\delta^{\rm LTE}}$, and ${\eta^{\rm APO}}$ on $C^*$.}
\label{fig:simu:D}
\vspace{-5mm}
\end{minipage}
\end{figure*}
Next we study the impact of $K$ on the LTE provider's expected payoff ${\bar \Pi}^{\rm LTE}\left(C^*\right)$. The settings of ${\delta^{\rm LTE}}$ and ${\eta^{\rm APO}}$, and the distribution of $r_k$ are the same as those in Fig. \ref{fig:simu:A}. We choose $R_{\rm LTE}=220{\rm~Mbps}$, $240{\rm~Mbps}$, and $260{\rm~Mbps}$, and plot the corresponding ${\bar \Pi}^{\rm LTE}\left(C^*\right)$ against $K$ in Fig. \ref{fig:simu:B}. We observe that ${\bar \Pi}^{\rm LTE}\left(C^*\right)$ increases with $K$ for these values of $R_{\rm LTE}$. Moreover, we choose $R_{\rm LTE}=90{\rm~Mbps}$, $110{\rm~Mbps}$, and $130{\rm~Mbps}$, and plot the corresponding ${\bar \Pi}^{\rm LTE}\left(C^*\right)$ against $K$ in Fig. \ref{fig:simu:C}. Different from Fig. \ref{fig:simu:B}, we find that ${\bar \Pi}^{\rm LTE}\left(C^*\right)$ does not significantly change with $K$ in Fig. \ref{fig:simu:C}. To understand the difference between Fig. \ref{fig:simu:B} and Fig. \ref{fig:simu:C}, we notice that the increase of $K$ has the following two opposite impacts on ${\bar \Pi}^{\rm LTE}\left(C^*\right)$: (i) the probability for the LTE provider to find an APO with a small bid increases, which potentially increases ${\bar \Pi}^{\rm LTE}\left(C^*\right)$; (ii) more APOs bid $``{\rm N}\textquotedblright$ instead of $C^*$ (Observation \ref{observation:1}), which potentially decreases ${\bar \Pi}^{\rm LTE}\left(C^*\right)$. In Fig. \ref{fig:simu:B}, the values of $R_{\rm LTE}$ are large, and the LTE provider can set large reserve rates $C^*$ to attract the APOs. In this situation, the interval of APO types that want to cooperate with the LTE provider is large, and impact (i) plays a dominant role. As a result, ${\bar \Pi}^{\rm LTE}\left(C^*\right)$ increases with $K$ in Fig. \ref{fig:simu:B}. On the other hand, the values of $R_{\rm LTE}$ are small in Fig. \ref{fig:simu:C}, and the LTE provider can only choose small reserve rates $C^*$. Hence, the interval of APO types that want to cooperate with the LTE provider is small. In this situation, impact (ii) becomes as important as impact (i). As a result, ${\bar \Pi}^{\rm LTE}\left(C^*\right)$ does not significantly change with $K$ in Fig. \ref{fig:simu:C}.
We summarize the following observations for Fig. \ref{fig:simu:B} and Fig. \ref{fig:simu:C}.
\begin{observation}\label{observation:2}
When the LTE provider has a large throughput $R_{\rm LTE}$, its expected payoff ${\bar \Pi}^{\rm LTE}\left(C^*\right)$ increases with $K$; otherwise, ${\bar \Pi}^{\rm LTE}\left(C^*\right)$ does not significantly change with $K$.
\end{observation}
\subsubsection{Influences of $R_{\rm LTE}$, ${\delta^{\rm LTE}}$, and ${\eta^{\rm APO}}$} We investigate the impacts of parameters $R_{\rm LTE}$, ${\delta^{\rm LTE}}$, and ${\eta^{\rm APO}}$ on $C^*$. We choose $K=4$, and the distribution of $r_k$ is the same as that in Fig. \ref{fig:simu:A}. We consider four pairs of data rate discounting factors: $\left({\delta^{\rm LTE}},{\eta^{\rm APO}}\right)=\left(0.4,0.7\right),\left(0.4,0.3\right),\left(0.4,0.1\right)$, and $\left(0.6,0.3\right)$. For each pair of $\left({\delta^{\rm LTE}},{\eta^{\rm APO}}\right)$, we change $R_{\rm LTE}$ from $10{\rm~Mbps}$ to $250{\rm~Mbps}$, and determine the corresponding $C^*$ numerically. In Fig. \ref{fig:simu:D}, we plot $C^*$ against $R_{\rm LTE}$ under the different pairs of $\left({\delta^{\rm LTE}},{\eta^{\rm APO}}\right)$.
Under all four settings, we observe that $C^*$ does not change with $R_{\rm LTE}$ when $R_{\rm LTE}$ is below $\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}$. In this case, the LTE provider does not have enough capacity to satisfy the APOs' requests. Based on Theorem \ref{theorem:optimalCcase}, it chooses a small reserve rate, and works in the \emph{competition mode}. When $R_{\rm LTE}$ is above $\frac{K-1+{\eta^{\rm APO}}}{K\left(1-{\delta^{\rm LTE}}\right)}r_{\min}$, $C^*$ increases with $R_{\rm LTE}$. This is because with a larger throughput $R_{\rm LTE}$, the LTE provider is able to allocate a larger data rate to the winning APO, and hence it increases the reserve rate $C^*$ to attract the APOs.
With ${\delta^{\rm LTE}}=0.4$, we find that $C^*$ increases with ${\eta^{\rm APO}}$ (see the top three curves). This is because under a larger ${\eta^{\rm APO}}$, the APOs are less heavily interfered by the LTE, and hence are less willing to cooperate with the LTE provider. As a result, the LTE provider needs to increase its reserve rate to attract the APOs.
With ${\eta^{\rm APO}}=0.3$, we find that $C^*$ under ${\delta^{\rm LTE}}=0.4$ is no smaller than that under ${\delta^{\rm LTE}}=0.6$. Under a smaller ${\delta^{\rm LTE}}$, the LTE provider is more heavily affected by the interference from Wi-Fi. In this case, the LTE provider chooses a larger reserve rate $C^*$ to motivate the cooperation with the APOs. Furthermore, compared with ${\eta^{\rm APO}}$, we find that the difference in ${\delta^{\rm LTE}}$ leads to a larger difference in $C^*$, which shows that ${\delta^{\rm LTE}}$ has a larger impact on $C^*$ than ${\eta^{\rm APO}}$.
We summarize the observations in Fig. \ref{fig:simu:D} as follows.
\vspace{-0.1cm}
\begin{observation}\label{observation:mono}
The optimal reserve rate $C^*$ is non-decreasing in $R_{\rm LTE}$, increasing in ${\eta^{\rm APO}}$, and non-increasing in ${\delta^{\rm LTE}}$. Moreover, ${\delta^{\rm LTE}}$ has a larger impact on $C^*$ than ${\eta^{\rm APO}}$.
\end{observation}
\vspace{-0.5cm}
\subsection{Comparison with {{The Benchmark Scheme}}}
In this section, we compare our auction-based spectrum sharing scheme with a benchmark scheme. Given a set ${\cal K}$ of APOs, the two schemes work as follows:
\begin{itemize}
\item \emph{Our auction-based scheme:} First, the LTE provider determines $C^*$ numerically based on the approach in Section \ref{subsec:stageI:3}. Second, each APO $k\in{\cal K}$ submits its bid based on the equilibrium strategy $b^*\left(r_k,C^*\right)$ in Section \ref{sec:stageII:APO}. Third, the LTE provider determines its working mode, the winning APO, and the allocated rate based on the auction rule in Section \ref{subsec:auctionframe}.
\item \emph{Benchmark scheme:} The LTE provider randomly shares a channel with one of the $K$ APOs.{\footnote{{{The existing studies focused on the LTE/Wi-Fi coexistence \cite{cano2015coexistence,zhangmodeling}, and there are no results studying the cooperation between the two types of networks. Hence, we represent the state-of-the-art solution by the benchmark scheme, where the LTE coexists with Wi-Fi. Since the LTE provider does not know the private information $r_k$, it cannot differentiate the channels. Therefore, in the benchmark scheme, the LTE provider will randomly pick a channel to coexist with the corresponding APO.}}}}
\end{itemize}
For a particular set of APOs, we denote the LTE provider's payoff under our auction-based and the benchmark schemes as $\pi_{\rm a}^{\rm LTE}$ and $\pi_{\rm b}^{\rm LTE}$, respectively.{\footnote{Note that ${\bar \Pi}^{\rm LTE}\left(C^*\right)$ is the expectation of $\pi_{\rm a}^{\rm LTE}$ with respect to the APO types.}} Furthermore, we denote the APOs' total payoff under our auction-based and the benchmark schemes as $\pi_{\rm a}^{\rm APO}$ and $\pi_{\rm b}^{\rm APO}$, respectively. For a given set of APOs, we compute the relative performance gains of our auction-based scheme over the benchmark scheme in terms of the LTE's payoff and the APOs' total payoff as
\begin{align}
\rho^{\rm LTE} \triangleq \frac{\pi_{\rm a}^{\rm LTE}-\pi_{\rm b}^{\rm LTE}}{\pi_{\rm b}^{\rm LTE}}{\rm ~and~} \rho^{\rm APO} \triangleq \frac{\pi_{\rm a}^{\rm APO}-\pi_{\rm b}^{\rm APO}}{\pi_{\rm b}^{\rm APO}}.
\end{align}
\subsubsection{Performance on Average $\rho^{\rm LTE}$ and $\rho^{\rm APO}$} We investigate the average $\rho^{\rm LTE}$ and $\rho^{\rm APO}$. We consider four pairs of data rate discounting factors: $\left({\delta^{\rm LTE}},{\eta^{\rm APO}}\right)=\left(0.4,0.1\right),\left(0.4,0.3\right),\left(0.4,0.7\right)$, and $\left(0.6,0.3\right)$,{\footnote{{In a practical implementation, the values of $\delta^{\rm LTE}$ and $\eta^{\rm APO}$ depend on the applied coexistence mechanism (\emph{e.g.}, LBT or CSAT) and the corresponding settings (\emph{e.g.}, LTE off time in CSAT).}}} and change $R_{\rm LTE}$ from $30{\rm~Mbps}$ to $370{\rm~Mbps}$. The other settings are the same as those in Fig. \ref{fig:simu:D}. Given a pair of $\left({\delta^{\rm LTE}},{\eta^{\rm APO}}\right)$ and a particular value of $R_{\rm LTE}$, we randomly choose $r_k$, $k\in{\cal K}$, based on the truncated normal distribution, implement our auction-based scheme and the benchmark scheme separately, and record the corresponding values of $\rho^{\rm LTE}$ and $\rho^{\rm APO}$. For each pair of $\left({\delta^{\rm LTE}},{\eta^{\rm APO}}\right)$ and each value of $R_{\rm LTE}$, we run the experiment $20,000$ times, and obtain the corresponding average values of $\rho^{\rm LTE}$ and $\rho^{\rm APO}$.
\begin{figure*}[t]
\centering
\begin{minipage}[t]{.32\linewidth}
\includegraphics[scale=0.3]{CQsimuE-eps-converted-to.pdf}
\caption{Comparison on LTE's Payoff.}
\label{fig:simu:E}
\vspace{-4mm}
\end{minipage}
\begin{minipage}[t]{.32\linewidth}
\centering
\includegraphics[scale=0.3]{CQsimuF-eps-converted-to.pdf}
\caption{Comparison on APOs' Payoffs.}
\label{fig:simu:F}
\vspace{-4mm}
\end{minipage}
\begin{minipage}[t]{.32\linewidth}
\centering
\includegraphics[scale=0.3]{QsimuG-eps-converted-to.pdf}
\caption{Comparison on Social Welfare.}
\label{fig:simu:G}
\vspace{-4mm}
\end{minipage}
\end{figure*}
In Fig. \ref{fig:simu:E}, we plot the average $\rho^{\rm LTE}$ against $R_{\rm LTE}$ for different $\left({\delta^{\rm LTE}},{\eta^{\rm APO}}\right)$ pairs. First, we observe that the average $\rho^{\rm LTE}$ increases with $R_{\rm LTE}$. In particular, all the average $\rho^{\rm LTE}$ with ${\delta^{\rm LTE}}=0.4$ are above $70\%$ for $R_{\rm LTE}=370{\rm~Mbps}$ (the maximum LTE throughput according to \cite{canousing}). That is to say, our auction-based scheme's performance gain on the LTE provider's payoff is more significant for a larger $R_{\rm LTE}$. The reason is that a larger $R_{\rm LTE}$ enables the LTE provider to set a larger reserve rate, which increases the probability for the cooperation between the LTE provider and the APOs. Second, when ${\eta^{\rm APO}}=0.3$ and ${\delta^{\rm LTE}}$ increases from $0.4$ to $0.6$, the average $\rho^{\rm LTE}$ decreases significantly. Since a larger ${\delta^{\rm LTE}}$ implies that the coexistence with Wi-Fi reduces the LTE's payoff less significantly, the cooperation with Wi-Fi is less beneficial to the LTE provider, which decreases the average $\rho^{\rm LTE}$. Third, when ${\delta^{\rm LTE}}=0.4$ and ${\eta^{\rm APO}}$ changes from $0.1$ to $0.7$, the change in the average $\rho^{\rm LTE}$ is small. Hence, ${\eta^{\rm APO}}$ has a smaller impact on the average $\rho^{\rm LTE}$ comparing with ${\delta^{\rm LTE}}$. We summarize the observations in Fig. \ref{fig:simu:E} as follows.
\begin{observation}
Compared with the benchmark scheme, our auction-based scheme improves the LTE's payoff by 70\% on average under a large $R_{\rm LTE}$ and a small ${\delta^{\rm LTE}}$. Moreover, the performance gain is not sensitive to ${\eta^{\rm APO}}$.
\end{observation}
In Fig. \ref{fig:simu:F}, we plot the average $\rho^{\rm APO}$ against $R_{\rm LTE}$ for different $\left({\delta^{\rm LTE}},{\eta^{\rm APO}}\right)$ pairs. First, we observe that the average $\rho^{\rm APO}$ increases with $R_{\rm LTE}$. Similar as the explanation for $\rho^{\rm LTE}$, this is because a larger $R_{\rm LTE}$ leads to a larger reserve rate, and creates more cooperation opportunities between the LTE provider and the APOs. Second, the average $\rho^{\rm APO}$ is large when both ${\delta^{\rm LTE}}$ and ${\eta^{\rm APO}}$ are small. In this case, there is a heavy interference between the LTE and the APOs in the \emph{competition mode}, and the both of them want to avoid the interference through the cooperation. Therefore, our auction-based scheme is much more efficient, and achieves a large $\rho^{\rm APO}$. We summarize the observations in Fig. \ref{fig:simu:F} as follows.
\begin{observation}
Compared with the benchmark scheme, our auction-based scheme is most beneficial to the APOs for a large $R_{\rm LTE}$ and small ${\delta^{\rm LTE}}$ and ${\eta^{\rm APO}}$.
\end{observation}
\vspace{-0.1cm}
\subsubsection{Performance on Social Welfare} We consider $\left({\delta^{\rm LTE}},{\eta^{\rm APO}}\right)=\left(0.4,0.3\right)$, and choose the same settings as Fig. \ref{fig:simu:E} and Fig. \ref{fig:simu:F} for the other parameters. In Fig. \ref{fig:simu:G}, we plot the average social welfares of the two schemes, and also show the average value of the maximum social welfare. To compute the maximum social welfare for a particular set of APOs, we assume that there is a \emph{centralized decision maker}, who allocates $K$ channels to the LTE provider and the $K$ APOs in a manner that maximizes the social welfare.{\footnote{Specifically, the \emph{centralized decision maker} can choose to: (i) keep the LTE idle, and allocate all channels to the APOs, (ii) keep one APO idle, and allocate all channels to the LTE and the remaining $K-1$ APOs, or (iii) let the LTE share one channel with one APO, and allocate the remaining channels to the remaining $K-1$ APOs.}} For each experiment, we randomly pick a set of APOs and record the social welfare achieved by the \emph{centralized decision maker}. We run the experiment $20,000$ times, and obtain the average value of the maximum social welfare.
When $R_{\rm LTE}$ increases, the social welfare gain of our auction-based scheme over the benchmark scheme increases, and the average social welfare under our auction-based scheme approaches the maximum social welfare.
This is because when $R_{\rm LTE}$ is large, it is always good for the LTE to exclusively occupy a channel to maximize the social welfare.
For our auction-based scheme, the increase of $R_{\rm LTE}$ improves the cooperation chance between the LTE and the APOs, and hence increases the probability for the LTE to exclusively occupy a channel.
The result in Fig. \ref{fig:simu:G} shows that in our auction-based scheme, even the LTE provider and APOs make decisions to maximize their own payoffs, and the LTE provider and each APO do not have the complete information on the other APOs' types, the eventual auction outcome leads to a close-to-optimal social welfare for a large $R_{\rm LTE}$. We summarize the observation in Fig. \ref{fig:simu:G} as follows.
\begin{observation}
Our auction-based scheme leads to a close-to-optimal social welfare when $R_{\rm LTE}$ is large.
\end{observation}
\vspace{-0.5cm}
{{
\section{Practical Implementation and Model Extension}\label{sec:discussion}
In this section, we first discuss the practical implementation of our auction framework. In particular, we explain the approach for the LTE provider and APOs to exchange information (\emph{e.g.}, reserve rate and bids).
Then we discuss some extensions of our model. Specifically, in Section \ref{subsec:discussion:APOshare}, we extend our model to the scenario where different APOs can share the same channel. In Section \ref{subsec:discussion:dumb}, we consider a scenario where some APOs' traffic cannot be onloaded to the LTE network. In Section \ref{subsec:discussion:multiLTE}, we extend our model to the scenario where there are multiple LTE providers.
In a practical implementation, a centralized broker (\emph{e.g.}, a private company or a company designated by the government) can coordinate the interactions between the LTE provider and APOs \cite{iosifidis2015iterative}.
Next we briefly introduce the centralized broker with an example from the TV white space networks, which are in the process of commercial trials in the US and UK. In the TV white space networks, a white space database operator (\emph{e.g.}, Google, Microsoft, and SpectrumBridge) serves as the broker to record and update the TV spectrum usage (by TV stations) as well as the secondary access (by non-TV devices) in the same area.
{{Moreover, the broker controls the spectrum allocated to different secondary service providers to avoid the interference between the secondary service providers' networks.}} This shows that it is possible to coordinate the spectrum sharing of different networks through a broker, even if these networks belong to different operators {{and have overlapping coverages}}. In our auction framework, the LTE provider can announce the reserve rate to the broker at the beginning of each time slot. The APOs that are interested in participating in the auction can communicate with the broker to obtain the reserve rate information and submit their bids to the broker.{\footnote{{In particular, when the APOs have multiple equilibrium strategies $b^*$ under the reserve rate (\emph{i.e.}, $M>1$ or $L>1$), the broker can coordinate the APOs' selection of the equilibrium strategy. Intuitively, the broker will suggest the equilibrium strategy that maximizes the social welfare to the APOs. We are interested in studying the details of this problem in our future work.}}} Then the broker determines the winning APO based on our auction rule, and broadcasts this result to the LTE provider and APOs. With the broker's help, the LTE provider does not need to directly communicate with all surrounding APOs.
\subsection{Extension: Channel Sharing Among APOs}\label{subsec:discussion:APOshare}
In this section, we discuss the extension of our framework to the scenario where different APOs can share the same channel. In this scenario, the LTE provider still determines at most one winning APO in each auction. The major challenge is that when there are other APOs in the winning APO's channel, the LTE provider has to coexist with these remaining APOs (based on the coexistence mechanisms like LBT and CSAT) after onloading the winning APO's traffic.
Therefore, we need to (i) extend the modeling of the LTE provider's payoff, the APOs' payoffs, and the APOs' types, and (ii) modify the auction rule. In the following, we briefly explain these two aspects.
For the modeling, we should first model the impact of the number of APOs in the same channel on the LTE provider's and the APOs' payoffs. Intuitively, the reductions in the LTE provider's and the APOs' payoffs are more severe when there are more APOs using the same channel. Second, we should model the multi-dimensional APO type. In Section \ref{sec:model}, we define the APO type as an APO's throughput without interference (\emph{i.e.}, $r_k$). Here, an APO's type should also include the information of the number of APOs in the same channel. In the equilibrium analysis, we can characterize the APOs' equilibrium strategies by a function that maps an APO type (\emph{i.e.}, throughput and number of APOs in the same channel) to a bid.
For the auction rule, the major modification is the rule of determining the winning APO. In Section \ref{sec:model}, the winning APO is always the APO with the lowest bid. However, when different APOs can share the same channel, such a rule is no longer optimal for the LTE provider. This is because the APO with the lowest bid may have many other APOs using the same channel, and hence the benefit for the LTE provider to cooperate with this APO may be small. Therefore, the LTE provider has to consider both the APOs' bids and the number of APOs in each channel to determine the winning APO.
\subsection{Extension: Complex APOs}\label{subsec:discussion:dumb}
In reality, some users' mobile devices, such as the laptops, do not have the LTE interfaces. The existence of these mobile devices prevents the corresponding APOs from participating in the auction and onloading all of their traffic to the LTE network. For ease of exposition, we use the \emph{simple} APOs to represent the APOs who can onload all of their traffic to the LTE network, and use the \emph{complex} APOs to represent the APOs who cannot onload all of their traffic to the LTE network. The \emph{complex} APOs will not participate in the auction and will simply use their original channels. In the following, we explain the impact of the consideration of \emph{complex} APOs on our analysis.
First, if all APOs occupy different channels (the assumption in Section \ref{sec:model}), our current analysis can be directly extended to the case where there are \emph{complex} APOs. Notice that even though the LTE provider can only cooperate with the \emph{simple} APOs in the cooperation mode, it can compete with both the \emph{simple} and \emph{complex} APOs in the competition mode. Therefore, the major change is that when the LTE provider works in the competition mode, the expected payoff of an APO depends on the number of all APOs (\emph{simple} and \emph{complex} APOs), instead of the number of APOs participating in the auction (\emph{simple} APOs).
Second, if different APOs can share the same channel (the scenario in Section \ref{subsec:discussion:APOshare}), it will be much more challenging to consider the \emph{complex} APOs in the analysis. This is because the \emph{complex} APOs may coexist with the \emph{simple} APOs in the same channel. In this situation, we need to characterize a \emph{simple} APO's equilibrium strategy based on the number of \emph{complex} APOs as well as the number of \emph{simple} APOs in the APO's channel.
\subsection{Extension: Multiple LTE Providers}\label{subsec:discussion:multiLTE}
In this section, we discuss the extension of our framework to the scenario where there are multiple LTE providers. According to \cite{forum}, the LTE networks of different providers can well coexist with each other in the same unlicensed channel. Hence, when there are multiple LTE providers, the focus of our auction framework is still onloading the Wi-Fi APOs' traffic to the LTE networks.
\begin{figure}[t]
\centering
\includegraphics[scale=0.37]{MultiLTE-eps-converted-to.pdf}\\
\caption{An Example of Two LTE Providers.}
\label{fig:majorrevision}
\vspace{-0.7cm}
\end{figure}
When there are multiple LTE providers, they can take turns to organize the auctions, which can be managed by the centralized broker. Suppose that there are two LTE small cell networks in the same area, and they are owned by LTE provider A and LTE provider B, respectively. We illustrate a protocol in Figure \ref{fig:majorrevision}.
During the odd number $\left(2k+1\right)$-th ($k\in\left\{0,1,\ldots\right\}$) time slot, LTE provider B directly operates in the competition mode, and LTE provider A can send a request to the broker and organize an auction. During the even number $\left(2k+2\right)$-th time slot, the two LTE providers switch their roles: LTE provider A operates in the competition mode, and LTE provider B can organize an auction. We can apply similar protocols to the situations with more than two LTE providers.
Since the protocol designs for the $\left(2k+1\right)$-th time slot and the $\left(2k+2\right)$-th time slot are symmetric, next we only introduce the protocol design for the $\left(2k+1\right)$-th time slot. At the beginning of the $\left(2k+1\right)$-th time slot, LTE provider B chooses the competition mode, \emph{i.e.}, it selects a channel and coexists with the corresponding APOs. Then LTE provider A organizes an auction: when no APO wants to cooperate with LTE provider A, LTE provider A works in the competition mode, selects a channel, and coexists with the corresponding networks; otherwise, LTE provider A works in the cooperation mode, onloads the winning APO's traffic, and accesses the corresponding channel. At the end of the $\left(2k+1\right)$-th time slot, both LTE provider A and LTE provider B release the channels they use. In particular, LTE provider A also needs to release the onloaded Wi-Fi users if LTE provider A works in the cooperation mode during the $\left(2k+1\right)$-th time slot.
Next we discuss the challenges of analyzing the scenario with multiple LTE providers under the protocol we introduced above. Briefly speaking, when a particular LTE provider organizes an auction, it needs to consider the number of other LTE providers in each channel. This is because the benefit for the LTE provider to cooperate with an APO decreases with the number of other LTE providers using the same channel. In the analysis, we should characterize an APO's equilibrium strategy based on its throughput $r_k$ and the number of LTE providers in the same channel. Furthermore, the auctioneer should consider both the APOs' bids and the number of LTE providers in each channel to determine the winning APO.
We provide a complete analysis of the scenario where there are multiple LTE providers in Section \ref{sec:supplementary} (supplementary materials).
\section{Conclusion}\label{sec:conclusion}
In this paper, we proposed a framework for LTE's coopetition with Wi-Fi in the unlicensed spectrum. We designed a reverse auction for the LTE provider to exclusively obtain the channel from the APOs by onloading their traffic.
{{Compared with the existing LTE/Wi-Fi coexistence mechanisms like LBT and CSAT, our auction can potentially avoid the interference between the LTE and APOs.}}
The analysis of the auction is quite challenging as the designed auction involves positive allocative externalities.
We characterized {{the unique form of the APOs' bidding strategies at the equilibrium}}, and analyzed the optimal reserve rate of the LTE provider.
Numerical results showed that our framework benefits both the LTE provider and the APOs, and it achieves a close-to-optimal social welfare under a large LTE throughput.
In our framework, the LTE provider announces the reserve rate and each APO then submits a bid at the beginning of each time slot, where the length of each time slot corresponds to several minutes.
Compared with the existing LTE/Wi-Fi coexistence mechanisms, our auction framework leads to more signaling overhead. However, if the LTE provider and an APO agree to cooperate, there is no more need for the LTE to frequently sense the channel activities, which removes the related operational overhead during the rest of the time slot.{\footnote{{{For example, in the LBT mechanism, the LTE senses the channel status (busy or idle) every $20$ microseconds; in the CSAT mechanism, the LTE senses the Wi-Fi activity on a time scale of $100$ milliseconds to determine the length of LTE off time {\cite{Qualcomm}}.}}}} Therefore, although our framework generates more signaling overhead initially, it can potentially significantly save the sensing cost (\emph{e.g.}, power) and improve the payoffs of both LTE and Wi-Fi.
{{An interesting observation of our framework}} is that sometimes even if the cooperation mutually benefits the LTE provider and the APOs, these two types of networks do not reach an agreement on the cooperation. The reason is that our framework considers an incomplete information setting. For example, the LTE provider determines the reserve rate to maximize its expected payoff by considering the distribution of ${\bm r}$ (the vector of APOs' types) instead of the actual value of ${\bm r}$. For some ${\bm r}$, such a reserve rate may not be optimal to the LTE provider and can make the LTE provider lose some cooperation chances that mutually benefit both types of networks (we provide an example in Appendix \ref{appendix:sec:example}). Similarly, the incomplete information among the APOs can also lead to the same inefficiency problem. In our future work, we will consider other mechanisms (\emph{e.g.}, bargaining) for the LTE/Wi-Fi coopetition to reduce such an inefficiency.
\bibliographystyle{IEEEtran}
|
1,314,259,996,557 | arxiv | \section{\bf{Introduction}}
A basic feature of quantum cosmology is that the universe starts with a quantum
character being dominated by quantum uncertainty and eventually it then becomes
large and completely classical. In quantum cosmology the universe is described by
a wavefunction $\Psi $ which satisfies the equation
\begin{equation}
\hat{H}\Psi=0
\end{equation}
where $\hat{H}$ is the Hamiltonian operator. The equation (1) is known as the
Wheeler-DeWitt (WD) equation. Equation (1) when compared with the Schroedinger
equation in quantum mechanics
\begin{equation}
i\hbar{\partial \over \partial t}= \hat{H}\Psi ,
\end{equation}
reveals that there is no "time" in quantum gravity and this is commonly referred
to as `the problem of time' in quantum gravity. It is now accepted as a broad
consensus [1,2,3,4] that the time in quantum gravity has an intrinsic character.
Recent trends suggest that one achieves an equation like (2) through a prescription
of time. The problem with equation (1) is not to find solutions but to find
a proper boundary condition that will not disturb the basic aspect of inflationary
cosmology. At present we have three boundary condition proposals. These are : (i).
Hartle-Hawking no boundary proposal [5] (ii). quantum tunneling proposal [6] and less
commonly known (iii). wormhole dominance proposal [7]. The third boundary condition
is more general in the sense that the proposals (i) and (ii) follow from (iii)
when the respective boundary conditions are introduced in it. The first two proposals produce
wavefunctions that are not normalized and have to rely on the concept of
conditional probability [8] for an interpretation of the wavefunction. The
proposal (iii) obtains wavefunction which is normalisable and the probabilistic
interpretation remains quite sensible, and workable as in ordinary quantum mechanics.
The problem with (2) is also to choose suitable initial conditions and to obtain
a reasonable connection with the three boundary conditions. In most works an
equation like (2) is derived from (1) and the equation is called
Schroedinger-Wheeler-DeWitt (SWD) equation [1,2,3,4].
\par
As mentioned the major problem in quantum gravity is not to find solutions of the WD and SWD
equations but to obtain suitable initial conditions such that the inflationary
scenario for the early universe and fruits emerging out of it are not changed. It
is now accepted that the inflation provides a natural mechanism for structure
formation and its origin is traced back to the quantum fluctuations in early
universe. These quantum fluctuations are related to a scalar field $\phi$ in
phase transition model and to the geometry itself in Starobinsky's spontaneous
transition model [9]. The idea of quantum universe necessitates an interpretation of the
wavefunction. For the orthodox ``Copenhagen interpretation'' one requires an
external classical observer but for a universe there is no observer external to
it. The success of classical Einstein equation along with the classical spacetime is
a reality, so we need along with an interpretation of the wavefunction, also a
mechanism from microscopic to macroscopic reduction. More specifically, we need
a mechanism from quantum to classical transitions. There have been many
discussions for a unified dynamics for microscopic and macroscopic systems [10].
Now it is known that classical properties emerges through the interactions of
the variables describing a quantum system such that the configuration variables are
divided in some way into macroscopic variables $M$ and microscopic variables $Q$
and quantum interference effects are suppressed by averaging out the microscopic
variations not distinguished by the associated observable. This process is known
as decoherence [11,12].
\par
In the context of quantum gravity, the Hamiltonian constraint leads to the
timeless WD equation and recovery of semiclassical time is carried out using two
main approaches. In one approach [13,14] a variable $t$ (depending on the
original position and momenta) whose conjugate momenta occurs linearly in the
Hamiltonian $H$ is brought to a form
\begin{equation}
H=H_r+p_t\,,
\end{equation}
through a canonical transformation. The quantization
$p_t\rightarrow -i\hbar {\partial \over {\partial t}}$
then brings (3) to the form (2) and obtains SWD equation from the Hamiltonian
constraint $H=0$. This approach succeeds in some cases like cylindrical
gravitational waves or eternal black holes but its general viability is far from
clear though the standard Hilbert space of quantum theory can be employed in such
an approach.
\par
The other approach starts from the WD equation (1) and treats all variables in
the same footing and tries to identify a sensible concept of time after
quantization. In this approach (i) the choice for an appropriate Hilbert space
structure is obscure, (ii) normalization of the wavefunction and probabilistic
interpretation remain awkward in absence of time and (iii) whether the
prescription of time parameter is an artifact and is related to Minskowskian time
are not clear. Though the concept of `conditional probabilities' is enforced for
an interpretation of the wavefunction, the driving quantum force guaranting the
validity of superposition principle in early universe and subsequently
enforcing decoherence remain unclear in the picture.
\par
The motivation of the present work is to investigate the role of time in quantum
gravity especially to understand the initial conditions of both the WD and SWD
equations and to obtain an inter-relation between them. In our approach we do
not enforce any canonical transformation to obtain an equation like (3) and do
not consider the Wheeler-DeWitt equation to obtain the SWD equation. If we look
at classical Einstein equation
\begin{equation}
G_{\mu\nu}\equiv R_{\mu\nu}- {1\over 2}g_{\mu\nu}R=kT_{\mu\nu}\,,
\end{equation}
we observe that `geometry and matter' get coupled through (4). It is also a
well known fact that the matter field is quantized and for that reason in equation
(4) one writes $<T_{\mu\nu}>$ on the right hand side of (4) and treats
$g_{\mu\nu}$ as classical background. Keeping this spirit of (4) in mind we introduce
Minskowski time $t$ through Schroedinger equation
\begin{equation}
i\hbar{{\partial \psi}\over {\partial t}}={\hat{H}}_m\psi\,,
\end{equation}
where ${\hat{H}}_m$ is now the matter field Hamiltonian. This $t$ now serves as an
external label. Without having the gravitational field quantized, we formulate a
time parameter $\sigma (x)$ such that (5) becomes equivalent to Einstein
equation with $\sigma(x)=t=const.$ acting as a global parameter. The geometry
itself acts as a generator of time and manifest only through matter field. We
discuss this recovery of semiclassical time in section II. In section III we
discuss the initial conditions for solution of SWD equation and its connection
to the three boundary condition proposals for the timeless Wheeler-DeWitt
equation mentioned earlier in the introduction.
Section IV contains a discussion of `quantum force' originated in the geometry
through wormhole picture.
\smallskip
\section{\bf{Semiclassical Time in Quantum Gravity}}
We consider a gravitational action with a minimally coupled scalar field $\phi$
in a Friedman-Robertson-Walker (FRW) background
\begin{equation}
I=M\int{dt\left[-{1\over 2}a{\dot{a}}^2+{{ka}\over 2}+{1\over M}
\{{1\over 2}{\dot{\phi}}^2-V(\phi)\}\;a^3\,\right]}\,,
\end{equation}
where $M={{3\pi}\over {2G}}={{3\pi m_{p}^{2}}\over 2},\, m_p$ being the Planck mass
and $k=0,\pm 1$ for flat, closed and open models respectively. The
$\left(\begin{array}{c}0\\0\end{array}\right)$ component of Einstein equation is
\begin{equation}
-{M\over 2}({{{\dot{a}}^2}\over a^2}+{k\over a^2})+{1\over 2}
{\dot{\phi}}^2+V(\phi)=0\,.
\end{equation}
Identifying
\begin{equation}
P_a=-Ma\dot{a}\,,\, P_{\phi}=a^3\dot{\phi}\,,
\end{equation}
(7) gives the Hamiltonian constraint
\begin{equation}
-{1\over {2M}}({P_{a}^{2}\over a})+{P_{\phi}^{2}\over {2a^3}}
-{M\over 2}ka+a^3V(\phi)=0\,.
\end{equation}
The dynamical equations are
\begin{equation}
-{{\ddot{a}}\over a}={{\dot{a}}^2\over {2a^2}}+{k\over {2a^2}}+ {3\over M}
\{{1\over 2}{\dot{\phi}}^2-V(\phi)\}\,,
\end{equation}
\begin{equation}
-\ddot{\phi}=3{\dot{a}\over a}\dot{\phi}+{{\partial V}\over {\partial \phi}}\,.
\end{equation}
The matter field Hamiltonian $H_m$ for the scalar field is
\begin{equation}
H_m={1\over {2a^3}}P_{\phi}^{2}+a^3V(\phi)
\end{equation}
as if $a^3({1\over 2}{\dot{\phi}}^2+V(\phi))=E$ is the energy of the scalar
field. We now define an action $S(a,\phi)$ such that
\begin{equation}
H_m=-{{\partial S}\over {\partial t}}
\end{equation}
and (12) reduces to
\begin{equation}
-{{\partial S}\over {\partial t}}={1\over {2a^3}}P_{\phi}^{2}+a^3V(\phi)\,.
\end{equation}
This $t$ is obviously a Newtonian time and acts as an external label.
Seemingly it appears that (14) has no connection with the gravitational field.
Using the Hamiltonian constraint (9), we write (14) as
\begin{equation}
-{{\partial S}\over {\partial t}}=+{1\over {2M}}{P_{a}^{2}\over a}
+{k\over 2}Ma\,.
\end{equation}
If we quantize in standard way with
$P_i=-i\hbar {\partial\over {\partial q_i}}$,
(14) and (15) are added we get the Wheeler-DeWitt equation and the time disappears from
the equation and this is the problem of time in quantum gravity. Our view is that
quantization is permitted in (14) but not in (15) as if (15) represent the
classical Einstein equation whereas (14) with
$p_t={{\partial S}\over {\partial t}}=-i\hbar {\partial\over {\partial t}}$
and $P_\phi=-i\hbar {\partial\over {\partial \phi}}$ acting as quantum equation.
It seems as if (14) and (15) have no dynamical content.
\par
We define therefore a time evolution parameter $\sigma$
\begin{equation}
{\partial\over {\partial \sigma}}=
{{\partial H}\over {\partial P_a}} {\partial\over {\partial a}}
+{{\partial H}\over {\partial P_\phi}} {\partial\over {\partial \phi}}
-{{\partial H}\over {\partial a}} {\partial\over {\partial P_a}}
-{{\partial H}\over {\partial \phi}} {\partial\over {\partial P_\phi}}\,.
\end{equation}
Using (9) and (16) one finds
\begin{eqnarray}
{\partial\over {\partial \sigma}}&=&-{1\over {Ma}}
({{\partial S}\over {\partial a}}){\partial\over {\partial a}}
+{1\over a^3}({{\partial S}\over {\partial \phi}}){\partial\over {\partial \phi}}\nonumber \\
&+&\left[{{kM}\over 2} -
{{({{\partial S}\over {\partial a}})^2}\over {2Ma^2}}
+3{{({{\partial S}\over {\partial \phi}})^2}\over {2a^4}}-3a^2V(\phi)\right]
{\partial\over {\partial P_a}}\nonumber \\
&-&a^3({{\partial V}\over {\partial \phi}}){\partial \over {\partial P_\phi}}\,.
\end{eqnarray}
In view of (14) and (15), we demand that $\sigma$ depends only upon geometry
(i.e., on $a$). This necessitates the co-efficients of
${\partial\over {\partial P_a}}$ and ${\partial\over {\partial P_\phi}}$ to
become zero in (17). This gives $V(\phi)=0$, and $S$ is a function of $a$
only with
\begin{equation}
{{({{\partial S_o}\over {\partial a}})^2}\over {2Ma^2}}-{kM\over 2}=0\,.
\end{equation}
The second term vanishes identically. We henceforth denote $S_o=S(a)$. Thus we have
\begin{equation}
{\partial\over {\partial \sigma}}=-{1\over {Ma}}
{{\partial S_o}\over {\partial a}}{\partial\over {\partial a}}\,.
\end{equation}
Let us suppose that $S$ defined in (14), (15) and (17) is related to $S_o(a)$
by the relation
\begin{equation}
S(a,\phi)=S_o(a)+S_1(\phi)
\end{equation}
with $S_1(\phi)<<S_o(a)$.
The reason for such an assumption will be clear as we
proceed through the text. From (19) we write
\begin{equation}
{{\partial S}\over {\partial \sigma}}=-{1\over {Ma}}
{{\partial S_o}\over {\partial a}}{{\partial S}\over {\partial a}}\,,
\end{equation}
and using (20) and (21), we find
\begin{equation}
{{\partial S}\over {\partial \sigma}}=-
{1\over {Ma}}
({{\partial S}\over {\partial a}})^2
+{1\over {Ma}}
{{\partial S_1}\over {\partial a}}
{{\partial S}\over {\partial a}}\,.
\end{equation}
Because of $S_1(\phi)<<S_o(a)$, we write the second term in (22), using (18) as
\begin{eqnarray}
{1\over {Ma}}
{{\partial S_1}\over {\partial a}}
{{\partial S_o}\over {\partial a}}&=&
{1\over {Ma}}
({{\partial S_o}\over {\partial a}})
({{\partial S}\over {\partial a}})-
{1\over {Ma}}
({{\partial S_o}\over {\partial a}})^2\,,\nonumber \\
&=&-{{\partial S}\over {\partial \sigma}}-
{1\over {Ma}}kM^2a^2\,,
\end{eqnarray}
so that (22) reduces to
\begin{equation}
{{\partial S}\over {\partial \sigma}}=-
{1\over {2Ma}}P_{a}^{2}-{{kMa}\over 2}\,.
\end{equation}
In arriving at (24) we have neglected
${1\over {Ma}}({{\partial S_1}\over {\partial a}})^2$
term in (23) and is quite obvious in large $a$ region. Comparing (24) with (15) we
find that
\begin{equation}
{{\partial S}\over {\partial \sigma}}=
{{\partial S}\over {\partial t}}\,.
\end{equation}
The prescription (19) and the condition (25) implants the geometry in the
quantum structure provided $\sigma=t$ is identified as time. Thus we have
avoided the quantization of gravitational field. This does not imply that the
gravitational field is not quantized. It manifests its quantum character only
through the matter field which is quantized. If we quantize the gravitational
field through the replacement
$P_a=-i\hbar{\partial\over {\partial a}}$,
time will disappear and this is why we get timeless character of the
Wheeler-DeWitt equation. Now upon quantization with
$P_i=i\hbar{\partial\over {\partial q_i}}$ with $q_i=t,\phi$, we get from (14)
\begin{equation}
i\hbar{{\partial \psi}\over {\partial t}}=\left[-{{\hbar^2}\over {2a^3}}
\partial_{\phi}^{2}+a^3V(\phi)\right]\,\psi \,.
\end{equation}
This is the desired Schroedinger-Wheeler-DeWitt equation of quantum gravity and
works with the standard Hilbert space structure. The derivative ${\partial\over
{\partial t}}$ is a directional derivative along each of the classical
spacetime and is viewed as classical `trajectories' in gravitational configuration space.
In conformity with classical Einstein equation, (26) describes quantized matter
field in a classical background of spacetime. In other works one starts from the
WD equation with the ansatz
\begin{equation}
\Psi({\it{G}},\phi)\simeq C[{\it{G}}]\,e^{{iS_o[{\it{G}}]}\over \hbar}
\psi [{\it{G}},\phi]\,,
\end{equation}
where ${\it{G}}$ denotes the gravitational field on a three dimensional space,
$\phi$ stands for non-gravitational field and $C$ is a slowly varying prefactor.
In obtaining (26) from the Wheeler-DeWitt equation, a WKB form has to be assumed
for $\Psi$ in (1) and
expand $S(a,\phi)$ as
\begin{equation}
S=MS_o+M^{-1}S_2+\ldots\,.
\end{equation}
Substituting all these in (1) and equating co-efficient of different orders of
$M$ to zero, one finds for $M^2$ order ${{\partial S_o}\over {\partial \phi}}=0,
\,M^1$ order gives source free Hamilton-Jacobi equation
\[ ({a^2\over 2})({{\partial S_o}\over {\partial a}})^2+{{ka^4}\over 2}=0\]
and the prefactor $C({\it{G}})$ is determined through a condition such that
\begin{equation}
f(a,\phi)=C(a)\exp{({{iS_1}\over \hbar})}
\end{equation}
satisfies (26). As the fundamental equation (1) is linear, it allows arbitrary
superposition of states like (27) and would thus forbid the derivation of (26).
Our approach remains free from all these defects. Our derivation itself suggests, (26)
is valid in semiclassical region because taking
\begin{equation}
\psi=\exp{\left[{i\over \hbar}S(a,\phi)\right]}\,,
\end{equation}
in (26), we find
\begin{equation}
-{{\partial S}\over {\partial t}}={1\over {2a^3}}P_{\phi}^{2}+a^3V(\phi)
-{{i\hbar}\over {2a^3}}{{\partial^2 S}\over {\partial \phi^2}}\,,
\end{equation}
which in the limit $\hbar\rightarrow 0$ gives back the classical equation (15).
Thus we conclude that the WD equation remains valid in high curvature region
(small $a$) and the SWD equation is effective in small curvature region
(large $a$). The emergence of (31) also implies that somehow the superposition has
been wiped out during the evolution and this mechanism is known as decoherence.
It is argued that the seed of this decoherence i.e., the non-occurrence of
interference terms lies in the early universe. More precisely, initial conditions
at early time (i.e., near the onset of inflation) somehow regulates
the behavioural pattern of the wavefunction necessitating decoherence. We now
discuss this initial condition for the SWD equation (26). The present trend
of investigation concentrates on the quantum to classical transition of the
universe especially in the light of decoherence mechanism. We discuss it in
the next section.
\smallskip
\section{\bf{Initial Conditions}}
The Wheeler-DeWitt equation is a quantum equation and the SWD equation is a
semiclassical equation. It is therefore necessary to prescribe an initial
condition for (26) so that decoherence can be effective in the framework
and investigate the nature of initial condition for WD equation that results
from such a choice, provided the inflation is guaranted in the description.
The usual assumption [4] is that at an early time, the modes are in their
adiabatic ground state and the initial adiabatic ground state is a Gaussian
state for the wavefunction since the Gaussian ansatz preserves the Gaussian
form during time evolution. We will now show that this initial condition is not
sufficient and requires a condition for the normalization of the wavefunctions.
For this purpose we start with equation (26) with a form of the potential
\begin{equation}
V(\phi)={\lambda\over 2}(1+m^2\phi^2)\,,
\end{equation}
$\lambda$ and $m^2$ being constants, and $m^2$ can also be chosen as negative.
We take for $\psi$, the Gaussian ansatz
\begin{equation}
\psi=N(t)\exp{\left[-{1\over 2}\Omega (t)\right]}\phi^2
\end{equation}
for the ground state of the wavefunction. Inserting (33) in (26) one finds the
set of equations
\begin{equation}
i{d\over {dt}}\ln N={\Omega\over a^3}+\lambda a^3\,,
\end{equation}
\begin{equation}
-i\dot{\Omega}={{\omega^2-\Omega^2}\over a^3}\,,
\end{equation}
with
\begin{equation}
\omega^2=m^2\lambda \,.
\end{equation}
It is worthwhile to point out that in (34) and (35)
\begin{equation}
\dot{\Omega}={{\partial \Omega}\over {\partial t}}\,,
\end{equation}
\begin{equation}
{1\over 2}{d\over {dt}}\ln N={\partial\over {\partial t}}\ln N
\end{equation}
because of (25) since we would evaluate $N$ considering multiple reflections at
the turning points arising out of the WD equation such that $N=N(t,\sigma)$.
Identification of `many fingered' time $\sigma$ as a global parameter leads to the
choice (38). We now introduce conformal time $\eta$ through the relation
$dt=a\,d\eta$ to reduce (37) to the form
\begin{equation}
y''+2{a'\over a}y'+m^2\lambda a^2y=0\,.
\end{equation}
In obtaining (39) we have taken
\begin{equation}
\Omega=-ia^2{y'\over y}\,,
\end{equation}
in which $y'={{\partial y}\over {\partial \eta}}$. As we require an exponential
expansion, we solve (39) with $a(\eta)=-{1\over {\sqrt{\lambda}\,\eta}}$. In this
case (39) reads
\begin{equation}
y(\eta)=\eta^{{3\over 2}\pm\sqrt{{9\over 4}-m^2}}\,.
\end{equation}
In the limit $m^2<{9\over 4}$ (which is usually assumed to be satisfied in
inflationary model), where the exponent in (41) can be approximated (taking
negative sign) as ${1\over {3m^2}}$ so that
\begin{equation}
\Omega\approx {{im^2a^3\sqrt{\lambda}}\over 3}\,.
\end{equation}
As $\Omega$ is imaginary, the state (33) will not be normalizable. Usually,
higher order modes of the scalar field are considered to obtain a real part in
$\Omega$, but here we take a different procedure. To be consistent with
standard notation we take $m^2\lambda=m_{o}^{2}$, the mass of the scalar field,
and $\lambda=H^2$ so that (42) reduces to
\begin{equation}
\Omega={{im_{o}^{2}a^3}\over {3H}}\,,
\end{equation}
where $H$ is now the Hubble constant. The imaginary $\Omega$ that contains now
the mass describes back reaction. Substituting (43) in (34) and taking
\begin{equation}
{d\over {dt}}=\sqrt{\lambda}a{d\over {da}}\equiv Ha{d\over {da}}\,,
\end{equation}
we find from (34)
\begin{equation}
N=N_o a^{m_{o}^{2}\over {3H}}\exp \left[{-{ia^3H}\over 3}\right]\,,
\end{equation}
so that
\begin{equation}
\psi=N_o a^{m_{o}^{2}\over {3H}}\exp \left[{-{ia^3H}\over 3}
(1+{1\over 2}m^2\phi^2)\right]\,.
\end{equation}
Since $m^2\phi^2<<1$, we write (46) absorbing a factor 2 in $V(\phi)$ to make
comparison with the standard result [6,7] as
\begin{equation}
\psi\simeq N_o a^{m_{o}^{2}\over {3H}}\exp \left[{-i\over {3V}}
(a^2V-1)^{3\over 2}\right]\,,
\end{equation}
where we have taken $a^2V>>1$ as expected and
$\sqrt{V(\phi)}=H(1+{1\over 2}m^2\phi^2)$. From the WD equation, we know that
$a^2V>>1$ and $a^2V<<1$ regions are respectively termed as classically allowed
and classically forbidden region. The points $a=0$ and $a={1\over \sqrt{V}}$ are
the turning points. According to wormhole dominance proposal [7], the normalization
constant $N_o$ is given by multiple reflections such that
\begin{equation}
N_o={{\exp{\left[S(a_o,0)\right]}}\over {1-\exp{\left[2S(a_o,0)\right]}}}\,,
\end{equation}
where
\begin{equation}
S(a_2,a_1)=\left[{{-i(a^2V-1)^{3\over 2}}\over {3V}}\right]_{a_1}^{a_2}\,.
\end{equation}
Evaluating (48) we find
\begin{equation}
N_o={{\exp{\left[{1\over {3V}}\right]}}\over {(1-\exp{\left[{2\over {3V}}\right]})}}\,.
\end{equation}
The wavefunction (47) now reads
\begin{equation}
\psi=C_1a^{m_{o}^{2}\over {3H}}
\exp\left[{1\over {3V}}(1-i(a^2V-1)^{3\over 2})\right]\,,
\end{equation}
where $C_1$ now refers to $(denominator)^{-1}$ in (50). Continuing in classical
forbidden region, we get from (51)
\begin{equation}
\psi(a^2V<1)=C_1a^{m_{o}^{2}\over {3H}}
\exp\left[{1\over {3V}}(1-(1-a^2V)^{3\over 2})\right]\,.
\end{equation}
We see from (52) that as $a\rightarrow 0$, the wavefunction behaves as
\begin{equation}
\psi\sim\exp({a^2\over 2})\,.
\end{equation}
Thus we find that Eqns.(51)-(53) all represent Hartle-Hawking wavefunction. Thus
we find that Gaussian ansatz at $a^2V>>1$ correctly reproduce the wavefunction
corresponding to the wormhole-dominance proposal, at least the Hartle-Hawking
proposal. Not only the Gaussian ansatz correctly describes the inflation, its
seed lie in the wormhole-dominance proposal.
\smallskip
\section{\bf{Discussion}}
With the inception of quantum cosmology and description through the
Wheeler-DeWitt equation attempts have been made to interpret the wavefunction of
the universe in terms of classical dynamics (i.e., Einstein equation) and
probabilistic concept. We have been able to show that a Gaussian ansatz for SWD
wavefunction correctly simulates the Hartle-Hawking wavefunction and is
normalized according to our prescription. The prescription obtains the
normalization constant through the contribution of wormholes
to the wavefunction and it contributes mainly around the zero scale factor
region. In otherwords, repeated reflections between the turning points and
superposition of states like $\exp{(iS)}$ and $\exp{(-iS)}$ are basically the
quantum character and owe its origin to the wormholes. Our work shows that the
times $`t'$ and $\sigma$ become equal at the semiclassical region and begin to
differ as we approach more and more to the classically forbidden region. We observe
that the quantization becomes worthy only through the matter Lagrangian i.e.,
the Newtonian time emerges through the matter field Hamiltonian in SWD equation.
This is in conformity with classical Einstein equation where in the righthand
side we use $<T_{\mu\nu}>$ to obtain the classical description i.e., the
geometry acts as a ``guidance field'' for matter. Akin to Madelung [15] and
Bohm [16] in our approach the initial positions are random and the quantum force
generating the randomness arises from wormhole contribution. In view of some
apathy towards the wormhole philosophy, we like to mention that the quantum
force generated at the initial stage (i.e., repeated reflections at the turning
points and the superposition) finds an interpretation through wormhole
contributions. This also guarantees the universal validity of superposition
principle. In otherwords, the geometry in the early universe initiates the
quantum randomness and with the beginning of nucleation and inflation, the
Newtonian time emerges through the SWD equation such that $S_1<<S_o$ and
$S_o=S_o(a)$, an aspect relevant to decoherence. The scale factor $a$ of a
Friedmann universe then becomes the relevant variable and attains the classical
character through continuous measurement. In the literature a question arises
that if the states $\exp{(iS)}$ and $\exp{(-iS)}$ describing an expanding and
collapsing universe decohere, can one recover an approximate time-dependent
Schroedinger equation from the timeless Wheeler-DeWitt equation and what are
the boundary conditions at small scales [17] that lead to quantum effects in
the vicinity of the turning point. In this work we answer all three questions.
We show that (i) an approximate time dependent Schroedinger equations follows
irrespective of the Wheeler-DeWitt equation, (ii) the adiabatic Gaussian
wavefunction for the SWD equation is consistent with the Hartle-Hawking
criterion plus the normalization prescription or the wormhole dominance proposal.
This solves the problem of small scale boundary condition.
(iii). Through the wormhole dominance proposal it has been explicitly shown that
the wormholes lead to quantum effects in the vicinity of the turning point (in
our case $a\simeq 0$) and classical turning point (in our case
$a\simeq H^{-1}$) behaves as an starting point for the arrow of time and is
manifested through the matter field as if the geometry itself looks into the
evolution through the matter field. (iv). In the standard derivation of SWD
equation, an expansion of the Wheeler-DeWitt wavefunction with respect to the
Planck mass leads to difficulties as discussed in [18]. Our derivation is
basically an expansion with respect to $\hbar$ and is justified through
equations (30) and (31). (iv). The decoherence mechanism in our approach is the
same as in ref. [3] and we do not repeat it here. (v). We also applied our
approach to Starobinsky type $R^2$-cosmology and find that the Gaussian
ansatz correctly reproduces the wavefunction corresponding to the wormhole
dominance proposal.
\smallskip
\begin{center}
{\bf{References}}
\end{center}
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\smallskip
\section{\bf{Acknowledgment}}
A. Shaw acknowledges the financial support from ICSC World Laboratory, LAUSSANE
during the course of the work.
\end{document}
|
1,314,259,996,558 | arxiv | \section{Introduction}
Since the discovery of superconductivity in LaFeAs(O$_{1-x}$F$_x$) \cite{Kam08}, high transition temperatures ($T_c$) up to 56~K have been reported in the doped Fe-based oxypnictides \cite{Tak08,Che08Ce,Ren08Pr,Ren08Nd,Kit08Nd,Che08Sm,Yan08Gd,Wan08Gd}. The new high-temperature superconductivity in Fe-pnictides has attracted tremendous interests both experimentally and theoretically. The `mother' materials have antiferromagnetic spin-density-wave order \cite{Cru08} and the superconductivity appears by doping charge carriers, either electrons or holes. Such carrier doping induced superconductivity resembles high-$T_c$ cuprates, but one of the most significant differences is the multiband electronic structure having electron and hole pockets in the Fe-based superconductors. The nature of superconductivity and the pairing mechanism in such systems are fundamental physical problem of crucial importance. The first experimental task to this problem is to elucidate the superconducting pairing symmetry, which is intimately related to the pairing interaction.
Unconventional superconducting pairings, most notably the sign-reversing $s_\pm$ state, have been suggested by several theories \cite{Maz08,Kur08,Seo08,Cve08,Ike08,Nom08} featuring the importance of the nesting between the hole and electron bands. This is also in sharp contrast to other multiband superconductors such as MgB$_2$, where the coupling between the different bands is very weak. Thus the most crucial is to clarify the novel multiband nature of superconductivity in this new class of materials.
In this context, identifying the detailed structure of superconducting order parameter, particularly the presence or absence of nodes in the gap function, is of primary importance. In the electron-doped $Ln$FeAs(O,F) or `1111' systems (where $Ln$ is Lanthanoide ions), while several experiments \cite{Nak08,Sha08} suggest nodes in the gap, the tunnelling measurements \cite{Che08} and angle resolved photoemission (ARPES) \cite{Kon08} suggest fully gapped superconductivity. In the hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$ or `122' system \cite{Rot08}, experimental situation is controversial as well: ARPES \cite{Zha08,Din08,Evt08} and lower critical field measurements \cite{Ren08} support nodeless gaps while $\mu$SR measurements \cite{Gok08} imply the presence of line nodes in the gap function.
One of the effective ways to judge the presence or absence of nodes in the gap is to investigate the properties of thermally excited quasiparticles at low temperatures. Measurements of the London penetration depth $\lambda$ are most suited for this, since the quasiparticle density is directly related to $\lambda(T)$. In $d$-wave superconductors with line nodes, for example, the low-temperature change in the penetration depth $\delta\lambda(T)=\lambda(T)-\lambda(0)$ shows a linear temperature dependence, as observed in high-$T_c$ cuprate superconductors \cite{Bon07}. In contrast, fully gapped superconductors without nodes exhibit an exponential temperature dependence of $\delta\lambda(T)$, reflecting the thermally activated behavior of quasiparticles. Also notable is that the penetration depth is typically hundreds of nm, much longer than lattice parameters, which enables us to discuss the bulk properties of Fe-based superconductors. This contrasts with other surface sensitive techniques such as ARPES and scanning tunneling spectroscopy.
Another important means to clarify not only the superconducting gap symmetry, but also the multiband nature of superconductivity is an accurate determination of the lower critical field $H_{c1}$. In particular, the two-gap superconductivity in MgB$_2$ manifests itself in the unusual temperature dependence of the anisotropy of $H_{c1}$ in the superconducting state \cite{Bou02,Lya04,Fle05}. However, the reliable measurement of the lower critical field is a difficult task, in particular when strong vortex pinning is present as in the case of Fe-arsenides. We also point out that to date the reported values of anisotropy parameter strongly vary \cite{Wey08,Mar08,Bal08,Kub08} spanning from 1.2 (Ref.~\onlinecite{Kub08}) up to $\sim 20$ (Ref.~\onlinecite{Wey08}) in 1111 systems. Although the anisotropy parameter may be different for different compounds, this apparent large spread may be partly due to the effects of strong vortex pinning, which lead to large ambiguity in some experiments.
Here we review our microwave surface impedance $Z_s$ measurements in single crystals of electron-doped PrFeAsO$_{1-y}$ ($y\sim0.1$) \cite{Has09} and hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$ ($x\approx 0.55$) \cite{Has08}. By using a sensitive superconducting cavity resonator, we can measure both real and imaginary parts of $Z_s$ pricesely in tiny single crystals, from which we can extract the in-plane penetration depth $\lambda_{ab}(T)$ as well as quasiparticle conductivity $\sigma_1(T)$ that yields information on the quasiparticle dynamics. We also measure $H_{c1}(T)$ in PrFeAsO$_{1-y}$ crystals by using an unambiguous method to avoid the difficulty associated with pinning \cite{Oka08}. We directly determine the field $H_p$ at which first flux penetration occurs from the edge of the crystal by measuring the magnetic induction just inside and outside the edge of the single crystals, with the use of a miniature Hall-sensor array. This allows us to extract the temperature dependent values of the lower critical fields parallel to the $c$-axis ($H_{c1}^c$) and to the $ab$-plane ($H_{c1}^{ab}$), as well as the anisotropy parameter $H_{c1}^c/H_{c1}^{ab}$ in single crystals of Fe-based superconductors.
The penetration depth in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ is found to be sensitive to disorder inherent in the crystals. The source of disorder may be microscopic inhomogeneous content of K, which is reactive with moisture or oxygen. The degree of disorder can be quantified by the quasiparticle scattering rate $1/\tau$. We find that $\lambda_{ab}(T)$ in crystals with large scattering rate shows strong temperature dependence which mimics that of superconductors with nodes in the gap. In the best crystal with the smallest $1/\tau$, however, $\lambda_{ab}(T)$ shows clear flattening at low temperatures, giving strong evidence for the nodeless superconductivity. $\lambda_{ab}(T)$ in PrFeAsO$_{1-y}$ also shows flat temperature dependence at low temperatures. The superfluid density can be consistently explained by the presence of two fully-opened gaps in both hole- and electron-doped systems. The lower critical field measurements consistently give the saturation behavior of superfluid density at low temperatures. We find that the anisotropy of the penetration depths $\gamma_{\lambda}\equiv \lambda_c/\lambda_{ab} \simeq H_{c1}^c/H_{c1}^{ab}$, where $\lambda_c$ and $\lambda_{ab}$ are out-of-plane and in-plane penetration depths, respectively, is smaller than the anisotropy of the coherence lengths $\gamma_{\xi}\equiv \xi_{ab}/\xi_c= H_{c2}^{ab}/H_{c2}^c$, where $\xi_{ab}$ and $\xi_c$ are in- and out-of-plane coherence lengths, and $H_{c2}^{ab}$ and $H_{c2}^c$ are the upper critical fields parallel and perpendicular to the $ab$-plane, respectively. This result provides further evidence for the multiband nature of the superconductivity. Finally, the quasiparticle conductivity exhibits a large enhancement in the superconducting state, which bears a similarity with high-$T_c$ cuprates and heavy fermion superconductors with strong electron scattering in the normal state above $T_c$.
\section{Experimental}
\subsection{Single crystals}
High-quality PrFeAsO$_{1-y}$ single crystals were grown at National Institute of Advanced Industrial Science and Technology (AIST) in Tsukuba by a high-pressure synthesis method using a belt-type anvil apparatus (Riken CAP-07). Powders of PrAs, Fe, Fe$_2$O$_3$ were used as the starting materials. PrAs was obtained by reacting Pr chips and As pieces at 500$^{\circ}$C for 10 hours, followed by a treatment at 850$^{\circ}$C for 5 hours in an evacuated quartz tube. The starting materials were mixed at nominal compositions of PrFeAsO$_{0.6}$ and ground in an agate mortar in a glove box filled with dry nitrogen gas. The mixed powders were pressed into pellets. The samples were then grown by heating the pellets in BN crucibles under a pressure of about 2~GPa at 1300$^{\circ}$C for 2 hours. Platelet-like single crystals of dimensions up $150 \times 150 \times 30$~$\mu$m$^3$ were mechanically selected from the polycrystalline pellets. The single crystalline nature of the samples was checked by Laue X-ray diffraction \cite{Has_JPSJ}. Our crystals, whose $T_c$ $(\approx35$~K) is lower than the optimum $T_c \approx$ 51~K of PrFeAsO$_{1-y}$ \cite{Ren08_PD}, are in the underdoped regime ($y\sim0.1$) \cite{Lee08}, which is close to the spin-density-wave order \cite{Zha08_PD}. The sample homogeneity was checked by magneto-optical (MO) imaging \cite{Oka08}. The crystal exhibits a nearly perfect Meissner state $\sim$~2~K below $T_c$; no weak links are observed, indicating a good homogeneity. The in-plane resistivity is measured by the standard four-probe method under magnetic fields up to 10 T. The electrical contacts were attached by using the W deposition technique in a Focused-Ion-Beam system.
Single crystals of Ba$_{1-x}$K$_x$Fe$_2$As$_2$ were grown at National Institute of Materials Science (NIMS) in Tsukuba by a self-flux method using high purity starting materials of Ba, K and FeAs. These were placed in a BN crucible, sealed in a Mo capsule under Ar atmosphere, heated up to 1190$^\circ$C, and then cooled down at a rate of 4$^\circ$C/hours, followed by a quench at 850$^\circ$C. Energy dispersive X-ray (EDX) analysis reveals the doping level $x=0.55(2)$ \cite{Has08}, which is consistent with the $c$-axis lattice constant $c=1.341(3)$~nm determined by X-ray diffraction \cite{Luo08}. Bulk superconductivity is characterized by the magnetization measurements using a commercial magnetometer \cite{Has08}. We find that the superconducting transition temperature varies slightly from sample to sample [see Table~\ref{table1}]. Since this is likely related to the microscopic inhomogeneity of K content near the surface, which can be enhanced upon exposure to the air, we carefully cleave both sides of the surface of crystal \#2 and cut into smaller size (crystal \#3). For \#3, the microwave measurements are done with minimal air exposure time.
\begin{table}[t]
\caption{\label{table1} Properties of the Ba$_{1-x}$K$_x$Fe$_2$As$_2$ crystals we studied. In this study, the transition temperature $T_c$ is evaluated from the extrapolation of superfluid density $n_s\rightarrow0$.}
\begin{ruledtabular}
\begin{tabular}{ccccccc}
sample &size ($\mu$m$^3$)&$T_c$ (K)& $1/\tau$(40~K) (s$^{-1}$)\\
\hline
\#1& $320\times 500\times 100$ & 26.4(3) & $27(3)\times10^{12}$ \\
\#2& $300\times 500\times 80$ & 25.0(4) & $21(2)\times10^{12}$ \\
\#3& $100\times 180\times 20$ & 32.7(2) & $7.8(5)\times10^{12}$
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{Microwave surface impedance}
In-plane microwave surface impedance $Z_s=R_s+{\rm i}X_s$, where $R_s$ ($X_s$) is the surface resistance (reactance), is measured in the Meissner state by using a 28~GHz TE$_{011}$-mode superconducting Pb cavity with a high quality factor $Q\sim10^6$ \cite{Shi94,Shi07}. To measure the surface impedance of a small single crystal with high precision, the cavity resonator is soaked in the superfluid $^4$He at 1.6~K and its temperature is stabilized within $\pm1$~mK. We place a crystal in the antinode of the microwave magnetic field ${\bf H}_\omega$ ($\parallel c$ axis) so that the shielding current ${\bf I}_\omega$ is excited in the $ab$ planes. The inverse of quality factor $1/Q$ and the shift in the resonance frequency are proportional to $R_s$ and the change in $X_s$, respectively. In our frequency range $\omega/2\pi\approx 28$~GHz, the complex conductivity $\sigma=\sigma_1-{\rm i}\sigma_2$ can be extracted from $Z_s(T)$ through the relation valid for the so-called skin-depth regime:
\begin{equation}
Z_s=R_s+{\rm i}X_s=\left(\frac{{\rm i}\mu_0\omega}{\sigma_1-{\rm i}\sigma_2}\right)^{1/2}.
\label{impedance}
\end{equation}
In our frequency range, the skin depth $\delta_{cl}$ is much shorter than the sample width, ensuring the skin-depth regime. In the superconducting state, the surface reactance is a direct measure of the superfluid density $n_s$ via $X_s(T)=\mu_0\omega\lambda_{ab}(T)$ and $\lambda_{ab}^{-2}(T)=\mu_0n_s(T)e^2/m^*$. In the normal state, $\sigma_1=ne^2\tau/m^*\gg\sigma_2$ gives $R_s(T)=X_s(T)=(\mu_0\omega/2\sigma_1)^{1/2}$ from Eq.~(\ref{impedance}), where $n$ is the total density of carriers with effective mass $m^*$. Below $T_c$, the real part of conductivity $\sigma_1$ is determined by the quasiparticle dynamics, and in the simple two-fluid model, which is known to be useful in cuprate superconductors \cite{Bon07,Bon94}, $\sigma_1$ is related to the quasiparticle scattering time $\tau$ through $\sigma_1=(n-n_s)e^2\tau/m^*(1+\omega^2\tau^2)$.
\subsection{Lower critical fields}
The local induction near the surface of the platelet crystal has been measured by placing the sample on top of a miniature Hall-sensor array tailored in a GaAs/AlGaAs heterostructure \cite{Shiba07}. Each Hall sensor has an active area of $3 \times 3$ $\mu$m$^2$; the center-to-center distance of neighboring sensors is 20~$\mu$m. The local induction at the edge of the crystal was detected by the miniature Hall sensor located at $\leq 10$~$\mu$m from the edge. The magnetic field $H_a$ is applied for {\boldmath $H$}$\parallel c$ and {\boldmath $H$}$\parallel ab$-plane by using a low-inductance 2.4~T superconducting magnet with a negligibly small remanent field.
\section{Results and Discussion}
\subsection{Surface impedance}
\begin{figure}[t]
\includegraphics[width=90mm]{Fig1_Zs_T.eps}%
\caption{Temperature dependence of the surface resistance $R_s$ and reactance $X_s$ at 28~GHz in a PrFeAsO$_{1-y}$ single crystal (a) and in a Ba$_{1-x}$K$_x$Fe$_2$As$_2$ crystal. In the normal state, $R_s=X_s$ as expected from Eq.~(\ref{impedance}). The low-temperature errors in $R_s$ are estimated from run-to-run uncertainties in $Q$ of the cavity. Inset in (a) shows the microwave resistivity $\rho_1(T)$ (red circles) compared with the dc resistivity in a crystal from the same batch (black squares). The dashed line represents a $T^2$ dependence. }
\label{Z_s}
\end{figure}
Figure~\ref{Z_s} shows typical temperature dependence of the surface resistance $R_s$ and $X_s$. In the normal state where $\omega\tau$ is much smaller than unity, the temperature dependence of microwave resistivity $\rho_1={2R_s^2/\mu_0\omega}$ is expected to follow $\rho(T)$ [see Eq.~(\ref{impedance})]. Such a behavior is indeed observed for PrFeAsO$_{1-y}$ as shown in the inset of Fig.~\ref{Z_s}(a). Below about 100~K $\rho_1(T)$ exhibits a $T^2$ dependence and it shows a sharp transition at $T_c\approx 35$~K. As shown in Fig.~\ref{Z_s}, the crystals have low residual $R_s$ values in the low temperature limit. These results indicate high quality of the crystals. We note that the transition in microwave $\rho_1(T)$ is intrinsically broader than that in dc $\rho(T)$, since the applied 28-GHz microwave (whose energy corresponds to 1.3~K) excites additional quasiparticles just below $T_c$. We also use $\lambda_{ab}(0)= 280$~nm, which is determined from the lower critical field measurements using a micro-array of Hall probes in the crystals from the same batch, and thus the absolute values of $R_s$ and $X_s$ are determined for PrFeAsO$_{1-y}$.
For Ba$_{1-x}$K$_x$Fe$_2$As$_2$, $Z_s(T)$ shows steeper temperature dependence in the normal state as shown in Fig.~\ref{Z_s}(b). This strong temperature dependence above $T_c$ allows us to determine precisely the offset of $X_s(0)/X_s$(40~K), since $R_s$ and $X_s$ should be identical in the normal state. From this, we are able to determine $\lambda_{ab}(T)/\lambda_{ab}(0)$ and $n_s(T)/n_s(0)=\lambda_{ab}^2(0)/\lambda_{ab}^2(T)$ without any assumptions \cite{Shi94}. This also gives us estimates of the normal-state scattering rate $1/\tau=1/\mu_0\sigma_1\lambda_{ab}^2(0)=2\omega(X_s(T)/X_s(0))^2$, which quantifies the degrees of disorder for the samples we used [see Table~\ref{table1}].
\subsection{Penetration depth and superfluid density}
\subsubsection{PrFeAsO$_{1-y}$}
\begin{figure}[t]
\includegraphics[width=90mm]{Fig2_dl_T_1111.eps}%
\caption{Temperature dependence of $\delta\lambda_{ab}(T)/\delta\lambda_{ab}(0)$ at low temperatures, for two crystals of PrFeAsO$_{1-y}$. The dashed lines represent $T$-linear dependence expected in clean $d$-wave superconductors with line nodes. The solid lines are low-$T$ fits to Eq.~(\ref{BCS}).}
\label{lambda_1111}
\end{figure}
\begin{figure}[t]
\includegraphics[width=90mm]{Fig3_ns_T_1111.eps}%
\caption{Temperature dependence of the superfluid density $\lambda_{ab}^2(0)/\lambda_{ab}^2(T)$ in PrFeAsO$_{1-y}$ crystals. The solid lines are the best fit results to the two-gap model [Eq.~(\ref{two-gap})], and the dashed and dashed-dotted lines are the single gap results for $\Delta_1$ and $\Delta_2$, respectively. $T_c$ is defined by the temperature at which the superfluid density becomes zero. Note that the experimental $X_s$ is limited by $\mu_0\omega\delta_{cl}/2$ above $T_c$, which gives apparent finite values of $n_s$ above $T_c$. }
\label{ns_1111}
\end{figure}
In Fig.~\ref{lambda_1111} shown are the normalized change in the in-plane penetration depth $\delta\lambda_{ab}(T)=\lambda_{ab}(T)-\lambda_{ab}(0)$ for two crystals of PrFeAsO$_{1-y}$. The overall temperature dependence of $\delta\lambda_{ab}(T)$ in these crystals is essentially identical, which indicates good reproducibility. It is clear from the figure that $\delta\lambda_{ab}(T)$ has flat dependence at low temperatures. First we compare our data with the expectations in unconventional superconductors with nodes in the gap. In clean superconductors with line nodes, thermally excited quasiparticles near the gap nodes give rise to the $T$-linear temperature dependence of $\delta\lambda_{ab}(T)$ at low temperatures, as observed in YBa$_2$Cu$_3$O$_{7-\delta}$ crystals with $d$-wave symmetry \cite{Har93}. In the $d$-wave case, $\delta\lambda_{ab}(T)/\lambda_{ab}(0)\approx \frac{\ln2}{\Delta_0}k_BT$ is expected \cite{Bon07}, where $\Delta_0$ is the maximum of the energy gap $\Delta({\bf k})$. This linear temperature dependence with an estimation $2\Delta_0/k_BT_c\approx4$ \cite{Nak08} [dashed lines in Fig.~\ref{lambda_1111}] distinctly deviates from our data. When the impurity scattering rate $\Gamma_{\rm imp}$ becomes important in superconductors with line nodes, the induced residual density of states changes the $T$-linear dependence into $T^2$ below a crossover temperature $T^*_{\rm imp}$ determined by $\Gamma_{\rm imp}$ \cite{Pro06}. This is also clearly different from our data in Fig.~\ref{lambda_1111}. If by any chance the $T^2$ dependence with a very small slope should not be visible by the experimental errors below $\sim10$~K, then we would require enormously high $T^*$. However, since no large residual density of states is inferred from NMR measurements even in polycrystalline samples of La-system with lower $T_c$ \cite{Nak08}, such a possibility is highly unlikely. These results lead us to conclude that in contradiction to the presence of nodes in the gap, the finite superconducting gap opens up all over the Fermi surface. We note that recent penetration depth measurements using MHz tunnel-diode oscillators in SmFeAsO$_{1-x}$F$_y$ \cite{Mal08} and NdFeAsO$_{0.9}$F$_{0.1}$ \cite{Mar08} are consistent with our conclusion of a full-gap superconducting state in 1111 system.
In fully gapped superconductors, the quasiparticle excitation is of an activated type, which gives the exponential dependence
\begin{equation}
\frac{\delta\lambda_{ab}(T)}{\lambda_{ab}(0)} \approx \sqrt{\frac{\pi\Delta}{2k_{\rm B}T}}\exp\left(-\frac{\Delta}{k_{\rm B}T}\right)
\label{BCS}
\end{equation}
at $T\lesssim T_c/2$ \cite{Hal71}. Comparisons between this dependence and the low-temperature data in Fig.~\ref{lambda_1111} enable us to estimate the minimum energy $\Delta_{\rm min}$ required for quasiparticle excitations at $T=0$~K; {\it i.e.} $\Delta_{\rm min}/k_BT_c=1.6\pm0.1$ for PrFeAsO$_{1-y}$.
We can also plot the superfluid density $n_s= \lambda_{ab}^2(0)/ \lambda_{ab}^2(T)$ as a function of temperature in Fig.~\ref{ns_1111}. Again, the low-temperature behavior is quite flat, indicating a full-gap superconducting state. We note that by using the gap $\Delta_{\rm min}$ obtained above alone, we are unable to reproduced satisfactory the whole temperature dependence of $n_s$ [see the dashed lines in Fig.~\ref{ns_1111}], although a better fit may be obtained by using a larger value of $\Delta=1.76k_BT_c$, the BCS value. Since Fe oxypnictides have the multi-band electronic structure \cite{Sin08,Liu08}, we also try to fit the whole temperature dependence with a simple two-gap model \cite{Bou01}
\begin{equation}
n_s(T)=xn_{s1}(T)+(1-x)n_{s2}(T).
\label{two-gap}
\end{equation}
Here the band 1 (2) has the superfluid density $n_{s1}$ ($n_{s2}$) which is determined by the gap $\Delta_1$ ($\Delta_2$), and $x$ defines the relative weight of each band to $n_{s}$. The temperature dependence of superfluid density $n_{si}(T)$ ($i=1,2$) for each band is calculated by assuming the BCS temperature dependence of superconducting gap $\Delta_i(T)$ \cite{Pro06}. This simple model was successfully used for the two-gap $s$-wave superconductor MgB$_2$ with a large gap ratio $\Delta_2/\Delta_1\approx2.6$ \cite{Fle05}. Here we fix $\Delta_1=\Delta_{\rm min}$ obtained from the low-temperature fit in Fig.~\ref{lambda_1111}, and excellent results of the fit are obtained for $\Delta_{2}/k_BT_c=2.0$ and $x=0.6$ [see Fig.~\ref{ns_1111}]. These exercises suggest that the difference in the gap value of each band in this system is less substantial than the case of MgB$_2$ \cite{Bou02,Lya04,Fle05}.
\subsubsection{Ba$_{1-x}$K$_x$Fe$_2$As$_2$}
\begin{figure}[t]
\includegraphics[width=90mm]{Fig4_compare_122.eps}%
\caption{(color online). (a) Temperature dependence of the normalized 28-GHz microwave resistivity $\rho_1(T)/\rho_1(40$~K) in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ single crystals. (b) Normalized superfluid density $\lambda_{ab}^2(0)/\lambda_{ab}^2(T)$ for 3 samples with different normal-state scattering rates (see Table~\ref{table1}).
}
\label{compare}
\end{figure}
In Fig.~\ref{compare}(a) we compare the temperature dependence of normal-state microwave resistivity $\rho_1=1/\sigma_1={2R_s^2/\mu_0\omega}$ [see Eq.~(\ref{impedance})] for 3 samples of Ba$_{1-x}$K$_x$Fe$_2$As$_2$. As mentioned in section 2.1, crystal \#3 was cleaved from \#2. We find that the cleavage dramatically improves the sample quality, and crystal \#3 exhibits the sharpest superconducting transition and the lowest normal-state scattering rate $1/\tau$ [see Table~\ref{table1}]. Figure~\ref{compare}(b) demonstrates the normalized superfluid density $n_s(T)/n_s(0)=\lambda_{ab}^2(0)/\lambda_{ab}^2(T)$ for these 3 samples. We find that crystals with large scattering rates exhibit strong temperature dependence of superfluid density at low temperatures, which mimics the power-law temperature dependence of $n_s(T)$ in $d$-wave superconductors with nodes. However, the data in cleaner samples show clear flattening at low temperatures. This systematic change indicates that the superfluid density is quite sensitive to disorder in this system and disorder promotes quasiparticle excitations significantly. It is tempting to associate the observed effect with unconventional superconductivity with sign reversal such as the $s_\pm$ state \cite{Maz08}, where impurity scattering may induce in-gap states in clear contrast to the conventional $s$-wave superconductivity \cite{Par08,Sen08}. Indeed, $T_c$ determined by $n_s \to 0$ is noticeably reduced for samples with large $1/\tau$ [Table \ref{table1}], consistent with theoretical studies \cite{Sen08}. At present stage, however, the microscopic nature of disorder inherent in our crystals is unclear, and a more controlled way of varying degrees of disorder is needed for further quantitative understanding of the impurity effects in Fe-based superconductors. In any case, our results may account for some of the discrepancies in the reports of superfluid density in Fe-arsenides \cite{Gok08,Gor08}.
\begin{figure}[t]
\includegraphics[width=80mm]{Fig5_dl_T_122.eps}%
\caption{$\delta\lambda_{ab}(T)/\lambda_{ab}(0)$ at low temperatures for the cleanest crystal \#3 of Ba$_{1-x}$K$_x$Fe$_2$As$_2$. The solid line is a fit to Eq.~(\ref{BCS}) with $\Delta=1.30k_BT_c$. The dashed line represents $T$-linear dependence expected in clean $d$-wave superconductors \cite{Bon07} with maximum gap $\Delta_0=2k_BT_c$ \cite{Nak08}.}
\label{lambda_122}
\end{figure}
\begin{figure}[t]
\includegraphics[width=80mm]{Fig6_ns_T_122.eps}%
\caption{$\lambda_{ab}^2(0)/\lambda_{ab}^2(T)$ for crystal \#3 of Ba$_{1-x}$K$_x$Fe$_2$As$_2$ fitted to the two-gap model Eq.~(\ref{two-gap}) (blue solid line) with $\Delta_1=1.17k_BT_c$ (dashed line) and $\Delta_2=2.40k_BT_c$ (dashed-dotted line). Green dotted line is the single-gap fit using $\Delta=1.30k_BT_c$. Above $T_c$, the normal-state skin depth contribution gives a finite tail.}
\label{ns_122}
\end{figure}
For the cleanest sample (\#3), the low-temperature change in the penetration depth $\delta\lambda_{ab}(T)=\lambda_{ab}(T)-\lambda_{ab}(0)$ is depicted in Fig.~\ref{lambda_122}, which obviously contradicts the $T$-linear dependence expected in clean $d$-wave superconductors with line nodes. The low-temperature data can rather be fitted to the exponential dependence Eq.~(\ref{BCS}) for full-gap superconductors with a gap value $\Delta=1.30k_BT_c$. This provides compelling evidence that the intrinsic gap structure in clean samples has no nodes in 122 system. Together with the fact that in 1111 the low-temperature $\delta\lambda_{ab}(T)$ also shows exponential behavior, we surmise that both electron and hole-doped Fe-arsenides are intrinsically full-gap superconductors.
Various theories have been proposed for the pairing symmetry in the doped Fe-based oxypnictides \cite{Maz08,Kur08,Seo08,Cve08,Ike08,Nom08,LeeWen08,Si08,Sta08,Wan08Lee}. As confirmed by recent ARPES measurements \cite{Liu08}, doped Fe-based oxypnictides have hole pockets in the Brillouin zone center and electron pockets in the zone edges \cite{Sin08}. It has been suggested that the nesting vector between these pockets is important, which favors an extended $s$-wave order parameter (or $s_{\pm}$ state) having opposite signs between the hole and electron pockets \cite{Maz08,Kur08,Seo08,Cve08,Ike08,Nom08}. Our penetration depth result of full gap is in good correspondence with such an $s_{\pm}$ state with no nodes in both gaps in these two bands.
As shown in Fig.~\ref{ns_122}, the overall temperature dependence of $n_s$ in crystal \#3 cannot be fully reproduced by the single gap calculations. Considering the multiband electronic structure in this system \cite{Zha08,Din08,Evt08}, we fit the data again to the two-gap model Eq.~(\ref{two-gap}). We obtain an excellent fit with $\Delta_1/k_BT_c=1.17$, $\Delta_{2}/k_BT_c=2.40$ and $x=0.55$. This leads us to conclude the nodeless multi-gap superconductivity having at least two different gaps in this system. It is noteworthy that the obtained gap ratio is comparable to the value $\Delta_2/\Delta_1\approx 2$ found in the ARPES studies \cite{Din08,Evt08} for different bands. A large value of $x=0.55$ implies that the Fermi surface with the smaller gap $\Delta_1$ has a relatively large volume or carrier number, which is also in good correspondence with the ARPES results.
\subsection{Lower critical fields}
\begin{figure}[t]
\includegraphics[width=85mm]{Fig7_M_H_1111.eps}
\caption{Local magnetization loops for {\boldmath $H$}$\parallel c$, measured by the miniature Hall sensor located at $\leq$~10~$\mu$m from the edge of the crystal. Arrows indicate the field sweep directions.}
\label{MH}
\end{figure}
In Fig.~\ref{MH} we show the field dependence of the ``local magnetization'', $M_{\rm edge} \equiv \mu_0^{-1}B_{\rm edge} - H_a$, at the edge of the crystal, for {\boldmath $H$}$\parallel c$, measured after zero field cooling. After the initial negative slope corresponding to the Meissner state, vortices enter the sample and $M_{\rm edge}(H_{a})$ shows a large hysteresis. The shape of the magnetization loops (almost symmetric about the horizontal axis) indicates that the hysteresis mainly arises from bulk flux pinning rather than from the (Bean-Livingston) surface barrier \cite{Kon99}. As shown in Fig.~\ref{MH}, the initial slope of the magnetization exhibits a nearly perfect linear dependence, $M_{\rm edge}=-\alpha H_a$. Since the Hall sensor is placed on the top surface, with a small but non-vanishing distance between the sensor and the crystal, the magnetic field leaks around the sample edge with the result that the slope $\alpha$ is slightly smaller than unity.
\begin{figure}[t]
\includegraphics[width=80mm]{Fig8_B_H_1111.eps}
\caption{(a) Typical curves of $\sqrt{B}$ (left axis) at the edge (circles) and at the center (squares) of the crystal and $\sqrt{\Delta j_{\rm edge}}$ (right axis) plotted as a function of $H_a$ for {\boldmath $H$}$\parallel c$ at $T$~=~22~K, in which $H_a$ is increased after ZFC. (b) The temperature dependence of the flux penetration fields $H_p$ at the edge and the center of the crystal. The insets are schematic illustrations of the experimental setup for {\boldmath $H$}$\parallel c$ and {\boldmath $H$}$\parallel ab$-plane. (c) Temperature dependence of the difference between $H_p$ in the center and at the edge (left axis), compared with the remanent magnetization $M_{\rm rem}$ (right axis). }
\label{BH}
\end{figure}
Figure~\ref{BH}(a) shows typical curves of $B^{1/2} \equiv \mu_{0}^{1/2}(M + \alpha H_a)^{1/2}$ at the edge (circles) and at the center (squares) of the crystal, plotted as a function of $H_a$; the external field orientation {\boldmath $H$}$\parallel c$ and $T$~=~22~K. The $\alpha H_a$-term is obtained by a least squares fit of the low-field magnetization. The first penetration field $H_p$ corresponds to the field $H_{p}$(edge), above which $B^{1/2}$ increases almost linearly, is clearly resolved. In Fig.~\ref{BH}(a), we show the equivalent curve, measured at the center of the crystal. At the center, $B^{1/2}$ also increases linearly, starting from a larger field, $H_p$(center).
We have measured the positional dependence of $H_p$ and observed that it increases with increasing distance from the edge. To examine whether $H_p$(edge), \em i.e. \rm $H_{p}$ measured at $\leq$~10~$\mu$m from the edge, truly corresponds to the field of first flux penetration at the boundary of the crystal, we have determined the local screening current density $j_{\rm edge} = \mu_0^{-1} (B_{\rm edge}-B_{\rm outside})/\Delta x$ at the crystal boundary. Here $B_{\rm edge}$ is the local magnetic induction measured by the sensor just inside the edge, and $B_{\rm outside}$ is the induction measured by the neighboring sensor just outside the edge. For fields less than the first penetration field, $j_{\rm edge} \simeq \beta H_{a}$ is the Meissner current, which is simply proportional to the applied field ($\beta$ is a constant determined by geometry). At $H_{p}$, the screening current starts to deviate from linearity. Figure~\ref{BH}(a) shows the deviation $\Delta j_{\rm edge} \equiv j_{\rm edge}-\beta H_a$ as a function of $H_a$. As depicted in Fig.~\ref{BH}(a), $\sqrt{\Delta j_{\rm edge}}$ again increases linearly with $H_a$ above $H_p$(edge). This indicates that the $H_p$(edge) is very close to the true field of first flux penetration.
In Fig.~\ref{BH}(b), we compare the temperature dependence of $H_{p}$(edge) and $H_p$(center). In the whole temperature range, $H_p$(center) well exceeds $H_p$(edge). Moreover, $H_p$(center) increases with decreasing $T$ without any tendency towards saturation. In sharp contrast, $H_p$(edge) saturates at low temperatures. Figure~\ref{BH}(c) shows the difference between $H_p$ measured in the center and at the edge, $\Delta H_p=H_p$(center) $-$ $H_p$(edge). $\Delta H_p$ increases steeply with decreasing temperature. Also plotted in Fig.~\ref{BH}(c) is the remanent magnetization $M_{\rm rem}$ (\em i.e. \rm the $H_{a} = 0$ value of $M_{\rm edge}$ on the decreasing field branch), measured at near the crystal center. This is proportional to the critical current density $j_c$ arising from flux pinning. The temperature dependence of $\Delta H_p$ is very similar to that of $j_{c}$, which indicates that $H_p$(center) is strongly influenced by pinning. Hence, the present results demonstrate that the lower critical field value determined by local magnetization measurements carried out at positions close to the crystal center is affected by vortex pinning effects and might be seriously overestimated.
The absolute value of $H_{c1}$ is evaluated by taking into account the demagnetizing effect. For a platelet sample, $H_{c1}$ is given by
\begin{equation}
H_{c1}=H_p/\tanh \sqrt{0.36b/a},
\label{Brandt}
\end{equation}
where $a$ and $b$ are the width and the thickness of the crystal, respectively \cite{Bra99}. In the situation where {\boldmath $H$}$\parallel c$, $a=$ 63 $\mu$m and $b=$ 18 $\mu$m, while $a=$ 18 $\mu$m and $b=$ 63 $\mu$m for {\boldmath $H$}$\parallel ab$-plane. These values yield $H_{c1}^c = 3.22 H_p$ and $H_{c1}^{ab} = 1.24 H_p$, respectively. In Fig.~\ref{Hc1}(a), we plot $H_{c1}$ as a function of temperature both for {\boldmath $H$} $\parallel c$ and {\boldmath $H$} $\parallel ab$-plane. The solid line in Fig.~\ref{Hc1}(a) indicates the temperature dependence of the superfluid density normalized by the value at $T=0$~K, which is obtained from the results in Fig.~\ref{ns_1111} of a sample from the same batch \cite{Has09}. $H_{c1}^c(T)$ is well scaled by the superfluid density, which is consistent with fully gapped superconductivity; it does not show the unusual behavior reported in Ref.~\onlinecite{Ren08_1111}. We note that recent $H_{c1}$ measurements on NdFeAs(O,F) by using Hall probes \cite{Klein} also show flat temperature dependence of $H_{c1}$, consistent with our results. To estimate the in-plane penetration depth at low temperatures, we use the approximate single-band London formula,
\begin{equation}
\mu_0H_{c1}^{c} = \frac{\Phi_0}{4\pi \lambda_{ab}^2}\left[
\ln\frac{\lambda_{ab}}{\xi_{ab}}+0.5 \right],
\label{GL}
\end{equation}
where $\Phi_0$ is the flux quantum. Using $\ln\lambda_{ab}/\xi_{ab} + 0.5 \sim 5$, we obtain $\lambda_{ab} \sim$~280~nm. This value is in close correspondence with the $\mu$SR results in slightly underdoped LaFeAs(O,F) \cite{Lut08}.
\begin{figure}[t]
\includegraphics[width=85mm]{Fig9_PD_anisotropy.eps}
\caption{(a) Lower critical fields as a function of temperature in PrFeAsO$_{1-y}$ single crystals (left axis). The solid line (right axis) presents the superfluid density $\lambda^2_{ab}(0)/\lambda^2_{ab}(T)$ in Fig.~\ref{ns_1111} determined by surface impedance measurements on crystals from the same batch. (b) Normalized temperature dependence of the anisotropies of $H_{c1}$ ($\gamma_{\lambda}$, closed circles) and $H_{c2}$ ($\gamma_{\xi}$, closed squares) in PrFeAsO$_{1-y}$ single crystals. The anisotropy of $H_{c2}$ in NdFeAsO$_{0.82}$F$_{0.18}$ ($\gamma_{\xi}$, open squares) measured by Y. Jia {\it et al.}\cite{Jia08Nd} is also plotted. The dashed line is a guide to the eye.}
\label{Hc1}
\end{figure}
Next, Fig.~\ref{Hc1}(b) shows the anisotropy of the lower critical fields, $\gamma_{\lambda}$ obtained from the results in Fig.~\ref{Hc1}(a). Here, since the penetration lengths are much larger than the coherence lengths for both {\boldmath $H$}$\parallel ab$ and {\boldmath $H$}$\parallel c$, the logarithmic term in Eq.(\ref{GL}) does not strongly depend on the direction of magnetic field. We thus assumed $H_{c1}^c/H_{c1}^{ab}\simeq \lambda_c/\lambda_{ab}$. The anisotropy $\gamma_{\lambda}\approx 2.5$ at very low temperature, and increases gradually with temperature. In Fig.~\ref{Hc1}(b), the anisotropy of the upper critical fields $\gamma_{\xi}$ is also plotted, where $\gamma_{\xi}$ is determined by the loci of 10\%, 50\% and 90\% of the normal-state resistivity measured up to 10~T in a crystal from the same batch \cite{Oka08}. Since $H_{c2}$ increases rapidly and well exceeds 10~T just below $T_c$ for {\boldmath $H$}$\parallel ab$, plotting $\gamma_{\xi}$ is restricted to a narrow temperature interval. In Fig.~\ref{Hc1}(b), we also plot the $H_{c2}$-anisotropy data from Ref.~\onlinecite{Jia08Nd} measured in NdFeAsO$_{0.82}$F$_{0.18}$. These indicate that the temperature dependence of $\gamma_{\lambda}$ is markedly different from that of $\gamma_{\xi}$.
According to the anisotropic Ginzburg-Landau (GL) equation in single-band superconductors, $\gamma_{\lambda}$ should coincide with $\gamma_{\xi}$ over the whole temperature range. Therefore, the large difference between these anisotropies provides strong evidence for multiband superconductivity in the present system. In a multiband superconductor, $\gamma_{\lambda}$ and $\gamma_{\xi}$ at $T_c$ are given as
\begin{equation}
\gamma_{\xi}^2(T_c)=\gamma_{\lambda}^2(T_c)=\frac{\langle \Omega^2
v_a^2 \rangle}{\langle \Omega^2 v_c^2 \rangle},
\end{equation}
where $\langle \cdot \cdot \cdot \rangle$ denotes the average over the Fermi surface, $v_a$ and $v_c$ are the Fermi velocities parallel and perpendicular to the $ab$-plane, respectively \cite{Kog02,Mir03}. $\Omega$ represents the gap anisotropy ($\langle\Omega^2\rangle = 1$), which is related to the pair potential $V(${\boldmath $v,v'$}$)=V_0\Omega(${\boldmath $v$})$\Omega(${\boldmath $v'$}). At $T=0$~K, the anisotropy of the penetration depths is
\begin{equation}
\gamma_{\lambda}^2(0)=\frac{\langle v_a^2 \rangle}{\langle v_c^2 \rangle}.
\end{equation}
The gap anisotropy does not enter $\gamma_{\lambda}(0)$, while $\gamma_{\xi}$ at $T=0$~K is mainly determined by the gap anisotropy of the active band responsible for superconductivity. Thus the gradual reduction of $\gamma_{\lambda}$ with decreasing temperature can be accounted for by considering that the contribution of the gap anisotropy diminished at low temperatures. This also implies that the superfluid density along the $c$-axis $\lambda_c^2(0)/\lambda_c^2(T)$ has steeper temperature dependence than that in the plane $\lambda_{ab}^2(0)/\lambda_{ab}^2(T)$. A pronounced discrepancy between $\gamma_{\xi}$ and $\gamma_{\lambda}$ provides strong evidence for the multiband nature of superconductivity in PrFeAsO$_{1-y}$, with different gap values in different bands. We note that similar differences between $\gamma_{\xi}(T)$ and $\gamma_{\lambda}(T)$, as well as $\lambda_c^2(0)/\lambda_c^2(T)$ and $\lambda_{ab}^2(0)/\lambda_{ab}^2(T)$, have been reported in the two-gap superconductor MgB$_2$ \cite{Lya04,Fle05}. We also note that ARPES \cite{Din08}, and Andreev reflection \cite{Sza08} have suggested multiband superconductivity with two gap values in Fe-based oxypnictides.
Band structure calculations for LaFeAsO$_{1-x}$F$_x$ yield an anisotropy of the resistivity of approximately 15 for isotropic scattering \cite{Sin08}, which corresponds to $\gamma_{\lambda}\sim 4$. This value is close to the observed value. The fact that $\gamma_{\xi}$ well exceeds $\gamma_{\lambda}$ indicates that the active band for superconductivity is more anisotropic than the passive band. According to band structure calculations, there are five relevant bands in LaFeAsO$_{1-x}$F$_x$. Among them, one of the three hole bands near the $\Gamma$ point and the electron bands near the M point are two-dimensional and cylindrical. The other two hole bands near the $\Gamma$ point have more dispersion along the $c$ axis \cite{Sin08}, although the shape of these Fermi surfaces is sensitive to the position of the As atom with respect to the Fe plane, which in turn depends on the rare earth \cite{Vil08}. Our results implying that the active band is more anisotropic is in good correspondence with the view that the nesting between the cylindrical hole and electron Fermi surfaces is essential for superconductivity. This is expected to make these two-dimensional bands the active ones, with a large gap, and the other more three-dimensional bands passive ones with smaller gaps.
\subsection{Quasiparticle conductivity}
\begin{figure}[t]
\includegraphics[width=80mm]{Fig10_s1_T.eps}%
\caption{(a) Temperature dependence of quasiparticle conductivity $\sigma_1$ normalized by its $T_c$ value at 28~GHz for two crystals of PrFeAsO$_{1-y}$. The solid line is a BCS calculation \cite{Zim91} of $\sigma_1(T)/\sigma_1(T_c)$ with $\tau=1.2\times10^{-13}$~s, which is estimated from $\rho(T_c)=77~\mu\Omega$cm and $\lambda_{ab}(0)=280$~nm. (b) $\sigma_1(T)$ at 28~GHz for crystal \#3 of Ba$_{1-x}$K$_x$Fe$_2$As$_2$ normalized at its 35-K value. The solid line is a BCS calculataion with $\tau(T_c)=4.4\times10^{-13}$~s.}
\label{sigma}
\end{figure}
Next let us discuss the quasiparticle conductivity $\sigma_1(T)$, which is extracted from $Z_s(T)$ through Eq.~(\ref{impedance}). The results for PrFeAsO$_{1-y}$ crystals are demonstrated in Fig.~\ref{sigma}(a). Although at low temperatures we have appreciable errors, it is unmistakable that $\sigma_1(T)$ shows a large enhancement just below $T_c$. This enhancement is considerably larger than the coherence peak expected in the BCS theory \cite{Zim91}.
\begin{figure}[tb]
\includegraphics[width=80mm]{Fig11_tau_ns.eps}%
\caption{Quasiparticle scattering rate $1/\tau$ as a function of the quasiparticle density $n_n(T)=n-n_s(T)$ with a comparison to the results in YBa$_2$Cu$_3$O$_{6.95}$ at 34.8~GHz \cite{Bon94}. The solid and dashed line are fits to the linear and cubic dependence, respectively.}
\label{tau}
\end{figure}
In Fig.~\ref{sigma}(b) we show the temperature dependence of the quasiparticle conductivity $\sigma_1(T)/\sigma_1(35$~K) in the cleanest sample \#3 of Ba$_{1-x}$K$_x$Fe$_2$As$_2$. It is evident again that below $T_c$, $\sigma_1(T)$ is enhanced from the normal-state values. Near $T_c$, the effects of coherence factors and superconducting fluctuations \cite{Bon07} are known to enhance $\sigma_1(T)$. The former effect, known as a coherence peak, is represented by the solid lines in Fig.~\ref{sigma}. We note that in the $s_\pm$ pairing state, the coherence peak in the NMR relaxation rate can be suppressed by a partial cancellation of total susceptibility $\sum_{\bf q}{\chi({\bf q})}$ owing to the sign change between the hole and electron bands \cite{Par08,Nag08}. For microwave conductivity, the long wave length limit ${\bf q} \rightarrow 0$ is important and the coherence peak can survive \cite{Dah08}, which may explain the bump in $\sigma_1(T)$ just below $T_c$.
At lower temperatures, where the coherence and fluctuation effects should be vanishing, the conductivity shows a further enhancement for the 122 crystal. This $\sigma_1(T)$ enhancement can be attributed to the enhanced quasiparticle scattering time $\tau$ below $T_c$. The competition between increasing $\tau$ and decreasing quasiparticle density $n_n(T)=n-n_s(T)$ makes a peak in $\sigma_1(T)$. This behavior is ubiquitous among superconductors having strong inelastic scattering in the normal state \cite{Bon07,Orm02,Shi07}. Following the pioneering work by Bonn {\it et al.} \cite{Bon94}, we employ the two-fluid analysis to extract the quasiparticle scattering rate $1/\tau(T)$ at low temperatures below $\sim 25$~K. In Fig.~\ref{tau}(b), the extracted $1/\tau(T)$ is plotted against the normailzed quasiparticle density $n_n(T)/n$ and compared with the reported results in the $d$-wave cuprate superconductor YBa$_2$Cu$_3$O$_{6.95}$ \cite{Bon94}. It is found that the scattering rate scales almost linearly with the quasiparticle density in our 122 system, which is distinct from $1/\tau$ in cuprates that varies more rapidly as $\sim n_n^3$. Such cubic dependence in cuprates is consistent with the $T^3$ dependence of spin-fluctuation inelastic scattering rate expected in $d$-wave superconcuctors, which have $T$-linear dependence of $n_n$ \cite{Qui94}. In $s$-wave superconductors without nodes, $n_n(T)$ and $1/\tau(T)$ are both expected to follow exponential dependence $\sim\exp({-\Delta/k_BT})$ at low temperatures \cite{Qui94}, which leads to the linear relation between $1/\tau(T)$ and $n_n(T)$. So this newly found relation further supports the fully gapped superconductivity in this system.
\section{Concluding remarks}
We have measured the microwave surface impedance of PrFeAsO$_{1-y}$ and Ba$_{1-x}$K$_x$Fe$_2$As$_2$ single crystals. We also developed an unambiguous method to determine lower critical fields by utilizing an array of miniature Hall sensors. The penetration depth and the lower critical field data both provide saturation behavior of superfluid density at low temperatures for PrFeAsO$_{1-y}$ crystals. The flat dependence of $\lambda_{ab}(T)$ and $n_s(T)$ at low temperatures demonstrates that the finite superconducting gap larger than $\sim1.6k_BT_c$ opens up all over the Fermi surface. For Ba$_{1-x}$K$_x$Fe$_2$As$_2$ single crystals we find that the temperature dependence of the superfluid density is sensitive to disorder and in the cleanest sample it shows exponential behavior consistent with fully opened two gaps.
The multiband superconductivity is also supported from the anisotropy of $H_{c1}$, which is found to decrease slightly with decreasing temperature. The lower values of $H_{c1}$ anisotropy than those of $H_{c2}$ suggest that the active band for superconductivity is more anisotropic than the passive band.
The microwave conductivity exhibits an enhancement larger than the BCS coherence peak, reminiscent of superconductors with strong electron scattering. The scattering rate analysis highlights the difference from the $d$-wave cuprates, which also supports the conclusion that the intrinsic order parameter in Fe-As superconductors is nodeless all over the Fermi surface. The present results impose an important constraint on the order parameter symmetry, namely the newly discovered Fe-based high-$T_c$ superconductors are fully gapped in contrast to the high-$T_c$ cuprate superconductors.
\section*{Acknowledgments}
This work has been done in collaboration with M. Ishikado, H. Kito, A. Iyo, H. Eisaki, S. Shamoto, who provided us PrFeAsO$_{1-y}$ crystals, S. Kasahara, H. Takeya, K. Hirata, T. Terashima, who grew Ba$_{1-x}$K$_x$Fe$_2$As$_2$ crystals, and K. Ikada, T. Kato, S. Tonegawa, H. Shishido, M. Yamashita, C.~J. van der Beek, M. Konczykowski, who contributed for several experiments. We also thank T. Dahm, H. Ikeda, K. Ishida, H. Kontani, and A. E. Koshelev for fruitful discussion, and M. Azuma, K. Kodama, V. Mosser, T. Saito, Y. Shimakawa, and K. Yoshimura for technical assistance. This work was supported by KAKENHI (No. 20224008) from JSPS, by Grant-in-Aid for the global COE program ``The Next Generation of Physics, Spun from Universality and Emergence'' from MEXT, Japan.
|
1,314,259,996,559 | arxiv | \section{Introduction}
For $\alpha>0$ we consider the Cauchy problem for the Navier-Stokes equations with fractional dissipation (of order $\alpha$) in three space dimensions
\begin{equation}\label{eq: NSalpha}
\begin{cases} \partial_t u + (u\cdot \nabla) u + \nabla p = - (-\Delta)^\alpha u \\
\div u =
\, .
\end{cases}
\end{equation}
Here $u: \mathbb{R}^3 \times [0,+\infty) \to \mathbb{R}^3$ is the velocity of an incompressible fluid, $p: \mathbb{R}^3 \times [0,+\infty) \to \mathbb{R}$ is the associated hydrodynamic pressure, $u( \cdot, 0)= u_0$ is a given, divergence-free initial datum, and the operator $(- \Delta)^\alpha$ corresponds to the Fourier symbol $|\xi|^{2\alpha}$.
The natural a priori bound associated with system \eqref{eq: NSalpha} is given by the total energy
$$\mathcal{E}(u; t) := \frac{1}{2} \int \lvert u \rvert^2 (x,t) \, \mathrm{d}x + \int_0^t \int \lvert (-\Delta)^{\alpha/2} u \rvert^2 (x,\tau) \, \mathrm{d}x \, \mathrm{d}\tau \,.$$
Moreover, the system \eqref{eq: NSalpha} has a natural scaling which preserves the equation: namely, given any solution $(u,p)$, also $(u_r, p_r)=(r^{2\alpha-1}u(rx,r^{2\alpha}t), r^{4\alpha-2}p(rx,r^{2\alpha}t))$ is a solution to the same equation.
Correspondingly, the total energy of the rescaled solution scales like $\mathcal{E}(u_r; t)= \frac{1}{r^{5-4\alpha}} \mathcal{E}(u;t)$. Consequently, the equation is called critical if the energy is scaling-invariant, namely for $\alpha= \frac 54$, subcritical for $\alpha>\frac 54$ and supercritical for $\alpha<\frac 54$.
Our main result shows that, given any (possibly large) initial datum $u_0$, the supercritical Navier-Stokes equation is globally well-posed at least for an open interval of orders below $\frac 54$.
\begin{theorem}[Global regularity below the critical order]\label{thm:main1} Let $\delta \in (0,1]$ arbitrary. Then for any divergence-free $u_0 \in H^{\delta}$ with $\norm{u_0}_{H^{\delta}}\leq M$ there exists $\epsilon= \epsilon(M, \delta)>0$ such that the Navier-Stokes equations of fractional order $\alpha \in (\frac{5}{4}-\epsilon, \frac{5}{4}]$ has a unique global smooth solution starting from $u_0$. Moreover, the dependence of $\epsilon$ on $M$ and $\delta$ is explicit through \eqref{eq: epsilonexplicit 54 ugly}
\end{theorem}
This result is related to a more general stability result on smooth solutions of the hyperdissipative Navier-Stokes equations, with respect to variations of both the initial datum and the fractional order. The following theorem quantifies the convergence of the initial data in a stronger norm since at difference from Theorem~\ref{thm:main1} it covers also ipodissipative orders.
\begin{theorem}\label{thm:main2} Let $p \in [1,2)$. The set of initial data $u_0$ and fractional orders $\alpha$ giving rise to global smooth solutions in $C([0,+\infty), H^1)$ of the fractional Navier-Stokes equations is an open set in
$$\left\{ u_0 \in L^p\cap H^{\frac 54}(\mathbb{R}^3; \mathbb{R}^3) : \div u_0=0\right\} \times \Big(\frac{3}{4}, \frac{5}{4}\Big]\, ,$$ endowed with the product topology
\end{theorem}
When $\alpha \geq \frac 54$ and {$u_0$ is smooth with sufficient decay at infinity}, the existence of global smooth solutions is well-known since \cite{Lions}. The attempt to build global smooth solutions for supercritical {Navier-Stokes} equations has been widely pursued. It has been done successfully for small initial data in scaling invariant norms, as in the classical results of Kato for $\alpha=1$.
In a different spirit, \cite{Tao} proved that the existence of a global regular solution still holds for any sufficiently regular initial datum when the right-hand side of the first equation in \eqref{eq: NSalpha} is replaced by a logarithmically supercritical operator; later, this result was generalized with a Dini-type condition in \cite{BMR}.
Recently, a result similar to Theorem~\ref{thm:main1} was obtained in \cite{CDM}, showing that for any $H^1$ initial datum there
are global smooth solutions of \eqref{eq: NSalpha} whenever $\alpha$ is sufficiently close to $\frac 54$ (and this closeness is uniform on bounded subsets of $H^1$). This was proven by means of an $\epsilon$-regularity result on a suitable notion of weak solutions, the known global regularity at $\alpha=\frac 54$ and a compactness argument. The present paper provides a simpler proof with respect to \cite{CDM}, and, not relaying on any contradiction argument, it has the advantage to provide an explicit $\varepsilon$ depending only on the size of the initial datum.
The paper is organized as follows. After recalling different notions of weak solutions to the system \eqref{eq: NSalpha} in Section~\ref{sec:prelim}, we prove a stability result on any finite time interval (of arbitrary length) in the fractional order and with respect of variations of the initial datum in Section~\ref{prop: locstab H3}. This estimate holds for any $\alpha \in (0,\frac32)$, but for low $\alpha$ we require stronger norms in the convergence of initial data. In Section~\ref{sec:leray}, we use the dissipation to pass from a local stability to a global result. In particular, following the ideas of Leray~\cite{Leray} (recently revisited in \cite{JiuWang} to cover the equation with fractional dissipation) we show the eventual regularization of the Navier-Stokes equation for $\alpha > \frac 56$. In turn, this kind of argument breaks down at $\alpha=\frac56$ since both norms $\|u(\cdot,t)\|_{L^2}$ and $\|(-\Delta)^{\frac\alpha2}u(\cdot,t)\|_{L^2}$ become scaling critical at this exponent.
Hence, we answer in Section~\ref{sec:reg-suit} an open question in \cite{JiuWang} by showing that, even for $\alpha\in (\frac34, \frac56)$, the eventual regularization holds, relying this time on partial regularity arguments.
A result of the type of Theorem~\ref{thm:main1} was recently obtained for the supercritical nonlinear wave euqation in \cite{CH}, the SQG equation in \cite{Cotivicol} and has been generalized to other active scalar equations \cite{SGS}; we expect that also a stability result as Theorem~\ref{thm:main2} could be suitably adapted to their context.
The long-time regularity relies in the case of the SQG equation on the scalar nature of the equation and on the maximum principle, indeed it works for any fractional dissipation $\alpha \in (0,1)$; a similar argument does not appear to apply to the Navier-Stokes equations.
\section{Preliminaries}\label{sec:prelim}
\subsection{Leray-Hopf solutions} We recall the classical concept of Leray-Hopf solutions introduced in the seminal papers of Leray \cite{Leray} and Hopf \cite{Hopf}.
\begin{definition}Let $u_0 \in L^2(\mathbb{R}^3)$ divergence-free. A Leray-Hopf solution is a distributional solution $(u,p)$ of \eqref{eq: NSalpha} on $\mathbb{R}^3 \times (0,T)$ such that
\begin{enumerate}[i)]
\item $u\in L^\infty((0,T), L^2(\mathbb{R}^3)) \cap L^2((0,T), H^\alpha(\mathbb{R}^3))$,
\item $p$ is the potential-theoretic solution of $-\Delta p = \div \div u\otimes u$,
\item For every $t\in (0,T)$, for $s=0$ and for almost every $0<s<t$ there holds the global energy inequality
\begin{align
\label{eq: energyineq2}
\frac{1}{2} \int \lvert u\rvert^2 (x,t) \, \mathrm{d}x + \int_s^t \int \lvert (-\Delta)^{\alpha/2} u \rvert^2(x, \tau) \, \mathrm{d}x \, \mathrm{d}\tau &\leq \frac{1}{2} \int \lvert u \rvert^2 (x,s)\ \, \mathrm{d}x \, .
\end{align}
\end{enumerate}
\end{definition}
From \eqref{eq: energyineq2} we deduce that up to changing $u$ on a set of measure $0$, we have $u\in C([0,T),L^2_w)$.
\begin{theorem}[Existence of Leray solutions]\label{thm: existLerayHopf} Let $\alpha>0$ and $u_0 \in L^2$ divergence-free. Then there exists a Leray-Hopf solution on $\mathbb{R}^3 \times (0,+\infty)$ to \eqref{eq: NSalpha}.
\end{theorem}
Let us recall from \cite{CDM, TY} that in the fractional case a Leray-Hopf solution can still be constructed following Leray's strategy in \cite{Leray}, that is as limit of solutions of the regularized system
\begin{equation}\label{eq: regularizedNS}
\begin{cases}
\partial_t u + ((u \ast \varphi_\epsilon) \cdot \nabla) u + \nabla p = -(-\Delta)^\alpha u \\
\div u =0
\, ,
\end{cases}
\end{equation} with the same initial datum $u_0$,
where $\{\varphi_\epsilon \}_{\epsilon>0}$ is a family of mollifiers in space. Indeed, \eqref{eq: regularizedNS} admits a unique solution $u_\epsilon \in C([0,+\infty), L^2) \cap L^2_{loc}([0,+\infty), \dot{H}^\alpha)$ which is smooth in the interior and satisfies the local and global energy equality associated to \eqref{eq: regularizedNS}. The pressure $p_\epsilon$ can be assumed to be the potential-theoretic solution of
\begin{equation}
-\Delta p_\epsilon = \div \big(((u_\epsilon \ast \varphi_\epsilon) \cdot \nabla) u_\epsilon\big) \, .
\end{equation}
Then there exists $u\in L^\infty([0,+\infty), L^2) \cap L^2([0,+\infty), \dot{H}^\alpha)$ such that for any $2 \leq p<\frac{6+4\alpha}{3}$ we have $u_\epsilon \rightarrow u$ strongly in $L^p_{loc}([0,+\infty) \times \mathbb{R}^3)$ and $p_\epsilon \rightarrow p$ strongly in $L^{p/2}_{loc}([0,+\infty) \times \mathbb{R}^3)$, where $p$ is now the potential theoretic solution of $-\Delta p=\div \div u\otimes u$. The pair $(u, p)$ is a Leray-Hopf solution of \eqref{eq: NSalpha}. Moreover, since $u_\epsilon$ satisfies even the local energy equality, the obtained weak solution $(u,p)$ is in fact even a suitable weak solution.
A point $(x, t)\in \mathbb{R}^3 \times (0,+\infty)$ is called a regular point of a Leray-Hopf
solution $(u,p)$ if there is a cylinder $Q_r(x, t)$ where $u$ is continuous. We denote by $Sing(u)$ the (relatively closed) set of points which are not regular. By classical boot-strap methods, we know that if $\alpha>\frac{1}{2}$ and $u \in L^\infty(Q_r(x,t))$ for some $r>0$, then $(x,t)$ is regular.
\subsection{Suitable weak solutions} Suitable weak solutions for the classical Navier-Stokes system have been introduced by Scheffer \cite{Scheffer1, Scheffer2} and Caffarelli-Kohn-Nirenberg \cite{CKN}. Only recently, the concept has been adapted to the ipodissipative range $\alpha \in (0,1)$ in \cite{TY} and to the hyperdissipative range $\alpha \in (1,2)$ in \cite{CDM}. The main ingredient is the Caffarelli-Silvestre extension for the fractional Laplacian \cite{CS} which allows to write a localized energy inequality also in the non-local setting. The existence of suitable weak solutions is obtained again through the regularization \eqref{eq: regularizedNS}, as shown in \cite{TY,CDM}. We recall here the notion of suitable weak solution for $\alpha \in (0,1)$, which will be essential to show the eventual regularization of solutions in Section~\ref{sec:reg-suit}.
\begin{definition}Let $\alpha \in (0,1)$. A Leray-Hopf solution $(u,p)$ on $\mathbb{R}^3 \times (0,T)$ is a suitable weak solution, if for every $\varphi\in C^\infty_c(\mathbb{R}^4_+\times (0,T))$ with $\partial_y \varphi(\cdot, 0, \cdot)=0$ on $\mathbb{R}^3 \times (0,T)$ and $\varphi \geq 0$ and almost every $t\in (0,T)$, there holds the localized energy inequality
\begin{align}\label{eq: localized energy inequality ipo}
\frac{1}{2}\int_{\mathbb{R}^3} &\lvert u \rvert^2 (x,t) \varphi(x,0,t) \, \mathrm{d}x+ c_\alpha \int_0^t \int_{\mathbb{R}^4_+} y^b \lvert \overline{\nabla} u^\ast \rvert^2 \varphi \, \mathrm{d}x \, \mathrm{d}y \, \mathrm{d}\tau \\
&\leq \int_0^t \int_{\mathbb{R}^3} \left(\frac{\lvert u \rvert^2}{2} \partial_t \varphi \vert_{y=0} + \left( \frac{\lvert u \rvert^2}{2}+ p\right) u\cdot \nabla \varphi\vert_{y=0} \right)\, \mathrm{d}x \, \mathrm{d}\tau +c_\alpha \int_0^t \int_{\mathbb{R}^4_+} y^b \lvert u^\ast \rvert^2 \overline{\Delta}_b \varphi \, \mathrm{d}x \, \mathrm{d}y \, \mathrm{d}\tau \, ,\nonumber
\end{align}
where $b:=1-2\alpha$, $u^\ast$ is the Caffarelli-Silvestre extension of order $\alpha$, the constant $c_\alpha$ the associated normalizing constant and $\overline{\Delta_b} u^\ast:= y^{-b}\div( y^b \overline{\nabla} u^\ast)$.
\end{definition}
\subsection{Weak-strong uniqueness}
Leray showed that Leray–Hopf solutions coincide with the classical solutions as long as the latter exist. Indeed, consider two Leray-Hopf solutions $u$ and $v$ with the same initial datum. Assuming smoothness, we can multiply the difference equation by $(u-v)$ and integrate in space. By incompressibility, we obtain
\begin{equation}\label{eq: diffeqLerayHopf}
\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \int \lvert u-v\rvert^2(x,t) \, \mathrm{d}x + \int \lvert D(u-v) \rvert^2 \, \mathrm{d}x = \int ((u-v) \cdot \nabla)(u-v) \cdot v \, \mathrm{d}x \, .
\end{equation}
Leray noticed that through regularization we can still derive \eqref{eq: diffeqLerayHopf} with an inequality, provided $v\in L^2((0,T), L^\infty)$ only.
Uniqueness then follows from a standard Gr\"onwall argument. As expected, $v\in L^2((0,T), L^\infty)$ still gives uniqueness in the hyperdissipative range $\alpha>1$ (see \cite{CDM}); this requirement can even be weakened using the stronger dissipation on the left-hand side (see forthcoming Proposition \ref{prop: weakstrong uniq}). In the ipodissipative case $0<\alpha<1$ however, it only holds $u-v \in L^2((0,T), H^\alpha)$ and the right-hand side of \eqref{eq: diffeqLerayHopf} is not meaningful, written in this form. Instead assuming smoothness, we observe that by incompressibility
\begin{equation}\label{eq: diffeqLerayHopf ipo}
\int ((u-v) \cdot \nabla)(u-v) \cdot v \, \mathrm{d}x = \int \div \left((u-v) \otimes (u-v)\right) \cdot v \, \mathrm{d}x \,.
\end{equation}
Integrating by parts $(-\Delta)^{(1-\alpha)/2}$-derivatives on $v$, this allows to deduce a weak-strong uniqueness criterion involving only assumptions on the integrability of $(-\Delta)^{(1-\alpha)/2}v$ and $v$. For $\alpha= 1$, this criterion recovers the classical one $v\in L^2((0,T), L^\infty)$.
\begin{prop}\label{prop: weakstrong uniq}
Let $0<\alpha<\frac{3}{2}$. Let $(u,p)$ and $(v,q)$ be Leray-Hopf solutions of \eqref{eq: NSalpha} on $\mathbb{R}^3\times (0,T)$ with common initial datum $u_0 \in L^2$ with $\div u_0=0$. If we additionally assume that
\begin{equation}\label{eq: WS}
\begin{cases}
v \in L^2((0,T), L^{\frac 3{\alpha-1}}) \, \qquad& \mbox{if $\alpha \in \big[ 1, \frac 32\big)$}\, ,\\
(-\Delta)^{(1-\alpha)/2} v \in L^2((0,T), L^\infty) \, \qquad& \mbox{if $\alpha \in \big[\frac 34,1\big)$}\, ,\\
(-\Delta)^{(1-\alpha)/2} v \in L^2((0,T), L^\infty) \mbox{ and } v \in L^2((0,T), L^\frac{3}{\alpha})\, \qquad& \mbox{if $\alpha \in \big(0,\frac 34\big)$}\, ,
\end{cases}
\end{equation}
then $u=v$ in $L^2$ on $(0,T)$.
\end{prop}
\begin{proof}
We can assume that $u,v \in C((0,T), L^2_w)$. Using the regularization of Leray and appropriate commutator estimates justifying the integration by parts of $(-\Delta)^{(1-\alpha)/2}$-derivatives on $v$, it is straight-forward to show that for every $t\in (0,T)$ it holds
\begin{align}\label{eq: diffeq leray sol}
\frac{1}{2} \int \lvert u-v\rvert^2 (x,t) \, &\mathrm{d}x + \int_0^t \int \lvert (-\Delta)^{\alpha/2} (u-v) \rvert^2(x, \tau) \, \mathrm{d}x \, \mathrm{d}\tau \nonumber \\ &\leq C \int_0^t \norm{(-\Delta)^{(1-\alpha)/2}v(\cdot, \tau)}_{L^\infty} \int \lvert (-\Delta)^{\alpha/2}(u-v) \rvert \lvert u-v\rvert (x, \tau)\, \mathrm{d}x \, \mathrm{d}\tau \, .
\end{align}
Observe that under the assumption \eqref{eq: WS}, the right-hand side of \eqref{eq: diffeq leray sol} is well-defined. The additional requirement $v\in L^2((0,T), L^\frac{3}{\alpha})$ in case $0<\alpha < \frac{3}{4}$ comes to ensure that the divergence terms vanish, that is that $\lvert v \rvert^2 \lvert u \rvert, \lvert p \rvert \lvert v \rvert, \lvert u \rvert \lvert q \rvert \in L^1\,.$ (See proof of Lemma \ref{lem: energyineq for diffeq H1} for a similar reasoning.)
Thus by reabsorbing on the left-hand side, we obtain for every $t\in (0,T)$
\begin{equation*}
\int \lvert u-v \rvert^2 (x,t) \, \mathrm{d}x \leq C \int_0^t \norm{(-\Delta)^{(1-\alpha)/2}v(\cdot, \tau)}_{L^\infty}^2 \int \lvert u-v \rvert^2 (x, \tau)\, \mathrm{d}x \, \mathrm{d}\tau \, .
\end{equation*}
Since $\tau \mapsto \norm{(-\Delta)^{(1-\alpha)/2}v(\cdot, \tau)}_{L^\infty}^2\in L^1((0,T))$ and $u(\cdot, 0)=v(\cdot, 0)$ in $L^2$, we conclude from Gr\"onwall that $u(\cdot, t)=v(\cdot, t)$ in $L^2$ for any $t \in (0,T)$. For $\alpha > 1$, we conclude analogously, using the right-hand side in the form of \eqref{eq: diffeqLerayHopf} and observing that $\norm{\nabla u}_{L^\frac{6}{5-2\alpha}} \leq C \norm{(-\Delta)^{\alpha/2} u}_{L^2}$.
\end{proof}
\subsection{Stability of constants in Sobolev and commutator estimates}
The stability of all constants in $\alpha$ will be crucial to close the argument, we will, whenever needed, keep track of them rigorously. If not stated otherwise, $C$ will denote a universal constant, independent on $\alpha$, which may change line by line. We denote by $\bar C_{\alpha}$ the constant from the embedding of $\dot{H}^\alpha(\mathbb{R}^3) \hookrightarrow L^\frac{6}{3-2\alpha}(\mathbb{R}^3)$ and we recall from its proof (for instance \cite{Stein}) that those constants are uniformly bounded in $\alpha$ away from the endpoint $\alpha=\frac{3}{2}$.
Let $0<s<2$, $s_1,s_2 \in [0,1)$, such that $s= s_1+s_2$. Let $p_1, p_2 \in (1,\infty)$ and $r \in [1,\infty)$ be such that $\frac 1 r = \frac 1{p_1}+\frac 1{p_2}$. We recall the following Leibnitz-type inequality from \cite[Theorem A.8]{KPV} and \cite{Dancona}
\begin{equation}
\label{eqn:commuta}
\| (-\Delta)^{s/2}(uv)-u(-\Delta)^{s/2}v- (-\Delta)^{s/2}u v\|_{L^r} \leq C(s, p_1,p_2) \| (-\Delta)^{s_1/2} u \|_{L^{p_1}}\| (-\Delta)^{s_2/2} u \|_{L^{p_2}}
\end{equation}
To track the dependence of our constants explicitly, throughout the paper we call $\bar C$ and $\bar D$ uniform upper bounds for the constant in the Sobolev inequality and in \eqref{eqn:commuta} respectively, that is
$$\bar C:= \sup_{\alpha \in [0,\frac 54]} \max\{1,\bar C_\alpha\}, \qquad
\bar D(s) := \sup_{p_1,p_2 \in [2, 12]}\max\{1,C(s, p_1,p_2)\}.$$
\section{Stability on finite time intervals for $\alpha>0$
}
\begin{prop}\label{prop: locstab H3}
Let $T>0$ arbitrary and $0< \alpha \leq \frac{5 }{4}$. Let $s=3$ if $\alpha \in (0, \frac{5}{4}]$, $s=2$ if $\alpha \in (\frac{1}{2}, \frac{5}{4}]$ and $s\in (\frac{5}{2}-2\alpha,1 ]$ if $\alpha\in (\frac{3}{4}, \frac{5}{4}]$. Suppose that there exists a smooth solution $u \in C([0,T], H^s) \cap L^2([0,T], H^{s+\alpha})$ to the Navier-Stokes equation of order $\alpha$ with divergence-free initial datum $u_0\in H^s(\mathbb{R}^3)$. We additionally assume that
\begin{equation}
u \in L^2([0,T], H^{s+\alpha+\delta}) \cap L^2([0, T], H^{s+1}) \, .
\end{equation}
for some $\delta \in (0, 1]$. Then there exists $\epsilon>0$ such that for any $v_0\in H^s(\mathbb{R}^3) $ divergence-free and for any $0< \beta \leq \frac{5}{4}$ satisfying
\begin{equation}\label{eqn:eps-choice}
\norm{u_0-v_0}_{H^s}^2 + \lvert \alpha-\beta \rvert^\delta < \epsilon,
\end{equation}
there exists a unique solution $v \in C([0,T], H^s) \cap L^2([0,T], H^{s+\beta})$ to the fractional Navier-Stokes equations of order $\beta$ with initial datum $v_0$ which is smooth in the interior.
\end{prop}
\subsection{An energy inequality for the difference equation} The stability argument employs an estimate of the difference of two fractional Laplacians of order $\alpha$ and $\beta$ respectively in terms of $\lvert \alpha-\beta \rvert$. This estimate also drives the additional regularity assumption $u_0 \in H^{\delta}$ for $\delta>0$.
\begin{lemma}\label{lem: fraclapRn} Let $s\geq 0$, $\delta \in \mathopen(0,1 \mathclose]$, $\frac{\delta}{2} \leq \beta \leq \alpha$, and $u \in H^{s+2\alpha+\delta}(\mathbb{R}^n)$. Then we can estimate
\begin{equation} \label{boundeasy fraclapRn}
\norm{[(-\Delta)^{\alpha}-(-\Delta)^{\beta}]u}_{H^s} \leq C (\alpha-\beta)^\delta \norm{u}_{H^{s+2\alpha+\delta}} \,,
\end{equation}
where $C>0$ is a universal constant independent of $s$, $\delta$, $\alpha$ and $\beta$.
\end{lemma}
\begin{proof}
Since the fractional Laplacian commutes with derivatives, it is enough to consider the case $s=0$. Let $\beta \in [\frac{\delta}{2}, \alpha)$. We write
\begin{equation*}
\norm{[(-\Delta)^{\alpha}-(-\Delta)^{\beta}] u}_{L^2}^2 = \int (\lvert \xi \rvert^{2\alpha}-\lvert \xi \rvert ^{2\beta})^2 \lvert \hat{ u}(\xi) \rvert^2 \, \mathrm{d}\xi = \rm I + II \, ,
\end{equation*}
where we split the integration domain into $\{\lvert\xi\rvert \leq 1\}$ and $\{\lvert \xi\rvert > 1\}$ respectively. Since $1-e^{-x} \leq x \text{ and } 1-\frac{1}{x} \leq \ln x \leq x-1$ for every $x>0$ and since $\left(x_1+x_2\right)^p \leq \max\{2^{p-1}, 1 \} ( x_1^p+x_2^p)$ for $ x_1, x_2>0$ and $p>0$, we have that
\begin{align*}
\rm I &= \int_{\lvert \xi \rvert \leq 1} (\lvert \xi \rvert^{2\beta}- \lvert \xi \rvert^{2\alpha})^{(2-2\delta)} \lvert \xi \rvert^{4\beta\delta}\left(1- e^{- 2(\alpha-\beta)\lvert \ln \lvert \xi \rvert\rvert }\right)^{2\delta} \lvert \hat{u}(\xi)\rvert^2 \, \mathrm{d}\xi \\
&\leq 2^{2\delta} (\alpha-\beta)^{2\delta} \int_{\lvert \xi \rvert \leq 1} (\lvert \xi \rvert^{2\beta}- \lvert \xi \rvert^{2\alpha})^{(2-{2\delta})} \lvert \xi \rvert^{4\beta \delta} \lvert \ln \lvert \xi \rvert \rvert^{2\delta} \lvert \hat{u}(\xi)\rvert^2 \, \mathrm{d}\xi \\
&\leq 2^{2\delta} (\alpha-\beta)^{2\delta} \max \{2^{1-{2\delta}}, 1\} \int_{\lvert \xi \rvert \leq 1} (\lvert \xi \rvert^{2\beta (2-{2\delta})} +\lvert \xi \rvert^{2\alpha (2-{2\delta})}) \lvert \xi \rvert^{4\beta {\delta}} \lvert \xi \rvert^{-{2\delta}} \lvert \hat{u}(\xi)\rvert^2 \, \mathrm{d}\xi \\
&\leq \max \{2, 2^{2\delta} \} (\alpha-\beta)^{2\delta} \int_{\lvert \xi \rvert \leq 1} (\lvert \xi \rvert^{4\beta-{2\delta}} +\lvert \xi \rvert^{4\alpha-{2\delta}(1+2(\alpha-\beta))}) \lvert \hat{u}(\xi)\rvert^2 \, \mathrm{d}x \, .
\intertext{Similarly, we estimate}
\rm II
&= \int_{\lvert \xi \rvert >1} (\lvert \xi \rvert^{2\alpha}-\lvert \xi \rvert^{2\beta})^{(2-{2\delta})} \lvert \xi \rvert^{4\alpha\delta} \left(1-e^{-2(\alpha-\beta)\ln\lvert\xi\rvert}\right)^{2\delta} \lvert \hat{u}(\xi)\rvert^2 \mathrm{d}\xi \\
&\leq 2^{2\delta} (\alpha-\beta)^{2\delta} \int_{\lvert \xi \rvert >1} (\lvert \xi \rvert^{2\alpha}-\lvert \xi \rvert^{2\beta})^{(2-{2\delta})} \lvert\xi\rvert^{4\alpha \delta} (\ln\lvert \xi\rvert)^{2\delta} \lvert \hat{u}(\xi)\rvert^2\mathrm{d}\xi\\
&\leq 2 (\alpha-\beta)^{2\delta} \int_{\lvert \xi \rvert >1} \lvert\xi\rvert^{4\alpha} \lvert \xi \rvert^{2\delta} \lvert \hat{u}(\xi)\rvert^2\mathrm{d}\xi \,.
\end{align*}
Collecting terms, we have obtained
\begin{equation*}
\norm{[(-\Delta)^{\alpha}-(-\Delta)^{\beta}]u}_{L^2}^2 \leq (\alpha-\beta)^{2\delta} \left( 2\norm{u}_{\dot{H}^{2\alpha+\delta}} + \max \{2, 2^{2\delta} \}( \norm{u}_{\dot{H}^{2\beta-\delta}}^2 + \norm{u}_{\dot{H}^{2\alpha- {2\delta}(\alpha-\beta)-\delta}}^2) \right) \, .
\end{equation*}
We conclude by observing that by the interpolation $\norm{u}_{\dot{H}^{2\beta-\delta}}, \norm{u}_{\dot{H}^{2\alpha-{2\delta}(\alpha-\beta)-\delta}} \leq \norm{u}_{H^{2\alpha + \delta}}\,.$
\end{proof}
\begin{lemma}\label{lem: energyineq for diffeq H1} Let $ \frac{3}{4} < \alpha \leq \frac{5}{4} $ and $\frac{5}{2}-2\alpha < s \leq 1$. Let $u_0 \in H^s$ divergence-free. Consider $u \in C(\left[0,T\right], H^s) \cap L^2([0,T], H^{s+\alpha})$, a smooth solution to the Navier-Stokes equation of order $\alpha$ starting from $u_0$. We additionally assume that
\begin{equation}\label{eq: addintdiffeq}
u \in L^2(\left[0, T\right], H^{s+\alpha+\delta}) \text{ and } Du \in L^2([0,T], H^s)
\end{equation}
for some positive $\delta \in (0, 1]$. Then, for any $\frac{3}{4} < \beta \leq \frac{5}{4}$ such that
\begin{equation}\label{eq: hypdiffeqHs}
\lvert \alpha-\beta \rvert \leq \frac{1}{2} \min \{ \delta, \frac 12 (s-(\frac{5}{2}-2\alpha)) \}
\end{equation}
and any smooth solution $v \in C([0,T], H^s) \cap L^2([0,T], H^{s+\beta})$ of the Navier-Stokes equation of order $\beta$, there holds with $f(t)=\norm{(v-u)(t)}_{H^s}^2$ for every $t\in [0,T]$ that
\begin{equation}\label{eq: diffineq}
f(t) \leq f(0) + \int_0^t C_0 f^\gamma(\tau) + C_1 \left( \norm{D u(\tau)}_{H^s}^2 + \norm{u(\tau)}_{H^{s+\alpha+\delta}}^2 \right) f(\tau) \, \mathrm{d}\tau + C_2 \lvert \alpha-\beta \rvert^{\delta} \int_0^t \norm{u(\tau)}_{H^{s+\alpha+\delta}}^2 \, \mathrm{d}\tau \, ,
\end{equation}
where $\gamma=\gamma(s, \beta)=\frac{6\beta-5+2s}{4\beta-5+2s}$, $C_0=C_0(s,\alpha)>0$, $C_1=C_1(s)>0$ and $C_2>0$ are universal. Moreover, the dependence of $C_0$ on $\alpha$ and $s$ is through $
C_0 \leq 2 (12(1+ \bar D) \bar C^2)^{\frac{5}{s-(\frac{5}{2}-2\alpha)}} \, .$
\end{lemma}
\begin{proof}
Set $w:=v-u$, $w_0:= v_0-u_0$ and call $p$ the difference of the pressure terms. Assume for now that $\beta \leq \alpha$ (the other case is handled analogously). By hypothesis, the difference $w \in C(\left[0,T\right], H^s)\cap L^2([0,T], H^{s+\beta})$ is divergence-free and solves the equation
\begin{equation}\label{eq: diffeq}
\partial_t w + (w\cdot \nabla)w+(u\cdot\nabla)w+(w\cdot\nabla)u+ \nabla p=-(-\Delta)^{\beta} w+ \left[(-\Delta)^{\alpha}-(-\Delta)^{\beta}\right]u
\end{equation}
with initial datum $w(\cdot, 0)=w_0$.
We multiply the equation by $w\psi_R$ for a cut-off $\psi_R \in C^\infty_c(\mathbb{R}^3)$ such that $\psi_R \equiv 1$ on $B_R(0)$, $0\leq \psi_R \leq 1$ and $\lvert \nabla \psi_R \rvert \leq \frac{C}{R}$. By incompressibility, we obtain
\begin{align*}
\frac{1}{2} \int &\lvert w \rvert^2(x,t) \psi_R \, \mathrm{d}x + \int_0^t \int (-\Delta)^{\beta} w \cdot w \psi_R \, \mathrm{d}x \, \mathrm{d}\tau \leq
\frac{1}{2} \int \lvert w_0 \rvert^2 \psi_R \, \mathrm{d}x + \left \lvert \int_0^t \int (w \cdot \nabla) u \cdot w \psi_R \, \mathrm{d}x \, \mathrm{d}\tau \right \rvert \\ &+ \left \lvert \int_0^t \int [(-\Delta)^\alpha-(-\Delta)^\beta] u \cdot w \, \psi_R \, \mathrm{d}x \, \mathrm{d}\tau \right \rvert + \left \lvert \int_0^t \int \left( w\left( \frac{\lvert w \rvert^2}{2} +p \right) + u \frac{\lvert u \rvert^2}{2} \right) \cdot \nabla \psi_R \, \mathrm{d}x \, \mathrm{d}\tau \right \rvert \, .
\end{align*}
Since $\lvert w \rvert^3 + \lvert w \rvert \lvert p \rvert + \lvert u \rvert^3 \in L^1([0, T], L^1)$ by Sobolev embeddings and Calderon-Zygmund estimates, we deduce that in the limit $R \to \infty$ the third line is negligible.
Passing to the limit $R \to \infty$, we have for $t\in [0,T]$ that
\begin{align*}
\frac{1}{2} \int \lvert w \rvert^2(x,t) \, \mathrm{d}x + \int_0^t \int \lvert (-\Delta)^{\beta/2} w \rvert^2 \, \mathrm{d}x \, \mathrm{d}\tau &\leq
\frac{1}{2} \int \lvert w_0 \rvert^2 \, \mathrm{d}x + \left \lvert \int_0^t \int (w \cdot \nabla) u \cdot w \, \mathrm{d}x \, \mathrm{d}\tau \right \rvert \\ &+ \left \lvert \int_0^t \int [(-\Delta)^\alpha-(-\Delta)^\beta] u \cdot w \, \mathrm{d}x \, \mathrm{d}\tau \right \rvert \, .
\end{align*}
By Gagliardo-Nirenberg-Sobolev inequality and Young, we estimate
\begin{align*}
\left \lvert \int (w \cdot \nabla) u \cdot w \, \mathrm{d}x \right \rvert &\leq \bar C \norm{w}_{L^\frac{6}{3-2\beta}} \norm{D u}_{L^\frac{6}{2\beta}} \norm{w}_{L^2} \leq \frac{1}{4} \int \lvert (-\Delta)^{\beta/2} w \rvert^2 \, \mathrm{d}x + \bar C^2 \norm{D u}_{L^\frac{6}{2\beta}}^2 \norm{w}_{L^2}^2 \, .
\end{align*}
To bound the last factor, we observe that as long as $\frac{3}{4} \leq \beta \leq \frac{3}{2}$ it holds by Gagliardo-Nirenberg-Sobolev and interpolation that
\begin{equation}\label{eq: est62beta}
\norm{f}_{L^\frac{6}{2\beta}} \leq \bar C \norm{f}_{\dot{H}^{\frac{3}{2}-\beta}} \leq \bar C \norm{(-\Delta)^{\beta/2} f}_{L^2}^{\frac{3}{2\beta}-1} \norm{f}_{L^2}^{2-\frac{3}{2\beta}} \, .
\end{equation}
By Plancherel, Young and Lemma \ref{lem: fraclapRn}, we estimate the dissipative term by
\begin{align}
\left \lvert \int [(-\Delta)^\alpha-(-\Delta)^\beta] u \cdot w \, \mathrm{d}x \right \rvert &\leq \frac{1}{4} \int \lvert (-\Delta)^{\beta/2} w \rvert^2 \, \mathrm{d}x + \int \lvert [(-\Delta)^{\alpha-\beta/2}-(-\Delta)^{\beta/2}] u \rvert^2 \, \mathrm{d}x \\\label{eqn:alphabeta}
&\leq \frac{1}{4} \int \lvert (-\Delta)^{\beta/2} w \rvert^2 \, \mathrm{d}x + C_2(\alpha-\beta)^\delta \norm{u}_{H^{\alpha+\delta}}^2 \, ,
\end{align}
where $C_2$ is the universal constant from Lemma \ref{lem: fraclapRn}. Reabsorbing in the left-hand side, we have
\begin{equation}\label{eq: diffineq L2}
\frac{\mathrm{d}}{\mathrm{d}t}\norm{ w(t)}_{L^2}^2+ \norm{(-\Delta)^{\beta/2} w(t)}_{L^2}^2 \leq 2 \bar C^4 \norm{ u(t)}_{\dot{H}^{\frac{5}{2}-\beta}}^2 \norm{w(t)}_{L^2}^2 + 2 C_2 (\alpha-\beta)^\delta \norm{u(t)}_{H^{\alpha+\delta}}^2 \, .
\end{equation}
Consider first $s<1$. Since $(\lvert w \rvert + \lvert u \rvert )\lvert (-\Delta)^{s/2} w \rvert^2 + \lvert (-\Delta)^{s/2} p \rvert \lvert (-\Delta)^{s/2} w \rvert \in L^1([0,T], L^1)$, we can argue as before to obtain the following energy inequality for the derivative of order $s$
\begin{align*}
\frac{1}{2}&\frac{\mathrm{d}}{\mathrm{d}t} \norm{(-\Delta)^{s/2} w(t)}_{L^2}^2 + \norm{(-\Delta)^{(s+\beta)/2} w (t)}_{L^2}^2
\\& \leq \left \lvert \int \left[
(-\Delta)^{s/2}\left[ (w \cdot \nabla) w \right] - (w \cdot \nabla) (-\Delta)^{s/2} w
\right] \cdot (-\Delta)^{s/2} w \, \mathrm{d}x \right \rvert \\
&+\left \lvert \int \left[ (-\Delta)^{s/2} \left[ (u \cdot \nabla) w\right] - (u \cdot \nabla) (-\Delta)^{s/2} w
\right] \cdot (-\Delta)^{s/2} w \, \mathrm{d}x \right \rvert \\
&+ \left \lvert \int \left[ (-\Delta)^{s/2} \left[ (w \cdot \nabla) u\right] - (w \cdot \nabla) (-\Delta)^{s/2} u + (w \cdot \nabla) (-\Delta)^{s/2} u \right] \cdot (-\Delta)^{s/2} w \, \mathrm{d}x \right \rvert \\
&+ \left \lvert \int \left[(-\Delta)^\alpha-(-\Delta)^\beta \right] (-\Delta)^{s/2} u \cdot (-\Delta)^{s/2} w \, \mathrm{d}x \right \rvert =: \rm I + II +III+ IV \, .
\end{align*}
We estimate line by line. Recall that by interpolation, we can bound as long as $1-\beta \leq s \leq 1$
\begin{equation}\label{eq: estgrad}
\norm{\nabla f}_{L^2} \leq \norm{(-\Delta)^{(s+\beta)/2} f}_{L^2}^\frac{1-s}{\beta} \norm{(-\Delta)^{s/2}f}_{L^2}^{1-\frac{(1-s)}{\beta}}
\end{equation}
with constant 1. Then the first term is estimated by Gagliardo-Nirenberg-Sobolev, \eqref{eqn:commuta} (with $s_1=s$, $p_1=\frac{6}{2\beta}$, $s_2=0$, $p_2=2$), \eqref{eq: est62beta} and \eqref{eq: estgrad} by
\begin{align*}
\text{I} &\leq \bar C \norm{ (-\Delta)^{s/2}\left[ (w \cdot \nabla) w \right] - (w \cdot \nabla) (-\Delta)^{s/2} w
}_{L^\frac{6}{3+2\beta}} \norm{(-\Delta)^{(s+\beta)/2} w}_{L^2} \\
&\leq \bar C (1+\bar D) \norm{(-\Delta)^{s/2}w}_{L^\frac{6}{2\beta}} \norm{\nabla w}_{L^2} \norm{(-\Delta)^{(s+\beta)/2} w}_{L^2} \\
&\leq \bar C^2 (1+\bar D) \norm{(-\Delta)^{(s+\beta)/2}w}_{L^2}^{\frac{5-2s}{2\beta}} \norm{(-\Delta)^{s/2} w}_{L^2}^\frac{6\beta-5+2s}{2\beta} \, .
\end{align*}
The hypothesis \eqref{eq: hypdiffeqHs} guarantees that $s > \frac{5}{2}-2\beta$,
and thus we can use Young with exponents $\frac{4\beta}{5-2s}$ and $\frac{4\beta}{4\beta-5+2s}$ to achieve
\begin{equation*}
\text{I} \leq \frac{1}{12} \norm{(-\Delta)^{(s+\beta)/2}w}_{L^2}^2 + C_0 \norm{(-\Delta)^{s/2}w}_{L^2}^{2 \gamma} \, ,
\end{equation*}
where we introduced $\gamma= \gamma(s, \beta)$ as in the statement and
\begin{align}
C_0&=C_0(s,\beta):=((1+\bar D) \bar C^2)^\frac{4\beta}{4\beta-5+2s}
\left( \frac{12(5-2s)}{4\beta}\right)^\frac{5-2s}{4\beta-5+2s}
\leq (12(1+ \bar D) \bar C^2)^\frac{5}{s-(\frac{5}{2}-2\alpha)}
\, . \label{eq: C0Hs}
\end{align}
The second term is estimated similarly (using now \eqref{eqn:commuta} with $s_1=s$, $p_1=\frac{6}{2(s+\beta)-2}$, $s_2=0$, $p_2=\frac{6}{5-2(s+\beta)}$) by
\begin{align*}
\text{II} &\leq (1+\bar D) \norm{(-\Delta)^{s/2} u}_{L^\frac{6}{2(s+\beta)-2}} \norm{\nabla w}_{L^\frac{6}{5-2(s+\beta)}} \norm{(-\Delta)^{s/2} w}_{L^2} \\
&\leq (1+\bar D) \bar C^2 \norm{u}_{\dot{H}^{\frac{5}{2}-\beta}} \norm{(-\Delta)^{(s+\beta)/2}w}_{L^2} \norm{(-\Delta)^{s/2}w }_{L^2} \\
&\leq \frac{1}{12} \norm{(-\Delta)^{(s+\beta)/2} w}_{L^2}^2 + 3 (1+\bar D)^2 \bar C^4 \norm{u}_{\dot{H}^{\frac{5}{2}-\beta}}^2 \norm{(-\Delta)^{s/2}w}_{L^2}^2 \, .
\end{align*}
We split the third line $\text{III}=\text{III}.1+\text{III}.2$, where $\text{III}.1$ contains the first two addends and $\text{III}.2$ stands for the last one. $\text{III}.1$ is again estimated using Gagliardo-Nirenberg-Sobolev, \eqref{eqn:commuta}(with $s_1=s$, $p_1=2$, $s_2=0$, $p_2=\frac{6}{2\beta}$) and \eqref{eq: est62beta} by
\begin{align*}
\text{III}.1&\leq \bar C^2 (1+\bar D) \norm{u}_{\dot{H}^{\frac{5}{2}-\beta}} \norm{(-\Delta)^{s/2}w}_{L^2} \norm{(-\Delta)^{(s+\beta)/2} w}_{L^2} \\
&\leq \frac{1}{24} \norm{(-\Delta)^{(s+\beta)/2} w}_{L^2} ^2 + 6 \bar C^4 (1+\bar D)^2 \norm{u}_{\dot{H}^{\frac{5}{2}-\beta}}^2\ \norm{(-\Delta)^{s/2}w}_{L^2}^2.
\end{align*}
To estimate $\text{III}.2$, we distinguish two cases. Assume first $s+\beta \geq \frac{3}{2}$. Then $\norm{f}_{\dot{H}^{\frac{3}{2}-\beta}} \leq \norm{f}_{H^s}$ with constant $1$ and thus
\begin{align*}
\text{III}.2 &= \left \lvert \int (w \cdot \nabla) (-\Delta)^{s/2} u \cdot (-\Delta)^{s/2} w \right \rvert \leq \frac{1}{24} \norm{(-\Delta)^{(s+\beta)/2} w}_{L^2} ^2 + 6\bar{C}^4 \norm{u}_{\dot{H}^{1+s}}^2 \norm{w}_{H^s}^2 \, .
\end{align*}
If now $s+\beta<\frac{3}{2}$, we use that $\norm{f}_{L^{\frac{6}{2(s+\beta)}}} \leq \bar C \norm{f}_{\dot{H}^{\frac{3}{2}-(s+\beta)}}$ to obtain
\begin{align*}
\text{III}.2 &\leq \frac{1}{24} \norm{(-\Delta)^{(s+\beta)/2} w}_{L^2} ^2 + 6 \bar C^6 \norm{ u}_{\dot{H}^{\frac{5}{2}-\beta}}^2 \norm{w}_{H^s}^2 \, .
\end{align*}
Finally, the dissipative term is estimated as before by
\begin{equation*}
\text{IV} \leq \frac{1}{4} \int \lvert (-\Delta)^{(s+\beta)/2} w \rvert^2 \, \mathrm{d}x + C_2(\alpha-\beta)^\delta \norm{(-\Delta)^{s/2}u}_{H^{\alpha+\delta}}^2 \, .
\end{equation*}
Collecting terms, we have obtained after reabsorption of $\frac{1}{2}\int \lvert (-\Delta)^{(s+\beta)/2} w \rvert^2 \, \mathrm{d}x$ on the left
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t} \int &\lvert (-\Delta)^{s/2} w \rvert^2 \, \mathrm{d}x \leq 2 C_0 \norm{(-\Delta)^{s/2} w}_{L^2}^{2 \gamma} + 2 C_2 (\alpha-\beta)^\delta \norm{(-\Delta)^{s/2} u}_{H^{\alpha+\delta}}^2\\
&+6 \left[3(1+\bar D)^2 \bar C^4 + 2
\bar C^6 \right] \norm{u}_{\dot{H}^{\frac{5}{2}-\beta}}^2 \norm{w}_{H^s} ^2
+ 12
\bar C^4 \norm{u}_{\dot{H}^{1+s}}^2 \norm{w}_{H^s} ^2 \, .
\end{align*}
Under the hypothesis \eqref{eq: hypdiffeqHs}, we can estimate $\norm{u}_{\dot{H}^{\frac{5}{2}-\beta}} \leq \norm{u}_{H^{s+\alpha+\delta}}$ with constant 1
\begin{align}
\frac{\mathrm{d}}{\mathrm{d}t} \norm{ w}_{H^s}^2 + \norm{(-\Delta)^{\beta/2} w}_{H^s}^2 &\leq 2 C_0 \norm{w}_{H^s}^{2 \gamma} + C_1 (\norm{u}_{H^{s+\alpha+\delta}}^2+\norm{Du}_{H^{s}}^2)\norm{w}_{H^s}^2 \nonumber \\
&+ 2 C_2(\alpha-\beta)^\delta \norm{u}_{H^{s+\alpha+\delta}}^2 \, ,
\end{align}
where $C_1:=C_1(s)= 6 \left[3(1+\bar D)^2 \bar C^4 + 2\bar C^6 \right]
$.
In case $s=1$, the estimates simplify considerably due to the absence of commutators and it is straight-forward to obtain the following energy inequality for the difference equation for $s=1$
\begin{align}\label{eq: diffineq H1}
\frac{\mathrm{d}}{\mathrm{d}t} \norm{w (t)}_{H^1}^2 + \norm{ (-\Delta)^{\beta/2} w(t)}_{H^1}^2 &\leq 2 C_0\norm{ w(t)}_{H^1}^{2\frac{3(2\beta-1)}{4\beta-3} } + 8 \bar C^4 \norm{w(t)}_{H^1}^2 \norm{D u(t)}_{H^1}^2 \nonumber \\
&+ 2C_2 (\alpha-\beta)^\delta \norm{Du(t)}_{H^{\alpha+ \delta}}^2\, ,
\end{align}
where now
\begin{equation*}\label{eq: C_0}
C_0=C_0(1, \beta):=\left(1-\frac{3}{4\beta} \right) \Big(\frac{6\bar C^\frac{3}{2\beta}}{\beta} \Big)^\frac{4\beta}{4\beta-3}
\leq (8 \bar C^2)^{\frac{5}{2(\alpha-\frac{3}{4})}}
\, .\qedhere
\end{equation*}
\end{proof}
\begin{lemma}\label{lem: energyineq for diffeq Hk} Let $ 0 < \alpha < \frac{3}{2} $, $T>0$, and $k\in \mathbb{N}$ such that $k > \frac{5}{2}-2\alpha$. Consider $u \in C(\left[0,T\right], H^k) \cap L^2([0,T], H^{k+\alpha})$, a smooth solution to the fractional Navier-Stokes equations of order $\alpha$ with initial datum $u_0\in H^k$. We additionally assume that
\begin{equation}\label{eq: addintegrab Hk}
u \in L^2(\left[0, T\right], H^{k+\alpha+\delta})\cap L^2([0,T], H^{k+1})
\end{equation}
for some positive $\delta \in (0, \min\{1, 2(k-\frac{5}{2}+2\alpha)\}]$. Then, for any $\frac{\delta}{2} \leq \beta < \frac{3}{2}$ such that
\begin{equation}\label{eq: smallnessreq Hk}
\lvert \alpha-\beta \rvert < \frac{\delta}{2}
\end{equation}
and any solution $v \in C([0,T], H^k) \cap L^2([0,T], H^{k+\beta})$ of the fractional Navier-Stokes equations of order $\beta$, there holds with $f(t)=\norm{(v-u)(t)}_{H^k}^2$ for every $t\in [0,T]$ that
\begin{equation}\label{eq: diff ineq H^k}
f(t) \leq f(0) + C_1 \int_0^t f^2(\tau) + \norm{Du(\tau)}_{H^k}^2 f(\tau) \, \mathrm{d}\tau + C_2 \lvert \alpha-\beta \rvert^{\delta} \int_0^T \norm{u(\tau)}_{H^{k+\alpha+\delta}}^2 \, \mathrm{d}\tau
\end{equation}
for $C_1=C_1(k)>0$ and a universal $C_2>0$.
\end{lemma}
\begin{proof}
For $\kappa=0, \dots, k$, we can differentiate the difference equation by $D^\kappa$ (by which we denote any derivative of order $\kappa$) and multiply it by $D^\kappa w$. Since $w$ is incompressible, we obtain
\begin{align*}
\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \lvert D^\kappa w \rvert^2 &+ \div\left( \frac{\lvert D^\kappa w \rvert^2}{2} w\right) + \sum_{j=1}^\kappa D^j w \star D^{\kappa+1-j} w \cdot D^\kappa w + \div \left( \frac{\lvert D^\kappa w \rvert^2}{2} u\right) \\
&+ \sum_{j=1}^{\kappa+1} D^j u \star D^{\kappa+1-j} w \cdot D^\kappa w = - (-\Delta)^\beta D^\kappa w \cdot D^\kappa w + [(-\Delta)^\alpha- (-\Delta)^\beta] D^\kappa u \cdot D^\kappa w \, ,
\end{align*}
where we denote by $\star$ any bilinear expression with constant coefficients. We then test the latter equality with a cut-off $\psi_R$ as in the proof of Lemma \ref{lem: energyineq for diffeq H1} and we note that since $\lvert D^\kappa w \rvert^2 w, \lvert D^\kappa w \rvert^2 u \in L^1([0,T], L^1)$, the contributions of the divergence term vanish in the limit $R\to \infty$. Thus,
\begin{align*}
\frac{1}{2}& \frac{\mathrm{d}}{\mathrm{d}t} \norm{D^\kappa w(t)}_{L^2}^2 + \norm{(-\Delta)^{\beta/2} D^\kappa w(t)}_{L^2}^2 \leq \sum_{j=1}^\kappa \left \lvert \int D^j w \star D^{k+1-j} w \cdot D^\kappa w \, (x,t) \, \mathrm{d}x \right \rvert \\
&+ \sum_{j=1}^{\kappa +1} \left \lvert \int D^j u \star D^{\kappa+1-j}w \cdot D^\kappa w \, (x,t) \, \mathrm{d}x \right \rvert + \left \lvert \int [(-\Delta)^\alpha-(-\Delta)^\beta] D^\kappa u \cdot D^\kappa w \, (x,t) \, \mathrm{d}x \right \rvert \, .
\end{align*}
By the Sobolev embeddings, it is straight-forward to check that as long as $k > \frac{5}{2}-2\beta$ it holds
\begin{align*}
\sum_{j=1}^\kappa \left \lvert \int D^j w \star D^{k+1-j} w \cdot D^\kappa w \, (x,t) \, \mathrm{d}x \right \rvert &\leq \frac{1}{8}\norm{(-\Delta)^{\beta/2} D^\kappa w(t)}_{L^2}^2 + C \norm{w(t)}_{H^k}^4 \,
\end{align*}
where $C=C(k, \beta)
. Notice that $k> \frac{5}{2}-2\beta$ is guaranteed through \eqref{eq: smallnessreq Hk} and the upper bound on $\delta$.
As for the second line, we have
\begin{align*}
\sum_{j=1}^{\kappa +1}\int \lvert D^j u \rvert \lvert D^{\kappa+1-j} w \rvert \lvert D^\kappa w \rvert (x,t) \, \mathrm{d}x
&\leq \bar C \sum_{j=1}^{\kappa+1} \norm{(-\Delta)^{\beta/2} D^\kappa w (t)}_{L^2} \norm{\lvert D^j u \rvert \lvert D^{\kappa+1-j}w\rvert(t)}_{L^\frac{6}{3+2\beta}} \\
&\leq \frac{1}{8} \norm{(-\Delta)^{\beta/2}D^\kappa w(t)}_{L^2}^2 + C \norm{w(t)}_{H^k}^2 \norm{Du(t)}_{H^k}^2 \, .
\end{align*}
We now estimate the final term as before, using Plancherel, H\"older, Young and Lemma \ref{lem: fraclapRn} as in \eqref{eqn:alphabeta}.
Combining the estimates, reabsorbing the contributions of $\norm{(-\Delta)^{\beta/2} D^\kappa w }_{L^2}^2$ on the left and summing over $\kappa=0, \dots, k$, we obtain
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t} \norm{w(t)}_{H^k}^2 + \norm{(-\Delta)^{\beta/2}w(t)}_{H^k}^2 &\leq C_1( \norm{w(t)}_{H^k}^4+ \norm{u(t)}_{H^{k+1}}^2\norm{w(t)}_{H^k}^2) + C_2(\alpha-\beta)^\delta \norm{u(t)}_{H^{k+\alpha+\delta}}^2 \, .
\end{align*}
with $C_1=C_1(k)>0$ and a universal $C_2>0$.
\end{proof}
{
\begin{proof}[Proof of Proposition~\ref{prop: locstab H3}] Consider first the case $\alpha \in (\frac{3}{4}, \frac{5}{4}]$. Let $s\in (\frac{5}{2}-2\alpha,1]$, $v_0 \in H^s$ divergence-free and $\beta \in (0, \frac{5}{4}]$ such that \eqref{eq: addintdiffeq} holds for an $\epsilon \in (0,1]$, yet to be determined. By choosing $\epsilon$ suitably small, we can always assume that $\lvert \alpha-\beta \rvert \leq \frac{1}{2} \min \{\delta, s-(\frac{5}{2}-2\alpha) \}$. Observe that then in particular $s>\frac{5}{2}-2\beta$, where the latter is the critical Sobolev regularity with respect to the natural scaling of \eqref{eq: NSalpha}. Classical arguments allow to build a maximal local solution $v\in C([0,T_{max}), H^s) \cap L^2([0,T_{max}), H^{s+\beta})$ to the Navier-Stokes euqations of order $\beta$ starting from $v_0$ which is smooth in the interior and unique among Leray-Hopf solutions by Proposition \ref{prop: weakstrong uniq}. Moreover, if $T_{max}< + \infty$ we have $\limsup_{t \uparrow T^\ast} \norm{v(t)}_{H^s}=+\infty$.
Let us define $f(t):= \norm{(u-v)(t)}_{H^s}$ for $t \in [0, \min \{T_{max}, T \})$. By Lemma \ref{lem: energyineq for diffeq H1}, $f$ satisfies the differential inequality \eqref{eq: diffineq} and hence,
for any $t\in [0, \min \{T_{max}, T \})$, we have an upper abound on $\max_{s\in [0,t]} f(s) \,.$
In particular, if
\begin{equation}\label{eq: localsmallnesreq}
(f(0)+C_2 \lvert \alpha-\beta\rvert^\delta \norm{u}_{L^2([0, T], H^{s+\alpha+\delta})}^2)^{\gamma-1} C_0 (\gamma-1) e^{C_1(\gamma-1)(\norm{u}_{L^2([0,T],H^{s+1})}^2 +\norm{u}_{L^2([0,T],H^{s+\alpha+\delta})}^2)}< \frac{1}{2T} \, ,
\end{equation}
where we recall from Lemma \ref{lem: energyineq for diffeq H1} that $\gamma-1=\frac{2\beta}{4\beta-5+2s}$, we have $$\max_{s\in [0,T]} f(s) \leq ((\gamma-1) C_0 T)^{-\frac{1}{\gamma-1}} \, .$$ We deduce that $\limsup_{t \uparrow T} \norm{(u-v)(t)}_{H^s} <+\infty$ and thus $T_{max}> T$. The condition \eqref{eq: localsmallnesreq} is thus satisfied, if we require that \eqref{eqn:eps-choice} is enforced with
\begin{align}\label{eq: localchoiceofepsilon}
\epsilon:= \min \{ &\frac{\delta}{2}, \frac{1}{4}(s-(\frac{5}{2}-2\alpha)), \\
&\max\{1, C_2 \norm{u}_{L^2([0,T], H^{s+\alpha+\delta})}^2 \} ( C_0 (\gamma-1)T )^{-\frac{1}{\gamma-1}} e^{-C_1 (\norm{u}_{L^2([0,T],H^{s+1})}^2+\norm{u}_{L^2([0,T],H^{s+\alpha+\delta})}^2)}\big \} \, .
\end{align}
Recall that $\gamma$ depends on $\beta$; however, by choosing $\beta$ close enough to $\alpha$, we can bound $\gamma-1$ uniformly away from $0$. This concludes the proof. The cases $\alpha \in (\frac{1}{2}, \frac{5}{4}]$ with $s=2$ and $\alpha \in (0, \frac{5}{4}]$ with $s=3$ follow analogously from Lemma \ref{lem: energyineq for diffeq Hk}. In the latter case, the local existence from $H^3$ initial data follows for instance from \cite[Theorem 3.4]{BM} (the proof there covers the classical Navier-Stokes
$\alpha=1$ and the Euler equations, and, being based on energy methods, can easily be adapted to the fractional Navier-Stokes equations of order $\alpha>0$).
\end{proof}
}
\section{Leray's estimate on singular times}\label{sec:leray}
In his seminal paper \cite{Leray}, Leray showed that if $u_0 \in H^1$, then the Leray-Hopf solution is unique and smooth for a short time with upper bound $T= C \norm{\nabla u_0}_{L^2}^{-4}$. Thanks to energy inequality, this bound can be iterated to get global existence provided that a Leray-Hopf solution exists and is smooth until a sufficiently large $T^\ast$. With minor modifications, Leray's argument applies to the fractional setting. Notice however, that for an eventual regularization of Leray-Hopf solutions in the ipodissipative range $\alpha \leq 1$, we need an upper bound of the form $T= C \norm{(-\Delta)^{\alpha/2} u_0}_{L^2}^{-\beta}$, for some $\beta>0$, since the energy inequality now only controls $(-\Delta)^{\alpha/2}u$ in $L^2(\mathbb{R}^3 \times [0, +\infty))$. Since $\norm{(-\Delta)^{\alpha/2} u_0 }_{L^2}$ is critical with respect to the natural scaling of \eqref{eq: NSalpha} at $\alpha=\frac{5}{6}$, such an estimate can only be expected in the subcritical range $\alpha> \frac{5}{6}$.
\begin{prop}\label{prop: LeraySingTime} Let $\frac{3}{4} < \alpha \leq \frac{5}{4}$ and $u_0 \in H^1(\mathbb{R}^3)$ divergence-free. Then there exists a universal $C_2=C_2(\alpha)$, uniformly bounded away from $\alpha=\frac{3}{4}$, such that, setting
\begin{equation} \label{eq: LeraySingTimeDef}
T(\norm{\nabla u_0}_{L^2}, \alpha):= C_2 \norm{\nabla u_0}_{L^2}^{-\frac{4\alpha}{4\alpha-3}} \, ,
\end{equation}
there exists a unique solution $u$ to \eqref{eq: NSalpha} on $(0,T)$ satisfying $u \in L^\infty([0,T), H^1) \cap L^2([0,T),H^{1+\alpha})$ which is smooth in $(0,T)$.
\end{prop}
\begin{proof}
Notice first that by Sobolev embeddings, we have $(-\Delta)^{(1-\alpha)/2} u \in L^2([0,T), H^{2\alpha}) \hookrightarrow L^2([0,T), L^\infty)$ for $\frac{3}{4}< \alpha<1$, whereas for $\alpha \geq 1$ it holds $u\in L^2([0,T), L^\frac{3}{\alpha-1})$. Thus the uniqueness of the Leray-Hopf solution on $[0,T)$ follows from Proposition \ref{prop: weakstrong uniq}. The smoothness in the interior follows from a standard boot-strap argument. We therefore focus on the Leray-Hopf solution $u$ which is attained as limit of the approximation scheme \eqref{eq: regularizedNS}. We perform all the estimates on the unique, smooth and global solutions $(u_\epsilon, p_\epsilon)$ of \eqref{eq: regularizedNS} and pass to the limit $\epsilon \to 0$ only at the very end.
By smoothness we may derive the equation by $\partial_j$ and multiply it by $\partial_j u_\epsilon$. To make the computation rigorous we employ a cutoff $\psi_R \in C^\infty_c(B_{2R}(0))$ and we then let $R\to \infty$; the pressure term can be neglected by Calderon-Zygmund estimates, $\norm{\partial_j(u_\epsilon \ast \varphi_\epsilon)}_{L^2_t L^\infty_x} \leq C \norm{\partial_j u_\epsilon}_{L^2_{t,x}}$ and $\norm{u_\epsilon \ast \varphi}_{L^\infty} \leq C \norm{u_\epsilon}_{L^\infty_t L^2_x}$, which give $\partial_j p_\epsilon \in L^2_{t, x}$.
We obtain for $t\in [0,+\infty)$
\begin{equation}\label{eq: energyineq H1}
\frac{1}{2} \int \lvert Du_\epsilon \rvert^2 (x,t) \, \mathrm{d}x + \int_0^t \int \lvert (-\Delta)^{\alpha/2} Du_\epsilon \rvert^2 \, \mathrm{d}x \, \mathrm{d}\tau \leq \frac{1}{2} \int \lvert Du_0 \rvert^2 \, \mathrm{d}x+ \int_0^t \int \lvert D(u_\epsilon \ast \varphi_\epsilon) \rvert \lvert Du_\epsilon \rvert^2 \, \mathrm{d}x \, \mathrm{d}\tau \, .
\end{equation}
We estimate the right-hand side for $\frac{3}{4}\leq \alpha < \frac{3}{2}$ using H\"older, Young's convolution inequality and the Gagliardo-Nirenberg-Sobolev inequality by
\begin{align}
\int \lvert D(u_\epsilon \ast \varphi_\epsilon) \rvert \lvert Du_\epsilon \rvert^2 \, \mathrm{d}x &\leq \norm{Du_\epsilon \ast \varphi_\epsilon}_{L^2} \norm{Du_\epsilon}_{L^4}^2 \leq \norm{Du_\epsilon}_{L^2} \norm{Du_\epsilon}_{L^\frac{6}{3-2\alpha}}^{2\theta} \norm{Du_\epsilon}_{L^2}^{2(1-\theta)} \nonumber \\
&\leq \bar C^{2\theta} \norm{(-\Delta)^{\alpha/2} Du_\epsilon}_{L^2}^{2\theta} \norm{Du_\epsilon}_{L^2}^{3-2\theta} \label{eq: estonL3norm}\, .
\end{align}
where $\theta=\theta(\alpha):= \frac{3}{4\alpha}$ solves $
\frac{1}{4} = \frac{\theta(3-2\alpha)}{6} + \frac{1-\theta}{2} \, .$
Hence, for $\alpha> \frac{3}{4}$, we may apply Young with exponents $\frac{1}{\theta} = \frac{4\alpha}{3}$ and $ \frac{4\alpha}{4\alpha-3}$ to obtain
\begin{align*}
\int \lvert Du_\epsilon \ast \varphi_\epsilon \rvert \lvert Du_\epsilon \rvert^2 \, \mathrm{d}x \leq \frac{3}{4\alpha}\norm{(-\Delta)^{\alpha/2} Du_\epsilon}_{L^2}^2 + \frac{(4\alpha-3) \bar C^{\frac{8\alpha \theta}{4\alpha-3}}}{4\alpha} \norm{Du_\epsilon}_{L^2}^{\frac{4\alpha(3-2\theta)}{4\alpha-3}} \, .
\end{align*}
Reabsorbing $\frac{3}{4\alpha}\norm{(-\Delta)^{\alpha/2} Du_\epsilon}_{L^2}^2$ on the left-hand side of \eqref{eq: energyineq H1} yields
\begin{equation}\label{eq: energyineq H1 final}
\frac{\mathrm{d}}{\mathrm{d}t} \int \lvert Du_\epsilon \rvert^2 \, \mathrm{d}x +\left(1-\frac{3}{4\alpha}\right) \int \lvert (-\Delta)^{\alpha/2} Du_\epsilon \rvert^2 \, \mathrm{d}x \leq \frac{2(4\alpha-3)C_2^{-1}}{4\alpha} \left( \int \lvert Du_\epsilon \rvert^2 \, \mathrm{d}x \right)^\beta \, ,
\end{equation}
where $\beta=\beta(\alpha):= \frac{3(2\alpha-1)}{4\alpha-3}$ and $C_2=C_2(\alpha)=\bar C^\frac{-6}{4\alpha-3}$. Setting
\begin{equation*}
T=T(\norm{\nabla u_0}_{L^2}, \alpha) := \frac{4\alpha}{2(4\alpha-3) \bar C^\frac{6}{4\alpha-3} (\beta-1) \norm{\nabla u_0}_{L^2}^{2(\beta-1)}} = C_2\norm{\nabla u_0}_{L^2}^{-\frac{4\alpha}{4\alpha-3}} \, ,
\end{equation*}
we have that for any $0 \leq t < T$ the estimate
\begin{equation}\label{eq: uniform bound on H1}
\norm{Du_\epsilon(t)}_{L^2}^2 \leq \frac{\norm{\nabla u_0}_{L^2}^2}{(1-C_2^{-1}t \norm{\nabla u_0}_{L^2}^{2(\beta-1)})^\frac{1}{\beta-1}} \, .
\end{equation}
Recalling \eqref{eq: energyineq H1 final}, we infer that $\{u_\epsilon\}_{\epsilon >0}$ is uniformly bounded in $L^\infty([0,T), H^1)\cap L^2([0,T), H^{1+\alpha})$. By a standard argument, we can now pass to the limit $\epsilon \to 0$ using weak lower semicontinuity and the strong convergence of $u_\epsilon \rightarrow u$ in $L^p_{loc}(\mathbb{R}^3\times [0,T])$ for $p<\frac{6+4\alpha}{3}$.
\end{proof}
\begin{prop}\label{prop: LeraySingTimeHalpha} Let $\frac{5}{6} < \alpha < 1$ and $u_0 \in W^{\alpha,2}(\mathbb{R}^3)$ divergence-free. Then there exists a universal $C_1=C_1(\alpha)$, uniformly bounded away from $\alpha=\frac{5}{6}$, such that, setting
\begin{equation} \label{eq: LeraySingTimeDefHalpha}
T(\norm{(-\Delta)^{\alpha/2} u_0}_{L^2}, \alpha):= C_1 \norm{(-\Delta)^{\alpha/2} u_0}_{L^2}^{-\frac{4\alpha}{6\alpha-5}} \, ,
\end{equation}
there exists a unique solution $u$ to \eqref{eq: NSalpha} on $(0,T)$ satisfying $u \in L^\infty([0,T), W^{\alpha,2}) \cap L^2([0,T), W^{2\alpha, 2})$ which is smooth on $(0,T)$.
\end{prop}
\begin{proof}
We argue as in the proof of Proposition \ref{prop: LeraySingTime} to obtain the energy inequality for the regularized system, use commutator estimates as in the proof of Lemma \ref{lem: energyineq for diffeq H1}, Gagliardo-Nirenberg-Sobolev and interpolation to obtain
\begin{align*}
\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int \lvert(-\Delta)^{\alpha/2} u_\epsilon\rvert^2 \, \mathrm{d}x &+ \int \lvert (-\Delta)^{\alpha} u_\epsilon \rvert^2 \, \mathrm{d}x \leq \left \lvert \int (-\Delta)^{\alpha/2} \left( (u_\epsilon \ast \varphi_\epsilon) \cdot \nabla) u_\epsilon\right) \cdot (-\Delta)^{\alpha/2} u_\epsilon \, \mathrm{d}x \, \right \rvert \\
&\leq (1+ \bar D) \bar C \norm{(-\Delta)^{\alpha} u_\epsilon}_{L^2} \norm{(-\Delta)^{\alpha/2} ( u_\epsilon \ast \varphi_\epsilon)}_{L^2} \norm{(-\Delta)^{\alpha/2} u_\epsilon}_{L^ \frac{6}{4\alpha-2}} \\
&\leq (1+ \bar D)\bar C ^\frac{5-2\alpha}{2\alpha} \norm{(-\Delta)^\alpha u_\epsilon}_{L^2}^\frac{5-2\alpha}{2\alpha} \norm{(-\Delta)^{\alpha/2} u_\epsilon}_{L^2}^\frac{8\alpha-5}{2\alpha} \, ,
\end{align*}
by \eqref{eqn:commuta} applied with $s_1=\alpha$, $p_1=2$, $s_2=0$ and $p_2=\frac{6}{5-4\alpha}$.
For $\alpha>\frac{5}{6}$, we use Young to obtain after reabsorption
\begin{equation}\label{eq: Halpha ipo}
\frac{\mathrm{d}}{\mathrm{d}t} \int \lvert (-\Delta)^{\alpha/2} u_\epsilon \rvert^2(x,t) \, \mathrm{d}x \leq \frac{2 (6\alpha-5)C_1^{-1}}{4\alpha} \left( \int \lvert (-\Delta)^{\alpha/2} u_\epsilon \rvert^2(x,t) \, \mathrm{d}x \right)^\frac{8\alpha-5}{6\alpha-5} \, ,
\end{equation}
where $C_1=C_1(\alpha):=(1+\bar D)^\frac{-4\alpha}{6\alpha-5} \bar C^\frac{-10+4\alpha}{6\alpha-5}\, .$
Defining $T$ by \eqref{eq: LeraySingTimeDefHalpha}, we conclude as before.
\end{proof}
\begin{corollary}[Leray's Estimate on Singular Times] \label{cor: LeraySingTime} Let $\frac{5}{6} < \alpha < \frac{5}{4}$ and $u_0 \in L^2(\mathbb{R}^3)$ divergence-free with $\norm{u_0}_{L^2} \leq M$. Then there exists $T^\ast = T^\ast (M, \alpha)>0$ and a Leray-Hopf solution $u$ to \eqref{eq: NSalpha} which is smooth on $[T^\ast, + \infty)$. Moreover, $T^\ast$ is uniformly bounded for $ \alpha \in (\frac{5}{6}, \frac{5}{4})$ and
\begin{equation}\label{eq: Tstar behavat54}
\lim_{\alpha \uparrow \frac{5}{4}} T^\ast(M, \alpha) =0 \, .
\end{equation}
\end{corollary}
\begin{proof} Let $T^\ast>0$. From the energy inequality, we infer that there exists $\bar{t} \in (0,T^\ast)$ such that
\begin{equation*}
\norm{(-\Delta)^{\alpha/2} u(\cdot, \bar{t})}_{L^2}^2 \leq \frac{\norm{u_0}_{L^2}^2}{2T^\ast} \leq \frac{M^2}{2T^\ast} \, .
\end{equation*}
Consider first $\alpha<1$. By Proposition \ref{prop: LeraySingTimeHalpha} the Leray-Hopf solution with initial datum $u(\cdot, \bar{t})$ is smooth and unique until
\begin{equation*}
{C_1}{\norm{(-\Delta)^{\alpha/2} u(\cdot, \bar{t})}_{L^2}^\frac{-4\alpha}{6\alpha-5}} \geq {C_1(2T^\ast)^\frac{2\alpha}{6\alpha-5}}{M^\frac{-4\alpha}{6\alpha-5}} \,
\end{equation*}
If we choose $T^\ast= T^\ast(M, \alpha)$ large enough, such that
\begin{equation}\label{eq: choiceofTstar ipo}
{C_1(2T^\ast)^\frac{2\alpha}{6\alpha-5}}{M^\frac{-4\alpha}{6\alpha-5}} > T^\ast \text{, or equivalently, } T^\ast > {M^\frac{4\alpha}{5-4\alpha}}{2^\frac{-2\alpha}{5-4\alpha}C_1^\frac{-6\alpha-5}{5-4\alpha}}\, ,
\end{equation}
then the Leray-Hopf solution is smooth and unique on $(\bar{t}, \bar{t}+T^\ast)$ and we can iterate this procedure thanks to the energy inequality.
If $\alpha\geq1$, we notice that by interpolation of Sobolev spaces
\begin{equation*}
\norm{\nabla u(\cdot, \bar{t})}_{L^2} \leq \norm{u(\cdot, \bar{t})}_{L^2}^{1-\frac{1}{\alpha}} \norm{(-\Delta)^{\alpha/2}u(\cdot, \bar{t})}_{L^2}^\frac{1}{\alpha} \leq M(2T^\ast)^{-\frac{1}{2 \alpha}} \, ,
\end{equation*}
and hence from Proposition \ref{prop: LeraySingTime} the Leray-Hopf solution starting at $\bar{t}$ is smooth and unique until
\begin{equation*}
C_2\norm{\nabla u(\cdot, \bar{t})}_{L^2}^{-\frac{4\alpha}{4\alpha-3}} \geq C_2(2T^\ast)^\frac{2}{4\alpha-3}M^{-\frac{4\alpha}{4\alpha-3}} \, .
\end{equation*}
As in the ipodissipative case, choosing $T^\ast=T^\ast(M, \alpha)$ large enough satisfying
\begin{equation}\label{eq: choiceofTstar hyp}
C_2(2T^\ast)^\frac{2}{4\alpha-3}M^{-\frac{4\alpha}{4\alpha-3}} > T^\ast \text{, or equivalently, } T^\ast > M^\frac{4\alpha}{5-4\alpha}2^{-\frac{2}{5-4\alpha}} C_2^{-\frac{4\alpha-3}{5-4\alpha}}
\end{equation}
allows to iterate the argument and build a global solution starting at $\bar{t}$. Observe now that the fact that $T^\ast(M, \alpha)$ is uniformly bounded away from $\alpha=\frac{5}{4}$ follows from its choice in \eqref{eq: choiceofTstar ipo} and \eqref{eq: choiceofTstar hyp} and the explicit expression of the constants $C_1$ and $C_2$. Let us now establish \eqref{eq: Tstar behavat54}. Take a non-regular time $T>0$ of a Leray-Hopf solution $(u, p)$ for the Navier-Stokes equations of order $\alpha\geq 1$. For almost every $0<t<T$, $u(\cdot, t) \in W^{\alpha,2}$ and thus by Proposition \ref{prop: LeraySingTime}, we must have
\begin{equation*}
T-t > C_2\norm{\nabla u(\cdot, t)}_{L^2}^{-\frac{4\alpha}{4\alpha-3}} \geq C_2\norm{(-\Delta)^{\alpha/2} u(\cdot, t)}_{L^2}^{-\frac{4}{4\alpha-3}} \norm{u_0}_{L^2}^{-\frac{4(\alpha-1)}{4\alpha-3}} \, .
\end{equation*}
The last inequality follows by interpolation of Sobolev spaces.
Recalling $C_2 = \bar C^{-\frac{6}{4\alpha-3}}$ and using the energy inequality, we deduce that
\begin{equation*}
\frac{1}{2} \norm{u_0}_{L^2}^2 \geq \int_0^T \norm{(-\Delta)^{\alpha/2} u(t)}_{L^2}^2 \, \mathrm{d}t \geq \frac{C_2^\frac{4\alpha-3}{2}}{\norm{u_0}_{L^2}^{2\alpha-2}} \int_0^T \frac{\mathrm{d}t}{(T-t)^{1-\frac{5-4\alpha}{2}}} = \frac{2 T^\frac{5-4\alpha}{2}}{\bar C^3\norm{u_0}_{L^2}^{2\alpha-2}(5-4\alpha)} \, .
\end{equation*}
Hence we have found the following improved upper bound for $T^\ast$ for $\alpha \geq 1$
\begin{equation}\label{eq: upperbound for Tstar hyp}
T^\ast(M, \alpha) \leq \left(\frac{(5-4\alpha)}{2}\bar C^3 M^{2\alpha} \right)^\frac{2}{5-4\alpha} \,.
\end{equation}
We deduce that for $M>0$ fixed
\begin{equation*}
\lim_{\alpha \uparrow \frac{5}{4}} T^\ast(M, \alpha) \leq \lim_{\alpha \uparrow \frac{5}{4}}(\bar C^3 M^{2\alpha} (5-4\alpha))^\frac{2}{5-4\alpha} = 0 \, .\qedhere
\end{equation*}
\end{proof}
\section{Eventual regularization of suitable weak solutions}\label{sec:reg-suit}
The eventual regularization property for $\frac{3}{4}<\alpha\leq\frac56$ can be obtained relying on partial regularity results.
The difference with Leray's estimate of Section~\ref{sec:leray} on the eventual regularization time $T^\ast$ consist in the fact that the dependence of $T^\ast$ on $M$ and $\alpha$ cannot be made explicit
\begin{prop}\label{prop: eventualreg} Let $\frac{3}{4}<\alpha\leq1$ and $p\in [1,2)$. Consider $u_0 \in L^p \cap L^2$ divergence-free with $\norm{u_0}_{L^2 \cap L^p} \leq M$. Then there exists $T^\ast= T^\ast(M, \alpha,p)>0$ and a suitable weak solution $u$ to \eqref{eq: NSalpha} which is smooth on $[T^\ast, \infty)$. Moreover, $T^\ast$ is uniformly bounded away from $\alpha=\frac{3}{4}$.
\end{prop}
\begin{proof}
Let us consider a global suitable weak solution $u$ of the Navier-Stokes equations of order $\alpha$ obtained as the limit of the regularized system \eqref{eq: regularizedNS}. Let us recall from \cite{TY, Scheffer1, CDM} that there exists $\epsilon=\epsilon(\alpha)>0$ and $\kappa=\kappa(\alpha)>0$ such that if for some $(x_0,t_0)$
\begin{equation}\label{eq: epsilonreg}
\int_{t_0-1}^{t_0}\int_{B_1(x_0)} \left( \mathcal{M} \lvert u \rvert^2 + \lvert p\rvert\right)^\frac{3}{2} < \epsilon,
\end{equation}
then $u$ is smooth in a neighborhood of $(x_0,t_0)$. Indeed, the maximal operator $\mathcal{M} $ accounts for the non-local effects of the fractional Laplacian and can be removed in the case $\alpha=1$, as in the classical $\epsilon$-regularity of Scheffer.
From \cite[Theorem 1.1]{JY}, we infer
\begin{equation}\label{eq: L2decay}
\norm{u(t)}_{L^2}^2 \leq C(1+t)^{-\frac{3}{2\alpha}(\frac{2}{p}-1)} \, ,
\end{equation}
where $C=C(M, \alpha)>0$. In \cite{JY}, this decay rate is obtained for the Leray-Hopf solution obtained through Garlekin approximation. However, as in \cite{Schonbek} for $\alpha=1$, the argument also applies to the Leray-Hopf solution obtained through the regularized system \eqref{eq: regularizedNS}. Since the decay rate coincides with the one of the fractional heat equation, it cannot, in general, be expected for $u_0 \in L^2$ only. Let $T^\ast:= T^\ast(\alpha, M,p)$ be such that $\norm{u(t)}_{L^2}^2 \leq \epsilon^\frac{1}{3}$ for $t\in [T^\ast-2^\alpha, + \infty)$
We estimate by H\"older, the maximal function estimate $\norm{\mathcal{M} f}_{L^p} \leq C \norm{f}_{L^p}$ for $1<p\leq + \infty$, Calderon-Zygmund and interpolation
\begin{align}\label{eq: suitsol smallnessreq}
\int_{t_0-1}^{t_0}\int_{B_1} \big (\mathcal{M} \lvert u \rvert^2 &+ \lvert p \rvert \big )^\frac{3}{2} \, \mathrm{d}x \, \mathrm{d}t
\leq \left( \int_{t_0-1}^{t_0}\int_{B_1} \left( \mathcal{M} \lvert u \rvert^2 + \lvert p \rvert\right)^\frac{3+2\alpha}{3} \, \mathrm{d}x \, \mathrm{d}t \right)^\frac{9}{2(3+2\alpha)}
\nonumber \\
&\leq C \left(\int_{t_0-1}^{t_0} \int_{\mathbb{R}^3} \lvert u \rvert^\frac{6+4\alpha}{3} \, \mathrm{d}x \, \mathrm{d}t \right)^\frac{9}{2(3+2\alpha)} \nonumber \\
&\leq C \left(\int_{t_0-1}^{t_0} \int_{\mathbb{R}^3} \lvert (-\Delta)^{\alpha/2} u \rvert^2 \, \mathrm{d}x \, \mathrm{d}t \right)^\frac{9}{3+2\alpha} \left( \esssup_{t \in [t_0-1, t_0]} \norm{u(t)}_{L^2}\right)^\frac{6\alpha}{3+2\alpha} \, .
\end{align}
This concludes the proof, since from \eqref{eq: epsilonreg} we then infer that $u$ is regular in $[T^\ast, \infty)$
\end{proof}
\section{Global-in-time stability}
\begin{prop}[Global-in-time stability]\label{prop:global} Let $\frac{3}{4}< \alpha \leq \frac{5}{4}$, $p\in [1,2)$ and $s\in (\frac{5}{2}-2\alpha, 1]$. Assume that there exists an a priori global smooth solution $u\in C([0,+\infty), H^s)$ to the fractional Navier-Stokes equations of order $\alpha$ with divergence-free initial datum $u_0\in H^s\cap L^p$. If additionally
\begin{equation}\label{eq: add integrability assump}
u\in L^2_{loc}([0,+\infty), H^{s+\alpha+\delta}) \text{ and } Du \in L^2_{loc}([0,+\infty), H^s)
\end{equation}
for some positive $\delta>0$, then there exists $\epsilon>0$ such that for all $v_0\in H^s \cap L^p$ divergence-free and all $\frac{3}{4}< \beta \leq \frac{5}{4}$ satisfying
\begin{equation}\label{hyp}
\norm{u_0-v_0}_{H^s\cap L^p}^2 + \lvert \alpha-\beta \rvert^\delta < \epsilon
\end{equation}
there exists a unique global smooth solution $v\in C([0,+\infty), H^s)$ to the fractional Navier-Stokes equations of order $\beta$ with initial datum $v_0$.
\end{prop}
\begin{remark}\label{rmk: dropLp} The requirement that $v_0 \in L^p$ for some $p\in [1,2)$ is in fact necessary only for $\alpha \in (\frac 34 , \frac 56)$ to obtain the decay of the $L^2$-norm of the solution in the proof of Proposition~\ref{sec:reg-suit}. Moreover, only boundedness of $v_0$ in $L^p$ (and not closeness to $u_0$) would suffice.
\end{remark}
\begin{proof} Let $v_0 \in H^s\cap L^p$ divergence-free and $\frac{3}{4}< \beta \leq \frac{5}{4}$. By Corollary \ref{cor: LeraySingTime} and Proposition \ref{prop: eventualreg}, there exists an eventual regularization time $T^\ast(\norm{v_0}_{L^2\cap L^p}, \beta,p)>0$ such that suitable weak solutions to the fractional Navier-Stokes equations of order $\beta$ starting from $v_0 $ are smooth after time $T^\ast$. By Proposition \ref{prop: eventualreg}, we may choose $T^\ast$ uniformly for $\beta$ and $v_0$ verifying \eqref{hyp} for some $0<\epsilon\leq 1$. The conclusion then follows from Proposition \ref{prop: locstab H3} applied with $T=T^\ast$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main2}] We check that the assumptions \eqref{eq: add integrability assump} of Proposition~\ref{prop:global} are satisfied for $s=1$. Let $u\in C([0,+\infty), H^1)$ be an a priori global smooth solution with initial datum $u_0$. Fix $T>0$. Then, differentiating the equation $k$-times and performing energy estimates, we see that $u \in L^\infty((0,T),H^k)\cap L^2((0,T),H^{k+\alpha})$ for every $k\geq 1$. Hence, if $u_0 \in H^s$ then $u\in L^2_{loc}([0,+\infty), H^r)$ for any $1 \leq r \leq s+\alpha$. We infer that in the ipodissipative case, the additional integrability assumption \eqref{eq: add integrability assump} is fulfilled with $\delta:=1-\alpha$ provided $u_0\in H^{2-\alpha}$. Since $2-\alpha \leq 2-\frac{3}{4}=\frac{5}{4}$ for $\alpha \in (\frac{3}{4},1)$, it is enough to ask $u_0 \in H^s$ for some $s\geq \frac{5}{4}$. Finally, in the hyperdissipative case, \eqref{eq: add integrability assump} is fulfilled for any $s>1$.
\end{proof}
\begin{remark}
In the hyperdissipative range $\alpha>1$, for any $s\in (0, \frac{1}{2}]$, we could adapt the proof of Theorem \ref{thm:main2} to deduce from Proposition~\ref{prop:global} - for instance - openness of initial data and fractional orders giving rise to global smooth solutions in
$$\left\{ u_0 \in H^s(\mathbb{R}^3; \mathbb{R}^3) : \div u_0=0\right\} \times \Big(\frac{5}{4}-\frac{s}{2}, \frac{5}{4}\Big]\, .$$
\end{remark}
\begin{proof}[Proof of Theorem~\ref{thm:main1}]
Let $u_0\in H^\delta$ divergence-free with $\norm{u_0}_{H^\delta} \leq M$ and let $u$ be the unique smooth Leray-Hopf solution to the fractional Navier-Stokes equations of order $\alpha=\frac{5}{4}$ starting from $u_0$. We claim the following a priori estimate
\begin{equation}\label{eq: aprioriest at 54}
\frac{\mathrm{d}}{\mathrm{d}t} \norm{u(t)}_{H^\delta}^2 + \norm{u(\tau)}_{H^{\delta+\alpha}}^2 \leq C_A \norm{(-\Delta)^{\alpha/2} u(t)}_{L^2}^2 \norm{u(t)}_{H^\delta}^2 \, ,
\end{equation}
where $C_A>0$ is independent of $\delta$. Indeed, arguing as in the proof of Lemma \ref{lem: energyineq for diffeq H1}, we obtain the following energy inequality for derivatives of order $\delta$
\begin{align*}
\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \norm{ (-\Delta)^{\delta/2} u(t)}_{L^2}^2 &+ \norm{ (-\Delta)^{(\alpha+\delta)/2} u(t)}_{L^2}^2 \leq \left \lvert \int (-\Delta)^{\delta/2}\left( u \cdot \nabla) u \right) \cdot (-\Delta)^{\delta/2} u \, \mathrm{d}x\right \rvert \\
&\leq (1+\bar D) \bar C ^2 \norm{(-\Delta)^{(\delta+\alpha)/2}u}_{L^2} \norm{u}_{\dot{H}^{\frac{5}{2}-\alpha}} \norm{(-\Delta)^{\delta/2} u}_{L^2}
\end{align*}
by \eqref{eqn:commuta} (applied with $p_1=\frac{6}{3-2\alpha}$, $p_2=\frac{6}{2\alpha}$, $s_1=s$, $s_2=0$). In particular, it does not depend on $\delta$. Observing that at $\alpha=\frac{5}{4}$ it holds $ \norm{u}_{\dot{H}^{\frac{5}{2}-\alpha}} = \norm{(-\Delta)^{\alpha/2} u}_{L^2}$, this proves \eqref{eq: aprioriest at 54} with $C_A = 2 (1+\bar D)^2 \bar C^4 \leq 8 \bar D^2 \bar C^4$. From \eqref{eq: aprioriest at 54}, we deduce by Gr\"onwall the estimates
\begin{align*}
\norm{u}_{L^\infty([0, T], H^\delta)}^2 &\leq \norm{u_0}_{H^\delta}^2 e^{C_A \int_0^T \int \lvert (-\Delta)^{\alpha/2} u(x,t) \rvert^2 \, \mathrm{d}x\, \mathrm{d}t} \leq M^2 e^{C_A M^2} \, ,\\
\int_0^T \norm{u(t)}_{H^{\delta+\frac{5}{4}}}^2 \, \mathrm{d}t &\leq \norm{u_0}_{H^\delta}^2 + \int_0^T \norm{(-\Delta)^{\alpha/2} u(t)}_{L^2}^2 \norm{u(t)}_{H^\delta}^2 \, \mathrm{d}t \leq M^2(1+M^4 e^{C_A M^2} )\, .
\end{align*}
In particular, $u$ satisfies the additional integrability assumption \eqref{eq: add integrability assump}. Thus, by Proposition \ref{prop:global}, there exists $\epsilon>0$ such that for any fractional order $\beta \in [1, \frac{5}{4}]$ satisfying $\lvert \frac{5}{4}-\beta \rvert < \epsilon$ there exists a unique global, smooth solution to the fractional Navier-Stokes equations of order $\beta$ starting from $u_0$. Notice that the additional assumption $u_0 \in L^p$ for some $p \in [1,2)$ can be dropped thanks to Remark \ref{rmk: dropLp}. We recall from the explicit choice of $\epsilon$ in \eqref{eq: localchoiceofepsilon} that
\begin{equation}\label{eq: epsilonexplicit 54}
\epsilon \leq \min \left \{ \left( \frac{\delta}{4} \right)^{\frac{1}{\delta}}, \left((C_0 T^\ast)^\frac{4\beta -5 + 2\delta}{2(\beta+\delta)}
e^{2C_1 \norm{u}_{L^2([0,T^\ast], H^{\frac{5}{4}+\delta})}^2} \max\{C_2 \norm{u}_{L^2([0,T^\ast], H^{\frac{5}{4}+\delta})}^2,1\}\right)^{-\frac{1}{\delta}} \right \} \,,
\end{equation}
where now $T^\ast$ is the eventual regularization time given by Corollary \ref{cor: LeraySingTime} and the constants $C_0$, $C_1$ and $C_2$ are coming from Lemma \ref{lem: energyineq for diffeq H1} applied with $s=\delta$. In the hyperdissipative range, we have an explicit upper bound on $T^\ast$ in terms of $\beta$ and $M$ through \eqref{eq: upperbound for Tstar hyp}. Indeed, it follows from \eqref{eq: upperbound for Tstar hyp} by a simple computation that $T^\ast$ can be bounded, uniformly in $\beta$, by
\begin{equation}
T^\ast \leq e^{M^\frac{5}{2} \bar C^3/e} \, ,
\end{equation}
Moreover, from the explicit expression of $C_0$ and $C_1$ in Lemma \ref{lem: energyineq for diffeq H1} (see \eqref{eq: C0Hs}), we have infer
\begin{align*}
C_0^\frac{4\beta -5 + 2\delta}{2(\beta+\delta)} \leq 24(1+ \bar D)\bar C^2 \leq 48 \bar D \bar C^2 \text{ and } C_1 \leq 6(3(1+\bar D)^2\bar C^2+2 \bar C^6) \leq 72 \bar D^2 C^6
\end{align*}
Hence we can estimate the $\epsilon$ in terms of $M$ and $\delta$ by
\begin{equation}\label{eq: epsilonexplicit 54 ugly}
\epsilon := \min \left \{ \left(\frac{\delta}{4}\right)^{\frac{1}{\delta}}, \left( 48 \bar D \bar C^2 e^{M^\frac{5}{2} \bar C^3/e} \max\{2 C_2 M^2(1+M^4 e^{C_A M^2}), 1 \} e^{144 \bar D^2 \bar C^6 M^2(1+M^4 e^{C_A M^2})} \right)^{-\frac{1}{\delta}} \right \}
\end{equation}
where $C_2$ is the universal constant from Lemma \ref{lem: fraclapRn}.
\end{proof}
\textbf{ Acknowledgements}. The authors were partially supported by
the Swiss national science foundation grant 182565 "Regularity issues for the Navier-Stokes equations and for other variational problems".
|
1,314,259,996,560 | arxiv | \section{Introduction} \label{Int}
General Theory of Relativity (GR) is the cornerstone of
astrophysics and cosmology, giving predictions with unprecedented success.
At astrophysical scales GR has been tested in, for example, the solar system, stellar dynamics, black hole formation and evolution, among others (see for instance\cite{FischbachTalmadge1999,Will1993,Kamionkowski:2007wv,*Matts,*Peebles:2013hla}).
However, GR is being currently tested with various phenomena that can be
significant challenges to the GR theory, generating important changes never seen before. Ones of the major challenges of modern cosmology are undoubtedly dark matter (DM) and dark energy. They comprise approximately $27\%$ for DM and $68\%$ for dark energy, of our
universe\cite{PlanckCollaboration2013} allowing the formation of large scale structures\cite{FW,Diaferio:2008jy}.
Dark matter has been invoked as the mechanism to stabilize spiral galaxies and to provide with a matter distribution component to explain the observed rotation curves.
Nowadays the best model of the universe we have is the
\emph{concordance} or $\Lambda$CDM model that has been successful in explaining the very large-scale structure formation, the statistics of the distribution of galaxy clusters, the temperature anisotropies of the cosmic microwave background radiation (CMB) and many other astronomical observations.
In spite of all successes we have mentioned,
this model has several problems, for example: predicts too much power on small scale\cite{Rodriguez-Meza:2012}, then it over predicts the number of observed satellite galaxies\cite{Klypin1999,Moore1999,Papastergis2011} and predicts halo profiles that are denser and
cuspier than those inferred observationally\cite{Navarro1997,Subramanian2000,Salucci}, and also predicts a population of massive and concentrated subclass that are inconsistent
with observations of the kinematics of Milky Way satellites\cite{BoylanKolchin2012}.
One of the first astronomical observations that brought attention on DM was the observation of rotation curves of spiral galaxies by Rubin and coworkers\cite{Rubin2001},
these observations turned out to be the main tool to investigate the role of DM at galactic scales: its role in determining the structure, how the mass is distributed, and the dynamics, evolution, and formation of spiral galaxies.
Remarkably, the corresponding rotation velocities of galaxies, can be explained with the density profiles of different Newtonian DM models like Pseudo Isothermal profile (PISO)\cite{piso}, Navarro-Frenk-White profile (NFW)\cite{Navarro1997}
or Burkert profile\cite{Burkert}, among
others\cite{Einasto}; except by the fact that until now it is unsolved the problem of cusp and core in the densities profiles. In this way, none of them have the last word because the main questions, about the density distribution and of course the \emph{nature} of DM, has not been resolved.
Alternative theories of gravity have been used to model DM. For instance a scalar field has been proposed to model DM\cite{Dick1996,Cho/Keum:1998},
and has been used to study rotation curves of spiral galaxies\cite{Guzman/Matos:2000}. This scalar field is coupled minimally to the metric, however, scalar fields coupled non minimally to the metric have also been used to study DM\cite{Rodriguez-Meza:2012,RodriguezMeza/Others:2001,RodriguezMeza/CervantesCota:2004,Rodriguez-Meza:2012b}. Equivalently $F(R)$ models exists in the literature that analyzes rotation curves\cite{Martins/Salucci:2007}.
On the other hand, one of the best candidates to extend GR is the brane theory, whose main characteristic is to add another dimension having a five dimensional bulk where it is embedded a four dimensional manifold called the brane\cite{Randall-I,*Randall-II}. This model is characterized by the fact that the standard model of particles is confined in the brane and only the gravitational interaction can travel in the bulk\cite{Randall-I,*Randall-II}. The assumption that the five dimensional Einstein's equations are valid, generates corrections in the four dimensional Einstein's equations confined in the brane bringing information from the extra dimension\cite{sms}.
These extra corrections in the Einstein's equations can help us to elucidate and solve the problems that afflicts the modern cosmology and astrophysics\cite{m2000,*yo2,*Casadio2012251,*jf1,*gm,*Garcia-Aspeitia:2013jea,*langlois2001large,*Garcia-Aspeitia:2014pna,*jf2,*PerezLorenzana:2005iv,*Ovalle:2014uwa,*Garcia-Aspeitia:2014jda,*Linares:2015fsa,*Casadio:2004nz}.
Before we start, let us mention here some experimental constraints on braneworld models, most of them about the so-called brane tension, $\lambda$, which appears explicitly as a free parameter in the corrections of the gravitational equations mentioned above. As a first example we have the measurements on the deviations from Newton's law of the gravitational interaction at small distances. It is reported that no deviation is observed for distances $l \gtrsim 0.1 \, {\rm mm}$, which then implies a lower limit on the brane tension in the Randall-Sundrum II model (RSII): $\lambda> 1 \, {\rm TeV}^{4}$\cite{Kapner:2006si,*Alexeyev:2015gja}; it is important to mention that these limits do not apply to the two-branes case of the Randall-Sundrum I model (RSI) (see \cite{mk} for details).
Astrophysical studies, related to gravitational waves and stellar stability, constrain
the brane tension to be $\lambda > 5\times10^{8} \, {\rm MeV}^{4}$\cite{gm,Sakstein:2014nfa}, whereas the existence of black hole X-ray binaries suggests that $l\lesssim 10^{-2} {\rm mm}$\cite{mk,Kudoh:2003xz,*Cavaglia:2002si}. Finally, from cosmological observations, the requirement of successful nucleosynthesis provides the lower limit $\lambda> 1\, {\rm MeV}^{4}$, which is a much weaker limit as compared to other experiments (another cosmological tests can be seen in: Ref. \cite{Holanda:2013doa,*Barrow:2001pi,*Brax:2003fv}).
In fact, this paper is devoted to study the main observable of brane theory which is the brane tension, whose existence delimits between the four dimensional GR and its high energy corrections. We are given to the task of perform a Newtonian approximation of the modified Tolman-Oppenheimer-Volkoff (TOV) equation, maintaining the effective terms provided by branes which cause subtle differences in the traditional dynamics.
In this way we test the theory at galactic scale using high resolution measurements of rotation curves of a sample of low surface brightness (LSB) galaxies with no photometry\cite{deBlok/etal:2001}
and a synthetic rotation curve built from 40 rotation curves of spirals of magnitude around $M_I=-18.5$ where was found that the baryonic components has a very small contribution\cite{Salucci1},
assuming PISO, NFW and Burkert DM profiles respectively and with that, we constraint the preferred value of brane tension with observables.
That the sample has no photometry means that the galaxies are DM dominated and then we have only two parameters related to the distribution of DM, a density scale and a length scale, adding the brane tension we have three parameters in total to fit.
The brane tension fitted values are compared among the traditional DM density profile models of spiral galaxies
(PISO, NFW and Burkert) and against the same models without the presence of branes and confronted with other values of the tension parameter coming from cosmological and astrophysical observational data.
This paper is organized as follows: In Sec.\ \ref{EM} we show the equations of motion (modified TOV equations) for a spherical symmetry and the appropriate initial conditions. In Sec.\ \ref{TOV MOD} we explore the Newtonian limit and we show the mathematical expression of rotation velocity with brane modifications; particularly we show the modifications to velocity rotation expressions of PISO, NFW and Burkert DM profiles and they are compared with models without branes.
In Sec.\ \ref{Results} we test the DM models plus brane with observations: we use a sample of high resolution measurements of rotation curves of LSB galaxies and a synthetic rotation curve representative of 40 rotation curves of spirals where the baryonic component has a very small contribution.
Finally in Sec.\ \ref{Disc}, we discuss the results obtained in the paper and make some conclusions.
In what follows, we work in units in which $c=\hbar=1$, unless explicitly written.
\section{Review of equations of motion for branes} \label{EM}
Let us start by writing the equations of motion for galactic stability in a brane embedded in a five-dimensional bulk according to the RSII model\cite{Randall-II}. Following an appropriate computation (for details see\cite{mk,sms}), it is possible to demonstrate that the modified four-dimensional Einstein's equations can be written as
\begin{equation}
G_{\mu\nu} + \xi_{\mu\nu} + \Lambda_{(4)}g_{\mu\nu} = \kappa^{2}_{(4)} T_{\mu\nu} + \kappa^{4}_{(5)} \Pi_{\mu\nu} +
\kappa^{2}_{(5)} F_{\mu\nu} , \label{Eins}
\end{equation}
where $\kappa_{(4)}$ and $\kappa_{(5)}$ are respectively the four and five- dimensional coupling constants, which are related in the form: $\kappa^{2}_{(4)}=8\pi G_{N}=\kappa^{4}_{(5)} \lambda/6$, where $\lambda$ is defined as the brane tension, and $G_{N}$ is the Newton constant. For purposes of simplicity, we will not consider bulk matter, which translates into $F_{\mu\nu}=0$, and discard the presence of the four-dimensional cosmological constant, $\Lambda_{(4)}=0$,
as we do not expect it to have any important effect at galactic scales (for a recent discussion about it see\cite{Pavlidou:2013zha}). Additionally, we will neglect any nonlocal energy flux, which is allowed by the static spherically symmetric solutions we will study below\cite{gm}.
The energy-momentum tensor, the quadratic energy-momentum tensor, and the Weyl (traceless) contribution, have the explicit forms
\begin{subequations}
\label{eq:4}
\begin{eqnarray}
\label{Tmunu}
T_{\mu\nu} &=& \rho u_{\mu}u_{\nu} + p h_{\mu\nu} \, , \\
\label{Pimunu}
\Pi_{\mu\nu} &=& \frac{1}{12} \rho \left[ \rho u_{\mu}u_{\nu} + (\rho+2p) h_{\mu\nu} \right] \, , \\
\label{ximunu}
\xi_{\mu\nu} &=& - \frac{\kappa^4_{(5)}}{\kappa^4_{(4)}} \left[ \mathcal{U} u_{\mu}u_{\nu} + \mathcal{P}r_{\mu}r_{\nu}+ \frac{ h_{\mu\nu} }{3} (\mathcal{U}-\mathcal{P} ) \right] \, .
\end{eqnarray}
\end{subequations}
Here, $p$ and $\rho$ are, respectively, the pressure and energy density of the stellar matter of interest, $\mathcal{U}$ is the nonlocal energy density, and $\mathcal{P}$ is the nonlocal anisotropic stress. Also, $u_{\alpha}$ is the four-velocity (that also satisfies the condition $g_{\mu\nu}u^{\mu}u^{\nu}=-1$), $r_{\mu}$ is a unit radial vector, and $h_{\mu\nu} = g_{\mu\nu} + u_{\mu} u_{\nu}$ is the projection operator orthogonal to $u_{\mu}$.
Spherical symmetry indicates that the metric can be written as:
\begin{equation}
{ds}^{2}= - B(r){dt}^{2} + A(r){dr}^{2} + r^{2} (d\theta^{2} + \sin^{2} \theta d\varphi^{2}) \, .\label{metric}
\end{equation}
If we define the reduced Weyl functions $\mathcal{V} = 6 \mathcal{U}/\kappa^4_{(4)}$, and $\mathcal{N} = 4 \mathcal{P}/\kappa^4_{(4)}$. First, we define the effective mass as:
\begin{equation}
\mathcal{M}^\prime_{eff} = 4\pi{r}^{2}\rho_{eff}. \label{eq:7a}
\end{equation}
Then, from Eqs. \eqref{Eins} and \eqref{eq:4} and after straightforward calculations we have the following equations of motion:
\begin{subequations}
\label{eq:7}
\begin{eqnarray}
p^\prime &=& -\frac{G_N}{r^{2}} \frac{4 \pi \, p_{eff} \, r^3 + \mathcal{M}_{eff}}{1 - 2G_N \mathcal{M}_{eff}/r} ( p + \rho ) \, , \label{eq:7b} \\
\mathcal{V}^{\prime} + 3 \mathcal{N}^{\prime} &=& - \frac{2G_N}{r^{2}} \frac{4 \pi \, p_{eff} \, r^3 + \mathcal{M}_{eff}}{1 - 2G_N \mathcal{M}_{eff}/r} \left( 2 \mathcal{V} + 3 \mathcal{N} \right)\nonumber\\
&& - \frac{9}{r} \mathcal{N} - 3 (\rho+p) \rho^{\prime} \, , \label{eq:7c}
\end{eqnarray}
\end{subequations}
where a prime indicates derivative with respect to $r$, $A(r) = [1 - 2G_N \mathcal{M}(r)_{eff}/r]^{-1}$, and the effective energy density and pressure, respectively, are given as:
\begin{subequations}
\label{eq:3}
\begin{eqnarray}
\rho_{eff} &=& \rho \left( 1 + \frac{\rho}{2\lambda} \right) + \frac{\mathcal{V}}{\lambda} \, , \label{eq:3a} \\
p_{eff} &=& p \left(1 + \frac{\rho}{\lambda} \right) + \frac{\rho^{2}}{2\lambda} + \frac{\mathcal{V}}{3\lambda} + \frac{\mathcal{N}}{\lambda} \, . \label{eq:3b}
\end{eqnarray}
\end{subequations}
Even though we will not consider exterior galaxy solutions, we must anyway take into account the information provided by the Israel-Darmois (ID) matching condition, which for the case under study can be written as\cite{gm}:
\begin{equation}
\label{eq:28}
(3/2) \rho^2(R) + \mathcal{V}^-(R) + 3 \mathcal{N}^-(R) = 0 \, .
\end{equation}
In this case, the superscript ($-$) indicates the interior value of the quantity at the halo surface\footnote{We denote the surface of the galaxy as a region where does not exist DM or baryons, \emph{i.e.}, the intergalactic space.} of the galaxy, assuming that $\rho(r>R)=0$ where $R$ denotes the maximum size of the galaxy. Also, the previous equation takes in consideration the fact that the exterior must be Schwarzschild which in general the following condition must be fulfilled $\mathcal{V}(r \geq R) = 0 =\mathcal{N}(r\geq R)$ (see\cite{Garcia-Aspeitia:2014pna} for details).
For completeness, we just note that the exterior solutions of the metric functions are given by the well known expressions $B(r) = A^{-1}(r) = 1 - 2G_N M_{eff}/r$.
Finally, we impose $\mathcal{N}=0$ (see\cite{Garcia-Aspeitia:2014pna}). Implying that Eq. \eqref{eq:28} is restricted as:
\begin{equation}
\label{eq:29}
-(3/2) \rho^2(R) = \mathcal{V}^-(R) \, ,
\end{equation}
with the aim of maintain a galaxy Schwarzschild exterior.
\section{Low energy limit and rotation curves} \label{TOV MOD}
To begin with, we observe, from Eq.\ \eqref{eq:7b} in the low energy (Newtonian) limit, that we have: $r^{2}p^{\prime}=-G_{N}\mathcal{M}_{eff}\rho$. Differentiating we found
\begin{equation}
\frac{d}{dr}\left(\frac{r^{2}}{\rho}\frac{dp}{dr}\right)=-4\pi r^{2}G_{N}\rho_{\rm eff}. \label{eqdiff9}
\end{equation}
From here it is possible to note that $d\Phi/dr=-\rho^{-1}(dp/dr)$ resulting in
\begin{equation}
\nabla^{2}\Phi_{\rm eff}=\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\Phi_{\rm eff}}{dr}\right)=4\pi G_{N}\rho_{\rm eff}, \label{Poisson}
\end{equation}
being necessary to define the energy density of DM together with the nonlocal energy density. Notice that the nonlocal energy density can be obtained easily from Eq.\ \eqref{eq:7c} in the galaxy interior and also the fluid behaves like dust, implying the condition $p=0$, always fulfilling the low energy condition $4\pi r^3p_{eff}\ll\mathcal{M}_{eff}$ and $2G_{N}\mathcal{M}_{eff}/r\ll1$, between effective quantities and in consequence $4G_{N}\mathcal{M}_{eff}\mathcal{V}/r^2\sim0$, is negligible.
In addition, the rotation curve is obtained from the contribution of the effective potential, this expression can be written as:
\begin{eqnarray}
V^2(r) &=& r\left\vert\frac{d\Phi_{\rm eff}}{dr}\right\vert=\frac{G_N \mathcal{M}_{eff}(r)}{r}
\nonumber \\
&=&
\frac{G_N }{r}
\left[
\mathcal{M}_{DM}(r) + \mathcal{M}_{Brane}(r)
\right]
, \label{rotvel}
\end{eqnarray}
where $\mathcal{M}_{DM}(r)$ is the contribution to the mass from DM, $\mathcal{M}_{Brane}(r)$ gives the modification to the DM mass that comes from the brane; and $\mathcal{M}_{eff}(r)$ must be greater than zero. From here, it is possible to study the rotation velocities of the DM, assuming a variety of density profiles.
Before we start let us define the following dimensionless variables: $\bar{r}\equiv r/r_{\rm s}$, $v_{0}^{2}\equiv4\pi G_{N}r_{\rm s}^{2}\rho_{\rm s}$ and $\bar{\rho}\equiv\rho_{\rm s}/2\lambda$ where $\rho_{\rm s}$, is the central density of the halo and $r_{s}$ is associated with the central radius of the halo.
\subsection{Pseudo isothermal profile for dark matter}
Here we consider that DM density profile is given by PISO\cite{piso} written as:
\begin{equation}
\rho_{\rm PISO}(r)=\frac{\rho_{\rm s}}{1+\bar{r}^{2}}. \label{PIP}
\end{equation}
From Eq. \eqref{rotvel}, together with Eq. \eqref{PIP}, it is possible to obtain:
\begin{eqnarray}
V_{\rm PISO}^{2}(\bar{r}) &=& v_{0}^{2}
\left\lbrace
\left(
1-\frac{1}{\bar{r}}\arctan\bar{r}
\right)
\right.
\nonumber \\
&&
\left. + \bar{\rho}
\left(
\frac{1}{1+\bar{r}^2}- \frac{1}{\bar{r}}\arctan\bar{r}
\right)
\right\rbrace.
\label{RCPISO}
\end{eqnarray}
In the limit $\bar{\rho}\to0$, we recover the classical rotation velocity associated with PISO for DM.
The effective density must be positive defined, then $\lambda > \rho_s$ must be fulfilled. The first right-hand term in parenthesis in Eq.\ \eqref{RCPISO} is PISO dark matter contribution and the second
is brane's contribution.
\subsection{Navarro-Frenk-White profile for dark matter}
Another interesting case (motivated by cosmological $N$-body simulations) is the NFW density profile, which is given by\cite{NFW}:
\begin{equation}
\rho_{\rm NFW}(r)=\frac{\rho_{\rm s}}{\bar{r}(1+\bar{r})^{2}}. \label{NFW}
\end{equation}
This is a density profile that diverges as $r \rightarrow 0$
and
it is not possible to say
that $\rho_s$ is related with the central density of the DM distribution.
Also density goes as $1/\bar{r}^3$ when $\bar{r} \gg 1$.
However, in this particular case, we will still be calling them the \emph{central} density and radius of the NFW matter distribution.
From Eq.\ \eqref{rotvel}, together with Eq.\ \eqref{NFW} we obtain the following rotation curve:
\begin{eqnarray}
V_{\rm NFW}^{2}(\bar{r}) &=& v_{0}^{2}\left\lbrace\left(\frac{(1+\bar{r})\ln(1+\bar{r})-\bar{r}}{\bar{r}(1+\bar{r})}\right)\right.\nonumber\\&&
\left.+\frac{2\bar{\rho}}{3\bar{r}}\left(\frac{1}{(1+\bar{r})^{3}}-1\right) \right\rbrace.
\label{RCNFW}
\end{eqnarray}
The first right-hand term in parenthesis in Eq.\ \eqref{RCNFW} is NFW dark matter contribution and the second one is the brane's contribution. Notice that we recover also the classical limit when $\bar{\rho}\to0$.
In addition, it is important to remark that the effective density must be positive defined, then $\lambda > \rho_s r_s /r$. Also, if $\mathcal{M}(r)$ must be greater than zero, then $r > r_{min}$ where $r_{min}$ is given by solving the following equation:
\begin{equation}
\frac{2}{3}\bar{\rho}=\frac{(\alpha+1)^2[(\alpha+1)\ln(\alpha+1)-\alpha]}{(\alpha+1)^3-1},\label{comp}
\end{equation}
where we define $\alpha\equiv r_{min}/r_s$ as a dimensionless quantity.
\subsection{Burkert density profile for dark matter}
Another density profile was proposed by Burkert\cite{Burkert}, which it has the form:
\begin{equation}
\rho_{\rm Burk}=\frac{\rho_{\rm s}}{(1+\bar{r})(1+\bar{r}^{2})}. \label{Burk}
\end{equation}
Again, from Eq.\ \eqref{rotvel}, together with Eq.\ \eqref{Burk} we obtain the following rotation curve:
\begin{eqnarray}
V_{\rm Burk}^{2}(\bar{r}) &=&\frac{v_{0}^{2}}{4\bar{r}} \left\lbrace \left( \ln[(1+\bar{r}^{2})(1+\bar{r})^{2}]-2\arctan(\bar{r}) \right)\right.
\label{RCBurkert}
\\&&
\left.+ \frac{1}{2}\bar{\rho}\left( \frac{1}{1+\bar{r}}+\frac{1}{1+\bar{r}^{2}}+\arctan(\bar{r})-2 \right)\right\rbrace.
\nonumber
\end{eqnarray}
In the limit $\bar{\rho}\to0$, we recover the classical rotation velocity associated with Burkert density profile\cite{Burkert}.
The effective density must be positive defined, then $\lambda > \rho_s$.
Again the first right-hand term in parenthesis in Eq.\ \eqref{RCBurkert} is
Burkert DM contribution and the second one comes from the
brane's contribution.
\section{Constrictions with galaxies without photometry} \label{Results}
To start with the analysis, we $\chi^{2}$ best fit the observational rotation curves of the sample with:
\begin{equation}
\chi^{2}=\sum_{i=1}^{N}\left(\frac{V_{theo}-V_{exp \; i}}{\delta V_{exp\; i}}\right)^{2},
\label{chi2Eq}
\end{equation}
where $i$ runs from one up to the number of points in the data, $N$; $V_{theo}$, is computed according to the velocity profile under consideration
and $\delta V_{exp\; i}$, is the error in the measurement of the rotational velocity. Notice that
the free parameters are only for DM-Branes: $r_{s}$, $\rho_{s}$ and $\lambda$.
In the tables below we show $\chi_{red}^{2} \equiv \chi^{2}/(N - n_p -1)$ where $n_p$ is the number of parameters to fit, being in our case, $n_p=3$.
The analyzed sample of galaxies are twelve high resolution rotation curves of LSB galaxies with no photometry (visible components, such as gas and stars, are negligible) as given in Ref.\cite{deBlok/etal:2001}. This sample was used to study DM equation of state (EoS) in Ref.\cite{Barranco/etal:2015}. We remark that in this part we use units such that $4 \pi G_{N}=1$, velocities are in km/s, and distances are given in kpc.
\subsection{Results: PISO profile + Branes}
We have estimated the parameters of the PISO+branes model
and were compared with PISO model without brane contribution, minimizing the appropriate $\chi^2$ for the sample of observed rotational curves, using Eq.\ (\ref{chi2Eq}) with Eq.\ (\ref{RCPISO}) and taking into account that $\lambda > \rho_s$ must be fulfilled.
In Fig.\ \ref{PISO1} we show, for each one of the galaxies in the sample,
the plots of the PISO theoretical rotation curve (solid line), that best fit of the corresponding observational data (orange symbols); also shown are the errors of the estimation (brown symbols).
For each galaxy we have plotted the contribution to the rotation velocity due only to the brane (red long-dashed curve) and only to the dark matter PISO density profile (blue short-dashed curve), see Eq.\ (\ref{RCPISO}).
Brane effects are very clear in galaxies:
ESO 2060140,
ESO 3020120,
U 11616,
U 11648,
U 11748,
U 11819.
In Table \ref{TablePiso} it is shown the central density, central radius and the brane tension which is the free parameter of the brane theory (only in PISO+branes). As a comparison, it is also shown the central density and radius without brane contribution.
The worst fitted galaxies were (high $\chi_{red}^2$ value):
U 11648,
U 11748.
The fitted brane tension values presents great dispersion, from the lower value:
$0.167\; M_{\odot}/\rm pc^3$, ESO 3020120 to the higher value:
$108.096\; M_{\odot}/\rm pc^3$, ESO 4880049.
It is useful to have $\lambda$ in eV, where the conversion from solar masses to eV is: $1 M_{\odot}/\rm pc^3 \sim 2.915\times10^{-4}eV^4$.
The brane tension parameter has an average value of $\langle\lambda\rangle_{\rm PISO} = 33.178 \; M_{\odot}/\rm pc^3$ with a standard deviation $\sigma_{\rm PISO} = 40.935 \; M_{\odot}/\rm pc^3$. Notice that we can't see a clear tendency to a $\lambda$ value or range of values.
\begin{figure}
\includegraphics[scale=0.33]{vceso30500900PISO+Branes}
\includegraphics[scale=0.33]{vceso0140040PISO+Branes} \\
\includegraphics[scale=0.33]{vceso2060140PISO+Branes}
\includegraphics[scale=0.33]{vcESO3020120PISO+Branes} \\
\includegraphics[scale=0.33]{vceso4250180PISO+Branes}
\includegraphics[scale=0.33]{vceso4880049PISO+Branes} \\
\includegraphics[scale=0.33]{vcf570_v1PISO+Branes}
\includegraphics[scale=0.33]{vcu11454PISO+Branes} \\
\includegraphics[scale=0.33]{vcu11616PISO+Branes}
\includegraphics[scale=0.33]{vcu11648PISO+Branes} \\
\includegraphics[scale=0.33]{vcu11748PISO+Branes}
\includegraphics[scale=0.33]{vcu11819PISO+Branes}
\caption{Group of analyzed galaxies using modified rotation velocity for PISO profile: ESO 3050090,
ESO 0140040, ESO 2060140, ESO 3020120, ESO 4250180, ESO 4880049, 570\_V1, U11454, U11616, U11648, U11748, U11819. We show in the plots: Total rotation curve (solid black line), only PISO curve (short dashed blue curve) and the rotation curve associated with the mass lost by the effect of the brane (red dashed curve).}
\label{PISO1}
\end{figure}
\subsection{Results: NFW profile + Branes}
For the NFW density profile case we have the following results:
We have estimated parameters with and without brane contribution by
minimizing the corresponding $\chi^2$, Eq.\ (\ref{chi2Eq}) with Eq.\ (\ref{RCNFW}), for the sample of observed rotation curves
and taking into account that
$\lambda > \rho_s r_s /r$ in order to have an effective density positive defined, always fulfilling Eq.\ (\ref{comp}).
In Fig.\ \ref{NFW1}, it is shown, for each galaxy in the sample of the LSB galaxies, the theoretical fitted curve to a preferred brane tension value (solid line),
the NFW curve and the rotation curve associated with the mass lost by the effects of branes, see Eq.\ (\ref{RCNFW}).
In Table \ref{TableNFW} it is shown, for the sample,
the central density, central radius and $\chi_{red}^2$ values without branes; and
the central density, central radius, brane tension
and $\chi_{red}^2$ values with branes contribution.
Galaxy U 11748 is the worst fitted case with $\chi_{red}^2 = 2.163$.
For galaxies:
ESO 4250180,
ESO 4880049,
and U 11648,
there are not clear brane effects.
Galaxy U 11648 is an \emph{outlier} with a brane tension value of $4323.28\; M_{\odot}/\rm pc^3$ that is out of the range of preferred values of the other galaxies in the sample.
Notice that we have found a preferred range of tension values, from $0.487$ to $9.232$ $M_{\odot}/\rm pc^3$. Without the outlier, the brane tension parameter has an average value of
$\langle\lambda\rangle_{\rm NFW}\simeq 2.51 \; M_{\odot}/\rm pc^3$
with a standard deviation $\sigma_{\rm NFW}\simeq 3.015 \; M_{\odot}/\rm pc^3$.
\begin{figure}
\includegraphics[scale=0.33]{vceso30500900NFW+Branes}
\includegraphics[scale=0.33]{vceso0140040NFW+Branes} \\
\includegraphics[scale=0.33]{vceso2060140NFW+Branes}
\includegraphics[scale=0.33]{vcESO3020120NFW+Branes} \\
\includegraphics[scale=0.33]{vceso4250180NFW+Branes}
\includegraphics[scale=0.33]{vceso4880049NFW+Branes} \\
\includegraphics[scale=0.33]{vcf570_v1NFW+Branes}
\includegraphics[scale=0.33]{vcu11454NFW+Branes} \\
\includegraphics[scale=0.33]{vcu11616NFW+Branes}
\includegraphics[scale=0.33]{vcu11648NFW+Branes} \\
\includegraphics[scale=0.33]{vcu11748NFW+Branes}
\includegraphics[scale=0.33]{vcu11819NFW+Branes}
\caption{Group of analyzed galaxies using modified rotation velocity for NFW profile: ESO 3050090,
ESO 0140040, ESO 2060140, ESO 3020120, ESO 4250180, ESO 4880049, 570\_V1, U11454, U11616, U11648, U11748, U11819. We show in the plots: Total rotation curve (solid black line), only NFW curve (short dashed blue curve) and the rotation curve associate with the mass lost by the effect of the brane (red dashed curve).}
\label{NFW1}
\end{figure}
\subsection{Results: Burkert+Branes profile}
In the case of Burkert DM density profile, we have also estimated the parameters of the Burkert+branes model
and were compared with Burkert model without branes, minimizing the appropriate $\chi^2_{red}$, Eq.\ (\ref{chi2Eq}) with Eq.\ (\ref{RCBurkert}), for the sample of observed rotation curves. We have considered that $\lambda > \rho_s$ must be fulfilled.
The results are shown in Fig.\ \ref{Burkert1}, where it is plotted the fit to a preferred brane tension value, remarking the total rotation curve (solid line), the Burkert DM density contribution curve (blue short-dashed line) and the rotation curve associated with the mass lost by the effects of branes (red dashed line), see Eq.\ (\ref{RCBurkert}).
In Table \ref{TableBurkert} it is shown the fitted values for the central density, central radius and the corresponding value of the $\chi_{red}^2$ without brane contribution; and the fitted values for
the central density, central radius, brane tension, and theirs $\chi_{red}^2$ values with brane contribution.
The worst fitted (high values of $\chi_{red}^2$) galaxies are: U 11648 and U 11748.
Galaxies ESO 3020120, U 11748, and
U 11819 show a clear brane effects and also are outliers. The main tendency is that $\lambda$ has values of the order of $10^3 \;M_{\odot}/\rm pc^3$ or above, approximately.
The brane tension parameter, without the outliers, for the DM Burkert profile case has an average value of
$\langle\lambda\rangle_{\rm Burk}\simeq 3192.02 \;M_{\odot}/\rm pc^3$,
and a standard deviation of
$\sigma_{\rm Burk}\simeq 2174.97 \; M_{\odot}/\rm pc^3$.
\begin{figure}
\includegraphics[scale=0.33]{vceso30500900Burkert+Branes}
\includegraphics[scale=0.33]{vceso0140040Burkert+Branes} \\
\includegraphics[scale=0.33]{vceso2060140Burkert+Branes}
\includegraphics[scale=0.33]{vcESO3020120Burkert+Branes} \\
\includegraphics[scale=0.33]{vceso4250180Burkert+Branes}
\includegraphics[scale=0.33]{vceso4880049Burkert+Branes} \\
\includegraphics[scale=0.33]{vcf570_v1Burkert+Branes}
\includegraphics[scale=0.33]{vcu11454Burkert+Branes} \\
\includegraphics[scale=0.33]{vcu11616Burkert+Branes}
\includegraphics[scale=0.33]{vcu11648Burkert+Branes} \\
\includegraphics[scale=0.33]{vcu11748Burkert+Branes}
\includegraphics[scale=0.33]{vcu11819Burkert+Branes}
\caption{Group of analyzed galaxies using modified rotation velocity for Burkert profile: ESO 3050090,
ESO 0140040, ESO 2060140, ESO 3020120, ESO 4250180, ESO 4880049, 570\_V1, U11454, U11616, U11648, U11748, U11819. We show in the plots: Total rotation curve (solid black line), only Burkert curve (short dashed blue curve) and the rotation curve associate with the mass lost by the effect of the brane (red dashed curve).}
\label{Burkert1}
\end{figure}
\subsection{Results: a synthetic rotation curve}
Finally, we show the fitting results of the DM models plus brane's contribution to a synthetic rotation curve. This synthetic rotation curve was made of 40 rotation curves of galaxies with magnitudes around $M_I = -18.5$\cite{Salucci1}.
These 40 rotation curves came out of 1100 galaxies that gave the universal rotation curve for spirals. For this sample of low luminosity galaxies, of $M_I = -18.5$, it was shown that the baryonic disk has a very small contribution (for details see reference\cite{Salucci1}).
In this subsection we are now using units such $G=R_{opt}=V(R_{opt})=1$, where $R_{opt}$ and $V(R_{opt})$ are the optical radius and the velocity at the optical radius, respectively. $R_{opt}$ is the radius encompassing 83 per cent of the total integrated light. For an exponential disk with a surface brightness given by: $I(r) \propto \exp(-r/R_D)$, we have that $R_{opt}=3.2 R_D$\cite{Salucci1}.
In figure \ref{SM185} we show the synthetic rotation curve and the fitting results using PISO, NFW and Burkert profiles with and without brane's contribution.
\begin{figure}
\includegraphics[scale=0.33]{vcM185PISO}
\includegraphics[scale=0.33]{vcM185PISO+Branes}
\includegraphics[scale=0.33]{vcM185NFW}
\includegraphics[scale=0.33]{vcM185NFW+Branes}
\includegraphics[scale=0.33]{vcM185Burkert}
\includegraphics[scale=0.33]{vcM185Burkert+Branes}
\caption{Synthetic rotation curve of galaxies with magnitud $M_I=-18.5$.
Left panels: rotation curves fitted without branes. Right panels: rotation curves fitted with branes.
First row is for PISO model; second row is for NFW model and third row is for Burkert model.
We show in the plots: Total rotation curve (solid black line), only DM model curve (short dashed blue curve) and the rotation curve associate with the mass lost by the effect of the brane (red dashed curve).}
\label{SM185}
\end{figure}
As we can see in Table \ref{TableSynthetic} the same trend is observed in the brane's tension values as compared with the results for the LSB catalog analyzed above using PISO, NFW and Burkert as a DM profiles: lower value is obtained for NFW model and higher values is obtained for Burkert density profile.
Given that this synthetic rotation curve is built from 40 rotation curves of real spirals, the values of the brane's tension in table \ref{TableSynthetic} is representative of all these rotation curves.
Then, for PISO model $\lambda=60.692$ $M_{\odot}/\rm pc^3$, a value that is greater than the average value of the tension shown in Table \ref{TablePiso} but inside the interval marked by the standard deviation.
For NFW model $\lambda=226.054$ $M_{\odot}/\rm pc^3$, this value is lower than the average value reported in Table \ref{TableNFW} and inside the range marked by the standard deviation.
For Burkert model $\lambda=1.58\times 10^5$ $M_{\odot}/\rm pc^3$ this value is well above than the average value shown in Table \ref{TableBurkert}; a value outside the range marked by the standard deviation.
\section{Discussion and conclusions} \label{Disc}
We have presented in this paper, the effects coming from the presence of branes in galaxy rotation curves for
three density profiles used to study the behavior of DM at galactic scales.
With this in mind, we were given to the task of study a sample of
high resolution measurements of rotation curves of galaxies without photometry\cite{deBlok/etal:2001}
and a synthetic rotation curve built from 40 rotation curves of galaxies of magnitude around $M_I=-18.5$
fitting
the values of $\rho_{s}$, $r_{s}$ and $\lambda$ through minimizing the $\chi^{2}_{red}$ value and we have compared with the standard results of $\rho_{s}$, $r_{s}$ for each DM density profile without branes.
The results for every observable in the three different profiles were summarized and compared in Tables \ref{TablePiso}-\ref{TableSynthetic}.
From here, it is possible to observe how the results show a weaker limit for the value of brane tension
($\sim10^{-3}\; \rm eV^4-46$ eV$^4$) for the three models, in comparison with other astrophysical and cosmological studies\cite{Kapner:2006si,Alexeyev:2015gja,mk,gm,Sakstein:2014nfa,Kudoh:2003xz,Cavaglia:2002si,Holanda:2013doa,Barrow:2001pi,Brax:2003fv}; for example, Linares \emph{et al.}\cite{Linares:2015fsa} show that weaker values than $\lambda \simeq 10^{4}$ MeV$^{4}$, present anomalous behavior in the compactness of a dwarf star composed by a polytropic EoS, concluding that a wide region of their bound
will show a non compactness stellar configuration, if it is applied to the study shown in\cite{Linares:2015fsa}.
It is important to notice that chosen a value of brane tension that not fulfill our bounds imposed through the paper, generate an anomalous behavior in the center of the galaxy which is characteristic of the model. Remarkable, for higher values of this bound, the modified rotation curves are in good agreement with
the observed rotation curves of the sample that we use,
presenting only the distinctive features of each density profile: For example,
NFW dark matter density profile prefers lower values of the brane tension (on the average $\lambda \sim 0.73\times 10^{-3}$ eV$^4$), implying clear effects of the brane;
PISO dark matter
case has an average value of $\lambda \sim 0.96\times 10^{-2}$ eV$^4$ and show relatively the maximum dispersion on the fitted values of the brane tension;
whereas Burkert DM density profile shows negligible brane effects, on the average $\lambda \sim 0.93$ eV$^4$ -- $46$ eV$^4$.
In addition, it is important to discuss briefly the changes caused by the presence of branes in the problem of cusp/core. Notice that in this case the part that play a role is the effective density which is written in terms of brane corrections as: $\rho_{eff}=\rho(1-\rho/\lambda)$; it is notorious how the small perturbations alleviate the cusp problem which afflicts NFW, albeit the excessive presence of these terms could generates a negative effective density profile; also PISO and Burkert show modifications when $r\to0$ but does not pledge its core behavior while the brane tension only takes small values. In this way, the possibility of having a core behavior, help us to constraint the value of brane tension and still keep in the game the NFW profile.
Summarizing, it is really challenging to establish bounds in a dynamical systems like rotation curves in galaxies due to the
low densities found in the galactic medium, giving only a weaker limits in comparison with other studies in a most energetic systems.
Our most important conclusion, is that despite the efforts, we think that it is not straightforward to do that the results fit with other astrophysical and cosmological studies, being impractical and not feasible to
find evidence of extra dimensions in galactic dynamics through the determination of the brane tension value,
even more, we think that exist too much dispersion in the fitted values of the brane tension using this method for some DM density profile models.
Also it is important to note that the value of the brane tension is strongly dependent of the characteristic of the galaxy studied, suggesting an average for the preferred value of the brane tension in each case.
In addition, we notice that the effects of extra dimensions are stronger in the galactic core, suggesting that the NFW model is not appropriate in the search of constraints in brane theory due to the divergence in the center of the galaxy (see Eq.\ \eqref{comp}); PISO and Burkert could be good candidates to explore the galactic core in this framework; however it is necessary a more extensive study before we obtain a definitive conclusion.
As a final note, we know that it is necessary to recollect more observational data to constraint
the models or even give the final conclusion about extra dimensions (for or against), supporting the brane constraints shown through this paper with a more profound study of galactic dynamic or other tests like cosmological evidences presented in CMB anisotropies.
However, this work is in progress and will be reported elsewhere.
\begin{acknowledgements}
MAG-A acknowledge support from SNI-M\'exico and CONACyT research fellow. Instituto Avanzado de Cosmolog\'ia (IAC) collaborations.
\end{acknowledgements}
\section{Introduction} \label{Int}
General Theory of Relativity (GR) is the cornerstone of
astrophysics and cosmology, giving predictions with unprecedented success.
At astrophysical scales GR has been tested in, for example, the solar system, stellar dynamics, black hole formation and evolution, among others (see for instance\cite{FischbachTalmadge1999,Will1993,Kamionkowski:2007wv,*Matts,*Peebles:2013hla}).
However, GR is being currently tested with various phenomena that can be
significant challenges to the GR theory, generating important changes never seen before. Ones of the major challenges of modern cosmology are undoubtedly dark matter (DM) and dark energy. They comprise approximately $27\%$ for DM and $68\%$ for dark energy, of our
universe\cite{PlanckCollaboration2013} allowing the formation of large scale structures\cite{FW,Diaferio:2008jy}.
Dark matter has been invoked as the mechanism to stabilize spiral galaxies and to provide with a matter distribution component to explain the observed rotation curves.
Nowadays the best model of the universe we have is the
\emph{concordance} or $\Lambda$CDM model that has been successful in explaining the very large-scale structure formation, the statistics of the distribution of galaxy clusters, the temperature anisotropies of the cosmic microwave background radiation (CMB) and many other astronomical observations.
In spite of all successes we have mentioned,
this model has several problems, for example: predicts too much power on small scale\cite{Rodriguez-Meza:2012}, then it over predicts the number of observed satellite galaxies\cite{Klypin1999,Moore1999,Papastergis2011} and predicts halo profiles that are denser and
cuspier than those inferred observationally\cite{Navarro1997,Subramanian2000,Salucci}, and also predicts a population of massive and concentrated subclass that are inconsistent
with observations of the kinematics of Milky Way satellites\cite{BoylanKolchin2012}.
One of the first astronomical observations that brought attention on DM was the observation of rotation curves of spiral galaxies by Rubin and coworkers\cite{Rubin2001},
these observations turned out to be the main tool to investigate the role of DM at galactic scales: its role in determining the structure, how the mass is distributed, and the dynamics, evolution, and formation of spiral galaxies.
Remarkably, the corresponding rotation velocities of galaxies, can be explained with the density profiles of different Newtonian DM models like Pseudo Isothermal profile (PISO)\cite{piso}, Navarro-Frenk-White profile (NFW)\cite{Navarro1997}
or Burkert profile\cite{Burkert}, among
others\cite{Einasto}; except by the fact that until now it is unsolved the problem of cusp and core in the densities profiles. In this way, none of them have the last word because the main questions, about the density distribution and of course the \emph{nature} of DM, has not been resolved.
Alternative theories of gravity have been used to model DM. For instance a scalar field has been proposed to model DM\cite{Dick1996,Cho/Keum:1998},
and has been used to study rotation curves of spiral galaxies\cite{Guzman/Matos:2000}. This scalar field is coupled minimally to the metric, however, scalar fields coupled non minimally to the metric have also been used to study DM\cite{Rodriguez-Meza:2012,RodriguezMeza/Others:2001,RodriguezMeza/CervantesCota:2004,Rodriguez-Meza:2012b}. Equivalently $F(R)$ models exists in the literature that analyzes rotation curves\cite{Martins/Salucci:2007}.
On the other hand, one of the best candidates to extend GR is the brane theory, whose main characteristic is to add another dimension having a five dimensional bulk where it is embedded a four dimensional manifold called the brane\cite{Randall-I,*Randall-II}. This model is characterized by the fact that the standard model of particles is confined in the brane and only the gravitational interaction can travel in the bulk\cite{Randall-I,*Randall-II}. The assumption that the five dimensional Einstein's equations are valid, generates corrections in the four dimensional Einstein's equations confined in the brane bringing information from the extra dimension\cite{sms}.
These extra corrections in the Einstein's equations can help us to elucidate and solve the problems that afflicts the modern cosmology and astrophysics\cite{m2000,*yo2,*Casadio2012251,*jf1,*gm,*Garcia-Aspeitia:2013jea,*langlois2001large,*Garcia-Aspeitia:2014pna,*jf2,*PerezLorenzana:2005iv,*Ovalle:2014uwa,*Garcia-Aspeitia:2014jda,*Linares:2015fsa,*Casadio:2004nz}.
Before we start, let us mention here some experimental constraints on braneworld models, most of them about the so-called brane tension, $\lambda$, which appears explicitly as a free parameter in the corrections of the gravitational equations mentioned above. As a first example we have the measurements on the deviations from Newton's law of the gravitational interaction at small distances. It is reported that no deviation is observed for distances $l \gtrsim 0.1 \, {\rm mm}$, which then implies a lower limit on the brane tension in the Randall-Sundrum II model (RSII): $\lambda> 1 \, {\rm TeV}^{4}$\cite{Kapner:2006si,*Alexeyev:2015gja}; it is important to mention that these limits do not apply to the two-branes case of the Randall-Sundrum I model (RSI) (see \cite{mk} for details).
Astrophysical studies, related to gravitational waves and stellar stability, constrain
the brane tension to be $\lambda > 5\times10^{8} \, {\rm MeV}^{4}$\cite{gm,Sakstein:2014nfa}, whereas the existence of black hole X-ray binaries suggests that $l\lesssim 10^{-2} {\rm mm}$\cite{mk,Kudoh:2003xz,*Cavaglia:2002si}. Finally, from cosmological observations, the requirement of successful nucleosynthesis provides the lower limit $\lambda> 1\, {\rm MeV}^{4}$, which is a much weaker limit as compared to other experiments (another cosmological tests can be seen in: Ref. \cite{Holanda:2013doa,*Barrow:2001pi,*Brax:2003fv}).
In fact, this paper is devoted to study the main observable of brane theory which is the brane tension, whose existence delimits between the four dimensional GR and its high energy corrections. We are given to the task of perform a Newtonian approximation of the modified Tolman-Oppenheimer-Volkoff (TOV) equation, maintaining the effective terms provided by branes which cause subtle differences in the traditional dynamics.
In this way we test the theory at galactic scale using high resolution measurements of rotation curves of a sample of low surface brightness (LSB) galaxies with no photometry\cite{deBlok/etal:2001}
and a synthetic rotation curve built from 40 rotation curves of spirals of magnitude around $M_I=-18.5$ where was found that the baryonic components has a very small contribution\cite{Salucci1},
assuming PISO, NFW and Burkert DM profiles respectively and with that, we constraint the preferred value of brane tension with observables.
That the sample has no photometry means that the galaxies are DM dominated and then we have only two parameters related to the distribution of DM, a density scale and a length scale, adding the brane tension we have three parameters in total to fit.
The brane tension fitted values are compared among the traditional DM density profile models of spiral galaxies
(PISO, NFW and Burkert) and against the same models without the presence of branes and confronted with other values of the tension parameter coming from cosmological and astrophysical observational data.
This paper is organized as follows: In Sec.\ \ref{EM} we show the equations of motion (modified TOV equations) for a spherical symmetry and the appropriate initial conditions. In Sec.\ \ref{TOV MOD} we explore the Newtonian limit and we show the mathematical expression of rotation velocity with brane modifications; particularly we show the modifications to velocity rotation expressions of PISO, NFW and Burkert DM profiles and they are compared with models without branes.
In Sec.\ \ref{Results} we test the DM models plus brane with observations: we use a sample of high resolution measurements of rotation curves of LSB galaxies and a synthetic rotation curve representative of 40 rotation curves of spirals where the baryonic component has a very small contribution.
Finally in Sec.\ \ref{Disc}, we discuss the results obtained in the paper and make some conclusions.
In what follows, we work in units in which $c=\hbar=1$, unless explicitly written.
\section{Review of equations of motion for branes} \label{EM}
Let us start by writing the equations of motion for galactic stability in a brane embedded in a five-dimensional bulk according to the RSII model\cite{Randall-II}. Following an appropriate computation (for details see\cite{mk,sms}), it is possible to demonstrate that the modified four-dimensional Einstein's equations can be written as
\begin{equation}
G_{\mu\nu} + \xi_{\mu\nu} + \Lambda_{(4)}g_{\mu\nu} = \kappa^{2}_{(4)} T_{\mu\nu} + \kappa^{4}_{(5)} \Pi_{\mu\nu} +
\kappa^{2}_{(5)} F_{\mu\nu} , \label{Eins}
\end{equation}
where $\kappa_{(4)}$ and $\kappa_{(5)}$ are respectively the four and five- dimensional coupling constants, which are related in the form: $\kappa^{2}_{(4)}=8\pi G_{N}=\kappa^{4}_{(5)} \lambda/6$, where $\lambda$ is defined as the brane tension, and $G_{N}$ is the Newton constant. For purposes of simplicity, we will not consider bulk matter, which translates into $F_{\mu\nu}=0$, and discard the presence of the four-dimensional cosmological constant, $\Lambda_{(4)}=0$,
as we do not expect it to have any important effect at galactic scales (for a recent discussion about it see\cite{Pavlidou:2013zha}). Additionally, we will neglect any nonlocal energy flux, which is allowed by the static spherically symmetric solutions we will study below\cite{gm}.
The energy-momentum tensor, the quadratic energy-momentum tensor, and the Weyl (traceless) contribution, have the explicit forms
\begin{subequations}
\label{eq:4}
\begin{eqnarray}
\label{Tmunu}
T_{\mu\nu} &=& \rho u_{\mu}u_{\nu} + p h_{\mu\nu} \, , \\
\label{Pimunu}
\Pi_{\mu\nu} &=& \frac{1}{12} \rho \left[ \rho u_{\mu}u_{\nu} + (\rho+2p) h_{\mu\nu} \right] \, , \\
\label{ximunu}
\xi_{\mu\nu} &=& - \frac{\kappa^4_{(5)}}{\kappa^4_{(4)}} \left[ \mathcal{U} u_{\mu}u_{\nu} + \mathcal{P}r_{\mu}r_{\nu}+ \frac{ h_{\mu\nu} }{3} (\mathcal{U}-\mathcal{P} ) \right] \, .
\end{eqnarray}
\end{subequations}
Here, $p$ and $\rho$ are, respectively, the pressure and energy density of the stellar matter of interest, $\mathcal{U}$ is the nonlocal energy density, and $\mathcal{P}$ is the nonlocal anisotropic stress. Also, $u_{\alpha}$ is the four-velocity (that also satisfies the condition $g_{\mu\nu}u^{\mu}u^{\nu}=-1$), $r_{\mu}$ is a unit radial vector, and $h_{\mu\nu} = g_{\mu\nu} + u_{\mu} u_{\nu}$ is the projection operator orthogonal to $u_{\mu}$.
Spherical symmetry indicates that the metric can be written as:
\begin{equation}
{ds}^{2}= - B(r){dt}^{2} + A(r){dr}^{2} + r^{2} (d\theta^{2} + \sin^{2} \theta d\varphi^{2}) \, .\label{metric}
\end{equation}
If we define the reduced Weyl functions $\mathcal{V} = 6 \mathcal{U}/\kappa^4_{(4)}$, and $\mathcal{N} = 4 \mathcal{P}/\kappa^4_{(4)}$. First, we define the effective mass as:
\begin{equation}
\mathcal{M}^\prime_{eff} = 4\pi{r}^{2}\rho_{eff}. \label{eq:7a}
\end{equation}
Then, from Eqs. \eqref{Eins} and \eqref{eq:4} and after straightforward calculations we have the following equations of motion:
\begin{subequations}
\label{eq:7}
\begin{eqnarray}
p^\prime &=& -\frac{G_N}{r^{2}} \frac{4 \pi \, p_{eff} \, r^3 + \mathcal{M}_{eff}}{1 - 2G_N \mathcal{M}_{eff}/r} ( p + \rho ) \, , \label{eq:7b} \\
\mathcal{V}^{\prime} + 3 \mathcal{N}^{\prime} &=& - \frac{2G_N}{r^{2}} \frac{4 \pi \, p_{eff} \, r^3 + \mathcal{M}_{eff}}{1 - 2G_N \mathcal{M}_{eff}/r} \left( 2 \mathcal{V} + 3 \mathcal{N} \right)\nonumber\\
&& - \frac{9}{r} \mathcal{N} - 3 (\rho+p) \rho^{\prime} \, , \label{eq:7c}
\end{eqnarray}
\end{subequations}
where a prime indicates derivative with respect to $r$, $A(r) = [1 - 2G_N \mathcal{M}(r)_{eff}/r]^{-1}$, and the effective energy density and pressure, respectively, are given as:
\begin{subequations}
\label{eq:3}
\begin{eqnarray}
\rho_{eff} &=& \rho \left( 1 + \frac{\rho}{2\lambda} \right) + \frac{\mathcal{V}}{\lambda} \, , \label{eq:3a} \\
p_{eff} &=& p \left(1 + \frac{\rho}{\lambda} \right) + \frac{\rho^{2}}{2\lambda} + \frac{\mathcal{V}}{3\lambda} + \frac{\mathcal{N}}{\lambda} \, . \label{eq:3b}
\end{eqnarray}
\end{subequations}
Even though we will not consider exterior galaxy solutions, we must anyway take into account the information provided by the Israel-Darmois (ID) matching condition, which for the case under study can be written as\cite{gm}:
\begin{equation}
\label{eq:28}
(3/2) \rho^2(R) + \mathcal{V}^-(R) + 3 \mathcal{N}^-(R) = 0 \, .
\end{equation}
In this case, the superscript ($-$) indicates the interior value of the quantity at the halo surface\footnote{We denote the surface of the galaxy as a region where does not exist DM or baryons, \emph{i.e.}, the intergalactic space.} of the galaxy, assuming that $\rho(r>R)=0$ where $R$ denotes the maximum size of the galaxy. Also, the previous equation takes in consideration the fact that the exterior must be Schwarzschild which in general the following condition must be fulfilled $\mathcal{V}(r \geq R) = 0 =\mathcal{N}(r\geq R)$ (see\cite{Garcia-Aspeitia:2014pna} for details).
For completeness, we just note that the exterior solutions of the metric functions are given by the well known expressions $B(r) = A^{-1}(r) = 1 - 2G_N M_{eff}/r$.
Finally, we impose $\mathcal{N}=0$ (see\cite{Garcia-Aspeitia:2014pna}). Implying that Eq. \eqref{eq:28} is restricted as:
\begin{equation}
\label{eq:29}
-(3/2) \rho^2(R) = \mathcal{V}^-(R) \, ,
\end{equation}
with the aim of maintain a galaxy Schwarzschild exterior.
\section{Low energy limit and rotation curves} \label{TOV MOD}
To begin with, we observe, from Eq.\ \eqref{eq:7b} in the low energy (Newtonian) limit, that we have: $r^{2}p^{\prime}=-G_{N}\mathcal{M}_{eff}\rho$. Differentiating we found
\begin{equation}
\frac{d}{dr}\left(\frac{r^{2}}{\rho}\frac{dp}{dr}\right)=-4\pi r^{2}G_{N}\rho_{\rm eff}. \label{eqdiff9}
\end{equation}
From here it is possible to note that $d\Phi/dr=-\rho^{-1}(dp/dr)$ resulting in
\begin{equation}
\nabla^{2}\Phi_{\rm eff}=\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\Phi_{\rm eff}}{dr}\right)=4\pi G_{N}\rho_{\rm eff}, \label{Poisson}
\end{equation}
being necessary to define the energy density of DM together with the nonlocal energy density. Notice that the nonlocal energy density can be obtained easily from Eq.\ \eqref{eq:7c} in the galaxy interior and also the fluid behaves like dust, implying the condition $p=0$, always fulfilling the low energy condition $4\pi r^3p_{eff}\ll\mathcal{M}_{eff}$ and $2G_{N}\mathcal{M}_{eff}/r\ll1$, between effective quantities and in consequence $4G_{N}\mathcal{M}_{eff}\mathcal{V}/r^2\sim0$, is negligible.
In addition, the rotation curve is obtained from the contribution of the effective potential, this expression can be written as:
\begin{eqnarray}
V^2(r) &=& r\left\vert\frac{d\Phi_{\rm eff}}{dr}\right\vert=\frac{G_N \mathcal{M}_{eff}(r)}{r}
\nonumber \\
&=&
\frac{G_N }{r}
\left[
\mathcal{M}_{DM}(r) + \mathcal{M}_{Brane}(r)
\right]
, \label{rotvel}
\end{eqnarray}
where $\mathcal{M}_{DM}(r)$ is the contribution to the mass from DM, $\mathcal{M}_{Brane}(r)$ gives the modification to the DM mass that comes from the brane; and $\mathcal{M}_{eff}(r)$ must be greater than zero. From here, it is possible to study the rotation velocities of the DM, assuming a variety of density profiles.
Before we start let us define the following dimensionless variables: $\bar{r}\equiv r/r_{\rm s}$, $v_{0}^{2}\equiv4\pi G_{N}r_{\rm s}^{2}\rho_{\rm s}$ and $\bar{\rho}\equiv\rho_{\rm s}/2\lambda$ where $\rho_{\rm s}$, is the central density of the halo and $r_{s}$ is associated with the central radius of the halo.
\subsection{Pseudo isothermal profile for dark matter}
Here we consider that DM density profile is given by PISO\cite{piso} written as:
\begin{equation}
\rho_{\rm PISO}(r)=\frac{\rho_{\rm s}}{1+\bar{r}^{2}}. \label{PIP}
\end{equation}
From Eq. \eqref{rotvel}, together with Eq. \eqref{PIP}, it is possible to obtain:
\begin{eqnarray}
V_{\rm PISO}^{2}(\bar{r}) &=& v_{0}^{2}
\left\lbrace
\left(
1-\frac{1}{\bar{r}}\arctan\bar{r}
\right)
\right.
\nonumber \\
&&
\left. + \bar{\rho}
\left(
\frac{1}{1+\bar{r}^2}- \frac{1}{\bar{r}}\arctan\bar{r}
\right)
\right\rbrace.
\label{RCPISO}
\end{eqnarray}
In the limit $\bar{\rho}\to0$, we recover the classical rotation velocity associated with PISO for DM.
The effective density must be positive defined, then $\lambda > \rho_s$ must be fulfilled. The first right-hand term in parenthesis in Eq.\ \eqref{RCPISO} is PISO dark matter contribution and the second
is brane's contribution.
\subsection{Navarro-Frenk-White profile for dark matter}
Another interesting case (motivated by cosmological $N$-body simulations) is the NFW density profile, which is given by\cite{NFW}:
\begin{equation}
\rho_{\rm NFW}(r)=\frac{\rho_{\rm s}}{\bar{r}(1+\bar{r})^{2}}. \label{NFW}
\end{equation}
This is a density profile that diverges as $r \rightarrow 0$
and
it is not possible to say
that $\rho_s$ is related with the central density of the DM distribution.
Also density goes as $1/\bar{r}^3$ when $\bar{r} \gg 1$.
However, in this particular case, we will still be calling them the \emph{central} density and radius of the NFW matter distribution.
From Eq.\ \eqref{rotvel}, together with Eq.\ \eqref{NFW} we obtain the following rotation curve:
\begin{eqnarray}
V_{\rm NFW}^{2}(\bar{r}) &=& v_{0}^{2}\left\lbrace\left(\frac{(1+\bar{r})\ln(1+\bar{r})-\bar{r}}{\bar{r}(1+\bar{r})}\right)\right.\nonumber\\&&
\left.+\frac{2\bar{\rho}}{3\bar{r}}\left(\frac{1}{(1+\bar{r})^{3}}-1\right) \right\rbrace.
\label{RCNFW}
\end{eqnarray}
The first right-hand term in parenthesis in Eq.\ \eqref{RCNFW} is NFW dark matter contribution and the second one is the brane's contribution. Notice that we recover also the classical limit when $\bar{\rho}\to0$.
In addition, it is important to remark that the effective density must be positive defined, then $\lambda > \rho_s r_s /r$. Also, if $\mathcal{M}(r)$ must be greater than zero, then $r > r_{min}$ where $r_{min}$ is given by solving the following equation:
\begin{equation}
\frac{2}{3}\bar{\rho}=\frac{(\alpha+1)^2[(\alpha+1)\ln(\alpha+1)-\alpha]}{(\alpha+1)^3-1},\label{comp}
\end{equation}
where we define $\alpha\equiv r_{min}/r_s$ as a dimensionless quantity.
\subsection{Burkert density profile for dark matter}
Another density profile was proposed by Burkert\cite{Burkert}, which it has the form:
\begin{equation}
\rho_{\rm Burk}=\frac{\rho_{\rm s}}{(1+\bar{r})(1+\bar{r}^{2})}. \label{Burk}
\end{equation}
Again, from Eq.\ \eqref{rotvel}, together with Eq.\ \eqref{Burk} we obtain the following rotation curve:
\begin{eqnarray}
V_{\rm Burk}^{2}(\bar{r}) &=&\frac{v_{0}^{2}}{4\bar{r}} \left\lbrace \left( \ln[(1+\bar{r}^{2})(1+\bar{r})^{2}]-2\arctan(\bar{r}) \right)\right.
\label{RCBurkert}
\\&&
\left.+ \frac{1}{2}\bar{\rho}\left( \frac{1}{1+\bar{r}}+\frac{1}{1+\bar{r}^{2}}+\arctan(\bar{r})-2 \right)\right\rbrace.
\nonumber
\end{eqnarray}
In the limit $\bar{\rho}\to0$, we recover the classical rotation velocity associated with Burkert density profile\cite{Burkert}.
The effective density must be positive defined, then $\lambda > \rho_s$.
Again the first right-hand term in parenthesis in Eq.\ \eqref{RCBurkert} is
Burkert DM contribution and the second one comes from the
brane's contribution.
\section{Constrictions with galaxies without photometry} \label{Results}
To start with the analysis, we $\chi^{2}$ best fit the observational rotation curves of the sample with:
\begin{equation}
\chi^{2}=\sum_{i=1}^{N}\left(\frac{V_{theo}-V_{exp \; i}}{\delta V_{exp\; i}}\right)^{2},
\label{chi2Eq}
\end{equation}
where $i$ runs from one up to the number of points in the data, $N$; $V_{theo}$, is computed according to the velocity profile under consideration
and $\delta V_{exp\; i}$, is the error in the measurement of the rotational velocity. Notice that
the free parameters are only for DM-Branes: $r_{s}$, $\rho_{s}$ and $\lambda$.
In the tables below we show $\chi_{red}^{2} \equiv \chi^{2}/(N - n_p -1)$ where $n_p$ is the number of parameters to fit, being in our case, $n_p=3$.
The analyzed sample of galaxies are twelve high resolution rotation curves of LSB galaxies with no photometry (visible components, such as gas and stars, are negligible) as given in Ref.\cite{deBlok/etal:2001}. This sample was used to study DM equation of state (EoS) in Ref.\cite{Barranco/etal:2015}. We remark that in this part we use units such that $4 \pi G_{N}=1$, velocities are in km/s, and distances are given in kpc.
\subsection{Results: PISO profile + Branes}
We have estimated the parameters of the PISO+branes model
and were compared with PISO model without brane contribution, minimizing the appropriate $\chi^2$ for the sample of observed rotational curves, using Eq.\ (\ref{chi2Eq}) with Eq.\ (\ref{RCPISO}) and taking into account that $\lambda > \rho_s$ must be fulfilled.
In Fig.\ \ref{PISO1} we show, for each one of the galaxies in the sample,
the plots of the PISO theoretical rotation curve (solid line), that best fit of the corresponding observational data (orange symbols); also shown are the errors of the estimation (brown symbols).
For each galaxy we have plotted the contribution to the rotation velocity due only to the brane (red long-dashed curve) and only to the dark matter PISO density profile (blue short-dashed curve), see Eq.\ (\ref{RCPISO}).
Brane effects are very clear in galaxies:
ESO 2060140,
ESO 3020120,
U 11616,
U 11648,
U 11748,
U 11819.
In Table \ref{TablePiso} it is shown the central density, central radius and the brane tension which is the free parameter of the brane theory (only in PISO+branes). As a comparison, it is also shown the central density and radius without brane contribution.
The worst fitted galaxies were (high $\chi_{red}^2$ value):
U 11648,
U 11748.
The fitted brane tension values presents great dispersion, from the lower value:
$0.167\; M_{\odot}/\rm pc^3$, ESO 3020120 to the higher value:
$108.096\; M_{\odot}/\rm pc^3$, ESO 4880049.
It is useful to have $\lambda$ in eV, where the conversion from solar masses to eV is: $1 M_{\odot}/\rm pc^3 \sim 2.915\times10^{-4}eV^4$.
The brane tension parameter has an average value of $\langle\lambda\rangle_{\rm PISO} = 33.178 \; M_{\odot}/\rm pc^3$ with a standard deviation $\sigma_{\rm PISO} = 40.935 \; M_{\odot}/\rm pc^3$. Notice that we can't see a clear tendency to a $\lambda$ value or range of values.
\begin{figure}
\includegraphics[scale=0.33]{vceso30500900PISO+Branes}
\includegraphics[scale=0.33]{vceso0140040PISO+Branes} \\
\includegraphics[scale=0.33]{vceso2060140PISO+Branes}
\includegraphics[scale=0.33]{vcESO3020120PISO+Branes} \\
\includegraphics[scale=0.33]{vceso4250180PISO+Branes}
\includegraphics[scale=0.33]{vceso4880049PISO+Branes} \\
\includegraphics[scale=0.33]{vcf570_v1PISO+Branes}
\includegraphics[scale=0.33]{vcu11454PISO+Branes} \\
\includegraphics[scale=0.33]{vcu11616PISO+Branes}
\includegraphics[scale=0.33]{vcu11648PISO+Branes} \\
\includegraphics[scale=0.33]{vcu11748PISO+Branes}
\includegraphics[scale=0.33]{vcu11819PISO+Branes}
\caption{Group of analyzed galaxies using modified rotation velocity for PISO profile: ESO 3050090,
ESO 0140040, ESO 2060140, ESO 3020120, ESO 4250180, ESO 4880049, 570\_V1, U11454, U11616, U11648, U11748, U11819. We show in the plots: Total rotation curve (solid black line), only PISO curve (short dashed blue curve) and the rotation curve associated with the mass lost by the effect of the brane (red dashed curve).}
\label{PISO1}
\end{figure}
\subsection{Results: NFW profile + Branes}
For the NFW density profile case we have the following results:
We have estimated parameters with and without brane contribution by
minimizing the corresponding $\chi^2$, Eq.\ (\ref{chi2Eq}) with Eq.\ (\ref{RCNFW}), for the sample of observed rotation curves
and taking into account that
$\lambda > \rho_s r_s /r$ in order to have an effective density positive defined, always fulfilling Eq.\ (\ref{comp}).
In Fig.\ \ref{NFW1}, it is shown, for each galaxy in the sample of the LSB galaxies, the theoretical fitted curve to a preferred brane tension value (solid line),
the NFW curve and the rotation curve associated with the mass lost by the effects of branes, see Eq.\ (\ref{RCNFW}).
In Table \ref{TableNFW} it is shown, for the sample,
the central density, central radius and $\chi_{red}^2$ values without branes; and
the central density, central radius, brane tension
and $\chi_{red}^2$ values with branes contribution.
Galaxy U 11748 is the worst fitted case with $\chi_{red}^2 = 2.163$.
For galaxies:
ESO 4250180,
ESO 4880049,
and U 11648,
there are not clear brane effects.
Galaxy U 11648 is an \emph{outlier} with a brane tension value of $4323.28\; M_{\odot}/\rm pc^3$ that is out of the range of preferred values of the other galaxies in the sample.
Notice that we have found a preferred range of tension values, from $0.487$ to $9.232$ $M_{\odot}/\rm pc^3$. Without the outlier, the brane tension parameter has an average value of
$\langle\lambda\rangle_{\rm NFW}\simeq 2.51 \; M_{\odot}/\rm pc^3$
with a standard deviation $\sigma_{\rm NFW}\simeq 3.015 \; M_{\odot}/\rm pc^3$.
\begin{figure}
\includegraphics[scale=0.33]{vceso30500900NFW+Branes}
\includegraphics[scale=0.33]{vceso0140040NFW+Branes} \\
\includegraphics[scale=0.33]{vceso2060140NFW+Branes}
\includegraphics[scale=0.33]{vcESO3020120NFW+Branes} \\
\includegraphics[scale=0.33]{vceso4250180NFW+Branes}
\includegraphics[scale=0.33]{vceso4880049NFW+Branes} \\
\includegraphics[scale=0.33]{vcf570_v1NFW+Branes}
\includegraphics[scale=0.33]{vcu11454NFW+Branes} \\
\includegraphics[scale=0.33]{vcu11616NFW+Branes}
\includegraphics[scale=0.33]{vcu11648NFW+Branes} \\
\includegraphics[scale=0.33]{vcu11748NFW+Branes}
\includegraphics[scale=0.33]{vcu11819NFW+Branes}
\caption{Group of analyzed galaxies using modified rotation velocity for NFW profile: ESO 3050090,
ESO 0140040, ESO 2060140, ESO 3020120, ESO 4250180, ESO 4880049, 570\_V1, U11454, U11616, U11648, U11748, U11819. We show in the plots: Total rotation curve (solid black line), only NFW curve (short dashed blue curve) and the rotation curve associate with the mass lost by the effect of the brane (red dashed curve).}
\label{NFW1}
\end{figure}
\subsection{Results: Burkert+Branes profile}
In the case of Burkert DM density profile, we have also estimated the parameters of the Burkert+branes model
and were compared with Burkert model without branes, minimizing the appropriate $\chi^2_{red}$, Eq.\ (\ref{chi2Eq}) with Eq.\ (\ref{RCBurkert}), for the sample of observed rotation curves. We have considered that $\lambda > \rho_s$ must be fulfilled.
The results are shown in Fig.\ \ref{Burkert1}, where it is plotted the fit to a preferred brane tension value, remarking the total rotation curve (solid line), the Burkert DM density contribution curve (blue short-dashed line) and the rotation curve associated with the mass lost by the effects of branes (red dashed line), see Eq.\ (\ref{RCBurkert}).
In Table \ref{TableBurkert} it is shown the fitted values for the central density, central radius and the corresponding value of the $\chi_{red}^2$ without brane contribution; and the fitted values for
the central density, central radius, brane tension, and theirs $\chi_{red}^2$ values with brane contribution.
The worst fitted (high values of $\chi_{red}^2$) galaxies are: U 11648 and U 11748.
Galaxies ESO 3020120, U 11748, and
U 11819 show a clear brane effects and also are outliers. The main tendency is that $\lambda$ has values of the order of $10^3 \;M_{\odot}/\rm pc^3$ or above, approximately.
The brane tension parameter, without the outliers, for the DM Burkert profile case has an average value of
$\langle\lambda\rangle_{\rm Burk}\simeq 3192.02 \;M_{\odot}/\rm pc^3$,
and a standard deviation of
$\sigma_{\rm Burk}\simeq 2174.97 \; M_{\odot}/\rm pc^3$.
\begin{figure}
\includegraphics[scale=0.33]{vceso30500900Burkert+Branes}
\includegraphics[scale=0.33]{vceso0140040Burkert+Branes} \\
\includegraphics[scale=0.33]{vceso2060140Burkert+Branes}
\includegraphics[scale=0.33]{vcESO3020120Burkert+Branes} \\
\includegraphics[scale=0.33]{vceso4250180Burkert+Branes}
\includegraphics[scale=0.33]{vceso4880049Burkert+Branes} \\
\includegraphics[scale=0.33]{vcf570_v1Burkert+Branes}
\includegraphics[scale=0.33]{vcu11454Burkert+Branes} \\
\includegraphics[scale=0.33]{vcu11616Burkert+Branes}
\includegraphics[scale=0.33]{vcu11648Burkert+Branes} \\
\includegraphics[scale=0.33]{vcu11748Burkert+Branes}
\includegraphics[scale=0.33]{vcu11819Burkert+Branes}
\caption{Group of analyzed galaxies using modified rotation velocity for Burkert profile: ESO 3050090,
ESO 0140040, ESO 2060140, ESO 3020120, ESO 4250180, ESO 4880049, 570\_V1, U11454, U11616, U11648, U11748, U11819. We show in the plots: Total rotation curve (solid black line), only Burkert curve (short dashed blue curve) and the rotation curve associate with the mass lost by the effect of the brane (red dashed curve).}
\label{Burkert1}
\end{figure}
\subsection{Results: a synthetic rotation curve}
Finally, we show the fitting results of the DM models plus brane's contribution to a synthetic rotation curve. This synthetic rotation curve was made of 40 rotation curves of galaxies with magnitudes around $M_I = -18.5$\cite{Salucci1}.
These 40 rotation curves came out of 1100 galaxies that gave the universal rotation curve for spirals. For this sample of low luminosity galaxies, of $M_I = -18.5$, it was shown that the baryonic disk has a very small contribution (for details see reference\cite{Salucci1}).
In this subsection we are now using units such $G=R_{opt}=V(R_{opt})=1$, where $R_{opt}$ and $V(R_{opt})$ are the optical radius and the velocity at the optical radius, respectively. $R_{opt}$ is the radius encompassing 83 per cent of the total integrated light. For an exponential disk with a surface brightness given by: $I(r) \propto \exp(-r/R_D)$, we have that $R_{opt}=3.2 R_D$\cite{Salucci1}.
In figure \ref{SM185} we show the synthetic rotation curve and the fitting results using PISO, NFW and Burkert profiles with and without brane's contribution.
\begin{figure}
\includegraphics[scale=0.33]{vcM185PISO}
\includegraphics[scale=0.33]{vcM185PISO+Branes}
\includegraphics[scale=0.33]{vcM185NFW}
\includegraphics[scale=0.33]{vcM185NFW+Branes}
\includegraphics[scale=0.33]{vcM185Burkert}
\includegraphics[scale=0.33]{vcM185Burkert+Branes}
\caption{Synthetic rotation curve of galaxies with magnitud $M_I=-18.5$.
Left panels: rotation curves fitted without branes. Right panels: rotation curves fitted with branes.
First row is for PISO model; second row is for NFW model and third row is for Burkert model.
We show in the plots: Total rotation curve (solid black line), only DM model curve (short dashed blue curve) and the rotation curve associate with the mass lost by the effect of the brane (red dashed curve).}
\label{SM185}
\end{figure}
As we can see in Table \ref{TableSynthetic} the same trend is observed in the brane's tension values as compared with the results for the LSB catalog analyzed above using PISO, NFW and Burkert as a DM profiles: lower value is obtained for NFW model and higher values is obtained for Burkert density profile.
Given that this synthetic rotation curve is built from 40 rotation curves of real spirals, the values of the brane's tension in table \ref{TableSynthetic} is representative of all these rotation curves.
Then, for PISO model $\lambda=60.692$ $M_{\odot}/\rm pc^3$, a value that is greater than the average value of the tension shown in Table \ref{TablePiso} but inside the interval marked by the standard deviation.
For NFW model $\lambda=226.054$ $M_{\odot}/\rm pc^3$, this value is lower than the average value reported in Table \ref{TableNFW} and inside the range marked by the standard deviation.
For Burkert model $\lambda=1.58\times 10^5$ $M_{\odot}/\rm pc^3$ this value is well above than the average value shown in Table \ref{TableBurkert}; a value outside the range marked by the standard deviation.
\section{Discussion and conclusions} \label{Disc}
We have presented in this paper, the effects coming from the presence of branes in galaxy rotation curves for
three density profiles used to study the behavior of DM at galactic scales.
With this in mind, we were given to the task of study a sample of
high resolution measurements of rotation curves of galaxies without photometry\cite{deBlok/etal:2001}
and a synthetic rotation curve built from 40 rotation curves of galaxies of magnitude around $M_I=-18.5$
fitting
the values of $\rho_{s}$, $r_{s}$ and $\lambda$ through minimizing the $\chi^{2}_{red}$ value and we have compared with the standard results of $\rho_{s}$, $r_{s}$ for each DM density profile without branes.
The results for every observable in the three different profiles were summarized and compared in Tables \ref{TablePiso}-\ref{TableSynthetic}.
From here, it is possible to observe how the results show a weaker limit for the value of brane tension
($\sim10^{-3}\; \rm eV^4-46$ eV$^4$) for the three models, in comparison with other astrophysical and cosmological studies\cite{Kapner:2006si,Alexeyev:2015gja,mk,gm,Sakstein:2014nfa,Kudoh:2003xz,Cavaglia:2002si,Holanda:2013doa,Barrow:2001pi,Brax:2003fv}; for example, Linares \emph{et al.}\cite{Linares:2015fsa} show that weaker values than $\lambda \simeq 10^{4}$ MeV$^{4}$, present anomalous behavior in the compactness of a dwarf star composed by a polytropic EoS, concluding that a wide region of their bound
will show a non compactness stellar configuration, if it is applied to the study shown in\cite{Linares:2015fsa}.
It is important to notice that chosen a value of brane tension that not fulfill our bounds imposed through the paper, generate an anomalous behavior in the center of the galaxy which is characteristic of the model. Remarkable, for higher values of this bound, the modified rotation curves are in good agreement with
the observed rotation curves of the sample that we use,
presenting only the distinctive features of each density profile: For example,
NFW dark matter density profile prefers lower values of the brane tension (on the average $\lambda \sim 0.73\times 10^{-3}$ eV$^4$), implying clear effects of the brane;
PISO dark matter
case has an average value of $\lambda \sim 0.96\times 10^{-2}$ eV$^4$ and show relatively the maximum dispersion on the fitted values of the brane tension;
whereas Burkert DM density profile shows negligible brane effects, on the average $\lambda \sim 0.93$ eV$^4$ -- $46$ eV$^4$.
In addition, it is important to discuss briefly the changes caused by the presence of branes in the problem of cusp/core. Notice that in this case the part that play a role is the effective density which is written in terms of brane corrections as: $\rho_{eff}=\rho(1-\rho/\lambda)$; it is notorious how the small perturbations alleviate the cusp problem which afflicts NFW, albeit the excessive presence of these terms could generates a negative effective density profile; also PISO and Burkert show modifications when $r\to0$ but does not pledge its core behavior while the brane tension only takes small values. In this way, the possibility of having a core behavior, help us to constraint the value of brane tension and still keep in the game the NFW profile.
Summarizing, it is really challenging to establish bounds in a dynamical systems like rotation curves in galaxies due to the
low densities found in the galactic medium, giving only a weaker limits in comparison with other studies in a most energetic systems.
Our most important conclusion, is that despite the efforts, we think that it is not straightforward to do that the results fit with other astrophysical and cosmological studies, being impractical and not feasible to
find evidence of extra dimensions in galactic dynamics through the determination of the brane tension value,
even more, we think that exist too much dispersion in the fitted values of the brane tension using this method for some DM density profile models.
Also it is important to note that the value of the brane tension is strongly dependent of the characteristic of the galaxy studied, suggesting an average for the preferred value of the brane tension in each case.
In addition, we notice that the effects of extra dimensions are stronger in the galactic core, suggesting that the NFW model is not appropriate in the search of constraints in brane theory due to the divergence in the center of the galaxy (see Eq.\ \eqref{comp}); PISO and Burkert could be good candidates to explore the galactic core in this framework; however it is necessary a more extensive study before we obtain a definitive conclusion.
As a final note, we know that it is necessary to recollect more observational data to constraint
the models or even give the final conclusion about extra dimensions (for or against), supporting the brane constraints shown through this paper with a more profound study of galactic dynamic or other tests like cosmological evidences presented in CMB anisotropies.
However, this work is in progress and will be reported elsewhere.
\begin{acknowledgements}
MAG-A acknowledge support from SNI-M\'exico and CONACyT research fellow. Instituto Avanzado de Cosmolog\'ia (IAC) collaborations.
\end{acknowledgements}
|
1,314,259,996,561 | arxiv | \section{Introduction}
Cluster analysis is the search for a priori unknown group structure in data.
Model-based clustering is increasingly becoming one of the most popular cluster analysis methods. Model-based clustering is based on Finite mixture models \citep{McLachlan:Peel:2000}, with each component density usually representing a cluster. For continuous data, Gaussian components are usually used to model clusters.
Model-based clustering as implemented in the \proglang{R} package \pkg{mclust} \citep{Fraley:Raftery:Murphy:Scrucca:2012} allows for automatic selection of the number of components, and selection of parsimonious covariance structures.
In cluster analysis, as in classification or other supervised learning tasks, the inclusion of noise variables, i.e. features without useful group information, can severely degrade the final results.
In fact, the presence of noise variables can negatively impact both the estimation of the number of clusters in the data and the recovery of those groups.
The new \pkg{clustvarsel} (version $\ge 2.0$) \proglang{R} package implements a wrapper method for automatic variable selection in model-based clustering (as implemented in the \pkg{mclust} package).
Thus, the addition of the \pkg{clustvarsel} package allows for automatic variable selection to be included in the estimation process.
\citet{Raftery:Dean:2006} introduced a stepwise variable selection methodology tailored to model-based clustering. An earlier version of \pkg{clustvarsel} (version 1) implemented this methodology. Variables designated as noise variables in this process were not required to be independent of the clustering variables. However, noise variables could be conditionally independent of the clustering, but still linearly dependent on the clustering variables. This linear dependency was modelled using linear regression.
\citet{Maugis:etal:2009a} extended the framework of \citet{Raftery:Dean:2006} by allowing the noise variables to depend on a (possibly null) subset of the clustering variables via stepwise variable selection in the linear regression \citep[see also][]{Maugis:etal:2009b}. This allows for a more parsimonious modelling of the relationship between the noise variables and the clustering variables. For more detail on the variable selection framework from both \citet{Raftery:Dean:2006} and \citet{Maugis:etal:2009a} see Section \ref{sec:methods}.
Section \ref{sec:clustvarsel} introduces the main function in the \pkg{clustvarsel} package, and discusses the options for the available arguments. In Section \ref{sec:examples}, several examples are presented by applying the methodology to both synthetic and real world datasets.
Algorithmic speedups are discussed in Section \ref{sec:speed}, including a description of a parallel implementation of the stepwise greedy search in Section \ref{sec:parallel}. The paper concludes with some discussion and final remarks in Section \ref{sec:discuss}.
\section{Methodology}
\label{sec:methods}
Model-based clustering assumes that the observed data are generated from a mixture of $G$ components, each representing the probability distribution for a different group or cluster \citep{McLachlan:Peel:2000, Fraley:Raftery:2002}.
For continuous data, the density of each mixture component can be described by the multivariate Gaussian distribution. Thus, the general form of a Gaussian finite mixture model is
$$
f(\boldsymbol{x}) = \sum_{g=1}^G \pi_g \phi(\boldsymbol{x}|\boldsymbol{\mu}_g,\boldsymbol{\Sigma}_g),
$$
where $\pi_g$ represents the mixing probabilities, so that $\pi_g > 0$ and $\sum_{g=1}^G\pi_g=1$, $\phi(\cdot)$ is the multivariate Gaussian density with parameters $(\boldsymbol{\mu}_g,\boldsymbol{\Sigma}_g)$ ($g=1,\ldots,G$).
Clusters are ellipsoidal, centred at the mean vector $\boldsymbol{\mu}_g$, with other geometric features, such as volume, shape and orientation, determined by $\boldsymbol{\Sigma}_g$.
Parsimonious parameterisation of covariance matrices is available through the eigenvalue decomposition $\boldsymbol{\Sigma}_g = \lambda_g \boldsymbol{D}_g \boldsymbol{A}_g \boldsymbol{D}^{\top}_g$, where $\lambda_g$ is a scalar controlling the volume of the ellipsoid, $\boldsymbol{A}_g$ is a diagonal matrix specifying the shape of the density contours, and $\boldsymbol{D}_g$ is an orthogonal matrix which determines the orientation of the corresponding ellipsoid \citep{Banfield:Raftery:1993, Celeux:Govaert:1995}. \citet[Table~1]{Fraley:Raftery:Murphy:Scrucca:2012} report some parameterisation of within-group covariance matrices available in the \proglang{R} package \pkg{mclust}, and the corresponding geometric characteristics.
\citet{Raftery:Dean:2006} discussed the problem of variable selection for model-based clustering by recasting the problem as a model selection procedure.
Their proposal is based on the use of BIC to approximate Bayes factors to compare mixture models fitted on nested subsets of variables. A generalisation of their approach was later discussed by \citet{Maugis:etal:2009a, Maugis:etal:2009b}.
Let us suppose that the set of available variables, $\mathcal{X}$, is partitioned into three disjoint parts: the set of previously selected variables, $\mathcal{X}_{\text{clust}}$, the variable under consideration for inclusion or exclusion from the active set, $X_i$, and the set of the remaining variables, $\mathcal{X}_{\text{other}} \equiv \mathcal{X} \setminus \{ \mathcal{X}_{\text{clust}} \cup X_i\}$.
\cite{Raftery:Dean:2006} showed that the inclusion (or exclusion) of variables can be assessed using the following BIC difference:
\begin{equation}
\BIC_{\text{diff}} = \BIC_{\text{clust}}(\mathcal{X}_{\text{clust}},X_i) - \BIC_{\text{not\,clust}}(\mathcal{X}_{\text{clust}},X_i),
\label{eq:BICdiff}
\end{equation}
where $\BIC_{\text{clust}}(\mathcal{X}_{\text{clust}},X_i)$ is the BIC value for the ``best'' clustering mixture model (i.e. assuming $G \ge 2$) fitted using the features set $\{\mathcal{X}_{\text{clust}} \cup X_i\}$, whereas $\BIC_{\text{not\,clust}}(\mathcal{X}_{\text{clust}},X_i)$ is the BIC value for no clustering for the same set of variables.
The latter can be written as
\begin{equation}
\BIC_{\text{not\,clust}}(\mathcal{X}_{\text{clust}},X_i) = \BIC_{\text{clust}}(\mathcal{X}_{\text{clust}}) + \BIC_{\text{reg}}(X_i|\mathcal{X}_{\text{clust}}),
\label{eq:BICnotclust}
\end{equation}
i.e. the BIC value for the ``best'' clustering model fitted using the set $\mathcal{X}_{\text{clust}}$ plus the BIC value for the regression of the candidate variable $X_i$ on the variables included in the set $\mathcal{X}_{\text{clust}}$.
The difference in BIC score \eqref{eq:BICdiff} is an approximation of the log of the Bayes factor comparing the model where the variable under consideration, $X_i$, is a clustering variable with the model where the variable is conditionally independent of the clustering. Large, positive values of $\BIC_{\text{diff}}$ can be taken as an evidence that variable $X_i$ is useful for clustering.
In all clustering models, the ``best'' model is identified with respect to the number of mixture components (assuming $G \ge 2$) and to model parameterisations.
In the linear regression model term, $X_i$ can depend on all the variables in $\mathcal{X}_{\text{clust}}$, a subset of them, or none (complete independence).
Finally, note that in both equations \eqref{eq:BICdiff} and \eqref{eq:BICnotclust} the set of remaining variables, $\mathcal{X}_{\text{other}}$, plays no role.
For more details of the methodology see \citet{Raftery:Dean:2006}, and for improvements due to the subset selection in the regression model see \citet{Maugis:etal:2009a}.
Practical implementation of the above methodology requires the use of an algorithm for checking single variables for inclusion/exclusion from the set of selected clustering variables.
A \textit{stepwise greedy search} algorithm checks
the inclusion of each single variable not currently selected into the current
set of selected clustering variables at each step.
The variable that has the highest evidence of inclusion is proposed and, if its clustering evidence is stronger than the evidence against clustering, it is included.
At every exclusion step, the algorithm checks the removal of each single variable in the currently selected set of clustering variables, and proposes the variable that has the lowest evidence of clustering. The proposed variable is removed if the evidence for its being a clustering variable is weaker than the
evidence against.
This is similar to the idea for stepwise regression and may suffer from the same instabilities mentioned in \citet{Miller:1990} inherent in that approach (although this has not been apparent in the simulations and examples tried thus far).
The stepwise algorithm can be implemented in a \textit{forward/backward} fashion, i.e. starting from the empty set of clustering variable and then continuing to add or remove features until there is no evidence of further clustering variables. It can also be implemented in a \textit{backward/forward} fashion, i.e. starting from the full set of features as clustering variables and then continuing to remove or add features until there is no evidence of further clustering variables.
Another possible algorithm is the \textit{headlong search}, which involves potentially checking less variables at each inclusion or exclusion step, and so may be quicker than the stepwise greedy search (at a possible price in terms of performance) for use on datasets with a large number of variables. At each inclusion step, the headlong algorithm only checks single variables not currently in the set of clustering variables until the difference between the BIC for clustering versus not clustering is above a pre-specified upper level (a default value of 0 implies that the evidence for clustering is greater than that for not clustering).
Because the algorithm stops once this criterion is satisfied, it will not necessarily check all the variables available, and the variable selected will not necessarily be the best possible feature at that step. Any variables that are checked during this step, whose difference between the BIC for clustering versus not clustering is below a pre-specified lower level, are removed from consideration for the rest of the algorithm (a default value of $-10$ means that there is a strong evidence against clustering). Because of this, possibly irrelevant variables can be removed early on in the algorithm and further reduce the number of variables checked at each step.
Similarly, at each exclusion, the algorithm only checks single variables currently in the set of clustering variables until the BIC difference for clustering versus not clustering is below the pre-specified upper level. The algorithm stops checking once a variable satisfies this criterion, and that variable is removed from the set. If the difference in BIC is smaller than the lower level, then the variable is removed from consideration for the rest of the algorithm, otherwise it can still be checked in future inclusion/exclusion steps.
See \citet{Badsberg:1992} for full details about the headlong algorithm.
\section[The R package clustvarsel]{The \proglang{R} package \pkg{clustvarsel}}
\label{sec:clustvarsel}
The \pkg{clustvarsel} package can be used to find the (locally) optimal subset of variables with group/cluster information in a dataset with continuous variables. In this section, usage of the main function \code{clustvarsel} and its arguments is described.
The \pkg{clustvarsel} package depends on other packages available on CRAN for model fitting (\pkg{mclust}, \citet{Rpkg:mclust}), or for providing some facilities, such as parallelisation (\pkg{parallel}, \citet{Rpkg:parallel}; \pkg{doParallel}, \citet{Rpkg:doParallel}; \pkg{foreach}, \citet{Rpkg:foreach}; \pkg{iterators}, \citet{Rpkg:iterators}), and subset selection in regression models (\pkg{BMA}, \citet{Rpkg:BMA}). By loading the package as usual with \code{library(clustvarsel)}, it will also take care of making all the other packages available for the current session.
Once the \pkg{clustvarsel} package has been loaded, the main function a user needs to invoke is the following:
\begin{quote}
\begin{Code}
clustvarsel(data,
G = 1:9,
search = c("greedy", "headlong"),
direction = c("forward", "backward"),
emModels1 = c("E", "V"),
emModels2 = mclust.options("emModelNames"),
samp = FALSE,
sampsize = round(nrow(data)/2),
hcModel = "VVV",
allow.EEE = TRUE,
forcetwo = TRUE,
BIC.diff = 0,
BIC.upper = 0,
BIC.lower = -10,
itermax = 100,
parallel = FALSE)
\end{Code}
\end{quote}
The available arguments are:
\begin{description}[itemsep=1ex, parsep=0pt]
\item[\code{data}] A numeric matrix or data frame where rows correspond to observations and columns correspond to variables. Categorical variables are not allowed.
\item[\code{G}] An integer vector specifying the numbers of mixture components (clusters) for which the BIC is to be calculated. The default is \code{G = 1:9}.
\item[\code{search}] A character vector indicating whether a \code{"greedy"} or, potentially quicker but less optimal, \code{"headlong"} algorithm is to be used in the search for clustering variables.
\item[\code{direction}] A character vector indicating the type of search: \code{"forward"} starts from the empty model and at each step of the algorithm adds/removes a variable until the stopping criterion is satisfied; \code{"backward"} starts from the model with all the available variables and at each step of the algorithm removes/adds a variable until the stopping criterion is satisfied. For the \code{"headlong"} search only the \code{"forward"} algorithm is available.
\item[\code{emModels1}] A vector of character strings indicating the models to be fitted in the EM phase of univariate clustering. Possible models are \code{"E"} and \code{"V"}, described in \code{help(mclustModelNames)}.
\item[\code{emModels2}] A vector of character strings indicating the models to be fitted in the EM phase of multivariate clustering. Possible models are those described in \code{help(mclustModelNames)}.
\item[\code{samp}] A logical value indicating whether or not a subset of observations should be used in the hierarchical clustering phase used to get starting values for the EM algorithm.
\item[\code{sampsize}] The number of observations to be used in the hierarchical clustering subset. By default, a random sample of approximately half of the sample size is used.
\item[\code{hcModel}] A character string specifying the model to be used in hierarchical clustering for choosing the starting values used by the EM algorithm. By default, the unconstrained \code{"VVV"} covariance structure is used.
\item[\code{allow.EEE}] A logical value indicating whether a new clustering will be run with equal within-cluster covariance for hierarchical clustering to get starting values, if the clusterings with variable within-cluster covariance for hierarchical clustering do not produce any viable BIC values.
\item[\code{forcetwo}] A logical value indicating whether at least two variables will be forced to be selected initially, regardless of whether BIC evidence suggests bivariate clustering or not.
\item[\code{BIC.diff}] A numerical value indicating the minimum BIC difference between clustering and no clustering used to accept the inclusion of a variable in the set of clustering variables in a forward step of the greedy search algorithm. Furthermore, minus \code{BIC.diff} is used to accept the exclusion of a selected variable from the set of clustering variable in a backward step of the greedy search algorithm. Default is $0$.
\item[\code{BIC.upper}] A numerical value indicating the minimum BIC difference between clustering and no clustering used to select a clustering variable in the headlong search. Default is $0$.
\item[\code{BIC.lower}] A numerical value indicating the level of BIC difference between clustering and no clustering below which a variable will be removed from consideration in the headlong algorithm. Default is $-10$.
\item[\code{itermax}] An integer value giving the maximum number of iterations (of addition and removal steps) the selected algorithm is allowed to run for.
\item[\code{parallel}] This argument allows to specify if the selected \code{"greedy"} algorithm should be run sequentially or in parallel. The possible values are:
\begin{enumerate}[itemsep=0pt, parsep=0pt, topsep=1ex, label=(\roman*)]
\item a logical value specifying if parallel computing should be used (\code{TRUE}) or not (\code{FALSE}, default) for running the algorithm;
\item a numerical value which gives the number of cores to employ (by default, this is obtained from function \code{detectCores} in the \pkg{parallel} package);
\item a character string specifying the type of parallelisation to use. The latter depends on system OS: on Windows OS only \code{"snow"} type functionality is available, whereas on Unix/Linux/Mac OSX both \code{"snow"} and \code{"multicore"} (default) functionalities are available.
\end{enumerate}
Options (ii) and (iii) imply that the search is performed in parallel. By default the algorithm is run sequentially.
\end{description}
A basic \code{clustvarsel} function call needs to input a matrix or data frame containing the data to analyse. Fine tuning is possible by specifying the arguments described above. The following section presents some examples of its usage in practice.
\section{Examples}
\label{sec:examples}
In this section we present some data analysis examples based on simulated data and on well-known real datasets.
\subsection{Simulated data}
\label{sec:simul}
We consider some of the synthetic data examples described in \citet{Maugis:etal:2009a}. Samples were simulated for a 10-dimensional feature vector where only the first two variables provide clustering information. These were generated from a mixture of four Gaussian distributions $\boldsymbol{X}_{[1:2]} \sim N(\boldsymbol{\mu}_k, \boldsymbol{I}_2)$ with $\boldsymbol{\mu}_1=(-2,-2)$, $\boldsymbol{\mu}_2=(-2,2)$, $\boldsymbol{\mu}_3 = -\boldsymbol{\mu}_2$, $\boldsymbol{\mu}_4 = -\boldsymbol{\mu}_1$, and mixing probabilities $\pi = (0.3,0.2,0.3,0.2)$. The remaining eight variables were simulated according to the model $\boldsymbol{X}_{[3:10]} = \boldsymbol{X}_{[1:2]}\boldsymbol{\beta} + \epsilon$, where $\epsilon \sim N(\boldsymbol{0}, \boldsymbol{\Omega})$. Different settings for $\boldsymbol{\beta}$ and $\boldsymbol{\Omega}$ define seven different scenarios \citep[see Table~1 in][]{Maugis:etal:2009a}. These range from independence of clustering variables on the other features (model 1 and 2) to cases of increasing degree of dependence of the irrelevant variables on the clustering ones (model 3 to 7).
In the following, we focus only on some of the scenarios. For ease of reading, the values of the parameters for such scenarios are reported in Table~\ref{tab:maugisetal:scenarios}.
\begin{table}[htb]
\caption[Parameter settings for the scenarios used in the simulation study]
{Parameter settings for the scenarios used to generated synthetic data: $\boldsymbol{\beta}$ defines the correlation of irrelevant variables on clustering variables, whereas $\boldsymbol{\Omega}$ is the covariance structure of the noise component. $\boldsymbol{0}_p$ indicates the $(2 \times p)$ matrix of zeroes, and $\boldsymbol{I}_p$ the $(p \times p)$ identity matrix.}
\label{tab:maugisetal:scenarios}
\centering
\begin{tabular*}{0.95\textwidth}{@{\extracolsep{\fill}}llcll}
\hline\noalign{\smallskip}
Scenario & Parameters && Scenario & Parameters \\
\hline\noalign{\smallskip}
\textit{Model 1} & $\boldsymbol{\beta} = \boldsymbol{0}_8$ &&
\textit{Model 5} &
$\boldsymbol{\beta} = \left(\begin{matrix} 0.5 & 0 & 2 & 0\\ 0 & 1 & 0 & 3\end{matrix}\;\boldsymbol{0}_4\right)$ \\[2ex]
& $\boldsymbol{\Omega} = \boldsymbol{I}_8$ &&&
$\boldsymbol{\Omega} = \diag(\boldsymbol{I}_2, 0.5\boldsymbol{I}_2, \boldsymbol{I}_4)$ \\[2ex]
\textit{Model 4} &
$\boldsymbol{\beta} = \left(\begin{matrix} 0.5 & 0 \\ 0 & 1 \end{matrix}\;\boldsymbol{0}_6\right)$ &&
\textit{Model 7} &
$\boldsymbol{\beta} = \left(\begin{matrix} 0.5 & 0 & 2 & 0 & 2 & 0.5 & 2 & 0\\ 0 & 1 & 0 & 3 & 0.5 & 1 & 0 & 3\end{matrix}\right)$ \\[2ex]
& $\boldsymbol{\Omega} = \boldsymbol{I}_8$ &&&
$\boldsymbol{\Omega} = \diag(\boldsymbol{I}_2, 0.5\boldsymbol{I}_4, \boldsymbol{I}_2)$ \\
\noalign{\smallskip}\hline
\end{tabular*}
\end{table}
The simulation results were evaluated using the following criteria:
\begin{itemize}[itemsep=0pt, parsep=0pt, topsep=1ex]
\item Variable Selection Error Rate (VSER) to assess \textit{variable selection performance}. VSER is defined as the ratio of the number of errors in selecting (or not selecting) variables to the total number of variables in the set. A perfect recovey of clustering variables gives VSER $= 0$, while VSER can be no greater than 1.
\item Adjusted Rand Index \citep[ARI,][]{Hubert:Arabie:1985} to measure \textit{classification accuracy}. A perfect classification gives ARI $= 1$, whereas ARI $= 0$ for a random classification.
\end{itemize}
Table~\ref{tab:maugis:sim1} shows the results from a simulation study for the above synthetic data using sample sizes $n=200$ and $n=1000$.
The methods compared are \code{MCLUST}, the best GMM using the full set of variables, \code{CLUSTVARSEL[fwd]} and \code{CLUSTVARSEL[bkw]}, the best GMM using the subset of relevant clustering variables selected by, respectively, the forward/backward and backward/forward greedy search, and \code{SPARSEKMEANS}, a sparse version of $k$-means algorithm \citep{Witten:Tibshirani:2010}.
Because the last method needs the number of clusters to be fixed in advance, we also included in the comparison versions of the methods based on GMMs with the number of components fixed at the true number of clusters, i.e. $G=4$.
Finally, note that true subset size is 2, so the optimal VSER should be 0, and the best average ARI value attainable, using the true clustering variables and fixed $G = 4$ components, is about $0.88$.
\begin{table}[htb]
\caption[Results of the simulation study based on the Maugis et al (2009) setup]
{Average values based on 100 simulation runs for some of the models in \citet{Maugis:etal:2009a}, where only two (out of ten) variables are relevant for clustering. The first four models use $G=4$ fixed number of clusters. \code{CLUSTVARSEL[fwd]} uses the forward/backward greedy search, whereas \code{CLUSTVARSEL[bkw]} employs the backward/forward greedy search. True subset size is 2. Smaller values of VSER and larger values of ARI are better. The best values for each experiment and each criterion are shown in bold font. For subset size, these are the values closest to 2, for VSER the smallest values, and for ARI the largest values.}
\label{tab:maugis:sim1}
\centering
\begin{tabular}{lrrrcrrr}
\hline\noalign{\smallskip}
Model & Subset size & VSER & ARI && Subset size & VSER & ARI \\
\hline\noalign{\smallskip}
Scenario 1 & \multicolumn{3}{c}{$n=200$} && \multicolumn{3}{c}{$n=1000$} \\
\cline{2-4}\cline{6-8}
\code{MCLUST,G=4} & 10.00 & .800 & .867 && 10.00 & .800 & .882 \\
\code{SPARSEKMEANS,G=4} & 9.91 & .791 & {\bf .872} && 9.37 & .737 & .881 \\
\code{CLUSTVARSEL[fwd],G=4} & {\bf 2.05} & {\bf .005} & {\bf .872} && {\bf 2.01} & {\bf .001} &{\bf .885} \\
\code{CLUSTVARSEL[bkw],G=4} & {\bf 2.05} & {\bf .005} & .873 && {\bf 2.01} & {\bf .001} &{\bf .885} \\
\code{MCLUST} & 10.00 & .800 & .770 && 10.00 & .800 & .882 \\
\code{CLUSTVARSEL[fwd]} & {\bf 2.05} & {\bf .005} & {\bf .872} && {\bf 2.01} & {\bf .001} &{\bf .885} \\
\code{CLUSTVARSEL[bkw]} & 3.86 & .278 & .681 && {\bf 2.01} & {\bf .001} &{\bf .885} \\
\hline\noalign{\smallskip}
Scenario 4 & \multicolumn{3}{c}{$n=200$} && \multicolumn{3}{c}{$n=1000$} \\
\cline{2-4}\cline{6-8}
\code{MCLUST,G=4} & 10.00 & .800 & .828 && 10.00 & .800 & .881 \\
\code{SPARSEKMEANS,G=4} & 9.89 & .789 & .834 && 9.35 & .735 & .842 \\
\code{CLUSTVARSEL[fwd],G=4} & {\bf 2.03} & {\bf .003} & {\bf .881} && {\bf 2.01} & {\bf .001} & {\bf .886} \\
\code{CLUSTVARSEL[bkw],G=4} & {\bf 2.03} & {\bf .003} & {\bf .881} && {\bf 2.01} & {\bf .001} & {\bf .886} \\
\code{MCLUST} & 10.00 & .800 & .698 && 10.00 & .800 & .881 \\
\code{CLUSTVARSEL[fwd]} & 2.05 & .007 & .877 && {\bf 2.01} & {\bf .001} & {\bf .886} \\
\code{CLUSTVARSEL[bkw]} & 2.98 & .162 & .752 && {\bf 2.01} & {\bf .001} & {\bf .886} \\
\hline\noalign{\smallskip}
Scenario 5 & \multicolumn{3}{c}{$n=200$} && \multicolumn{3}{c}{$n=1000$} \\
\cline{2-4}\cline{6-8}
\code{MCLUST,G=4} & 10.00 & .800 & .847 && 10.00 & .809 & .879 \\
\code{SPARSEKMEANS,G=4} & 9.34 & .736 & .801 && 7.09 & .509 & .857 \\
\code{CLUSTVARSEL[fwd],G=4} & {\bf 2.00} & {\bf .014} & {\bf .881} && {\bf 2.01} & {\bf .001} & {\bf .884} \\
\code{CLUSTVARSEL[bkw],G=4} & 2.03 & .035 & .880 && 2.03 & .003 & .884 \\
\code{MCLUST} & 10.00 & .800 & .461 && 10.00 & .800 & .879 \\
\code{CLUSTVARSEL[fwd]} & 1.99 & .017 & .873 && {\bf 2.01} & {\bf .001} & {\bf .884} \\
\code{CLUSTVARSEL[bkw]} & 2.06 & .030 & .868 && 2.03 & .003 & {\bf .884} \\
\hline\noalign{\smallskip}
Scenario 7 & \multicolumn{3}{c}{$n=200$} && \multicolumn{3}{c}{$n=1000$} \\
\cline{2-4}\cline{6-8}
\code{MCLUST,G=4} & 10.00 & .800 & .838 && 10.00 & .800 & .880 \\
\code{SPARSEKMEANS,G=4} & 9.04 & .704 & .847 && 9.00 & .700 & .865 \\
\code{CLUSTVARSEL[fwd],G=4} & {\bf 2.00} & .032 & {\bf .874} && {\bf 2.00} & {\bf .000} & {\bf .885} \\
\code{CLUSTVARSEL[bkw],G=4} & 2.03 & .021 & .872 && 2.01 & .003 & {\bf .885} \\
\code{MCLUST} & 10.00 & .800 & .447 && 10.00 & .800 & .880 \\
\code{CLUSTVARSEL[fwd]} & {\bf 2.00} & .024 & {\bf .874} && {\bf 2.00} & {\bf .000} & {\bf .885} \\
\code{CLUSTVARSEL[bkw]} & 2.02 & {\bf .014} & .868 && 2.01 & .003 & {\bf .885} \\
\hline\noalign{\smallskip}
\multicolumn{8}{l}{\footnotesize Standard errors for VSER and ARI are all $\le .030$.}
\end{tabular}
\end{table}
Compared to the performance of \code{CLUSTVARSEL} reported in Table~1 of \citet{Maugis:etal:2009a}, the new version of the algorithm is able to correctly discard irrelevant variables, both when they are independent of the clustering ones and when they are correlated.
When $G$ is fixed at the true number of clusters, \code{MCLUST} gives slightly less accurate results for $n=200$, except in the case of complete independence (scenario 1). \code{CLUSTVARSEL} provides equivalent accuracy, both if a forward/backward search or a backward/forward search is used.
\code{SPARSEKMEANS} shows results equivalent to greedy search in term of accuracy, but it tends to select (i.e. assigns weights different from zero to) too many variables. Consequently, the VSER of \code{SPARSEKMEANS} is always worse than that of \code{CLUSTVARSEL}.
When $G$ is unknown, \code{MCLUST} often provides inaccurate clustering, in particular when $n=200$. On the contrary, \code{CLUSTVARSEL} is generally able to select the true clustering variables (i.e. VSER is near or exactly zero), and also provides very accurate clustering (i.e. ARI is close to $0.88$). The only exceptions are scenarios 1 and 4, for the backward/forward search when $n=200$. In these cases the number of selected variables is slightly larger, which in turn causes a small degradation of clustering accuracy. However, for $n=1,000$ the forward/backward and backward/forward greedy searches are equivalent.
\afterpage{\clearpage}
\subsection{Crabs data}
The crabs dataset in the \pkg{MASS} package contains five morphological measurements on 200 specimens of Leptograpsus variegatus crabs recorded on the shore in Western Australia \citep{Campbell:Mahon:1974}. Crabs are classified according to their color (blue and orange) and sex, giving four groups. Fifty specimens are available for each combination of colour and sex.
\begin{CodeInput}
R> data(crabs, package = "MASS")
R> X = crabs[,4:8]
R> Class = with(crabs, paste(sp, sex, sep = "|"))
R> table(Class)
\end{CodeInput}
\begin{CodeOutput}
Class
B|F B|M O|F O|M
50 50 50 50
\end{CodeOutput}
First we look at the result obtained using the function \code{Mclust} from the \pkg{mclust} package, with the best model selected by BIC for clustering on all the variables, allowing all possible parameterisations and the number of groups to range over 1 to 5:
\begin{CodeInput}
R> mod1 = Mclust(X, G = 1:5)
R> summary(mod1)
\end{CodeInput}
\begin{CodeOutput}
----------------------------------------------------
Gaussian finite mixture model fitted by EM algorithm
----------------------------------------------------
Mclust EEV (ellipsoidal, equal volume and shape) model with 4 components:
log.likelihood n df BIC ICL
-1241.006 200 68 -2842.298 -2854.29
Clustering table:
1 2 3 4
60 55 39 46
\end{CodeOutput}
The estimated MAP classification is obtained from \code{mod1$classification}, so a table comparing the estimated and the true classifications, the corresponding misclassification error rate and the adjusted Rand index (ARI), can be obtained as follows:
\begin{CodeInput}
R> table(Class, mod1$classification)
\end{CodeInput}
\begin{CodeOutput}
Class 1 2 3 4
B|F 49 0 0 1
B|M 11 0 39 0
O|F 0 5 0 45
O|M 0 50 0 0
\end{CodeOutput}
\begin{CodeInput}
R> classError(Class, mod1$classification)$errorRate
\end{CodeInput}
\begin{CodeOutput}
[1] 0.085
\end{CodeOutput}
\begin{CodeInput}
R> adjustedRandIndex(Class, mod1$classification)
\end{CodeInput}
\begin{CodeOutput}
[1] 0.793786
\end{CodeOutput}
The algorithm for selecting the variables that are useful for clustering can be run with the following code:
\begin{CodeInput}
R> result = clustvarsel(X, G = 1:5)
R> result
\end{CodeInput}
\begin{CodeOutput}
'clustvarsel' model object:
Stepwise (forward) greedy search:
Variable proposed BIC BIC difference Type of step Decision
1 CW -1408.710 -6.21775 Add Accepted
2 RW -1908.964 127.38583 Add Accepted
3 FL -2357.252 81.24626 Add Accepted
4 FL -2357.252 81.24074 Remove Rejected
5 BD -2609.777 56.08094 Add Accepted
6 BD -2609.777 71.39446 Remove Rejected
7 CL -2609.777 -31.07119 Add Rejected
8 BD -2609.777 71.39446 Remove Rejected
Selected subset: CW, RW, FL, BD
\end{CodeOutput}
By default, a greedy forward/backward search is used.
The printed output shows the trace of the algorithm:
at each step the most important variable is considered for addition or deletion from the set of clustering variables, with each proposal which can be accepted or rejected. In this case, the final subset contains four out of five morphological features:
\begin{CodeInput}
R> result$subset
\end{CodeInput}
\begin{CodeOutput}
CW RW FL BD
4 2 1 5
\end{CodeOutput}
The same subset is also obtained by using a backward/forward greedy search:
\begin{CodeInput}
R> clustvarsel(X, G = 1:5, direction = "backward")
\end{CodeInput}
\begin{CodeOutput}
'clustvarsel' model object:
Stepwise (backward) greedy search:
Variable proposed BIC BIC difference Type of step Decision
1 CL -2609.777 -31.07119 Remove Accepted
2 BD -2609.777 56.08094 Remove Rejected
Selected subset: FL, RW, CW, BD
\end{CodeOutput}
The identified subset can be used for fitting the final clustering model as follows:
\begin{CodeInput}
R> Xs = X[,result$subset]
R> mod2 = Mclust(Xs, G = 1:5)
R> summary(mod2)
\end{CodeInput}
\begin{CodeOutput}
----------------------------------------------------
Gaussian finite mixture model fitted by EM algorithm
----------------------------------------------------
Mclust EEV (ellipsoidal, equal volume and shape) model with 4 components:
log.likelihood n df BIC ICL
-1180.378 200 47 -2609.777 -2624.892
Clustering table:
1 2 3 4
53 60 40 47
\end{CodeOutput}
The accuracy of the clustering obtained on the selected subset of variables is obtained as:
\begin{CodeInput}
R> table(Class, mod2$classification)
\end{CodeInput}
\begin{CodeOutput}
Class 1 2 3 4
B|F 0 50 0 0
B|M 0 10 40 0
O|F 3 0 0 47
O|M 50 0 0 0
\end{CodeOutput}
\begin{CodeInput}
R> classError(Class, mod2$classification)$errorRate
\end{CodeInput}
\begin{CodeOutput}
[1] 0.065
\end{CodeOutput}
\begin{CodeInput}
R> adjustedRandIndex(Class, mod2$classification)
\end{CodeInput}
\begin{CodeOutput}
[1] 0.8399679
\end{CodeOutput}
\subsection{Coffee data}
Data on twelve chemical constituents of coffee for 43 samples were collected from 29 countries around the world \citep{Streuli:1973}. Each coffee sample is either of the Arabica or Robusta variety. The dataset is available in the \proglang{R} package \pkg{pgmm}.
\begin{CodeInput}
R> data(coffee, package = "pgmm")
R> X = as.matrix(coffee[,3:14])
R> Class = factor(coffee$Variety, levels = 1:2, labels = c("Arabica", "Robusta"))
R> table(Class)
\end{CodeInput}
\begin{CodeOutput}
Class
Arabica Robusta
36 7
\end{CodeOutput}
\begin{CodeInput}
R> mod1 = Mclust(X)
R> summary(mod1)
\end{CodeInput}
\begin{CodeOutput}
----------------------------------------------------
Gaussian finite mixture model fitted by EM algorithm
----------------------------------------------------
Mclust VEI (diagonal, equal shape) model with 3 components:
log.likelihood n df BIC ICL
-392.9397 43 52 -981.4619 -981.6379
Clustering table:
1 2 3
22 14 7
\end{CodeOutput}
Model-based clustering applied to this dataset selects the \code{VEI} model with 3 components.
The clustering table and the corresponding adjusted Rand index (ARI) are the following:
\begin{CodeInput}
R> table(Class, mod1$classification)
\end{CodeInput}
\begin{CodeOutput}
Class 1 2 3
Arabica 22 14 0
Robusta 0 0 7
\end{CodeOutput}
\begin{CodeInput}
R> adjustedRandIndex(Class, mod1$classification)
\end{CodeInput}
\begin{CodeOutput}
[1] 0.3833116
\end{CodeOutput}
The Arabica variety appears to be split into two sub-varieties, whereas the Robusta is correctly identified as a single cluster. As a result, a small value of ARI is obtained.
Now, we may try variable selection to drop irrelevant features, and see if we can improve upon the above model.
The following code uses the backward/forward greedy search for variable selection, which by default is performed over all the covariance decomposition models and numbers of mixture components from 1 up to 9:
\begin{CodeInput}
R> result = clustvarsel(X, direction = "backward")
R> result
\end{CodeInput}
\begin{CodeOutput}
'clustvarsel' model object:
Stepwise (backward) greedy search:
Variable proposed BIC BIC difference Type of step Decision
1 Extract Yield -788.3021 -10.930431 Remove Accepted
2 Neochlorogenic Acid -852.5413 -9.982637 Remove Accepted
3 Chlorogenic Acid -805.6227 -11.065351 Remove Accepted
4 Extract Yield -999.1101 -9.315106 Add Rejected
5 Isochlorogenic Acid -816.8139 -13.958685 Remove Accepted
6 Extract Yield -936.2985 66.325955 Add Accepted
7 Extract Yield -936.2985 66.325955 Remove Rejected
8 Isochlorogenic Acid -999.1101 -90.028182 Add Rejected
Selected subset:
Water, Bean Weight, ph Value, Free Acid, Mineral Content, Fat, Caffine,
Trigonelline, Extract Yield
\end{CodeOutput}
Then, the clustering model estimated on the selected subset of variables is:
\begin{CodeInput}
R> mod2 = Mclust(X[,result$subset])
R> summary(mod2)
\end{CodeInput}
\begin{CodeOutput}
----------------------------------------------------
Gaussian finite mixture model fitted by EM algorithm
----------------------------------------------------
Mclust EEI (diagonal, equal volume and shape) model with 3 components:
log.likelihood n df BIC ICL
-443.2269 43 38 -1029.379 -1030.937
Clustering table:
1 2 3
22 14 7
\end{CodeOutput}
\begin{CodeInput}
R> table(Class, Cluster = mod2$class)
\end{CodeInput}
\begin{CodeOutput}
Cluster
Class 1 2 3
Arabica 22 14 0
Robusta 0 0 7
\end{CodeOutput}
\begin{CodeInput}
R> table(Class, Cluster = mod2$class)
\end{CodeInput}
\begin{CodeOutput}
Cluster
Class 1 2 3
Arabica 22 14 0
Robusta 0 0 7
\end{CodeOutput}
Both the covariance parameterisation (EEE) and the number of mixture components (3) used with the selected features subset agree with those from the model using all the variables. The final clustering confirms the structure we already discussed, in particular the two sub-varieties of Arabica coffee.
To show graphically these findings, we may project the data onto a dimension reduced subspace by using the methodology described in \citet{Scrucca:2010}:
\begin{CodeInput}
R> mod2dr = MclustDR(mod2)
R> plot(mod2dr, what = "scatterplot", symbols = c("A", "a", "R"))
\end{CodeInput}
From Figure~\ref{fig1:coffee} there is an evident separation between Arabica and Robusta coffee samples along the first direction. Moreover, it seems to confirm the non homogeneous group of Arabica samples, which splits in two sub-varieties along the second direction.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{coffee_fig1}
\caption[Clustering of coffee data samples]
{Projection of coffee data samples marked according to the clustering obtained from the variables selected using the forward/backward greedy search. The symbol \textcolor{green3}{\sf R} indicates Robusta coffees, \textcolor{dodgerblue2}{\sf A} and \textcolor{red3}{\sf a} the sub-varieties of Arabica coffees.}
\label{fig1:coffee}
\end{figure}
\subsection{Simulated high-dimensional data}
\label{sec:highdim}
\citet[Sec. 3.3.2]{Witten:Tibshirani:2010} discussed an example where five clustering variables are conditionally independent given the cluster memberships, whereas the remaining twenty features are simply independent standard normal variables, also independent from the clustering ones.
The first five variables are distributed according to a spherical Gaussian distribution with mean $\boldsymbol{\mu}_1=(\mu, \mu, \ldots, \mu)$, $\boldsymbol{\mu}_2 = \boldsymbol{0}$, $\boldsymbol{\mu}_3 = -\boldsymbol{\mu}_1$, where $\mu=1.7$, and common unit standard deviation. We replicated this experiment (denoted by WT) with varying sample sizes ($n_g$ cases for each group) and for a set of different techniques.
Table~\ref{tab:WTsim} reports the variable selection error rate (VSER) and the classification error rate (CER).
As already mentioned in Section~\ref{sec:examples}.1, the VSER is defined as the ratio of the number of errors in selecting (or not selecting) variables with respect to the total number of variables considered.
The CER between two partitions, which is equivalent to one minus the Rand index \citep{Rand:1971}, is equal to 0 in the case of perfect agreement, and becomes larger for increasing disagreement \citep[for a formal definition see][]{Witten:Tibshirani:2010}.
Smaller values of both VSER and CER are better.
These two measures have been chosen for the purpose of comparison with the results in \citet[Tab. 4]{Witten:Tibshirani:2010} and \citet[Sect. 3.1]{Celeux:etal:2013}.
The model with uniformly better performance is the \code{MCLUST} model with the first five variables, i.e. the model which most resembles the data generation mechanism. However, this model is not available when we do not know the
clustering variables, as we are assuming here.
\code{SPARSEKMEANS} performs well in term of accuracy, but the number of selected variables increases with group sample size, and all the features are selected for $n_g=50$.
\code{MCLUST} using all the variables and EII parameterisation, both at the hierarchical initialisation step and for the mixture modelling, has quite good accuracy and improves as group sample size increases.
\code{CLUSTVARSEL} using backward/forward greedy search, with EII parameterisation both for modelling and initialisation also has good accuracy, and improves as group sample size gets larger. For $n_g=50$ it is equivalent to the best models. However, the VSER is better than for the other methods, and improves as $n_g$ gets larger. Thus, for increasing sample size it converges to the true subset size.
Note that forward/backward greedy search is clearly less accurate than the backward/forward search, with the VSER which is also larger, so the performance is overall worst than that of backward/forward search.
Finally, results for \code{CLUSTVARSEL} using backward/forward greedy search are essentially equivalent to those obtained with the \code{SELVARCLUSTINDEP} software of \citet{Maugis:etal:2009b}.
\begin{table}[htb]
\centering
\caption[Results of the simulation study based on the Witten and
Tibshirani (2010) setup]
{Average values based on 100 simulation runs for the WT simulation scheme with group sample size $n_g$. All models assume the number of clusters $G=3$ to be known. For all the \code{MCLUST} models, including those fitted by the variables subset selection algorithm, the EII parameterisation was used for both the hierarchical initialization and the mixture modeling. \code{CLUSTVARSEL[fwd]} uses the forward/backward greedy search, whereas \code{CLUSTVARSEL[bkw]} uses the backward/forward greedy search. Smaller values of both VSER and CER are better.}
\label{tab:WTsim}
\begin{tabular}{lrrrcrrr}
\hline\noalign{\smallskip}
& \multicolumn{3}{c}{VSER} & & \multicolumn{3}{c}{CER} \\
Model & $n_g=10$ & $n_g=20$ & $n_g=50$ & & $n_g=10$ & $n_g=20$ & $n_g=50$ \\
\hline\noalign{\smallskip}
\code{K-MEANS} & .800 & .800 & .800 && .258 & .248 & .224 \\
\code{SPARSEKMEANS} & .438 & .678 & .800 && .070 & .066 & .055 \\
\code{MCLUST[}$X_1,\ldots,X_5$\code{]} & .000 & .000 & .000 && .063 & .064 & .054 \\
\code{MCLUST[}$X_1,\ldots,X_{25}$\code{]} & .800 & .800 & .800 && .129 & .093 & .060 \\
\code{CLUSTVARSEL[fwd]} & .268 & .200 & .054 && .370 & .275 & .090 \\
\code{CLUSTVARSEL[bkw]} & .180 & .081 & .033 && .151 & .082 & .053 \\
\code{SELVARCLUSTINDEP} & .216 & .098 & .032 && .162 & .089 & .057 \\
\hline\noalign{\smallskip}
\multicolumn{8}{l}{\footnotesize Standard errors are all $< .030$.}
\end{tabular}
\end{table}
\section{Adjustments for speeding up the algorithm}
\label{sec:speed}
\subsection{Sub-sampling at hierarchical initialisation step}
\label{sec:subsample}
The EM algorithm is initialised in \pkg{mclust} using the partitions obtained from model-based agglomerative hierarchical clustering. Efficient numerical algorithms for approximately maximizing the classification likelihood with multivariate normal models have been discussed by \citet{Fraley:1998}. However, for datasets having a large number of observations this step can be computationally expensive.
When the number of observations is large, we may allow \code{clustvarsel} to use only a subset of the observations at the model-based hierarchical stage of clustering, to speed up the algorithm.
This is easily done by setting the argument \code{samp = TRUE}, and by specifying the number of observations to be used in the hierarchical clustering subset with \code{sampsize}.
Consider the following simulation scheme which constructs a medium sized dataset on five dimensions. Only the first two variables contain clustering information, the third is highly correlated with the first one, whereas the remaining features are simply noise variables.
\begin{CodeInput}
R> library(MASS)
R> n = 1000 # sample size
R> pro = 0.5 # mixing proportion
R> mu1 = c(0,0) # mean vector for the first cluster
R> mu2 = c(3,3) # mean vector for the second cluster
R> sigma1 = matrix(c(1,0.5,0.5,1),2,2) # covar matrix for the first cluster
R> sigma2 = matrix(c(1.5,-0.7,-0.7,1.5),2,2) # covar matrix for the second cluster
R> X = matrix(0, n, 5); colnames(X) = paste("X", 1:ncol(X), sep ="")
R> # generate the clustering variables
R> set.seed(123)
R> u = runif(n)
R> Class = ifelse(u < pro, 1, 2)
R> X[u < pro, 1:2] = mvrnorm(sum(u < pro), mu = mu1, Sigma = sigma1)
R> X[u >= pro, 1:2] = mvrnorm(sum(u >= pro), mu = mu2, Sigma = sigma2)
R> # generate the non-clustering variables
R> X[,3] = X[,1] + rnorm(n)
R> X[,4] = rnorm(n, mean = 1.5, sd = 2)
R> X[,5] = rnorm(n, mean = 2, sd = 1)
R> clPairs(X, Class, gap = 0.2)
\end{CodeInput}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{sim1_fig1}
\caption{Scatterplot matrix of simulated data with points marked according to the known groups.}
\label{fig1:sim1}
\end{figure}
We may compare the procedure which uses sampling at the hierarchical stage with the default call to \code{clustvarsel}, both in term of computing time, using the function \code{system.time}, and in term of clustering accuracy.
\begin{CodeInput}
R> system.time(result1 <- clustvarsel(X, G = 1:5, samp = TRUE, sampsize = 200))
\end{CodeInput}
\begin{CodeOutput}
user system elapsed
6.033 0.009 6.044
\end{CodeOutput}
\begin{CodeInput}
R> system.time(result2 <- clustvarsel(X, G = 1:5))
\end{CodeInput}
\begin{CodeOutput}
user system elapsed
9.739 0.044 9.820
\end{CodeOutput}
Thus, by using sub-sampling for the initial hierarchical clustering we get an algorithm that is 38\% faster with the same accuracy:
\begin{CodeInput}
R> result1$subset
\end{CodeInput}
\begin{CodeOutput}
X2 X1
2 1
\end{CodeOutput}
\begin{CodeInput}
R> result2$subset
\end{CodeInput}
\begin{CodeOutput}
X2 X1
2 1
\end{CodeOutput}
To investigate the effect of sampling as the number of observations increase we conducted a small simulation study by replicating the above simulation setting with different sample sizes and fixed size at 200 observations for choosing the initial starting points. Figure~\ref{fig2:sim1} shows the results averaged over 10 replications.
Panel (a) reports the computing time required as the sample size grows, whereas panel (b) shows the relative gain from using a subset of observations at the initial hierarchical stage. As can be seen, efficiency improves roughly exponentially as the number of observations increases, with sampling being about 40 times faster at $10,000$ cases. As the system time required increases linearly for sampling, when no sampling is used at the initial stage the time required increases approximately exponentially.
Furthermore, in all the replications the first two variables have been selected by both methods. Hence, the improvement in terms of computational efficiency has not caused any deterioration in terms of accuracy.
\begin{figure}
\centering
\subfloat[][system time vs sample size]{\includegraphics[width=0.48\linewidth]{sim1_fig2a}}
\quad
\subfloat[][relative gain of sub-sampling vs sample size]{\includegraphics[width=0.48\linewidth]{sim1_fig2b}}
\caption[Computing time vs sample size for subsampling method]
{Comparison of computing time vs sample size. Panel (a) shows the average over 10 replications for \code{clustvarsel} using sub-sampling with fixed size at 200 observations, and no sampling. Panel (b) shows the relative gain of sub-sampling, calculated as the ratio of system times for no sampling and sub-sampling. All axes are on the logarithmic scale.}
\label{fig2:sim1}
\end{figure}
\subsection{Headlong search}
\label{sec:headlong}
When a dataset contains a large number of variables we may find that using the headlong search algorithm option (\code{search = "headlong"}) is faster than the default greedy search. To show an example we simulated a dataset analogous to the previous one for the clustering variables, then six more irrelevant variables were added, some correlated with the clustering ones, some independent and some correlated among themselves. Then, we may compare the time required by using the headlong method and using the greedy method.
\begin{CodeInput}
R> library(MASS)
R> n = 400 # sample size
R> pro = 0.5 # mixing proportion
R> mu1 = c(0,0) # mean vector for the first cluster
R> mu2 = c(3,3) # mean vector for the second cluster
R> sigma1 = matrix(c(1,0.5,0.5,1),2,2) # covar matrix for the first cluster
R> sigma2 = matrix(c(1.5,-0.7,-0.7,1.5),2,2) # covar matrix for the second cluster
R> X = matrix(0, n, 10); colnames(X) = paste("X", 1:ncol(X), sep ="")
R> # generate the clustering variables
R> set.seed(1234)
R> u = runif(n)
R> Class = ifelse(u < pro, 1, 2)
R> X[u < pro, 1:2] = mvrnorm(sum(u < pro), mu = mu1, Sigma = sigma1)
R> X[u >= pro, 1:2] = mvrnorm(sum(u >= pro), mu = mu2, Sigma = sigma2)
R> # generate the non-clustering variables
R> X[,3] = X[,1] + rnorm(n)
R> X[,4] = X[,2] + rnorm(n)
R> X[,5] = rnorm(n, mean = 1.5, sd = 2)
R> X[,6] = rnorm(n, mean = 2, sd = 1)
R> X[,7:8] = mvrnorm(n, mu = mu1, Sigma = sigma1)
R> X[,9:10] = mvrnorm(n, mu = mu2, Sigma = sigma2)
\end{CodeInput}
\begin{CodeInput}
R> system.time(result1 <- clustvarsel(X, G = 1:5, search = "headlong"))
\end{CodeInput}
\begin{CodeOutput}
user system elapsed
5.818 0.029 5.847
\end{CodeOutput}
\begin{CodeInput}
R> system.time(result2 <- clustvarsel(X, G = 1:5))
\end{CodeInput}
\begin{CodeOutput}
user system elapsed
10.107 0.033 10.139
\end{CodeOutput}
In situations where there are many observations and a large number of variables, sub-sampling at the hierarchical initialisation step and the headlong search can be used concurrently to improve computational efficiency.
A small simulation study was conducted by replicating the previous simulation scheme with different sample sizes. The methods compared are greedy and headlong searches, without and with sampling using \code{sampize = 200}.
The results averaged over 10 replications are shown in Figure~\ref{fig1:sim2}. Without sampling, headlong search is faster than greedy search with a constant relative gain of about $1.7$. The use of sampling at the initial hierarchical stage enables us to achieve an exponential relative gain as the sample size increases for both type of searches. Also in this case, the headlong search appears to be about twice as fast as the greedy search.
\begin{figure}
\centering
\subfloat[][system time vs sample size]{\includegraphics[width=0.48\linewidth]{sim2_fig1a}}
\quad
\subfloat[][relative gain of sub-sampling vs sample size]{\includegraphics[width=0.48\linewidth]{sim2_fig1b}}
\caption[Computing time vs. sample size for greedy and headlong methods]
{Comparison of computing time vs sample size. Panel (a) shows the average over 10 replications for \code{clustvarsel} using \code{search = "greedy"} with and without sampling, and \code{search = "headlong"} with and without sampling. A fixed value \code{sampsize = 200} is used throughout. Panel (b) shows the relative gain, calculated as the ratio of system times, of each strategy against the default greedy search with no sampling. All axes are on the logarithmic scale.}
\label{fig1:sim2}
\end{figure}
The speed/optimally tradeoff in a headlong search can be changed by increasing or decreasing the different levels, e.g. by setting the upper level to 10 instead of 0 we would require a variable to have stronger evidence of clustering before it is included, and by setting the lower level to 0 we would remove variables that at any stage have evidence of clustering weaker by any amount than evidence against clustering.
\subsection{Parallel computing}
\label{sec:parallel}
Parallel computing is a form of computation in which the required calculations are performed simultaneously, either on a single multi-core processors machine or on a cluster of multiple computers.
Direct support of parallelism in \proglang{R} is available since version 2.14.0 (released in October 2011) through the package \pkg{parallel} \citep{Rpkg:parallel}. This is essentially a merger of the \pkg{multicore} package \citep{Rpkg:multicore}
and the \code{snow} package \citep{Rpkg:snow}.
The \code{multicore} functionality supports parallelism via forking, which is a concept from POSIX operating systems, so it is available on all \proglang{R} platforms except Windows.
In contrast, \pkg{snow} supports different transport mechanisms (e.g. socket connections) to communicate between the master and the workers, and it is available on all operating systems.
Other approaches to parallel computing in \proglang{R} are available as described in \citet{McCallum:Weston:2011}. For an extensive list of packages see CRAN task view on \textit{High-Performance and Parallel Computing with R} at \url{http://cran.r-project.org/web/views/HighPerformanceComputing.html}.
The greedy search discussed in Section~\ref{sec:methods} constitutes an embarrassingly parallel problem, i.e. one for which little or no effort is required to separate the problem into a number of parallel tasks. Essentially, the sequential evaluation of candidate variables for inclusion or exclusion, which is the most time consuming task, can be done in parallel.
For the actual implementation in \pkg{clustvarsel} we used the \pkg{doParallel} package \citep{Rpkg:doParallel},
a ``parallel backend'' which acts as an interface between the \pkg{foreach} package \citep{Rpkg:foreach} and the \pkg{parallel} package. Essentially, it provides a mechanism needed to execute for loops in parallel.
To specify if parallel computing should be used in the evaluation of the $\BIC_{\text{diff}}$ criterion in \eqref{eq:BICdiff}, the optional argument \code{parallel} must be set to \code{TRUE} in the \code{clustvarsel} function call. In this case all the available cores, as returned by the \code{detectCores} function, are used. A numeric value specifying the number of cores to employ can also be specified in the optional argument \code{parallel}. Finally, the parallelisation functionality depends on system OS: on Windows only 'snow' type functionality is available, whereas on Unix/Linux/Mac OSX both 'snow' and 'multicore' (default) functionalities are available.
As an example, consider a sample of $n=200$ observations generated according to the simulation scheme described in Section~\ref{sec:headlong}. We may compare the sequential greedy backward/forward search with a parallel version of the algorithm with the default maximum cores available and by specifying 2 cores (on a MacBook Pro with i5 Intel\textsuperscript{\textregistered} CPU with 4 cores running at $2.3$GHz and with $4$GB of RAM):
\begin{CodeInput}
R> system.time(result1 <- clustvarsel(X, G = 1:9, direction = "backward"))
\end{CodeInput}
\begin{CodeOutput}
user system elapsed
103.860 0.509 104.774
\end{CodeOutput}
\begin{CodeInput}
R> system.time(result2 <- clustvarsel(X, G = 1:9, direction = "backward",
parallel = TRUE))
\end{CodeInput}
\begin{CodeOutput}
user system elapsed
158.578 5.685 51.254
\end{CodeOutput}
\begin{CodeInput}
R> system.time(result2 <- clustvarsel(X, G = 1:9, direction = "backward",
parallel = 2))
\end{CodeInput}
\begin{CodeOutput}
user system elapsed
119.161 2.887 67.221
\end{CodeOutput}
In this case, by using 4 cores we were able to halve the computing time, whereas a 35\% speed up is achieved using 2 cores.
By using a machine with $P$ processors instead of just one, we would like to obtain an increase in calculation speed of $P$ times. As shown above, this is not the case because in the implementation of a parallel algorithm there are some inherent non-parallelizable parts and communication costs between tasks \citep{Nakano:2012}.
Amdahl's Law \citep{Amdahl:1967} is often used in parallel computing to predict the theoretical maximum speedup when using multiple processors. If $f$ is the fraction of non-parallelizable task and $P$ is the number of processors in use, then the maximum speedup achievable on a parallel computing platform is given by
\begin{equation}
S_P = \frac{1}{f + (1-f)/P} .
\label{eq:AmdahlLaw}
\end{equation}
In the limit, the above ratio converges to $S_{\max} = 1/f$, which represents the maximum increase of speed achievable in theory, i.e. by a machine with an infinite number of processors.
To investigate the performance of our parallel algorithm implementation, we conducted a small simulation study using the above simulation setting for increasing numbers of cores. The study was performed on a 24 cores Intel$^{\textregistered}$ Xeon\textsuperscript{\textregistered} CPU X5675 running at $3.07$GHz and with $128$GB of RAM.
Figure~\ref{fig:AmdahlLaw} shows the results averaged over 10 replications. The points represent the observed speedup factor (obtained as $s_P=t_1/t_P$, where $t_P$ is the system time employed using $P$ cores) for running the backward algorithm with up to 10 cores. The curve represents the Amdahl's Law \eqref{eq:AmdahlLaw} with $f$ estimated by non-linear least squares. It turns out that the estimated fraction of sequential part of the backward/forward search for variable selection is $\hat{f} = 0.13$, which yields a maximum speedup of $S_{\max} = 7.7$.
\begin{figure}[htb]
\centering
\includegraphics[width=0.7\linewidth]{AmdahlLaw}
\caption[Speedup by number of cores in the parallel algorithm]
{Graph of speedup factor vs the number of cores employed in the parallel algorithm for backward/forward subset selection in model-based clustering. The estimated fraction of non-parallelizable task is estimated as $f=0.13$, which gives a maximum speedup achievable by parallelisation of around $7.7$ times the sequential algorithm.}
\label{fig:AmdahlLaw}
\end{figure}
\section{Conclusions and future work}
\label{sec:discuss}
This paper has presented the \proglang{R} package \pkg{clustvarsel} which provides a convenient set of tools for variable selection in model-based clustering using a finite mixture of Gaussian densities. Stepwise greedy search and headlong algorithm are implemented in order to find the (locally) optimal subset of variables with cluster information.
The computational burden of such algorithms can be decreased by some ad hoc modifications in the algorithms, or via the use of parallel computation as implemented in the package.
Examples illustrating the use of the package in practical applications have been presented.
Finally, given the vast solution space, other optimisation techniques could be usefully employed. For instance, the use of genetic algorithms as described in \citet{Scrucca:2014} will be included in a future release of the package.
\baselineskip=15pt
\bibliographystyle{chicago}
|
1,314,259,996,562 | arxiv | \section{Bigger Picture}
\begin{abstract}
Molecular discovery is a multi-objective optimization problem that requires identifying a molecule or set of molecules that balance multiple, often competing, properties. %
Multi-objective molecular design is commonly addressed by combining properties of interest into a single objective function using scalarization, which imposes assumptions about relative importance and uncovers little about the trade-offs between objectives. In contrast to scalarization, Pareto optimization does not require knowledge of relative importance and reveals the trade-offs between objectives. However, it introduces additional considerations in algorithm design. %
In this review, %
we describe pool-based and \textit{de novo} generative approaches to multi-objective molecular discovery with a focus on Pareto optimization algorithms. We show how pool-based molecular discovery is a relatively direct extension of multi-objective Bayesian optimization and how the plethora of different generative models extend from single-objective to multi-objective optimization in similar ways using non-dominated sorting in the reward function (reinforcement learning) or to select molecules for retraining (distribution learning) or propagation (genetic algorithms). %
Finally, we discuss some remaining challenges and opportunities in the field, emphasizing the opportunity to adopt Bayesian optimization techniques into multi-objective \textit{de novo} design. %
\end{abstract} %
\section{Introduction}
\begin{sloppypar}
Molecular discovery is inherently a constrained multi-objective optimization problem. Almost every molecular design application requires multiple properties to be optimized or constrained. For example, for a new drug to be successful, it must simultaneously be potent, bioavailable, safe, and synthesizable. Multi-objective optimization, also referred to as multi-parameter optimization (MPO), pertains to other applications as well, including solvent design \cite{chong_design_2022, ten_computer-aided_2021, papadopoulos_multiobjective_2006, mah_design_2019}, personal care products \cite{yee_optimization_2022, ooi_design_2022}, electronic materials \cite{karasuyama_computational_2020, devereux_chapter_2021, hautier_finding_2019, hachmann_harvard_2011, ling_high-dimensional_2017}, functional polymers \cite{jablonka_bias_2021, mannodi-kanakkithodi_multi-objective_2016}, and other materials \cite{hanaoka_bayesian_2021, solomou_multi-objective_2018, khatamsaz_multi-objective_2022}.
Redox-active species in redox flow batteries must maximize redox potential and solubility to ensure a high cell voltage \cite{kowalski_recent_2016, winsberg_redoxflow_2017}.
Sustainability of new materials (e.g., emissions caused during production and disposal \cite{fleitmann_cosmo-suscampd_2021}) is also an increasingly important design objective \cite{wilson_accelerating_2022, melia_materials_2021}, which is particularly important for working fluids \cite{raabe_molecular_2019, kazakov_computational_2012,fleitmann_cosmo-suscampd_2021}. Multi-objective optimization can address multiple design criteria simultaneously, allowing for the discovery of molecules that are most fit for a specific application.
When many objectives must be optimized simultaneously, a common approach is to aggregate the objectives into a single objective function, which requires quantifying the relative importance of each objective. This method, also known as \emph{scalarization}, reduces a multi-objective molecular optimization problem into one that is solvable with single-objective algorithms, but the ability to explore trade-offs between objectives is limited. Further, the optimization procedure must be repeated each time the scalarization function is adjusted. In contrast, Pareto optimization, which discovers a set of solutions that reveal the trade-offs between objectives, relies on no prior measure of the importance of competing objectives. This approach allows an expert to modify the relative importance of objectives without sacrificing optimization performance or repeating the optimization procedure. The solution set of a Pareto optimization contains the solution to every scalarization problem with any choice of weighting factors. For these reasons, we believe that Pareto optimization is the most robust approach to multi-objective molecular discovery.
The discovery of optimal molecules can be framed as either a search for molecules from an enumerated library or generation of novel molecules (i.e., \textit{de novo} design) \cite{sridharan_modern_2022, meyers_novo_2021}. The extension of both discovery approaches from single-objective to multi-objective optimization has been reviewed for molecular discovery \cite{segall_multi-parameter_2012, nicolaou_molecular_2007} and more specifically drug discovery \cite{ekins_evolving_2010, nicolaou_multi-objective_2013}. However, recent developments, specifically in \textit{de novo} design using deep learning, warrant further discussion and organization of new methods.
In this review, we organize established and emerging multi-objective molecular optimization (MMO) techniques. %
After defining MMO and introducing relevant mathematical concepts, we describe key design choices during the formulation of an optimization scheme. Then, we provide a thorough discussion of relevant methods and case studies, first in library-based optimization and then in \textit{de novo} design. %
Finally, we share some open challenges in MMO and propose future work that we believe would most advance the field.
\end{sloppypar}
\section{Defining Multi-Objective Molecular Optimization}
The molecular discovery literature is riddled with approaches to solve the inverse problem of property $\rightarrow$ structure, many of which are labeled ``multi-objective''. However, the line between multi-objective molecular optimization (MMO) and single-objective or constrained optimization is quite blurred. To organize the field's communication of MMO methodologies, we classify MMO as follows: %
\begin{enumerate}
\item Multiple objectives, which are not aggregated into a single scalar objective, are considered. Some trade-off exists between objectives (i.e., they are not perfectly correlated).
\item The domain over which to optimize (``design space'') is a chemical space. Molecules in this space may be defined either implicitly (e.g., as latent variables that can be decoded using generative models) or explicitly (i.e., as a molecular library).
\item The goal of the optimization task is to identify molecules that maximize or minimize some molecular properties. We consider tasks that aim to identify molecules with properties within some specified range to be constrained generation, not multi-objective optimization.
\end{enumerate}
Any definitive scope of MMO is bound to be somewhat subjective. Yet, we believe the preceding definition captures all relevant implementations of MMO and excludes methods that are better categorized elsewhere (e.g., as a single-objective optimization or constrained optimization).
Exhaustive screening for multiple optimized properties, typically referred to as virtual screening \cite{rizzuti_chapter_2020}, can be viewed as an inefficient approach to MMO. This approach has been used to identify multi-target inhibitors \cite{wei_multiple-objective_2019, ramsay_perspective_2018, kim_two-track_2022} as well as selective inhibitors \cite{kuck_novel_2010}. In the interest of summarizing efficient optimization algorithms, we do not discuss enumeration and exhaustive screening approaches in this review.
\section{Preliminary Mathematical Concepts in MMO}
\subsection{The Pareto front}
In MMO problems, two or more desirable molecular properties compete with one another. For \emph{Pareto optimal} solutions, an improvement in one objective is detrimental to at least one other objective. For instance, when a selective drug is designed, strong affinity to the target and weak affinity to off-targets are both desired. However, when the binding affinities to on- and off-targets are highly correlated (i.e., they bind strongly to similar molecules), an increase in potency to the target often necessitates a decrease in selectivity. The \emph{Pareto front} quantifies (and, in the 2- or 3-objective case, visualizes) these types of trade-offs. Figure~\ref{fig:basic_pf_acq}A illustrates a Pareto front for two objectives which are to be maximized, with points in red representing the \emph{non-dominated points}, which form the Pareto front and define the set of optimal solutions for the multi-objective optimization problem. For these points, an improvement in one objective necessitates a detriment to the other objective. One can imagine that each objective is a desired property and that each point on the plot represents one molecule. For simplicity and ease of visualization, we always consider that objectives are maximized for the remainder of the review. Pareto fronts for minimized objectives would instead appear in the lower left corner, as opposed to the upper right.
\begin{figure}
\centering
\includegraphics[trim={0cm 0 0cm 0},clip]{images/review_prelim.png}
\caption{Terminology and acquisition functions in pareto optimization. (A) Visual depiction of common Pareto terminology including the Pareto front, dominated and non-dominated points, and dominated region. The area of the dominated region is the hypervolume. (B) Non-dominated sorting, also referred to as Pareto ranking. (C) Hypervolume improvement for one candidate point over the current hypervolume defined by the set of previously acquired points in the absence of uncertainty. }
\label{fig:basic_pf_acq}
\end{figure}
The \emph{hypervolume} of a set is the volume spanned by the Pareto front with respect to a reference point. In the 2-dimensional case, the hypervolume is the area that is dominated by the Pareto front (the red shaded region in Figure~\ref{fig:basic_pf_acq}AC). This metric can evaluate how ``good'' a Pareto front is: a larger hypervolume indicates a larger dominated region (i.e., a ``better'' Pareto front).
Progress in new materials development is often reported and visualized by the advancement of a Pareto front. As an example, in gas separation applications, membrane selectivity and permeability are two competing objectives which are both to be maximized. The trade-offs for this optimization can be visualized as a Pareto front. Figure~\ref{fig:membrane_PF} shows the improving upper bound for the two maximized objectives, which can be understood as an expansion of the Pareto front from 1991 to 2015 \cite{swaidan_fine-tuned_2015}.
\begin{figure}
\centering
\includegraphics[scale=1.5]{images/gas_membrane_PF.jpeg}
\caption{Progress in membranes for gas separation as revealed by the movement of a Pareto front. Reproduced from \citet{swaidan_fine-tuned_2015}.}
\label{fig:membrane_PF}
\end{figure}
\subsection{Single-objective Bayesian optimization} Bayesian optimization (BO) is a strategy for black box optimization where the scalar function to be optimized, sometimes referred to as the \emph{oracle}, may be non-differentiable or difficult to measure (costly) \cite{frazier_bayesian_2018}. The workflow of Bayesian optimization applied to single-objective molecular discovery is summarized in Figure~\ref{fig:BO}A. %
\begin{figure}
\centering
\includegraphics[scale=1, trim={0cm 0cm 0cm 0cm},clip ]{images/Review_Bayesian_both.png}
\caption{Overview of the Bayesian Optimization workflow and the commonalities between the (A) single-objective and (B) multi-objective settings.}
\label{fig:BO}
\end{figure}
BO is an iterative optimization procedure that begins by defining some prior model to map the design space to the objective. This model is called a \textit{surrogate model} and, in the molecular setting, is equivalent to a quantitative %
structure-property relationship (QSPR) model.
The surrogate model is used to predict the objective values of hypothetical candidates in the design space, which an \textit{acquisition function} uses (along with the surrogate model uncertainty) to prioritize which candidates %
to sample next. The newly sampled, or \emph{acquired}, molecules are then evaluated, or \emph{scored}, against the oracle, and this new data is used to refine the surrogate model. The process is repeated until some stopping criterion is met: the objective value of the acquired molecules converges, resources are expended, or some objective value threshold is attained.
The acquisition function is central to BO. This function quantifies the ``utility'' of performing a given experiment and can be broadly understood to balance both the \emph{exploitation} and \emph{exploration} of the design space \cite{shahriari_taking_2016}.
In molecular BO, exploration prevents stagnation in local optima and can encourage acquisition of more diverse molecules.
However, the acquisition function must also exploit, selecting candidates predicted to optimize the objective, which enables the algorithm to converge upon an optimum and identify the best-performing molecules. A few acquisition functions for the case where a single objective ($f$) is maximized are worth mentioning:
\begin{enumerate}
\item Expected improvement (EI): \begin{equation}
\text{EI}(x) = \mathbb{E}[\max\{0,f(x)-f^*\}], \end{equation}
in which $f(x)$ represents the objective value for some molecule $x$, $\mathbb{E}$ is the expectation operator, and $f^*$ is the best objective value attained so far from the acquired molecules \cite{frazier_bayesian_2018, shahriari_taking_2016}.
\item Probability of improvement (PI):
\begin{equation} \text{PI}(x) = \mathbb{E}[(f(x)-f^*)>0] \end{equation} The PI metric estimates how likely a new molecule $x$ is to outperform the current best molecule \cite{shahriari_taking_2016}.
\item Greedy acquisition (G):
\begin{equation} \text{G}(x) = \hat f(x) \end{equation} Here, the acquisition function is simply the predicted value for the objective function, regardless of uncertainty and what has been observed so far \cite{pyzer-knapp_bayesian_2018}.
\item Upper confidence bound (UCB):
\begin{equation}
\text{UCB}(x) = \hat f(x) + \beta \sigma(x), \end{equation}
in which $\sigma$ is the surrogate model prediction uncertainty and $\beta$ is a hyperparameter \cite{shahriari_taking_2016}.
\end{enumerate}
\begin{sloppypar}
While the BO literature thoroughly discusses and tests many acquisition functions, we have only described a few which are most popular in MMO. We refer readers interested in single-objective acquisition functions to \citeauthor{frazier_bayesian_2018}'s tutorial \cite{frazier_bayesian_2018} or \citeauthor{shahriari_taking_2016}'s review \cite{shahriari_taking_2016}.
\end{sloppypar}
\subsection{Multi-objective Bayesian optimization}
Pareto optimization problems, in which multiple objectives are considered simultaneously without quantification of relative objective importance, must be handled with a slightly modified set of tools, although the core BO ideology remains the same (Figure~\ref{fig:BO}B). First, all oracle functions must be approximated either with multiple surrogate models, a multi-task surrogate model \cite{shahriari_taking_2016}, or some combination thereof.
Second, the acquisition function must account for all objectives without explicitly assigning a relative importance weight to each of them. Here, the goal is to
expand the Pareto front, or increase the dominated hypervolume, as much as possible. We focus on three multi-objective acquisition functions:
\begin{enumerate}
\item Expected hypervolume improvement (EHI):
\begin{equation}
\text{EHI}(x) = \mathbb{E}[\max(0, \text{HV}(\mathcal{X}_{acq} \cup \{x\} )-\text{HV}(\mathcal{X}_{acq}))],
\end{equation}
in which HV is the hypervolume and $\mathcal{X}_{acq}$ is the set of previously acquired candidates. EHI is best understood as an analog to the single-objective expected improvement which measures improvement in hypervolume instead of objective value.
\item Probability of hypervolume improvement (PHI):
\begin{equation}
\text{PHI}(x) = \mathbb{E}[(\text{HV}(\mathcal{X}_{acq}\cup \{x\})-\text{HV}(\mathcal{X}_{acq})) > 0]
\end{equation}
PHI, comparable to probability of improvement, is the probability that an acquired point will improve the hypervolume by any amount.
\item Non-dominated sorting (NDS): NDS assigns an integer rank to each molecule by sorting the set of molecules into separate fronts. One can imagine identifying a Pareto front from a finite set of molecules (denoted first rank), removing that Pareto front, and subsequently identifying the next Pareto front (denoted second rank), as shown in Figure~\ref{fig:basic_pf_acq}B. The assigned Pareto rank to each molecule is taken to be its acquisition score. %
NDS does not consider uncertainty, and a candidate's assigned Pareto rank is taken to be its acquisition score. The first rank candidates are equivalent to the set of points that would be acquired from using greedy acquisition with every set of possible scalarization weights, so NDS can be thought of as a multi-objective analog of greedy acquisition.
\end{enumerate}
\subsection{Batching and batch diversity}
While the canonical BO procedure evaluates candidates sequentially by acquiring the single candidate with the highest acquisition score at each iteration, many molecular oracles can be evaluated in batches. %
Experiments performed in well plates are naturally run in parallel, and expensive computations are often distributed in batches to make the best use of computational resources. In the BO workflow, this means that an acquisition function should be used to select \emph{a set} of molecules, instead of just one. A naïve approach, \emph{top-$k$} batching, scores molecules normally and acquires the $k$ candidates with the highest acquisition scores. The utility of the entire set is thus implicitly taken to be the sum of individual acquisition scores. However, the information gained from acquiring one molecule that is highly similar to another molecule in the batch is likely to be small.
In batched multi-objective optimization, the acquisition function should maximize the utility of scoring the \emph{entire batch}. %
For the case of acquisition with EHI, this refers to the improvement in hypervolume after \emph{all} molecules in a batch are acquired. %
One can imagine that acquiring a set of candidates very near each other on the Pareto front would not maximize this utility. An ideal batching algorithm would consider all possible batches, predict the utility of each, and select the batch with greatest utility. However, solving this combinatorial optimization exactly is intractable. Instead, approximations are used to construct batches iteratively: identify the most promising molecule, assume it has been observed, select the next most promising molecule, and repeat this until the desired batch size is achieved \cite{hiot_kriging_2010}.
Batched optimization is more often approached with heuristics that promote some measure of \emph{diversity} within a batch while selecting molecules with high acquisition scores. For example, the objective space can be split into regions (Figure~\ref{fig:diversity}A) with a limit on the number of candidates acquired in each region \cite{konakovic_lukovic_diversity-guided_2020, deb_evolutionary_2014}; likewise, candidates in less crowded regions along the Pareto front can be more strongly favored \cite{deb_fast_2002}. Such approaches to promote \emph{Pareto diversity} have been incorporated into multi-objective molecular design \cite{verhellen_graph-based_2022, agarwal_discovery_2021, grantham_deep_2022}.
Diversity of the \emph{design space} can also be considered during acquisition, which is distinct from Pareto diversity and can also be applied to single-objective optimization \cite{gonzalez_new_2022}.
In MMO, design space diversity is equivalent to the the \emph{structural}, or \emph{molecular}, diversity of a batch (Figure~\ref{fig:diversity}B). %
Molecular diversity can be measured with metrics like Tanimoto similarity using fingerprint representations, which characterize a specific kind of structural similarity. As with Pareto diversity, structural diversity constraints can be imposed during acquisition \cite{janet_accurate_2020, nicolaou_novo_2009}.
While one might predict that Pareto front diversity also indicates molecular diversity, this is not necessarily true. It is possible for two structurally similar molecules to have different properties and therefore lie in different regions of the objective space; conversely, molecules with similar properties are not necessarily structurally similar. %
\begin{figure}[ht]
\centering
\includegraphics[scale=1]{images/DIVERSITY.png}
\caption{Comparing (A) Pareto diversity and (B) molecular/structural diversity for batch acquisition. Promoting one form of diversity does not necessarily improve the other.}
\label{fig:diversity}
\end{figure}
\section{Formulating Molecular Optimization Problems}
A molecular optimization task always begins with some statement of desired properties. Some of the subsequent formulation decisions are listed in Figure~\ref{fig:formulating}. First, the individual properties must be converted to mathematical objectives. Then, the means of proposing candidate molecules, either \textit{de novo} or library-based, must be selected. If more than one objective exists, they must either be aggregated into a single objective or treated with an appropriate multi-objective formulation. Finally, an acquisition function, or selection criterion in the case of \emph{de novo} design, must be selected. In this section, we explore some of these design choices in detail.
\begin{figure}
\centering
\includegraphics[scale=1, trim={0cm 0cm 0cm 0cm},clip ]{images/list.png}
\caption{Decisions when formulating MMO problems. As discussed further in later sections, iterative generative models employ selection criteria for retraining or propagation, which are analogous to acquisition functions in Bayesian optimization. Conditional generation, although capable of proposing molecules with a specified property profile, is non-iterative and therefore does not utilize selection criteria or an acquisition function. Single-objective acquisition functions can only consider molecular diversity, while Pareto acquisition functions can consider both molecular and Pareto diversity.}
\label{fig:formulating}
\end{figure}
\subsection{Converting a desired property to a mathematical objective function}
In the formulation of any MMO task, after properties of interest are identified by a subject matter expert, the individual objectives must be quantitatively defined (Figure~\ref{fig:formulating}, Panel 2). While this seems like an easy task, framing the objectives can be subjective in nature.
If one property of interest for a molecular optimization task is estimated by a score $S(x)$, there are still multiple ways to represent the corresponding value to be maximized ($J(x)$), including but not limited to:
\begin{enumerate}
\item A continuous, strictly monotonic treatment, where a greater value is strictly better:
\begin{equation}
J(x) = S(x)
\end{equation}
\item A thresholded, monotonic treatment, where some minimum $T$ is required:
\begin{equation}
J(x) = \left\{
\begin{array}{ll}
S(x) & \quad S(x) \geq T \\
-\infty & \quad S(x) \leq T
\end{array}
\right.
\end{equation}
\item A Boolean treatment, where some minimum $T$ is required and no preference is given to even higher values: \begin{equation}
J(x) = \left\{
\begin{array}{ll}
1 & \quad S(x) \geq T \\
0 & \quad S(x) \leq T
\end{array}
\right.
\end{equation}
\end{enumerate}
The most appropriate representation depends on the property of interest and the application, demonstrated here for common properties of interest for novel drug molecules. If $S$ predicts a ligand's binding affinity to a target protein, a higher affinity is often better, so the first representation may be most appropriate. If $S$ predicts solubility, there may be no additional benefit of greater solubility once a certain solubility is met that allows for sufficient delivery and bioavailability. In this case, the third representation, which is most consistent with a property constraint instead of an optimized objective, would be most fitting. In a similar manner, remaining components of Lipinski's Rule of 5 \cite{lipinski_experimental_2001} define some threshold, and no extra benefit is attained once the threshold is met. These heuristics may be most appropriately defined as constraints and not optimized objectives. %
The perspectives of domain experts during objective formulation are extremely valuable to ensure that molecules identified as optimal are suitable for the application. However, in cases where expertise is not available or a specific threshold is unknown, we argue that solving the problem with a simple continuous representation (Representation 1) is most robust because it requires no predefined hyperparameters or assumptions. This way, constraints can later be imposed on the solution set without needing to repeat the optimization from scratch.
\subsection{Choosing between library-based selection and \textit{de novo} design}
Once the objectives are defined, an approach to chemical space exploration must be chosen. The scope of exploration can be limited to an explicitly defined molecular library, which can be constructed to bias exploration toward chemical spaces relevant to a specific task. Alternatively, a \emph{\textit{de novo}} design tool can be used to ideate novel molecules not previously seen or enumerated.
The type of generative model influences the area of chemical space that is explored \cite{coley_defining_2021}. %
For example, the chemical space explored by genetic algorithms %
may be constrained by the molecules used as the initial population and the set of evolutionary operators that are applied to the population. In a more general sense, the molecules that can be generated by any \textit{de novo} model will be determined by the training set and many other design choices. %
Care can be taken to ensure that the chemical space explored is sufficient for the given task. %
\subsection{Defining the relationship between different objectives}%
Once individual objective functions are defined and the chemical space approach is chosen, the next challenge is to decide how to consider all objectives simultaneously.
The most naive choice %
is to simply combine the objective functions into one aggregated objective function, referred to as \emph{scalarization}. The scalarized objective function is most commonly a weighted sum of objectives \cite{gomez-bombarelli_automatic_2018, winter_grunifai_2020, fu_mimosa_2021, hartenfeller_concept_2008, s_v_multi-objective_2022, ooi_integration_2018, liu_data-driven_2022}, with weighting factors indicating the relative importance of different objectives. A weighted sum of multiple binding affinities has been used to identify multi-target as well as selective inhibitors \cite{winter_efficient_2019}.
Nonlinear scalarization approaches are also utilized in MMO problems \cite{urbina_megasyn_2022, firth_moarf_2015, hoffman_optimizing_2022}. For example, \citeauthor{gajo_multi-objective_2018} divide predicted drug activity by toxicity to yield a scalarized objective function \cite{gajo_multi-objective_2018}. The objective function can also be framed as a product of Booleans \cite{chen_helix-mo_2022}, each of which denotes whether a given threshold is met. This scalarization approach has been utilized to identify multi-target kinase inhibitors \cite{jin_multi-objective_2020}. %
Booleans can also be summed to define an objective function, commonly referred to as multi-property optimization \cite{barshatski_multi-property_2021}.
As with the definition of individual objectives, the scalarization function must be justified by the use case.
There are alternatives to scalarization that also reduce a multi-objective optimization into one that can be solved with single-objective algorithms, such as defining a hierarchy of objective importance \cite{hase_chimera_2018} or using alternating rewards to maximize each objective in turn \cite{goel_molegular_2021, pereira_optimizing_2021}.
However, the solution to a scalarized multi-objective problem is equivalent to just a single point out of the many non-dominated solutions that exist on the Pareto front. %
Scalarization is overly simplistic and requires a user to quantify the relative importance of different objective. It therefore fails to inform a user about the trade-offs between objectives. Even when the relative importance of objectives is known or can be approximated a priori, scalarization is strictly less informative than Pareto optimization which identifies the full set of molecules that form a Pareto front. We focus exclusively on Pareto optimization approaches to molecular discovery throughout the remainder of this review.
\section{Examples of MMO from Virtual Libraries}
Library-based multi-objective molecular optimization aims to identify the Pareto front (or a set close to the Pareto front) of a large molecular library while scoring few molecules with the objectives. The well-established Bayesian optimization workflow (Figure~\ref{fig:BO}B) is exemplified by the retrospective studies of \citet{del_rosario_assessing_2020} and \citet{gopakumar_multi-objective_2018}. In general, the iterative optimization scheme entails training a surrogate model to predict properties of interest, selecting molecules for acquisition using surrogate model predictions and uncertainties, scoring the acquired molecules with the ground-truth objectives, and retraining the surrogate model. %
\citet{janet_accurate_2020} apply this methodology to discover transition metal complexes for redox flow battery applications with maximized solubility and redox potential. Ideal complexes must be soluble in polar organic solvents commonly used for flow batteries and have high redox potentials to yield sufficient cell voltage. The design space the authors explore is a combinatorial library of almost 3 million complexes. A neural network surrogate model predicts solubilities and redox potentials from feature vector representations of complexes \cite{janet_resolving_2017}. DFT calculations served as the oracle for both solubility and redox potential, and the expected hypervolume improvement acquisition function was used. To encourage exploration of structurally diverse complexes, the top 10,000 performers according to EHI were clustered in feature space to identify and evaluate 100 medoids. Improvements of over three standard deviations from the initial random set of complexes were observed for both objectives in just five iterations, which the authors estimate to represent a 500x reduction in simulations compared to a random search.
In a similar vein, \citet{agarwal_discovery_2021} use library-based Pareto optimization to search for redox-active materials with minimized reduction potential and solvation free energy. A third objective penalized deviation from a target peak absorption wavelength of 375nm. Candidates were scored with expected hypervolume improvement, while crowding distance constraints ensured acquisition of a diverse set along the Pareto front. When retrospectively applied to a dataset of 1400 molecules, a random search required 15 times more evaluations than did Bayesian optimization to acquire molecules dominating 99\% of the total possible hypervolume. Then, a prospective search was performed on a set of 1 million molecules, with the prior dataset serving as the first set of acquired molecules. Of the 100 molecules acquired during prospective BO iterations, 16 new Pareto-optimal molecules were identified.
Most pool-based MMO problems follow this exact workflow with minor variability in the choice of acquisition function and consideration of diversity. This approach works effectively and is almost guaranteed to outperform random search baselines. While there is certainly room for algorithmic improvement (e.g., increasing sample efficiency of surrogate models, exploring the effects of batch size and diversity), we expect that future work will largely focus on additional applications incorporating more meaningful objective functions and experimental validation.
\section{Examples of MMO using Generative Models}
The primary drawback of pool-based MMO is the explicit constraint on the chemical space that can be accessed. \textit{De novo} design relaxes this constraint and can, in principle, explore a wider (and in some cases, arguably infinite) region of chemical space. In many generative models, molecules are proposed as SMILES/SELFIES strings, graphs, or synthetic pathways. Some generate novel molecules by decoding continuous embeddings into discrete molecular structures while others modify those already identified with discrete actions. %
We focus not on the details of each model, but instead on how certain categories of models aid in the molecular optimization task. A reader interested in a detailed discussion of generative models, which is outside the scope of this review, is directed to other publications \cite{bilodeau_generative_2022, sanchez-lengeling_inverse_2018, alshehri_deep_2020, mouchlis_advances_2021}. %
The myriad of multi-objective \textit{de novo} design approaches noticeably lack standardization. Unlike library-based discovery where multi-objective optimization is a modest extension of Bayesian optimization, the adaptation of generative models to MMO is not nearly as straightforward. We therefore introduce another categorization scheme for case studies in this section.
\begin{figure}
\centering
\includegraphics[scale=1]{images/generative.png}
\caption{Optimization workflows for various generative model categories. Note that all model classes, except conditional generation, involve a scoring step and are designed to be iterative. The reward calculation step in reinforcement learning and the selection step in distribution learning and genetic algorithms are analogous to an acquisition function in multi-objective Bayesian optimization. While the termination criterion is not explicitly shown for distribution learning, genetic algorithms, and reinforcement learning, these iterative loops can accommodate various stopping criteria. We also emphasize that while an autoencoder architecture is depicted in both distribution learning and conditional generation, these generators can also be recurrent neural networks or other generative architectures.}
\label{fig:generative}
\end{figure}
\subsection{Iterative retraining for distribution learning} Generative models that are designed for \emph{distribution learning} are intended to ideate molecules exhibiting a distribution of structures similar to those of the training set \cite{flam-shepherd_language_2022}. A very basic approach to optimization with an unsupervised generative model is to sample a set of molecules, evaluate their properties, and identify those that optimize the objective function; to extend this to multi-objective optimization, the Pareto front of the sampled set can be identified by evaluating all oracles \cite{frey_fastflows_2022}. This approach essentially uses a generative model to define a virtual library suitable for exhaustive screening. Optimization schemes can use distribution learning \emph{iteratively} to progressively shift the distribution of generated molecules and push the Pareto front. %
To achieve this, generative models are iteratively retrained on the increasingly promising (e.g., closest to the Pareto front) subsets of the molecules they propose. %
This process is akin to a simulated design-make-test loop, in which \emph{design} is analogous to sampling, \emph{make} to decoding to a molecule, and \emph{test} to evaluating the oracles. %
The iterative distribution learning workflow for single-objective optimization is exemplified by the library generation strategy defined by \citet{segler_generating_2018} to identify inhibitors predicted to be active against the 5-HT$_{2\text{A}}$ receptor. Here, a subset of molecules from the ChEMBL database, with corresponding experimental pIC$_{50}$ values against 5-HT$_{2\text{A}}$, was used to train both a SMILES-based recurrent neural network and a QSAR classifier to predict whether a molecule inhibits 5-HT$_{2\text{A}}$. Then, sequences of characters were randomly sampled from the RNN to generate SMILES representations of novel molecules. Molecules predicted by the QSAR classifier to be active were used to retrain the model, progressively biasing the generator to propose active molecules. After four iterations of retraining, 50\% of sampled molecules were predicted to be active, a significant increase from only 2\% in the initial random library. The same procedure has also been employed using a variational autoencoder to generate molecules with high docking scores to the DRD3 receptor\cite{boitreaud_optimol_2020}.
The extension of the method to multiple objectives is best illustrated by \citet{yasonik_multiobjective_2020} for the generation of drug-like molecules. As before, a recurrent neural network was pretrained to generate valid molecular SMILES strings.
Five oracles associated with drug-likeness were then minimized: ClogP (estimated lipophilicity), molecular weight, number of hydrogen bond acceptors, number of hydrogen bond donors, and number of rotatable bonds. A set of about 10k novel, unique, and valid molecules were sampled and scored according to the five properties. Non-dominated sorting was used to select half of these molecules for retraining. The use of NDS distinguishes this Pareto optimization from \citeauthor{segler_generating_2018}'s single-objective optimization. %
Although continuous objective values were used during selection of molecules for retraining
, constraints associated with the oracles, derived from the ``Rule of Three'' \cite{congreve_rule_2003} (an extension of Lipinski's Rule of 5\cite{lipinski_experimental_2001}), were used to evaluate the generator's performance. After five retraining iterations, the fraction of molecules that fulfilled all five constraints increased from 2\% to 33\%. While there is no evidence that the Pareto front was shifted outwards (i.e., that the dominated hypervolume increased) after retraining iterations, this study demonstrates that a generative model's property distributions for multiple objectives can be shifted simultaneously.
In addition to recurrent neural networks, as in the prior two examples, variational autoencoders and other generative models can be iteratively retrained to simultaneously fulfill multiple property constraints \cite{iovanac_actively_2022}. \citet{abeer_multi-objective_2022} describe one such approach to generate drugs with high predicted binding affinity to the DRD2 receptor, high ClogP, and low synthesizability score using a VAE as the unsupervised generator. After initial training, sampling, and scoring, the best molecules were selected according to their Pareto rank, but some random molecules were also included in the retraining set. Importantly, the authors show a progression of the 2-dimensional Pareto fronts beyond those of the original training set: they identified molecules that are strictly superior to (i.e., that ``dominate'' in a Pareto optimality sense) the best molecules in the training set. Two such plots are shown in Figure~\ref{fig:abeer}. Here, it is clear that this method is capable of increasing the dominated hypervolume and identifying novel molecules that have property values outside of the objective space spanned by the training set.
\begin{figure}
\centering
\includegraphics[scale=0.5]{images/Abeer_et_al.png}
\caption{Advancement of the Pareto front from \citeauthor{abeer_multi-objective_2022} using iterative retraining for distribution learning. Both (a) and (b) are from the same optimization task, with each set only showing two objectives for ease of visualization. The first and second columns are the distribution of the training molecules and the first batch of sampled molecules, respectively. The following 3 columns depict molecules sampled from the model after 1, 5, and 10 iterations. Reproduced from \citet{abeer_multi-objective_2022}.
}
\label{fig:abeer}
\end{figure}
\subsection{Genetic algorithms}
In contrast to many deep learning architectures, genetic algorithms (GAs) do not rely on a mapping between continuous and discrete spaces. Instead, molecules are iteratively transformed into new ones using \emph{evolutionary operators} like mutations and crossovers. Molecular \emph{mutations} may include the addition or removal of atoms, bonds, or molecular fragments, while molecular \emph{crossover} involves molecular fragment exchange between two parent molecules. GAs begin with a starting population of molecules that are scored by the oracle function(s). Selection criteria are imposed to determine which molecules in the population are chosen as \emph{parents} to be propagated. This selection step is what guides a GA to optimized molecules and, like an acquisition function in BO, determines whether an optimization is a Pareto optimization or not. Evolutionary operators are randomly chosen and applied to the parents, and the population is updated with the resulting molecules.
Genetic algorithms were the first popularized polymer \cite{venkatasubramanian_computer-aided_1994} and small molecule \cite{sheridan_using_1995} %
generators. In 1995, \citet{sheridan_using_1995} proposed generating small molecules by iteratively evolving integer sequence representations of molecules. That same year, \citet{weber_optimization_1995} used a GA to find optimal molecules from a synthetically-enumerated library. Since then, GAs have adopted evolutionary operators which function directly on molecular graphs \cite{ pegg_genetic_2001, brown_graph-based_2004, jensen_graph-based_2019} or SMILES strings \cite{nigam_parallel_2022}. Some genetic algorithms even mutate molecules using chemical reaction templates to encourage synthesizability \cite{weber_optimization_1995, durrant_autogrow_2013, daeyaert_pareto_2017}. Multiple objectives can be scalarized during selection to frame a multi-objective GA as a single-objective one \cite{devi_multi-objective_2021, pegg_genetic_2001, jensen_graph-based_2019, herring_evolutionary_2015}.
As with any generative model, if the selection criteria consider multiple objectives simultaneously without imposing assumptions about relative importance, a GA can advance the population's Pareto front. One such GA was proposed by \citet{brown_graph-based_2004} to generate ``median molecules'', which maximize Tanimoto similarity \cite{bender_molecular_2004} to two different molecules simultaneously. In each iteration, molecules in a population are manipulated with either mutations (add/delete atoms, add/delete bonds) or crossovers (molecular fragment exchange between two parent molecules). Non-dominated sorting, using the two Tanimoto similarities as objectives, determine which molecules are selected for propagation. The critical adaptation for the multi-objective case is the use of Pareto ranking---specifically, NDS---as a selection criterion, instead of using a single property estimate or a scalarization of multiple properties.
A comparable multi-objective GA, presented by \citet{nicolaou_novo_2009}, generates ligands with maximized docking scores for a target receptor (Estrogen Receptor $\beta$, or ER$\beta$) and minimized scores for a negative but closely related target (Estrogen Receptor $\alpha$, or ER$\alpha$). As an extension from the prior example, the non-dominated sorting selection criterion was modified to include \emph{niching} and \emph{elitism}. Niching encourages structurally diverse populations by grouping candidates into \emph{niches} based on their structural similarity during selection, and only a set number of molecules may be acquired in each niche. Promoting diversity can be especially beneficial to GA performance, as GAs are constrained by their starting set and set of modification operators \cite{filipic_diversity_2020, zhou_counteracting_2008}. When elitism is imposed, all Pareto-dominant molecules found during prior iterations are appended to the population before selection to prevent good molecules from being ``forgotten.'' The authors report that both elitism and niching improve optimization performance. The depicted progression of the Pareto front is replicated here (Figure~\ref{fig:nicolau-GA}). %
The notion of optimizing against a negative target can be generalized into a ``selectivity score'' that aggregates affinity to multiple off-target controls \cite{van_der_horst_multi-objective_2012}.
\begin{figure}
\centering
\includegraphics[scale=1]{images/NICOLAU_2009.png}
\caption{Pareto front for the identification of selective inhibitors. The $\Delta$G values represent docking scores. Note that the Pareto front in this plot is located in the bottom left. The Pareto front is shown after 1, 20, 50, and 100 iterations. It clearly shifts to the bottom left with each iteration. Here, niching is used but elitism is not. Redrawn from \citet{nicolaou_novo_2009}.}
\label{fig:nicolau-GA}
\end{figure}
The effect of diversity-aware acquisition is further explored by \citet{verhellen_graph-based_2022}, wherein the effectiveness of two different multi-objective GAs that promote \emph{Pareto} diversity are compared. Both GAs use non-dominated sorting to select the population members to be propagated as parents of the next generation. %
The first, NSGA-II \cite{deb_fast_2002}, promotes selection of molecules with a larger distance from other molecules in the objective space and has precedent in application to a synthesizability-constrained molecular GA \cite{daeyaert_pareto_2017}. The second, NSGA-III \cite{deb_evolutionary_2014}, enforces diversity by requiring at least one molecule to be acquired in each of a set of reference regions in the objective space (Figure~\ref{fig:diversity}A). Both genetic algorithms are applied to seven molecular case studies, each with a different set of objectives including affinity to a target, selectivity, and/or molecular weight. Using the dominated hypervolume as an evaluation metric, both multi-objective optimization approaches outperform a weighted-sum scalarization baseline, but there is no clear winner among the two NSGA algorithms. %
A measure of internal similarity indicates that the structural diversity decreased with each evolutionary iteration. Nonetheless, the selection criteria promoted Pareto diversity, demonstrating that Pareto diversity can be achieved without necessarily requiring molecular, or structural, diversity.
\subsection{Reinforcement learning} Reinforcement learning (RL)-based generative models are trained to create molecules by learning to maximize a reward function quantifying the desirability of generated molecules. In molecular reinforcement learning, a \emph{policy} determines which molecules are generated and can be iteratively updated to maximize the reward as new molecules are generated and scored. The set of actions or choices available to the policy is denoted the \emph{action space}. The framing of the reward function, analogous to the BO acquisition function and GA selection criteria, determines whether an RL method utilizes Pareto optimization.
When the learned policy generates molecules by modifying a previous population of molecules, the action space may be comprised of atom- and bond-level graph modifications \cite{ zhou_optimization_2019, leguy_evomol_2020, khemchandani_deepgraphmolgen_2020} or a set of fragment-level graph modifications \cite{erikawa_mermaid_2021}. In a similar manner, graph modifications resulting from chemical reactions can constitute the action space to promote synthesizability \cite{horwood_molecular_2020}. When the policy is a deep learning generator that designs molecules from scratch, any \textit{de novo} generator that decodes latent variables to a molecule, such as SMILES recurrent neural networks, can be considered the policy \cite{olivecrona_molecular_2017, popova_deep_2018, pereira_diversity_2021, neil_exploring_2018, blaschke_reinvent_2020}. Typically, these policies are trained using policy gradient algorithms (e.g., REINFORCE) \cite{williams_simple_1992}.
Most RL approaches to molecular discovery, and specifically to drug design \cite{tan_reinforcement_2022}, optimize a reward that considers a single property \cite{olivecrona_molecular_2017, popova_deep_2018, pereira_diversity_2021} or a scalarized objective \cite{neil_exploring_2018, de_cao_molgan_2018, blaschke_application_2018, zhou_optimization_2019, wei_multiple-objective_2019, abeer_multi-objective_2022, leguy_evomol_2020, horwood_molecular_2020, erikawa_mermaid_2021, khemchandani_deepgraphmolgen_2020, mcnaughton_novo_2022, you_graph_2018, ishitani_molecular_2022, s_v_multi-objective_2022, abbasi_multiobjective_2021}. %
We are aware of only one molecular RL approach whose reward function directly encourages %
molecules to be generated along a Pareto front. In DrugEx v2, presented by \citet{liu_drugex_2021}, RL is used to generate multi-target drug molecules. To promote the discovery of molecules along the Pareto front, NDS is used to calculate the reward. The authors test their algorithm with both this Pareto reward function and a weighted sum reward function. In the weighted-sum benchmark, the weighting factors were set as dynamic parameters which were altered during inference to encourage the model to find solutions at different locations on the Pareto front, analogous to the alternating reward approach to scalarization. For the multi-target discovery case, the fraction of generated molecules deemed desirable (defined as having all properties above some threshold value) was 81\% with the Pareto scheme and 97\% with the weighted sum scheme. The two approaches were only compared in this constraint-style evaluation, not in terms of a Pareto optimization criterion such as hypervolume improvement, so it is not clear if the lackluster performance of the Pareto optimizer is merely due to this misalignment of evaluation criteria. %
\subsection{Conditional generation} Conditional generators produce molecules that are meant to achieve some set of user-defined properties instead of directly maximizing or minimizing them in an iterative manner. Although our focus in this review is on multi-objective optimization, we feel that discussing the role of conditional generators in MMO is necessary due to their prevalence in the field and the ease of extending from single-objective (single-constraint) conditional generators to multi-objective (multi-constraint) conditional generators.
Many conditional generators are autoencoders that map molecules to latent embeddings and vice versa. In order to generate molecules with specific properties, the latent variables of these generators can be manipulated during training such that they represent the properties of interest. One such manipulation applied to variational autoencoders is to recenter the prior distribution around the associated molecule's property value $c$ instead of the origin, encouraging the latent distribution to match $\mathcal{N}(c,\sigma^2)$ instead of $\mathcal{N}(0,\sigma^2)$ \cite{richards_conditional_2022, makhzani_adversarial_2016, kang_conditional_2019}. This approach can be expanded to multiple objectives by centering each latent dimension along a different property of interest \cite{richards_conditional_2022}. Then, during inference, sampled latent variables are chosen according to the desired property values with at least partial success.
Autoencoders can also be manipulated for conditional generation by directly feeding the property value(s) of training molecules to the decoder during training \cite{polykovskiy_entangled_2018, simonovsky_graphvae_2018}. As one example, \citet{lim_molecular_2018} use this approach to fulfill certain ``drug-like'' property criteria. During CVAE (conditional VAE) training, a condition vector including molecular weight, ClogP, number of hydrogen bond donors, number of hydrogen acceptors, and topological polar surface area is appended to the latent space during decoding. Then, during generation, a manually specified conditional vector influences the decoder to generate molecules with the stated properties. In all case studies, less than 1\% of generated molecules have properties within 10\% of the values set in the condition vector. Another study using a similar architecture \cite{lee_mgcvae_2022} demonstrates that it is possible for the properties of up to 33\% of generated molecules, ``when rounded up'', to reflect the specified properties. In this case, it appears that this fraction strongly correlates with how many training molecules also fulfilled those constraints.
Some conditional generators modify existing molecular graphs or scaffolds provided as input instead of generating molecules from scratch. These models are typically trained with matched molecular pairs: pairs of molecules with only one well-defined structural transformation that causes a change in molecular properties \cite{leach_matched_2006, tyrchan_matched_2017}. One such single-objective generative model is intended to ``translate'' molecules that are inactive as DRD2 inhibitors to active inhibitor molecules \cite{jin_learning_2019}, wherein activity is predicted by a trained classifier. The generative model is presumed to learn graphical translations that most contribute to inhibitory strength.
This methodology can be extended to the multi-constraint case if improvements in multiple properties are desired \cite{wang_retrieval-based_2022, irwin_chemformer_2022, he_transformer-based_2022}. For example, MolGPT, a conditional generator proposed by \citet{bagal_molgpt_2021}, accepts a scaffold and desired property values. It then outputs a molecule that it believes to fulfill the input constraints. Molecules are completed from scaffolds as SMILES strings, and the model is trained on sets of \{scaffold, molecule, properties\}. %
The success of MolGPT in meeting target properties relies on having molecules with that property be well-represented in the training set.
While MolGPT is able to generate molecules conditioned on multiple properties, the authors do not report whether their model is capable of generating molecules with combinations of property values not present in the training set.
The effectiveness of conditional molecule generators depends not only on their ability to generate valid and unique molecules, but also on the accuracy of the implicit molecule-property model. If this model is inaccurate, the generator will suggest molecules that do not actually exhibit the desired properties. We further emphasize that, in order to identify Pareto-optimal molecules, the model must be able to extrapolate past the training set because, by definition, Pareto-optimal molecules have properties (or combinations of properties) that are not dominated by members of the training set.
Therefore, we find it unlikely that these non-iterative conditional generators will succeed in advancing the Pareto front. This is in contrast to iterative optimization methods, wherein the predictive capability of the generators is improved for newly explored regions of chemical space with each iteration.
Further, the nature of conditional generators requires that a user know what property value ranges are feasible. Based on the discussed and other case studies \cite{kotsias_direct_2020, he_molecular_2021}, conditional generators perform well primarily when attempting to generate novel molecules with property combinations spanned by the training set. A pIC50-conditioned model would propose some set of molecules if asked to achieve a pIC50 value of 100, even though such a value is unrealistic. Their behavior in these settings is not well understood, so a user may need to know which property constraints are valid or possible.
Due to these concerns, we caution the reader that conditional generators may not be most appropriate for Pareto optimization tasks.
\subsection{Hybrid approaches} %
The case studies that we have shared so far fall neatly into our defined categories. However, certain other approaches that combine methods from multiple categories or otherwise deviate from this classification are worth mentioning.
\citet{grantham_deep_2022} introduce one such hybrid approach, in which latent representations of molecules are mutated with a genetic algorithm and decoded to generate new molecules. %
A variational autoencoder is first trained to encode molecules into latent vectors. After encoding the starting population, mutations are applied to their corresponding latent vectors, which are then decoded. From this new set of evolved molecules, non-dominated sorting with a crowding distance constraint (specifically, NSGA-II \cite{deb_fast_2002}) is used to select new molecules to use for retraining the autoencoder. %
The proposed method outperforms two Bayesian optimization baselines in terms of the hypervolume of the final Pareto front when applied to an optimization of ClogP, QED, and synthesizability score. A similar methodology was used to optimize both drug-likeness properties and binding affinity (estimated via docking scores) to carbonic anhydrase IX \cite{mukaidaisi_multi-objective_2022}.
Iterative retraining has also been used to improve the performance of a conditional generator. In one example, a conditional graph generator is fine-tuned with molecules that are active against both JNK3 and GSK-3$\beta$ \cite{li_multi-objective_2018}. This workflow essentially follows the iterative retraining of distribution learning algorithms, but uses conditional generation to provide an extra bias toward sampling molecules with favorable properties.
In a similar manner, reinforcement learning methods can be considered conditional generation if the reward function favors molecules with a target property profile \cite{ domenico_novo_2020, wang_multi-constraint_2021, stahl_deep_2019}. Two such methods \cite{jin_multi-objective_2020, chen_fragment-based_2021} use RL to generate molecules that are predicted to be dual inhibitors of GSK3$\beta$ and JNK3 receptors according to pretrained surrogate models. In the final populations in both studies, 100\% of molecules are active against both inhibitors. However, the dataset used in both studies for training already includes a small fraction of dual inhibitors. Therefore, discovering ``active inhibitors'' in this case is equivalent to discovering the chemical space that is classified as active according to the surrogate models, and this task is easier than extrapolating with a continuous oracle. In general, the reported success of generators conditioned on Boolean values (instead of continuous ones) can be overoptimistic, as the degree of optimization success is harder to quantify with metrics such as the hypervolume. %
\section{Discussion}
In the description of library-based MMO, we %
explained that these methods are a natural extension of Bayesian optimization. In contrast, \textit{de novo} methods stray farther from classic BO, although some aspects of BO acquisition functions are present in generative workflows. In particular, NDS is often used as the selection criterion for retraining (distribution learning) or propagation (genetic algorithms). Other conventional BO acquisition functions, such as EHI and PHI, are rarely incorporated into optimization with generative models. These acquisition functions use the uncertainty in surrogate model predictions, which aids in the balance between exploration and exploitation. But most generative optimization architectures score molecules with the ground truth objectives during selection, thus bypassing uncertainty quantification and making EHI and PHI unusable as acquisition functions. An opportunity exists to incorporate Bayesian principles into \textit{de novo} design by including a separate surrogate model that predicts objective function values and can be retrained as new data are acquired to guide selection . These and other adjustments to \textit{de novo} optimization approaches may help bridge the gap between generation and model-guided optimization.
We have also observed that the performance of Pareto optimization approaches is often evaluated using individual property values or constraints. These metrics, however, reveal little about the \emph{combination of properties} of discovered molecules, which is of foremost interest in MMO. Hypervolume improvement can indicate the shift in the Pareto front, but other qualities of the discovered molecules related to the Pareto front \cite{collette_three_2005, li_quality_2020} can be of equal importance, including the density of the Pareto front or the average Pareto rank of the molecules. In molecular discovery, imperfect property models are often used as oracles. In these cases, it is beneficial to discover a dense Pareto front and many %
close-to-optimal molecules
\emph{according to QSPR predictions}, even if not all increase the hypervolume. Naturally, some molecules that are predicted to perform well will not validate experimentally, and having a denser population to sample from will increase the probability of finding true hits. For the same reason, promoting structural diversity and not just Pareto diversity is a way to hedge one's bets and avoid the situation where none of the Pareto-optimal molecules validates.
In batched multi-objective optimization, Pareto diversity can be considered during acquisition to promote exploration. In molecular optimization, structural diversity similarly encourages exploration of a wider region of chemical space. Thus, in MMO, both potential measurements of diversity are relevant, and either or both can be used during optimization. At this point, neither diversity metric has been shown to outperform the other in MMO tasks, and the question of how best to incorporate both into acquisition (or whether this actually benefits optimization) remains. At present, diversity-aware acquisition is most commonly incorporated into multi-objective genetic algorithms rather than other generative architectures. Acquisition that promotes diversity may improve performance of generators using reinforcement learning or iterative distribution learning, although this has yet to be demonstrated.
We have argued that Pareto optimization is a more practical approach to many molecular discovery tasks than scalarization or constrained optimization, but the ability of Pareto optimization to scale to several dimensions must also be addressed. Non-dominated sorting increasingly fails to differentiate the optimality of solutions with more objectives, as more and more points are non-dominated in a higher-dimensional space \cite{maltese_scalability_2018}. The numerical estimation of hypervolume has a computational cost that scales exponentially with the number of objectives, making EHI and PHI acquisition functions also increasingly difficult to use in high dimensions \cite{maltese_scalability_2018}. The increased computational costs associated with fine-tuning many surrogate models and scoring candidates for every objective contribute to scalability issues as well. Considering the challenges faced with Pareto optimization of many (more than three) objectives, scalarizing certain objectives
or converting some to constraints to make the problem solvable may be the most practical approach, especially when some objectives are known to be more important than others. The question of whether Pareto optimization can robustly scale to many objectives is a worthwhile one only if a problem cannot be feasibly reduced. The visualization of the Pareto front is an additional consideration; objective trade-offs are more easily conveyed with a Pareto front of two or three objectives. Ultimately, the optimal formulation of an MMO problem will depend on the use case, and collaboration with subject matter experts can ensure that the problem formulation is feasible but does not impose unrealistic assumptions.
Beyond these unique challenges posed by multi-objective optimization, many challenges from single-objective optimization remain relevant \cite{bilodeau_generative_2022, renz_failure_2019, meyers_novo_2021}. The first is the need for realistic oracle functions that can be evaluated computationally but meaningfully describe experimental performance; this is closely related to the need for more challenging benchmarks to mimic practical applications. Optimizing QED, ClogP, or a Boolean output from a classifier are easy tasks and are not good indicators of robustness or generality. Generative models specifically must also prove effective with fewer oracle calls, which is often the bottleneck when molecules must be scored with experiments or high-fidelity simulations \cite{gao_sample_2022}. For experimental applications, the synthesizability of generated molecules is an additional factor that must be considered \cite{gao_synthesizability_2020} and can be cast as a continuous objective or a rigid constraint. Experimental prospective validation is essential to demonstrate the viability of molecular discovery algorithms, though algorithmic advances can be made more rapidly with purely computational studies. %
\section{Conclusion}
Though many approaches to computer-aided molecular design have been developed with just single-objective optimization in mind, molecular discovery is a multi-objective optimization problem.
In certain situations, such as optimization from a library (BO-accelerated virtual screening), the extension from single-objective to multi-objective requires only minor modifications, e.g., to the acquisition function and to the number of surrogate models. In contrast, \textit{de novo} design workflows vary more in methodology and are less directly analogous to Bayesian optimization. %
The use of Pareto rank as a reward (for RL) or the use of non-dominated sorting to select sampled molecules to include in subsequent populations (for GAs) or training sets (for iterative distribution learning) replaces greedy acquisition functions. Yet, there is an opportunity to define new generative workflows which more directly incorporate model-guided optimization methods with consideration of model uncertainty. Batching in MMO can encourage chemical space exploration by rewarding structural diversity, Pareto diversity, or both, but best practices around diversity-aware batching are not well established. %
Emerging workflows will benefit from the adoption of challenging benchmarks and evaluation metrics that measure the dominated hypervolume or Pareto front density. As newly proposed molecular discovery tools increasingly emphasize multi-objective optimization, emerging methods must address the algorithmic complexities introduced by Pareto optimization. %
\section{Acknowledgment}
The authors thank Wenhao Gao, Samuel Goldman, and David Graff for commenting on the manuscript. This work was funded by the DARPA Accelerated Molecular
Discovery program under contract HR00111920025. %
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